Vibration and Shock Handbook (Mechanical Engineering)

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Vibration and Shock Handbook

Mechanical Engineering Series Frank Kreith - Series Editor Published Titles Distributed Generation: The Power Paradigm for the New Millennium Anne-Marie Borbely & Jan F. Kreider Elastoplasticity Theor y Vlado A. Lubarda Energy Audit of Building Systems: An Engineering Approach Moncef Krarti Engineering Experimentation Euan Somerscales Entropy Generation Minimization Adrian Bejan Finite Element Method Using MATLAB, 2nd Edition Young W. Kwon & Hyochoong Bang Fluid Power Circuits and Controls: Fundamentals and Applications John S. Cundiff Fundamentals of Environmental Discharge Modeling Lorin R. Davis Heat Transfer in Single and Multiphase Systems Greg F. Naterer Introductor y Finite Element Method Chandrakant S. Desai & Tribikram Kundu Intelligent Transportation Systems: New Principles and Architectures Sumit Ghosh & Tony Lee Mathematical & Physical Modeling of Materials Processing Operations Olusegun Johnson Ilegbusi, Manabu Iguchi & Walter E. Wahnsiedler Mechanics of Composite Materials Autar K. Kaw Mechanics of Fatigue Vladimir V. Bolotin Mechanics of Solids and Shells: Theories and Approximation Gerald Wempner & Demosthenes Talaslidis Mechanism Design: Enumeration of Kinematic Structures According to Function Lung-Wen Tsai Nonlinear Analysis of Structures M. Sathyamoorthy Practical Inverse Analysis in Engineering David M. Trujillo & Henry R. Busby Pressure Vessels: Design and Practice Somnath Chattopadhyay Principles of Solid Mechanics Rowland Richards, Jr. Thermodynamics for Engineers Kau-Fui Wong Vibration and Shock Handbook Clarence W. de Silva Viscoelastic Solids Roderic S. Lakes


Edited by

Clarence W. de Silva

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8493-1580-8 (Hardcover) International Standard Book Number-13: 978-0-8493-1580-0 (Hardcover) Library of Congress Card Number 2004065847 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Vibration and shock handbook / editor-in-chief, Clarence W. de Silva. p. cm. Includes bibliographical references and index. ISBN 0-8493-1580-8 (alk. paper) 1. Vibration--Handbooks, manuals, etc. 2. Shock (Mechanics)--Handbooks, manuals, etc. I. De Silva, Clarence W. TA355.V5197 2005 620.3--dc22

2004065847

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of T&F Informa plc

and the CRC Press Web site at http://www.crcpress.com

Preface

With individual chapters authored by distinguished leaders and experienced professionals in their respective topics, this handbook provides engineers, technicians, designers, researchers, educators, and students with a convenient, thorough, up to date, and authoritative reference source on techniques, tools, and data for modeling, analysis, design, monitoring, instrumentation, modification, and control of vibration, shock, noise, and acoustics. The handbook consists of 45 chapters, grouped into nine sections. In these sections, the following are considered: fundamentals and analytical techniques; computer techniques, tools, and signal analysis including finite element methods and wavelet analysis and the use of MATLABw toolboxes; shock and vibration methodologies, particularly for civil and mechanical engineering systems; instrumentation and testing methods, including sensors, exciters, signal acquisition, conditioning, and recording, and LabVIEWw tools for virtual instrumentation; vibration suppression, damping, and control; monitoring and diagnosis; design for and control of seismic vibration, and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. Important information and results are summarized as windows, tables, graphs, and lists throughout the chapters for easy reference and information tracking. References are given at the end of each chapter for further information and study. An extensive glossary, which defines useful technical terms in the fields of vibration, shock, and acoustics, is provided at the end of the handbook. Cross referencing is used throughout to indicate other places in the handbook where further information on a particular topic is provided. In the handbook, equal emphasis is given to theory and practical application. The chapters are grouped into fundamentals, basic theory, advanced theory, analytical techniques, numerical techniques, experimental techniques, design methodology, practical problems and solutions, applications, regulatory considerations, and useful data. Analytical formulations, numerical methods, design approaches, control techniques, and commercial software tools are presented and illustrated. Commercial equipment, computer hardware, and instrumentation are described, analyzed, and demonstrated for field application, practical implementation, and experimentation. Examples and case studies are given throughout the handbook to illustrate the use and application of the included information. The material is presented in a format that is convenient for easy reference and recollection. Mechanical vibration is a manifestation of oscillatory behavior in mechanical systems, as a result of either the repetitive interchange of kinetic and potential energies among components in the system, or a forcing excitation that is oscillatory. Such oscillatory responses are not limited to purely mechanical systems, and are found in electrical and fluid systems as well. In purely thermal systems, however, free natural oscillations are not possible, and an oscillatory excitation is needed to obtain an oscillatory response. Shock is vibration caused by brief, abrupt, and typically high intensity excitations. Sound, noise, and acoustics are manifestations of pressure waves, sources of which are often vibratory dynamic systems. Low levels of vibration mean reduced noise and an improved work environment. Vibration modification and control can be crucial in maintaining high performance and production efficiency, and v

vi

Preface

prolonging the useful life in industrial machinery. Consequently, a considerable effort is devoted today to studying and controlling the vibration and shock generated by machinery components, machine tools, transit vehicles, impact processes, civil engineering structures, fluid flow systems, and aircraft. Noise and acoustic problems can originate from undesirable vibrations and fluid structure interactions, as found, for example, in automobile engines. Engine noise, environmental noise, and noise from high speed and high temperature exhaust gases in a vehicle will not only cause passenger discomfort and public annoyance, they will also result in damaging effects to the vehicle itself. Noise suppression methods and devices, and sound absorption material and structures are crucial in such situations. Before designing or controlling a system for good vibratory or acoustic performance, it is important to understand, analyze, and represent the dynamic characteristics of the system. This may be accomplished through purely analytical means, computer analysis of analytical models, testing and analysis of test data, or by a combination of these approaches. It follows that modeling, analysis, testing, and design are all important aspects of study in vibration, shock, and acoustics. In recent years, educators, researchers, and practitioners have devoted considerable effort toward studying and controlling vibration, shock, and noise in a range of applications in various branches of engineering, particularly in civil, mechanical, aeronautical and aerospace, and production and manufacturing engineering. Specific applications are found in machine tools, transit vehicles, impact processes, civil engineering structures, construction machinery, industrial processes, product qualifica tion and quality control, fluid flow systems, ships, and aircraft. This handbook is a contribution toward these efforts. In view of the analytical methods, practical considerations, design issues, and experimental techniques that are presented throughout the handbook, and in view of the simplified and snap shot style presentation of formulas, data, and advanced theory, the handbook serves as a useful reference tool and an extensive information source for engineers and technicians in industry and laboratories, researchers, instructors, and students in the areas of vibration, shock, noise, and acoustics.


Editor in Chief Vancouver

Acknowledgments

I wish to express my gratitude to the authors of the chapters for their valuable and highly professional contributions. The assistance of my research engineer, Jian Zhang, in some graphics and formatting of the handbook was quite valuable. I am very grateful to Cindy Renee Carelli, Acquisitions Editor Engineering, CRC Press, for her enthusiasm, characteristic vigor, and support throughout the project. The editorial and production staff at CRC Press and its affiliates, particularly Jessica Vakili, and Carol Cooper have done an excellent job in getting this handbook in print. Finally, I wish to lovingly acknowledge the patience and understanding of my family.

vii

Editor-in-Chief

Clarence W. de Silva, P. Eng., Fellow ASME and Fellow IEEE, is Professor of Mechanical Engineering at the University of British Columbia, Vancouver, Canada, and has occupied the NSERC Research Chair in Industrial Automation since 1988. He has earned Ph.D. degrees from Massachusetts Institute of Technology (1978) and Cambridge University, England (1998). De Silva has also occupied the Mobil Endowed Chair Professorship in the Department of Electrical and Computer Engineering at the National University of Singapore. He has served as a consultant to several companies, including IBM and Westinghouse in the U.S., and has led the development of six industrial machines. He is recipient of the Killam Research Prize, Outstanding Engineering Educator Award of the Institute of Electrical and Electronics Engineers (IEEE) Canada, Education Award of the Dynamic Systems and Control Division of the American Society of Mechanical Engineers (ASME), Lifetime Achievement Award of the World Automation Congress, IEEE Third Millennium Medal, Meritorious Achievement Award of the Association of Professional Engineers of BC, and the Outstanding Contribution Award of the Systems, Man, and Cybernetics Society of the IEEE. As well as editing or coediting 12 volumes, he has authored or coauthored 16 technical books, over 150 journal papers, and about 200 conference papers and book chapters. He has served on the editorial boards of 12 international journals, in particular as the Editor in Chief of the International Journal of Control and Intelligent Systems, Editor in Chief of the International Journal of Knowledge Based Intelligent Engineering Systems, Senior Technical Editor of Measurements and Control, and Regional Editor, North America, of Engineering Applications of Artificial Intelligence the International Journal of Intelligent Real Time Automation. He is a Lilly Fellow, Senior Fulbright Fellow to Cambridge University, Fellow of the Advanced Systems Institute of British Columbia, and a Killam Fellow.

ix

Contributors

S. Akishita

Ritsumeikan University Kusatsu, Japan

Haym Benaroya Rutgers University Piscataway, NJ

Su Huan Chen

Jilin University Changchun, P.R. China

M. Dabestani

Furlong Research Foundation London, UK

Marie D. Dahleh

Harvard University Cambridge, MA

Kourosh Danai

University of Massachusetts Amherst, MA

Chris “Flip” DeFilippo National Instruments Austin, TX


Giuseppe Failla

Universita` degli Studi Mediterranea di Reggio Calabria Italy

Mohamed S. Gadala

The University of British Columbia Vancouver, BC, Canada

Seon M. Han

Texas Tech University Lubbock, TX

P.S. Heyns

University of Pretoria Pretoria, South Africa

Cheng Huang

University of Missouri at Rolla Rolla, MO

T.H. Lee

National University of Singapore Singapore

Y.P. Leow

Singapore Institute of Manufacturing Technology Singapore

S.Y. Lim

S. Huang

Jiahao Lin


Dalian University of Technology Liaoning, P.R. China

Hirokazu Iemura

W. Lin

Kyoto University Kyoto, Japan

Sarvesh Kumar Jain

Ibrahim Esat

Nader Jalili

University of Ontario Oshawa, ON, Canada

Robert G. Landers


Madhav Institute of Technology and Science Madhya Pradesh, India

Ebrahim Esmailzadeh

Engineering Consultant Jalles, France

National Research Council of Canada Vancouver, Canada

The University of British Columbia Vancouver, BC, Canada

Brunel University Middlesex, UK

Christian Lalanne

Clemson University Clemson, SC

Takayuki Koizumi Doshisha University Kyoto Hu, Japan


L.Y. Lu

National Kaohsiung First University of Science and Technology Kaohsiung, Taiwan

Chris K. Mechefske Queen’s University Kingston, ON, Canada

Priyan Mendis

University of Melbourne Melbourne, Victoria, Australia

xi

xii

Contributors

Y.L. Mo

Mulyo Harris Pradono

Kiyoshi Nagakura

Nikolaos P. Politis

University of Houston Houston, TX

Railway Technical Research Institute Tokyo To, Japan

Tuan Ngo

University of Melbourne Melbourne, Victoria, Australia

Teruo Obata Teikyo University Totigi Ken, Japan

Kiyoshi Okura

Mitsuboshi Belting Ltd. Hyogo Ken, Japan

Randall D. Peters Mercer University Macon, GA

Kyoto University Kyoto, Japan

Rice University Houston, TX

H. Sam Samarasekera Sulzer Pumps (Canada), Inc. Burnaby, BC, Canada

C. Scheffer

University of Stellenbosch Pretoria, South Africa

Michael Sedlak National Instruments Austin, TX

Zhong-Sheng Liu


Datong Song


Pol D. Spanos Rice University Houston, TX

K.K. Tan


K.Z. Tang


Y.B. Yang

National Taiwan University Taipei, Taiwan

J.D. Yau

Tamkang University Taipei, Taiwan

Yahui Zhang

Dalian University of Technology Liaoning, P.R. China

Contents

SECTION I

Fundamentals and Analysis

1

Time-Domain Analysis Clarence W. de Silva .............................................................................. 1 1

2

Frequency-Domain Analysis Clarence W. de Silva .................................................................... 2 1

3

Modal Analysis Clarence W. de Silva .............................................................................................. 3 1

4

Distributed-Parameter Systems Clarence W. de Silva ............................................................... 4 1

1.1 1.2 1.3 1.4 1.5 1.6

2.1 2.2 2.3 2.4 2.5 2.6

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

4.1 4.2

Introduction ................................................................................................................................ 1 1 Undamped Oscillator ................................................................................................................. 1 2 Heavy Springs ........................................................................................................................... 1 12 Oscillations in Fluid Systems .................................................................................................. 1 14 Damped Simple Oscillator ...................................................................................................... 1 16 Forced Response ....................................................................................................................... 1 27

Introduction ................................................................................................................................ 2 1 Response to Harmonic Excitations ........................................................................................... 2 2 Transform Techniques .............................................................................................................. 2 14 Mechanical Impedance Approach ........................................................................................... 2 25 Transmissibility Functions ....................................................................................................... 2 31 Receptance Method .................................................................................................................. 2 37 Appendix 2A Transform Techniques ...................................................................................... 2 40

Introduction ................................................................................................................................ 3 1 Degrees of Freedom and Independent Coordinates ............................................................... 3 2 System Representation ............................................................................................................... 3 4 Modal Vibrations ..................................................................................................................... 3 10 Orthogonality of Natural Modes ............................................................................................ 3 14 Static Modes and Rigid Body Modes ..................................................................................... 3 15 Other Modal Formulations ..................................................................................................... 3 22 Forced Vibration ...................................................................................................................... 3 28 Damped Systems ...................................................................................................................... 3 32 State Space Approach .............................................................................................................. 3 36 Appendix 3A Linear Algebra ................................................................................................... 3 41

Introduction ................................................................................................................................ 4 1 Transverse Vibration of Cables ................................................................................................. 4 2 xiii

xiv

Contents 4.3 4.4 4.5 4.6 4.7

5

Longitudinal Vibrations of Rods ............................................................................................ Torsional Vibration of Shafts .................................................................................................. Flexural Vibration of Beams .................................................................................................... Damped Continuous Systems ................................................................................................. Vibration of Membranes and Plates .......................................................................................

7

8

13 19 26 50 52

Random Vibration Haym Benaroya .............................................................................................. 5 1 5.1 5.2 5.3 5.4 5.5 5.6

Random Vibration ..................................................................................................................... 5 1 Single Degree of Freedom: The Response to Random Loads ................................................ 5 2 Response to Two Random Loads .............................................................................................. 5 7 Multi Degree of Freedom Vibration ...................................................................................... 5 12 Multi Degree of Freedom: The Response to Random Loads .............................................. 5 17 Continuous System Random Vibration ................................................................................. 5 29

SECTION II

6

4 4 4 4 4

Computer Techniques

Numerical Techniques 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Marie D. Dahleh ...................................................................................... 6 1 Introduction ................................................................................................................................ 6 1 Single Degree of Freedom System ............................................................................................ 6 2 Systems with Two or More Degrees of Freedom ..................................................................... 6 8 Finite Difference Method for a Continuous System ............................................................. 6 11 Matrix Methods ........................................................................................................................ 6 14 Approximation Methods for the Fundamental Frequency ................................................... 6 18 Finite Element Method ............................................................................................................ 6 20 Appendix 6A Introduction to MATLABq ............................................................................. 6 24

Vibration Modeling and Software Tools Zhong 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Datong Song, Cheng Huang, and Sheng Liu ....................................................................................................................................... 7 1 Introduction ................................................................................................................................ 7 1 Formulation ................................................................................................................................ 7 2 Vibration Analysis ...................................................................................................................... 7 9 Commercial Software Packages ............................................................................................... 7 13 The Basic Procedure of Vibration Analysis ............................................................................ 7 16 An Engineering Case Study ..................................................................................................... 7 19 Comments ................................................................................................................................. 7 21

Computer Analysis of Flexibly Supported Multibody Systems Ibrahim Esat and M. Dabestani ..................................................................................................................................... 8 1 8.1 Introduction ................................................................................................................................ 8 1 8.2 Theory ......................................................................................................................................... 8 2 8.3 A Numerical Example ................................................................................................................ 8 7 8.4 An Industrial Vibration Design Problem ............................................................................... 8 11 8.5 Programming Considerations ................................................................................................. 8 16 8.6 VIBRATIO ................................................................................................................................. 8 17 8.7 Analysis ...................................................................................................................................... 8 24 8.8 Comments ................................................................................................................................. 8 31 Appendix 8A VIBRATIO Output for Numerical Example in Section 8.3 .......................... 8 32

Contents

xv

9

Finite Element Applications in Dynamics Mohamed S. Gadala .......................................... 9 1

10

Vibration Signal Analysis Clarence W. de Silva ........................................................................ 10 1

11

Wavelets — Concepts and Applications

9.1 9.2 9.3 9.4 9.5 9.6

10.1 10.2 10.3 10.4 10.5 10.6 10.7

Problem and Element Classification ......................................................................................... 9 2 Types of Analysis ...................................................................................................................... 9 20 Modeling Aspects for Dynamic Analysis ................................................................................ 9 23 Equations of Motion and Solution Methods ......................................................................... 9 27 Various Dynamic Analyses ...................................................................................................... 9 33 Checklist for Dynamic FE Analysis ........................................................................................ 9 41

Introduction .............................................................................................................................. 10 1 Frequency Spectrum ................................................................................................................. 10 2 Signal Types .............................................................................................................................. 10 7 Fourier Analysis ........................................................................................................................ 10 7 Analysis of Random Signals .................................................................................................. 10 18 Other Topics of Signal Analysis ............................................................................................ 10 26 Overlapped Processing ........................................................................................................... 10 28

Pol D. Spanos, Giuseppe Failla, and Nikolaos P. Politis ................................................................................................................................... 11 1 11.1 Introduction .............................................................................................................................. 11 1 11.2 Time Frequency Analysis ....................................................................................................... 11 2 11.3 Time Dependent Spectra Estimation of Stochastic Processes ............................................ 11 11 11.4 Random Field Simulation ..................................................................................................... 11 14 11.5 System Identification .............................................................................................................. 11 15 11.6 Damage Detection .................................................................................................................. 11 17 11.7 Material Characterization ...................................................................................................... 11 18 11.8 Concluding Remarks .............................................................................................................. 11 19

SECTION III

Shock and Vibration

12

Mechanical Shock

13

Vibration and Shock Problems of Civil Engineering Structures Priyan Mendis and

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11

Christian Lalanne .......................................................................................... 12 1 Definitions ................................................................................................................................ 12 2 Description in the Time Domain ........................................................................................... 12 3 Shock Response Spectrum ....................................................................................................... 12 4 Pyroshocks .............................................................................................................................. 12 17 Use of Shock Response Spectra ............................................................................................. 12 18 Standards ................................................................................................................................. 12 24 Damage Boundary Curve ...................................................................................................... 12 26 Shock Machines ...................................................................................................................... 12 28 Generation of Shock Using Shakers ..................................................................................... 12 44 Control by a Shock Response Spectrum .............................................................................. 12 52 Pyrotechnic Shock Simulation .............................................................................................. 12 58

Tuan Ngo ................................................................................................................................................ 13 1 13.1 Introduction .............................................................................................................................. 13 2 13.2 Earthquake Induced Vibration of Structures ........................................................................ 13 3

xvi

Contents 13.3 13.4 13.5 13.6 13.7

14

Dynamic Effects of Wind Loading on Structures ................................................................ Vibrations Due to Fluid Structure Interaction .................................................................. Blast Loading and Blast Effects on Structures ..................................................................... Impact Loading ...................................................................................................................... Floor Vibration .......................................................................................................................

13 13 13 13 13

22 33 34 47 51

Reinforced Concrete Structures Y.L. Mo ................................................................................. 14 1 14.1 14.2 14.3 14.4

Introduction .............................................................................................................................. 14 1 Analytical Models ..................................................................................................................... 14 6 Beams under Harmonic Excitations ..................................................................................... 14 18 Design for Explosions/Shocks ............................................................................................... 14 21

SECTION IV Instrumentation and Testing

15

Vibration Instrumentation Clarence W. de Silva ..................................................................... 15 1 15.1 15.2 15.3 15.4 15.5 15.6

Introduction .............................................................................................................................. 15 1 Vibration Exciters ..................................................................................................................... 15 3 Control System ....................................................................................................................... 15 15 Performance Specification ..................................................................................................... 15 21 Motion Sensors and Transducers .......................................................................................... 15 27 Torque, Force, and Other Sensors ........................................................................................ 15 50 Appendix 15A Virtual Instrumentation for Data Acquisition, Analysis, and Presentation ..................................................................................................................... 15 73

16

Signal Conditioning and Modification

17

Vibration Testing Clarence W. de Silva ........................................................................................ 17 1

18

16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9

17.1 17.2 17.3 17.4 17.5

Clarence W. de Silva .............................................. 16 1 Introduction .............................................................................................................................. 16 2 Amplifiers .................................................................................................................................. 16 2 Analog Filters .......................................................................................................................... 16 15 Modulators and Demodulators ............................................................................................. 16 29 Analog Digital Conversion ................................................................................................... 16 37 Bridge Circuits ........................................................................................................................ 16 43 Linearizing Devices ................................................................................................................ 16 49 Miscellaneous Signal Modification Circuitry ....................................................................... 16 56 Signal Analyzers and Display Devices .................................................................................. 16 62

Introduction .............................................................................................................................. 17 1 Representation of a Vibration Environment .......................................................................... 17 3 Pretest Procedures .................................................................................................................. 17 24 Testing Procedures .................................................................................................................. 17 37 Some Practical Information .................................................................................................. 17 52

Experimental Model Analysis Clarence W. de Silva ................................................................ 18 1 18.1 18.2 18.3 18.4

Introduction .............................................................................................................................. 18 1 Frequency Domain Formulation ............................................................................................ 18 2 Experimental Model Development ......................................................................................... 18 8 Curve Fitting of Transfer Functions ..................................................................................... 18 10

Contents

xvii

18.5 18.6

Laboratory Experiments ........................................................................................................ 18 18 Commercial EMA Systems .................................................................................................... 18 24

SECTION V

Vibration Suppression and Control

19

Vibration Damping

20

Damping Theory

19.1 19.2 19.3 19.4 19.5

20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 20.14 20.15 20.16 20.17 20.18 20.19

Clarence W. de Silva ................................................................................... 19 1 Introduction .............................................................................................................................. 19 1 Types of Damping .................................................................................................................... 19 2 Representation of Damping in Vibration Analysis ............................................................... 19 9 Measurement of Damping ..................................................................................................... 19 16 Interface Damping .................................................................................................................. 19 26

Randall D. Peters ............................................................................................. 20 1 Preface ....................................................................................................................................... 20 2 Introduction .............................................................................................................................. 20 4 Background ............................................................................................................................. 20 12 Hysteresis More Details .................................................................................................... 20 19 Damping Models .................................................................................................................... 20 20 Measurements of Damping ................................................................................................... 20 23 Hysteretic Damping ............................................................................................................... 20 27 Failure of the Common Theory ............................................................................................ 20 29 Air Influence ........................................................................................................................... 20 30 Noise and Damping ............................................................................................................... 20 31 Transform Methods ................................................................................................................ 20 34 Hysteretic Damping ............................................................................................................... 20 36 Internal Friction ..................................................................................................................... 20 41 Mathematical Tricks Linear Damping Approximations ................................................ 20 43 Internal Friction Physics ........................................................................................................ 20 44 Zener Model ........................................................................................................................... 20 45 Toward a Universal Model of Damping ............................................................................... 20 48 Nonlinearity ............................................................................................................................ 20 58 Concluding Remark ............................................................................................................... 20 65

21

Experimental Techniques in Damping Randall D. Peters .................................................... 21 1

22

Structure and Equipment Isolation Y.B. Yang, L.Y. Lu, and J.D. Yau .................................. 22 1

21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10

22.1 22.2

Electronic Considerations ........................................................................................................ 21 2 Data Processing ........................................................................................................................ 21 3 Sensor Choices .......................................................................................................................... 21 7 Damping Examples .................................................................................................................. 21 8 Driven Oscillators with Damping ......................................................................................... 21 19 Oscillator with Multiple Nonlinearities ............................................................................... 21 21 Multiple Modes of Vibration ................................................................................................ 21 24 Internal Friction as Source of Mechanical Noise ................................................................ 21 28 Viscous Damping Need for Caution ............................................................................... 21 29 Air Influence ........................................................................................................................... 21 31

Introduction .............................................................................................................................. 22 2 Mechanisms of Base Isolated Systems .................................................................................... 22 4

xviii

Contents 22.3 22.4 22.5 22.6

Structure Equipment Systems with Elastomeric Bearings .................................................. 22 9 Sliding Isolation Systems ....................................................................................................... 22 17 Sliding Isolation Systems with Resilient Mechanism .......................................................... 22 36 Issues Related to Seismic Isolation Design .......................................................................... 22 50

23

Vibration Control

24

Helicopter Rotor Tuning

23.1 23.2 23.3 23.4

24.1 24.2 24.3 24.4 24.5 24.6

Nader Jalili and Ebrahim Esmailzadeh ......................................................... 23 1 Introduction .............................................................................................................................. 23 1 Vibration Control Systems Concept ....................................................................................... 23 4 Vibration Control Systems Design and Implementation ................................................... 23 12 Practical Considerations and Related Topics ....................................................................... 23 38 Kourosh Danai .................................................................................. 24 1 Introduction .............................................................................................................................. 24 1 Neural Network Based Tuning ................................................................................................ 24 4 Probability Based Tuning ........................................................................................................ 24 5 Adaptive Tuning ....................................................................................................................... 24 8 Case Study ............................................................................................................................... 24 12 Conclusion .............................................................................................................................. 24 17

SECTION VI Monitoring and Diagnosis

25

26

27

Machine Condition Monitoring and Fault Diagnostics Chris K. Mechefske ................. 25 1 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 25.10 25.11

Introduction .............................................................................................................................. 25 2 Machinery Failure .................................................................................................................... 25 2 Basic Maintenance Strategies .................................................................................................. 25 4 Factors which Influence Maintenance Strategy ..................................................................... 25 7 Machine Condition Monitoring ............................................................................................. 25 8 Transducer Selection .............................................................................................................. 25 10 Transducer Location ............................................................................................................... 25 14 Recording and Analysis Instrumentation ............................................................................. 25 14 Display Formats and Analysis Tools ..................................................................................... 25 16 Fault Detection ....................................................................................................................... 25 21 Fault Diagnostics .................................................................................................................... 25 25

Vibration-Based Tool Condition Monitoring Systems C. Scheffer and P.S. Heyns ........ 26 1 26.1 26.2 26.3 26.4 26.5 26.6

Introduction .............................................................................................................................. 26 1 Mechanics of Turning .............................................................................................................. 26 2 Vibration Signal Recording ..................................................................................................... 26 7 Signal Processing for Sensor Based Tool Condition Monitoring ...................................... 26 11 Wear Model/Decision Making for Sensor Based Tool Condition Monitoring ................ 26 15 Conclusion .............................................................................................................................. 26 20

Fault Diagnosis of Helicopter Gearboxes Kourosh Danai ................................................... 27 1 27.1 27.2 27.3 27.4

Introduction .............................................................................................................................. 27 1 Abnormality Scaling ................................................................................................................. 27 5 The Structure Based Connectionist Network ........................................................................ 27 8 Sensor Location Selection ...................................................................................................... 27 11

Contents

xix

27.5 27.6

A Case Study ........................................................................................................................... 27 14 Conclusion .............................................................................................................................. 27 23

28

Vibration Suppression and Monitoring in Precision Motion Systems K.K. Tan, T.H. Lee, K.Z. Tang, S. Huang, S.Y. Lim, W. Lin, and Y.P. Leow ........................................................ 28 1 28.1 Introduction .............................................................................................................................. 28 1 28.2 Mechanical Design to Minimize Vibration ............................................................................ 28 2 28.3 Adaptive Notch Filter ............................................................................................................. 28 10 28.4 Real Time Vibration Analyzer ............................................................................................... 28 17 28.5 Practical Insights and Case Study ......................................................................................... 28 29 28.6 Conclusions ............................................................................................................................. 28 35

SECTION VII

Seismic Vibration

29

Seismic Base Isolation and Vibration Control Hirokazu Iemura, Sarvesh Kumar Jain,

30

Seismic Random Vibration of Long-Span Structures

31

Seismic Qualification of Equipment Clarence W. de Silva ................................................... 31 1

and Mulyo Harris Pradono .................................................................................................................... 29 1 29.1 Introduction .............................................................................................................................. 29 1 29.2 Seismic Base Isolation .............................................................................................................. 29 4 29.3 Seismic Vibration Control ..................................................................................................... 29 33

30.1 30.2 30.3 30.4 30.5 30.6

31.1 31.2 31.3

Jiahao Lin and Yahui Zhang ...... 30 1 Introduction .............................................................................................................................. 30 2 Seismic Random Excitation Fields ........................................................................................ 30 11 Pseudoexcitation Method for Structural Random Vibration Analysis .............................. 30 16 Long Span Structures Subjected to Stationary Random Ground Excitations .................. 30 27 Long Span Structures Subjected to Nonstationary Random Ground Excitations ........... 30 34 Conclusions ............................................................................................................................. 30 39

Introduction .............................................................................................................................. 31 1 Distribution Qualification ....................................................................................................... 31 1 Seismic Qualification ............................................................................................................... 31 6

SECTION VIII

32

Design and Applications

Vibration Design and Control Clarence W. de Silva .............................................................. 32 1 32.1 32.2 32.3 32.4 32.5 32.6 32.7 32.8

Introduction .............................................................................................................................. 32 2 Specification of Vibration Limits ............................................................................................ 32 3 Vibration Isolation ................................................................................................................... 32 5 Balancing of Rotating Machinery ......................................................................................... 32 15 Balancing of Reciprocating Machines .................................................................................. 32 26 Whirling of Shafts .................................................................................................................. 32 33 Design through Modal Testing .............................................................................................. 32 39 Passive Control of Vibration ................................................................................................. 32 45

xx

Contents 32.9 32.10

33

Active Control of Vibration .................................................................................................. 32 61 Control of Beam Vibrations .................................................................................................. 32 67 Appendix 32A MATLAB Control Systems Toolbox ........................................................... 32 73

Structural Dynamic Modification and Sensitivity Analysis Su Huan Chen ................. 33 1 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9

Introduction .............................................................................................................................. 33 2 Structural Dynamic Modification of Finite Element Model ................................................ 33 2 Perturbation Method of Vibration Modes ............................................................................. 33 4 Design Sensitivity Analysis of Structural Vibration Modes ................................................. 33 8 High Accuracy Modal Superposition for Sensitivity Analysis of Modes .......................... 33 11 Sensitivity of Eigenvectors for Free Free Structures .......................................................... 33 13 Matrix Perturbation Theory for Repeated Modes .............................................................. 33 14 Matrix Perturbation Method for Closely Spaced Eigenvalues ........................................... 33 16 Matrix Perturbation Theory for Complex Modes .............................................................. 33 22

34

Vibration in Rotating Machinery

35

Regenerative Chatter in Machine Tools

36

Fluid-Induced Vibration

34.1 34.2 34.3 34.4 34.5

35.1 35.2 35.3 35.4 35.5 35.6 35.7

36.1 36.2 36.3

H. Sam Samarasekera ...................................................... 34 1 Introduction .............................................................................................................................. 34 1 Vibration Basics ........................................................................................................................ 34 6 Rotordynamic Analysis .......................................................................................................... 34 18 Vibration Measurement and Techniques ............................................................................. 34 39 Vibration Control and Diagnostics ...................................................................................... 34 39 Robert G. Landers ................................................. 35 1 Introduction .............................................................................................................................. 35 1 Chatter in Turning Operations ............................................................................................... 35 3 Chatter in Face Milling Operations ........................................................................................ 35 9 Time Domain Simulation ..................................................................................................... 35 14 Chatter Detection ................................................................................................................... 35 18 Chatter Suppression ............................................................................................................... 35 20 Case Study ............................................................................................................................... 35 24 Seon M. Han ..................................................................................... 36 1 Description of the Ocean Environment ................................................................................. 36 1 Fluid Forces ............................................................................................................................. 36 16 Examples ................................................................................................................................. 36 23

SECTION IX Acoustics

37

38

Sound Levels and Decibels S. Akishita ...................................................................................... 37 1 37.1 37.2 37.3

Introduction .............................................................................................................................. 37 1 Sound Wave Characteristics .................................................................................................... 37 1 Levels and Decibels .................................................................................................................. 37 3

Hearing and Psychological Effects 38.1 38.2 38.3

S. Akishita ........................................................................ Introduction .............................................................................................................................. Structure and Function of the Ear .......................................................................................... Frequency and Loudness Response ........................................................................................

38 38 38 38

1 1 1 2

Contents

xxi

38.4 38.5

Hearing Loss ............................................................................................................................. 38 4 Psychological Effects of Noise ................................................................................................. 38 4

39

Noise Control Criteria and Regulations

40

Instrumentation Kiyoshi Nagakura .............................................................................................. 40 1

41

Source of Noise S. Akishita ........................................................................................................... 41 1

42

Design of Absorption Teruo Obata ............................................................................................ 42 1

43

39.1 39.2 39.3 39.4 39.5

40.1 40.2 40.3

41.1 41.2

42.1 42.2 42.3 42.4 42.5 42.6 42.7 42.8

S. Akishita ............................................................. Introduction .............................................................................................................................. Basic Ideas behind Noise Policy .............................................................................................. Legislation ................................................................................................................................. Regulation ................................................................................................................................. Measures of Noise Evaluation .................................................................................................

39 39 39 39 39 39

1 1 1 2 4 5

Sound Intensity Measurement ................................................................................................ 40 1 Mirror Microphone System ................................................................................................... 40 4 Microphone Array .................................................................................................................... 40 6

Introduction .............................................................................................................................. 41 1 Radiation of Sound .................................................................................................................. 41 1

Introduction .............................................................................................................................. 42 1 Fundamentals of Sound Absorption ...................................................................................... 42 2 Sound Absorbing Materials ..................................................................................................... 42 3 Acoustic Characteristic Computation of Compound Wall .................................................. 42 6 Attenuation of Lined Ducts ................................................................................................... 42 10 Attenuation of Dissipative Mufflers ...................................................................................... 42 12 General Considerations .......................................................................................................... 42 15 Practical Example of Dissipative Muffler ............................................................................. 42 17

Design of Reactive Mufflers 43.1 43.2 43.3 43.4 43.5 43.6

Teruo Obata ................................................................................. 43 1 Introduction .............................................................................................................................. 43 1 Fundamental Equations ........................................................................................................... 43 2 Effects of Reactive Mufflers ...................................................................................................... 43 3 Calculation Procedure .............................................................................................................. 43 5 Application Range of Model ................................................................................................... 43 6 Practical Example ................................................................................................................... 43 13

44

Design of Sound Insulation

45

Statistical Energy Analysis Takayuki Koizumi ........................................................................... 45 1

44.1 44.2

45.1 45.2 45.3 45.4

Kiyoshi Okura .............................................................................. 44 1 Theory of Sound Insulation .................................................................................................... 44 1 Application of Sound Insulation .......................................................................................... 44 13

Introduction .............................................................................................................................. Power Flow Equations ............................................................................................................. Estimation of Sea Parameters .................................................................................................. Application in Structures .........................................................................................................

Glossary

45 45 45 45

1 2 4 7

............................................................................................................................................ G 1

Fundamentals and Analysis

I

I1

1

Time-Domain Analysis 1.1 1.2

Clarence W. de Silva The University of British Columbia

Introduction............................................................................. 1 1 Undamped Oscillator .............................................................. 1 2 Energy Storage Elements Energy † Free Response

†

The Method of Conservation of

1.3

Heavy Springs ........................................................................ 1 12

1.4 1.5

Oscillations in Fluid Systems................................................ 1 14 Damped Simple Oscillator.................................................... 1 16

1.6

Forced Response .................................................................... 1 27

Kinetic Energy Equivalence

Case 1: Underdamped Motion ðz , 1Þ † Logarithmic Decrement Method † Case 2: Overdamped Motion ðz . 1Þ † Case 3: Critically Damped Motion ðz ¼ 1Þ † Justification for the Trial Solution † Stability and Speed of Response Impulse Response Function † Forced Response Response to a Support Motion

†

Summary This chapter concerns the modeling and analysis of mechanical vibrating systems in the time domain. Useful procedures of time domain analysis are developed. Techniques of analyzing both free response and forced response are given. Examples of the application of Newton’s force motion approach, energy conservation approach, and Lagrange’s energy approach are given. Particular emphasis is placed on the analysis of an undamped oscillator and a damped oscillator. Associated terminology, which is useful in vibration analysis, is defined.

1.1

Introduction

Vibrations can be analyzed both in the time domain and in the frequency domain. Free vibrations and forced vibrations may have to be analyzed. This chapter provides the basics of the time domain representation and analysis of mechanical vibrations. Free and natural vibrations occur in systems because of the presence of two modes of energy storage. When the stored energy is repeatedly interchanged between these two forms, the resulting time response of the system is oscillatory. In a mechanical system, natural vibrations can occur because kinetic energy (which is manifested as velocities of mass [inertia] elements) may be converted into potential energy (which has two basic types: elastic potential energy due to the deformation in spring like elements, and gravitation potential energy due to the elevation of mass elements under the Earth’s gravitational pull) and back to kinetic energy, repetitively, during motion. An oscillatory excitation (forcing function) is able to make a dynamic system respond with an oscillatory motion (at the same frequency as the forcing excitation) even in the absence of two forms of energy storage. Such motions are forced responses rather than natural or free responses. 11

12


An analytical model of a mechanical system is a set of equations. These may be developed either by the Newtonian approach, where Newton’s Second Law is explicitly applied to each inertia element, or by the Lagrangian or Hamiltonian approach which is based on the concepts of energy (kinetic and potential energies). A time domain analytical model is a set of differential equations with respect to the independent variable time ðtÞ: A frequency domain model is a set of input output transfer functions with respect to the independent variable frequency ðvÞ: The time response will describe how the system moves (responds) as a function of time. The frequency response will describe the way the system moves when excited by a harmonic (sinusoidal) forcing input, and is a function of the frequency of excitation.

1.2

Undamped Oscillator

Consider the mechanical system that is schema System Dynamic System tically shown in Figure 1.1. The inputs (or Outputs . State Variables (y,y) (Response) excitation) applied to the system are represented System Parameters (m,k,b) y Inputs by the force f ðtÞ: The outputs (or response) of b (Excitation) the system are represented by the displacement y: m f(t) Environment The system boundary (real or imaginary) demarcates the region of interest in the analysis. k What is outside the system boundary is the System environment in which the system operates. An Boundary analytical model of the system may be given by one or more equations relating the outputs to FIGURE 1.1 A mechanical dynamic system. the inputs. If the rates of changes of the response (outputs) are not negligible, the system is a dynamic system. In this case the analytical model, in the time domain, becomes one or more differential equations rather than algebraic equations. System parameters (e.g., mass, stiffness, damping constant) are represented in the model, and their values should be known in order to determine the response of the system to a particular excitation. State variables are a minimum set of variables, which completely represent the dynamic state of a system at any given time t: These variables are not unique (more than one choice of a valid set of state variables is possible). For a simple oscillator (a single degree of freedom (DoF) mass spring damper system) an appropriate set of state variables would be the displacement y and the velocity y_ : An alternative set would be y_ and the spring force. In the present section, we will first show that many types of oscillatory systems can be represented by the equation of an undamped simple oscillator. In particular, mechanical, electrical, and fluid systems will be considered. The conservation of energy is a straightforward approach for deriving the equations of motion for undamped oscillatory systems (which fall into the class of conservative systems). The equations of motion for mechanical systems may be derived using the free body diagram approach, with the direct application of Newton’s Second Law. An alternative and rather convenient approach is the use of Lagrange’s equations. The natural (free) response of an undamped simple oscillator is a simple harmonic motion. This is a periodic, sinusoidal motion.

1.2.1

Energy Storage Elements

Mass (inertia) and spring are the two basic energy storage elements in mechanical systems. The concept of state variables may be introduced as well through these elements. 1.2.1.1

Inertia (m)

Consider an inertia element of lumped mass m; excited by force f ; as shown in Figure 1.2. The resulting velocity is v:

Time Domain Analysis

13

Newton’s Second Law gives

v

dv ¼f m dt

ð1:1Þ

f

m

Kinetic energy stored in the mass element is equal to the work done by the force f on the mass. Hence, ð ð dx ð ð m dv Energy E ¼ f dx ¼ f dt ¼ fv dt ¼ v dt dt dt ð ¼ m v dv

FIGURE 1.2

A mass element.

or 1 Kinetic energy KE ¼ mv2 ð1:2Þ 2 Note: v is an appropriate state variable for a mass element because it can completely represent the energy of the element. Integrate Equation 1.1: 1 ðt vðtÞ ¼ vð0 Þ þ f dt ð1:3Þ m 02 Hence, with t ¼ 0þ ; we have vð0þ Þ ¼ vð0 Þ þ

þ 1 ð0 f dt m 02

ð1:4Þ

Since the integral of a finite quantity over an almost zero time interval is zero, these results tell us that a finite force will not cause an instantaneous change in velocity in an inertia element. In particular, for a mass element subjected to finite force, since the integral on the right hand side of Equation 1.4 is zero, we have vð0þ Þ ¼ vð0 Þ

1.2.1.2

ð1:5Þ

Spring (k)

Consider a massless spring element of lumped stiffness k; as shown in Figure 1.3. One end of the spring is fixed and the other end is free. A force f is applied at the free end, which results in a displacement (extension) x in the spring. Hooke’s Law gives df f ¼ kx or ¼ kv dt

x f = kx

k FIGURE 1.3

A spring element.

ð1:6Þ

Elastic potential energy stored in the spring is equal to the work done by the force on the spring. Hence, ð dx ð ð 1 df ð ð 1ð 1 2 dt ¼ fv dt ¼ f dt ¼ f f df ¼ Energy E ¼ f dx ¼ kx dx ¼ 12 kx2 ¼ f dt k dt k 2k or Elastic potential energy PE ¼

1 2 1 f2 kx ¼ 2 2 k

ð1:7Þ

Note: f and x are both appropriate state variables for a spring, because both can completely represent the energy in the spring.

14


Integrate Equation 1.6: f ðtÞ ¼ f ð0 Þ þ

1 ðt v dt k 02

ð1:8Þ

Set t ¼ 0þ : We have f ð0þ Þ ¼ f ð0 Þ þ

þ 1 ð0 v dt k 02

ð1:9Þ

From these results, it follows that at finite velocities there cannot be an instantaneous change in the force of a spring. In particular, from Equation 1.9 we see that when the velocities of a spring are finite: f 0þ ¼ f ð0 Þ

ð1:10Þ

x 0þ ¼ xð0 Þ

ð1:11Þ

Also, it follows that

1.2.1.3

Gravitation Potential Energy

The work done in raising an object against the gravitational pull is stored as gravitational poten tial energy of the object. Consider a lumped mass m; as shown in Figure 1.4, which is raised to a height y from some reference level. The work done gives ð ð Energy E ¼ f dy ¼ mg dy

f = mg m y

FIGURE 1.4

Hence,

A mass element subjected to gravity.

Gravitational potential energy PE ¼ mgy

1.2.2

mg

ð1:12Þ

The Method of Conservation of Energy

There is no energy dissipation in undamped systems which contain energy storage elements only. In other words, energy is conserved in these systems, which are known as conservative systems. For mechanical systems, conservation of energy gives

KE þ PE ¼ const:

ð1:13Þ

These systems tend to be oscillatory in their natural motion, as noted before. Also, analogies exist with other types of systems (e.g., fluid and electrical systems). Consider the six systems sketched in Figure 1.5. 1.2.2.1

System 1 (Translatory)

Figure 1.5(a) shows a translatory mechanical system (an undamped oscillator) which has just one degree of freedom x: This may represent a simplified model of a rail car that is impacting against a snubber. The conservation of energy (Equation 1.13) gives

1 2 1 2 m_x þ kx ¼ const: 2 2

ð1:14Þ


15

x

l

m

k

θ

mg

(d)

(a)

A

K

J

y θ

y

h Mass density = r

(b)

(e)

l

x m

vL

i L

k1

k = k1+ k2

k2

+

(c)

(f)

C

−

vC

FIGURE 1.5 Six examples of single degree of freedom oscillatory systems: (a) translatory; (b) rotatory; (c) flexural; (d) pendulous; (e) liquid slosh; (f) electrical.

Here, m is the mass and k is the spring stiffness. Differentiate Equation 1.14 with respect to time t: We obtain m_xx€ þ kx_x ¼ 0 Since generally x_ – 0 at all t; we can cancel it out. Hence, we obtain the equation of motion: x€ þ 1.2.2.2

System 2 (Rotatory)

k x¼0 m

ð1:15Þ

Figure 1.5(b) shows a rotational system with the single DoF u: It may represent a simplified model of a motor drive system. As before, the conservation energy gives 1 _2 1 J u þ K u 2 ¼ const: 2 2

ð1:16Þ

In this equation, J is the moment of inertia of the rotational element and K is the torsional stiffness of the shaft. Then, by differentiating Equation 1.16 with respect to t and canceling u_; we obtain the equation of motion: K u€ þ u ¼ 0 J 1.2.2.3

ð1:17Þ

System 3 (Flexural)

Figure 1.5(c) is a lateral bending (flexural) system, which is a simplified model of a building structure. Again, a single DoF x is assumed. Conservation of energy gives 1 2 1 2 m_x þ kx ¼ const: 2 2

ð1:18Þ

16


Here, m is the lumped mass at the free end of the support and k is the lateral bending stiffness of the support structure. Then, as before, the equation of motion becomes x€ þ 1.2.2.4

k x¼0 m

ð1:19Þ

System 4 (Pendulous)

Figure 1.5(d) shows a simple pendulum. It may represent a swinging type building demolisher or a ski lift, and has a single DoF u. We have KE ¼

1 mðlu_Þ2 2 mgl cos u

Gravitational PE ¼ Eref

Here, m is the pendulum mass, l is the pendulum length, and g is the acceleration due to gravity. Hence, conservation of energy gives 1 2 _2 ml u 2

mgl cos u ¼ const:

ð1:20Þ

Differentiate with respect to t after canceling the common ml: lu_u€ þ g sin u u_ ¼ 0 Since, u_ – 0 at all t; we have the equation of motion: g u€ þ sin u ¼ 0 l

ð1:21Þ

This system is nonlinear, in view of the term sin u: For small u; sin u is approximately equal to u: Hence, the linearized equation of motion is g u€ þ u ¼ 0 l 1.2.2.5

ð1:22Þ

System 5 (Liquid Slosh)

Consider a liquid column system shown in Figure 1.5(e). It may represent two liquid tanks linked by a pipeline. The system parameters are: area of cross section of each column ¼ A; mass density of liquid ¼ r; length of liquid mass ¼ l: We have 1 ðrlAÞ_y 2 2 g Gravitational PE ¼ rAðh þ yÞ ðh þ yÞ þ rAðh 2 KE ¼

g yÞ ðh 2

yÞ

Hence, conservation of energy gives 1 rlA_y 2 þ 12 rAgðh þ yÞ2 þ 12 rAgðh 2

yÞ2 ¼ const:

Differentiate: l_yy€ þ gðh þ yÞ_y

gðh

yÞ_y ¼ 0

But, we have y_ – 0 for all t Hence, y€ þ gðh þ yÞ

gðh

yÞ ¼ 0

ð1:23Þ


17

or 2g y¼0 l

y€ þ

1.2.2.6

ð1:24Þ

System 6 (Electrical)

Figure 1.5(f) shows an electrical circuit with a single capacitor and a single inductor. Again, conservation of energy may be used to derive the equation of motion. Voltage balance gives vL þ vC ¼ 0

ð1:25Þ

where vL and vC are voltages across the inductor and the capacitor, respectively. Constitutive equation for the inductor is L

di ¼ vL dt

ð1:26Þ

C

dvC ¼i dt

ð1:27Þ

Constitutive equation for the capacitor is

Hence, by differentiating Equation 1.26, substituting Equation 1.25, and using Equation 1.27, we obtain L

d2 i dv ¼ L ¼ dt 2 dt

dvC ¼ dt

i C

or LC

d2 i þi¼0 dt 2

ð1:28Þ

Now consider the energy conservation approach for this electrical circuit, which will give the same result. Note that power is given by the product vi: 1.2.2.7

Capacitor Electrostatic energy E ¼

ð

vi dt ¼

Here, v denotes vC : Also, v¼

ð

vC

ð dv Cv2 dt ¼ C v dv ¼ dt 2

1 ð i dt C

ð1:29Þ

ð1:30Þ

Ðþ Since the current i is finite for a practical circuit, we have 002 i dt ¼ 0: Hence, in general, the voltage across a capacitor cannot change instantaneously. In particular, vð0þ Þ ¼ vð0 Þ

1.2.2.8

Inductor Electromagnetic energy E ¼

ð

vi dt ¼

ð di ð Li2 L i dt ¼ L i di ¼ dt 2

ð1:31Þ

18


Here, v denotes vL : Also,

1 ð v dt L

ð1:32Þ

ið0þ Þ ¼ ið0 Þ

ð1:33Þ

i¼

Ðþ Since v is finite in a practical circuit, we have 002 v dt ¼ 0: Hence, in general, the current through an inductor cannot change instantaneously. In particular,

Since the circuit in Figure 1.5(f) does not have a resistor, there is no energy dissipation. As a result, energy conservation gives Cv2 Li2 þ ¼ const: 2 2

ð1:34Þ

Differentiate Equation 1.34 with respect to t: Cv

dv di þ Li ¼0 dt dt

Substitute the capacitor constitutive equation 1.27: iv þ Li

di ¼0 dt

Since i – 0 in general, we can cancel it. Now, by differentiating Equation 1.27, we have di=dt ¼ Cðd2 v=dt 2 Þ: Substituting this in the above equation, we obtain LC

d2 v þv ¼0 dt 2

ð1:35Þ

LC

d2 i þi¼0 dt 2

ð1:36Þ

Similarly, we obtain

1.2.3

Free Response

The equation of free motion (i.e., without an excitation force) of the six linear systems considered above (Figure 1.5) is of the same general form: x€ þ v2n x ¼ 0

ð1:37Þ

This is the equation of an undamped, simple oscillator. The parameter vn is the undamped natural frequency of the system. For a mechanical system of mass m and stiffness k; we have sffiffiffiffi k vn ¼ ð1:38Þ m To determine the time response x of this system, we use the trial solution: x ¼ A sinðvn t þ fÞ

ð1:39Þ

in which A and f are unknown constants, to be determined by the initial conditions (for x and x_ Þ; say, xð0Þ ¼ x0 ; x_ ð0Þ ¼ v0 Substitute the trial solution into Equation 1.37. We obtain ð Av2n þ Av2n Þsinðvn t þ fÞ ¼ 0

ð1:40Þ


19

This equation is identically satisfied for all t: Response x x Hence, the general solution of Equation 1.37 is A ω indeed Equation 1.39, which is periodic and φ sinusoidal. This response is sketched in Figure 1.6. Note φ 0 π − φ 2π − f Time t −ω ω ω that this sinusoidal oscillatory motion has a frequency of oscillation of v (radians/sec). Hence, a system that provides this type of natural motion is called a simple oscillator. In other words, the FIGURE 1.6 Free response of an undamped simple response exactly repeats itself in time periods of T oscillator. or a cyclic frequency f ¼ 1=T (Hz). The frequency v is in fact the angular frequency given by v ¼ 2p f : Also, the response has an amplitude A; which is the peak value of the sinusoidal response. Now, suppose that we shift this response curve to the right through f =v: Consider the resulting curve to be the reference signal (with signal value ¼ 0 at t ¼ 0; and increasing). It should be clear that the response shown in Figure 1.6 leads the reference signal by a time period of f=v: This may be verified from the fact that the value of the reference signal at time t is the same as that of the signal in Figure 1.6 at time t f=v: Hence, f is termed the phase angle of the response, and it is a phase lead. The left hand side portion of Figure 1.6 is the phasor representation of a sinusoidal response. In this representation, an arm of length A rotates in the counterclockwise direction at angular speed v: This is the phasor. The arm starts at an angular position f from the horizontal axis, at time t ¼ 0: The projection of the arm onto the vertical ðxÞ axis is the time response. In this manner, the phasor representation can conveniently indicate the amplitude, frequency, phase angle, and the actual time response (at any time t) of a sinusoidal motion. A repetitive (periodic) motion of the type 1.39 is called a simple harmonic motion, meaning it is a pure sinusoidal oscillation at a single frequency. Next, we will show that the amplitude A and the phase angle f both depend on the initial conditions. Substitute the initial conditions (Equation 1.40) into Equation 1.39 and its time derivative. We obtain x0 ¼ A sin f

ð1:41Þ

v0 ¼ Avn cos f

ð1:42Þ

Now divide Equation 1.41 by Equation 1.42, and also use the fact that sin2 f þ cos2 f ¼ 1: We obtain x tan f ¼ vn 0 v0 x0 A Hence,

2

þ

2

v0 Avn

¼1

sffiffiffiffiffiffiffiffiffiffiffi v2 Amplitude A ¼ x02 þ 02 vn Phase f ¼ tan

1

vn x0 v0

ð1:43Þ ð1:44Þ

Example 1.1 A simple model for a tracked gantry conveyor system in a factory is shown in Figure 1.7. The carriage of mass ðmÞ moves on a frictionless track. The pulley is supported on frictionless bearings, and its axis of rotation is fixed. Its moment of inertia about this axis is J: The motion of the carriage is restrained by a spring of stiffness k1 ; as shown. The belt segment that drives the carriage runs over the

1 10


pulley without slippage, and is attached at the other end to a fixed spring of stiffness k2 : The displacement of the mass is denoted by x; and the corresponding rotation of the pulley is denoted by u: When x ¼ 0 (and u ¼ 0) the springs k1 and k2 have an extension of x10 and x20 , respectively, from their unstretched (free) configurations. Assume that the springs will remain in tension throughout the motion of the system.

k2 x k1

ðiÞ

F2 ¼ k2 ðx20

ðiiÞ

xÞ F1

ðiiiÞ

J u€ ¼ rF2

rF

ðivÞ

m

k2

r

A tracked conveyor system.

F2

F2

FIGURE 1.8 system.

θ

J m

F1

k1

Frictionless Bearings

Frictionless No-Slip Belt Rollers And Pulleys

x

Newton’s Second Law for the inertia elements: m€x ¼ F

J

FIGURE 1.7

Solution Using Newton’s Second Law A free body diagram for the system is shown in Figure 1.8. Hooke’s Law for the spring elements: F1 ¼ k1 ðx10 þ xÞ

θ

r

F

A free body diagram for the conveyor

Compatibility: x ¼ ru

ðvÞ

Straightforward elimination of F1 ; F2 ; F; and u in Equation i to Equation v, using algebra, gives mþ

J x€ þ ðk1 þ k2 Þx ¼ k2 x20 r2

k1 x10

ðviÞ

It follows that J r2 Equivalent stiffness keq ¼ k1 þ k2 Equivalent mass meq ¼ m þ

Solution Using Conservation of Energy 1 2 1 _2 m_x þ J u 2 2 1 1 Potential energy ðelasticÞ V ¼ k1 ðx10 þ xÞ2 þ k2 ðx20 2 2 Kinetic energy T ¼

xÞ2

Total energy in the system: E ¼TþV ¼

1 1 1 1 m_x 2 þ J u_ 2 þ k1 ðx10 þ xÞ2 þ k2 ðx20 2 2 2 2

Differentiate with respect to time: m_xx€ þ J u_u€ þ k1 ðx10 þ xÞ_x

k2 ðx20

xÞ_x ¼ 0

Substitute the compatibility relation, x_ ¼ ru_; to obtain m_xx€ þ

J x_ x€ þ k1 ðx10 þ xÞ_x r2

k2 ðx20

xÞ_x ¼ 0

xÞ2 ¼ const:


1 11

Eliminate the common velocity variable x_ (which cannot be zero for all t). We obtain mþ


k1 x10

which is the same result as before.

Solution Using Lagrange’s Equations Lagrangian L ¼ T

V¼

1 2 1 _2 m_x þ J u 2 2

1 k ðx þ xÞ2 2 1 10

1 k ðx 2 2 20

xÞ2

Now, substituting for u in terms of x (Equation v), we obtain L¼

1 J m þ 2 x_ 2 2 r

1 k ðx þ xÞ2 2 1 10

1 k ðx 2 2 20

xÞ2

Use Lagrange’s equation (see Box 1.1): d ›L dt ›q_ i

›L ¼ Qi ›qi

for i ¼ 1; 2; …; n

where the generalized coordinate qi ¼ x and the corresponding generalized force Qi ¼ 0 because there are no nonconservative and external forces. We obtain

›L J ¼ m þ 2 x_ ›x_ r ›L ¼ ›x

k1 ðx10 þ xÞ þ k2 ðx20

xÞ

From this, we obtain the equation of motion: mþ


k1 x10

which is identical to what we obtained before.

Natural Frequency From the equivalent translational system, the natural frequency (undamped) of the system is obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi J vn ¼ keq =meq ¼ ðk1 þ k2 Þ m þ 2 ðviiÞ r

=

Substitute for x and its derivatives into Equation vi using the compatibility condition (Equation v). We obtain the equivalent rotational system: ðr 2 m þ JÞu€ þ r2 ðk1 þ k2 Þu ¼ rk2 x20 rk1 x10 The equivalent moment of inertia: Jeq ¼ r2 m þ J The equivalent torsional stiffness: Keq ¼ r2 ðk1 þ k2 Þ Therefore, the corresponding natural frequency is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J vn ¼ r2 k1 þ k2 = r 2 m þ J ¼ ðk1 þ k2 Þ m þ 2 r

=

ðviiiÞ

This result is identical to the previous result (vii). This is to be expected, as the system has not changed (only the response variable was changed).

1 12


Common methods of developing equations of motion for mechanical systems are summarized in Box 1.1.

Box 1.1 APPROACHES FOR DEVELOPING EQUATIONS OF MOTION 1. Conservative Systems (No Nonconservative Forces/No Energy Dissipation): Kinetic energy ¼ T Potential energy ¼ V Conservation of energy: T þ V ¼ const: Differentiate with respect to time t 2. Lagrange’s Equations: Lagrangian L ¼ T V d ›L ›L ¼ Qi dt ›q_ i ›qi

for i ¼ 1; 2; …; n

n ¼ number of DoFs Qi ¼ generalized force corresponding to generalized coordinate qi : Find Qi using: X dW ¼ Qi dqi where dW ¼ work done by nonconservative forces in a general incremental motion ðdq1 ; dq2 ; …; dqn Þ: 3. Newtonian Approach: X dX Forces ¼ Linear Momentum dt X dX Torques ¼ Angular Momentum dt (About centroid or a fixed point)

1.3

Heavy Springs

A heavy spring has its mass and flexibility properties continuously distributed throughout its body. In that sense it has an infinite number of DoF(s), and a single coordinate cannot represent its motion. However, for many practical purposes, a lumped parameter approximation with just one lumped mass to represent the inertial characteristics of the spring may be sufficient. Such an approximation may be obtained by using the energy approach. Here, we represent the spring by a lumped parameter “model” such that the original spring and the model have the same net kinetic energy and potential energy. This energy equivalence is used in deriving a lumped mass parameter for the model. Even though damping (energy dissipation) is neglected in the present analysis, it is not difficult to incorporate that as well in the model.


1.3.1

1 13

Kinetic Energy Equivalence

Consider the uniform, heavy spring shown in Figure 1.9, with one end fixed and the other end moving at velocity v: Note that:

v

k, ms x

δx l

k ¼ stiffness of spring ms ¼ mass of spring l ¼ length of spring

FIGURE 1.9

A uniform heavy spring.

Local speed of element dx of the spring is given by ðx=lÞv: Element mass ¼ ðms =lÞdx: Hence, Element KE ¼

1 ms x dx v 2 l l

2

In the limit, we have dx ! dx: Then, Total KE ¼

ðl 1 m x 2 1 m s v 2 ðl 2 1 ms v 2 s x dx ¼ dx v ¼ l 2 l3 2 3 0 2 l 0

ð1:45Þ

Hence, the equivalent lumped mass concentrated at the free end ¼ ð1=3Þ £ spring mass: Note: This derivation assumes that one end of the spring is fixed. Furthermore, the conditions are assumed to be uniform along the spring. An example of utilizing this result is shown in Figure 1.10. Here, a system with a heavy spring and a lumped mass is approximated by a light spring (having the same stiffness) and a lumped mass. Another example is shown in Figure 1.11. In this case, it is not immediately clear which of the approximations shown on the right hand side is most appropriate.

Example 1.2 A uniform heavy spring of mass ms and stiffness k is attached at one end to a mass m that is free to roll on a frictionless horizontal plane. The other end is anchored to a vertical post. A schematic diagram of this arrangement is shown in Figure 1.12. k, ms

FIGURE 1.10

m1

m

k

m+

=

ms 3

Lumped parameter approximation for an oscillator with heavy spring.

k, ms

m2

=

m 1+

ms 6

k

m1+

ms 3

k

m1

k

m2 +

ms 6

m2

m2 +

ms 3

etc. FIGURE 1.11

An example where lumped parameter approximation for spring is ambiguous.

1 14


The unstretched length of the spring is l: Assume that, when the velocity of the connected mass is v; the velocity distribution along the spring is given by px vs ðxÞ ¼ v sin 2l where x is the distance of a point along the spring, as measured from the fixed end. We will determine an equivalent lumped mass located at the moving end of the spring (i.e., at the moving mass m) to represent the inertia effects of the spring. Consider an element of length dx at location x of the spring. Since the spring is uniform, we have element mass ¼ ðms =lÞdx: Also, according to the given assumption, element velocity ¼ vsinðpx=2lÞ: Hence, kinetic energy of the spring is ðl 1 px v sin 2l 0 2 ¼

1 ms 2 v x 4 l

2

Velocity v l ms, k Mass m Heavy Spring x

FIGURE 1.12 stock.

A heavy spring connected to a rolling

ms 1 ms 2 ðl 1 m s 2 ðl px dx ¼ v sin2 v 1 dx ¼ l 2 l 4 l 2l 0 0

px l sin l p

l 0

¼

cos

px dx l

1 ms 2 1 ms 2 v l¼ v 4 l 2 2

It follows that the equivalent lumped mass to be located at the moving end of the spring is ms =2: This result is valid only for the assumed velocity distribution, and corresponds to the first mode of motion only. In fact, a linear velocity distribution would be more realistic in this low frequency (quasi static motion) region, which will give an equivalent lumped mass of ð1=3Þmsp ; as we have seen before. Such ffiffiffiffiffiffi approximations will not be valid for high frequencies say, higher than k=ms .

1.4

Oscillations in Fluid Systems

Fluid systems can undergo oscillations (vibrations) quite analogous to mechanical and electrical systems. Again, the reason for their natural oscillation is the ability to store and repeatedly interchange two types of energy kinetic energy and potential energy. The kinetic energy comes from the velocity of fluid particles during motion. The potential energy arises primarily from the following three main sources: 1. Gravitational potential energy 2. Compressibility of the fluid volume 3. Flexibility of the fluid container A detailed analysis of these three effects is not undertaken here. However, we have seen from the example in Figure 1.5(e) how a liquid column can oscillate due to repeated interchange between kinetic energy and gravitational potential energy. Now, let us consider another example.

Example 1.3 Consider a cylindrical wooden peg of uniform cross section and height h; floating in a tank of water, as in Figure 1.13(a). It is pushed by hand until completely immersed in water, in an upright orientation. When released, the object will oscillate up and down while floating in the tank. Let r b and r l be the mass


1 15

(a)

M

y C

R = mg

mg (b)

(c)

C = body centroid M = metacenter

FIGURE 1.13 (a) A buoyancy experiment; (b) upright oscillations of the body; (c) restoring buoyancy couple due to a stable metacenter.

densities of the body (peg) and the liquid (water), respectively. The natural oscillations and the stability of this system may be studied as below. Suppose that, under equilibrium in the upright position of the body, the submersed length is l: The mass of the body is m ¼ Ahr b

ðiÞ

where A is the area of cross section (uniform). By the Archimedes principle, the buoyancy force R is equal to the weight of the liquid displaced by the body. Hence, R ¼ Alr l g

ðiiÞ

R ¼ mg

ðiiiÞ

For equilibrium, we have

or Alr l g ¼ Ahr b g Hence, l¼

rb h for r b , r l rl

ðivÞ

For a vertical displacement y from the equilibrium position, the equation of motion is (Figure 1.13(b)) m€y ¼ mg

Aðl þ yÞr l g

Substitute Equation ii and Equation iii. We obtain m€y ¼

Ar l gy

Substitute Equation i: Ahr b y€ þ Ar l gy ¼ 0

1 16


or y€ þ

rl g y¼0 r bh

The natural frequency of oscillations is

vn ¼

rffiffiffiffiffiffi rl g rb h

Note that this result is independent of the area of the cross section of the body. Assumptions made: 1. The tank is very large compared to the body. The change in liquid level is negligible as the body is depressed into the water. 2. Fluid resistance (viscous effects, drag, etc.) is negligible. 3. Dynamics of the liquid itself is negligible. Hence, “added inertia” due to liquid motion is neglected. To study stability of the system, note that the buoyancy for R acts through the centroid of the volume of displaced water (Figure 1.13(c)). Its line of action passes through the central axis of the body at point M. The point is known as the metacenter. Let C be the centroid of the body. If M is above C; then, when tilted, there will be a restoring couple that will tend to restore the body to its upright position. Otherwise the body will be in an unstable situation, and the buoyancy couple will tend to tilt it further towards a horizontal configuration.

1.5

Damped Simple Oscillator

Now we will consider free (natural) response of a simple oscillator in the presence of energy dissipation (damping). Assume viscous damping, and consider the oscillator shown in Figure 1.14. The free body diagram of the mass is shown separately. We use the following notation:

vn ¼ undamped natural frequency vd ¼ damped natural frequency vr ¼ resonant frequency v ¼ frequency of excitation

x k m b

.. x kx . bx

m

The concept of resonant frequency will be addressed in Chapter 2. FIGURE 1.14 A damped simple oscillator and its free Usually, the viscous damping constant of a single body diagram. DoF is denoted by b (but, sometimes c is used instead of b; particularly for multi DoF systems). Apply Newton’s Second Law. From the free body diagram in Figure 1.14, we have the equation of motion m€x ¼ kx b_x or m€x þ b_x þ kx ¼ 0

ð1:46Þ

x€ þ 2zvn x_ þ v2n x ¼ 0

ð1:47Þ

or


1 17

This is a free (or unforced, or homogeneous) equation of motion. Its solution ispthe free (natural) response ffiffiffiffiffi of the system and is also known as the homogeneous solution. Note that vn ¼ k=m; which is the natural frequency when there is no damping, and 2zvn ¼

b m

ð1:48Þ

Hence,

z¼

1 2

rffiffiffiffi m b k m

or

z¼

1 b pffiffiffiffi 2 km

ð1:49Þ

Note that z is called the damping ratio. The formal definition of, and the rationale for, this terminology will be discussed later. Assume an exponential solution: x ¼ C elt

ð1:50Þ

This is justified by the fact that linear systems have exponential or oscillatory (i.e., complex exponential) free responses. A more detailed justification will be provided later. Substitute Equation 1.50 into Equation 1.47 to obtain ½l2 þ 2zvn l þ v2n C elt ¼ 0 Note that C elt is not zero in general. It follows that, when l satisfies the equation

l2 þ 2zvn l þ v2n ¼ 0

ð1:51Þ

then Equation 1.50 will represent a solution of Equation 1.47. Equation 1.51 is called the characteristic equation of the system. This equation depends on the natural dynamics of the system, not on forcing excitation or initial conditions. The solution of Equation 1.51 gives the two roots: qffiffiffiffiffiffiffiffi ð1:52Þ l ¼ zvn ^ z2 1vn ¼ l1 and l2 These are called eigenvalues or poles of the system. When l1 – l2 ; the general solution is x ¼ C1 el1 þ C2 el2 t

ð1:53Þ

The two unknown constants C1 and C2 are related to the integration constants and can be determined by two initial conditions which should be known. If l1 ¼ l2 ¼ l; we have the case of repeated roots. In this case, the general solution (Equation 1.53) does not hold because C1 and C2 would no longer be independent constants to be determined by two initial conditions. The repetition of the roots suggests that one term of the homogenous solution should have the multiplier t (a result of the double integration of zero). Then the general solution is x ¼ C1 elt þ C2 t elt

ð1:54Þ

We can identify three categories of damping level, as discussed below, and the nature of the response will depend on the particular category of damping.

1 18

1.5.1


Case 1: Underdamped Motion (z < 1)

In this case it follows from Equation 1.52 that the roots of the characteristic equation are qffiffiffiffiffiffiffiffi l ¼ zvn ^ j 1 z2 vn ¼ zvn ^ jvd ¼ l1 and l2

ð1:55Þ

where the damped natural frequency is given by

qffiffiffiffiffiffiffiffi vd ¼ 1 z2 vn

ð1:56Þ

Note that l1 and l2 are complex conjugates. The response (Equation 1.53) in this case may be expressed as h i ð1:57Þ x ¼ e zvn t C1 e jvd t þ C2 e jvd t The term within the square brackets of Equation 1.57 has to be real, because it represents the time response of a real physical system. It follows that C1 and C2 have to be complex conjugates. Note: e jvd t ¼ cos vd t þ j sin vd t e

jvd t

¼ cos vd t

j sin vd t

So, an alternative form of the general solution would be x¼e

zvn t

½A1 cos vd t þ A2 sin vd t

ð1:58Þ

Here, A1 and A2 are the two unknown constants. By equating the coefficients it can be shown that A1 ¼ C1 þ C2 and A2 ¼ jðC1

C2 Þ

ð1:59Þ

Hence, C1 ¼ 1.5.1.1

1 ðA 2 1

jA2 Þ and C2 ¼

1 ðA þ jA2 Þ 2 1

ð1:60Þ

Initial Conditions

Let xð0Þ ¼ x0 ; x_ ð0Þ ¼ v0 as before. Then, x0 ¼ A1

and

v0 ¼

zvn A1 þ vd A2

ð1:61Þ

or A2 ¼

v0 zvn x0 þ vd vd

ð1:62Þ

zvn t

ð1:63Þ

Yet, another form of the solution would be x¼Ae

Here, A and f are the unknown constants with qffiffiffiffiffiffiffiffiffiffi A ¼ A21 þ A22 and

sinðvd t þ fÞ A1 sin f ¼ pffiffiffiffiffiffiffiffiffiffi 2 A1 þ A22

ð1:64Þ

Also, A2 cos f ¼ pffiffiffiffiffiffiffiffiffiffi A21 þ A22

and

tan f ¼

A1 A2

Note that the response x ! 0 as t ! 1: This means the system is asymptotically stable.

ð1:65Þ


1.5.2

1 19

Logarithmic Decrement Method

The damping ratio z may be experimentally determined from the free response by the logarithmic decrement method. To illustrate this approach, note from Equation 1.63 that the period of damped oscillations is T¼

2p vd

Ae

zvn t

ð1:66Þ

Also, from Equation 1.63, we have xðtÞ ¼ xðt þ nTÞ Ae

zvn ðtþnTÞ

sinðvd t þ fÞ sin½vd ðt þ nTÞ þ f

But, sin½vd ðt þ nTÞ þ f ¼ sinðvd t þ f þ 2npÞ ¼ sinðvd þ fÞ: Hence, xðtÞ e zvn t ¼ zv ðtþnTÞ ¼ ezvn nT xðt þ nTÞ e n

ð1:67Þ

Take the natural logarithm of Equation 1.67, the logarithmic decrement,

zvn nT ¼ ln

xðtÞ xðt þ nTÞ

However,

vn T ¼ vn

vn 2p 2p 2p ¼ pffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffi 2 vd 1 z vn 1 z2

Hence, with xðtÞ=xðt þ nTÞ ¼ r we have the logarithmic decrement. 2pnz pffiffiffiffiffiffiffiffi ¼ ln r 1 z2 Note that ð1=nÞln r is the “per cycle” logarithmic decrement and ð1=2pnÞln r is the “per radian” logarithmic decrement. The latter is

z 1 pffiffiffiffiffiffiffiffi ¼ ln r ¼ a 2 p n 2 1 z Then, we have

sffiffiffiffiffiffiffiffiffiffi a2 z¼ 1 þ a2

ð1:68Þ

ð1:69Þ

This is the basis of the logarithmic decrement method of measuring damping. Start by measuring a point xðtÞ and another point xðt þ nTÞ at n cycles later. For high accuracy, pick the peak points of the response curve for the measurement of xðtÞ and xðt þ nTÞ: From Equation 1.68 it is clear that, for small damping, z ¼ a ¼ per radian logarithmic decrement.

1.5.3

Case 2: Overdamped Motion (z > 1)

In this case, roots l1 and l2 of the characteristic equation (Equation 1.51) are real. Specifically, we have qffiffiffiffiffiffiffiffi ð1:70Þ l1 ¼ zvn þ z2 1vn , 0

l2 ¼

zvn

qffiffiffiffiffiffiffiffi z2 1vn , 0

ð1:71Þ

1 20


and the response equation (Equation 1.53) is nonoscillatory. Also, it should be clear from Equation 1.70 and Equation 1.71 that both l1 and l2 are negative. Hence, x ! 0 as t ! 1: This means the system is asymptotically stable. From the initial conditions xð0Þ ¼ x0 ; x_ ð0Þ ¼ v0 we obtain x0 ¼ C1 þ C2

ðiÞ

v0 ¼ l1 C1 þ l2 C2

ðiiÞ

and Multiply the first initial condition (Equation i) by l1 :

l1 x0 ¼ l1 C1 þ l1 C2

ðiiiÞ

Subtract Equation iii from Equation ii: v0

l1 x0 ¼ C2 ðl2

l1 Þ

We obtain C2 ¼

v0 l1 x0 l2 l1

ð1:72Þ

Similarly, multiply the first initial condition (Equation i) by l2 and subtract from Equation ii. We obtain v0

l2 x0 ¼ C1 ðl1

l2 Þ

Hence, C1 ¼

1.5.4

v0 l2 x0 l1 l2

ð1:73Þ

Case 3: Critically Damped Motion (z 5 1)

Here, we have repeated roots, given by

l1 ¼ l2 ¼

vn

ð1:74Þ

The response for this case is given by (see Equation 1.54) x ¼ C1 e

vn t

þ C2 t e

vn t

vn t

goes to zero faster than t goes to infinity, we have t e Since the term e system is asymptotically stable. Now use the initial conditions xð0Þ ¼ x0 ; x_ ð0Þ ¼ v0 : We obtain

ð1:75Þ vn t

! 0 as t ! 1: Hence, the

x0 ¼ C1 v0 ¼

vn C1 þ C2

Hence, C1 ¼ x0

ð1:76Þ

C2 ¼ v0 þ vn x0

ð1:77Þ

Note: When z ¼ 1 we have the critically damped response because below this value the response is oscillatory (underdamped) and above this value, the response is nonoscillatory (overdamped). It follows


1 21

that we may define the damping ratio as

z ¼ damping ratio ¼

1.5.5

damping constant damping constant for critically damped conditions

ð1:78Þ

Justification for the Trial Solution

In the present analysis, the trial solution (Equation 1.50) has been used for the response of a linear system having constant parameter values. A justification for this is provided now. 1.5.5.1

First-Order System

Consider a first order (homogeneous, no forcing input) linear system given by d dt

l x ¼ x_

lx ¼ 0

ð1:79Þ

This equation can be written as dx ¼ l dt x Integrate: ln x ¼ lt þ ln C Here, ln C is the constant of integration. Hence, x ¼ C elt

ð1:80Þ

This is then the general form of the free response of a first order system. It incorporates one constant of integration, and hence will need one initial condition. 1.5.5.2

Second-Order System

We can write the equation of a general second order (homogeneous, unforced) system in the operational form: d dt

l1

d dt

l2 x ¼ 0

ð1:81Þ

By reasoning as before, the general solution would be of the form x ¼ C1 el1 t þ C2 el2 t : Here, C1 and C2 are the constants of integration, which are determined using two initial conditions. 1.5.5.3

Repeated Roots

The case of repeated roots deserves a separate treatment. First, consider d2 x ¼0 dt 2

ð1:82Þ

Integrate twice: dx ¼ C; x ¼ Ct þ D dt Note the term with t in this case. Hence, a suitable trial solution for the system d dt

l

d dt

l x¼0

would be x ¼ C1 elt þ C2 t elt : The main results for the free (natural) response of a damped oscillator are given in Box 1.2.

ð1:83Þ

ð1:84Þ

1 22


Box 1.2 FREE (NATURAL ) RESPONSE OF A DAMPED SINGLE OSCILLATOR System Equation: m€x þ b_x þ kx ¼ 0 orpx€ffiffiffiffiffi þ 2zvn x_ þ v2n x ¼ 0 Undamped natural frequency pffiffiffiffi vn ¼ k=m Damping ratio z ¼ b=2 km Characteristic Equation: l2 þ 2zvn l þ v2n ¼ 0 pffiffiffiffiffiffiffiffi Roots (Eigenvalues or Poles): l1 and l2 ¼ zvn ^ z2 1vn Response: x ¼ C1 el1 t þ C2 el2 t for unequal roots ðl1 – l2 Þ x ¼ ðC1 þ C2 tÞelt for equal roots ðl1 ¼ l2 ¼ lÞ Initial Conditions: xð0Þ ¼ x0 and x_ ð0Þ ¼ v0 Case 1: Underdamped (z < 1) Poles are complex conjugates: zv n ^ jvd pffiffiffiffiffiffiffiffi Damped natural frequency vd ¼ 1 z2 vn x¼e

zvn t

½C1 e jvd t þ C2 e

j vd t

¼e

zvn t

A1 ¼ C1 þ C2 C1 ¼

½A1 cos vd t þ A2 sin vd t ¼ A e

and

A2 ¼ jðC1

1 2

C2 Þ

1 2

ðA1 jA2 Þ and C2 ¼ ðA1 þ jA2 Þ qffiffiffiffiffiffiffiffiffiffi A A ¼ A21 þ A22 and tan f ¼ 1 A2

Initial conditions give: A1 ¼ x0 and A2 ¼

v0 þ zvn x0 vd

Logarithmic Decrement per Radian:

a¼

z 1 ln r ¼ pffiffiffiffiffiffiffiffi 2pn 1 z2

where r ¼ xðtÞ=½xðt þ nTÞ ¼ decay ratio over n complete cycles. For small z: z ø a Case 2: Overdamped (z > 1) pffiffiffiffiffiffiffiffi Poles are real and negative: l1 ; l2 ¼ zvn ^ z2 1vn x ¼ C1 el1 t þ C2 el2 t

C1 ¼

v0 l2 x0 l1 l2

and

C2 ¼

v0 l1 x0 l2 l1

Case 3: Critically Damped (z 5 1) Two identical poles: l1 ¼ l2 ¼ l ¼ vn x ¼ ðC1 þ C2 tÞe vn t with C1 ¼ x0 and C2 ¼ v0 þ vn x0

zvn t

sinðvd t þ fÞ


1.5.6

1 23

Stability and Speed of Response

The free response of a dynamic system, particularly a vibrating system, can provide valuable information concerning the natural characteristics of the system. The free (unforced) excitation may be obtained, for example, by giving an initial condition excitation to the system and then allowing it to respond freely. Two important characteristics which can be determined in this manner are: 1. Stability 2. Speed of response The stability of a system implies that the response will not grow without bounds when the excitation force itself is finite. This is known as bounded input bounded output (BIBO) stability. In particular, if the free response eventually decays to zero, in the absence of a forcing input, the system is said to be asymptotically stable. We have seen that a damped simple oscillator is asymptotically stable, but an undamped oscillator, while being stable in a general (BIBO) sense, is not asymptotically stable. The speed of response of a system indicates how fast the system responds to an excitation force. It is also a measure of how fast the free response (1) rises or falls if the system is oscillatory; or (2) decays, if the system is nonoscillatory. Hence, the two characteristics, stability and speed of response, are not completely independent. In particular, for nonoscillatory (overdamped) systems these two properties are very closely related. It is clear then that stability and speed of response are important considerations in the analysis, design, and control of vibrating systems. The level of stability of a linear dynamic system depends on the real parts of the eigenvalues (or poles), which are the roots of the characteristic equation. Specifically, if all the roots have real parts that are negative, then the system is stable. Also, the more negative the real part of a pole, the faster the decay of the free response component corresponding to that pole. The inverse of the negative real part is the time constant. The smaller the time constant, the faster the decay of the corresponding free response, and hence, the higher the level of stability associated with that pole. We can summarize these observations as follows: Level of stability. Depends on decay rate of free response (and hence on time constants or real parts of poles) Speed of response. Depends on natural frequency and damping for oscillatory systems and decay rate for nonoscillatory systems Time constant. Determines stability and decay rate of free response (and speed of response in nonoscillatory systems) Now let us consider the specific case of a damped simple oscillator given by Equation 1.47.

Case 1 (z < 1) The free response is given by x ¼ A e

zvn t

sinðvd t þ fÞ

Time constant t ¼

1 zvn

ð1:85Þ

The system is asymptotically stable. The larger zvn ; the more stable the system. Also, the speed of response increases with both vd and zvn :

Case 2 (z > 1) The response is nonoscillatory, and is given by

where l1 ¼

x ¼ A1 el1 t ðdecays slowerÞ þ A2 el2 t ðdecays fasterÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi zvn þ z2 1vn and l2 ¼ zvn z2 1vn :

1 24


This system has two time constants:

t1 ¼

1 1 and t2 ¼ ll1 l ll2 l

ð1:86Þ

Note that t1 is the dominant (slower) time constant. The system is also asymptotically stable. The larger the ll1 l the faster and more stable the system. Consider an underdamped system and an overdamped system with damping ratios zu and zo ; respectively. We can show that the underdamped system is more stable than the overdamped system if and only if qffiffiffiffiffiffiffiffi ð1:87aÞ zo z2o 1 , zu or equivalently,

zo .

z2u þ 1 2zu

ð1:87bÞ

where zo . 1 . zu . 0 by definition.

Proof To be more stable, we should have the underdamped pole located farther away than the dominant overdamped pole from the imaginary axis of the pole plane; thus qffiffiffiffiffiffiffiffi zu vn . zo vn z2o 1vn Hence,

zu . zo

qffiffiffiffiffiffiffiffi z2o 1

Now, bring the square root term to the left hand side and square it:

z2o

1 . ðzo

zu Þ2 ¼ z2o

2zo zu þ z2u

Hence, 2zo zu . z2u þ 1 or

zo . This completes the proof.

z2u þ 1 2zu

A

To explain this result further, consider an undamped ðz ¼ 0Þ simple oscillator of natural frequency vn : Its poles are at ^jvn (on the imaginary axis of the pole plane). Now let us add damping and increase z from 0 to 1. Then the complex conjugates poles zvn ^ jvd will move away from the imaginary axis as z increases (because zvn increases) and hence the level of stability will increase. When z reaches the value 1 (critical damping) we obtain two identical and real poles at vn : When z is increased beyond 1, the poles will be real and unequal, with one pole having a magnitude smaller than vn and the other having a magnitude larger than vn : The former (closer to the “origin” of zero) is the dominant pole, and will determine both stability and the speed of response of the overdamped system. It follows that, as z increases beyond 1, the two poles will branch out from the location vn ; one moving towards the origin (becoming less stable) and the other moving away from the origin. It is now clear that as z is increased beyond the point of critical damping, the system becomes less stable. Specifically, for a given value of zu , 1; there is a value of zo . 1; governed by Equation 1.87, above which the overdamped system is less stable and slower than the underdamped system.


1 25

Example 1.4 Consider the simple oscillator shown in Figure 1.14, with parameters m ¼ 4 kg; k ¼ 1:6 £ 103 N=m; and the two cases of damping: 1. b ¼ 80 N=m=sec 2. b ¼ 320 N=m=sec We will study the nature of the free response in each case. The undamped natural frequency of the system is sffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi k 1:6 £ 103 ¼ vn ¼ rad=sec ¼ 20:0 rad=sec m 4

Case 1 2zvn ¼ Then,

b m

or 2z £ 20 ¼

80 4

zu ¼ 0:5 The system is underdamped in this case.

Case 2 2z £ 20 ¼ Then,

320 4

zo ¼ 2:0 The system is overdamped in this case.

Case 3 The characteristic equation is l2 þ 2 £ 0:5 £ 20l þ 202 ¼ 0 or

l2 þ 20l þ 202 ¼ 0 The roots (eigenvalues or poles) are

l¼

pffiffiffiffiffiffiffiffiffiffiffiffi 10 ^ j 202 102 ¼

The free (no force) response is given by x¼Ae

10t

pffiffi 10 ^ j10 3

pffiffi sin 10 3t þ f

The amplitude A and the phase angle f can be determined using initial conditions. 1 ¼ 0:1 sec Time constant t ¼ 10

Case 4 The characteristic equation is

l2 þ 2 £ 2 £ 20l þ 202 ¼ 0

1 26


or

Response

l2 þ 80l þ 202 ¼ 0 The roots are

l¼ ¼

pffiffiffiffiffiffiffiffiffiffiffiffi 40 ^ 402 202 ¼ 5:36;

1

pffiffi 40 ^ 20 3

74:64

e−5.36t

The free response is given by x ¼ C1 e

5:36t

þ C2 e

74:64t

The constants C1 and C2 can be determined using initial conditions. The second term on the right hand side goes to zero much faster than the first term, as shown in Figure 1.15. Hence, the first term will dominate and will determine the dominant time constant, level of stability, and speed of response. Specifically, the response may be approximated as x ø C1 e

e−74.64t 0

t

FIGURE 1.15 The free (homogeneous) response components of an overdamped system.

5:36t

Hence, 1 ¼ 0:19 sec 5:36 This value is double that of Case 1. Consequently, it is clear that the underdamped system (Case 1) decays faster than the overdamped system (Case 2). In fact, according to Equation 1.87b, with zu ¼ 0:5 we have Time constant t ¼

0:52 þ 1 ¼ 1:25 2 £ 0:5 Hence, an overdamped system of damping ratio greater than 1.25 will be less stable than the underdamped system of damping ratio 0.5.

zu .

Table 1.1 summarizes some natural characteristics of a damped simple oscillator under three different levels of damping. The nature of the natural response for these three cases is sketched in Figure 1.16. In general, the natural response of a system is governed by its eigenvalues (or poles), which are the roots of the characteristic equation. The poles may be marked on a complex plane (s plane), with the horizontal axis representing the real part and the vertical axis representing the imaginary part. The nature of the free response depending on the pole location of the system is shown in Figure 1.17. TABLE 1.1

Natural Characteristics of a Damped Oscillator Damping Ratio

Level of damping Oscillatory response Stability

Speed of response Time constant

z,1

z.1

z¼1

Underdamped Yes Asymptotically stable (less stable than z ¼ 1 case but not necessarily less stable than the overdamped case) Better than overdamped 1=ðzvn Þ

Overdamped No Asymptotically stable (less stable than the critically damped case)

Critically damped No Asymptotically stable (most stable)

Lower than critical pffiffiffiffiffiffiffiffi 1=ðzvn ^ z2 2 1vn Þ

Good 1=vn


1 27

x(m) 0.20

ζ1

0.10 0.00

0

−0.10

0.5

1.0 t(s)

−0.20 FIGURE 1.16

1.6

Free response of a damped oscillator: (a) underdamped; (b) critically damped; (c) overdamped.

Forced Response

The free response of a vibratory system is the response to some initial excitation and in the absence of any subsequent forcing input. This corresponds to the “natural” response of the system. Mathematically, it is the homogeneous solution, because it is obtained by solving the homogeneous equation of the system (i.e., without the input terms). The natural response, the free response, and the homogeneous solution are synonymous, in the absence of a forcing input to the system. The forced response of a dynamic (vibratory) system is the response of the system to a forcing input. When there is a forcing excitation (i.e., an input) on a system, the equation of motion will be nonhomogeneous (i.e., the right hand side will not be zero). Then, the total solution (total response T) will be given by the sum of the homogeneous solution ðHÞ and the particular integral ðPÞ; subject to the system initial conditions. This may be determined by the mathematical solution of the equation of motion. The total response can be separated into the terms that depend on the initial conditions ðXÞ and the terms that depend on the forcing excitation ðFÞ: This is in fact the physical interpretation of the total solution. Note that X is called the “free response,” “initial condition response,” or the “zero input

1 28


Im

s-Plane (Eigenvalue Plane) E

A

C

B

A

Re

D

C E

FIGURE 1.17 Dependence of free response (or stability) on the pole location. (A and B are stable; C is marginally stable; D and E are unstable.)

response.” The term F is called the “forced response” or the “zero initial condition response” or the “zero state response.” In general H is not identical to X and P is not identical to F: But, when there is no forcing excitation (no input), by definition H and X will be identical. Furthermore, under steady state conditions the homogeneous part or initial condition response will die down (assuming that there is some damping, and the system is stable). Then, F will become equal to P: Note that, even when the initial conditions are zero, F and P may not be identical because F may contain a natural response component that is excited by the forcing input. This component will die out with time, however. The total response will depend on the natural characteristics of the system (as for the free response) and also on the nature of the forcing excitation. Mathematically, then, the total response will be determined by both the homogeneous solution and the particular solution. The complete solution will require a knowledge of the input (forcing excitation) and the initial conditions. The behavior of a dynamic system when subjected to a certain forcing excitation may be studied by analyzing a model of the system. This is commonly known as system response analysis. System response may be studied either in the time domain, where the independent variable of the system response is time, or in the frequency domain, where the independent variable of the system response is frequency. Time domain analysis and frequency domain analysis are equivalent. Variables in the two domains are connected through the Fourier (integral) transform. The preference of one domain over the other depends on such factors as the nature of the excitation input, the type of the available analytical model, the time duration of interest, and the quantities that need to be determined. The frequency domain analysis will be addressed in Chapter 2.

1.6.1

Impulse-Response Function

Principle of superposition. Consider a linear dynamic (vibratory) system. The principle of superposition holds for a linear system. More specifically, if y1 is the system response to excitation u1 ðtÞ and y2 is the response to excitation u2 ðtÞ; then ay1 þ by2 is the system response to input au1 ðtÞ þ bu2 ðtÞ for any


1 29

u(t)

u(t)

1 ∆t

(a)

0 FIGURE 1.18

t

τ t + ∆t

τ

(b) 0

t

Illustrations of (a) unit pulse and (b) unit impulse.

constants a and b and any excitation functions u1 ðtÞ and u2 ðtÞ: This is true for both time variant parameter linear systems and constant parameter linear systems. Unit impulse. A unit pulse of width Dt starting at time t ¼ t is shown in Figure 1.18(a). Its area is unity. A unit impulse is the limiting case of a unit pulse when Dt ! 0: Unit impulse acting at time t ¼ t is denoted by dðt tÞ and is graphically represented as in Figure 1.18(b). In mathematical analysis, this is known as the Dirac delta function. It is mathematically defined by the two conditions: ( 0 for t – t ð1:88Þ dðt tÞ ¼ 1 for t ¼ t and ð1 1

dðt

tÞdt ¼ 1

ð1:89Þ

The Dirac delta function has the following well known and useful properties: ð1 1

f ðtÞdðt

tÞdt ¼ f ðtÞ

ð1:90Þ

and ð1 dn f ðtÞ n dðt 1 dt

tÞdt ¼

dn f ðtÞ dt n

t¼t

ð1:91Þ

for any well behaved time function f ðtÞ: Impulse response function. The system response (output) to a unit impulse excitation (input) acted at time t ¼ 0 is known as the impulse response function and is denoted by hðtÞ:

1.6.2

Forced Response

The system output to an arbitrary input may be expressed in terms of its impulse response function. This is the essence of the impulse response approach to determining the forced response of a dynamic system. Without loss of generality, assume that the system input uðtÞ starts at t ¼ 0; that is, uðtÞ ¼ 0

for t , 0

ð1:92Þ

For a physically realizable system, the response does not depend on the future values of the input. Consequently, yðtÞ ¼ 0

for t , 0

ð1:93Þ

1 30


y(t)

u(t)

0

t

u(t−τ)

0

0

t

y(t−τ)

τ

t FIGURE 1.19

0

τ

t

Response to a delayed input.

and hðtÞ ¼ 0 for t , 0

ð1:94Þ

where yðtÞ is the response of the system to any general excitation uðtÞ: Furthermore, if the system is a constant parameter system, then the response does not depend on the time origin used for the input. Mathematically, this is stated as follows: if the response to input uðtÞ satisfying Equation 1.92 is yðtÞ; which satisfies Equation 1.93, then the response to input uðt tÞ; which satisfies, is yðt

uðt

tÞ ¼ 0 for t , t

ð1:95Þ

yðt

tÞ ¼ 0 for t , t

ð1:96Þ

tÞ; and it satisfies

This situation is illustrated in Figure 1.19. It follows that the delayed impulse input dðt tÞ; having time delay t; produces the delayed response hðt tÞ: Convolution integral. A given input uðtÞ can be u(t) divided approximately into a series of pulses of width Dt and magnitude uðtÞ Dt: In Figure 1.20, Area = u(t).∆t as Dt ! 0; the pulse shown by the shaded area becomes an impulse acting at t ¼ t; having the magnitude ut dt: The value of this impulse is given by dðt tÞuðtÞdt: In a linear, constant parameter system, it produces the response hðt tÞuðtÞdt: By integrating over the entire time duration of the input uðtÞ; the overall response t t+∆t 0 t yðtÞ is obtained as ð1 hðt tÞuðtÞdt yðtÞ ¼ ð1:97Þ FIGURE 1.20 A general input treated as a continuous 0

series of impulses.


1 31

Equation 1.97 is known as the convolution integral. This is in fact the forced response, under zero initial conditions. In view of Equation 1.94, it follows that hðt tÞ ¼ 0 for t . t: Consequently, the upper limit of integration in Equation 1.97 could be made equal to t without affecting the result. Similarly, in view of Equation 1.92, the lower limit of integration in Equation 1.97 could be made 1. Furthermore, by introducing the change of variable t ! t t; an alternative version of the convolution integral is obtained. Several valid versions of the convolution integral (or response equation) for a linear, constant parameter system are given below: ð1 yðtÞ ¼ hðtÞuðt tÞdt ð1:97aÞ 0

yðtÞ ¼ yðtÞ ¼ yðtÞ ¼ yðtÞ ¼

ð1 1

ð1 1

ðt 1

ðt

yðtÞ ¼ yðtÞ ¼

1

ðt 0

ðt 0

tÞuðtÞdt

hðt

hðtÞuðt hðt

tÞdt

ð1:97cÞ

tÞuðtÞdt

ð1:97dÞ

hðtÞuðt

hðt

tÞdt

tÞuðtÞdt

hðtÞuðt

ð1:97bÞ

tÞdt

ð1:97eÞ ð1:97f Þ ð1:97gÞ

In fact, the lower limit of integration in the convolution integral could be any value satisfying t # 0; and the upper limit could be any value satisfying t $ t: The use of a particular pair of integration limits depends on whether the functions hðtÞ and uðtÞ implicitly satisfy the conditions given by Equation 1.93 and Equation 1.94, or whether these conditions have to be imposed on them by means of the proper integration limits. It should be noted that the two versions given by Equation 1.97f and Equation 1.97g take these conditions into account explicitly and therefore are valid for all inputs and impulse response functions. It should be emphasized that the response given by the convolution integral assumes a zero initial state, and is known as the zero state response, because the impulse response itself assumes a zero initial state. As we have stated, this is not necessarily equal to the “particular solution” in mathematical analysis. Also, as t increases ðt ! 1Þ; this solution approaches the steady state response denoted by yss ; which is typically the particular solution. The impulse response of a system is the inverse Laplace transform of the transfer function. Hence, it can be determined using Laplace transform techniques. This aspect will be addressed in Chapter 3. Some useful concepts of forced response are summarized in Box 1.3.

1.6.3

Response to a Support Motion

An important consideration in vibration analysis and in the testing of machinery and equipment is the response to a support motion. To illustrate the method of analysis, consider the linear, single DoF system consisting of mass m; spring constant k; and damping constant b; subjected to support motion (displacement) uðtÞ: Vertical and horizontal configurations of this system are shown in Figure 1.21. Both configurations possess the same equation of motion, provided the support motion uðtÞ and the mass response (displacement) y are measured from the fixed points that correspond to the initial, static equilibrium position of the system. In the vertical configuration, the compressive force in the spring exactly balances the weight of the mass when it is in static equilibrium. In the horizontal configuration, the spring is unstretched when in static equilibrium. It may be easily verified that the equation of motion

1 32


Box 1.3 CONCEPTS OF FORCED RESPONSE Total Response ðTÞ ¼ Homogeneous Solution ðHÞ þ Particular Integral ðPÞ ¼ Free Response ðXÞ þ Forced Response ðFÞ ¼ Initial Condition Response ðXÞ þ Zero Initial Condition Response ðFÞ ¼ Zero Input Response ðXÞ þ Zero State Response ðFÞ Note: In general, H – X and P – F With no input (no forcing excitation), by definition, H ; X At steady state, F becomes equal to P: Ð Ð Convolution Integral: Response y ¼ t0 hðt tÞuðtÞdt ¼ t0 hðtÞuðt tÞdt; where u ¼ excitation (input) and h ¼ impulse response function (response to a unit impulse input). Damped Simple Oscillator: y€ þ 2zvn y_ þ v2n y ¼ v2n uðtÞ 8 qffiffiffiffiffiffiffiffi < zv ^ z2 1v for z $ 1 n n Poles ðeigenvaluesÞ l1 ; l2 ¼ : for z a 1 zvn ^ jvd natural frequency, vd ¼ damped natural frequency, z ¼ damping ratio. vn ¼ undamped pffiffiffiffiffiffiffiffi Note: vd ¼ 1 z2 vn : Impulse-Response Function (Zero Initial Conditions): 8 vn > pffiffiffiffiffiffiffiffi expð zvn tÞsin vd t for z a 1 > > 1 z2 > > < vn hðtÞ ¼ ½exp l1 t exp l2 t for z s 1 > pffiffiffiffiffiffiffiffi > 2 z2 1 > > > : 2 for z ¼ 1 vn t expð vn tÞ Unit Step Response (Zero Initial Conditions): 8 1 > > 1 pffiffiffiffiffiffiffiffi expð zvn tÞsinðvd t þ fÞ > > > z2 1 > < 1 ystep ðtÞ ¼ > pffiffiffiffiffiffiffiffi ½l1 exp l2 t l2 exp l1 t 1 > 2 > 1v n 2 z > > > : 1 ðvn t þ 1Þexpð vn tÞ with cos f ¼ z Note: Impulse Response ¼

d ðStep ResponseÞ dt

for z a 1 for z s 1 for z ¼ 1


1 33

y(t) Equipment

m

Static Equilibrium Level

u(t)

y(t) k

k

b

m

u(t) b

Support Static Equilibrium Level

(a) FIGURE 1.21

(b)

A system subjected to support motion: (a) vertical configuration; (b) horizontal configuration.

is given by m€y þ b_y þ ky ¼ kuðtÞ þ b_uðtÞ

ð1:98Þ

in which ð_Þ ¼ d=dt and ð€Þ ¼ d2 =dt 2 : The two parameters vn and z are undamped natural frequency and pffiffiffiffiffi damping ratio, respectively, given by vn ¼ k=m and 2zvn ¼ b=m; as usual. This results in the equivalent equation of motion: y€ þ 2zvn y_ þ v2n y ¼ v2n uðtÞ þ 2zvn u_ ðtÞ

ð1:99Þ

There are several ways to determine the response y from Equation 1.99 once the excitation function uðtÞ is specified. The procedure used below is to first solve the modified equation: y€ þ 2zvn y_ þ v2n y ¼ v2n uðtÞ

ð1:100Þ

This can be identified as the equation of motion of the single DoF system shown in Figure 1.15. Once this response is known, the response of the system (Equation 1.99) is obtained by applying the principle of superposition. 1.6.3.1

Impulse Response

Many important characteristics of a system can be studied by analyzing the system response to an impulse or a step input excitation. Such characteristics include system stability, speed of response, time constants, damping properties, and natural frequencies. In this way, a knowledge of the system response to an arbitrary excitation is gained. A unit impulse or a unit step are baseline inputs or test inputs. Responses to such inputs can also serve as the basis for system comparison. In particular, it is usually possible to determine the degree of nonlinearity in a system by exciting it at two input intensity levels, separately, and checking whether proportionality is retained at the output; or by applying a harmonic excitation and checking whether limit cycles are encountered by the response. The response of the system (Equation 1.100) to a unit impulse uðtÞ ¼ dðtÞ may be conveniently determined by the Laplace transform approach. Here, we will use a time domain approach, instead. First, integrate Equation 1.100 over the almost zero time interval from t ¼ 0 to t ¼ 0þ : We obtain y_ ð0þ Þ ¼ y_ ð0 Þ

2zvn ½yð0þ Þ

yð0 Þ

v2n

ð 0þ 02

y dt þ v2n

ð 0þ 02

uðtÞdt

ð1:101Þ

Suppose that the system starts from rest. Hence, yð0 Þ ¼ 0 and y_ ð0 Þ ¼ 0: Also, when an impulse is applied over an infinitesimal time period ½0 ; 0þ the system will not be able to move through a finite distance during that period. Hence, yð0þ Þ ¼ 0 as well, and furthermore, the integral of y on the right hand side of Equation 1.101 will also be zero. Now, by the definition of a unit impulse, the integral of u

1 34


on the right hand side of Equation 1.101 will be unity. Hence, we have y_ ð0þ Þ ¼ v2n : It follows that as soon as a unit impulse is applied to the system (Equation 1.100) the initial conditions will become yð0þ Þ ¼ 0

and

y_ ð0þ Þ ¼ v2n

ð1:102Þ

þ

Also, beyond t ¼ 0 the excitation uðtÞ ¼ 0; according to the definition of an impulse. Then, the impulse response of the system (Equation 1.100) is obtained by its homogeneous solution (as carried out before, under free response), but with the initial conditions given in Equation 1.102. The three cases of damping ratio (z , 1; z . 1; and z ¼ 1) should be considered separately. We obtain the following results: vn ð1:103aÞ yimpulse ðtÞ ¼ hðtÞ ¼ pffiffiffiffiffiffiffiffi expð zvn tÞsin vd t for z , 1 1 z2 vn yimpulse ðtÞ ¼ hðtÞ ¼ pffiffiffiffiffiffiffiffi ð1:103bÞ ½exp l1 t exp l2 t for z . 1 2 z2 1 yimpulse ðtÞ ¼ hðtÞ ¼ v2n t expð vn tÞ for z ¼ 1

ð1:103cÞ

An explanation concerning the dimensions of hðtÞ is appropriate here. Note that yðtÞ has the same dimensions as uðtÞ: Since hðtÞ is the response to a unit impulse, dðtÞ; it follows that they have the same dimensions. The magnitude of dðtÞ is represented by a unit area in the uðtÞ versus t plane. Consequently, dðtÞ has the dimensions of (1/time) or (frequency). Clearly then, hðtÞ also has the dimensions of (1/time) or (frequency). 1.6.3.2

The Riddle of Zero Initial Conditions

For a second order system, zero initial conditions correspond to yð0Þ ¼ 0 and y_ ð0Þ ¼ 0: It is clear _ from Equations 1.103 that hð0Þ ¼ 0; but hð0Þ – 0; which appears to violate the zero initial conditions assumption. This situation is characteristic in a system response to an impulse and its derivatives. This may be explained as follows. When an impulse is applied to a system at rest (zero initial state), the highest derivative of the system differential equation momentarily becomes infinity.

FIGURE 1.22

Impulse response functions of a damped oscillator.


1 35

As a result, the next lower derivative becomes finite (nonzero) at t ¼ 0þ : The remaining lower derivatives maintain their zero values at that instant. When an impulse is applied to the system given by Equation 1.100, for example, the acceleration y€ ðtÞ becomes infinity, and the velocity y_ ðtÞ takes a nonzero (finite) value shortly after its application ðt ¼ 0þ Þ: The displacement yðtÞ; however, would not have sufficient time to change at t ¼ 0þ : The impulse input is therefore equivalent to a velocity initial condition, in this case. This initial condition is determined by using the integrated version (Equation 1.101) of the system (Equation 1.100), as has been done. The impulse response functions given by Equations 1.103 are plotted in Figure 1.22 for some representative values of the damping ratio. It should be noted that for 0 , z , 1 the angular frequency of damped vibrations is vd ; the damped natural frequency, which is smaller than the undamped natural frequency vn : 1.6.3.3

Step Response

A unit step excitation is defined by

( uðtÞ ¼

1

for t . 0

0

for t # 0

ð1:104Þ

Unit impulse excitation dðtÞ may be interpreted as the time derivative of uðtÞ; duðtÞ dðtÞ ¼ ð1:105Þ dt Note that Equation 1.105 re establishes the fact that for a nondimensional uðtÞ; the dimension of dðtÞ is (time) 1. Then, since a unit step is the integral of a unit impulse, the step response can be obtained directly as the integral of the impulse response; thus, ðt ystep ðtÞ ¼ hðtÞdt ð1:106Þ 0

This result also follows from the convolution integral (Equation 1.97g) because, for a delayed unit step, we have ( 1 for t , t ð1:107Þ uðt tÞ ¼ 0 for t $ t Thus, by integrating Equations 1.103 with zero initial conditions, the following results are obtained for step response: ystep ðtÞ ¼ 1 ystep ¼ 1

1 pffiffiffiffiffiffiffiffi expð zvn tÞsinðvd t þ fÞ 1 z2 1 pffiffiffiffiffiffiffiffi ½l1 exp l2 t 2 1 z2 vn

ystep ¼ 1

l2 exp l1 t

ðvn t þ 1Þexpð vn tÞ

for z , 1 for z . 1

for z ¼ 1

ð1:108aÞ ð1:108bÞ ð1:108cÞ

with cos f ¼ z

ð1:109Þ

The step responses given by Equations 1.108 are plotted in Figure 1.23 for several values of damping ratio. Note that, since a step input does not cause the highest derivative of the system equation to approach infinity at t ¼ 0þ ; the initial conditions that are required to solve the system equation remain unchanged at t ¼ 0þ ; provided that there are no derivative terms on the input side of the system equation. If there are derivative terms in the input, then, for example, a step can become an impulse and the situation changes. Now, the response of the system in Figure 1.21, when subjected to a unit step of support excitation (see Equation 1.99), is obtained by using the principle of superposition, as the sum of the unit step

1 36


FIGURE 1.23

Unit step response of a damped simple oscillator.

response and ð2z=vn Þ times the unit impulse response of Equation 1.100. Thus, from Equation 1.103 and Equation 1.108, we obtain the step response of the system in Figure 1.21 as yðtÞ ¼ 1

expð zvn tÞ pffiffiffiffiffiffiffiffi ½sinðvd t þ fÞ 1 z2

1 yðtÞ ¼ 1 þ pffiffiffiffiffiffiffiffi ½l2 exp l2 t 2 1 z2 vn yðtÞ ¼ 1 þ ðvn t

1.6.3.4

2z sin vd t

l1 exp l1 t

for z , 1 for z . 1

1Þexpð vn tÞ for z ¼ 1

ð1:110aÞ ð1:110bÞ ð1:110cÞ

Liebnitz’s Rule

The time derivative of an integral whose limits of integration are also functions of time may be obtained using Liebnitz’s rule. It is expressed as d ðbðtÞ dbðtÞ daðtÞ ðbðtÞ ›f ðt; tÞdt ð1:111Þ f ðt; tÞdt ¼ f ½bðtÞ; t f ½aðtÞ; t þ dt aðtÞ dt dt aðtÞ ›t By repeated application of Liebnitz’s rule to Equation 1.97g, we can determine the ith derivative of the response variable; thus, " # " # di yðtÞ dhðtÞ di 1 hðtÞ dhðtÞ di 2 hðtÞ duð0Þ uð0Þ þ hðtÞ þ ¼ hðtÞ þ þ · · · þ þ · · · þ dt dt dt dt i dt i 1 dt i 2 þ · · · þ hðtÞ

di 1 uð0Þ ðt di uðt tÞ þ hðtÞ dt i 1 dt dt i 0

ð1:112Þ

From this result, it follows that the zero state response to input ½di uðtÞ =dt i is ½di yðtÞ =dt i ; provided that all lower order derivatives of uðtÞ vanish at t ¼ 0: This result verifies the fact that, for instance, the first derivative of the unit step response gives the impulse response function. It should be emphasized that the convolution integral (Equation 1.97) gives the forced response of a system, assuming that the initial conditions are zero. For nonzero initial conditions, the homogeneous


1 37

solution (e.g., Equation 1.54 or Equation 1.58) should be added to this zero initial condition response and then the unknown constants should be evaluated by using the initial conditions. Care should be exercised in the situation in which there is an initial velocity in the system and to this an impulsive excitation is applied. In this case, one approach would be to first determine the velocity at t ¼ 0þ by adding to the initial velocity at t ¼ 0 ; the velocity change in the system due to the impulse. The initial displacement will not change, however, due to the impulse. Once the initial conditions at t ¼ 0þ are determined in this manner, the complete solution can be obtained as usual.

Bibliography Benaroya, H. 1998. Mechanical Vibration, Prentice Hall, Upper Saddle River, NJ. Crandall, S.H., Karnopp, D.C., Kurtz, E.F., and Prodmore Brown, D.C. 1968. Dynamics of Mechanical and Electromechanical Systems, McGraw Hill, New York. den Hartog, J.P. 1956. Mechanical Vibrations, McGraw Hill, New York. de Silva, C.W. 2000. VIBRATION Fundamentals and Practice, CRC Press, Boca Raton, FL. de Silva, C.W. 2004. MECHATRONICS An Integrated Approach, CRC Press, Boca Raton, FL. Dimarogonas, A. 1996. Vibration for Engineers, 2nd ed., Prentice Hall, Upper Saddle River, NJ. Inman, D.J. 1996. Engineering Vibration, Prentice Hall, Englewood Cliffs, NJ. Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood Cliffs, NJ. Meirovitch, L. 1986. Elements of Vibration Analysis, 2nd ed., McGraw Hill, New York. Rao, S.S. 1995. Mechanical Vibrations, 3rd ed., Addison Wesley, Reading, MA. Shearer, J.L. and Kulakowski, B.T. 1990. Dynamic Modeling and Control of Engineering Systems, MacMillan, New York. Shearer, J.L., Murphy, A.T., and Richardson, H.H. 1971. Introduction to System Dynamics, Addison Wesley, Reading, MA. Steidel, R.F. 1979. An Introduction to Mechanical Vibrations, 2nd ed., Wiley, New York. Thomson, W.T. and Dahleh, M.D. 1998. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper Saddle River, NJ. Volterra, E. and Zachmanoglou, E.C. 1965. Dynamics of Vibrations, Charles E. Merrill Books, Columbus, OH.

2

Frequency-Domain Analysis 2.1 2.2


Introduction ........................................................................... 2 1 Response to Harmonic Excitations ...................................... 2 2 Response Characteristics (Q Factor Method)

†

Measurement of Damping Ratio

2.3

Transform Techniques ......................................................... 2 14

2.4

Mechanical Impedance Approach ...................................... 2 25

2.5

Transmissibility Functions .................................................. 2 31

2.6

Receptance Method ............................................................. 2 37

Transfer Function † Frequency Response Function (Frequency Transfer Function) Interconnection Laws

Force Transmissibility † Motion Transmissibility † General Case † Peak Values of Frequency Response Functions Application of Receptance

Appendix 2A Transform Techniques ................................. 2 40

Summary This chapter presents the frequency domain analysis of mechanical vibrating systems. In the frequency domain, the independent variable is frequency. The response of a vibrating system to harmonic excitations under various levels of damping (overdamped, underdamped, and critically damped) is analyzed. Frequency transfer function techniques including impedance, mobility, force transmissibility, motion transmissibility, and receptance are studied. Transform techniques (Fourier and Laplace) are applicable in the frequency domain analysis. The Q factor method of measuring damping is derived. Component interconnection laws are established for frequency domain analysis. The emphasis is on single degree of freedom (single DoF) systems.

2.1 Introduction In many vibration problems, the primary excitation force typically has a repetitive periodic nature, and in some cases this periodic forcing function may be even purely sinusoidal. Examples are excitations due to mass eccentricity and misalignments in rotational components, tooth meshing in gears, and electromagnetic devices excited by AC or periodic electrical signals. In basic terms, the frequency response of a dynamic system is the response to a pure sinusoidal excitation. As the amplitude and the frequency of the excitation are changed, the response also changes. In this manner, the response of the system over a range of excitation frequencies can be determined. This represents the frequency response. In this case, frequency ðvÞ is the independent variable and hence we are dealing with the frequency domain. 21

22


Frequency domain considerations are applicable even when the signals are not periodic. In fact, a time signal can be transformed into its frequency spectrum through the Fourier transform. For a given time signal, an equivalent Fourier spectrum, which contains all the frequency (sinusoidal) components of the signal, can be determined either analytically or computationally. Hence, a time domain representation and analysis has an equivalent frequency domain representation and analysis, at least for linear dynamic systems. For this reason, and also because of the periodic nature of typical vibration signals, frequency response analysis is extremely useful in the subject of mechanical vibrations. The response to a particular form of “excitation” is what is considered in the frequency domain analysis. Hence, we are specifically dealing with the subject of “forced response” analysis, albeit in the frequency domain.

2.2 Response to Harmonic Excitations Consider a simple oscillator with an excitation force f ðtÞ; as shown in Figure 2.1. The equation of motion is given by m€x þ b_x þ kx ¼ f ðtÞ

Spring k

Mass m

ð2:1Þ

Suppose that f ðtÞ is sinusoidal (i.e., harmonic). Pick the time reference such that f ðtÞ ¼ f0 cos vt

x

Viscous Damper b

ð2:2Þ

where

FIGURE 2.1

A forced simple oscillator.

v ¼ excitation frequency f0 ¼ forcing excitation amplitude For a system subjected to a forcing excitation, we have Total Response ¼ Homogeneous Response xh þ Particular Response xp T

H ðNatural ResponseÞ

¼

Free Response

X ðDepends only on initial conditionsÞ does not contain enforced response and depends entirely on the natural=homogeneous response

P ðEnforced ResponseÞ

þ

Forced Response

F ðDepends only on f0 Þ but contains a natural=homogeneous component

Using these concepts, we analyze the forced problem, which may be written as b k f x_ þ x ¼ 0 cos vt ¼ uðtÞ m m m

ð2:3Þ

x€ þ 2zvn x_ þ v2n x ¼ a cos vt ¼ uðtÞ

ð2:4Þ

x€ þ or

where uðtÞ is the modified excitation. Also,

vn ¼ undamped natural frequency z ¼ damping ratio The total response is given by x ¼ xh þ xp

ð2:5Þ

23

Frequency Domain Analysis

with xh ¼ C1 el1 þ C2 e l2 t

ð2:6Þ

The particular solution xp ; by definition, is one solution that satisfies Equation 2.4. It should be intuitively clear that this will be of the form xp ¼ a1 cos vt þ a2 sin vt {except for the case: z ¼ 0 and v ¼ vn }

ð2:7Þ

where the constants a1 and a2 are determined by substituting Equation 2.7 into the system Equation 2.4 and equating the like coefficient. This is known as the method of undetermined coefficients. We will consider several important cases.

2.2.1

Response Characteristics

Case 1: Undamped oscillator with excitation frequency

natural frequency

We have x€ þ v2n x ¼ a cos vt

with v – vn

ð2:8Þ

Homogeneous solution: xh ¼ A1 cos vn t þ A2 sin vn t

ð2:9Þ

Particular solution: xp ¼

ðvn2

a

cos vt

v2 Þ

ð2:10Þ

It can be easily verified that xp given by Equation 2.10 satisfies the forced system Equation 2.4, with z ¼ 0: Hence, it is a particular solution. Complete solution: a cos vt ðv2n v2 Þ |fflfflfflfflfflffl{zfflfflfflfflfflffl}

x ¼ A1 cos vn t þ A2 sin vn t þ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} H Satisfies the homogeneous equation

ð2:11Þ

P Satisfies the equation with input

Now A1 and A2 are determined using the initial conditions: xð0Þ ¼ x0 and x_ ð0Þ ¼ v0

ð2:12Þ

Specifically, we obtain x0 ¼ A1 þ

v2n

a

v0 ¼ A2 vn

v2

ð2:13aÞ

ð2:13bÞ

24


Hence, the complete response is v a cos vn t þ v0 sin vn t þ 2 a 2 cos vt x ¼ x0 n ðv2n v2 Þ vn v |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl ffl{zfflfflfflfflfflffl} H Homogeneous solution

v ¼ x0 cos vn t þ v0 sin vn t n |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

P Particular solution

þ

X Free response ðdepends only on initial conditionsÞ comes from xh pSinusodal at vn

ð2:14aÞ

ðv2n

a

½cos vt cos v t v2 Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflnffl} 2 sin

ðvn þvÞ

t sin

ðv n v Þ

ð2:14bÞ t

2 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl ffl}

F Forced response ðdepends on inputÞ comes from both xh and xp pWill exhibit a beat phenomenon for vn v; i:e:; ðvn þvÞ=2 wave modulated by ðvn vÞ=2 wave

This is a stable response in the sense of bounded input bounded output (BIBO) stability, as it is bounded and does not increase steadily. Note: If there is no forcing excitation, the homogeneous solution H and the free response X will be identical. With a forcing input, the natural response (the homogeneous solution) will be influenced by it in general, as is clear from Equation 2.14b.

Case 2: Undamped oscillator with v 5 vn (resonant condition) In this case, the xp that was used before is no longer valid. This is the degenerate case, because otherwise the particular solution cannot be distinguished from the homogeneous solution and the former will be completely absorbed into the latter. Instead, in view of the “double integration” nature of the forced system equation when v ¼ vn ; we use the particular solution ðPÞ: xp ¼

at sin vt 2v

ð2:15Þ

This choice of particular solution is strictly justified by the fact that it satisfies the forced system equation. Complete solution: x ¼ A1 cos vt þ A2 sin vt þ

at sin vt 2v

ð2:16Þ

Initial conditions: xð0Þ ¼ x0 and x_ ð0Þ ¼ v0 : We obtain x0 ¼ A1

ð2:17aÞ

v0 ¼ vA2

ð2:17bÞ

The total response is v x ¼ x0 cos vt þ v0 sin vt |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

X Free response ðdepends on initial conditionsÞ pSinusodal at v

þ

at sin vt 2vffl{zfflffl |fflffl ffl}

ð2:18Þ

F Forced response ðdepends on inputÞ pAmplitude increases linearly

Since the forced response increases steadily, this is an unstable response in the BIBO sense. Furthermore, the homogeneous solution H and the free response X are identical, and the particular solution P is identical to the forced response F in this case.

25


0

Time t

(a)

0

Time t

(b)

0

Time t

(c) FIGURE 2.2 Forced response of a harmonically excited undamped simple oscillator: (a) for a large frequency difference; (b) for a small frequency difference (beat phenomenon); (c) response at resonance.

Note that the same system (undamped oscillator) gives a bounded response for some excitations, while producing an unstable (steady linear increase) response when the excitation frequency is equal to its natural frequency. Hence, the system is not quite unstable, but is not quite stable either. In fact, the undamped oscillator is said to be marginally stable. When the excitation frequency is equal to the natural frequency it is reasonable for the system to respond in a complementary and steadily increasing manner because this corresponds to the most “receptive” excitation. Specifically, in this case, the excitation complements the natural response of the system. In other words, the system is “in resonance” with the excitation, and the condition is called a resonance. We will address this aspect for the more general case of a damped oscillator, in the sequel. Figure 2.2 shows typical forced responses of an undamped oscillator when there is a large difference between the excitation and the natural frequencies (Case 1); a small difference between the excitation and the natural frequencies when a beat phenomenon is clearly manifested (also Case 1); and for the resonant case (Case 2).

Case 3: Damped oscillator The equation of forced motion is x€ þ 2zvn x_ þ v2n x ¼ a cos vt

ð2:19Þ

Particular Solution (Method 1): Since derivatives of both odd order and even order are present in this equation, the particular solution should have terms corresponding to odd and even derivatives of the

26


forcing function (i.e., sin vt and cos vt). Hence, the appropriate particular solution will be of the form: xp ¼ a1 cos vt þ a2 sin vt

ð2:20Þ

Substitute Equation 2.20 into Equation 2.19. We obtain

v2 a1 cos vt

v2 a2 sin vt þ 2zvn ½ va1 sin vt þ va2 cos vt þ v2n ½a1 cos vt þ a2 sin vt ¼ a cos vt

Equate like coefficients:

v2 a1 þ 2zvn va2 þ v2n a1 ¼ a v2 a2

2zvn va1 þ v2n a2 ¼ 0

Hence, we have ðv2n

v2 Þa1 þ 2zvn va2 ¼ a

2zvn va1 þ ðv2n

v2 Þa2 ¼ 0

This can be written in the matrix vector form: " 2 #" # a1 ðvn v2 Þ 2zvn v ðv2n

2zvn v The solution is

"

a1

#

a2

¼

1 D

"

v2 Þ

ðv2n

a2

v2 Þ

¼

2zvn v ðv2n

2zvn v

ð2:21aÞ ð2:21bÞ " # a 0 #" # a

v2 Þ

0

ð2:21cÞ

ð2:22Þ

with the determinant D ¼ ðv2n

v2 Þ2 þ ð2zvn vÞ2

ð2:23Þ

On simplification, we obtain a1 ¼

ðv2n

a2 ¼

v2 Þ D

a

2zvn v a D

ð2:24aÞ ð2:24bÞ

This is the method of undetermined coefficients.

Particular Solution (Method 2): Complex Function Method Consider x€ þ 2zvn x_ þ v2n x ¼ ae jvt

ð2:25Þ

where the excitation is complex. (Note: e jvt ¼ cos vt þ j sin vt.) The resulting complex particular solution is xp ¼ XðjvÞe jvt Note that we should take the real part of this solution as the true particular solution. First substitute Equation 2.26 into Equation 2.25: X

v2 þ 2zvn jv þ v2n e jvt ¼ ae jvt

ð2:26Þ

27


Hence, since e jvt – 0 in general, X¼

a ½ v2 þ 2zvn jv þ v2n

ð2:27Þ

The characteristic polynomial of the system is DðlÞ ¼ l2 þ 2zvn l þ v2n

ð2:28aÞ

DðsÞ ¼ s2 þ 2zvn s þ v2n

ð2:28bÞ

or, with the Laplace variable s; If we set s ¼ jv; we have

v2 þ 2zvn jv þ v2n

DðjvÞ ¼ as

ð2:28cÞ

Note that Equation 2.28c is the denominator of Equation 2.27. Hence, Equation 2.27 can be written X¼

a DðjvÞ

ð2:29Þ

It follows from Equation 2.26 that the complex particular solution is a xp ¼ e jvt DðjvÞ

ð2:30Þ

Next let DðjvÞ ¼ lDle jf

ð2:31Þ

Then, by substituting Equation 2.31 in Equation 2.30, we obtain a jðvt fÞ e xp ¼ lDl

ð2:32Þ

where it is clear from Equation 2.31 that lDl ¼ magnitude of D ðjvÞ

f ¼ phase angle of DðjvÞ The actual real particular solution is the real part of Equation 2.32 and is given by a cosðvt fÞ xp ¼ lDl

ð2:33Þ

It can easily be verified that this result is identical to what was obtained previously by Method 1, as given by Equation 2.20 together with Equations 2.23 and 2.24. In passing, we will note here that the frequency domain transfer function (i.e., response and excitation in the frequency domain) of the system Equation 2.19 is: GðjvÞ ¼

1 ¼ D

1 s2 þ 2zvn s þ v2n

s¼jv

ð2:34Þ

This frequency transfer function (also known as the frequency response function) is obtained from the Laplace transfer function GðsÞ by setting s ¼ jv: We will discuss this aspect in more detail later. The particular section ðPÞ is equal to the steady state solution, because the homogeneous solution dies out due to damping. The particular solution (Equation 2.33) has the following characteristics: 1. The frequency is the same as the excitation frequency v: 2. The amplitude is amplified by the magnitude 1 lDl ¼ lGðjvÞl:

28


3. The response is “lagged” by the phase angle f of D (or “led” by the phase angle of GðjvÞ; denoted by ,GðjvÞÞ: 4. Since the homogeneous solution of a stable system decays to zero, the particular solution is also the steady state solution.

Resonance The amplification lGðjvÞl ¼ 1 lDl is maximum (i.e., resonance) when lDl is a minimum or lDl2 is a minimum. As noted earlier, this condition of peak amplification of a system when excited by a sinusoidal input is called resonance and the associated frequency of excitation is called resonant frequency. We can determine the resonance of the system (Equation 2.19) as follows: Equation 2.28c is D ¼ v2n v2 þ 2zvn v j Hence, lDl2 ¼ ðv2n v2 Þ2 þ ð2zvn vÞ2 ¼ D ð2:35Þ The resonance corresponds to a minimum value of D; or dD ¼ 2ðv2n dv

v2 Þð 2vÞ þ 2ð2zvn Þ2 v ¼ 0 For a minimum:

ð2:36Þ

Hence, with straightforward algebra, the required condition for resonance is

v2n þ v2 þ 2z2 v2n ¼ 0 or

v2 ¼ ð1

or

2z2 Þv2n

qffiffiffiffiffiffiffiffiffiffi v ¼ 1 2z 2 v n This is the resonant frequency, and is denoted as qffiffiffiffiffiffiffiffiffi vr ¼ 1 2z2 vn

ð2:37Þ pffiffiffiffiffiffiffiffi Note that vr # vd # vn , where vd is the damped natural frequency given by vd ¼ 1 z2 vn : These three frequencies (resonant frequency, damped natural frequency, and undamped natural frequency) are almost equal for small z (i.e., for light damping). The magnitude and the phase angle plots of GðjvÞ are shown in Figure 2.3. These curves correspond to the amplification and the phase change of the particular response (the steady state response) with respect to the excitation input. This pair, the magnitude and phase angle plots of a transfer function with respect to frequency, is termed a Bode plot. Usually, logarithmic scales are used for both magnitude (e.g., decibels) and frequency (e.g., decades). In summary, the steady state response of a linear system to a sinusoidal excitation is completely determined by the frequency transfer function of the system. The total response is determined by adding H to P and substituting initial conditions, as usual. For an undamped oscillator ðz ¼ 0Þ; we notice from Equation 2.34 that the magnitude of GðjvÞ becomes infinity when the excitation frequency is equal to the natural frequency ðvn Þ of the oscillator. This frequency ðvn Þ is clearly the resonant frequency (as well as natural frequency) of the oscillator. This fact has been further supported by the nature of the corresponding time response (see Equation 2.18 and Figure 2.2(c)), which grows (linearly) with time.

2.2.2

Measurement of Damping Ratio (Q-Factor Method)

The frequency transfer function of a simple oscillator (Equation 2.19) may be used to determine the damping ratio. This frequency domain method is also termed the half power point method, for reasons that should become clear from the following development.

29


Amplification a G= D

Frequency w Phase Lead (−f) 0

wr

wn

w

−90

−180 FIGURE 2.3

Magnitude and phase angle curves of a simple oscillator (A Bode plot).

pffiffi pffiffi First we assume that z , 1= 2: Strictly speaking, we should assume that z , 1=2 2: Without loss of generality, consider the normalized (or, non dimensionalized) transfer function " # v2n v2n GðjvÞ ¼ 2 ð2:38Þ ¼ 2 2 2 s þ 2zvn s þ vn s¼jv vn v þ 2zvn vj As noted before, the transfer function GðsÞ; where s is the Laplace variable, can be converted into the corresponding frequency transfer function simply by setting s ¼ jv: Its value at the undamped natural frequency is GðjvÞlv¼vn ¼

1 2zj

ð2:39aÞ

Hence, the magnitude of GðjvÞ (amplification) at v ¼ vn is lGðjvÞlv¼vn ¼

1 2z

ð2:39bÞ

For small z we have vr ø vn : Hence, 1=2z is approximately the peak magnitude at resonance (the resonant peak). The actual peak is slightly larger. It is clear from Equation 2.39a that the phase angle of GðjvÞ at v ¼ vn is p=2: When power is half of the peak power value (e.g., because the displacement squared is proportional to potential energy), then the velocity squared is proportional to kinetic energy, and power is the rate of

2 10


pffiffi change of energy. Therefore, when the amplification is 1= 2 of the peak value we have half power points, given by:

v2n ¼ v2 þ 2jzvn v

1 1 pffiffi ¼ 2 vn 2 2z

1 2

v vn

1

þ 2jz

v vn

ð2:40Þ

Square Equation 2.40: 1 ¼ 2 £ 4z 2

1

2 2

v vn

1

þ 4z 2

v vn

2

Hence,

v vn

4

2

v vn

2

þ 1 þ 4z 2

v vn

2

¼ 8z 2

or

v vn

4

2ð1

2z 2 Þ

v vn

2

þ ð1 8z 2 Þ ¼ 0 |fflffl{zfflffl}

ð2:41Þ

.0

pffiffi Now assume that z2 , 1=8 or z , 1=2 2: Otherwise, we will not obtain two positive roots for ðv=vn Þ2 . Solve for ðv=vn Þ2, which will give two roots v21 and v22 for v2 : Next, assume v22 . v21 : Compare ðv2 v21 Þðv2 v22 Þ ¼ 0 with Equation 2.41. Sum of roots:

v22 þ v21 ¼ 2ð1 v2n

2z 2 Þ

ð2:42Þ

Product of roots:

v22 v21 ¼ ð1 v4n Hence,

v2

vn

v1

2

8z 2 Þ

ð2:43Þ

! qffiffiffiffiffiffiffiffiffi v22 þ v21 2v 2 v1 2 ¼ 2ð1 2 z Þ 2 1 8z 2 ¼ v2n 1 £ 8z2 þ Oðz 4 Þ ¼ 2 4z 2 2 1 2 ø 2 4z2 2 þ 8z2 {because Oðz 4 Þ ! 0 for small z } ø 4z 2

or

v2

vn

v1

ø 2z

Hence, the damping ratio

zø

ðv2 v1 Þ Dv v v1 ¼ ø 2 2vn 2vn v2 þ v1

ð2:44Þ

It follows that, once the magnitude of the frequency response function GðjvÞ is experimentally determined, the damping ratio can be estimated from Equation 2.44, as illustrated in Figure 2.4.

2 11


Amplification 1 2z 2 2z

1

0

w1 wnw 2

Frequency w

∆w FIGURE 2.4

The Q factor method of damping measurement.

The Q factor, which measures the sharpness of resonant peak, is defined by Q factor ¼

1 vn ¼ 2z Dv

ð2:45Þ

The term originated from the field of electrical tuning circuits where a sharp resonant peak is a desirable thing (quality factor). Some useful results on the frequency response of a simple oscillator are summarized in Box 2.1.

Example 2.1 A dynamic model of a fluid coupling system is shown in Figure 2.5. The fluid coupler is represented by a rotatory viscous damper with damping constant b: It is connected to a rotatory load of moment of inertia J; restrained by a torsional spring of stiffness k; as shown. We now obtain the frequency transfer function of the system that relates the restraining torque t of the spring to the angular displacement excitation aðtÞ that is applied at the free end of the fluid coupler. If aðtÞ ¼ a0 sin vt; what is the magnitude (i.e., amplitude) of t at steady state?

Solution Newton’s Second Law gives J u€ ¼ bða_

u_Þ

ku

Hence, J u€ þ bu_ þ ku_ ¼ ba_

ðiÞ

bs u ¼ 2 Js þ bs þ k a

ðiiÞ

Motion transfer function is

Note that the frequency transfer function is obtained simply by setting s ¼ jv: The restraining torque of the spring is t ¼ ku: Hence,

t ku kbs ¼ ¼ 2 ¼ GðsÞ a a Js þ bs þ k

ðiiiÞ

2 12


Box 2.1 HARMONIC RESPONSE OSCILLATOR

OF A

SIMPLE

Undamped Oscillator: x€ þ v2n x ¼ a cos vt; xð0Þ ¼ x0 ; x_ ð0Þ ¼ v0 v x ¼ x0 cos vn t þ v0 sin vn t þ 2 a 2 ½cos vt cos vn t n vn v |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl} X

x ¼ Same X þ

for v – vn

F

at sin vt 2v

for v ¼ vn ðresonanceÞ

Damped Oscillator: x€ þ 2zvn x_ þ v2n x ¼ a cos vt x¼Hþ where

a cosðvt fÞ lv2n v2 þ 2jzvn vl |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} P

tan f ¼

2zvn v ; v2n v2

f ¼ phase lag:

Particular solution P is also the steady state response. Homogeneous solution H ¼ A1 el1 t þ A2 el2 t where l1 and l2 are roots of l2 þ 2zvn l þ v2n ¼ 0 ðcharacteristic equationÞ

A1 and A2 are determined from initial conditions: xð0Þ ¼ x0 ; x_ ð0Þ ¼ v0 pffiffiffiffiffiffiffiffiffi Resonant Frequency: vr ¼ 1 2z2 vn The magnitude of P will peak at resonance. Damping Ratio:

z¼

v v1 Dv ¼ 2 for low damping 2v n v2 þ v1

where, Dv ¼ half power bandwidth ¼ v2 v1 v 1 for low damping. Note: Q factor ¼ n ¼ Dv 2z Then, the corresponding frequency response function (frequency transfer function) is kbjv GðjvÞ ¼ ðk J v2 Þ þ bjv

ðivÞ

For a harmonic excitation of

a ¼ a0 e jvt

ðvÞ

2 13


we have

t ¼ lGðjvÞla0 e

ð jvtþfÞ

Fluid Coupling b

ðviÞ

at steady state. Here, the phase lead of t with respect to a is

f ¼ /GðjvÞ ¼ /jv ¼

p 2

tan

1

/ðk

a (t)

J v2 þ bjvÞ

bv ðk J v2 Þ

lk

t

q

Load Inertia J

ðviiÞ

The magnitude of the restraining torque, at steady state, is

t0 ¼ a0 lGðjvÞl ¼ a0

Restraining Spring k

FIGURE 2.5

A fluid coupling system.

kbv J v2 þ bjvl

a0 kbv ffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk J v2 Þ2 þ b2 v2 Hence, J a0 v2n v £ 2zvn ffi t0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv2n v2 Þ2 þ ð2zvn vÞ2

ðviiiÞ

a0 k2zvn v ffi t0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðvn v2 Þ2 þ ð2zvn vÞ2

ðixÞ

or

with k=J ¼ v2n ; b=J ¼ 2zvn ; and vn equal to the undamped natural frequency of the load. Now, define the normalized frequency r¼

v vn

ðxÞ

Then, from (ix) we have 2ka0 zr t0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 r2 Þ2 þ ð2zrÞ2

ðxiÞ

For r ¼ 1:

t0 ¼

2ka0 z ¼ ka0 2z

ðxiiÞ

This means, at resonance, the applied twist is directly transmitted to the load spring. For small r:

t0 ¼

2ka0 zr 1

ðxiiiÞ

which is small, and becomes zero at r ¼ 0: Hence, at low frequencies, the transmitted torque is small. For larger r:

t0 ¼

2ka0 zr 2ka0 z ¼ 2 r r

ðxivÞ

which is small and goes to zero. Hence, at high frequencies as well, the transmitted torque is small. The variation of t0 with the frequency ratio r is sketched in Figure 2.6.

2 14


Magnitude of Restraining Torque t 0 ka 0

0 FIGURE 2.6

1

Excitation Frequency Ratio r

Variation of the steady state transmitted torque with frequency.

2.3 Transform Techniques Concepts of frequency response analysis originate from the nature of the response of a dynamic system to a sinusoidal (i.e., harmonic) excitation. These concepts can be generalized because the time domain analysis, where the independent variable is time ðtÞ; and the frequency domain analysis, where the independent variable is frequency ðvÞ; are linked through the Fourier transformation. Analytically, it is more general and versatile to use the Laplace transformation, where the independent variable is the Laplace variable ðsÞ which is complex (i.e., non real). This is true because analytical Laplace transforms may exist even for time functions that do not have “analytical” Fourier transforms. But with compatible definitions, the Fourier transform results can be obtained from the Laplace transform results simply by setting s ¼ jv: In the present section, we will formally introduce the Laplace transformation and the Fourier transformation, and will illustrate how these techniques are useful in the response analysis of vibrating systems. The preference of one domain over another will depend on such factors as the nature of the excitation input, the type of the analytical model available, the time duration of interest, and the quantities that need to be determined.

2.3.1

Transfer Function

The Laplace transform of a piecewise continuous function f ðtÞ is denoted here by FðsÞ and is given by the Laplace transformation ð1 FðsÞ ¼ f ðtÞ expð stÞdt ð2:46Þ 0

in which s is a complex independent variable known as the Laplace variable, expressed as ð2:47Þ s ¼ s þ jv pffiffiffiffi and j ¼ 1: Laplace transform operation is denoted as Lf ðtÞ ¼ FðsÞ: The inverse Laplace transform operation is denoted by f ðtÞ ¼ L 1 FðsÞ and is given by 1 ðsþj1 f ðtÞ ¼ FðsÞ expðstÞds ð2:48Þ 2p j s j1 The integration is performed along a vertical line parallel to the imaginary (vertical) axis, located at s from the origin in the complex Laplace plane (s plane). For a given piecewise continuous function

2 15


f ðtÞ; the Laplace transform exists if the integral in Equation 2.46 converges. A sufficient condition for this is ð1 ð2:49Þ lf ðtÞl expð stÞdt , 1 0

Convergence is guaranteed by choosing a sufficiently large and positive s: This property is an advantage of the Laplace transformation over the Fourier transformation. (For a more complete discussion of the Fourier transformation, see later in this chapter and in Chapter 4.) By the use of Laplace transformation, the convolution integral equation can be converted into an algebraic relationship. To illustrate this, consider the convolution integral which gives the response yðtÞ of a dynamic system to an excitation input uðtÞ; with zero initial conditions. By definition (Equation 2.46), its Laplace transform is written as ð1 ð1 YðsÞ ¼ hðtÞuðt tÞdt expð stÞdt ð2:50Þ 0

0

Note that hðtÞ is the impulse response function of the system. Since the integration with respect to t is performed while keeping t constant, we have dt ¼ dðt tÞ: Consequently, ð1 ð1 uðt tÞ exp½ sðt tÞ dðt tÞ hðtÞ expð stÞdt YðsÞ ¼ t

0

The lower limit of the first integration can be made equal to zero, in view of the fact that uðtÞ ¼ 0 for t , 0: Again using the definition of Laplace transformation, the foregoing relation can be expressed as YðsÞ ¼ HðsÞUðsÞ

ð2:51Þ

in which HðsÞ ¼ LhðtÞ ¼

ð1 0

hðtÞ expð stÞdt

ð2:52Þ

Note that, by definition, the transfer function of a system, denoted by HðsÞ; is given by Equation 2.51. More specifically, the system transfer function is given by the ratio of the Laplace transformed output and the Laplace transformed input, with zero initial conditions. In view of Equation 2.52, it is clear that the system transfer function can be expressed as the Laplace transform of the impulse response function of the system. The transfer function of a linear and constant parameter system is a unique function that completely represents the system. A physically realizable, linear, constant parameter system possesses a unique transfer function, even if the Laplace transforms of a particular input and the corresponding output do not exist. This is clear from the fact that the transfer function is a system model and does not depend on the system input itself. Note: The transfer function is also commonly denoted by GðsÞ: However, in the present context we use HðsÞ in view of its relation to hðtÞ: Some useful Laplace transform relations are given in Table 2.1. Also, note that the Fourier transform (set s ¼ jvÞ is given by Forward: ð1 FðjvÞ ¼ f ðtÞ expð jvtÞdt 0

Inverse: f ðtÞ ¼

ðj1 j1

FðjvÞ expðjvtÞdv

Consider the nth order linear, constant parameter dynamic system given by an

dn y þ an dt n

1

dn 1 y duðtÞ dm uðtÞ þ · · · þ bm n 1 þ · · · þ a0 y ¼ b0 u þ b1 dt dt dt m

ð2:53Þ

2 16


TABLE 2.1

Important Laplace Transform Relations

L 1 FðsÞ ¼ f ðtÞ Ð 1 sþj1 FðsÞ expðstÞds 2pj s j1

Ð1

k1 f1 ðtÞ þ k2 f2 ðtÞ

k1 F1 ðsÞ þ k2 F2 ðsÞ

expð2atÞf ðtÞ

Fðs þ aÞ

f ðt 2 tÞ

expð2tsÞFðsÞ

f

ðnÞ

Ðt

0

dn f ðtÞ ðtÞ ¼ dt n

1

Conversion Lf ðtÞ ¼ FðsÞ f ðtÞ expð2stÞdt

sn FðsÞ 2 sn 1 f ð0þ Þ 2 sn 2 f 1 ð0þ Þ 2 · · · 2 f n 1 ð0þ Þ Ð0 FðsÞ 1 f ðtÞdt þ s s

f ðtÞdt

Impulse function, dðtÞ

1 1 s n! snþ1 1 sþa

Step function, UðtÞ tn expð2atÞ sin vn t

vn s2 þ v2n

cos vn t

s s2 þ v2n

for physically realizable systems, m # n: By applying a Laplace transformation and then integrating by parts, it may be verified that L

dk f ðtÞ ^ ¼ sk FðsÞ dt k

sk 1 f ð0Þ

sk

2

df ð0Þ dt

··· þ

dk 1 f ð0Þ dt k 1

ð2:54Þ

By definition, the initial conditions are set to zero in obtaining the transfer function. This results in HðsÞ ¼

b0 þ b1 s þ · · · þ bm sm a0 þ a1 s þ · · · þ an sn

ð2:55Þ

for m # n: Note that Equation 2.55 contains all the information that is contained in Equation 2.53. Consequently, the transfer function is an analytical model of a system. The transfer function may be employed to determine the total response of a system for a given input, even though it is defined in terms of the response under zero initial conditions. This is quite logical because the analytical model of a system is independent of the system’s initial conditions. The denominator polynomial of a transfer function is the system’s characteristic polynomial. Its roots are the poles or the eigenvalues of the system. If all the eigenvalues have negative real parts, the system is stable. The response of a stable system is bounded (that is, remains finite) when the input is bounded (which is the BIBO stability). The zero input response of an asymptotically stable system approaches zero with time.

2.3.2

Frequency-Response Function (Frequency Transfer Function)

The Fourier integral transform of the impulse response function is given by ð1 Hð f Þ ¼ hðtÞ expð j2p ftÞdt 1

ð2:56Þ

where f is the cyclic frequency (measured in cps or hertz). This is known as the frequency response function (or frequency transfer function) of a system. The Fourier transform operation is denoted as F hðtÞ ¼ Hð f Þ:

2 17


In view of the fact that hðtÞ ¼ 0 for t , 0; the lower limit of integration in Equation 2.56 could be made zero. Then, from Equation 2.52, it is clear that Hð f Þ is obtained simply by setting s ¼ j2pf in HðsÞ: Hence, strictly speaking, we should use the notation Hð j2pf Þ and not Hð f Þ. However, for notational simplicity, we denote Hðj2pf Þ by Hð f Þ: Furthermore, since the angular frequency v ¼ 2pf ; we can express the frequency response function by Hð jvÞ; or simply by HðvÞ for notational convenience. It should be noted that the frequency response function, like the Laplace transfer function, is a complete representation of a linear, constant parameter system. In view of the fact that both uðtÞ ¼ 0 and yðtÞ ¼ 0 for t , 0; we can write the Fourier transforms of the input and the output of a system directly by setting s ¼ j2pf ¼ jv in the corresponding Laplace transforms. Specifically, we have, according to the notation used here Uð f Þ ¼ Uðj2p f Þ ¼ UðjvÞ and Yð f Þ ¼ Yð j2p f Þ ¼ Yð jvÞ: Then, from Equation 2.51, we have Yð f Þ ¼ Hð f ÞUð f Þ

ð2:57Þ

Note: Sometimes, for notational convenience, the same lowercase letters are used to represent the Laplace and Fourier transforms as well as the original time domain variables. If the Fourier integral transform of a function exists, then its Laplace transform also exists. The converse is not generally true, however, because of the poor convergence of the Fourier integral in comparison to the Laplace integral. This arises from the fact that the factor expð stÞ is not present in the Fourier integral. For a physically realizable, linear, constant parameter system, Hð f Þ exists even if Uð f Þ and Yð f Þ do not exist for a particular input. The experimental determination of Hð f Þ; however, requires system stability. For the nth order system given by Equation 2.53, the frequency response function is determined by setting s ¼ j2pf in Equation 2.55 as Hð f Þ ¼

b0 þ b1 j2pf þ · · · þ bm ðj2pf Þm a0 þ a1 j2pf þ · · · þ an ðj2pf Þn

ð2:58Þ

This, generally, is a complex function of f that has a magnitude denoted by lHð f Þl and a phase angle denoted by /Hð f Þ: A further interpretation of the frequency response function can be given in view of the developments given in Section 2.2. Consider a harmonic input having cyclic frequency f ; expressed by uðtÞ ¼ u0 cos 2p ft

ð2:59aÞ

In analysis, it is convenient to use the complex input uðtÞ ¼ u0 ðcos 2p ft þ j sin 2p ftÞ ¼ u0 expðj2p ftÞ

ð2:59bÞ

and take only the real part of the final result. Note that Equation 2.59b does not implicitly satisfy the requirement of uðtÞ ¼ 0 for t , 0: Therefore, an appropriate version of the convolution integral, where the limits of integration automatically account for this requirement, should be used. For instance, we can write ðt yðtÞ ¼ Re hðtÞu0 exp½j2p f ðt tÞ dt ð2:60aÞ 1

or yðtÞ ¼ Re u0 expðj2p ftÞ

ðt 1

hðtÞ expð j2p f tÞdt

ð2:60bÞ

in which Re[·] denotes the real part. As t ! 1; the integral term in Equation 2.60b becomes the frequency response function Hð f Þ; and the response yðtÞ becomes the steady state response yss : Accordingly, yss ¼ Re½Hð f Þu0 expðj2p ftÞ

ð2:61aÞ

2 18


or yss ¼ u0 lHð f Þl cosð2p ft þ fÞ

ð2:61bÞ

for a harmonic excitation, in which the phase lead angle f ¼ /Hð f Þ: It follows from Equation 2.61b that, when a harmonic excitation is applied to a stable, linear, constant parameter dynamic system having frequency response function Hð f Þ; its steady state response will also be harmonic at the same frequency, but with an amplification factor of lHð f Þl in its amplitude and a phase lead of /Hð f Þ: This result has been established previously, in Section 2.2. Consequently, the frequency response function of a stable system can be experimentally determined using a sine sweep test or a sine dwell test. With these methods, a harmonic excitation is applied as the system input, and the amplification factor and the phase lead angle in the corresponding response are determined at steady state. The frequency of excitation is varied continuously for a sine sweep and in steps for a sine dwell. The sweep rate should be slow enough, and the dwell times should be long enough, to guarantee steady state conditions at the output. The pair of plots of lHð f Þl and /Hð f Þ against f completely represent the complex frequency response function, and are Bode plots or Bode diagrams, as noted earlier. In Bode plots, logarithmic scales are normally used for both frequency f and magnitude lHð f Þl:

Impulse Response The impulse response function of a system can be obtained by taking the inverse Laplace transform of the system transfer function. For example, consider the damped simple oscillator given by the normalized transfer function: HðsÞ ¼

v2n s2 þ 2zvn s þ v2n

ð2:62Þ

The characteristic equation of this system is given by s2 þ 2zvn s þ v2n ¼ 0

ð2:63Þ

The eigenvalues (poles) are given by its roots. Three possible cases exist, as given below.

Case 1: ðz < 1Þ This is the case of complex eigenvalues l1 and l2 : Since the coefficients of the characteristic equation are real, the complex roots should occur in conjugate pairs. Hence, l1 ; l2 ¼ in which

zvn ^ jvd

ð2:64Þ

qffiffiffiffiffiffiffiffi vd ¼ 1 z2 vn

ð2:65Þ

is the damped natural frequency.

Case 2: ðz > 1Þ This case corresponds to real and unequal eigenvalues, qffiffiffiffiffiffiffiffi l1 ; l2 ¼ zvn ^ z2 1vn ¼

a;

b

ð2:66Þ

with a – b; in which ab ¼ v2n

ð2:67Þ

2 19


and a þ b ¼ 2zvn

Case 3: ðz 5 1Þ In this case, the eigenvalues are real and equal: l1 ¼ l2 ¼

ð2:68Þ

vn

ð2:69Þ

In all three cases, the real parts of the eigenvalues are negative. Consequently, these second order systems of a damped simple oscillator are always stable. The impulse response functions hðtÞ corresponding to these three cases are determined by taking the inverse Laplace transform (Table 2.1) of Equation 2.62 for z , 1; z . 1; and z ¼ 1; respectively. The following results are obtained: vn yimpulse ðtÞ ¼ hðtÞ ¼ pffiffiffiffiffiffiffiffi ð2:70aÞ expð zvn tÞ sin vd t for z , 1 1 z2 ab yimpulse ðtÞ ¼ hðtÞ ¼ ð2:70bÞ ½expð atÞ expð btÞ for z . 1 ðb aÞ yimpulse ðtÞ ¼ hðtÞ ¼ v2n t expð vn tÞ

for

z¼1

ð2:70cÞ

Step Response Unit step function is defined by

( UðtÞ ¼

1

for t . 0

0

for t # 0

ð2:71Þ

Unit impulse function dðtÞ may be interpreted as the time derivative of UðtÞ; thus, dUðtÞ ð2:72Þ dt Note that Equation 2.72 re establishes the fact that for non dimensional UðtÞ; the dimension of dðtÞ is (time) 1. Since LUðtÞ ¼ 1=s (see Table 2.1), the unit step response of the dynamic system (Equation 2.62) can be obtained by taking the inverse Laplace transform of

dðtÞ ¼

Ystep ðsÞ ¼

1 v2n 2 s ðs þ 2zvn s þ v2n Þ

ð2:73Þ

which follows from Equation 2.73. To facilitate using Table 2.1, partial fractions of Equation 2.73 are determined in the form a1 a þ a3 s þ 2 2 s ðs þ 2zvn s þ v2n Þ in which the constants a1 ; a2 ; and a3 are determined by comparing the numerator polynomial; thus,

v2n ¼ a1 ðs2 þ 2dvn s þ v2n Þ þ sða2 þ a3 sÞ Then a1 ¼ 1; a2 ¼ 2zvn ; and a3 ¼ 1: The following results are obtained: ystep ðtÞ ¼ 1 ystep ¼ 1

1 pffiffiffiffiffiffiffiffi expð zvn tÞ sinðvd t þ fÞ for z , 1 1 z2 1 ½b expð atÞ a expð btÞ for z . 1 ðb aÞ

ystep ¼ 1

ðvn t þ 1Þ expð vn tÞ

for z ¼ 1

ð2:74aÞ ð2:74bÞ ð2:74cÞ

In Equation 2.74c, cos f ¼ z

ð2:75Þ

2 20


Transfer Function Matrix Consider the state space model of a linear dynamic system as given by x_ ¼ Ax þ Bu

ð2:76aÞ

y ¼ Cx þ Du

ð2:76bÞ

where, x ¼ the nth order state vector, u ¼ the rth order input vector, y ¼ the mth order output vector, A ¼ the system matrix, B ¼ the input gain matrix, C ¼ the output (measurement) gain matrix, and D ¼ the feedforward gain matrix. We can express the input output relation between u and y; in the Laplace domain, by a transfer function matrix of the order m £ r: To obtain this relation, let us Laplace transform the Equations 2.76a and 2.76b and use zero initial conditions for x; thus, sXðsÞ ¼ AXðsÞ þ BUðsÞ

ð2:77aÞ

YðsÞ ¼ CXðsÞ þ DUðsÞ

ð2:77bÞ

From Equation 2.77a it follows that, XðsÞ ¼ ðsI

AÞ 1 B þ UðsÞ

ð2:78Þ

in which I is the nth order identity matrix. By substituting Equation 2.78 into Equation 2.77b, we obtain the transfer relation YðsÞ ¼ ½CðsI

AÞ 1 B þ D UðsÞ

ð2:79aÞ

or YðsÞ ¼ GðsÞUðsÞ

ð2:79bÞ

The transfer function matrix GðsÞ is an m £ r matrix given by GðsÞ ¼ CðsI

AÞ 1 B þ D

ð2:80Þ

In practical systems with dynamic delay, the excitation uðtÞ is not fed forward into the response y: Consequently, D ¼ 0 for systems that we normally encounter. For such systems GðsÞ ¼ CðsI

AÞ 1 B

ð2:81Þ

Several examples are given now to illustrate the approaches of obtaining transfer function models when the time domain differential equation models are given, and to indicate some uses of a transfer function model. Some useful results in the frequency domain are summarized in Box 2.2.

Example 2.2 Consider the simple oscillator equation given by m€y þ b_y þ ky ¼ kuðtÞ

ðiÞ

Note that uðtÞ can be interpreted as a displacement input (e.g., support motion) or kuðtÞ can be interpreted as the input force applied to the mass. Take the Laplace transform of the system equation (i) with zero initial conditions; thus, ðms2 þ bs þ kÞYðsÞ ¼ kUðsÞ

ðiiÞ

The corresponding transfer function is GðsÞ ¼

YðsÞ k ¼ UðsÞ ms2 þ bs þ k

ðiiiÞ

2 21


Box 2.2 USEFUL FREQUENCY-DOMAIN RESULTS Laplace Transform (L): FðsÞ ¼

ða 0

f ðtÞ expð stÞdt

Fourier Transform ðF F ): FðjvÞ ¼

ða 1

f ðtÞ expð jvtÞdt

Note: May use FðvÞ to denote FðjvÞ Note: Set s ¼ jv ¼ j2p f to convert Laplace results into Fourier results.

v ¼ angular frequency (rad/sec) f ¼ cyclic frequency (cps or Hz) Transfer function HðsÞ ¼ output input in Laplace domain; with zero initial conditions: Frequency transfer function (or frequency response function) ¼ HðjvÞ Note: Notation ðGðsÞÞ is also used to denote a system transfer function Note: HðsÞ ¼ LhðtÞ hðtÞ ¼ impulse response function ¼ response to a unit impulse input. Frequency response: YðjvÞ ¼ HðjvÞUðjvÞ where UðjvÞ ¼ Fourier spectrum of input uðtÞ YðjvÞ ¼ Fourier spectrum of output yðtÞ Note: lHðjvÞl ¼ response amplification for a harmonic excitation of frequency v /HðjvÞ ¼ response phase “lead” for a harmonic excitation Multivariable Systems: State Space Model:

x_ ¼ Ax þ Bu y ¼ Cx þ Du

Transfer Matrix Model: YðsÞ ¼ GðsÞUðsÞ where GðsÞ ¼ CðsI

AÞ 1 B þ D

2 22


or, in terms of the undamped natural frequency vn and the damping ratio z; where v2n ¼ k=m and 2zvn ¼ b=m; the transfer function is given by GðsÞ ¼

v2n s þ 2zvn s þ v2n

ðivÞ

2

This is the transfer function corresponding to the displacement output. It follows that the output velocity transfer function is sYðsÞ sv2n ¼ sGðsÞ ¼ 2 s þ 2zvn s þ v2n UðsÞ

ðvÞ

and the output acceleration transfer function is s2 YðsÞ s2 v2n ¼ s2 GðsÞ ¼ 2 s þ 2zvn s þ v2n UðsÞ

ðviÞ

In the output acceleration transfer function, we have m ¼ n ¼ 2: This means that if the acceleration of the mass that is caused by an applied force is measured, the input (applied force) is instantly felt by the acceleration. This corresponds to a feedforward action of the input excitation or a lack of dynamic delay. For example, this is the primary mechanism through which road disturbances are felt inside a vehicle that has very hard suspension.

Example 2.3 Again let us consider the simple oscillator differential equation y€ þ 2zvn y_ þ v2n y ¼ v2n uðtÞ

ðiÞ

By defining the state variables as x ¼ ½x1 ; x2

T

¼ ½y; y_

T

ðiiÞ

a state model for this system can be expressed as " # " # 0 1 0 uðtÞ x_ ¼ xþ 2 2zvn vn v2n

ðiiiÞ

If we consider both displacement and velocity as outputs, we have y¼x

ðivÞ

Note that the output gain matrix C is the identity matrix in this case. From Equation 2.79b and Equation 2.81 it follows that: " # 1" # s 1 0 YðsÞ ¼ UðsÞ ðvÞ 2 vn s þ 2zvn v2n 1 YðsÞ ¼ 2 ½s þ 2zvn s þ v2n

"

s þ 2zvn

v2n

1 YðsÞ ¼ 2 ½s þ 2zvn s þ v2n The transfer function matrix is

" GðsÞ ¼

sv2n =DðsÞ

#"

v2n sv2n

0

v2n

s

"

v2n =DðsÞ

1

# UðsÞ

# UðsÞ

ðviÞ

# ðviiÞ

2 23


in which DðsÞ ¼ s2 þ 2zvn s þ v2n is the characteristic polynomial of the system. The first element in the only column in GðsÞ is the displacement response transfer function and the second element is the velocity response transfer function. These results agree with the expressions obtained in the previous example. Now, let us consider the acceleration y€ as an output and denote it by y3 : It is clear from the system equation (i) that 2zvn y_

y3 ¼ y€ ¼

v2n y þ v2n uðtÞ

ðviiiÞ

or, in terms of the state variables, y3 ¼

v2n x1 þ v2n uðtÞ

2zvn x2

ðixÞ

Note that this output explicitly contains the input variable. The feedforward situation implies that the matrix D is non zero for the output y3 : Now, 2zvn X2 ðsÞ

Y3 ðsÞ ¼

v2n X1 ðsÞ þ v2n UðsÞ ¼

2zvn

sv2n UðsÞ DðsÞ

v2n

v2n UðsÞ þ v2n UðsÞ DðsÞ

which simplifies to Y3 ðsÞ ¼

s2 v2n UðsÞ DðsÞ

ðxÞ

This again confirms the result for the acceleration output transfer function that was obtained in the previous example.

Example 2.4 (a) Briefly explain an approach that you could use to measure the resonant frequency of a mechanical system. Do you expect this measured frequency to depend on whether displacement, velocity, or acceleration is used as the response variable? Justify your answer. (b) A vibration test setup is schematically shown in Figure 2.7. In this experiment, a mechanical load is excited by a linear motor and its acceleration response is measured by an accelerometer and charge amplifier combination. The force applied to the load by the linear motor is also measured, using a force sensor (strain gauge type). The frequency response function acceleration/force is determined from the sensor signals, using a spectrum analyzer. Instrumentation

Mechanical Load

Charge Amplifier

a Spectrum Analyzer f

FIGURE 2.7

Acceleration a

Accelerometer Force f(t)

Linear Motor Force Sensor

Mass m

k

b

Measurement of the acceleration spectrum of a mechanical system.

2 24


Suppose that the mechanical load is approximated by a damped oscillator with mass m; stiffness k; and damping constant b; as shown in Figure 2.7. If the force applied to this load is f ðtÞ and the displacement in the same direction is y; show that the equation of motion of the system is given by m€y þ b_y þ ky ¼ f ðtÞ Obtain an expression for the acceleration frequency response function GðjvÞ in the frequency domain, with excitation frequency v as the independent variable. Note that the applied force f is the excitation input and the acceleration a of the mass is the response, in this case. Express GðjvÞ in terms of (normalized) frequency ratio r ¼ v=vn ; where vn is the undamped natural frequency. Giving all the necessary steps, determine an expression for r at which the acceleration frequency response function will exhibit a resonant peak. What is the corresponding peak magnitude of lGl? For what range of values of damping ratio z would such a resonant peak be possible?

Solution (a) For a single DoF system, apply a sinusoidal forcing excitation at the DoF and measure the displacement response at the same location. Vary the excitation frequency v in small steps, and for each frequency at steady state determine the amplitude ratio of the (displacement response/forcing excitation). The peak amplitude ratio will correspond to the resonance. For a multi DoF system, several tests may be needed, with excitations applied at different locations of freedom and the response measured at various locations as well (see Chapters 10 and 11). In the frequency domain we have, lVelocity response spectruml ¼ vxlDisplacement response spectruml lAcceleration response spectruml ¼ vxlVelocity response spectruml It follows that the shape of the frequency response function will depend on whether the displacement, velocity, or acceleration is used as the response variable. Hence it is likely that the frequency at which the peak amplification occurs (i.e., resonance) will also depend on the type of response variable that is used. (b) A free body diagram of the mass element is shown in Figure 2.8. Newton’s Second Law gives m€y ¼ f ðtÞ

b_y

ky

ðiÞ

m€y þ b_y þ ky ¼ f ðtÞ

ðiiÞ

y 1 ¼ 2 f ðms þ bs þ kÞ

ðiiiÞ

Hence, the equation of motion is The displacement transfer function is

Note that, for notational convenience, the same lowercase letters are used to represent the Laplace transforms as well as the original time domain variables (y and f ). The acceleration transfer function is obtained by multiplying Equation iii by s2 : (From Table 2.1, the Laplace transform of d=dt is s; with zero initial conditions). Hence a s2 ¼ GðsÞ ¼ 2 ðms þ bs þ kÞ f

ðivÞ

y

ky f(t)

m by

FIGURE 2.8

Free body diagram.

2 25


In the frequency domain, the corresponding frequency response function is obtained by substituting jv for s: Hence, GðjvÞ ¼

ð

mv2

v2 þ bjv þ kÞ

ðvÞ

Divide throughout by m and use b=m ¼ 2zvn and k=m ¼ v2n ; where vn ¼ undamped natural frequency and z ¼ damping ratio. Then, GðjvÞ ¼

v2 =m ¼ v þ 2jzvn vÞ 1

ðv2n

2

r2 =m r þ 2jzrr 2

ðviÞ

where r ¼ v=vn : The magnitude of GðjvÞ gives the amplification of the acceleration signal with respect to the forcing excitation: r2 =m lGðjvÞl ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðviiÞ ð1 r2 Þ2 þ ð2zrÞ2 Its peak value corresponds to the peak value of pðrÞ ¼

ð1

r4 r2 Þ2 þ ð2zrÞ2

ðviiiÞ

and gives the resonance. This occurs when dp=dr ¼ 0: Hence, ½ð1

r2 Þ2 þ ð2zrÞ2 4r3

r4 ½2ð1

r2 Þð 2rÞ þ 8z2 r ¼ 0

The solution is r¼0

or

½ð1

r2 Þ2 þ 4z2 r2 þ r2 ½1

r2

2z2 ¼ 0

The first result ðr ¼ 0Þ corresponds to static conditions and is ignored. Hence, the resonant peak occurs when ð1 which has the valid root

r 2 Þ 2 þ 4z 2 r 2 þ r 2

r4

2z 2 r 2 ¼ 0

1 r ¼ pffiffiffiffiffiffiffiffiffiffi 1 2z 2

ðixÞ

Note that r has to be real and positive. It follows that, for a resonance to occur we need 1 0 , z , pffiffi 2 Substitute in (vii), the resonant value of r; to obtain 1

mð1 2z2 Þ lGlpeak ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 4z 2 1 þ 2 1 2z 1 2z 2 or lGlpeak ¼

1 pffiffiffiffiffiffiffiffi 2mz 1 z2

ðxÞ

2.4 Mechanical Impedance Approach Any type of force or motion variable may be used as input and output variables in defining a system transfer function. In vibration studies, three particular choices are widely used. The corresponding

2 26


frequency transfer functions are named impedance functions, mobility functions, and transmissibility functions. These are described in the present section and in the subsequent section, and their use is illustrated. Through variables (force) and across variables (velocity), when expressed in the frequency domain (as Fourier spectra), are used in defining the two important frequency transfer functions: mechanical impedance and mobility. In the case of impedance functions, velocity is considered the input variable and the force is the output variable; whereas in the case of mobility functions, the converse applies. Specifically, M¼

1 Z

ð2:82Þ

It is clear that mobility ðMÞ is the inverse of impedance ðZÞ: Either transfer function may be used in a given problem. One can define several other versions of frequency transfer functions that might be useful in the modeling and analysis of mechanical systems. Some of the relatively common ones are listed in Table 2.2. Note that in the frequency domain, since acceleration¼ jv £ velocity; and displacement ¼ velocity/ðjvÞ; the alternative types of transfer functions as defined in Table 2.2, are related to mechanical impedance and mobility through a factor of jv; specifically, dynamic inertia ¼ force/acceleration ¼ impedance/ðjvÞ acceleration ¼ acceleration/force ¼ mobility £ ðjvÞ dynamic stiffness ¼ force/displacement ¼ impedance £ jv receptance ¼ displacement ¼ mobility/ðjvÞ In these definitions, the variable (force, acceleration, and displacement) should be interpreted as the corresponding Fourier spectra. The time domain constitutive relations for the mass, spring, and the damper elements are well known. The corresponding transfer relations are obtained by replacing the derivative operator d=dt by the Laplace operator s: The frequency transfer functions are obtained by substituting jv or j2p for s: These results are derived below. Mass Element: dv m ¼f dt In the frequency domain, f mjvv ¼ f ¼ mjv or v Hence, Zm ¼ mjv ð2:83aÞ and Mm ¼ TABLE 2.2

1 mjv

Definitions of Useful Mechanical Transfer Functions Transfer Function

Dynamic stiffness Receptance, dynamic flexibility, or compliance Impedance ðZÞ Mobility ðMÞ Dynamic inertia Accelerance Force transmissibility ðTf Þ Motion transmissibility ðTm Þ

Definition (in the Frequency Domain) Force/displacement Displacement/force Force/velocity Velocity/force Force/acceleration Acceleration/force Transmitted force/applied force Transmitted velocity/applied velocity

ð2:83bÞ

2 27


TABLE 2.3

Impedance and Mobility Functions of Basic Mechanical Elements Frequency Transfer Function (Set s ¼ jv ¼ j2p f )

Element

Impedance Mass, m

Zm ¼ ms

Spring, k

Zk ¼

Damper, b

Zb ¼ b

Mobility 1 ms s Mk ¼ k 1 Mb ¼ b Mm ¼

k s

Spring Element: f ¼ kx

Receptance 1 ms2 1 Rk ¼ k 1 Rb ¼ bs

Rm ¼

df ¼ kv dt

or

In the frequency domain, jvf ¼ kv

f k ¼ v jv

or

Hence, Zk ¼

k jv

ð2:84aÞ

Mk ¼

jv k

ð2:84bÞ

and

Damper Element: f ¼ bv

f ¼b v

or

Then, Zb ¼ b

ð2.85aÞ

1 b

ð2.85bÞ

or Mb ¼ These results are summarized in Table 2.3.

2.4.1

Interconnection Laws

Any general impedance element or a mobility element may be interpreted as a two port element in which, under steady conditions, energy (or power) transfer into the device takes place at the input port and energy (or power) transfer out of the device takes place at the output port. Each port of a two port element has a through variable, such as force or current, and an across variable, such as velocity or voltage, associated with it. Through variables are called flux variables, and across variable are called potential variables. Through variables are not always the same as flow variables (velocity and current). Similarly, across variables are not the same as effort variables (force and voltage). For example, force is an effort variable, but it is also a through variable. Similarly, velocity is a flow variable and is also an across variable. The concept of effort and flow variables is useful in giving unified definitions for electrical and mechanical impedance. However, in component interconnecting and circuit analysis, mechanical impedance is not analogous to electrical impedance. The definition of mechanical impedance is force/velocity in the frequency domain. This is a ratio of (through variable)/(across variable); whereas electrical impedance, defined as voltage/current in the frequency domain, is a ratio of

2 28 TABLE 2.4


Interconnection Laws for Impedance and Mobility Series Connections

f

Z1

Parallel Connections

f1

Z2

M1

v1

M2 v

Z1

f f2

v1

M1 Z2 M2

v ¼ v1 þ v2 v v v ¼ 1 þ 2 f f f M ¼ M1 þ M2 1 1 1 þ ¼ Z Z1 Z2

v

f ¼ f1 þ f2 f f f ¼ 1 ¼ 2 v v v Z ¼ Z1 þ Z2 1 1 1 þ ¼ M M1 M2

(across variable)/(through variable). Since both force and voltage are “effort” variables, and velocity and current are “flow” variables, it is then convenient to use the definition Impedance ðelectrical or mechanicalÞ ¼

Effort Flow

In other words, impedance measures how much effort is needed to drive a system at unity flow. Nevertheless, this definition does not particularly help us to analyze interconnected systems with mechanical impedance, because mechanical impedance cannot be manipulated using the rules for electrical impedance. For example, if two electric components are connected in series, the current (through variable) will be the same for both components, and the voltage (across variable) will be additive. Accordingly, the impedance of a series connected electrical system is just the sum of the impedances of the individual components. Now consider two mechanical components connected in series. Here the force (through variable) will be the same for both components, and velocity (across variable) will be additive. Hence, it is mobility, not impedance, that is additive in the case of series connected mechanical components. It can be concluded that, in circuit analysis, mobility behaves like electrical impedance and mechanical impedance behaves like electrical admittance. Hence, the “generalized series element” is electrical impedance or mechanical mobility, and the “generalized parallel element” is electrical admittance or mechanical impedance. The corresponding interconnection laws are summarized in Table 2.4. Now, two examples are given to demonstrate the use of impedance and mobility methods in frequency domain problems.

Example 2.5 Consider the simple oscillator shown in Figure 2.9(a). A schematic mechanical circuit is given in Figure 2.9(b). Note here that in this circuit, the broken line from the mass to the ground represents how the inertia force of the mass is felt by, or virtually transmitted to, the ground. This is the case because the net force that generates the acceleration in the mass (i.e., the inertia force) has to be transmitted to the ground at the reference point of the force source. This is the same reference with respect to which the velocity of the mass is expressed. If the input is the force f ðtÞ; the source element is a force source. The corresponding response is the velocity v; and in this situation, the transfer function Vð f Þ=Fð f Þ is a mobility function. On the other hand, if the input is the velocity vðtÞ; the source element is a velocity source. Then, f is the output, and the transfer function Fð f Þ=Vð f Þ is an impedance function.

2 29


f(t)

v

m

v f(t)

k

b

Suspension 0

(a)

fs

(b)

fs

v

Zm

(c) FIGURE 2.9

Zk

Zb

f(t)

0

(a) A ground based mechanical oscillator; (b) schematic mechanical circuit; (c) impedance circuit.

Suppose that a known forcing function is applied to this system (with zero initial conditions) using a force source, and the velocity is measured. Now, if we were to move the mass exactly at this predetermined velocity (using a velocity source), the force generated at the source will be identical to the originally applied force. In other words, mobility is the reciprocal (inverse) of impedance, as noted earlier. This reciprocity should be intuitively clear because we are dealing with the same system and same initial conditions. Owing to this property, we may use either the impedance representation or the mobility representation, depending on whether the elements are connected in parallel or in series, and irrespective of whether the input is a force or a velocity. Once the transfer function is determined in one form, its reciprocal gives the other form. In the present example, the three elements are connected in parallel, as is clear from the mechanical circuit shown in Figure 2.9(c). Hence, the impedance representation is appropriate. The overall impedance function of the system is Zð f Þ ¼

Fð f Þ k ms2 þ bs þ k ¼ Zm þ Zk þ Zb ¼ ms þ þ b ¼ Vð f Þ s s s¼j2p f

s¼j2p f

ð2:86Þ

Then, the mobility function is Mð f Þ ¼

Vð f Þ s ¼ Fð f Þ ms2 þ bs þ k

s¼j2p f

ð2:87Þ

Note that, if in fact the input is the force, the mobility function will govern the system behavior. In this case, the characteristic polynomial of the system is s2 þ bs þ k; which corresponds to a simple oscillator, and accordingly the (dependent) velocity response of the system would be governed by this. If, on the other hand, the input is the velocity, the impedance function will govern the system behavior. The characteristic polynomial of the system, in this case, is s; which corresponds to a simple integrator. The (dependent) force response of the system would be governed by an integrator type behavior. To explore

2 30


this behavior further, suppose that the velocity source has a constant value. The inertia force will be zero. The damping force will be constant. The spring force will increase linearly. Hence, the net force will have an integration (linearly increasing) effect. If the velocity source provides a linearly increasing velocity (constant acceleration), the inertia force will be constant, the damping force will increase linearly, and the spring force will increase quadratically.

Example 2.6 Consider the system shown in Figure 2.10(a). In this example, the motion of the mass, m, is not associated with an external force. The support motion, v; however, is associated with a force, f : The schematic mechanical circuit representation shown in Figure 2.10(b), and the corresponding impedance circuit shown in Figure 2.10(c), indicate that the spring and the damper are connected in parallel and that the mass is connected in series with this pair. By impedance addition for parallel elements and mobility addition for series elements, it follows that the mobility function is Vð f Þ 1 1 1 ¼ Mm þ ¼ þ Fð f Þ ðZk þ Zb Þ ms ðk=s þ bÞ

s¼j2pf

¼

ms2 þ bs þ k msðbs þ kÞ

s¼j2pf

ð2:88Þ

It follows that, when the support force is the input (force source) and the support velocity is the output, the system characteristic polynomial is msðbs þ kÞ, which is known to be inherently unstable due to the presence of a free integrator, and has a non oscillatory transient response. The impedance function that corresponds to support velocity input (velocity source) is the reciprocal of the previous mobility function; thus, Fð f Þ msðbs þ kÞ ¼ Vð f Þ ms2 þ bs þ k

ð2:89Þ

s¼j2pf

vm vm

m

k

b

+

v(t) (a)

f

– (b)

v(t)

f

Zm vm Zs,Ms + – (c) FIGURE 2.10

v(t)

o

(a) An oscillator with support motion; (b) schematic mechanical circuit; (c) impedance circuit.

2 31


and Vm ð f Þ 1 ¼ Fð f Þ ms

s¼j2pf

ð2:90Þ

The resulting impedance function Fð f Þ=Vm ð f Þ is not admissible and is physically non realizable because Vm cannot be an input as there is no associated force. This is confirmed by the fact that the corresponding transfer function is a differentiator, which is not physically realizable. The mobility function Vm ð f Þ=Fð f Þ corresponds to a simple integrator. Physically, when a force, f , is applied to the support, it is transmitted to the mass, unchanged, through the parallel spring damper unit. Accordingly, when f is constant, a constant acceleration is produced at the mass causing its velocity to increase linearly (an integration behavior). Maxwell’s principle of reciprocity can be demonstrated by noting that the mobility function Vm ð f Þ=Fð f Þ; obtained in this example, will be identical to the mobility function when the locations of f and Vm are reversed (i.e., when a force, f , is applied to the mass m and the resulting motion, Vm , of the support, which is not restrained by a force, is measured), with the same initial conditions. The reciprocity property is valid for linear, constant parameter systems in general and is particularly useful in vibration analysis and the testing of multi DoF systems; for example, to determine a transfer function that is difficult to measure, by measuring its symmetrical counterpart in the transfer function matrix.

2.5 Transmissibility Functions Transmissibility functions are transfer functions that are particularly useful in the analysis of vibration isolation in machinery and other mechanical systems. Two types of transmissibility function, force transmissibility and motion transmissibility, can be defined (see Table 2.2). Due to a reciprocity characteristic in linear systems, it can be shown that these two transfer functions are equal and, consequently, it is sufficient to consider only one of them. Let us, however, consider both types first and show their equivalence.

2.5.1

Force Transmissibility

Consider a mechanical system that is supported on a rigid foundation through a suspension system. If a forcing excitation is applied to the system, the force is not directly transmitted to the foundation. The suspension system acts as a vibration isolation device. Force transmissibility determines the fraction of the forcing excitation that is transmitted to the support structure (foundation) through the suspension, at different excitation frequencies, and is defined as Force Transmissibility; Tf ¼

Force Transmitted to Support; Fs Applied Force; F

ð2:91Þ

Note that this is defined in the frequency domain, and that accordingly Fs and F should be interpreted as the Fourier spectra of the corresponding forces. A schematic representation of the force transmissibility mechanism is shown in Figure 2.11(a). The reason for the suspension force fs being not equal to the applied force f is attributed to the inertia paths (broken line in the figure) that are present in a mechanical system.

2.5.2

Motion Transmissibility

Consider a mechanical system that is supported by suspension on a structure that may be subjected to undesirable motions (e.g., guideway deflections, vehicle motions, seismic disturbances). Motion transmissibility determines the fraction of the support motion that is transmitted to the system through

2 32


f (t)

Forcing Excitation

Mechanical System

vm Mechanical System

Zm

Mm =

1 Zm

fs vm = v f Suspension (a)

Zs

Suspension

fs

v (t) Ms =

1 Zs

(b)

FIGURE 2.11 (a) An inertial system with ground based suspension; (b) the counterpart (complementary) system with support motion.

its suspension at different frequencies. It is defined as Motion Transmissibility; Tm ¼

System Motion; Vm Support Motion; V

ð2:92Þ

The velocities Vm and V are expressed in the frequency domain, as Fourier spectra. A schematic representation of the motion transmissibility mechanism is shown in Figure 2.11(b). Typically, the motion of the system is taken as the velocity of one of its critical masses. Different transmissibility functions are obtained when different mass points (or DoFs) of the system are considered. Next, two examples are given to show a reciprocity property that makes the force transmissibility and the motion transmissibility functions identical in complementary (reciprocal) systems. System Suspended on a Rigid Base (Force Transmissibility) Consider the system suspended on a rigid base and excited by force f ðtÞ; as shown earlier in Figure 2.9(a). Here the system is the inertia element m, and the suspension is the parallel spring and damper combination. We have noted that the three elements m; k; and b are all in parallel. Here, Zm ¼ mjv; Zb ¼ b; and Zk ¼ k=jv; as given in Table 2.3. Now, because the elements are corrected in parallel, we have (see Table 2.4) f ¼ Zm þ Zb þ Zk v

ð2:93Þ

v 1 ¼ f Zm þ Zb þ Zd

ð2:94Þ

fs ¼ Zb þ Zk ¼ Zs v

ð2:95Þ

Hence,

Also, suspension impedance is

where fs is the force transmitted to the support structure (foundation). Then, fs Zb þ Zk Zs ¼ ¼ f Zm þ Zb þ Zk Zm þ Zs

ð2:96Þ

2 33


Transmissibility T 1+ 1 2 4z

1

0

FIGURE 2.12

Frequency r Ratio

1

Transmissibility curve for a simple mechanical oscillator.

This result should be immediately clear since force is divided among parallel branches in proportion to their impedance values (because the velocity is common). Now, Force Transmissibility Magnitude; lTf l ¼

Force Transmitted to Support Zs ¼ Applied Force to System Zm þ Zs

ð2:97Þ

Substitute parameters f lTf l ¼ s ¼ f

bþ

k jv

¼ k k jv sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2zvn vÞ2 þ v4n ¼ 2 ðvn v2 Þ2 þ ð2zvn vÞ2 mjv þ b þ

On simplification, we obtain

bjv þ k ¼ mv2 þ bjv

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4z2 r2 lTf l ¼ ð1 r2 Þ2 þ 4z2 r2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 v2 þ k2 ðk mv2 Þ2 þ b2 v2

ð2:98Þ

where the normalized frequency is

v ¼r vn At r ¼ 0; we have lTf l ¼ 1. At r ¼ 1;

sffiffiffiffiffiffiffiffiffiffiffi 1 lTf l ¼ 1 þ 2 4z

ð2:99Þ

This transmissibility magnitude curve is shown in Figure 2.12. System with Support Motion (Motion Transmissibility) Consider again the system suspended on a moving platform as shown in Figure 2.10(a). For this system, we have Zm ¼ mjv and Mm ¼ 1=mjv for the mass element. Since the damper and the spring are

2 34


connected in parallel, the corresponding impedances are additive. Hence, we have Zs ¼ Zb þ Zk ¼ b þ

k jv

and

Ms ¼

1 bþ

k jv

Motion of system inertia Mm ¼ Mm þ Ms Applied support motion

Motion Transmissibility Magnitude; Tm ¼

ð2:100Þ

This directly follows from the fact that the velocity is divided among series elements in proportion to their mobilities (because the force is common). However, Mm ¼ Mm þ Ms Hence,

1

M 1þ s Ms

lTm l ¼ It follows that

¼

1

¼

Z 1þ m Zs

Zs Zm þ Zs

ð2:101Þ

Zs Zm þ Zs

ð2:102Þ

Tf ¼ Tm

ð2:103Þ

This establishes the reciprocity property.

2.5.3

General Case

Consider an inertial system with a ground based suspension, as shown in Figure 2.11(a), and its counterpart with a moving support, as shown in Figure 2.11(b). The corresponding impedance circuits are shown in Figure 2.13. For system (a), we have fs Zs ¼ f Zm þ Zs For system (b), we have Vm Mm ¼ ¼ v Mm þ Ms

1

M 1þ s Mm

¼

1

Z 1þ m Zs

Zs

Zs Zs þ Zm

Zm , Mm

fs Zm

¼

vm f(t)

Zs, Ms + −

(a)

v(t)

(b)

FIGURE 2.13 Impedance circuits: (a) inertial system with ground based suspension; (b) system and suspension with support motion.

2 35


J k

. θ

Zb . θ

b Velocity + Source . α (t) 0

(a) FIGURE 2.14

+ . α(t) Zk

Zm

(b)

(a) Schematic mechanical circuit of the fluid coupling system; (b) impedance circuit.

It follows that fs V ¼ m f v

and

Tf ¼ Tm

ð2:104Þ

for this more general situation.

Example 2.7 Consider again the problem of a fluid coupling system shown in Figure 2.5 and studied in Example 2.1. A schematic mechanical circuit for the system is shown in Figure 2.14(a). The corresponding impedance circuit is shown in Figure 2.14(b). Let us use the impedance method to solve the same problem. For this problem, we have: input angular velocity ¼ a_ ðtÞ; angular velocity of the load ¼ u_; and load impedance Zl ¼ Zm þ Zk : In view of the series connection of Zb and Zl , we have

u_ Ml 1=Mb Zb ¼ ¼ ¼ a_ Ml þ Mb 1=Mb þ 1=Ml Zb þ Zm þ Zk

ðiÞ

t ¼ Zk u_

ðiiÞ

For the torsional spring,

Multiply Equation (i) and Equation (ii) together. Zk Zb t ¼ a_ ðZb þ Zm þ Zk Þ

ðiiiÞ

Since the time derivative corresponds to multiplication by jv in the frequency domain, we can write (iii) in the form

t Zk Zb ¼ jv a ðZb þ Zm þ Zk Þ Substitute Zb ¼ b; Zm ¼ jvJ; and Zk ¼ k=jv: We obtain t=a ¼ kbjv=ððk identical to what we obtained in Example 2.1.

2.5.4

ðivÞ J v2 Þ þ bjvÞ; which is

Peak Values of Frequency-Response Functions

The peak values of a frequency transfer function correspond to the resonances. The frequencies at these points are called resonant frequencies. Since a transfer function is the ratio of a response variable to an input variable, it is reasonable to obtain different peak frequencies for the same excitation input, if the response variable that is considered is different. Some results obtained for a damped oscillator model are summarized in Table 2.5.

2-36

TABLE 2.5

Some Practical Frequency Response Functions and Their Peaks System

Response/Excitation

Frequency Response Function (Normalized) 1 ð1 2 r 2 Þ þ 2jzr

Normalized Frequency ðrp Þ pffiffiffiffiffiffiffiffiffi 1 2 2z2

Peak Magnitude (Normalized) 1 pffiffiffiffiffiffiffiffi 2z 1 2 z2

Displacement/force

Simple oscillator with velocity response

Velocity/force

jr ð1 2 r2 Þ þ 2jzr

1

1 2z

Simple oscillator with acceleration response

Acceleration/force

2r 2 ð1 2 r 2 Þ þ 2jzr

1 pffiffiffiffiffiffiffiffiffiffi 1 2 2z2

1 pffiffiffiffiffiffiffiffi 2z 1 2 z2

Fluid coupling system

Torque/displacement

2jzr ð1 2 r 2 Þ þ 2jzr

1

1

Force transmissibility

Force/force

1 þ 2jzr ð1 2 r2 Þ þ 2jzr

qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffi 1 þ 8z2 2 1 2z

sffiffiffiffiffiffiffiffiffiffi 1 1 þ 2 (for small z) 4z

Motion transmissibility

Velocity/velocity

Same

Same

Same


Simple oscillator

2 37


2.6 Receptance Method Receptance is another name for dynamic flexibility or compliance, and is given by the transfer function (output displacement)/(input force) in the frequency domain (see Table 2.2). Also, as has been mentioned previously, it is directly related to mobility through Mobility ð2:105Þ jv and this relationship should be clear due to the fact that velocity ¼ jv £ displacement, with zero initial conditions, in the frequency domain. Hence, the receptance functions for the basic elements mass ðmÞ; spring ðkÞ; and damper ðbÞ can be derived from the mobility functions of these elements, as given in Table 2.3. As a result, the interconnection laws given for mobility ðMÞ will be valid for receptance ðRÞ as well. Specifically, for two receptance elements R1 and R2 connected in series, we have the combined receptance Receptance ¼

Series: R ¼ R1 þ R2

ð2:106Þ

because the displacement is additive and the force is common. For two receptance elements connected in parallel, the combined receptance R is given by Parallel:

1 1 1 þ ¼ R R1 R2

ð2:107Þ

because the forces are additive and the displacement is common. The inverse of receptance is dynamic stiffness.

2.6.1

Application of Receptance

The receptance method is widely used in the frequency domain analysis of multi DoF systems. This is true particularly because the receptance of a multicomponent system can be expressed in a convenient form in terms of the receptances of its constituent components. In deriving such relations, we use the conditions of continuity (forces balance at points of interconnection, or nodes) and compatibility (relative displacements in a loop add to zero). In fact, Equation 2.106 and Equation 2.107 are special cases of receptance relations for multicomponent systems. It should be clear from Table 2.3 that the receptance Rm of an inertia element ð 1=ðv2 mÞÞ; and the receptance Rk of a spring element ð1=kÞ; are real quantities, unlike the corresponding mobility functions. The receptance Rb of a damper ð1=ðbjvÞÞ is imaginary, however. It follows that receptance functions of undamped systems are real, and we will have to deal with real quantities only in the receptance analysis of undamped systems. This makes the analysis quite convenient. Also, since the displacement response of an undamped system becomes infinite when excited by a harmonic force at its natural frequency, we see that the receptance function of an undamped system goes to infinity (or its inverse becomes zero) at its natural frequencies. This property can be utilized in determining an undamped natural frequency (say, the fundamental natural frequency) of a system using the receptance method. In particular, the characteristic equations for a system with two interconnected components are 1 ¼0 R1 þ R2

ð2:108Þ

Parallel: R1 þ R2 ¼ 0

ð2:109Þ

Series:

and their solutions will give the undamped natural frequencies of the combined system. Now, we will consider two examples to illustrate the application of receptance techniques. Undamped Simple Oscillator Consider the simple oscillator shown in Figure 2.9, but assume that the damper is not present. As was noted earlier, the mass and the spring elements are connected in parallel. Hence, the characteristic

2 38


equation of the undamped system is or

or

Rm þ Rk ¼ 0

ð2:110Þ

1 1 þ ¼0 v2 m k

ð2:111Þ

k þ v2 m ¼ 0

whose positive solution is

sffiffiffiffi k v ¼ vn ¼ m

which gives the undamped natural frequency. Dynamic Absorber Dynamic absorbers are commonly used for vibration suppression in machinery over narrow frequency ranges. Specifically, a dynamic absorber can “absorb” the vibration energy from the main system (machine) at a specific frequency (the tuned frequency) and thereby completely balance the vibration excitation in the system. Consider a machine of equivalent mass M and equivalent stiffness K that is mounted on a rigid foundation, as modeled in Figure 2.15(a). A dynamic absorber, which is a lightly damped oscillator, of mass m and stiffness k is mounted on the machine. The damping is neglected in the model. The machine receives a vibration excitation f ðtÞ; and the objective of the absorber is to counteract this excitation. A schematic mechanical circuit of the system is shown in Figure 2.15(b). The overall system can be considered to consist of two subsystems: the subsystem a, representing the machine, has M and K connected in parallel with the excitation source; and the subsystem b; representing the vibrating absorber,

m

Dynamic Absorber (Subsystem b)

Machine (Subsystem a)

k

m

Subsystem b

k

f(t) Excitation M

M Subsystem a

K Ground

(a)

f(t) K

Source 0

(b)

a

b Rk RK

RM

f(t)

Rb

Source

Ra

f(t) Source

Rm (c)

(d)

FIGURE 2.15 (a) A machine with a vibration absorber; (b) schematic mechanical circuit; (c) component receptance circuit; and (d) subsystem receptance circuit.

2 39


Box 2.3 CONCEPTS OF RECEPTION Receptance; R

ðCompliance; Dynamic FlexibilityÞ

¼

Displacement Force

Receptance ¼ Mobility=jv Series Connection: R ¼ R1 þ R2 Parallel Connection: 1=R ¼ 1=R1 þ 1=R2 Note: R is real for undamped systems. Natural Frequency: R ! 1 (undamped case) Characteristic Equation: P For system with series components: 1= Ri ¼ 0 P For system with parallel components: 1=Ri ¼ 0

has m and k connected in series, as is clear from Figure 2.15(b). The corresponding receptance circuit, indicating the two subsystems with receptances Ra and Rb ; is shown in Figure 2.15(d). Since M and K are connected in parallel, from Equation 2.107 we have 1 ¼ Ra

v2 M þ K

ð2:112Þ

Since m and k are connected in series, from Equation 2.106 we have Rb ¼

1 1 þ v2 m k

ð2:113Þ

Now, since the subsystems a and b are connected in parallel, from Equation 2.109, the characteristic equation of the overall system is given by Ra þ Rb ¼ 0 ð2:114Þ Substitute Equations 2.112 and 2.113 in Equation 2.114. We obtain 1 þ v MþK 2

1 1 þ ¼0 2 vm k

ð2:115Þ

On simplification, after multiplying throughout by the common denominator, we obtain the characteristic equation mM v4

ðkM þ Km þ kmÞv2 þ kK ¼ 0

ð2:116Þ

This will give two positive roots for v; which are the two undamped natural frequencies of the system. Typically, the natural frequency of the vibration absorber has to be tuned to the frequency of excitation in order to achieve effective vibration suppression, as discussed in Chapter 12 (Enunciations 1 26). Here, we have only considered direct receptance functions, where the considered excitation and response are both for the same node. For more complex, multicomponent, multi DoF systems, we will need to consider cross receptance functions, where the response is considered at a node other than where the excitation force is applied. Such situations are beyond the scope of the present, introductory material. Some concepts of receptance are summarized in Box 2.3.

2 40


Bibliography Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, Wiley, New York. den Hartog, J.P. 1956. Mechanical Vibrations, McGraw Hill, New York. de Silva, C.W. 2004. MECHATRONICS An Integrated Approach, CRC Press, Boca Raton, FL. de Silva, C.W. 2000. VIBRATION Fundamentals and Practice, CRC Press, Boca Raton, FL. Inman, D.J. 1996. Engineering Vibration, Prentice Hall, Englewood Cliffs, NJ. Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood Cliffs, NJ. Randall, R.B. 1977. Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark. Rao, S.S. 1995. Mechanical Vibrations, 3rd ed., Addison Wesley, Reading, MA. Shearer, J.L., Murphy, A.T., and Richardson, H.H. 1971. Introduction to System Dynamics, Addison Wesley, Reading, MA. Thomson, W.T. and Dahleh, M.D. 1998. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper Saddle River, NJ.

Appendix 2A Transform Techniques 2A.1

Introduction

Many people use “transforms” without even knowing it. A transform is simply a number, variable, or function in a different form. For example, since 102 ¼ 100, one can use the exponent (2) to represent the number 100. Doing this for all numbers (i.e., using their exponent to the base 10) results in a “table of logarithms.” One can perform mathematical computations using only logarithms. The logarithm transforms all numbers into their exponential equivalents; a table of such transforms (i.e., a log table) enables a user to quickly transform any number into its exponent, do the computations using exponents (where a product becomes an addition and a division becomes a subtraction), and transform the result back into the original form (i.e., by an inverse logarithm). It is seen that the computations become simpler by using logarithms, but at the cost of the time and effort for transformation and inverse transformation. Other common transforms include the Laplace transform, Fourier transform, and Z transform. In particular, the Laplace transform provides a simple, algebraic way to solve (i.e., integrate) a linear differential equation. Most functions that we use are of the form t n ; sin vt; or et ; or some combination of them. Thus, in the expression y ¼ f ðtÞ the function y is quite likely to be a power, a sine, or an exponential function. Also, often, we must work with derivatives and integrals of these functions and differential equations containing these functions. These tasks can be greatly simplified by the use of the Laplace transform. Concepts of frequency response analysis originate from the nature of the response of a dynamic system to a sinusoidal (i.e., harmonic) excitation. These concepts can be generalized because the time domain analysis, where the independent variable is time (t), and the frequency domain analysis, where the independent variable is frequency (v), are linked through the Fourier transformation. Analytically, it is more general and versatile to use the Laplace transformation, where the independent variable is the Laplace variable (s), which is complex (nonreal). This is true because analytical Laplace transforms may exist even for time functions that do not have “analytical” Fourier transforms. However, with compatible definitions, the Fourier transform results can be obtained from the Laplace transform results simply by setting s ¼ jv: In the present appendix, we will formally introduce the Laplace transformation and the

2 41


Fourier transformation, and will illustrate how these techniques are useful in the analysis of mechatronic systems. The preference of one domain over another will depend on such factors as the nature of the excitation input, the type of the analytical model available, the time duration of interest, and the quantities that need to be determined.

2A.2

Laplace Transform

The Laplace transformation relates the time domain to the Laplace domain (also called s domain or complex frequency domain). The Laplace transform, Y(s), of a piecewise continuous function or signal, y(t), is given, by definition, as ð1 YðsÞ ¼ yðtÞexpð stÞdt ð2A:1Þ 0

and is denoted using the Laplace operator, L, as YðsÞ ¼ LyðtÞ

ð2A:1Þp

Here, s is a complex independent variable known as the Laplace variable, defined by s ¼ s þ jv

ð2A:2Þ

where s is a real valued pffiffiffiffi constant that will make the transform (Equation 2A.1) finite, v is simply frequency, and j ¼ 1: The real value (a) can be chosen sufficiently Ð large so that the integral in Equation 2A.1 is finite even when the integral of the signal itself (i.e., yðtÞdt) is not finite. This is the reason, for example, the Laplace transform functions better than the Fourier transform, which will be defined later, from the analytical point of view. The symbol s can be considered to be a constant, when integrating with respect to t, in Equation 2A.1. The inverse relation (i.e., obtaining y from its Laplace transform) is 1 ðsþjv yðtÞ ¼ YðsÞexpðstÞds ð2A:3Þ 2pj s jv and is denoted using the inverse Laplace operator, L 1, as yðtÞ ¼ L 1 YðsÞ

ð2A:3Þp

The integration in Equation 2A.3 is performed along a vertical line parallel to the imaginary (vertical) axis, located at s from the origin in the complex Laplace plane (the s plane). For a given piecewise continuous function, y(t), the Laplace transform exists if the integral in Equation 2A.1 converges. A sufficient condition for this is ð1 lyðtÞlexpð stÞdt , 1 ð2A:4Þ 0

Convergence is guaranteed by choosing a sufficiently large and positive s: This property is an advantage of the Laplace transformation over the Fourier transformation. 2A.2.1

Laplace Transforms of Some Common Functions

Now we determine the Laplace transform of some useful functions using the definition (Equation 2A.1). Usually, however, we use Laplace transform tables to obtain these results. Laplace Transform of a Constant Suppose our function yðtÞ is a constant, B: Then the Laplace transform is LðBÞ ¼ YðsÞ ¼

ð1 0

Be

st

dt ¼ B

e

st 1

s

0

¼

B s

2 42


Laplace Transform of the Exponential If yðtÞ is eat ; its Laplace transform is Lðeat Þ ¼ Note: If yðtÞ is e

at

ð1

st

e

0

eat dt ¼

ð1 0

eða

sÞt

dt ¼

1 ða

sÞ

eða

sÞt

1 0

¼

1 s

a

; it is obvious that the Laplace transform is

Lðe

at

Þ¼

ð1 0

e

st

e

at

dt ¼

ð1

e

0

ðaþsÞt

dt ¼

1 ða

sÞ

e

ðaþsÞt

1 0

¼

This result can be obtained from the previous result simply by replacing a with

1 sþa a.

Laplace Transform of Sine and Cosine pffiffiffiffi In the following, the letter j ¼ 1: If yðtÞ is sin vt; the Laplace transform is ð1 e st ðsin vtÞdt Lðsin vtÞ ¼ 0

Consider the identities: ejvt ¼ cos vt þ j sin vt jvt

e

¼ cos vt

j sin vt

If we add and subtract these two equations, respectively, we obtain the expressions for the sine and the cosine in terms of ejvt and e jvt : cos vt ¼

1 jvt ðe þ e 2

sin vt ¼

1 jvt ðe 2i

jwt

Þ

j vt

e

Þ

Lðcos vtÞ ¼

1 1 Lðejvt Þ þ Lðe 2 2

Lðsin vtÞ ¼

1 Lðejvt Þ 2

1 Lðe 2

jvt

Þ

jvt

Þ

We have just seen that Lðeat Þ ¼

1 s

;

Lðe

at

1 ; jvt

Lðe

jvt

a

Þ¼

1 sþa

Hence, Lðejvt Þ ¼

s

Þ¼

1 s þ jvt

Substituting these expressions, we obtain: Lðcos vtÞ ¼

Lðsin vtÞ ¼ ¼

1 1 1 1 þ 2 s jv 2 s þ jv

1 Lðejvt 2j

e

1 2jv 2j s2 þ v2

jvt

¼

Þ¼

1 1 2j s jv

v s2 þ v2

¼

1 s þ jv s jv þ 2 s ðjvÞ2 2 s2 ðjvÞ2 1 1 2j s þ jv

¼

¼

s s2 þ v 2

1 s þ jv s jv 2 þ 2 2 ðjvÞ s ðjvÞ2 2j s

2 43


Transform of a Derivative Let us transform a derivative of a function. Specifically, the derivative of a function y of t is denoted by y_ ¼ dy=dt: Its Laplace transform is given by ð1 ð1 dy dt ð2A:5Þ Lð_yÞ ¼ e st y_ dt ¼ e st dt 0 0 Now, we integrate by parts, to eliminate the derivative within the integrand. Integration by parts: From calculus we Ð know that dðuvÞ ¼ u dv þ v du: Ð By integrating we obtain uv ¼ u dv þ v du Hence, ð ð u dv ¼ uv v du

ð2A:6Þ

This is known as integration by parts. In Equation 2A.5, let u¼e

st

and

v¼y

Then, dv ¼ dy ¼

du ¼

dy dt ¼ y_ dt dt

du dt ¼ dt

se

st

dt:

Substitute in Equation 2A.5 to integrate by parts: Lð_yÞ ¼

ð1 0

e

¼ sLðyÞ

st

dy ¼

ð

u dv ¼ uv

ð

v du ¼ e

st

1

yðtÞ 0

ð1 0

se

st

yðtÞ dt ¼

yð0Þ þ sL½yðtÞ

yð0Þ

where y(0) ¼ initial value of y. This says that the Laplace transform of a first derivative, y_ ; equals s times the Laplace transform of the function y minus the initial value of the function (the initial condition). Note: We can determine the Laplace transforms of the second and higher derivatives by repeated application of the result for the first derivative. For example, the transform of the second derivative is given by L½€yðtÞ ¼ L

d_yðtÞ dt

L½€yðtÞ ¼ s2 L½yðtÞ

2A.2.2

¼ sL½_yðtÞ syð0Þ

y_ð0Þ ¼ s{sL½yðtÞ

yð0Þ}

y_ ð0Þ

or

y_ ð0Þ

Table of Laplace Transforms

Table 2A.1 shows the Laplace transforms of some common functions. Specifically, the table lists functions as yðtÞ; and their Laplace transforms (on the right) as YðsÞ or LyðtÞ: If one is given a function, one can obtain its Laplace transform from the table. Conversely, if one is given the transform, one can obtain the function from the table. Some general properties and results of the Laplace transform are given in Table 2A.2. In particular, note that with zero initial conditions, differentiation can be interpreted as multiplication by s. Also, integration can be interpreted as division by s.

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TABLE 2A.1

Laplace Transform Pairs

yðtÞ ¼ L 1 ½YðsÞ

L½yðtÞ ¼ YðsÞ

B

B=s

e

1 sþa

at

eat

1 s2a

sinh at

a s2 2 a2

cosh at

s s2 2 a2

sin vt

v s2 þ v2

cos vt

s s2 þ v2

e

at

sin vt

v ðs þ aÞ2 þ v2

e

at

cos vt

sþa ðs þ aÞ2 þ v2 1 s2

Ramp t e

at

s ðs þ aÞ2

ð1 2 atÞ

yðtÞ

YðsÞ

dy ¼ y_ dt

sYðsÞ 2 yð0Þ

d2 y ¼ y€ dt 2

s2 YðsÞ 2 syð0Þ 2 y_ ð0Þ

d3 y ... ¼y dt 3 Ðt a yðtÞdt

s3 YðsÞ 2 s2 yð0Þ 2 s_yð0Þ 2 y€ð0Þ 1 1 Ða YðsÞ 2 yðtÞdt s s 0

af ðtÞ þ bgðtÞ

aFðsÞ þ bGðsÞ (

Unit step U(t) ¼

1 for

t$0

0 for

otherwise

Delayed step cUðt 2 bÞ

1 s

C 0

0

b

c e s

t

C

Pulse c½UðtÞ 2 Uðt 2 bÞ

0

c 0

b

t

Impulse function dðtÞ

1

_ 2 bÞ Delayed impulse dðt 2 bÞ ¼ Uðt

e

Sine pulse

0

1 0

0

p/w

t

b

bs

12e s

bs

bs

t

v ð1 þ e s2 þ v2

ðp s=vÞ

Þ

2 45


TABLE 2A.2

Important Laplace Transform Relations

L 1 FðsÞ ¼ f ðtÞ 1 ðsþj1 FðsÞexpðstÞds 2pj s j1 k1 f1 ðtÞ þ k2 f2 ðtÞ

0

Lf ðtÞ ¼ FðsÞ f ðtÞexpð2stÞdt

k1 F1 ðsÞ þ k2 F2 ðsÞ

expð2atÞf ðtÞ

Fðs þ aÞ

f(t 2 t)

exp(2 ts)F(s)

f ðnÞ ðtÞ ¼

Ðt 1

dn f ðtÞ dt n

sn FðsÞ 2 sn 1 f ð0þ Þ 2 sn 2 f 1 ð0þ Þ 2 · · · 2 f n 1 ð0þ Þ Ð0 FðsÞ 1 f ðtÞdt þ s s n! snþ1 n! ðs þ aÞnþ1

f ðtÞdt

tn tn e

2A.3

Ð1

at

Response Analysis

The Laplace transform method can be used in the response analysis of dynamic systems, mechatronic and control systems in particular. We will give examples for the approach.

Example 2A.1 The capacitor charge equation of the RC circuit shown in Figure 2A.1 is e ¼ iR þ v

i

R

ðiÞ e

For the capacitor, i¼C

dv dt

C

v

ðiiÞ

Substitute Equation (ii) in (i) to get the circuit equation dv e ¼ RC þv dt

ðiiiÞ

FIGURE 2A.1 An RC circuit with applied voltage, e, and voltage, v, across the capacitor.

Take the Laplace transform of each term in Equation (iii), with all initial conditions ¼ 0 EðsÞ ¼ RCsVðsÞ þ VðsÞ The transfer function expressed as the output/input ratio (in the transform form) is VðsÞ VðsÞ 1 1 ¼ ¼ ¼ EðsÞ sRCVðsÞ þ VðsÞ sRC þ 1 ts þ 1

ðivÞ

where t ¼ RC: The actual response can now be found from Table 2A.1 for a given input E. The first step is to get the transform into proper form (like Line 2) 1 1=t a 1 ¼ ¼ ¼a ts þ 1 s þ ð1=tÞ sþa sþa where a ¼ 1=t: Suppose that input (excitation), e, is a unit impulse. Its Laplace transform (see Table 2A.1)

2 46


is E ¼ 1. Then from Equation (iv) VðsÞ ¼

1 ts þ 1

From Line 2 of Table 2A.1, the response is at

v¼ae

¼

1 e t

t=t

¼

1 e RC

t=RC

A common transfer function for an overdamped second order system (e.g., one with two RC circuit components of Figure 2A.1) would be VðsÞ 1 ¼ EðsÞ ð1 þ t1 sÞð1 þ t2 sÞ This can be expressed as “partial fractions” in the form A B þ 1 þ t1 s 1 þ t2 s and solved in the usual manner.

Example 2A.2 The transfer function of a thermal system is given by GðsÞ ¼

2 ðs þ 1Þðs þ 3Þ

If a unit step input is applied to the system, with zero initial conditions, what is the resulting response?

Solution Input: UðsÞ ¼ 1=s (for a unit step) Since YðsÞ 2 ¼ UðsÞ ðs þ 1Þðs þ 3Þ the output (response) is YðsÞ ¼

2 sðs þ 1Þðs þ 3Þ

Its inverse Laplace transform gives the time response. For this, first convert the expression into partial fractions as 2 A B C ¼ þ þ sðs þ 1Þðs þ 3Þ s ðs þ 1Þ ðs þ 3Þ

ðiÞ

The unknown, A, is determined by multiplying Equation (i) throughout by s and then setting s ¼ 0: We obtain A¼

2 2 ¼ ð0 þ 1Þð0 þ 3Þ 3

Similarly, B is obtained by multiplying Equation (i) throughout by ðs þ 1Þ and then setting s ¼ We obtain B¼

2 ¼ ð 1Þð 1 þ 3Þ

1

1:

2 47


Next, C is obtained by multiplying Equation (i) throughout by ðs þ 3Þ and then setting s ¼ We obtain C¼

2 ð 3Þð 3 þ 1Þ

¼

3:

1 3

Hence, YðsÞ ¼

2 3s

1 1 þ ðs þ 1Þ 3ðs þ 3Þ

Take the inverse transform using Line 2 of Table 2A.1 yðtÞ ¼

2 3

e

t

þ

1 e 3

3t

Example 2A.3 The transfer function of a damped simple oscillator is known to be of the form YðsÞ v2n ¼ 2 ðs þ 2zvn s þ v2n Þ UðsÞ where vn is the undamped natural frequency and z the damping ratio. Suppose that a unit step input (i.e., UðsÞ ¼ 1=s) is applied to the system. Using Laplace transform tables, determine the resulting response with zero initial conditions.

Solution YðsÞ

v2n 1 s ðs2 þ 2zvn s þ v2n Þ

The corresponding partial fractions are of the form: YðsÞ

A Bs þ C þ 2 s ðs þ 2zvn s þ v2n Þ

v2n sðs2 þ 2zvn s þ v2n Þ

ðiÞ

We need to determine A, B, and C. Multiply Equation (i) throughout by s and set s ¼ 0: We obtain: A¼1 Next, note that the roots of the characteristic equation, s2 þ 2zvn s þ v2n ¼ 0 are s¼

qffiffiffiffiffiffiffiffi zvn ^ z2 1vn ¼

zvn ^ jvd

These are the poles of the system and are complex conjugates. Two equations for Bpand C are obtained pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi 2 by multiplying Equation (i) by s þ zvnpffiffiffiffiffiffiffiffi z2 1vn and setting s ¼ zvnpþ z ffiffiffiffiffiffiffiffi 1vn ; and by z2 1vn : We obtain multiplying Equation (i) by s þ zvn þ z2 1vn and setting s ¼ zvn B ¼ 1 and C ¼ 0: Consequently, 1 s 1 s þ zvn z vd ¼ þ pffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 ½ðs þ zvn Þ þ vd s ðs þ 2zvn s þ vn Þ s 1 z ½ðs þ zvn Þ þ vd pffiffiffiffiffiffiffiffi where vd ¼ 1 z2 vn is the damped natural frequency. YðsÞ ¼

2 48


Now, use Table 2A.1 to obtain the inverse Laplace transform: ystep ðtÞ ¼ 1 ¼1

e

zvn t

cos vd t

z pffiffiffiffiffiffiffiffi e 1 z2

zvn t

e zvn t pffiffiffiffiffiffiffiffi ½sin f cos vd t þ cos f sin vd t 1 z2

sin vd t ¼ 1

e zvn t pffiffiffiffiffiffiffiffi sinðvd t þ fÞ 1 z2

where

qffiffiffiffiffiffiffiffi sin f ¼ 1 z2

cos f ¼ z ¼ damping ratio;

Example 2A.4 The open loop response of a plant to a unit impulse input, with zero initial conditions, was found to be 2 e t sin t: What is the transfer function of the plant?

Solution By linearity, since a unit impulse is the derivative of a unit step, the response to a unit impulse is given by the derivative of the result given in the previous example; thus,

zvn yimpulse ðtÞ ¼ pffiffiffiffiffiffiffiffi e 1 z2 vn e ¼ pffiffiffiffiffiffiffiffi 1 z2

zvn t

zvn t

vd pffiffiffiffiffiffiffiffi e 1 z2

sinðvd t þ fÞ

½cos f sinðvd t þ fÞ

zvn t

sin f cosðvd t þ fÞ

or

vn yimpulse ðtÞ ¼ pffiffiffiffiffiffiffiffi e 1 z2

zvn t

sin vd t

Compare this with the given expression. We have

vn pffiffiffiffiffiffiffiffi ¼ 2; 1 z2

zvn ¼ 1;

vd ¼ 1

However,

v2n ¼ ðzvn Þ2 þ v2d ¼ 1 þ 1 ¼ 2 Hence,

pffiffi vn ¼ 2

Hence, 1 z ¼ pffiffi 2 The system transfer function is ðs2

cosðvd t þ fÞ

2 v2n ¼ 2 þ 2zvn s þ v2n Þ s þ 2s þ 2

2 49


Example 2A.5 Express the Laplace transformed expression XðsÞ ¼

s3 þ 5s2 þ 9s þ 7 ðs þ 1Þðs þ 2Þ

as partial fractions. From the result, determine the inverse Laplace function, xðtÞ:

Solution XðsÞ

sþ2þ

2 sþ1

1 sþ2

From Table 2A.1, we get the inverse Laplace transform xðtÞ

d dðtÞ þ 2dðtÞ þ 2 e dt

t

e

2t

where dðtÞ is the unit impulse function. 2A.4

Transfer Function

By the use of Laplace transformation, a convolution integral equation can be converted into an algebraic relationship. To illustrate this, consider the convolution integral which gives the response, y(t), of a dynamic system to an excitation input, u(t), with zero initial conditions, as discussed in Chapter 2. By definition (Equation 2A.1), its Laplace transform is written as ð1 ð1 ð2A:7Þ YðsÞ ¼ hðtÞuðt tÞdt expð stÞdt 0

0

Note that h(t) is the impulse response function of the system. Since integration with respect to t is performed while keeping t constant, we have dt ¼ dðt tÞ: Consequently, ð1 ð1 uðt tÞexp½ sðt tÞ dðt tÞ hðtÞexpð stÞdt YðsÞ ¼ t

0

The lower limit of the first integration can be made equal to zero, in view of the fact that uðtÞ ¼ 0 for t , 0: Again, by using the definition of Laplace transformation, the foregoing relation can be expressed as YðsÞ ¼ HðsÞUðsÞ

ð2A:8Þ

in which HðsÞ ¼ LhðtÞ ¼

ð1 0

hðtÞexpð stÞdt

ð2A:9Þ

Note that, by definition, the transfer function of a system, denoted by HðsÞ, is given by Equation 2A.8. More specifically, the system transfer function is given by the ratio of the Laplace transformed output and the Laplace transformed input with zero initial conditions. In view of Equation 2A.9, it is clear that the system transfer function can be expressed as the Laplace transform of the impulse response function of the system. The transfer function of a linear and constant parameter system is a unique function that completely represents the system. A physically realizable, linear, constant parameter system possesses a unique transfer function, even if the Laplace transforms of a particular input and the corresponding output do not exist. This is clear from the fact that the transfer function is a system model and does not depend on the system input itself. Note: The transfer function is also commonly denoted by G(s). However, in the present context, we use H(s) in view of its relation to h(t).

2 50


Consider the nth order linear, constant parameter dynamic system given by an

dn y þ an dt n

1

dn 1 y duðtÞ dm uðtÞ þ · · · þ bm n 1 þ · · · þ a0 y ¼ b0 u þ b1 dt dt dt m

ð2A:10Þ

for a physically realizable system, m # n. By applying Laplace transformation and then integrating by parts, it may be verified that L

dk f ðtÞ ^ ¼ sk FðsÞ dt k

sk 1 f ð0Þ

sk

2

df ð0Þ dt

··· þ

dk 1 f ð0Þ dt k 1

ð2A:11Þ

By definition, the initial conditions are set to zero in obtaining the transfer function. This results in HðsÞ ¼

b0 þ b1 s þ · · · þ bm sm a0 þ a1 s þ · · · þ an sn

ð2A:12Þ

for m # n. Note that Equation 2A.12 contains all the information that is contained in Equation 2A.10. Consequently, transfer function is an analytical model of a system. The transfer function may be employed to determine the total response of a system for a given input, even though it is defined in terms of the response under zero initial conditions. This is logical because the analytical model of a system is independent of the initial conditions of the system.

2A.5

Fourier Transform

The Fourier transform, Y( f ), of a signal, y(t), relates the time domain to the frequency domain. Specifically, ðþ1 ðþ1 Yð f Þ ¼ yðtÞexpð j2p ftÞdt ¼ yðtÞe jvt dt ð2A:13Þ 1

1

Using the Fourier operator, F, terminology: Yð f Þ ¼ FyðtÞ

ð2A:14Þ

Note that if yðtÞ ¼ 0 for t , 0; as in the conventional definition of system excitations and responses, the Fourier transform is obtained from the Laplace transform by simply changing the variable according to s ¼ j2p f or jv: The Fourier is a special case of the Laplace, where, in Equation 2A.2, s ¼ 0: Yð f Þ ¼ YðsÞjs¼j2p f

ð2A:15Þ

YðvÞ ¼ YðsÞjs¼jv

ð2A:16Þ

or The (complex) function Y( f) is also termed the (continuous) Fourier spectrum of the (real) signal, y(t). The inverse transform is given by: ðþ1 ð2A:17Þ yðtÞ ¼ Yð f Þexpðj2p ftÞdf 1

or yðtÞ ¼ F

1

Yð f Þ

Note that according to the definition given by Equation 2A.13, the Fourier spectrum, Y( f ), is defined for the entire frequency range f( 1, þ1) which includes negative values. This is termed the two sided spectrum. Since, in practical applications it is not possible to have “negative frequencies,” the one sided spectrum is usually defined only for the frequency range f(0, 1). In order that a two sided spectrum has the same amount of power as a one sided spectrum, it is necessary to make the one sided spectrum double the two sided spectrum for f . 0. If the signal is not sufficiently transient (fast decaying or damped), the infinite integral given by Equation 2A.13 might not exist, but the corresponding Laplace transform might still exist.

2 51


2A.5.1 Frequency-Response Function (Frequency Transfer Function) The Fourier integral transform of the impulse response function is given by ð1 hðtÞexpð j2p ftÞdt Hð f Þ ¼ 1

ð2A:18Þ

where f is the cyclic frequency (measured in cycles/sec or hertz). This is known as the frequency response function (or frequency transfer function) of a system. Fourier transform operation is denoted as FhðtÞ ¼ Hð f Þ: In view of the fact that hðtÞ ¼ 0 for t , 0; the lower limit of integration in Equation 2A.18 can be made zero. Then, from Equation 2A.9, it is clear that H( f) is obtained simply by setting s ¼ j2p f in H(s). Hence, strictly speaking, we should use the notation Hðj2p f Þ and not H( f). However, for notational simplicity, we denote Hðj2p f Þ by H( f). Furthermore, since the angular frequency v ¼ 2p f , we can express the frequency response function as HðjvÞ; or simply by HðvÞ for notational convenience. It should be noted that the frequency response function, like the Laplace transfer function, is a complete representation of a linear, constant parameter system. In view of the fact that both uðtÞ ¼ 0 and yðtÞ ¼ 0 for t , 0, we can write the Fourier transforms of the input and the output of a system directly by setting s ¼ j2p f ¼ jv in the corresponding Laplace transforms. Then, from Equation 2A.8, we have Yð f Þ ¼ Hð f ÞUð f Þ

ð2A:19Þ

Note: Sometimes, for notational convenience, the same lowercase letters are used to represent the Laplace and Fourier transforms as well as the original time domain variables. If the Fourier integral transform of a function exists, then its Laplace transform also exists. The converse is not generally true, however, because of the poor convergence of the Fourier integral in comparison to the Laplace integral. This arises from the fact that the factor expð stÞ is not present in the Fourier integral. For a physically realizable, linear, constant parameter system, H( f ) exists even if U( f ) and Y( f) do not exist for a particular input. The experimental determination of H( f ), however, requires system stability. For the nth order system given by Equation 2A.10, the frequency response function is determined by setting s ¼ j2p f in Equation 2A.12 as Hð f Þ ¼

b0 þ b1 j2p f þ · · · þ bm ðj2p f Þm a0 þ a1 j2p f þ · · · þ an ðj2p f Þn

ð2A:20Þ

This, generally, is a complex function of f, which has a magnitude denoted by lHð f Þl and a phase angle denoted by /Hð f Þ:

2A.6 The s-Plane We have noted that the Laplace variable s is a complex variable, with a real part and an imaginary part. Hence, to represent it we will need two axes at right angles to each other the real axis and the imaginary axis. These two axes form a plane, which is called the s plane. Any general value of s (or any variation or trace of s) may be marked on the s plane. 2A.6.1 An Interpretation of Laplace and Fourier Transforms In the Laplace transformation of a function, f ðtÞ; we multiply the function by e st and integrate with respect to t. This process may be interpreted as determining the “components” FðsÞ of f ðtÞ in the “direction” e st ; where s is a complex variable. All such components FðsÞ should be equivalent to the original function, f ðtÞ: In the Fourier transformation of f ðtÞ we multiply it by e jvt and integrate with respect to t. This is the same as setting s ¼ jv: Hence, the Fourier transform of f ðtÞ is FðjvÞ: Furthermore, FðjvÞ represents the components of f ðtÞ that are in the direction of e jvt : Since e jvt ¼ cos vt j sin vt; in the Fourier transformation we determine the sinusoidal components of frequency v, of a time

2 52


function f ðtÞ: Since s is complex, FðsÞ is also complex and so is Fð jvÞ: Hence, they all will have a real part and an imaginary part.

2A.6.2 Application in Circuit Analysis The fact that sin vt and cos vt are 908 out of phase is further confirmed in view of ejvt ¼ cos vt þ j sin vt

ð2A:21Þ

Consider the R L C circuit shown in Figure 2A.2. For the capacitor, the current, i, and the voltage, v, are related through i¼C

dv dt

ð2A:22Þ

If the voltage v ¼ v0 sin vt; the current i ¼ v0 vC cos vt: Note that the magnitude of v=i is 1=vC (or 1=2p fC where v ¼ 2p f ; f is the cyclic frequency and v is the angular frequency). But v and i are out of phase by 908. In fact, in the case of a capacitor, i leads v by 908. The equivalent circuit resistance of a capacitance is called reactance, and is given by XC ¼ ¼

1 2p fC

ð2A:23Þ

1 vC

ð2A:24Þ

Note that this parameter changes with the frequency. We cannot add the reactance of the capacitor and the resistance of the resistor algebraically; we must add them vectorially because the voltages across a capacitor and resistor in series are not in phase, unlike in the case of a resistor. Also, the resistance in a resistor does not change with frequency. In a series circuit, as in Figure 2A.2, the current is identical in each element, but the voltages differ in both amplitude and phase; in a parallel circuit, the voltages are identical, but the currents differ in amplitude and phase. Similarly, for an inductor, v¼L

di dt

ð2A:25Þ

The corresponding reactance is XL ¼ vL ¼ 2p f L

ð2A:26Þ

If the voltage (E) across R in Figure 2A.2(a) is in the direction shown in Figure 2A.2(b) (i.e., pointing to the right), then the voltage across the inductor, L, must point up (908 leading) and the voltage across the capacitor, C, must point down (908 lagging). Since the current (I) is identical in each component of a EL or IXL R ER or IR

L

E

q Z

R XL − XC

(c) C

(b)

EC or IXC

(a) FIGURE 2A.2

(a) Series RLC circuit; (b) phases of voltage drops; (c) impedance triangle.

2 53


series circuit, we see the directions of IR, IXL and IXC as in Figure 2A.2(b), giving the impedance triangle shown in Figure 2A.2(c). To express these reactances in the s domain, we simply substitute s for jv : 1 sC jXL ¼ sL jXC ¼

The series impedance of the RLC circuit can be expressed as 1 Z ¼ R þ jXL jXC ¼ R þ sL þ sC pffiffiffiffi In this discussion, note the use of 1 or j to indicate a 908 phase change.

3

Modal Analysis 3.1 3.2

Introduction .......................................................................... 3 1 Degrees of Freedom and Independent Coordinates .......... 3 2

3.3

System Representation ......................................................... 3 4

The University of British Columbia

Stiffness and Flexibility Matrices † Inertia Matrix Approach for Equations of Motion

†

Direct

3.4 3.5

Modal Vibrations ................................................................ 3 10 Orthogonality of Natural Modes ...................................... 3 14

3.6

Static Modes and Rigid Body Modes ............................... 3 15

3.7

Other Modal Formulations ............................................... 3 22

3.8


Nonholonomic Constraints

Modal Mass and Normalized Modal Vectors

Static Modes † Linear Independence of Modal Vectors † Modal Stiffness and Normalized Modal Vectors † Rigid Body Modes † Modal Matrix † Configuration Space and State Space Nonsymmetric Modal Formulation Modal Formulation

†

Transformed Symmetric

Forced Vibration ................................................................. 3 28 First Mode (Rigid Body Mode) (Oscillatory Mode)

†

Second Mode

3.9

Damped Systems ................................................................. 3 32

3.10

State Space Approach ......................................................... 3 36

Proportional Damping

Modal Analysis † Mode Shapes of Nonoscillatory Systems Mode Shapes of Oscillatory Systems

†

Appendix 3A Linear Algebra ............................................ 3 41

Summary This chapter presents the modal analysis of lumped parameter mechanical vibrating systems. In the considered systems, inertia, flexibility, and damping characteristics are lumped at a finite number of discrete points in the system. Techniques for determining the natural frequencies and mode shapes of vibration are given. The orthogonality of mode shapes is established. The existence of natural modes in damped systems is investigated. Proportional damping is discussed. Both free vibration and forced vibration of multi degree of freedom (multi DoF) systems are analyzed.

3.1

Introduction

Complex vibrating systems usually consist of components that possess distributed energy storage and energy dissipative characteristics. In these systems, inertial, stiffness, and damping properties vary (piecewise) continuously with respect to the spatial location. Consequently, partial differential equations, with spatial coordinates (e.g., Cartesian coordinates x; y; z) and time t as independent variables are necessary to represent their vibration response. 31

32


A distributed (continuous) vibrating system may be approximated (modeled) by an appropriate set of lumped masses properly interconnected using discrete spring and damper elements. Such a model is termed lumped parameter model or discrete model. An immediate advantage resulting from this lumped parameter representation is that the system equations become ordinary differential equations. Often, linear springs and linear viscous damping elements are used in these models. The resulting linear ordinary differential equations can be solved by the modal analysis method. The method is based on the fact that these idealized systems (models) have preferred frequencies and geometric configurations (or natural modes) in which they tend to execute free vibration. An arbitrary response of the system can be interpreted as a linear combination of these modal vibrations, and as a result its analysis may be conveniently done using modal techniques. Modal analysis is an important tool in vibration analysis, diagnosis, design, and control. In some systems, mechanical malfunction or failure can be attributed to the excitation of their preferred motion such as modal vibrations and resonances. By modal analysis, it is possible to establish the extent and location of severe vibrations in a system. For this reason, it is an important diagnostic tool. For the same reason, modal analysis is also a useful method for predicting impending malfunctions or other mechanical problems. Structural modification and substructuring are techniques of vibration analysis and design that are based on modal analysis. By sensitivity analysis methods using a modal model, it is possible to determine which degrees of freedom (DoFs) of a mechanical system are most sensitive to addition or removal of mass and stiffness elements. In this manner, a convenient and systematic method can be established for making structural modifications to eliminate an existing vibration problem, or to verify the effects of a particular modification. A large and complex system can be divided into several subsystems which can be independently analyzed. By modal analysis techniques, the dynamic characteristics of the overall system can be determined from the subsystem information. This approach has several advantages, including: (1) subsystems can be developed by different methods such as experimentation, finite element method, or other modeling techniques and assembled to obtain the overall model; (2) the analysis of a high order system can be reduced to several lower order analyses; and (3) the design of a complex system can be carried out by designing and developing its subsystems separately. These capabilities of structural modification and substructure analysis which are possessed by the modal analysis method make it a useful tool in the design development process of mechanical systems. Modal control, a technique that employs modal analysis, is quite effective in the vibration control of complex mechanical systems.

3.2

Degrees of Freedom and Independent Coordinates

The geometric configuration of a vibrating system can be completely determined by a set of independent coordinates. This number of independent coordinates, for most systems, is termed the number of DoFs of the system. For example, a particle freely moving on a plane requires two independent coordinates to completely locate it (e.g., x and y Cartesian coordinates or r and u polar coordinates); its motion has two DoF. A rigid body that is free to take any orientation in (three dimensional) space needs six independent coordinates to completely define its position. For instance, its centroid is positioned using three independent Cartesian coordinates ðx; y; zÞ: Any axis fixed in the body and passing through its centroid can be oriented by two independent angles ðu; fÞ: The orientation of the body about this body axis can be fixed by a third independent angle ðcÞ: Altogether, six independent coordinates have been utilized; the system has six DoF. Strictly speaking, the number of DoF is equal to the number of independent, incremental, generalized coordinates that are needed to represent a general motion. In other words, it is the number of incremental independent motions that are possible. For holonomic systems (i.e., systems possessing holonomic constraints only), the number of independent incremental generalized coordinates is equal to the number of independent generalized coordinates; hence, either definition may be used for the number of DoF. If, on the other hand, the system has nonholonomic

Modal Analysis

33

constraints, the definition based on incremental coordinates should be used, because in these systems the number of independent incremental coordinates is in general less than the number of independent coordinates that are required to completely position the system.

3.2.1

Nonholonomic Constraints

Constraints of a system that cannot be represented by purely algebraic equations in its generalized coordinates and time are termed nonholonomic constraints. For a nonholonomic system, more coordinates than the number of DoF are required to completely define the position of the system. The number of excess coordinates is equal to the number of nonalgebraic relations that define the nonholonomic constraints in the system. Examples for nonholonomic systems are afforded by bodies rolling on surfaces and bodies whose velocities are constrained in some manner.

Example 3.1 A good example for a nonholonomic system is provided by a sphere rolling, without slipping, on a plane surface. In Figure 3.1, the point O denotes the center of the sphere at a given instant, and P is an arbitrary point within the sphere. The instantaneous point of contact with the plane surface is denoted by Q, so that the radius of the sphere is OQ ¼ a. This system requires five independent generalized coordinates to position it. For example, the center O is fixed by the Cartesian coordinates x and y: Since the sphere is free to roll along any arbitrary path on the plane and return to the starting point, the line OP can assume any arbitrary orientation for any given position for the center O. This line can be oriented by two independent coordinates u and f; defined as in Figure 3.1. Furthermore, since the sphere is free to spin about the z axis and is also free to roll on any trajectory (and return to its starting point), it follows that the sphere can take any orientation about the line OP (for a specific location of point O and line OP). This position can be oriented by the angle c: These five generalized coordinates x; y; u; f; and c are independent. The corresponding incremental coordinates dx; dy; du; df; and dc are, however, not independent, as a result of the constraint of rolling without slipping. It can be shown that two independent differential equations can be written for this constraint, and that consequently there exist only three independent incremental coordinates; the system actually has only three DoF. To establish the equations for the two nonholonomic constraints note that the incremental displacements dx and dy of the center O about the instantaneous point of contact Q can be written dx ¼ a db;

a da

dy ¼

z ψ φ

P

y β

θ

O a

α

x

Q

FIGURE 3.1

Rolling sphere on a plane (an example of a nonholonomic system).

34


in which the rotations of a and b are taken as positive about the positive directions of x and y; respectively (Figure 3.1). Next, we will express da and db in terms of the generalized coordinates. Note that du is directed along the z direction and has no components along the x and y directions. On the other hand, df has the components df cos u in the positive y direction and df sin u in the negative x direction. Furthermore, the horizontal component of dc is dc sin f: This in turn has the components ðdc sin fÞcos u and ðdc sin fÞsin u in the positive x and y directions, respectively. It follows that da ¼

df sin u þ dc sin f cos u

db ¼ df cos u þ dc sin f sin u Consequently, the two nonholonomic constraint equations are dx ¼ aðdf cos u þ dc sin f sin uÞ dy ¼ aðdf sin u

dc sin f cos uÞ

Note that these are differential equations that cannot be directly integrated to give algebraic equations. A particular choice for the three independent incremental coordinates associated with the three DoF in the present system of a rolling sphere would be du; df; and dc: The incremental variables da; db; and du will form another choice. The incremental variables dx; dy; and du will also form a possible choice. Once three incremented displacements are chosen in this manner, the remaining two incremental generalized coordinates are not independent and can be expressed in terms of these three incremented variables using the constraint differential equations.

Example 3.2 A relatively simple example for a nonholonomic system is a single dimensional rigid body (a straight line) moving on a plane such that its velocity is always along the body axis. The idealized motion of a ship in calm water is a practical situation representing such a system. This body needs three independent coordinates to completely define all possible configurations that it can take. For example, the centroid of the body can be fixed by two Cartesian coordinates x and y on the plane, and the orientation of the axis through the centroid may be fixed by a single angle u: Note that, for a given location ðx; yÞ of the centroid, any arbitrary orientation ðuÞ for the body axis is feasible, because, as in the previous example, any arbitrary trajectory can be followed by this body and return the centroid to the starting point, but with a different orientation of the axis of the body. Since the velocity is always directed along the body axis, a nonholonomic constraint exists and it is expressed as dy ¼ tan u dx It follows that there are only two independent incremental variables; the system has only two DoF. Some useful definitions and properties that were discussed in this section are summarized in Box 3.1.

3.3

System Representation

Some damped systems do not possess real modes. If a system does not possess real modes, modal analysis could still be used, but the results would only be approximately valid. In modal analysis it is convenient to first neglect damping and develop the fundamental results, and then subsequently extend the results to damped systems, for example, by assuming a suitable damping model that possesses real modes. Since damping is an energy dissipation phenomenon, it is usually possible to determine a model that possesses real modes and also has an energy dissipation capacity equivalent to that of the actual system.

Modal Analysis

35

Box 3.1 SOME DEFINITIONS AND PROPERTIES OF MECHANICAL SYSTEMS Holonomic constraints

Constraints that can be represented by purely algebraic relations

Nonholonomic constraints

Constraints that require differential relations for their representation

Holonomic system

A system that possesses holonomic constraints only

Nonholonomic system Number of DoFs

A system that possesses one or more nonholonomic constraints The number of independent incremental coordinates that are needed to represent general incremental motion of a system ¼ number of independent incremental motions ¼ 2 £ number of DoF (typically)

Order of a system For a holonomic system Number of independent incremental coordinates For a nonholonomic system Number of independent incremental coordinates

¼ Number of independent coordinates ¼ number of DoF ,Number of independent coordinates

Consider the three undamped system representations (models) shown in Figure 3.2. The motion of system (a) consists of the translatory displacements y1 and y2 of the lumped masses m1 and m2 : The masses are subjected to the external excitation forces (inputs) f1 ðtÞ and f2 ðtÞ and the restraining forces of the discrete, tensile compressive stiffness (spring) elements k1 ; k2 ; and k3 : Only two independent y1

(a)

Translatory System

Flexural System

k1

k1

m1

m1

y2 k2 f1(t)

f1t)

k2

k1

m1

f2(t)

k2

1 FIGURE 3.2

k3

f2(t)

m2

k3 y2

y1 (c)

f2(t)

y2

f1(t) Torsional System

m2

y1

(b)

k3

m2

2

Three types of two DoF systems.

36


incremental coordinates (dy1 and dy2 ) are required to completely define the incremental motion of the system subject to its inherent constraints. It follows that the system has two DoF. In system (b), shown in Figure 3.2, the elastic stiffness to the transverse displacements y1 and y2 of the lumped masses is provided by three bending ( flexural) springs that are considered massless. This flexural system is very much analogous to the translatory system (a) even though the physical construction and the motion itself are quite different. System (c) in Figure 3.2 is the analogous torsional system. In this case, the lumped elements m1 and m2 should be interpreted as polar moments of inertia about the shaft axis, and k1 ; k2 ; and k3 as the torsional stiffness in the connecting shafts. Furthermore, the motion coordinates y1 and y2 are rotations and the external excitations f1 ðtÞ and f2 ðtÞ are torques applied at the inertia elements. Practical examples where these three types of vibration system models may be useful are: (a) a two car train, (b) a bridge with two separate vehicle loads, and (c) an electric motor and pump combination. The three systems shown in Figure 3.2 are analogous to each other in the sense that the dynamics of all three systems can be represented by similar equations of motion. For modal analysis, it is convenient to express the system equations as a set of coupled second order differential equations in terms of the displacement variables (coordinates) of the inertia elements. Since in modal analysis we are concerned with linear systems, the system parameters can be given by a mass matrix and a stiffness matrix, or by a flexibility matrix. Lagrange’s equations of motion directly yield these matrices; however, we will now present an intuitive method for identifying the stiffness and mass matrices. The linear, lumped parameter, undamped systems shown in Figure 3.2 satisfy the set of dynamic equations " #" # " #" # " # m11 m12 k11 k12 y€1 f1 y1 þ ¼ m21 m22 k21 k22 y€2 f2 y2 or M€y þ Ky ¼ f

ð3:1Þ

Here, M is the inertia matrix which is the generalized case of mass matrix, and K is the stiffness matrix. There are many ways to derive Equation 3.1. Below, we will describe an approach, termed the influence coefficient method, which accomplishes the task by separately determining K and M.

3.3.1

Stiffness and Flexibility Matrices

In the systems shown in Figure 3.2 suppose the accelerations y€ 1 and y€ 2 are both zero at a particular instant, so that the inertia effects are absent. The stiffness matrix K is given under these circumstances by the constitutive relation for the spring elements: " # " #" # f1 k11 k12 y1 ¼ f2 k21 k22 y2 or f ¼ Ky

ð3:2Þ

T

T

in which f is the force vector ½f1 ; f2 and y is the displacement vector ½y1 ; y2 : Both are column vectors. The elements of the stiffness matrix, in this two DoF case, are explicitly given by " # k11 k12 K¼ k21 k22 Suppose that y1 ¼ 1 and y2 ¼ 0 (i.e., give a unit displacement to m1 while holding m2 at its original position). Then k11 and k21 are the forces needed at location 1 and location 2, respectively, to maintain this static configuration. For this condition it is clear that f1 ¼ k1 þ k2 and f2 ¼ k2 : Accordingly, k11 ¼ k1 þ k2 ;

k21 ¼

k2

Modal Analysis

37

Similarly, suppose that y1 ¼ 0 and y2 ¼ 1: Then k12 and k22 are the forces needed at location 1 and location 2, respectively, to maintain the corresponding static configuration. It follows that k12 ¼

k2 ;

k22 ¼ k2 þ k3

Consequently, the complete stiffness matrix can be expressed in terms of the stiffness elements in the system as " # k1 þ k2 k2 K¼ k2 k2 þ k3 From the foregoing development, it should be clear that the stiffness parameter kij represents the force that is needed at the location i to obtain a unit displacement at location j: Hence, these parameters are termed stiffness influence coefficients. Observe that the stiffness matrix is symmetric. Specifically, kij ¼ kji

for i – j

or KT ¼ K

ð3:3Þ

Note, however, that K is not diagonal in general (kij – 0 for at least two values of i – j). This means that the system is statically coupled (or flexibly coupled). Flexibility matrix L is the inverse of the stiffness matrix L¼K

1

ð3:4Þ

To determine the flexibility matrix using the influence coefficient approach, we have to start with a constitutive relation of the form y ¼ Lf

ð3:5Þ

Assuming that there are no inertia forces at a particular instant, we then proceed as before. For the systems in Figure 3.2, for example, we start with f1 ¼ 1 and f2 ¼ 0: In this manner, we can determine the elements l11 and l21 of the flexibility matrix " # l11 l12 L¼ l21 l22 However, here, the result is not as straightforward as in the previous case. For example, to determine l11 , we will have to find the flexibility contributions from either side of m1 : The flexibility of the stiffness element k1 is 1=k1 : The combined flexibility of k2 and k3 ; which are connected in series, is 1=k2 þ 1=k3 because the displacements (across variables) are additive in series. The two flexibilities on either side of m1 are applied in parallel at m1 : Since the forces (through variables) are additive in parallel, the stiffness will also be additive. Consequently, 1 1 1 ¼ þ l11 ð1=k1 Þ ð1=k2 þ 1=k3 Þ After some algebraic manipulation we get l11 ¼

k2 þ k3 k1 k2 þ k2 k3 þ k3 k1

38


TABLE 3.1

Combination Rules for Stiffness and Flexibility Elements

Connection

Graphical Representation

Combined Stiffness

Combined Flexibility

Series

1 ð1=k1 þ 1=k2 Þ

l1 þ l2

Parallel

k1 þ k2

1 ð1=l1 þ 1=l2 Þ

Since there is no external force at m2 in the assumed loading configuration, the deflections at m2 and m1 are proportioned according to the flexibility distribution along the path. Accordingly, l21 ¼

1=k3 l 1=k3 þ 1=k2 11

or l21 ¼

k2 k1 k2 þ k2 k3 þ k3 k1

l12 ¼

k2 k1 k2 þ k2 k3 þ k3 k1

l22 ¼

k1 þ k2 k1 k2 þ k2 k3 þ k3 k1

Similarly, we can obtain

and

Note that these results confirm the symmetry of flexibility matrices lij ¼ lji

for i – j

or LT ¼ L

ð3:6Þ

Also, we can verify the fact that L is the inverse of K. The series parallel combination rules for stiffness and flexibility that are useful in the present approach are summarized in Table 3.1. The flexibility parameters lij represent the displacement at the location i when a unit force is applied at location j: Hence, these parameters are termed flexibility influence coefficients.

3.3.2

Inertia Matrix

The mass matrix, which is used in the case of translatory motions, can be generalized as inertia matrix M in order to include rotatory motions as well. To determine M for the systems shown in Figure 3.2, suppose the deflections y1 and y2 are both zero at a particular instant so that the springs are in their static equilibrium configuration. Under these conditions, the equation of motion 3.1 becomes f ¼ M€y For the present two DoF case, the elements of M are denoted by " # m11 m12 M¼ m21 m22

ð3:7Þ

Modal Analysis

39

To identify these elements, first set y€ 1 ¼ 1 and y€ 2 ¼ 0: Then, m11 and m21 are the forces needed at the locations 1 and 2, respectively, to sustain the given accelerations; specifically, f1 ¼ m1 and f2 ¼ 0: It follows that m11 ¼ m1 ;

m21 ¼ 0

Similarly, by setting y€1 ¼ 0 and y€ 2 ¼ 1; we get m12 ¼ 0; Then, the mass matrix is obtained as

" M¼

m22 ¼ m2 m1

0

0

m2

#

It should be clear now that the inertia parameter mij represents the force that should be applied at the location i in order to produce a unit acceleration at location j: Consequently, these parameters are called inertia influence coefficients. Note that the mass matrix is symmetric in general; specifically mij ¼ mji

for i – j

or MT ¼ M

ð3:8Þ

Furthermore, when the independent displacements of the lumped inertia elements are chosen as the motion coordinates, as is typical, the inertia matrix becomes diagonal. If not, it can be made diagonal by using straightforward algebraic substitutions so that each equation contains the second derivative of just one displacement variable. Hence, we may assume mij ¼ 0 for i – j

ð3:9Þ

Then the system is said to be inertially uncoupled. This approach to finding K and M is summarized in Box 3.2. It can be conveniently extended to damped systems for determining the damping matrix C.

Box 3.2 INFLUENCE COEFFICIENT METHOD OF DETERMINING SYSTEM MATRICES (UNDAMPED CASE ) Stiffness Matrix (K)

Mass Matrix (M)

1. Set y€ ¼ 0 f ¼ Ky

1. Set y ¼ 0 f ¼ M€y

2. Set yj ¼ 1 and yi ¼ 0 for all i – j

2. Set y€j ¼ 1 and y€i ¼ 0 for all i – j

3. Determine f from the system diagram, that is needed to main equilibrium ¼ jth column of K

3. Determine f to maintain this condition ¼ jth column of M

Repeat for all j

Repeat for all j

3 10


y2

y1

k1y1

m1

k2 (y2 − y1) f1(t)

FIGURE 3.3

3.3.3

k2 (y2 − y1)

m2

k3y2 f2(t)

Free body diagram of the two DoF system.

Direct Approach for Equations of Motion

The influence coefficient approach that was described in the previous section is a rather indirect way of obtaining the equations of motion 3.1 for a multi DoF system. The most straightforward approach, however, is to sketch a free body diagram for the system, mark the forces or torques on each inertia element, and finally, apply Newton’s Second Law. This approach is now illustrated for the system shown in Figure 3.2(a). The equations of motion for the systems in Figures 3.2(b) and (c) will follow analogously. The free body diagram of the system in Figure 3.2(a) is sketched in Figure 3.3. Note that all the forces on each inertia element are marked. Application of Newton’s Second Law to the two mass elements separately gives m1 y€1 ¼

k1 y1 þ k2 ðy2

m2 y€ 2 ¼

k2 ðy2

y1 Þ

y1 Þ þ f1 ðtÞ k3 y2 þ f2 ðtÞ

The terms can be rearranged to obtain the following two coupled, second order, linear, ordinary differential equations: m1 y€ 1 þ ðk1 þ k2 Þy1 m2 y€ 2

k2 y2 ¼ f1 ðtÞ

k2 y1 þ ðk2 þ k3 Þy2 ¼ f2 ðtÞ

which may be expressed in the vector matrix form as " #" # " #" # " # m1 0 y€ 1 k1 þ k2 k2 y1 f1 ðtÞ þ ¼ 0 m2 y€ 2 k2 k2 þ k3 y2 f2 ðtÞ Observe that this result is identical to what we obtained by the influence coefficient approach. Another convenient approach that would provide essentially the same result is the energy method through the application of Lagrange’s equations. Two common types of models used in vibration analysis and applications are summarized in Box 3.3.

3.4

Modal Vibrations

Among the infinite number of relative geometric configurations the lumped masses in a multi DoF system could assume under free motion (i.e., with fðtÞ ¼ 0), when excited by an arbitrary initial state, there is a finite number of configurations that are naturally preferred by the system. Each of these configurations will have an associated frequency of motion. These motions are termed modal motions. By choosing the initial displacement y(0) proportional to a particular modal configuration, with zero initial velocity, y_ ð0Þ ¼ 0; that particular mode can be excited at the associated natural frequency of motion. The displacements of different DoF retain this initial proportion at all times. This constant proportion in displacement can be expressed as a vector c for that mode, and represents the mode shape. Note that each modal motion is a harmonic motion executed at a specific frequency v known as the natural frequency (undamped). In view of these general properties of modal motions,

Modal Analysis

3 11

Box 3.3 MODEL TYPES Linear

Nonlinear

Coupled second order equations M€y þ C_y þ Ky ¼ fðtÞ Response vector: y ¼ ½y1 ; y2 ; …; yp T ; p ¼ number of DoF Excitation vector: fðtÞ ¼ ½f1 ; f2 ; …; fp T M ¼ mass matrix C or B ¼ damping matrix K ¼ stiffness matrix

M€y ¼ hðy; y; fðtÞ)

Coupled first order equations (state space models) x_ ¼ Ax þ Bu y ¼ Cx State vector x ¼ ½x1 ; x2 ; …; xn T ; n ¼ order of the system Input (excitation) vector u ¼ ½u1 ; u2 ; …; um T Output (response) vector y ¼ ½y1 ; y2 ; …; yp T

x_ ¼ aðx; uÞ y ¼ yðxÞ

Notes: 1. Definition of state: If xðt0 Þ; and u from t0 to t1 ; are known, xðt1 Þ can be determined completely 2. State vector x contains a minimum number (n) of elements 3. State model is not unique (different state models are possible " # for the same system) y 4. One approach to obtaining a state model is to use x ¼ y_

they can be expressed by y ¼ c cos vt

ð3:10Þ

or, in the complex form, for ease of analysis, as y ¼ c ejvt

ð3:11Þ

When Equation 3.11 is substituted into the equation of unforced (free) motion, M€y þ Ky ¼ 0

ð3:12Þ

as required by the definition of modal motion, the following eigenvalue problem results: ½v2 M

Kc¼0

ð3:13Þ

For this reason, natural frequencies are sometimes called eigenfrequencies, and mode shapes are termed eigenvectors. The feasibility of modal motions for a given system is determined by the existence of nontrivial solutions for c (i.e., c – 0). Specifically, nontrivial solutions for c are possible if and only if the determinant of the system of linear homogeneous equation 3.13 vanishes; thus det½v2 M

K ¼0

ð3:14Þ

Equation 3.14 is known as the characteristic equation of the system. For an n DoF system, M and K are both n £ n matrices. It follows that the characteristic equation has n roots for v2 : For physically realizable systems these n roots are all nonnegative and they yield the n natural frequencies v1 ; v2 ; …; vn of the system. For each natural frequency vi ; when substituted into Equation 3.13 and solved for c, there

3 12


results a mode shape vector ci that determines up to one unknown parameter which can be used as a scaling parameter. Extra care should be exercised, however, when determining mode shapes for zero natural frequencies (i.e., rigid body modes) and repeated natural frequencies (i.e., for systems with a dynamic symmetry). We shall return to these considerations in later sections.

Example 3.3 Consider a mechanical system modeled as in Figure 3.4. This is obtained from the systems in Figure 3.2 by setting m1 ¼ m; m2 ¼ am; k1 ¼ k; k2 ¼ bk; and k3 ¼ 0: The corresponding mass matrix and the stiffness matrix are " M¼

m

0

0

am

# ;

" K¼

ð1 þ bÞk

bk

bk

Mass 1

k

βk

m

FIGURE 3.4

Mass 2 αm

A modal vibration example.

#

bk

The natural frequencies are given by the roots of the characteristic equation " 2 # v m ð1 þ bÞk bk det ¼0 bk v2 am bk By expanding the determinant, this can be expressed as ½v2 m

ð1 þ bÞk ½v2 am

b 2 k2 ¼ 0

bk

or

v4 am2

v2 km½b þ að1 þ bÞ þ bk2 ¼ 0 pffiffiffiffiffi Let us define a frequency parameter v0 ¼ k=m: This parameter is identified as the natural frequency of an undamped simple oscillator (single DoF mass spring system) with mass m and stiffness k: Consequently, the characteristic equation of the given 2 DoF system can be written as a

v v0

;

v2 v0

4

v v0

ða þ b þ abÞ

2

þ b¼0

whose roots are

v1 v0

2

2

¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 {a þ b þ ab} 1 7 1 ðaþ4bab þabÞ2 2a

The mode shapes are obtained by solving for c in " 2 v m ð1 þ bÞk

bk or

2

v 6 6 v0 6 6 4

#

bk v2 am

bk

c¼0

3

2

ð1 þ bÞ

b

b a

v v0

2

b

7 7 7c ¼ 0 7 5

In a mode shape vector, only the ratio of the elements is needed. This is because, in determining a mode shape, we are concerned about the relative motions of the lumped masses, not the absolute motions.

Modal Analysis

3 13

Nondimensional Frequency w 5.0 w0 4.0

w2 Curves (Mode 2)

Mass Ratio a

0.1 0.2

3.0

0.5 0.7 1.0 2.0 10.0

2.0 1.0

0.1 0.2 0.5 0.7 1.0

0.8 0.6

2.0

0.4 0.2 0.0 0.0

FIGURE 3.5

10.0

w1 Curves (Mode 1)

a 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 1.8 2.0 Stiffness Ratio b

Dependence of natural frequencies ðv=v0 Þ on mass ratio ðaÞ and stiffness ratio ðbÞ:

From the above equation, it is clear that this ratio is given by

c2 ¼ c1

ð1 þ bÞ

b

v v0

2

b

¼

b

a

v v0

2

which is evaluated by substituting the appropriate value for ðv=v0 Þ; depending on the mode, into any one of the right hand side expressions above. The dependence of the natural frequencies on the parameters a and b is illustrated by the curves in Figure 3.5. Some representative values of the natural frequencies and mode shape ratios are listed in Table 3.2. Note that, when b ¼ 0; the spring connecting the two masses does not exist and the system reduces to two separate systems: a simple oscillator of natural frequency v0 and a single mass particle (of zero natural frequency). It is clear that in this case v1 =v0 ¼ 0 and v2 =v0 ¼ 1: This fact can be established from the expressions for natural frequencies of the original system by setting b ¼ 0: The mode corresponding to v1 =v0 ¼ 0 is a rigid body mode in which the free mass moves indefinitely (zero frequency) and the other mass (restrained mass) stands still. It follows that the mode shape ratio ðc2 =c1 Þ1 ! 1: In the second mass, the free mass stands still and the restrained mass moves. Hence, ðc2 =c1 Þ1 ¼ 0: These results are also obtained from the general expressions for the mode shape ratios of the original system. TABLE 3.2

The Dependence of Natural Frequencies and Mode Shapes on Inertia and Stiffness

a

0.5

b

v1 =v0

v2 =v0

ðc2 =c1 Þ1

0 0.5 1.0 2.0 5.0 1

0 0.71 0.77 0.79 0.81 0.82

1.0 1.41 1.85 2.52 3.92 1

1 2.0 1.41 1.19 1.07 1.0

1.0 ðc2 =c1 Þ2 0 21.0 21.41 21.69 21.87 22.0

v1 =v0

v2 =v0

ðc2 =c1 Þ1

0 0.54 0.62 0.66 0.69 0.71

1.0 1.31 1.62 2.14 3.24 1

1 2.41 1.62 1.28 1.10 1.0

2.0 ðc2 =c1 Þ2 0 20.41 20.62 20.78 20.91 21.0

v1 =v0

v2 =v0

ðc2 =c1 Þ1

0 0.40 0.47 0.52 0.55 0.57

1.0 1.26 1.51 1.93 2.86 1

1 2.69 1.78 1.37 1.14 1.0

ðc2 =c1 Þ2 0 20.19 20.28 20.37 20.44 20.5

3 14


When b ! 1; the spring connecting the two masses becomes rigid and the two masses act as a single mass ð1 þ aÞm restrained by a spring of stiffness k: This simple oscillator has a squared natural frequency of v20 =ð1 þ aÞ: This is considered the smaller natural frequency of the corresponding system: ðv1 =v0 Þ2 ¼ 1=ð1 þ aÞ: The larger natural frequency v2 approaches 1 in this case and it corresponds to the natural frequency of a massless spring. These limiting results can be derived from the general expressionspfor the ffiffiffiffiffiffiffi natural frequencies of the original system by using the fact that for small lxl p 1; the expression 1 x is approximately equal to 1 ð1=2Þx: (Proof: Use the Taylor series expansion.) In the first mode, the two masses move as one unit and hence the mode shape ratio ðc2 =c1 Þ1 ¼ 1: In the second mode, the two masses move in opposite directions restrained by an infinitely stiff spring. This is considered the static mode which results from the redundant situation of associating two DoF to a system that actually has only one lumped mass ð1 þ aÞm: In this case, the mode shape ratio is obtained from the general result as follows: For large b; we can neglect a in comparison. Hence,

v2 v0

2

¼

1 b {b þ ab}{1 þ 1} ¼ ð1 þ aÞ 2a a

By substituting this result in the expression for the mode shape ratio, we obtain

c2 c1

2

b

¼

b

v a 2 v0

2

b

¼

b að1 þ aÞ a

b

¼

1 a

Finally, consider the case a ¼ 0 (with b – 0). In this case, only one mass m restrained by a spring of stiffness k is present. The spring of stiffness bk has an open end. The first mode corresponds to a simple oscillator of natural frequency v0 : Hence, v1 =v0 ¼ 1: The open end has the same displacement as the point mass. Consequently, ðc2 =c1 Þ1 ¼ 1: These results can be derived from the general expressions for the original system. In the second mode the simple oscillator stands still and the open ended spring oscillates (at an infinite frequency). Hence v2 =v0 ¼ 1; and this again corresponds to a static mode situation which arises by assigning two DoF to a system that has only one DoF associated with its inertia elements. Since the lumped mass stands still, we have ðc2 =c1 Þ2 ¼ 1: Note that, when a ¼ 0 and b ¼ 0, the system reduces to a simple oscillator and the second mode is completely undefined. Hence, the corresponding results cannot be derived from the general results for the original system.

3.5

Orthogonality of Natural Modes

Let us write Equation 3.13 explicitly for the two distinct modes i and j: Distinct modes are defined as those having distinct natural frequencies (i.e., vi – vj ).

Premultiply Equation 3.15 by

cTj

v2i Mci

Kci ¼ 0

ð3:15Þ

v2j Mcj

Kcj ¼ 0

ð3:16Þ

and Equation 3.16 by

cTi

v2i cTj Mci

cTj Kci ¼ 0

ð3:17Þ

v2j cTi Mcj

cTi Kcj

ð3:18Þ

¼0

Take the transpose of Equation 3.18, which is a scalar:

v2j cTj MT ci

cTj KT ci ¼ 0

This, in view of the symmetry of M and K as expressed in Equation 3.8 and Equation 3.3, becomes

v2j cTj Mci

cTj Kci ¼ 0

Modal Analysis

3 15

By subtracting this result from Equation 3.17, we get ðv2i v2j ÞcTi Mcj ¼ 0 Now, because vi – vj ; it follows that ( 0 for i – j T ci Mcj ¼ Mi for i ¼ j

y1

m

k

θ 1 (y +y ) 2 1 2 Centroid

y2

m

k

ð3:19Þ

l Equation 3.19 is a useful orthogonality condition for natural modes. Even though the foregoing condition of FIGURE 3.6 A simplified vehicle model for heave and M orthogonality was proved for distinct (unequal) pitch motions. natural frequencies it is generally true, even if two or more modes have repeated (equal) natural frequencies. Indeed, if a particular natural frequency is repeated r times, there will be r arbitrary elements in the modal vector. As a result ,we are able to determine r independent mode shapes that are orthogonal with respect to the M matrix. An example is given later in the problem of Figure 3.6. Note further that any such mode shape vector corresponding to a repeated natural frequency will also be M orthogonal to the mode shape vector corresponding to any of the remaining distinct natural frequencies. Consequently, we conclude that the entire set of n mode shape vectors is M orthogonal even in the presence of various combinations of repeated natural frequencies.

3.5.1

Modal Mass and Normalized Modal Vectors

Note that, in Equation 3.19, a parameter Mi has been defined to denote cTi Mci : This parameter is termed the generalized mass or modal mass for the ith mode. Since the modal vectors ci are determined for up to one unknown parameter, it is possible to set the value of Mi arbitrarily. The process of specifying the unknown scaling parameter in the modal vector, according to some convenient rule, is called the normalization of modal vectors. The resulting modal vectors are termed normal modes. A particularly useful method of normalization is to set each modal mass to unity ðMi ¼ 1Þ: The corresponding normal modes are said to be normalized with respect to the mass matrix, or M normal. Note that, if ci is normal with respect to M, then it follows from Equation 3.18 that ci is also normal with respect to M. Specifically, ð ci ÞT Mð ci Þ ¼ cTi Mci ¼ 1 ð3:20Þ It follows that M normal modal vectors are still arbitrary up to a multiplier of 1. A convenient practice for eliminating this arbitrariness is to make the first element of each normalized modal vector positive.

3.6 3.6.1

Static Modes and Rigid-Body Modes Static Modes

Modes corresponding to infinite natural frequencies are known as static modes. For these modes, the modal mass is zero; in the normalization process with respect to M static modes cannot be included. If we assign a DoF for a massless location, the resulting mass matrix M becomes singular ðdet M ¼ 0Þ and a static mode arises. We have come across two similar situations in a previous example; in one case the stiffness of the spring connecting two masses is made infinite so that they act as a single mass in the limit, and in the other case one of the two masses is made equal to zero so that only one mass is left. We should take extra precautions to avoid such situations by using proper modeling practices; the presence of a static mode amounts to assigning a DoF to a system that it does not actually possess. In a static mode, the system behaves like a simple massless spring. In the literature of experimental modal analysis, the static modes are represented by a residual flexibility term in the transfer functions. Note that, in this case, modes of natural frequencies that are

3 16


higher than the analysis bandwidth or the maximum frequency of interest are considered static modes. Such issues of experimental modal analysis will be discussed in Chapter 18.

3.6.2

Linear Independence of Modal Vectors

In the absence of static modes (i.e., modal masses Mi – 0), the inertia matrix M will be nonsingular. Then the orthogonality condition 3.19 implies that the modal vectors are linearly independent, and consequently, they will form a basis for the n dimensional space of all possible displacement trajectories y for the system. Any vector in this configuration space (or displacement space), therefore, can be expressed as a linear combination of the modal vectors. Note that we have assumed in the earlier development that the natural frequencies are distinct (or unequal). Nevertheless, linearly independent modal vectors are possessed by modes of equal natural frequencies as well. An example is the situation where these modes are physically uncoupled. These modal vectors are not unique, however; there will be arbitrary elements in the modal vector equal in number to the repeated natural frequencies. Any linear combination of these modal vectors can also serve as a modal vector at the same natural frequency. To explain this point further, without loss of generality suppose that v1 ¼ v2 : Then, from Equation 3.15, we have

v21 Mc1

Kc1 ¼ 0

v21 Mc2

Kc2 ¼ 0

Multiply the first equation by a; the second equation by b, and add the resulting equations. We get

v21 Mðac1 þ bc2 Þ

Kðac1 þ bc2 Þ ¼ 0

This verifies that any linear combination ac1 þ bc2 of the two modal vectors c1 and c2 will also serve as a modal vector for the natural frequency v1 : The physical significance of this phenomenon should be clear in Example 3.4.

3.6.3

Modal Stiffness and Normalized Modal Vectors

It is possible to establish an alternative version of the orthogonality condition given as Equation 3.19 by substituting it into Equation 3.18. This gives ( 0 for i – j cTi Kcj ¼ ð3:21Þ Ki for i ¼ j This condition is termed K orthogonality. Since the M orthogonality condition (Equation 3.19) is true even for the case of repeated natural frequencies, it should be clear that the K orthogonality condition (Equation 3.21) is also true, in general, even with repeated natural frequencies. The newly defined parameter Ki represents the value of cTi Kci and is known as the generalized stiffness or modal stiffness corresponding to the ith mode. Another useful way to normalize modal vectors is to choose their unknown parameters so that all modal stiffnesses are unity (Ki ¼ 1 for all i). This process is known as normalization with respect to the stiffness matrix. The resulting normal modes are said to be K normal. These normal modes are still arbitrary up to a multiplier of 1. This can be eliminated by assigning positive values to the first element of all modal vectors. Note that it is not possible to normalize a modal vector simultaneously with respect to both M and K, in general. To understand this further, we may observe that v2i ¼ Ki =Mi and consequently we are unable to pick both Ki and Mi arbitrarily. In particular, for the M normal case Ki ¼ v2i and for the K normal case Mi ¼ 1=v2i :

Modal Analysis

3.6.4

3 17

Rigid-Body Modes

Rigid body modes are those for which the natural frequency is zero. Modal stiffness is zero for rigid body modes, and as a result it is not possible to normalize these modes with respect to the stiffness matrix. Note that when rigid body modes are present the stiffness matrix becomes singular ðdet K ¼ 0Þ: Physically, removal of a spring that connects two DoF results in a rigid body mode. In Example 3.3 we came across a similar situation. In experimental modal analysis applications, low stiffness connections or restraints, which might be present in a test object, could result in approximate rigid body modes that would become prominent at low frequencies. Some important results of modal analysis that we have discussed thus far are summarized in Table 3.3.

Example 3.4 Consider a light rod of length l having equal masses m attached to its ends. Each end is supported by a spring of stiffness k as shown in Figure 3.6. Note that this system may represent a simplified model of a vehicle in heave and pitch motions. Gravity effects can be eliminated by measuring the displacements y1 and y2 of the two masses about their respective static equilibrium positions. Assume small front to back rotations u in the pitch motion and small up and down displacements ð1=2Þðy1 þ y2 Þ of the centroid in its heave motion. 3.6.4.1

Equation of Heave Motion

From Newton’s Second Law for rigid body motion, we get 1 2m ð€y1 þ y€ 2 Þ ¼ ky1 2 3.6.4.2

ky2

Equation of Pitch Motion

Note that, for small angles of rotation, u ¼ ðy1 y2 Þ=l: The moment of inertia of the system about the centroid is 2mðl=2Þ2 ¼ ð1=2Þml2 : Hence, by Newton’s Second Law for rigid body rotation, we have 1 2 y€ 1 y€ 2 ml 2 l

¼

l l ky þ ky 2 1 2 2

These two equations of motion can be written as y€ 1 þ y€ 2 þ v20 ðy1 þ y2 Þ ¼ 0 y€1 TABLE 3.3

y€ 2 þ v20 ðy1

y2 Þ ¼ 0

Some Important Results of Modal Analysis

System

M€y þ Ky ¼ fðtÞ

Symmetry

MT ¼ M and KT ¼ K

Modal problem

½v2 M 2 K c ¼ 0

Characteristic equation (gives natural frequencies)

det½v2 M 2 K ¼ 0 ( 0 for i – j cTi Mcj ¼ Mi for i ¼ j ( 0 for i – j cTi Kcj ¼ Ki for i ¼ j

M orthogonality K orthogonality Modal mass (generalized mass)

Mi

Modal stiffness (generalized stiffness)

Ki

Natural frequency

pffiffiffiffiffiffiffi vi ¼ Ki =Mi

M normal case

Mi ¼ 1; Ki ¼ v2i

K normal case

Ki ¼ 1; Mi ¼ 1=v2i

Presence of rigid body modes

det K ¼ 0; Ki ¼ 0; and vi ¼ 0

Presence of static modes

det M ¼ 0; Mi ¼ 0; and vi ! 1

3 18


pffiffiffiffiffi in which v0 ¼ k=m: By straightforward algebraic manipulation, a pair of completely uncoupled equations of motion are obtained; thus y€ 1 þ v20 y1 ¼ 0 y€ 2 þ v20 y2 ¼ 0 It follows that the resulting mass matrix and the stiffness matrix are both diagonal. In this case, there is an infinite number of choices for mode shapes, and any two linearly independent second order vectors can serve as modal vectors for the system. Two particular choices are shown in Figure 3.7. Any of these mode shapes will correspond to the same natural frequency v0 : In each of these two choices, the mode shapes have been chosen so that they are orthogonal with respect to both M and K. This fact is verified below. Note that, in the present example " # " 2 # 1 0 v0 0 M¼ and K ¼ 0 1 0 v20 For Case 1:

" ½1

For Case 2: ½1

1 M

1

#

" ¼0

1

" # 0 0 M ¼0 1

and

and

½1

½1

1 K

0 K

1 1

" # 0 1

# ¼0

¼0

In general, since both elements of each eigenvector can be picked arbitrarily, we can write " # " # 1 1 c1 ¼ and c2 ¼ a b where a and b are arbitrary, limited only by the M orthogonality requires " 1 ½1 a 0 and K orthogonality requires

" ½1

a

orthogonality requirement for c1 and c2 : The 0

#" # 1

¼0

1

b

v20

0

#" # 1

0

v20

Case 1

b

¼0

Case 2

Mode 1

y1 = 1 1 

y1 = 1 0  

Mode 2

1 y2 =   −  1

0 y2 =   1  

FIGURE 3.7

Two possibilities of mode shapes for the symmetric heave pitch vehicle.

Modal Analysis

3 19

Both conditions give 1 þ ab ¼ 0; which corresponds to ab ¼ 1: Note that Case 1 corresponds to a ¼ 1 and b ¼ 1 and Case 2 corresponds to a ¼ 0 and b ! 1: More generally, we can pick as modal vectors

c1 ¼

" # 1

" and

a

c2 ¼

#

1 1=a

such that the two mode shapes are both M orthogonal and K orthogonal. In fact, if this particular system is excited by an arbitrary initial displacement, it will undergo free vibrations at frequency v0 while maintaining the initial displacement ratio. Hence, if M orthogonality and K orthogonality are not required, any arbitrary second order vector may serve as a modal vector to this system.

Example 3.5

y1

An example for a system possessing a rigid body mode is shown in Figure 3.8. This system, a crude model of a two car train, can be derived from the system shown in Figure 3.4 by removing the end spring (inertia restraint) and setting a ¼ 1 and b ¼ 1: The equation for unforced motion of this system is "

m

0

0

m

#"

y€ 1

#

y€ 2

y2 k

m

FIGURE 3.8

"

k

þ

k

k

#"

k

y1 y2

m

A simplified model of a two car train.

# ¼

" # 0 0

Note that det M ¼ m2 – 0 and hence the system does not possess static modes. This should also be obvious from the fact that each DoF (y1 and y2 ) has an associated, independent mass element. On the other hand, det K ¼ k2 k2 ¼ 0 which signals the presence of rigid body modes. The characteristic equation of the system is " 2 # vm k k ¼0 det v2 m k k or ðv2 m

kÞ2

k2 ¼ 0

pffiffiffiffiffiffi The two natural frequencies are given by the roots: v1 ¼ 0 and v2 ¼ 2k=m: Note that the zero natural frequency corresponds to the rigid body mode. The mode shapes can reveal further interesting facts. 3.6.4.3

First Mode (Rigid-Body Mode)

In this case, we have v ¼ 0: Consequently, from Equation 3.15, the mode shape is given by " #" # " # 0 k k c1 ¼ 0 k k c2 which has the general solution c1 ¼ c2 ; or

"

c1 c2

# ¼ 1

" # a a

The parameter a can be chosen arbitrarily. The corresponding modal mass is " #" # m 0 a M1 ¼ ½ a a ¼ 2ma2 0 m a

3 20


pffiffiffiffi If the modal vector is normalized with respect to M, we have M1 ¼ 2ma2 ¼ 1: Then, a ¼ ^1= 2m and the corresponding normal mode vector would be 2 1 3 2 1 3 " # p ffiffiffiffi p ffiffiffiffi 6 2m 7 6 c1 2m 7 6 6 7 7 ¼6 or 6 7 7 4 1 5 1 5 c2 1 4 pffiffiffiffi pffiffiffiffi 2m 2m which is arbitrary up to a multiplier of 1. If the first element of the normal mode is restricted to be positive, the former vector (one with positive elements) should be used. We have already noted that it is not possible to normalize a rigid body mode with respect to K. Specifically, the modal stiffness for the rigid body mode is " #" # k k a K1 ¼ ½ a a ¼0 k k a for any choice for a; as expected. 3.6.4.4

Second Mode pffiffiffiffiffiffi For this mode, v2 ¼ 2k=m: By substituting into Equation 3.15 we get " #" # " # 0 k k c1 ¼ 0 k k c2 2 the solution of which gives the corresponding modal vector (mode shape). The general solution is c2 ¼ c1 ; or " # " # a c1 ¼ a c2 2 in which a is arbitrary. The corresponding modal mass is given by " #" # m 0 a M2 ¼ ½ a a ¼ 2ma2 0 m a and the modal stiffness is given

"

k

k

#"

a

#

¼ 4ka2 a pffiffiffiffi Then, for M normality we must have 2ma2 ¼ 1 or a ¼ ^1= 2m: It follows that the M normal mode vector would be 2 2 3 1 1 3 " # p ffiffiffiffi p ffiffiffiffi 6 6 c1 2m 7 2m 7 6 6 7 7 ¼6 7 or 6 7 4 4 5 5 1 1 c2 1 pffiffiffiffi pffiffiffiffi 2m 2m K2 ¼ ½ a

a

k

k

The corresponding value of the modal stiffness is K2 ¼pffiffiffiffi 2k=m; which is equal to v22 ; as expected. Similarly, for K normality we must have 4Ka2 ¼ 1; or a ¼ ^1= 4K : Hence, the K normal modal vector would be 2 1 3 2 1 3 " # p ffiffiffi p ffiffiffi 7 6 6 c1 4k 7 4k 7 6 6 7 ¼6 7 or 6 7 4 4 5 5 1 1 c2 2 pffiffiffi pffiffiffi 4k 4k The corresponding value of the modal mass is M2 ¼ m=ð2kÞ which is equal to 1=v22 ; as expected.

Modal Analysis

3 21

The mode shapes of the system are shown in Figure 3.9. Note that in the rigid body mode both masses move in the same direction through the same distance, with the connecting spring maintained in the unstretched configuration. In the second mode, the two masses move in opposite directions with equal amplitudes. This results in a node point halfway along the spring. A node is a point in the system that remains stationary under a modal motion. It follows that, in the second mode, the system behaves like an identical pair of simple oscillators, each posses sing twice the stiffness of the original spring (see Figure p 3.10). ffiffiffiffiffiffi The corresponding natural frequency is 2k=m; which is equal to v2 : Orthogonality of the two modes may be verified with respect to the mass matrix as " ½1

1

m

0

0

m

#"

1 1

Mode 1

Mode 2

Node

FIGURE 3.9 example.

Node

Mode shapes of the two car train

2k m

# ¼0

and, with respect to the stiffness matrix, as ½1

1

FIGURE 3.10 Equivalent system for mode 2 of the two car train example.

"

k k

k

#"

k

1 1

# ¼0

Since K is singular, due to the presence of the rigid body mode, the first orthogonality condition (Equation 3.19), and not the second (Equation 3.21), is the useful result for this system. In particular, since M is nonsingular, the orthogonality of the modal vectors with respect to the mass matrix implies that they are linearly independent vectors by themselves. This is further verified by the nonsingularity of the modal matrix; specifically " # 1 1 det½ c1 ; c2 ¼ det –0 1 1 Since M is a scalar multiple of the identity matrix, we note that the modal vectors are in fact orthogonal, as is clear from " # 1 ¼0 cT1 c2 ¼ ½ 1 1 1

3.6.5

Modal Matrix

An n DoF system has n modal vectors c1 ; c2 ; …; cn ; which are independent. The n £ n square matrix C whose columns are the modal vectors is known as the modal matrix C ¼ ½c1 ; c2 ; …; cn

ð3:22Þ

Since the mass matrix M can always be made nonsingular through proper modeling practices (in choosing the DoF), it can be concluded that the modal matrix is nonsingular det C – 0

ð3:23Þ

3 22


and the inverse C 1 exists. Before showing this fact, note that the orthogonality conditions (Equation 3.19 and Equation 3.21) can be written in terms of the modal matrix as CT MC ¼ diag½M1 ; M2 ; …; Mn ¼ M T

C KC ¼ diag½K1 ; K2 ; …; Kn ¼ K

ð3:24Þ ð3:25Þ

in which M and K are the diagonal matrices of modal masses and modal stiffnesses, respectively. Next, we use the result from linear algebra, which states that the determinant of the product of two square matrices is equal to the product of the determinants. Also, a square matrix and its transpose have the same determinant. Then, by taking the determinant of both sides of Equation 3.24, it follows that det CT MC ¼ ðdet CÞ2 det M ¼ det M ¼ M1 ; M2 ; …; Mn

ð3:26Þ

Here, we have also used the fact that in Equation 3.24 the RHS matrix is diagonal. Now, Mi – 0 for all i since there are no static modes in a well posed modal problem. It follows that det C – 0

ð3:27Þ

which implies that C is nonsingular.

3.6.6

Configuration Space and State Space

All solutions of the displacement response y span a Euclidean space known as the configuration space. This is an n Euclidean space ðLn Þ: This is also the displacement space. The trace of the displacement vector y is not a complete representation of the dynamic response of a vibrating system because the same y can correspond to more than one dynamic state of the system. Hence, y is not a state vector. However, " # y y_

2n

is a state vector, because it includes both displacement and velocity and completely represents the state of the system. This state vector spans the state space ðL2n Þ which is a 2n Euclidean space. 3.6.6.1

State Vector

This is a vector x consisting of a minimal set of response variables of a dynamic system such that, with knowledge of the initial state xðt0 Þ and the subsequent input u½t0 ; t1 to the system over a finite time interval ½t0 ; t1 ; the end state xðt1 Þ can be uniquely determined. Each point in a state space uniquely (and completely) determines the state of the dynamic system under these conditions. Note: Configuration space can be thought of as a subspace of the state space, which is obtained by projecting the state space into the subspace formed by the axes of the y vector. For an n DoF vibrating system (see Equation 3.1), the displacement response vector y is of order n. If we know the initial condition y(0) and the forcing excitation fðtÞ; it is not possible to completely determine yðtÞ in general. However, if we know y(0) and y_ ð0Þ as well as fðtÞ; then it is possible to completely determine yðtÞ and y_ ðtÞ: This says what we have noted before; y alone does not constitute a state vector, but y and y_ together do. In this case, the order of the state space is 2n; which is twice the number of DoF.

3.7

Other Modal Formulations

The modal problem (eigenvalue problem) studied in the previous sections consists of the solution of

v2 Mc ¼ Kc

ð3:28Þ

which is identical to Equation 3.13. The natural frequencies (eigenvalues) are given by solving the characteristic equation 3.14. The corresponding mode shape vectors (eigenvectors) ci are determined by

Modal Analysis

3 23

substituting each natural frequency vi into Equation 3.13 and solving for a nontrivial solution. This solution will have at least one arbitrary parameter. Hence, c represents the relative displacements at the various DoF of the vibrating system and not the absolute displacements. Now, two other formulations are given for the modal problem. The first alternative formulation given below involves the solution of the eigenvalue problem of a nonsymmetric matrix ðM 1 KÞ: The other formulation given consists of first transforming the original problem into a new set of motion coordinates, then solving the eigenvalue problem of a symmetric matrix ðM 1=2 KM 1=2 Þ; and then transforming the resulting modal vectors back to the original motion coordinates. Of course, all three of these formulations will give the same end result for the natural frequencies and mode shapes of the system, because the physical problem would remain the same regardless of what formulation and solution approach are employed. This fact will be illustrated using an example.

3.7.1

Nonsymmetric Modal Formulation

Consider the original modal formulation given by Equation 3.28 that we have studied. Since the inertia matrix M is nonsingular, its inverse M 1 exists. The premultiplication of Equation 3.28 by M 1 gives

v2 c ¼ M 1 Kc

ð3:29Þ

lc ¼ Sc

ð3:30Þ

This vector matrix equation is of the form where l ¼ v2 and S ¼ M 1 K: Equation 3.30 represents the standard matrix eigenvalue problem for matrix S. It follows that Squared natural frequencies ¼ eigenvalues of M 1 K Mode shape vectors ¼ eigenvectors of M 1 K

3.7.2

Transformed Symmetric Modal Formulation

Now consider the free (unforced) system equations M€y þ Ky ¼ 0

ð3:31Þ

whose modal problem needs to be solved. First, we define the square root of matrix M, as denoted by M1=2 ; such that M1=2 M1=2 ¼ M

ð3:32Þ

Since M is symmetric, M1=2 also has to be symmetric. Next, we define M Specifically, M

1=2

M1=2 ¼ M1=2 M

1=2

¼I

1=2

as the inverse of M1=2 : ð3:33Þ

1=2

where I is the identify matrix. Note that M is also symmetric. Once M 1=2 is defined in this manner, we transform the original problem 3.31 using the coordinate transformation y¼M

1=2

q

ð3:34Þ

3 24


Here, q denotes the transformed displacement vector, which is related to the actual displacement vector y through the matrix transformation using M 1=2 : By differentiating Equation 3.34 twice, we get 1=2

y€ ¼ M

q€

ð3:35Þ

Substitute Equation 3.34 and Equation 3.35 into Equation 3.31. This gives MM Premultiply this result by M

1=2

1=2

1=2

q€ þ KM

q¼0

and use the fact that

M

1=2

MM

1=2

¼M

1=2

M1=2 M1=2 M

1=2

¼I

which follows from Equation 3.32 and Equation 3.33. We get q€ þ M

1=2

KM

1=2

q¼0

ð3:36Þ

Equation 3.36 is the transformed problem, whose modal response may be given by q ¼ e jvt f

ð3:37Þ

where v represents a natural frequency and f represents the corresponding modal vector, as usual. Then, in view of Equation 3.34, we have y ¼ e jvt M

1=2

f ¼ e j vt c

ð3:38Þ

It follows that the natural frequencies of the original problem 3.31 are identical to the natural frequencies of the transformed problem 3.36, and the modal vectors c of the original problem are related to the modal vectors f of the transformed problem through c¼M

1=2

f

ð3:39Þ

Substitute the modal response 3.37 into Equation 3.36. We get

lf ¼ Pf 2

1=2

ð3:40Þ

1=2

KM : where l ¼ v and P ¼ M Equation 3.40, just like Equation 3.30, represents a standard matrix eigenvalue problem. But now matrix P is symmetric. As a result, its eigenvectors f will not only be real but also orthogonal. The solution steps for the present, transformed, and symmetric modal problem are: 1. Determine M 1=2 : 2. Solve for eigenvalues l and eigenvectors f of M 1=2 KM 1=2 : Eigenvalues are squares of the natural frequencies of the original system. 3. Determine the modal vectors c of the original system by using c ¼ M 1=2 f: The three approaches of modal analysis which we have studied are summarized in Table 3.4. TABLE 3.4

Three Approaches of Modal Analysis

Approach

Standard

Nonsymmetric Matrix Eigenvalue

Symmetric Matrix Eigenvalue

Modal formulation Squared natural frequencies ðv2i Þ Mode shape vectors ðci Þ

½v2 M 2 K c ¼ 0 Roots of det½v2 M 2 K ¼ 0

v2 c ¼ M 1 Kc Eigenvalues of M 1 K

v2 f ¼ M 1=2 KM Eigenvalues of M

Nontrivial solutions of ½v2i M 2 K c ¼ 0

Eigenvectors of M 1 K

Determine eigenvectors fi of M 1=2 KM 1=2 : Then ci ¼ M

1=2 1=2

f KM

1=2

1=2

fi

Modal Analysis

3 25

Example 3.6 We will use the 2 DoF vibration problem given in Figure 3.4 (Example 3.3) to demonstrate the fact that all three approaches summarized in Table 3.4 will lead to the same results. Consider the special case of a ¼ 0:5 and b ¼ 0:5: Then we have 2 3 3 k 2 3 m 0 6 2k 27 6 7 M¼4 m 5 and K ¼ 6 7 4 k 0 k 5 2 2 2

Approach 1 Using the standard approach, we obtain the modal results pffiffiffiffiffi given in Table 3.3. Specifically, we get the natural frequencies (normalized with respect to v0 ¼ k=m)

v1 1 ¼ pffiffi v0 2

pffiffi v2 ¼ 2 v0

and

and the mode shapes

c2 c1

1

¼2

c2 c1

and

2

¼

1

Let us now obtain these results using the other two approaches of modal analysis. Approach 2 M

2 1

6 ¼4

1 m

0

7 2 5; m

0

2 6 M 1K ¼ 4

3

1 m

0

2 3 3 6 k 76 2 5 2 6 4 k m 2

k 2 k 2

0

3

2

3 k 7 62 m 7 6 7¼6 5 4 k m

1 k 2 m k m

3

2 3 7 6 7 7 ¼ v20 4 2 5 1

Note that this is not a symmetric matrix. We solve the eigenvalue problem of 2 3 3 1 6 2 27 4 5 1 1 Eigenvalues l are given by

2 6l det4

3 2

3

1 2

1

l

3 ðl 2

1Þ

1

7 5¼0

or

l

1 ¼0 2

or

l2

5 lþ1¼0 2

3 1 27 5 1

3 26


the roots of which are

l1 ; l2 ¼

1 ;2 2

pffiffi pffiffi It follows that v1 =v0 ¼ 1= 2 and v2 =v0 ¼ 2 as before. The eigenvector corresponding to l1 (mode 1) is given by 21 3 1 3" # " # 0 c 62 2 2 7 6 7 1 ¼ 4 5 1 0 c2 1 1 1 2 The solution is ðc2 =c1 Þ1 ¼ 2 as before. The eigenvector corresponding to l2 (mode 2) is given by 2 3 " # " # 3 1 0 c1 2 6 7 2 2 5 ¼ 4 0 c2 2 1 2 1 The solution is ðc2 =c1 Þ2 ¼

1 as before.

Approach 3 Since M is diagonal, it is easy to obtain M1=2 : We simply take the square root of the diagonal elements; thus, 2 pffiffiffi 3 m 0 6 rffiffiffiffi 7 M1=2 ¼ 4 m5 0 2 Its inverse is given by inverting the diagonal elements; thus 2 1 3 pffiffiffi 0 6 m 7 6 7 M 1=2 ¼ 6 7 r ffiffiffiffi 4 2 5 0 m Now,

M

1=2

KM

1=2

32

2 1 pffiffiffi 6 m 6 ¼6 4 0 2 6 6 ¼ v20 6 4

0 76 76 6 rffiffiffiffi 7 2 54 m 3 2 1 pffiffi 2

3 k 2 k 2 3

k 2 k 2

32 1 pffiffiffi 76 m 76 76 54 0

3

2

0 7 6 7 6 ¼6 rffiffiffiffi 7 6 2 5 4 m

3 k 2 m 1 k pffiffi 2 m

3 1 k pffiffi 7 2 m7 7 7 k 5 m

1 pffiffi 7 27 7 5 1

Note that, as expected, this is a symmetric matrix. We solve for eigenvalues and eigenvectors of 2 3 1 3 p ffiffi 7 6 2 27 6 6 7 4 5 1 pffiffi 1 2

Modal Analysis

3 27

Eigenvalues are given by 2

3 6l 2 6 det6 4 1 pffiffi 2

1 3 pffiffi 7 2 7 7¼0 5 l 1

or 3 ðl 2

l

1Þ

1 ¼0 2

or 5 lþ1¼0 2

l2

which is identical to the characteristic equation obtained in the first two approaches. It follows that the same two natural frequencies are obtained by this method. The eigenvector f1 for mode 1 is given by 21 3 1 3 p ffiffi 7" # " # 62 0 2 2 7 f1 6 ¼ 7 6 5 f 4 1 1 0 2 1 pffiffi 1 2 2 hpffiffii which gives ðf2 =f1 Þ1 ¼ 2 : Accordingly, we may use " # " # 1 f1 ¼ pffiffi 2 f2 1 The eigenvector f2 for mode 2 is given by 2 3 62 2 6 6 4 1 pffiffi 2 pffiffi which gives ðf2 =f1 Þ2 ¼ 1= 2. Accordingly, we use "

1 3 pffiffi 7" # " # 0 2 7 f1 ¼ 7 5 f 0 2 2 2 1

f1 f2

2

#

6 ¼4

2

3 1 7 1 5 pffiffi 2

Now, we transform these eigenvectors back to the original coordinate system using Equation 3.39. We get "

c1

2 1 pffiffiffi 6 m 6 ¼6 4 1 0

#

c2

which gives ðc2 =c1 Þ1 ¼ 2 as before.

2 3 # 0 7" 6 6 7 1 pffiffi ¼ 6 rffiffiffiffi 7 4 5 2 2 m

1 3 pffiffiffi 7 m7 7 2 5 pffiffiffi m

3 28


Also, "

c1 c2

which gives c2 c1 2 ¼

3.8

2 1 pffiffiffi 6 m 6 ¼6 4 2 0

#

0

3

2

7 76 4 rffiffiffiffi 7 2 5 m

2 3 1 6 7 6 1 5¼6 4 pffiffi 2

1 3 pffiffiffi 7 m7 7 1 5 pffiffiffi m

1 as before:

Forced Vibration

The forced motion of a linear, n DoF, undamped system is given by the nonhomogeneous equation of motion 3.1: M€y þ Ky ¼ fðtÞ

ð3:41Þ

Even though the following discussion is based on this undamped model, the results can be easily extended to the damped case. We have observed that the modal vectors form a basis for the configuration space. In other words, it is possible to express the response y as a linear combination of the modal vectors ci : y ¼ q1 c1 þ q2 c2 þ · · · þ qn cn

ð3:42Þ

The parameters qi are a set of generalized coordinates and are functions of time t: Equation 3.42 is written in the vector matrix form 2 3 q1 6 7 6q 7 6 27 6 7 y ¼ ½c1 ; c2 ; …; cn 6 . 7 6 . 7 6 . 7 4 5 qn or y ¼ Cq

ð3:43Þ

and can be viewed as a coordinate transformation from the trajectory space to the canonical space of generalized coordinates (principal coordinates or natural coordinates). Note that the inverse transformation exists because the modal matrix C is nonsingular. On substituting Equation 3.43 into Equation 3.41, we obtain MCq€ þ KCq ¼ fðtÞ This result is premultiplied by CT and the orthogonality conditions (Equation 3.24 and Equation 3.25) are substituted to obtain the canonical form of the system equation: Mq€ þ Kq ¼ fðtÞ

ð3:44Þ

in which M and K are the diagonal matrices given by Equation 3.24 and Equation 3.25, and the transformed forcing vector is given by fðtÞ ¼ CT fðtÞ

ð3:45Þ

Modal Analysis

3 29

Since M and K are diagonal matrices, Equation 3.44 corresponds to the set of n uncoupled simple oscillator equations: Mi q€ i þ Ki qi ¼ fi ðtÞ

for i ¼ 1; 2; …; n

ð3:46Þ

In other words, the coordinate transformation 3.43 using the modal matrix has uncoupled the equations of forced motion. It follows from Equation 3.46 that the natural frequencies of the system are given by

v2i ¼ Ki =Mi

for i ¼ 1; 2; …; n

ð3:47Þ

as we have noted before. It is particularly convenient to employ M normal modal vectors. In this case, M becomes the nth order identity matrix, and K the diagonal matrix having v2i as its diagonal elements; thus M¼I i h K ¼ diag v21 ; v22 ; …; v2n

ð3:48Þ ð3:49Þ

The corresponding uncoupled equations 3.46 take the form q€ i þ v2i qi ¼ fi ðtÞ

for i ¼ 1; 2; …; n

ð3:50Þ

in which the forcing terms fi ðtÞ are given by Equation 3.45, C being M normal. Typically, the initial conditions for the original system are provided as the initial position yð0Þ and the initial velocity y_ ð0Þ: The corresponding initial conditions for the transformed equations of motion are obtained using Equation 3.43 as qð0Þ ¼ C 1 yð0Þ

ð3:51Þ

q_ ð0Þ ¼ C 1 y_ ð0Þ

ð3:52Þ

The complete response of the original system can be conveniently obtained by first solving the simple oscillator equation 3.50 and then transforming the results back into the trajectory space using Equation 3.43. The complete solution to this linear system can be viewed as the sum of the initial condition response in the absence of the forcing function and the forced response with zero initial conditions, as discussed in Chapter 2. For the simple oscillator equation 3.50, the initial condition response qiI is given by qiI ¼ qi ð0Þ cos vi t þ

q_ i ð0Þ sin vi t vi

ð3:53Þ

The impulse response function (i.e., response to a unit impulse excitation) for the undamped oscillator equation is hi ðtÞ ¼

1 sin vi t vi

for t $ 0

ð3:54Þ

The forced response qiF ; with zero initial conditions, is obtained using the convolution integral method; specifically ðt qiF ¼ fi ðtÞhi ðt tÞdt ð3:55Þ 0

The complete solution in the canonical domain is qi ¼ qiI þ qiF ¼ qi ð0Þcos vi t þ

q_ i ð0Þ 1 ðt sin vi t þ f ðtÞsin vi ðt vi vi 0 i

tÞdt i ¼ 1; 2; …; n

ð3:56Þ

Each of the n responses is a modal response that is the contribution from that mode to the actual response y. This approach is summarized in Box 3.4. Next, we will illustrate this approach of solving for the forced vibration by means of an example.

3 30


Box 3.4 MODAL APPROACH TO FORCED RESPONSE Forced system: M€y þ Ky ¼ fðtÞ Number of DoFs ¼ n Modal transformation: y ¼ Cq where modal matrix C ¼ ½c1 ; c2 ; …; cn ¼ matrix of mode shape vectors. We obtain the diagonalized system Mq€ þ Kq ¼ fðtÞ or Mi q€ i þ Ki qi ¼ fi ðtÞ;

i ¼ 1; 2; …; n

where M ¼ CT MC ¼ diag½M1 ; M2 ; …; Mn K ¼ CT KC ¼ diag½K1 ; K2 ; …; Kn fðtÞ ¼ CT fðtÞ ¼ ½f1 ; f2 ; …; fn

T

Initial conditions: qð0Þ ¼ C 1 yð0Þ q_ ð0Þ ¼ C 1 y_ ð0Þ Steps: Use M normal case: Mi ¼ 1; Ki ¼ v2i Then, q€ i þ v2i qi ¼ fi ðtÞ;

i ¼ 1; 2; …; n

1. Free, initial condition response (zero input response) q_ ð0Þ qiI ¼ qi ð0Þcos vi t þ i sin vi t vi 2. Forced, zero initial condition response 1 ðt qiF ¼ f ðtÞsin vi ðt vi 0 i 3. qi ¼ qiI þ qiF 4. Transform back to y using y ¼ Cq

tÞdt

Modal Analysis

3 31

Example 3.7 Let us consider again the system shown in Figure 3.8. A step input force f ðtÞ given by ( f0 for t $ 0 f ðtÞ ¼ 0 for t , 0 is applied to the left hand mass (DoF y1 ). Assume that the system starts from rest ðyð0Þ ¼ 0 and y_ ð0Þ ¼ 0Þ: As before, the M normal modal matrix of the system is 2 1 3 1 " # p ffiffiffiffi p ffiffiffiffi 6 2m 1 1 2m 7 1 6 7 C¼6 7 ¼ pffiffiffiffi 4 1 1 5 2m 1 1 pffiffiffiffi pffiffiffiffi 2m 2m No forcing input is applied to the second DoF ðy2 Þ: Hence, the overall forcing input vector is " # f0 fðtÞ ¼ for t $ 0 0 From Equation 3.45, the transformed forcing input vector is obtained as " #" # " # 1 1 f0 1 1 f0 T ¼ pffiffiffiffi fðtÞ ¼ C fðtÞ ¼ pffiffiffiffi 2m 1 2m 1 1 0

for t $ 0

From Equation 3.51 and Equation 3.52, the initial conditions for the modal (canonical) variables are obtained as q_ ð0Þ ¼ 0 and qð0Þ ¼ 0: The modal responses q1 ðtÞ and q2 ðtÞ are obtained using Equation 3.56.

3.8.1

First Mode (Rigid-Body Mode)

Note that lim

vi !0

sin vi ðt vi

tÞ

t

¼t

It follows that q1 ðtÞ ¼

3.8.2

ðt f ðt tÞ f t2 0 pffiffiffiffi dt ¼ p0 ffiffiffiffi 0 2m 2 2m

Second Mode (Oscillatory Mode) q2 ðtÞ ¼

1 ðt f0 pffiffiffiffi sin v2 ðt v2 0 2m

tÞdt ¼

v22

f p0ffiffiffiffi ð1 2m

cos v2 tÞ

The overall response in the physical trajectory space is obtained by transforming the modal responses using Equation 3.43; thus 2 2 2 3 3 t 1 f t2 " #6 þ ð1 cos tÞ v p0 ffiffiffiffi 2 7 6 2 7 1 1 6 v22 6 7 7 2 2m 1 6 7 7 ¼ f0 6 y ¼ pffiffiffiffi 6 6 7 7 2 2m 2m 1 4t 5 1 4 f0 ð1 cos v2 tÞ 5 1 p ffiffiffiffi ð1 cos v2 tÞ v22 2m 2 v22

3 32


Response y 2000 (Normalized f with 0 2 ) mw 2 1500

Quadratic Growth y1 y2

1000

500

0

0

FIGURE 3.11

10

20

30

40

50

60

70 Time t (Normalized with w 2)

Forced response obtained through modal analysis.

with v22 ¼ 2k=m: The response of both masses grows (unstable) quadratically in an oscillatory manner, as shown in Figure 3.11.

3.9

Damped Systems

Let us examine the possibility of extending the modal analysis results to damped systems. Damping is an energy dissipation phenomenon. In lumped parameter models of vibrating systems, damping force can be represented by a resisting force at each lumped mass. In view of Equation 3.1, then, the system equations for a damped system can be written as M€y þ Ky ¼ fðtÞ

d

ð3:57Þ

in which d is the damping force vector. Modeling of damping is usually quite complicated. Often, a linear model, whose energy dissipation capacity is equivalent to that of the actual system, is employed. Such a model is termed as an equivalent damping model. The most popular model for damping is the linear viscous model, in which the damping force is proportional to the relative velocity. In lumped parameter dynamic models, (linear) viscous damping elements may be assigned across pairs of DoF or across a DoF and a fixed reference. The damping force for such a model may be expressed as d ¼ C_y

ð3:58Þ

in which C is the damping matrix. The resulting damped system equation is M€y þ C_y þ Ky ¼ fðtÞ

ð3:59Þ

To determine the elements of C for a given system model by the influence coefficient approach, the same procedure outlined previously for obtaining the elements of K may be used, except that the velocities y_ i should be used in place of the displacements yi : The coefficients cij are termed as damping influence coefficients.

Modal Analysis

3.9.1

3 33

Proportional Damping

Because the modal vectors are orthogonal with respect to both M and K, the transformation 3.43 will decouple the undamped system equation 3.41. It should be clear, however, that the same transformation will not diagonalize the C matrix in general. As we have observed in the previous section, decoupling or modal decomposition, is a convenient tool in response analysis. This is because each uncoupled modal equation is a simple oscillator equation with a well known solution that can be transformed back (or, recombined) to obtain the total response. This simple procedure cannot be used for damped systems unless the modal vectors are orthogonal with respect to C as well. In modal vibration, all DoF move in the same displacement proportion, as given by the modal vector. This type of synchronous motion may not be possible in damped systems. Another way to state this is that most damped systems do not possess real modes. If we try to excite such a damped system at one of its natural frequencies, we will notice that the constant proportion, given by the modal vector (for the undamped system), is violated during motion. Furthermore, if the undamped system has a (fixed) node point in the particular mode, that point would not remain fixed but would move in a cyclic manner when set to vibrate at the natural frequency of the mode. Note that viscous damping is just a model for energy dissipation. If we become rather restrictive in choosing the parameters of viscous damping, we will be able to develop an equivalent damping matrix, with respect to which, the modal vectors of the undamped system would be orthogonal. In other words, we require that the transformed damping matrix C be a diagonal matrix; thus CT CC ¼ diag½C1 ; C2 ; …; Cn ¼ C

ð3:60Þ

In this case, the corresponding viscous damping model is termed proportional damping or Rayleigh damping (after the person who first identified this simplification). Modal decomposition of Equation 3.59, assuming proportional damping using the transformation 3.43, results in the canonical form (uncoupled modal equation) M€q þ C_q þ Kq ¼ fðtÞ

ð3:61Þ

or Mi q€ i þ Ci q_ i þ Ki qi ¼ fi ðtÞ for i ¼ 1; 2; …; n Equation 3.61 can be written in the standard form for a damped simple oscillator: q€ i þ 2zi vi q_ i þ v2i qi ¼ fi ðtÞ for i ¼ 1; 2; …; n

ð3:62Þ

in which, the modal matrix is assumed M normal (i.e., Mi ¼ 1). It can be concluded that a proportionally damped system possesses real modal vectors that are identical to the modal vectors of its undamped counterpart. The damped natural frequency, however, is smaller than the undamped natural frequency and is given by qffiffiffiffiffiffiffiffi ð3:63Þ vdi ¼ 1 z2i vi One way to guarantee proportional damping is to pick a damping matrix that satisfies C ¼ cm M þ ck K From modal transformation (see Equation 3.61), it follows that Equation 3.64 corresponds to Modal damping constant Ci ¼ cm Mi þ ck Ki or 2zi vi ¼ cm þ ck v2i

ð3:64Þ

3 34


or Modal damping ratio zi ¼

1 cm þ ck vi 2 vi

ð3:64aÞ

Equation 3.64, however, is not the only way to achieve real modes in a damped system (Equation 3.60). The first term on the right hand side (RHS) of Equation 3.64 is termed the inertial damping matrix. The corresponding damping force on each lumped mass in the model will be proportional to the momentum. This term may represent the energy loss associated with a momentum and is termed momentum damping. Physically, this is incorporated by assigning a viscous damper between each DoF and its fixed reference, with the damping constant proportional to the mass concentrated at that location. The second term in Equation 3.64 is termed the stiffness damping matrix. The corresponding damping force is proportional to the rate of change of the local deformation forces (stresses) in flexible structural members and joints. It may be interpreted as a simplified model for structural damping. Physically, this model is realized by assigning a viscous damper across every spring element in the model, with the damping constant proportional to the stiffness. It is known that rate of change of stresses or rate of change of strains will give rise to viscoelastic damping, which is associated with plasticity and viscoelasticity. This type of damping is known as strain rate damping. Usually, structural damping is most appropriately modeled as being present across (lumped) stiffness elements. Coulomb damping is modeled as acting between an inertia element and its fixed reference point. These intuitive observations also support the damping model given by Equation 3.64. Some terminology and properties of damped systems are summarized in Box 3.5.

Example 3.8 Let us examine the lumped parameter damped model shown in Figure 3.12 to determine whether it has real modes. Note that if we modify the model as in Figure 3.13, the damping matrix will become proportional to the stiffness matrix. This will be a case of proportional damping and the modified (damping) system will possess real modes with modal vectors that are identical to those for the corresponding undamped system. It should be verified that the equations of motion for this modified system (Figure 3.13) are "

m

0

0

m

# y€ þ

"

2c

c

c

2c

# y_ þ

"

2k

k

k

2k

# y¼

"

0

#

f ðtÞ

Notice the similarity between the stiffness matrix and the damping matrix. Another damped system that possesses classical modes is shown in Figure 3.14. In this model, the damping matrix is proportional to the mass (inertia) matrix. It may be easily verified that the system equations are " # " # " # " # m 0 c 0 2k k 0 y€ þ y_ þ y¼ 0 m 0 c k 2k f ðtÞ Here, notice the similarity between the inertia matrix and the damping matrix. Returning to the original system shown in Figure 3.12, the equations of motion can be written as " # " # " # " # 2k k 0 m 0 c1 þ c2 c2 y€ þ y_ þ y¼ k 2k f ðtÞ 0 m c2 c2 From these equations, it is not obvious whether this system possesses real modes. The undamped natural frequencies are given by the roots of the characteristic equation " 2 # v m 2k k det ¼0 v2 m 2k k

Modal Analysis

3 35

Box 3.5 TERMINOLOGY AND PROPERTIES OF DAMPED SYSTEMS Characteristic equation: det½Ms2 þ Cs þ K ¼ 0 Roots s are the eigenvalues li Imðli Þ ¼ vdi ¼ damped natural frequencies lli l ¼ vi ¼ undamped natural frequencies Reðli Þ=vi ¼ zi ¼ modal damping ratios qffiffiffiffiffiffiffiffiffiffi 1 2z2i vi ¼ vri ¼ resonant frequencies qffiffiffiffiffiffiffiffi 1 z2i vi ¼ vdi Existence of real modes: *

Condition for existence of modes that are identical to undamped modes CT CC ¼ diagonal matrix where C ¼ modal matrix of undamped modes

*

Another (equivalent) condition M 1 CM 1 K ¼ M 1 KM 1 C

*

ðProveÞ

i.e., M 1 C and M 1 K must commute Special case: Proportional damping C ¼ cm M þ ck K ) Modal damping constant Ci ¼ cm Mi þ ck Ki Modal damping ratio zi ¼

1 cm þ ck vi 2 vi

pffiffiffiffiffi pffiffiffiffiffiffi The natural frequencies are v1 ¼ k=m and v2 ¼ 3k=m: The corresponding mode shapes are given by the nontrivial solution of " 2 #" # " # 0 c1 vi m 2k k ¼ for i ¼ 1; 2 2 0 c2 i vi m 2k k Also, let us normalize the modal vectors by choosing the first element of each vector to be unity. Then, by following the usual procedure of modal analysis, we will obtain the normalized modal vectors " # " # 1 1 c1 ¼ and c2 ¼ 1 1

3 36


The modal matrix is

"

C ¼ ½c1 ; c2 ¼ Consequently, we obtain

"

CT MC ¼ M ¼ CT KC ¼ K ¼ " CT CC ¼ C ¼

1

1

1

1

2m

0

" 0 2k

#

c1 þ 4c2 f ðtÞ

c

f ðtÞ

Notice that the transformed damping matrix C is not diagonal in general and, consequently, real modes will not exist. This is to be expected without the condition of proportional damping. Proportional damping is realized in this model if c1 ¼ 0: Then, C will be diagonal and the transformed systems equations will be uncoupled. The uncoupled model equations are

2m€q2 þ 4c2 q_ 2 þ 6kq2 ¼

f ðtÞ

m ⇒ f(t)

k

A system with linear viscous damping.

y2 y1

k m

k

#

2m€q1 þ 2kq1 ¼ f ðtÞ

c2

c

#

c1

CT fðtÞ ¼ fðtÞ ¼

FIGURE 3.12

6k c1

"

m

k

#

c1

and

y1

c1

2m# 0

0

y2

k

c

m ⇒ f(t)

k

FIGURE 3.13 A system with proportional damping in proportion to stiffness.

y2

c c k

y1

k m

f1(t)

m

f2(t)

k

FIGURE 3.14 A system with proportional damping in proportion to inertia.

The first mode is always undamped for this choice of damping model. This confirms that, in the case of proportional damping, it is not generally possible to pick an arbitrary structure for the damping matrix. For this reason, proportional damping is sometimes an analytical convenience rather than a strict physical reality. Modal analysis and response analysis of a system with general viscous damping can also be accomplished using the state space concepts. In this case, modal analysis is carried out in terms of the eigenvalues and eigenvectors of the system matrix of a suitable state variable model. These “eigen results,” if complex, will occur in complex conjugate pairs and then the modes are said to be complex. This approach is outlined next.

3.10

State-Space Approach

The state space approach to modal analysis may be used for any linear dynamic system. The starting point is to formulate a state space model of the system, which is a set of coupled first order differential equations: x_ ¼ Ax þ Bu where x ¼ state vector; u ¼ input vector; A ¼ system matrix; and B ¼ input gain matrix.

ð3:65Þ

Modal Analysis

3 37

There are many approaches to formulating a vibration problem as a state space model 3.65. One simple method is to first obtain the conventional, coupled, second order differential equations: M€y þ C_y þ Ky ¼ fðtÞ Next, the state vector and the input vector are defined as " # y x¼ and u ¼ fðtÞ y_

ð3:66Þ

Note that Equation 3.59 may be written as y€ ¼

M 1 Ky

M 1 C_y þ M 1 fðtÞ

which is identical to 1

1

y€ ¼ ½ M K

M C

" # y y_

ð3:67Þ

þ fðtÞ

ð3:67aÞ

This, together with the identity y_ ¼ y_ ; can be expressed in the form # " # " #" # " y_ 0 I y 0 fðtÞ ¼ þ y€ M 1K M 1C y_ M 1 which is in the state space form 3.65 where " 0 A¼ M 1K

#

I M 1C

" and

B¼

#

0 M

ð3:68Þ

1

ð3:69Þ

Note that, in Equation 3.68 and Equation 3.69, I denotes an identity matrix of an appropriate size.

3.10.1

Modal Analysis

Consider the free motion ðu ¼ 0Þ of the nth order system given by Equation 3.65. Its solution is given by x ¼ FðtÞxð0Þ

ð3:70Þ

It is known that the state transition matrix FðtÞ is given by the matrix exponential expansion equation. FðtÞ ¼ expðAtÞ ¼ I þ At þ

1 2 2 A t þ ··· 2!

ð3:71Þ

To discuss the rationale for this exponential response further, we begin by assuming a homogeneous solution of the form x ¼ X expðltÞ

ð3:72Þ

By substituting Equation 3.72 in the homogeneous equation of motion (that is, Equation 3.65 with u ¼ 0), the following matrix eigenvalue problem results: ðA

sIÞX ¼ 0

ð3:73Þ

We shall assume that the n eigenvalues ðl1 ; l2 ; …; ln Þ of A are distinct. Then, the corresponding eigenvectors X1 ; X2 ; …; Xn are linearly independent vectors; that is, any one eigenvector cannot be expressed as a linear combination of the rest of the eigenvectors in the set. Thus, the general solution for free dynamics is xðtÞ ¼ X1 expðl1 tÞ þ X2 expðl2 tÞ þ · · · þ Xn expðln tÞ

ð3:74Þ

3 38


Each of the n eigenvectors has an unknown parameter. The total of n unknowns is determined using n initial conditions: xð0Þ ¼ x0

3.10.2

ð3:75Þ

Mode Shapes of Nonoscillatory Systems

Since the eigenvectors are independent, if the initial state is set at x0 ¼ Xi ; then the subsequent motion should not have any Xj terms with j – i in Equation 3.74. Otherwise, when we set t ¼ 0; Xi becomes a linear combination of the remaining eigenvectors, which contradicts the linear independence. Hence, the motion due to this eigenvector initial condition is given by xðtÞ ¼ Xi expðli tÞ; which vector is parallel to Xi throughout the motion. Thus, Xi gives the mode shape of the system corresponding to the eigenvalue li :

3.10.3

Mode Shapes of Oscillatory Systems

The analysis in the preceding section is valid for real eigenvalues and eigenvectors. In vibratory systems, li and Xi generally are complex. Let

li ¼ si þ jvi

ð3:76Þ

Xi ¼ Ri þ jIi

ð3:77Þ

For real systems, corresponding complex conjugates exist:

l i ¼ si

jvi

ð3:78Þ

Xj ¼ Ri

jIi

ð3:79Þ

Equation 3.76 to Equation 3.79 represent the ith mode of the system. The corresponding damped natural frequency is vi and the damping parameter is si : The net contribution of the ith mode to the solution (Equation 3.74) is ðRi cos vi t

Ii sin vi tÞ2 expðsi tÞ

It should be clear, for instance from Equation 3.66, that only some of the state variables in xðtÞ correspond to displacements of the masses (or spring forces). These can be extracted through an output relationship of the form y ¼ Cx

ð3:80Þ

The contribution of the ith mode to displacement variables is Yi ¼ C½Ri cos vi t

Ii sin vi t 2 expðsi tÞ

ð3:81Þ

If Equation 3.81 can be expressed in the form Yi ¼ Si sinðvi t þ fi Þexpðsi tÞ

ð3:82Þ

in which Si is a constant vector that is defined up to one unknown, then it is possible to excite the system so that every independent mass element undergoes oscillations in phase (hence, passing through the equilibrium state simultaneously) at a specific frequency vi : We have noted that this type of motion is known as normal mode motion. The vector Si gives the mode shape corresponding to the (damped) natural frequency vi : A normal mode motion is possible for undamped systems and for certain classes of damped systems. The initial state that is required to excite the ith mode is x0 ¼ Ri : The corresponding displacement and velocity initial conditions are obtained from Equation 3.81; thus Yi ð0Þ ¼ CRi Y_ i ð0Þ ¼ CðRi si

ð3:83Þ Ii vi Þ

ð3:84Þ

Note that the constant factor 2 has been ignored, because Xi is known up to one unknown complex parameter.

Modal Analysis

3 39

Example 3.9 A torsional dynamic model of a pipeline segment is shown in Figure 3.15(a). Free body diagrams in Figure 3.15(b) show internal torques acting at sectioned inertia junctions for free motion. A state model is obtained using the generalized velocities (angular velocities Vi ) of the inertia elements and the generalized forces (torques Ti ) as state variables. A minimum set that is required for complete representation determines the system order. There are two inertia elements and three spring elements a total of five energy storage elements. The three springs are not independent, however. The motion of two springs completely determines the motion of the third. This indicates that the system is a fourth order system. We obtain the model as follows. Newton’s Second Law gives

I1 k1 T1 (a)

(b)

I2

k2 T2 W2

W1

T1

T2

T2 T3

T1

T2

T2 T3

k3 T3

FIGURE 3.15 (a) Dynamic model of a pipeline segment; (b) free body diagrams.

I1 V_ 1 ¼

T1 þ T2

ðiÞ

I2 V_ 2 ¼

T2

ðiiÞ

T3

and Hooke’s Law gives T_ 1 ¼ k1 V1 T_ 2 ¼ k2 ðV2

ðiiiÞ

V1 Þ

ðivÞ

Torque T3 is determined in terms of T1 and T2 ; using the displacement relation for the inertia I2 : T1 T T þ 2 ¼ 3 k1 k2 k3

ðvÞ

The state vector is chosen as x ¼ ½V1 ; V2 ; T1 ; T2

T

ðviÞ

The corresponding system matrix is 2

6 0 6 6 6 6 0 A¼6 6 6 6 6 k1 4 k2

1 I1

0 0

1 I2

k3 k1

1 I1 1 k 1þ 3 I2 k2

0

0

0

k1

0

0

3 7 7 7 7 7 7 7 7 7 7 5

ðviiÞ

The output displacement vector is y¼

T1 T1 T ; þ 2 k1 k1 k2

ðviiiÞ

3 40


which corresponds to the output 2 1 6 0 0 k1 6 C¼6 4 1 0 0 k1

gain matrix 3 0 7 7 ðixÞ 7 1 5 k2 1st Mode

For the special case given by I1 ¼ I2 ¼ I and k1 ¼ k3 ¼ k; the system eigenvalues are sffiffiffi k ðxÞ l1 ; l1 ¼ ^jv1 ¼ ^j I sffiffiffiffiffiffiffiffiffiffi k þ 2k2 ðxiÞ l2 ; l2 ¼ ^jv2 ¼ ^j I

Node

FIGURE 3.16

2nd Mode

Mode shapes of the pipeline segment.

and the corresponding eigenvectors are X1 ; X1 ¼ R1 ^ j I1 ¼ X2 ; X2 ¼ R2 ^ j I2 ¼

a1 ½v ; v ; 7jk1 ; 0 2 1 1

T

a2 ½v ; v2 ; 7jk1 ; ^2jk2 2 2

ðxiiÞ T

In view of Equation 3.81, the modal contributions to the displacement vector are " # " # 1 1 Y1 ¼ a1 sin v1 t and Y2 ¼ a2 sin v2 t 1 1

ðxiiiÞ

ðxivÞ

This Equation xiv is of the form given by Equation 3.82. The mode shapes are given by the vectors S1 ¼ ½1; 1 T and S2 ¼ ½1; 1 T ; which are illustrated in Figure 3.16. In general, each modal contribution introduces two unknown parameters, ai and fi ; into the free response (homogeneous solution) where fi are the phase angles associated with the sinusoidal terms. For an n DoF (order 2n) system, this results in 2n unknowns, which require the 2n initial conditions x(0).

Bibliography Beards, C.F. 1996. Engineering Vibration Analysis with Application to Control Systems, Halsted Press, New York. Benaroya, H. 1998. Mechanical Vibration, Prentice Hall, Upper Saddle River, NJ. Crandall, S.H., Karnopp, D.C., Kurtz, E.F., and Prodmore Brown, D.C. 1968. Dynamics of Mechanical and Electromechanical Systems, McGraw Hill, New York. den Hartog, J.P. 1956. Mechanical Vibrations, McGraw Hill, New York. de Silva, C.W., On the modal analysis of discrete vibratory systems, Int. J. Mech. Eng. Educ., 12, 35 44, 1984. de Silva, C.W. 2000. Vibration Fundamentals and Practice, CRC Press, Boca Raton, FL. de Silva, C.W. 2004. Mechatronics An Integrated Approach, CRC Press, Boca Raton, FL. Dimarogonas, A. 1996. Vibration for Engineers, 2nd ed., Prentice Hall, Upper Saddle River, NJ. Inman, D.J. 1996. Engineering Vibration, Prentice Hall, Englewood Cliffs, NJ. Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood Cliffs, NJ. Meirovitch, L. 1986. Elements of Vibration Analysis, 2nd ed., McGraw Hill, New York. Noble, B. 1969. Applied Linear Algebra, Prentice Hall, Englewood Cliffs, NJ. Rao, S.S. 1995. Mechanical Vibrations, 3rd ed., Addison Wesley, Reading, MA.

Modal Analysis

3 41

Steidel, R.F. 1979. An Introduction to Mechanical Vibrations, 2nd ed., Wiley, New York. Thomson, W.T. and Dahleh, M.D. 1998. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper Saddle River, NJ. Volterra, E. and Zachmanoglou, E.C. 1965. Dynamics of Vibrations, Charles E. Merrill Books, Columbus, OH.

Appendix 3A Linear Algebra 3A.1

Introduction

Linear algebra, the algebra of sets, vectors, and matrices, is useful in the study of mechanical vibrating systems in general, and modal analysis in particular. In practical mechanical systems, interactions among various components are inevitable. There are many response variables associated with many excitations. It is convenient to consider all excitations (inputs) simultaneously as a single variable, and also all responses (outputs) as a single variable. Use of linear algebra makes the analysis of such a system convenient. Modal analysis using such a vector matrix representation will provide natural frequencies and mode shapes, as discussed in Chapter 3 and other places in the handbook. The subject of linear algebra is complex and is based on a rigorous mathematical foundation. In this appendix, we will review the basics of vectors and matrices.

3A.2

Vectors and Matrices

In the analysis of mechatronic systems, vectors and matrices will be useful in both time and frequency domains. First, consider the time domain formulation of a mechanical system. For a single DoF system with a single forcing excitation, f ðtÞ; and a corresponding single displacement response, y; the dynamic equation would be m€y þ c_y þ ky ¼ f ðtÞ

ð3A:1Þ

Note that, in this single DoF case, the quantities f ; y; m; c; and k are scalars. If the system has n DoF, with excitation forces ðx; y þ zÞ ¼ ðx; yÞ þ ðx; zÞ and associated displacement responses, y1 ; y2 ; …; yn ; the equations of motion may be expressed as kx þ yk # kxk þ kyk in which 2

y1

3

6 7 6y 7 6 27 6 7 y ¼ 6 . 7 ¼ displacement vector ðnth order column vectorÞ 6 . 7 6 . 7 4 5 yn 2

f1

3

6 7 6f 7 6 27 6 7 f ¼ 6 . 7 ¼ forcing excitation vector ðnth order column vectorÞ 6.7 6.7 4 5 fn

ð3A:2Þ

3 42


2

m11 6 6m 6 21 6 M¼6 . 6 . 6 . 4 mn1 2 c11 6 6c 6 21 6 C¼6 . 6 . 6 . 4 cn1 2 k11 6 6k 6 21 6 K¼6 . 6 . 6 . 4 kn1

m12

···

m22

···

mn2

···

c12 c22

cn2 k12 k22

kn2

···

m1n

c1n

3

7 m2n 7 7 7 7 ¼ mass matrix ðn £ n square matrixÞ 7 7 5 mnn 3

7 c2n 7 7 7 7 ¼ damping matrix ðn £ n square matrixÞ 7 7 5 · · · cnn 3 · · · k1n 7 · · · k2n 7 7 7 7 ¼ stiffness matrix ðn £ n square matrixÞ 7 7 5 · · · knn ···

In this manner, vectors and matrices are introduced into the formulation of a multi DoF mechanical system. Further vector matrix concepts will enter into the picture in subsequent analysis of the system (e.g., in modal analysis). Next consider the frequency domain formulation. In the single input single output (SISO) case, the system equation may be given as y ¼ Gu

ð3A:3Þ

in which u ¼ frequency spectrum (Fourier spectrum) of the forcing excitation (input) y ¼ frequency spectrum (Fourier spectrum) of the response (output) G ¼ frequency transfer function (frequency response function) of the system The quantities u; y; and G are scalars because each one is a single quantity and not a collection of several quantities. Next, consider a multi input multi output (MIMO) system having two excitations, u1 and u2 ; and two responses, y1 and y2 ; each yi now depends on both u1 and u2 : It follows that we need four transfer functions to represent all the excitation response relationships that exist in this system. We use the four transfer functions (G11 ; G12 ; and G22 ). For example, the transfer function G12 relates the excitation u2 to the response y1 : The associated two equations that govern the system are y1 ¼ G11 u1 þ G12 u2 y2 ¼ G21 u1 þ G22 u2

ð3A:4Þ

Instead of considering the two excitations (two inputs) as two separate quantities, we can consider them as a single “vector” u having the two components u1 and u2 : As before, we can write this as a column consisting of the two elements: " # u1 u¼ u2 In this case we have a “column vector.” Alternately, we can write a “row vector” as u ¼ ½u1 ; u2 The column vector representation is more common.

Modal Analysis

3 43

Similarly, we can express the two outputs, y1 and y2 ; as a vector y: Consequently, we have the column vector " # y1 y¼ y2 or the row vector y ¼ ½y1 ; y2 It should be kept in mind that the order in which the components (or elements) are given is important since the vector ½u1 ; u2 is not equal to the vector ½u2 ; u1 : In other words, a vector is an “ordered” collection of quantities. Summarizing, we can express a collection of quantities, in an orderly manner, as a single vector. Each quantity in the vector is known as a component or an element of the vector. What each component means will depend on the particular situation. For example, in a dynamic system it may represent a quantity such as voltage, current, force, velocity, pressure, flow rate, temperature, or heat transfer rate. The number of components (elements) in a vector is called the order, or dimension of the vector. Next let us introduce the concept of a matrix using the frequency domain example given above. Note that we needed four transfer functions to relate the two excitations to the two responses. Instead of considering these four quantities separately we can express them as a single matrix G having four elements. Specifically, the transfer function matrix for the present example is " # G11 G12 G¼ G21 G22 Note that our matrix has two rows and two columns. Hence the size or order of the matrix is 2 £ 2. Since the number of rows is equal to the number of columns in this example, we have a square matrix. If the number of rows is not equal to the number of columns, we have a rectangular matrix. Actually, we can interpret a matrix as a collection of vectors. Hence, in the previous example, the matrix G is an assembly of the two column vectors " # " # G11 G12 and G21 G22 or alternatively, an assembly of the two row vectors ½G11 ; G12 and ½G21 ; G22

3A.3

Vector– Matrix Algebra

The advantage of representing the excitations and the responses of a mechatronic system as the vectors u and y; and the transfer functions as the matrix G is clear from the fact that the excitation response (input output) equations can be expressed as the single equation y ¼ Gu

ð3A:5Þ

instead of the collection of scalar equations (Equation 3A.4). Hence, the response vector y is obtained by premultiplying the excitation vector u by the transfer function matrix G: Of course, certain rules of vector matrix multiplication have to be agreed upon in order that this single equation is consistent with the two scalar equations given by Equation 3A.4. Also, we have to agree upon rules for the addition of vectors or matrices.

3 44


A vector is a special case of a matrix. Specifically, a third order column vector is a matrix having three rows and one column. Hence, it is a 3 £ 1 matrix. Similarly, a third order row vector is a matrix having one row and three columns. Accordingly, it is a 1 £ 3 matrix. It follows that we only need to know matrix algebra; the vector algebra will follow from the results for matrices. 3A.3.1

Matrix Addition and Subtraction

Only matrices of the same size can be added. The result (sum) will also be a matrix of the same size. In matrix addition, we add the corresponding elements (i.e., the elements at the same position) in the two matrices, and write the results at the corresponding places in the resulting matrix. As an example, consider the 2 £ 3 matrix " # 1 0 3 A¼ 2 6 2 and a second matrix

" B¼

2

1

0

The sum of these two matrices is given by

3 "

AþB¼

#

5 2

1 1 2 3

2

#

0

The order in which the addition is done is immaterial. Hence AþB¼BþA

ð3A:6Þ

In other words, matrix addition is commutative. Matrix subtraction is defined just like matrix addition, except the corresponding elements are subtracted. An example is given below: 2 3 2 3 2 3 1 2 4 2 5 0 6 7 6 7 6 7 6 3 07 6 2 6 7 17 4 5 4 5 ¼ 4 1 15 4

3A.3.2

1

3

0

1

1

Null Matrix

The null matrix is a matrix whose elements are all zeros. Hence, when we add a null matrix to an arbitrary matrix, the result is equal to the original matrix. We can define a null vector in a similar manner. We can write Aþ0¼A As an example, the 2 £ 2 null matrix is

3A.3.3

"

0

0

0

0

ð3A:7Þ

#

Matrix Multiplication

Consider the product AB of the two matrices A and B. Let us write this as C ¼ AB

ð3A:8Þ

Modal Analysis

3 45

We say that B is premultiplied by A or, equivalently, A is postmultiplied by B. For this multiplication to be possible, the number of columns in A has to be equal to the number of rows in B. Then, the number of rows of the product matrix C is equal to the number of rows in A, and the number of columns in C is equal to the number of columns in B. The actual multiplication is done by multiplying the elements in a given row (say, the ith row) of A by the corresponding elements in a given column (say, the jth column) of B and summing these products. The result is the element cij of the product matrix C. Note that cij denotes the element that is common to the ith row and the jth column of matrix C. So, we have X cij ¼ aik bkj ð3A:9Þ k

As an example, suppose

" A¼

1

2

3

2

1 6 B¼6 42

1

3 1

4 2

3

5

4

3

7 27 5

4

3

#

1

0

Note that the number of columns in A is equal to three and the number of rows in B is also equal to three. Hence, we can perform the premultiplication of B by A. For example c11 ¼ 1 £ 1 þ 2 £ 2 þ ð 1Þ £ 5 ¼ 0 c12 ¼ 1 £ ð 1Þ þ 2 £ 3 þ ð 1Þ £ ð 3Þ ¼ 8 c13 ¼ 1 £ 2 þ 2 £ ð 4Þ þ ð 1Þ £ 1 ¼

7

c14 ¼ 1 £ 4 þ 2 £ 2 þ ð 1Þ £ 0 ¼ 8 c21 ¼ 3 £ 1 þ ð 3Þ £ 2 þ 4 £ 5 ¼ 17 c22 ¼ 3 £ ð 1Þ þ ð 3Þ £ 3 þ 4 £ ð 3Þ ¼ and so on. The product matrix is

" C¼

0

8

17

24

7

8

22

6

24

#

It should be noted that both products AB and BA are not always defined and, even when they are defined, the two results are not equal in general. Unless both A and B are square matrices of the same order, the two product matrices will not be of the same order. Summarizing, matrix multiplication is not commutative: AB – BA

3A.3.4

ð3A:10Þ

Identity Matrix

An identity matrix (or unity matrix) is a square matrix whose diagonal elements are all equal to 1 and all the remaining elements are zeros. This matrix is denoted by I. For example, the third order identity matrix is 2 3 1 0 0 6 7 7 I¼6 40 1 05 0

0

1

3 46


It is easy to see that when any matrix is multiplied by an identity matrix (provided, of course, that the multiplication is possible) the product is equal to the original matrix; thus AI ¼ IA ¼ A

3A.4

ð3A:11Þ

Matrix Inverse

An operation similar to scalar division can be defined in terms of the inverse of a matrix. A proper inverse is defined only for a square matrix and, even for a square matrix, an inverse may not exist. The inverse of a matrix is defined as follows. Suppose that a square matrix A has the inverse B. Then, these must satisfy the equation AB ¼ I

ð3A:12Þ

BA ¼ I

ð3A:13Þ

or equivalently where I is the identity matrix, as defined before. The inverse of A is denoted by A 1. The inverse exists for a matrix if and only if the determinant of the matrix is nonzero. Such matrices are termed nonsingular. We shall discuss the determinant in Section 3A.6. Before explaining a method for determining the inverse of a matrix, let us verify that " # 2 1 1 is the inverse of

"

1

1

1

1

#

2

To show this, we simply multiply the two matrices and show that the product is the second order unity matrix. Specifically, " #" # " # 1 1 2 1 1 0 ¼ 1 2 1 1 0 1 or

"

2 1 1 1

3A.4.1

#"

1

1

1

2

#

" ¼

1

0

0

1

#

Matrix Transpose

The transpose of a matrix is obtained by simply interchanging the rows and the columns of the matrix. The transpose of A is denoted by A T. For example, the transpose of the 2 £ 3 matrix " # 1 2 3 A¼ 2 2 0 is the 3 £ 2 matrix

2 6 AT ¼ 6 4

1 2 3

2

3

7 2 7 5 0

Modal Analysis

3 47

Note that the first row of the original matrix has become the first column of the transposed matrix, and the second row of the original matrix has become the second column of the transposed matrix. If AT ¼ A; then we say that the matrix A is symmetric. Another useful result on the matrix transpose is expressed by ðABÞT ¼ BT AT

ð3A:14Þ

It follows that the transpose of a matrix product is equal to the product of the transposed matrices, taken in the reverse order. 3A.4.2

Trace of a Matrix

The trace of a square matrix is given by the sum of the diagonal elements. The trace of matrix A is denoted by trðAÞ: X trðAÞ ¼ aii ð3A:15Þ i

For example, the trace of the matrix

2

2

6 A¼6 4 4

1

3 4 0

0

3

7 17 5

3

is given by trðAÞ ¼ ð 2Þ þ ð 4Þ þ 3 ¼

3A.4.3

3

Determinant of a Matrix

The determinant is defined only for a square matrix. It is a scalar value computed from the elements of the matrix. The determinant of a matrix A is denoted by detðAÞ or lAl: Instead of giving a complex mathematical formula for the determinant of a general matrix in terms of the elements of the matrix, we now explain a way to compute the determinant. First, consider the 2 £ 2 matrix " # a11 a12 A¼ a21 a22 Its determinant is given by detðAÞ ¼ a11 a22 Next, consider the 3 £ 3 matrix

2

a11

a12

a12 a21 a13

3

6 A¼6 4 a21

a22

7 a23 7 5

a31

a32

a33

Its determinant can be expressed as detðAÞ ¼ a11 M11

a12 M12 þ a13 M13

where the minors of the associated matrix elements are defined as " # " # " a22 a23 a21 a22 a21 M11 ¼ det ; M12 ¼ det ; M13 ¼ det a32 a33 a31 a32 a31

a22 a32

#

3 48


Note that Mij , the determinant of the matrix, is obtained by deleting the ith row and the jth column of the original matrix. The quantity Mij is known as the minor of the element aij of the matrix A. If we attach a proper sign to the minor depending on the position of the corresponding matrix element, we have a quantity known as the cofactor. Specifically, the cofactor, Cij ; corresponding to the minor, Mij ; is given by Cij ¼ ð 1Þiþj Mij

ð3A:16Þ

Hence, the determinant of the 3 £ 3 matrix may be given by detðAÞ ¼ a11 C11 þ a12 C12 þ a13 C13 Note that in the two formulas given above for computing the determinant of a 3 £ 3 matrix, we have expanded along the first row of the matrix. We get the same answer, however, if we expand along any row or any column. Specifically, when expanded along the ith row, we have detðAÞ ¼ ai1 Ci1 þ ai2 Ci2 þ ai3 Ci3 Similarly, if we expand along the jth column, we have detðAÞ ¼ a1j C1j þ a2j C2j þ a3j C3j These ideas of computing a determinant can be easily extended to 4 £ 4 and higher order matrices in a straightforward manner. Hence, we can write X X detðAÞ ¼ aij Cij ¼ aij Cij ð3A:17Þ j

3A.4.4

i

Adjoint of a Matrix

The adjoint of a matrix is the transpose of the matrix whose elements are the cofactors of the corresponding elements of the original matrix. The adjoint of matrix A is denoted by adjðAÞ: As an example, in the 3 £ 3 case, we have 3 2 3 2 C11 C21 C31 C11 C12 C13 T 7 6 7 6 7 6 7 adjðAÞ ¼ 6 4 C21 C22 C23 5 ¼ 4 C12 C22 C32 5 C31

C32

C13

C33

C23

In particular, it is easily seen that the adjoint of the matrix 2 3 1 2 1 6 7 7 A¼6 40 3 2 5 1

1

1

is given by 2 6 adjðAÞ ¼ 6 4 Accordingly, we have

2

1

2

3

3

7 1 7 5

2

7

2

3

1

3

7

6 adjðAÞ ¼ 6 4 2 3

3T

2 1

3

7 27 5

3

C33

Modal Analysis

3 49

Hence, in general adjðAÞ ¼ ½Cij

3A.4.5

T

ð3A:18Þ

Inverse of a Matrix

At this juncture, it is appropriate to give a formula for the inverse of a square matrix. Specifically, 1

A

¼

adjðAÞ detðAÞ

ð3A:19Þ

Hence, in the 3 £ 3 matrix example given before, since we have already determined the adjoint, it remains only to compute the determinant in order to obtain the inverse. Now, expanding along the first row of the matrix, the determinant is given by detðAÞ ¼ 1 £ 1 þ 2 £ 2 þ ð 1Þ £ ð 3Þ ¼ 8 Accordingly, the inverse is given by 2 A

1

¼

1

16 6 2 84 3

3

7

3

7 27 5

2 1

3

For two square matrices A and B we have ðABÞ

1

¼ B 1A

1

ð3A:20Þ

As a final note, if the determinant of a matrix is zero, the matrix does not have an inverse. Then we say that the matrix is singular. Some important matrix properties are summarized in Box 3A.1.

Box 3A.1 SUMMARY OF MATRIX PROPERTIES Addition: Am£n þ Bm£n ¼ Cm£n Multiplication: Am£n Bn£r ¼ Cm£r Identity: AI ¼ IA ¼ A ) I is the identity matrix Note: AB ¼ 0 ) ⁄ A ¼ 0 or B ¼ 0 in general Transposition: CT ¼ ðABÞT ¼ BT AT Inverse: AP ¼ I ¼ PA ) A ¼ P 1 and P ¼ A 1 ðABÞ

1

¼ B 1A

1

Commutativity: AB – BA in general Associativity: ðABÞC ¼ AðBCÞ Distributivity: CðA þ BÞ ¼ CA þ CB Distributivity: ðA þ BÞD ¼ AD þ BD

3 50


3A.5

Vector Spaces

3A.5.1

Field (F)

Consider a set of scalars. For any a and b from the set, if a þ b and ab are also elements in the set; if 1. a þ b ¼ b þ a and ab ¼ ba 2. ða þ bÞ þ g ¼ a þ ðb þ gÞ and ðabÞg ¼ aðbgÞ 3. aðb þ gÞ ¼ ab þ ag

(commutativity) (associativity) (distributivity)

are satisfied; and if 1. Identity elements 0 and 1 exist in the set such that a þ 0 ¼ a and 1a ¼ a 2. Inverse elements exist in the set such that a þ ð aÞ ¼ 0 and a·a 1 ¼ 1 then the set is a field. For example, the set of real numbers is a field. 3A.5.2

Vector Space (L)

Properties: 1. 2. 3. 4.

Vector addition ðx þ yÞ and scalar multiplication ðaxÞ are defined. Commutativity: x þ y ¼ y þ x and associativity: ðx þ yÞ þ z ¼ x þ ðy þ zÞ are satisfied. Unique null vector 0 and negation ð xÞ exist such that x þ 0 ¼ x, x þ ð xÞ ¼ 0: Scalar multiplication satisfies

aðbxÞ ¼ ðabÞx

ðassociativityÞ

aðx þ yÞ ¼ ax þ by

) ðdistributivityÞ

ða þ bÞx ¼ ax þ bx 1x ¼ x;

0x ¼ 0

Special case: Vector space Ln has vectors with n elements from the field F: Consider 2

x1

3

2

6 7 6x 7 6 27 6 7 x ¼ 6 . 7; 6 . 7 6 . 7 4 5 xn

y1

3

6 7 6y 7 6 27 6 7 y¼6 . 7 6 . 7 6 . 7 4 5 yn

Then 2 6 6 xþy ¼6 4

x1 þ y1 .. . xn þ yn

3 7 7 7¼yþx 5

Modal Analysis

3 51

and

2

ax1

3

6 7 6 7 ax ¼ 6 ... 7 4 5

axn 3A.5.3

Subspace S of L

1. If x and y are in S then x þ y is also in S: 2. If x is in S and a is in F then ax is also in S: 3A.5.4

Linear Dependence

Consider the set of vectors: x1 ; x2 ; …; xn :They are linearly independent if any one of these vectors cannot be expressed as a linear combination of one or more remaining vectors. Necessary and sufficient condition for linear independence:

a1 x1 þ a2 x2 þ · · · þ an xn ¼ 0 gives a ¼ 0 (trivial solution) as the only solution. For example, 2 3 2 3 1 2 6 7 6 7 7 7 x1 ¼ 6 x2 ¼ 6 4 2 5; 4 1 5; 3

ð3A:21Þ

2 3 5 6 7 6 x3 ¼ 4 0 7 5

1

5

These vectors are not linearly independent because x1 þ 2x2 ¼ x3 : 3A.5.5

Bases and Dimension of a Vector Space

1. If a set of vectors can be combined to form any vector in L then that set of vectors is said to span the vector space L (i.e., a generating system of vectors). 2. If the spanning vectors are all linearly independent, then this set of vectors is a basis for that vector space. 3. The number of vectors in the basis ¼ the dimension of the vector space. Note: The dimension of a vector space is not necessarily the order of the vectors. For example, consider two intersecting third order vectors. They will form a basis for the plane (two dimensional) that contains the two vectors. Hence, the dimension of the vector space ¼ 2, but the order of each vector in the basis ¼ 3. Note: L n is spanned by n linearly independent vectors ) dimðLn Þ ¼ n For example,

2 3 2 3 2 3 1 0 0 6 7 6 7 6 7 607 617 607 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 0 7; 6 0 7; · · ·; 6 ... 7 6 7 6 7 6 7 6.7 6.7 6 7 6.7 6.7 6 7 6.7 6.7 607 4 5 4 5 4 5 0 0 1

3 52


3A.5.6

Inner Product ðx; yÞ ¼ yH x

ð3A:22Þ

where H denotes the Hermitian transpose (i.e., complex conjugate and transpose). Hence, yH ¼ ðyp ÞT where ( )p denotes complex conjugation. Note: 1. 2. 3. 4.

ðx; xÞ $ 0 and ðx; xÞ ¼ 0 if and only if (iff) x ¼ 0 ðx; yÞ ¼ ðy; xÞp ðlx; yÞ ¼ lðx; yÞðx; lyÞ ¼ lp ðx; yÞ ðx; y þ zÞ ¼ ðx; yÞ þ ðx; zÞ

3A.5.7

Norm

Properties kxk $ 0 and kxk ¼ 0 iff x ¼ 0 klxk ¼ lllkxk for any scalar l kx þ yk # kxk þ kyk For example, the Euclidean norm: H

kxk ¼ x x ¼

n X i¼1

xi2

!1=2 ð3A:23Þ

Unit vector kxk ¼ 1 Normalization x ¼ x^ kxk Angle between vectors We have cos u ¼

ðx; yÞ ¼ ð^x; y^ Þ kxkkyk

ð3A:24Þ

where u is the angle between x and y: Orthogonal vectors iff ðx; yÞ ¼ 0

ð3A:25Þ

Note: n orthogonal vectors in L n are linearly independent and span Ln ; and form a basis for Ln : 3A.5.8

Gram –Schmidt Orthogonalization

Given a set of vectors, x1 ; x2 ; …; xn ; that are linearly independent in Ln ; we construct a set of orthonormal (orthogonal and normalized) vectors, y^ 1 ; y^ 2 ; …; y^ n ; which are linear combinations of x^ i : Start y^ 1 ¼ x^ 1 ¼

x1 kx1 k

Modal Analysis

3 53

Then y i ¼ xi

i 1 X j¼1

ðxi ; y^ j Þ^yj

for i ¼ 1; 2; …; n

Normalize yi to produce y^ i : 3A.5.9

Modified Gram –Schmidt Procedure

In each step, compute new vectors that are orthogonal to the just computed vector. Initialization: x y^ 1 ¼ 1 kx1 k as before. Then xð1Þ i ¼ xi y^ i ¼

ð^y1 ; xi Þ^y1 xð1Þ i kxð1Þ i k

for i ¼ 2; 3; …; n

for i ¼ 2; 3; …n

and ð1Þ xð2Þ i ¼ xi

ð^y2 ; xð1Þ y2 ; i Þ^

i ¼ 3; 4; …; n

and so on.

3A.6

Determinants

Now, let us address several analytical issues of the determinant of a square matrix. Consider the matrix 3 2 a11 · · · a1n 7 6 7 6 A ¼ 6 ... 7 5 4 an1

· · · ann

The minor of aij ¼ Mij ¼ the determinant of matrix formed by deleting the ith row and the jth column of the original matrix. The cofactor of aij ¼ Cij ¼ ð 1Þiþj Mij cof ðAÞ ¼ cofactor matrix of A adjðAÞ ¼ adjoint A ¼ ðcof AÞT 3A.6.1 1. 2. 3. 4. 5.

Properties of Determinant of a Matrix

Interchange two rows (columns) ) determinant’s sign changes. Multiply one row (column) by a ) a det( ). Add a [a £ row(column)] to a second row(column) ) determinant unchanged. Identical rows(columns) ) zero determinant. For two square matrices A and B; detðABÞ ¼ detðAÞdetðBÞ:

3A.6.2

Rank of a Martix

Rank A ¼ number of linearly independent columns ¼ number of linearly independent rows ¼ dim(column space) ¼ dim(row space) Here “dim” denotes the “dimension of.”

3 54

3A.7


System of Linear Equations

Consider the following set of linear algebraic equations: a11 x1 þ a12 x2 þ · · · þ a1n xn ¼ c1 a21 x1 þ a22 x2 þ · · · þ a2n xn ¼ c2 .. . am1 x1 þ am2 x2 þ · · · þ amn xn ¼ cm We need to solve for x1 ; x2 ; …; xn : This problem can be expressed in the vector matrix form: Am£n xn ¼ cm

B ¼ ½A; c

A solution exists iff rank½A; c ¼ rank½A Two cases can be considered: Case 1: If m $ n and rank½A ¼ n ) unique solution for x: Case 2: If m # n and rank½A ¼ m ) infinite number of solutions for x x ¼ AH ðAAH Þ 1 C ( minimum norm form Specifically, out of the infinite possibilities, this is the solution that minimizes the norm, xH x: Note that the superscript H denotes the Hermitian transpose, which is the transpose of the complex conjugate of the matrix. For example, " # 1 þ j 2 þ 3j 6 A¼ 3 j 5 1 2j Then

2

1

6 AH ¼ 6 42

6

j

3þj

3j

5

3 7 7 5

1 þ 2j

If the matrix is real, its Hermitian transpose is simply the ordinary transpose. In general, if rank½A # n ) infinite number of solutions. The space formed by solutions Ax ¼ 0 ) is called the null space dimðnull spaceÞ ¼ n

3A.8

k where rank½A ¼ k

Quadratic Forms

Consider a vector, x; and a square matrix, A: Then the function QðxÞ ¼ ðx; AxÞ is called a quadratic form. For a real vector x and a real and symmetric matrix A; QðxÞ ¼ xT Ax Positive definite matrix: If ðx; AxÞ . 0 for all x – 0; then A is said to be a positive definite matrix. Also, the corresponding quadratic form is also said to be positive definite.

Modal Analysis

3 55

Positive semidefinite matrix: If ðx; AxÞ $ 0 for all x – 0; then A is said to be a positive semidefinite matrix. Note that in this case the quadratic form can assume a zero value for a nonzero x: The corresponding quadratic form is also said to be positive semidefinite. Negative definite matrix: If ðx; AxÞ , 0 for all x – 0; then A is said to be a negative definite matrix. The corresponding quadratic form is also said to be negative definite. Negative semidefinite matrix: If ðx; AxÞ # 0 for all x – 0; then A is said to be a negative semidefinite matrix. Note that, in this case, the quadratic form can assume a zero value for a nonzero x: The corresponding quadratic form is also said to be negative semidefinite. Note: If A is positive definite, then A is negative definite. If A is positive semidefinite, then A is negative semidefinite. Principal minors: Consider the matrix 2 3 a11 a12 · · · a1n 6 7 6a 7 6 21 a22 · · · a2n 7 6 7 A¼6 . 7 6 . 7 6 . 7 4 5 an1 an2 · · · ann Its principal minors are the determinants of the various matrices along the principal diagonal, as given by 2 3 a11 a12 a13 " # a11 a12 6 7 7 D1 ¼ a11 ; D2 ¼ det ; D3 ¼ det6 4 a21 a22 a23 5; and so on a21 a22 a31 a32 a33 Sylvester’s theorem: A matrix is positive if definite if all its principal minors are positive.

3A.9 3A.9.1

Matrix Eigenvalue Problem Characteristic Polynomial

Consider a square matrix A: The polynomial DðsÞ ¼ det½sI

A

is called the characteristic polynomial of A: 3A.9.2

Characteristic Equation

The polynomial equation DðsÞ ¼ det½sI

A ¼0

is called the characteristic equation of the square matrix A: 3A.9.3

Eigenvalues

The roots of the characteristic equation of a square matrix A are the eigenvalues of A: For an n £ n matrix, there will be n eigenvalues. 3A.9.4

Eigenvectors

The eigenvalue problem of a square matrix A is given by Av ¼ lv

3 56


where the objective is to solve for l and the corresponding nontrivial (i.e., nonzero) solutions for v: The problem can be expressed as ðlI

AÞv ¼ 0

Note: If v is a solution of this equation, then any multiple of it, av; is also a solution. Hence, an eigenvector is arbitrary up to a multiplication factor. For a nontrivial (i.e., nonzero) solution to be possible for v; one must have det½lI

A ¼0

Since this is the characteristic equation of A; as defined above, it is clear that the roots of l are the eigenvalues of A: The corresponding solutions for v are the eigenvectors of A: For an n £ n matrix, there will be n eigenvalues and n corresponding eigenvectors.

3A.10

Matrix Transformations

3A.10.1

Similarity Transformation

Consider a square matrix, A; and a nonsingular square matrix, T: Then, the matrix obtained according to B ¼ T 1 AT is the similarity transformation of A by T: The transformed matrix B has the same eigenvalues as the original matrix A: Also, A and B are said to be similar. 3A.10.2

Orthogonal Transformation

Consider a square matrix A and another square matrix T: Then, the matrix obtained according to B ¼ TT AT is the orthogonal transformation of A by T: If T 1 ¼ TT then the matrix T is said to be an orthogonal matrix. In this case, the similarity transformation and the orthogonal transformation become identical.

3A.11

Matrix Exponential

The matrix exponential is given by the infinite series expðAtÞ ¼ I þ At þ

1 2 2 A t þ ··· 2!

ð3A:26Þ

expðltÞ ¼ 1 þ lt þ

1 2 2 l t þ ··· 2!

ð3A:27Þ

exactly like the scalar exponential

The matrix exponential may be determined by reducing the infinite series given in Equation 3A.26 into a finite matrix polynomial of order n 1 (where, A is n £ n) by using the Cayley Hamilton theorem. 3A.11.1

Cayley –Hamilton Theorem

This theorem states that a matrix satisfies its own characteristic equation. The characteristic polynomial of A can be expressed as DðlÞ ¼ detðA

lIÞ ¼ an ln þ an 1 ln

1

þ · · · þ a0

ð3A:28Þ

Modal Analysis

3 57

in which det( ) denotes determinant. The notation DðAÞ ¼ an An þ an 1 An

1

þ · · · þ a0 I

ð3A:29Þ

is used. Then, by the Cayley Hamilton theorem, we have 0 ¼ an An þ an 1 An

3A.11.2

1

þ · · · þ a0 I

ð3A:30Þ

Computation of Matrix Exponential

Using the Cayley Hamilton theorem, we can obtain a finite polynomial expansion for expðAtÞ: First, we express Equation 3A.26 and Equation 3A.27 as expðAtÞ ¼ SðAÞ·DðAÞ þ an 1 An

1

n 1

expðltÞ ¼ SðlÞ·DðlÞ þ an 1 l

þ an 2 An

2

n 2

þ an 2 l

þ · · · þ a0 I

ð3A:31Þ

þ · · · þ a0

ð3A:32Þ

in which Sð·Þ is an appropriate infinite series, which is the result of dividing the exponential (infinite) series by the characteristic polynomial Dð·Þ: Next, since DðAÞ ¼ 0 by the Cayley Hamilton theorem, Equation 3A.31 becomes expðAtÞ ¼ an 1 An

1

þ an 2 An

2

þ · · · þ a0 I

ð3A:33Þ

Now it is just a matter of determining the coefficients, a0 ; a1 ; …; an 1 ; which are functions of time. This is done as follows. If l1 ; l2 ; …; ln are the eigenvalues of A; however, then, by definition Dðli Þ ¼ detðA

li IÞ ¼ 0 for i ¼ 1; 2; …; n

ð3A:34Þ

Thus, from Equation 3A.32, we obtain expðli tÞ ¼ an 1 lni

1

þ an 2 lni

2

þ · · · þ a0

for i ¼ 1; 2; …; n

ð3A:35Þ

If the eigenvalues are all distinct, Equation 3A.35 represents a set of n independent algebraic equations from which the n unknowns a0 ; a1 ; …; an 1 could be determined. If some eigenvalues are repeated, the derivatives of the corresponding equations (Equation 3A.35) have to be used as well.

4

Distributed-Parameter Systems


4.1 4.2

Introduction ....................................................................... Transverse Vibration of Cables .........................................

4.3

Longitudinal Vibrations of Rods ..................................... 4 13

4.4

Torsional Vibration of Shafts ........................................... 4 19

Wave Equation † General (Modal) Solution † Cable with Fixed Ends † Orthogonality of Natural Modes † Application of Initial Conditions Equation of Motion

†

41 42

Boundary Conditions

Shaft with Circular Cross Section Noncircular Shafts

†

Torsional Vibration of

4.5

Flexural Vibration of Beams ............................................. 4 26

4.6

Damped Continuous Systems .......................................... 4 50

4.7

Vibration of Membranes and Plates ................................ 4 52

Governing Equation for Thin Beams † Modal Analysis † Boundary Conditions † Free Vibration of a Simply Supported Beam † Orthogonality of Mode Shapes † Forced Bending Vibration † Bending Vibration of Beams with Axial Loads † Bending Vibration of Thick Beams † Use of the Energy Approach † Orthogonality with Inertial Boundary Conditions Modal Analysis of Damped Beams

Transverse Vibration of Membranes † Rectangular Membrane with Fixed Edges † Transverse Vibration of Thin Plates † Rectangular Plate with Simply Supported Edges

Summary This chapter presents the analysis of continuous (or distributed parameter) mechanical vibrating systems. In these systems, inertial, elastic, and dissipative effects are found continuously distributed in one, two, or three dimensions. Examples such as strings, rods, shafts, beams, membranes, and plates are studied. Modal analysis is carried out using the separation of time and space. The orthogonality property of mode shapes is established. Boundary conditions are derived. Free vibration and forced vibration are analyzed.

4.1

Introduction

Often in vibration analysis, it is assumed that inertial (mass), flexibility (spring), and dissipative (damping) characteristics can be “lumped” as a finite number of “discrete” elements. Such models are termed lumped parameter or discrete parameter systems. Generally, in practical vibrating systems, inertial, elastic, and dissipative effects are found continuously distributed in one, two, or three dimensions. Correspondingly,

41

42


we have line structures, surface/planar structures, or spatial structures. They will possess an infinite number of mass elements, continuously distributed in the structure, and integrated with some connecting flexibility (elasticity) and energy dissipation. In view of the connecting flexibility, each small element of mass will be able to move out of phase (or somewhat independently) with the remaining mass elements. It follows that a continuous system (or a distributed parameter system) will have an infinite number of degrees of freedom (DoFs) and will require an infinite number of coordinates to represent its motion. In other words, extending the concept of a finite degree of freedom system as analyzed previously, an infinite dimensional vector is needed to represent the general motion of a continuous system. Equivalently, a one dimensional continuous system (a line structure) will need one independent spatial variable, in addition to time, to represent its response. In view of the need for two independent variables in this case, one for time and the other for space, the representation of system dynamics will require partial differential equations (PDEs) rather than ordinary differential equations (ODEs). Furthermore, the system will depend on the boundary conditions as well as the initial conditions. Strings, cables, rods, shafts, beams, membranes, plates, and shells are example of continuous members. In special cases, closed form analytical solutions can be obtained for the vibration of these members. A general structure may consist of more than one such member, and furthermore, boundary conditions (BCs) could be various, individual members may be nonuniform, and the material characteristics may be inhomogeneous and anisotropic. Closed form analytical solutions would not be generally possible in such cases. Nevertheless, the insight gained by analyzing the vibration of standard members will be quite beneficial in studying the vibration behavior of more complex structures. The concepts of modal analysis may be extended from lumped parameter systems to continuous systems. In particular, since the number of principal modes is equal to the number of DoFs of the system, a distributed parameter system will have an infinite number of natural modes of vibration. A particular mode may be excited by deflecting the member so that its elastic curve assumes the shape of the particular mode, and then releasing from this initial condition. When damping is significant and nonproportional, however, there is no guarantee that such an initial condition could accurately excite the required mode. A general excitation consisting of a force or an initial condition will excite more than one mode of motion. However, as in the case of discrete parameter systems, the general motion may be analyzed and expressed in terms of modal motions, through modal analysis. In a modal motion, the mass elements will move at a specific frequency (the natural frequency), and bearing a constant proportion in displacement (i.e., maintaining the mode shape), and passing the static equilibrium of the system simultaneously. In view of this behavior, it is possible to separate the time response and spatial response of a vibrating system in a modal motion. This separability is fundamental to modal analysis of a continuous system. Furthermore, in practice an infinite number of natural frequencies and mode shapes are not significant and typically the very high modes may be neglected. Such a modal truncation procedure, even though carried out by continuous system analysis, is equivalent to approximating the original infinite degree of freedom system by a finite degree of freedom one. Vibration analysis of continuous systems may be applied in the modeling, analysis, design, and evaluation of such practical systems as cables; musical instruments; transmission belts and chains; containers of fluid; animals; structures including buildings, bridges, guideways, and space stations; and transit vehicles, including automobiles, ships, aircraft, and spacecraft.

4.2

Transverse Vibration of Cables

The first continuous member which we will study is a string or cable in tension. This is a line structure whose geometric configuration can be completely defined by the position of its axial line with reference to a fixed coordinate line. We will study the transverse (lateral) vibration problem; that is, the vibration in a direction perpendicular to its axis and in a single plane. Applications will include stringed musical instruments, overhead transmission lines (of electric power or telephone signals), drive systems (belt drives, chain drives, pulley ropes, etc.), suspension bridges, and structural cables carrying cars (e.g., ski lifts, elevators, overhead sightseeing systems, and cable cars).

Distributed Parameter Systems

43

As usual, we will make some simplifying assumptions for analytical convenience. However, the results and insight obtained in this manner will be useful in understanding the behavior of more complex systems containing cable like structures. The main assumptions are: 1. The system is a line structure. The lateral dimensions are much smaller compared with the longitudinal dimension (normally in the x direction). 2. The structure stays in a single plane and the motion of every element of the structure will be in a fixed transverse direction ð yÞ: 3. The cable tension ðTÞ remains constant during motion. In other words, the initial tension is sufficiently large that the variations during motion are negligible. ›v 4. Variations in slope ðuÞ along the structure are small. Hence, for example, u ø sin u ø tan u ¼ : ›x A general configuration of a cable (or string) is shown in Figure 4.1(a). Consider a small element of length dx of the cable at location x; as shown in Figure 4.1(b). The equation (Newton’s Second Law) of motion (transverse) of this element is given by T sin u þ T sinðu þ duÞ ¼ mðxÞdx

f ðx; tÞdx

›2 vðx; tÞ ›t 2

ð4:1Þ

in which vðx; tÞ ¼ transverse displacement of the cable f ðx; tÞ ¼ lateral force per unit length of cable mðxÞ ¼ mass per unit length of cable T ¼ cable tension u ¼ cable slope at location x: Note that the dynamic loading f ðx; tÞ may arise due to such causes as aerodynamic forces, fluid drag, and electromagnetic forces, depending on the specific application. y Force per unit length = f(x,t)

T x x+dx

0

l

x

Mass per unit length = m(x)

(a) y

f.dx

T

θ+

∂q dx ∂x

m.dx

θ

T

(b) FIGURE 4.1

x

x+dx

x

(a) Transverse vibration of a cable in tension; (b) motion of a general element.

44


Using the small slope assumption we have sin u ø u and sinðu þ duÞ ø u þ du with u ¼ ›v=›x and du ¼ ð›2 v=›x2 Þdx as dx ! 0: On substitution of these approximations into Equation 4.1 and canceling out dx, we obtain mðxÞ

›2 vðx; tÞ ›2 vðx; tÞ ¼T þ f ðx; tÞ 2 ›t ›x2

ð4:2Þ

Now consider the case of free vibration where f ðx; tÞ ¼ 0: We have 2 ›2 vðx; tÞ 2 › vðx; tÞ ¼ c ›t 2 ›x 2

with

ð4:3Þ

pffiffiffiffiffiffi c ¼ T=m

ð4:4Þ

Also, assume that the cable is uniform so that m is constant.

4.2.1

Wave Equation

The solution to any equation of the form (Equation 4.3) will appear as a wave, traveling either in the forward (positive x) or in the backward (negative x) direction at speed c: Hence, Equation 4.3 is called the wave equation and c is the wave speed. To prove this fact, first, we show that a solution to Equation 4.3 can take the form vðx; tÞ ¼ v1 ðx First, let x

ct ¼ z: Hence, v1 ðx

ctÞ

ð4:5Þ

ctÞ ¼ v1 ðzÞ: Then,

›v1 dv ›z › v1 dv ›z ¼ 1 and ¼ 1 dz ›x dz ›t ›x ›t with

›z ›z ¼ 1 and ¼ ›x ›t

c

It follows that

› 2 v1 ›2 v1 ¼ v001 and ¼ c2 v001 2 ›x ›t 2 where v001 ¼

d2 v1 dz 2

Clearly, then, v1 satisfies Equation 4.3. Now, let us examine the nature of the solution v1 ðx ctÞ: It is clear that v1 will be constant when x ct ¼ constant: However, the equation x ct ¼ constant corresponds to a point moving along the x axis in the positive direction at speed c: What this means is that the shape of the cable at t ¼ 0 will “appear” to travel along the cable at speed c: This is analogous to the waves we observe in a pond when excited by dropping a stone. Note that the particles of the cable do not travel along x: it is the deformation “shape” (the wave) that travels. Similarly, it can be shown that vðx; tÞ ¼ v2 ðx þ ctÞ

ð4:6Þ


45

is also a solution to Equation 4.3 and this corresponds to a wave that travels backward (negative x direction) at speed c: The general solution, of course, will be of the form vðx; tÞ ¼ v1 ðx

ctÞ þ v2 ðx þ ctÞ

ð4:7Þ

which represents two waves, one traveling forward and the other backward.

4.2.2

General (Modal) Solution

As usual, we look for a separable solution of the form vðx; tÞ ¼ YðxÞqðtÞ

ð4:8Þ

for the cable/string vibration problem given by the wave equation 4.3. If a solution in the form of Equation 4.8 is obtained, it will be essentially a modal solution. This should be clear from the separability itself of the solution. Specifically, at any given time t; the time function qðtÞ will be fixed and the structure will have a shape given by YðxÞ: Hence, at all times the structure will maintain a particular shape YðxÞ and this will be a mode shape. Also, at a given point x of the structure, the space function YðxÞ will be fixed and the structure will vibrate according to the time response qðtÞ: It will be shown that qðtÞ will obey the simple harmonic motion of a specific frequency. This is the natural frequency of vibration corresponding to that particular mode. Note that, for a continuous system, there will be an infinite number of solutions of the form of Equation 4.8 with different natural frequencies. The corresponding functions YðxÞ will be “orthogonal” in some sense. Hence, they are called normal modes (normal meaning perpendicular). The systems will be able to move independently in each mode and this collection of solutions in the form of Equation 4.8 will be a complete set. With this qualitative understanding, let us now seek a solution of the form of Equation 4.8 for the system Equation 4.3. Substitute Equation 4.8 in Equation 4.3. We obtain YðxÞ

2 d2 qðtÞ 2 d YðaÞ ¼ c qðtÞ dt 2 dx2

or 1 d2 YðxÞ 1 d2 qðtÞ ¼ 2 ¼ 2 YðxÞ dx c qðtÞ dt 2

l2

ð4:9Þ

In Equation 4.9, since the left hand terms are a function of x only and the right hand terms are a function of t only, for the two sides to be equal in general, each function should be a constant (that is independent of both x and t). This constant is denoted by l2, which is called the separation constant and is designated to be negative. There are two reasons for this. If this common constant were positive, the function qðtÞ would be nonoscillatory and transient, which is contrary to the nature of undamped vibration. Furthermore, it can be shown that a nontrivial solution for YðxÞ would not be possible if the common constant were positive. The unknown constant l is determined by solving the space equation (mode shape equation) of Equation 4.9; specifically d2 YðxÞ þ l2 YðxÞ ¼ 0 dx2

ð4:10Þ

and then applying the BCs of the problem. There will be an infinite number of solutions for l; with corresponding natural frequencies v and mode shapes YðxÞ: The characteristic equation of 4.10 is p2 þ l2 ¼ 0

ð4:11Þ

which has the characteristic roots (or eigenvalues) p ¼ ^jl

ð4:12Þ

46


The general solution is YðxÞ ¼ A1 e jlx þ A2 e

jlx

¼ C1 cos lx þ C2 sin lx

ð4:13Þ

Note that, since YðxÞ is a real function representing a geometric shape, the constants A1 and A2 have to be complex conjugates and C1 and C2 have to be real. Specifically, in view of the fact that cos lx ¼ ðejlx þ e jlx = Þ2 and sin lx ¼ ðejlx e jlx =2j Þ, we can show that A1 ¼

1 ðC 2 1

jC2 Þ

and

A2 ¼

1 ðC þ jC2 Þ 2 1

For analytical convenience, we will use the real parameter form of Equation 4.13. Note that we cannot determine both constants C1 and C2 using BCs. Only their ratio is determined and the constant multiplier is absorbed into qðtÞ in Equation 4.8 and then determined using the appropriate initial conditions (at t ¼ 0). It follows that the ratio of C1 and C2 and the value of l are determined using the BCs. Two BCs will be needed. Some useful situations and appropriate relations are given in Table 4.1.

4.2.3

Cable with Fixed Ends

Let us obtain the complete solution for the free vibration of a taut cable that is fixed at both ends. The applicable BCs are Yð0Þ ¼ YðlÞ ¼ 0

ð4:14Þ

where l is the length of the cable. Substitution into Equation 4.13 gives C1 £ 1 þ C2 £ 0 ¼ 0 C1 cos ll þ C2 sin ll ¼ 0 Hence, we have C1 ¼ 0

and C2 sin ll ¼ 0

ð4:15Þ

A possible solution for Equation 4.15 is C2 ¼ 0: However, this is the trivial solution, which corresponds to YðxÞ ¼ 0 (i.e., a stationary cable with no vibration). It follows that the applicable, nontrivial solution is sin ll ¼ 0 which produces an infinite number of solutions for l given by

li ¼

ip l

with i ¼ 1; 2; …; 1

ð4:16Þ

As mentioned earlier, the corresponding infinite number of mode shapes is given by Yi ðxÞ ¼ Ci sin

ipx l

ð4:17Þ

Note: If we had used a positive constant l2 instead of l2 in Equation 4.9, only a trivial solution (with C1 ¼ 0 and C2 ¼ 0) would be possible for YðxÞ: This further justifies our decision to use l2 : Substitute Equation 4.16 into Equation 4.9 to determine the corresponding time response (generalized coordinates) qi ðtÞ; thus d2 qi ðtÞ þ v2i qi ðtÞ ¼ 0 dt 2

ð4:18Þ

in which

vi ¼ li c ¼

ip l

rffiffiffiffi T m

for i ¼ 1; 2; …; 1

ð4:19Þ

Some Useful Boundary Conditions for the Cable Vibration Problem

Type of End Condition

Nature of End x ¼ x0

Fixed

Boundary Condition vðx0 ; tÞ ¼ 0

x

x Free

T

Modal Boundary Condition Yi ðx0 Þ ¼ 0

T

›vðx0 ; tÞ ¼0 ›x

dYi ðx0 Þ ¼0 dx

T

›vðx0 ; tÞ 2 kvðx0 ; tÞ ¼ 0 ›x

T

dYi ðx0 Þ 2 kYi ðx0 Þ ¼ 0 dx

T

›vðx0 ; tÞ ›2 vðx0 ; tÞ 2 kvðx0 ; tÞ ¼ M ›x ›t 2

T

dYi ðx0 Þ 2 ðk 2 v2i MÞYi ðx0 Þ ¼ 0 dx

∂v ∂x

Distributed-Parameter Systems

TABLE 4.1

x Flexible

T

∂v ∂x x

k xo Flexible and inertial

T M

∂v ∂x x

k xo

4-7

48


Equation 4.18 represents a simple harmonic motion with the modal natural frequencies vi given by Equation 4.19. It follows that there are an infinite number of natural frequencies, as mentioned earlier. The general solution of Equation 4.19 is given by qi ðtÞ ¼ ci sinðvi t þ fi Þ

ð4:20Þ

where the amplitude parameter ci and the phase parameter fi are determined using two of the initial conditions of the system. It should be clear that it is redundant to use a separate constant Ci for Yi ðxÞ in Equation 4.17, and that this may be absorbed into the amplitude constant in Equation 4.20 to express the general free response of the cable as vðx; tÞ ¼

X ipx ci sin sinðvi t þ fi Þ l

ð4:21Þ

In this manner, the complete solution has been expressed as a summation of the modal solutions. This is known as the modal series expansion. Such a solution is quite justified because of the fact that the mode shapes are orthogonal in some sense, and what we obtained above were a complete set of normal modes (normal in the sense of perpendicular or orthogonal). The system is able to move in each mode independently, with a unique spatial shape, at the corresponding natural frequency, because each modal solution is separable into a space function, Yi ðxÞ, and a time function (generalized coordinate), qi ðtÞ: Of course, the system will be able to simultaneously move in a linear combination of two modes (say, C1 Y1 ðxÞq1 ðtÞ þ C2 Y2 ðxÞq2 ðtÞ), since this combination satisfies the original system Equation 4.3 because of its linearity and because each modal component satisfies the equation. However, clearly, this solution, with two modes, is not separable into a product of a space function and a time function. Hence, it is not a P modal solution. In this manner, it can be argued that the infinite sum of modal solutions ci Yi ðxÞqi ðtÞ is the most general solution to the system (Equation 4.3). The orthogonality of mode shapes plays a key role in this argument and, furthermore, it is useful in the analysis of the system, as we shall see. In particular, in Equation 4.21, the unknown constants ci and fi are determined using the system initial conditions, and the orthogonality property of modes is useful in that procedure.

4.2.4

Orthogonality of Natural Modes

A cable can vibrate at frequency vi while maintaining a unique natural shape Yi ðxÞ; called the mode shape of the cable. We have shown that, for the fixed ended cable, the natural mode shapes are given by sinðipx=lÞ with the corresponding natural frequencies, vi : It can be easily verified that 8 > l l 0 : 2

ðl

for i – j for i ¼ j

ð4:22Þ

In other words, the natural modes are orthogonal. Equation 4.22 represents the principle of orthogonality of natural modes in this case. Orthogonality makes the modal solutions independent and the corresponding mode shapes “normal.” It also makes the infinite set of modal solutions a complete set, or a basis, so that any arbitrary response can be formed as a linear combination of these normal mode solutions. Orthogonality holds for other types of BCs as well. To show this, we observe from Equation 4.9 that d2 Yi ðxÞ þ l2i Yi ðxÞ ¼ 0 dx2

for mode i

ð4:23Þ

d2 Yj ðxÞ þ l2j Yj ðxÞ ¼ 0 dx2

for mode j

ð4:24Þ


49

Multiply Equation 4.23 by Yj ðxÞ; Equation 4.24 by Yi ðxÞ; subtract that second result from the first, and integrate with respect to x along the cable length from x ¼ 0 to l: We obtain " # ðl ðl d2 Yj d2 Yi ð4:25Þ Yj 2 Yi 2 dx þ ðl2i l2j Þ Yi Yj dx ¼ 0 dx dx 0 0 Integrating by parts, we obtain the results ðl 0

ðl 0

Yj Yi

d2 Yi dY dx ¼ Yj i dx2 dx

l

d2 Y j dYj dx ¼ Yi dx2 dx

l

ðl dY dYj i dx 0 dx dx

0

ðl dY dYj i dx 0 dx dx

0

Hence, the first term of Equation 4.25 becomes Yj

dYi dx

Yi

dYj dx

l 0

which will vanish for common BCs. Then, since li – lj for i – j, we have ðl 0

Yi ðxÞYj ðxÞdx ¼ 0

for i – j

We can pick the value of the multiplication constant in the general solution for YðxÞ; given by Equation 4.13, so as to normalize the mode shapes such that ðl 0

Yi2 ðxÞdx ¼

l 2

which is consistent with the result 4.22. Hence, the general condition of orthogonality of natural modes may be expressed as 8 > < 0 for i – j ðl ð4:26Þ Yi ðxÞYj ðxÞdx ¼ l > 0 : for i ¼ j 2 4.2.4.1

Nodes

When vibrating in a particular mode, one or more points of the system (cable) that are not physically fixed may remain stationary at all times. These points are called the nodes of that mode. For example, in the second mode of a cable with its ends fixed, there will be a node at the midspan. This should be clear from the fact that the mode shape of the second mode is sin 2px=l which becomes zero at x ¼ l=2: Similarly, in the third mode, with mode shape sin 3px=l; there will be nodes at x ¼ l=3 and 2l=3:

Example 4.1 If the cable tension varies along the length x; what is the corresponding equation of free lateral vibration? A hoist mechanism has a rope of freely hanging length l in a particular equilibrium configuration and carrying a load of mass M; as shown in Figure 4.2(a). Determine the equation of lateral vibration and the applicable BCs for the rope segment. Solution With reference to Figure 4.1(b), Equation 4.1 may be modified for the case of variable T as T sin u þ ðT þ dTÞ sinðu þ duÞ ¼ m dx

›2 v ›t 2

ð4:27Þ

4 10


where f ðx; tÞ ¼ 0 for free vibration. Now, with the assumption of small u; and by neglecting the second order product term dT du; we obtain T du þ u dT ¼ m

x l

2

›v dx ›x 2

Next, using

u¼

›v ; ›x

du ¼

›2 v ›T ; and dT ¼ dx 2 ›x ›x

and canceling dx; we obtain the equation of lateral vibration of a cable as m

›2 v ›2 v ›T ›v ¼T 2 þ 2 ›t ›x ›x ›x

Longitudinal (axial) dynamics of the rope are negligible for the case of a stationary hoist. Then, longitudinal equilibrium (in the x direction) of the small element of rope shown in Figure 4.2(b) gives ðT þ dTÞ cosðu þ duÞ

T cos u

0

y

ð4:28Þ

Mg

(a) T+dT θ+dθ

mg dx ¼ 0

For small u; we have cos u ø 1 and cosðu þ duÞ ø 1 up to the first order term in the Taylor series expansion. Hence, dT ¼ mg dx

x+dx x θ

ð4:29Þ (b)

Integration gives T ¼ T0 þ mgx

ð4:30Þ

with the end condition

T

mgdx

FIGURE 4.2 (a) Free segment of a stationary hoist; (b) a small element of the rope.

T ¼ Mg at x ¼ 0 Hence, T ¼ Mg þ mgx

ð4:31Þ

Note from Equation 4.29 that ›T=›x ¼ dT=dx ¼ mg for this problem. Substitute in (Equation 4.28) this fact and Equation 4.31 to obtain m

›2 v ›2 v ›v ¼ ðM þ mxÞg 2 þ mg 2 ›t ›x ›x

or

›2 v ¼ ›t 2

M ›2 v ›v þx g 2 þg m ›x ›x

ð4:32Þ

The BC at x ¼ 0 is obtained by applying Newton’s Second Law to the end mass in the lateral ðyÞ direction. This gives T0

›vð0; tÞ ›2 vð0; tÞ ¼M ›x ›t 2

Now, using the fact that T0 ¼ Mg, we have the boundary condition g

›vð0; tÞ ›2 vð0; tÞ ¼ ›x ›t 2


4 11

For mode i:

›vð0; tÞ dYi ð0Þ ¼ q ðtÞ ›x dx i and

›2 vð0; tÞ d2 qi ðtÞ ¼ Y ð0Þ ¼ i ›t 2 dt 2

v2i Yi ð0Þqi ðtÞ

which holds for all t and where vi is the ith natural frequency of vibration. Hence, the modal BC at x ¼ 0 is g

dYi ð0Þ þ v2i Yi ð0Þ ¼ 0 dx

for i ¼ 1; 2; …

ð4:33Þ

The BC at x ¼ l is vðl; tÞ ¼ 0

ð4:34Þ

which holds for all t: Hence, the corresponding modal BC is Yi ðlÞ ¼ 0

4.2.5

for i ¼ 1; 2; …

ð4:35Þ

Application of Initial Conditions

The general solution to the cable vibration problem is given by X vðx; tÞ ¼ ci Yi ðxÞ sinðvi t þ fi Þ

ð4:36Þ

where Yi ðxÞ are the normalized mode shapes which satisfy the orthogonality property (Equation 4.26). The unknown constants ci and fi are determined using the initial conditions vðx; 0Þ ¼ dðxÞ

ð4:37Þ

›vðx; 0Þ ¼ sðxÞ ›t

ð4:38Þ

By substituting Equation 4.36 into Equation 4.37 and Equation 4.38, we obtain X dðxÞ ¼ ci Yi ðxÞ sin fi X sðxÞ ¼ ci vi Yi ðxÞ cos fi

ð4:39Þ ð4:40Þ

Multiply Equation 4.39 and Equation 4.40 by Yj ðxÞ and integrate with respect to x from 0 to l; making use of the orthogonality condition (Equation 4.26). We obtain ðl 0

ðl 0

dðxÞYj ðxÞdx ¼ cj

l sin fj 2

sðxÞYj ðxÞdx ¼ cj vj

l cos fj 2

Solving these two equations, we obtain ðl tan fj ¼ vj ð0l 0

dðxÞYj ðxÞdx sðxÞYj ðxÞdx

for j ¼ 1; 2; 3; …

ð4:41Þ

4 12


Once fj is determined in this manner, we can obtain cj by using cj ¼

ðl 2 dðxÞYj ðxÞdx l sin fj 0

Example 4.2 Consider a taut horizontal cable of length l and mass m per unit length, as shown in Figure 4.3, excited by a transverse point force f0 sin vt at location x ¼ a; where v is the frequency of (harmonic) excitation and f0 is the forcing amplitude. Determine the resulting response of the cable under general end conditions and initial conditions. For the special case of fixed ends, what is the steady state response of the cable?

for j ¼ 1; 2; 3; …

ð4:42Þ

fo sinwt

y

a

0

FIGURE 4.3

l

x

A cable excited by a point harmonic force.

Solution We have shown that the forced transverse response of a cable is given by Equation 4.2: f ðx; tÞ ›2 vðx; tÞ ›2 vðx; tÞ ¼ c2 þ 2 ›t ›x2 m

ð4:2Þ

where vðx; tÞ is the transverse displacement and f ðx; tÞ is the external force per unit length of the cable. For the point force F at x ¼ a; an analytical representation of the equivalent distributed force per unit length is f ðx; tÞ ¼ F dðx

aÞ

ð4:43Þ

where the Dirac delta function (unit impulse function) dðxÞ is such that ð a2 gðxÞdðx aÞdx ¼ gðaÞ

ðiÞ

a1

for an arbitrary function gðxÞ; provided that the point a is within the interval of integration ½a1 ; a2 : We seek a “modal superposition” solution of the form X vðx; tÞ ¼ Yi ðxÞqi ðtÞ ð4:44Þ where qi ðtÞ are the generalized coordinates of the forced response solution (which are generally different from those for the free solution; i.e., qi ðtÞ). Substitute the solution (Equation 4.44) into the system Equation 4.2 and make use of the governing equation of the mode shapes (see Equation 4.10) d2 Yi ðxÞ ¼ dx2 we obtain m

X Yi ðxÞq€ i ðtÞ ¼

T

X

l2i Yi ðxÞ

l2i Yi ðxÞqi ðtÞ þ f0 sin vt dðx

ð4:45Þ

aÞ

ðiiÞ

Multiply Equation (ii) by Yj ðxÞ; and integrate from x ¼ 0 to l using the orthogonality property (Equation 4.26) and also Equation (i). We obtain l m q€ i ðtÞ ¼ 2

l T l2j qj ðtÞ þ f0 Yj ðaÞ sin vt 2


4 13

pffiffiffiffiffiffi Now since vj ¼ lj T=m (see Equation 4.19), we obtain q€ j ðtÞ þ v2j qj ðtÞ ¼

2f0 Y ðaÞ sin vt lm j

for j ¼ 1; 2; 3; …

ð4:46Þ

This has the familiar form of a simple oscillator excited by a harmonic force and its solution is well known. The initial conditions qj ð0Þ and q_ j ð0Þ are needed. Suppose that the initial transverse displacement and the speed of the cable are vðx; 0Þ ¼ dðxÞ

v_ ðx; 0Þ ¼ sðxÞ

and

Then, in view of Equation 4.45, we can write X

Yi ðxÞqi ð0Þ ¼ dðxÞ X Yi ðxÞq_ i ð0Þ ¼ sðxÞ

ð4:47Þ ð4:48Þ

Multiply Equation 4.47 and Equation 4.48 by Yj ðxÞ; and integrate from x ¼ 0 to l using the orthogonality property 4.26. We obtain the necessary initial conditions 2 ðl dðxÞYj ðxÞdx l 0 2 ðl q_ j ð0Þ ¼ sðxÞYj ðxÞdx l 0

qj ð0Þ ¼

ð4:49Þ ð4:50Þ

which will provide the complete solution for Equation 4.46 and hence will completely determine Equation 4.44. For a fixed ended cable, we have Yi ðxÞ ¼ sin

ipx l

ðiiiÞ

and, at steady state, the time response qj ðtÞ will be harmonic at the same frequency as the excitation frequency v: Hence, we have qj ðtÞ ¼ q0j sinðvt þ fj Þ

ð4:51Þ

We see that, for Equation 4.51 to satisfy Equation 4.46 in this undamped problem, we must have fj ¼ 0: Direct substitution gives ½ v2 þ v2j q0j ¼

2f0 Y ðaÞ lm j

which determines q0j : Hence, from Equation 4.45, the complete solution for the fixed ended problem, at steady state, is vðx; tÞ ¼

X sin ipa=l 2f0 ipx sin vt sin lm ðv2i v2 Þ l

ð4:52Þ

Some important results for transverse vibration of strings and cables are summarized in Box 4.1.

4.3

Longitudinal Vibrations of Rods

It can be shown that the governing equation of the longitudinal vibration of line structures such as rods and bars is identical to that of the transverse vibration of cables and strings. Hence, it is not necessary to repeat the complete analysis here. We will first develop the equation of motion, then consider BCs, next identify the similarity with the cable vibration problem, and will conclude with an illustrative example.

4 14


Box 4.1 TRANSVERSE VIBRATION CABLES

OF

STRINGS

Equation of motion: mðxÞ

›2 vðx; tÞ ›2 vðx; tÞ ¼ T þ f ðx; tÞ ›t 2 ›x2

Separable (modal) solution for free vibration: vðx; tÞ ¼

X Yi ðxÞqi ðtÞ

with d2 Yi ðxÞ þ l2i Yi ðxÞ ¼ 0 ðneeds two boundary conditionsÞ dx2 and d2 qi ðtÞ þ v2i qi ðtÞ ¼ 0 ðneeds two initial conditionsÞ dt 2 Natural frequency: vi ¼ li c rffiffiffiffi T Wave speed: c ¼ m Traveling-wave solution (long cable, independent of end conditions): vðx; tÞ ¼ v1 ðx ctÞ þ v2 ðx þ ctÞ Orthogonality:

8 > 0 : 2

ðl

for i – j for i ¼ j

Initial conditions: (for initial displacement dðxÞ and speed sðxÞ) 2 ðl qi ð0Þ ¼ dðxÞYi ðxÞdx l 0 2 ðl q_ i ð0Þ ¼ sðxÞYi ðxÞdx l 0 Variable-tension problem: m

›2 v ›2 v ›T ›v ¼T 2 þ 2 ›t ›x ›x ›x

AND


4.3.1

4 15

Equation of Motion

Consider a rod that is mounted horizontally (so that the gravitational effects can be neglected) as shown in Figure 4.4(a). A small element of length dx (the limiting case of dx) at position x is shown in Figure 4.4(b). The longitudinal strain at x is given by 1¼

›u ›x

ð4:53Þ

where uðx; tÞ ¼ longitudinal displacement of the rod at x from a fixed reference. Note that the fixed reference may be chosen arbitrarily but, if the assumption of small u is needed, the relaxed (unstrained) position of the element must be chosen as the reference. The longitudinal stress at the cross section at x is s ¼ E1 and, hence, the longitudinal force is P ¼ EA

›u ›x

ð4:54Þ

where E ¼ Young’s modulus of the rod A ¼ area of cross section It is not necessary at this point to assume a uniform rod. Hence, A may depend on x: The equation of motion for the small element shown in Figure 4.4(b) is

rA dx

›2 uðx; tÞ ¼ P þ dP ›t 2

P þ f ðx; tÞdx

or

rA

›2 u dx ¼ dP þ f ðx; tÞdx ›t 2

ð4:55Þ

Now, from Equation 4.54, we have dP ¼

› ›u EAðxÞ dx ›x ›x

ð4:56Þ

Force per unit length = f (x,t)

0

x x

Mass density = r

(a)

l x+dx

u(x,t) 0

P+dP

P

x x x+dx

(b) FIGURE 4.4

Area of cross section = A(x)

(a) A rod with distributed loading and in longitudinal vibration; (b) a small element of the rod.

4 16


which when substituted into Equation 4.55 gives

rA

›2 uðx; tÞ › ›uðx; tÞ ¼ EAðxÞ þ f ðx; tÞ 2 ›t ›x ›x

ð4:57Þ

For the case of a uniform rod (constant A) in free vibration ðf ðx; tÞ ¼ 0Þ; we have 2 ›2 u 2 › uðx; tÞ ¼ c ›t 2 ›x2

ð4:58Þ

which is identical to the cable vibration equation 4.3, but with the wave speed parameter given by sffiffiffi E c¼ ð4:59Þ r which should be compared with Equation 4.4. The analysis of the present problem may be carried out exactly as for the cable vibration. In particular, the traveling wave solution will hold. Mode shape orthogonality will hold also. Even the BCs are similar to those of the cable vibration problem.

4.3.2

Boundary Conditions

As for the cable vibration problem, two BCs will be needed along with two initial conditions in order to obtain the complete solution to the longitudinal vibration of a rod. Both free and forced vibration may be analyzed as before. For a fixed end at x ¼ x0 ; we will have no deflection. Hence, uðx0 ; tÞ ¼ 0

ð4:60Þ

with the corresponding modal end condition Xi ðx0 Þ ¼ 0

for i ¼ 1; 2; 3; …

ð4:61Þ

For a free end at x ¼ x0 ; there will not be an end force. Hence, in view of Equation 4.54, the applicable BC will be

›uðx0 ; tÞ ¼0 ›x

ð4:62Þ

with the corresponding modal boundary condition dXi ðx0 Þ ¼0 dx

for i ¼ 1; 2; 3; …

The mode shapes Xi ðxÞ will satisfy the orthogonality property ( ðl 0 for i – j Xi ðxÞXj ðxÞdx ¼ 0 lj for i ¼ j

ð4:63Þ

ð4:64Þ

as before. It can be easily verified, for example, that, for a rod with both ends fixed Xi ðxÞ ¼ sin

ipx l

ð4:65Þ

Example 4.3 A uniform structural column of length l; mass M; and area of cross section A hangs from a rigid platform and is supported on a flexible base of stiffness k: A model is shown in Figure 4.5. Initially, the system remains stationary, in static equilibrium. Suddenly an axial (vertical) speed of u0 is imparted uniformly


4 17

on the entire column due to a seismic jolt. Determine the subsequent vibration motion of the column from its initial equilibrium configuration.

0

Solution

x

The gravitational force corresponds to a force per unit length f ðx; tÞ ¼

Mg l

and Equation 4.57 becomes 2

l

2

Mg › uðx; tÞ › uðx; tÞ ¼ c2 þ ›t 2 ›x2 rAl

k

Since M ¼ rAl, we have 2 ›2 uðx; tÞ 2 › uðx; tÞ ¼ c þg ›t 2 ›x2

ð4:66Þ

Boundary conditions are uð0; tÞ ¼ 0

FIGURE 4.5 A column suspended from a fixed plat form and supported on a flexible base.

ð4:67Þ EA

›uðl; tÞ þ kuðl; tÞ ¼ 0 ›x

ð4:68Þ

Initial conditions are uðx; 0Þ ¼ 0

ð4:69Þ

›uðx; 0Þ ¼ u0 ›t

ð4:70Þ

We seek the modal summation solution uðx; tÞ ¼

X Xi ðxÞqi ðtÞ

ð4:71Þ

where the mode shapes Xi ðxÞ satisfy d2 Xi ðxÞ þ l2i Xi ðxÞ ¼ 0 dx2

ð4:72Þ

Xi ðxÞ ¼ C1 sin li x þ C2 cos li x

ð4:73Þ

whose solution is

According to Equation 4.67 and Equation 4.68, the modal BCs are Xi ð0Þ ¼ 0 EA

dXi ðlÞ þ kXi ðlÞ dx

ð4:74Þ ð4:75Þ

Substitute Equation 4.74 into Equation 4.73. We have C2 ¼ 0: Next, use Equation 4.75. We obtain EAli C1 cos li l þ kC1 sin li l ¼ 0 Since, C1 – 0 for a nontrivial solution, the required condition is EAli cos li l þ k sin li l ¼ 0

4 18


which may be expressed as tan li l þ

EA l ¼0 k i

ð4:76Þ

This transcendental equation has an infinite number of solutions li ; which correspond to the modes of vibration. The solution may be made computationally and the corresponding natural frequencies are obtained using sffiffiffiffiffiffi sffiffiffi E EAl ð4:77Þ ¼ li vi ¼ li c ¼ li M r Substitute Equation 4.71 into Equation 4.66 and use Equation 4.72 to obtain X X Xi ðxÞ€qi ðtÞ ¼ c2 l2i Xi ðxÞqi ðtÞ þ g

ð4:78Þ

Multiply Equation 4.78 by Xj ðxÞ and integrate from x ¼ 0 to l; using the orthogonality property (Equation 4.64), to obtain lj q€ j ðtÞ þ c2 l2j lj qj ðtÞ ¼ g

ðl 0

Xj ðxÞdx

ð4:79Þ

We normalize the mode shapes as Xi ðxÞ ¼ sin li x

ð4:80Þ

where the constant multiplier ðC1 Þ has been absorbed into qi ðtÞ in Equation 4.71. Then, " #l " # ðl 1 ðl 1 1 1 1 2 ½1 cos 2lj x dx ¼ x l sin 2lj x ¼ sin 2lj l ð4:81Þ lj ¼ sin lj x dx ¼ 2 2lj 2 2lj 0 2 0 0 and ðl 0

sin lj x dx ¼

1 ½1 lj

cos lj l

Accordingly, Equation 4.79 becomes q€ j ðtÞ þ v2j qj ðtÞ ¼

g ½1 lj lj

cos lj l

ð4:82Þ

where the right hand side is a constant and is completely known from Equation 4.81 and Equation 4.76, and vj is given by Equation 4.77. Now Equation 4.82, which corresponds to a simple oscillator with a constant force input, may be solved using any convenient approach. For example, the particular solution is qjp ¼

g ½1 v2j lj lj

cos lj l

ð4:83Þ

and the overall solution is qj ðtÞ ¼ Aj sin vj t þ Bj cos vj t þ qjp

ð4:84Þ

The constants Aj and Bj are determined using the initial conditions qj ð0Þ and q_ j ð0Þ: These are obtained by substituting Equation 4.71 into Equation 4.67 and Equation 4.68, multiplying by Xj ðxÞ; and integrating from x ¼ 0 to l; making use of the orthogonality property (Equation 4.64). Specifically, we obtain qj ð0Þ ¼ 0

ð4:85Þ


4 19

and q_ j ð0Þ ¼

4.4

u0 ðl u sin lj x dx ¼ 0 ½1 lj 0 lj lj

cos lj l

ð4:86Þ

Torsional Vibration of Shafts

Torsional vibrations are oscillating angular motions of a device about some axis of rotation. Examples are vibration in shafts, rotors, vanes, and propellers. The governing PDE of the torsional vibration of a shaft is quite similar to that we previously encountered of the transverse vibration of a cable in tension and the longitudinal vibration of a rod. However, in the present case, the vibrations are rotating (angular) motions with resulting shear strains, shear stresses, and torques in the torsional member. Furthermore, the parameters of the equation of motion will take different meanings. When bending and torsional motions occur simultaneously, there can be some interaction, thereby making the analysis more difficult. Here, we neglect such interactions by assuming that only the torsional effects are present or that the motions are quite small. Since the form of the torsional vibration equation is similar to forms we have studied before, the same procedures of analysis may be employed and, in particular, the concepts of modal analysis will be similar. However, the torsional parameters will be rather complex for members with noncircular cross sections. Nevertheless, vast majority of torsional devices have circular cross sections.

4.4.1

Shaft with Circular Cross Section

Here, we will formulate the problem of the torsional vibration of a shaft having a circular cross section. The general case of a nonuniform cross section along the shaft is considered, but the usual assumptions such as homogeneous, isotropic, and elastic material are made. First, we will obtain a relationship between torque ðTÞ and angular deformation or twist ðuÞ for a circular shaft. Consider a small element of length dx along the shaft axis and the cylindrical surface at a general radius r (in the interior of the shaft segment), as shown in Figure 4.6(a). During vibration, this element will deform (twist) through a small angle du: A point on the circumference will deform through r du as a result, and a longitudinal line on the cylindrical surface will deform through angle g; as shown in Figure 4.6(a). From the strength of materials and elasticity theory of solid mechanics, we know that g is the shear strain. Hence, Shear strain g ¼

r du dx

r+dr r T

(a)

γ

r dθ

dx

Shear stress τ

T+dT

(b)

FIGURE 4.6 (a) Small element of a circular shaft in torsion; (b) shear stresses in a small annular cross section carrying torque.

4 20


However, allowing for the fact that the angular shift u is a function of t as well as x in the general case of dynamics, we use partial derivatives and write

g¼r

›u ›x

ð4:87Þ

The corresponding shear stress at the deformed point at radius r is

t ¼ Gg ¼ Gr

›u ›x

ð4:88Þ

where G ¼ shear modulus. This shear stress acts tangentially. Consider a small annular cross section of width dr at radius r of the shaft, as shown in Figure 4.6(b). By symmetry, the shear stress will be the same throughout this region and will form a torque of rt £ 2pr dr ¼ 2pr2 t dr: Hence, the overall torque at the shaft cross section is ð T ¼ 2pr 2 t dr which, in view of Equation 4.88, is written as T¼G

›u ð 2pr3 dr ›x

ð4:89Þ

It is clear that the integral term is the polar moment of area of the shaft cross section: ð J ¼ 2pr 3 dr

ð4:90Þ

In particular, for a solid shaft of radius r J¼

pr4 2

ð4:91Þ

and, for a hollow shaft of inner radius r1 and the outer radius r2 p J ¼ ðr24 r14 Þ 2

ð4:92Þ

So, we write Equation 4.89 as T ¼ GJðxÞ

›u ›x

ð4:93Þ

The combined parameter GJ is termed the torsional rigidity of the shaft. We have emphasized that the shaft may be nonuniform and hence J is a function of x: Consider a uniform shaft segment of length l; with associated overall angular deformation u: Equation 4.93 can be written as Torsional stiffness K ¼

T GJ ¼ l u

ð4:94Þ

Note: For a shaft with noncircular cross section, replace J by Jt in this equation. It follows that, the larger the torsional rigidity GJ; the higher the torsional stiffness K; as expected. Furthermore, longer members have a lower torsional stiffness (and smaller natural frequencies). Now, we apply Newton’s Second Law for rotatory motion of the Ð Ð small element dx; Ð shown in Figure 4.6(a). The polar moment of inertia of the element is r 2 dm ¼ r2 r dx dA ¼ r dx r2 dA ¼ rJ dx; where J is the polar moment of area, as discussed before. Also, suppose that a distributed external torque of tðx; tÞ per unit length is applied along the shaft. Hence, the equation of motion is

rJ dx

›2 u ¼ T þ dT ›t 2

T þ tðx; tÞdx ¼

›T dx þ tðx; tÞdx ›x


4 21

Substitute Equation 4.93 and cancel dx to obtain the equation of torsional vibration of a circular shaft as

rJ

›2 uðx; tÞ › ›uðx; tÞ ¼ GJðxÞ þ tðx; tÞ ›t 2 ›x ›x

ð4:95Þ

For the case of a uniform shaft (constant J) in free vibration ðtðx; tÞ ¼ 0Þ; we have 2 ›2 uðx; tÞ 2 › uðx; tÞ ¼ c ›t 2 ›x2

with

sffiffiffiffi G c¼ r

ð4:96Þ

ð4:97Þ

Note that Equation 4.96 is quite similar to that for transverse vibration of a cable in tension and the longitudinal vibration of a rod. Hence, the same concepts and procedures of analysis may be used. In particular, two boundaries conditions will be needed in the solution; for example Fixed end at x ¼ x0: uðx0 ; tÞ ¼ 0

ð4:98Þ

›uðx0 ; tÞ ¼0 ›x

ð4:99Þ

Free end at x ¼ x0: The orthogonality property of mode shapes Qi ðxÞ: ðl 0

4.4.2

(

Qi ðxÞQj ðxÞdx ¼

0 for i – j lj

for i ¼ j

ð4:100Þ

Torsional Vibration of Noncircular Shafts

Unlike the longitudinal and transverse vibrations of rods and beams, when considering the torsional vibration of shafts, the equation of motion for circular shafts (Equation 4.95 and Equation 4.96) cannot be used for shafts with noncircular cross sections. The reason is that the shear stress distributions in the two cases can be quite different, and Equation 4.88 does not hold for noncircular sections. Hence, the parameter J in the torque deflection relations (e.g., Equation 4.93 and Equation 4.94) is not the polar moment of area in the case of noncircular sections. In the noncircular case, we write T ¼ GJt

›u ›x

ð4:101Þ

where Jt ¼ torsional parameter. The Saint Venant theory of torsion and the related membrane analogy, developed by Prandtl, have provided equations for Jt in special cases. For example, for a thin hollow section Jt ¼

4tA2s p

ð4:102Þ

where As ¼ enclosed (contained) area of the hollow section p ¼ perimeter of the section t ¼ wall thickness of the section For a thin, solid section, we have

Jt ¼

t3a 3

ð4:103Þ

4 22


where a ¼ length of the narrow section t ¼ thickness of the narrow section Torsional parameters for some useful sections are given in Table 4.2.

TABLE 4.2

Torsional Parameters for Several Sections

Section

Shape

Torsional Parameter Jt

Solid circular

p 4 r 2

r

p 4 ðr 2 r14 Þ 2 2

Hollow circular

r2

r1

4tA2s p

Thin closed

p

As t

Thin open

t3a 3

t

a 0:1406a4

Solid square

a a 0:1406ða42 2 a41 Þ

Hollow square

a1 a1 a2

a2


4 23

Example 4.4 Consider a thin, rectangular hollow section of thickness t; height a; and width a=2; as shown in Figure 4.7(a). Suppose that the section is opened by making a small slit as in Figure 4.7(b). Study the affect on the torsional parameter Jt and torsional stiffness K of the member due to the opening. Solution (a) Closed section: The contained area of the section is As ¼ a2 =2 The perimeter of the section p ¼ 3a Using Equation 4.102, the torsional parameter is Jtc ¼

ta3 3

(b) Open section: The solid length of the section ¼ 3a Using Equation 4.103, the torsional parameter is

Jt0 ¼ at 3 The ratio of the torsional parameters is Jt0 3t 2 ¼ 2 Jtc a For members of equal length, torsional stiffness will also be in the same ratio as is given by this expression. Since t is small compared with a; there will be a significant drop in torsional stiffness due to the opening (cutout).

Example 4.5 An innovative automated transit system uses an elevated guideway with cars whose suspension is attached to (and slides on) the side of the guideway. Owing to this eccentric loading on the guideway, a/2

t a

(a)

(b) FIGURE 4.7

(a) A thin closed section; (b) a thin open section.

4 24


xj vj

Tj

Car Guideway Support Pier

l FIGURE 4.8

Support Pier

A torsional guideway transit system.

there is a significant component of torsional dynamics in addition to bending. Assume that the torque Tj acting on the guideway due to the jth suspension of the vehicle to be constant, acting at a point xj as measured from one support pier, and moving at speed vj : A schematic representation is given in Figure 4.8. The guideway span shown has a length L and a cross section which is a thin walled rectangular box of height a; width b; and thickness t: The ends of the guideway span are restrained for angular motion (i.e., fixed): (a) Formulate and analyze the torsional (angular) motion of the guideway. (b) For a single point vehicle entering a guideway that is at rest, what is the resulting dynamic response of the guideway? What is the critical speed that should be avoided? (c) Given the parameter values l ¼ 60 ft ð18:3 mÞ abt ¼ 5 ft £ 2:2 ft £

1 ft ð1:52 m £ 0:67 m £ 0:15 mÞ 2

r ¼ 4:66 slugs=ft3 ð2:4 £ 103 kg=m3 Þ G ¼ 1:55 £ 106 lb=in:2 ð1:07 £ 1010 N=m2 Þ and vehicle speed v ¼ 60 mi=h ð26:8 m=secÞ Compute the crossing frequency ratio given by vc ¼

Rate of span crossing Fundamental natural frequency of guideway

and discuss its implications.

Solution (a) For a uniform guideway with distributed torque load tðx; tÞ and a noncircular cross section having torsional parameter Jt ; the governing equation is rJ

›2 uðx; tÞ ›2 u ¼ GJ þ tðx; tÞ t ›t 2 ›x2

ð4:104Þ

As usual, the mode shapes are obtained by solving d2 Q þ l2 Q ¼ 0 dx2 and the corresponding natural frequencies are given by pffiffiffiffiffiffiffiffi vi ¼ li GJ=rJt

ð4:105Þ

ð4:106Þ


4 25

The general solution of Equation 4.105 is

QðxÞ ¼ A1 sin lx þ A2 cos lx where Ai ði ¼ 1; 2Þ are the constants of integration. The torsional BCs corresponding to the fixed ends (no twist) are

Qð0Þ ¼ QðlÞ ¼ 0 where l ¼ guideway span length. For a nontrivial solution, we need A2 ¼ 0 and

li ¼

p l

i ¼ 1; 2; …

ð4:107Þ

The solution corresponds to an infinite set of eigenfunctions Qi ðxÞ satisfying Equation 4.105. Each of these represents a natural mode in which the beam can undergo free torsional vibrations. The actual motion consists of a linear combination of the normal modes depending on beam initial conditions and the forcing term tðx; tÞ: The integration constant A1 may be incorporated (partially) into the generalized coordinate q; which is still unknown and is determined through initial conditions. Here, we use the normalized eigenfunctions pffiffi ipx Qi ðxÞ ¼ 2 sin l The orthogonality condition given by 1 ðl Q Q dx ¼ l 0 i j

i ¼ 1; 2; … (

0

for i – j

1

for i ¼ j

ð4:108Þ

ð4:109Þ

is satisfied. In view of the relations (Equation 4.106 and Equation 4.107) the natural frequencies corresponding to different eigenfunctions (natural mode shapes) are

vi ¼

ip pffiffiffiffiffiffiffiffi GJ=rJt l

i ¼ 1; 2; …

ð4:110Þ

For n number of vehicle’s suspensions located on the analyzed span

tðx; tÞ ¼

n X j¼1

Tj dðx

x0j

vj tÞ

ð4:111Þ

where Tj ¼ torque exerted on guideway by the jth suspension vj ¼ speed of the jth suspension x0j ¼ initial position (at t ¼ 0) along the guideway of the jth suspension dð·Þ ¼ Dirac delta function The forced motion can be represented in terms of the normalized eigenfunction as

uðx; tÞ ¼

1 X i¼1

qi ðtÞQi ðxÞ

ð4:112Þ

where qi ðtÞ ¼ the generalized coordinate for forced motion in the ith mode. On substituting the relations (Equation 4.111 and Equation 4.112) into Equation 4.104, and integrating the result over the span length, after multiplying by a general eigenfunction while making use of the

4 26


orthogonality relation (Equation 4.109), one obtains n d2 q i 1 X þ v2i qi ¼ T Q ðx þ vj tÞ 2 dt rJt l j¼1 j i 0j

i ¼ 1; 2; …

ð4:113Þ

(b) For a single suspension entering the guideway at t ¼ 0; with the guideway initially at rest ðqi ð0Þ ¼ q_ i ð0Þ ¼ 0Þ; we have n ¼ 1 and x01 ¼ 0: Then, the complete solution of Equation 4.113 is pffiffi ipvt 2lT sin 2vc sin vi t l qi ðtÞ ¼ GJt p2 i2 ð1 vC2 Þ

i ¼ 1; 2; …

where the crossing frequency ratio vC is given by v vC ¼ lv1

ð4:114Þ

ð4:115Þ

Note from Equation 4.114 that the critical speed corresponds to vC ¼ 1 and should be avoided. In typical transit systems, vC is considerably less than 1. (c) For the given numerical values, by straightforward computations, it can be shown that Jt ¼ 3:485 £ 105 in:4 ð1:451 £ 107 cm4 Þ J ¼ 4:574 £ 105 in:4 ð2:736 £ 107 cm4 Þ

v1 ¼ 263:8 rad=sec ¼ 42:0 Hz Note: We used the expression for a thin, hollow section, given in Table 4.2, in computing Jt : Finally, the crossing frequency ratio is computed to be vC ¼ 0:017, which is much less than 1.0, as expected.

4.5

Flexural Vibration of Beams

In this section, we will study a beam (or rod or shaft) in flexural vibration. The vibration is in the transverse or lateral direction, which is accompanied by bending (or flexure) of the member. Hence, the vibrations are perpendicular to the main axis of the member, as in the case of a cable or string, which we studied in Section 4.2. However, a beam, unlike a string, can support shear forces and bending moments at its cross section. In the initial analysis of bending vibration, we will assume that there is no axial force at the ends of the beam. We will make further simplifying assumptions that will be clear in the development of the governing equation of motion. The analysis procedure will be quite similar to that we have followed in the previous sections. The study of the bending vibration (or lateral or transverse vibration) of beams is very important in a variety of practical situations. Noteworthy are the vibration analyses of structures like bridges, vehicle guideways, tall buildings, and space stations; the ride quality and structural integrity analysis of buses, trains, ships, aircraft and spacecraft; the dynamics and control of rockets, missiles, machine tools and robots; and the vibration testing, evaluation, and qualification of products with continuous members.

4.5.1

Governing Equation for Thin Beams

Now, we will develop the Bernoulli Euler equation, which governs the transverse vibration of thin beams. Consider a beam bending in the x y plane, with x as the longitudinal axis and y as the transverse axis of bending deflection, as shown in Figure 4.9. We will develop the required equation by considering the moment deflection relation, rotational equilibrium, and transverse dynamics of a beam element.


4 27

δx

x

δx

δA

w B

A

σ Neutral Axis

w

R y δθ

FIGURE 4.9

A thin beam in bending.

(1) Moment deflection relation A small beam element of length dx subjected to bending moment M is shown. Neglect any transverse deflections due to shear stresses. Consider a lateral area element dA in the cross section A of the beam element, at a distance w (measure parallel to y) from the neutral axis of bending. Normal strain (at dA) 1¼

ðR þ wÞdu R du R du

Note that the neutral axis joins the points along the beam when the normal strain and stress are zero. Hence, 1¼

w R

ð4:116Þ

where R ¼ radius of curvature of the bent element. Normal stress in the axial direction

s ¼ E1 ¼ E

w R

ð4:117Þ

where E ¼ Young’s modulus (of elasticity). Then, bending moment ð ð E E ð 2 EI w dA ¼ M¼ ws dA ¼ w2 dA ¼ R R R A where I ¼ second moment of area of the beam cross section about the neutral axis. So, we have M¼

EI R

ð4:118Þ

Slope at A ¼ ›v=›x; slope at B ¼ ð›v=›xÞ þ ð›2 v=›x2 Þdx; where v ¼ lateral deflection of the beam at element dx: Hence, the change in slope ¼ ð›2 v=›x2 Þdx ¼ du; where du is the arc angle of bending for the beam element dx; as shown in Figure 4.9.

4 28


Also, we have dx ¼ R du: Hence, ð›2 v=›x2 ÞR du ¼ du: Cancel du: We obtain

›2 v 1 ¼ 2 ›x R

ð4:119Þ

Substitute Equation 4.119 into Equation 4.118. We obtain M ¼ EI

›2 v ›x 2

ð4:120Þ

(2) Rotatory dynamics (equilibrium) Again, consider the beam element dx; as shown in Figure 4.10, where forces and moments acting on the element are indicated. Here, f ðx; tÞ ¼ excitation force per unit length acting on the beam, in the transverse direction, at location x: Disregard the rotatory inertia of the beam element. Hence, the equation of angular motion is given by the equilibrium condition of moments: M þ Q dx

Mþ

›M ›x ¼ 0 ›x

or

›M › ›2 v Q¼ ¼ EI 2 ›x ›x ›x

! ð4:121Þ

where the previously obtained result (Equation 4.120) for M has been substituted. Note that we have not assumed a uniform beam and hence I ¼ IðxÞ will be variable along the beam length. (3) Transverse dynamics The equation of transverse motion (Newton’s Second Law) for element dx is ðrA dxÞ

›2 v ¼ f ðx; tÞdx þ Q ›t 2

Qþ

›Q dx ›x

Here, r ¼ mass density of the beam material. So, we obtain

rA

›2 v ›Q þ ¼ f ðx; tÞ ›t 2 ›x

or, in view of Equation 4.121, we have the governing equation of forced transverse motion for the beam as ! ›2 v ›2 ›2 v ð4:122Þ rA 2 þ 2 EI 2 ¼ f ðx; tÞ ›t ›x ›x

x + δx

x

x f (x,t)δx M+

M Q

y FIGURE 4.10

Q+

∂Q ∂x

∂M δx ∂x δx

Dynamics of a beam element in bending.


4.5.2

4 29

Modal Analysis

The solution to the flexural vibration problem given by Equation 4.122 may be obtained exactly as for other continuous members. Specifically, we first obtain the natural frequencies and mode shapes and express the general solution as a summation of the modal responses. The approach is similar for both free and forced problems, but the associated generalized coordinates will be different. This approach is followed here. For modal (natural) vibration, consider the free motion described by ! ›2 ›2 v ›2 v EI þ ¼0 ð4:123Þ r A ›x 2 ›x 2 ›t 2 For a uniform beam, EI will be constant and Equation 4.123 can be expressed as

›2 vðx; tÞ ¼ ›t 2 where

c2

›4 vðx; tÞ ›x4

sffiffiffiffiffi EI c¼ rA

ð4:124Þ

ð4:125Þ

Observe from Equation 4.124 that it is fourth order in x and second order in t; whereas the governing equations for the transverse vibration of a cable, longitudinal vibration of a rod, and torsional vibration of a shaft, are all identical in fourth and second order in x: So the behavior of transverse vibrations of beams will not be exactly identical to that of these other three types of continuous system. In particular, the traveling wave solution 4.7 will not be satisfied. However, there are also many similarities. In each mode the system will vibrate in a fixed shape ratio. Hence, the time and space functions will be separable for a modal motion; we seek a solution of the form vðx; tÞ ¼ YðxÞqðtÞ

ð4:126Þ

This separable solution for a modal response has been justified previously. Note that, even in the lumped parameter case, we make the same assumption, except in that case we have a modal vector 2 3 Y1 6 7 6Y 7 6 27 6 7 Y¼6 . 7 6 . 7 6 . 7 4 5 Yn instead of a mode shape function YðxÞ: For a given mode of a lumped parameter system, Yi values denote the relative displacements of various inertia elements mi ; as shown in Figure 4.11. Hence, the vector Y corresponds to the mode shape. Note that Yi can be either positive or negative. Also, qðtÞ is the harmonic function corresponding to the natural frequency. It should be clear that Y and qðtÞ are separable in this lumped parameter case of modal motion. Then, in the limit, YðxÞ and qðtÞ should also be separable for the distributed parameter case. Substitute Equation 4.124 into Equation 4.123, and bring the terms containing x to the left hand side (LHS) and terms containing t to the right m1 m2 m3 m4 hand side (RHS). ! 1 d2 d2 Y 1 d2 q ¼ ¼ v2 2 EI 2 dx qðtÞ dt 2 rAY dx ð4:127Þ Y1q(t) Y2q(t) Y3q(t) Y4q(t) Since a function of x cannot be equal to a function of t in general, unless each function is equal to the FIGURE 4.11 Modal motions of a lumped parameter system. same constant, we have defined v2 as a constant.

4 30


We have not shown that this separation constant ðv2 Þ should be positive. This requirement can be verified due to the nature of the particular vibration problem; that is, qðtÞ should have an oscillatory solution in general. It is also clear that the physical interpretation of v is a natural frequency of the system. Equation 4.127 corresponds to the two ODEs, one in t and the other in x; as d2 qðtÞ þ v2 qðtÞ ¼ 0 dt 2 d2 d2 YðxÞ EI dx2 dx2

ð4:128Þ

v2 rAYðxÞ ¼ 0

ð4:129Þ

The solution of these two equations will provide the natural frequencies v and the corresponding mode shapes YðxÞ of the beam. For further analysis of the modal behavior, assume a uniform beam. Then, EI will be constant and Equation 4.129 may be expressed as d4 YðxÞ dx4 where

l4 YðxÞ ¼ 0

ð4:130Þ

sffiffiffiffiffi EI v¼l c¼l rA 2

2

ð4:131Þ

The positive parameter l is yet to be determined, and will come from the mode shape analysis. The characteristic equation corresponding to Equation 4.130 is p4

l4 ¼ 0;

ðp2

or;

l2 Þðp2 þ l2 Þ ¼ 0

ð4:132Þ

The roots are p ¼ ^l; ^jl

ð4:133Þ

Hence, the general solution for a mode shape (eigenfunction) is given by YðxÞ ¼ A1 elx þ A2 e

lx

A3 eþjlx þ A4 e

jlx

¼ C1 cosh lx þ C2 sinh lx þ C3 cos lx þ C4 sin lx ð4:134Þ

There are five unknowns (C1 ; C2 ; C3 ; C4 ; and l) here. The mode shapes can be normalized and one of the first four unknowns can be incorporated into qðtÞ as usual. The remaining four unknowns are determined by the end conditions of the beam. So, four BCs will be needed. Note: cosh lx ¼ cos lx ¼

e

elx þ e 2

lx

jlx

jlx

þe 2

; ;

d cosh lx ¼ l sinh x; dx

4.5.3

sinh lx ¼ sin lx ¼

e

elx

e

lx

e

jlx

2 jlx

2j

d sinh x ¼ l cosh x dx

Boundary Conditions

The four modal BCs that are needed can be derived in the usual manner, depending on the conditions at the two ends of the beam. The procedure is to apply the separable (modal) solution Equation 4.126 to the end relation with the understanding that this relation has to be true for all possible values of qðtÞ: The relation (Equation 4.128) may be substituted as well, if needed.


4 31

For example, consider an end x ¼ x0 that is completely free. Then, both bending moment and shear force have to be zero at this end. From Equation 4.120 and Equation 4.121, we have

›2 vðx0 ; tÞ ¼0 ›x2 ! › ›2 vðx0 ; tÞ EI ¼0 ›x 2 ›x EI

ð4:135Þ ð4:136Þ

Substitute Equation 4.126 into Equation 4.135 and Equation 4.136: EI

d2 Yðx0 Þ qðtÞ ¼ 0 dx2

d d2 Yðx0 Þ EI qðtÞ ¼ 0 dx dx2 which are true for all qðtÞ: Hence, the following modal BCs result for a free end: d2 Yðx0 Þ ¼0 dx2

ð4:137Þ

d d2 Yðx0 Þ EI ¼0 dx dx2

ð4:138Þ

For a uniform beam, Equation 4.137 becomes d3 Yðx0 Þ ¼0 dx3 Some common conditions and the corresponding modal BC equations for the bending vibration of a beam are listed in Box 4.2.

4.5.4

Free Vibration of a Simply Supported Beam

To illustrate this approach, consider a uniform beam of length l that is pinned (simply supported) at both ends. In this case, both displacement and the bending moment will be zero at each end. Accordingly, we have the modal boundary conditions Yð0Þ ¼ 0 ¼ YðlÞ

ð4:139Þ

d2 Yð0Þ d2 YðlÞ ¼0¼ 2 dx dx2

ð4:140Þ

where l ¼ length of the beam. Substitute Equation 4.134 into Equation 4.139: C1 þ C3 ¼ 0

ð4:141Þ

C1 cosh ll þ C2 sinh ll þ C3 cos ll þ C4 sin ll ¼ 0

ð4:142Þ

To apply the bending moment BCs, first differentiate Equation 4.134 to obtain dY ¼ lC1 sinh lx þ lC2 cosh lx dx d2 Y ¼ l2 C1 cosh lx þ l2 C2 sinh lx dx2

lC3 sin lx þ lC4 cos lx l2 C3 cos lx

l2 C4 sin lx

and the substitute these into the bending moment BCs (Equation 4.140). We obtain C1 C1 cosh ll þ C2 sinh ll

C3 ¼ 0 C3 cos ll

ð4:143Þ C4 sin ll ¼ 0

ð4:144Þ

4 32


Box 4.2 BOUNDARY CONDITIONS FOR TRANSVERSE VIBRATION OF BEAMS 1.

Simply supported (pinned)

Deflection ¼ 0 ) Y ¼ 0 Bending moment ¼ 0 )

2.

Deflection ¼ 0 ) Y ¼ 0

Clamped (fixed)

Slope ¼ 0 )

3.

Free

dY ¼0 dx

Bending moment ¼ 0 ) Shear force ¼ 0 )

4.

Sliding

Slope ¼ 0 )

Dynamic (flexible, inertial, etc.)

d2 Y ¼0 dx2

d d2 Y EI 2 ¼ 0 dx dx

dY ¼0 dx

Shear force ¼ 0 ) 5.

d2 Y ¼0 dx2

d d2 Y EI 2 ¼ 0 dx dx

Transverse equation of motion (or force balance) Substitute Equation 4.128, if needed Rotatory equation of motion (or moment balance)

as l – 0 in general, due to the oscillatory nature of most modes. Equation 4.141 and Equation 4.143 give C1 ¼ 0 ¼ C3 : Then, Equation 4.142 becomes: Equation 4.144 becomes: Add

C2 sinh ll þ C4 sin ll ¼ 0 C2 sinh ll

C4 sin ll ¼ 0

C2 sinh ll ¼ 0 However, sinh ll ¼ 0 if and only if l ¼ 0: This corresponds to zero frequency conditions (no oscillations), and is rejected as it is not true in general. Hence, we have C2 ¼ 0: Accordingly, we are left


4 33

with the remaining equation: C4 sin ll ¼ 0

ð4:145Þ

However, if C4 ¼ 0 then YðxÞ ¼ 0; which corresponds to a stationary beam with no oscillations, and is rejected as the trivial solution. Hence, the valid solution is given by sin ll ¼ 0 which gives the infinite set of solutions:

li l ¼ ip

for i ¼ 1; 2; 3; …

ð4:146Þ

Note that we must have i . 0 because l has to be nonzero, thereby giving nonzero natural frequencies according to Equation 4.131, as required for the given problem. 4.5.4.1

Normalization of Mode Shape Functions

For absorbing the yet unknown constant C4 into qðtÞ; we will normalize the mode shape functions. The commonly used normalization condition is ðl l Yi2 dx ¼ ð4:147Þ 2 0 Hence, ðl ðl l ipx ipx C2 ¼ C42 sin2 dx ¼ C42 sin2 dx ¼ 4 l 2 2 l l 0 0 Note that we used cos 2u ¼ 1 2 sin2 u prior to integration. Then, for normalized mode shape functions, we have C4 ¼ 1: Hence, the normalized eigenfunctions (mode shape functions) for various modes are given by Yi ðxÞ ¼ sin

ipx l

for i ¼ 1; 2; 3; …

ð4:148Þ

Using the result (Equation 4.146) in Equation 4.131, the natural frequencies of the ith mode are i2 p2 vi ¼ 2 l

sffiffiffiffiffi EI rA

for i ¼ 1; 2; 3; …

ð4:149Þ

In this manner, we have obtained an infinite set of mode shape functions Yi ðxÞ for a simply supported beam. Hence, according to the solution (Equation 4.126), we have a corresponding infinite set of generalized coordinates qi ðtÞ; i ¼ 1; 2; 3; …; which satisfy Equation 4.128. It follows that the overall response of the beam is X vðx; tÞ ¼ Yi ðxÞqi ðtÞ ð4:150Þ 4.5.4.2

Initial Conditions

We have yet to solve Equation 4.128 for determining qi ðtÞ: To do so, we need to know the initial conditions qi ð0Þ and q_ i ð0Þ: These are determined from the beam initial conditions of displacement and speed, which have to be known: vðx; 0Þ ¼ dðxÞ

ð4:151Þ

›vðx; 0Þ ¼ sðxÞ ›t

ð4:152Þ

Substitute Equation 4.150 into Equation 4.151 and Equation 4.152 to obtain X Yi ðxÞqi ð0Þ ¼ dðxÞ X Yi ðxÞ_qi ð0Þ ¼ sðxÞ

ð4:153Þ ð4:154Þ

4 34


Multiply by Yj ðxÞ and integrate from x ¼ 0 to l; using the orthogonality property of Yi ðxÞ ¼ sinðipx=lÞ; namely 8 > < 0 for i – j ðl ipx jpx ð4:155Þ sin sin dx ¼ l > l l 0 : for i ¼ j 2 We obtain qj ð0Þ ¼

2 ðl dðxÞYj ðxÞdx l 0

ð4:156Þ

q_ j ð0Þ ¼

2 ðl sðxÞYj ðxÞdx l 0

ð4:157Þ

In this manner, qi ðtÞ is completely determined for each vi by solving Equation 4.128, using the initial conditions (Equation 4.156 and Equation 4.157). Hence, the complete solution (Equation 4.150) is determined for the free bending vibration of a simply supported beam.

4.5.5

Orthogonality of Mode Shapes

We have seen that the mode shapes of simply supported beams in bending vibrations are orthogonal (see Equation 4.155). This property is not limited to simply supported beams but holds for most BCs, as we will show now. First, from integration by parts, twice, we have ðl 0

Yi

l

d4 Yj d3 Yj 4 dx ¼ Yi dx dx3

0

" ðl dY d3 Yj d3 Yj i dx ¼ Yi 3 3 dx 0 dx dx

dYi d2 Yj dx dx2

#l þ 0

ðl d2 Y d2 Yj i dx 2 dx2 0 dx

ð4:158Þ

Now consider two separate modes, i and j; which have the modal equations Mode i :

d4 Y i ¼ l4i Yi dx4

ðaÞ

Mode j :

d4 Yj ¼ l4j Yj dx4

ðbÞ

Multiply Equation (a) by Yj, multiply Equation (b) by Yi, integrate both with respect to x from 0 to l; make use of Equation 4.158 and subtract the second result from the first. We obtain ðl4i

l4j Þ

ðl 0

Yi Yj dx ¼

ðl "

0

¼ Yj

d4 Y Yj 4i dx d3 Yi dx3

! d4 Yj Yi 4 dx dx #l " dYj d2 Yi d3 Yj Yi 3 2 dx dx 0 dx

dYi d2 Yj dx dx2

#l

ð4:159Þ

0

Clearly, the two RHS terms are zero for typical BCs, such as pinned, fixed, free, and sliding. Now, since li – lj for i – j (unequal modes), we have ðl 0

Yi Yj dx ¼ 0

for i – j

lj

for i ¼ j

Note that normalized mode shape functions may be used here to obtain the constant lj ¼ l=2:

ð4:160Þ


4.5.5.1

4 35

Case of Variable Cross Section

The orthogonality of mode shapes holds for nonuniform beams as well. Here, EI is not constant. We use integration by parts: 2 3l 2 3l 2 2 ðl ðl 6 d 7 6 dYj d2 Yj d2 Yi 2 2 d dY 7 Yj 2 EI 2i dx ¼ 6 Yj dx EI ddxY2i 7 EI ddxY2i 5 þ EI 2 dx ð4:161Þ 4 4 5 dx dx dx |ffl{zffl} dx dx2 0 |fflffl{zfflffl} 0 Q

M

0

0

Again, the first two terms on the RHS are zero for typical BCs. Then, as before, we use the modal equations (Equation 4.129) for two different modes, i and j: d2 d2 Y i ¼ v2i rAYi 2 EI dx2 dx d2 Yj d2 EI ¼ v2j rAYj dx2 dx2 Multiply the first equation by Yj ; the second equation by Yi ; subtract the second result from the first, integrate the result form x ¼ 0 to l; and finally use Equation 4.161 to cancel the equal terms. We obtain "

ðl

d d2 Yi rAYi Yj dx Yj EI dx dx 0 " #l 2 2 dYj d Yi dYi d Yj þ EI 2 EI 2 ¼ 0 dx dx 0 dx dx

ðv2i

v2j Þ

d2 Yj d Yi EI dx dx

#l 0

ð4:162Þ

Now, as before, for common BCs, the second and third boundary terms in Equation 4.162 will vanish. Hence, after canceling the term v2i v2j ; which is – 0 i – j; we obtain the orthogonality condition for nonuniform beams as ðl 0

(

rAYi Yj dx ¼

0

for i – j

aj

for i ¼ j

ð4:163Þ

The general steps for the modal analysis of a distributed parameter vibrating system are summarized in Box 4.3.

4.5.6

Forced Bending Vibration

The equation of motion is

›2 ›2 v ›2 v ¼ f ðx; tÞ 2 EI 2 þ rA ›x ›x ›t 2

ð4:164Þ

Assume a separable, forced response: vðx; tÞ ¼

X i

qi ðtÞYi ðxÞ

ð4:165Þ

where qi ðtÞ are the generalized coordinates in the forced case. Substitute Equation 4.165 into Equation 4.164 in the beam equation: X i

qi ðtÞ

X d2 d2 Yi ðxÞ EI þ rA q€ i ðtÞYi ðxÞ ¼ f ðx; tÞ 2 2 dx dx i

4 36


Box 4.3 MODAL ANALYSIS OF CONTINUOUS SYSTEMS Equation of free (unforced) motion: Lðx; tÞvðx; tÞ ¼ 0

ðiÞ

where vðx; tÞ ¼ system response Lðx; tÞ ¼ partial differential operator in space ðxÞ and time ðtÞ Model solution: Assume a separable solution vðx; tÞ ¼ YðxÞqðtÞ

ðiiÞ

because a modal response is separable in time and space. Note: YðxÞ ¼ mode shape qðtÞ ¼ generalized coordinate for free response Note: For two and three dimensional space systems, time and space will still be separable for a modal response. However, the space function itself may not be separable along each coordinate direction. Steps: 1. Substitute Equation (ii) in Equation (i) and separate the space function (of x) and the time function (of t ), each of which should be equal to the same constant. 2. Solve the resulting ODE for YðxÞ using system boundary conditions. We obtain an infinite set of mode shapes Yi ðxÞ; up to one unknown (removed by normalization), and natural frequencies vi : 3. Solve the ODE for qðtÞ using system initial conditions to determine qi ðtÞ for mode i: (The orthogonality of Yi ðxÞ will be needed to establish the initial conditions for qi ðtÞ.) 4. Overall response X vðx; tÞ ¼ Yi ðxÞqi ðtÞ i

The first term on the LHS, on using the mode shape equation 4.129, becomes qi ðtÞrAv2i Yi ðxÞ: Multiply the result by Yj ðxÞ and integrate with respect to x½0; l : We obtain

v2j qj ðtÞ

ðl 0

rAYj2 ðxÞdx þ q€ j ðtÞ

ðl

ðl rAYj2 ðxÞdx ¼ Yj ðxÞf ðx; tÞdx 0 0 |fflfflfflfflfflffl ffl{zfflfflfflfflfflfflffl} fj ðtÞ

Each of the two integrals on the LHS evaluates aj according to Equation 4.163. Hence, q€ j ðtÞ þ v2j qj ðtÞ ¼

1 f ðtÞ aj j

for j ¼ 1; 2; 3; …

ð4:166Þ


4 37

We can then solve this equation to determine the generalized coordinates qj ðtÞ; using the knowledge of the forcing function fj ðtÞ and the initial conditions qj ð0Þ and ðdqj =dtÞð0Þ: Specifically, if the initial displacement and speed of the beam are given by Equation 4.151 and Equation 4.152, respectively, by following the procedure that was adopted to obtain the results (Equation 4.156 and Equation 4.157), we determine 1 ðl qj ð0Þ ¼ rAdðxÞYj ðxÞdx ð4:167Þ aj 0 dqj ð0Þ 1 ðl rAsðxÞYj ðxÞdx ð4:168Þ ¼ aj 0 dt Finally, we obtain the overall response of the forced system as X vðx; tÞ ¼ qj ðtÞYj ðxÞ

ð4:169Þ

i

The main steps in the forced response analysis are summarized in Box 4.4.

Box 4.4 FORCED RESPONSE SYSTEMS

OF

CONTINUOUS

Equation of forced motion: Lðx; tÞvðx; tÞ ¼ L1 ðx; tÞf ðx; tÞ

ðiÞ

where vðx; tÞ ¼ forced response of the system f ðx; tÞ ¼ distributed force per unit space L and L1 are partial differential operators in space and time Steps: 1. Substitute the modal expansion vðx; tÞ ¼

X Yi ðxÞqi ðtÞ

in Equation (i), where Yi ðxÞ ¼ mode shapes qi ðtÞ ¼ generalized coordinates for forced motion 2. Multiply by Yj ðxÞ and integrate with respect to space ðxÞ using orthogonality ðl mðxÞYi ðxÞYj ðxÞdx ¼ 0 for i – j 0

ðiiÞ

ðiiiÞ

Note: Additional boundary terms are present in Equation (iii) when there are lumped elements at the system boundary. 3. Determine the initial conditions for qi ðtÞ: We require Equation (iii) for this. 4. Solve the ODE for qi ðtÞ using the initial conditions. 5. Substitute the results into Equation (ii).

4 38


Pump l l/2

0

x

f0 cos wt

y FIGURE 4.12

A pump mounted on a simply supported beam.

Example 4.6 A pump is mounted at the midspan of a simply supported thin beam of uniform cross section and length l: The pump rotation generates a transverse force f0 cos vt, as schematically shown in Figure 4.12. Initially, the system starts from rest, from the static equilibrium position of the beam, such that vðx; tÞ ¼ 0

›vðx; tÞ ¼ 0 at t ¼ 0 ›t

and

It is necessary to obtain the transverse response vðx; tÞ of the beam in the form of a modal summation during operation of the pump. First, determine qj ðtÞ in terms of f0 ; v; vj and the beam parameters r; A; and l; assuming that the beam is completely undamped. Are all modes of the beam excited by the pump? If the beam is lightly damped, what would be its steady state response? In particular, what is the steady state response of the beam at the pump location? Sketch its amplitude as a function of the excitation frequency v:

Solution Using the Dirac delta function dðxÞ; we can express the equation of forced motion of the beam as EI

›4 v ›2 v þ ¼ f0 cos vt d x r A ›x4 ›t 2

l 2

ð4:170Þ

P Substitute vðx; tÞ ¼ i Yi ðxÞqi ðtÞ; where normalized mode shapes for the simply supported beam are Yi ðxÞ ¼ sin ipx=l, multiply by Yj ðxÞ, integrate over x ¼ ½0; l and use the orthogonality of mode shapes. We obtain EI

ip l

4

l l jp q ðtÞ þ rA q€ j ¼ f0 cos vt sin 2 j 2 2

Note: ð aþ a2

pðxÞdðx

aÞdx ¼ pðaÞ

Hence, from Equation 4.149, we obtain q€ j þ v2j qj ¼ aj cos vt where

aj ¼

2f0 p sin j 2 rAl

The given initial conditions are satisfied if and only if qj ð0Þ ¼ 0 and q_ j ð0Þ ¼ 0 for all j:

ð4:171Þ


4 39

Hence, the complete solution is qj ðtÞ ¼

ðv2j

aj

v2 Þ

½cos vt

cos vj t

ð4:172Þ

It follows that the total response is vðx; tÞ ¼

2f0 X sin jp=2 px sin j ½cos vt rAl ðv2j v2 Þ l

cos vj t

ð4:173Þ

Clearly, sinðjp=2Þ ¼ 0 for even values of j: We see that even modes of the beam are not excited by the pump. This is to be expected because, for even modes, the midspan is a node point and has no motion. If the beam is lightly damped, its natural response terms ðcos vj tÞ will decay to zero with time. Hence, the steady state response will be X sin jp=2 sin jpx=l 2f vss ðx; tÞ ¼ 0 cos vt ð4:174Þ ðv2j v2 Þ rAl At the pump location ðx ¼ l=2Þ; the steady state response is " # X sin2 jp=2 2f0 2f0 1 1 1 cos vt cos vt vss ðl=2; tÞ ¼ ¼ þ 2 þ 2 þ ··· ðv2j v2 Þ ðv21 v2 Þ ðv3 v2 Þ ðv5 v2 Þ rAl rAl Note again that only the odd modes contribute to the response. Furthermore, for a simply supported beam

vj ¼ j2 v1

for j ¼ 1; 2; 3; …

where 2

p l

v1 ¼

sffiffiffiffiffi EI rA

Nondimensionalize the midspan response at steady state as v¼

rAlv21 v ðl=2; tÞ ¼ 2f0 ss

1 ð14

v2 Þ

þ

1 ð34

v2 Þ

þ

1 ð54

v2 Þ

þ · · · cos vt

Its amplitude is v0 ðvÞ ¼

1 ð14

v2 Þ

þ

1 ð34

v2 Þ

þ

1 ð54

v2 Þ

þ ···

The characteristic of the amplitude as a function of the nondimensional excitation frequency is sketched in Figure 4.13.

Example 4.7 Perform a modal analysis to determine natural frequencies and mode shapes of transverse vibration of a thin cantilever (i.e., a beam with “fixed free” or “clamped free” end conditions). The coordinate system and the beam parameters are as shown in Figure 4.14. Solution As usual, the mode shapes are given by YðxÞ ¼ C1 cosh lx þ C2 sinh lx þ C3 cos lx þ C4 sin lx

ð4:134Þ

4 40


Steady-State Response Amplitude (Nondimensional)

2.0

1.5

1.0

0.5

0.0 0.1

FIGURE 4.13

9 10 25 49 100 Excitation Frequency w /w1

1

Amplitude of the steady response at the pump as a function of excitation frequency.

Its first three derivatives are dYðxÞ ¼ C1 l sinh x þ C2 l cosh lx dx d2 YðxÞ ¼ C1 l2 cosh lx þ C2 l2 sinh lx dx2

C3 l sin lx þ C4 l cos lx

ð4:175Þ

C3 l2 cos lx

C4 l2 sin lx

ð4:176Þ

d3 YðxÞ ¼ C1 l3 sinh lx þ C2 l3 cosh lx þ C3 l3 sin lx dx3

C4 l3 cos lx

ð4:177Þ

The BCs of the beam are At x ¼ 0 : vð0; tÞ ¼ 0 At x ¼ l : EI

and

›2 vðl; tÞ ¼0 ›x2

and

›vð0; tÞ ¼0 ›x EI

›3 vðl; tÞ ¼0 ›x3

The corresponding modal BCs are dYð0Þ d2 YðlÞ ¼ 0; ¼ 0; dx dx2 d3 YðlÞ ¼0 dx3

Yð0Þ ¼ 0;

E, I, r, A

ð4:178Þ

l x

0

Substitute Equation 4.134, Equation 4.175 to Equation 4.177 into Equation 4.178. We obtain C1 þ C3 ¼ 0

ðiÞ

y, v(x,t)

C2 þ C4 ¼ 0

ðiiÞ

FIGURE 4.14

A cantilever in bending vibration.


4 41

C1 cosh ll þ C2 sinh ll

C3 cos ll

C4 sin ll ¼ 0

ðiiiÞ

C1 sinh ll þ C2 cosh ll þ C3 sin ll

C4 cos ll ¼ 0

ðivÞ

Eliminate C1 and C2 in Equation (iii) and Equation (iv) by substituting Equation (i) and Equation (ii). We obtain ½cosh ll þ cos ll C3 þ ½sinh ll þ sin ll C4 ¼ 0

ðvÞ

½sinh ll

ðviÞ

sin ll C3 þ ½cosh ll þ cos ll C4 ¼ 0

or, in the vector matrix form " cosh ll þ cos ll sinh ll

sin ll

sinh ll þ sin ll cosh ll þ cos ll

#"

C3

#

C4

¼

" # 0 0

ð4:179Þ

The trivial solution of Equation 4.179 is C3 ¼ 0 ¼ C4 : Then, from Equation (i) and Equation (ii), we also have C1 ¼ 0 ¼ C2 : This solution corresponds to YðxÞ ¼ 0 and is not acceptable in general for a vibrating system. Hence, the matrix in Equation 4.179 must be noninvertible (i.e., singular). Hence, the determinant of the matrix must vanish (see Appendix 3A); thus ðcosh ll þ cos llÞ2

or

ðsinh ll þ sin llÞðsinh ll

cosh2 ll þ 2 cosh ll cos ll þ cos2 ll

sin llÞ ¼ 0

sinh2 ll þ sin2 ll ¼ 0

However, it is well known that cosh2 ll

Hence,

sinh2 ll ¼ 1

cos2 ll þ sin2 ll ¼ 1

and

cos ll cosh ll ¼

1

ð4:180Þ

This equation has an infinite number of solutions li for i ¼ 1; 2; 3; … giving an infinite number of natural frequencies: sffiffiffiffiffi EI 2 ð4:181Þ vi ¼ li rA The corresponding mode shapes are given by Equation 4.134 subject to Equation (i), Equation (ii), and Equation (v) or Equation (vi). This gives Yi ðxÞ ¼ C3 ðcos li x cosh li xÞ þ C4 ðsin li x sinh li xÞ with C3 ¼

½sinh li l þ sin li l C ½cosh li l þ cos li l 4

It follows that Yi ðxÞ ¼ C4 ½sin li x

sinh li x þ C4

sinh li l þ sin li l ½ cos li x þ cosh li x cosh li l þ cos li l

The unknown multiplier C4 simply scales the mode shape and is absorbed into the generalized coordinate qi ðtÞ as usual. In fact, this is a process of normalization of mode shapes, where C4 ¼ 1 is used. So, we have the normalized mode shapes: Yi ðxÞ ¼ a sin li x þ b sinh li x þ ai ½c cos li x þ d cosh li x

ð4:182Þ

with a ¼ 1;

b¼

1;

c¼

1;

d¼1

and

ai ¼

sinh li l þ sin li l cosh li l þ cos li l

The first three roots of Equation 4.180 are

l1 l ¼ 1:875104;

l2 l ¼ 4:694091;

l3 l ¼ 7:854757

ð4:183Þ

4 42


Mode Shape Yi(x)

0

FIGURE 4.15

Location Along Beam

x/l

1

First three modes of a cantilever (fixed free beam) in transverse vibration.

The corresponding three mode shapes are sketched in Figure 4.15. Note, in particular, that the node points are not physically fixed but remain stationary during a modal motion. This completes the solution. The modal information corresponds to the infinite set of natural frequencies that are obtained by solving Equation 4.180 subject to Equation 4.181, and the mode shapes are given by Equation 4.182 subject to Equation 4.183. Modal information corresponding to other common BCs may also be put in this form. Table 4.3 summarizes such data. Table 4.4 provides numerical values corresponding to this modal information for the first three modes. TABLE 4.3

Modal Information for Bending Vibration of Beams

End Conditions

Pinned pinned Fixed fixed Free free Fixed pinned

Natural Frequencies pffiffiffiffiffiffiffi vi ¼ l2i EI=rA where li are roots of sin li l ¼ 0 i ¼ 1; 2; 3; … cosh li l cos li l ¼ 1 i ¼ 1; 2; 3; … cosh li l cos li l ¼ 1 i ¼ 0; 1; 2; … tanh li l ¼ tan li l i ¼ 1; 2; 3; …

Mode Shapes Yi ðxÞ ¼ a sin li x þ b sinh li þ ai ½c cos li x þ d cosh li x a

b

c

ai

d

1

0

0

0

0

1

21

21

1

sinh li l 2 sin li l cosh li l 2 cos li l

1

1

21

21

Same

1

21

21

1

Same sinh li l þ sin li l cosh li l þ cos li l

Fixed free

cosh li l cos li l ¼ 21

1

21

21

1

Fixed sliding

tanh li l ¼ 2tan li l

1

21

21

1

Pinned free

tanh li l ¼ tan li l

1

ai

0

0

cosh li l þ cos li l sin li l 2 sin li l sin li l sinh li l


4 43

TABLE 4.4 Roots of the Frequency Equation for Bending Vibration of Beams First Three Roots li l

End Conditions Pinned pinned

4.5.7

p 2p 3p

Fixed fixed

4.730041 7.853205 10.995608

Free free

0 4.730041 7.853205 10.995608

Fixed pinned

3.926602 7.068583 10.210176

Fixed free

1.875104 4.694091 7.854757

Fixed sliding

2.365020 5.497804 8.639380

Pinned free

0 3.926602 7.068583 10.210176

Bending Vibration of Beams with Axial Loads

In practice, beam type members that undergo flexural (transverse) vibrations may carry axial forces. Examples are structural members such as columns, struts, and towers. Generally, a tension will increase the natural frequencies of bending and compression will decrease them. Hence, one way to avoid the excitation of a particular natural frequency (and mode) of bending vibration is to use a suitable tension or compression in the axial direction. The equation of motion for the transverse motion of a thin beam subjected to an axial tension P may be easily derived by following the procedure that led to Equation 4.122. For simplicity, assume a thin beam subjected to a constant tensile force P: A small element dx of the beam is shown in Figure 4.16. The vertical component (in the positive y direction) of the axial force is P sin u þ P sinðu þ duÞ ø

Pu þ Pðu þ duÞ ¼ P du

because the slope u ¼ ›v=›x is small. Also, the change in slope is du ¼

›2 v dx ›x 2

It is clear that the previous equation of transverse dynamics of element dx has to be modified simply by adding the term Pð›2 v=›x2 Þdx to the f ðx; tÞdx side of the equation. Then, the resulting equation of transverse vibration will be ! ›2 v ›2 ›2 v ›2 v P 2 ¼ f ðx; tÞ ð4:184Þ rA 2 þ 2 EI 2 ›t ›x ›x ›x

4 44


dx

x

P θ

q+dq P

y, v(x,t) FIGURE 4.16

Beam element in transverse vibration and subjected to axial tension.

For modal analysis of a uniform beam, then, we use the equation of free motion

rA

›2 v ›4 v þ EI 4 2 ›t ›x

P

›2 v ¼0 ›x2

ð4:185Þ

With a separable (modal) solution of the form vðx; tÞ ¼ YðxÞqðtÞ

ð4:186Þ

we have EI ðd4 Y=dx4 Þ P ðd2 Y=dx2 Þ ¼ rAY

q€ ðtÞ ¼ qðtÞ

v2

ð4:187Þ

which gives, as before, the time response equation of the generalized coordinates q€ ðtÞ þ v2 qðtÞ ¼ 0

ð4:188Þ

and the mode shape equation EI

d4 Y dx4

P

d2 Y dx2

rAv2 Y ¼ 0

ð4:189Þ

Note that the mode shape equation is still fourth order, but is different. The analysis, however, may be done as before by using four BCs at the two ends of the beam to determine the natural frequencies (an infinite set) and the corresponding normalized mode shapes.

4.5.8

Bending Vibration of Thick Beams

In our derivation of the governing equation for the lateral vibration of thin beams (known as the Bernoulli Euler beam equation), we neglected the following effects in particular: 1. Deformation and associated lateral motion due to shear stresses 2. Moment of inertia of beam elements in rotatory motion Note, however, that we did use the fact that shear forces ðQÞ are present in a beam cross section, even though the resulting deformations were not taken into account. Also, in writing the equation for rotational motion of a beam element dx; we simply summed the moments to zero, without including the inertia moment. These assumptions are valid for a beam whose cross sectional dimensions are small compared with its length. However, for a thick beam, the effect of shear deformation and rotatory inertia must be included in deriving the governing equation. The resulting equation is known as the Timoshenko beam equation. Important steps in the derivation of the equation of motion for the forced transverse vibration of beams, including the effects of shear deformation and rotatory inertia, are given now.


4 45

x

Bent Only

x

f(x,t) dx

x+d x ∂2 θ

M

θ

Q

q +f =

(a) y, v(x,t)

x+dx

x

x

∂ t2 ∂ 2v ∂t2

M + Q+

∂M ∂x

δx

∂Q δx ∂x

∂v ∂x

Bent and Sheared

(b) y

FIGURE 4.17 A Timoshenko beam element: (a) combined effect of bending and shear; (b) dynamic effects including rotatory inertia.

Consider a small element dx of a beam. Figure 4.17(a) illustrates the contribution of the bending of an element and the deformation due to transverse shear stresses towards the total slope of the beam neutral axis. Let

u ¼ angle of rotation of the beam element due to bending f ¼ increase in slope of the element due to shear deformation in transverse shear (this is equal to shear strain) Then, the total slope of the beam element is

›v ¼uþf ›x

ð4:190Þ

Here, v and x take the usual meanings, as for a thin beam. Figure 4.17(b) shows an element dx of the beam with the forces, moments, and the linear and angular accelerations marked. With the sign convention shown in Figure 4.17(b), the linear shear stress shear strain relation can be stated as Q¼

kGAf

ð4:191Þ

where the Timoshenko shear coefficient k¼

Average shear stress on beam cross section at x Shear stress at the neutral axis at x

and G ¼ shear modulus The equation for translatory motion is f ðx; tÞdx þ Q

Qþ

›Q ›2 v dx ¼ ðrA dxÞ 2 ›x ›t

Hence, f ðx; tÞ

›Q ›2 v ¼ rA 2 ›x ›t

ð4:192Þ

4 46


The equation for the rotatory motion of the element, taking into account the rotatory inertia, is Mþ

›M dx ›x

M

Q dx ¼ ðI r dxÞ

›2 u ›t 2

which becomes

›M ›x

Q ¼ Ir

›2 u ›t 2

ð4:193Þ

From the elementary theory of bending, as before M ¼ EI

›u ›x

ð4:194Þ

The relationship between the shear modulus and the modulus of elasticity is known to be E ¼ 2ð1 þ nÞG

ð4:195Þ

where n ¼ Poisson’s ratio. This relation may be substituted if desired. Manipulation of these equations yields EI

›4 v ›2 v þ r A ›x4 ›t 2 ¼ f ðx; tÞ

rI 1 þ

›4 v r2 I ›4 v E þ kG ›t 4 kG ›x2 ›t 2

EI ›2 f ðx; tÞ rI ›2 f ðx; tÞ þ kGA ›x2 kGA ›t 2

ð4:196Þ

This is the Timoshenko beam equation for forced transverse motion. Note that this equation is fourth order in time, whereas the thin beam equation is second order in time. The modal analysis may proceed as before, by using the free ðf ¼ 0Þ equation and a separable solution. However, the resulting differential equation for the generalized coordinates will be fourth order in time and, as a result, additional natural frequency bands will be created. The reason is the independent presence of shear and bending motions. The differential equation of mode shapes will be fourth order in x and the solution procedure will be as before, through the use of four BCs at the two ends of the beam.

4.5.9

Use of the Energy Approach

So far, we have used only the direct, Newtonian approach in deriving the governing equations for continuous members in vibration. Of course, the same results may be obtained by using the Lagrangian (energy) approach. The general approach here is to first express the Lagrangian L of the system as L ¼ Tp

V

ð4:197Þ

where Tp ¼ total kinetic co energy (equal to kinetic energy T for typical systems) V ¼ total potential energy Then, for a virtual increment (variation) of the system through incrementing the system variable, the following condition will hold: ðt 2 ½dL þ dW dt ¼ 0 ð4:198Þ t1

where dL is the increment in the Lagrangian and dW is the work done by the external forces on the system due to the increment. Finally, using the arbitrariness of the variation, the equation of motion, along with the BCs, can be obtained. This approach is illustrated now.


4 47

First, consider the free motion. We can easily extend the result to the case of forced motion. Kinetic energy is given by T¼

›v 1 ðl rA dx ›t 2 0

2

¼

›v 1 ðl rA ›t 2 0

2

dx

ð4:199Þ

Potential energy due to bending results from work needed to bend the beam (angle denoted by u ¼ ›v=›x as usual) under bending moment M; thus !2 ›2 v 1 ðEnd 2 1 ðl V¼ dx ð4:200Þ M df ¼ EI ›x 2 2 End 1 2 0 Before proceeding further, note the following steps of variation and integration by parts with respect to t: d

ðt2 1 ›v t1 2 ›t

2

dt ¼

ðt2 ›v › dv ðt2 ›v ›v d dt ¼ dt ¼ ›t t1 ›t ›t t1 ›t

›v dv ›t

t2 t1

ðt2 ›2 v 2 dv dt t1 ›t

ð4:201Þ

Note that we interchanged the operations d and ›=›t prior to integrating by parts. Also, by convention, we assume that no variations are performed at the starting and ending times (t1 and t2 ) of integration. Hence, dvðt1 Þ ¼ 0

and

dvðt2 Þ ¼ 0

ð4:202Þ

Similarly, variation and integration by parts with respect to x are done as follows: !2 ! ðl 1 ðl ðl ›2 v ›2 v ›2 v ›2 v › ›v EI d dx ¼ EI 2 d dx ¼ EI 2 d dx 2 2 ›x ›x ›x ›x ›x ›x 0 2 0 0 " ¼ EI "

›2 v ›v d ›x 2 ›x

›2 v ›v ¼ EI 2 d ›x ›x

#l 0

#l 0

" #l ðl › ›2 v ›v ›2 v ›v EI 2 d dx ¼ EI 2 d ›x ›x ›x ›x 0 0 ›x "

› ›2 v EI 2 dv ›x ›x

#l þ 0

ðl ›2 ›2 v EI dv dx 2 ›x2 0 ›x

ðl › ›2 v › dv EI 2 dx ›x ›x 0 ›x ð4:201aÞ

Now, for the case of free vibration ðdW ¼ 0Þ, substitute Equation 4.201 and Equation 4.201a in dT; and dV of Equation 4.199 and Equation 4.200, to obtain " # ðt2 ðl ðt2 ðl t2 ›2 v ›2 ›2 v ›v dv dL dt ¼0 ¼ dt dx rA 2 þ 2 EI 2 dv þ dxrA ›t ›x ›x ›t 0 t1 t1 0 t1 ð4:203Þ " #l " #l 2 2 ð ðt2 t 2 › v ›v › ›v þ EI dV dt dt EI 2 d ›x ›x 0 ›x ›x2 t1 t1 0 Since Equation 4.203 holds for all arbitrary variations dvðtÞ; its coefficient should vanish. Hence,

rA

›2 v ›2 ›2 v ¼0 2 þ 2 EI ›t ›x ›x 2

ð4:204Þ

which is the same Bernoulli Euler beam equation for free motion as we had derived before. The second integral term on the RHS of Equation 4.203 has no consequence. We conventionally pick dvðt1 Þ ¼ 0 and dvðt2 Þ ¼ 0 at the time points t1 and t2 : The third integral term on the RHS of Equation 4.203 gives some BCs. Specifically, if the slope BC ›v=›x is zero (i.e., fixed end), then the corresponding bending moment at the end is arbitrary, as expected. However, if the slope at the boundary is arbitrary, then the bending moment EIð›2 v=›x2 Þ at the end should be zero (i.e., pinned or free end). The last integral term on the RHS of Equation 4.203 gives some other BCs. Specifically, if the displacement BC v is zero (i.e., pinned or fixed end) then the corresponding shear force at the

4 48


end is arbitrary. However, if the displacement at the boundary is arbitrary, then the shear force ð›=›xÞEIð›2 v=›x2 Þ at that end should be zero (i.e., free or sliding end). Next, consider a forced beam with force per unit length given by f ðx; tÞ: Then, the work done by f ðx; tÞdx in a small element dx of the beam, when moved through a displacement of dv; is f ðx; tÞdx dv

ð4:205Þ

Then, by combining Equation 4.205 with Equation 4.203, for arbitrary variation dv; we obtain the forced vibration equation

rA

›2 v ›2 ›2 v ¼ f ðx; tÞ 2 þ 2 EI ›t ›x ›x 2

ð4:206Þ

Note that external forces and moments applied at the ends of the beam can be incorporated into the BCs in the same manner.

4.5.10

Orthogonality with Inertial Boundary Conditions

It can be verified that the conventional orthogonality condition (Equation 4.163) holds for beams in transverse vibration, under common noninertial BCs. When an inertia element (rectilinear or rotatory) is present at an end of the beam, this condition is violated. A modified and more general orthogonality condition can be derived for application to beams with inertial boundary conditions. To illustrate the procedure, consider a beam with a mass m attached at the end x ¼ l; as shown in Figure 4.18(a). A free body diagram giving the sign convention for shear force Q acting on m is shown in Figure 4.18(b). The BCs at x ¼ l are: 1. Bending moment vanishes, because there is no rotatory inertia at the end that is free. Hence, EI

›2 vðl; tÞ ¼0 ›x 2

l

EI(x), ρ A(x) 0

(a)

ð4:207Þ

Point Mass m

x

y, v(x,t) Q m

(b)

Q=

v(l,t)

∂ ∂2v (l,t) EI 2 ∂x ∂x

FIGURE 4.18 (a) A beam with an end mass in transverse vibration; (b) free body diagram showing the shear force acting on the end mass.


4 49

2. Equation of rectilinear motion of the end mass m

›2 vðl; tÞ › ›2 vðl; tÞ ¼ EI 2 ›t ›x 2 ›x

ð4:208Þ

In the usual manner, by substituting vðx; tÞ ¼ Yi ðxÞqi ðtÞ for mode i; along with q€ i ðtÞ ¼ v2i qi ðtÞ; into Equation 4.207 and Equation 4.208, with the understanding that the result should hold for any qi ðtÞ; we obtain the corresponding modal boundary conditions: d2 Yi ðlÞ ¼0 dx2

ð4:209Þ

d d2 Yi ðlÞ EI þ v2i mYi ðlÞ ¼ 0 dx dx2

ð4:210Þ

for mode i: Now, we return to Equation 4.162: " ðl d d2 Y 2 2 ðvi vj Þ rAYi Yj dx Yj EI 2i dx dx 0

d2 Yj d Yi EI 2 dx dx

#l " 0

dYj d2 Yi EI 2 dx dx

dYi d2 Yj EI 2 dx dx

#l ¼0 0

ð4:162Þ The second and the third terms of Equation 4.162 will vanish at x ¼ 0 for noninertial BCs, as usual. At x ¼ l; the third term will vanish in view of Equation 4.209. So, we are left with the second term at x ¼ l: Substitute Equation 4.210: d2 Y j d Yi EI 2 dx dx

d d2 Y Yj EI 2i dx dx

! ¼ l

ðv2i

v2j ÞmYi ðlÞYj ðlÞ

Substitute Equation 4.211 into Equation 4.162 and cancel v2i v2j – 0 for i – j: We obtain ( ðl 0 for i – j rAYi Yj dx þ mYi ðlÞYj ðlÞ ¼ 0 aj for i ¼ j

ð4:211Þ

ð4:212Þ

This is the modified and more general orthogonality property. If the mass is at x ¼ 0; the direction of Q that acts on m will reverse and, hence, the second term in Equation 4.212 will become mYi ð0ÞYj ð0Þ: 4.5.10.1

Rotatory Inertia

If there is a free rotatory inertia at x ¼ l; without an associated rectilinear inertia, then the shear force will vanish, giving d d2 Yi ðlÞ EI ¼0 dx dx2

ð4:213Þ

The equation of rotational motion of J will give EI

d2 Yi ðlÞ dx2

v2i J

dYi ðlÞ ¼0 dx

ð4:214Þ

Here, the second term in Equation 4.162 will vanish in view of Equation 4.213. Then, by substituting Equation 4.214 into the third term of Equation 4.162, we obtain the modified orthogonality relation ðl

dY ðlÞ dYj ðlÞ rAYi Yj dx þ J i ¼ dx dx 0

(

0

for i – j

aj

for i ¼ j

ð4:215Þ

4 50

4.6


Damped Continuous Systems

All practical mechanical systems have some form of energy dissipation (damping). When the level of dissipation is small, damping is neglected, as we have done thus far in this chapter. Yet, some effects of damping (e.g., the fact that at steady state the natural [modal] vibration components decay to zero leaving only the steady forcing component) are tacitly assumed even in undamped analysis. The natural behavior of a system is expected to change due to the presence of damping. In particular, the system’s natural frequencies will decrease (and be called damped natural frequencies) as a result of damping. Furthermore, it is quite possible that a damped system would not possess “real” modes in which it could independently vibrate. Mathematically, in that case, the modes will become complex (as opposed to real) and, physically, all the points of the system will not move, maintaining a constant phase at a given damped natural frequency. In other words, a real solution that is separable in space ðxÞ and time ðtÞ may not be possible for the free vibration problem of a damped system. Also, node points of an undamped system may vary with time as a result of damping. With light damping, of course, such effects of damping will be negligible. Since there are damped systems that do not possess real natural modes of vibration, care should be exercised when extending the results of modal analysis from an undamped system to a damped one. However, in some cases, the mode shapes will remain the same after including damping (even though the natural frequencies will change). This is analogous to the case of proportional damping, which was discussed in the section on lumped parameter (multi degree of freedom) vibrating systems. The modal analysis of a damped system will become significantly easier if we assume that the mode shapes will remain the same as those for the undamped system. Even when the actual type of damping in the system results in complex modes, for analytical convenience, an equivalent damping model that gives real modes is used in simplified analysis. This is analogous to the use of proportional damping in lumped parameter systems.

4.6.1

Modal Analysis of Damped Beams

Consider the problem of free damped transverse vibration of a thin beam, given by

›2 ›2 v ›v ›2 v þ rA 2 ¼ 0 2 EI 2 þL ›x ›x ›t ›t

ð4:216Þ

where L is a spatial differential operator (in x). Consider the following two possible models of damping: 1: L ¼

›2 p ›2 E I 2 dx2 ›x

2: L ¼ c

ð4:217Þ ð4:218Þ

Model 1 corresponds to the Kelvin Voigt model of material (internal) damping given by the stress strain relation

s ¼ E1 þ E p

›1 ›t

ð4:219Þ

where Ep is the damping parameter of the beam material. Hence, we obtain the damped beam equation simply by replacing E in the undamped beam equation by E þ Ep ð›=›tÞ: Also, Ep is independent of the frequency of vibration for the viscoelastic damping model, but will be frequency dependent for the hysteretic damping model. Modal analysis is done regardless of any frequency dependence of Ep and, in the final modal result for a particular modal frequency vi ; the appropriate frequency function for EpðvÞ is used with v ¼ vi if the damping is of the hysteretic type. It can be easily verified that the mode shapes of the damped system with model 4.217 are identical to those of the undamped system, regardless of whether the beam cross section is uniform or not. In Model 2 (Equation 4.218), the operator is a constant c: This corresponds to external damping of the linear viscous type, distributed along the beam length. For example, imagine a beam resting on a


4 51

foundation of viscous damping material. For Model 2, it can be shown that the damped mode shapes are identical to the undamped ones, assuming that the beam cross section is uniform. If the beam is nonuniform, the damped and the undamped mode shapes are identical if we assume that the damping constant c varies along the beam in proportion to the area of cross section AðxÞ of the beam. We shall show this in the example given below.

Example 4.8 Perform the modal analysis for transverse vibration of a thin nonuniform beam with linear viscous damping distributed along its length and satisfying the beam equation

›2 ›2 v ›v ›2 v EIðxÞ þ þ ¼0 r AðxÞb r AðxÞ ›x2 ›x2 ›t ›t 2

ð4:220Þ

Determine damped natural frequencies, modal damping ratios, and the response vðx; tÞ as a modal series expansion, given vðx; 0Þ ¼ dðxÞ and v_ ðx; 0Þ ¼ sðxÞ:

Solution Substitute the separable solution vðx; tÞ ¼ YðxÞqðtÞ

ð4:221Þ

into Equation 4.220. We obtain d2 d2 YðxÞ EI qðtÞ þ rAðxÞbYðxÞ_qðtÞ þ rAðxÞYðxÞ€qðtÞ ¼ 0 2 dx dx2 Group the functions of x and t separately and equate to the same constant v2 ; as usual: d2 d2 YðxÞ 2 EI dx dx2 ¼ rAðxÞYðxÞ

q€ ðtÞ þ b_qðtÞ ¼ v2 qðtÞ

ð4:222Þ

v2 rAYðxÞ ¼ 0

ð4:223Þ

We have d2 d2 YðxÞ EI 2 dx dx2 and q€ ðtÞ þ b_qðtÞ þ v2 qðtÞ ¼ 0

ð4:224Þ

Note that Equation 4.223 is identical to that for the undamped beam. Hence, with known BCs, we will obtain the same mode shapes Yi ðxÞ and the same undamped natural frequencies vi in the usual manner. However, the equation of modal generalized coordinates qðtÞ given by Equation 4.224 is different from that for the undamped case ðb ¼ 0Þ: We write, for mode i q€ i ðtÞ þ 2zi vi q_ i ðtÞ þ v2i qi ðtÞ ¼ 0

ð4:225Þ

where

zi ¼

b 2vi

is the modal damping ratio for mode i: Damped natural frequencies are qffiffiffiffiffiffiffiffi vdi ¼ 1 z2i vi

ð4:226aÞ

ð4:226bÞ

Equation 4.225 can be solved in the usual manner, with initial conditions qi ð0Þ and q_ i ð0Þ determined a priori, using known vðx; 0Þ and v_ ðx; 0Þ:

4 52


The modal series solution is vðx; tÞ ¼ The initial conditions are

X Yi ðxÞqi ðtÞ

X Yi ðxÞqi ð0Þ ¼ dðxÞ X Yi ðxÞ_qi ð0Þ ¼ sðxÞ

ð4:227Þ

ð4:228Þ ð4:229Þ

Multiply Equation 4.228 and Equation 4.229 by rAðxÞYj ðxÞ and integrate from x ¼ 0 to l using the orthogonality condition ( ðl 0 for i – j ð4:230Þ rAðxÞYi ðxÞYj ðxÞdx ¼ 0 aj for i ¼ j We obtain qj ð0Þ ¼

1 ðl dðxÞrAðxÞdx aj 0

ð4:231Þ

q_ j ð0Þ ¼

1 ðl sðxÞrAðxÞdx aj 0

ð4:232Þ

This completes the solution for the free damped beam. The forced damped case can be analyzed in the same manner as for the forced undamped case because the mode shapes are the same.

4.7

Vibration of Membranes and Plates

The cables, rods, shafts, and beams whose vibration we have studied thus far in this chapter are one dimensional members or line structures. These continuous members need one spatial variable ðxÞ; in addition to the time variable ðtÞ; as an independent variable to represent their governing equation of motion. Membranes and plates are two dimensional members or planar structures. They need two independent spatial variables (x and y) in addition to time ðtÞ; for representing their dynamics. A membrane may be interpreted as a two dimensional extension of a string or cable. In particular, it has to be in tension and cannot support any bending moment. A plate is a two dimensional extension of a beam. It can support a bending moment. Their governing equations will, therefore, resemble two dimensional versions of their respective one dimensional counterparts. Modal analysis will also follow the familiar steps, after accounting for the extra dimension. In this section, we will give an introduction to the modal analysis of membranes and plates. For simplicity, only special cases of rectangular members with relatively simple BCs will be considered. Analysis of more complicated boundary geometries and conditions will follow analogous procedures, but requires a greater effort and produces more complicated results.

4.7.1

Transverse Vibration of Membranes

Consider a stretched membrane (in tension) that lies on the x y plane, as shown in Figure 4.19. Transverse vibration vðx; y; tÞ in the z direction is of interest. By following a procedure that is somewhat analogous to the derivation of the cable equation, we can obtain the governing equation as " # ›2 vðx; y; tÞ ›2 ›2 2 ¼c þ 2 vðx; y; tÞ ð4:233Þ ›t 2 ›x2 ›y


4 53

a

0

x

b z, v(x,t)

y FIGURE 4.19

A membrane or a plate in Cartesian coordinates.

with

sffiffiffiffiffi T0 c¼ m0

ð4:234Þ

where T 0 ¼ tension per unit length of membrane section (assumed constant) m0 ¼ mass per unit surface area of membrane For modal analysis, we seek a separable solution of the form vðx; y; tÞ ¼ YðxÞZðyÞqðtÞ Substitute Equation 4.235 into Equation 4.233. We obtain " # 1 d2 YðxÞ 1 d2 ZðyÞ 2 YðxÞZðyÞ€qðtÞ ¼ c þ YðxÞ dx2 ZðyÞ dy2

ð4:235Þ

ð4:236Þ

Since Equation 4.236 is true for all possible values of t; x; and y that are independent, the three function groups should separately equal the constants; thus 1 d2 Y ¼ YðxÞ dx2

a2

or

d2 YðxÞ þ a2 YðxÞ ¼ 0 dx2

ð4:237Þ

1 d2 ZðyÞ ¼ ZðyÞ dy 2

b2

or

d2 ZðyÞ þ b2 ZðyÞ ¼ 0 dy2

ð4:238Þ

q€ ðtÞ ¼ qðtÞ

v2

or

q€ ðtÞ þ v2 qðtÞ ¼ 0

ð4:239Þ

with

v2 ¼ c2 ða2 þ b2 Þ

ð4:240Þ

The argument for using positive constants a2 ; b2 ; and v2 is similar to that we gave for the one dimensional case. Next, Equation 4.237 and Equation 4.238 have to be solved using two end conditions for each direction, as usual. This will provide an infinite number of solutions ai and bj ; and the corresponding natural frequencies

v2ij ¼ cða2i þ b2j Þ1=2

for i ¼ 1; 2; 3; … and j ¼ 1; 2; 3; …

along with the mode shape components Yi ðxÞ and Zj ðyÞ for the two dimensions.

ð4:241Þ

4 54

4.7.2


Rectangular Membrane with Fixed Edges

Consider a rectangular membrane of length a and width b as shown in Figure 4.19 and with the four edges fixed. The BCs are vð0; y; tÞ ¼ 0;

vða; y; tÞ ¼ 0;

vðx; 0; tÞ ¼ 0;

vðx; b; tÞ ¼ 0

Using these in solving Equation 4.237 and Equation 4.238, as usual, we obtain ip a jp with bj ¼ b

Yi ðxÞ ¼ sin ai x

with ai ¼

Zj ðyÞ ¼ sin bj y "

vij ¼

c½a2i

þ

b2j 1=2

i2 j2 ¼ pc 2 þ 2 a b

#1=2

for i ¼ 1; 2; 3; … and j ¼ 1; 2; 3

ð4:242Þ ð4:243Þ ð4:244Þ

Note that the spatial mode shapes are given by Yj ðxÞZj ðyÞ ¼ sin

4.7.3

ipx jpy sin a b

ð4:245Þ

Transverse Vibration of Thin Plates

Consider a thin plate of thickness h in a Cartesian coordinate system as shown in Figure 4.19. The usual assumptions as for the derivation of the Bernoulli Euler beam equation are used. In particular, h is assumed small compared with the surface dimensions (a and b for a rectangular plate). Then, shear deformation and rotatory inertia can be neglected, and also normal stresses in the transverse direction ðzÞ can be neglected. Furthermore, any end forces in the planar directions (x and y) are neglected. The governing equation is " #2 ›2 vðx; y; tÞ ›2 ›2 2 þc þ 2 vðx; y; tÞ ¼ 0 ð4:246Þ ›t 2 ›x 2 ›y with c2 ¼

E0 I 0 Eh2 0 ¼ rA 12ð1 n2 Þr

ð4:247Þ

where E0 ¼ I0 ¼

E ð1

n2 Þ

h3 ¼ second moment of area per unit length of section 12

ð4:248Þ ð4:249Þ

A0 ¼ h ¼ area per unit length of section r ¼ mass density of material E ¼ Young’s modulus of elasticity of the plate material n ¼ Poisson’s ratio of the plate material If we attempt modal analysis by assuming a completely separable solution of the form vðx; y; tÞ ¼ YðxÞZðyÞqðtÞ in Equation 4.246, a separable grouping of functions of x and y will not be achieved in general. However, the space and the time will be separable in modal motions. Hence, we seek a solution of the form vðx; y; tÞ ¼ Yðx; yÞqðtÞ

ð4:250Þ


4 55

x

y

FIGURE 4.20

Mode 1 i=1 , j=1

Mode 2 i=2 , j=1

Mode 3 i=1 , j=2

Mode 4 i=2 , j=2

Mode 5 i=3 , j=1

Mode 6 i=3 , j=2

Mode shapes of transverse vibration of a simply supported rectangular plate.

We will obtain q€ ðtÞ þ v2 qðtÞ ¼ 0

ð4:251Þ

and 72 72 Yðx; yÞ

l4 Yðx; yÞ ¼ 0

ð4:252Þ

with the natural frequencies v given by

v ¼ l2 c

ð4:253Þ

and 72 is the Laplace operator given by 72 ¼

›2 ›2 2 þ ›x ›y2

ð4:254Þ

4 56


and, hence, 72 72 is the biharmonic operation 74 given by 74 ¼

›4 ›4 ›4 4 þ2 2 2 þ ›x ›x ›y ›y 4

ð4:255Þ

The solution of Equation 4.252 will require two sets of BCs for each edge of the plate (as for a beam), but will be mathematically involved. Instead of a direct solution, a logical trial solution that satisfies Equation 4.252 and the BCs is employed next for a simply supported rectangular plate. The solution tried is in fact the correct solution for the particular problem.

4.7.4

Rectangular Plate with Simply Supported Edges

As a special case, we now consider a thin rectangular plate of length a; width b; and thickness h; as shown in Figure 4.19, whose edges are simply supported. For each edge, the BCs are the displacement in zero and the bending moment about the edge zero. Specifically, we have ! 2 ›2 v 0 0 › v þ n 2 ¼ 0 for x ¼ 0 and a; 0 # y # b vðx; y; tÞ ¼ 0 and Mx ¼ E I ›x 2 ›y ! 2 ›2 v 0 0 › v vðx; y; tÞ ¼ 0 and My ¼ E I þ n 2 ¼ 0 for y ¼ 0 and b; 0 # x # a ð4:256Þ ›y 2 ›x where E0 and I 0 are as given in Equation 4.248 and Equation 4.249. In this case, the mode shapes are found to be Yij ðx; yÞ ¼ sin

ipx jpy sin a b

for i ¼ 1; 2; … and j ¼ 1; 2; …

ð4:257Þ

which clearly satisfy the BCs (Equation 4.256) and the governing model equation 4.252, with an infinite set of solutions for l given by ! 2 j2 2 2 i þ 2 ð4:258Þ lij ¼ p a2 b Hence, from Equation 4.253, the natural frequencies are i2 j2 vij ¼ p c 2 þ 2 a b

!

2

ð4:259Þ

where c is given by Equation 4.247. The overall response, then, is given by vðx; y; tÞ ¼

1 X 1 X i¼1 j¼1

Aij sin vij t þ Bij cos vij t sin

ipx ipy sin a b

ð4:260Þ

The unknown constants Aij and Bij are determined by the system initial conditions vðx; y; 0Þ and v_ ðx; y; 0Þ: The first six mode shapes of transverse vibration of a rectangular plate are sketched in Figure 4.20.

Bibliography Benaroya, H. 1998. Mechanical Vibration, Prentice Hall, Upper Saddle River, NJ. Cao, Y., Modi, V.J., de Silva, C.W., and Misra, A.K., On the control of space platform based variable geometry manipulator, Adv. Astronaut. Sci., 103, Part 1, 305 326, 1999. Caron, M., Modi, V.J., Pradhan, S., de Silva, C.W., and Misra, A.K., Planar dynamics of flexible manipulators with slewing deployable links, J. Guidance Control Dyn., 21, 4, 572 580, 1998. Chen, Y., Modi, V.J., de Silva, C.W., and Misra, A.K., On the performance of variable geometry manipulators: part I three dimensional order N formulation, Acta Astronaut., 49, 1, 1 12, 2001.


4 57

Chen, Y., Modi, V.J., de Silva, C.W., and Misra, A.K., On the performance of variable geometry manipulators: part II computer implementation and dynamical study, Acta Astronaut., 49, 1, 13 30, 2001. Crandall, S.H., Karnopp, D.C., Kurtz, E.F., and Prodmore Brown, D.C. 1968. Dynamics of Mechanical and Electromechanical Systems, McGraw Hill, New York. den Hartog, J.P. 1956. Mechanical Vibrations, McGraw Hill, New York. de Silva, C.W., A technique to model the simply supported Timoshenko beam in the design of mechanical vibrating systems, Int. J. Mech. Sci., 17, 389 393, 1975. de Silva, C.W., Dynamic beam model with internal damping, rotatory inertia and shear deformation, AIAA J., 14, 5, 676 680, 1976. de Silva, C.W. 2000. VIBRATION Fundamentals and Practice, CRC Press, Boca Raton, FL. de Silva, C. W., Buyukozturk, O., and Wormley, D. N., Postcracking compliance of RC beams, J. Struct. Div., Trans. ASCE, 105, ST1, 35 51, 1979. de Silva, C.W., and Wormley, D. N., Material optimization in a torsional guideway transit system, J. Adv. Transp., 13, 3, 41 60, 1979. de Silva, C.W., and Wormley, D.N., Torsional analysis of cutout beams, J. Struct. Div., Trans. ASCE, 106, ST9, 1933 1946, 1980. de Silva, C.W., Wormley, D.N. 1983. Automated Transit Guideways: Analysis and Design, D.C. Heath and Co., Lexington, KY. Dimarogonas, A. 1996. Vibration for Engineers, 2nd ed., Prentice Hall, Upper Saddle River, NJ. Inman, D.J. 1996. Engineering Vibration, Prentice Hall, Englewood Cliffs, NJ. Kim, S.W., Misra, A.K., Modi, V.J., Cyril, X., and De Silva, C.W., Modelling of contact dynamics of two flexible multi body systems, Acta Astronaut., 45, 11, 669 677, 1999. Meirovitch, L. 1986. Elements of Vibration Analysis, 2nd ed., McGraw Hill, New York. Rao, S. S. 1995. Mechanical Vibrations, 3rd ed., Addison Wesley, Reading, MA. Thomson, W.T., and Dahleh, M.D. 1998. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper Saddle River, NJ. Volterra E. and Zachmanoglou, E.C. 1965. Dynamics of Vibrations, Charles E. Merrill Books, Columbus, OH. Zhang, J., Modi, V.J., and De Silva, C.W., Parametric dynamical study of a novel manipulator, Adv. Astronaut. Sci., 110, 115 136, 2001.

5

Random Vibration 5.1 5.2

Random Vibration ............................................................. Single Degree of Freedom: The Response to Random Loads .................................................................... Formulation Correlations

† †

Derivation of Equations † Response Response Spectral Density

Rutgers University

52

5.3 5.4

Response to Two Random Loads ...................................... 5 7 Multi Degree of Freedom Vibration ................................ 5 12

5.5

Multi Degree of Freedom: The Response to Random Loads .................................................................... 5 17

Deterministic Vibration † Solution by Frequency Response Function † Modal Analysis

Response due to a Single Random Force † Response to Multiple Random Forces † Impulse Response Approach Modal Analysis Approach

Haym Benaroya

51

5.6

†

Continuous System Random Vibration ........................... 5 29 Transverse Vibration of Beams Vibration

†

Random Transverse

Summary This chapter summarizes the key ideas of linear random vibration. This discipline focuses on determining the response statistics of an oscillator or structure to input forces that are definable only in terms of their statistics. Typical problems include the following: (1) given the power spectrum of the force, find the power spectrum of the response; (2) given the mean value and variance of the force, find the mean value and variance of the response. The methodology is built upon the linear theory of vibration for discrete single and multi degree of freedom (DoF) systems, and continuous systems. The approaches are essentially the direct method and the modal analysis method. The direct method may also be called a transfer matrix method (see Chapter 2). Modal analysis (see Chapters 3 and 4) has the same benefit in random vibration as is done in deterministic vibration studies: it can be computationally more efficient. A number of examples are given, as are a list of representative references.

5.1

Random Vibration

The discipline of random vibration of structures was borne of the need to understand how structures respond to dynamic loads that are too complex to model deterministically. Examples include aerodynamic loading on aircraft and earthquake loading of structures. Essentially, the question that must be answered is: given the statistics (read: uncertainties) of the loading, what are the statistics (read: most likely values with bounds) of the response? Generally, for engineering applications the statistics of greatest concern are the mean, or average value, and the variance, or scatter. These concepts are discussed in detail subsequently. Suppose that we are aircraft designers currently working on the analysis and design of a wing for a new airplane. As engineers, we are very familiar with the mechanics of solids and can size the wing for static loads. Also, we have vibration experience and can evaluate the response of the wing to a harmonic 51

52


or impulsive forcing. However, this wing provides Force lift to an airplane flying through a turbulent atmosphere. Even though we are not fluid dynamicists, we know that turbulence is a very complicated physical process. In fact, the fluid (air) motion is so complicated that probabilistic models are required to model the behavior. Here, a plausibly deterministic but very complicated Time dynamic process is taken to be random for the purposes of modeling. Wing design requires force FIGURE 5.1 Turbulent force history. data resulting from the interaction between fluid and structure. Such data can be shown as the time history in Figure 5.1. The challenge is to make sense of such intricate fluctuations. The analyst and designer must run scale model tests. A wing section is set up in the wind tunnel and representative aerodynamic forces are generated. Data on wind forces and structural response are gathered and analyzed. With additional data analysis, it is possible to estimate the force magnitudes. Estimates of the mean values of these forces can be calculated, as well as of the range of possible forces. With these estimates, it is possible to study the behavior of the wing under a variety of realistic loading scenarios using the tools of probability and statistics to model this complex physical problem. This text introduces the use of probabilistic information in mechanical systems, primarily structural and dynamic systems. These tools are applicable to all the sciences and engineering, even though this text focuses on the mechanical sciences and engineering.

5.2

Single Degree of Freedom: The Response to Random Loads

Consider the second order differential equation1 governing the linear motion of an oscillator € þ 2zvn XðtÞ _ þ v2n XðtÞ ¼ FðtÞ XðtÞ

ð5:1Þ

where the input force per unit mass is given by stationary random process FðtÞ and the output displacement by random process XðtÞ: Note that it is customary to use upper case letters XðtÞ to represent random processes, and lower case letters xðtÞ to represent the realizations of a random process. The notation here is standard: 2zvn ¼ c=m where c ¼ viscous damping and m ¼ mass; and v2n ¼ k=m where k ¼ stiffness. We assume that the reader understands the concepts of impulse response and convolution. We would also like to present a priori the results which are derived in the subsequent section; the mean value of the response mX and the spectral density of the output SXX ðvÞ :

mX ¼ Hð0ÞmF SXX ðvÞ ¼ lHðvÞl2 SFF ðvÞ " #2 1=v2n 2 lHðvÞl ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 v2 =v2n Þ2 þ ð2zv=vn Þ2 These equations tell us that the mean value of the response mx is proportional to the mean value of the force mF , and that the response spectral density SXX ðvÞ is proportional to the force spectral density SFF ðvÞ: For both results, the proportionality constants depend on the structural or system parameters. 1 In this instance, the system parameters z and vn are deterministic and, therefore, so is the governing equation of motion. If z and vn are either random variables or processes then the governing equation is random. The case of random parameters is much more complicated because the system itself is random rather than just the forcing. We would have to solve the problem for the “many randomly prescribed systems” rather than for just one system with randomly prescribed forces.

Random Vibration

53

By previewing the end results, the reader will hopefully be able to better follow the mathematical manipulations which follow.

5.2.1

Formulation

Consider the linear system defined by Equation 5.1 and assume random process input FðtÞ to be stationary, with mean mF and power spectrum SFF ðvÞ: The stationarity assumption for the forcing means that transient dynamic behavior cannot be directly considered here.2 Thus, the initial loading transients of an earthquake, a wind gust, or an extreme ocean wave cannot be considered as stationary. Assuming that the character of the loading does not change, steady state behavior can be assumed to be statistically stationary.

5.2.2

Derivation of Equations

Begin with the convolution equation for the deterministic response of a linear oscillator ð1 XðtÞ ¼ gðtÞFðt tÞdt 1

ð5:2Þ

where gðtÞ is the impulse response given by gðtÞ ¼

1 e vd

zvn t

sin vd t

and stationary random load per unit mass FðtÞ is applied at t ¼ 1; that is, long before the present time. This ensures that the impulse response is stationary. Beginning with Equation 5.2, take the expected value of both sides and use the linear property of mathematical expectation to interchange it with the integral: ð1 ð1 E{XðtÞ} ¼ gðtÞE{Fðt tÞ}dt ¼ E{FðtÞ} gðtÞdt 1

¼ mF

ð1 1

1

gðtÞdt ¼ mF Hð0Þ

where the fact that FðtÞ is stationary was utilized in the second and third equations,3 and HðvÞ is the frequency response function; Hð0Þ ¼ HðvÞlv¼0 : Therefore, we arrive at the first important result

mX ¼ Hð0ÞmF ¼

1 m v2n F;

which shows us that since mF is time independent, then so must be E{xðtÞ}: In order to derive the output spectral density, we must work through the intermediate results involving the correlation function.

5.2.3

Response Correlations

For a stationary process, the autocorrelation function is given by ð1 xðtÞxðt þ tÞfX ðxÞdx RXX ðtÞ ¼ 1

where fX ðxÞ is the probability density function of the process. We cannot take the Fourier transform of this equation to find the response spectral density because we do not know the response density 2

There are, however, clever ways by which stationary solutions can be utilized in nonstationary cases. One possibility is to multiply the stationary process by a deterministic time function such that the product is an evolutionary, or nonstationary, process. For example, use AðtÞFðtÞ as the forcing function where AðtÞ is a deterministic transient function and FðtÞ is stationary. 3 The force is stationary and has a constant mean value.

54


function fX ðxÞ: There are two other equivalent ways to proceed: (i) utilize the ergodic definition of the autocorrelation,4 or (ii) use Equation 5.2, utilizing available information on FðtÞ; as follows. First, derive the cross correlation between FðtÞ and XðtÞ: Multiply both sides of Equation 5.2 by Fðt a1 Þ and take the expected values of both sides: ð1 E{XðtÞFðt a1 Þ} ¼ gðt1 ÞE{Fðt t1 ÞFðt a1 Þ}dt1 1

where E{Fðt t1 ÞFðt a1 Þ} ¼ RFF ðt1 a1 Þ and E{XðtÞFðt between loading FðTÞ and response XðtÞ: Thus, ð1 RXF ða1 Þ ¼ gðt1 ÞRFF ðt1 1

a1 Þ} ¼ RXF ða1 Þ; the cross correlation a1 Þdt1

ð5:3Þ

and RFF is known from experimental data. Next, multiply both sides of Equation 5.2 by Xðt þ a2 Þ and take expected values of both sides: ð1 E{Xðt þ a2 ÞXðtÞ} ¼ gðt2 ÞE{Xðt þ a2 ÞFðt t2 Þ}dt2 1

RXX ða2 Þ ¼

ð1

1

gðt2 ÞRXF ðt2 þ a2 Þdt2

Substitute Equation 5.3 into Equation 5.4 to find ð1 ð1 RXX ðtÞ ¼ gðaÞgðbÞRFF ðt þ b 1

1

ð5:4Þ

aÞda db

ð5:5Þ

which is a double convolution. To evaluate the variance, we use

s2X ¼ E{XðtÞ2 }

E2 {XðtÞ} ¼ RXX ð0Þ

½Hð0ÞE{FðtÞ}

2

While this information on the correlation is of interest, a more important result is the response spectral density that we derive in the next section.

5.2.4

Response Spectral Density

Begin with the Fourier transform relation between power spectrum and correlation function 1 ð1 SXX ðvÞ ¼ RXX ðtÞe ivt dt; 2p 1 and substitute Equation 5.5 for RXX ðtÞ; with l ¼ t þ b 1 ð1 e 2p 1 ð1 ¼ gðaÞe

SXX ðvÞ ¼

1

ivt iva

ð1 ð1 da

1 ð1

¼ HðvÞH p ðvÞSFF ðvÞ

1

1

a:

gðaÞgðbÞRFF ðlÞda db dt

gðbÞeþivb db £

1 ð1 RFF ðlÞe 2p 1

ivl

dl

by definition, where p denotes complex conjugate and therefore SXX ðvÞ ¼ lHðvÞl2 SFF ðvÞ

ð5:6Þ

This is the fundamental result for random vibration and linear systems theory that allows us to evaluate the output spectral density, given the input spectral density and the system frequency response. 4

Rxx ðtÞ ¼ limT!1

1 ðþT xðtÞxðt þ tÞdt: 2T T

Random Vibration

55

It is emphasized here that the derivation of Equation 5.6 made use of the convolution equation that is valid for linear systems and structures. Any generalization for nonlinear behavior requires problem specific approaches.

Example 5.1

Oscillator Response to White Noise

Consider a simple application of the above ideas to an oscillator. What is the response of a damped oscillator to a force with white noise density?

Solution The governing equation of motion is € þ 2zvn XðtÞ _ þ v2n XðtÞ ¼ FðtÞ XðtÞ where FðtÞ is the external force per unit mass, the system transfer function is given by HðvÞ ¼

1 v2n þ i2zvn v þ ðivÞ2

and the squared magnitude of HðvÞ is given by lHðvÞl2 ¼

ðv2n

1 v Þ þ ð2zvn vÞ2 2 2

Therefore, given any input spectral density SFF ðvÞ; the response spectral density is SXX ðvÞ ¼ lHðvÞl2 SFF ðvÞ ¼

ðv2n

SFF ðvÞ 2 v Þ2 þ ð2zv

n vÞ

2

Suppose, for mathematical simplicity, that the forcing is white noise, SFF ðvÞ ¼ S0 : Then, SXX ðvÞ ¼

ðv2n

S0 v2 Þ2 þ ð2zvn vÞ2

ð5:7Þ

and the mean square response is given by ð1 pm2 S0 pS0 ¼ E{XðtÞ2 } ¼ SXX ðvÞdv ¼ kc 2v3n z 1 The mean square response can also be written in terms of a one sided spectrum pm2 S0 m2 W0 E{XðtÞ2 } ¼ ¼ kc 4kc where the one sided density W0 is related to the two sided density by S0 ¼ pW0 =4: This integral is not standard, but can be found in texts on random vibration.5 Even though infinite mean square energy is input to the system,6 it responds with finite mean square energy. See Figure 5.2 for plots of the components of Equation 5.7. Only the positive frequency ranges are plotted as they are 5

For example, the integral of this example problem is a specialized version of 2 ð1 B0 þ ivB1 pðA0 B21 þ A2 B20 Þ dv ¼ 2 A0 A1 A2 1 A0 þ ivA1 2 v A2

where A0 ¼ v2n ; 6

A1 ¼ 2zvn ;

A2 ¼ 2;

B0 ¼ 1;

B1 ¼ 0:

The energy input equals to the area under the spectral density, which for white noise is ð1 S0 dv ¼ 1: 1

56


symmetric about the abscissae. White noise is useful and frequently used, even though it is nonphysical, because it leads to good approximate results.

SFF

Input Spectrum So

ω

Example 5.2 Response to Colored Noise Suppose the same system as in the last example is subjected to more complex loading, where the spectral density of the forcing is not a constant, but a function of v: How would the above analysis change?

|H(iω)|2 System function

ω Solution ωn The output spectral density becomes a more complex function of frequency, for example, if SXX the loading density is similar to those found for Response spectrum wind loads. Then, the mean square response must be evaluated numerically. The applied problems are always solved numeri cally, although hopefully after some significant ω analytical exposition. There are numerical ωn methods that are specifically geared to handling uncertainties. Of particular note is the group FIGURE 5.2 SFF ðvÞ; lHðivÞl2 ; SXX ðvÞ: known as Monte Carlo methods. These methods utilize the massive computational power available today to account for uncertainties. This is accomplished by generating random numbers that are used to represent random parameters. For each of these generated random values, the program recalculates the problem. After running enough cycles to ensure the convergence of the statistics, these numerical realizations are averaged to find the mean value and variance of the relevant parameters. We summarize the key results for the random vibration of a single DoF linear oscillator in Table 5.1 to Table 5.3. Figure 5.3 shows some of the most important correlation and spectral density pairs.

TABLE 5.1

Mean Value Response

mX ¼ Hð0ÞmF 2 6 lHðvÞl2 ¼ 4 q ð1 2

1=v2n 2 2 2 v =vn Þ þ

32 ð2zv=vn Þ2

7 5

mF is known from force data

TABLE 5.2 RXX ðtÞ ¼

s2X

Output Correlation Function/Variance

Ð1 Ð1 1

1

gðaÞgðbÞRFF ðt þ b 2 aÞda db

¼ RXX ð0Þ 2 ½Hð0ÞE{FðtÞ}

2

RFF ðt þ b 2 aÞ is known from force data

Random Vibration

57 TABLE 5.3

Output Spectral Density

SXX ðvÞ ¼ lHðvÞl2 SFF ðvÞ lHðvÞl2 from structural parameters SFF ðvÞ is the input force power spectrum

FIGURE 5.3

5.3

Correlation functions and corresponding power spectral densities.

Response to Two Random Loads

Previously, the system responses were due to a single randomly varying force. In general, the situation is more complicated, because more than one load may act on a system and the resulting response depends not only on the properties of each force, but also on the correlation between the two forces.

58


Consider the response of the system to two Q(t) P(t) random loadings, PðtÞ and QðtÞ; acting simul taneously but at different points on the system, as X(t) shown in Figure 5.4. We are interested in calculating the response statistics of the displace ment XðtÞ at an arbitrary point on the system. Assume that E{PðtÞ} ¼ 0 and E{QðtÞ} ¼ 0: Also, by utilizing available data, we are able to estimate FIGURE 5.4 Response to two random loads. RPP ðtÞ and RQQ ðtÞ: Our interest is in evaluating RXX ðtÞ and its Fourier transform SXX ðvÞ: Using linear superposition and the convolution integral, the response due to both forces is given by ð1 XðtÞ ¼ gXP ðt1 ÞPðt t1 Þ þ gXQ ðt1 ÞQðt t1 Þ dt1 1

Similarly, for Xðt þ tÞ :

ð1

Xðt þ tÞ ¼

1

gXP ðt2 ÞPðt þ t

t2 Þ þ gXQ ðt2 ÞQðt þ t

t2 Þ dt2

where gXP is the impulse response function at coordinate X due to force P and gXQ is the impulse response function at X due to Q: Then, ð1 gXP ðt1 ÞPðt t1 Þ þ gXQ ðt1 ÞQðt t1 Þ dt1 RXX ðtÞ ¼ E{XðtÞXðt þ tÞ} ¼ E 1

ð1 1

gXP ðt2 ÞPðt þ t

t2 Þ þ gXQ ðt2 ÞQðt þ t

t2 Þ dt2

Now expand the product, and then move the expectation operator to the random processes, as follows: ð1 ð1 RXX ðtÞ ¼ gXP ðt2 ÞE{Pðt t1 ÞPðt þ t t2 Þ}dt2 dt1 gXP ðt1 Þ þ þ þ

1

1

ð1

ð1

gXP ðt1 Þ

1

ð1

gXQ ðt1 Þ

1

ð1

gXQ ðt1 Þ

1

1

ð1 1

ð1 1

gXQ ðt2 ÞE{Pðt

t1 ÞQðt þ t

t2 Þ}dt2 dt1

gXP ðt2 ÞE{Qðt

t1 ÞPðt þ t

t2 Þ}dt2 dt1

gXQ ðt2 ÞE{Qðt

t1 ÞQðt þ t

t2 Þ}dt2 dt1

In this expression E{Pðt

t1 ÞPðt þ t

t2 Þ} ¼ RPP ðt þ t1

t2 Þ

E{Qðt

t1 ÞQðt þ t

t2 Þ} ¼ RQQ ðt þ t1

t2 Þ

The expectations in the second and third terms are cross correlations of the form RPQ ðtÞ ¼ E{PðtÞQðt þ tÞ}: Therefore, the autocorrelation of the response becomes RXX ðtÞ ¼

ð1 þ þ þ

gXP ðt1 Þ

1 ð1

1

ð1

1

ð1

1

ð1

gXP ðt1 Þ gXQ ðt1 Þ gXQ ðt1 Þ

gXP ðt2 ÞRPP ðt þ t1

1 ð1

1

ð1

1

ð1

1

t2 Þdt2 dt1

gXQ ðt2 ÞRPQ ðt þ t1

t2 Þdt2 dt1

gXP ðt2 ÞRQP ðt þ t1

t2 Þdt2 dt1

gXQ ðt2 ÞRQQ ðt þ t1

t2 Þdt2 dt1

Random Vibration

59

The importance of this result is primarily in the observation that we cannot derive RXX ðtÞ unless the cross correlations RPQ ðtÞ and RQP ðtÞ are also known. Using the Fourier transform relation between RXX ðtÞ and SXX ðvÞ; that is 1 ð1 RXX ðtÞe ivt dt SXX ðvÞ ¼ 2p 1 we obtain p p ðvÞHXP ðvÞSPP ðvÞ þ HXP ðvÞHXQ ðvÞSPQ ðvÞ SXX ðvÞ ¼ HXP

p p ðvÞHXQ ðvÞSQQ ðvÞ ðvÞHXP ðvÞSQP ðvÞ þ HXQ þ HXQ

ð5:8Þ

where p HXP ðvÞHXP ðvÞ ¼ lHXP ðvÞl2 p HXQ ðvÞHXQ ðvÞ ¼ lHXQ ðvÞl2

and SPQ ðvÞ ¼

1 ð1 RPQ ðtÞe 2p 1

ivt

dt

As expected, the evaluation of the output spectral density requires knowledge about the cross spectra SPQ ðvÞ and SQP ðvÞ: If there are more than two forces, then we will have additional cross spectra between each pair of forces. Examining Equation 5.8 closely, we find that we can write SXX ðvÞ as " #" # i SPP ðvÞ SPQ ðvÞ h HXP ðvÞ p p SXX ðvÞ ¼ HXP ðvÞ HXQ ðvÞ ð5:9Þ SQP ðvÞ SQQ ðvÞ HXQ ðvÞ

Example 5.3 Conjugates of Cross Spectra It was briefly mentioned that RPQ ðtÞ RQP ð tÞ: How are SPQ ðvÞ and SQP ðvÞ related? Solution By the definition of spectral density SPQ ðvÞ ¼ Replacing RPQ ðtÞ with RQP ð tÞ SPQ ðvÞ ¼ Letting

1 ð1 RPQ ðtÞexpð ivtÞdt 2p 1 1 ð1 RQP ð tÞexpð ivtÞdt 2p 1

t¼t SPQ ðvÞ ¼

1 ð1 RPQ ðtÞexpðivtÞdt 2p 1

Then, SPQ ðvÞ ¼ SQP ð vÞ ¼ SpQP ðvÞ

Example 5.4

Response Spectrum due to Two Random Loads

Consider a mass spring damper system in Figure 5.5 subject to two random forces PðtÞ and QðtÞ: Find the response spectrum SXX ðvÞ assuming that SPP ðvÞ ¼ SP ;

SPQ ðvÞ ¼ 0;

SQQ ðvÞ ¼ SQ

5 10


Solution The equation of motion for this system is given by mX€ þ cX_ þ kX ¼ PðtÞ þ QðtÞ

X(t) k

First, assume that QðtÞ ¼ 0 in order to first obtain HXP ðvÞ: Taking the Fourier transform, the equation of motion is given by

Q(t)

m c

P(t)

ð mv2 þ civ þ kÞXðvÞ ¼ PðvÞ Then, the frequency response function HXP ðvÞ is 1 HXP ðvÞ ¼ 2 ð mv þ civ þ kÞ

FIGURE 5.5 Single DoF system subjected to two random loads.

or 1 mð v2 þ 2vn ziv þ v2n Þ

HXP ðvÞ ¼

Similarly, HXQ ðvÞ is obtained by setting PðtÞ ¼ 0 and is also given by 1 HXQ ðvÞ ¼ 2 mð v þ 2vn ziv þ v2n Þ Then, the spectral density of the response is given by SXX ðvÞ ¼

h

m 2 4

v2n

v

2

2vn ziv

1

ðmðv2n

v2 þ 2vn zivÞÞ

ðmðv2n

v2 þ 2vn zivÞÞ

m 1

v2n

v

2vn ziv

3

1

2 i SPP ðvÞ 4 SQP ðvÞ

SPQ ðvÞ

3 5

SQQ ðvÞ

5 ¼ SPP ðvÞ þ SPQ ðvÞ þ SQP ðvÞ þ SQQ ðvÞ m2 ðv2n v2 Þ2 þ ð2vn zÞ2 1

In our case, the spectral density is reduced to SXX ðvÞ ¼

2

SP þ SQ m2

v2n

2

v2 þ 2vn z

2

Spectral densities are rather complicated expressions to evaluate in general. A number of special cases will help better understand the effects of the cross terms: 1. PðtÞ and QðtÞ arise from independent sources and are, therefore, uncorrelated.7 Then, RPQ ðtÞ ¼ 0; RQP ðtÞ ¼ 0 and SPQ ðvÞ ¼ 0; SQP ðvÞ ¼ 0: 2. PðtÞ and QðtÞ are directly correlated; that is, QðtÞ ¼ kPðtÞ where k is a constant. 3. PðtÞ and QðtÞ are exponentially correlated, E{PðtÞQðt þ tÞ} ¼ kPQ exp{ at} where kPQ is a constant. 4. PðtÞ and QðtÞ are correlated in a “simplified” exponential; that is, with a triangular correlation defined by E{PðtÞQðt þ tÞ} ¼ kPQ ð1 t=t1 Þ; t1 # t # t1 :

7

Independence implies that E{Pðt1 ÞQðt2 Þ} ¼ E{Pðt1 Þ}E{Qðt2 Þ}

They are uncorrelated if CovðPðt1 ÞQðt2 ÞÞ ¼ E{Pðt1 ÞQðt2 Þ} 2 E{Pðt1 Þ}E{Qðt2 Þ} ¼ 0 Independent processes are always uncorrelated whereas uncorrelated processes may not be independent.

Random Vibration

5 11

We consider in more detail the first two cases listed above. Where the loads are independent, because the cross correlations are identically zero, the output spectral density is just the sum of the two respective spectral densities obtained with the forces acting separately: p p SXX ðvÞ ¼ HXP ðvÞHXP ðvÞSPP ðvÞ þ HXQ ðvÞHXQ ðvÞSQQ ðvÞ

¼ lHXP ðvÞl2 SPP ðvÞ þ lHXQ ðvÞl2 SQQ ðvÞ

ð5:10Þ

We note that the output spectral density for a linear system follows a strict interpretation of the principle of linear superposition only when the forces are uncorrelated. For the case where QðtÞ ¼ kPðtÞ, we have RPQ ðtÞ ¼ E{PðtÞkPðt þ tÞ} ¼ kRPP ðtÞ RQP ðtÞ ¼ E{kPðtÞPðt þ tÞ} ¼ kRPP ðtÞ RQQ ðtÞ ¼ E{kPðtÞkPðt þ tÞ} ¼ k2 RPP ðtÞ Then, we can obtain the respective spectral densities SPQ ðvÞ ¼ SQP ðvÞ ¼ kSPP ðvÞ SQQ ðvÞ ¼ k2 SPP ðvÞ leading to the spectral density of the response p p p ðvÞHXP ðvÞSPP ðvÞ þ HXP ðvÞHXQ ðvÞkSPP ðvÞ þ HXQ ðvÞHXQ ðvÞkSPP ðvÞ SXX ðvÞ ¼ HXP p p p þ kHXQ ÞSPP ðvÞ þ HXQ ðvÞHXQ ðvÞk2 SPP ðvÞ ¼ ðHXP þ kHXQ ÞðHXP

ð5:11Þ

2

¼ lHXP ðvÞ þ kHXQ ðvÞl SPP ðvÞ This last expression is related to the relative phase between the two functions HXP ðvÞ and HXQ ðvÞ: The addition of two frequency response functions HXP þ kHXQ is shown graphically in Figure 5.6. Suppose SPP ðvÞ ¼ SQQ ðvÞ ¼ SðvÞ Then, from Equation 5.10, for independent loadings SXX ðvÞ ¼ ½lHXP l2 þ lHXQ l2 SðvÞ

ð5:12Þ

If the forces are directly correlated with parameter k ¼ 1; Equation 5.11 yields SXX ðvÞ ¼ lHXP þ HXQ l2 SðvÞ ¼ ½lHXP l2 þ lHXQ l2 þ 2lHXP llHXQ lcos f SðvÞ

ð5:13Þ

where f is the phase difference between HXP and HXQ as shown in Figure 5.6. The law of cosines is used for the second relation. Therefore, a comparison of Equation 5.12 with Equation 5.13 shows that the results of an uncorrelated loading will be identical to those that are correlated where Imaginary cos f ¼ 0; that is, f ¼ ^p=2: This is when the two vectors in Figure 5.6 are perpendicular to each other. For other values of f, the spectral density in the correlated case may have any value in the HXP +kHXQ range defined by ½lHXP l2 ^ lHXQ l2 SðvÞ depending kHXQ on the value of f: If, at some frequency, HXP ¼ HXQ , the spectral HXP density at that frequency for the correlated case φ with k ¼ 1 will be zero. For any case where HXP ¼ HXQ , the spectral density with correlation Real will be twice that obtained without correlation. Another specialized result is where QðtÞ follows FIGURE 5.6 Addition of frequency response PðtÞ after a lag of t0 so that QðtÞ ¼ Pðt þ t0 Þ: functions.

5 12


Then, RPQ ðtÞ ¼ E{PðtÞPðt þ t0 þ tÞ} ¼ RPP ðt0 þ tÞ with the respective spectral density 1 ð1 1 ð1 SPQ ðvÞ ¼ RPQ ðtÞe ivt dt ¼ RPP ðt0 þ tÞe ivt dt 2p 2p 1 1 1 ivt0 ð1 RPP ðt0 þ tÞe ivðt0 þtÞ dðt0 þ tÞ ¼ eivt0 SPP ðvÞ e ¼ 2p 1 Since SPQ ðvÞ ¼ SQP ðvÞ are complex conjugates SQP ðvÞ ¼ e

ivt0

SPP ðvÞ

Also, SPP ðvÞ ¼ SQQ ðvÞ: Then, p p ðvÞ þ eivt0 HXP ðvÞHXQ ðvÞ þ e SXX ðvÞ ¼ HXP ðvÞHXP

5.4

ivt0

p p HXQ ðvÞHXP ðvÞ þ HXQ ðvÞHXQ ðvÞ SPP ðvÞ

Multi-Degree-of-Freedom Vibration

In this section, a review of deterministic vibration is provided. These results are used in the next section, where the forcing is taken to be a random process.

5.4.1

Deterministic Vibration

Consider a case where there is more than one DoF. The equations of motion can be written as

F1(t) c1

½m {€xðtÞ} þ ½c {_xðtÞ} þ ½k {xðtÞ} ¼ {FðtÞ}

F2(t)

x1(t)

x2(t)

c2

m2 For an N DoF system, the matrices ½m ; ½c and ½k k1 k2 are of dimension N £ N; and the response {xðtÞ} and force {FðtÞ} vectors are of dimension N £ 1: For purposes of demonstration and discussion, FIGURE 5.7 Two DoF mass spring damper system. the necessary concepts will be introduced primar ily by working through the solution of a two DoF system. All these ideas transfer to larger systems, but with the two DoF model we can demonstrate the key ideas without the complications of the major algebraic and numerical demands made by the larger systems. For the system shown in Figure 5.7, we can derive the coupled equations of motion using either Newton’s Second Law of motion applied to a free body diagram for each mass, as shown in Figure 5.8, or by Lagrange’s equation. In either case, we find the governing equations to be m1

m1 x€ 1 ðtÞþðc1 þ c2 Þ_x1 ðtÞ þ ðk1 þ k2 Þx1 ðtÞ m2 x€ 2 ðtÞ þ c2 x_ 2 ðtÞ

c2 x_ 2 ðtÞ

c2 x_ 1 ðtÞ

k2 x2 ðtÞ ¼ F1 ðtÞ;

k2 x1 ðtÞ þ k2 x2 ðtÞ ¼ F2 ðtÞ

or in matrix form "

m1

0

0

m2

#(

x€ 1 ðtÞ x€ 2 ðtÞ

"

) þ

c1 þ c2 c2

c2 c2

#(

x_ 1 ðtÞ x_ 2 ðtÞ

"

) þ

k1 þ k2 k2

k2 k2

#(

x1 ðtÞ x2 ðtÞ

)

( ¼

F1 ðtÞ F2 ðtÞ

) ð5:14Þ

Random Vibration

5 13

F1(t)

k1 x1 c1 x1

m1

FIGURE 5.8

F2(t)

x1(t) k2(x2−x1)

x2(t)

m2

c2(x2−x1)

Free body diagram of a two DoF system.

In this case, the mass, damping, and stiffness matrices are given by " # " # " c1 þ c2 c2 k1 þ k2 m1 0 ; ½c ¼ ; ½k ¼ ½m ¼ 0 m2 c2 c2 k2

5.4.2

k2

#

k2

Solution by Frequency Response Function

Begin by taking the Fourier transform of the equations of motion, obtaining ð v2 ½m þ iv½c þ ½k Þ{XðvÞ} ¼ {FðvÞ} where {XðvÞ} and {FðvÞ} are Fourier transforms of {xðtÞ} and {FðtÞ}; respectively. The matrix ð v2 ½m þ iv½c þ ½k Þ is denoted as ½ZðvÞ : Then, {XðvÞ} ¼ ½ZðvÞ The matrix ½ZðvÞ

1

1

{FðvÞ}

is identical to the frequency response matrix denoted as ½HðvÞ : {XðvÞ} ¼ ½HðvÞ {FðvÞ}

For the two DoF system in Figure 5.7 " m2 v2 þ ivc2 þ k2 1 ½HðvÞ ¼ det½Z ivc2 þ k2 det½Z ¼ v4 m1 m2

ð5:15Þ ivc2 þ k2

#

m1 v2 þ ivðc1 þ c2 Þ þ ðk1 þ k2 Þ

iv3 ðm1 c2 þ c1 m2 þ c2 m2 Þ þ ivðc1 k2 þ k1 c2 Þ

v2 ðm1 k2 þ c1 c2 þ k1 m2 þ k2 m2 Þ þ k1 k2 Note that the frequency response function of a two DoF system is given as a matrix with dimension of 2 £ 2. To clarify the meaning of each element, let us expand Equation 5.15: X1 ðvÞ ¼ H11 ðvÞF1 ðvÞ þ H12 ðvÞF2 ðvÞ X2 ðvÞ ¼ H21 ðvÞF1 ðvÞ þ H22 ðvÞF2 ðvÞ Now, the meaning of each element is clear. Each element Hij ðvÞ is the frequency response function for coordinate i due to a force at j: In general, ½HðvÞ is a symmetric matrix since ½m ; ½c and ½k are symmetric. For a deterministic response, recall that the impulse response function is related to the frequency response function by 1 ð1 gij ðtÞ ¼ Hij ðvÞexpðivtÞdv 2p 1

5 14


which is written for each element in the matrix. For the two DoF system 1 ð1 m2 v2 þ ivc2 þ k2 g11 ðtÞ ¼ expðivtÞdv 2p det½Z 1 1 ð1 ivc2 þ k2 expðivtÞdv g12 ðtÞ ¼ 2p 1 det½Z 1 ð1 ivc2 þ k2 expðivtÞdv ¼ g12 ðtÞ g21 ðtÞ ¼ 2p 1 det½Z 1 ð1 m1 v2 þ ivðc1 þ c2 Þ þ ðk1 þ k2 Þ expðivtÞdv g22 ðtÞ ¼ det½Z 2p 1 Again, ½gðtÞ is a symmetric matrix. Using the convolution integral, the response xi ðtÞ due to Fj is ð1 xi ðtÞ ¼ gij ðtÞFj ðt tÞdt 1

The total response xi ðtÞ of mass mi is then equal to the sum of the individual responses to each of the forces. In our case, the responses are ð1 x1 ðtÞ ¼ ½g11 ðtÞF1 ðt tÞ þ g12 ðtÞF2 ðt tÞ dt 1

x2 ðtÞ ¼

ð1

1

½g21 ðtÞF1 ðt

tÞ þ g22 ðtÞF2 ðt

tÞ dt

We can extend our analysis to a system with N DoF. The equations of motion are written in matrix form as ½m {€xðtÞ} þ ½c {_xðtÞ} þ ½k {xðtÞ} ¼ {FðtÞ} where {xðtÞ} and {FðtÞ} are response and force vectors and the frequency response matrix is given by ½HðvÞ ¼ ð v2 ½m þ iv½c þ ½k Þ

1

The impulse response matrix ½gðtÞ is given by ½gðtÞ ¼

1 ð1 ½HðvÞ expðivtÞdv 2p 1

and the response matrix is given by {xðtÞ} ¼

ð1 1

½gðtÞ {Fðt

tÞ}dt

Note that the impulse response vector is not trivial to evaluate, much less the convolution integrals. More importantly, this method becomes much harder for each additional DoF. Therefore, this method is rarely used, and we often rely on modal analysis.

5.4.3

Modal Analysis

Let us first consider an undamped system in free vibration with equations of motion ½m {€xðtÞ} þ ½k {xðtÞ} ¼ 0 If we assume that {xðtÞ} is harmonic with frequency v; {xðtÞ} ¼ A0 {u}eivt the equations of motion are reduced to the eigenvalue problem given by ð v2 ½m þ ½k Þ{u} ¼ {0}

ð5:16Þ

Random Vibration

5 15

The goal is to find the nonzero vector {u}: Therefore, equations must be linearly dependent,8 or there should be no inverse for ð v2 ½m þ ½k Þ: This can be satisfied if the determinant of ð v2 ½m þ ½k Þ is zero:

l

v2 ½m þ ½k l ¼ 0

There are several values of v2 which can make the determinant equal zero, and these values of v2 are called eigenvalues. For each nonrepeating eigenvalue, there is a corresponding vector {u} that satisfies Equation 5.16. This vector is called the eigenvector. Physically, the eigenvalues of this problem are the natural frequencies squared, and the eigenvectors are the mode shapes, since they represent the respective motion of each modal coordinate. By solving this eigenvalue problem, we can obtain N sets of eigenvalues v2i with corresponding eigenvectors {u}i for an N DoF system.9 It is customary that the eigenvalues are arranged in ascending order

v21 , v22 , · · · , v2N The eigenvectors can be normalized with respect to the mass matrix, {u}Ti ½m {u}i ¼ 1; where

i ¼ 1; 2; …; N

9 8 u1i > > > > > > = < .. {u}i ¼ . > > > > > > ; : uNi

For the ith and jth set of eigenvalues and eigenvectors, we can write

v2i ½m {u}i þ ½k {u}i ¼ {0} v2j ½m {u}j þ ½k {u}j ¼ {0} Multiplying the first equation by {u}Tj and the second equation by {u}Ti ; we obtain

v2i {u}Tj ½m {u}i þ {u}Tj ½k {u}i ¼ {0}

ð5:17Þ

v2j {u}Ti ½m {u}j þ {u}Ti ½k {u}j ¼ {0}

ð5:18Þ

Take the transpose of Equation 5.18,

v2j {u}Tj ½m T {u}i þ {u}Tj ½k T {u}i ¼ {0} Since ½m and ½k are symmetric, ½m ¼ ½m

T

and ½k ¼ ½k T ; and

v2j {u}Tj ½m {u}i þ {u}Tj ½k {u}i ¼ {0} 8

or

An example of a linear dependent equation is

"

1

2

2

4

#(

u1 u2

) ¼

ð5:19Þ

( ) 0 0

u1 þ 2u2 ¼ 0 2u1 þ 4u2 ¼ 0 In this case, u1 and u2 need not be zeros. Instead, the equations are satisfied as long as u1 ¼ 2u2 : Therefore, there is an infinite number of solutions. 9 An eigenvalue problem will be demonstrated in the next example.

5 16


Subtracting Equation 5.19 from Equation 5.17, we obtain ðv2j

v2i Þ{u}Tj ½m {u}i ¼ 0

If the eigenvalues are unique ðv2j – v2i Þ; then {u}Tj ½m {u}i ¼ 0;

i–j

{u}Tj ½k {u}i ¼ 0;

i–j

From Equation 5.19, similarly

That is, the eigenvectors are orthogonal (or normal) to each other with respect to the mass and the stiffness matrices. If the eigenvectors are normalized with respect to the mass matrix, Equation 5.19 with i ¼ j can be written as

v2i {u}Ti ½m {u}i þ {u}Ti ½k {u}i ¼ {0} and therefore {u}Ti ½k {u}i ¼ v2i In summary, we just found that the eigenvectors have the following properties: ( 1; i ¼ j {u}Ti ½m {u}j ¼ 0; i – j and

( {u}Ti ½k

{u}j ¼

v2i ; i ¼ j 0;

i–j

If we construct a matrix composed of eigenvectors ½P ¼ ½ {u}1

···

{u}N

ð5:20Þ

we can write ½p T ½m ½p ¼ ½I 2 2 3 v1 0 0 6 7 6 7 .. 7 ¼ diagðv2 Þ ½p T ½k ½p ¼ 6 6 0 7 . 0 4 5 2 0 0 vN where ½I is the identity matrix and [P] is called the modal matrix. Now, let us return to our original equations of motion given by ½m {€xðtÞ} þ ½c {_xðtÞ} þ ½k {xðtÞ} ¼ {FðtÞ} We restrict our problem to the proportional damping case, where ½c is a linear combination of ½m and ½k : ½c ¼ a½m þ b½k with constant a and b: Let us define a new set of coordinates {zðtÞ}; so that {xðtÞ} ¼ ½P {zðtÞ} where ½p is the modal matrix defined in Equation 5.20. The equations of motion then become ½m ½P {€zðtÞ} þ ½c ½P {_zðtÞ} þ ½k ½P {zðtÞ} ¼ {FðtÞ}

ð5:21Þ

Random Vibration

5 17

Multiplying by ½P T ; ½P T ½m ½P {€zðtÞ} þ ½P T ½c ½P {_zðtÞ} þ ½P T ½k ½P {zðtÞ} ¼ ½P T {FðtÞ} which reduces to {€zðtÞ} þ ða½I þ b £ diagðv2 ÞÞ{_zðtÞ} þ diagðv2 Þ{zðtÞ} ¼ ½P T {FðtÞ} or z€1 ðtÞ þ a þ bv21 z_1 ðtÞ þ v21 zðtÞ ¼ {u}T1 {FðtÞ} .. . z€N ðtÞ þ a þ bv2N z_N ðtÞ þ v2N zðtÞ ¼ {u}TN {FðtÞ} It is clear now why proportional damping is considered. The damping matrix ½c becomes diagonalized so that the equations of motion are decoupled. The coordinates zi ðtÞ are called the modal coordinates, and {u}Ti {FðtÞ} are called the modal forces. We often let

a þ bv2i ¼ 2vi zi {u}Ti {FðtÞ} ¼ qi ðtÞ to simplify the equations to z€1 ðtÞ þ 2v1 z1 z_1 ðtÞ þ v21 z1 ðtÞ ¼ q1 ðtÞ .. .

ð5:22Þ

z€N ðtÞ þ 2vN zN z_N ðtÞ þ v2N zðtÞ ¼ qN ðtÞ The solution to these equations can be obtained using the impulse response function and the convolution integral for each coordinate ð1 zi ðtÞ ¼ gi ðtÞgi ðt tÞdt; i ¼ 1; 2; …; N 1

where

and

8 1 > < e gi ðtÞ ¼ vdi > : 0;

zvi t

sin vdi t; t $ 0 t,0

qffiffiffiffiffiffiffiffi vdi ¼ vi 1 z2i

Once we find all the zi ðtÞ; we recover the physical coordinates by using the transformation in Equation 5.21 or {xðtÞ} ¼ ½P {zðtÞ}

5.5

Multi-Degree-of-Freedom: The Response to Random Loads

From the analysis of the Single DoF system, we know that knowledge of the frequency response function HðvÞ is crucial in obtaining the response statistics to random loads. For a system with more than one DoF, there is more than one frequency response function HðvÞ: For an N DoF system, the frequency response functions will be given as an N £ N symmetric matrix HðtÞ:

5 18


In the single DoF system, the response statistics were given in terms of the mean response mX and the spectral density of the response SXX ðvÞ: We will find here that the response statistics of an N DoF system are given in terms of the mean response for each DoF, mX1 ; …; mXN ; and a matrix of spectral densities of the response SðvÞ; which is also of dimension N £ N: We present a priori the results which are derived in the subsequent section; the mean response vector mX and the matrix spectral density of the output SXX ðvÞ: mX ¼ HXX ð0ÞmF SXX ðvÞ ¼ HpXX ðvÞSFF ðvÞHTXX ðvÞ Here, we will first obtain the frequency response function of a typical two DoF system. We will then consider a two DoF system subjected to a single random load. Finally, we will expand our analysis to an N DoF system subjected to N random loads.

5.5.1

Response due to a Single Random Force

When a body vibrates, points on the body move relative to each other. To describe this relative motion of two points on a body subjected to random loading, we must know both their cross spectral densities and cross correlations, and also the spectral densities and the autocorrelation function of the individual motions of each point. This requirement is true even though there is only one force acting on the system. This is similar to the problem of the single mass loaded by two random loads. Consider a body excited by a single randomly varying force PðtÞ with autocorrelation RTT ðtÞ and spectral density SPP ðvÞ. Owing to this loading, two points on the body have displacements XðtÞ and YðtÞ with means mX and mY ; autocorrelations RXX ðtÞ and RYY ðtÞ; and spectral densities SXX ðvÞ and SYY ðvÞ; respectively. Note again that upper case letters are used to signify random processes. We know the following from our earlier studies E{XðtÞ} ¼ HXP ð0ÞmP E{YðtÞ} ¼ HYP ð0ÞmP and SXX ðvÞ ¼ lHXP ðvÞl2 SPP ðvÞ

ð5:23Þ

SYY ðvÞ ¼ lHYP ðvÞl2 SPP ðvÞ

ð5:24Þ

where HXP ðvÞ ¼ HYP ðvÞ ¼

ð1 1

ð1 1

gXP ðtÞe

ivt

dt

gYP ðtÞe

ivt

dt

and XðtÞ ¼ YðtÞ ¼

ð1 1

ð1 1

gXP ðtÞPðt

tÞdt

gYP ðtÞPðt

tÞdt

Random Vibration

5 19

With these relations, we can now find the expressions for the cross correlation RXY ðtÞ and cross spectral density SXY ðvÞ: Using the definition of a cross correlation function, we obtain RXY ðtÞ ¼ E{XðtÞYðt þ tÞ} ¼ E ¼ ¼

ð1 1

ð1 1

gXP ðt1 Þ gXP ðt1 Þ

ð1 1

ð1 1

ð1 1

gXP ðt1 ÞPðt

gYP ðt2 ÞE{Pðt

t1 Þdt1

t1 ÞPðt þ t

gYP ðt2 ÞRPP ðt þ t1

ð1 1

gYP ðt2 ÞPðt þ t

t2 Þdt2

t2 Þ}dt2 dt1

t2 Þdt2 dt1

Even though we have an explicit expression, it is generally not easy to interpret. The cross spectral density can be derived by using the Fourier transform relation between RXY ðtÞ and SXY ðvÞ: Thus, ð1 1 ð1 1 ð1 ð1 gYP ðt2 ÞRPP ðt þ t1 t2 Þdt2 dt1 e RXY ðtÞe ivt dt ¼ gXP ðt1 Þ 2p 1 2p 1 1 1 ð1 ð1 1 ð1 gXP ðt1 Þeþivt1 ¼ gYP ðt2 Þe ivt2 RPP ðt þ t1 t2 Þe ivðtþt1 t2 Þ dt dt2 dt1 2p 1 1 1

SXY ðvÞ¼

ivt

dt ð5:25Þ

In order to integrate this last expression, it is easier to proceed by transforming the variables in the innermost integral according to

l ; t þ t1 t2 and dl ¼dt where t1 and t2 are dummy variables. Then, Equation 5.25 becomes SXY ðvÞ¼

ð1 1 ð1 gXP ðt1 Þeþivt1 dt1 gYP ðt2 Þe 2p 1 1

ivt2

dt2

ð1 1

RPP ðlÞe

ivl

dl

p ðvÞHYP ðvÞSPP ðvÞ ¼HXP

ð5:26Þ

Where points X and Y coincide, this result reduces to the classical fundamental relation SXX ðvÞ¼lHXP l2 SPP ðvÞ: Equation 5.23, Equation 5.24 and Equation 5.26 can be written in matrix form "

or

"

SXX ðvÞ

SXY ðvÞ

SYX ðvÞ

SYY ðvÞ

SXX ðvÞ SXY ðvÞ SYX ðvÞ

SYY ðvÞ

#

#

" ¼

" ¼

p HXP ðvÞ p ðvÞ HYP

# SPP ðvÞ HXP ðvÞ

HYP ðvÞ

p p HXP ðvÞSXX ðvÞHXP ðvÞ HXP ðvÞSXY ðvÞHYP ðvÞ p ðvÞSYX ðvÞHXP ðvÞ HYP

ð5:27Þ

#

p ðvÞSYY ðvÞHYP ðvÞ HYP

Example 5.5 Response Spectra for a Two-Degree-of-Freedom System Subjected to a Single Random Force Consider the mass spring damper system shown in Figure 5.9. Assume that the random force F1 ðtÞ is a white noise random process with SF1 F1 ðvÞ ¼ S0 and F2 ðtÞ ¼ 0: (i) Find the frequency response functions H11 ðvÞ and H21 ðvÞ; and (ii) find the spectral densities SX1 X1 ðvÞ; SX1 X2 ðvÞ and SX2 X2 ðvÞ:

5 20


F1(t) X1(t) c1

X2(t)

c2 m2

m1 k1

k2

FIGURE 5.9

Two DoF system excited by a single random force.

Solution The equation of motion for this system has been obtained as Equation 5.14 with F2 ðtÞ ¼ 0: We found previously that the frequency response function for this system is given by " # ivc2 þ k2 m2 v2 þ ivc2 þ k2 1 ½HðvÞ ¼ det½Z m1 v2 þ ivðc1 þ c2 Þ þ ðk1 þ k2 Þ ivc2 þ k2 where det½Z ¼ v4 m1 m2

iv3 ðm1 c2 þ c1 m2 þ c2 m2 Þ þ ivðc1 k2 þ k1 c2 Þ

v2 ðm1 k2 þ c1 c2 þ k1 m2 þ k2 m2 Þ þ k1 k2 Since F2 ¼ 0; the frequency response functions due to F2 ; H12 ðvÞ and H22 ðvÞ; are zero. The frequency response functions due to F1 are given by H11 ðvÞ ¼ H21 ðvÞ ¼

m2 v2 þ ivc2 þ k2 det½Z ivc2 þ k2 det½Z

Using Equation 5.27, the response spectra are given by p ðvÞH11 ðvÞSF1 F1 ðvÞ ¼ S0 SX1 X1 ðvÞ ¼ H11 p SX2 X2 ðvÞ ¼ H22 ðvÞH22 ðvÞSF1 F1 ðvÞ ¼ S0 p SX1 X2 ðvÞ ¼ H11 ðvÞH21 ðvÞSF1 F1 ðvÞ ¼ S0 p ðvÞH11 ðvÞSF1 F1 ðvÞ ¼ S0 SX2 X1 ðvÞ ¼ H21

ðk2

m2 v2 Þ2 þ ðvc2 Þ2 ðdet½Z Þ2

k22 þ ðvc2 Þ2 ðdet½Z Þ2 ð m2 v2

ivc2 þ k2 Þðivc2 þ k2 Þ ðdet½Z Þ2

ð m2 v2 þ ivc2 þ k2 Þð ivc2 þ k2 Þ ðdet½Z Þ2

Note that SX2 X1 ðvÞ ¼ SX1 X2 ð vÞ ¼ SpX1 X2 ðvÞ:

5.5.2

Response to Multiple Random Forces

Consider a multi DoF system subjected to multiple random forces. The goal is to obtain the relation equivalent to Equation 5.27 for the general response for N DoF by first considering the response of a Single DoF to N forces, and then generalizing to N DoF. This will be done in two ways. First, by extending the impulse response method leading to the convolution integral, and then by utilizing the modal

Random Vibration

5 21

analysis approach where the coupled physical equations of motion are decoupled in a new modal coordinate system.

5.5.3

Impulse-Response Approach

Let us start by expressing the response Xk ðtÞ using the impulse response functions gki ðtÞ: Recall that gki ðtÞ is the impulse response of mass mk to force Fi ðtÞ: By linear superposition, the total response Xk ðtÞ of mass mk is equal to the sum of the individual responses to each of the N forces Xk ðtÞ ¼

N X i¼1

Xki ðtÞ ¼

N ð1 X 1

i¼1

gki ðtÞFi ðt

tÞdt

Assuming that the forces are stationary with respective means mFi and cross correlations RFi Fj ðtÞ; the mean and correlation of the response can be found as follows: N ð1 X

E{Xk ðtÞ} ¼

1

i¼1 N X

mXk ¼

i¼1

mFi

gki ðtÞE{Fi ðt

ð1 1

tÞ}dt

gki ðtÞdt ¼

N X i¼1

mFi Hki ð0Þ

In matrix form mX ¼ HXX ð0ÞmF This is a static response and, therefore, can be ignored here and added on at the end of the computations. We next evaluate the expressions for the response correlations and spectral densities, from which we can evaluate the mean square response. By definition, the correlations are given by ( N ) N X X Xjn ðt þ tÞ Xkm ðtÞ RXk Xj ðtÞ ¼ E{Xk ðtÞXj ðt þ tÞ} ¼ E ( ¼E

m¼1

N ð1 X m¼1

1

gkm ðz ÞFm ðt

N X N ð1 ð1 X

¼

m¼1 n¼1

1

1

N X N ð1 ð1 X

¼

m¼1 n¼1

1

1

z Þdz

n¼1

N ð1 X n¼1

1

) gjn ðj ÞFn ðt þ t

gkm ðz Þgjn ðj ÞE{Fm ðt

z ÞFn ðt þ t

gkm ðz Þgjn ðj ÞRFm Fn ðt

j þ z Þdz dj

j Þdj j Þ}dz dj

The response spectral density is by definition SXk Xj ðvÞ ¼

1 ð1 RXk Xj ðtÞexpð ivtÞdt 2p 1

Before substituting the correlation function into this equation, multiply it by expð iv½z j Þ £ expðiv½z j Þ: Also, define n ¼ t j þ z with dn ¼ dt: All of these manipulations allow us to put the spectral density in the following form: SXk Xj ðvÞ ¼ ¼

N X N ð1 X m¼1 n¼1 N X N X m¼1 n¼1

1

gkm ðzÞexpðivzÞdz

p Hkm ðvÞHjn ðvÞSFm Fn ðvÞ

ð1 1

gjn ðjÞexpð ivjÞdj £

1 ð1 RFm Fn ðnÞexpð ivnÞdn 2p 1

5 22


where the star denotes complex conjugate. In matrix form SXk Xj ðvÞ ¼ Hpk ðvÞSFF ðvÞHTj ðvÞ

ð5:28Þ

In this notation, Hpk ðvÞ is a row vector of dimension 1 £ N p Hpk ðvÞ ¼ Hk1 ðvÞ · · ·

HjT ðvÞ is a column vector of dimension N £ 1

2

6 6 HTj ðvÞ ¼ 6 6 4 and SFF ðvÞ is a matrix of dimension N £ N 2 6 6 SFF ðvÞ ¼ 6 6 4

p ðvÞ HkN

Hj1 ðvÞ .. . HjN ðvÞ

3 7 7 7 7 5

SF1 F1 ðvÞ

···

SF1 FN ðvÞ

.. .

..

.. .

SFN F1 ðvÞ

.

· · · SFN FN ðvÞ

3 7 7 7 7 5

Equation 5.28 can now be generalized for any Xk and Xj SXX ðvÞ ¼ Hp ðvÞSFF ðvÞHT ðvÞ where

2 6 6 HðvÞ ¼ 6 4

and

2 6 6 SXX ðvÞ ¼ 6 6 4

H11 ðvÞ

···

H1N ðvÞ

.. .

..

.. .

HN1 ðvÞ

···

.

ð5:29Þ 3 7 7 7 5

HNN ðvÞ

SX1 X1 ðvÞ

···

SX1 XN ðvÞ

.. .

..

.. .

SXN X1 ðvÞ

.

· · · SXN XN ðvÞ

3 7 7 7 7 5

Let us recall Equation 5.6, Equation 5.9, and Equation 5.27. Equation 5.6 is the response spectral density of a single DoF system to a single random force, Equation 5.9 is the response spectral density of a single DoF system to two random forces, and Equation 5.27 is the response spectral densities of a two DoF system due to a single force. Comparing them with Equation 5.29, we realize that they are all special cases of this equation. For a zero mean, the mean square response of the jth coordinate is given by the relation ð1 s2Xj ¼ ð5:30Þ SXj Xj ðvÞdv 1

Example 5.6

Spacecraft Design Problem (based on Wirsching et al., p. 228)

A rocket mounting is to be designed. The rocket of mass m1 ¼ 181 kg is mounted on a supporting structure of mass m2 ¼ 362 kg using structural brackets that can be modeled as springs of constant k ¼ 1:40 MN/m. There is also damping with coefficient c ¼ 3:50 kNsec/m. When the engine fires, the thrust can be interpreted as being composed of a mean value thrust and a random fluctuating thrust.

Random Vibration

5 23

The random variations are due to factors such as fluctuations in fuel mix and eccentricities in thrust direction. The mean thrust is given by mT ¼ 16 kN and the random component is given by T, which is a random function of time. T is characterized as white noise of intensity 1980 N2/Hz over the frequency range 0 50 Hz. The mean thrust can be viewed as a static load while the random thrust as the dynamic load. The design must consider that components in the supporting structure will be damaged if the acceleration is too large. Also, the structural brackets will fail if the relative displacement between the rocket and the supporting structure is too large. Both these design issues are difficult because the input is random. Therefore, there is a trade off between the cost of making a component stronger and making it more reliable. The probabilistic framework allows the analyst to quantitatively understand the trade off.

Solution The goal of the analysis is to determine the acceleration of the supporting mass X€ 2 and the relative displacement Z ¼ X2 X1 : First we solve the mean value problem. The mean value acceleration is a¼

F 16;000 29:46 ¼ ¼ 29:46 m=sec2 ¼ ¼ 3:00g m1 þ m2 181 þ 362 9:81

The mean force in the bracket is found by applying the equation of motion to m2

mFb ¼ m2 a2 ¼ 362 £ 29:46 ¼ 10665 N ¼ 10:66 kN The mean bracket deflection is

mZ ¼

mFb 10:66 £ 103 ¼ ¼ 7:61 £ 10 k 1:40 £ 106

3

m ¼ 7:61 mm

While the mean value is the single most important descriptor of randomness, the variance is important because it is a measure of the scatter of possible results. In order to calculate the variance, we need to derive the response spectral density and the frequency response function (see Equation 5.30). This we compute next. The equations of motion are given by m1 X€ 1

kðX2

m2 X€ 2 þ kðX2

X_ 1 Þ ¼ FðtÞ

cðX_ 2

X1 Þ

X_ 1 Þ ¼ 0

X1 Þ þ cðX_ 2

Given the problem statement, the variables of interest are ðZ; X2 Þ: Transform variables using Z ¼ X2 X1 : The equations of motion become m1 X€ 2

m1 Z€

kZ

cZ_ ¼ FðtÞ

m2 X€ 2 þ kZ þ cZ_ ¼ 0 To derive the frequency response function, assume harmonic excitation " # F0 i vt FðtÞ ¼ e 0 and harmonic response of the form

"

ZðtÞ

#

X€ 2

" ¼

Z0 X€ 0

# eivt

Substitute these equations into the matrix equation of motion to find m1 X€ 0 þ m1 v2 Z0

kZ0

icvZ0 ¼ F0

m2 X€ 0 þ kZ0 þ icvZ0 ¼ 0

5 24


The frequency response function is then found to be " m2 m1 1 HðvÞ ¼ DðvÞ k icv m1 v2 k 2

DðvÞ ¼ m1 m2 v

kðm1 þ m2 Þ

# icv

icvðm1 þ m2 Þ

There is only one applied force, and the spectral density function matrix is " # S0 0 SF ðvÞ ¼ 0 0 Apply Equation 5.29 to find " SXX ðvÞ ¼

SZZ

SZ X€

SXZ €

SX€ X€

#

S0 ¼ lDðvÞl2

"

m22

km2 þ icm2 v

km2 þ icm2 v

#

k2 þ ðcvÞ2

To determine the root mean square of relative displacement ZðtÞ; note that the spectral density function of ZðtÞ is the top term on the diagonal of SXX ðvÞ: SZZ ðvÞ ¼ S0

m22 lDðvÞl2

The mean square value of ZðtÞ is then

s2Z ¼

ð1 1

SZZ ðvÞ ¼

pS0 m22 ckðm1 þ m2 Þ2

The above are for a two sided spectrum. The given spectrum is one sided, therefore, S0 ¼ W0 =4p: Then, with appropriate substitutions, we find sZ . 0:212 mm: In summary, the relative displacement has mean and root mean square

mZ ¼ 7:61 mm;

sZ ¼ 0:212 mm

The force in the spring has mean and root mean square

mFb ¼ kmZ ¼ 1:40 £ 106 £ 7:61 ¼ 10:65 kN sFb ¼ ksZ ¼ 1:40 £ 106 £ 0:212 ¼ 0:297 kN Finally, the spectral density of the acceleration is the second diagonal term of SX ðvÞ: SX€ X€ ¼ S0

s2X€ ¼

k2 þ ðcvÞ2

ð1

lDðvÞl2

1

SX€ X€ ðvÞdv

and

s2X€ ¼ pS0

k c þ cðm1 þ m2 Þ2 ðm1 þ m2 Þm1 m2

Upon making the appropriate substitutions, we find

sX€ ¼ 0:852 m=sec2 ¼ 0:087g In summary, the mean and standard deviation of X€ 2 are mX€ 2 ¼ 3g;

sX€ 2 ¼ 0:087g

See Example 5.1 for a discussion on the above nonstandard integrals. A designer would now use these mean and standard deviation numbers to decide how reliable to make the components. For example, if we assume that Fb and X€ 2 are normal density, we can specify a design based

Random Vibration

5 25

fX (x)

on m ^ 3s, which tells us the probability of exceeding this 6s range is less than 0.1%. If this design is too expensive, then the designer will try a design based on m ^ 3s; one that will have a higher probability of failure. In this way, a trade off study is performed weighing cost against probability of failure. Figure 5.10 demonstrates a design based on 5% failure probability.

0.95

mX −1.96σX mX FIGURE 5.10 probability.

5.5.4

mX + 1.96σX

Normal density with 5% failure

Modal Analysis Approach

The response of a multi DoF discrete system loaded by random forces is generally a very complicated problem to formulate and solve. Here, to demonstrate one possible approach, we study the modal analysis of such a structure and work through some of the intricacies. We will start the analysis at the point where the modal equations of motion have been formulated and the assumption of proportional damping has been made. From Equation 5.22, for a random vibration, the modal equations in indicial notation are Z€ i þ 2zi vi Z_ i þ v2i Zi ¼ Qi ðtÞ;

i ¼ 1; 2; …; N

ð5:31Þ

where the parameters are familiar. We further assume that the modal forces Qi ðtÞ are ergodic random excitations. The transformation between physical and modal spaces is Xj ðtÞ ¼

N X i¼1

uji Zi ðtÞ

and the transformation between physical and modal forces is Qj ðtÞ ¼

N X i¼1

where ui is the modal vector for the ith DoF and

uij Fi ðtÞ ¼ uTj F

2

u1i

ð5:32Þ

3

6 7 6 7 ui ¼ 6 ... 7 4 5 uNi In matrix form, we can write X ¼ PZ Q ¼ PT F where P is the modal matrix given by P ¼ ½ u1

···

uN

For a two DoF system, we can be specific Xj ðtÞ ¼ Qj ðtÞ ¼

2 X i¼1 2 X i¼1

uji Zi ðtÞ ¼ uj1 Z1 ðtÞ þ uj2 Z2 ðtÞ;

j ¼ 1; 2 ð5:33Þ

uij Fi ðtÞ ¼ u1j F1 ðtÞ þ u2j F2 ðtÞ;

j ¼ 1; 2

5 26


for each DoF j of a two DoF structure. Our goal in this analysis is to evaluate the statistics of the response, that is, to find autocorrelations and cross correlations, RXk Xj ðtÞ; and its Fourier transform, the power spectrum, SXk Xj ðvÞ: Begin with the definition of the autocorrelation and substitute Equation 5.33 for Xj ðtÞ: (n n ) X X ukl ujm Zl ðtÞZm ðt þ tÞ ð5:34Þ RXk Xj ðtÞ ¼ E{Xj ðtÞXj ðt þ tÞ} ¼ E l¼1 m¼1

where Zi ðtÞ is the solution to Equation 5.31: Zi ðtÞ ¼ gi ðtÞ ¼

ðt 0

Qi ðtÞgi ðt

tÞdt

ð5:35Þ

1 expð zi vi tÞsin vdi t vdi

vdi ¼ vi ð1

ð5:36Þ

z2i Þ1=2

ð5:37Þ

Note that, since the impulse response function gðtÞ is zero for t , 0; the lower limit on the integral defining ZðtÞ can be made 1 without changing the value of the integral. Substituting Equation 5.35 to Equation 5.37 into Equation 5.34 and letting the expectation operate only on the stochastic terms, we have RXk Xj ðtÞ ¼

n X n ðtþt ðt X 1

l¼1 m¼1

1

ukl ujm E{Ql ðu1 ÞQm ðu2 Þ}gl ðt

u1 Þgm ðt þ t

u2 Þdu1 du2

ð5:38Þ

where u1 and u2 are dummy time variables, and RQl Qm ðu2

u1 Þ ¼ E{Ql ðu1 ÞQm ðu2 Þ}

owing to the assumed ergodicity (and thus stationarity) of the forcing. If the system is lightly damped and has well separated modal frequencies, as is the case in many engineering structures, the response due to Ql ðtÞ is almost statistically independent of the response due to Qm ðtÞ: The cross correlation terms that arise in Equation 5.38 are then almost zero, with the only nonzero terms arising for m ¼ l : RQl Ql ðu2

u1 Þ ¼ E{Ql ðu1 ÞQl ðu2 Þ}

Now that we have the correlation function for the response in terms of the correlation function for the random forcing, we can proceed to evaluate the response spectral density, from which we can evaluate probabilities of occurrence. To do this, the following transformation of variables is first necessary: u1 ; t

u1 ;

du1 ¼

du1 ;

u2 ; t þ t du2 ¼

u2 ;

du2

resulting in the response correlation RXk Xj ðtÞ ¼

n X n ð1 ð1 X l¼1 m¼1

1

1

ukl ujm RQl Qm ðu1

u2 þ tÞgl ðu1 Þgm ðu2 Þdu1 du2

The power spectral density for response XðtÞ is equal to the Fourier transform of the correlation function SXk Xj ð21vÞ ¼ ¼

1 ð1 RXk Xj ðtÞe ivt dt 2p 1 (n n X ð1 ð1 1 ð1 X 2p

1

l¼1 m¼1

0

0

) ukl ujm RQl Qm ðu1

u2 þ tÞgl ðu1 Þgm ðu2 Þdu1 du2 e

ivt

dt

Random Vibration

5 27

or, recalling and utilizing the assumption that the processes are ergodic, and, therefore, by averaging in time SXk Xj ðvÞ ¼

n X n X l¼1 m¼1

ukl ujm lim

T!1

1 ðT 1 ðT gl ðu1 Þdu1 lim gm ðu2 Þdu2 T!1 2p 2p T T

1 ðT RQl Qm ðu1 lim T!1 2p T

u2 þ tÞe

ivt

dt

where the lower limits on the integrals were set to T since gðtÞ is zero for t , 0; and thus the change in lower limit does not affect the values of the integrals. Using the change of variables

g ; u1

u2 þ t;

dg ¼ dt

we obtain SXk Xj ðvÞ ¼

n X n X l¼1 m¼1

lim

T!1

ukl ujm lim

T!1

1 ðT gl ðu1 Þexpðivu1 Þdu1 2p T

1 ðT 1 ðT u2 þu1 RQl Qm ðgÞe gm ðu2 Þexpð ivu2 Þdu2 lim T!1 2p 2p T u2 þu1 T

ivg

dg

In the last integral, we make the physical argument that RQj Qm ðgÞ ! 0 as lgl increases and, therefore, the limits can be replaced by T and T; respectively.10 Then, Hl ð vÞ ¼ lim

T!1

Hm ðvÞ ¼ lim

T!1

SQl Qm ðvÞ ¼ lim

T!1

1 ðT gl ðu1 Þexpðivu1 Þdu1 2p T 1 ðT gm ðu2 Þexpð ivu2 Þdu2 2p T 1 ðT u2 þu1 RQl Qm ðgÞe 2p T u2 þu1

ivg

dg

with the resulting response spectral density SXk Xj ðvÞ ¼

n X n X l¼1 m¼1

ukl ujm Hl ð vÞHm ðvÞSQl Qm ðvÞ

For a two DoF system, we would have #" # " p #" #" uj1 SQ1 Q1 ðvÞ SQ1 Q2 ðvÞ H1 ðvÞ 0 H1 ðvÞ 0 SXk Xj ðvÞ ¼ uk1 uk2 p uj2 SQ2 Q1 ðvÞ SQ2 Q2 ðvÞ 0 H2 ðvÞ 0 H2 ðvÞ ¼ uk1 uj1 H1p ðvÞH1 ðvÞSQ1 Q1 ðvÞ þ uk1 uj2 H1p ðvÞH2 ðvÞSQ1 Q2 ðvÞ þ uk2 uj1 H2p ðvÞH1 ðvÞSQ2 Q1 ðvÞ þ uk2 uj2 H2p ðvÞH2 ðvÞSQ2 Q2 ðvÞ Suppose that we wish to find SXX for a two DoF system. Then " SXX ¼ 10

u11

u12

u21

u22

#"

H1p ðvÞ

0

0

H2p ðvÞ

#"

SQ1 Q1 ðvÞ

SQ1 Q2 ðvÞ

SQ2 Q1 ðvÞ

SQ2 Q2 ðvÞ

#"

H1 ðvÞ

0

0

H2 ðvÞ

#"

u11

u21

u12

u22

#

Physically, we are stating that, as time difference (trial mode) increases, there will be an exponentially decaying correlation. This is borne out by experiments on physical systems.

5 28


In general, we can write SXX ðvÞ ¼ PHp ðvÞSQQ ðvÞHðvÞPT where

2 6 6 HðvÞ ¼ 6 4

H1 ðvÞ

0

0

..

0

0

0 .

0

ð5:39Þ

3 7 7 7 5

HN ðvÞ

1 v2 þ i2zi vi v þ v2i

Hi ðvÞ ¼

The spectral densities of the modal forces SQl Qm ðvÞ can be obtained from SFF ðvÞ using Equation 5.32. The autocorrelation RQl Qm ðtÞ is defined as 8 9 N N N X N <X = X X RQl Qm ðtÞ ¼ E{Ql ðtÞQm ðt þ tÞ} ¼ E uil Fi ðtÞ ujm Fj ðt þ tÞ ¼ u u E{Fi ðtÞFj ðt þ tÞ} :i¼1 ; i¼1 j¼1 il jm j¼1 ¼

N N X X i¼1 j¼1

uil ujm RFi Fj ðtÞ

Taking the Fourier transform SQl Qm ðvÞ ¼

N X N X i¼1 j¼1

uil ujm SFi Fj ðvÞ

or SQl Qm ðvÞ ¼ uTl SFF um or SQQ ðvÞ ¼ PT SFF P

ð5:40Þ

Substituting Equation 5.40 into Equation 5.39, we obtain SXX ðvÞ ¼ PHp ðvÞPT SFF ðvÞPHðvÞPT Comparing this result with Equation 5.29, we find that HpXX ðvÞ ¼ PHp ðvÞPT HXX ðvÞ ¼ PHðvÞPT which is a fully populated transfer function matrix of XðtÞ: For lightly damped systems with well spaced modal frequencies, the cross terms in the double summation, those where l – m; contribute very little to the mean square response given by Ð1 1 SXi Xj ðvÞdv: In this case, we can use the approximation SXj Xj ðvÞ . where

2

n X l¼1

u2jl lHl ðvÞl2 SQl Ql ðvÞ

32 2 v 1= 6 7 l lHl ðvÞl2 ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 ð1 v2 =v2l Þ2 þ ð2zl v=vl Þ2

Random Vibration

5 29

Suppose that we only account for the first three modes out of 1000 DoF. Then, the spectral density SXX based on three modes can be obtained by SXX ðvÞ ¼ Pl1000£3 Hp ðvÞl3£3 PT l3£1000 SFF l1000£1000 Pl1000£3 HðvÞl3£3 PT l3£1000 where Pl1000£3 ¼ ½ u1 2 6 HðvÞl3£3 ¼ 6 4

u2

u3

H1 ðvÞ

0

0

0

H2 ðvÞ

0

0

0

H3 ðvÞ

3 7 7 5

Thus, we can easily see that the modal approach allows the analyst and designer to retain only those modes that are key to the problem at hand. If, as above, a thousand DoF system can be adequately modeled as a three DoF system because that is where all the energy is located, then a great deal of unnecessary computation is avoided.

5.6

Continuous System Random Vibration

Continuous models of vibrating systems are viewed as more realistic because the structural properties are distributed rather than being concentrated at discrete points. The price to pay for increased realism is increased complexity. The governing equations of motion go from being ordinary differential equations for the discrete models, to partial differential equations for the continuous ones, since displacement is a function of both time and position. For our elementary applications, deriving and solving the partial differential equations does not pose a great challenge. But, when we need to model nonuniform structures with varying cross sections and material properties, approximate techniques will be needed. Here our introduction to continuous systems will focus on strings and vibrating beams. These are found in most structures and machines, and understanding how they behave is of great use to modeling more complex systems. We will approach these studies using the direct and modal solutions.

5.6.1

Transverse Vibration of Beams

Begin with the equation of motion of a damped, forced, simply supported beam with uniform properties EI

›4 y ›y ›2 y þ m 2 ¼ pðx; tÞ 4 þc ›x ›t ›t

ð5:41Þ

where it is assumed the natural frequencies and the mode shapes have already been evaluated. To solve this problem, we will use the normal mode method, which depends on the modal orthogonality properties. First, expand the applied force pðx; tÞ in terms of the modes,11 then do the

11

The modes Yj ðxÞ are the eigenfunctions of the problem EI

›4 y ›2 y þm 2 ¼0 ›x 4 ›t

That is, they satisfy EI and the boundary conditions at hand.

d4 Yj 2 v2 mYj ¼ 0; dx4

j ¼ 1; 2; …

5 30


same with the structural displacement pðx; tÞ ¼

1 X j¼1

pj ðtÞYj ðxÞ

Multiply both sides of this equation by mYk ðxÞ and then integrate over the beam span ðL 0

mpðx; tÞYk ðxÞdx ¼

1 X j¼1

pj ðx; tÞ

ðL 0

mYj ðxÞYk ðxÞdx

On the right hand side of this equation, we note that, by the orthogonality properties of the modes, the integral equals zero for k – j; and otherwise is normalized to 1 for k ¼ j: Therefore, we are left with ðL mpðx; tÞYj ðxÞdx ¼ pj ðtÞ ð5:42Þ 0

Apply the same procedure to the response yðx; tÞ ¼

1 X j¼1

yj ðtÞYj ðxÞ

ð5:43Þ

and multiply and integrate as above to find ðL 0

myðx; tÞYj ðxÞdx ¼ yj ðtÞ

yðx; tÞ can be differentiated with respect to x and t so that all the terms in the equation of motion (Equation 5.41) can be put into the modal expansion form " # 1 1 X X d4 Yj dyj d2 yj EIyj ðtÞ 4 þ c pj ðtÞYj ðxÞ ð5:44Þ Yj þ m 2 Yj ¼ dx dt dt j¼1 j¼1 We know that the modes are defined by and satisfy the equation EI d4 Yj =dx4 ¼ v2j mYj : Substituting this relation into Equation 5.44 leads to 1 X j¼1

½v2j myj ðtÞ þ c_yj ðtÞ þ m€yj ðtÞ

pj ðtÞ Yj ðxÞ ¼ 0

Since the eigenfunctions Yj ðxÞ are not zero except for unrestrained motion, the expression in the square brackets must vanish for every j; y€ j ðtÞ þ

c 1 y_ ðtÞ þ v2j yj ðtÞ ¼ pj ðtÞ; m j m

j ¼ 1; 2; …

ð5:45Þ

As for single DoF systems, let c=m ¼ 2zj vj ; that is,

zj ¼

c 2mvj

Equation 5.52 is the damped, forced, harmonic oscillator, with the well known convolution solution, 1 ðt p ðtÞgj ðt tÞdt hj ðtÞ ¼ ð5:46Þ m 0 j where gj ðtÞ is the impulse response function for a damped oscillator. Once solved, yj ðtÞis substituted into the expansion equation yðx; tÞ ¼

1 X j¼1

yj ðtÞYj ðxÞ ¼

1 ðt 1 X Yj ðxÞ pj ðtÞgj ðt m j¼1 0

tÞdt

Random Vibration

5 31

and substituting Equation 5.42 for pj ðtÞ results in yðx; tÞ ¼

1 X j¼1

Yj ðxÞ

ðt ðL 0

0

pðj; tÞYj ðjÞdj gj ðt

tÞdt

ð5:47Þ

This is the complete steady state time domain solution.

5.6.2

Random Transverse Vibration

Assume that the loading Pðx; tÞ is random and represents an ensemble of functions of space and time. We need only the first two statistics of the load to proceed with the analysis of the response. Its mean value is denoted E{Pðx; tÞ} ¼ mP ðx; tÞ with autocorrelation E{Pðx1 ; t1 ÞPðx2 ; t2 Þ} ¼ RPP ðx1 ; x2 ; t1 ; t2 Þ These statistics are available through data gathering experiments performed on the loading to be experienced by the structure. Our goal in this analysis is to derive the statistics for the response, mY and RYY ; which are functions of the statistics of the loading and the structural parameters. For steady state vibration problems, we may reasonably assume that the statistics of the loading are stationary. Begin with Equation 5.47 but now, for random loading Pðx; tÞ yðx; tÞ ¼

1 X j¼1

Yj ðxÞ

ðt ðL 0

0

Pðj; t

tÞYj ðjÞdj gj ðtÞdt

where the shift t has been placed within pðx; tÞ and take the expectation of both sides E{Yðx; tÞ} ¼

1 X j¼1

Yj ðxÞ

ðt ðL 0

0

E{Pðj; t

tÞ}Yj ðjÞdj gj ðtÞdt

The impulse response function is gðtÞ, and for stationary loading, the expected value is a constant, E{Pðj; t tÞ} ¼ mP ðjÞ: In order to simplify the following analysis, we assume that the loading has zero mean. This does not overly simplify the problem since a nonzero mean can be introduced later by simply shifting the response by the (constant) mean value. For zero mean loading, mP ðxÞ ¼ 0, and the response is also zero mean, E{Yðx; tÞ} ¼ mY ¼ 0: We need to relate the autocorrelation of the response to that for the loading. Begin with Equation 5.42: ðL ðL RPj Pk ðt1 ; t2 Þ ¼ E{Pj ðt1 ÞPk ðt2 Þ} ¼ m2 E{Pðx1 ; t1 ÞPðx2 ; t2 Þ}Yj ðx1 ÞYk ðx2 Þdx1 dx2 0

¼ m2

ðL ðL 0

0

0

RPP ðx1 ; x2 ; t1 ; t2 ÞYj ðx1 ÞYk ðx2 Þdx1 dx2

where, for a stationary loading, RPP ðx1 ; x2 ; t1 ; t2 Þ ¼ RPP ðx1 ; x2 ; tÞ; where t ¼ t2 RPj Pk ðt1 ; t2 Þ ¼ RPj Pk ðtÞ: Using Equation 5.43, the response autocorrelation is RYY ðx1 ; x2 ; t1 ; t2 Þ ¼ E{Yðx1 ; t1 ÞYðx2 ; t2 Þ} ¼ ¼

1 X 1 X j¼1 k¼1

1 X 1 X j¼1 k¼1

RYj Yk ðt1 ; t2 ÞYj ðx1 ÞYk ðx2 Þ

ð5:48Þ t1 and therefore,

E{Yj ðt1 ÞYk ðt2 Þ}Yj ðx1 ÞYk ðx2 Þ

5 32


where, using Equation 5.46, with upper and lower limits extended to ^1 without changing the final value of the integral due to the fact that the impulse response is zero outside the original limits ð1 ð1 1 RYj Yk ðt1 ; t2 Þ ¼ 2 E Pj ðt1 uÞgj ðuÞdu Pk ðt2 kÞgk ðkÞdk m 1 1 1 ð1 ð1 E{Pj ðt1 uÞPk ðt2 kÞ}gj ðuÞgk ðkÞdu dk ¼ 2 m 1 1 1 ð1 ð1 ¼ 2 RPj Pk ðt1 u; t2 kÞgj ðuÞgk ðkÞdu dk m 1 1

u; t2

Since Pðx; tÞ is stationary, RPj Pk ðt1 t1 Þ: Therefore,

kÞ ¼ RPj Pk ðt2

1 ð1 ð1 RPj Pk ðt2 m2 1 1

RYj Yk ðt1 ; t2 Þ ¼

t1 þ u t1 þ u

kÞ and RYj Yk ðt1 ; t2 Þ ¼ RYj Yk ðt2 kÞgj ðuÞgk ðkÞdu dk

ð5:49Þ

and RYY ðx1 ; x2 ; t2

t1 Þ ¼

1 1 X X j¼1 k¼1

RYj Yk ðt2

t1 ÞYj ðx1 ÞYk ðx2 Þ

This last equation is a relation between the forcing and response autocorrelations. The spectral density of the response, SYY ðvÞ; is given by the Fourier transform of RYY ðtÞ SYY ðvÞ ¼

1 ð1 1 X X 1

j¼1 k¼1

RYj Yk ðtÞe

ivt

dt Yj ðx1 ÞYk ðx2 Þ

ð5:50Þ

where t ¼ t2 t1 ; and the term in the square brackets equals SYj Yk ðvÞ; which now needs to be evaluated in terms that have already been derived. By definition, we know the Fourier transform relation ð1 SYj Yk ðvÞ ¼ RYj Yk ðtÞe ivt dt 1

We can evaluate RYj Yk by beginning with Equation 5.49 RYj Yk ð t Þ ¼

1 ð1 ð1 RPj Pk ðt þ u m2 1 1

Take the Fourier transform of both sides, letting l ¼ t þ u SYj Yk ðvÞ ¼

ð1 1

e

ivðl uþkÞ

kÞgj ðuÞgk ðkÞdu dk k; to find

1 ð1 ð1 RPj Pk ðlÞgj ðuÞgk ðkÞdu dk dl m2 1 1

ð5:51Þ

where t has been replaced by l u þ k and dt by dl: Rewrite Equation 5.51 in a more useful form by separating the integrals according to dummy variables SYj Yk ðvÞ ¼

ð1 1 ð1 gk ðkÞe gj ðuÞeivu du 2 m 1 1

ivk

dk

ð1 1

RPj Pk ðlÞe

ivl

dl

ð5:52Þ

The Fourier transform of the impulse response function gðtÞ is the frequency response function HðvÞ: Therefore, Equation 5.52 becomes SYj Yk ðvÞ ¼

1 p H ðvÞHk ðvÞSPj Pk ðvÞ m2 j

ð5:53Þ

Random Vibration

5 33

where Hj ðvÞ ¼ ½v2j v2 þ 2zj vj v 1 and SPj Pk ðvÞ; the spectral density of the modal force components, is derived assuming that the Fourier transform for Pj ðtÞ exists, as it does for most physical processes 1 ð1 Pj ðtÞ ¼ Pj ðvÞeivt dv 2p 1 Take the Fourier transform of Equation 5.48, to find SPj Pk ðvÞ ¼ m2

ðL ðL 0

0

SPP ðvÞYj ðx1 ÞYk ðx2 Þdx1 dx2

ð5:54Þ

where SPP ðvÞ is the spectral density of the loading, a quantity that is estimated from data. Substituting Equation 5.54 into Equation 5.53, we now have the expression for SYj Yk ðvÞ; which can be substituted into Equation 5.50, resulting in SYY ðx1 ; x2 ; vÞ ¼

1 X 1 1 X H p ðvÞHk ðvÞSPj Pk ðvÞYj ðx1 ÞYk ðx2 Þ 2 m j¼1 k¼1 j

One value of having such an equation is that the mean square displacement can be evaluated ð1 YMS ðxÞ ¼ RYY ðx; x; 0Þ ¼ SYY ðx; x; vÞdv 1

s2Y ðxÞ;

Recall that if mY ðxÞ ¼ 0 then yMS ðxÞ ¼ the variance. The derivations are now complete, but what do they mean and what do they do for us? One of the functions of a probabilistic analysis is to help us bound our uncertainties so that we can understand how randomness in the forcing results in a scatter of possible structural responses. Furthermore, this scatter is not haphazard, but is defined by a standard deviation and possibly a density function. It is the variance that is used to bound the mean value response.

Nomenclature Symbol

Quantity

Symbol

Quantity

E{·} HðvÞ RXX ðtÞ RXY ðtÞ SXX ðvÞ SXY ðvÞ

mathematical expectation frequency response function autocorrelation function cross correlation function power spectral density power cross spectral density

{u}i W0 zðtÞ m s2

structural mode one sided spectral density modal coordinates mean value (first moment) variance (second moment)

Acknowledgments I am pleased to acknowledge my collaboration over the past few years with Dr. Seon Mi Han of Wood Hole Oceanographic Institute. Seon has had input to this document and has prepared the figures. It is also a great pleasure to acknowledge the support provided by the Office of Naval Research and by our program manager Dr. Thomas Swean under Grant No. N00014 97 10017. Finally, I appreciate the invitation by Professor Clarence W. de Silva to prepare this chapter.

Bibliography The discipline of random vibration includes thousands of references. Applications include wind and earthquake engineering, aerospace engineering, and ocean engineering to name the broad areas.

5 34


More recently, applications in materials engineering and biomechanical engineering have also broadened to include random effects. Therefore, we provide here a brief list of references. Augusti, G., Baratta, A., and Casciati, F. 1984. Probabilistic Methods in Structural Engineering, Chapman & Hall, London. Benaroya, H. 1998. Mechanical Vibration: Analysis, Uncertainties, and Control, Prentice Hall, Upper Saddle River, NJ. Bolotin, V.V. 1969. Statistical Methods in Structural Mechanics, Holden Day, San Francisco, CA. Bolotin, V.V. 1984. Random Vibrations of Elastic Systems, Martinus Nijhoff, The Hague. Dimentberg, M.F. 1988. Statistical Dynamics of Nonlinear and Time Varying Systems, Research Studies Press, England. Ibrahim, R.A. 1985. Parametric Random Vibration, Research Studies Press, England. Lin, Y.K. 1976. Probabilistic Theory of Structural Dynamics, Krieger, Malabar, FL. Madsen, H.O., Krenk, S., and Lind, N.C. 1986. Methods of Structural Safety, Prentice Hall, Englewood Cliffs, NJ. To, C.W.S. 2000. Nonlinear Random Vibration: Analytical Techniques and Applications, Swets & Zeitlinger, Lisse, The Netherlands. Vanmarcke, E. 1983. Random Fields: Analysis and Synthesis, MIT Press, Cambridge, MA. Wirsching, P.H., Paez, T.L., and Ortiz, K. 1995. Random Vibrations: Theory and Practice, Wiley Interscience, New York.

Computer Techniques

II

II 1

6

Numerical Techniques 6.1 6.2 6.3

Introduction ....................................................................... Single Degree of Freedom System ................................... Forced Single Degree of Freedom System Single Degree of Freedom System

†

Summary of

Systems with Two or More Degrees of Freedom ............ Example

†

Summary of Two Degree of Freedom System

61 62 68

6.4

Finite Difference Method for a Continuous System ...... 6 11

6.5

Matrix Methods ................................................................. 6 14

6.6

Approximation Methods for the Fundamental Frequency ........................................................................... 6 18

Bar † Beam † Summary of Finite Difference Methods for a Continuous System Example: Three Degree of Freedom System † Bisection Method † Directly Calculating the Eigenvalues and Eigenvectors from the Matrix Equation † Summary of Matrix Methods

Rayleigh Method † Dunkerley’s Formula † Summary of Approximations for the Fundamental Frequency

Marie D. Dahleh Harvard University

6.7

Finite Element Method ..................................................... 6 20 Bar Element

†

Beam

†

Summary of Finite Element Method

Appendix 6A Introduction to MATLABw ...................... 6 24

Summary This chapter gives an overview of numerical techniques for vibration analysis. The centered difference approximation for the first, second, and fourth derivative are given. These form the basis for the finite difference approximation of both spring mass systems and the continuous problem. The fourth order Runge Kutta method is presented. Both of these approaches are used to solve the single degree of freedom (single DoF) system. In order to demonstrate these techniques for the multiple degree of freedom (multi DoF) system a two degree of freedom (two DoF) system is explored. Finite element and finite difference methods are presented as solution techniques for the continuous problem (also see Chapter 9). The bar and beam are used for examples. The Rayleigh method and Dunkerley’s formula are presented. These are methods for computing the fundamental frequency.

6.1

Introduction

This chapter presents numerical techniques for three classes of vibration problems; the single spring mass, the multiple spring mass, and the continuous model. The simplest oscillatory problem results when the problem can be modeled by the single spring mass system. This system is modeled by a second order constant coefficient differential equation, which can be solved analytically when either there is no forcing, or the forcing is harmonic. When the forcing is not harmonic, the finite difference method can be used to

61

62


solve this second order equation or to solve the system of first order equations, which is equivalent to the second order equation. A larger computational problem arises when there are multiple springs and masses subjected to nonharmonic forcing. Again, the finite difference method can be used to compute the full solution. It is not always necessary to compute the full solution. Often, all that is needed is the fundamental frequency. This can be computed from the eigenvalues of a matrix problem. Both the numerical solution to the full problem and the matrix problem will be discussed. Finally, one may want to look at the vibration of a continuous element like a beam. Here, both the finite difference method and the finite element method will be discussed.

6.2

Single-Degree-of-Freedom System

The simple undamped spring mass system oscil lates with a frequency known as the natural frequency. The natural frequency is determined by the spring constant pffiffiffiffiffiand the mass in the following way, v ¼ k=m: This relationship is derived by applying Newton’s second law to the basic spring mass system. The resulting equation is given by m€x ¼ kx: The solution to this equation is harmonic with the frequency specified pffiffiffiffiffi FIGURE 6.1 Single spring mass system. by the expression v ¼ k=m: A more realistic vibration model of a simple oscillatory system includes a mass, a spring and a damper (see Figure 6.1). For simplicity, the mass is concentrated at the center of mass of the object, the spring is assumed to be of negligible mass, and, for the purposes of this discussion, the damping will be modeled by viscous damping. This is described by a force proportional to the velocity, denoted by x_ : In the case of no external forcing, the system can be described by the following equation: m€x þ c_x þ kx ¼ 0 where m; c; and k are constants and m is the mass, c is the damping coefficient, and k is the spring constant. This equation can be solved analytically by assuming a solution of the form x ¼ est where s is a constant. Substitution into the differential equation yields the following quadratic equation: ðms2 þ cs þ kÞest ¼ 0 This equation is satisfied for all values of t when the following quadratic equation, known as the characteristic equation, is satisfied: s2 þ The characteristic equation has two roots

c k sþ ¼0 m m

s1 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ ðc=2mÞ2 ðk=mÞ 2m

s1 ¼

c 2m

and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc=2mÞ2 ðk=mÞ

The general solution is x ¼ A e s1 t þ B e s2 t The constants A and B are determined by the initial conditions xð0Þ and x_ ð0Þ: The single spring mass system exhibits three types of behavior, overdamped, underdamped and critically damped.

Numerical Techniques

63

The overdamped state exists when k=m is larger than ðc=2mÞ2 : No oscillations exist in this state. The underdamped case is oscillatory and results when ðc=2mÞ2 is larger than k=m: The limiting case between oscillatory and nonoscillatory motion occurs when ðc=2mÞ2 ¼ k=m: When this condition, is met, the system is said to be critically damped. The next level of complexity results when the system is forced harmonically. The following equation serves as a model for this system: m€x þ c_x þ kx ¼ F sin vt The solution of this equation is found by first computing the complementary function, which is the solution of the homogeneous equation discussed above, and then the particular solution. The details for computing a particular solution can be found in books on differential equations or mechanical vibrations (Thomson and Dahleh, 1998).

6.2.1

Forced Single-Degree-of-Freedom System

In general, when the oscillatory system is forced in a nonharmonic way, the resulting differential equation cannot be solved in closed form, and numerical methods must be employed to predict the behavior of the system. In this section, we consider two finite difference methods chosen for their simplicity. For a more detailed discussion of numerical methods for ordinary differential equations see Atkinson (1978) and Isaacson (1966). The spring mass system that is subjected to general forcing, Fðx; x_ ; tÞ; can be modeled by the following differential equation: m€x þ c_x þ kx ¼ f ðx; x_ ; tÞ x0 ¼ xð0Þ x_ 0 ¼ x_ ð0Þ where m; c; and k are constants and F is an arbitrary function. The two initial conditions, x0 and x_ 0 , are known. In the first method, the second order equation is integrated without change in form; in the second method, the second order equation is rewritten as a system of two first order equations and then the system of equations is integrated. Both methods approximate the first and second derivatives with the centered difference approximation for the derivatives. Finite difference methods are based on the Taylor expansion. The centered difference method for the first and second derivative results from a combination of the forward and backward Taylor expansion about the point xi : To get the forward expansion, one writes the Taylor expansion for xiþ1 : Similarly, the backward expression is obtained from the Taylor expansion about xi 1 : These are given by xiþ1 ¼ xi þ h_xi þ

h2 h3 x€ i þ ffl x þ ··· 2 6

xi

h2 x€ 2 i

1

¼ xi

h_xi þ

h3 ffl x þ ··· 6

where h ¼ Dt: A second order approximation is one which matches the Taylor expansion exactly up to and including terms of order h2 ; that is, to determine a second order expansion one neglects terms of order h3 and higher. A second order approximation for the first derivative is obtained by subtracting the backward difference from the forward difference. The resulting centered difference approximation is given by 1 h2 x_ i ¼ ðxiþ1 xi 1 Þ ffl x þ ··· 2h 6 Errors result when this expression is truncated after the first term. These errors depend on h2 and the third derivative of xi : If the error in the computed derivative is larger than order h2 it may well arise from the neglected third derivative term.

64


The centered difference approximation for the second derivative is found by adding the forward and backward difference expansions and ignoring terms of order h4 and higher. The resulting approximation is x€ i ¼

1 ðx h2 i

2xi þ xiþ1 Þ þ

1

h2 iv x þ ··· 12

The truncation error that occurs for this expression depends on h2 and the fourth derivative of xi : Both the first and second derivative centered difference approximations are order h2 : They can be used together to create a second order approximation to a second order ordinary differential equation such as that describing the spring mass system. 6.2.1.1

Centered Difference Approximation

After substituting these two centered difference approximations into the differential equation and rearranging terms, one gets the following recurrence relation for the single spring mass system: xiþ1 ¼ h2 f ðxi ; ti Þ

ð fh2

2mÞxi

ðm

ch=2Þxi

1

with the initial conditions x0 ¼ xð0Þ x_ 0 ¼ x_ ð0Þ The recurrence relation should be used to compute all values of x from the initial condition. By letting i ¼ 1 in the recurrence relation, we get x2 ¼ h2 f ðx1 ; t1 Þ

ðkh2

2mÞx1

ðm

ch=2Þx0

In order to compute x2 ; both x0 and x1 are needed. The initial conditions provide x0 : However, to start the calculation, we need an additional equation for x1 : This equation is derived by substituting i ¼ 0 into the Taylor expansion for xiþ1 ; using the initial conditions and ignoring terms of order h2 and higher. Since the centered difference approximation is of order h2 ; it is consistent to compute x1 with an error of order h2 : For the point i ¼ 0; the forward difference is given by x1 ¼ x0 þ h_x0 This equation allows one to find x1 in terms of the two initial conditions, after which the recurrence relation can be used to find all subsequent discrete values of x: 6.2.1.2

Pseudocode for Centered Difference Approximation

The following is an example of the MATLABw routine for the solution to the centered difference approximation of the general single DoF spring mass equations. The function f needs to be specified in a separate function file: %initial conditions (index has to start at 1) xð1Þ ¼ x0 xd ¼ x00 %specify a time step h¼H %specify the constants m; k; c: m¼M k¼K c¼C %compute xð2Þ from the Taylor expansion xð2Þ ¼ xð1Þ þ h p xd %specify the total number of steps TEND ¼ tfinal


65

%compute the solution for all remaining times using the recurrence relation tð1Þ ¼ h tð2Þ ¼ 2 p h for I ¼ 3:TEND Xi ¼ h2 p f ðxi 1 ; ti 1 Þ ðk p h2 2 p mÞ p xi 1 ðm c p h=2Þ p xi 2 TðiÞ ¼ tðI 1Þ þ h end The method just presented has ignored terms of order h2 and higher. This is known as the truncation error. The calculation will contain other errors such as round off error due to the loss of significant figures. This loss is related to both the machine and the language used for the calculations. The round off errors are also related to the time increment h ¼ Dt in a complicated way that is beyond the scope of this chapter. Better accuracy can be obtained by choosing a smaller Dt: However, the smaller the Dt; the larger the number of computations needed to reach the solution in a fixed time T: The increased number of computations affects both the total time of the calculation and the overall accuracy. 6.2.1.3

Runge–Kutta Methods

The centered difference approximation is not a self starting method. In other words, the calculation that determines x1 from the initial conditions does not come from the discretized equation, but rather from the Taylor expansion directly. An alternate way to solve this equation is to use what is known as a Runge Kutta method. All of these methods approximate the differential equation with Taylor series expansions. The advantage of this class of methods is that they are self starting. The disadvantage is that they only work on first order equations (or systems of first order equations). Before the Runge Kutta method is discussed in detail, the second order spring mass equation has to be written as a system of two first order equations. The equation for the single DoF system, subjected to arbitrary forcing f is m€x þ c_x þ kx ¼ f ðx; x_ ; tÞ It can be rewritten as the following system of first order equations x_ ¼ u u_ ¼ x€ ¼

1 ½f ðx; u; tÞ m

cu þ kx

These equations are coupled and need to be solved together. Any Runge Kutta method for the system of equations requires a Taylor series expansion of x and u about xi and ui : The Taylor expansion is given by xiþ1 ¼ xi þ x_ i h þ x€

h2 þ ··· 2

uiþ1 ¼ ui þ u_ i h þ u€ i

h2 þ ··· 2

As above, the time increment will be denoted by h ¼ Dt: The first order Runge Kutta method (also known as Euler’s method) is obtained by retaining terms of first order and lower, i.e., the Euler approximation is given by xiþ1 ¼ xi þ x_ i h

oðhÞ

uiþ1 ¼ ui þ u_ i h

oðhÞ

The truncation error for Euler’s method is order h: The error depends linearly on h: Matching more terms in the Taylor expansion generates higher order Runge Kutta methods. The most commonly used Runge Kutta method is the fourth order method that matches the Taylor expansion up to terms of order h4 : This is a significant reduction in the error. For a derivation of this method see Cheney and Kincaid (1999).

66


TABLE 6.1

Fourth Order Runge Kutta Method for Spring Mass Equation

t T1 T2 T3 T4

X_ ¼ u

x

¼ ti ¼ ti þ h=2 ¼ ti þ h=2 ¼ ti þ h

X1 X2 X3 X4

¼ xi ¼ xi þ U1 h=2 ¼ xi þ U2 h=2 ¼ x i þ U3 h

U1 U2 U3 U4

¼ ui ¼ ui þ G1 h=2 ¼ ui þ G2 h=2 ¼ ui þ G3 h

X€ ¼ G G1 G2 G3 G4

¼ GðT1 ; X1 ; U1 Þ ¼ GðT2 ; X2 ; U2 Þ ¼ GðT3 ; X3 ; U3 Þ ¼ GðT4; ; X4 ; U4 Þ

Source: Thomson and Dahleh 1998. Theory of Vibration Applications, 5th ed. With permission.

For the system of first order equations given above, the fourth order Runge Kutta method requires four values of t; x; u, and, G where G ¼ 1=m½f ðx; u; tÞ cu þ kx : It can be computed for each point i as follows in Table 6.1. Combining these quantities in the following method gives the fourth order Runge Kutta method: xiþ1 ¼ xi þ h=6ðU1 þ 2U2 þ 2U3 þ U4 Þ uiþ1 ¼ ui þ h=6ðG1 þ 2G2 þ 2G3 þ G4 Þ where it is recognized that the four values of U divided by six represent the average slope dx=dt and the four values of G divided by six result in an average of du=dt: A way to check the accuracy of a Runge Kutta method is to Taylor expand G1 ; G2 ; G3 ; and G4 and collect like terms. One will find that the above combination produces an expansion which is exact up to order h4 : 6.2.1.4

Pseudocode for the Fourth-Order Runge –Kutta Method

For simplicity, the code is given for the single first order equation x_ ¼ Gðt; xÞ with the initial data xð0Þ ¼ x0 : The function G should be specified in a function file. %initial conditions (index has to start at 1) xð1Þ ¼ x0 ; %time step needs to be specified h ¼ H; h2 ¼ 2 p H; %TEND is the total number of time steps TEND ¼ Tfinal; T ¼ h; for I ¼ 1:TEND G1 ¼ H p Gðt; xÞ; G2 ¼ H p Gðt þ h2; x ¼ 0:5 p G1Þ; G3 ¼ H p Gðt þ h2; x ¼ 0:5 p G2Þ; G4 ¼ h p Gðt þ h; x þ G3Þ; X ¼ x þ ðG1 þ 2 p G2 þ 2 p G3 þ G4Þ=6:0; t ¼ ðI þ 1Þ p h; end 6.2.1.5

Example

Solve numerically the differential equation 4€x þ 2000x ¼ FðtÞ with the initial conditions x0 ¼ x_ 0 ¼ 0 The forcing is as shown in Figure 6.2.

FIGURE 6.2 The forcing function for Example 6.2.1.5. (Source: Thomson and Dahleh 1998. Theory of Vibration Applications, 5th ed. With permission.)


67

6.2.1.5.1 Centered Difference Using the centered difference approximation for the second derivative, one gets the following discrete equation xiþ1 ¼ h2 =4FðtÞ

500h2 xi

xi

1

þ 2xi

This equation is valid for i $ 1: From the initial conditions x0 ¼ x_ 0 ¼ 0; x1 is computed by the following x1 ¼ x0 þ x_ 0 ¼ 0 þ 0: A time step of h ¼ 0.02 sec has been used in the calculation. The numerical solution as compared with the exact solution is given in Table 6.2. Table 6.3 contains the absolute errors for the two discrete solutions. 6.2.1.5.2 Runge–Kutta In order to use the Runge Kutta method, the second order equation needs to be written as a system of first order equations. Let u ¼ x_ ; then u_ ¼ Gðx; tÞ ¼ :25 p FðtÞ

500x

where xð0Þ ¼ 0 and uð0Þ ¼ 0: This is now in a form which can be directly input into a fourth order Runge Kutta solver. The Runge Kutta method is more accurate then the centered difference approximation. This can be seen in Table 6.3, which gives the absolute value of the difference between the exact solution and the computed solutions. This is known as the absolute error. Moreover, the Runge Kutta method is self starting and can be used for a single variable or a system of variables as in the above example. The price that is paid is that the fourth order Runge Kutta method requires four function evaluations for the first derivative for each time step. This is offset by the fact that this method has higher accuracy and has a large stability region so one can take larger time steps. TABLE 6.2 Time t 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48

Solution to Example 6.2.1.5 Exact Solution 0 0.00492 0.01870 0.03864 0.06082 0.08086 0.09451 0.09743 0.08710 0.06356 0.02949 20.01005 20.04761 20.07581 20.08910 20.08486 20.06393 20.03043 0.00906 0.04677 0.07528 0.08898 0.08518 0.06436 0.03136

Central Difference 0 0.00500 0.01900 0.03920 0.06159 0.08167 0.09541 0.09807 0.08712 0.06274 0.02782 20.01267 20.05063 20.07846 20.09059 20.08461 20.06171 20.02646 0.01407 0.05180 0.07916 0.09069 0.08409 0.06066 0.02511

Runge Kutta 0 0.00492 0.01869 0.03862 0.06076 0.08083 0.09447 0.09741 0.08709 0.06359 0.02956 20.00955 20.04750 20.07571 20.08903 20.08485 20.06400 20.03056 0.00887 0.04656 0.07509 0.08886 0.08516 0.06423 0.03157

68

Vibration and Shock Handbook TABLE 6.3 Absolute Error for Centered Difference and for the Fourth Order Runge Kutta Time t 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48

Error for Central Difference

Error for Runge Kutta All Times 1.0 £ 10 3

0 0.0001 0.0003 0.0006 0.0008 0.0008 0.0009 0.0006 0.0000 0.0008 0.0017 0.0026 0.0030 0.0026 0.0015 0.0003 0.0022 0.0040 0.0050 0.0050 0.0039 0.0017 0.0011 0.0037 0.0062

0 0.0 0.0100 0.0180 0.0600 0.0300 0.0400 0.0200 0.0100 0.0300 0.0700 0.0500 0.1100 0.1000 0.0700 0.0100 0.0700 0.1300 0.1900 0.2100 0.1900 0.1200 0.0200 0.1300 0.2100

Stability is a measure of how quickly errors in the computed solution grow or decay. There are very few numerical methods that are stable for all choices of time step. For most methods, there is a range of time steps which produce a stable method. The fourth order Runge Kutta method is stable for larger values of the time step than are the lower order Runge Kutta methods. A numerical method will not converge to a solution if the time step does not produce a stable method. Often, the stability criterion places a stricter limitation on the time step than accuracy does. For a more complete discussion of stability, see Strang (1986). In Table 6.3, the absolute error for both methods grows but it does not grow exponentially. Controlled error growth is the signature of a stable time step.

6.2.2 *

*

*

*

*

6.3

Summary of Single-Degree-of-Freedom System Unforced equation of motion m€x þ c_x þ kx ¼ 0: Forced equation of motion m€x þ c_x þ kx ¼ f ðx; x_ ; tÞ: 1 ðx Centered difference approximation for the first derivative x_ i ¼ 2h 1 iþ1 Centered difference approximation for the second derivative x€ i ¼ 2 ðxi h Fourth order Runge Kutta method.

xi 1 Þ: 1

2xi þ xiþ1 Þ:

Systems with Two or More Degrees of Freedom

A system that requires more than one coordinate to describe its motion is a multi DoF system. These systems differ from single DoF systems in that n DoF are described by n simultaneous differential equations and have n natural frequencies. When these systems are written in matrix notation, they look


69

just like the single DoF system. The equations of motion of a viscously damped multi DoF system can be written as follows: ½M x€ þ ½C x_ þ ½K x ¼ f where ½M ; ½C ; and ½K are the mass, damping, and stiffness matrices, respectively; x is the displacement vector; and f is the force vector. Both the central difference method and Runge Kutta can be applied to the matrix equation. The method follows exactly that given above where the scalar quantities are replaced by the matrix quantities.

6.3.1

Example

As an example, consider the two DoF system shown in Figure 6.3. In this example, k1 ¼ k2 ¼ 36 KN=m; m1 ¼ 100 kg; m2 ¼ 25 kg; and f ¼ 4000 N for t . 0 and 0 for t # 0: The initial conditions are all zero, i.e., x1 ¼ x_ 1 ¼ x2 ¼ x_ 2 ¼ 0: The equation of motion for this system is 100€x1 þ 36;000x1

36;000ðx2

25€x2 þ 36;000ðx2

x1 Þ ¼ 0

x1 Þ ¼ f

This can be written in matrix notation as " # " # 100 0 2 1 x€ þ 36;000 ¼f 0 25 1 1 where x ¼ ðx1 ; x2 Þt and f ¼ ð0; f Þt : 6.3.1.1

FIGURE 6.3 Two DoF system. (Source: Thomson and Dahleh 1998. Theory of Vibration Applications, 5th ed. With permission.)

Centered Difference

Using the centered difference approximation for the second derivatives, one obtains the following two recurrence relations x1iþ1 ¼ ð 720x1i þ 360x2i ÞDt 2 þ 2x1i x2iþ1 ¼ ð1440ðx1i

x1i

x2i Þ þ 160ÞDt 2 þ 2x2i

1

x2i

1

These two equations are only valid for i $ 3: From the single DoF system, we know that this method is not self starting. In order to compute x12 and x22 ; one needs to use an additional equation, which one obtains from the Taylor expansion. First one needs to compute the initial acceleration for the system. This is obtained from the differential equation and the initial data. For this problem, the initial acceleration is given by x€ 1 ¼ 0 and x€ 2 ¼ 160: Next, values for x1 1 and x2 1 are computed from the Taylor expansion x1 1 ¼ x10

Dt x_ 1 þ

Dt 2 x€ ¼ 0 2 1

x2 1 ¼ 80Dt 2 Now, the recurrence relations can be used to compute the rest of the terms. An example of the MATLAB code for this calculation is the following.

6 10

6.3.1.2


Pseudocode for Example 6.3.1

The following is a MATLAB m file for Example 6.3.1. clear deltat ¼ 0.01; deltsq ¼ deltat p deltat; x2ð1Þ ¼ 0; x1ð1Þ ¼ 0; x2(2) ¼ 160/2 p deltsq; x1(2) ¼ (60 px2(2)pdeltsq)/(1 þ 120 p deltsq); x2ddot(2) ¼ 1440p(x1(2) x2(2)) þ 160; x1ddot(2) ¼ 720px1(2) þ 360px2(2); for i ¼ 3:51 x2(i) ¼ x2ddot(i 1)pdeltsq þ 2 p x2(i 1) x1(i) ¼ x1ddot(i 1)pdeltsq þ 2px1(i 1) x2ddot(i) ¼ 1440p(x1(i) x2(i)) þ 160; x1ddot(i) ¼ 720 p x1(i) þ 360 p x2(i); end

x2(i x1(i

2); 2);

Figure 6.4 gives the displacement of x1 and x2 over time for Example 6.3.1. 6.3.1.3

Runge –Kutta

Alternately, the system of second order equations can be integrated using the fourth order Runge Kutta method. In order to use Runge Kutta, the second order system has to be written as a system of first order equations. The following second order system x€ 1 ¼

720x1 þ 360x2

x€ 2 ¼ 1440ðx1

x2 Þ þ 160

0.45 0.4 0.35

x2

0.3 0.25 x1

0.2 0.15 0.1 0.05 0 − 0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

FIGURE 6.4 Displacement versus time. (Source: Thomson and Dahleh 1998. Theory of Vibration Applications, 5th ed. With permission.)


6 11

becomes the following first order system x_ 1 ¼ x3 x_ 2 ¼ x4 x_ 3 ¼

720x1 þ 360x2

x_ 4 ¼ 1400ðx1

x2 Þ þ 160

Two new variables have been introduced. 6.3.1.4

Integrating Ordinary Differential Equations Using MATLAB

There are several ODE solvers in MATLAB, especially in version six. The discussion here will be limited to the least complicated solvers. These are ode23 and ode45, which are implementations of a second and third order, and a fourth and fifth order solver, respectively. Like all Runge Kutta methods, these solvers work on first order equations. If the equation is of higher order, it needs to be converted to a system of first order equations. Then the equation should be written in vector form, i.e., x0 ¼ fðx; tÞ: The user must write a MATLAB function routine which computes the values of f ¼ ðf1 ; f2 ; …fn Þ given the values of ðx; tÞ: Once this function files exists, it can be used as input in either ode23 or ode45. The function call details are given in the on line help facility in MATLAB. See the Appendix for an introduction to MATLAB.

6.3.2 1. 2. 3. 4.

6.4 6.4.1

Summary of Two-Degree-of-Freedom System Equation of motion: ½M x€ þ ½C x_ þ ½K x ¼ f: Centered difference for systems of equations. Runge Kutta methods. MATLAB commands for solution of ordinary differential equations

ode23 and ode45.

Finite Difference Method for a Continuous System Bar

A thin uniform rod is the first example of a continuous system that will be examined. The equation of motion for such a system is given by

›2 u=›x2 ¼ 1=c2 ›2 u=›t 2

pffiffiffiffi where c ¼ E=r: E is the modulus of elasticity and r is the density of the rod. A complete derivation of this equation can be found in many books (see Thomson and Dahleh 1998). Separation of variables can be used to find a solution of the form uðx; tÞ ¼ UðxÞGðtÞ: This substitution gives the equation. d2 U 1 d2 GðtÞ GðtÞ ¼ UðxÞ dx2 c2 dt 2 Upon rearrangement, this becomes 1 d2 U 1 d2 GðtÞ 2 ¼ 2 UðxÞ dx c GðtÞ dt 2 The left hand side of the equation is independent of t and the right hand side is independent of x: Therefore, each side must equal a constant. Let this constant be denoted by a and a ¼ ðv=cÞ2 : There are now two differential equations, the displacement equation d2 U=dx2 þ a2 U ¼ 0

6 12


and the temporal equation d2 G=dt 2 þ v2 G ¼ 0 The displacement equation can be solved using a finite difference approximation. The finite difference approximation requires dividing the FIGURE 6.5 Discretized bar. bar of length L into n 1 pieces each of length h ¼ L=ðn 1Þ: The finite difference mesh consists of n points, 1; 2; 3; …; n: The first node (labeled 1) is the left hand endpoint of the bar, and the last node, labeled n; is the right hand endpoint of the bar (see Figure 6.5). The value of U at each point i is denoted by Ui : Using the centered difference approximation for the second derivative, the discretized version of the displacement equation becomes ðUiþ1

2Ui þ Ui 1 Þ=h2 þ aUi ¼ 0

Collecting like terms, the equation becomes Uiþ1

ð2

lÞUi þ Ui

1

¼0

where l ¼ h2 a2 : As is typical for the finite difference approximation, this equation holds at the interior mesh points ði ¼ 2; 3; …; n 1Þ and not at the endpoints. To see that this equation does not hold at the endpoints, substitute i ¼ 1 into the recurrence relationship. The resulting equation is U2

ð2

lÞU1 þ U0 ¼ 0

The problem is that there is no value U0 : Recall that U1 represents the displacement of the left hand endpoint. Since U0 represents a point even further to the left, it is not on the bar. Similarly, the recurrence relation does not work for i ¼ n: When the relationship is evaluated for i ¼ n; the value Unþ1 is needed and it does not exist. The displacement equation evaluated for i ¼ n is Unþ1

ð2

lÞUn þ Un

1

¼0

When the relationship is evaluated for ði ¼ 2; 3; …; n 1Þ, the resulting matrix problem is tri diagonal. This is characteristic of the centered difference approximation. It is given by the following matrix: 2 32 3 2 3 1 2 l 1 0 … 0 U1 0 6 76 7 6 7 6 0 7 6 7 6 1 0 … 0 76 U2 7 6 0 7 1 2 l 6 7 6 76 7 6 7 6 76 7 6 7 6 0 6 7 6 7 1 … 0 7 0 1 2 l 6 76 U3 7 6 0 7 6 76 7 6 7 6 76 7 ¼ 6 7 6 76 7 6 7 6 76 7 6 7 6 76 7 6 7 6 76 7 6 7 6 76 7 6 7 6 76 7 6 7 6 76 7 6 7 4 54 5 4 5 1 0 1 2 l Un 0 This matrix problem does not have a unique solution since there are n unknowns and there are only n 2 equations. The boundary conditions for the bar provide the needed two additional constraints. For a bar that is fixed at both ends, the deflection is zero at the ends (x ¼ 0 and x ¼ L). Under these conditions, U1 ¼ Un ¼ 0: If one of the ends is free then the stress is zero at that end, which means that dU=dx ¼ 0: For the purposes of illustration, let us assume that the end at x ¼ 0 is fixed and the other end is free. This means that U1 ¼ 0 and dU=dxln ¼ 0: If we use the centered difference approximation for the derivative, we get dU=dxln ¼ ðUnþ1 which means that Unþ1 ¼ Un 1 :

Un 1 Þ=2h ¼ 0


6 13

Unþ1 is the displacement of a point off the bar. We never compute Unþ1 ; however, as one can see from a previous calculation, this value is used when the recurrence relationship is evaluated for i ¼ n; Unþ1 ð2 lÞUn þ Un 1 ¼ 0: Under the assumption that the stress is zero at the right hand endpoint, the remaining constraint becomes ð2

lÞUn

2Un

1

¼0

For a bar that has one fixed left hand endpoint and a free right hand end, the matrix problem is the following: 2 32 3 2 3 22l 21 0 0 … 0 U2 0 7 6 7 6 76 6 7 6 U3 7 6 0 7 6 21 2 2 l 21 7 0 … 0 7 6 7 6 7 6 76 7 6 7 6 76 6 U4 7 6 0 7 6 0 7 6 7 21 2 2 l 21 … 0 7 6 76 7 6 7 6 76 6 7¼6 7 7 6 7 6 76 6 7 6 7 6 76 7 6 7 6 76 7 6 7 6 76 7 6 7 6 0 7 6 7 6 7 21 2 2 l 21 7 607 4 56 U n 1 4 5 4 5 Un 0 22 2 2 l This matrix problem has a unique solution because there are n there are n 1 unknowns. 6.4.1.1

1 linearly independent equations and

Example

Consider a bar of unit length that is divided into four equal pieces. The left hand end is fixed so that U1 ¼ 0; and the right hand end is free. The eigenvalue problem for this system is the following: 2 32 3 2 3 1 0 0 U2 0 2 l 6 76 7 6 7 6 1 2 l 7 6 7 6 1 0 76 U3 7 6 0 7 6 7 6 76 7 ¼ 6 7 6 76 7 6 7 6 0 6 7 6 7 1 7 1 2 l 4 54 U4 5 4 0 5 0 0 2 2 l U5 0 The eigenvalues, l; are l ¼ ½0:1522; 1:2346; 2:7654; 3:8478 : The natural frequencies can be recovered pffiffiffiffi from the eigenvalues since l ¼ h2 v=c and c ¼ E=r is determined by the material properties of the bar.

6.4.2

Beam

A second example of finite difference approximation is given by looking at a centered difference approximation to compute the transverse vibration of a uniform beam. We will consider a special case of the Euler equation for the beam. The following fourth order equation results when the flexural rigidity, ðEIÞ; is constant: d4 W=dx4

b4 W ¼ 0

where b contains all of the material properties of the beam (i.e., b4 ¼ rAv2 =EI). The centered difference formula for the fourth derivative is given by f iv ¼ 1=h4 ðfiþ2

4fiþ1 þ 6fi

4fi

1

þ fi 2 Þ

In order to see that this expansion is in fact second order, combine the Taylor expansions for fiþ2 ; fiþ1 ; fi ; fi 1 ; and fi 2 ; and notice that you obtain an approximation for the fourth derivative which exactly matches the combined Taylor approximations up to order h2 : Using this approximation, the transverse beam equation at the interior mesh point becomes Wiþ2

4WIþ1 þ ð6

lÞWi

4Wi

1

þ Wi

2

¼0

6 14


where l ¼ h4 b4 : This equation is valid for i ¼ 3; n 2: As before, the boundary conditions have to be used in order to have sufficent conditions to determine all n values for the deflection, W: There will be two additional constraints placed at each end. In this case, all of the types of boundary condition require fictitious points. In other words, they require that the beam be extended beyond the physical boundaries to include W 1 ; W0 ; Wnþ1 ; and Wnþ2 : Let us consider three types of boundary conditions: fixed end, free end, and simply supported end. For simplicity, we will derive all of the conditions for the right hand endpoint x ¼ L: Fixed end: When the end of the beam is fixed, both the deflection W ¼ 0 and dW=dx ¼ 0: As above, the centered difference approximation for the first derivative requires the introduction of a fictitious point. These conditions translate into Wn ¼ 0 and dW=dxln ¼ 1\h2 ðWnþ1

Wn 1 Þ ¼ 0

These constraints reduce to the following: Wnþ1 ¼ Wn

1

Free end: The bending moment and the shear force are zero at the free end. These are given by a second and third derivative, respectively. Assuming the free end is located at N; the two fictitious points introduced are at n þ 1 and n þ 2: The discrete version of the two constraints is: d2 W=dx2 ln ¼ 1=h2 ðWn d3 W=dx3 ln ¼ 1=ð2h3 ÞðWnþ2

1

2Wn þ Wnþ1 Þ ¼ 0

2Wnþ1 þ 2Wn

1

Wn 2 Þ ¼ 0

Simply supported end: The boundary conditions of this type require that the deflection and the moment are zero. When these are applied to the right hand endpoint Wn ¼ 0 d2 W=dx2 ¼ 1=h2 ðWnþ1 2Wn þ Wn 1 Þ ¼ 0 Combining these two equations, one sees that the value of W at the fictitious point is equal to the value on the beam, i.e., Wnþ1 ¼ Wn 1 : All three of these boundary condition types allow one to replace the fictitious values in the recurrence relations with values on the beam. The procedures followed are very similar to those outlined for the longitudinal vibration of a beam.

6.4.3

Summary of Finite Difference Methods for a Continuous System

Equation of motion for the bar: ›2 u=›x2 ¼ 1=c2 ›2 u=›t 2 : Displacement equation for the bar: d2 U=dx2 þ a2 U ¼ 0: Finite difference approximation: Uiþ1 ð2 lÞUi þ Ui 1 ¼ 0: Equation of motion for the beam: d4 W=dx4 b4 W ¼ 0: Centered difference approximation for the fourth derivative: f iv ¼ 1=h4 ð fiþ2 4fiþ1 þ 6fi 4fi 1 þ fi 2 Þ: 6. Finite difference approximation for the beam equation: Wiþ2 4Wiþ1 þ ð6 lÞWi 4Wi 1 þ Wi 2 ¼ 0: 1. 2. 3. 4. 5.

6.5

Matrix Methods

In the preceding section, the equations of motion are solved for the system. Often, one is not interested in the complete solution. Rather, one is interested in the natural frequencies and the normal modes. When the number of DoF is very high, only the lowest natural frequency needs to be computed. Several methods to find some, but not all, of the eigenvalues of the system will be presented.


6.5.1

6 15

Example: Three-Degree-of-Freedom System

A three DoF system will be used as an example. The larger DoF systems work exactly in the same manner but produce larger calculations. Let us consider the following spring mass system in Figure 6.6. The equation of motion for this system is 2 32 3 2 32 3 2 0 0 x€ 1 3 1 0 x1 6 76 7 6 76 7 6 76 7 6 76 7 M 6 0 1 0 76 x€ 2 7 þ K 6 1 2 1 76 x 2 7 4 54 5 4 54 5 0 0 1 2 3 0 6 7 6 7 ¼ 607 4 5

x€ 3

0

1

1

x3

FIGURE 6.6 Three spring mass system. (Source: Thomson and Dahleh 1998. Theory of Vibration Applications, 5th ed. With permission.)

0 Assuming a harmonic solution of the form x ¼ A sin ðvtÞ gives the following eigenvalue problem: 3 2 312 3 2 3 0 2 2 0 0 3 1 0 x1 0 7 6 7C6 7 6 7 B 6 B l6 0 1 0 7 þ k6 1 2 C6 7 6 7 17 5 4 5A4 x 2 5 ¼ 4 0 5 @ 4 0 0 1

0

1

1

x3

0

where l ¼ v2 m=k: The eigenvalues are found by setting the characteristic equation of the determinant equal to zero. This gives the following polynomial:

l3

4:5l2 þ 5l

1¼0

It is a simple matter to graph this polynomial and see that there are three real roots to this equation.

6.5.2

Bisection Method

One method for calculating the roots of a nonlinear function is called the bisection method. This method finds the zeros of nonlinear functions by bracketing the zero in an interval ½a; b : The interval is chosen so that f ðaÞ and f ðbÞ are of opposite sign. If f is a continuous function and it is positive at one endpoint, say f ðaÞ . 0; and it is negative at the other endpoint, f ðbÞ , 0; then it has had to go through zero at some point in the interval. It is possible that it has gone through zero more then once in the interval. From Figure 6.7, one can see that one root is between [0, 0.5]; another is between [1, 1.5]; and a third root is in [2.5, 3]. In order to find all three roots, the bisection method has to be used three times; one for each interval. As an example, the MATLAB code for the first root is given below. 6.5.2.1

MATLAB Code for the Bisection Method

clear %set an acceptable tolerance for the root tol ¼ 10e 4 %endpoints of the interval a¼0 b ¼ 0:5 for i ¼ 1:20 c ¼ 0:5pða þ bÞ if abs( f(c)) , tol, break, end

6 16


6 4 2 0 −2 −4 −6 −8 −10 −12 −1

−0.5

0

0.5

1

2

1.5

FIGURE 6.7 Graph of the cubic polynomial l3 4:5l2 þ 5l of Vibration Applications, 5th ed. With permission.)

2.5

3

3.5

1 ¼ 0: (Source: Thomson and Dahleh 1998. Theory

if f ðaÞpf ðcÞ , 0 b¼c else a¼c end end The root 0.2554 is found after ten iterations. As long as the initial interval is chosen so that the function has different signs at the endpoints, this method will always converge. There are nonlinear root finding methods, such as Newton’s method, which when they converge do so more quickly. However, they may or may not converge. 6.5.2.2

MATLAB Function for Finding the Roots of a Polynomial

MATLAB has a built in root finding method which requires the creation of a vector c that contains the coefficients of the polynomial in descending order. The MATLAB command roots (c), produces the roots of the polynomial. For our example, c ¼ ½1; 4:5; 5; 1 : The roots of this equation are given by l ¼ 2.8892, 1.3554, 0.2554. 6.5.2.3

Mode Shapes

The mode shapes are determined by substituting each eigenvalue into the matrix problem and computing the corresponding eigenvector. As an example of how to compute the eigenvector for a given eigenvalue, we will compute the eigenvector associated with l ¼ 2:8892: The first step is to substitute this value of l into the eigenvalue equation. This gives the following problem: 2 32 3 2 3 2:489 1 0 x1 0 6 76 7 6 7 6 1 1:745 6 7 6 7 1 7 4 54 x 2 5 ¼ 4 0 5 0

1

0:744

x3

0

The solution vector found by Gaussian elimination is given by x ¼ ðx1 ; x2 ; x3 Þt ¼ ð0:2992; 0:7446; 1:0Þt


6 17

6.5.3 Directly Calculating the Eigenvalues and Eigenvectors from the Matrix Equation Ideally one would like to use a computer program such as MATLAB to compute both the eigenvalues and the eigenvectors directly from the matrix equation. In order to do this, the matrix equation needs to be rewritten slightly. In general, the matrix equation for the normal mode vibration is given by ½ lM þ K x ¼ 0 where M and K are the mass and stiffness matrices, respectively, both are square symmetric matrices, and l is the eigenvalue related to the natural frequency by l ¼ nv2 : Premultiplying the preceding equation by M 1 , we have another form of the equation: ½ lI þ A x ¼ 0 where A ¼ M 1 K and A is called the dynamic matrix. In general, A is not symmetric. In order to use MATLAB to compute the eigenvectors and eigenvalues, the matrix must be symmetric. There is a standard transformation of coordinates that converts the matrix problem into a standard eigenvalue problem which can be solved by a computer. Assume that we have a transformation of coordinates which has the following form: x¼U 1 When this transformation is substituted into the equation, we get ½ lMU Premultiplying the equation by U

t

1

þ KU

1

1

þ U t KU

y¼0

gives

½ lU t MU

1

y¼0

From this equation, one can see that if either M or K equals Ut U; then the preceding equation is reduced to standard form, at which time it is possible to compute both the eigenvalues and eigenvectors directly. To see how this works, let us assume that M ¼ Ut U: The equation becomes ½ lI þ U t KU

1

y¼0

where l ¼ v2 : This equation is in standard form since U t KU 6.5.3.1

1

is symmetric.

Example

To illustrate the use of the dynamic matrix and the standard computer form, we can use MATLAB to calculate the eigenvalues and eigenvectors for Example 6.5.1. The first step is to convert the problem into standard form. For this example, the dynamic matrix is given by 2 3 1:5 0:5 0 6 7 A¼6 1:0 7 4 1:0 2:0 5 0

1:0

1:0

We can compute both the eigenvalues and the eigenvectors in MATLAB using the command ½U; D ¼ eigðAÞ: The result of this command is two matrices, U and D: Matrix U contains the eigenvectors as column vectors and D is a diagonal matrix which has the eigenvalues on the diagonal. Continuing with our example, we get the following two matrices: 2 3 0:7569 0:3031 0:2333 6 7 U¼6 0:8422 0:5808 7 4 0:2189 5 0:6158

0:4458

0:7799

6 18


and 2 6 D¼6 4

1:3554

0

0

0

2:8892

0

0

0

0:2544

3 7 7 5

In order to use this transformation then, one has to be able to write M or K as Ut U: This is known as the Cholesky decomposition. In MATLAB, if one has a positive definite matrix M; the matrix can be Cholesky decomposed with the built in function chol. The command U ¼ cholðMÞ produces an upper triangular matrix U such that Ut U ¼ M: Cholesky decomposition is a special case of LU factorization where L is a lower triangular matrix and U is an upper triangular matrix. The computational procedure begins by writing the algebraic equations that result from the following calculation: 2 32 3 2 3 u11 0 0 u11 u12 u13 m11 m12 m13 6 76 7 6 7 6 u12 u22 0 76 0 u22 u23 7 ¼ 6 m21 m22 m23 7 4 54 5 4 5 u13

u23

u33

0

0

u33

m31

m32

m33

If the matrix M is positive definite, then the above matrix multiplication results in six linearly independent equations.

6.5.4

Summary of Matrix Methods

1. Bisection method for the roots of a polynomial. 2. MATLAB command roots for finding the roots of a polynomial. 3. Cholesky decomposition.

6.6

Approximation Methods for the Fundamental Frequency

The smallest natural frequency, known as the fundamental frequency, of a multi DoF system is often of greater interest than the high natural frequencies because its forced response in many cases is the largest. One approach to this problem is to extend the Rayleigh method to matrix problems. We will see that the Rayleigh frequency approaches the fundamental frequency from the high side.

6.6.1

Rayleigh Method

Let M and K be the mass and stiffness matrices, respectively, and x is the assumed displacement vector for the amplitude of vibration. For harmonic motion, the maximum kinetic energy is Tmax ¼ 1=2vxt Mx and the maximum potential energy is Umax ¼ 1=2xt Kx Since the maximum kinetic energy equals the maximum potential energy, these two quantities are equal. Equating these two and solving for v2 gives the Rayleigh quotient:

v2 ¼

xt Kx xt Mx

It can be shown (Thomson and Dahleh, 1998) that this quotient approaches the lowest natural frequency from above and it is somewhat insensitive to the choice of amplitudes.


6.6.2

6 19

Dunkerley’s Formula

Dunkerley’s formula produces a lower bound for the fundamental frequency and can be used in conjunction with the Rayleigh method to get a good approximation for the fundamental frequency. Dunkerley’s formula is based on the characteristic equation for the flexibility coefficients. The flexibility influence coefficient, aii , is defined as the displacement at i due to a unit force applied at j with all other forces equal to zero. This concept is most easily understood through an example. 6.6.2.1

Computation of the Flexibility Matrix

The procedure for computing the flexibility matrix and in particular, the computation of the flexibility matrix for the three spring mass matrix systems shown in Figure 6.6, are discussed.

Example First, one applies a unit force to mass 1 with no other forces present, i.e., f1 ¼ 1; f2 ¼ f3 ¼ 0: The displacements are located in the first column of the flexibility matrix. This gives 2 3 2 32 3 x1 1=k1 0 0 1 6 7 6 76 7 6 x2 7 ¼ 6 1=k1 0 0 76 0 7 4 5 4 54 5 x3

1=k1

0

0

0

In this case, springs k2 and k3 are not stretched and are displaced equally with mass 1. Now, a unit force is applied to mass 2 and there are no other forces. This allows us to write the second column of the matrix to get 32 3 2 3 2 0 1=k1 0 x1 0 76 7 6 7 6 6 x2 7 ¼ 6 0 1=k1 þ 1=k2 0 76 1 7 54 5 4 5 4 0 1=k1 þ 1=k2

x3

0

0

This time, the unit force is transmitted through k1 and k2 : The spring k3 is not stretched. Finally, the force is applied to mass 3 and there are no other forces present. This gives the third column of the matrix: 32 3 2 3 2 0 0 1=k1 x1 1 76 7 6 7 6 76 0 7 6 x2 7 ¼ 6 0 0 1=k1 þ 1=k2 54 5 4 5 4 0

x3

0

1=k1 þ 1=k2 þ 1=k3

0

Since the flexibility matrix is the sum of the three previous matrices, it is given by 2

x1

3

2

1=k1

6 7 6 6 x2 7 ¼ 6 1=k1 4 5 4 x3

1=k1

32 3 f1 76 7 76 f 2 7 54 5

1=k1

1=k1

1=k1 þ 1=k2

1=k1 þ 1=k2

1=k1 þ 1=k2

1=k1 þ 1=k2 þ 1=k3

f3

An interesting feature of the flexibility matrix is that it is symmetric about the diagonal. For simplicity of notation, let the ij element of the flexibility matrix be given by aij mj : Dunkerley’s formula is obtained from the characteristic equation of the flexibility matrix, which is obtained by computing the determinant of the following matrix: a11 m1 a21 a31

1=v2

a12 a22 m2 a32

a13 1=v

2

¼0

a23 a33 m3

2

1=v

6 20


A third degree equation in 1=v2 is obtained by expanding the determinant. One obtains the following cubic equation: ð1=v2 Þ3

ða11 m1 þ a22 m2 þ a33 m3 Þð1=v2 Þ2 þ · · · ¼ 0

A cubic equation has three roots that are denoted by ð1=v2i Þ for i ¼ 1; 2; 3: This allows the cubic equation to be factored: ð1=v2

1=v21 Þð1=v2

1=v22 Þð1=v2

1=v23 Þ ¼ 0

The highest two powers of this equation are given by ð1=v2 Þ3

ð1=v21 þ 1=v22 þ 1=v23 Þð1=v2 Þ2 þ · · · ¼ 0

The coefficient of the second highest power is equal to the sum of the roots of the characteristic equation, which is also equal to the sum of the diagonal elements of the matrix A 1 : This relationship is not just true for n ¼ 3 but is more generally true for n greater than or equal to 3. For the general n DoF system 1=v21 þ 1=v22 þ · · ·1=v2n ¼ a11 m1 þ a22 m2 þ · · · þ ann mn The fundamental frequency is the smallest natural frequency. Since v2 ; v3 ; … are larger than v1 ; the reciprocal of these frequencies is smaller then the reciprocal of the fundamental frequency. An estimate for the fundamental frequency is obtained by neglecting all of the higher modes in the left hand side of the above equation. This estimate gives a value for v1 that is smaller then the true value of the fundamental frequency. Dunkerley’s formula is a lower bound for the fundamental frequency and it is given by 1=v21 , a11 m1 þ a22 m2 þ · · · þ ann mn

6.6.3

Summary of Approximations for the Fundamental Frequency

1. Rayleigh method

v2 ¼

xt Kx xt Mx

2. Dunkerley’s formula 1=v21 , a11 m1 þ a22 m2 þ · · · þ ann mn :

6.7

Finite Element Method

In the finite element method, complex structures are replaced by assemblages of simple structural elements known as finite elements. The elements are connected by joints or nodes. The force and moments at the ends of the elements are known from structural theory, the joints between the elements are matched for compatibility of displacement, and the force and moment at the joints are established by imposing the condition of equilibrium. The accuracy obtainable from the finite element method depends on being able to duplicate the vibration mode shapes. Using only one finite element between structure joints or corners gives good results for the first lowest mode, because the static deflection curve is a good approximation to the lowest dynamic mode shape. For higher modes, several elements are necessary between structural joints. This leads to large matrices. The eigenvalues and eigenvectors need to be computed numerically. This section introduces the basic idea of the finite element method as it applies to the simple vibration problem. The basic idea behind the finite element method is to break up the structure into simple component structures. The structural elements for the bar and the beam are discussed here.


6.7.1

6 21

Bar Element

The force displacement relationship for a uni form rod is F ¼ ðEA=LÞU where E is the young’s modulus, A is the cross sectional area, L is the length of the element and U is the displacement. Figure 6.8 shows this one dimensional element. The two endpoints are the nodes. For simplicity, assume the axial displacement at any point 6 ¼ x=L is linear:

FIGURE 6.8

One dimensional element.

Uðx; tÞ ¼ aðtÞ þ bðtÞx and U1 ðtÞ ¼ Uð0; tÞ; U2 ðtÞ ¼ UðL; tÞ: These two conditions uniquely determine the coefficients aðtÞ and bðtÞ: They are given by the following: aðtÞ ¼ U1 ðtÞ and bðtÞ ¼ ðU2 ðtÞ

FIGURE 6.9

U1 ðtÞÞ=L

Linear mode shapes.

Using the linear element, the displacement anywhere along the beam is given by Uðx; tÞ ¼ ð1

x=LÞU1 ðtÞ þ x=LU2 ðtÞ

¼ w1 U1 ðtÞ þ w2 U2 ðtÞ where

w1 ¼ 1

x=L and w2 ¼ x=L

FIGURE 6.10

Superposition of the linear mode shapes.

w1 and w2 are known as the mode shape, which can be seen in Figure 6.9. These two mode shapes can be superimposed to create a linear function. An example of such a function is given in Figure 6.10. The kinetic energy of the bar is given by T ¼ :5 6.7.1.1

ðl 0

u_ 2 mdx ¼ :5m

ðl 0

ðð1

6Þ_u1 ðtÞ þ 6u_ 2 ðtÞÞ2 ld6 ¼ :5mlð1=3_u21 þ 1=3_u1 u_ 2 þ 1=3_u22 Þ

Mass Matrix

The generalized mass matrix is derived from the Lagrange equations using the following: D ›T Dt ›u_ 1 Given the kinetic energy for the bar, the Lagrange equations become D ›T ¼ mLð1=3€u1 þ 1=6€u2 Þ Dt ›u_ 1 D ›T ¼ mLð1=3€u1 þ 1=6€u2 Þ Dt ›u_ 2

6 22


The mass matrix for an axial element with a uniform mass distribution per unit length is given by " # mL 2 1 6 6.7.1.2

1

2

Stiffness Matrix

The force displacement relationship for a uniform bar is given by " #" # " # 1 u1 F1 EA 1 ¼ L 1 1 u1 F2 6.7.1.3

Variable Properties

A simple approach to problems with variable properties is to use a large number of elements of short length. The variation of mass or stiffness over each element is small and can be neglected. In this model, the mass and stiffness for each element is constant and can be placed outside the integral. If a large number of elements are needed to capture the behavior, this will lead to a large matrix problem.

Example A tapered rod is modeled as two uniform sections where EA1 ¼ 2EA2 and m1 ¼ 2m2 ; k2 ¼ 2EA2 =L: The displacement equation for this system is given by " # " #" # " # 1 u1 0 v2 m2 2ð2 þ 1Þ 1 2EA2 2 þ 1 þ ¼ 6 L 1 2 1 1 u2 0 This can be written as a standard eigenvalue problem where l ¼ v2 m2 L=12EA2 . The eigenvalue problem is the following 3

6l

ð1 þ lÞ

ð1 þ lÞ ð1

2lÞ

¼0

The solution of the determinant requires computing the roots of a quadratic equation. The two roots are l ¼ ½0:6140; 1:1088 : The two natural frequencies can be computed from l. They are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v1 ¼ 1:4029 ððEA2 Þ=ðM2 LÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ¼ 3:6477 ððEA2 Þ=ðM2 LÞÞ The modes shapes are calculated by solving the eigenvalue problem. The two eigenvectors are x1 ¼ ½0:5773; 1

t

x2 ¼ ½ 5258; 1

t

and

6.7.2

Beam

When the ends of the element are rigidly connected to the adjoining structure, the elements act like a beam with the moments and lateral forces acting at the joints. Generally, the axial displacement u2 u1 is small compared to the lateral displacement V of the beam. The local coordinates of the beam element are lateral displacements, V; and rotation, u, at the two ends. This results in four coordinates V1 ; u1; V2; u2 : Four constraints uniquely determine a cubic polynomial. Therefore, the lateral displacement of a beam is assumed to be described by a cubic


6 23

polynomial VðxÞ ¼ p1 þ p2 6 þ p3 62 þ p4 64 where 6 ¼ x=L and pi ; the coefficients of the polynomials, are related to the lateral displacement and the rotation. The lateral displacement at the left hand side determines V1 : Vð0; tÞ ¼ V1 ðtÞ ¼ p1 The rotation at the left hand end determines the second coefficient, u1 :

›V ð0; tÞ ¼ u1 ¼ p2 ›x The remaining two coefficients are combinations of the variables and they are given by the following two constraints: VðL; tÞ ¼ V2 ðtÞ and

›V ðL; tÞ ¼ u2 ›x

Applying these two conditions, one obtains P3 ¼ 1=L2 ð 3v1 ðtÞ

2u1 ðtÞL þ 3v2 ðtÞ

3

P4 ¼ 1=L ð2v1 ðtÞ þ u1 ðtÞL

u2 ðtÞLÞ

2v2 ðtÞ þ u2 ðtÞLÞ

These coefficients can be represented by the following matrix equation: 2

p1

3

2

1

6 7 6 6p 7 6 0 6 27 6 6 7¼6 6 7 6 6 p3 7 6 3 4 5 4 p4 2

0

0

1

0

2 1

3 2

0

32

v1

3

7 76 6 7 07 76 Lu2 7 7 76 7 76 6 7 17 5 4 v2 5 Lu 1

The shape functions for the beam elements are determined by equating a single displacement to one, and all other displacement to zero. The first shape function is derived by letting V1 ¼ 1 and the remaining three variables zero; i.e., V2 ¼ u1 ¼ u2 ¼ 0: This gives p1 ¼ 1

p2 ¼ 0

p3 ¼

3

p4 ¼ 2

The first shape function becomes 36 2 þ 2 6 3

w1 ¼ 1

A similar calculation for u1 ¼ 1 and V2 ¼ V1 ¼ u2 ¼ 0 yields p1 ¼ 0

p2 ¼ l

p3 ¼

w2 ¼ l6

2l62 þ l63

and the shape function becomes

2l

p4 ¼ l

The remaining two shape functions are determined similarly. The four shape functions for the beam are

w1 ¼ 1

36 2 þ 2 6 3

w 2 ¼ l6

2l62 þ l63

w 3 ¼ 36 2 w4 ¼

26 3

l62 þ l63

6 24

6.7.2.1


Mass Matrix

Just as in the case of the bar, the generalized mass mij is given by ðl Mij ¼ wi wj mdx 0

Substitution of the above shape functions and integration yields the following mass matrix for the uniform beam element in terms of the end displacements: 2 3 156 22L 54 13L 6 7 6 22L 2L2 13L 3L2 7 7 mL 6 6 7 6 7 420 6 54 13L 156 22L 7 4 5 13L

6.7.3

3L2

4L2

22L

Summary of Finite Element Method

1. Linear finite element for a bar w1 ¼ 1 2. Mass matrix for the bar

x=L and w2 ¼ x=L ML 6

3. Stiffness matrix for the bar

"

F1 F2

4. Cubic finite elements for the beam

#

EA ¼ L

"

"

2

1

1

2

#

1

1

1

#"

1

w1 ¼ 1

362 þ 263

w 2 ¼ l6

2l62 þ l63

w3 ¼ 362 w4 ¼

u1

#

u2

263

l6 2 þ l6

Appendix 6A Introduction to MATLABw MATLAB is a software package for numerical computation, visualization, and symbolic manipulation (also see Appendix 32A). It is an interactive environment with hundreds of built in functions, which are in essence subroutines. These functions range from plotting commands, to those for finding the eigenvalues and eigenvectors of a matrix, to those for solving an ordinary differential equation, and much more. In addition to the built in functions, MATLAB contains a programming language, which allows the user to write their own functions. The name MATLAB is an abbreviation of matrix laboratory. The original versions of MATLAB concentrated on numerical analysis of linear systems of equation. MATLAB is available from the Mathworks (www.mathworks.com). The best way to learn MATLAB is by playing around with the different functions (Pratap, 2001). The first thing to note when you launch MATLAB is that it is a window based environment. There are three windows: the command window, the graphics window, and the edit window. The command


6 25

window is the main window and it is the one in which you run all functions (built in or user created). This is the window which appears when the program is launched, and it has the symbol q as a prompt. The graphics window is where all of the graphics are displayed, and the edit window is where users create and save all of their own programs, known as m files. MATLAB provides routines for all of the basic areas of numerical analysis (numerical linear algebra, data analysis, Fourier transform, and interpolation), curve fitting, root finding, numerical solution of ordinary differential equations, integration, and graphics. There are specialized tool boxes for signal processes and control systems to name two. With these hundreds of functions, it is imperative to have a good help facility. In the command window, the command help functionname provides online help. For example, type help help to get more information about help. There are three other commands for information: lookfor, helpwin, and helpdesk. Lookfor gives a list of functions with the keyword in their description. Helpwin gives a help window. Helpdesk is a web browser based help. Given the early history of MATLAB as a matrix laboratory, it should not be surprising that one of its strengths is its ability to manipulate vectors and matrices very well. A row vector is created by typing in the command window the following: qx ¼ ½2; 3; 6; 4 This command will produce x¼ 2

3

6

4

A common vector is created by entering the following qx ¼ ½2; 1; 3; 4 This produces x ¼ 2 1 3 4 The elements of a vector or matrix are separated by commas or by spaces, and the rows are separated by semicolons. Printing is suppressed by ending the line with a semicolon. The following command will produce a 2 £ 2 matrix but it will not print it: " # 2 4 A¼ 1 5 Providing that the operations make sense, it is easy to do operations on vectors and matrices in MATLAB. For example, if A and B are two matrices of the same size, the command A þ B adds the matrices. There are several MATLAB commands given throughout the text of this chapter. Several books are available to get you started with MATLAB; for example, Hanselman and Littlefield (2001), Palm (2001), Pratap (2001), and Recktenwald (2000). The best way to learn more about these commands is to type help and the command name. This way, you get the most up to date information concerning the function.

References Atkinson, K. 1978. An Introduction to Numerical Analysis, 2nd ed., Wiley, New York. Cheney, E. and Kincaid, D. 1999. Numerical Mathematics and Computing, 4th ed., Brooks Cole, Monterey, CA.

6 26


Hanselman, D. and Littlefield, B. 2001. Mastering MATLAB 6: A Comprehensive Tutorial and Reference, Prentice Hall, Upper Saddle River, NJ. Isaacson, E. and Keller, H. 1966. Analysis of Numerical Methods, Wiley, New York. Palm, W. 2001. Introduction to MATLAB 6 for Engineers, McGraw Hill, Boston. Pratap, R. 2001. Getting Started with MATLAB 6: A Quick Introduction for Scientists and Engineers, Oxford University Press, New York. Recktenwald, G. 2000. Numerical Methods with MATLAB: Implementation and Application, Prentice Hall, Upper Saddle River, NJ. Strang, G. 1986. Introduction to Applied Mathematics, Wellesley Cambridge Press, Wellesley, MA. Thomson, W. and Dahleh, M. 1998. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper Saddle River, NJ.

7

Vibration Modeling and Software Tools 7.1 7.2

7.3 7.4

Datong Song

7.5

National Research Council of Canada

Cheng Huang


Zhong-Sheng Liu


7.6 7.7

Introduction ....................................................................... Formulation .......................................................................

71 72

Vibration Analysis .............................................................

79

Differential Formulation † Integral Formulation and Rayleigh Ritz Discretization † Finite Element Method † Lumped Mass Matrix † Model Reduction Natural Vibration † Harmonic Response Response † Response Spectrum

†

Transient

Commercial Software Packages ........................................ 7 13

ABAQUS † ADINA † ALGOR † ANSYS † COSMOSWorks † MSC.Nastran † ABAQUS/Explicit † DYNA3D † LS DYNA

The Basic Procedure of Vibration Analysis ..................... 7 16

Planning † Preprocessing † Solution † Postprocessing † Engineering Judgment

An Engineering Case Study .............................................. 7 19

Objectives Conditions

† †

Modeling Strategy † Boundary Material † Results

Comments .......................................................................... 7 21

Summary In this chapter, several aspects of vibration modeling are addressed. They include the formulation of the equations of motion both in differential form and integral form, the Rayleigh Ritz method and the finite element methods, and model reduction. Natural vibration analysis and response analysis are discussed in detail. Several commercial finite element analysis (FEA) software tools are listed and their capabilities for vibration analysis are introduced. The basic procedure in using the commercial FEA software packages for vibration analysis is outlined (also see Chapter 6 and Chapter 9). The vibration analysis of a gearbox housing is presented to illustrate the procedure.

7.1

Introduction

Vibration phenomenon, common in mechanical devices and structures [2,9], is undesirable in many cases, such as machine tools. But this phenomenon is not always unwanted; for example, vibration is needed in the operation of vibration screens. Thus, reducing or utilizing vibration is among the challenging tasks that mechanical or structural engineers have to face. Vibration modeling has been used extensively for a better understanding of vibration phenomena. The vibration modeling here implies a process of converting an engineering vibration problem into a mathematical model, whereby the major vibration characteristics of the original problem can be accurately predicted. The mathematical model of vibration in its general sense consists of four components: a mass (inertia) term; a stiffness term; an 71

72


excitation force term; and a boundary condition term. These four terms are represented in differential equations of motion for discrete (or, lumped parameter) systems, or boundary value problems for continuous systems. A damping term is included if damping effects are of concern. Depending on the nature of the vibration problem, the complexity of the mathematical model varies from simple spring mass systems (see Chapter 1) to multi degree of freedom (DoF) systems (see Chapter 3); from a continuous system (see Chapter 4) for a single structural member (beam, rod, plate, or shell) to a combined system for a built up structure; from a linear system to a nonlinear system. The success of the mathematical model heavily depends on whether or not the four terms mentioned before can represent the actual vibration problem. In addition, the mathematical model must be sufficiently simplified in order to produce an acceptable computational cost. The construction of such a representative and simple mathematical model requires an in depth understanding of vibration principles and techniques, extensive experience in vibration modeling, and ingenuity in using vibration software tools. Furthermore, it also requires sufficient knowledge of the vibration problem itself in terms of working conditions and specifications. Except for few special cases that promise exact and explicit analytical solutions, vibration models have to be studied by means of approximate numerical methods such as the finite element method. The finite element method has been very successfully used for vibration modeling for the past two decades. Its success is attributed to the development of sophisticated software packages and the rapid growth of computer technology. In this chapter, several aspects of the construction of mathematical models of linear vibration problems without damping will be addressed. The capabilities of the available software packages for vibration analysis are listed and the basic procedure for vibration analysis is summarized. As an illustration, an engineering example is given.

7.2 7.2.1

Formulation Differential Formulation

In a majority of engineering vibration problems, the amplitude of vibrations is very small, so that the following assumptions hold: (1) a linear form of strain displacement relationships, and (2) a linear form of stress strain relationships (Hooke’s Law). If the energy losses are negligible, it is straightforward to apply Newton’s (second) law and Hooke’s Law to derive the equations of motion, which appear as differential equations. Consider a single DoF spring mass system, as shown in Figure 7.1. The two laws are given by ( m€uðtÞ ¼ f ; f ¼ kuðtÞ;

u(t) k m

FIGURE 7.1

Newton’s law Hooke’s Law

Single DoF spring mass system.

ð7:1Þ

The first equation describes the inertia force, and the second equation describes the elastic force. These two forces are essential for mechanical vibration to exist (see Chapter 1). In a similar way, the differential equations are given directly when Newton’s law plus Hooke’s Law is applied to a multiple DoF spring mass system, shown in Figure 7.2 ( € ¼ F; Newton’s law MuðtÞ ð7:2Þ F ¼ KuðtÞ; Hooke’s Law where M is the (diagonal) mass matrix, and K is the stiffness matrix.

Vibration Modeling and Software Tools

73

u1(t) k1

u2(t) kn

k2 m1

un(t)

m2

FIGURE 7.2

mn

Multiple DoF spring mass system.

In the case of continua, the differential equations of motion can be derived by means of Newton’s law and Hooke’s Law in the same way as above. But in this case the boundary conditions have to be specified in order to make the problem statement complete (see Chapter 4). For example, as a direct consequence of Newton’s law and Hooke’s Law, the differential equation of bending vibration of a clamped clamped Euler beam may be given as (Chapter 4) 8 2 › uðx; tÞ > > ¼ f; Newton’s law > r > > dt 2 > > < ›4 uðx; tÞ ð7:3Þ ; Hooke’s Law f ¼ EI > > ›x4 > > > > > : uð0; tÞ ¼ uðl; tÞ ¼ ›uð0; tÞ ¼ ›uðl; tÞ ¼ 0; Boundary conditions ›x ›x where r represents the mass per unit length, l the beam length, and EI the bending stiffness (flexural rigidity).

7.2.2

Integral Formulation and Rayleigh– Ritz Discretization

Besides the approach in which Newton’s law and Hooke’s Law are directly used to establish equations of motion, there are alternatives: Hamilton’s principle, the minimum potential energy principle, and the virtual work principle; all of which appear in integral form. From a mathematical standpoint, the differential equations and the integral equations are equivalent in that one can be derived from another. However, they are very different in that the integral equations facilitate the application of the discretization schemes such as the finite element method, an element wise application of Rayleigh Ritz method. Therefore, Hamilton’s principle, as one of the integral formulations, and its Rayleigh Ritz discretization are briefly introduced here in order to provide a better understanding of the finite element method. Denote T as the system kinetic energy, V the system potential energy, and dW the virtual work done by nonconservative forces. Hamilton’s principle [11] states that the variation of the Lagrangian ðT VÞ Standard terminology plus the line integral of the virtual work done by the nonconservative forces during any time interval must be equal to zero. If the time interval is denoted by ½t1 ; t2 ; then Hamilton’s principle can be expressed as ðt2 ðt2 d ðT VÞdt þ dW dt ¼ 0 ð7:4Þ t1

t1

In the case of a continuum, we look for an approximate solution uðx; y; z; tÞ in the form of uðx; y; z; tÞ ¼

n X i¼1

wi ðx; y; zÞqi ðtÞ

ð7:5Þ

where wi ðx; y; zÞ is called a Rayleigh Ritz shape function and qi ðtÞ is called a generalized coordinate. In this way, the system kinetic energy and the system potential energy can be, respectively, expressed

74


as follows: T¼

n X n 1X 1 _ m q_ q_ ; u_ T ðtÞMuðtÞ 2 i¼1 j¼1 ij i j 2

ð7:6Þ

n X n 1X 1 kij qi qj ; uT ðtÞKuðtÞ 2 i¼1 j¼1 2

ð7:7Þ

and V¼

where uT ðtÞ ; ½q1 ; q2 ; …; qn ; M ; ½mij ; K ; ½kij : The virtual work done by the generalized forces is dW ¼

n X i¼1

fi ðtÞdqi ¼ FT duðtÞ

ð7:8Þ

where FT ; ½ f1 ðtÞ; f2 ðtÞ; …; fn ðtÞ and fi ðtÞ is the generalized force corresponding to the nonconservative force f ðx; y; z; tÞ ð ð7:9Þ fi ðtÞ ¼ wi ðx; y; zÞf ðx; y; z; tÞdv Substituting Equation 7.6, Equation 7.7 and Equation 7.8 into Hamilton’s principle (Equation 7.4) and conducting a routine variation operation, one has ðt2 ðu_ T M du_ uT K du þ FT duÞdt ¼ 0 ð7:10Þ t1

Applying the separation integration to the first term of the above equation and noting that the variations of the generalized coordinate du at times t1 and t2 equal zero, Equation 7.10 is rewritten as ðt2 € ð MuðtÞ KuðtÞ þ FÞdu dt ¼ 0 ð7:11Þ t1

Because du; the variation of the generalized coordinate vector, is arbitrary and independent, from the above equation one obtains € þ Ku ¼ F MuðtÞ

ð7:12Þ

which is the vibration equation resulting from a Rayleigh Ritz discretization.

7.2.3

Finite Element Method

In the finite element method (FEM) [7,10,12], a continuum is divided into a number of relatively small regions called elements that are interconnected at selected nodes. This procedure is called discretization. The deformation within each element is expressed by interpolating polynomials. The coefficients of these polynomials are defined in terms of the element nodal DoF that describe the displacements and slopes of selected nodes on the element. By using the connectivity between elements, the assumed displacement field can then be written in terms of the nodal DoF by means of the element shape function. Using the assumed displacement field, the kinetic energy and the strain energy of each element are expressed in the form of the element mass and stiffness matrices. The energy expressions for the entire continua can be obtained by adding the energy expressions of its elements. This leads to the assembled mass matrix and the assembled stiffness matrix, and finally to the finite element vibration equation. The displacement in the interior of an element e is determined by a polynomial uðx; y; z; tÞ ¼ Nue

ð7:13Þ


75

where the matrix N is called the shape function matrix of the element e; and ue the vector of the nodal DoF. Based on the element displacement expression Equation 7.13, one can obtain the strain and the stress in the element e and finally the strain energy. The strain and the stress in the element e are 1 ¼ ›uðx; y; z; tÞ ¼ ›Nue ¼ Bue

ð7:14Þ

s ¼ D1 ¼ DBue ¼ Sue

ð7:15Þ

and

respectively, where › is the differential operator matrix, B ¼ ›N is the element strain matrix, D is the elastic matrix, and S ¼ DB is called the element stress matrix. The strain energy in the element e is given by the element strain 1 and stress s as 1ð T 1 Ve ¼ 1 s dv ¼ ðue ÞT Ke ue ð7:16Þ 2 2 where Ke ¼

ð

BT DB dv

ð7:17Þ

is called the element stiffness matrix. The velocity at a point ðx; y; zÞ in the element e can be obtained from Equation 7.13 as u_ ðx; y; z; tÞ ¼ Nu_ e

ð7:18Þ

So the kinetic energy of the element e is Te ¼

1ð T 1 ru_ u_ dv ¼ ðu_ e ÞT Me u_ e 2 2

ð7:19Þ

where Me ¼

ð

rNT N dv

ð7:20Þ

is called the element mass matrix. The equivalent nodal force Fe corresponding to the force f e applied onto the element e is determined by equaling the work done by Fe to the work done force by f e along any virtual displacement. This leads to the following: ð ð ðdue ÞT Fe ¼ duT f e dv ¼ ðdue ÞT NT f e dv ð7:21Þ Note that as the variation of the nodal displacement is arbitrary, one can obtain the expression of the equivalent nodal force Fe from Equation 7.21 as ð Fe ¼ NT f e dv ð7:22Þ Now we have the kinetic energy, the strain energy, and the equivalent nodal force of the element e: But these quantities are expressed in the local coordinate system ðX e ; Y e ; Z e Þ of the element e; not in the global coordinate system ðX; Y; ZÞ: In order to calculate the corresponding counterparts for the whole structure, it is necessary to transform the expressions of the kinetic energy, the strain energy, and the equivalent nodal force of the element e from the local coordinate system into the global one. Let L be the transformation matrix from the global coordinate system to the local coordinate system. Then the nodal displacement vector ue in the local coordinate system is related to the nodal displacement

76


vector ue in the global coordinate system by the following: ue ¼ Lue

ð7:23Þ

Similarly, the equivalent nodal force vector Fe in the local coordinate system is related to the equivalent nodal force vector Fe in the global coordinate system by Fe ¼ LFe

ð7:24Þ

Substituting Equation 7.23 into Equation 7.16 and Equation 7.19, and noting that L is a normal matrix ðLT ¼ L 1 Þ; the element stiffness and mass matrices in the global coordinate system can be, respectively, expressed as Ke ¼ LT Ke L

ð7:25Þ

Me ¼ LT Me L

ð7:26Þ

and

The equivalent nodal force vector in the global coordinate system is solved from Equation 7.24 Fe ¼ LT Fe

ð7:27Þ

In this way, we can obtain the total strain energy of the structure as V¼

X e

Ve ¼

1X eT e e 1 ðu Þ K u ¼ uT Ku 2 e 2

where the matrix K¼

X e

Ke

ð7:28Þ

ð7:29Þ

is called the global stiffness matrix of the structure. The vector u is the global nodal displacement vector of the structure. Similarly, the total kinetic energy of all of the elements can be written as X e 1X eT e e 1 T¼ T ¼ ðu_ Þ M u_ ¼ u_ T Mu_ ð7:30Þ 2 2 e e where the matrix M¼

X e

Me

ð7:31Þ

is called the global mass matrix. The vector u_ is the global nodal velocity vector. The total virtual work done by the external forces is dW ¼

X e

dW e ¼

X e

ðdue ÞT Fe ¼ ðduÞT F

ð7:32Þ

where the vector F¼

X e

is a global generalized force vector.

Fe

ð7:33Þ


77

Substituting Equation 7.28, Equation 7.30, and Equation 7.32 into Hamilton’s principle (Equation 7.4) and conducting the routine variation operation, one has Mu€ þ Ku ¼ F

ð7:34Þ

which is the vibration equation resulting from the finite element discretization.

7.2.4

Lumped Mass Matrix

The element mass matrix given by Equation 7.20 is normally a full symmetric matrix, because the element shape functions are not orthogonal with each other. It is desirable to reduce this full matrix into a diagonal matrix. In practice, this is achieved by lumping the element mass at its nodes. For example, the consistent element mass matrix of a beam element is 2 6 6 rAl 6 6 M ¼ 420 6 6 4 e

156

22l

54

22l

4l2

13l

54

13l

156

13l

3l2

22l

13l

3

7 3l2 7 7 7 7 22l 7 5 2 4l

ð7:35Þ

When the inertia effect associated with the rotational DoF is negligible, the element lumped mass matrix can be obtained by lumping one half of the total beam element mass at each of the two nodes along the translation DoF: 2

1

0

0 0

3

6 7 60 0 0 07 7 rAl 6 6 7 M ¼ 6 2 60 0 1 07 7 4 5 0 0 0 0 e

ð7:36Þ

When the inertia effect associated with the rotational DoF is not negligible, the mass moment of inertia of one half of the beam element about each node can be computed and included at the diagonal locations corresponding to the rotational DoF: 2

1

6 60 rAl 6 6 M ¼ 2 6 60 4 0 e

7.2.5

0

0

0

l2 =12

0

0

0

1

0

0

0

l2 =12

3 7 7 7 7 7 7 5

ð7:37Þ

Model Reduction

The finite element discretization of an engineering vibration problem usually generates a very large number of DoF. In particular, when automatic meshing schemes are not properly applied, or three dimensional elements must be used, the number of elements created could become too great to be cost effectively handled with limited computer capabilities. To solve this problem, modelers have to pay close attention to how the meshing is done in commercial software packages. Very often, simplification and idealization based on the nature of the problem of concern can tremendously reduce the number

78


of elements. For example, there could be two ways of generating the finite elements of a clamped free steel beam with a metal block attached to its free end. One way is to mesh both the beam and the block using three dimensional elements; the other way is to mesh the beam with one dimensional beam elements and treat the block as a lumped mass, zero dimensional element. It is obvious that the first approach will result in many more elements than the second approach. However, both approaches will give very similar results for the first several natural frequencies and the associated mode shapes. Another technique for reducing the number of elements comes from deleting the detailed features. The detailed features here imply those geometrical details, such as filets, chamfers, small holes, and so on, which do not have significant contributions to the vibration behavior of the entire structure, but increase the number of elements. Generally these detailed features can be deleted without any visible effect on the results, if the global behavior of the vibration problem is of concern. Note that such detailed features may have to be kept if the localized behavior such as fatigue (stress) induced by vibration is to be evaluated. When further model reduction is necessary, Guyan reduction [3] is considered. It was proposed two decades ago when computer capabilities were much more limited than today. In fact, Guyan reduction is still in use today and has been cast into many commercial software packages. In Guyan reduction, the model scale is reduced by removing those DoF (called slave DoF) that can be approximately expressed by the rest of the DoF (called master DoF) through a static relation. The DoF associated with zero mass or relatively small mass are likely candidates for slave DoF. By rearranging the DoF u so that those to be removed, denoted by u2 ; appear last in the vector, and partitioning the mass and the stiffness matrices accordingly, one obtains "

M11

M12

M21

M22

#(

u€ 1

"

)

u€ 2

þ

K11

K12

K21

K22

#(

u1 u2

) ¼

( ) F 0

ð7:38Þ

If we assume M22 ¼ 0; and M21 ¼ 0; then the second equation in Equation 7.38 can be written as u2 ¼

K221 K21 u1

ð7:39Þ

Define the transformation u ¼ Qu1

ð7:40Þ

where the transformation matrix Q is " Q¼

I

#

K221 K21

ð7:41Þ

and I is the unit (identity) matrix. Substituting Equation 7.40 into Equation 7.38 and premultiplying the resulting equation by QT ; one obtains a new reduced order model QT MQu€ 1 þ QT KQu1 ¼ F

ð7:42Þ

where QT MQ ¼ M11

M12 K221 K21 þ KT21 K221 M22 K221 K21

ð7:43Þ

and QT KQ ¼ K11

K12 K221 K21

ð7:44Þ


79

Newton’s law € ¼ MuðtÞ

F

Hooke’s Law F ¼ KuðtÞ Hamilton’s principle d

ðt2 t1

ðT

VÞdt þ

ðt2 t1

dW dt ¼ 0

Finite element equation without damping M¼

X e

T

e

L M L;

Mu€ þ Ku ¼ F X T e K¼ L K L; e

F¼

X e

LT Fe

Guyan reduction scheme QT MQu€ 1 þ QT KQu1 ¼ F " # I Q¼ K221 K21

7.3

Vibration Analysis

According to the vibration characteristics to be extracted, vibration analysis can be categorized into the following two types: natural vibration analysis, including modal analysis (see Chapter 1 and Chapter 3), and (forced) response analysis (see Chapter 2). Natural vibration analysis can extract natural vibration frequencies and the associated mode shapes, which is a matrix eigenvalue problem (see Chapter 3), and can result from a finite element discretization. The response analysis refers to the calculation of the response, which can be displacements, strain, or stress, when the system is subjected to time varying excitation forces. The response analysis can be further divided into any one a combination of harmonic response analysis, transient response analysis, and response spectrum analysis, depending on the nature of excitation forces.

7.3.1

Natural Vibration

As noted in previous chapters, the natural vibration frequencies (or simply natural frequencies) and the associated mode shapes of a vibrating system are independent of excitation forces. In other words, they are intrinsic characteristics of the vibration problem. Therefore, they constitute an important part of vibration theory and vibration engineering. When vibration engineers specify design requirements in terms of vibration, they normally do so by restricting natural frequencies, and sometimes restricting mode shapes as well. For instance, in order to enhance the passenger comfort, vehicle designers have to ensure that the first few natural frequencies of the vehicle are not within a certain range; in order to avoid vibration resonance, the natural frequencies of a transmission shaft should be designed not to be identical or even close to the rotating speeds of the shaft; in order to effectively control vibration, vibration sensors and actuators have to be located at those places where the dominant mode shapes have large displacements.

7 10


From a mathematical standpoint, the natural vibration analysis of a multi DoF system requires the solution of a matrix eigenvalue problem. According to the theory of second order ordinary differential equations, the solution of Equation 7.34, when F ¼ 0; can be expressed as u ¼ v eivt : By substituting u ¼ v eivt into Equation 7.34 and letting F ¼ 0; one can obtain

v2 Mv ¼ Kv

ð7:45Þ

Equation 7.45 represents a generalized matrix eigenvalue problem. For an N dimensional matrix pair ðM; KÞ; there exist N pairs of solutions ðvi ; vi Þ; v2i Mv ¼ Kv i ði ¼ 1; 2; …; NÞ, where vi and v i are called the ith natural frequency and the associated ith mode shape, respectively. The numerical methods for solving the matrix eigenvalue problem given by Equation 7.45 have been well developed with the symmetric and sparse features of ðM; KÞ being fully considered. Those that have been used by commercial finite element software packages include the power method, the subspace iteration method, the LR method, the QR method, the Givens method, the Householder method, and the Lanczos method. When conducting vibration modeling, modelers need to understand how idealization and simplification will affect the resulting natural frequencies and the associated mode shapes. Idealization and simplification cause a difference between the actual mass matrix M and the resulting mass matrix Mr ðM ¼ Mr þ DMÞ; and a difference between the actual stiffness matrix K and the resulting stiffness matrix Kr ðK ¼ Kr þ DKÞ: Rayleigh’s quotient [9,11] can be used to determine the effect of DK and DM on a particular natural frequency. Rayleigh’s quotient is defined as RðxÞ ¼

xT Kx xT Mx

ð7:46Þ

Note that Rayleigh’s quotient RðxÞ becomes the square of the ith natural vibration frequency, RðxÞ ¼ v2i ; when x ¼ vi : Thus, Rayleigh’s quotient can be expressed as

v2i þ Dv2i ¼

ðvTi þ DvTi ÞðKr þ DKÞðvi þ Dvi Þ ðv Ti þ DvTi ÞðMr þ DMÞðv i þ Dvi Þ

ð7:47Þ

where Dv2i and Dvi are the increase of the ith natural frequency and the variation of the ith mode shape, respectively, induced by DK and DM: Because of the fact that RðxÞ reaches the stationary value when x is equal to the eigenvector vi ; Equation 7.47 can be simplified as [1,4] Dv2i ¼ v Ti ðDK

DMÞvi

ð7:48Þ

Equation 7.48 indicates that an increase in stiffness leads to a rise in a natural frequency, but an increase in mass causes a decrease in a natural frequency, as is intuitively clear.

7.3.2

Harmonic Response

Harmonic response analysis determines the response of a vibration system (model) to harmonic excitation forces. A typical output is a plot showing response (usually displacement of a certain DoF) versus frequency. This plot indicates how the response at a certain DoF, as a function of excitation frequency, changes with excitation frequency. The harmonic response can also be used to calculate the response to a general periodic excitation force, if it can be satisfactorily approximated by a summation of its major harmonic components. Consider a harmonic excitation force, FðtÞ ¼ F0 eivt : Substituting it into Equation 7.34, we have € þ KuðtÞ ¼ eivt F0 MuðtÞ

ð7:49Þ

According to the theory of differential equations, its steady solution can be written as uðtÞ ¼ eivt U: After substitution of uðtÞ ¼ eivt U into Equation 7.49, one obtains ð v2 M þ KÞU ¼ F0

ð7:50Þ


7 11

Harmonic response analysis will solve Equation 7.50 for U against v: There are many numerical methods available for solving Equation 7.50. The most efficient one is the modal superposition method. In the modal superposition method, the response is expressed as a linear combination given by uðtÞ ¼

j X i¼1

_ ðtÞ v i_ ui ðtÞ ¼ Fu

ð7:51Þ

where F ¼ ½v1 ; v 2 ; …; v j is a modal matrix that contains the dominant mode shapes, _ _ _ and _T u ðtÞ ¼ ½u 1 ðtÞ; u2 ðtÞ; …; uj ðtÞ are called modal coordinates. Substituting Equation 7.51 into Equation 7.49 and premultiplying the result by FT ; we obtain € _

_ u ðtÞ þ Lu ðtÞ ¼ FT eivt F0

ð7:52Þ

Note that the modal matrix F has already been normalized against the mass matrix ðFT MF ¼ IÞ; and that L ¼ diagðv1 ; v2 ; …; vj Þ: Equation 7.52 represents a set of decoupled modal equations with a much smaller dimension than Equation 7.49. After solving Equation 7.52 for _ u ðtÞ and transforming _ u ðtÞ back to uðtÞ through Equation 7.51, we obtain uðtÞ:

7.3.3

Transient Response

Transient response analysis (sometimes called time history analysis) determines the dynamic response of a structure under the action of time varying excitation. Excitation forces are explicitly defined in the time domain. The computed response usually includes the time varying displacements, accelerations, strains, and stresses. Consider Equation 7.34 in its general form ( € þ KuðtÞ ¼ FðtÞ MuðtÞ ð7:53Þ _ uð0Þ ¼ u0 ; uð0Þ ¼ u_ 0 where FðtÞ is the excitation force, u0 is the initial displacement, and u_ 0 is the initial velocity. As in the harmonic response analysis, Equation 7.53 can be solved by the modal superposition method. Substituting Equation 7.51 into Equation 7.53, premultiplying the result of the first equation by FT ; and premultiplying the result of the initial condition by FT M; we obtain ( _€ _ u ðtÞ þ Lu ðtÞ ¼ FT FðtÞ u ð0Þ ¼ FT Mu0 ; _ u_ ð0Þ ¼ FT Mu_ 0

_

ð7:54Þ

Equation 7.54 represents a set of decoupled modal equations, which can be solved by means of numerical integration techniques. After solving Equation 7.54 for _ u ðtÞ and transforming _ u ðtÞ back to uðtÞ through Equation 7.51, we can obtain uðtÞ: To implement the numerical integration techniques, the overall time period being studied has to be divided into a number of smaller time steps. If the time step is too large, portions of the response (such as spikes) could be missed or truncated. On the other hand, if the time step is too small, the analysis will take an excessively long time or even a prohibitive amount of time.

7.3.4

Response Spectrum

The excitation forces, resulting from earthquakes, winds, ocean waves, jet engine thrust, uneven roads, and so on, do not have repeated patterns, for a variety of reasons, and thus it is difficult to describe them using a deterministic time history. Such excitations are normally treated as random excitations. The assumption that such excitation forces are random is recognition of our lack of knowledge of the detailed

7 12


characteristics of the excitation forces. Some excitation forces, like those resulting from an uneven road, could be measured to any desired accuracy, and thus they would become deterministic rather than random. But it is not cost effective and not convenient to do so. Therefore, engineers prefer to characterize these excitation forces by a statistical description that can be easily measured on any particular representative length of time history. Of the statistical descriptions, the autocorrelation function and the power spectral density function are the most important. Denote f ðtÞ as a stationary random excitation force, and Rf ðtÞ as its autocorrelation function and Sf ðvÞ as its power spectral density function. Their relations [6] are Rf ðtÞ ¼ lim Sf ðvÞ ¼

1 ðT f ðtÞf ðt þ tÞdt T 0

ð1 1

Rf ðtÞe

ivt

dt

ð7:55Þ ð7:56Þ

The response spectrum analysis here calculates the power spectral density function of the response of a vibration model to a random excitation force, described by its power spectral density function (see Chapter 5). For the single DoF system given in Figure 7.1 m€uðtÞ þ kuðtÞ ¼ f ðtÞ

ð7:57Þ

The power spectral density function of the response uðtÞ is given by Su ðvÞ ¼ lHðvÞl2 Sf ðvÞ

ð7:58Þ

where HðvÞ ¼ ð mv2 þ kÞ

1

ð7:59Þ

is the frequency response function representing the natural vibration characteristic of the system. In the case of the multiple DoF system given by Equation 7.34, the random excitation force FðtÞ is a column vector. For the sake of simplicity, we assume all of the components in the vector FðtÞ are stationary and statistically independent. Accordingly, all of the components in the vector of the response uðtÞ are stationary. Under this assumption, the power spectral density function of the response uðtÞ is determined by Su ðvÞ ¼ HðvÞSf ðvÞHT ðvÞ

ð7:60Þ

where Sf ðvÞ is a diagonal matrix with its ith element as the power spectral density function of the ith element in FðtÞ; and HðvÞ is the frequency response function matrix defined by HðvÞ ¼ ð Mv2 þ KÞ

1

ð7:61Þ

From Equation 7.60 and Equation 7.61 one can see that the power spectral density function matrix of the response is correlated to the power spectral density function matrix of the excitation force by means of the frequency response matrix of the system HðvÞ: In commercial finite element software packages, HðvÞ is often calculated by the truncated modal method in which HðvÞ is approximately expressed by the dominant natural frequencies and the associated mode shapes, neglecting the contributions of the other mode shapes to HðvÞ; as given below. HðvÞ
> ai > > 6 7< = 6 7 0 xpi 5 bi ð8:13Þ ai £ rpi ¼ 4 zpi > > > : > ; ypi xpi 0 gi

Computer Analysis of Flexibly Supported Multibody Systems

and

2 6 aj £ rpi ¼ 6 4

Let us choose the matrix notation Rpi as

2

6 Rpi ¼ 6 4

0

zpj

zpj

0

ypj

xpj

0

zpi

zpi

0

ypi and Rpj as

2 6 Rpj ¼ 6 4

0 zpj ypj

38 9 ypj > > aj > > 7< = 7 xpj 5 bj > > > : > ; gj 0 ypi

zpj

ypj

ð8:14Þ

3

7 xpi 7 5 0

xpi

85

ð8:15Þ

3

7 xpj 7 5 0

ð8:16Þ

38 9 ypi > > > ai > 7< = xpi 7 b 5> i > > ; : > 0 gi

ð8:17Þ

0 xpj

Therefore, di ¼ xi þ ai £ rpi which is

8 9 2 0 xi > > > = 6 < > z ¼ yi þ 6 > 4 pi > > ; : > ypi xi

zpi 0 xpi

Using the new notation, we have di ¼ xi þ Rpi ai

ð8:18Þ

dj ¼ xj þ aj £ rpj

ð8:19Þ

Now dj is given by and can be written as

8 9 2 0 xj > > > = 6 < > 6 dj ¼ yj þ 4 zpj > > > ; : > ypj xj

zpj 0 xpj

38 9 ypj > > > aj > 7< = 7 xpj 5 bj > > > ; : > 0 gj

and therefore in matrix notation, we have dj ¼ xj þ Rpj aj

ð8:20Þ

To calculate the reactions acting on each body, the relative displacements between the connecting points (stretch) should be calculated. The relative displacements are given by d ¼ dj

di

ð8:21Þ

The reaction forces due to the relative displacements on each body are given by Fsi ¼ Kp d

ð8:22Þ

For equal but opposite directions, we have Fsj ¼

Kp d

ð8:23Þ

86


Moments for spring forces acting at points ri and rj on bodies i and j; respectively, are given by Mi ¼ ri £ Fsi

ð8:24Þ

Mj ¼ rj £ Fsj

ð8:25Þ

On body j; we have The cross products may be expressed in matrix form as 2 0 zpi 6 0 Mi ¼ ri £ Fsi ¼ 6 4 zpi ypi xpi 2 0 zpj 6 6 0 Mj ¼ rj £ Fsj ¼ 4 zpj ypj xpj

9 38 Fsxi > > > > = 7< xpi 7 Fs 5> yi > > > ; : 0 Fszi 9 38 ypj > Fsxj > > > = 7< xpj 7 Fs yj 5> > > > ; : 0 Fszj ypi

ð8:26Þ

ð8:27Þ

Note that the matrices in Equation 8.26 and Equation 8.27 are transposed forms of the matrices in Equation 8.15 and Equation 8.16 3 2 0 zpi ypi 7 6 0 xpi 7 ð8:28Þ RTpi ¼ 6 5 4 zpi ypi xpi 0 3 2 0 zpj ypj 7 6 0 xpj 7 ð8:29Þ RTpj ¼ 6 5 4 zpj ypj xpj 0 Now the equations of motion can be compiled for the translation of body i mi x€ i þ Kp di

Kp dj ¼ Fi

ð8:30Þ

In this case, Fi is the vector of external forces acting on body i: Substituting di and dj into Equation 8.30, from Equation 8.18 and Equation 8.20 we have mi x€ i þ Kp ðxi þ Rpi ai Þ

Kp ðxj þ Rpj aj Þ ¼ Fi

ð8:31Þ

Kp x j

ð8:32Þ

Expanding this, we get mi x€ i þ Kp xi þ Kp Rpi ai

Kp Rpj aj ¼ Fi

Similarly, for body j; substituting the expressions for di and dj ; we get mj x€ j þ Kp di

Kp dj ¼ Fj

ð8:33Þ

Again, Fj in this case is the vector of external forces acting on body j: mj x€ j þ Kp ðxi þ Rpi ai Þ mj x€ j þ Kp xi þ Kp Rpi ai

Kp ðxj þ Rpj aj Þ ¼ Fj

ð8:34Þ

Kp x j

ð8:35Þ

Kp Rpj aj ¼ Fj

With Equation 8.32 and Equation 8.35, the force acceleration equations are complete. Moment Equations The moment equation may be written for body i as shown in Equation 8.36, where Mi is the external moment acting on body i: Ji a€i þ ri £ ðKp di

Kp dj Þ ¼ Mi

ð8:36Þ


87

Substituting expressions for di and dj and converting the cross product to the matrix form, we get Ji a€i þ RTpi ðKp ðxi þ Rpi ai Þ

Kp ðxj þ Rpj aj ÞÞ ¼ Mi

ð8:37Þ

Expanding this, we get Ji a€i þ RTpi Kp xi þ RTpi Kp Rpi ai

RTpi Kp xj

RTpi Kp Rpj aj ¼ Mi

ð8:38Þ

The moment equation may be written for body j as given in Equation 8.39, where Mj is the external moment acting on body j: Thus, Jj a€j

rj £ ðKp di

Kp dj Þ ¼ Mj

ð8:39Þ

Substituting di and dj and converting the cross product to the matrix form, we get Jj a€j

RTpj ðKp ðxi þ Rpi ai Þ

Kp ðxj þ Rpj aj ÞÞ ¼ Mj

ð8:40Þ

Expanding this, we get Jj a€j

RTpj Kp xi

RTpj Kp Rpi ai þ RTpj Kp xj þ RTpj Kp Rpj aj ¼ Mj

ð8:41Þ

If we then collect Equation 8.32 and Equation 8.38 for body I; then Equation 8.32 becomes mi x€ i þ Kp xi þ Kp Rpi ai

Kp x j

Kp Rpj aj ¼ Fi

ð8:42Þ

and Equation 8.38 becomes Ji a€i þ RTpi Kp xi þ RTpi Kp Rpi ai

RTpi Kp xj

RTpi Kp Rpj aj ¼ Mi

Expressing Equation 8.42 and Equation 8.43 in matrix form, we have 3( ) 2 3( ) ( ) " #( ) 2 Kp Kp Rpj Kp Kp Rpi xj mi 0 x€ i xi Fi 5 4 5 þ4 T ¼ T T T aj 0 Ji a€ i ai Mi Rpi Kp Rpi Kp Rpi Rpi Kp Rpi Kp Rpj

ð8:43Þ

ð8:44Þ

Similarly, if we collect Equation 8.34 and Equation 8.41 for body j; then Equation 8.34 becomes mj x€ j

Kp xi

Kp Rpi ai þ Kp xj þ Kp Rpj aj ¼ Fj

ð8:45Þ

and Equation 8.41 becomes Jj a€j

RTpj Kp xi

RTpj Kp Rpi ai þ RTpj Kp xj þ RTpj Kp Rpj aj ¼ Mj

Expressing Equation 8.45 and Equation 8.46 in the matrix form, we have 3( ) 2 3( ) ( ) " #( ) 2 Kp Kp Rpj Kp Kp Rpi mj 0 x€ j Fj xj xi 4 5 5 þ4 T ¼ T T T 0 Jj Mj a€j aj ai Rpj Kp Rpj Kp Rpi Rpj Kp Rpj Kp Rpj

ð8:46Þ

ð8:47Þ

Overall, the equations of motion are now complete. Equation 8.44 and Equation 8.47 provide all that is needed to complete the final equations of motion. It is worth restating that the stiffness and damping matrices are identical in their structure. To obtain a damping matrix, all one needs to do is to replace the stiffness coefficients with the corresponding damping coefficients.

8.3 A Numerical Example In order to illustrate the use of the equations given before, let us consider a rigid body flexibly supported by a number of springs. For this, the simplest starting point would be Equation 8.44 3( ) ( ) 3( ) 2 " #( ) 2 Kp Kp Rpj Kp Kp Rpi xj Fi xi mi 0 x€ i 4 5 5 ¼ ð8:48Þ þ4 T aj Mi ai 0 Ji a€ i RTpi Kp RTpi Kp Rpj Rpi Kp RTpi Kp Rpi

88


Since body j does not exist, all the terms relevant to body j will disappear. Furthermore, since we are dealing with a single mass, the suffix i is not needed either. However, for n number of springs, the stiffness matrices need to be summed up. Summation has to be carried out for each stiffness p attached at a position on the body. We then have 2 X n Kp " #( ) 6 6 p¼1 m 0 x€ 6 þ6 n 6X a€ 0 J 4 RTp Kp p¼1

3 Kp Rp 7( ) ( ) 7 x F p¼1 7 ¼ 7 n 7 a X M 5 RTp Kp Rp n X

ð8:49Þ

p¼1

For a situation where the axes of the springs are parallel to the global coordinate system, no transformation of the stiffness matrix is needed. Hence, kp ¼ Kp : To obtain the submatrices of the stiffness matrix given in Equation 8.49, start with the stiffness matrix for spring p: Specifically, 2

kpx

6 Kp ¼ 6 4 0 0

0

0

3

7 0 7 5 kpz

kpy 0

ð8:50Þ

Now, Kp Rp is given by 2

kpx 6 Kp Rp ¼ 6 4 0 0

0

0

32

0 76 6 0 7 5 4 zp kpz yp

kpy 0

zp

yp

0 xp

3

7 xp 7 5 0

ð8:51Þ

Expanding this, we get 2

0

6 Kp Rp ¼ 6 4 kpy zp

kpx zp

kpx yp

7 kpy xp 7 5 0

0

kpz yp

3

kpz xp

ð8:52Þ

For RTp Kp ; we have 2

0

6 RTp Kp ¼ 6 4 zp yp

zp

yp

32

kpx

0

76 6 xp 7 54 0

xp

0

0 kpy

0

0

0

3

7 0 7 5 kpz

ð8:53Þ

Expanding this, we get 2

0

6 RTp Kp ¼ 6 4 kpx zp kpx yp

kpy zp

kpz yp

3

7 kpz xp 7 5

0 kpy xp

ð8:54Þ

0

Finally, RTp Kp Rp is given by 2

0

6 RTp Kp Rp ¼ 6 4 kpx zp kpx yp

kpy zp 0 kpy xp

kpz yp

32

76 6 kpz xp 7 54 0

0

zp

zp

0

yp

xp

yp

3

7 xp 7 5 0

ð8:55Þ


2 6 6 RTp Kp Rp ¼ 6 4

kpz yp2 þ kpy zp2

kpz xp yp

kpy xp zp

kpx yp zp

3

ky xp zp

kpz xp2 þ kpx zp2

kpz xp yp

89

kpx yp zp kpy xp2 þ kpx yp2

7 7 7 5

ð8:56Þ

The overall stiffness matrix from Equation (8.48) for a single spring is given by 2

kpx

6 6 0 6 6 6 6 0 6 6 6 0 6 6 6 6 k z 6 px p 4 kpx yp

0

0

kpy

0

0

kpz

kpy zp 0

0 kpy zp

kpz xp

þ

kpy xp

kpz xp

kpy zp2

0

kpz xp yp

kpy xp zp

kpz xp2 þ kpx zp2

kpz xp yp

0

kpx yp

0

kpz yp kpz yp2

kpz yp

kpy xp

kpx zp

kpy xp zp

kpx yp zp kpy xp2

kpx yp zp

þ kpx yp2

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

ð8:57Þ

The mass matrix is diagonal and, for the inertia matrix, it is assumed that the principal axes of the body coincide with the global coordinate system. Specifically, 2

m

0

0

3

6 m¼6 40

m

7 07 5

0

0

m

Ixx 6 J¼6 4 0

0

0

2

0

ð8:58Þ

3

7 0 7 5

Iyy 0

ð8:59Þ

Izz

Now, the overall equations of motion 2 m 6 60 6 6 60 6 6 6 60 6 6 60 4

may be assembled for n springs. 38 9 0 0 0 0 0 > x€ > > > > 7> > > > y€ > m 0 0 0 0 7 > 7> > > > 7> > > > 7 = < 0 m 0 0 0 7 z€ > 7 7> > þ 0 0 Ixx 0 0 7 > > a€ > 7> > > > 7> > > € > > b 0 0 0 Iyy 0 7 > 5> > > > ; : > g€ 0 0 0 0 0 Izz

2

kpx

6 6 0 6 6 6 n 6 X 6 0 6 6 0 p¼1 6 6 6 6 k z 6 px p 4 kpx yp

0

0

kpy

0

0

kpz

kpy zp 0 kpy xp

kpz yp kpz xp 0

0

kpx zp

kpy zp kpz yp kpz yp2

þ

kpy zp2

kpz xp yp kpy xp zp

0 kpz xp kpz xp yp kpz xp2 þ kpx zp2 kpx yp zp

38 9 8 9 x > > Fx > > > > > 7> > > > 7> > > > > > > > kpy xp 7> F y > > > > y 7> > > > > > > 7> > > > > > > > > 7 0 7< z = < F z = 7 ¼ > > kpy xp zp 7 7> > > Mx > > > > >a> 7> > > > > > > > 7> > > > > 7 > > > > M b kpx yp zp 7> > > > y > > > > > > > 5> ; : ; : 2 2 Mz g kpy xp þ kpx yp kpx yp

ð8:60Þ

8 10

8.3.1


A Uniform Rectangular Prism

A rectangular prism is supported by four springs as z shown in Figure 8.3. Springs have stiffness values in all three directions (kpx ; kpy ; kpz ; where p is the spring number). The axes of each spring in which the stiffness values are measured are parallel to the y principal axes of the springs, which in turn are parallel to the global coordinate system of the rectangular prism. Thus, no transformation is x needed. The end of spring p is located at ðxp ; yp ; zp Þ; measured relative to the COG of the body. The mass of the prism is m and the principal moments of inertia are Ixx; Iyy; and Izz: A simplified equation of motion of the system in 3 D space may FIGURE 8.3 A rectangular prism supported on springs. be obtained from Equation 8.59. If one attempts to carry this out, one will realize that some terms will disappear because the z components of the positions are all zero and some will disappear because of the symmetry of points. The body shown in Figure 8.3 corresponds to m ¼ 1000 kg; moments of inertia Ixx ¼ 10 kg m2 ; Iyy ¼ 20 kg m2 ; and Izz ¼ 30 kg m2 ; supported by four identical (thus, point suffix p is dropped) springs with stiffness values ðkx ¼ 10;000 N=m; ky ¼ 20;000 N=m; kz ¼ 30;000 N=mÞ. The positions of the springs are given as follows: P1ð1; 2; 0Þ P2ð1;

2; 0Þ

P3ð 1; 2; 0Þ P4ð 1;

2; 0Þ

The coordinates imply that the COG is on the bottom plane of the prism. The system has six degrees of freedom and all six natural frequencies will be calculated. Since stiffness parameters are on the Oxy plane, no coupling will occur between (x and b) and (y and a). Similarly, the vertical motion is also uncoupled from the others due to symmetry. Thus, vffiffiffiffiffiffiffiffiffi uX u 4 u kpx rffiffiffiffiffiffiffiffiffiffi u t p¼1 40;000 ¼ ¼ 6:32 rad=sec ¼ 1:0065 Hz vx ¼ m 1000 vffiffiffiffiffiffiffiffiffi uX u 4 u kpy rffiffiffiffiffiffiffiffiffiffi u t p¼1 80;000 vy ¼ ¼ ¼ 8:94 rad=sec ¼ 1:4235 Hz m 1000 vffiffiffiffiffiffiffiffiffi uX u 4 u kpz rffiffiffiffiffiffiffiffiffiffiffi u t p¼1 120;000 ¼ 10:95 rad=sec ¼ 1:743 Hz vz ¼ ¼ m 1000 vffiffiffiffiffiffiffiffiffiffiffiffi u 4 uX u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 k u u p¼1 p pz 4 £ 22 £ 30;000 t ¼ 219:09 rad=sec ¼ 34:87 Hz va ¼ ¼ Ixx 10


8 11

vffiffiffiffiffiffiffiffiffiffiffiffi uX u 4 2 u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xp kpz u u p¼1 4 £ 12 £ 30;000 t vb ¼ ¼ 77:45 rad=sec ¼ 12:32 Hz ¼ Iyy 20 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 uX u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 k þ yp2 kpx Þ u u p¼1 p py 4ð12 £ 20;000 þ 22 £ 10;000Þ t ¼ 89:44 rad=sec ¼ 14:23 Hz ¼ vg ¼ Izz 30

8.3.2

VIBRATIO Output

For the numerical problem given above, the output obtained from the software package VIBRATIO (see Appendix 8A for the full listing) is tabulated below. Dominant Oscillation Direction Frequency Frequency Frequency Frequency Frequency Frequency

Frequencies

in X in Y in Z in alpha in beta in gamma

1.01 Hz (60 CPM) 1.42 Hz (85 CPM) 1.74 Hz (105 CPM) 34.87 Hz (2092 CPM) 12.33 Hz (740 CPM) 14.24 Hz (854 CPM)

8.4 An Industrial Vibration Design Problem In the analysis of vibration characteristics of an engine or an engine generator set, flexibly supported rigid body representations are commonly used. In the case study given here, an engine assembly system is considered. In designing an engine mounting system, engineers consider a number of issues. Particularly useful are: static deflection; natural frequencies and spread of these frequencies relative to the engine speed range; time response under shock or other transient loads; and frequency response of the system, especially when subjected to unbalanced engine generated excitations.

8.4.1

Static Deflection

This problem involves the selection of mounting geometry and mount stiffness parameters. Here, the design engineer seeks to achieve a pure deflection with as little tilt as possible. In principle, there is nothing wrong with some mounts deflecting more than the others, but an excessive tilt normally indicates other problems such as a high degree of coupling between the modes of motion. However, achieving a pure vertical deflection, at least mathematically, is relatively easy and involves the selection of spring positions/stiffness values to satisfy the following conditions: n-spring X p¼1

yp kpz ¼ 0

and n-spring X p¼1

xp kpz ¼ 0:

8 12


Here, xp and yp are spring mount positions and kpz is the spring stiffness in the z direction. In addition, the designer needs to ensure that the static deflection is well within the allowable deflection range, especially since it needs to account for any additional deflection due to vibration.

8.4.2

Natural Frequencies

Normally, design application demands that the natural frequencies are kept away from the running speed of an engine. Such a requirement is easy to satisfy for single DoF systems. However, in real life situations, the number of DoF(s) is six just for a single rigid body. Any change to the mounting configuration or stiffness parameters (or mass distribution) will affect all six natural frequencies. In order to be able to modify a natural frequency corresponding to a particular mode shape (e.g., resonance in the vertical direction), a vibration design engineer will attempt to decouple various modes of oscillation. Such a design requirement, however, is unrealistic for many engineering problems, even for a single rigid body, due to space considerations and is practically impossible for multiple rigid body systems. However, it is possible to achieve partial decoupling. For example, the condition given above for pure vertical deflection will also provide decoupling between the rocking motion about two coordinate systems (about Ox and Oy) in the horizontal plane and vertical motion.

8.4.3

Transient Response Analysis

Static deflection analysis and modal analysis are normally essential (see Chapter 3). However, there are two more problem specific analyses that a design engineer may need to perform depending on the problem. For example, in many engineering applications, the response of a flexibly supported system to a shock loading is very important. In this case, coupling among modes as well as stiffness of the system play an important role in determining the levels of shock transmission to the engine system. As a rule of thumb, the softer the spring the smaller the shock transferred to the flexibly mounted structure. Making mounting stiffness elements softer may not be a realistic option (see Chapter 32) as this can result in unacceptable static deflections and may even be in conflict with the requirements discussed above in relation to the positioning of a natural frequency relative to the operating speed of the engine and the static deflection.

8.4.4

Frequency Analysis

Frequency analysis is probably one of the most useful among the various types of analysis listed above. If the vibration problem is not transient, then it is a problem involving a steady state vibration. The analysis of a steady state vibration problem under sinusoidal excitation is normally referred to as frequency analysis (see Chapter 2). In mathematical terms, this is the particular solution of the differential equation of motion. Under a sinusoidal excitation, a vibrating system reaches a steady state vibration and frequency analysis provides information of the amplitude of vibration as a function of excitation frequency. If there is more than one excitation force, then the resulting motion will be a combination of the results due to each excitation. Multiexcitation force analysis is sometimes referred to as harmonics analysis (see Chapter 2). Here, the designer needs to ensure that the highest amplitude of oscillation under harmonic excitation does not exceed the safe limits for the system and, in particular, for the mounts. Although design considerations linked to static deflection, positioning of natural frequencies, decoupling of modes, and response to shock represent a large proportion of vibration design problems, there are other and more complex design specifications. For example, minimizing vibration at a point on a flexibly supported body may be considered a design objective. Such a requirement may then cause problems where the engineer intends to place a drive shaft coupling or an additional mounting at this position. Because the vibration (or deflection) is at its minimum at the assembly point, a coupling or additional mount will add a minimum constraint to the system.


8 13

Obviously, all the design objectives discussed above have to be satisfied within the physical constraints relating to the problem at hand. In general, these constraints are associated with space and the stiffness range of industrially available mountings.

8.4.5

A Flexibly Supported Engine — A Numerical Problem

The example considered here involves a study of an engine mounting configuration. Mounting configurations are restricted by the geometry and little flexibility exists in modifying these positions. They are located by the engine manufacturer. To perform vibration analysis, the mass/moments of inertia values of the system, the coordinates of the mounting positions relative to the COG, and the forces acting on the system are needed. Although the mass is normally given or easy to obtain, moments of inertia and the COG are not always supplied by engine manufacturers. Even though it is possible to build a solid model of an engine in order to calculate moments of inertia, this is a rather tedious and costly task. The alternative is to define engine moments of inertia in an approximate manner. This can be done by assuming that the main assemblies of the engine are made of regular geometrical primitives representing approximate shapes without going into exact geometrical models. Such an approach works reasonably well, especially since the mass values of these primitives can be obtained in an exact manner. When calculating the moments of inertia and the overall mass of the assembly, its COG can also be calculated. Once these are obtained, the mount positions can be calculated relative to the COG. Having obtained the mass, the moments of inertia, the COG, and the mount position coordinates relative to the COG, the main step of analysis may be started. This involves selecting the mount stiffness parameters in such a way that the various conditions and objectives described above are met. The vibration design, like all engineering problems, involves reconciling many conflicting requirements. Engine mass and moments of inertia (symmetry of mass distribution is assumed) m ¼ 250 kg and moments of inertia Ixx ¼ 45 kg m2 ; Iyy ¼ 80 kg m2 ; and Izz ¼ 110 kg m2 : Mounts stiffness values: ðkx ¼ 150;000 N=m; ky ¼ 150;000 N=m; kz ¼ 300;000 N=mÞ: Mount positions (all in mm): 1 2 3 4

8.4.5.1

0.000 250.000 0.000 2170.000

100.000 0.000 2100.000 0.000

2 75.000 275.000 275.000 275.000

Satisfying Static Deflection

On running a static analysis under a vertical load of 2500 N (weight), the following results are obtained: (displacements are in mm and angles are in rad) X

Y

Z

Alpha

Beta

Gamma

xc

yc

zc

0.1392

0.0000

2.1205

0.0000

0.0019

0.0000

1142.5000

0.0000

275.0000

In relation to the static deflection, there are two considerations: (i) overall deflection should not be more than what is allowed by the deflection range of the springs selected for the design, and (ii) the static position and orientation of the engine should not be outside what is allowed by spatial and other constraints; i.e., it should not tilt to one side excessively. In either case, stiffer springs will tend to solve

8 14


the problem. However, such a choice may not be the best for transients, shocks, and vibration transmission from the supporting frame. “Tilt” level calculated above is assumed to be small (0.0019 rad). The results from the program also list the deflection at the mount positions (in mm) as: mass no. ¼ 1 Mount no.

Position X

1 2 3 4

Y

0.000 250.000 0.000 2170.000

100.000 0.000 2100.000 0.000

Deflection Z

x

y

z

275.000 275.000 275.000 275.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

2.120 1.656 2.120 2.436

The maximum deflection is at the fourth mount position and is 2.436 mm. The mount selected should give a deflection range that would extend well beyond this to ensure that any additional deflection due to vibration could be accommodated. The coordinates xc, yc, zc give the instantaneous center of rotation for this particular static deflection result. This point (as discussed above) may be used in some design applications as it remains stationary during the deflection. 8.4.5.2

Eigenvalue Analysis

The eigenvalue analysis (see Chapter 3 and Appendix 3A) will help ensure that the natural frequencies are not in the vicinity of the idling speed of the engine. It may also help to minimize the number of natural frequencies in the speed range of the engine. Since the spring positions and their locations are already specified to satisfy the considerations for static deflection, it becomes difficult to modify them to satisfy the “natural frequency” requirements as well. However, all stiffness values could be increased together in the same proportion. This ensures that the “no tilt” condition is maintained. Of course, stiffening now reduces the static deflection and is likely to increase the vibration transmission to the frame. The eigenvalue analysis results are listed below. Here, the natural frequencies spread from 1.98 to 12.09 Hz. The widest gap between these frequencies is between 3.17 and 8.69 Hz. It would be desirable to have the idling speed in the middle of this range. It is equally important that the cruising speed does not coincide with the two higher frequencies. X

Beta

Gamma

0.0000

Frequency in X ¼ 8.63 Hz (518 CPM) 20.0111 0.0000

20.2709

0.0000

0.0000

1.0000

Frequency in Y ¼ 8.69 Hz (521 CPM) 0.0000 0.4403

0.0000

0.0484

20.0051

0.0000

Frequency in Z ¼ 12.09 Hz (725 CPM) 21.0000 0.0000

0.0685

0.0000

0.0000

0.0797

Frequency in alpha ¼ 1.98 Hz (119 CPM) 0.0000 21.0000

0.0000

20.0226

0.0869

0.0000

Frequency in beta ¼ 3.17 Hz (190 CPM) 0.0215 0.0000

1.0000

0.0000

0.0000

20.0163

Frequency in gamma ¼ 2.13 Hz (128 CPM) 0.0000 20.0625

0.0000

1.0000

1.0000

Y

Z

Alpha

In order to achieve decoupling between different motions, one practical technique is to minimize the distance between the COG and the “center of stiffness.” Center of stiffness is a crude term used in industry to ensure that the coupling between different motions of body is minimized. The definition of center of stiffness is similar to that of the COG. The Ox axis is located in such a


8 15

P way that p yp kpz ¼ 0 holds. Then this axis will pass through the center of stiffness. Similarly, P p xp kpz ¼ 0 will hold for the Oy axis passing through the center of stiffness. If the center of stiffness coincides with the COG and the axes defined by the first moment of stiffness coincide with the principal axis of mass, then full decoupling can be achieved. As far as the horizontal plane is concerned, this also ensures that the assembly is leveled. Note that the relationships are the same as the “no tilt” condition described above for static deflection. Normally, it may not be possible to achieve this in all three planes and the designer may choose to achieve this in one plane where the excitation forces are the greatest. However, it is common among designers to focus on the horizontal plane alone, purely due to deflection under gravity considerations. 8.4.5.3

Time Domain Analysis — Analysis of the System under a Shock Loading

Suppose that a 10 g, 10 msec, half sine shock is applied in the vertical, z; direction. The shock response in the z direction is shown in Figure 8.4. The shock response in the y direction is shown in Figure 8.5. The results show that, in addition to static deflection, if the mounts were to withstand the applied shock, they should be able to deflect 9 mm in shear and more than 6 mm in the vertical directions. 8.4.5.4

Frequency Analysis

The frequency analysis specifications are given below. The engine is subjected to an unbalanced force which is known to be proportional to the square of the engine running speed. In other words, this is given as Aw2 : It is measured that when the engine speed is 300 rpm the unbalanced force is 250 N. A simple calculation shows that A ¼ 1:013: The force menu option 20 in VIBRATIO provides the required excitation, which increases with the square of the running speed. The option 20 allows the A value to be linearly increased between the start and end frequencies during which the excitation is active. In our case, this is to be taken to be the same (frequency is independent of the A value). The vertical amplitude vs. frequency results are given in Figure 8.6. The amplitudes in other directions are much smaller and are not shown here. The analysis is not carried out beyond 15 Hz as we know already that the maximum resonance is at 12.09 Hz. According to the result, the selected mount should allow a 16 mm deflection on top of static deflection. Now, the designer should be in a position to make a decision on whether the selected

FIGURE 8.4

Shock response in the z direction.

8 16


FIGURE 8.5

Shock response in the y direction.

spring type is acceptable or not. If the answer is no, then the whole analysis process has to be repeated to find an acceptable solution.

8.5 Programming Considerations Equations developed in this chapter are formulated and structured in such a way that they can be used for developing general vibration analysis software. As the equations given above refer to bodies i and j; they can be placed in the global coordinates accordingly. Four submatrices (6 £ 6) will be placed as follows: 2 3 Kr Kr Rpi 4 5 will be placed; starting from position ð6ði 1Þ þ 1; 6ði 1Þ þ 1Þ RTpi Kr RTpi Kr Rpi

FIGURE 8.6

Frequency response in the z direction for unbalanced engine excitation.


2

Kr Rpj

RTpi Kr

RTpi Kr Rpj

Kr

Kr Rpi

RTpj Kr

RTpj Kr Rpi

Kr

Kr Rpj

RTpj Kr

RTpj Kr Rpj

2 4 2 4

3

Kr

4

8 17

5 will be placed; starting from position ð6ði

1Þ þ 1; 6ð j

1Þ þ 1Þ

1Þ þ 1; 6ði

1Þ þ 1Þ

3 5will be placed; starting from position ð6ð j

3 5will be placed; starting from position ð6ð j

1Þ þ 1; 6ð j

1Þ þ 1Þ

Entry to the global stiffness matrix must be additive (a new entry is added to the previously entered values to account for any contribution coming from other springs). In programming terms, the global stiffness matrix is constructed based on the spring count loop. Therefore, the data structure for a spring must be such that the end of the spring refers to the mass number. Hence, for a spring stiffness, submatrices could be located in the global matrix according to the mass numbers to which the ends of that spring are attached. To start assembling the equations, the first step is to express the stiffness matrices in global coordinates. Here, it is assumed that the stiffness values are given in three orthogonal axes. These axes will be assumed to be the principal axes of the springs (in practice, this is not strictly correct and the assumption holds only for small deflections). The orientation of the principal axes relative to the global coordinates can be described in terms of Euler angles (or using direction cosines). Once the transformation matrix is obtained, then the stiffness can be expressed in the global coordinate system Equation 8.4. In developing one’s own software, it is important to bear in mind the way the transformation matrix is constructed. Euler angles have to be applied one by one and the order in which they are processed is important. For example, to orient a given body, one may assume that, initially, its local coordinates are parallel to the principal coordinate frame. From here, the first Euler angle will rotate the body to a new orientation. Relative to this position, the next rotation will be applied, resulting in yet another orientation. Lastly, the third rotation will be applied relative to this orientation, giving the final position of the body under consideration. Assuming that these rotations are a b g in the given order, the transformation matrix is built as T ¼ TðaÞTðbÞTðgÞ In other words, any vector in the local coordinates will be transformed to the global coordinates in reverse order. To explain further, any vector in the local coordinates is first transformed by TðgÞ, then the resulting vector is transformed by TðbÞ; and finally the resulting vector is transformed by TðaÞ: The expression X ¼ Tx processes these matrices in that order. Transformation will apply to spring stiffness matrices and inertia matrices. Therefore, the input to the vibration program has to account for Euler angles for each body and spring, and the order in which these rotations are performed. Once the transformation is completed, all that remains is to perform the appropriate matrix algebra in order to construct the stiffness matrix for each spring and its location in the global stiffness matrix. The inertia matrix for each mass, after expressing it in the global coordinate frame by performing matrix transformation, will need to be placed in the appropriate location in the global inertia matrix.

8.6 VIBRATIO 8.6.1

Capabilities

VIBRATIO is a vibration analysis program designed to model flexibly supported multibody systems. An educational version is made available free of charge by the author of this chapter for the readers of

8 18


this handbook. The executable file could be downloaded from the website, www.signal research.com. The educational version will allow analysis of a flexibly supported system not exceeding more than 10 masses (60 dof). For the purpose of supporting the theory presented in this handbook, four types of analysis are activated: static; eigenvalue (both damped and undamped); frequency domain; and time domain. Standard excitations for time and frequency domains will be based on the excitation functions available in the menu. Numerically defined excitations, and all the other modules, such as signal processing, finite element shaft analysis, fatigue analysis, mass design, etc. are not available in this version.

8.6.2

Modeling on VIBRATIO

Click vibratio2002 from start/programs/vibratio2002 menu. The VIBRATIO screen shown in Figure 8.7 will be displayed. The top part of the frame gives inputs for application details. The date/time box can be double clicked to produce current date and time. File options are listed in the file box. Only files with a VIB extension are filtered for the listing. You may also use the file menu to load VIBRATIO files. If you use any other extension to save your data, you will have to use the file menu to open it. VIB files are text files and can be read by any text editor. It is not advisable, however, to create these manually by using any editor other than VIBRATIO. 8.6.2.1

Entering Spring Data

The second section on this screen, titled “Spring Type Description,” is for entering spring stiffnesses. Here, the types of springs are defined, not the definition of individual springs.

FIGURE 8.7

VIBRATIO main window.


8 19

Order should always be set to 1. In general, it is assumed that the spring deflections can be described as polynomials, and “order” refers to order of the polynomial. Only linear capability is offered in the present version; thus, order must always be set to 1. 8.6.2.2

Entering Mass Data

When VIBRATIO runs, it automatically loads default mass data. To create a two mass system, change “total mass” to 2. This is shown in Figure 8.8. Normally, if these two masses are not the same then you will need to modify accordingly. Also, you may copy mass data from one previously defined mass to another by clicking the “copy from” button. CogX, CogY, CogZ are the centers of mass of the current mass relative to the COG of mass 1. It means that the program will not allow you to change the COG coordinates of mass 1. “Frame” shows that you have massX; massY; massZ values. For normal applications, mass does not have a directional property and values in each direction will be the same. There are applications where directional “effective mass” may be appropriate, such as modeling the ground mass for earthquake analysis. Moment of inertia elements are calculated relative to the COG of current mass and about coordinates that are parallel to the coordinates of the global frame. The center of the global coordinate system coincides with the center of mass 1. If the principal coordinate system of the current body is not parallel to the global coordinate system, then cross inertias Ixy; Ixz; Iyz need to be calculated. The full version of

FIGURE 8.8

Mass data entry.

8 20


FIGURE 8.9

Entering spring data.

VIBRATIO offers a mass calculator that, among other things, can calculate the necessary transformation for given sets of Euler angles. Control points: Here, you may enter the coordinates of points at which the linear deflection module will calculate the displacements. For time and frequency domain analyses this module is not used. Graphics programs come with options to create point tables for analyzing motions at those points.

FIGURE 8.10

Spring attachment designer window.


8 21

Initial conditions: For each mass, initial conditions may be assigned (displacements and velocities) for time domain analysis. Now you have a two mass system where both masses are identical. 8.6.2.3

Entering Spring Attachment Data

The process of entering spring data is illustrated in Figure 8.9 to Figure 8.11. The first step is to decide how these bodies are connected to the ground and to each other. Connection can be by a spring or a buffer. Buffers are not covered in this chapter. One needs to make sure that the spring option button is selected (spring option is selected as the default). Consider the option buttons. The “text box” refers to the total number of connections to be entered in this frame. Do not modify this one for now. What you enter in this box will depend on how you create these connections. The next very important action is to select “connected to” combo box. This refers to the mass number to which the current mass is connected. If the connection is to the ground, then “0” should be selected. Ground is called mass 0. Once you select a connection mass, “þ” will appear in front of the mass number. This makes it easy to identify the masses to which a connection is made. If a “þ” character had not been added, then the user would have to search all the mass numbers one by one to view the data. Even in a 10 mass system, this could involve checking 55 possible connections. Note that “connected to” lists mass numbers that are less than the current mass. For example, if the current mass number is 5, then the “connected to” list will have 4, 3, 2, 1, 0.

FIGURE 8.11

Spring data entry.

8 22


FIGURE 8.12

Four spring attachments generated by “spring attachment designer.”

To start entering connections, make sure that the current mass is 1, the “spring” rather than the “buffer” option is selected, and, in this case, “connected to” would be 0 (the only option). Also, make sure that you have pressed the “display springs/buffers” (on/off) button. Now you have three options: 1. Enter the total number of spring connections and start entering the data one by one, but make sure that you enter no more than the total connections you had specified. (This is rather tedious and most of the spring attachments will have identical angle values.) 2. Once you have selected the total number of springs, click the “initialize” button. This will fill all the cells with default numbers and you can simply modify these to create your own data. 3. An even easier approach is to follow the steps given below. Enter values as shown in Figure 8.11. Point number is used as a reference in this frame only. Coordinates X; Y; Z are in mm and are relative to the body displayed currently and connecting to the “connected to” mass. If you want to attach springs to the next mass, you must display the mass in your main window (attachment is to the current mass). “Type” is the spring type. Spring type numbers were previously described. Double symmetry means the point (point number 1, referring to the point above) will be duplicated by its symmetry about the Oxz plane. Then these two points will be doubled (four altogether) by taking their symmetry about Oyz: Click “ok” (for double symmetry) to complete the operation. The points shown in Figure 8.12 are now generated. Note that all the points have the same spring attachments (the same type). Remember that these four springs are between mass 1 and mass 0 (ground) and the coordinates are relative to mass 1. To enter spring attachments between the second mass and the first mass, you need to click “þ” to increase the Current Mass No. to 2. Having done that, you need to select the “connected to” combo box. When you click “2”, option 1 will appear: 0 and 1, see Figure 8.13. This figure shows how to choose mass number 1 to start creating a new set of spring attachment (between mass 2 and mass 1). As explained previously, mass 2 can connect to mass 1 and mass 0. If we are connecting mass 2 to

FIGURE 8.13

Selecting mass 2 and clicking the “connected to” combo box.


FIGURE 8.14

8 23

Use of “spring attachment designer” to create four attachments by double symmetry.

mass 1, then select 1. You may again click the “spring attachment designer” button to create attachment information. The spring attachment frame will appear with exactly the last set of information (assuming that you have not cleared the text boxes before exiting the frame). Figure 8.14 shows how to use “spring attachment designer” to create four attachments by double symmetry. Here, click the “ok” button on the “create double symmetry” subframe. Spring attachment information is now generated, between mass 2 and mass 1. Again, coordinates are relative to the current mass 2. 8.6.2.4

Entering Force Data

Make sure that your current mass is 1. When you run VIBRATIO, it assigns a single force of type 1 on mass 1. To see the force data, click “display forces” (on/off). You will see that the “Total Force No.” is 1, “current force” is 1, and “type” is also selected to be 1, as shown in Figure 8.15. In this case, “force” is a term that has a wider meaning. Specifically, “flags” on top of “force” magnitudes identify what is meant by “force.” Note that “flags” can take values between (and including) 0 and 4. Here, 0 means force does not exist; 1 means ordinary force, measured in N; 2 means prescribed displacement, measured in m; 3 means prescribed velocity, measured in m/sec; and 4 means prescribed acceleration, measured in m/sec2.

FIGURE 8.15

Default force vector.

8 24


FIGURE 8.16

Analysis options and analysis control parameters.

Note that time 1 and time 2 describe the period in which this force is active. Also, x; y; z; and (a) alpha, (b) beta, (g) gamma are “force” magnitudes. Now we have created a two mass system. Mass 1 is connected to the ground with four springs and mass 2 is connected to mass 1 with four springs. Springs attached to each mass are located at the four corners of the respective masses. There is a constant force acting on mass 1 for 2 sec (although this will be modified according to analysis). Now we are ready to perform analysis.

8.7 Analysis 8.7.1

Analysis Options

To see the analysis frame click the “display analysis options” button. The analysis option frame as shown in Figure 8.16 will appear. Analysis options offered here are those relevant to the theory presented in this chapter and elsewhere in the handbook (see Chapters 1 to 3). The theory concerns rigid bodies FIGURE 8.17 Analysis options available for the connected to each other by flexible springs (and provided version of software. dampers). Since flexible shafting analysis is not available in the version provided to the reader, the “rigid” option has to be selected. The frame also contains a number of analysis parameters. Only some of the parameters are needed for a given analysis. For example, for deflection, eigenvalue, and eigenvalue (with damping), no parameters are needed. Once you select your analysis, parameters relevant to the analysis will remain visible and anything else will disappear. Among analysis options, those shown in Figure 8.17 will be functional in your copy of the software. Starting from the left, the first button is for linear deflection, the third button is for eigenvalue analysis (ignores damping) ,the fourth button is for eigenvalue analysis (damping included), the fifth button is for frequency analysis, and the sixth and final button is for time domain analysis. When you place your cursor on a button, the information box will identify its use.

8.7.2

Eigenvalue Analysis

It is recommended that the first analysis you do is eigenvalue analysis. This will reveal whether you have any inconsistency/error in your data entry and if the created system is physically viable or not. To perform eigenvalue analysis, click the third button as shown in Figure 8.18. All the analysis parameter data boxes will disappear. The color of the selected button will change to red. This allows the user to enter any appropriate analysis parameter (in this case no parameter is needed).


FIGURE 8.18

FIGURE 8.19

8 25

Eigenvalue analysis button.

Postprocessing options.

When the button is clicked a second time, the analysis will be performed. To perform eigenvalue analysis for damped system, the procedure is the same but click the fourth button for this analysis. To see results, you need to click the “display postprocessing options” (on/off) button. The postprocessing options will appear as shown in Figure 8.19. VIBRATIO has a basic text editor that may be used to display textual results. Click the “DISP TEXT” button to view eigenvalue results. 8.7.2.1

Eigenvalue/Vector Analysis Results

The editor window will open, and from the file menu you can then open “eigftext.txt.” You should obtain the results given below X

Y

Relative eigenvector values for mass ¼ 1 1.0000 0.0000 Relative eigenvector values for mass ¼ 2 20.6180 0.0000 Relative eigenvector values for mass ¼ 1 0.6180 0.0000 Relative eigenvector values for mass ¼ 2 1.0000 0.0000 Relative eigenvector values for mass ¼ 1 0.0000 20.6180 Relative eigenvector values for mass ¼ 2 0.0000 21.0000 Relative eigenvector values for mass ¼ 1 0.0000 21.0000 Relative eigenvector values for mass ¼ 2 0.0000 0.6180

Z

Alpha

Beta

Gamma

Frequency in X ¼ 1.26 Hz (76 CPM) 0.0000

0.0000

0.5025

0.0000

0.0000

0.0000

20.3106

0.0000

Frequency in X ¼ 0.48 Hz (29 CPM) 0.0000

0.0000

0.3106

0.0000

0.0000

0.0000

0.5025

0.0000

Frequency in Y ¼ 0.48 Hz (29 CPM) 0.0000

0.3098

0.0000

0.0000

0.0000

0.5012

0.0000

0.0000

0.0000

0.0000

Frequency in Y ¼ 1.26 Hz (76 CPM) 0.0000

0.5012

0.0000

20.3098

0.0000 0.0000 (continued on next page)

8 26


(continued) X

Y

Relative eigenvector values for mass ¼ 1 0.0000 0.0000 Relative eigenvector values for mass ¼ 2 0.0000 0.0000 Relative eigenvector values for mass ¼ 1 0.0000 0.0000 Relative eigenvector values for mass ¼ 2 0.0000 0.0000

Z

Alpha

Beta

Gamma

Frequency in Z ¼ 1.78 Hz (107 CPM) 1.0000

0.0000

0.0000

0.0000

20.6180

0.0000

0.0000

0.0000

Frequency in Z ¼ 0.68 Hz (41 CPM) 0.6180

0.0000

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

Frequency in alpha ¼ 25.26 Hz (1516 CPM) Relative eigenvector values for mass ¼ 1 0.0000 20.0050 0.0000 21.0000 Relative eigenvector values for mass ¼ 2 0.0000 0.0031 0.0000 0.6180

0.0000

0.0000

0.0000

0.0000

Frequency in alpha ¼ 9.65 Hz (579 CPM) Relative eigenvector values for mass ¼ 1 0.0000 20.0031 0.0000 20.6180 Relative eigenvector values for mass ¼ 2 0.0000 20.0050 0.0000 21.0000

0.0000

0.0000

0.0000

0.0000

Frequency in beta ¼ 17.89 Hz (1073 CPM) Relative eigenvector values for mass ¼ 1 0.0100 0.0000 0.0000 0.0000 Relative eigenvector values for mass ¼ 2 20.0062 0.0000 0.0000 0.0000

21.0000

0.0000

0.6180

0.0000

Relative eigenvector values for mass ¼ 1 20.0062 0.0000 Relative eigenvector values for mass ¼ 2 20.0100 0.0000

Frequency in beta ¼ 6.83 Hz (410 CPM) 0.0000

0.0000

0.6180

0.0000

0.0000

0.0000

1.0000

0.0000

Frequency in gamma ¼ 14.57 Hz (874 CPM) Relative eigenvector values for mass ¼ 1 0.0000 0.0000 0.0000 0.0000 Relative eigenvector values for mass ¼ 2 0.0000 0.0000 0.0000 0.0000

0.0000

21.0000

0.0000

0.6180

Frequency in gamma ¼ 5.56 Hz (334 CPM) Relative eigenvector values for mass ¼ 1 0.0000 0.0000 0.0000 0.0000 Relative eigenvector values for mass ¼ 2 0.0000 0.0000 0.0000 0.0000

0.0000

0.6180

0.0000

1.0000

This is the result file created as a result of executing eigenvalue analysis (if you had performed eigenvalue analysis with damping you need to open the “eigdtext.txt” file). A successful eigenvalue analysis normally implies physically feasible data (of course, not always). Now we are ready to perform other analysis options.

8.7.3

Linear Deflection Analysis

The linear deflection equations may be obtained from vibration equations by simply removing the acceleration and velocity terms. If the excitation (force) vector is made of constant values, then the


FIGURE 8.20

8 27

Linear deflection analysis; no control parameter is needed.

problem is a static deflection problem. By clicking the linear deflection button, you will see that no analysis parameters are needed. Next, click the red button. This will perform a static deflection analysis, as indicated in Figure 8.20. Numerical results can be displayed using the “DISP TEXT” button from postprocessing options. The deflection results are saved in the “distext.txt” file. This file holds the results given in the next section. Note that, since force is acting on mass 1, mass 2 does not deflect relative to mass 1. 8.7.3.1

Linear Deflection Analysis Results

X

Y

0.0000 64,000.0000

Alpha

Deflections of mass no. ¼ 1 25.0000 0.0000

0.0000

Beta

Gamma

0.0000

0.0000

0.0000

0.0000

Center of rotation of mass no. ¼ 1 64,000.0000

64,000.0000

0.0000 64,000.0000

Z

Deflections of mass no. ¼ 2 25.0000 0.0000

0.0000

Center of rotation of mass no. ¼ 2 64,000.0000

64,000.0000

Deflection at control points (mm) Mass no. ¼ 1

Position

Point no. Mass no. ¼ 2

X

Point no.

X

Y

Deflection Z

x

y

Position Y

z

Deflection Z

x

y

z

Deflections at coupling/mount positions Mount no. Mass no. ¼ 1 1 2 3 4

Position

Deflection

X

Y

Z

x

y

z

1000.000 21000.000 21000.000 1000.000

1000.000 1000.000 21000.000 21000.000

21000.000 21000.000 21000.000 21000.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

25.000 25.000 25.000 25.000

(continued on next page)

8 28


(continued) Mount no.

Position

Mass no. ¼ 2 5 6 7 8

Deflection

X

Y

Z

x

y

z

1000.000 21000.000 21000.000 1000.000

1000.000 1000.000 21000.000 21000.000

21000.000 21000.000 21000.000 21000.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

25.000 25.000 25.000 25.000

Global and Local Deflections of Couplings/Mounts Coordinates mount no.

Global Deflection

Local Deflection

xp

yp

zp

X

Y

Z

Mass no. ¼ 1 1 2 3 4

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

25.00 25.00 25.00 25.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

25.00 25.00 25.00 25.00

Mass no ¼ 2 5 6 7 8

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

8.7.4

Frequency Analysis

In order to perform a frequency analysis, the force selection must be relevant to the frequency analysis. Force options between 16 and 21 are for frequency analysis. Option 16 ðA Sin WtÞ is chosen to demonstrate the frequency analysis, as shown in Figure 8.21. Here, default excitation is in the z direction. Flag ¼ 1 means this is an ordinary sinusoidal force with 1000 N excitation amplitude. It exists in the frequency range from 0 to 2 Hz (freq 1 and freq 2) acting in

FIGURE 8.21

FIGURE 8.22

Frequency analysis options 16 to 21.

Constant force amplitude between 0 and 2 Hz.


FIGURE 8.23

8 29

Frequency analysis parameters.

the z direction, as shown in Figure 8.22. You may have more than one force, with different amplitudes and different frequency ranges. Sweep time, in seconds (any number other than zero), refers to the time taken to sweep from freq 1 to freq 2 (see Chapter 17). If the sweep time is infinitely slow, then a zero entry signifies this (exactly the opposite meaning!). There are two amplitudes for each direction, the top entry refers to the amplitude at freq 1 and the bottom one refers to the amplitude at freq 2. Amplitudes between these frequencies vary linearly. To execute a frequency analysis, select analysis options and click the button shown in Figure 8.23. Frequency analysis parameters: start frequency, end frequency, and frequency steps are shown in this figure. Specifically, the options Freq 1 and Freq 2 will appear here. These relate to the frequency range to be analyzed. Del freq is the frequency step. Your choice of frequency range should not be arbitrary. Either the problem dictates the range or you may be interested in amplitudes of the system at resonant frequencies. In the latter case, you may look at the eigenvalue results to identify which resonant frequencies will be exhibited in the selected range. For the example considered, the excitation force is in the z direction. Therefore, one needs to find out the resonant frequencies in the z direction (or coupled modes which include the z direction). If the eigenvalue results are studied, one can see that the resonant frequencies in the z direction are given as: 0.68 and 1.78 Hz. Therefore, analysis beyond 2 Hz in the z direction will not reveal any other resonances. Clicking the red button a second time will execute the analysis. To see the results you need to click the “display postprocessing” button, and from the postprocessing frame click the “plot freq” button. The frequency plotting window will appear. The frequency motion in six directions will be plotted for mass 1. “þ” can be used to move to the next mass. You may also display motion curves of all bodies together by choosing the “plot all” option. You will end up with the results shown in Figure 8.24.

8.7.5

Time-Domain Analysis

Forces 1 to 14 are for the time domain, although 13 and 14 are not available in this version of VIBRATIO. When option 1 is selected in the force “type” option, the default force would be 1000 N in the z direction, remaining active between 0 and 2 sec. Modify Time 2 to 0.01 sec (10 msec). You can use option 1 for a rectangular shock but there are also other shock options. Figure 8.25 shows a constant force, magnitude 10,000 N acting in the z direction between 0 and 0.1 sec. To perform a time domain analysis, the button shown in Figure 8.26 needs to be clicked. Relevant analysis parameters will appear as shown in this figure. Time 1 and Time 2 give the range of the time domain analysis. If Time 1 is chosen not to start from 0, even though the resulting data will not be collected until Time 1 reaches, still the solution will be executed starting from start time ¼ 0. In other words, if Time 1 is not zero then this will be the time when data recording starts. Analysis itself, irrespective of Time 1, always starts from 0 sec. del t0 is the initial step length for integration

8 30


FIGURE 8.24

Frequency analysis results.

(The time domain analysis uses variable step length Runge Kutta and the time integration step where an algorithm automatically modifies the integration step to achieve the required accuracy). Accr is the accuracy of integration and del t is the time step at which data is sampled. To execute, click the button again. 8.7.5.1

Time-Domain Results

Time domain results will be displayed by clicking the plot time button on the postprocessing frame. The results shown in Figure 8.27 will be obtained, again from the “options” menu, when “plot all” is selected.

FIGURE 8.25

Constant force representation.


FIGURE 8.26

8 31

Time domain analysis button and analysis control parameters.

FIGURE 8.27

Time domain results window.

8.8 Comments This chapter presented a method of analyzing general multiple rigid body systems interconnected by linear springs and linear dampers. The mathematical modeling was presented for small vibrations where nonlinear geometry effects and gyroscopic couplings were negligible and the deflection characteristics of the mountings were linear. There is a shortage of published material on general mathematical modeling of flexibly supported rigid multibody systems for vibration analysis. Detailed mathematics of time domain, frequency domain and eigenvalue/eigenvector analyses can be found in

8 32


standard vibration textbooks. Some are listed in the references section, but this list is not exhaustive. This chapter also gives a precise and clear formulation suitable for computational implementation. The formulation presented in this chapter forms the core of the vibration analysis suite “VIBRATIO”, a version of which is made available to the users of this handbook, at www.signal research.com, as indicated in the text.

Bibliography Bishop, R.F.D. and Johnson, D.C. 1960. The Mechanics of Vibration, Cambridge University Press, New York. Caughey, T.K., Classical normal modes in damped linear dynamic systems, J. Appl. Mech., 27, 269 271, 1960. den Hartog, J.P. 1956. Mechanical Vibration, 4th ed., McGraw Hill, New York. de Silva, C.W. 2000. VIBRATION: Fundamentals and Practice, CRC Press, Boca Raton, FL. Jacobsen, L.S. and Ayre, R.S. 1958. Engineering Vibrations, McGraw Hill, New York. Meirovitch, L. 1986. Elements of Vibration Analysis, 2nd ed., McGraw Hill, New York. Meirovitch, L. 1967. Analytical Methods in Vibrations, Macmillan, New York. Ralston, A. 1965. A First Course in Numerical Analysis, McGraw Hill, New York. Thomson, W.T. 1988. Theory of Vibrations with Applications, 3rd ed., Prentice Hall, Englewood Cliffs, NJ. Timoshenko, S., Young, D.H., and Weaver, W. Jr. 1974. Vibration Problems in Engineering, 4th ed., Wiley, New York. VIBRATIO, information and download for the readers of this chapter, www.signal research.com. Wilkinson, J.H. 1965. The Algebraic Eigenvalue Problem, Clarendon Press, Oxford.

Appendix 8A VIBRATIO Output for Numerical Example in Section 8.3 X

Y

Beta

Gamma

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Frequency in alpha ¼ 34.87 Hz (2092 CPM) Relative eigenvector values for mass ¼ 1 0.0000 0.0000 0.0000 1.0000

0.0000

0.0000

Frequency in beta ¼ 12.33 Hz (740 CPM) Relative eigenvector values for mass ¼ 1 0.0000 0.0000 0.0000 0.0000

1.0000

0.0000

Frequency in gamma ¼ 14.24 Hz (854 CPM) Relative eigenvector values for mass ¼ 1 0.0000 0.0000 0.0000 0.0000

0.0000

1.0000

Relative eigenvector values for mass ¼ 1 1.0000 0.0000 Relative eigenvector values for mass ¼ 1 0.0000 1.0000 Relative eigenvector values for mass ¼ 1 0.0000 0.0000

Z

Alpha

Frequency in X ¼ 1:01 Hz (60 CPM) 0.0000

0.0000

Frequency in Y ¼ 1:42 Hz (85 CPM) 0.0000

0.0000

Frequency in Z ¼ 1:74 Hz (105 CPM) 1.0000

0.0000

9

Finite Element Applications in Dynamics 9.1

9.2 9.3 9.4 9.5

Mohamed S. Gadala The University of British Columbia

9.6

Problem and Element Classification ................................. 9 2

Geometric Modeling † Discrete Element Types in FE Programs † Truss and Beam Problems † Two Dimensional Problems † Shell and Plate Problems † Three Dimensional Solid Problems † Synopsis of Problem Classification and Element Choice

Types of Analysis ................................................................. 9 20 Modeling Aspects for Dynamic Analysis .......................... 9 23

Model Size and Choice of Master Degrees of Freedom † Lumped and Consistent Mass Modeling † Use of Symmetry

Equations of Motion and Solution Methods ................... 9 27

Equation of Motion † Direct Integration Method Superposition Method † Damping Formulation

†

Modal

Various Dynamic Analyses ................................................. 9 33

Modal Analysis † Transient Dynamic Analysis † Harmonic Response Analysis † Response Spectrum Analysis

Checklist for Dynamic FE Analysis ................................... 9 41

Summary This chapter discusses the use of the finite element (FE) method in problems of vibrations and structural dynamics. The first three sections outline the main steps in modeling a physical problem for a specific dynamic analysis. Section 9.1 concentrates on the finite element aspect of the modeling process. The basis for geometric modeling is outlined and an overview of commonly used types of elements in typical commercial programs is presented. A summary of element capabilities is given at the end of the section (Table 9.1). Section 9.2 discusses the basis for classifying and choosing a particular type of dynamic analysis, including modal, harmonic or frequency response, transient and shock, and random analysis. Section 9.3 discusses some special aspects in modeling that are pertinent to dynamic system analysis; namely, choice of master and slave degrees of freedom (DoF), lumped vs. consistent mass modeling, and use of symmetry in dynamic analysis. The second part of the chapter discusses solution methods and damping considerations, and outlines basic steps for performing various dynamic analyses. Section 9.4 briefly presents the theory and equations for various dynamic analyses. The analysis types included in the discussion are direct integration and modal superposition. In the direct integration analysis, both implicit and explicit schemes are discussed and an example of each is presented. Most of the theory and equations are presented in a summarized form without rigorous mathematical proofs. Section 9.5 describes details of various dynamic analyses and provides a brief discussion of the choice of the solution method for each analysis. Emphasis is placed on the basic steps required to perform a particular analysis type. Modal analysis, transient analysis (direct integration and mode superposition approaches), 91

92


frequency response harmonic analysis, and random response analysis are presented. The various methods of combination of modal responses are also discussed. Finally, Section 9.6 provides some general guidelines for a typical dynamic analysis using the FE method.

9.1

Problem and Element Classification

The first step in any finite element (FE) analysis is to build a model of the physical structure to be analyzed. This is an important step and normally requires extensive time and interaction with the analyst. The adequacy of the model, assumptions involved, and types of elements used for a specific structure and analysis type establish the accuracy level of the FE analysis. Much research effort is being devoted to automating this process by providing options for automatic mesh generation, automatic mesh refinement, error estimation, and error bounds. Such research has resulted in a significant reduction in the time needed for this step but the role of the analyst is still dominant and vital in obtaining a good mesh and model for the problem. The modeling process can be divided into steps as follows: *

*

Build a geometric database for the structure. This includes description of the characteristic geometric features of the structure, such as boundaries, holes, intersections, curvatures, etc., in the finite element program database. The level of geometric detail has an important effect on the accuracy and the size of the model. Build a FE model for the geometric model. This may include important aspects such as: Establishing the type of analysis to be performed, Choosing the appropriate element or elements for building the model, Considering aspects of symmetry in the structure, and Establishing critical areas for increasing mesh density. Apply constraints and loading boundary conditions. Establish the material model(s) to be used in the analysis. Perform various options of model checks. Solve sample load cases and compare the results with hand calculations or experimental results in order to check the behavior and response of the model. Fine tune the model based on the results obtained from sample load cases.

*

*

*

*

*

*

*

*

*

Most of the time, the above steps are rather linked together, and an overall knowledge of the problem is required to perform a specific step. An important decision that should be made at the beginning of the analysis is to identify the category of the problem. This will have an impact on the first three steps mentioned above. In the first part of the present section, we provide a brief discussion on the geometric modeling aspects of the problem. The rest of the section then provides general guidelines on classifying the problem into one of the main categories available in typical commercial finite element programs; namely, truss, beam, two dimensional, shell, or three dimensional problems. We also provide simple examples for each type of problem. The concepts of element choice and problem classification are summarized in a table at the end of the section.

9.1.1

Geometric Modeling

Geometric modeling simply means transforming a physical problem into a geometric database in a FE program. The process is very similar to creating an engineering drawing or a model in a CAD program. Most FE programs have built in preprocessors that are dedicated to generating the geometry database. Generally, FE preprocessors have similar capabilities to those available in CAD programs. In many cases an existing geometric database may be available for the structure in a CAD program and may be imported into the FE program database. However, there can be translation problems between the two databases, especially in three dimensional and shell structures. In most cases, it is faster to regenerate the geometric database for the problem directly using the FE program.

Finite Element Applications in Dynamics

FIGURE 9.1

93

Geometric modeling entities.

Finite element programs use specific building blocks or entities to build the geometry of a structure (Kamel, 1991; NISA User’s Manual, 1992; ANSYS User’s Manual, 2003). The theory on which geometry generation is based is quite simple and basically relies on parametric cubic modeling of curves and surfaces in space. Figure 9.1 shows the basic entities used by most programs for building a geometric model. We define these basic geometric entities and briefly discuss their use in practical modeling of structures. 9.1.1.1

Key Point

A key point is a coordinate location in space. In two dimensional space, a key point is uniquely defined by two coordinates (e.g., x; y; r; u). In three dimensional space, a key point is uniquely defined by three coordinates (e.g., x; y; z; r; u; z). Key points are normally the starting building blocks in the geometry. Connection between key points will generate lines, surfaces, or volumes. On the other hand, most programs will be capable of extracting key points from the end points or corners of lines, surfaces, and volumes. Key points should not be confused with nodes, and should be considered only as spatial locations in space. One may place a node at the same location as a key point or leave the location without node generation. 9.1.1.2

Line or Line Segment

A line or line segment is a portion of a cubic spline curve bounded on both ends by a key point. A line may be straight or curved. The curvature of a line is limited only by its parametric cubic equation in the program. If a physical curve in the structure is presented by up to and including a third order parametric cubic equation, it may be modeled exactly by a single line. In many practical situations, however, this is not the case. As an example, the parametric equation of a circle is not cubic and the recommended way to model a circle accurately is to break it into at least four lines. Breaking a space curve into many lines will always increase the accuracy of the geometric modeling but entails increasing the complexity of the model. The analyst should be careful in situations where the order of the line to be modeled is not known. A common case is the generation of the line of intersection between two surfaces. In such situations, it is important to break the intersection line into a few separate lines. Most programs provide extensive methods for line generation. These may include generation by joining two grid points, cubic spline fitting of four grid points, best fitting a curve between several grid points, extracting the edges of a surface or a volume, intersection of surfaces, and mirroring and copying other lines. 9.1.1.3

Area or Patch

An area or a patch is a portion of a bi cubic surface completely bounded by three line segments (for triangular areas) or four line segments (for quadrilateral areas). The same limitations discussed above for using line segments may be extended here by realizing that the area is modeled by parametric cubic equations in two directions representing the two edges of the area. Most programs provide extensive methods for generating areas or patches. These include generation by sweeping the space between two lines, filling in the area between four edge lines, rotating a line about an axis, extracting the boundary surfaces of a volume, and intersection between volumes.

94


9.1.1.4

Volume or Hyperpatch

A volume or a hyperpatch is a portion of tri cubic solid completely bounded by four areas (for tetrahedron volumes) or six areas (for brick volumes). The equations used to model the edge lines of a volume are still parametric cubic equations and the same restrictions discussed for lines can be extended to a volume in the three directions of the volume. As for line and area generation, most programs provide a wide variety of methods for generating volumes. These may include generation by sweeping the volume between two surfaces, filling the volume between bounding surfaces, rotating an area about an axis, and copying and mirror imaging the existing volume. An important addition in the generation of volumes is the ability of many programs to use solid primitives as building blocks (Kamel, 1991; ANSYS User’s Manual, 2003). These solid primitives may include tetrahedrons, cubes, cylinders, conical volumes, spheres, torus elements, and other standard volumes. These may be used as building blocks that may be combined, subtracted, or intersected with each other. Many programs provide simple Boolean operations to use for such processes.

REMARKS *

*

*

*

9.1.2

In most modeling cases, it is faster to regenerate the geometric database of the problem directly using the FE program, especially in complicated and three dimensional models. The basic building blocks in geometric modeling are key points, lines, areas, and volumes. Many programs provide the ability to create solid primitives as building blocks. These solid primitives may include tetrahedrons, cubes, cylinders, conical volumes, spheres, toruses, and other standard volumes. Boolean operations are normally used to combine, subtract, or intersect various geometric entities.

Discrete Element Types in FE Programs

Most commercial FE programs have extensive element libraries that may be used in static and dynamic analyses. For dynamic analysis, it may be convenient to classify elements into discrete and continuum types. Discrete types include concentrated (lumped) mass and inertia, spring, and damper elements, whereas continuum (distributed) types include all other one dimensional (1 D), two dimensional (2 D), and three dimensional (3 D) deformable elements. In this section, we briefly discuss the discrete type of elements whereas the continuum type will be discussed in detail in subsequent sections. 9.1.2.1

Concentrated Mass/Inertia Element

A concentrated mass/inertia element represents a structural mass and moment of inertia concentrated at one point and has six DoF: three translational and three rotational. The mass and rotary (moment of) inertia may be assigned different values in the three coordinate directions (see Figure 9.2), even though, typically, the mass is the same in all three directions (see Chapter 8). The element is rigid with no geometrical properties and it only contributes to the global mass matrix of the structure. In building up a model, the element may be attached to a structural node of other deformable or elastic elements or be positioned in space and attached to structure nodes through rigid elements or elastic spring and/or damper elements. Most FE programs provide rotary inertia quantities for various components of the geometric model as part of the standard preprocessing data. These can be used to model parts of the structure as lumped mass and inertia that may be connected to the structure through elastic elements.


An example of the use of a concentrated mass and inertia element is the modeling of a heavy, rigid machine mounted on an elastic support. The mass and inertia effects of the machine may be represented by a concentrated or lumped mass/inertia element at the center of the machine. Care must be taken in connecting this element to the deformable elements of the structure or the support. In general, a rigid link may be used to connect the mass/inertia element to the nearest node of a deformable element rather than placing the mass/inertia element directly on that node (Cook et al., 1989). This will generally account for proper interaction between transla tional and rotational DoF of the mass and the structure. It should be noted that some computational algorithms might have problems with zero diagonal values of the mass matrix. This happens if the inertia terms of the mass/inertia element are assigned as zero. This can be avoided by always assigning an arbitrary and small value to the rotary inertia terms. 9.1.2.2

95

uy qy

Y uz

ux qx

qz

Mx , My, Mz Ixx , Iyy, Izz

X Z FIGURE 9.2

Three dimensional mass/inertia element.

Y

J

u K

Z FIGURE element.

X 9.3

I One dimensional

C spring/damper

Spring and Damper Elements

Most programs provide 1 D spring and damper elements that can be used to model hydraulic cylinders, discrete dampers, and shock absorbers. The element normally has one spring constant and one damping coefficient (along the element axis defined by the nodes I and J as shown in Figure 9.3). It is easy to model 3 D stiffness and damping characteristics by replicating the element in the required directions. A mass may be attached to one or two nodes of the element. The force transmitted through the element nodes is the sum of the spring and the damping forces along the element axis.

REMARKS *

*

*

9.1.3

A concentrated mass/inertia element may be used to model lumped mass/inertia at specific points in the structure. Likewise, spring or damper elements may be used to model stiffness and damping characteristics between two points in the structure. A concentrated mass/inertia element represents a structural mass and inertia at a point and has six DoF: three translational and three rotational. Different mass and inertia values may be assigned in different directions. Three dimensional stiffness and damping characteristics between two nodes may be modeled by replicating 1 D spring or damper elements in the required directions.

Truss and Beam Problems

Three dimensional truss and beam elements are shown in Figure 9.4. The main difference between the two elements is the fact that beams may support bending moments and have rotations as extra DoF at

96


FIGURE 9.4 for node j.)

Three dimensional truss and beam elements. (Nodal rotations for the beam element are only shown

each node. This means that a beam element may have loads along the beam axis and not necessary only at the end points. The following conditions should be satisfied for a structure to be classified as a truss or a beam: *

Geometry: Thin, slender, straight bars or rods with pin joints at both ends. Depending on the application, the element may be considered as a truss, a link, a cable, a spring, etc. If the joints are fixed or the bar is curved, the problem will be classified as a beam type. Fixed joints may be created by completely fixing the beam ends to a wall or support, or to another member. This is normally realized if the joint is built in, welded, or fully bolted to the support or to other members. Geometry may be 1 D, 2 D, or 3 D. Loading: For truss problems, loading may only be by tension or compression of the members. This may be achieved by having only concentrated loading at the joints (no bending moment) and no loadings along the member. For beam problems, loading may be in any direction and may be applied along the member axis. This will generally create a bending moment and the member should have rotations as DoF. Thermal loading may also be applied to the member. Body or inertia forces will generally create a bending moment and violate the truss condition and so may be applied only to beam problems. Such loads may be applied, however, to truss elements under the simplification of assuming the inertia effects to be applied only at the nodes or the joints and ignoring the bending moment that will be created on the member.

*

*

*

*

*

*

Degrees of freedom: For 3 D truss elements, the DoF per node are the displacement in the three coordinate directions: ux ; uy ; and uz : Two dimensional truss members only have ux and uy as DoF. Three dimensional beam elements have all six DoF: three translational, ux ; uy ; and uz ; as well as three rotational, ux ; uy ; and uz : Two dimensional beam elements have ux ; uy ; and uz as DoF. Element shapes: Various element shapes commonly available in commercial finite element programs are shown in Figure 9.5. The cross section of the element may be a solid or hollow prismatic section, e.g., a rectangular, circular, trapezoidal, or thin walled section or a channel, thin walled tubular, I section, etc.

2-node (linear) Truss and beam FIGURE 9.5

3-node (quadratic) Beam only

4-node (cubic) Beam only

Various element shapes for truss and beam elements.


P1

P2

97

P3

P4

Cable: tension-only truss elements Heavy I-beam: beam elements

Truss structure P P1

P2

Springs may be modeled as truss members. The cross member should be modeled as beam.

Frames should be modeled as beam structures

Cantilever with distributed load Beam model

P

M

P

Beam with fixed ends FIGURE 9.6

Beam with moment load

Thin arches (beam model)

Typical applications for truss and beam elements.

The analyst will generally be required to identify the orientation of the beam cross section by specifying local axes or identifying key points for the program to locate the local axes. Most programs are capable of modeling variable cross section beam elements to avoid excessive subdivision of the beam into smaller elements. Example applications: Figure 9.6 shows sample applications for the use of truss and beam elements.

REMARKS *

*

*

*

Truss elements have translational DoF whereas beam elements have both translational and rotational DoF at the nodes. Three dimensional truss elements have 3 DoF/node (ux ; uy ; and uz ), whereas 3 D beam elements have 6 DoF/node (ux ; uy ; uz ; ux ; uy ; and uz ). Truss structures may only carry loads at the end of each truss whereas beam structures may also carry loads along the beam length. Curved members should be modeled with beam elements. Problems with body and inertia forces should generally be modeled as beam elements. If lumping of the body forces is assumed and the effect of the bending moment created by body forces is neglected, the effect may be modeled with truss elements.

98


9.1.4

Two-Dimensional Problems

9.1.4.1

Two-Dimensional Plane Stress Problems

Figure 9.7 shows a typical structure, which may be considered as a 2 D plane stress problem. The following geometry and load conditions should be satisfied for a structure to be categorized as a 2 D plane stress structure (Boresi and Chong, 2000). Conditions: Geometry: A flat, thin surface in one plane (e.g., xy plane), simply connected or multiply connected with a small thickness in the third direction (e.g., z direction). The boundary of the structure in the xy plane may be straight or curved. Loading: Loading is restricted to the plane of the structure (e.g., xy plane). Thermal loading may also be applied to the plane (i.e., T ¼ Tðx; yÞ). Body or inertia forces may be due to linear or angular acceleration in the plane of the structure. *

*

*

*

*

*

*

Degrees of freedom: For 2 D elements, the DoF per node are the displacements in the two coordinate directions, ux and uy : Typical element shapes (see Figure 9.8): Elements may be triangular or quadrilateral. Most programs provide the option of having 3, 6, or 9 nodes for triangular elements and 4, 8, or 12 nodes for quadrilateral elements. Increasing the number of nodes will increase the element accuracy at the expense of having more DoF and requiring more CPU time to solve the problem. This is normally called a “p conversion” approach in finite elements. The same goal may be achieved by increasing the number of elements while fixing the number of nodes per element, which is usually called “h conversion” (Zeng et al., 1992; Babuska and Guo, 1996). The choice of method to obtain greater accuracy is rather arbitrary and mostly depends on the availability of the option in the program. Only a limited number of programs provide extensive “p conversion” options. Example applications: Figure 9.9 shows some typical examples that may be modeled using a plane stress assumption. It should be noted such models cannot capture any out of plane modes of vibration. If such vibration modes are of concern, the model should be capable of capturing lateral displacement DoF and out of plane rotations. This may be realized by using shell or 3 D solid elements as will be discussed in the following sections. p1 F1 δ1 Y

Y

X δ2

Free surface

Z F2

p2 FIGURE 9.7

A general 2 D plane stress structure.

free


3 or 4-node (linear)

99

6 or 8-node (quadratic)

FIGURE 9.8

9 or 12-node (cubic)

Typical plane stress elements.

p Y X

Y X Thin bracket

Thin plate with a central hole Y

M X

Y X

Thin arch FIGURE 9.9

9.1.4.2

Through thickness crack in a plate Typical plane stress examples.

Two-Dimensional Plane Strain Problems

Figure 9.10 shows a typical structure that may be considered as a 2 D plane strain. The following geometry and load conditions should be satisfied for a structure to be categorized as a plane strain structure (Boresi and Chong, 2000). Conditions: Geometry: A flat surface in a plane (e.g., xy plane), simply or multiply connected with large thickness in the third or z direction (with much larger dimensions than in the xy plane). The boundary of the structure in the xy plane may be straight or curved. Loading: Loading is restricted to the plane of the structure (e.g., xy plane) with possible uniform loading or constraint in the z direction. The loading in the z direction will remain only as a function of the x and y coordinates; i.e., Pz ¼ Pz ðx; yÞ: Thermal loading may also be applied to the plane (i.e., T ¼ Tðx; yÞ). Body or inertia forces may be applied to the plane. *

*

*

*

*

*

*

Degrees of freedom: For 2 D plane strain elements, the DoF per node are the displacement in the two coordinate directions ux and uy : The plane strain assumption means that the strain in z direction will be zero and that the stress will be nonzero, 1z ¼ 0 and sz – 0:

9 10


p1 F1 May be free, loaded or constrained

δ1 Y

Y X

Surface may be free, loaded or constrained

δ2

F2

Z May be free, loaded or constrained

p2 FIGURE 9.10

A general 2 D plane stress structure.

Typical element shapes: The same element shapes as in the case of plane stress are available in most commercial FE programs (see Figure 9.8). The comment above on p conversion and h conversion also applies here. Example applications: Figure 9.11 shows some typical examples that may be modeled using a plane strain assumption. Once again, it should be noted that such a model cannot caputre any out of plane modes of vibration. If such vibration modes are of concern, the model should be capable of capturing lateral displacement DoF and out of plane rotations. This may be realized by using shell or 3 D solid elements as will be discussed in the following sections. 9.1.4.3

Two-Dimensional Axisymmetric Problems

Axisymmetric problems are characterized by having an axis of symmetry or axis of revolution for geometry and loading. Referring to Figure 9.12, any arbitrary plane that passes by the axis of symmetry will be a plane of symmetry. Symmetry means that the two halves of the structure on each side of the plane of symmetry are mirror images of each other. Symmetry must be satisfied for all aspects affecting the response of the structure including geometry, load, constraint, and material properties. The following conditions summarize the requirements for categorizing a problem as axisymmetric (Boresi and Chong, 2000). Conditions: Geometry: A solid of revolution formed by rotating a flat area (e.g., in the rz plane) around an axis of symmetry (e.g., the z axis). The boundary of the flat area in the rz plane may be straight or curved and the area may be simply or multiply connected. Loading: Loading is restricted to the rz plane with no variation of loading in the u direction. No loading in the u direction. Thermal loading may be applied in the rz plane (i.e., T ¼ Tðr; zÞ). Body or inertia forces may be applied in the rz plane. *

*

*

*

*

*

*

Degrees of freedom: For 2 D axisymmetric elements, the DoF per node are the displacements in the two coordinate directions ur and uz : Some programs use x and y coordinates to replace the r and z axes, respectively.


FIGURE 9.11

axis of symmetry

symmetry plane (arbitrary)

uq = 0

9 11

Typical plane strain examples.

ur

ur uq = 0

Sectional view FIGURE 9.12

Axial symmetry conditions.

z- axis of symmetry

r

9 12


FIGURE 9.13

Typical axisymmetric elements.

z

z

z

r

Cylinder under axisymmetric loading

r Pressure vessel FIGURE 9.14

r

Loading of a conical washer

Typical axisymmetric examples.

Typical element shapes: Figure 9.13 shows typical axisymmetric element shapes that are commonly available in commercial FE programs. The elements are shaped as a torus with various cross sections as shown in the figure. Application examples: Figure 9.14 shows typical axisymmetric examples.

9.1.5

Shell and Plate Problems

Shell and plate problems are quite similar to plane stress problems (see Figure 9.15). We first recall the conditions for a plane stress problem: (1) that the geometry has to be flat in one plane with a small thickness, and (2) that the load has to be in the same plane. If either of these two conditions is violated, the problem becomes a shell problem. For example, if the load is a moment about any direction other than the z direction then it has an out of plane component, or if the geometry is not flat then the problem ceases to be a plane stress problem and should be modeled as a shell problem. Normally, shell structures would still maintain a small thickness in the direction normal to the surface of the shell. This maintains the condition that the stress normal to the shell surface will be zero, although there may still be a pressure applied to the surface of the shell. The following summarizes the conditions for a shell structure. *

Geometry: Thin surfaces or plates that have a thickness in the direction normal to the surface. The midsurface or midplane of the structure may be flat or curved. The boundary or edges of the structure may be straight or curved. Loading: Both in plane and out of plane loadings are permitted. Thermal loading may also be applied in all x ; y ; and z directions. Body or inertia forces may be due to linear or angular acceleration in all three directions.

*

*

*

*

*

*

Degrees of freedom: Three dimensional shell elements have six DoF: three translational, ux ; uy ; and uz ; and three global rotational, ux ; uy ; and uz :


9 13

REMARKS *

*

*

*

*

*

Structures with a flat planar surface of a small thickness and with loading only in the plane (e.g., xy) of the structure (no out of plane loading or constraints) are categorized as plane stress structures. Such structures will have three nonzero stress components: sxx ; syy ; and txy : Structures with a flat planar surface (e.g., xy) but a very large dimension in the third direction and with loading only in the plane of the structure (with possible uniform loading or constraints in the third direction) are categorized as plane strain structures. Such structures will have four nonzero stress components: sxx ; syy ; szz ; and txy : Axisymmetric problems are characterized by having an axis of symmetry or axis of revolution (e.g., z axis) for geometry and loading. Such structures will have four nonzero stress components: srr ; suu ; szz ; and trz : Two dimensional problems may be modeled by elements having two translational DoF per node. The elements may be triangular with 3, 6, or 9 nodes or quadrilateral with 4, 8, or 12 nodes. Axisymmetric elements are shaped as a torus with triangular or quadrilateral cross sections. Increasing the number of elements while fixing the nodes per element is called “h conversion,” whereas increasing the number of nodes per element while fixing the number of elements is called “p conversion.” Two dimensional models cannot capture out of plane vibration modes. If such vibration modes are of concern, the model should be capable of capturing lateral displacement DoF and out of plane rotations.

Element shapes: Figure 9.16 shows some typical shell elements available in most commercial FE programs. To properly model a curved shell with linear elements that are flat requires using a large number of elements to capture the curvature. Quadratic and cubic elements, on the other hand, may have curvature in two directions and a much reduced number of elements will be needed to model a curved shell. Most programs are capable of modeling variable thickness shell and plate elements. p1

M

F1 δ1

Y Y

δ2

X Surface may be free, loaded or constrained

F2

Z

P3

F3 p2

FIGURE 9.15

M Shell and plate structures.

9 14


Some programs offer shell elements that only have membrane capabilities and others that have both membrane and bending capabilities. In shell and plate analyses, it is important to note that linear elements, i.e., three node triangles and four node quadrilaterals, may behave in an unrealistically stiff manner in shear deformation when the element thickness to size ratio is very small. This phenomenon is normally called “shear Linear Quadratic Cubic locking” (Bathe and Dvorkin, 1985; Bathe, 1996; Luo, 1998; Cesar de Sa et al., 2002). The problem FIGURE 9.16 Typical shell elements. occurs when in plane displacements are coupled with section rotations in the governing equation of the element and when low order interpolations (linear elements) are adopted. The same phenomenon is also evident in beam elements. Similarly, when in plane displacements are coupled with section rotations in the governing equations and low order interpolations are used, “membrane locking” will be evident. Most programs provide various remedies for shear and membrane locking problems. These include selective/reduced integration, enhanced assumed strain method, mixed field method, etc. Depending on the availability of a particular method in the program, the user should initiate the remedy. The problem may also be alleviated by switching to higher order elements, such as quadratic or cubic elements. Example applications: Figure 9.17 shows typical examples of applications for shell problems.

REMARKS *

*

*

*

9.1.6

Shell structures are general 3 D surfaces of small thickness. Loading can be in plane and out of plane. All stress components except those normal to the surface will be nonzero. The shell element may be triangular with 3, 6, or 9 nodes or quadrilateral with 4, 8, or 12 nodes. Commercial programs offer shell elements with variable thickness and with membrane and/or bending capabilities. Linear elements may experience “shear locking”: nonphysical, high stiffness in shear. Various remedies are available in most programs.

Three-Dimensional Solid Problems

By default, if the problem is not one of those discussed in Section 9.1.3 to Section 9.1.5 then it will be classified as a 3 D problem. Three dimensional problems are easily identified by their 3 D geometry and loading, as shown in Figure 9.18. Any of the categories of problems discussed above may be solved by using 3 D elements. The critical drawback is the substantial increase in analyst time for modeling and in CPU time in processing the solution (the time needed may be one order of magnitude larger than for a corresponding 2 D problem). The following summarizes the conditions for a 3 D problem. *

Geometry: A 3 D object with no apparant thickness or uniformity in any direction. The boundary of the object may be straight or curved.

*

*


9 15

Thin-walled cylinder with 3D loading

C-Channel with 3D loading

P

Y X

V t Cantliver under in-plane and out-of-plane loading FIGURE 9.17 *

Z

Typical shell and plate examples.

Loading: Three dimensional loading. An import ant note should be added here: 3 D elements normally have three displace ment DoF and no rotational DoF. This means that moments cannot be directly applied to such elements and the moment effect should be simulated via concen trated forces or couples. Other loading, including thermal loading and body or inertia forces, may be applied in any direction.

F2

δ1

*

*

Thin-walled T-junction under internal pressure

y

P

δ2

F1

x z FIGURE 9.18

Three dimensional problems.

Degrees of freedom: Three dimensional solid elements have three translational DoF: ux ; uy ; and uz : As mentioned above, concentrated moments should be modeled by an equivalent system of concentrated forces or couples. Element shapes: Figure 9.19 shows some typical 3 D element shapes available in most commercial FE programs. Example applications: Figure 9.20 shows some typical examples of 3 D structures requiring 3 D solid modeling and elements. It should be noted that if out of plane vibration modes of a 2 D structure are of concern, then a 3 D solid or shell model should be used even if the loading is in one plane.

Remarks *

*

Structures that are not classified as truss, beam, 2 D, or shell may be modeled with 3 D solid elements. Three dimensional elements have three translational DoF per node (ux ; uy ; and uz ) and may have 4 32 nodes per element.

9 16


4-node tetrahedron, 6-node wedge and 8-node cube (linear)

16-node tetrahedron 21-node wedge and 32-node cube (cubic)

10-node tetrahedron 15-node wedge and 20-node cube (quadratic)

FIGURE 9.19

Typical 3 D solid element shapes.

Shaft with torque, bending moment and shear forces

Welded joint with three dimensional loading

Y X

Z Thumb nail crack in a thick block FIGURE 9.20

9.1.7

Thick T-junction Typical 3 D examples.

Synopsis of Problem Classification and Element Choice

Table 9.1 summarizes the concepts discussed above (Section 9.1.2 to Section 9.1.6) for element choice and problem classification. The table also shows displacement and stress variations within each element as well as standard element output quantities. The displacement variation within the element represents the basic element assumption in FE analysis and is called the “shape function” or the “approximation function” assumption. The strain variation within the element follows by differentiating the displacement variation according to the strain displacement relations. The number of nodes per element is linked to the shape function assumption. For example, in 2 D elements three node triangles have linear shape functions and in 3 D elements eight node bricks have linear shape functions. Elements with linear shape function are sometimes called low order or linear elements. Such elements may be used quite efficiently in both linear and nonlinear analyses.

Summary of Problem Classification and Corresponding Element Characteristics

Characteristics

Element Truss

Beam

2

Two Dimensional Plane Elements

UY

Node-3

2

UY qy UZ

1

UX

uy 1

UZ

UX qz

qx

Node-1 ux

Node-3

Node-2

Y

uy Node-1 ux Y

X Dimension Applications Degree of freedom Geometry

1 , 2 , or 3 D Trusses only axial stiffness (acts as a spring between two nodes) UX, UY, UZ Linear

Displacement

Linear

Stress/strain

Constant

Loading

Only at end nodes (also possible mass) No distributed load Constant thermal Axial force or stress

Stress output

Node-2

X

2 or 3 D Frames with axial and bending stiffness

2D 2 D plane stress or plane strain problems

2D 2 D plane stress or plane strain problems

UX, UY, UZ, ROTX, ROTY, ROTZ Linear (principal axes found from third node) Axial: linear Bending: cubic Axial: constant Bending: linear Line load Self weight Centrifugal force Thermal Axial force

UX, UY Linear

UX, UY Quadratic

Linear

Quadratic

Constant (triangular) Linear/constant (Quad) Edge pressure Self weight Thermal Centrifugal force 2 D stresses: sxx ; syy ; txy ; szz (plane strain) Stress intensities and principal stresses

Linear (triangular) Quadratic/linear (Quad) Edge pressure Self weight Thermal Centrifugal force 2 D stresses: sxx ; syy ; txy ; szz (plane strain) Stress intensities and principal stresses

Bending moment


9-17

Shear in two perpendicular axes or alternatively outer fiber stresses


TABLE 9.1

9-18

TABLE 9.1 (continued) Characteristics

Element Axisymmetric Solid

Three Dimensional Solids

z

z

r

r

Z Dimension Applications Degree of freedom Geometry Displacement Stress/strain

Stress output

X

Z

X

2D Axisymmetric structures with possible nonaxisymmetric loads UR, UZ Linear Linear Constant (triangular) Linear constant (quad)

2D Axisymmetric structures with possible nonaxisymmetric loads

3D 3 D solids with general loading

3D 3 D solids with general loading

UR, UZ Quadratic Quadratic Linear (triangular) Quadratic/linear (quad)

Pressure Self weight Thermal Centrifugal force Axisymmetric stresses: srr ; szz ; suu ; trz Stress intensities and principal stresses

Pressure Self weight Thermal Centrifugal force Axisymmetric stresses: srr ; szz ; suu ; trz Stress intensities and principal stresses

UX, UY, UZ Linear Linear Constant (tetrahedron) Linear/constant (hexahedron) Pressure Self weight Thermal Centrifugal force 3 D stresses: sxx ; syy ; szz ; txy ; tyz ; tzx Stress intensities and principal stresses

UX, UY, UZ Quadratic Quadratic Linear (tetrahedron) Quadratic/linear (hexahedron) Pressure Self weight Thermal Centrifugal force 3 D stresses: sxx ; syy ; szz ; txy ; tyz ; tzx Stress intensities and principal stresses


Distributed loads

Y

Y

Characteristics

Element Three Dimensional Shell Structures

uy uy qz

Y Z

Dimension Applications Degree of freedom Geometry Displacement Stress/strain Distributed loads

Stress output

X

uz

qy

qz

ux qx

3D 3 D shell structures with general loading UX, UY, UZ, ROTX, ROTY, ROTZ Linear In shell local coordinates: linear on midsurface and linear through the thickness On the midsurface and in shell local coordinates: Triangle: constant Quad: linear/constant (linear through thickness) Surface and edge pressure Self weight Thermal Centrifugal force 3 D stresses sxx ; syy ; szz ; txy ; tyz ; tzx (global) Local stresses Stress intensities and principal stresses

Y Z

qy ux qx

X

uz

3D 3 D shell structures with general loading UX, UY, UZ, ROTX, ROTY, ROTZ Quadratic In shell local coordinates: quadratic on midsurface and linear through the thickness On the midsurface and in shell local coordinates: Triangle: linear Quad: quadratic/linear (linear through thickness) Surface and edge pressure Self weight Thermal Centrifugal force 3 D stresses sxx ; syy ; szz ; txy ; tyz ; tzx (global) Local stresses Stress intensities and principal stresses


TABLE 9.1 (continued)

9-19

9 20

9.2


Types of Analysis

If the applied loading on the structure is to change with time, the designer should make a decision on whether or not a dynamic analysis is required. To be able to make this decision, information about the loading and the natural frequencies of the structure are required. From the loading point of view, we classify a general loading on a structure into one of four categories: steady, cyclic, transient, and random. Figure 9.21 shows a schematic presentation of the four loading categories. Figure 9.21b shows two types of cyclic loading, one is a simple harmonic loading with amplitude of oscillation Fo ; period T ¼ 2p; and frequency f ¼ 1=T: The second is a periodic loading with a period T: Using Fourier analysis, any periodic function with a period, T; may be decomposed into a series of harmonic sines and cosines with frequencies f1 ¼ 1=T; 2f1 ; 3f1 ; …; nf1 : Figure 9.21c shows a transient load with a duration T: Some forcing functions do not lend themselves to simple frequency or time specifications. Figure 9.21d shows a typical time history of a forcing function of such a category that may be considered as random excitation. Transient and cyclic forcing functions are normally specified as force vs. time or frequency, respectively. On the other hand, random forcing functions are commonly specified as the magnitude of the input acceleration squared vs. frequency. This input data is normally called a power spectral density (PSD) input curve. Time history input may be used in the analysis of random excitation and the random input may be treated as transient. This would normally require extensive computer resources and CPU time due to the very large number of time steps that would be required to capture the peak response. Random excitation may occur from random sources such as road undulation on vehicles, noise, earthquakes, and seismic events on buildings, and wind and turbulent loading on airplanes. For practical purposes, PSD curves have been compiled for various random events and are normally available in most FE commercial, programs. In addition to the load specification, the second factor affecting the decision on the type of analysis is the natural frequency of the structure. Structures with mass and stiffness characteristics are capable of free vibrations after removing the initial excitation on the structure. Depending on the initial conditions of excitation, the structure may vibrate in one natural frequency or in a combination of more than one

Force

Force

Harmonic with period = 2π Force

Periodic with period = T

Fo

Fo

Time Time (a) Steady

T = 2π

T

Time (b) Cyclic

Force

Force

Time T (c) Transient

Time (d) Random FIGURE 9.21

Various loading types.


9 21

natural frequency. In the latter case, the vibration will result in a complex periodic motion that may be decomposed into a series of harmonic motions vibrating at the first natural frequency (fundamental one) and its multiples. As indicated, these frequencies are called the natural or resonant frequencies of the system and they depend on the mass and elastic or stiffness characteristics of the system. In other words, the natural frequencies of a system are those frequencies that the system tends to vibrate under conditions of free vibration. Theoretically, a continuous system or structure has infinite DoF and infinite natural frequencies. From a practical modeling point of view, the structure will have a number of natural frequencies equal to the number of DoF used to model the structure. The mode shapes give the relative displacements of each point in the structure when it vibrates in one of the natural frequencies. Each natural frequency has a corresponding mode shape. The first mode shape corresponding to the first natural frequency of a structure represents the most flexible way in which the structure may deform or vibrate and corresponds to the least strain energy level in all modes. It is important to note that, a symmetric structure will have symmetric and antisymmetric mode shapes. This may be realized by considering the mode shapes of a simple beam as shown in Figure 9.22 where all odd number modes are symmetric and all even number modes are antisymmetric. The information provided by the natural frequencies and mode shapes of a structure is vital in understanding the behavior of the structure under general excitation. If a structure is excited in one of its natural frequencies, then theoretical analysis of the structure response as an undamped system shows that the amplitude of the resulting vibration response reaches infinity. In practice, all structures possess a certain amount of damping that will limit the amplitude of vibration. Determining the type of analysis required for a structure depends on the nature of the applied load and the magnitude of the first natural frequency or the fundamental frequency of the structure. The two main categories of analysis types are static and dynamic analyses. A steady load, i.e., a load that does not change with time (Figure 9.21a), would require simple static analysis. If the load varies with time, it does not necessarily mean that dynamic analysis needs to be performed. For example, if the loading is harmonic with a frequency less than approximately one third of the first natural frequency of the structure then static analysis will provide an accurate solution and we only need to solve the static problem for the peak load values. (For this frequency range, static analysis may provide a maximum difference of 12.5% in the response for undamped structures and for a forcing frequency less than one fourth of the natural frequency, the difference is only 6.7%.) In this case, the load is normally called “quasi static.” We may apply the same rule to periodic loading after decomposing it into its harmonic components. In transient loading, we should consider the time of application of the load or the “rise time.” If the longest natural period corresponding to the first natural frequency of the structure is more than about twice the rise time, the loading should be classified as shock or impact loading and transient dynamic analysis would be required. If the longest natural period of a system is less than about one third of the rise time, it would be sufficient to perform static analysis and consider the loading to be quasi static. If the longest natural period falls between the quasi static and shock conditions then a transient dynamic analysis would be required and the load would be classified as transient loading. If the loading cannot be categorized as frequency dependent or time dependent, as for example the one shown schematically in Figure 9.21d, then it should be considered random.

Mode-1

Mode-3

Simple Beam Mode-2 FIGURE 9.22

First four mode shapes of a simple beam.

Mode-4

9 22


From the above discussion, the following categories of dynamic analysis may be classified (Mirovitch, 1980): Modal and natural frequency analysis: This is normally performed before any other type of dynamic analysis and will produce the natural frequencies and mode shapes of the structure. This information is vital for understanding the dynamic behavior of the structure and provides data for decoupling the dynamic equations in other analyses. One objective of this analysis is to make sure that the structure is not operating at a frequency close to one of its natural frequencies. A comfortable range for an operating frequency is three times higher than the nearest natural frequency from the lower side and three times smaller than the nearest natural frequency from the higher side. If the operating frequency is higher than the first natural frequency, then large vibration amplitudes or “shudder” will be evident upon passing the natural frequency and certain startup procedures may have to be devised. Frequency response analysis: This type of analysis is performed if the loading on the structure is harmonic or periodic. In periodic loading, the loading function is first decomposed into its sine and cosine components using Fourier analysis. In this analysis, the response will be harmonic with the same loading frequency and with a possible phase shift. The output of such analysis would be displacements, velocities, and accelerations that may be used to calculate forces and stresses in the structure. The displacement output of this analysis defines the deformed shape of the structure, which, in general, is different from the structural mode shapes and may be used to calculate stresses and strains in the structure. Assuming linear conditions, the response to multiple frequency inputs may be simply summed up using the superposition technique. Transient response analysis: This type of analysis is performed if the loading on the structure is classified as transient or shock. There are two general approaches to solving the equation of motion in transient analysis. One is the “direct integration approach” and the other is the “modal superposition approach.” In the direct integration approach, the system equations of motion are integrated directly in the time domain. The required number of time steps depends on the period and the assumptions specified for displacement, velocities, and accelerations within the time step. This number may be quite large and the solution to large size problems may become a computationally difficult task. In the second method, the modal superposition approach, the equations of motion are first transformed to modal generalized displacements. In order to perform this transformation, the mode shapes of the structure should first be determined through a modal analysis. The transformation yields a set of decoupled second order differential equations for the system that are easier to solve. The basic assumption in this approach is that the superposition of the structural response due to the first few lower mode shapes adequately represents the dynamic response of the structure under general transient loading. In practice, this means that only a fraction of the mode shapes of the structure are needed to accurately represent the dynamic behavior. This approach will be generally less accurate than the direct integration approach but will provide substantial savings in computation time for large size problems. The output of transient analysis is time histories of displacements, velocities, and accelerations of the system that may be used to calculate forces and stresses. Random response analysis: If the loading on the structure cannot be classified as frequency or time dependent, it is considered random. Problems with random loading may be solved using a “time history approach” or a “power spectral density (PSD) approach.” The time history approach treats the random input as a transient one and performs a step by step integration over the excitation period. This is normally very costly and requires intensive use of computational resources. If the structure has multiple random inputs in more than one direction, then the transient time history approach is more accurate and may be the preferred approach to solve the problem. Structures with random inputs in more than one direction may be analyzed by a modal superposition approach with special procedures to combine the modal responses (see Section 9.5.4 for details). In the PSD approach, the input will be acceleration as a function of frequency. This is normally specified as a discrete or continuous spectrum by providing values of the ðG2 =f Þ vs. f ; where G is the input acceleration and f is the frequency. This approach is most appropriate for a single input in one direction.


9 23

As mentioned above, most programs will have standard time history inputs or PSD inputs to simulate commonly used random excitations such as vibrations due to road irregularities, earthquakes and seismic inputs, vibration due to wind loads for aerospace parts, and wind loads for wind tunnel tests.

REMARKS *

*

*

*

*

*

9.3

Structural loading may be classified into four categories: steady (constant with time), cyclic (harmonic with period T ¼ 2p and frequency f ¼ 1=T or general periodic that may be decomposed into harmonics by Fourier analysis), transient with a duration T; and random. Random forcing functions are commonly specified as the magnitude of the input acceleration squared vs. frequency (PSD). Typical PSD curves have been compiled for various random events and are normally available in commercial FE programs. The natural frequencies of a system are those frequencies that the system tends to vibrate under conditions of free vibration. The mode shapes give the relative displacements of each point in the structure when it vibrates in one of the natural frequencies. Symmetric structures will generally have symmetric and antisymmetric mode shapes. If the loading is harmonic with a frequency of less than approximately one third of the first natural frequency of the structure, static analysis for the peak load levels suffices. In transient loading, if the longest natural period of a system is less than about one third of the rise time, loading is quasi static and static analysis suffices. In transient loading, if the longest natural period corresponding to the first natural frequency of the structure is more than about twice the rise time of the load, it is necessary to classify the loading as shock or impact loading and transient dynamic analysis would be required. If the longest natural period falls between the quasi static and shock conditions, a transient dynamic analysis would be required.

Modeling Aspects for Dynamic Analysis

Section 9.1 and Section 9.2 provide guidelines for problem and analysis classification. Once a decision has been made about the type of problem and the method of analysis to be used, the designer still has to finalize many details to create a working model for the structure. This section provides a brief discussion of the main aspects required to do this. This includes choice of model size and master DoF, lumped and consistent mass modeling, and use of symmetry. Another important aspect of modeling, damping, will be dealt with in Section 9.4.

9.3.1

Model Size and Choice of Master Degrees of Freedom

Using the same model for both static and dynamic analysis will make the FE solution to the dynamic problem more complicated and therefore much more memory and CPU resources will be required. Also, the results may not be necessarily more useful or accurate. Therefore, FE models for dynamic analysis should, in general, be kept simple. Depending on the type of dynamic analysis, it may be possible to construct a much simpler model for the dynamic problem, possibly using spring, beam, and mass elements. If we consider modal analysis, for example, the objective is to find the mode shapes of the structure and the corresponding natural frequencies. In practice, we seldom need more than the very few lower natural frequencies and mode

9 24


FIGURE 9.23

Static and dynamic models for a shaft.

shapes for a structure. Using a static model that may have thousands of DoF in a modal analysis is very time consuming and does not serve the purpose of the mode shape analysis. It may be better and more efficient to construct a mass spring/beam model for the modal analysis. Figure 9.23 shows a static analysis model for a shaft and a corresponding much simpler mode shape analysis model. In the mass/beam model for modal analysis, the beam properties may change to accommodate specific variations in the shaft geometry. It is also possible to extract accurate values for the spring and beam stiffnesses in 3 D using a detailed static model. Most programs provide means for condensing or reducing the DoF in a dynamic analysis. A common condensation method used is the “Guyan reduction” (see e.g., Freed and Flanigan, 1991; Bouhaddi and Fillod, 1992; Deiters and Smith, 2000). In these condensation methods, the analyst specifies certain DoF to be masters and to be retained by the program, while all other DoF would be considered slaves and would be eliminated or condensed. The dynamic analysis is then carried out using the master DoF. Information pertaining to slave DoF is still available and may be extracted in subsequent runs that may involve stress or other detailed analyses or postprocessing. Reduction has the same effect as modal superposition in terms of reducing the problem size and the required CPU time. Fundamental differences between the two methods may be realized due to the nature of reducing the system of equations. In the modal superposition technique, reduction is accomplished by assuming that the superposition of the structural response due to the first few lower mode shapes adequately represents the response of the structure. In other reduction methods, such as the Guyan reduction technique, we assume that the response of the system is accomplished from the response of certain predetermined master DoF. In the modal superposition technique, a modal analysis must be performed first and the final equations of motion are decoupled, whereas in the other reduction methods no modal analysis is necessary and the final equations of motion are generally full and coupled. Reduction of the total DoF to a set of master DoF has the effect of imposing displacement constraints on the system that increase the stiffness and subsequently overestimate the natural frequencies. The least affected modes are the lower ones. Therefore, reduction and modal superposition techniques should not be used, in general, for shock and impact loading (the modal superposition technique entirely ignores the response from higher modes). The choice of master DoF may be performed automatically by the program. There is no assurance, however, that this will always produce the best choice of masters. The analyst may like to use the program for a first choice of master DoF and then augment the choice manually to improve the accuracy. Simple rules may be used to identify the master DoF (see ANSYS User’s Manual, 2003). The general rule is to choose DoF with large mass and small stiffness or large mass to stiffness ratios as masters. Master DoF should be chosen in the direction of the expected response of the structure, e.g., in a beam or shell model, lateral DoF would be more appropriate than axial or membrane DoF, assuming that the load produces lateral deflections. Rotational DoF are seldom used as masters. It is also important to spread out and not cluster the master DoF so that they will be capable of producing the structural response. Finally, DoF with concentrated mass, a specified input force, or displacement should be retained as masters.


9 25

The question of how many master DoF are adequate is not an easy one to answer. The rule of thumb is to have having at least double the number of master DoF compared with the number of modes of interest. The other guideline is to use 10 to 50% of the total DoF as masters. In practice, the only sure way to test the adequacy of the number of master DoF is to run the analysis with two different choices and compare the results to make sure that the modes of interest are not substantially different.

REMARKS *

*

*

*

9.3.2

In general, dynamic analysis models should be simpler and much smaller in terms of the number of DoF (than static analysis ones). In modal analysis, it is better and more efficient to construct a mass spring/beam model for the structure. “Guyan reduction” may be used to reduce the model size by selecting master DoF for the analysis and condensing out slave DoF. Master DoF may initially be chosen by the program. The analyst should, however, ensure that the choice includes nodes with a large mass to stiffness ratio and with DoF in the direction of the expected response of the structure. Also, it is important to spread out and not cluster the master DoF. As a rule of thumb, the number of master DoF should be at least double the number of modes of interest or about 10 to 50% of the total DoF.

Lumped and Consistent Mass Modeling

The mass characteristic of the structure may be modeled as a lumped or consistent mass matrix. Lumped mass simply means dividing the total mass of the element by the number of nodes. This produces a diagonal mass matrix and may significantly reduce the numerical effort required to solve the dynamic equations. Considering rotational inertia terms in lumped mass assumptions is arbitrary and is normally ignored. If the lumped mass matrix has zero diagonal terms, the matrix will be positive semidefinite and the natural periods corresponding to the zero diagonal terms will be zero. This will have serious implications on the choice of the time step in explicit direct integration solution methods and will be discussed in Section 9.4. A lumped mass assumption implies a discontinuous acceleration field in the element and it usually gives good accuracy for lower modes and frequencies (Chan et al., 1993; Jensen, 1996). A consistent mass matrix is derived by introducing inertia forces in the virtual work formulation of a dynamic FE problem. This would normally lead to a full mass matrix making contributions to both translational and rotational DoF as well as coupling terms. Consistent mass representation usually provides better accuracy for higher modes and frequencies (Kim, 1993). It may be concluded that, if no reduced integration is used to compute element stiffness, the element is compatible, and a consistent mass matrix is used, then the calculated natural frequencies will be an upper bound of the model frequencies (Cook et al., 1989). In some applications such as plates, shells, and 3 D solid elements, it has been found that the convergence rate of the natural frequency calculations for the lumped mass discretization is the same as that for a consistent mass formulation (Chan et al., 1993; Jensen, 1996). Some FE programs provide a modified lumped mass modeling in which the diagonal translational terms are proportional to the diagonal terms of the corresponding consistent mass matrix. This yields more accurate results, in general, than the traditional lumped mass assumption. Several studies have been published on the merits of combining consistent and lumped mass formulations into some kind of modified lumped mass matrix (Chan et al., 1993; Kim, 1993; Jensen, 1996). The modified lumped mass matrix is generally a linear combination of the lumped and the consistent mass matrices. For an improved accuracy, the consistent mass should be weighted more than the lumped mass.

9 26


In building a lumped mass model for the structure, the analyst should realize that one of the objectives is to capture the few lower natural frequencies and mode shapes of the structure. To achieve this, the model should minimize the strain energy and maximize the kinetic energy of the structure. Minimizing the strain energy of the structure may be realized by accurately modeling the soft links in the load path. Structural supports are particularly difficult in this regard. Changing the support model from completely rigid to slightly flexible may shift the first natural frequency by as much as 40% (although it will not have much impact on the mode shape). Maximizing the kinetic energy is simply realized by accurate modeling of the system mass, especially in areas where the mass is large and the stiffness is small. Care should be exercised when modeling masses attached to the system, e.g., a large “nonstructural” mass on top of a building (e.g., a mass of a machine, reservoir, etc.) or the mass of an engine in a car model. If the stiffness characteristic of the nonstructural mass is not important, it may be simply modeled as a rigid lumped mass with an offset mass center from the nearest node. Most programs have a rigid link capability to connect this mass to the structure. This kind of model is transformed internally to a nondiagonal mass matrix and may only be used with a consistent mass formulation for the structural mass. If the structural mass is modeled as lumped, the nonstructural mass will simply be added to the nearest node, which may not be an accurate representation of the physical model.

REMARKS A lumped mass matrix is a diagonal mass matrix that may be obtained by simply dividing the total mass by the number of DoF or alternatively by certain combinations of lumped and consistent mass terms. A consistent mass matrix is derived from FE virtual work equations and is normally a full mass matrix having translational, rotational, and coupling terms. Lumped mass may significantly reduce the numerical effort required to solve the dynamic equations without impacting the accuracy. Zero diagonal terms in the mass matrix should be avoided as much as possible. A lumped mass assumption usually gives good accuracy for lower modes and frequencies, whereas consistent mass representation usually gives better accuracy for higher modes and frequencies.

*

*

*

*

9.3.3

Use of Symmetry

Use of symmetry to reduce the analysis cost and computer time is rather tricky in modal analysis. As shown in Table 9.2, symmetric structures have both symmetric and nonsymmetrical mode shapes. If we only use a symmetric model with symmetric boundary conditions, all nonsymmetrical modes will be undetected. This will change the interpretation of the results and will have an impact on subsequent analyses using the modal data. If the objective is to perform only a modal analysis, it may be advisable to have a less detailed full model rather than a detailed half or quarter model. Also, if the TABLE 9.2

Symmetric and Antisymmetric Boundary Conditions

Plane of symmetry

Symmetric Boundary Conditions ux

XY XZ YZ

0

uy 0

uz

ux

uy

0

0 0

0 0

Antisymmetric Boundary Conditions

uz 0 0

ux

uy

0 0

0 0

uz 0 0

ux

0

uy 0

uz 0


9 27

objective is to perform frequency response or transient analysis using the modal data, it may be advisable to rely on a full model. Experienced analysts may use symmetric models and run the modal analysis with symmetric and antisymmetric boundary conditions to capture the response of all modes. Table 9.2 summarizes symmetric and antisymmetric boundary conditions for translational and rotational DoF.

9.4

Equations of Motion and Solution Methods

In this section, an overview of the equation of motion and the various solution methods available are provided. In most cases, detailed derivation of the equations is not given. The intention is to highlight various solution schemes and the practical aspects of choosing a particular one for a given system or a given dynamic analysis. A consideration of damping and its idealization are also discussed in this section.

9.4.1

Equation of Motion

The equations of motion describing the dynamic behavior of a structural system may be obtained from the extended Hamilton’s principle for elastodynamics: ðtf ðtf d ðU TÞdt dW dt ¼ 0 ð9:1Þ where

to

to

d: first variation to ; tf : two arbitrary time points at which the first variation vanishes U: strain energy of the system T: kinetic energy of the system dW: virtual work of the external forces acting on the system during virtual displacements The external forces may be conservative or nonconservative. Nonconcentrative forces, e.g., damping and follower forces, are deformation dependent, i.e., their magnitudes and/or directions depend on the deformations. Introducing the finite element discretization assumption and substituting expressions for the virtual work and the kinetic and strain energy terms into the above equation, we obtain the Lagrange equation which finally leads to the equations of motion of the system that may be written in the following form: Mu€ þ Cu_ þ Ku ¼ pðtÞ ð9:2Þ where M, C, and K are, respectively, the global mass, global damping, and global stiffness matrices (made € u; _ up by proper assembly of the element matrices); pðtÞ is the time dependent applied force vector; and u; and u are the nodal acceleration, nodal velocity, and nodal displacement vectors, respectively. The global mass and damping matrices are assembled from the element matrices that are given by (Cook et al., 1989; Bathe, 1996): ð MðeÞ ¼ rðeÞ NT N dV ð9:3Þ V

CðeÞ ¼

ð V

csðeÞ NT N dV

ð9:4Þ

where rðeÞ and csðeÞ are the mass density and the viscous damping coefficient for an element e; and N is the matrix of the element shape functions. As indicated above, there are two general approaches to solving Equation 9.1. One is the direct integration approach and the other is the modal superposition approach. With the direct integration approach, the equations of motion are integrated directly in the time domain. In the second approach, the modal superposition method, the equations of motion are first transformed to modal generalized displacements through the use of the mode shapes of the structure. The transformation yields a set of

9 28


decoupled second order differential equations for the system that are easier to solve. In the remaining part of this section, we describe the two solution approaches in more detail.

REMARKS *

9.4.2

Equation 9.2 is the general equation of motion for a dynamic system. Two solution methods are available: the direct integration approach and the modal superposition approach.

Direct Integration Method

With this method, the coupled equations of motion are integrated directly without special transformation. Direct integration schemes seek to satisfy the equilibrium equations of motion at € u; _ discrete time points, 0; Dt; 2Dt; …; t; t þ Dt; … etc. This requires assumptions for the variation of u; and u within the time step. These assumptions will determine the accuracy and convergence characteristics of a particular time integration scheme. They also govern the variation of these quantities with time and are different from the shape function assumptions for the element, which govern the displacement variation within the element, i.e., with spatial coordinates. Direct integration schemes may be categorized as “explicit” and “implicit” schemes. In explicit schemes, the equilibrium equations at time t are used to solve the unknowns at time t þ Dt: These schemes are usually fast and efficient and require no factorization of the effective stiffness matrix provided that the mass and damping matrices can be formulated as diagonal matrices. The main drawback of these schemes is the requirement that the time step for the analysis has to be smaller than a critical value. On the other hand, in implicit schemes, the equilibrium equations at time t þ Dt are used to solve unknowns at time t þ Dt: These schemes require factorization of the effective stiffness matrix of the structure at each time step but do not require the condition of a critical time step. In implicit schemes, the number of operations at each time step is proportional to the matrix order times half of the band width, m ¼ anb; where m is the number of operations, n is the number of equations, a is a constant $ 2, and b is half the band width. Many integration schemes are available in commercial FE programs. These include Euler forward, Euler backward, central difference, Houbolt, Wilson u, and Newmark schemes (Belytschko and Hughes, 1983; Zeng et al., 1992). In the following, one example of an explicit scheme and one example of an implicit scheme are discussed. 9.4.2.1

The Central Difference Method

This is an explicit scheme in which the velocity and acceleration assumptions are given by 9 1 tþDt t > > u t Dt u u_ ¼ = 2Dt t

u€ ¼

1 ðDtÞ2

tþDt

u

t

2 uþ

t Dt

ð9:5Þ

> ; u >

Using Equation 9.5 in the equilibrium equation at time (t) to obtain a solution at time ðt þ DtÞ gives 1 1 Mþ C ðDtÞ2 2ðDtÞ

tþDt

u ¼ tp

K

2 M ðDtÞ2

t

u

1 M ðDtÞ2

1 C 2ðDtÞ

t Dt

u

ð9:6Þ

If the mass and damping matrices are diagonal, Equation 9.6 requires no assembly and no factorization of the effective stiffness matrix of the structure is required. The scheme is, however, conditionally stable and requires a time step smaller than a critical value given by Dtcr # Tn =p

ð9:7Þ


9 29

where Tn is the smallest period in the FE assembly with n DoF. This is an important condition and should be examined carefully. If the mass matrix has a zero diagonal value, the corresponding period will be zero and Equation 9.7 cannot be satisfied. Introducing very small diagonal values may not provide a practical solution since the time step will be very small and the number of steps will be excessive. The scheme is generally applied when a lumped mass assumption with no zero diagonal terms and velocity dependent damping are assumed. The initial conditions required for this scheme are the displacements, velocities, and accelerations at time t ¼ 0: 9.4.2.2

The Newmark Method

This is a commonly used implicit scheme with the following assumptions: h i tþDt u_ ¼ t u_ þ ð1 aÞt u€ þ atþDt u€ Dt tþDt

u ¼ t u þ t u_ Dt þ

1 2

b

t

u€ þ btþDt u€ ðDtÞ2

ð9:8Þ ð9:9Þ

€ where a and b are user input parameters. Equation 9.8 and Equation 9.9 are solved for tþDt u_ and tþDt u; then substituted into the equilibrium equations at time ðt þ DtÞ to obtain: Kþ

a 1 Cþ M bðDtÞ2 bDt

tþDt

u ¼ tþDt p

ð9:10Þ

_ and t u: € Equation 9.10 where the right hand side is a function of the parameters a; b; Dt; K; M; C; t u; t u; indicates that the scheme requires assembly and factorization of the effective stiffness matrix of the structure. The choice of parameters a and b will determine the stability of the scheme and may reduce the scheme to be equivalent to other known ones as follows: *

*

For 2b $ a $ 1=2; the scheme is unconditionally stable. This does not guarantee accuracy and only ensures that the results will not grow out of bounds (Belytschko and Hughes, 1983). A slightly more stringent criterion for unconditional stability that provides artificial damping in higher modes is given by using (Belytschko and Hughes, 1983):

a$ *

*

1 1 1 and b $ aþ 2 4 2

2

A commonly used option gives a ¼ 1=2 and b ¼ 1=4 which corresponds to a trapezoidal rule with constant average acceleration. Another option that corresponds to a linear acceleration assumption is given by a ¼ 1=6 and b ¼ 1=2: This method is very good for small Dt but tends to be unstable for large Dt:

Table 9.3 summarizes the above choices for a and b: TABLE 9.3 a Value 2b $ a $ 1=2 1 2 1 a$ 2

a$

a$

1 2

a ¼ 1=6

Typical Choices of a and b Parameters for the Newmark Scheme b Value

Comments

2b $ a $ 1=2 1 1 aþ 4 2 1 1 b$ aþ 4 2

b$

b$

1 4

b ¼ 1=2

No guarantee of accuracy 2

Improved accuracy Provides artificial damping in higher modes

2

Improved accuracy Corresponds to a trapezoidal rule with constant average acceleration Improved accuracy Provides artificial damping in higher modes Good for small Dt Improved accuracy Corresponds to a linear acceleration Very good for small Dt but tends to be unstable for large Dt

9 30


REMARKS *

*

*

*

9.4.3

In the direct integration approach, the equations of motion are integrated directly without special transformation. Such schemes may be categorized as explicit and implicit schemes. Explicit schemes are usually fast, efficient, and require no factorization of the effective stiffness matrix provided that the mass and damping are diagonal (refer to Equation 9.6 for an example). However, explicit schemes require the time step to be smaller than a critical value (Equation 9.7) that normally leads to very large number of steps. Implicit schemes require factorization of the effective stiffness matrix of the structure at each time step (refer to Equation 9.10 for an example) but do not require the condition of a critical time step. Although implicit schemes are unconditionally stable, accuracy is not always guaranteed without careful consideration of the step size (refer to Table 9.3).

Modal Superposition Method

In this method, the nodal displacement response is expressed in terms of the normal modes that may be found in an eigenvalue analysis. The coupled equations of motion, Equation 9.2, are first transformed into a set of independent or decoupled differential equations cast in modal generalized coordinates (Mirovitch, 1980). The dynamic response of the original system is then obtained by superimposing the responses of the uncoupled modal equations. The generalized coordinates (modal coordinates) are introduced by the following coordinate transformation m X u ¼ Fq and ui ¼ ð9:11Þ wij qj j¼1

where F is an n £ m matrix called the eigenvector or mode shape matrix, m is the number of eigenvectors ðm # nÞ; n is the number of DoF of the system, and q is a vector of size m £ 1 representing the number of mode amplitude generalized or normal coordinates. The transformation given by Equation 9.11 represents a change of basis from nodal displacements (u) to modal generalized coordinates (q). In order for the equations to be decoupled in the modal generalized coordinate system, the triple product FT CF has to be a diagonal matrix and the following orthogonality property is assumed for damping: FT CF ¼ diagð2ji vi Þ

ð9:12Þ

where ji is the damping ratio for mode i and diagð2ji vi Þ indicates a diagonal matrix with the ith diagonal component of 2ji vi : This form has been assumed by generalization of damping for a single DoF system. It is practically more convenient to define the damping by the damping ratios of each mode than it is to evaluate the damping matrix explicitly. Introducing Equation 9.11 and Equation 9.12 into the equation of motion, the following decoupled equations of motion are obtained: q€ þ diagð2ji vi Þq_ þ diagðv2i Þq ¼ FT pðtÞ

ð9:13Þ

If the components of the applied force vector have the same time function, so that pðtÞ can be expressed as pðtÞ ¼ pgðtÞ

ð9:14Þ

where p is a constant vector and gðtÞ is a function giving the time change of the load vector, then the modal load can be written as fr ðtÞ ¼ ðwTr pÞgðtÞ ¼ Gr gðtÞ

ð9:15Þ


9 31

where wTr is a row modal vector corresponding to mode r: The quantity Gr is referred to as the modal participation factor (NISA User’s Manual, 1992). A particular use of this quantity is in ground motion types of excitation. Once the generalized coordinates, q, are evaluated, the physical response of the original system in terms of nodal displacements, velocities, accelerations, and stresses are recovered from Equation 9.11. For the stress recovery, it is noted that (NISA User’s Manual, 1992; Kim et al., 1994): m X ð9:16Þ sðtÞ ¼ DBuðeÞ ðtÞ ¼ DB FðeÞ q ¼ ½DB FðeÞ q ¼ Lq or si ðtÞ ¼ Lir qr r¼1

where si ðtÞ represents a stress component in an element (i.e., sxx ; syy ; sxy ; etc.), Lir lists the stress components corresponding to the rth mode at a typical point. That is, Lir ; r ¼ 1; 2; …; m are modal stresses which should be interpreted as stress shapes rather than absolute values of stress and the matrix D is the material stress strain matrix. The above derivation of the mode superposition method shows the decoupling advantage of the normal coordinates, whereby the change of basis from nodal displacements to normal modes yields a set of uncoupled modal equations, with each equation cast in the form of a single DoF oscillator. As indicated above, another major advantage of the normal mode method is that a good approximation to the response may be obtained using a drastically reduced number of coordinates ðm p nÞ: For most types of loading, with the exclusion of shock and impact loading, the contributions of the lowest few modes are generally more pronounced than the higher modes. Furthermore, practical finite element idealization approximates the lower modes better and tends to be less reliable for higher modes of vibration.

REMARKS *

*

*

*

9.4.4

In the modal superposition approach, the equations of motion are first transformed into a set of decoupled equations cast in modal generalized coordinates. The response of the system is then obtained by superimposing the solutions of the decoupled modal equations (Equation 9.13 and Equation 9.16). The modal superposition approach may only be achieved for linear systems with proportional or directly assumed modal damping (refer to Section 9.4.4). In the modal superposition approach, a good approximation to the response may be obtained using a drastically reduced number of modes. For most types of loading, with the exclusion of shock and impact loading, the contributions of the lowest few modes are generally more pronounced than the higher modes. Practical FE idealization approximates the lower modes better and tends to be less reliable for higher modes of vibration.

Damping Formulation

Damping is a source of energy dissipation in the structure and it leads to decay of the free vibration amplitude with time. Damping sources in a structure include internal friction in the material, Coulomb friction in sliding and pin joints, and other viscous friction forces due to fluid friction (Beards, 1982; Fretzen, 1986). The overall damping of a system is normally quite difficult to obtain and, in general, must be determined experimentally (Kareem and Gurley, 1996). In free vibrations and modal analysis, the damping may be neglected or an overall small value for the system may be assumed. This is a reasonable assumption since in practice damping is small enough and can usually be assumed to be viscous. In forced vibrations, however, and when the forcing frequency is close to one of the system’s natural frequencies, the response of the system is dominated by the specified damping values.

9 32


Commonly, damping is described in viscous or structural form. Both descriptions are used for their mathematical convenience but may not truly represent the actual damping behavior and mechanism. For viscous damping, the damping force is proportional to and opposes the velocity. With structural damping, the damping force is proportional in magnitude to the internal elastic force (i.e., to the displacement) and is in the opposite direction to the velocity. In practical FE analysis, most programs provide the following damping representation that may be used for dynamic analysis: 1. Discrete viscous damper elements (dashpots). As discussed in Section 9.1.2, these elements are damping counterparts of the spring elements and are discrete idealization of viscous damping in the structure. The damping matrix resulting from these elements, in general, cannot be decoupled as in Equation 9.12. Hence, these elements may be used only with direct integration solution methods. 2. Proportional viscous damping (Rayleigh damping). In this type, the following arbitrary form for the damping matrix is assumed: C ¼ c1 K þ c2 M

ð9:17Þ

where c1 and c2 are constants to be determined and supplied by the user. A commonly used method in determining the constants c1 and c2 is to define two damping rations j1 and j2 corresponding to two unequal natural frequencies v1 and v2 ; respectively. Since C is proportional to K and/or M, it satisfies the orthogonality property and we have c1 v2i þ c2 ¼ 2ji vi

ð9:18Þ

c1 vi c þ 2 2 2v i

ð9:19Þ

which leads to

ji ¼

Equation 9.19 is then used to calculate the coefficient c1 and c2 from the knowledge of the specified damping rations j1 and j2 and their corresponding natural frequencies v1 and v2 : For the case when C is only proportional to K ðc2 ¼ 0Þ; the damping ratio is proportional to the natural frequency, and thus the higher modes will be damped more than the lower modes. Similarly, if C is only proportional to M ðc1 ¼ 0Þ; the damping ratio is inversely proportional to the natural frequency and the higher modes will have less damping than the lower modes. The relationship of Equation 9.19 is diagrammatically illustrated in Figure 9.24. The above method for calculating the coefficients c1 and c2 should not restrict the use of proportional damping to only modal analysis. Proportional damping may be used in harmonic, modal, and transient analyses, as well as substructure and reduced types of analyses. For direct transient dynamic analysis, different sets of constants c1 and c2 may be assigned to different parts of the model. This may result in a nonorthogonal global damping matrix. The damping orthogonality condition, however, is not required in direct transient dynamic analysis since the governing equations are directly integrated. x stiffness proportional (c2 = 0, xi = x2 x1

c1wi ) 2

combined mass proportional (c1 = 0, xi =

w2

w1 FIGURE 9.24

Proportional damping.

w

c2 ) 2wi

Finite Element Applications in Dynamics TABLE 9.4

9 33

Typical Damping Ratios for Various Systems and Materials System/Material

Various metals in elastic range Small diameter piping systems Large building during earthquake Large diameter piping systems Welded joints and rigid metal structures Prestressed concrete structures Metal structures with joints Transmission lines (aluminum or steel) Reinforced concrete structures Rubber Bolted joints Shock absorbers

Damping Ratio ,0.01 0.01 0.02 0.01 0.05 0.02 0.03 0.02 0.04 0.02 0.05 0.03 0.07 0.04 0.04 0.07 0.05 0.07 0.30

3. Modal viscous damping, in which the damping ratios are directly specified for the participating modes. This type of damping can be used in modal dynamic analyses and usually provides good results for systems with small damping. In this type, the damping matrix is assumed to be diagonal with values of 2ji vi ; where i is the ith diagonal and ji is the damping ratio for mode i with frequency vi : Typical values of overall damping ratios for various systems are given in Table 9.4 (Adams and Askenazi, 1999).

REMARKS *

*

*

*

*

*

9.5 9.5.1

Damping is a source of energy dissipation in the structure and is generally difficult to quantify. Damping sources in a structure include internal, Coulomb, and viscous friction. The most common forms of damping provided by FE programs are discrete, proportional, and modal viscous damping. Discrete damping is modeled by special spring and damping elements in FE applications. The damping matrix resulting from these elements cannot, in general, be decoupled as in Equation 9.12. Hence, these elements may be used only with direct integration solution methods. Proportional viscous or Rayleigh damping (Equation 9.17) leads to a decoupled damping matrix and may be used in both direct integration as well as modal superposition approaches. A commonly used method to determine the coefficient of proportionality is through defining two modal damping ratios (Equation 9.19). In modal viscous damping, the damping ratios are directly specified for the participating modes and the damping matrix is assumed to be diagonal.

Various Dynamic Analyses Modal Analysis

This is normally the first, and in certain cases the only, type of analysis required for a structure. It is used to obtain the dynamic characteristics of a system in terms of its natural frequencies or eigenvalues

9 34


and associated undamped free vibration natural modes or eigenvectors. These eigenmodes may then be used to decouple the general equations of motion in other types of dynamic analyses. Modal analysis is the first step required before performing mode superposition harmonic or transient analysis as well as spectrum analysis. The natural frequencies of the system may be obtained by solving the eigenvalue problem: ðK

v2 MÞw ¼ 0

ð9:20Þ

For a nontrivial solution, the coefficient matrix ðK v2 MÞ is singular and its determinant is equal to zero, which yields the values of the natural frequencies. For each eigenvalue, vni , a corresponding mode shape, wi , can be obtained from Equation 9.20. If the mass matrix has some zero diagonal values, the number of eigenpairs will be less than the total number of DoF by the number of zero entries on the diagonal of the mass matrix. For properly constrained structures, K is positive definite and all natural frequencies are positive. If the structure is unconstrained or partially constrained, however, K is positive semidefinite and the eigenvalues will contain zero frequencies representing the rigid body modes. The calculated mode shapes from Equation 9.20 are normalized and they satisfy the orthogonality properties: wTi Mwj ¼ 0 and wTi Kwj ¼ 0 for i – j and wTi Mwi ¼ 1:0 and wTi Kwi ¼ v2i ; where wi ¼ pffiffiffi wi = mi and mi ¼ wTi Mwi : Several eigenvalue extraction methods are available. Among the most commonly used are the block Lanczos, Subspace Iteration, Power Dynamics, and Reduced Householder methods (Mirovitch, 1980; Belytschko and Hughes, 1983; Bathe, 1996). The Lanczos method is recommended for large models with many modes to be extracted (about 50 or more). It is also better suited for models containing different types of elements such as solid, shell, and beam elements with the possibility that some elements are distorted. The method is fast and efficient but requires more memory than the Subspace Iteration method. In contrast, the Subspace Iteration method is well suited for extracting lower numbers of modes (less than 50) in models with well shaped elements and it requires less memory than the Lanczos method. The Power Dynamics method is used for very large models, with more than 100,000 DoF, and is especially useful for obtaining a solution for the first several modes of the model. The Reduced Householder method is recommended for finding all the modes in small to medium models of less than 10,000 DoF (ANSYS Users Manual, 2003). The information obtained from an eigenvalue or modal analysis is vital in the analysis of dynamic systems. The designer’s goal is to ensure that the loading frequency is not close to any natural frequency of the system and that the mode shapes do not produce excessive deformation in weak sections of the structure. With respect to the mode shapes, it is desirable to avoid mode shapes that are similar to deformation patterns obtained from the static loading on the structure. As stated previously, the comfortable range for an operating frequency is three times higher than the nearest natural frequency from the lower side and three times smaller than the nearest natural frequency from the higher side. The natural frequencies may be altered by changing the mass distribution or adjusting the geometry of the system. If the loading frequency is higher than one or more of the first natural frequencies then large vibrations or “shudder” may occur during startup and in certain cases special startup procedures may have to be devised. If the loading frequency is close to one of the natural frequencies, a large factor of safety should be used. For example, an operating frequency 20% away from a natural frequency may call for doubling the nominal loads applied on a shaft. In some very special applications, it may be desirable actually to operate the system at one of its natural frequencies and utilize the amplification in the response, e.g., ultrasonic welding equipment and certain nanoscale measuring instruments. Modal analysis requires no special boundary conditions and the structure may be completely free or partially constrained. Special attention should be given to the application of constraints, since overconstraining a system will result in overprediction of the first modes, which are generally nonconservative in a modal analysis.


9 35

REMARKS *

*

*

9.5.2

Modal dynamic analysis is used to obtain the dynamic characteristics of a system in terms of its natural frequencies (eigenvalues) and natural modes (eigenvectors). It is normally the first (and may be the only) dynamic analysis to be performed and its results are required to decouple the equations of motion for other analyses. Common eigenvalue extraction methods include: Lanczos (recommended for large models with many modes to be extracted; about 50 or more); Subspace Iteration (recommended for extracting lower number of modes, less than 50); Power Dynamics (recommended for very large models of more than 100,000 DoF); and Householder (recommended to find all modes in small to medium sized models of less than 10,000 DoF). Modal analysis requires no special boundary conditions and the structure may be completely free.

Transient Dynamic Analysis

9.5.2.1

Direct Integration Method

As indicated above, with the direct integration method, the general equation of motion is integrated directly without transformation. The input loads are specified as a function of time and the response is calculated in the time domain. Transient analysis could be viewed as solving an equivalent static analysis problem for each time step in the domain. The response quantities may include the time history of displacements, velocities, and accelerations at each point in the structure. Most programs will also provide a stress history output that is calculated in a postprocessing phase. The output results are straightforward and easy to interpret. As in static analysis, structures undergoing transient analysis should be fully constrained and all rigid body motion should be eliminated. Initial conditions of displacement and velocity are also required to perform the analysis. As discussed in Section 9.4.2, direct integration may be performed through explicit or implicit schemes. It should be noted that while implicit schemes do not require a critical time step, the size of the time step will affect the accuracy of the solution. Most programs are capable of automatic time stepping, which should be seriously considered. If automatic time stepping is not available in the program, the rule of thumb is to use a time step that is at least one tenth of the load period. Smaller time steps may be required to capture peak response and sharp changes in the input load. The equations of motion (Equation 9.2) may first be reduced in size before the start of the integration process by choosing certain master DoF as discussed above. The reduced equations are then solved by one of the direct integration schemes and the results may then be expanded to the full set of DoF. The following steps may be used as general guidelines in performing direct integration transient dynamic analysis: *

*

*

Build the FE model: This will include identifying the problem type and the type(s) of elements to be used in the model. The mesh should be fine enough to capture the highest mode required for the analysis. If stresses are required, the intensity of the mesh should be adequate for stress calculations. Apply the loads and boundary conditions: All rigid body modes should be constrained and the model should be fully constrained. Time dependent loads are normally specified in a table format. Specify initial conditions: Initial conditions of displacement and velocity are required for transient analysis. Most programs assume zero initial conditions for acceleration. If initial accelerations are not zero, the analyst may apply appropriate acceleration loads over a small time interval. It should be noted that initial conditions are required for all unconstrained DoF.

9 36

Vibration and Shock Handbook *

Set solution control parameters, start solution and review results: This includes mass and damping formulation, damping parameters, time integration, and time stepping parameters as well as parameters to control output and postprocessing options.

9.5.2.2

Mode Superposition Method

In this method, the uncoupled modal equation 9.13, repeated here for convenience, q€ þ diagð2jr vr Þ_q þ diagðv2r Þq ¼ {fr ðtÞ} ¼ FT pðtÞ

ð9:21Þ

consist of m independent second order differential equations with constant coefficients, where m is the number of the participating modes used in the mode superposition process. At time t ¼ 0; each modal equation is subject to the initial conditions qr ð0Þ ¼ wTr Mu0 and q_ r ð0Þ ¼ T wr Mu_ 0 : For the underdamped case ðjr , 1Þ; the solution to a typical modal equation (excluding rigid body modes) is ðt ð9:22Þ qr ðtÞ ¼ e jr vr t ½ar sin vr t þ br cos vr t þ fr ðtÞhr ðt tÞdt 0

where hr ðt

tÞ is the unit impulse response function, defined by 1 jr vr ðt tÞ e hr ðt tÞ ¼ sin vr ðt vr

tÞ

ð9:23Þ

The first term in Equation 9.23 represents the free vibration response (homogenous solution) and the second term represents the particular solution (Duhamel integral). ar and br are constants evaluated from the initial conditions and vr is the damped natural frequency given by qffiffiffiffiffiffiffiffiffi vr ¼ vr 1 jr2 ð9:24Þ The integral in Equation 9.22 may be evaluated in closed form if the applied loading pðtÞ; and consequently fr ðtÞ; is a step input, ramp input, piecewise linear function of time, or harmonic input; otherwise, numerical integration is utilized to obtain the response. The general steps of a typical modal transient analysis include building the model, extracting the required modes for the analysis, obtaining the modal transient analysis and, finally, expanding the modal superposition solution. It should be noted that most programs do not allow change of displacement constraints after the mode extraction step is performed; i.e., all displacement constraints should be specified before performing the modal analysis to extract the required modes. Some programs may require the full loading conditions to be specified in the modal extraction step. Initial conditions should be specified before performing the modal transient solution. This may include initial displacement and velocities.

REMARKS *

*

*

*

In transient analysis, the input loads are specified as a function of time and the response (displacement, velocity, acceleration, and stress) is calculated in the time domain. The structure should be fully constrained and initial conditions are required. Direct integration transient analysis may be performed with explicit (conditionally stable) or implicit (unconditionally stable) schemes. Prior to performing integration, the size of the equations may be reduced by choosing master DoF and condensing out slave DoF. Transient analysis with mode superposition requires an initial modal analysis run followed by the superposition of modal responses (Equation 9.22) obtained from the solution of the decoupled equations. The integral in Equation 9.22 may be evaluated in closed form for step, ramp, piecewise linear, and harmonic inputs; otherwise, numerical integration is utilized.


9.5.3 9.5.3.1

9 37

Harmonic Response Analysis Direct Integration Method

Harmonic response analysis involves computing the steady state response of a structure to a set of harmonic concentrated loads, pressure loads, and harmonic ground motion. The harmonic loads may be defined in terms of different amplitude and phase spectra. In general, the load vector may be represented in the form: pi ðtÞ ¼ pi ðVÞsin½Vt þ Ci ðVÞ

ð9:25Þ

where pi ðtÞ is a component of the forcing function having a magnitude, pi ðVÞ; phase shift, ci ðVÞ, and forcing frequency, V: Substituting Equation 9.25 into Equation 9.2, we obtain: Mu€ þ Cu_ þ Ku ¼ pðtÞ ¼ pðVÞsin½Vt þ CðVÞ

ð9:26Þ

Using direct integration schemes to solve Equation 9.26 is theoretically viable but not recommended practically for a few reasons. A large number of time steps would be normally required to obtain a meaningful response to a harmonic excitation and the equations have to be resolved for each forcing frequency. Also, if the damping is zero and a particular forcing frequency coincides with one of the natural frequencies, the matrix becomes singular and the solution process fails. If the forcing frequency is close to one of the natural frequencies, the matrix becomes ill conditioned and poses solution problems even if the damping is not zero. For these reasons, it is recommended to perform harmonic response analyses using the mode superposition method. 9.5.3.2

Mode Superposition Method

In the mode superposition method, Equation 9.26 is transformed into generalized modal coordinates and decoupled. For the rth mode, the equation takes the form: q€ r þ 2jr vr q_ r þ v2r qr ¼ fr eiVt

ð9:27Þ

where fr is the amplitude of modal load given by fr ¼ wTr p: The steady state modal response is given by qr ðtÞ ¼ qr eiVt

ð9:28Þ

which when substituted into Equation 9.27 gives qr ðVÞ ¼ Hr ðVÞ·fr ðVÞ

ð9:29Þ

and "

Hr ðVÞ ¼

v2r

1

2

V v2r

1 !

V þ i 2j r vr

#

ð9:30Þ

Equation 9.29 in the frequency domain is equivalent to Equation 9.22 in the time domain and Hr ðVÞ is the Fourier transform of the unit impulse response function hr ðtÞ: The physical response is recovered from the generalized modal response through the transformation u ¼ Fq: The velocity and acceleration components are obtained from the corresponding displacement component through multiplication by V and V2 ; respectively. Finally, the stress components are obtained by the use of modal stresses (Equation 9.16). Most programs provide the option that the user can obtain the responses in either an amplitude phase format or real imaginary format and the actual value of all responses can also be obtained for a given value of Vt:

9 38


The input to harmonic analysis constitutes the forcing frequency and the magnitude of the load vector at various points of load applications. For each loading case, all forces should have the same frequency. A change in the forcing frequency gives rise to a new load case. In harmonic analysis, as is the case in transient and static analyses, the structure should be fully constrained and all rigid body motions should be eliminated. If the structure has rigid body modes in one of the DoF, the analyst may use soft springs to ground the structure or utilize symmetry to provide constraints. The general steps of a typical modal harmonic analysis are quite similar to those for the modal transient analysis, i.e., they include building the model, extracting the required modes for the analysis, obtaining the modal harmonic analysis and, finally, expanding the modal superposition solution. It should also be noted that most programs do not allow changes in displacement constraints after the mode extraction step, i.e., all displacement constraints should be specified before performing the modal analysis to extract the required modes. Some programs may require the full loading conditions to be specified in the modal extraction step.

REMARKS *

*

*

*

9.5.4

Harmonic response analysis involves computing the steady state response of a structure to a set of harmonic loads and ground motion (Equation 9.26). Using direct integration schemes in harmonic analysis is theoretically viable but not recommended or used practically. Harmonic analysis with mode superposition requires an initial modal analysis run followed by the superposition of modal responses (Equation 9.28) obtained from the solution of the decoupled equations. The input to harmonic analysis constitutes the frequency and the magnitude of the load vector at all points of load applications. For each load case, all forces should have the same frequency. The structure should be fully constrained.

Response Spectrum Analysis

This analysis is an efficient alternative to transient dynamic analysis for estimating the maximum response under support excitations without regard to the time at which the maximum occurs. If the input loading is classified as a shock or impulse, this analysis is termed “shock spectrum analysis.” In theory, one may use a direct integration scheme to find the response as a function of time and then use the maximum value of the response. This may be prohibitive because of the very large number of time steps that will be required to capture the peak response. As a more efficient alternative, a modal analysis should be performed first and then a sufficient number of modes should be retained for the response spectrum analysis. Then the solution of the individual decoupled modal equation (Equation 9.13) for a viscous underdamped system under ground motion wðtÞ is 1 ðt qr ðtÞ ¼ ð9:31Þ wð € tÞe jr vr ðt tÞ sin½vr ðt tÞ dt vr 0 The response function in Equation 9.31 is scanned over time to find the maximum scalar value ðqr Þmax : Then the physical value of the response of the ith DoF due to ðqr Þmax may be given by uir ¼ Fir ðqr Þmax

ð9:32Þ

in which Fir is the component of the modal vector, r; in direction i: Equation 9.32 is different from Equation 9.11 in which all the components of the modal vector, Fi, and the generalized response, q i, are used and, therefore, actual and accurate physical displacements are obtained.


9 39

It can be readily seen that the quantities ðqr Þmax ; and hence uir ; are only functions of the natural frequency and damping (in addition to the ground excitation). A shock (or response) spectrum curve for a certain value of damping may be defined as the maximum responses of all such single DoF systems with a given damping and plotted as a function of natural frequency. Similarly, the maximum acceleration and maximum velocity may also be determined and are termed as spectral acceleration and spectral velocity, respectively. The mass matrix in response spectrum analysis may be lumped or modified lumped based on the consistent mass matrix. Response spectrum analysis using consistent mass generally lowers the resulting internal forces and moments in the structure and therefore produces less stresses and may be considered generally nonconservative (Gregory, 1990). Unlike transient dynamic analysis, the contributions of the physical responses for each of the modes cannot be directly summed to obtain the total response. This is because the maxima for each mode occur at different times and the information on the time of maxima is not available in the shock spectra. Reasonable, but arbitrary, estimates of the maxima may be obtained by using one of the following commonly used modal combination methods (Wilson et al., 1981; NISA User’s Manual, 1992; Roussel, 1994; Joshi and Gupta, 1998): 1. The sum of absolute magnitudes (ABS or PEAK): The absolute sum of the modal responses is given by m X lRr l ð9:33Þ Rtot ¼ r¼1

in which Rr is the physical response (displacement, velocity, or acceleration) due to mode r and m is the number of modes considered. This method is conservative and is used if the natural frequencies are closely spaced (within 10% of each other) and/or when damping is large. It is also shown that the method may lead to unrealistically high calculated responses, e.g., in the coupled analysis of light secondary systems attached to heavy primary structures or in the decoupled analysis of systems when the centers of mass and stiffness do not coincide (Mertens, 1994). 2. The square root of the sum of the squares (SRSS or RMS): The square root of the sum of the squares of the modal response is given by vffiffiffiffiffiffiffi uX um ð9:34Þ Rtot ¼ t R2r r¼1

This method is applicable if the modes are statistically independent, which is the case when the natural frequencies are far apart and/or when damping is small. The SRSS method is commonly used in a wide range of applications. Some studies indicate, however, that the method may underestimate the response of structures with high frequency modes, e.g., long span bridges exposed to high wind velocities (Joshi and Gupta, 1998). 3. The complete quadratic combination (CQC): The complete quadratic combination is given by the following formula (Wilson et al., 1981; Der Kiureghian and Nakamura, 1993): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX m um X Rr ·Rs Prs Rtot ¼ t ð9:35Þ r¼1 s¼1

in which Prs ¼

ð1

pffiffiffiffiffi 8 jr js ðjr þ gjs Þg 3=2 g2 Þ þ 4jr js ð1 þ g 2 Þ þ 4g 2 ðjr2 þ js2 Þ

ð9:36Þ

where jr and js are modal damping ratios and g ¼ vs =vr : This method encompasses the SRSS and ABS procedures for jr ¼ js : If g ¼ 0, the CQC method reduces to the former, and if g ¼ 1, it reduces to the latter. Certain modifications of the CQC method exist, aimed at improving response estimates for structures with high frequency modes (Der Kiureghian and Nakamura, 1993).

9 40


TABLE 9.5

Common Methods of Combining Modal Responses

Combinational Method

Comments

Sum of absolute magnitudes (ABS or PEAK)

Generally gives conservative results Used if the natural frequencies are closely spaced (within 10% of each other) and/or when damping is large Use when natural frequencies are far apart and/or when damping is small May underestimate the response of structures with high frequency modes Combination of the SRSS and the ABS methods Certain modifications exist to correct the response of structures with high frequency modes Another form of combination of the SRSS and the ABS methods employed by simply adding the response of the two methods

Square root of the sum of the squares (SRSS or RMS) Complete quadratic combination (CQC) Combination of ABS and SRSS

4. Combination of ABS and SRSS: The absolute maxima of the modal responses is added to the square root of the sum of the squares of the remaining modal responses as follows (NISA User’s Manual, 1992): vffiffiffiffiffiffiffiffiffiffiffi u X u m Rmax ¼ lRj l þ t ð9:37Þ R2r r¼1;r–j

where Rj ¼ max for all r lRr l

ð9:38Þ

The above superposition rules are employed to get the response maxima due to ground excitation in one direction. Similarly, the maximum responses due to ground motions in other directions are computed separately. These maxima are then superimposed by using the SRSS or ABS procedures to get the total response. A final note is given here on the use of spectrum analysis in earthquake resistant design. Most building codes require the designer to consider the response of the structure due to simultaneous action of three translational components of ground motion: two in the horizontal plane and one in the vertical direction. Standard design practice is to calculate the peak responses of these inputs independently and then combine these peak responses according to rules similar to those discussed above (Menun and Der Kiureghian, 1998; Lopez et al., 2001). Table 9.5 summarizes the above modal combination methods, their use and merits.

REMARKS *

*

*

Response spectrum and shock analysis are efficient alternatives to transient dynamic analysis for estimating the maximum response under support excitations without regard to the time at which the maximum occurs. In response spectrum analysis, the solution of the individual decoupled modal equation (Equation 9.13) for ground motion, wðtÞ; is obtained (Equation 9.31) and the maximum value of the response is recorded (with no reference to the time it occurs). Reasonable, but arbitrary, estimates of the maxima for the overall structure may be obtained by using various methods of combining the modal maximums (Table 9.5).


9.6

9 41

Checklist for Dynamic FE Analysis

1. Study the physical system and decide on the type of problem, i.e., beam, 2 D, shell, or 3 D, and the type of elements to be used. Typical questions to be addressed are: Can the geometry be idealized as 2 D or shell or do we have to use 3 D analysis? (Section 9.1.2 to Section 9.1.6). Can the loading and constraints be idealized as 2 D? Are we concerned about responses that are not compatible with the loading type? Are we interested in displacements, stresses, or natural frequencies? Can we have a simple spring/mass model? What type of element might be used in the analysis, i.e., linear, quadratic, or cubic (Section 9.1.7)? Generally, linear and quadratic elements are used for the majority of analyses. The choice between linear and quadratic elements is rather arbitrary and may depend on certain advanced analysis aspects. For linear analysis, there is little difference between the response of the two elements if the overall model has the same or similar total DoF. 2. Study all types of loading to identify the required analysis type. Important questions to be addressed are: Is the load steady, cyclic, transient, or random (Section 9.2)? What is the frequency content of the load? A simple Fourier analysis may be required to define the main harmonics of a transient load input. What are the first few natural frequencies of the structure? This question will be answered in detail by performing a modal analysis. It is important, however, to have an approximate idea of the range of frequencies of the structure and to compare this to the loading frequency in order to identify the proper type of analysis (Section 9.2). In order to answer this question adequately, a very simplified mass/spring model and quick hand calculations may be considered. What kind of data is required from the analysis? Data required may be in terms of displacements, velocities, accelerations, stresses, strains, etc. Also, identify if peak values or time histories are required. This decision may have an impact on what analysis type is required. What is the type of analysis required (Section 9.2)? 3. Identify other data required to perform the analysis. This may include: Material properties. Damping consideration, modeling, and values (Section 9.4.4). Identifying all load cases and all constraints of the system. Special attention must be paid to the modeling of constraints and nonstructural masses (Section 9.3). If modal superposition is going to be used, how many modes are required for the analysis? 4. Start the actual modeling process, testing and verification of the model. Important issues to be considered may include: A decision should be made of whether or not a simple or coarse model is required. In general, dynamic analysis of large models is quite costly. Dynamic analysis results (such as natural frequencies, mode shapes, and to some extent displacements, velocities, and accelerations) may be accurately obtained from a coarse model or even a mass/spring model. The main concern would be in obtaining stress and strain responses. In many cases, it may be advisable to have two different models one for dynamic analysis and the other for stress analysis. The output from the dynamic model may be used as input to the stress model. In any case, if the model is large, it is advisable to have a coarser model for testing and basic understanding of the behavior of the structure (Section 9.3). Choose a program to perform the analysis. The key issue in choosing a program would be the availability of the required analysis type. Other issues that may be considered include ease of model generation, ease of postprocessing, and cost effectiveness. It is always advisable to have the same program for all phases of the analysis, i.e., modeling, solution, and postprocessing. *

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

9 42


Build the model geometry (Section 9.1.1). In 2 D problems, the model geometry may be imported from a CAD program. In 3 D and shell problems, however, this process entails many problems and it may be faster for the analyst to build the model directly in a FE database. Build the finite element model and apply load cases and constraints (Section 9.2). Test the model. Most FE programs provide extensive testing capabilities that may be invoked to test the quality of the mesh and the continuity of the model. It is important to note that additional testing is required. The response of the system to simple load cases should be considered. Testing should involve load cases that may be easily verified. This may include computing structure response due to unit impulse, step input, or sudden release of an initial displacement. Also, simple static analysis runs can be very useful in this regard and may provide initial conditions for the final dynamic run. 5. Analyze the model for the actual load cases. The following important issues should be considered: Identify solution parameters. This includes mass and damping formulation, damping parameters (Section 9.4.4), time integration, and time stepping parameters, as well as parameters to control output and postprocessing options. If modal analysis is considered, identify the modal extraction method (Section 9.5). Check and apply restart capabilities in the program. Large size dynamic analyses should, in general, be done in steps and the restart option should be invoked. If modal analysis is performed, a careful check of mode shapes will provide important information about the validity of the model and the appropriateness of solution steps to follow. Identify solution control parameters that invoke the storage of specific data required for postprocessing. 6. Postprocess the results. This is the final stage of the analysis and may involve: Graphical and contour plots of various response quantities. This may be done in the form of time history plots, envelopes of maximum responses, or contour plots of stress maxima. The amount of output data associated with large dynamic analysis runs is vast and such graphical display of results is essential. Most programs have extensive capabilities in results animation, search facilities, preparing output summaries, and some can also assist in report writing. Such capabilities may become important in processing the results of large problems. *

*

*

*

*

*

*

*

*

References Adams, V. and Askenazi, A. 1999. Building Better Products with Finite Element Analysis, On Word Press, Santa Fe. ANSYS User’s Manual, Revision 7.0, Swanson Analysis Systems Inc., Houston, 2003. Babuska, I. and Guo, B.Q., Approximation properties of the h p version of the finite element method, Comput. Methods Appl. Mech. Eng., 133, 319 346, 1996. Bathe, K.J. 1996. Finite Element Procedures in Engineering Analysis, Prentice Hall, New York. Bathe, K.J. and Dvorkin, E.N., A four node plate bending element based on Mindlin Reissner plate theory and a mixed interpolation, Int. J. Numer. Methods Eng., 21, 367 383, 1985. Beards, C.F., Damping in structural joints, Sound Vib. Dig., 14, 9 11, 1982. Belytschko, B. and Hughes, T.J.R., Eds. 1983. Computational Methods for Transient Analysis, North Holland, Amsterdam. Boresi, A.P. and Chong, K.P. 2000. Elasticity in Engineering Mechanics, 2nd ed., Wiley, New York. Bouhaddi, N. and Fillod, R., Method for selecting master DoF in dynamic substructuring using the Guyan condensation method, Comput. Struct., 45, 941 946, 1992. Cesar de Sa, J.M.A., Natal, J.R.M., Fontes, R.A., and Almeida, P.M., Development of shear locking free shell elements using an enhanced assumed strain formulation, Int. J. Numer. Methods Eng., 53, 1721 1750, 2002.


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Chan, H.C., Cai, C.W., and Cheung, Y.K., Convergence studies of dynamic analysis by using the finite element method with lumped mass matrix, J. Sound Vib., 165, 193 207, 1993. Cook, R.D., Malkus, D.S., and Plesha, M.E. 1989. Concepts and Applications of Finite Element Analysis, 3rd ed., Wiley, New York. Deiters, T.A. and Smith, K.S., Faster, easier finite element model reduction, Exp. Tech., 24, 35 40, 2000. Der Kiureghian, A. and Nakamura, Y., CQC modal combination rule for high frequency modes, Earthquake Eng. Struct. Dyn., 22, 943 956, 1993. Freed, A.M. and Flanigan, C., Comparison of test analysis model reduction methods, Sound Vib., 25, 30 35, 1991. Fretzen, C.P., Identification of mass, damping, and stiffness matrices of mechanical systems, ASME J. Vib. Acoust. Stress Reliab. Des., 108, 9 16, 1986. Gregory, F.M., Response spectra analysis. Consistent mass versus lumped mass, ASME Pressure Vessels Piping Div. (Publ.) PVP Seismic Eng., 197, 85 89, 1990. Jensen, M.S., High convergence order finite elements with lumped mass matrix, Int. J. Numer. Methods Eng., 39, 1879 1888, 1996. Joshi, R.G. and Gupta, I.D., On the relative performance of spectrum superposition methods considering modal interaction effects, Soil Dyn. Earthquake Eng., 17, 357 369, 1998. Kamel, H.A., Integration of solid modeling and finite element generation, Comput. Methods Appl. Mech. Eng., 89, 485 496, 1991. Kareem, A. and Gurley, K., Damping in structures: its evaluation and treatment of uncertainty, J. Wind Eng. Ind. Aerodyn., 59, 131 157, 1996. Kim, K., Review of mass matrices for eigenproblems, Comput. Struct., 46, 1041 1048, 1993. Kim, H.M., Bartkowicz, T.J., and Van Horn, D.A., Data recovery and model reduction methods for large structures, Finite Elem. Anal. Des., 16, 85 98, 1994. Lopez, O.A., Chopra, A.K., and Hernandez, J.J., Evaluation of combination rules for maximum response calculation in multicomponent seismic analysis, Earthquake Eng. Struct. Dyn., 30, 1379 1398, 2001. Luo, Y., Explanation and elimination of shear locking and membrane locking with field consistence approach, Comput. Methods Appl. Mech. Eng., 162, 249 269, 1998. Menun, C. and Der Kiureghian, A., Replacement for the 30%, 40%, and SRSS rules for multicomponent seismic analysis, Earthquake Spectra, 14, 153 164, 1998. Mertens, P.G., New algebraic white noise modal combination rule GAC(A), Nucl. Eng. Des., 147, 299 309, 1994. Mirovitch, L. 1980. Computational Methods in Structural Dynamics, Sijthoff and Noordhoff, Alphen a/d Rijn. NISA User’s Manual, Engineering Mechanics Research Corp., Michigan, USA, 1992. Roussel, G., Algebraic modal combination method in seismic response spectrum analysis, ASME Pressure Vessels Piping Div. (Publ.) PVP Seismic Eng., 275 1, 211 216, 1994. Wilson, E.L., Der Kiureghian, A., and Bayo, E.P., A replacement for the SRSS method in seismic analysis, Earthquake Eng. Struct. Dyn., 9, 187 194, 1981. Zeng, L.F., Wiberg, N. E., and Bernspang, L., Adaptive finite element procedure for 2D dynamic transient analysis using direct integration, Int. J. Numer. Methods Eng., 34, 997 1014, 1992.

10

Vibration Signal Analysis 10.1 Introduction ........................................................................ 10 1 10.2 Frequency Spectrum ........................................................... 10 2 Frequency † Amplitude Spectrum † Phase Angle † Phasor Representation of Harmonic Signals † RMS Amplitude Spectrum † One Sided and Two Sided Spectra † Complex Spectrum

10.3 Signal Types ......................................................................... 10 7 10.4 Fourier Analysis .................................................................. 10 7 Fourier Integral Transform † Fourier Series Expansion Discrete Fourier Transform † Aliasing Distortion † Another Illustration of Aliasing

†

10.5 Analysis of Random Signals ............................................... 10 18 Ergodic Random Signals † Correlation and Spectral Density † Frequency Response Using Digital Fourier Transform † Leakage (Truncation Error) † Coherence † Parseval’s Theorem † Window Functions † Spectral Approach to Process Monitoring † Cepstrum

10.6 Other Topics of Signal Analysis ......................................... 10 26


Bandwidth † Transmission Level of a Band Pass Filter † Effective Noise Bandwidth † Half Power (or 3 dB) Bandwidth † Fourier Analysis Bandwidth † Resolution in Digital Fourier Results

10.7 Overlapped Processing ....................................................... 10 28 Order Analysis

Summary This chapter considers the nature and analysis of vibration signals. Both time domain techniques and frequency domain techniques are investigated, which are related through Fourier transform. Topics covered include signal types, signal sampling, aliasing error, truncation error, window functions, spectral analysis, bandwidth issues, and order analysis. Several applications of signal analysis in mechanical vibration are discussed.

10.1 Introduction Numerous examples can be drawn from engineering applications for vibrating (dynamic) systems. A steam generator of a nuclear power plant that undergoes flow induced vibration; a high rise building subjected to seismic motions at its foundation; an incinerator tower subjected to aerodynamic disturbances; an airplane excited by atmospheric turbulence; a gate valve under manual operation; and a heating, ventilating, and air conditioning (HVAC) control panel stressed due to vibrations in its support structure are such examples. 10 1

10 2


Body Response Aerodynamic Excitations FIGURE 10.1

Aerodynamic Excitations

Engine Excitations

Control Surface Excitations

In flight excitations and responses of an aircraft.

Consider an aircraft in flight, as schematically shown in Figure 10.1. There are many excitations on this dynamic system. For example, jet engine forces and control surface movements are intentional excitations, whereas aerodynamic disturbances are unintentional (and unwanted) excitations. The primary response of the aircraft to these excitations will be the motions in various degrees of freedom (DoF), including rigid body and flexible (vibratory) mode motions. Even though the inputs and outputs (excitations and responses) are functions of time, they can also be represented as functions of frequency, through Fourier transformation. The resulting Fourier spectrum of a signal can be interpreted as the set of frequency components which the original signal contains. This frequency domain representation of a signal can highlight many salient characteristics of the signal and also those of the corresponding system. For this reason, frequency domain methods, particularly Fourier analysis, are used in a wide variety of applications such as data acquisition and interpretation, experimental modeling and modal analysis, diagnostic techniques, signal/image processing and pattern recognition, acoustics and speech research, signal detection, telecommunications, and dynamic testing for design development, quality control, and qualification of products. Many such applications involve the study of mechanical vibrations.

Excitations (inputs) to a dynamic system progress with time, thereby producing responses (outputs) which themselves vary with time. These are signals that can be recorded or measured. A measured signal is a time history. Note that in this case the independent variable is time and the signal is represented in the time domain. A limited amount of information can be extracted by examining a time history. As an example, consider the time history record shown in Figure 10.2. It can be characterized by parameters such as the following:

Acceleration

10.2 Frequency Spectrum ap ap/2

0 −ap/2

Te Tp /2 Time Ts(Nz = 14)

FIGURE 10.2

A time history record.

ap ¼ peak amplitude Tp ¼ period in the neighborhood of the peak ¼ 2 £ interval between successive zero crossings near the peak Te ¼ duration of the record Ts ¼ duration of strong response (i.e., the time interval beyond which no peaks occur that are larger than ap =2Þ Nz ¼ number of zero crossings within Ts ðNz ¼ 14 in Figure 10.2) It is obviously cumbersome to keep track of so many parameters and, furthermore, not all of them are equally significant in a given application. Note, however, that all the parameters listed above are directly

Vibration Signal Analysis

FIGURE 10.3

10 3

Time domain representation of a periodic signal.

or indirectly related to either the amplitude or the frequency of zero crossings within a given time interval. This signifies the importance of a frequency variable in representing a time signal. This is probably the fundamental motivation for using frequency domain representations. In this context, however, more rigorous definitions are needed for the parameters: amplitude and frequency. A third parameter, known as phase angle, is also needed for unique representation of a signal in the frequency domain.

10.2.1

Frequency

Let us further examine the basis of frequency domain analysis. Consider the periodic signal of period T that is formed by combining two harmonic (or sinusoidal) components of periods T and T=2 and amplitudes a1 and a2 as shown in Figure 10.3. The cyclic frequency (cycles/sec, or hertz, or Hz) of the two components are f1 ¼ 1=T and f2 ¼ 2=T: Note that in order to obtain the angular frequency (radians/sec), the cyclic frequency has to multiplied by 2p:

10.2.2

Amplitude Spectrum

An alternative graphical representation of the periodic signal shown in Figure 10.3 is given in Figure 10.4. In this representation, the amplitude of each harmonic component of the signal is plotted against the corresponding frequency. This is known as the amplitude spectrum of the signal, and it forms the basis of the frequency domain representation. Note that this representation is often more compact, and can be far more useful than the time domain representation. Note further that in the frequency domain representation, the independent variable is frequency.

Phase Angle

In its present form, Figure 10.4 does not contain all the information of Figure 10.3. For instance, if the high frequency component in Figure 10.3 is shifted through half its period ðT=4Þ; the resulting signal is shown in Figure 10.5. This signal is quite different from that in Figure 10.3 but since the amplitudes and the frequencies of the two harmonic components are identical for both signals, they possess the same amplitude spectrum. So, what is lacking in Figure 10.3 in order to make

Amplitude

10.2.3

a1 a2 0

FIGURE 10.4 signal.

f1 = 1/T

f2 = 2/T

Frequency f

The amplitude spectrum of a periodic

10 4


FIGURE 10.5

A periodic signal with an identical amplitude spectrum as for Figure 10.2.

it a unique representation of a signal, is the information concerning the exact location of the harmonic components with respect to the time reference or origin ðt ¼ 0Þ: This is known as the phase information. For example, the distance of the first positive peak of each harmonic component from the time origin can be expressed as an angle (in radians) by multiplying it by 2p=T: This is termed the phase angle of the particular component. In both signals (shown in Figure 10.3 and Figure 10.5) the phase angle of the first harmonic component is the same and equals p=2 according to the present convention. The phase angle of the second harmonic component is p=2 in Figure 10.3 and zero in Figure 10.5.

10.2.4

Phasor Representation of Harmonic Signals

A convenient geometric representation of a harmonic signal of the form yðtÞ ¼ a cosðvt þ fÞ

ð10:1Þ

is possible by means of a phasor. This representation is illustrated in Figure 10.6. Specifically, consider a rotating arm of radius a; rotating in the counter clockwise (ccw) direction at an angular speed of v rad/sec. Suppose that the arm starts (i.e., at t ¼ 0) at an angular position f with respect to the y axis (vertical axis) in the ccw sense. Then, it is clear from Figure 10.6(a) that the projection of the rotating arm on the y axis gives the time signal yðtÞ: This is the phasor representation, where we have Signal amplitude ¼ length of the phasor Signal frequency ¼ angular speed of the phasor Signal phase angle ¼ initial position of the phasor with respect to the y axis It should be clear that a phase angle makes practical sense only when two or more signals are compared. This is so because for a given harmonic signal we can pick any point as the time reference ðt ¼ 0Þ: However, when two harmonic signals are compared, as in Figure 10.6(b), we may consider one of those signals that starts (at t ¼ 0) at its position peak as the reference signal. This will correspond to a phasor whose initial configuration coincides with the positive y axis. As is clear from Figure 10.6(b), for this reference signal we have, f ¼ 0: Then the phase angle f of any other harmonic signal would correspond to the angular position of its phasor with respect to the reference phasor. Note that, in this example, the time shift between the two signals is f=v; which is also a direct representation of the phase. It should be clear, then, that the phase difference between two signals is also a representation of the time lead or time lag (delay) of one signal with respect to the other. Specifically, the phase that is ahead of the reference phasor is considered to “lead” the reference signal. In other words, the signal a cosðvt þ fÞ has a phase lead of f or a time lead of f=v with respect to the signal of a cos vt:


10 5

y

y = a cos (w t+f)

y

a

a

f w

x

t

0

Period 2p/w

(a) y

y a f

t

0

x

f/w

(b)

a cos w t a cos (w t+f)

FIGURE 10.6 Phasor representation of a harmonic signal. (a) A phasor and the corresponding signal; (b) representation of a phase angle (phase lead) f:

Another important observation may be made with regard to the phasor representation of a harmonic signal. A phasor may be expressed as the complex quantity yðtÞ ¼ a e jðvtþfÞ ¼ a cosðvt þ fÞ þ ja sinðvt þ fÞ

ð10:2Þ

whose real part is a cosðvt þ fÞ; which is in fact the signal of interest. It is clear from Figure 10.6 that, if we take the y axis to be real and the x axis to be imaginary, the complex representation 10.2 is indeed a complete representation of a phasor. By using the complex representation 10.2 for a harmonic signal, significant benefits of mathematical convenience could be derived in vibration analysis. It suffices to remember that practical vibrations are “real” signals, and regardless of the type of mathematical analysis that is used, only the real part of a complex signal of the form 10.2 will make physical sense.

10.2.5

RMS Amplitude Spectrum

If a harmonic signal yðtÞ is averaged over one period T; the negative portion cancels out the positive portion, giving zero. Consider a harmonic signal of angular frequency v (or cyclic frequency f ), phase angle f; and amplitude a; as given by yðtÞ ¼ a cosðvt þ fÞ ¼ a cosð2pft þ fÞ

ð10:3Þ

Its average (mean) value is ymean ¼

1 ðT yðtÞdt ¼ 0 T 0

ð10:4Þ


which can be verified by direct integration, while noting that ð10:5Þ T ¼ 1=f ¼ 2p=v For this reason, mean value is not a measure of the “strength” of a signal in general. Now let us define the root mean square (RMS) value of a signal. This is the square root of the mean value of the square of the signal. By direct integration, it can be shown that for a sinusoidal (or harmonic) signal; the RMS value is given by yRMS ¼

RMS Amplitude

10 6

a1/√1 a1/√2

Frequency f f1 = 1/T

f2 = 2/T

FIGURE 10.7 The RMS amplitude spectrum of a periodic signal.

1 ðT 2 y ðtÞdt T 0

1=2

a ¼ pffiffi 2

ð10:6Þ

pffiffi It follows that the RMS amplitude spectrum is obtained by dividing the amplitude spectrum by 2: For example, for the periodic signal formed by combining two harmonic components as in Figure 10.3, the RMS amplitude spectrum is shown in Figure 10.7. This again is a frequency domain representation of a signal, and the independent variable is frequency.

10.2.6

One-Sided and Two-Sided Spectra

Mean squared amplitude spectrum of a signal (sometimes called power spectrum because the square of a variable such as voltage and velocity is a measure of quantities such as power and energy, even though it is not strictly the spectrum of power in the conventional sense) is obtained by plotting the mean squared amplitude of the signal against frequency. Note that these are one sided spectra because only the positive frequency band is considered. This is a realistic representation because one cannot talk about negative frequencies for a real system. But, from a mathematical point of view, we may consider negative frequencies as well. In a spectral representation it is at times convenient to consider the entire frequency band (consisting of both negative and positive values of frequency). It then becomes a two sided spectrum. In this case the spectral component at each frequency value should be equally divided between the positive and the negative frequency values (hence, the spectrum is symmetric), such that the overall mean squared amplitude (or power or energy) remains the same. We have seen that for a harmonic signal component of amplitude a and frequency f (e.g., a cosð2pf þ fÞ) the RMS amplitude is a2 =2 at frequency f ; whereas the two side spectrum has a magnitude of a2 =4 at both the frequency values f and þf : Note that, even though it is possible to interpret the meaning of a negative time (which represents the past, previous to the starting point), it is not possible to give a realistic meaning to a negative frequency. This concept is introduced primarily for analytical convenience.

10.2.7

Complex Spectrum

We have shown that for unique representation of a signal in the frequency domain, both amplitude and phase information should be provided for each frequency component. Alternatively, the spectrum can be expressed as a complex function of frequency, having a real part and an imaginary part. For instance, for a harmonic component given by a cosð2pfi þ fÞ the two sided complex spectrum can be expressed as a a Yð fi Þ ¼ i ðcos fi þ j sin fi Þ ¼ i e jfi 2 2 and, a a ð10:7Þ Yð fi Þ ¼ i ðcos fi j sin fi Þ ¼ i e jfi 2 2 pffiffiffiffi in which j is the imaginary unity as given by j ¼ 1: Note that the spectral component at the negative frequency is the complex conjugate of that at the positive frequency. This concept of complex spectrum is


10 7

the basis of (complex) Fourier series expansion (FSE), which we shall consider in detail in a later section. It should be clear that the complex conjugate of a spectrum is obtained by changing either j to j or v to v (or f to f ).

10.3 Signal Types Signals can be classified into different types y0 sin (wt+f) depending on their characteristics. Note that the signal itself is a time function, but its frequency domain representation can bring up some of its salient features. Signals particularly important to us here are the excitations and responses of vibrating systems. These can be divided into two broad classes: deterministic signals and random signals depending on whether we are dealing with deterministic vibrations or random vibrations. Consider a damped cantilever beam that is u 0 sin w t subjected to a sinusoidal base excitation of frequency v and amplitude u0 in the lateral FIGURE 10.8 Response to base excitations of a tall direction (Figure 10.8). In the steady state, the tip structure (cantilever). of the beam will also oscillate at the same frequency, but with a different amplitude y0 and, furthermore, there will be a phase shift by an angle f: For a given frequency and known beam properties, the quantities y0 and f can be completely determined. Under these conditions the tip response of the cantilever is a deterministic signal in the sense that when the experiment is repeated, the same response is obtained. Furthermore, the response can be expressed as a mathematical relationship in terms of parameters whose values are determined with 100% certainty, and probabilities are not associated with these parameters (such parameters are termed deterministic parameters). Random signals are nondeterministic (or stochastic) signals. Their mathematical representation requires probability considerations. Furthermore, if the process were to be repeated there would always be some uncertainty as to whether an identical response signal could be obtained again. Deterministic signals can be classified as periodic, quasi periodic, and transient. Periodic signals repeat exactly at equal time periods. The frequency (Fourier) spectrum of a periodic signal constitutes a series of equally spaced impulses. Furthermore, a periodic signal will have a Fourier series representation. This implies that a periodic signal can be expressed as the sum of sinusoidal components whose frequency ratios are rational numbers (not necessarily integers). Quasi periodic (or almost periodic) signals also have discrete Fourier spectra, but the spectral lines are not equally spaced. Typically, a quasi periodic signal can be generated by combining two or more sinusoidal components, provided that at least two of the components have as their frequency ratio an irrational number. Transient signals have continuous Fourier spectra. These types of signals cannot be expressed as a sum of sinusoidal components (or a Fourier series). All signals that are not periodic or quasi periodic can be classified as transient. Most often, highly damped (overdamped) signals with exponentially decaying characteristics are termed transient, even though various other forms of signals such as exponentially increasing (unstable) responses, sinusoidal decays (underdamped responses), and sinesweeps (sinewaves with variable frequency) also fall into this category. Table 10.1 gives examples for these three types of deterministic signals. The corresponding amplitude spectra are sketched in Figure 10.9. A general classification of signals, with some examples is given in Box 10.1.

10.4 Fourier Analysis Fourier analysis is the key to frequency analysis of vibration signals. The frequency domain representation of a time signal is obtained through the Fourier transform. One immediate advantage

10 8 TABLE 10.1

Vibration and Shock Handbook Deterministic Signals

Primary Classification

Nature of the Fourier Spectrum

Example

Periodic

Discrete and equally spaced

Quasi periodic

Discrete and irregularly spaced

5 vt þ f 3 pffiffi y0 sin vt þ y1 sinð 2vt þ fÞ

Transient

Continuous

y0 expð2ltÞsinðvt þ fÞ

y0 sin vt þ y1 sin

Magnitude |Y( f )|

(a)

0

Frequency f

Magnitude |Y( f )|

(b)

0

Frequency f

Magnitude |Y( f )|

(c) FIGURE 10.9 (c) transient.

0

Frequency f

Magnitude spectra for three types of deterministic signals. (a) Periodic; (b) quasi periodic;

of the Fourier transform is that, through its use, differential operations (differentiation and integration) in the time domain are converted into simpler algebraic operations (multiplication and division). Transform techniques are quite useful in mathematical applications. For example, a simple, yet versatile transformation from products into sums is accomplished through the use of the logarithm. Three versions of Fourier transform are important to us. The Fourier integral transform (FIT) can be applied to any general signal, whereas the FSE is applicable only to periodic signals, and the discrete Fourier


10 9

Box 10.1 SIGNAL CLASSIFICATION Signal Types

Deterministic

Random (Future not precisely known through finite observations or analysis)

Periodic

Transient

E.g., * Blade-passing signal of a turbine at constant speed * Counter-rotating-mass shaker signal at constant speed * Step response of an undamped oscillator * Steady-state response of a damped system to a sine excitation

E.g., * Shock wave generated from an impact test with known impulse * Step response of a damped oscillator * Response of a variable-speed rotor * Excitation of a variablefrequency shaker

E.g., * Machine tool vibration * Jet engine noise * Aerodynamic gusts * Road irregularity disturbances * Atmospheric temperature * Earthquake motions * Electrical line noise

transform (DFT) is used for discrete signals. As we shall see, all three versions of transform are interrelated. In particular, we have to use the DFT in digital computation of both FIT and FSE.

10.4.1

Fourier Integral Transform

The Fourier spectrum Xð f Þ of a time signal xðtÞ is given by the forward transform relation Xð f Þ ¼

ð1 1

xðtÞ expð j2p ftÞdt

ð10:8Þ

pffiffiffiffi with j ¼ 1 and f the cyclic frequency variable. When Equation 10.8 is multiplied by expð j2pf tÞ and integrated with respect to f using the orthogonality property (which can be considered as a definition of the Dirac delta function d) ð1 1

exp½j2p f ðt

tÞ dt ¼ dðt

tÞ

ð10:9Þ

10 10


we get the inverse transform relation xðtÞ ¼

ð1 1

Xð f Þ expð2jp ftÞdf

ð10:10Þ

The forward transform is denoted by the operator F and the inverse transform by F 1. Hence, the Fourier transform pair is given by Xð f Þ ¼ F xðtÞ and xðtÞ ¼ F 1 Xð f Þ

ð10:11Þ

Note that for real systems, xðtÞ is a real function but Xð f Þ is a complex function in general. Hence, the Fourier spectrum of a signal can be represented by the magnitude lXð f Þl and the phase angle /Xð f Þ of the (complex) Fourier spectrum Xð f Þ: Alternatively, the real part Re Xð f Þ and the imaginary part Im Xð f Þ together can be used to represent the Fourier spectrum. According to the present definition, the Fourier spectrum is defined for negative frequency values as well as positive frequencies (i.e., a two sided spectrum). The complex conjugate of a complex value is obtained by simply reversing the sign of the imaginary part; in other words, replacing j with j. By noting that replacing j with j in the forward transform relation is identical to replacing f with f ; it should be clear that the Fourier spectrum (of real signals) for negative frequencies is given by the complex conjugate X p ð f Þ of the Fourier spectrum for positive frequencies. As a result, only the positive frequency spectrum needs to be specified and the negative frequency spectrum can be conveniently derived from it, through complex conjugation. The Laplace transform is similar to the FIT. Laplace transform is defined by the forward and inverse relations XðsÞ ¼

ð1 0

xðtÞ expð stÞdt

ð10:12Þ

and xðtÞ ¼

1 ðsþj1 XðsÞ expðstÞds 2p j s j1

ð10:13Þ

Since the signal itself is zero for t , 0; it is seen that for all practical purposes, Fourier transform results can be deduced from the Laplace transform analysis, simply by substituting s ¼ j2pf ¼ jv and s ¼ 0:

10.4.2

Fourier Series Expansion

For a periodic signal xðtÞ of period T; the FSE is given by 1 X

xðtÞ ¼ DF

n¼ 1

An expð j2pnt=TÞ

ð10:14Þ

with DF ¼ 1=T: Strictly speaking (see FIT relations) this is the inverse transform relation. The scaling factor DF is not essential but is introduced so that the Fourier coefficients An will have the same units as the Fourier spectrum. The Fourier coefficients are obtained by multiplying the inverse transform relation by expð j2pmt=TÞ and integrating with respect to t from 0 to T using the orthogonality condition 1 ðT exp½j2pðn T 0 Note that the Kronecker delta dmn is defined as

dmn ¼

(

mÞt=T dt ¼ dmn

1

for m ¼ n

0

for m – n

ð10:15Þ

ð10:16Þ


10 11

for integer values of m and n: The forward transform that results is given by ðT xðtÞ expð j2pnt=TÞdt An ¼ ð10:17Þ

| An|

0

Note that An are complex quantities in general. It can be shown that for periodic signals, FSE is a special case of FIT, as expected. Consider a Fourier spectrum consisting of a sum of equidistant impulses separated by the frequency interval DF ¼ 1=T: Xð f Þ ¼ DF

1 X n¼ 1

An dð f

n · DFÞ

ð10:18Þ

−3∆F −2∆F −∆F 0

∆F 2∆F 3∆F

f

FIGURE 10.10 Fourier spectrum of a periodic signal and its relation to Fourier series.

This is shown in Figure 10.10 (only the magnitudes lAn l can be plotted in this figure because An is complex in general). If we substitute this spectrum into the inverse FIT relation given earlier, we get the inverse FSE relation 10.14. Furthermore, this shows that the Fourier spectrum of a periodic signal is a series of equidistant impulses.

10.4.3

Discrete Fourier Transform

The DFT relates an N element sequence of sampled (discrete) data signal {xm } ¼ ½x0 ; x1 ; …; xN

1

ð10:19Þ

to an N element sequence of spectral results {Xn } ¼ ½X0 ; X1 ; …; XN

1

ð10:20Þ

through the forward transform relation Xn ¼ DT

NX1 m¼0

xm expð j2pmn=NÞ

ð10:21Þ

with n ¼ 0; 1; …; N 1: The values Xn are called the spectral lines. It can be shown that these quantities approximate the values of the Fourier spectrum (continuous) at the corresponding discrete frequencies. Let us identify DT as the sampling period (i.e., the time step between two adjacent points of sampled data). The inverse transform relation is obtained by multiplying the forward transform relation by expð j2pnr=NÞ and summing over n ¼ 0 to N 1; using the orthogonality property 1 NX1 exp½j2pnðr N n¼0

mÞ=N ¼ drm

ð10:22Þ

Note that this orthogonality relation can be considered as a definition of Kronecker delta. The inverse transform is xm ¼ DF

NX1 n¼0

Xn expð j2pmn=NÞ

ð10:23Þ

The data record length is given by T ¼ N · DT ¼ 1=DF

ð10:24Þ

The DFT is a transform in its own right, independent of the FIT. It is possible, however, to interpret this transformation as the trapezoidal integration approximation of FIT. We have deliberately chosen appropriate scaling factors DT and DF in order to maintain this equivalence, and it is very useful in computing the Fourier spectrum of a general signal or the Fourier coefficients of a periodic signal using a digital computer. Proper interpretation of the digital results is crucial, however, in using DFT to compute (an approximate) Fourier spectrum of a (continuous) signal. In particular, two types of error: aliasing

10 12


TABLE 10.2

Unified Definitions for Three Fourier Transform Types

Relation Name Forward transform Inverse transform Orthogonality Notes

Fourier Integral Transform Discrete Fourier Transform (DFT) Ð1 P 1 Xð f Þ ¼ 1 xðtÞ expð2j2p ftÞdt Xn ¼ DT N m 0 xm expð2j2pnm=NÞ; n ¼ 0; 1; …; N 2 1 Ð P 1 xm ¼ DF N xðtÞ ¼ 11 Xð f Þ expð j2p ftÞdf n 0 Xn expð j2pnm=NÞ; m ¼ 0; 1; …; N 2 1

Fourier Series Expansion (FSE) Ð An ¼ T0 xðtÞ expð2j2pnt=TÞdt n ¼ 0; 1; … P xðtÞ ¼ DF 1 n 1 An £ expð j2pnt=TÞ

Ð1

1 PN N n

1 ÐT exp½ j2pðr 2 nÞt=T dt ¼ drn T 0

T ¼ NDT

DF ¼ 1=T

exp½ j2p f ðt 2 tÞ df ¼ dðt 2 tÞ 1

TABLE 10.3

1 0

exp½ j2pnðr 2 mÞ=N ¼ drm

Xð f Þ ¼ DF

P1 n

1

An dð f 2 n=TÞ

Important Properties of the Fourier Transform

Function of Time

Fourier Spectrum

xðtÞ k1 x1 ðtÞ þ k2 x2 ðtÞ xðtÞ expð2j2p taÞ xðt þ tÞ dn xðtÞ dt n Ðt 1 xðtÞdt

Xð f Þ k1 X1 ð f Þ þ k2 X2 ð f Þ Xð f þ aÞ Xð f Þ expð j2p f tÞ ð j2p f Þn Xð f Þ Xð f Þ j2p f

and leakage (or truncation error) should be considered. This subject will be addressed later. The three transform relations, corresponding inverse transforms, and the orthogonality relations are summarized in Table 10.2. The link between the time domain signals and models and the corresponding frequency domain equivalents is the FIT. Table 10.3 provides some important properties of the FIT and the corresponding time domain relations that are useful in the analysis of signals and system models. These properties may be easily derived from the basic FIT relations (Equation 10.8 through Equation 10.10). It should be noted that, inherent in the definition of the DFT given in Table 10.2 is the N point periodicity of the two sequences; that is, Xn ¼ XnþiN and xm ¼ xmþiN ; for i ¼ ^1; ^2; …: The definitions given in Table 10.2 may differ from the versions available in the literature by a multiplicative constant. However, it is observed that according to the present definitions, the DFT may be interpreted as the trapezoidal integration of the FIT. The close similarity between the definitions of the FSE and the DFT is also noteworthy. Furthermore, according to the last row in Table 10.2, the FSE can be expressed as a special FIT consisting of an equidistant set of impulses of magnitude An T located at f ¼ n=T:

10.4.4

Aliasing Distortion

Recalling that the primary task of digital Fourier analysis is to obtain a discrete approximation to the FIT of a piecewise continuous function, it is advantageous to interpret the DFT as a discrete (digital computer) version of the FIT rather than an independent discrete transform. Accordingly, the results from a DFT must be consistent with the exact results obtained if the FIT were used. The definitions given in Table 10.2 are consistent in this respect because the DFT is given as the trapezoidal integration of the FIT. However, it should be clear that if Xð f Þ is the FIT of xðtÞ; then the sequence of sampled values {Xðn · DFÞ} is not exactly the DFT of the sampled data sequence {xðm · DTÞ}: Only an approximate relationship exists. A further advantage of the definitions given in Table 10.2 is apparent when dealing with the FSE. As we have noted, the FIT of a periodic function is a set of impulses. We can avoid dealing with impulses by


10 13

relating the complex Fourier coefficients to the DFT sequence of sampled data from the periodic function via the present definitions. Aliasing distortion is an important consideration when dealing with sampled data from a continuous signal. This error may enter into computation in both the time domain and the frequency domain, depending on the domain in which the results are presented. We will address this issue next. 10.4.4.1

Sampling Theorem

The basic relationships between the FIT, the DFT, and the FSE are summarized in Table 10.4. By means of straightforward mathematical procedures, the relationship between the FIT and the DFT can be established. Even though {Xðn · DFÞ} is not the DFT of {xðm · DTÞ}; the results in Table 10.4 show that ~ · DFÞ} is the DFT of {~xðm · DTÞ} where the periodic functions Xð ~ f Þ and x~ ðtÞ are as defined as in {Xðn Table 10.4. This situation is illustrated in Figure 10.11. It should be recalled that Xð f Þ is a complex function in general, and as such it cannot be displayed as a single curve in a two dimensional coordinate system. Both the magnitude and the phase angle variations with respect to frequency f are needed. For brevity, only the magnitude lXð f Þl is shown in Figure 10.11(a). Nevertheless, the argument presented applies to the phase angle /Xð f Þ as well. ~ · DFÞ is only It is obvious that in the time interval ½0; T ; xðtÞ ¼ x~ ðtÞ and xm ¼ x~ m : However, Xðn approximately equal to Xðn · DFÞ in the frequency interval ½0; F : This is known as the aliasing distortion ~ f Þ will become closer to Xð f Þ in the in the frequency domain. As DT decreases (i.e., as F increases) Xð frequency interval ½0; F=2 ; as is clear from Figure 10.11(c). Furthermore, due to the F periodicity of ~ f Þ; its value in the frequency range ½F=2; F will approximate Xð f Þ in the frequency range ½ F=2; 0 : Xð It is clear from the preceding discussion that if a time signal xðtÞ is sampled at equal steps of DT; no information regarding its frequency spectrum Xð f Þ is obtained for frequencies higher than fc ¼ 1=ð2DTÞ: This fact is known as Shannon’s sampling theorem, and the limiting (cut off) frequency is called the Nyquist frequency. In vibration signal analysis, a sufficiently small sample step DT should be chosen in order to reduce aliasing distortion in the frequency domain, depending on the highest frequency of interest in the analyzed signal. This however, increases the signal processing time and the computer storage requirements, which is undesirable, particularly in real time analysis. It can also result in stability problems in numerical computations. The Nyquist sampling criterion requires that the sampling rate ð1=DTÞ for a signal should be at least twice the highest frequency of interest. Instead of making the sampling rate very high, a moderate value that satisfies the Nyquist sampling criterion is used in practice, together with an antialiasing filter to remove the distorted frequency components. It should be noted that the DFT results in the frequency interval ½fc ; 2fc are redundant because they merely approximate the frequency spectrum in the negative frequency interval ½ fc ; 0 which is known for real signals. This fact is known as the Hermitian property. The last column of Table 10.4 presents the relationship between the FSE and the DFT. It is noted that the sequence {A~ n } rather than the sequence of complex Fourier series coefficients {An } represents the DFT TABLE 10.4

Unified Fourier Transform Relationships

Description

Relationship DFT and FIT

Given

FIT

DFT and FSE FSE

Form

xðtÞ ! Xð f Þ P x~ ðtÞ ¼ 1 k 1 xðt þ kTÞ P1 ~ Xð f Þ ¼ k 1 Xð f þ kFÞ

xðtÞ ! {An } P A~ n ¼ 1 k 1 AnþkN

Then

DFT {~xm } ! {X~ n }

DFT {~xm } ! {A~ n }

Where

~ · DFÞ x~ m ¼ x~ ðm · DTÞ; X~ n ¼ Xðn

xm ¼ xðm · DTÞ

F ¼ 1=DT; T ¼ 1=DF

N ¼ T=DT

10 14


X (f)

x(t)

T Time t

(a) 0

−F/2

0

F/2 Frequency f

~ x(t)

(b)

−2T

−T

0

T

2T

t

~ X ( f)

X( f ) (c)

(d)

−3F/2

−2T

−F

−F/2

0

F/2

(−fc)

~ xm

( fc)

0

−T

∆T

F

T = N∆T

3F/2 f

2T

t

~ Xm

(e) −3F/2 −F = −N∆F −F/2

0 ∆F

(−fc)

F/2

F = 1/∆T 3F/2 f

( fc) Xn = X(n . ∆F)

(f)

−F/2

0 ∆F

F/2

f

FIGURE 10.11 Relationship between FIT and DFT, with an illustration of aliasing error. (a) Fourier integral transformation (FIT) of a signal; (b) periodicity arranged time signal; (c) periodicity of the frequency spectrum; (d) sampled time signal; (e) sampled frequency spectrum (with aliasing error); (f) sampled original spectrum (no aliasing error).

of the sampled data sequence {xðm · DTÞ}: In practice, however, An ! 0 as n ! 1: Consequently, A~ n is a good approximator to An in the range ½ N=2 # n # N=2 for sufficiently large N: This basic result is useful in determining the Fourier coefficients of a periodic signal using discrete data that are sampled at time steps of DT ¼ 1=F; in which F is the fundamental frequency of the periodic signal. Again the aliasing error ðA~ n An Þ may be reduced by increasing the sampling rate (i.e., by decreasing DT or increasing N).


10.4.4.2

10 15

Aliasing Distortion in the Time Domain

In vibration applications it is sometimes required to reconstruct the signal from its Fourier spectrum. Inverse DFT is used for this purpose and is particularly applicable in digital equalizers in vibration testing. Due to sampling in the frequency domain, the signal becomes distorted. The aliasing error ð~xm xðmDTÞÞ is reduced by decreasing the sample period DF: It should be noted that no information regarding the signal for times greater than T ¼ 1=DF is obtained from the analysis. By comparing Figure 10.11(a) with (c), or (e) with ( f), it should be clear that the aliasing error in X~ in comparison with the original spectrum X is caused by “folding” of the high frequency segment of X beyond the Nyquist frequency into the low frequency segment of X: This is illustrated in Figure 10.12. 10.4.4.3

Antialiasing Filter

It should be clear from Figure 10.12 that, if the original signal is low pass filtered at a cut off frequency equal to the Nyquist frequency, then the aliasing distortion would not occur due to sampling. A filter of this type is called an antialiasing filter. In practice, it is not possible to achieve perfect filtering. Hence, some aliasing could remain even after using an antialiasing filter. Such residual errors may be reduced by using a filter cut off frequency that is slightly less than the Nyquist frequency. The resulting spectrum would then only be valid up to this filter cut off frequency (and not up to the theoretical limit of Nyquist frequency).

Example 10.1 Consider 1024 data points from a signal, sampled at 1 msec intervals. Sample rate fs ¼ 1=0:001 samples=sec ¼ 1000 Hz ¼ 1 kHz Nyquist frequency ¼ 1000=2 Hz ¼ 500 Hz Due to aliasing, approximately 20% of the spectrum (i.e., spectrum beyond 400 Hz) will be distorted. Here we may use an antialiasing filter.

Spectral Magnitude

(a)

0

fc

Frequency f fc = Nyquist frequency

Spectral Magnitude

Aliasing Folded High-Frequency Spectrum (b) FIGURE 10.12 aliasing.

0

Original Spectrum

fc

Frequency f

Aliasing distortion of frequency spectrum. (a) Original spectrum; (b) distorted spectrum due to

10 16


Box 10.2 SIGNAL SAMPLING CONSIDERATIONS The maximum useful frequency in digital Fourier results is half the sampling rate. Nyquist frequency or cut off frequency or computational bandwidth: 1 fc ¼ £ sampling rate 2 Aliasing distortion: High frequency spectrum beyond Nyquist frequency folds on to the useful spectrum, thereby distorting it. Summary: 1. Pick a sufficiently small sample step DT in the time domain, to reduce the aliasing distortion in the frequency domain. 2. The highest frequency for which the Fourier transform (frequency spectrum) information would be valid, is the Nyquist frequency fc ¼ 1=ð2DTÞ: 3. DFT results that are computed for the frequency range ½fc ; 2fc merely approximate the frequency spectrum in the negative frequency range ½ fc ; 0 :

Suppose that a digital Fourier transform computation provides 1024 frequency points of data up to 1000 Hz. Half of this number is beyond the Nyquist frequency and will not give any new information about the signal. Spectral line separation ¼ 1000=1024 Hz ¼ 1 Hz ðapproximatelyÞ Keep only the first 400 spectral lines as the useful spectrum. Note: Almost 500 spectral lines may be retained if an accurate antialiasing filter is used. Some useful information on signal sampling is summarized in Box 10.2.

10.4.5

Another Illustration of Aliasing

A simple illustration of aliasing is given in Figure 10.13. Here, two sinusoidal signals of frequency, f1 ¼ 0:2 Hz and f2 ¼ 0:8 Hz; are shown (Figure 10.13(a)). Suppose that the two signals are sampled at the rate of fs ¼ 1 sample/sec. The corresponding Nyquist frequency is fc ¼ 0:5 Hz: It is seen that, at this sampling rate, the data samples from the two signals are identical. In other words, the high frequency signal cannot be distinguished from the low frequency signal. Hence, a high frequency signal component of frequency 0.8 Hz will appear as a low frequency signal component of frequency 0.2 Hz. This is aliasing, as is clear from the signal spectrum shown in Figure 10.13. Specifically, the spectral segment of the signal beyond the Nyquist frequency ð fc Þ cannot be recovered. It is apparent from Figure 10.11(e) that the aliasing error becomes increasingly prominent for ~ f Þ in frequencies of the spectrum closer to the Nyquist frequency. With reference to the expression for Xð Table 10.4, it should be clear that when the true Fourier spectrum Xð f Þ has a steep roll off prior to F=2 ð¼ fc Þ; the influence of the Xð f nFÞ segments for n $ 2 and n # 1 is negligible in the discrete spectrum in the frequency range ½0; F=2 : Hence the aliasing distortion in the frequency band ½0; F=2 comes primarily from Xð f FÞ; which is the true spectrum shifted to the right through F: Therefore,


10 17

f1 = 0.2 Hz f2 = 0.8 Hz ×

Signal

× ×

1

2

3

4

×

× 5 Time (s)

× Sampling rate

fs = 1 sample/s

Nyquist frequency fc = 0.5 Hz

(a) Amplitude Spectrum

f1 0

(b)

fc

0.2

f2

0.5

0.8

Frequency (Hz)

FIGURE 10.13 A simple illustration of aliasing. (a) Two harmonic signals with identical sampled data; (b) Frequency spectra of the two harmonic signals.

a reasonably accurate expression for the aliasing error is en ¼ Xð ðF

n · DFÞÞ ¼ Xðn · DF

FÞ ¼ X

ðN nÞ

¼ Xn

N

n ¼ 0; 1; 2; …; N=2

ð10:25Þ

Note from Equation 10.8 that the spectral value obtained when f in the complex exponential is replaced by f is the same as the spectral value obtained when jis replaced by j. Since the signal xðtÞ is real, it follows that the Fourier spectrum for the negative frequencies is simply the complex conjugate of the Fourier spectrum for the positive frequencies; thus Xð f Þ ¼ X p ð f Þ

ð10:26Þ

or, in the discrete case Xn

N

¼ XNp

n

ð10:27Þ

It follows from Equation 10.25 that the aliasing distortion is given by en ¼ XNp

n

for n ¼ 0; 1; 2; …; N=2

ð10:28Þ

This result confirms that aliasing can be interpreted as folding of the complex conjugate of the true spectrum beyond the Nyquist frequency fc ð¼ F=2Þ over to the original spectrum. In other words, due to aliasing, frequency components higher than the Nyquist frequency appear as lower frequency components (due to folding). These aliasing components enter into the digital Fourier results in the useful frequency range ½0; fc : Aliasing reduces the valid frequency range in digital Fourier results. Typically, the useful frequency limit is fc 1:28 so that the last 20% of the spectral points near the Nyquist frequency should be neglected.

10 18


It should be clear that if a low pass filter with its cut off frequency set at fc is used on the time signal prior to sampling and digital Fourier analysis, the aliasing distortion can be virtually eliminated. Analog hardware filters may be used for this purpose. They are the antialiasing filters. Note that sometimes fc 1:28ðø 0:8fc Þ is used as the filter cut off frequency. In this case, the computed spectrum is accurate up to 0:8fc and not up to fc : The buffer memory of a typical commercial Fourier analyzer can store N ¼ 210 ¼ 1024 samples of data from the time signal. This is the size of the data block analyzed in each digital Fourier transform calculation. This will result in N=2 ¼ 512 spectral points (spectral lines) in the frequency range ½0; fc : Out of this only the first 400 spectral lines (approximately 80%) are considered free of aliasing distortion.

Example 10.2 Suppose that the frequency range of interest in a particular vibration signal is 0 200 Hz. We are interested in determining the sampling rate (digitization speed) and the cut off frequency for the antialiasing (low pass) filter. The Nyquist frequency fc is given by fc 1:28 ¼ 200 Hence; fc ¼ 256 Hz The sampling rate (or digitization speed) for the time signal that is needed to achieve this range of analysis is F ¼ 2fc ¼ 512 Hz: With this sampling frequency, the cut off frequency for the antialiasing filter could be set at a value between 200 and 256 Hz.

10.5 Analysis of Random Signals Random (stochastic) signals are generated by some random mechanism. Each time the mechanism is operated a new time history (sample function) is generated. The chance of any two sample functions becoming identical is governed by some probabilistic law. If all sample functions are identical (with unity ~ probability), then the corresponding signal is a deterministic signal. A random process is denoted by XðtÞ; ~ while any sample function of it is denoted by xðtÞ: No numerical computations can be performed on XðtÞ because it is not known for certain. Its Fourier transform, for instance, can be written down as an analytical expression, but cannot be numerically computed. However, once the signal is generated, numerical computations can be performed on that sample function xðtÞ because it is a completely known function of time.

10.5.1

Ergodic Random Signals

~ 1 Þ is a random variable which has a certain probability distribution. Consider a At any given time t1 ; Xðt ~ well behaved function f {XðtÞ} of this random variable (which is also a random variable). Its expected ~ value (statistical mean) is E½ f {XðtÞ} : This is also known as the ensemble average because it is equivalent to the average value at t of a collection (ensemble) of a large number of sample functions xðtÞ: Consider the function f {xðtÞ} of one sample function xðtÞ: Its temporal (time) mean is expressed by 1 ðT lim f {xðtÞ}dt T!1 2T T Now, if ðT ~ 1 Þ} ¼ lim 1 E½ f {Xðt f {xðtÞ}dt T!1 2T T

ð10:29Þ


10 19

then the random signal is said to be ergodic. It should be noted that the right hand side of Equation 10.29 does not depend on time. Consequently, the left hand side should also be independent of the time point t1 : For analytical convenience, random vibration signals are usually assumed to be ergodic (the ergodic hypothesis). Using this hypothesis, the properties of a random signal could be determined by performing computations on a sufficiently long record (sample function) of the signal. Since the ergodic hypothesis is not exactly satisfied for vibration signals, and since it is impossible to analyze infinitely long data records, the accuracy of the numerical results depends on various factors such as the record length, sampling rate, frequency range of interest, and the statistical nature of the random signal (e.g., closeness to a deterministic signal, frequency content, periodicity, damping characteristics). Accuracy can be improved in general, by averaging the results for more than one data record.

10.5.2

Correlation and Spectral Density

~ ~ 1 Þ and Xðt ~ 2 Þ depend on the time difference If for a random signal XðtÞ; the joint statistical properties of Xðt ðt2 t1 Þ and not on t1 itself, then the signal is said to be stationary. Consequently, the statistical ~ will be independent of t: It is noted from Equation 10.29 that ergodic properties of a stationary XðtÞ random signals are necessarily stationary. However, in general the converse is not true. ~ and YðtÞ ~ is given by E½XðtÞ ~ Yðt ~ þ tÞ : If the The cross correlation function of two random signals XðtÞ signals are stationary, this expected value is a function of t (not t) and is denoted by fxy ðtÞ: In view of the ergodic hypothesis, the cross correlation function may be expressed as

fxy ðtÞ ¼ lim

T!1

1 ðT xðtÞyðt þ tÞdt T 0

ð10:30Þ

The FIT of fxy ðtÞ is the cross spectral density function which is denoted by Fxy ð f Þ: When the two signals are identical, we have the autocorrelation function fxx ðtÞ in the time domain and the power spectral density (PSD) Fxx ð f Þ in the frequency domain. The continuous and the discrete versions of the correlation theorem are given in the first row of Table 10.5. It follows that the cross spectral density may be estimated using the DFT (FFT) of the two signals, as ½Xn p Yn T in which T is the record length and ½Xn p is the complex conjugate of ½Xn : Parseval’s theorem (second row of Table 10.5) follows directly from the correlation theorem. Consequently, the mean square value of a random signal may be obtained from the area under the PSD curve. This suggests a hardware based method of estimating the PSD as illustrated by the functional diagram in Figure 10.14(a). Alternatively, a software based digital Fourier analysis could be used (Figure 10.14(b)). A single sample function would not give the required accuracy, and averaging is usually needed. In real time digital analysis, the running average as well as the current estimate is usually computed. In the running average, it is desirable to give a higher weighting to the more recent estimates. The fluctuations in the PSD estimate about the local average could be reduced by selecting a large filter TABLE 10.5

Some Useful Fourier Transform Results

Description Correlation theorem

Ð1

If zðtÞ ¼

1

Continuous xðtÞyðt þ tÞdt

zm ¼ DT

Then Zð f Þ ¼ ½Xð f Þ p Yð f Þ Parseval’s theorem

FIT

Convolution theorem

Ð1

If yðtÞ ¼

1

DFT

{ym } ! {Yn }

y2 ðtÞdt ¼

Ð1

1

1 r 0 xr yrþm

Zn ¼ ½Xn p Yn

If yðtÞ ! Yð f Þ Then

Discrete

PN

Ð1

1

lYð f Þl2 df

hðtÞuðt 2 tÞdt ¼

Then Yð f Þ ¼ Hð f ÞUð f Þ

Ð1

1

DT hðt 2 tÞuðtÞdt

PN

1 2 m 0 ym

ym ¼ DT

PN

Yn ¼ Hn Un

¼ DF

1 r 0

PN

hr um

1 n 0 r

lYn l2

¼ DT

PN

1 r 0

hm r ur

10 20


Signal (a) Signal (b)

Tracking Filter Bandwidth ∆f

Analog-to-Digital Conversion (ADC)

Approx. psd

Squaring Hardware

Averaging Network

1 ∆f

Digital Correlation Function

Digital Fourier Transform

Averaging Software

Approx. psd

FIGURE 10.14 Power spectral density computation. (a) Narrow band filtering method; (b) correlation and Fourier transformation method.

bandwidth Df and a large record length T: A measure of these fluctuations is given by 1 1 ¼ pffiffiffiffiffiffi Df T

ð10:31Þ

It should be noted that a large Df results in reduction of the precision of the estimates while improving the appearance. To offset this, T has to be increased further.

10.5.3

Frequency Response Using Digital Fourier Transform

Vibration test programs usually require a resonance search type pretesting. In order to minimize the damage potential, it is carried out at a much lower intensity than the main test. The objective of such exploratory tests is to determine the significant frequency response functions of the test specimen. These provide the natural frequencies, damping ratios, and mode shapes of the test specimen. Such frequency response data are useful in planning and conducting the main test. For example, more attention is required when testing in the vicinity of the resonance points (slower sweep rates, larger dwell periods, etc.). Also, the frequency response data are useful in determining the most desirable test input directions and intensities. The degree of nonlinearity and time variance of the test object can be determined by conducting more than one frequency response test at different input intensities. If the deviation of the frequency response function is sufficiently small, then linear, time invariant analysis is considered to be satisfactory. Often, frequency response tests are conducted at full test intensity. In such cases, it is considered as a part of the main test rather than a prescreening test. Other uses of the frequency response function include the following: it can be employed as a system model (experimental model) for further analysis of the test specimen (experimental modal analysis). Most desirable frequency range and sweep rates for vibration testing can be estimated by examining frequency response functions. The time response hðtÞ to a unit impulse is known as the impulse response function. For each pair of input and output locations (A,B) of the test specimen, a corresponding single response function would be obtained (assuming linearity and time invariance). Entire collection of these responses would determine the response of the test specimen to an arbitrary input signal. The response yðtÞ at B to an arbitrary input uðtÞ applied at A, is given by yðtÞ ¼

ð1 1

hðtÞuðt

tÞdt ¼

ð1 1

hðt

tÞuðtÞdt

ð10:32Þ

The right hand side of Equation 10.32 is the convolution integral of hðtÞ and uðtÞ and is denoted by hðtÞ p uðtÞ: By substituting the inverse FIT relations (Table 10.2) in Equation 10.32, the frequency response function (frequency transfer function) Hð f Þ is obtained as the ratio of the (complex) FITs of the output and the input. It exists for physically realizable (casual) systems even when the individual FITs of the input and output signals do not converge. The continuous convolution theorem and the discrete


10 21

counterpart are given in the last row of Table 10.5. The discrete convolution can be interpreted as the trapezoidal integration of Equation 10.32. Frequency response function is a valid representation (model) for linear, time invariant systems. It is related to the system transfer function GðsÞ (ratio of the Laplace transforms of output and input with zero initial conditions) through Hð f Þ ¼

Yð f Þ ¼ Gð2pfjÞ Uð f Þ

ð10:33Þ

But for notational convenience, the frequency response function corresponding to GðsÞ may be denoted by either Gð f Þ or GðvÞ; where v is the angular frequency and f is the cyclic frequency. Using Fourier transform theory, three methods of determining Hð f Þ can be established. First, using any transient excitation signal to a system at rest and the corresponding output, Hð f Þ is determined from their FITs (Table 10.5). Second, if the input is sinusoidal, the signal amplification of the steady state output is the magnitude lHð f Þl at the input frequency, and the phase lead of the steady state output is the corresponding phase angle /Hð f Þ: Third, using a random input signal and the corresponding input and output spectral density functions, Hð f Þ is determined as the ratio Hð f Þ ¼ Fuy ð f Þ=Fuu ð f Þ

10.5.4

ð10:34Þ

Leakage (Truncation Error)

In digital processing of vibration signals (e.g., accelerometer signals), sampled data are truncated to eliminate less significant parts. This is of course essential in real time processing because, in that case, only sufficiently short segments of continuously acquisitioned data are processed at a time. The computer memory (and buffer) limitations, the speed and cost of processing, the frequency range of importance, the sampling rate, and the nature of the signal (level of randomness, periodicity, decay rate, etc.) should be taken into consideration in selecting the truncation point of data. The effect of direct truncation of a signal xðtÞ on its Fourier spectrum is shown in Figure 10.15. In the time domain, truncation is accomplished by multiplying xðtÞ by the box car function bðtÞ: This is equivalent to a convolution ðXð f Þ p Bð f ÞÞ in the frequency domain. This procedure introduces ripples (side lobes) into the true spectrum. The resulting error ðXð f Þ Xð f Þ p Bð f ÞÞ is known as leakage or truncation error. Similar leakage effects arise in the time domain, as a result of truncation of the frequency spectrum. The truncation error may be reduced by suppressing the side lobes, which requires modification of the truncation function (window) from the box car shape bðtÞ to a more desirable shape. Commonly used windows are the Hanning, Hamming, Parzen, and Gaussian windows.

10.5.5

Coherence

~ and YðtÞ ~ are said to be statistically independent if their joint probability Random vibration signals XðtÞ distribution is given by the product of the individual distributions. A special case of this is the uncorrelated signals which satisfy ~ 1 ÞYðt ~ 2 Þ ¼ E½Xðt ~ 1 Þ E½Yðt ~ 2Þ E½Xðt

ð10:35Þ

~ ~ In the stationary case, the means mx ¼ E½XðtÞ and my ¼ E½YðtÞ are time independent. The autocovariance functions are given by ~ cxx ðtÞ ¼ E½{XðtÞ

~ þ tÞ mx }{Xðt

mx } ¼ fxx ðtÞ

m2x

ð10:36Þ

~ cyy ðtÞ ¼ E½{YðtÞ

~ þ tÞ my }{Yðt

mx } ¼ fyy ðtÞ

m2y

ð10:37Þ

mx my

ð10:38Þ

and the cross covariance function is given by ~ cxy ðtÞ ¼ E½{XðtÞ

~ þ tÞ mx }{Yðt

my } ¼ fxy ðtÞ

10 22


x(t)

|X( f )|

TRUE SIGNAL (a)

0

T

t

0

f |B( f )|

b(t) 1 TRUNCATION WINDOW (b)

0

T

t

−1/T

1/T

f

|X( f )*B( f )|

x(t)b(t) RESULT

0 (c)

T

t

TIME

0 FREQUENCY

f

FIGURE 10.15 Illustration of truncation error. (a) Signal and its frequency spectrum; (b) a rectangular (box car) window and its frequency spectrum; (c) truncated signal and its frequency spectrum.

For uncorrelated signals fxy ðtÞ ¼ mx my and cxy ðtÞ ¼ 0: The correlation function coefficient is defined by

cxy ðtÞ rxy ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cxx ð0Þcyy ð0Þ

ð10:39Þ

which satisfies 1 # rxy ðtÞ # 1: For uncorrelated signals we have rxy ðtÞ ¼ 0: This function measures the degree of correlation of the two signals. In the frequency domain, the correlation is determined by its (ordinary) coherence function

g2xy ð f Þ ¼

lFxy ð f Þl2 Fxx ð f ÞFyy ð f Þ

ð10:40Þ

which satisfies the condition 0 # g2xy ð f Þ # 1: In this definition, the signals are assumed to have zero means. Alternatively, the FIT of the covariance functions may be used. If the signals are uncorrelated (or ~ is the response of a better, independent) the coherence function vanishes. On the other hand, if YðtÞ ~ linear, time invariant system to an input XðtÞ; then

Fxy ð f Þ ¼ Fxx ð f ÞHð f Þ

ð10:41Þ

Fyy ð f Þ ¼ Fxx ð f ÞlHð f Þl2

ð10:42Þ

Consequently, the coherence function becomes unity for this ideal case. In practice, however, the coherence function of an excitation and the corresponding response is usually less than unity. This is due to deviations such as measurement noise, system nonlinearities, and time variant effects. Consequently, the coherence function is commonly used as a measure of the accuracy of frequency response estimates.


10.5.6

10 23

Parseval’s Theorem

For a pair of rapidly decaying (aperiodic) deterministic signals xðtÞ and yðtÞ; the cross correlation function is given by ð1 fxy ðtÞ ¼ xðtÞyðt þ tÞdt ð10:43Þ 1

This is equivalent to Equation 10.30, for a pair of ergodic, random (stochastic) signals xðtÞ and yðtÞ: By using the definition of the inverse FIT (see Table 10.2) in Equation 10.43, and by following straightforward mathematical manipulation, it may be shown that

Fxy ð f Þ ¼ ½Xð f Þ p Yð f Þ in which the cross spectral density Fxy ð f Þ is the FIT of fxy ðtÞ; as given by ð1 Fxy ð f Þ ¼ fxy ðtÞ expð j2pf tÞdt 1

ð10:44Þ

ð10:45Þ

and [ ]p denotes the complex conjugation operation. This result, which is known as the correlation theorem (see Table 10.5) has applications in the evaluation of the correlation functions and PSD functions of finite record length data. The inverse FIT relation corresponding to Equation 10.45 is ð1 fxy ðtÞ ¼ Fxy ð f Þ expð j2pf tÞdf ð10:46Þ 1

From Equation 10.44, we have

fxy ðtÞ ¼

ð1 1

½Xð f Þ p Yð f Þ expð j2pf tÞdf

If we set t ¼ 0 and x ¼ y in Equation 10.47, we get ð1 lYð f Þl2 df fxy ð0Þ ¼ 1

ð10:47Þ

ð10:48Þ

Similarly, from Equation 10.43, we get

fxy ð0Þ ¼

ð1

y2 ðtÞdt

1

By comparing Equation 10.48 and Equation 10.49, we obtain Parseval’s theorem and thus ð1 ð1 y2 ðtÞdt ¼ lYð f Þl2 df 1

1

ð10:49Þ

ð10:50Þ

By using the discrete correlation theorem is an analogous manner, we can establish the discrete version of Equation 10.50: DT

NX1 m¼0

2 ¼ DF ym

NX1 n¼0

lYn l2

ð10:51Þ

These results are listed in the second row of Table 10.5.

10.5.7

Window Functions

Consider the unit box car function wðtÞ; defined as ( 1 for 0#t,T wðtÞ ¼ 0 otherwise

ð10:52Þ

10 24 TABLE 10.6

Vibration and Shock Handbook Some Common Window Functions

Function Name

Time Domain Representation ½wðtÞ

( Box car

1 for 0 # t , T

Hanning

(

Parzen

2 0

þ

pt 1 cos T 2

for ltl , T

2

ltl T

ltl þ6 T

3

0

ltl T

for ltl ,

T 2

T , ltl # T 2 otherwise

3

for

0 12

T sin 2p f T 2p f T½1 2 ð2 f TÞ2

otherwise

t 126 T 2 12

( Bartlett

1 ½1 2 cos 2p f T þ j sin 2p f T j2p f

0 otherwise

(1

Frequency Domain Representation ½Wðf Þ

for ltl # T

T

otherwise

34

2

3 6 sin p f T=2 7 7 T6 5 4 4 1 pfT 2 sin p f T pfT

4

This is shown in Figure 10.15(b). The FIT of wðtÞ is Wð f Þ ¼

1 ½1 j2pf

cos 2p f T þ j sin 2p f T

ð10:53Þ

Clearly, this (rectangular) window function produces side lobes (leakage) in the frequency domain. In spectral analysis of vibration signals, it is often required to segment the time history into several parts, and then perform spectral analysis on the individual results to observe the time development of the spectrum. If segmenting is done by simple (a) truncation (multiplication by the box car win w(t) dow), the process would introduce rapidly fluctu Box-Car ating side lobes into spectral results. Window 1 functions, or smoothing functions other than the Hanning box car function, are widely used to suppress the side lobes (leakage error). Some common smooth Bartlett ing functions are defined in Table 10.6. A graphical comparison of these four window Parzen types is given in Figure 10.16. Hanning windows are very popular in practical applications. A related 0 window is the Hamming window, which is simply a Time T t Hanning window with rectangular cut offs at the (b) two ends. A Hamming window will have charac W( f ) teristics similar to those of a Hanning window, Box-Car 2 except that the side lobe fall off rate at higher frequencies is less in the Hamming window. From Figure 10.16(b), we observe that the Hanning & Bartlett frequency domain weight of each window varies 1 with the frequency range of interest. Obviously, Parzen the box car window is the worst. In practical applications, the performance of any window 0 could be improved by simply increasing the Frequency f window length T. In addition, a window in the shape of a Gaussian function may be used when a FIGURE 10.16 Some common window functions. (a) Time domain function; (b) frequency spectrum. rapid roll off without side lobes is desired.

Vibration Signal Analysis TABLE 10.7

10 25

Signal Types and Appropriate Windows Signal Type

Window

Periodic with period ¼ T Rapid transients within ½0; T Periodic with period – T Quasi periodic Slow transients beyond ½0; T Nonstationary random Beat like signals with period < T Narrow band random Stationary random Important low level components mixed with widely spaced high level spectral components Broad band random (white noise, pink noise, etc.)

Rectangular Gaussian Flat top cosine Hamming Hanning Gaussian Bartlett (triangular) Rectangular Hamming Parzen Gaussian

Characteristics of the signal that is being analyzed and also the nature of the system that generates the signal should be considered in choosing an appropriate truncation window. In particular, the Hanning window is recommended for signals generated by heavily damped systems and the Hamming window is recommended for use with lightly damped systems. Table 10.7 lists some useful signal types and appropriate window functions.

10.5.8

Spectral Approach to Process Monitoring

Fourier Spectrum Magnitude

In mechanical systems, component degradation dB At Start may be caused by vibrating excitations, which can With Degradation result in malfunction or failure. In this sense, continuous monitoring during testing of mechanical deterioration in various critical com ponents of a vibratory system is of prime importance. This usually cannot be done by simple visual observation, unless malfunction is detected by operability monitoring of the system. However, 0 10 20 30 40 50 f since mechanical degradation is always associated Frequency (Hz) with a change in vibration level, by continuously monitoring the development of Fourier spectra in time (during system operation) at various critical FIGURE 10.17 Effect of mechanical degradation on a monitored Fourier spectrum. locations of the system, it is possible to con veniently detect any mechanical deterioration and impending failure. In this respect, real time Fourier analysis is very useful in process monitoring and failure detection and prediction. Special purpose real time analyzers with the capability of spectrum comparison (often done by an external command) are available for this purpose. Various mechanical deteriorations manifest themselves at specific frequency values. A change in spectrum level at a particular frequency (and its multiples) would indicate a specific type of mechanical degradation or component failure. An example is given in Figure 10.17, which compares the Fourier spectrum at a monitoring location of a vibratory system at the start of test with the Fourier spectrum after some mechanical degradation has taken place. To facilitate spectrum comparison within a narrow frequency band, it is customary to plot such Fourier spectra on a linear frequency axis. It is seen that the overall spectrum levels have increased as a result of mechanical degradation. Also, a significant change has occurred near 30 Hz. This information is useful in diagnosing the cause of degradation or malfunction. Figure 10.17 might indicate, for example, impending failure of a component having resonant frequency close to 30 Hz.

10 26

10.5.9


Cepstrum

A function known as the cepstrum is sometimes used to facilitate the analysis of Fourier spectrum in detecting mechanical degradation. The cepstrum (complex) CðtÞ of a Fourier spectrum Yð f Þ is defined by CðtÞ ¼ F

1

log Yð f Þ

ð10:54Þ

The independent variable t is known as quefrency, and it has the units of time. An immediate advantage of cepstrum arises from the fact that the logarithm of the Fourier spectrum is taken. From Equation 10.33 it is clear that, for a system having frequency transfer function Hð f Þ; and excited by a signal having Fourier spectrum Uð f Þ; the response Fourier spectrum Yð f Þ could be expressed in the logarithmic form: log Yð f Þ ¼ log Hð f Þ þ log Uð f Þ

ð10:55Þ

Since the right hand side terms are added rather than multiplied, any variation in Hð f Þ at a particular frequency will be less affected by a possible low spectrum level in the excitation Uð f Þ at that frequency, when considering log Yð f Þ rather than Yð f Þ: Consequently, any degradation will be more conspicuous in the cepstrum than in the Fourier spectrum. Another advantage of cepstrum is that it is better capable of detecting variation in phenomena that manifest themselves as periodic components in the Fourier spec trum ( for example, harmonics and sidebands). Such phenomena, which appear as repeated peaks in the Fourier spectrum, occur as a single peak in the cepstrum, and so any variations can be detected more easily.

10.6 Other Topics of Signal Analysis In this section, we will briefly address some other important topics of signal analysis. We will start by discussing bandwidth in different contexts then we will present several practically useful analysis procedures and results on vibration signals.

10.6.1

Bandwidth

Bandwidth has different meanings depending on the particular context and application. For example, when studying the response of a dynamic system, the bandwidth relates to the fundamental resonant frequency, and correspondingly to the speed of response for a given excitation. In band pass filters, the bandwidth refers to the frequency band within which the frequency components of the signal are allowed through the filter, the frequency components outside the band being rejected by it. With respect to measuring instruments, bandwidth refers to the range frequencies within which the instrument measures a signal accurately. Note that these various interpretations of bandwidth are somewhat related. For example, if a signal passes through a band pass filter, then we know that its frequency content is within the bandwidth of the filter; but we cannot determine the actual frequency content of the signal through |G( f )|2

|G( f )|

Reference Level

G2 r

G2 /2 r

(a)

0

FIGURE 10.18

Frequency f

(b)

0

Bp

Equivalent Ideal Filter Actual Filter

Be

f

Characteristics of (a) an ideal band pass filter; (b) a practical band pass filter.


10 27

such an observation. In this context, the bandwidth appears to represent a frequency uncertainty in the observation (i.e., the larger the bandwidth of the filter, the less certain the actual frequency content of a signal that is allowed through the filter).

10.6.2

Transmission Level of a Band-Pass Filter

Practical filters can be interpreted as dynamic systems. In fact all physical, dynamic systems (e.g., mechanical structures) are analog filters. It follows that the filter characteristic can be represented by the frequency transfer function Gð f Þ of the filter. A magnitude squared plot of such a filter transfer function is shown in Figure 10.18. In a logarithmic plot the magnitude squared curve is obtained by simply doubling the corresponding magnitude (Bode plot) curve. Note that the actual filter transfer function (Figure 10.18(b)) is not flat like the ideal filter shown in Figure 10.18(a). The reference level Gr is the average value of the transfer function magnitude in the neighborhood of its peak.

10.6.3

Effective Noise Bandwidth

Effective noise bandwidth of a filter is equal to the bandwidth of an ideal filter that has the same reference level and that transmits the same amount of power from a white noise source. Note that white noise has a constant (flat) PSD. Hence, for a noise source of unity PSD, the power transmitted by the practical filter is given by ð1 lGð f Þl2 df 0

which, by definition, is equal to the power effective noise bandwidth Be is given by

G2r Be

Be ¼

10.6.4

ð1 0

lGð f Þl2 df =G2r

ð10:56Þ

Half-Power (or 3 dB) Bandwidth

Half of the power from a unity PSD noise source, as transmitted by an ideal filter, is G2r Br 2: Hence, pffiffi Gr 2 is referred to as the half power level. This is also pffiffi known as a 3 dB level because 20 log10 2 ¼ 10 log10 2 ¼ 3 dB: (Note: 3 dB refers topffiffia power ratio of 2 or an amplitude ratio of 2: Furthermore, 20 dB corresponds to an amplitude ratio of 10 or a power ratio of 100). The 3 dB (or half power) bandwidth corresponds to the width of the filter transfer function at the half power level. This is denoted by Bp in Figure 10.18(b). Note that Be and Bp are different in general. However, in an ideal case where the magnitude squared filter characteristic has linear rise and fall off segments, these two bandwidths are equal (see Figure 10.19).

10.6.5

transmitted by the equivalent ideal filter. Hence, the

|G( f )|2 Gr2

G2r / 2

0 FIGURE 10.19

Be= Bp

Frequency f An idealized filter with linear segments.

Fourier Analysis Bandwidth

In Fourier analysis, bandwidth is interpreted, again, as the frequency uncertainty in the spectral results. In analytical FIT results, which assume that the entire signal is available for analysis, the spectrum is

10 28


continuously defined over the entire frequency range ½ 1; 1 and the frequency increment df is infinitesimally small ðdf ! 0Þ: There is no frequency uncertainty in this case, and the analysis bandwidth is infinitesimally narrow. In digital Fourier transform, the discrete spectral lines are generated at frequency intervals of DF: This finite frequency increment DF; which is the frequency uncertainty, is therefore the analysis bandwidth B for this analysis. Note that DF ¼ 1=T; where T is the record length (or window length for a rectangular window). It follows also that the minimum frequency that has a meaningful accuracy is the bandwidth. This interpretation for analysis bandwidth is confirmed by noting the fact that harmonic components of frequency less than DF (or period greater than T) cannot be studied by observing a signal record of length less than T: Analysis bandwidth carries information regarding distinguishable minimum frequency separation in computed results. In this sense bandwidth is directly related to the frequency resolution of analyzed results. The accuracy of analysis increases by increasing the record length T (or decreasing the analysis bandwidth B). When a time window other than the rectangular window is used to truncate a measured vibration signal, then reshaping of data occurs according to the shape of the window. This reshaping reduces leakage due to suppression of side lobes of the Fourier spectrum of the window. At the same time, however, an error is introduced due to the information lost through data reshaping. This error is proportional to the bandwidth of the window itself. The effective noise bandwidth of a rectangular window is only slightly less than 1=T; because the main lobe of its Fourier spectrum is nearly rectangular. Hence, for all practical purposes, the effective noise bandwidth can be taken as the analysis bandwidth. Note that data truncation (multiplication in the time domain) is equivalent to convolution of the Fourier spectrum (in the frequency domain). The main lobe of the spectrum uniformly affects all spectral lines in the discrete spectrum of the data signal. It follows that a window main lobe having a broader bandwidth (effective noise bandwidth) introduces a larger error into the spectral results. Hence, in digital Fourier analysis, bandwidth is taken as the effective noise bandwidth of the time window that is employed.

10.6.6

Resolution in Digital Fourier Results

Resolution is the frequency separation between spectral lines in digital Fourier analysis results. For a data record of length T; the resolution is DF ¼ 1=T irrespective of the type of window used. There is a noteworthy distinction between analysis bandwidth and resolution. Suppose that we have a data record of length T: If we double the length by augmenting it with trailing zeros, digital Fourier analysis of the resulting record of length 2T will yield a spectral line separation of 1=ð2TÞ: Thus, the resolution is halved. But, unless the true signal value is also zero in the second time interval t½T; 2T ; no new information is presented in the augmented record of duration ½0; 2T in comparison to the original record of duration ½0; T : So, the analysis bandwidth (a measure of accuracy) will remain unchanged. If, on the other hand, the signal itself was sampled over ½0; 2T and the resulting 2N data points were used in digital Fourier analysis, the bandwidth as well as the resolution would be halved. Some relations that are useful in the digital computation of spectral results for signals are summarized in Box 10.3.

10.7 Overlapped Processing Digital Fourier analysis is performed on blocks of sampled data (e.g., 210 ¼ 1024 samples at a time). In overlapped processing, each data block is made to include part of the previous data block that was analyzed. After completing a computation, the overlapped data at the end of the computed block is moved to the beginning of the block, and the leading vacancy is filled with new data so that the end data in one block is identical to the beginning data in the next block, in the overlapped region. In other words, the overlapped portions of each data block (the two end portions) are processed twice. It follows that if there is 50% (or more) overlapping then the entire data block is processed twice. Three main reasons can


10 29

Box 10.3 USEFUL RELATIONS COMPUTATIONS F yðtÞ

DFT ! Yð f Þ ðFFTÞ

FOR

DIGITAL SPECTRAL

Fourier spectrum

1 p Y ð f ÞYð f Þ T

¼ Power spectral density (PSD)

Power spectrum

¼ B £ Power spectral density

¼

Energy spectrum

¼ T £ Power spectrum

¼ BY p ð f ÞYð f Þ

Energy spectral density RMS spectra (always shown for þve frequencies only)

1 £ Energy spectrum B 1=2 2 Ðf þB ¼ lYð f 0 Þl2 df 0 B f ¼

B p Y ð f ÞYð f Þ T

¼ Y p ð f ÞYð f Þ p one sided p like lYðf Þl but smoother p no phase information p increase B ! high bandwidth

Note: T ¼ Record length B ¼ Bandwidth of digital analysis (minimum frequency for which meaningful results are obtained) ! includes window effect Periodic or stationary signals Use power spectra (infinite energy) Transient signals (finite energy) Energy spectra can be used One sided spectrum ¼ 2 £ (þve frequency part of two sided spectrum) Coherent output power ¼ coherence g2uy £ output power ˆ could be power spectrum (spectrum or spectral density) or PSD of the output

be given for using overlapped processing in digital Fourier analysis: 1. It is an effective means of averaging spectral results. 2. It reduces the waiting time for assembling the data buffer. 3. It reduces the error caused by the end shaping effect of time windows (when a window other than the rectangle window is used). From reasons 1 and 2, it is clear that, due to overlapping, the statistical error of computations is reduced for the same speed of computation, and the computing power is more efficiently used. To explain reason 3, let us examine Figure 10.20. This example shows a 50% overlap in data. It is seen that the window function can be assumed to be relatively flat, at least over 50% of the window length (record length). Then the entire data block will correspond to the flat part of the window in three successive analyses. Consequently, the shaping error (or the error due to increased analysis bandwidth) that is caused by a nonrectangular time window is virtually eliminated by overlapped processing. The flatness of a time window is determined by its effective noise bandwidth Be : The effective record length Te is defined as Te ¼

1 Be

ð10:57Þ

10 30


FIGURE 10.20

Overlapped processing of windowed signals.

which provides a measure for the flat segment of the window. The percentage effective record length is given by Te as a percentage of the actual record length T: The degree of overlapping is chosen using the relation %overlap ¼ 100 1

Te T

ð10:58Þ

Example 10.3 For a Hamming window, Be ¼ 1:4=T: Hence, a typical value for the percentage overlap is 100 1

1 1:4

¼ 29%

We might want to use a conservative overlap and even go up to 50% in this case because the window is not quite flat.

10.7.1

Order Analysis

Speed related vibrations in rotation machinery may be analyzed through order analysis. Machinery vibrations under start up (accelerating) and shut down (decelerating) conditions are analyzed in this manner. Orders represent the rotating speed related frequency components in a response signal. The ratio of the response frequency to the rotating speed is termed “order.” Order analysis is done essentially through digital Fourier analysis of a rotating speed related response signal. Practically, this may be accomplished in many ways. The format in which the spectral results are presented will depend on the procedure used in order analysis. Some of the typical formats of data presentation are given below. 10.7.1.1

Speed-Spectral Map

As the rotating speed of a machine is changing in a given range, the Fourier spectrum of the response signal is determined for equal increments of speed. The results are presented as a speed spectral map which is a three dimensional cascade diagram (or waterfall display). The two base axes of the plot are spectral frequency and rotating speed. The third axis gives the spectral magnitude (see Figure 10.21). These types of plots are useful in identifying order related components during start up or coast down conditions. Note that for each speed the frequency band of digital Fourier analysis is kept the same (i.e., fixed sampling rate). Each distinct crest trace denotes an order related resonance. The fact that these traces are almost straight lines indicates the significance of order (the ratio, frequency/rotating speed) in exciting these resonances.


10 31

40.0 Spectrum Magnitude (dB) 0.0

Calibration: 100 mV/g

4000

3000 Speed (rpm) 2000

1.0

Frequency (Hz) FIGURE 10.21

10.7.1.2

1000 400.0

A speed spectral map obtained from order analysis.

Time-Spectral Map

Under variable speed conditions (not necessarily accelerating or decelerating) the response signal is Fourier analyzed at equal increments of time. The results are plotted in a cascade diagram, with frequency and time as the base axes. The third axis again represents the magnitude of the Fourier spectrum (see Figure 10.22). In this case, the crest traces are not necessarily straight, and can change their orientation arbitrarily. This variation in crest orientation is determined by the degree of speed variation. 40.0 Spectrum Magnitude (dB) 0.0

Calibration: 100 mV/g

15.0

10.0

5.0

0.5


Time (s)

0.0 200.0

A time spectral map obtained from order analysis.

10 32

10.7.1.3


Order Tracking

In order tracking, a “tracking frequency multiplier” monitors the rotating speed of the machine (as for a speed spectral map). But, in the present case, the sampling rate of the response signal (for Fourier analysis) is changed in proportion to the rotating speed. Note that, in this manner, the maximum useful frequency (approximately 400/512 £ Nyquist Frequency) is increased as the rotating speed increases, so that the aliasing effects are reduced. If the same sampling rate is used for high speeds (as in the Speed Spectral Map discussed above), aliasing error can be significant at high rotating speeds. In presenting order tracking spectral results, the frequency axis is typically calibrated in orders. Both speed spectral maps and time spectral maps may be presented in this manner. Other types of data presentation may be used as well in order analysis. For example, instead of the Fourier spectrum of the response signal, power spectrum or composite power spectrum (in which the total signal power is computed in specified frequency bands and presented as a function of the rotating speed) may be used in the schemes described in this section. Order analysis provides information on most severe operating speeds with respect to vibration (and dynamic stress). For example, suppose that, for a given speed of operation, two major resonances occur, one at 10 Hz and the other at 80 Hz. Then, the structure of the system (rotating machine and its support fixtures) should be modified to change and preferably damp out these resonances. Furthermore, the most desirable operating speed can be chosen in terms of the lowest resonant peaks by observing a speed spectral map.

Bibliography Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, Wiley Interscience, New York. Brigham, E.O. 1974. The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ. Broch, J.T. 1980. Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum. de Silva, C.W. 1983. Dynamic Testing and Seismic Qualification Practice, D.C. Heath and Co., Lexington, MA. de Silva, C.W., Optimal estimation of the response of internally damped beams to random loads in the presence of measurement noise, J. Sound Vib., 47, 4, 485 493, 1976. de Silva, C.W., The digital processing of acceleration measurements for modal analysis, Shock Vib. Dig., 18, 10, 3 10, 1986. de Silva, C.W. 2000. Vibration Fundamentals and Practice, CRC Press, Boca Raton, FL. de Silva, C.W. 2004. Mechatronics An Integrated Approach, CRC Press, Boca Raton, FL. Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd, Letchworth. MATLAB Control Systems Toolbox, The Math Works, Inc., Natick, MA, 2004. Meirovitch, L. 1980. Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, Rockville, MD. Randall, R.B. 1977. Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum.

11

Wavelets — Concepts and Applications 11.1 Introduction ....................................................................... 11.2 Time Frequency Analysis ................................................ Gabor Transform Families

Pol D. Spanos Rice University

Giuseppe Failla Universita` degli Studi Mediterranea di Reggio Calabria

Nikolaos P. Politis Rice University

†

Wavelet Transform

†

Wavelet

11.3 Time Dependent Spectra Estimation of Stochastic Processes ............................................................................. 11.4 Random Field Simulation ................................................ 11.5 System Identification ......................................................... 11.6 Damage Detection ............................................................. 11.7 Material Characterization ................................................. 11.8 Concluding Remarks .........................................................

11 1 11 2

11 11 11 11 11 11

11 14 15 17 18 19

Summary Section 11.1 provides a brief introduction to wavelet concepts in vibration related applications. Aspects of time frequency analysis are discussed in Section 11.2. Specifically, the Gabor and wavelet transforms are outlined. Further, several wavelet families commonly used in vibration related applications are presented. Estimation of time dependent spectra of stochastic processes is considered in Section 11.3. Section 11.4 to Section 11.7 discuss applications of wavelet analysis in vibration related applications. In particular, applications in random field simulation, system identification, damage detection, and material characterization are examined. Section 11.8 provides an overview and concluding remarks on the applicability and usefulness of the wavelet analysis in vibration theory. To enhance the usefulness of this chapter, a list of readily available references in the form of books and archival articles is provided.

11.1

Introduction

Wavelets based representations offer an important option for capturing localized effects in many signals. This is achieved by employing representations via double integrals (continuous transforms), or via double series (discrete transforms). Seminal to these representations are the processes of scaling and shifting of a generating (mother) function. Over a period of several decades, wavelet analysis has been set on a rigorous mathematical framework and has been applied to quite diverse fields. Wavelet families associated with specific mother functions have proven quite appropriate for a variety of problems. In this context, fast decomposition and reconstruction algorithms ensure computational efficiency, and rival classical spectral analysis algorithms such as the fast Fourier transform (FFT). The field of vibration analysis has benefited from this remarkable mathematical development in conjunction with vibration monitoring, system identification, damage detection, and several other tasks. There is a voluminous body 11 1

11 2


of literature focusing on wavelet analysis. However, this chapter has the restricted objective of, on one hand, discussing concepts closely related to vibration analysis, and on the other hand, citing sources that can be readily available to a potential reader. In view of this latter objective, almost exclusively books and archival articles are included in the list of references. First, theoretical concepts are briefly presented; for more mathematical details, the reader may consult references [1 23]. Next, the theoretical concepts are supplemented by vibration analysis related sections on time varying spectra estimation, random field synthesis, structural identification, damage detection, and material characterization. It is noted that most of the mathematical developments pertain to the interval [0,1] relating to dimensionless independent variables derived by normalization with respect to the spatial or temporal “lengths” of the entire signals.

11.2

Time–Frequency Analysis

A convenient way to introduce the wavelet transform is through the concept of time frequency representation of signals. In the classical Fourier theory, a signal can be represented either in the time or in the frequency domain, and the Fourier coefficients define the average spectral content over the entire duration of the signal. The Fourier representation is appropriate for signals that are stationary, in terms of parameters which are deemed important for the problem in hand, but becomes inadequate for nonstationary signals, in which important parameters may evolve rapidly in time. The need for a time frequency representation is obvious in a broad range of physical problems, such as acoustics, image processing, earthquake and wind engineering, and a plethora of others. Among the time frequency representations available to date, the wavelet transform has unique features in terms of efficacy and versatility. In mathematical terms, it involves the concept of scale as a counterpart to the concept of frequency in the Fourier theory. Thus, it is also referred to as time scale representation. Its formulation stems from a generalization of a previous time frequency representation, known as the Gabor transform. For completeness, and to underscore the significant advantages achieved by the development of the wavelet transform, the Gabor transform is briefly discussed in Section 11.2.1. Section 11.2.2 is entirely devoted to the wavelet transform, and the most commonly used wavelet families are described in Section 11.2.3.

11.2.1

Gabor Transform

The first steps in time frequency analysis trace back to the work of Gabor [24], who applied in signal analysis fundamental concepts developed in quantum mechanics by Wigner a decade earlier [25]. Given a function f ðtÞ belonging to the space of finite energy one dimensional functions, denoted by L 2(R), Gabor introduced the transform ð1 f ðtÞgðt t0 Þ e iv ðt t0 Þ dt ð11:1Þ Gf ðv; t0 Þ ¼ 1

where gðtÞ is a window and the bar ð Þ denotes complex conjugation. This transform, generally referred to as the continuous Gabor transform (CGT) or the short time Fourier transform of f ðtÞ; is a complete representation of f ðtÞ: That is, the original function f ðtÞ can be reconstructed as ð1 ð1 1 f ðtÞ ¼ ð11:2Þ Gf ðv; t0 Þgðt t0 Þeivðt t0 Þ dv dt0 2 1 1 2p g Ð 2 where g ¼ 11 gðtÞ 2 dt: The Gabor transform (Equation 11.1) may be seen as the projection of f ðtÞ onto the family {gðv;t0 Þ ðtÞ; v; t0 [ R} of shifted and modulated copies (atoms) of gðtÞ expressed in the form gðv;t0 Þ ðtÞ ¼ eivðt

t0 Þ

gðt

t0 Þ

ð11:3Þ

These time frequency atoms, also referred to as Gabor functions, are shown in Figure 11.1 for three different values of v: Clearly, if gðtÞ is an appropriate window function, then Equation 11.1 may be regarded as the standard Fourier transform of the function f ðtÞ; localized at the time t0 : In this context, t0 is the

Wavelets

Concepts and Applications

11 3

w1

w2

w3

FIGURE 11.1 Plots of Gabor function gðv; t0 Þ versus the independent variable x for three values of the frequency v; the effective support is the same for the three values of the frequency.

time parameter which gives the center of the window, and v is the frequency parameter which is used to compute the Fourier transform of the windowed signal. As intuition suggests, the accuracy of the CGT representation (Equation 11.2) of f ðtÞ depends on the window function gðtÞ; which must exhibit good localization properties both in the time and the frequency domains. As discussed in Ref. [6], a measure of the localization properties may be obtained by the average and the standard deviation of the density gðtÞ 2 in the time domain. That is, ð1 2 ktl ¼ t gðtÞ dt ð11:4aÞ

s2t ¼

ð1 1

1

ðt

2

ktlÞ2 gðtÞ dt

The counterparts of Equation 11.4a and Equation 11.4b in the frequency domain are ð1 ^ vÞ 2 dv kvl ¼ v Gð

s2v ¼

ð1 1

1

ðv

2

^ vÞ dv kvlÞ2 Gð

^ vÞ denotes the Fourier transform of gðtÞ given by the equation where Gð ð1 ^ vÞ ¼ p1ffiffiffiffi gðtÞe ivt dt Gð 2p 1

ð11:4bÞ

ð11:5aÞ ð11:5bÞ

ð11:6Þ

The well known Heisenberg uncertainty principle is in actuality a mathematically proven property and states that the values st and sv cannot be independently small [6]. Specifically, for an arbitrary 2 window normalized so that g ¼ 1; it can be shown that

st sv $

1 2

ð11:7Þ

Thus, high resolution in the time domain (small value of st ) may be generally achieved only at the expense of a poor resolution (bigger than a minimum value sv ) in the frequency domain and vice versa. Note that the optimal time frequency resolution, that is st sv ¼ 1=2; may be attained when the Gaussian window ! 1 t2 gðtÞ ¼ pffiffiffiffiffiffiffi exp ð11:8Þ 4 2ps2 4st2 t is selected. Clearly, as a time frequency representation, the Gabor transform exhibits considerable limitations. The time support, governed by the window function gðtÞ; is equal for all of the Gabor functions (Equation 11.3) for all frequencies (see Figure 11.1). In order to achieve good localization of high frequency components, narrow windows are required; as a result of that, low frequency components are poorly represented. Thus, a more flexible representation with nonconstant windowing is quite desirable,

11 4


to enhance the time resolution for short lived high frequency phenomena and frequency resolution for long lasting low frequency phenomena.

11.2.2

Wavelet Transform

The preceding shortcomings of the Gabor transform have been overcome with significant effectiveness and efficiency by wavelets based signal representation. Its two formulations, continuous and discrete, are described in the ensuing sections. Because of the numerous applications of wavelets beyond time frequency analysis, the t time domain will be replaced by a generic x space domain. For succinctness, the formulation will be developed for scalar functions only, but generalization for multidimensional spaces is well established in the literature [1 22]. 11.2.2.1

Continuous Wavelet Transform

The concept of wavelet transform was introduced first by Goupillaud et al. for seismic records analysis [26,27]. In analogy to the Gabor transform, the idea consists of decomposing a function f ðxÞ into a two parameter family of elementary functions, each derived from a basic or mother wavelet, c ðxÞ: The first parameter, a; corresponds to a dilation or compression of the mother wavelet that is generally referred to as scale. The second parameter, b; determines a shift of the mother wavelet along the x domain. In mathematical terms 1 ð1 x b f ðxÞc dx Wf ða; bÞ ¼ pffiffi a 1 a

ð11:9Þ

where a [ Rþ ; b [ R: In the literature, Equation 11.9 is generally referred to as continuous wavelet transform (CWT). Note that the factor a 1=2 is a normalization factor, included to insure that the mother wavelet and any dilated wavelet a 1=2 c ðx=aÞ have the same total energy [26]. Clearly, alternative normalizations may also be chosen [1]. An example of wavelet functions is shown in Figure 11.2 for different values of the scale parameter a: As a result of scaling, all the wavelet functions exhibit the same number of cycles within the x support of the mother wavelet. Obviously, the spatial and frequency localization properties of the wavelet transform depend on the value of the parameter a: As a approaches zero, the dilated wavelet a 1=2 cðx=aÞ is highly concentrated at the point x ¼ 0; the wavelet transform, Wf ða; bÞ; then gives increasingly sharper spatial resolution displaying the small scale/higher frequency features of the function f ðxÞ; at various locations b: However, as a approaches þ 1, the wavelet transform Wf ða; bÞ gives increasingly coarser spatial resolution, displaying the large scale/low frequency features of the function f ðxÞ: For the function f ðxÞ to be reconstructable from the set of coefficients (Equation 11.9), in the form f ðxÞ ¼

a1

1 ð1 ð1 da Wf ða; bÞca;b ðxÞ 2 db pcc 0 a 1

a2

ð11:10Þ

a3

FIGURE 11.2 Plots versus time of wavelet functions corresponding to three different values of a scale a of the same mother function; the effective time support increases with the magnitude of the scale.

Wavelets


where ca;b ðxÞ ¼ a

1=2

c½ðx

11 5

bÞ=a ; the wavelet function cð·Þ must satisfy the admissibility condition cc ¼

ð 1 C^ ðvÞ 2 dv , 1 v 1

ð11:11Þ

where C^ ðvÞ denotes the Fourier transform of cðxÞ: As pointed out in Ref. [26], the condition 11.11 includes a set of subconditions, such as: 1. The analyzing wavelet cð·Þ is absolutely integrable and square integrable. That is, ð1 cðxÞ dx , 1 1

ð1 1

2

cðxÞ dx , 1

ð11:12aÞ ð11:12bÞ

2. The Fourier transform C^ ðvÞ must be sufficiently small at the vicinity of the origin v ¼ 0; or in mathematical terms ð 1 C^ ðvÞ dv , 1 v 1

ð11:13Þ

Ð Subcondition 2, then, implies that C^ ð0Þ ¼ 0; that is, 11 cðxÞdx ¼ 0: Therefore, for an analyzing wavelet to be admissible, its real and imaginary parts must both be symmetric with respect to the x axis. From the reconstruction formula (Equation 11.10), it can be shown that 1 ð1 ð1 2 da 2 ð11:14Þ Wf ða; bÞ 2 db f ¼ pcc 0 a 1 Based on Equation 11.14, the square modulus of the wavelet transform (Equation 11.9) is often taken as an energy density in a spatial scale domain. Extensive use of this concept has been made for spectra estimation purposes, as discussed in Section 11.3. Note that the reconstruction wavelet in Equation 11.10 can be different from the analyzing wavelet used in Equation 11.9. That is, under some admissibility conditions on xðxÞ [1], the original function f ðxÞ may be reconstructed as 1 ð1 ð1 da f ðxÞ ¼ Wf ða; bÞxa;b ðxÞ 2 db ð11:15Þ ccx 0 a 1 where xa;b ðxÞ ¼ a 1=2 x½ðx bÞ=a and ccx is a constant parameter depending on the Fourier transforms of both cðxÞ and xðxÞ: This property, referred to as redundancy in mathematical terms, may be advantageous in some applications for reducing the error due to noise in signal reconstruction [28,29], but highly undesirable for signal coding or compression purposes [1]. Further, under certain conditions [1], the following simplified reconstruction formula holds 1 ð1 da f ðxÞ ¼ ð11:16Þ W ða; xÞ 3=2 kc 0 f a where kc is a constant parameter given by the equation pffiffiffiffi ð 1 C^ ðvÞ k c ¼ 2p dv v 0

ð11:17Þ

Use of this formula has been made, in a discrete version, in the approximation theory of functional spaces [1] and also in structural identification applications, as discussed in Section 11.5.

11 6

11.2.2.2


Discrete Wavelet Transform

For numerical applications, where fast decomposition or reconstruction algorithms are generally required, a discrete version of the CWT is to prefer. In this sense, a natural way to define a discrete wavelet transform (DWT) is 1 ð1 j f ðxÞcða0 x kb0 Þdx; j; k [ Z ð11:18Þ Wf ð j; kÞ ¼ qffiffiffi j a0 1 Equation 11.18 is derived from a straightforward discretization of the CWT (Equation 11.9) by j j considering the discrete lattice a ¼ a0 ; a0 . 1; b ¼ kb0 a0 ; b0 – 0: In developing Equation 11.18, however, the main mathematical concern is to ensure that sampling the CWT on a discrete set of points does not lead to a loss of information about the wavelet transformed function f ðxÞ: Specifically, the original function f ðxÞ must be fully recoverable from a discrete set of wavelet coefficients. That is, X f ðxÞ ¼ Wf ðj; kÞcj;k ðxÞ ð11:19Þ j;k[Z

j=2

j

where cj;k ðxÞ ¼ a0 cða0 x kb0 Þ. Another crucial aspect in Equation 11.18 involves selecting the wavelet functions cj;k ðxÞ such that Equation 11.19 may be regarded as the expansion of f ðxÞ in a basis, thus eliminating the redundancy of the CWT. This issues are addressed by using the theory of Hilbert space frames, introduced in 1952 by Duffin and Schaeffer in context with non harmonic Fourier series [30]. In general, if hl ðxÞ [ L2 ðRÞ and L is a countable set, a family of functions {hl ðxÞ; l [ L} constitutes a frame, if for any f ðxÞ [ L2 ðRÞ X 2 2 2 ð11:20Þ kf ; hl l # B f A f # Ð1

l[L

where k f ; hl l ¼ 1 f ðxÞhl ðxÞ dx and A . 0; B , 1; the so called frame bounds, are independent of f ðxÞ [1]. The concept of frame may be interpreted as an extension of the concept of basis, in the sense that the reconstruction of the original function is possible via stable numerical expressions in terms of the set {hl ðxÞ; l [ L}: For instance, if the frame is tight, that is A ¼ B; the simple formula f ðxÞ ¼

X l[L

k f ; hl lhl ðxÞ

ð11:21Þ

holds [29]. In contrast to a basis, however, the vectors of a frame may be linearly dependent and, for this, a certain degree of redundancy is still retained in the reconstruction formula (Equation 11.21) [29,31]. The concept of frame has played a crucial role in the formulation of the DWT. The first wavelet frames were constructed by Daubechies et al. [32]. Later, Battle [33] constructed orthonormal bases with an exponential decay. The ensemble of these results has demonstrated the advantages of the wavelet transform over the Gabor transform. In fact, it has been shown that discrete versions of Gabor transform are not capable of generating orthonormal bases [32] due to the so called Balian Low phenomenon [1]. Mallat [34] has shown that the orthonormal wavelet bases proposed by Battle can all be derived by a multiresolution analysis. The latter involves representing an arbitrary f ðxÞ [ L2 ðRÞ as a limit of successive approximations, at different resolutions. That is, if {Vj }j[Z is a sequence of subspaces of L2 ðRÞ; and fj is the orthogonal projection of f ðxÞ on Vj ; in a multiresolution analysis the following conditions hold lim fj ¼ f

ð11:22aÞ

lim fj ¼ 0

ð11:22bÞ

j! 1 j !1

Each approximation fj ; then, represents a smoothed version of f ðxÞ and, in the limit, more and more localized smoothing functions lead to the function f ðxÞ: From a mathematical point of view [29,31], a

Wavelets


11 7

multiresolution analysis requires that 1. The subspaces Vj ’s are closed and embedded, that is · · · , V2 , V1 , V0 , V

1

,V

2

, ···

ð11:23Þ

where V m ! L2 ðRÞ for m ! 1 and f [ Vm , f ð2·Þ [ Vm 1 : 2. A scaling function fðxÞ [ L2 ðRÞ exists, such that, for each j; the family of functions

fj;k ðxÞ ¼ 2

j=2

fð2 j x

kÞ; k [ Z

ð11:24Þ

spans the subspace Vj and constitutes a Riesz basis for Vj ; that is, there exists 0 , C 0 # C 00 , 1 such that 2 X 2 ð1 X X 2 C0 ck # ck fj;k ðxÞ dx # C 00 ck ð11:25Þ 1

k

k

k

for all sequences of numbers ðck Þk[Z : Equation 11.25 is a more stringent condition of Equation 11.20 and includes the latter as a special case. The concept of multiresolution analysis offers a straightforward and mathematically coherent approach to discrete wavelet analysis. Given a scaling function fðxÞ as in 2, in fact, families of orthonormal wavelet bases

cj;k ðxÞ ¼ 2

j=2

cð2 j x

kÞ; j; k [ Z

ð11:26Þ

can be developed by appropriate algorithms. For this, Mallat has used the frequency response of a high pass filter [35], while Daubechies has devised a systematic approach to build orthonormal wavelet bases with compact support in the x domain [36]. Specifically, for each even integer 2M; the Daubechies scaling function fðxÞ can be computed as X1 pffiffi 2M fðxÞ ¼ 2 hkþ1 fð2x

kÞ

ð11:27Þ

k¼0

where hk ’s are 2M coefficients obtained by imposing M orthogonality conditions and M accuracy conditions to enhance the rate of convergence of the approximation to the original function f ðxÞ: In turn, the mother wavelet is related to the scaling function fðxÞ by the equation X1 pffiffi 2M cðxÞ ¼ 2 gkþ1 fð2x

kÞ

ð11:28Þ

k¼0

where gk ’s are the same as hk ’s but reversed in order and with alternate signs. Numerical values of both series hk ’s and gk ’s are readily available in the literature [16]. Also based on multiresolution analysis concepts, a wavelet decomposition algorithm for image analysis has been developed [34,35]. If associated to Daubechies wavelets, the algorithm becomes quite efficient from a computational point of view, since no numerical integration is involved to compute wavelet and scale coefficients. It relies on the projection of f ðxÞ onto a sufficiently fine scale j of the set 11.24. That is, X j f ðxÞ < fj ðxÞ ¼ ck fj;k ðxÞ ð11:29Þ k

where, for orthogonal wavelets, j

ck ¼

ð1 1

f ðxÞfj;k ðxÞdx

ð11:30Þ

Based on Equation 11.22a and Equation 11.22b, the projection fj ðxÞ can be rewritten in terms of the projection fjþ1 ðxÞ onto the coarser scale ð j þ 1Þ and the incremental detail djþ1 ðxÞ; that is the pieces of

11 8


information contained in the subspace Vj and lost when “moving” to the subspace Vjþ1 : Therefore, fj ðxÞ ¼ fjþ1 ðxÞ þ djþ1 ðxÞ ¼ fjþl ðxÞ þ djþ1 ðxÞ þ · · · þ djþl ðxÞ < djþ1 ðxÞ þ · · · þ djþl ðxÞ

ð11:31Þ

As a fundamental result of multiresolution analysis, the details dj ðxÞ can be decomposed in terms of the set of wavelet functions at the same scale. That is, X j dk cj;k ðxÞ ð11:32Þ dj ðxÞ ¼ k

j where dk ’s are the wavelet coefficients of f ðxÞ: Based on Equation

can be computed recursively by the closed form expressions j

ck ¼ j

dk ¼

2M X1 l¼0 2M X1 l¼0

j 1

hlþ1 c2kþl j 1

glþ1 c2kþl

11.28, both wavelet and scale coefficients

1

ð11:33aÞ

1

ð11:33bÞ

Similarly, the reconstruction algorithm can be implemented by the formula X j 1 j j hk 2lþ2 cl þ gk 2lþ2 dl ck ¼

ð11:34Þ

l

The reconstruction algorithm described by Equation 11.34 lends itself to interpretation as a scale linear system [37,38]. Based on this concept, applications have also been developed for random field simulation [39].

11.2.3

Wavelet Families

A great number of wavelet families with various properties are available. Selecting an optimal family for a specific problem is not, in general, an easy task and there are properties that prove more important to certain fields of application. For instance, symmetry may be of great help for preventing dephasing in image processing, while regularity is critical for building smooth reconstructed signals or accurate nonlinear regression estimates. Compactly supported wavelets, either in the time or in the frequency domain, may be preferable for enhanced time or frequency resolution. The number of vanishing moments, M; that is the highest integer m for which the equation ð1 xm c ðxÞdx ¼ 0; m ¼ 0; 1; …; M 1 ð11:35Þ 1

holds is important in signal processing for compression, or in damage detection for enhancement of singularities in the vibration modes. Also, in some cases, wavelets may be required to be progressive. In mathematical terms, this means that their Fourier transform is defined only for positive frequencies. That is,

C^ ðvÞ ¼ 0; for v , 0

ð11:36Þ

The progressive wavelet transform of a real valued signal f ðtÞ and the associated analytic signal zf ðtÞ ¼ f ðtÞ þ iH½f ðtÞ

ð11:37Þ

are related by the equation Wf ða; bÞ ¼

1 W ða; bÞ 2 zf

ð11:38Þ

where H½· denotes the Hilbert transform operator [40]. Equation 11.38 is quite useful for structural identification. Note also that significant reduction of computational costs is generally achieved if orthogonal wavelets in the frequency or in the x domain are used.

Wavelets


11 9

A brief description of the most used families is given below. A distinction is made between real and complex wavelets, and the most relevant properties for application purposes are discussed. A more exhaustive review may be in found in Ref. [15]. 11.2.3.1

Real Wavelets

1. Daubechies orthonormal wavelets A family of bases, each corresponding to a particular value of the parameter M in Equation 11.27 and Equation 11.28 [36]. Closed form expressions for fðxÞ in Equation 11.27 are available only for M ¼ 1; to which the well known Haar basis corresponds. In this case, the scaling function and the mother wavelet are 8 1 > > 1; 0 # x , ; > > ( 2 > < 1; 0 # x , 1; 1 ð11:39Þ fðxÞ ¼ cðxÞ ¼ > 1; # x , 1; > 0; elsewhere; > 2 > > : 0; elsewhere: Various algorithms, however, are available in the literature for determining fðxÞ and cðxÞ numerically for M . 1: Daubechies wavelets support both CWT and DWT, although the latter is most generally performed due to the fast decomposition and reconstruction algorithm mentioned in Section 11.2.2.2. Both fðxÞ and cðxÞ are compactly supported in the x domain, and the support is equal to the segment ½0; 2M 1 : Also, M is equal to the number of vanishing moments of the wavelet function. Note that most Daubechies wavelets are not symmetric; regularity and harmonic like shape increases with M: 2. Meyer wavelets Families of wavelets [10], each defined for a particular choice of an auxiliary function vðvÞ which appears in the following expression for the Fourier transform of the mother wavelet: 8 1 3 p > > pffiffiffiffi eiv=2 sin v v > > 2 2 p 2 p > > > < 1 3 p C^ ðvÞ ¼ pffiffiffiffi eiv=2 cos v v > 4p 2 2 p > > > > > > : 0;

1

;

2 4 p # v # p; 3 3

1

;

4 8 p # v # p; 3 3

v

2 8 p; p ; 3 3

for vðvÞ to be an admissible auxiliary function it is required that ( 0; v # 0; vðvÞ ¼ 1; v $ 1; vðvÞ þ vð1

ð11:40Þ

vÞ ¼ 1; 0 # v # 1

ð11:41aÞ ð11:41bÞ

The most common form of vðvÞ in the literature is vðvÞ ¼ v4 ð35

84v þ 70v2

20v3 Þ; 0 # v # 1

ð11:42Þ

The mother wavelet, for which only numerical expressions are available, is then constructed by inverse Fourier transforming Equation 11.40. Meyer wavelets are suitable for both CWT and DWT. Unlike Daubechies wavelets, they are compact in the frequency domain but not in the x domain. Because of their fast decay, however, an effective x support [ 8,8] is generally assumed. Appealing features of Meyer wavelets are orthogonality, infinite regularity, and symmetry. 3. Mexican Hat wavelets A family of wavelets in the x domain [15] related to a mother function that is proportional to the second derivative of the Gaussian probability density function.

11 10


That is, 2 cðxÞ ¼ pffiffi p 3

1=4

ð1

x2 Þe

x2 =2

ð11:43Þ

The Mexican Hat wavelets allow CWT only. Unlike Daubechies or Meyer wavelets, Mexican Hat wavelets are not compact both in the frequency and in the x domain, although an effective support [ 5,5] may be considered for practical calculations. They are infinitely regular and symmetric. 4. Biorthogonal wavelets Families of wavelets derived by generalizing the ordinary concept of wavelet bases, and creating a pair of dual wavelets, say ðc ðxÞ; c~ ðxÞÞ; satisfying the following properties [41,42]: ð1 cj;k ðxÞc~j0 ;k0 ðxÞdx ¼ djj0 dkk 0 ð11:44Þ 1

where the symbol dmn denotes the Kronecker delta. One wavelet, say cðxÞ; may be used for reconstruction and the dual one, c~ðxÞ; for decomposition. Therefore, Equation 11.18 and Equation 11.19 can be rewritten as ð1 Wf ð j; kÞ ¼ 2 j=2 f ðxÞcj;k ð2 j x kÞdx; j; k [ Z ð11:45Þ 1

f ðxÞ ¼

X

j;k[Z

Wf ð j; kÞc~j;k ðxÞ

ð11:46Þ

Biorthogonal wavelets support both CWT and DWT. The properties of a biorthogonal basis are specified in terms of a pair of integers ðNd ; Nr Þ: These integers, in analogy with the Daubechies wavelets, govern the regularity and the number of vanishing moments Nd of the decomposition wavelet c ðxÞ; and the regularity and the number of vanishing moments Nr of the reconstruction wavelet c~ ðxÞ: Obviously, this feature allows a greater number of choices for signal decomposition and reconstruction. Both wavelet functions c ðxÞ and c~ ðxÞ are symmetric. 11.2.3.2

Complex Wavelets

5. Harmonic wavelets [16,43,44]:

A Family of bases defined in the frequency domain by the formula

C^ m;n ðvÞ ¼

8 > < > :

1 ; 2p ðn mÞ

mp # v # np;

0;

elsewhere;

ð11:47Þ

where m and n are positive numbers but not necessarily integers. The pair of values m, n is referred to as level m, n and represents, for harmonic wavelets, the scale index j: A harmonic wavelet basis thus corresponds to a complete set of adjacent levels m, n, spanning all the positive frequency axis. By inverse Fourier transforming Equation 11.47, the corresponding wavelet functions at a generic step k on the x domain take the complex form:

cm;n;k ðxÞ ¼

exp in2p x

k n i2p ðn

exp im2p x

m mÞ x

k n

k n

m

ð11:48Þ

m

A common choice for the pairs m; n is m; n ¼ 0; 1; 2; 4; …; 2j ; 2jþ1 ; … . In this case, all the wavelets have octave bands, except for the first one. Harmonic wavelets have been devised in context with a DWT, for which extremely fast decomposition and reconstruction algorithms exist. They exhibit a compact support in the frequency domain (see Equation 11.47), while in the x domain their rate of decay away from

Wavelets


11 11

the wavelet’s center is relatively low and proportional to x 1 : Further, they satisfy relevant orthogonality properties [16]. From Equation 11.48, it is seen that the real part of the wavelet is even, while the imaginary part is odd. For signal processing, this ensures that harmonic components in a signal can be detected regardless of the phase. Note that this feature cannot be achieved by real wavelets such as the Meyer wavelets, which are all self similar, being derived from a unique mother wavelet by scaling and shifting. Also, note that orthonormal basis of real wavelets can be generated by considering either the real or the imaginary parts only of Equation 11.48. For instance, the well known Shannon wavelets correspond to the imaginary parts of Equation 11.48, for m; n ¼ 1; 2; 2; 4; 4; 8; … : Harmonic wavelets are used in many mechanics applications such as acoustics, vibration monitoring, and damage detection [45 49]. 6. Complex Gaussian wavelets Families of wavelets, each corresponding to a pth order derivative of a complex Gaussian function. That is,

cp ðxÞ ¼ Cp

dp ðe dx p

ix

e

x2 =2

Þ; p ¼ 1; 2; …

ð11:49Þ

where Cp is a normalization constant such that cðxÞ 2 ¼ 1: Complex Gaussian wavelets support the CWT only. They have no finite support in the x domain, although the interval [ 5,5] is generally taken as effective support. Despite their lack of orthogonality, they are quite popular in image processing applications due to their regularity [1]. 7. Complex Morlet wavelets Families of [50], each obtained as the derivative of the classical Morlet 2 wavelet c0 ðxÞ ¼ e x =2 eiv0 x : That is,

cp ðxÞ ¼

dp ðe dxp

x2 =2

eiv0 x Þ; p ¼ 1; 2; …:

ð11:50Þ

Except for c0 ðxÞ; which does not satisfy the admissibility condition (Equation 11.11) in a strict sense, all the other members of the family are proper wavelets. For practical purposes, however, c0 ðxÞ is generally considered admissible for v0 $ 5: Complex Morlet wavelets support the CWT only and are not orthogonal. However, they are all progressive, that is, they satisfy the condition posed by Equation 11.36. Further, for the Morlet wavelet c0 ðxÞ; there exists a relation between the scale parameter a and the frequency v at which its Fourier transform focuses. That is, v ð11:51Þ a¼ 0 v Complex Morlet wavelets are then applied for structural identification purposes, as shown in Section 11.5.

11.3

Time-Dependent Spectra Estimation of Stochastic Processes

Wavelets based approaches are significant tools for joint time frequency analysis of problems related to vibrations of mechanical and structural systems. This applies both to the characterization of the system excitation, the system identification, and the system response determination. Several examples exist in nature of stochastic phenomena with a time dependent frequency content. The frequency content of earthquake records, for instance, evolves in time due to the dispersion of the propagating seismic waves [51,52]. Further, sudden changes in the wave frequency at a given location of the sea surface are often induced by fast moving meteorological fronts [53]. Also, a rapid change in the frequency content is generally associated with waves at the breaking stage. Similarly, turbulent gusts of time varying frequency content are often embedded in wind fields. Appropriate description of such phenomena is obviously crucial for design and reliability assessments. In an early attempt, concepts of traditional Fourier spectral theory were generalized to provide spectral estimates, such as the Wigner Ville method [25,54] or the CGT of Equation 11.1. However, it soon

11 12


became clear that the extension of the traditional concept of a spectrum is not unique, and proposed time varying spectra could have contradictory properties [6,55]. Wavelet analysis is readily applicable for estimating time varying spectral properties, and a significant effort has been devoted to formulating “wavelet energy principles” that work as alternatives to classical Fourier methods. Measures of a time varying frequency content were first obtained by “sectioning,” at different time instants, the wavelet coefficients mean square map [49,56 58]. Developing consistent spectral estimates from such sections, however, is not straightforward. From a theoretical point of view, it either requires an appropriate wavelet based definition of time varying spectra, or it must relate to well established notions of time varying spectra. From a numerical point of view, it involves certain difficulties in converting the scale axis to a frequency axis, especially when the wavelet functions are not orthogonal in the frequency domain; that is, when the frequency content of wavelet functions at adjacent scales do overlap. Early investigations on wavelet based spectral estimates may be found in references such as [44,59 64], where wavelet analysis was applied in the context of earthquake engineering problems. In a particular approach, a modified Littlewood Paley (MLP) wavelet basis can be introduced, whose mother wavelet is defined in the frequency domain by the equation 8 1 > < pffiffiffiffiffiffiffiffiffiffiffiffiffi ; p # v # sp; ^ 2ð s 1Þp ð11:52Þ CðvÞ ¼ > : 0; elsewhere In Equation 11.52, the symbol s denotes a scalar factor, to be adjusted depending on the desired frequency resolution. The MLP wavelets are orthogonal in the frequency domain, that is, wavelets at adjacent scales span nonoverlapping intervals. The MLP wavelets have been used in conjunction with a discretized version of the CWT proposed by Alkemade [65] for a finite energy process f ðtÞ X KDb f ðtÞ ¼ ð11:53Þ Wf ðaj ; bi Þcaj ;bi ðtÞ aj i; j where aj ¼ s j ; Db is a time step, and K is a constant parameter depending on s: In many instances, Equation 11.53 can be construed as representing realizations of a stochastic process, and in this case, the following estimate of its instantaneous mean square value of f ðtÞ has been constructed E½f 2 ðtÞ

t¼bi

¼K

X E½Wf ðaj ; bi Þ aj j

2

ð11:54Þ

where E½· is the mathematical expectation operator over the ensemble of realizations. From Equation 11.54, and based on the orthogonality properties of the MLP wavelets, the following quantity Sf ðvÞ t¼bi ¼

X j

K

E½Wf ðaj ; bi Þ aj

2

2 C^ aj ;bi ðvÞ

ð11:55Þ

where the symbol C^ aj ; bi ðvÞ denotes the Fourier transform of the wavelet function caj ; bi ðtÞ, can be taken as a measure of the time varying power spectral density (PSD) of the process f ðtÞ: Based on Equation 11.55, closed form expressions can be derived between the input and the output PSDs [63]. In this context, linear response statistics, such as the instantaneous rate of crossings of the zero level or the instantaneous rate of occurrence of the peaks, have been estimated with considerable accuracy. Analysis of nonlinear systems has also been attempted by an equivalent statistical linearization procedure [61,66]. Wavelet analysis for spectral estimation has also been pursued by Kareem et al., who have used the squared wavelet coefficients of a DWT to estimate the PSD of stationary processes [56]. To improve the frequency resolution of the DWT, where only adjacent octave bands can be accounted for, a CWT can be implemented based on a complex Morlet wavelet basis. The latter is preferable due to the one to one

Wavelets


11 13

correspondence between the scale a and the center frequency (Equation 11.51), which allows minimizing the overlap between spectral estimates at adjacent scales. Further, the product of wavelet coefficients can be used as a measure of the cross correlation between two nonstationary signals xðtÞ and yðtÞ [56]. This concept can be refined by the introduction of a wavelet coherence measure [57,58] expressed by the equation 2

SW xy ða; bÞ ðc ða; bÞÞ ¼ W Sxx ða; bÞSW yy ða; bÞ W

2

ð11:56Þ

In this equation, the local spectrum SW ij ða; bÞ is defined as SW ij ða; bÞ ¼

ð T

W i ða; bÞWj ða; bÞdt

ð11:57Þ

where the time integration window T depends on the desired time resolution. The local spectrum (Equation 11.57), owing to the time average over T; allows smoothing of potential measurement noise effects. Measures of higher order correlation can also be introduced [56,58], such as the wavelet bicoherence 2

2 ðbW xxy ða1 ; a2 ; bÞÞ ¼ ð T

where 1=~a ¼ 1=a1 þ 1=a2 ; and BW xxy ða1 ; a2 ; bÞ ¼

BW xxy ða1 ; a2 ; bÞ ð 2 2 Wx ða1 ; tÞWx ða2 ; tÞ dt Wy ð~a; tÞ dt

ð11:58Þ

T

ð T

Wx ða1 ; tÞWx ða2 ; tÞWy ð~a; tÞdt

ð11:59Þ

Related remedies can be adopted to suppress spurious correlations induced by statistical noise, based on a reference noise map created from artificially simulated signals [58]. Signal energy representation concepts have been examined in Ref. [67] by using quasi orthogonal Daubechies wavelets in the frequency domain to simulate earthquake ground motion accelerations. Further, Massel has used wavelet analysis to capture time varying frequency composition of sea surface records due to fast moving atmospheric fronts in deep water, wave growth, and breaking or disintegration of mechanically generated wave trains [68]. In this regard, absolute value wavelet maps and a spectral measure called global wavelet energy spectrum, defined by the equation ð1 E3 ðaÞ ¼ E1 ða; bÞdb ð11:60Þ 0

are used. The symbol E1 ðt; bÞ denotes a time scale energy density Wf ða; bÞ a

E1 ða; bÞ ¼

2

ð11:61Þ

The scale in Equation 11.61 is readily translated into frequency by selecting the Morlet wavelet basis. Spanos and Failla [69] have applied wavelet analysis to estimate the evolutionary power spectral density (EPSD) of nonstationary oscillatory processes defined as [70] ð1 ð11:62Þ f ðtÞ ¼ Aðv; tÞeivt dZðvÞ 1

The symbol Aðv; tÞ denotes a slowly varying time and frequency dependent modulating function, and ZðvÞ is a complex random process with orthogonal increments such that E½ dZðvÞ 2 ¼ Sf0 f0 ðvÞdv; where Sf0 f0 ðvÞ is the two sided PSD of the zero mean stationary process f0 ðtÞ ¼

ð1 1

eivt dZðvÞ

ð11:63Þ

11 14


The two sided EPSD of f ðtÞ is then taken as 2

Sff ðv; tÞ ¼ Aðv; tÞ Sf0 f0 ðvÞ

ð11:64Þ

Due to its localization properties, the wavelet transform of f ðtÞ (Equation 11.62) may be approximated as an oscillatory stochastic process. That is, ð1 Wf ða; bÞ < Aðv; bÞeivb dZ 0 ðvÞ ð11:65Þ 1

pffiffiffiffiffi where dZ ðvÞ ¼ 2paC^ ðvaÞdZðvÞ: Based on Equation 11.65, the following integral relation is found between the mean squared wavelet coefficients at each scale a and the EPSD of f ðtÞ: That is, ð1 2 E½Wf ða; bÞ2 ¼ 4pa C^ ðvaÞ Sff ðv; bÞdv ð11:66Þ 0

0

A sufficient number of Equations 11.66, one for each scale a; can be solved by a standard solution algorithm, applicable for both orthogonal and nonorthogonal bases in the frequency domain. This procedure has proved quite accurate using both the Littlewood Paley and the real Morlet wavelet bases.

11.4

Random Field Simulation

The use of wavelets for random field synthesis can be examined within the more general framework of scale type methods. The latter have been developed to improve the computational performances of Monte Carlo simulations. Classical methods such as the spectral approach [71] or the autoregressive moving average (ARMA) [72] are not readily applicable for this purpose, especially when using nonuniform meshes or when enhancement of local resolution is desirable. To address these shortcomings, Fournier et al. [73] have proposed a “random midpoint method” to synthesize fractional Brownian motion; that is, a scale type method where values of the random field for points within a coarser scale are generated first, and then the generated samples are used to determine values for a finer scale. This approach has been extended by Lewis [74] into a “generalized stochastic subdivision method,” suitable for a broad class of stationary processes, and by Fenton and Vanmarcke [75] into a “local average subdivision method,” which includes a random field smoothing procedure producing averages of the field for an increasingly finer scale. An interpretation of scale type approaches in the context of random field synthesis has been given by Zeldin and Spanos [39] using compactly supported Daubechies wavelets. Specifically, a synthesis algorithm has been developed that includes the previous methods proposed by Lewis [74] and Fenton and Vanmarcke [75] as a particular case. To synthesize a sample of a given process, the closed form expressions h i ð1 ð1 j;i j rk;l ¼ E dk dli ¼ Rf ðx1 ; x2 Þcj;k ðx1 Þci;l ðx2 Þdx1 dx2

ð11:67Þ

i ð1 ð1 h j;i j Rf ðx1 ; x2 Þfj;k ðx1 Þci;l ðx2 Þdx1 dx2 bk;l ¼ E ck dli ¼

ð11:68Þ

h i ð1 ð1 j;i j ak;l ¼ E ck cli ¼ Rf ðx1 ; x2 Þfj;k ðx1 Þfi;l ðx2 Þdx1 dx2

ð11:69Þ

1

1

1

1

1

1

given in Refs. [21,39] are considered to relate the autocorrelation function Rf ðx1 ; x2 Þ of the process to the coefficients of its wavelet transform, which in this case are random variables. The synthesis algorithm is based on the wavelet reconstruction algorithm developed by Mallat [34,35], which proceeds from coarse to fine scales to determine the wavelet coefficients. Some relevant properties of wavelet ensure the computational efficiency of the algorithm. Specifically, using the quasi differential properties of j j wavelets showed by Belkin [76], the coefficients dk ’s are derived directly from ck’s by the approximate

Wavelets


11 15

linear combination j

dk ¼

X l

j j j ak;l cl þ bk uk

ð11:70Þ j

where uk ’s are uncorrelated, zero mean, unit variance random variables, statistically independent of ck ’s. For a wide class of stochastic processes, wavelet coefficients prove weakly correlated as the difference k l increases and, for this, the summation in Equation 11.70 is generally restricted to adjacent elements only. The algorithm is completed by an error assessment procedure which allows refining of the triggering scale j in order to fit the sought target statistical properties of the synthesized field. Further studies on the role of wavelet analysis in stochastic mechanics applications may be found in Ref. [21], which has showed how wavelet bases can be used in approximate Karhunen Loe`ve expansions. Any stationary process can then be represented as X j dk cj;k ðtÞ f ðtÞ ¼ ð11:71Þ j;k

where

11.5

j dk ’s

are uncorrelated random variables and cj;k ðtÞ are a nonorthogonal wavelet like basis.

System Identification

Wavelet analysis lends itself to system identification applications. For instance, frequency localization properties allow detection and decoupling of individual vibration modes of multi degree of freedom (multi DoF) linear systems. The wavelet representation of the system response can be truncated to an appropriate scale parameter in order to filter measurement noise. Also, the wavelet transform coefficients can be related directly to the system parameters, as long as specific bases are used. Early investigations trace back to the work by Robertson et al. [77], who have used the DWT for the estimation of the impulse response function of multi DoF systems. Compared with alternative time domain techniques, the DWT based extraction procedure offers significant advantages. It is robust since singularities in the procedure related matrices can generally be avoided by selecting orthonormal wavelet functions. Further, the reconstructed impulse response function captures the low frequency com ponents, referred to as static modes and mode shape errors, which ordinarily are difficult to estimate. An important application of wavelet analysis to structural identification is due to Staszewski [78], who has used complex Morlet wavelets for modal damping estimation. Specifically, Staszewski has interpreted in terms of the wavelet transform some concepts already used in well established methods, where the Hilbert transform has been applied to a free vibration linear response [79]. In the case of light damping, the free response in each mode xj ðtÞ may be approximated in the complex plane by an analytical signal, given by Equation 11.37. The modulus of the Morlet wavelet transform of xj ðtÞ can be expressed as Wxj ða; bÞ < Aj e

zj vj b

qffiffiffiffiffiffiffiffi C^ ðîaj vj 1 z2j Þ

ð11:72Þ

where Aj is the residue magnitude, and vj and zj are the mode natural frequency and damping ratio, respectively. In Equation 11.72, the symbol aj denotes the specific scale value, related to the mode natural frequency vj by the closed form relation 11.51, typical of Morlet wavelets. Assuming that the natural frequency vj has been previously computed, the damping ratio zj can then be estimated as the slope of a straight line, representing the cross section wavelet modulus (Equation 11.72) plotted in a semilogarithmic scale. That is, qffiffiffiffiffiffiffiffi ð11:73Þ ln ðWxj ðaj ; bÞÞ < zj vj b þ ln Aj C^ ðîaj vj 1 z2j Þ Staszewski has also proposed an alternative damping estimation method based on the ridge and skeletons of the wavelet transform. A ridge is a curve of local maxima in the mean square wavelet map, and the

11 16


corresponding skeleton is given by the values of the wavelet transform restricted to the ridge. As a result of the localization properties of the wavelet transform, the ridges and skeletons of the wavelet transform can be detected separately for each mode. Specifically, the real part of the skeleton of the wavelet transform gives the impulse response function for each single mode from which a straightforward estimate of the damping ratio is obtained from a logarithmic equation analogous to Equation 11.73. A generalization of the method for nonlinear systems can also be formulated [80]. Ruzzene et al. [81] have also presented a damping estimation algorithm based on the same concepts and leading to analogous results. Certain issues have been addressed in detail concerning the frequency resolution of the adopted wavelet basis, crucial for detecting coupled modes, and appropriate algorithms for ridges extraction [50]. Lardies and Gouttebroze [82] have estimated modal parameters via ambient records without input measurements. To this end, the random decrement method (see Ref. [83] and the references therein) has been used to convert ambient vibration response into a free vibration response. Also, a modified Morlet wavelet basis has been developed with enhanced properties for modal parameters estimation. The method devised by Staszewski and by Ruzzene et al. has also been implemented by Slavic et al. [84] by replacing the Morlet wavelets by Gabor wavelets, whose time and frequency resolution may be adjusted by an appropriate parameter. Explicit conditions have been given on the frequency bandwidths of the Gabor wavelet transform, in order to estimate the instantaneous frequencies of two adjacent modes. Damping coefficients have been estimated using a logarithmic decrement formula, where the ratio of the wavelet transform at two subsequent extremes of the pseudo period Tj ¼ 2p=vj of the response in each mode is involved for a selected wavelet transform scale [85,86]. For the procedure to estimate the damping coefficient associated with the fundamental mode, it is sufficient to adapt the analyzing scale so that the higher frequency modes are filtered. For an arbitrary mode j; low pass filtering is used to cancel the fundamental and the first j 1 modes. Ghanem and Romeo [87] have formulated a wavelet Galerkin method for time varying systems, where both damping and stiffness parameters are computed by solving a matrix equation. The latter is built by a standard Galerkin method by projecting the solution of the differential equation of motion onto a subspace described by the wavelet scaling functions of a compactly supported Daubechies wavelet basis. The method is accurate for both free and forced vibration responses. A formulation for nonlinear systems has also been proposed [88]. Another application is due to Yu and Xiao [89], who have used wavelet transform to identify the parameters of a Preisach model of hysteresis (see Refs. [69,90] and the references therein). The output function of the Preisach model is expanded in terms of the scaling functions of a given wavelet basis. Then, the coefficients of such an expansion are determined by fitting a number of experimental data points with a minimum energy method. From the output function, the so called Preisach function can be determined in a closed form. A comprehensive application of wavelet analysis concepts to system identification problems has been given by Le and Argoul [91]. They have developed closed form expressions to compute the damping ratio, the natural frequency and the shape of each mode, based on ridges and skeletons of the wavelet transformed free vibration response. As an alternative, Yin et al. [92] have proposed to apply the wavelet transform to the frequency response function (FRF) of the system. Specifically, given the FRF of an N DoF system in the form

HðvÞ ¼

N X r¼1

" iv

Ar

lr

þ

#

Ar iv

ð11:74Þ

lr

where lr is the rth complex pole and Ar the rth residue, a complex fractional function

cy ðxÞ ¼

1 ¼e ð1 þ ixÞyþ1

ðyþ1Þlogð1þixÞ

;

y [ Rþ

ð11:75Þ

Wavelets


11 17

is selected as a wavelet basis. Based on Equation 11.75, a closed form expression can be established for the pffiffi CWT of Equation 11.74 multiplied by ð aÞ y : Specifically, ð1 v b HðvÞc y Hy ða; bÞ ¼ a ðyþ1Þ=2 dv a 1 ! N X Ar Ar ðyþ1Þ=2 þ ð11:76Þ ¼ 2pa lr Þyþ1 ða þ ib lr Þyþ1 r¼1 ða þ ib Natural frequencies and damping ratios can be estimated by locating the maxima of Equation 11.76 in the ða; bÞ plane.

11.6

Damage Detection

Properties of the wavelet transform are also quite appealing for damage detection purposes. Early investigations in this field [93,94] used wavelet analysis to detect local faults in machineries. Specifically, visual inspection of the modulus and phase of the wavelet transform was used to localize the fault [93]. Further, it was shown that transient vibrations due to developing damage are disclosed by the local maxima of the mean square wavelet map [94]. These investigations gave a qualitative approach to damage detection as no estimate of the damage amplitude was provided. Additional studies have confirmed the correlation between local maxima of the wavelet transform and damage in beams and plates [95 97], and a first attempt to estimate the damage amplitude was made by Okafor and Dutta [98]. Specifically, Daubechies wavelets were used to wavelet transform the mode shapes of a damaged cantilever beam, and a regression analysis by a least square method was conducted to correlate the peaks of the wavelet coefficients with the corresponding damage amplitude. A consistent mathematical framework for wavelet analysis of damaged beams is due to Hong et al. [99]. The focal concept is that defects in structures, even if small, may affect significantly the vibration mode shapes, depending on the location and the kind of damage. Such variations may not be apparent in the measured data but become detectable as singularities if wavelet analysis is used due to its high resolution properties. Specifically, Hong et al. have shown that the singularity of the vibration modes can be described in terms of Lipschitz regularity, a concept also encountered in the theory of differential equations, widely used in image processing where object contours correspond to irregularities in the intensity [100,101]. In mathematical terms, a function f ðxÞ is Lipschitz a $ 0 at x ¼ x0 if there exists K . 0, and a polynomial of order m (m is the largest integer satisfying m # a), pm ðxÞ, such that and a polynomial of order m; pm ðxÞ; such that f ðxÞ ¼ pm ðxÞ þ 1ðxÞ 1ðxÞ # K x

x0

ð11:77Þ

a

ð11:78Þ

The wavelet transform of Lipschitz a functions enjoys some properties. Mallat and Hwang [100] have shown that for a wavelet basis with a number of vanishing moments a # n; a local Lipschitz singularity at x0 corresponds to maxima lines of the wavelet transform modulus. That is, local maxima with asymptotic decay across scales. Near the cone of influence x ¼ x0 ; such moduli satisfy the equation Wf ða; xÞ # Aaaþ1=2 ;

A.0

ð11:79Þ

from which the Lipschitz exponent is computed as log2 Wf ða; xÞ # log2 A þ a þ

1 log2 a 2

ð11:80Þ

By plotting the wavelet coefficients on a logarithmic scale, A and a may be computed by setting the equality sign in Equation 11.80 and minimizing the error in the least square sense. Hong et al. have

11 18


applied Equation 11.79 to the first mode shape of a damaged cantilever beam via a Mexican Hat wavelet transform. The first mode shape is preferable since it is the most accurately determined by modal testing; it features the lowest curvature; and sets off the singularity better. A correlation between damage size and the magnitude of the Lipschitz exponent has been found from a number of beams with different damage parameters. Some of the ideas presented by Hong et al. may also be found in the work by Douka et al., who have pursued crack identification in beams and plates using Daubechies wavelets [102,103]. The first mode vibration response has been considered and the singularity induced by local defects has been characterized in terms of Equation 11.79. The Lipschitz exponent has been used to describe the kind of singularity, and the parameter A has been taken as the factor relating the depth of the crack to the amplitude of the wavelet transform. Specifically, a second order polynomial law has been found for the intensity factor as a function of the crack depth. The work by Douka et al. has pointed out the importance of the number of vanishing moments M of the chosen wavelet basis. It is intuitive that the capability of setting off singularities in a regular function increases with M: However, wavelet functions with high M exhibit a long support and lack space resolution. A compromise, then, must be achieved, depending on the application in hand. Further insight into some mathematical details of both the methods developed by Hong et al. and Douka et al. may be found in Haase and Widjajakusuma [50]. Specifically, a fast algorithm to determine the maxima lines of the wavelet transform has been devised. Also, the performance of various wavelet bases, such as the Gaussian family of wavelets, has been assessed versus Daubechies wavelets used by Douka et al. Another approach for damage detection problems has been proposed by Yam et al. [104]. Clearly, detection of small and incipient damage cannot be pursued by computing modal parameters that change only if the amount of damage is significant. Thus, a method has been devised based on the energy variation of the vibration response due to the occurrence of damage. The method is implemented in two steps. The first involves the construction of damage feature proxy vectors using the energy at various scales of the wavelet transformed vibration response. Then, classification and identification of the structural damage status is pursued by using artificial neural networks (ANNs), which offer significant advantages compared with genetic algorithms (GAs) developed by Moslem and Nafaspour for damage identification purposes [105]. Genetic algorithm based damage detection, in fact, requires repeatedly searching among numerous damage parameters to find the optimal solution of the objective function. Yet another approach for applications of wavelet analysis to damage detection has been discussed by Paget et al. [106], who have developed a procedure to detect impact damage in composite plates. It is based on Lamb waves generated and received by embedded piezoceramic transducers. The Lamb waves can be quite effective since they can propagate over long distances in the composite material and can interfere with damage. To characterize the damage, the Lamb waves are wavelet transformed using an original wavelet basis, devised from the recurrent waveforms of the Lamb waves. The changes in the Lamb waves interacting due to the occurrence of damage are captured by the amplitude change of the wavelet coefficients. From this effect, an estimate of the impact energy and the damage level is obtained based on experimental results.

11.7

Material Characterization

Material properties description is another application for wavelet analysis. Intuition suggests that multiscale analysis is a natural way of describing microstructure or material heterogeneity. Various, in fact, are the examples of multiscale microstructures, such as porosity distributions in ceramics, defects, dislocations, grain boundaries, and pores. It is important, however, to understand how information at different scales is related, and whether large or small scales affect macroscopic material properties such as deformation, toughness, and electrical conductance. Further interest towards a multiscale description of material properties is motivated by the need of alternatives to the standard finite element method (FEM).

Wavelets


11 19

The latter, although capable in principle, cannot simulate the actual behavior of materials such as aluminum alloys, where pores may attain a size up to 500 mm and inclusions may attain a size of 3 to 6 mm in diameter. Further, in FEM based methods, the constitutive response of the material at increasing scales is not the result of microstructural analysis at smaller scales, but it is rather assumed on the basis of macroscopic experiments. Willam et al. [107] have performed multiresolution homogenization based on a recursive Schur reduction method in conjunction with the Haar wavelet transform. The method allows coarse grained parameters, such as Young’s modulus of elasticity, to be extracted from fine grained properties at the meso and microscales. Also, progressive elastic degradation can be modeled, which initiates at a quite fine scale and evolves into a macroscopic zero stiffness at the continuum level. Frantziskonis [108] has focused on stationary and isotropic porous media. The geometry of porous media is generally described in terms of a fundamental function, defined as unity for spatial locations in the matrix, and as zero for locations in the pores or flaws. At a solid flaw interface, the porous medium is represented mathematically through a local jump in the fundamental function. It has been found that such a jump can be captured by a wavelet transform, as long as the finest scale is small enough relative to the size of the pores. From this fact, a relationship between the energy of the wavelet transform of the porous medium, and the variance and the correlation distance of the solid phase can be derived. In the presence of heterogeneous materials, with multiscale porosity, the role of porosity at each scale has been identified through the variation of the energy of the wavelet transform as a function of scale. Peaks of the energy reveal the dominant scale in determining macroscopic properties of the materials, such as mechanical failure. Specifically, a biorthogonal spline with four vanishing moments has been employed as a wavelet basis. The results obtained have been subsequently extended in a second study, addressing the crack formation in an aluminum alloy with distributed pores and inclusions [109]. The problem, implemented for a one dimensional solid, is tackled by wavelet transforming the flexibility function, assumed to vary along the longitudinal axis of the one dimensional solid. The relationship between the energy of the wavelet transform and the variance of the flexibility is used to detect the dominant scale in the crack formation process. Note that an application of a two dimensional wavelet transform has been described in Ciliberto et al. [110] for porosity classification on carbon fiber reinforced plastics.

11.8

Concluding Remarks

Concepts of wavelets based continuous and discrete representations have been reviewed. Further, an overview of vibration related applications for evolutionary spectrum estimation, random field simulation, system identification, damage detection, and material characterization has been included. The list of references is not exhaustive. However, these references can serve as readily available resources for canvassing the multitude of concepts and applications of this remarkable tool for capturing and representing localization features of many physical phenomena. Wavelets based algorithms and commercial codes are an indispensable family of tools of vibration analysis and offer, in many cases, a potent improvement over the classical Fourier transform based approaches.

Acknowledgments The support of this work through a grant from the U.S. Department of Energy is gratefully acknowledged.

11 20


Nomenclature Symbol

Quantity

Symbol

Quantity

E½·

the operator of mathematical expectation correlation function Hilbert transform operator inner product inclusion set of all elements with a specified property absolute value norm the set of integer numbers the set of complex numbers complex conjugate

v z a b cðxÞ fðxÞ x W i dmn

frequency damping ratio scale shift mother wavelet scale function spatial variable wavelet pffiffiffiffi transform 1 Kronecker delta defined as 0 for m – n dmn ¼ 1 for m ¼ n

Rð·; ·Þ H½· k·; ·l , {·; ·} · · Z C ð·Þ

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48. Newland, D.E., Ridge and phase identification in the frequency analysis of transient signals by harmonic wavelets, J. Vib. Acoust., 121, 149, 1999. 49. Newland, D.E. and Butler, G.D. 1999. Time varying cross spectra by harmonic wavelets for soil motion damage, Proceedings of the ASME DETC: 17th ASME Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, p. 1335. 50. Haase, M. and Widjajakusuma, J., Damage identification based on ridges and maxima lines of the wavelet transform, Int. J. Eng. Sci., 41, 1423, 2003. 51. Trifunac, M.D., Response envelope spectrum and interpretation of strong earthquake ground motion, Bull. Seismol. Soc. Am., 61, 343, 1971. 52. Spanos, P.D., Donley, M., and Roesset, J. 1987. Evolutionary power spectrum estimation of earthquake accelerograms, September 19, 1985, Mexico. In Stochastic Approaches in Earthquake Engineering: U.S. Japan Joint Seminar, Boca Raton, Florida, USA, May 6 7, Y.K. Lin and R. Minai, Eds., p. 322. Springer, Berlin. 53. Massel, S.R. 1996. Ocean Surface Waves: Their Physics and Prediction, World Scientific, River Edge, NJ. 54. Ville, J., Theorie et applications de la notion de signal analytique, Cables Transm., 2, 61, 1948. 55. Loynes, R.M., On the concept of the spectrum for nonstationary processes, J. R. Stat. Soc. Ser. B Methodol., 30, 1, 1968. 56. Gurley, K. and Kareem, A., Applications of wavelet transforms in earthquake, wind, and ocean engineering, Eng. Struct., 21, 149, 1999. 57. Gurley, K., Kijewski, T., and Kareem, A., First and higher order correlation detection using wavelet transforms, J. Eng. Mech., 129, 188, 2003. 58. Kareem, A. and Kijewski, T., Time frequency analysis of wind effects on structures, J. Wind Eng. Ind. Aerodyn., 90, 1435, 2002. 59. Basu, B. and Gupta, V.K., Non stationary seismic response of multi DoF systems by wavelet transform, Earthquake Eng. Struct. Dyn., 26, 1243, 1997. 60. Basu, B. and Gupta, V.K., Seismic response of single DoF systems by wavelet modeling of nonstationary processes, J. Eng. Mech., 124, 1142, 1998. 61. Basu, B. and Gupta, V.K., Stochastic seismic response of single DoF systems through wavelets, Eng. Struct., 22, 1714, 2000. 62. Basu, B. and Gupta, V.K., Wavelet based non stationary response analysis of a friction base isolated structure, Earthquake Eng. Struct. Dyn., 29, 1659, 2000. 63. Tratskas, P. and Spanos, P.D., Linear multi DoF system stochastic response by using the harmonic wavelet transform, J. Appl. Mech., 70, 724, 2003. 64. Spanos, P.D. and Failla, G., Evolutionary spectra estimation using wavelets, J. Eng. Mech., 130, 2004. 65. Alkemade, J.A.H. 1993. The finite wavelet transform with an application to seismic processing. In Wavelets: An Elementary Treatment of Theory and Applications, T.H. Koornwinder, Ed., p. 183. World Scientific, Hackensack, NJ. 66. Basu, B. and Gupta, V.K., On equivalent linearization using the wavelet transform, J. Vib. Acoust., 121, 429, 1999. 67. Iyama, J. and Kuwamura, H., Application of wavelets to analysis and simulation of earthquake motions, Earthquake Eng. Struct. Dyn., 28, 255, 1999. 68. Massel, S.R., Wavelet analysis for processing of ocean surface wave records, Ocean Eng., 28, 957, 2001. 69. Spanos, P. D., Cacciola, P., and Redhorse, J., Random vibration of SMA systems via Preisach formalism, Nonlinear Dyn., 36, 405, 2004. 70. Priestley, M.B. 1981. Spectral Analysis and Time Series, Academic Press, New York. 71. Shinozuka, M. and Deodatis, G., Stochastic process models for earthquake ground motion, Probab. Eng. Mech., 3, 114, 1988. 72. Spanos, P.D. and Mignolet, M.P., Simulation of homogeneous two dimensional random fields: Part II MA and ARMA models, J. Appl. Mech., 114, 270, 1992.

Wavelets


11 23

73. Fournier, A., Fussell, D., and Carpenter, L., Computer rendering of stochastic models, Commun. ACM, 25, 371, 1982. 74. Lewis, J.P., Generalized stochastic subdivision, ACM Trans. Graph., 6, 167, 1987. 75. Fenton, G.A. and Vanmarcke, E.H., Simulation of random fields via local average subdivision, J. Eng. Mech., 116, 1733, 1990. 76. Belkin, G., On the representation of operators in bases of compactly supported wavelets, J. Numer. Anal., 6, 1716, 1993. 77. Robertson, A.N., Park, K.C., and Alvin, K.F., Extraction of impulse response data via wavelet transform for structural system identification, Proc. Des. Eng. Tech. Conf. DE 84.1 ASME 95, 1323, 1995. 78. Staszewski, W.J., Identification of damping in multi DoF systems using time scale decomposition, J. Sound Vib., 203, 283, 1997. 79. Jones, D.I.G. 1988. Application of damping treatments. In Shock and Vibration Handbook, 3rd ed., C.M. Harris, Ed. McGraw Hill, New York. 80. Staszewski, W.J., Identification of non linear systems using multi scale ridges and skeletons of the wavelet transform, Shock Vib. Dig., 214, 639, 1998. 81. Ruzzene, M., Fasana, A., Garibaldi, L., and Piombo, B., Natural frequencies and dampings identification using wavelet transform: application to real data, Mech. Syst. Signal Process., 11, 207, 1997. 82. Lardies, J. and Gouttebroze, S., Identification of modal parameters using the wavelet transform, Int. J. Mech. Sci., 44, 2263, 2002. 83. Spanos, P.D. and Zeldin, B.A., Generalized random decrement method for analysis of vibration data, J. Vib. Acoust., Trans. ASME, 120, 806, 1998. 84. Slavic, J., Simonovski, I., and Boltezar, M., Damping identification using a continuous wavelet transform: application to real data, J. Sound Vib., 262, 291, 2003. 85. Lamarque, C.H., Pernot, S., and Cuer, A., Damping identification in multi DoF systems via a wavelet logarithmic decrement. Part 1: Theory, J. Sound Vib., 235, 361, 2000. 86. Hans, S., Ibraim, E., Pernot, S., Boutin, C., and Lamarque, C.H., Damping identification in multi DoF system via a wavelet logarithmic decrement. Part 2: Study of a civil engineering building, J. Sound Vib., 235, 375, 2000. 87. Ghanem, R. and Romeo, F., A wavelet based approach for the identification of linear time varying dynamical systems, J. Sound Vib., 234, 555, 2000. 88. Ghanem, R. and Romeo, F., Wavelet based approach for model and parameter identification of non linear systems, Int. J. Non Linear Mech., 36, 835, 2001. 89. Yu, Y., Xiao, Z., Lin, E. B., and Naganathan, N., Analytic and experimental studies of a wavelet identification of Preisach model of hysteresis, J. Magn. Magn. Mater., 208, 255, 2000. 90. Mayergoyz, I.D. 2003. Mathematical Models of Hysteresis and Their Applications, 1st ed., Academic Press, Boston. 91. Le, T. P. and Argoul, P., Continuous wavelet transform for modal identification using free decay response, J. Sound Vib., 277, 73, 2004 92. Yin, H.P., Duhamel, D., and Argoul, P., Natural frequencies and damping estimation using wavelet transform of a frequency response function, J. Sound Vib., 271, 999, 2004. 93. Staszewski, W.J. and Tomlinson, G.R., Application of the wavelet transform to fault detection in a spur gear, Mech. Syst. Signal Process., 8, 289, 1994. 94. Wang, W.J. and McFadden, P.D., Application of orthogonal wavelets to early gear damage detection, Mech. Syst. Signal Process., 9, 497, 1995. 95. Liew, K.M. and Wang, Q., Application of wavelet theory for crack identification in structures, J. Eng. Mech., 124, 152, 1998. 96. Deng, X. and Wang, Q., Crack detection using spatial measurements and wavelet analysis, Int. J. Fracture, 91, L23, 1998.

11 24


97. Wang, Q. and Deng, X., Damage detection with spatial wavelets, Int. J. Solids Struct., 36, 3443, 1999. 98. Okafor, A.C., Chandrashekhara, K., and Jiang, Y.P., Location of impact in composite plates using waveform based acoustic emission and Gaussian cross correlation techniques, Proc. SPIE Int. Soc. Opt. Eng., 2718, 291, 1996. 99. Hong, J. C., Kim, Y.Y., Lee, H.C., and Lee, Y.W., Damage detection using the Lipschitz exponent estimated by the wavelet transform: applications to vibration modes of a beam, Int. J. Solids Struct., 39, 1803, 2002. 100. Mallat, S. and Hwang, W.L., Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory, 38, 617, 1992. 101. Grossmann, A. 1986. Wavelet transform and edge detection. In Stochastic Processes in Physics and Engineering, M. Hazewinkel, Ed. Reidel, Dodrecht. 102. Douka, E., Loutridis, S., and Trochidis, A., Crack identification in beams using wavelet analysis, Int. J. Solids Struct., 40, 3557, 2003. 103. Douka, E., Loutridis, S., and Trochidis, A., Crack identification in plates using wavelet analysis, J. Sound Vib., 270, 279, 2004. 104. Yam, L.H., Yan, Y.J., and Jiang, J.S., Vibration based damage detection for composite structures using wavelet transform and neural network identification, Compos. Struct., 60, 403, 2003. 105. Moslem, K. and Nafaspour, R., Structural damage detection by genetic algorithms, AIAA J., 40, 1395, 2002. 106. Paget, C.A., Grondel, S., Levin, K., and Delebarre, C., Damage assessment in composites by Lamb waves and wavelet coefficients, Smart Mater. Struct., 12, 393, 2003. 107. Willam, K., Rhee, I., and Beylkin, G., Multiresolution analysis of elastic degradation in heterogeneous materials, Meccanica, 36, 131, 2001. 108. Frantziskonis, G., Wavelet based analysis of multiscale phenomena: application to material porosity and identification of dominant scales, Probab. Eng. Mech., 17, 349, 2002. 109. Frantziskonis, G., Multiscale characterization of materials with distributed pores and inclusions and application to crack formation in an aluminum alloy, Probab. Eng. Mech., 17, 359, 2002. 110. Ciliberto, A., Cavaccini, G., Salvetti, O., and Chimenti, M., Porosity detection in composite aeronautical structures, Infrared Phys. Technol., 43, 139, 2002.

Shock and Vibration

III

III 1

12

Mechanical Shock 12.1 Definitions ............................................................................ 12 2 Shock † Simple (or Perfect) Shock † Half Sine Shock Versed Sine (or Haversine) Shock † Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock † Rectangular Shock

†

12.2 Description in the Time Domain ...................................... 12 3 12.3 Shock Response Spectrum .................................................. 12 4 Need † Shock Response Spectrum Definition † Response of a Linear One Degree of Freedom System † Definitions † Standardized Shock Response Spectrum † Choice of Damping † Shock Response Spectra Domains † Algorithms for Calculation of the Shock Response Spectra † Choice of the Digitization Frequency of the Signal † Use of Shock Response Spectra for the Study of Systems with Several Degrees of Freedom

12.4 Pyroshocks ........................................................................... 12 17 12.5 Use of Shock Response Spectra .......................................... 12 18 Severity Comparison of Several Shocks † Test Specification Development from Real Environment Data

12.6 Standards .............................................................................. 12 24 Types of Standards † Installation Conditions of Test Item † Uncertainty Factor † Bump Test

12.7 Damage Boundary Curve ................................................... 12 26 Definition

†

Analysis of Test Result

12.8 Shock Machines ................................................................... 12 28 Main Types † Impact Shock Machines † Shock Simulators (Programmers) † Limitations † Pneumatic Machines † High Impact Shock Machines † Specific Test Facilities

12.9 Generation of Shock Using Shakers .................................. 12 44 Principle Behind the Generation of a Simple Shape Signal versus Time † Main Advantages † Pre and Postshocks † Limitations of Electrodynamic Shakers † The Use of Electrohydraulic Shakers

12.10 Control by a Shock Response Spectrum ........................... 12 52 Principle † Principal Shapes of Elementary Signals † Comparison of WAVSIN, SHOC Waveforms, and Decaying Sinusoid † Criticism of Control by a Shock Response Spectrum

12.11 Pyrotechnic Shock Simulation ........................................... 12 58

Christian Lalanne Engineering Consultant

Simulation Using Pyrotechnic Facilities † Simulation Using Metal to Metal Impact † Simulation Using Electrodynamic Shakers † Simulation Using Conventional Shock Machines

12 1

12 2


Summary Transported or on board equipment is very frequently subjected to mechanical shocks in the course of its useful lifetime (in material handling, transportation, etc.). This kind of environment, although of extremely short duration (from a fraction of a millisecond to a few dozen milliseconds), is often severe and cannot be neglected. What is presented in this chapter is summarized here. After a brief recapitulation of the shock shapes most widely used in tests, the shock response spectrum (SRS) is presented with its numerous definitions and calculation methods. The main properties of the spectrum are described, showing that important characteristics of the original signal can be drawn from it, such as its amplitude or the velocity change associated with the movement during a shock. The SRS is the ideal tool for comparing the severity of several shocks and for drafting specifications. Recent standards require writing test specifications from real environment measurements associated with the life profile of the material (test tailoring). The process that makes it possible to transform a set of recorded shocks into a specification of the same severity is detailed. Packages must protect the equipment contained within them from various forms of disturbance related to handling and possible free fall drop and impact onto a floor. A method to characterize the shock fragility of the packaged product, using the “damage boundary curve” (DBC), and to choose the characteristics of the cushioning material constituting the package is described. The principle of shock machines that are currently most widely used in laboratories is described. To reduce costs by restricting the number of changes in test facilities, specifications expressed in the form of a simple shock (half sine, rectangle, sawtooth with a final peak) can occasionally be tested using an electrodynamic exciter. The problems encountered, which stress the limitations of such means, are set out together with the consequences of modifications, that have to be made to the shock profile on the quality of the simulation. Determining a simple shape shock of the same severity as a set of shocks on the basis of their response spectrum is often a delicate operation. Thanks to progress in computerization and control facilities, this difficulty can sometimes be overcome by expressing the specification in the form of a response spectrum and by controlling the exciter directly from that spectrum. In practical terms, as the exciter can only be driven with a signal that is a function of time, the software of the control rack determines a time signal with the same spectrum as the specification displayed. The principles of composition of the equivalent shock are described with the shapes of the basic signals commonly used, while their properties and the problems that can be encountered, both in the generation of the signal and with respect to the quality of the simulation obtained, are emphasized. Pyrotechnic devices or equipment (cords, valves, etc.) are frequently used in satellite launchers due to the very high degree of accuracy that they provide in operating sequences. Shocks induced in structures by explosive charges are extremely severe, with very specific characteristics. It is shown that they cannot be correctly simulated in the laboratory by conventional means and that their simulation requires specific tools.

12.1 12.1.1

Definitions Shock

Shock occurs when a force, a position, a velocity, or an acceleration is abruptly modified and creates a transient state in the system considered. The modification is normally regarded as abrupt if it occurs in a time period that is short compared with the natural period concerned (AFNOR, 1993). Shock is defined as a vibratory excitation having a duration between the natural period of the excited mechanical system and two times that period (Figure 12.1).

12.1.2

Simple (or Perfect) Shock

A shock whose signal can be represented exactly in simple mathematical terms is called a simple (or perfect) shock. Standards generally specify one of the three following: half sine (approached by a versed sine waveform), terminal peak sawtooth, and rectangular shock (approached by a trapezoidal waveform).

Mechanical Shock

12.1.3

12 3

Half-Sine Shock

This is a simple shock for which the acceleration time curve has the form of a half period (part positive or negative) of a sinusoid.

Acceleration (m/s2)

150

12.1.4 Versed-Sine (or Haversine) Shock This is a simple shock for which the acceleration time curve has the form of one period of the curve representative of the function ½1 cosð Þ ; with this period starting from the zero value of this function. It is thus a signal ranging between two minima of a sine wave.

100 50 0 −50 −100 −150

0

2

4

6 8 10 Time (ms)

12

14

FIGURE 12.1 Example of a shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12.1.5 Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock This is a simple shock for which the acceleration time curve has the shape of a triangle, where acceleration increases linearly up to a maximum value and then instantly decreases to zero.

12.1.6

Rectangular Shock

This is a simple shock for which the acceleration time curve increases instantaneously up to a given value, remains constant throughout the signal, and decreases instantaneously to zero. In practice, what is carried out are trapezoidal shocks.

12.1.6.1

Trapezoidal Shock

This is a simple shock for which the acceleration time curve grows linearly up to a given value, remains constant during a certain time period, after which it decreases linearly to zero.

12.2

Description in the Time Domain

Three parameters are necessary to describe a shock in the time domain: its amplitude, its duration, t; and its form. The physical parameter expressed in terms of time is generally an acceleration, x€ ðtÞ, but can be also a velocity, vðtÞ; a displacement, xðtÞ, or a force, FðtÞ: In the first case, which we will consider in particular in this chapter, the velocity change corresponding to the shock movement is equal to (Table 12.1)

DV ¼

ðt 0

x€ ðtÞdt

ð12:1Þ

12 4

Vibration and Shock Handbook TABLE 12.1

Main Simple Shock Waveforms (Amplitude, xm ; Duration, t; Velocity Change, DV) Waveform

12.3

Function

Half sine

xðtÞ ¼ xm sin

Versed sine

xðtÞ ¼

Rectangle

xðtÞ ¼ xm

Terminal peak sawtooth

xðtÞ ¼ xm

p t t

xm 2p 1 2 cos t t 2

DV 2 x t p m 1 x t 2 m xm t

t t

1 x t 2 m

Shock Response Spectrum

12.3.1

Need

Very often, the problem is to evaluate the relative severity of several shocks (shocks measured in the real environment, measured shocks with respect to standards, establishment of a specification, etc.). A shock is an excitation of short duration, which induces transitory dynamic stress in structures. These stresses are a function of the following: *

*

The characteristics of the shock (amplitude, duration, and shape). The dynamic properties of the structure (resonant frequencies, Q factors; see Chapter 19).

The severity of a shock can thus be estimated only according to the characteristics of the system that undergoes it. The evaluation of this severity requires in addition the knowledge of the mechanism leading to a degradation of the structure. The two most common mechanisms are as follows: *

*

The exceeding of a value threshold of the stress in a mechanical part can lead to either a permanent deformation (acceptable or not) or a fracture, or at any rate, a functional failure. If the shock is repeated many times (e.g., the shock recorded on the landing gear of an aircraft, the operation of an electromechanical contactor), the fatigue damage accumulated in the structural elements can lead in the long term to fracture (Lalanne, 2002c).

The comparison would be difficult to carry out if one used a fine model of the structure, and in any case this is not always available, particularly at the stage of the development of the specification of dimensioning. One searches for a method of general nature, which leads to results that can be extrapolated to any structure.

12.3.2

Shock Response Spectrum Definition

In a thesis on the study of earthquakes’ effects on buildings, Biot (1932) proposed a method consisting of applying the shock under consideration to a “standard” mechanical system, which thus does not claim to be a model of the real structure. It is composed of a support and of N linear one degree of freedom (one DoF) resonators, comprising each one are a mass, mi a spring of stiffness, p kiffiffiffiffiffi and a damping device, ci ; chosen such that the fraction of critical damping (damping ratio) j ¼ ci 2 ki mi is the same for all N resonators. A model for the shock response spectrum (SRS) is shown in Figure 12.2 (also see Chapter 17). When the support is subjected to the shock,pffiffiffiffiffiffi each mass, mi ; has a specific movement response according to its natural frequency, f0i ¼ ð1=2pÞ ki =mi and to the chosen damping ratio, j; while a

Mechanical Shock

12 5

z(t) t f1

ξ

f2

ξ

fN−1

ξ

fN

ξ .. x(t) t

FIGURE 12.2 permission.)

Model of the SRS. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With

stress, si ; is induced in the elastic element. The analysis consists of seeking the largest stress, smi ; observed at each frequency in each spring. For applications deviating from the assumptions of definition of the SRS (linearity, only one DoF), it is desirable to observe a certain prudence if one wishes to estimate quantitatively the response of a system starting from the spectrum (Bort, 1989). The response spectra are more often used to compare the severity of several shocks. It is known that the tension static diagram of many materials comprises a more or less linear arc on which the stress is proportional to the deformation. In dynamics, this proportionality can be allowed within certain limits for the peaks of the deformation. If a mass spring damper system is supposed to be linear, it is then appropriate to compare two shocks by the maximum response stress, sm , that they induce or by the maximum relative displacement, zm ; that they generate. This occurs since it is supposed

sm ¼ Kzm

ð12:2Þ

zm is a function only of the dynamic properties of the system, whereas sm is also a function, via K; of the properties of the materials which constitute it. The curve giving the largest relative displacement, zsup multiplied by v20 (v0 ¼ 2p f0 ) according to the natural frequency, f0 ; for a given damping ratio j; is the SRS.

12.3.3

Response of a Linear One-Degree-of-Freedom System

12.3.3.1

Shock Defined by a Force

Consider a mass spring damping system subjected to a force, FðtÞ; applied to the mass (Figure 12.3). The differential equation of the movement is written as m

2

dz dz þc þ kz ¼ FðtÞ dt 2 dt

Force Mass m

ð12:3Þ

where zðtÞ is the relative displacement of the mass, m; relative to its support in response to the shock, FðtÞ: This equation can be expressed in the form (Lalanne, 2002b): d2 z dz FðtÞ þ 2jv0 þ v20 z ¼ ð12:4Þ dt 2 dt m pffiffiffiffi where pffiffiffiffiffi j ¼ c=2 km (damping ratio) and v0 ¼ k=m (natural frequency).

Damping constant c

Stiffness k

Fixed support FIGURE 12.3 Linear one Dof system subjected to a force. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12 6


12.3.3.2

Shock Defined by an Acceleration

Let us set x€ ðtÞ as an acceleration applied to the base of a linear one DoF mechanical system, with y€ðtÞ the absolute acceleration response of the mass, m; and zðtÞ the relative displacement of the mass, m; with respect to the base (Figure 12.4). The equation of the movement is written as above: m

d2 y ¼ dt 2

kðy

xÞ

c

dy dt

dx dt

Mass m

Damping constant c

Stiffness k

Moving base

ð12:5Þ

that is d2 y dy dx þ 2jv0 þ v20 y ¼ v20 xðtÞ þ 2jv0 dt 2 dt dt ð12:6Þ or while setting zðtÞ ¼ yðtÞ

xðtÞ

d2 z dz þ 2jv0 þ v20 z ¼ dt 2 dt

d2 x dt 2

Absolute reference FIGURE 12.4 Linear one DoF system subjected to acceleration. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

ð12:7Þ

The differential equation (Equation 12.7) can be integrated by parts or by using the Laplace transformation. If the excitation is an acceleration of the support, the response relative displacement is given, for zero initial conditions, by an integral called Duhamel’s integral: qffiffiffiffiffiffiffiffi ðt 1 pffiffiffiffiffiffiffiffi zðtÞ ¼ ð12:8Þ x€ ðaÞe jv0 ðt aÞ sin v0 1 j2 ðt aÞda v 0 1 j2 0 where a is an integration variable homogeneous with time. The absolute acceleration of the mass is given by qffiffiffiffiffiffiffiffi ðt v0 x€ ðaÞe jv0 ðt aÞ ½ð1 2j2 Þsin v0 1 j2 ðt y€ ðtÞ ¼ pffiffiffiffiffiffiffiffi 1 j2 0 qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi þ 2j 1 j2 cos v0 1 j2 ðt aÞ da

12.3.4

Definitions

12.3.4.1

Response Spectrum

aÞ ð12:9Þ

This is a curve representative of the variations of the largest response of a linear one DoF system subjected to a mechanical excitation, plotted against its natural frequency, f0 ¼ v0 =2p; for a given value of its damping ratio (see Chapter 17). 12.3.4.2

Absolute Acceleration Shock Response Spectrum

In the most usual cases where the excitation is defined by an absolute acceleration of the support or by a force applied directly to the mass, the response of the system can be characterized by the absolute acceleration of the mass (which can be measured using an accelerometer fixed to this mass). The response spectrum is then called the absolute acceleration SRS. 12.3.4.3

Relative Displacement Shock Spectrum

In similar cases, we often calculate the relative displacement of the mass with respect to the displacement of the base of the system. This displacement is proportional to the stress created in the

Mechanical Shock

12 7

spring (since the system is regarded as linear). In practice, one generally expresses in ordinates the quantity v20 zsup ; which is called the equivalent static acceleration (Biot, 1941). This product has the dimensions of acceleration, but does not represent the absolute acceleration of the mass, except when damping is zero. However, when damping is close to the current values observed in mechanics, and in particular when j ¼ 0:05; as a first approximation one can assimilate v20 zsup to the absolute acceleration y€ sup of the mass, m (Lalanne, 1975, 2002b). The quantity v20 zsup is termed pseudo acceleration. In the same way, one terms the product v0 zsup pseudo velocity. The spectrum giving v20 zsup vs. the natural frequency is named the relative displacement shock spectrum. In each of these two important categories, the response spectrum can be defined in various ways according to how the largest response at a given frequency is characterized. 12.3.4.4 Primary Positive Shock Response Spectrum or Initial Positive Shock Response Spectrum This is the highest positive response observed during the shock. 12.3.4.5

Primary (or Initial) Negative Shock Response Spectrum

This is the highest negative response observed during the shock. 12.3.4.6

Secondary (or Residual) Shock Response Spectrum

This is the largest response observed after the end of the shock. Here also, the spectrum can be positive or negative.

Example An example giving standardized primary and residual relative displacement SRS curves for a half sine pulse is shown in Figure 12.5. 12.3.4.7

Positive (or Maximum Positive) Shock Response Spectrum

This is the largest positive response due to the shock, without reference to the duration of the shock. It thus corresponds to the envelope of the positive primary and residual spectra. Half-sine (1 m/s2 - 1 s)

2.0

Primary positive spectrum

w 20 zsup (m/s2)

1.5 1.0

Residual positive spectrum

0.5 0.0 −0.5

Primary negative spectrum

−1.0 −1.5 0.0

Residual negative spectrum 0.5

1.0

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

x = 0.05 4.0

4.5

5.0

FIGURE 12.5 Standardized primary and residual relative displacement SRS of a half sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12 8

12.3.4.8


Negative (or Maximum Negative) Shock Response Spectrum

This is the largest negative response due to the shock, without reference to the duration of the shock. As before, it corresponds to the envelope of the negative primary and residual spectra. 12.3.4.9

Maximax Shock Response Spectrum

This is the envelope of the absolute values of the positive and negative spectra. 12.3.4.10

Choice of Shock Response Spectrum

Which spectrum must be used? Absolute acceleration SRS can be useful when absolute acceleration is the parameter easiest to compare with a characteristic value (as in a study of the effects of a shock on a man, a comparison with the specification of an electronics component, etc.). In practice, it is very often the stress (and thus the relative displacement) which seems the most interesting parameter. The spectrum is primarily used to study the behavior of a structure, to compare the severity of several shocks (the stress created is a good indicator), to write test specifications (as it is also a comparison between the real environment and the test environment), or to dimension a suspension (relative displacement and stress are then useful). The damage is assumed to be proportional to the largest value of the response, i.e., to the amplitude of the spectrum at the frequency considered, and it is of little importance for the system whether this maximum, zm ; takes place during or after the shock. The most interesting spectra are thus the positive and negative spectra that are most frequently used in practice, with the maximax spectrum. The distinction between positive and negative spectra must be made each time the system, if dissymmetrical, behaves differently, for example under different tension and compression. It is, however, useful to know these various definitions so as to be able to correctly interpret the curves published.

The Shock Response Spectrum is a curve representative of the variations of the largest response of a linear one DoF system subjected to a mechanical excitation, plotted against its natural frequency, for a given value of its damping ratio. The response can be defined by the pseudo acceleration, v20 zsup (relative displacement shock spectrum) or by the absolute acceleration of the mass (absolute acceleration SRS). For the usual values of Q; the spectra are very close. The most interesting spectra are the positive and negative spectra, which are most frequently used in practice, with the maximax spectrum.

The relation between the various types of SRS that have been discussed here is shown in Figure 12.6.

Primary (initial) positive SRS Primary (initial) negative SRS

Positive SRS

Secondary (residual) positive SRS

Negative SRS

Maximax SRS

Secondary (residual) negative SRS FIGURE 12.6

Relation between the different types of SRS.

Mechanical Shock

12 9

12.3.5

Standardized Shock Response Spectrum

12.3.5.1

Definition

*

*

1.4 1.2 SRS / xm

For a given shock, the spectra plotted for various values of the duration and the amplitude are similar in shape. It is thus useful, for simple shocks, to have a standardized or reduced spectrum plotted in dimensionless coordinates, while plotting on the abscissa the product f0 t (instead of f0 ) or v0 t and on the ordinate the spectrum/shock pulse ampli tude ratio, v20 zm =€xm ; which, in practice, amounts to tracing the spectrum of a shock of duration equal to 1 sec and amplitude 1 m/sec2. This is shown in Figure 12.7. These standardized spectra can be used for two purposes:

1.0 0.8 0.6 0.4 Q = 10

0.2 0.0 0.0

1.0

2.0

f0t

3.0

4.0

5.0

FIGURE 12.7 Standardized positive SRS of a terminal peak sawtooth pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Plotting of the spectrum of a shock of the same form, but of arbitrary amplitude and duration. Investigating the characteristics of a simple shock of which the spectrum envelope is a given spectrum (resulting from measurements from the real environment).

12.3.5.2

Standardized Shock Response Spectra of Simple Shocks

Figure 12.8 to Figure 12.15 give the reduced SRSs for various pulse forms, with unit amplitude and unit duration, for several values of damping. To obtain the spectrum of a particular shock of arbitrary amplitude, x€ m ; and duration, t (different from 1) from these spectra, it is enough to regraduate the scales as follows: *

*

For the amplitude, multiply the reduced values by x€ m : For the abscissae (x axis values), replace each value fð¼ f0 tÞ by f0 ¼ f=t:

We will see later on how these spectra can be used for the calculation of test specifications.

2.0 1.5

Half-sine (1 m/s2 - 1 s) 0

0.05

1.0 w 20 zsup (m/s2)

0.025

0.1 0.25 0.5

Positive spectra

0.5 0.5

0.0 −0.5

0.25

−1.0

0.1

−1.5 −2.0 0.0

0

0.5

0.05 0.025

1.0

1.5

Negative spectra 2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 12.8 Standardized positive and negative relative displacement SRS of a half sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12 10


Half-sine (1 m/s2 - 1 s)

Absolute acceleration ysup (m/s2)

2.0

0.025

0

1.5

0.1

1.0

0.5

0.05

Positive spectra

0.25

0.5 0.0

0.5

−0.5

0.25

−1.0

0.1 0.05

−1.5

0

−2.0 0.0

Negative spectra

0.025

0.5

1.0

1.5


3.5

4.0

4.5

5.0

FIGURE 12.9 Standardized positive and negative absolute acceleration SRS of a half sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12.3.5.2.1 Half-Sine Pulse Figure 12.8 and Figure 12.9 show the standardized SRS curves in this case. 12.3.5.2.2 Versed Sine Pulse Figure 12.10 shows the standardized SRS curves in this case. 12.3.5.2.3 Terminal Peak Sawtooth Pulse Figure 12.11 and Figure 12.12 show the standardized SRS curves for terminal peak sawtooth (TPS) pulse. 12.3.5.2.4 Rectangular Pulse Figure 12.13 gives the standardized SRS curves for a rectangular pulse shock. 12.3.5.2.5 Trapezoidal Pulse Figure 12.14 presents the standardized SRS curve for a trapezoidal pulse. A comparison of various SRS curves is given in Figure 12.15. Versed-sine (1 m/s2 - 1 s)

2.0 1.5

0.025

0

0.05

1.0 w 20 zsup (m/s2)

Positive spectra

0.1 0.25 0.5

0.5 0.0

0.5

−0.5

0.25

−1.0

0.1 0.05

−1.5 −2.0 0.0

0

0.5

1.0

Negative spectra

0.025

1.5


3.5

4.0

4.5

5.0

FIGURE 12.10 Standardized positive and negative relative displacement SRS of a versed sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Mechanical Shock

12 11

1.5

T.P.S. (1 m/s2 - 1 s) 0.025 0.05

0

w 20 zsup (m/s2)

1.0 0.5

0.1

0.25 0.5

0.0

0.5 0.25

−0.5

0.1 0.05

−1.0 −1.5 0.0

Positive spectra

Negative spectra

0.025

0

0.5

1.0

1.5


3.5

4.0

4.5

5.0

FIGURE 12.11 Standardized positive and negative relative displacement SRS of a TPS pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12.3.6

Choice of Damping

The choice of damping should be carried out according to the structure subjected to the shock. When this is not known, or studies are being carried out with a view to comparison with other already calculated spectra, the outcome is that one plots the shock response spectra with a damping ratio equal to 0.05 (i.e., Q ¼ 10; see Chapter 19). It is an approximately average value for the majority of structures. Unless otherwise specified, as noted on the curve, it is the value chosen conventionally. With the spectra varying relatively little with damping, this choice is often not very important. To limit possible errors, the selected value should, however, be systematically noted on the diagram.

12.3.7

Shock Response Spectra Domains

Three domains can be schematically distinguished for shock spectra. 1. An impulse domain at low frequencies, in which the amplitude of the spectrum (and thus of the response) is lower than the amplitude of the shock: The system reduces the effects of the shock. It is thus

1.5

T.P.S. (1 m/s2−1s−td = 0.1s) 0

w 20 zsup (m/s2)

1.0 0.5

0.1 0.25 0.5

0.0

Positive spectra

0.5 0.25

−0.5

0.1 0.05

−1.0

Negative spectra

0.025

0

−1.5 0.0

0.025 0.05

0.5

1.0

1.5


3.5

4.0

4.5

5.0

FIGURE 12.12 Standardized positive and negative relative displacement SRS of a TPS pulse with nonzero decay time. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)


w 20 zsup (m/s2)

12 12

Rectangle (1 m / s2 - 1s)

2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 0.0

0

0.025

0.05

Positive spectra

0.1 0.25 0.5

0.5 0.25 0.1 0.05 0.025

0

0.5

1.0

Negative spectra

1.5


3.5

4.0

4.5

5.0

FIGURE 12.13 Standardized positive and negative relative displacement SRS of a rectangular pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

in this impulse region that it would be advisable to choose the natural frequency of an isolation system to the shock, from which we can deduce the stiffness envisaged of the insulating material: k ¼ mv20 ¼ 4p2 f02 m ð12:10Þ with m being the mass of the material to be protected. The shock here is of very short duration with respect to the natural period of the system. In this impulse region ð0 # f0 t # 0:2Þ: *

*

The form of the shock has little influence on the amplitude of the spectrum. Only (for a given damping value) the velocity change DV associated with the shock, equal to the algebraic surface under the curve x€ ðtÞ is important. The slope p at the origin of the spectrum plotted for zero damping in linear scales is proportional to the velocity change DV corresponding to the shock pulse (Lalanne, 2002b): dðv20 zsup Þ ¼ 2pDV ð12:11Þ p¼ df0 This relation is approximate if damping is small. Trapezoid (1 m/s2−1 s−t r = 0.1 s−t d = 0.1 s)

2.5

0

2.0

0.025

w 02 zsup (m/s2)

1.5

Positive spectra

0.05 0.1 0.25 0.5

1.0 0.5 0.0

0.5

−0.5

0.25

−1.0

0.1

−1.5 −2.0 −2.5 0.0

0.05 0.025

0

0.5

1.0

Negative spectra

1.5


3.5

4.0

4.5

5.0

FIGURE 12.14 Standardized positive and negative relative displacement SRS of a trapezoidal pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Mechanical Shock

12 13

2.0 1.5

w 20 zsup (m/s2)

1.0 ξ = 0.05

0.5 0.0 −0.5 −1.0 −1.5 −2.0 0.0

FIGURE 12.15 *

1.0

2.0 3.0 Frequency (Hz)

4.0

5.0

Comparison between the SRS of the three main simple shock waveforms.

The positive and negative spectra are in general the residual spectra (it is sometimes necessary that the frequency of spectrum is very small, and there can be exceptions for certain long shocks in particular). They are nearly symmetrical so long as damping is small.

2. A static domain in the range of the high frequencies, where the positive spectrum tends towards the amplitude of the shock whatever the damping: All occurs here as if the excitation were a static acceleration (or a very slowly varying acceleration), as the natural period of the system is small compared with the duration of the shock. This does not apply to rectangular shocks or to the shocks with zero rise time. Real shocks having necessarily a rise time different from zero, this restriction remains theoretical. 3. An intermediate domain in which there is dynamic amplification of the effects of the shock, the natural period of the system being close to the duration of the shock: This amplification, which is more or less significant depending on the shape of the shock and the damping of the system, does not exceed 1.77 for shocks of traditional simple shape (half sine, versed sine, TPS). Much larger values are reached in the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid. Various domains of an SRS are illustrated in Figure 12.16.

Example Consider a half sine shock pulse, amplitude x€ m ¼ 50 m=sec2 ; duration t ¼ 11 msec; positive SRS (relative displacement) for a damping ratio j ¼ 0:05: The slope p of the SRS (Figure 12.16) at the origin is equal to p ¼ 30:6=15 ¼ 2:04 m=sec; yielding DV
0 > ; 1d =z10 # 1 FIGURE 14.16 Compression softening models with > < zf c 2 z10 z10 sd ¼ Hognestad curve. > > 1d =z10 1 2 > : zf 0c 1 ; 1d =z10 . 1 2=z 1 ð14:51aÞ 0:9 ð14:51bÞ z ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 6001r Tensile stress strain relationship of concrete. From the tests involving panels subjected to shear, it was clear that the tensile stress of concrete, sr ; is not zero as assumed in the simple truss model. Based on the tests of 35 full size panels (Hsu, 1993), a set of formulas were recommended as follows: If 1r # 1cr ; sr ¼ Ec 1y If 1r . 1cr ; sr ¼ fcr

1cr 1r

ð14:52Þ 0:4

ð14:53Þ

where

pffiffiffi pffiffiffi Ec ¼ 47;000 f 0c ; and both f 0c and f 0c are in pounds per square inch 1cr ¼ strain qffiffiat ffi cracking of concrete ¼ 0.00008 fcr ¼ 3:75 f 0c

Stress strain relationship of steel. The stress strain curve of a steel bar in concrete relates the average stress to the average strain of a long bar crossing several cracks, whereas the stress strain curve of a bare bar relates the stress to the strain at a local point (Okamura and Maekawa, 1991). In other words, a steel bar in concrete is stiffened by the tensile stress of the concrete. If the tensile strength of concrete is neglected, as it is in the most of truss models, the following equations are used: If 1t # 1ty ; fl ¼ Es 1l

ð14:54Þ

If 1l . 1ly ; fl ¼ fly

ð14:55Þ

where Es ¼ modulus of elasticity of steel bars fly ¼ yield stress of longitudinal steel bars 1ly ¼ yield strain of longitudinal steel bars It was recommended by Belarbi and Hsu (1995) that both the tensile strength of concrete, presented in the previous section, and the average stress strain curve of steel stiffened by concrete, be taken into account. In this model, the following equations are used for describing the stress strain relationship of steel: If 1l Es # f 0ly ; fl ¼ Es 1l

ð14:56Þ

Reinforced Concrete Structures

14 15

2

If 1l Es # f 0ly ; fl ¼ 1 where

a=458 ½ð0:91 1000rl

2BÞfly þ ð0:02 þ 0:25BÞEs 1l

B ¼ ð1=rl Þðfcr =fly Þ1:5 f 0ly ¼ ½1 14.2.3.5

ð14:58Þ

a=458Þ=1000rl ð0:93

ð2

ð14:57Þ

2BÞfly

ð14:59Þ

Solution Procedures

Figure 14.17 shows a framed shear wall. This kind of shear wall will be analyzed in this section. As discussed by Hsu and Mo (1985), in the design of low rise structural walls, the boundary elements are reinforced to resist the applied bending moment, while the webs are designed to resist the applied shear force. The size and shape of the boundary elements do not have a significant influence on the shear behavior, as long as they are sufficient to carry the required bending moment. The effect of the boundary elements on structural walls has been studied by Mo and Kuo (1998). Owing to the restriction of the boundary elements, the strain of transverse steel in low rise framed shear walls can be neglected, as verified by the PCA tests; i.e., 1t ¼ 0: Therefore, adding Equation 14.48 and Equation 14.49 gives 1r ¼ 1l 2

Inserting 1r sin a ¼ 1r

1d

ð14:60Þ

2

1r cos a into Equation 14.20 gives 1 1l cos2 a ¼ r 1r 1d

ð14:61Þ

fl ¼

1 s rl l

VER. DIR.

Substituting Equation (14.60) and Equation (14.61) into Equation 14.45 results in

HOR. DIR.

sd

ð 1d Þ ð1l 21d Þ

1d Þ 21d Þ

ð14:62Þ

σdb cos α τb

(a) General view

A

I (b) Wall element

α τ(1)

α

I

co s

A

cot α

I

ð1l ð1l

τb cot α

V hw

sr

tf

b A

C

B

D

d (c) Section I-I FIGURE 14.17

A framed shear wall.

14 16


Neglecting the tensile strength of concrete, i.e., sr ¼ 0; gives fl ¼

1 s rl l

sd

ð 1d Þ ð1l 21d Þ

ð14:63Þ

For low rise framed shear walls, the average shear stress t on the horizontal cross section is defined as

t¼

V bd

ð14:64Þ

where d is the effective depth, which is defined as the distance between the centroids of the longitudinal bars in the two flanges, b is the width of the web, and V is the horizontal shear force. The deflection at the top of the shear wall, d; is determined by

d ¼ gh

ð14:65Þ

where h ¼ height of the shear wall. Based on the softened truss model theory presented above, the algorithm is shown in Figure 14.18 (Mo and Jost, 1993; Mo and Shiau, 1993) and is explained below. 1. 2. 3. 4. 5. 6. 7. 8. 9a. 9b.

Select a given 1d : Assume a value of 1l : Calculate 1r from Equation 14.60. Calculate z using Equation 14.51b. Calculate sd from Equation 14.51a. Calculate sr from Equation 14.52 or Equation 14.53. Calculate fl from Equation 14.62 or Equation 14.63. Check fl using Equation 14.54 and Equation 14.55 or from Equation 14.56 to Equation 14.59. If the calculated value for fl determined in Step 8 is not sufficiently close to the value shown in step 7, repeat steps 2 to 7. If the calculated value for fl deter mined in Step 8 is sufficiently close to the value shown in Step 7, proceed to calculate t (or V) and g (or d) from Equation 14.47 (or Equation 14.64) and Equation 14.49 (or Equation 14.65), respectively. This will provide one set of solutions.

Select other values of 1d and repeat Steps 1 to 9 for each 1d : This will provide a number of sets of quantities. From these sets of quantities, one can plot the shear stress vs. distortion curve (or shear force vs. deflection curve), the longitudinal steel strain vs. deflection curve, and the longitudinal steel strain vs. concrete strain curve. In general, the maximum 1d value can be chosen as 0.003 with an increment of 0.00005.

Select ed Estimate el Calculate er Calculate z Calculate sd Calculate sr Calculate fl

check if the error for fl is acceptable

NO

Yes Calculate t, g, n, d check if ed > 0.003

NO

Yes END FIGURE 14.18 analysis.

Algorithm for framed shear wall


14.2.4

14 17

Time-History Analysis

To accurately determine the dynamic behavior of concrete structures, the time history analysis (Clough and Penzien, 1993; Paz, 1997) is preferred. This section will show an example for single DoF systems, such as simple beams, torsional box tubes, spandrel beams, continuous beams, one story frames and one story framed shear walls. All of these structures will be discussed later. In time history analysis, a framed shear wall can be modeled as a nonlinear single DoF system (Figure 14.19). The dynamic incremental equili brium is shown in Figure 14.19(c). The equation of the equilibrium is mD€yi þ ci D_yi þ ki Dyi ¼ DFi

ð14:66Þ

where m is the mass at the top. ci and ki are calculated for values of velocity and displacement corresponding to time t and assumed to remain constant during the increment of time Dt: Incremental acceleration, incremental velocity, and incremental displacement are D€yi ; D_yi ; and Dyi ; respectively. To perform the step by step integration of Equation 14.66, the linear acceleration method is employed. In this method, it is assumed that the acceleration may be expressed by a linear function of time during the time interval Dt: Let ti and tiþ1 ¼ ti þ Dt be, respectively, the designation for the time at the beginning and at the end of the time interval Dt: When the acceleration is assumed to be a linear function of time for the interval of time ti to tiþ1 ¼ ti þ Dt; as shown in Figure 14.20, the acceleration may be expressed as y€ ðtÞ ¼ y€ i þ

D€yi ðt Dt

ti Þ

m

F(t)

(a) y

k

m

c

(b)

m∆y··i

ki∆yi c ∆y· i

(c)

F(t)

∆Fi

i

FIGURE 14.19 (a) Framed shear wall with a mass m at the top; (b) model for a nonlinear single DoF system, and (c) free body diagram showing the incremental inertial force, the incremental damping force, the incremental spring force and the incremental external force.

ÿ

∆ÿi ÿi

ÿi+1

∆t ti

FIGURE 14.20 time interval.

t i+1

t

Linear variation of acceleration during

ð14:67Þ

Integrating Equation 14.67 twice with respect to time between the limits ti and t ¼ ti þ Dt and using the incremental displacement Dy as the basic variable gives D€yi ¼

6 Dy Dt 2 i

D_yi ¼

3 Dy Dt i

6 y Dt i

3€yi

ð14:68Þ

Dt y€ 2 i

ð14:69Þ

and 3_yi

14 18


The substitution of Equation 14.68 and Equation 14.69 into Equation 14.66 leads to the following form of the equation of motion: m

6 Dy Dt 2 i

6 y_ Dt i

3€yi þ ci

3 Dy Dt i

Dt y€ þ ki Dyi ¼ DFi 2 i

3_yi

ð14:70Þ

Transferring all the terms containing the unknown incremental displacement, Dyi ; to the left hand side gives ki Dyi ¼ DFi

ð14:71Þ

where ki ¼ ki þ

6m 3c þ i Dt Dt 2

ð14:72Þ

and DFi ¼ DFi þ m

6 Dt y_ i þ 3€yi þ ci 3_yi þ y€ Dt 2 i

ð14:73Þ

It should be noted that Equation 14.71 is equivalent to the static incremental equilibrium equation and may be solved for the incremental displacement by simply dividing the equivalent incremental load, DFi ; by the equivalent spring constant ki : The displacement yiþ1 and the velocity y_ iþ1 at time tiþ1 ¼ ti þ Dt are yiþ1 ¼ yi þ Dyi

ð14:74Þ

y_ iþ1 ¼ y_ i þ D_yi

ð14:75Þ

and The acceleration y€ iþ1 at the end of the time step is obtained directly from the differential equation of motion to avoid the errors that generally might tend to accumulate from step to step. It follows y€ iþ1 ¼

1 ½Fðtiþ1 Þ m

ciþ1_yi þ1

kiþyi þ1

ð14:76Þ

where the coefficients ciþ1 and kiþ1 are now evaluated at time tiþ1 . After the displacement, velocity, and acceleration have been determined at time tiþ1 ¼ ti þ Dt; the procedure just outlined is repeated to calculate these quantities at the following time step, tiþ2 ¼ tiþ1 þ Dt: In general, sufficiently accurate results can be obtained if the time interval is taken to be no longer than one tenth of the natural period of the structure (Clough and Penzien, 1993; Paz, 1997).

14.3 14.3.1

Beams under Harmonic Excitations Mechanical Properties

A 6 m long beam is simply supported. The cross section and material properties are shown in Figure 14.2(b). f 0c ¼ 39 N=mm2 A0s ¼ 142 mm2 d ¼ 305

12:7

6:35

1 ð19:05Þ ¼ 276 mm 2

As ¼ 852 mm2 Find the moment ðMÞ curvature ðf Þ relationship of the cross section at the midspan and the force ðPÞ displacement ðdÞ relationship of the beam.


14 19

Solution 1. Moment curvature relationship qffiffiffi E ¼ 1500 f 0c ðN=mm2 Þ ¼ 29614:9 N=mm2

(563, 86.1) (122, 82.5)

M (kN m)

Using the derived equations in Section 14.2.2 gives

(8.66, 9.26)

Mu ¼ 86;122;000 N mm;

cu ¼ 5:63 £ 10

5

y (×10−7 rad/mm)

rad=mm FIGURE 14.21 example beam.

My ¼ 82;491;700 N mm;

cy ¼ 1:22 £ 10

5

rad=mm

Mc ¼ 9;263;000 N mm

cc ¼ 8:66 £ 10

Moment curvature curve for the

7

rad=mm

They are plotted in Figure 14.21, and the corresponding loads are determined below. 2. Force displacement relationship

PU ¼

4 £ 86;122;000 ¼ 57;415 N 6000

PY ¼

4 £ 82;491;700 ¼ 54;994 N 6000

PC ¼

4 £ 9;263;000 ¼ 6175 N 6000

The corresponding deflections can be determined as follows:

A

C

B

1. Cracking state (Figure 14.22) ψc

dc ¼ ¼

Mc L2 12EI 9;263;000£ð6000Þ2 1 12£ð29;614:9Þ£ £ð152:4Þ£ð305Þ3 12

L FIGURE 14.22

¼ 2:6 mm

L1

L2

A

2. Yielding state (Figure 14.23) 9;263;000 L1 ¼ 3000£ ¼ 336:9 mm 82;491;700

RA

Curvature diagram at cracking state.

Yc L3

FIGURE 14.23

C

B

Yy Curvature diagram at yielding state.

14 20


L2 ¼ 3000 L3 ¼ 336:9 þ

8:66 £ 10

336:9 ¼ 2663:1mm

2663:1 2663:1 2 þ ð1:22 £ 10 5 8:66 £ 10 7 Þ £ £ £ 2663:1 2 2 3 1 5 5 7 8:66 £ 10 Þ 8:66 £ 10 £ 2663:1 þ £ 2663:1 £ ð1:22 £ 10 2 7

£ 2663:1 £

¼ 2053 mm

dy ¼ 13 £ 8:66 £ 10

7

£ ð342:2Þ2 þ 12 £ ð8:66 £ 10

7

þ 1:22 £ 10 5 Þ £ ð2663:1Þ £ 2053 ¼ 35:75 mm L1

3. Ultimate state (Figure 14.24)

82;491;600 86;122;000 ¼ 2550:8 mm

L4 ¼ 3000

L3 ¼ 322:7 þ

322:7

¼ 1967:3 mm

L5 ¼ 322:7þ2550:8þ

Y L3 c

322:7

L5

C

B

Yy Yu

2550:8 ¼ 126:5 mm

8:66 £ 10

L4

A

9;263;000 ¼ 322:7 mm L1 ¼ 3000 £ 86;122;000 L2 ¼ 3000 £

L2

FIGURE 14.24

Curvature diagram at yielding state.

2550:8 2550:8 2 þ ð1:22 £ 10 5 8:66 £ 10 7 Þ £ £ £ 2550:8 2 2 3 1 7 5 7 8:66 £ 10 £ 2550:8 þ £ 2550:8 £ ð1:22 £ 10 8:66 £ 10 Þ 2 7

£ 2550:8 £

126:5 126:5 2 þð5:63£10 5 1:22£10 5 Þ£ £ £126:5 2 2 3 1 5 5 5 1:22£10 £126:5þ £126:5£ð5:63£10 1:22£10 Þ 2

1:22£10

¼ 2950:4 mm

5

£126:5£

1 du ¼ £8:66£10 7 £ð322:7Þ2 3 1 þ £ð8:66£10 7 þ1:22£10 5 Þ 2 £ 2550:8£1967:3 1 þ £ð5:63£10 5 þ1:22£10 5 Þ 2 £ 126:5£2950:4 ¼ 45:6 mm

(45.6, 57415) (35.75, 54994) P (N) (2.6, 6175) d (mm)

The load deflection Figure 14.25.

curve

is

shown

in

FIGURE 14.25

Load deflection curve.


14.3.2

14 21

Design for Machine Vibration

The same beam as described in Section 14.3.1 is supporting a machine with a rotating frequency of 9 Hz at the midspan of the beam. The mass of the rotating machine and the beam is 1 N sec2/mm. Check if resonance will occur.

Solution Find the stiffness, k; the natural frequency, f ; and the frequency ratio, r; for each of the three states, namely the elastic, cracking, and yield states. 1. Elastic state 617:5 ¼ 2375 N=mm 2:6 rffiffiffiffiffiffiffi 1 2375 ¼ 7:76 Hz f ¼ 2p 1 7:76 r¼ ¼ 0:862 9 k¼

Since 0:8 , r , 1:2; it is in the resonance zone. The beam needs to be redesigned to avoid resonance. 2. Cracking state k¼

5499:4 35:75

617:5 ¼ 1472:7 N=mm 2:6 rffiffiffiffiffiffiffiffiffiffi 1 1472:7 f ¼ ¼ 6:11 Hz 2p 1 6:11 ¼ 0:68 , 0:8 r¼ 9

OK 3. Yield state k¼

5741:5 45:6

5499:4 ¼ 245:8 N=mm 35:75 rffiffiffiffiffiffiffiffi 1 245:8 ¼ 2:50 Hz f ¼ 2p 1 2:50 r¼ ¼ 0:28 , 0:8 9

OK

14.4 14.4.1

Design for Explosions/Shocks Column

A 3 m tall viaduct column supporting a mass of 50 N sec2/cm is subjected to a shock acceleration that is the same as the 1940 El Centro seismogram. 14.4.1.1

Load–Displacement Relationship

Using the same procedures presented in Section 14.3, the moment curvature relationship of the cross section and the force displacement relationship of the column can be determined. It is assumed that the force displacement relationship of the column was found to be the same as that shown in Figure 14.25.

14 22


14.4.1.2

Dynamic Response

To perform the dynamic response analysis, the computer program in Mo (1994) can be used. The required input data may be determined as follows: Stiffness 1 ¼

0:6175 ¼ 0:2375 T=mm 2:6

Stiffness 2 ¼

5:4994 0:6175 ¼ 0:1473 T=mm 35:75 2:6

Stiffness 3 ¼

5:7415 45:6

5:4994 ¼ 0:246 T=mm 35:75

dc ¼ 2:6 mm dy ¼ 35:75 mm du ¼ 45:6 mm Result: Maximum deflection ¼ 38.33 mm

14.4.2

Shear Walls

14.4.2.1

Design Approach

14.4.2.1.1 Design Considerations To insure ductile failure, the two design limitations for overreinforcement and minimum reinforcement can be derived in terms of the inclination angle, a; of the diagonal concrete struts. 1. Overreinforcement. Mo (1987) shows the inclination angle of the diagonal compression struts in the “lower balanced reinforcement,” alb ; to be 8 2 0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi139 u 2 = ke ke 0 xe > me > > > > > > > > > > < = 6 < = < = 7 6 7 ms x€ s þ 4 ke ks þ ke ks 5 xs ¼ ms Xg v2 eivt ð22:15Þ > > > > > > > > > > : ; : > ; > : ; mb x€ b 0 ks ks þ kb xb mb Correspondingly, the steady state responses of the system can be expressed as 8 9 8 9 Xe > xe > > > > = = > < > < > xs ¼ Xs eivt > > > > > ; ; > : > : > Xb xb

ð22:16Þ

where ðXe ; Xs ; and Xb Þ represent the amplitudes of the equipment, structure, and base, respectively. Substituting Equation 22.16 into Equation 22.15 yields 9 2 38 9 8 me > ke 0 ke me v2 > > Xe > > > > > < = < = 6 7 2 6 7 Xs ¼ ms Xg v2 ð22:17Þ v k k k þ k m s e s e s 4 5> > > > > > > ; > : ; 2 : Xb mb 0 ks ks þ kb mb v

22 12


from which the amplitudes ðXe ; Xs ; and Xb Þ for the system can be solved as follows: Xs þ Xg fe2 1 fe2

ð22:18aÞ

Xs þ 1b fs2 Xg 1 þ kb =ks 1b fs2

ð22:18bÞ

me mb X v2 þ 2 1 fe 1 þ kb =ks 1b fs2 g kb mb v2 me ms þ v2 1 þ kb =ks 1b fs2 1 fe2

ð22:18cÞ

Xe ¼ Xb ¼

Xs ¼

ms þ

where the amplitudes of the equipment and base, that is, Xe and Xb ; have been expressed in terms of the amplitude of the base, Xs : The parameters in Equation 22.18 are defined as fe ¼ v=ve

ð22:19aÞ

fs ¼ v=vs

ð22:19bÞ

1b ¼ mb =ms pffiffiffiffiffiffiffi ve ¼ ke =me pffiffiffiffiffiffiffi vs ¼ ks =ms

ð22:19cÞ ð22:19dÞ ð22:19eÞ

Finally, the state steady absolute acceleration responses of the structure, equipment, and base can be expressed in terms of the ground acceleration x€ g as as ¼ x€ s þ x€ g ¼

DðvÞ ¼ ðkb

kb x€ g DðvÞ

ð22:20aÞ

ðXs þ Xg Þv2 eivt kb x€ g ¼ 1 fe2 ð1 fe2 ÞDðvÞ

ð22:20bÞ

½Xs þ Xg þ ðkb =ks ÞXg v2 eivt ðD 1 ðvÞ þ ks 1 Þkb x€ g ¼ 1 þ kb =ks 1b fs2 1 þ kb =ks 1b fs2

ð22:20cÞ

ae ¼ x€ e þ x€ g ¼ ab ¼ x€ b þ x€ g ¼

ðXs þ Xg Þv2 eivt ¼

mb v2 Þ

ð1 þ kb =ks

1b fs2 Þ ms þ

1

me v2 fe2

ð22:20dÞ

As can be seen, the acceleration response of each component in the structure equipment system depends mainly on the stiffness of the elastic bearing, kb : In particular, the use of a smaller bearing stiffness, kb ; can result in significant reduction of the shear forces transmitted to the superstructure, as indicated by Equation 22.20a. This explains why an elastic bearing can be effectively used as an isolator for reducing the base shear of the structure equipment system. In contrast, if the bearing stiffness, kb ; is made to be infinitely large, that is, by letting kb ! 1; the acceleration responses in Equation 22.20, reduce to as ¼ ae ¼

ð1 ð1

fe2 Þð1

ð1

fe2 Þð1

fe2 Þ€xg fs2 Þ x€ g

ab ¼ x€ g

fs2 Þ

1e fs2 1e fs2

ð22:21aÞ ð22:21bÞ ð22:21cÞ

with the use of L’Hospital’s Rule, where 1e ¼ me =ms : As can be seen from Equation 22.21c, the acceleration of the structural base is equal to the ground acceleration. Clearly, the present problem has been reduced to a two DoF system with a rigid base, for which the solutions have been given in Equation 22.21a and Equation 22.21b.

Structure and Equipment Isolation

22 13

Some important high tech facilities, such as semiconductor factories and medical devices, are very sensitive to vibrations, especially to those caused by resonance. To consider the effect of resonance, we shall let the ground excitation frequency, v; coincide with the equipment frequency, ve ; that is, by letting fe ¼ 1 or ve ¼ v: For this case, the acceleration responses of the system in Equation 22.20 reduce to as ¼ 0 ae ¼

ð22:22aÞ

kb x€ g ke ½1 þ ðkb mb v2e Þ=ks

ð22:22bÞ

kb x€ g ks þ kb mb v2e

ð22:22cÞ

ab ¼

Because of the coincidence of the ground excitation frequency with the equipment frequency, the equipment behaves like a vibration absorber of the structure. For this reason, the response of the equipment is greatly amplified, as implied by Equation 22.22b, while the response of structure is completely suppressed, as indicated by Equation 22.22a. Moreover, if the frequency of the equipment is equal to the fundamental frequency of the structure equipment isolation system, that is, ve ð< v1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kb =ðme þ ms þ mb Þ; then the responses of the system in Equation 22.22 can further be expressed as follows: as ¼ 0 ae ¼

ð22:23aÞ

kb x€ g ke ½1 þ ðms þ me Þv2e =ks

ð22:23bÞ

kb x€ g ks þ ðms þ me Þv2e

ð22:23cÞ

ab ¼

Since the equipment mass is generally much smaller than the structural mass, the preceding equation can be further reduced to as ¼ 0

ð22:24aÞ

ae ¼

kb x€ g ke ð1 þ v2e =v2s Þ

ð22:24bÞ

ab ¼

kb x€ g ks ð1 þ v2e =v2s Þ

ð22:24cÞ

As indicated by Equation 22.24b, the acceleration response of the equipment depends on the stiffness ratio, kb =ke ; of the base to the equipment. For the resonance condition of ve ¼ v; mentioned previously, let us consider the case when the structural frequency is equal to the equipment frequency, that is, ve ¼ vs : For this case, the responses of the system in Equation 22.22 reduce to as < 0 ae
> > > = < ð22:28Þ xðtÞ ¼ xs ðtÞ > > > > ; : xb ðtÞ The mass, damping, and stiffness matrices in Equation 22.27 are defined as 3 2 me 0 0 7 6 7 M¼6 4 0 ms 0 5; 0 2 6 C¼6 4 2 6 K¼6 4

ce ce

0

mb ce

ce þ cs

7 cs 7 5;

cs

cs

ke

ke

0

0

ke þ ks ks

FIGURE 22.15 Schematic for an isolated structure equipment system with sliding bearing.

ce

ks

me

ke

cs

μ

3

0

0

ke

Sliding Bearing

xe

ms mb

xs

xb

.. xg

ð22:29Þ

3

FIGURE 22.16 Model for an isolated structure equipment system with sliding support.

7 ks 7 5

ks

and the force distribution vectors as 8 9 1> > > < > = L1 ¼ 1 ; > > > : > ; 1

8 9 0> > > < > = L2 ¼ 0 > > > : > ; 1

ð22:30Þ

Note that in Equation 22.27, the isolator frictional force, f ðtÞ; which is not a constant, is moved to the right hand side of the equation. This nonlinear force requires a special treatment in an analysis procedure, as will be explained later on. For a systematic treatment, the above equation of motion can be further written in a state space form as shown below (Meirovitch, 1990): z_ ðtÞ ¼ AzðtÞ þ E€xg ðtÞ þ Bf ðtÞ

ð22:31Þ


22 19

where the state vector zðtÞ and the system matrix A are defined as " # " # x_ ðtÞ M 1C M 1K zðtÞ ¼ ; A¼ xðtÞ I 0 and the excitation and friction distribution vectors as " # " # L1 M 1 L2 ; B¼ E¼ 0 0

22.4.1.2

ð22:32Þ

ð22:33Þ

Conditions for Stick and Sliding States

As mentioned above, the motion of a sliding structure at any instant has two possible states, namely, the stick (or nonsliding) and sliding states. The following are the conditions that must be satisfied by the sliding structure: (1) In stick state lf ðtÞl , fmax ¼ mW

ð22:34aÞ

x_ b ðtÞ ¼ 0

ð22:34bÞ

where m is the coefficient of friction and W is the total weight of the structure. According to the preceding equations, the frictional force in the stick state is an unknown with a magnitude less than the maximum frictional force, fmax ; which equals the product of m and W; while the sliding velocity of the structure is simply zero. Whenever the frictional force satisfies Equation 22.34a, the sliding system remains in the stick state, otherwise it changes into the sliding state. (2) In sliding state f ðtÞ ¼

sgnð_xb ðtÞÞfmax ¼

sgnð_xb ðtÞÞmW

x_ b ðtÞ – 0

ð22:35aÞ ð22:35bÞ

where the function sgnðxÞ denotes the sign of the variable x: According to Equation 22.35a and Equation 22.35b, the frictional force in the sliding state has a magnitude equal to the maximum frictional force, but directed in a sense opposite to that of the sliding velocity. On the other hand, the sliding velocity of the isolator remains as an unknown.

22.4.2

Methods for Numerical Analysis

Two numerical methods commonly used for the analysis of sliding isolated structural systems, the shear balance method and fictitious spring method, will be introduced in this section. By employing the discrete time state space formula, both methods can be cast in an incremental form that is suitable for the analysis of sliding systems with multiple DoF. 22.4.2.1

Shear Balance Method

Consider the state space equation, Equation 22.31, and assume that both the ground acceleration and frictional force vary linearly within each time interval, as shown in Figure 22.17. Equation 22.31 may be written in the following incremental form (Meirovitch, 1990) z½k þ 1 ¼ Ad z½k þ E0 x€ g ½k þ E1 x€ g ½k þ 1 þ B0 f ½k þ B1 f ½k þ 1

ð22:36Þ

where the symbol x½k denotes that the variable x is evaluated at the kth time step. The other coefficient

22 20


matrices in equation 22.36 are defined as

B0 ¼ ðAÞ 1 Ad þ B1 ¼

ðAÞ

1

þ

1 X Dt i i A i! i¼0

ð22:37Þ

1 ðAÞ 2 ðI Dt

Ad Þ B

ð22:38aÞ

1 ðAÞ 2 ðAd Dt

IÞ B

ð22:38bÞ

Ground acceleration

Ad ¼ eADt ¼

.. xg[k + 1] .. xg[k]

Time step where Dt denotes the size of the time step k k+1 considered for analysis. The matrices E0 and E1 can be computed in a way similar to B0 and B1 ; FIGURE 22.17 Force integration scheme with linear except that the matrix B in Equation 22.38a and interpolation. Equation 22.38b should be replaced by the matrix E. In some applications, the system matrix A may not be invertible, that is, A 1 does not always exist. If this is the case, one may compute B0 and B1 (and similarly E0 and E1 ) by the following formulas:

B0 ¼ Apd B;

B1 ¼ ðA^ d

Apd ÞB

ð22:39Þ

where Apd ¼

1 X ðDtÞiþ1 i A; i!ði þ 2Þ i¼0

A^ d ¼

1 X ðDtÞiþ1 i A ði þ 1Þ! i¼0

ð22:40Þ

Note that, on the right hand side of Equation 22.36, the only unknown at the kth time step is the frictional force f ½k þ 1 ; therefore, f ½k þ 1 must be determined before the next time step response z½k þ 1 is computed. Wang et al. (1998) proposed the shear balance method for computing the frictional force f ½k þ 1 : By this method, the sliding structure is first assumed to be in the stick state at the ðk þ 1Þth step, for which the condition given in Equation 22.34b must be satisfied x_ b ½k þ 1 ¼ Dz½k þ 1 ¼ 0

ð22:41Þ

where D is a relation matrix, equal to D ¼ ½0 0 1 0 0 0 for the model shown in Figure 22.16. Substituting z½k þ 1 in Equation 22.36 into Equation 22.41, one may solve for the estimated frictional force at the ðk þ 1Þth time step as f ½k þ 1 ¼

ðDB1 Þ 1 DðAd z½k þ B0 f ½k þ E0 x€ g ½k þ E1 x€ g ½k þ 1 Þ

ð22:42Þ

where f ½k þ 1 with an overbar signifies that the frictional force is an estimate obtained by assuming the sliding structure to be at the stick state. Such a value may not be the actual one if the system is not in the stick state. The physical meaning for f ½k þ 1 is that it represents the balanced shear force required at the ðk þ 1Þth time step for the structure to remain in the stick state. Therefore, the sign of f ½k þ 1 indicates the direction of the resistant force provided by the isolation system. In spite of the fact that f ½k þ 1 may not be the actual frictional force, it plays an important role for determining the actual state (stick or sliding) and the actual frictional force of the sliding isolated structure, as will be described below based on Equation 22.34a and Equation 22.35a. (1) The system is in the “stick state” if lf ½k þ 1 l , fmax and the frictional force is f ½k þ 1 ¼ f ½k þ 1

ð22:43Þ

(2) The system is in the “sliding state” if lf ½k þ 1 l $ fmax and the frictional force is f ½k þ 1 ¼ sgnð f ½k þ 1 Þfmax

ð22:44Þ


22 21

As can be seen, the term sgnð_xb ½k þ 1 Þ in Equation 22.35a is replaced by sgnðf ½k þ 1 Þ in Equation 22.44. Such a replacement is justified since the sign of f ½k þ 1 indicates the direction of the resistant force at the ðk þ 1Þth time step. Once the correct frictional force, f ½k þ 1 ; is determined by using either Equation 22.43 or Equation 22.44, it can be substituted into Equation 22.36 to obtain the response z½k þ 1 for the next time step. The computational flow chart for the shear balance method has been given in Figure 22.18. 22.4.2.2

Fictitious Spring Method

The fictitious spring method was first proposed by Yang et al. (1990) for the analysis of a sliding structure. Later, Lu and Yang (1997) reformulated the method into a state space form for the analysis of equipment Start

k=0

Assume stick state and compute estimated friction force f [k + 1]

No

|f [k + 1]| < fmax

Sliding State

Yes

Stick State

f [k+1] = |f [k+1]|

f [k + 1] = sgn( f [k + 1]) fmax

Compute response z[k + 1] = Ad z[k] + E0 xg [k] + E1 xg [k + 1] + B0 f [k] + B1 f [k + 1]

k=k+1

No

End of time steps Yes Stop

FIGURE 22.18

Computational flow chart for shear balance method.

22 22


mounted on a sliding structure. By this method, a fictitious spring, kf ; is introduced between the base mat and the ground, as in Figure 22.19, to represent the mechanism of sliding or friction. The stiffness, kf ; of the fictitious spring is taken as zero for the sliding state and as a very large value for the stick state. With the introduction of the fictitious spring, the stiffness matrix, K; in Equation 22.29 should be modified as follows: 2 3 ke ke 0 6 7 K ¼ Kðkf Þ ¼ 6 ks 7 4 ke ke þ ks 5 ð22:45Þ 0

ks

ce

xb0

kf

ks

me

ke

cs

xe

ms

mb

xs

xb

xg

FIGURE 22.19 Model for isolated structures with fictitious spring.

ks þ kf

Accordingly, the state space dynamic equation, Equation 22.31, should be modified as ~ z_ ðtÞ ¼ Aðkf ÞzðtÞ þ E€xg ðtÞ þ BfðtÞ where

" A ¼ Aðkf Þ ¼

M 1C

M 1 Kðkf Þ

I

ð22:46Þ #

0

ð22:47Þ

Depending on the current state of the sliding system, the fictitious stiffness, kf ; and the modified friction ~ in Equation 22.46 may take one of the following two sets of values: term, fðtÞ; (1) In the stick state kf ¼ a ks ;

~ ¼ kf xb0 fðtÞ

ð22:48Þ

(2) In the sliding state kf ¼ 0;

~ ¼ fðtÞ

sgnð_xb ðtÞÞmW

ð22:49Þ

In Equation 22.48, the symbol a represents a constant of very large value, and xb0 the initial elongation of the fictitious spring in the current stick state (computation of xb0 will be explained later). Note that the modified friction term, f~ ðtÞ; may not be the actual frictional force. The actual frictional force can be determined as follows: (1) In the stick state f ðtÞ ¼ kf ðxb ðtÞ

xb0 Þ

ð22:50Þ

(2) In the sliding state ~ f ðtÞ ¼ fðtÞ

ð22:51Þ

According to Equation 22.50 and Equation 22.51, the actual frictional force of the isolation system in the stick state is equal to the internal force of the fictitious spring, while in the sliding state it is equal to the ~ The frictional force computed from the preceding two equations should modified frictional force, fðtÞ: obey the conditions given in Equation 22.34a and Equation 22.35a as well. With the conditions imposed for the stick and sliding states in Equation 22.48 and Equation 22.49, respectively, the equation of motion in Equation 22.46 actually represents two different sets of equations.


22 23

Ground acceleration

Specifically, Equation 22.46 and Equation 22.48 collectively describe the motion of the structure in the stick state, while Equation 22.46 and Equation .. 22.49 represent the motion of the structure in the xg [k] sliding state. Owing to the fact that a sliding system may switch between the two states at certain instants, the behavior of the entire system should undoubtedly be regarded as a nonlinear one. Nevertheless, within each particular state, the behavior of the system as represented either by Time step k k+1 Equation 22.46 and Equation 22.48 or Equation 22.46 and Equation 22.49 is a linear one. FIGURE 22.20 Constant force integration scheme. In the following, a numerical solution scheme based on the concept of fictitious spring will be introduced. Let Dt denote a time increment, which is usually taken as a very small value, and assume that the ground excitation and frictional force are constant within each time increment, Dt (see Figure 22.20). Accordingly, the discrete time solution of Equation 22.46 can be rewritten in an incremental form (Meirovitch, 1990) as z½k þ 1 ¼ Ad z½k þ Ed x€ g ½k þ Bd f~ ½k

ð22:52Þ

where Ad ¼ Ad ðkf Þ ¼ eAðkf ÞDt ¼

1 X Dt i Aðkf Þi i! i¼0

ð22:53Þ

Ed ¼ Ed ðkf Þ ¼ Aðkf Þ 1 ðAd ðkf Þ

IÞE

ð22:54aÞ

Bd ¼ Bd ðkf Þ ¼ Aðkf Þ 1 ðAd ðkf Þ

IÞB

ð22:54bÞ

For the case where the system matrix A is invertible, Bd and Ed may be computed instead using the following formulas: " # 1 X Dt i i 1 Ed ¼ Ed ðkf Þ ¼ Aðkf Þ E ð22:55Þ i! i¼0 "1 # X Dt i i 1 Bd ¼ Bd ðkf Þ ¼ Aðkf Þ B i! i¼0

ð22:56Þ

Equation 22.52 is the solution of the sliding system given in incremental form, because the response, z½k þ 1 ; can be computed from the solution of the previous step, z½k : Note that, in Equation 22.53 and Equation 22.54, the coefficient matrices Ad ; Ed ; and Bd have two possible sets of values, as the fictitious spring constant, kf ; may take different values for the sliding and stick states. Nevertheless, once the time step size, Dt; is chosen, the coefficient matrices Ad ; Ed ; and Bd ; remain constant for each state. As such, they need only be calculated once at the beginning of the incremental procedure. The computational flow chart for the fictitious spring method described above has been given in Figure 22.21. The dynamic equation and its discrete time solution for the sliding structure in the two states have been presented above. In the following, we shall describe how to determine the transition time for the sliding structure to switch from one state to the other. Once the transition time is determined, the original step size should be scaled down accordingly to reflect the transition point (Yang et al., 1990).

22 24


Start

k = 0; Assume stick state

yes

no

Check sliding state?

Sliding state

Stick state

~ kf = 0 and f [k] = -sgn(xb[k]) fmax

~ kf ≠ 0 and f [k] = kf xb0

Compute response: ~ z[k + 1] = Ad z[k] + Ed xg[k] + Bd f [k]

End of time steps?

yes

Stop

no Determine the state of the next time step

k=k+1 FIGURE 22.21

Computational flow chart for fictitious spring method.

Transition from stick to sliding state. As stated in Equation 22.34a, the condition for a sliding isolated structure to remain in the stick state is that the static frictional force under the base mat be less than its maximum value, fmax . Once the frictional force exceeds this maximum value, the system starts to slide. With the fictitious spring method, the static frictional force is computed as the internal force of the fictitious spring, as in Equation 22.50. Based on the above considerations and Equation 22.50, the condition for the sliding system to transfer from the stick to the sliding state is lf ðt0 Þl ¼ lkf ðxb ðt0 Þ

xb0 Þl ¼ fmax ¼ mW

ð22:57Þ

where t0 denotes the transition time at which the structure starts to slide and kf is the spring constant given in Equation 22.48 for the stick state. Because a very large value has been used for the fictitious spring constant, kf ; in the stick state, the deviation in base displacement due to spring elongation is very small in this state, which can just be neglected. In practice, the transition time, t0 may not occur precisely


22 25

at the discrete points of time considered. It is likely that the spring force is less than fmax at the current time step, say, the kth step, but exceeds fmax at the following time step. If this is the case, numerical methods such as the bisection method should be employed to locate the transition time, t0 within the time interval ðkDt; ðk þ 1ÞDtÞ considered based on Equation 22.57. Transition from sliding state to stick state. The structure in the sliding state may return to the stick state whenever the following two conditions are satisfied. (1) The relative velocity of the base mat to the ground reaches zero, that is, j_b ðt0 Þ ¼ 0 where t0 is the transition time; (2) the estimated static frictional force, denoted by f ðt0 Þ; is less than the maximum static frictional force, that is, f ðt0 Þ , fmax : Here, the estimated static frictional force, f ðt0 Þ; is defined as the shear force required to balance the motion of the superstructure if the system is assumed to be in the stick state, similar to the one given in Equation 22.42 for f ½k þ 1 : By letting the relative velocity and relative acceleration between the base mat and the ground be equal to zero, that is, j€b ðt0 Þ ¼ j_b ðt0 Þ ¼ 0; the estimated frictional force can be calculated from the free body diagram of the base mat as f ðt0 Þ ¼ mb x€ g ðt0 Þ

ks ðxs ðt0 Þ

xb ðt0 ÞÞ

cs x_ s ðt0 Þ

ð22:58Þ

For the sliding structure to transfer from the sliding to the stick state, both the aforementioned conditions must be satisfied simultaneously. Once the structure enters the stick state, the term f ðt0 Þ should be set equal to f ðt0 Þ and used as the initial frictional force. For the sake of equilibrium, the initial base mat displacement, xb0 ; should be computed as xb0 ¼ xb ðt0 Þ

ð f ðt0 Þ=kf Þ

ð22:59Þ

where the value of kf is the one given for the stick state in Equation 22.48. Concerning the two conditions mentioned above, it may happen that only the first condition, j_b ¼ 0 is satisfied, while the computed f ðt0 Þ is still larger than fmax : If this is the case, the sliding system should not be regarded as a transition to the stick state. Rather, the situation should be regarded as an indication for reversing the direction of sliding in the next time step. Correspondingly, the frictional force, f ðt0 Þ; should be set equal to the sliding frictional force, rather than the estimated one, f ðt0 Þ:

22.4.3

Simulation Results for Sliding Isolated Systems

22.4.3.1

Numerical Model and Ground Excitations

In this section, the dynamic behavior of the sliding isolated structure equipment system shown in Figure 22.16 will be analyzed using the shear balance method. Although the sliding structure and equipment considered are both of single DoF, there exists no difficulty for use of the method to solve problems with multi DoF systems. In Table 22.3, the material properties adopted for the present model have been listed, which are intended to simulate a small, five story, reinforced concrete frame. For the present purposes, two types of ground excitation are considered, namely, harmonic and earthquake excitations. The harmonic excitation is considered primarily for studying the frequency response of the sliding system, while the earthquake excitation is considered for the effect of earthquake intensity. For the TABLE 22.3

System Parameters Used in Simulation (Section 22.4.3) Equipment

Parameter Mass me Frequency ve Damping ratio ze

Superstructure

Isolation System

Value

Parameter

Value

Parameter

Value

3 t ð¼ ms =100Þ 5vs or a variable 5%

Mass ms Frequency vs Damping ratio zs

300 t 1.67 Hz 5%

Mass mb Frictional coefficient m

100 t ð¼ ms =3Þ 0.05, 0.1, 0.25

22 26


harmonic excitation, a sinusoidal ground acceleration of the following form is adopted: x€ g ðtÞ ¼ 0:5g sin vg t

ð22:60Þ

where vg denotes the excitation frequency and g is the gravitational acceleration. For the earthquake excitation, the 1940 El Centro earthquake (NS component) is considered, for which the waveform has been given in Figure 22.10. The PGA level of the earthquake will be adjusted for reasons of research. Concerning the effectiveness of response reduction, three quantities are chosen as the indices, namely, the base displacement (base drift), structural acceleration, and equipment acceleration. For all the figures shown in Section 22.4.3 below, the symbol m in the legend is used to denote the frictional coefficient of the sliding isolation system and the word “fixed” denotes the response of the corresponding fixed base structure. 22.4.3.2

Harmonic Response of Structure

0.5

Displacement (m)

Displacement (m)

m = 0.1 Time history. For the isolated system subjected to a m = 0.05 harmonic excitation of vg ¼ 1 Hz, the base 0 displacements computed for different coefficients of friction were plotted in Figure 22.22 and −0.5 Figure 22.23. As can be seen, the base displace ments quickly reach the steady state response −1 within the first few cycles. Meanwhile, the use of a smaller frictional coefficient results in a larger −1.5 permanent displacement before the steady state is reached. From the structural accelerations plotted –2 0 5 10 15 in Figure 22.24, one observes that, for a sliding Time (s) structure with a smaller coefficient of friction, the steady state response is achieved in a faster way, FIGURE 22.22 Comparison of base displacements accompanied by a larger reduction on structural ðvg ¼ 1 Hz; ve ¼ 5vs Þ: acceleration. Of interest in Figure 22.24 is that the response of the fixed base case shows a clear period 0.2 of 1 sec, while in the sliding case, the response is m = 0.1 contaminated by high frequency signals caused by m = 0.25 0 the sliding stick transitions. Hysteretic behavior. In order to understand the −0.2 mechanical characteristics of a nonlinear device used for vibration control, it is common to present a diagram showing the force deformation relation −0.4 of the device, also referred to as the hysteretic diagram (Soong and Dargush, 1997). Figure 22.25 −0.6 shows the hysteresis loops of the sliding isolation system (the sliding layer) for m ¼ 0:1 and 0.25, −0.8 0 5 10 15 when the system is subjected to a harmonic Time (s) excitation of vg ¼ 1 Hz. In the figure, the horizontal and vertical axes, respectively, represent FIGURE 22.23 Comparison of base displacements the base displacement and shear force, that is, the ðv ¼ 1 Hz; v ¼ 5v Þ: g e s frictional force, under the mat. Just like many other frictional elements or devices, the shape of the hysteresis loop of a sliding bearing is rectangular. The height of the rectangle is equal to the maximum frictional force that depends on the coefficient of friction, while the width of the hysteresis loop is determined by the base sliding displacement. As the coefficient of friction decreases, the height of the loop decreases, while the width increases. The total area of the hysteresis loop is equivalent to the portion of the energy dissipated by the sliding bearing.


22 27

10

Acceleration (m/s2)

5

0

−5

−10 m = 0.1

Fixed −15

0

5

Time (s)

10

15

10

Acceleration (m/s2)

5 0

−5

−10 m = 0.25

−15

Fixed 0

FIGURE 22.24

5

Time (s)

10

15

Comparison of structural accelerations ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

Frequency response. Figure 22.26 shows the maximum structural acceleration with respect to the excitation frequency for four different frictional coefficients, m ¼ 0:05; 0.1, 0.25 and 1 (for the fixed base case). Here, the maximum acceleration means the steady state acceleration response. The following observations can be made from Figure 22.26: (1) Compared with the fixed base case, the use of a smaller frictional coefficient can reduce the structural acceleration for the frequency range considered. (2) The sliding mechanism can effectively suppress the main resonant response, associated with the natural frequency of 1.67 Hz of the superstructure system. (3) As the coefficient of friction, m; decreases from 1 to 0.05, the main resonant frequency associated with the structural natural frequency drifts from the fixed base frequency of 1.67 Hz toward a higher value. (4) For the sliding cases of m ¼ 0:05; 0.1, 0.25, there exist some minor peaks in the range of lower excitation frequencies, besides the main resonant peak. Such a phenomenon is called the subharmonic resonance. For a large frictional coefficient, say, with m ¼ 0:25; the subharmonic resonant response may be even larger than that of the main resonance.

22 28


1000

m = 0.1 m = 0.25

Total base shear (kN)

500

0

−500

−1000 − 0.8

FIGURE 22.25

−0.6

− 0.4 − 0.2 Displacement (m)

Acceleration (m/s2)

0.2

Hysteresis loops of a sliding bearing ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

10 2

m = 0.05 m = 0.1 m = 0.25

Fixed 101

100

10−1 10−1

FIGURE 22.26

0

100 Excitation Freq. (Hz)

101

Maximum structural acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

Figure 22.27 shows the frequency responses of the maximum base displacement for various frictional coefficients. The following are observed: (1) The larger the frictional coefficient, the smaller the base drift is. (2) The base displacement has very large magnitudes in the lower excitation frequencies and decreases monotonically as the excitation frequency increases. (3) The extremely large base drift exhibited in the lower excitation frequency range is due to the initial permanent displacement observed in Figure 22.22. 22.4.3.3

Harmonic Response of Equipment

Time history. Consider an equipment item of a natural frequency equal to five times of the structural frequency, that is, ve ¼ 5vs : For the case of a harmonic excitation of vg ¼ 1 Hz, the accelerations solved for the equipment mounted on the structure with m ¼ 0:1 and 0.25, along with the fixed base case, have


22 29

104

m = 0.05 m = 0.1 m = 0.25

Displacement (m)

102 100 10−2 10−4 10−6 10−8 10−1

FIGURE 22.27


101

Maximum base displacement vs. ground excitation frequency ðve ¼ 5vs Þ:

been plotted in Figure 22.28. As can be seen, the equipment quickly reaches the steady state within a few cycles of oscillation. The equipment response is effectively suppressed for the case with a smaller frictional coefficient. Additionally, the waveforms shown in Figure 22.28 for the equipment appear to be marginally higher than those of the primary structure shown in Figure 22.24, which can be attributed to the use of a relatively stiff equipment, that is, with ve ¼ 5vs : Frequency responses. For an equipment item of the frequency ve ¼ 5vs (¼ 8.34 Hz), the maximum acceleration response has been plotted as a function of the excitation frequency in Figure 22.29. By comparing Figure 22.29 with Figure 22.26, one observes that the frequency response curves of the equipment and primary structure are generally similar, except that a secondary resonant peak occurs around the equipment natural frequency of 8.34 Hz in Figure 22.29. The other observations from Figure 22.29 are as follows: (1) In comparison with the fixed base case, the sliding isolation alleviates both the structural and equipment resonant peaks around the frequencies of 1.67 and 8.34 Hz, respectively. However, the level of alleviation is more apparent for the former than for the latter. (2) The equipment also exhibits the same subharmonic resonance behavior as that of the primary structure, in terms of the resonance peaks and frequencies. (3) As the frictional coefficient m decreases from 1 (for the fixed base case) to 0.05, the main resonant frequency associated with the structure drifts toward a higher value. However, the resonant frequency associated with the equipment remains the same. Effect of equipment tuning. The effect of equipment tuning refers to the case when the equipment frequency is tuned to the structural frequency, that is, ve ¼ vs : Figure 22.30 shows the frequency response of the equipment when the equipment tuning occurs. Compared with Figure 22.29, this figure shows the following: (1) When equipment tuning occurs, the sliding isolation system can still mitigate the main resonant peak of the equipment, but the effectiveness of mitigation is drastically reduced. (2) Although the subharmonic resonance can still be observed, the relevant frequencies of the equipment are different from those of the primary structure. (3) The frequency of the maximum resonant response remains equal to the tuned equipment’s natural frequency of 1.67 Hz, regardless of the change in the frictional coefficient, m; from 1 to 0.05. 22.4.3.4

Earthquake Response of Structure

Time history. For the isolated system subjected to the El Centro earthquake with a PGA of 0:5g; the structural acceleration and base displacement of the sliding system have been plotted in Figure 22.31 and Figure 22.32, respectively, together with the response for the fixed base case in Figure 22.31. As can be

22 30


10

Acceleration (m/s2)

5

0

−5

−10 m = 0.1

Fixed −15

0

5

Time (s)

10

15

10

Acceleration (m/s2)

5 0

−5

−10

m = 0.25

Fixed −15

0

FIGURE 22.28

5

Time (s)

10

15

Comparison of equipment accelerations ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

seen from Figure 22.31, the main shock response occurring between 0 and 10 sec for the fixed base structure has been effectively suppressed by the sliding isolators with m ¼ 0:1 and 0.25. About 80 and 60% of the maximum structural acceleration have been suppressed by the isolators with m ¼ 0:1 and 0.25, respectively. On the other hand, Figure 22.32 demonstrates that the better suppression effect for the case with m ¼ 0:1 is achieved at the expense of a larger base displacement. It is interesting to note that the horizontal segments in the curves of Figure 22.32 actually represent the stick state of the sliding system, which is useful for unveiling the sliding stick mechanism involved. Effect of earthquake intensity. The maximum structural acceleration and base displacement vs. the PGA have been plotted in Figure 22.33 and Figure 22.34, respectively. As can be seen from Figure 22.33, the maximum structural acceleration for the fixed base case is proportional to the earthquake intensity, while in all the sliding cases it remains essentially as a constant after the PGA reaches a certain level. In other words, the reduction in structural maximum response and the efficiency of isolation have increased


22 31

102

m = 0.05 m = 0.1 m = 0.25

Acceleration (m/s2)

Fixed 101

100

10 −1 10−1

FIGURE 22.29


Maximum equipment acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

103

m = 0.05 m = 0.1 m = 0.25

Fixed

Acceleration (m/s2)

102

101

100

10 −1 10−1

FIGURE 22.30

101


101

Maximum equipment acceleration vs. ground excitation frequency for equipment tuning ðve ¼ vs Þ:

with the increase in earthquake intensity. For example, for the case of m ¼ 0:1; Figure 22.33 shows that the structural acceleration is reduced by around 60% at PGA ¼ 0:2g; while it is reduced by more than 90% at PGA ¼ 1:0g: However, as indicated by Figure 22.34, the above reduction in structural response has been achieved at the expense of increased base displacements. For the same PGA level, a sliding system with a smaller frictional coefficient has a better effect of vibration reduction, but this is accompanied by a larger base displacement. Residual base displacement. The residual base displacement is defined as the permanent base displacement of the structure after it stops vibrating. This quantity is important in the study of sliding structures. Figure 22.35 shows the residual base displacement as a function of the earthquake PGA level. A first look at the figure reveals that no clear relation exists between the earthquake intensity and residual

22 32


15

m = 0.1

Fixed

Acceleration (m/s2)

10 5 0 −5

−10 −15

0

10

20

30 Time (s)

40

15

60

m = 0.25

Fixed

10 Acceleration (m/s2)

50

5 0 −5

−10 −15

0

FIGURE 22.31

10

20

30 Time (s)

40

50

60

Comparison of structural accelerations ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

displacement, because a larger PGA may lead to a smaller residual base displacement. Nevertheless, after taking the average of the residual displacements over the PGA range of 0.1 to 1g; we obtain xres ¼ 0:083; 0.084, 0:084 m for the case of m ¼ 0:05; 0.1, 0.25, respectively. These values indicate that a smaller frictional coefficient leads to a larger residual base displacement in general. However, when the frictional coefficient, m; approaches zero, the residual displacement approaches a constant equal to the permanent ground displacement.

22.4.3.5

Earthquake Response of Equipment

Time history. Consider an equipment item with a natural frequency equal to five times the structural frequency, that is, ve ¼ 5vs (¼ 8.34 Hz). For the isolated system subjected to the El Centro earthquake


22 33

0.15

m = 0.1 m = 0.25

Displacement (m)

0.1

0.05

0

−0.05

0

FIGURE 22.32

10

20

30 Time (s)

40

50

60

Comparison of base displacements ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

m = 0.05 m = 0.1 m = 0.25


Fixed 15

10

5

0.2

FIGURE 22.33

0.4 0.6 0.8 Peak Ground Acceleration (g)

1

Maximum structural acceleration vs. PGA ðve ¼ 5vs Þ:

with PGA ¼ 0:5g; the time histories computed for the equipment acceleration for the cases with m ¼ 0:1 and 0.25, along with the fixed base case, have been plotted in Figure 22.36a and b. As can be seen, the main shock response of the fixed base structure occurring for the first 10 sec has been effectively suppressed through installation of the sliding isolator with m ¼ 0:1 and 0.25. A higher level of reduction can be achieved if a smaller frictional coefficient is chosen. Effect of earthquake intensity. Figure 22.37 shows the maximum equipment acceleration as a function of the PGA level. Because of the use of a relatively stiff equipment ðve ¼ 5vs Þ; the curves shown in Figure 22.37 are similar to those for the primary structure in Figure 22.33, but with slightly higher values. Therefore, the observations made previously for Figure 22.33 are applicable to Figure 22.37. The maximum response of equipment items with other frequencies will be discussed below.

22 34


m = 0.05 m = 0.1 m = 0.25

Displacement (m)

0.5 0.4 0.3 0.2 0.1

0.2

FIGURE 22.34

1

Maximum base displacement vs. PGA ðve ¼ 5vs Þ:

m = 0.05 m = 0.1 m = 0.25

0.25

Displacement (m)


0.2 0.15 0.1 0.05

0.2

FIGURE 22.35


1

Residual base displacement vs. PGA ðve ¼ 5vs Þ:

Effect of equipment tuning. In order to study the equipment tuning effect, the maximum equipment acceleration has been plotted as a function of the equipment frequency in Figure 22.38. As can be seen, for all the values of m considered, the equipment response is amplified when the equipment frequency moves close to the structural frequency of 1.67 Hz for the fixed base case. Note that, since the resonant frequency of a sliding structure shifts to a higher value as the frictional coefficient decreases (see Figure 22.26), the frequency for which the most severe tuning effect occurs in Figure 22.38 also shifts from 1.67 Hz to a higher value as m decreases. Nevertheless, it is concluded that, by choosing a smaller m; the amplification of the equipment response due to tuning effect can be effectively suppressed.

22.4.4

Concluding Remarks

The dynamic behavior of a sliding isolated structural system with an attached equipment item was investigated in this section. A sliding isolated structure is classified as a nonlinear dynamic system, as the


22 35

15

m = 0.1 Fixed

Acceleration (m/s2)

10 5 0 −5 −10 −15

0

10

20

30 Time (s)

40

15

60

m = 0.25 Fixed


50

5 0 −5 −10 −15

FIGURE 22.36

0

10

20

30 Time (s)

40

50

60

Comparison of equipment accelerations ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

frictional forces induced on the sliding surface do not remain constant. To deal with such nonlinear systems, two analysis methods were formulated, the shear balance method and the fictitious spring method, both of which were presented in an incremental form that is suitable for direct implementation. Through the selection of a sliding isolated structure equipment model, the responses of the structure and equipment subjected to both harmonic and earthquake excitations were analyzed. For the case of harmonic excitation, the results showed that the resonant responses of both the structure and attached equipment can be effectively suppressed, which remains good even when the equipment frequency is tuned to the structural frequency. For the case of seismic excitation, the results indicated that the level of reduction on the structural and equipment responses increases as the PGA level of the earthquake increases. Moreover, a sliding system with a smaller frictional coefficient has a higher isolation efficiency, at the expense of a larger base displacement.

22 36


25

m = 0.05 m = 0.1 m = 0.25 Fixed

Acceleration (m/s2)

20

15

10

5

0.2

FIGURE 22.37

Acceleration (m/s2)

102


1

Maximum equipment acceleration vs. PGA ðve ¼ 5vs Þ:

m = 0.05 m = 0.1 m = 0.25 Fixed

101

100

10−1 10−1

100

101

Equipment Freq. (Hz) FIGURE 22.38

22.5

Maximum equipment acceleration vs. equipment frequency ðPGA ¼ 0:5gÞ:

Sliding Isolation Systems with Resilient Mechanism

In Section 22.4, the relevant equations of analysis have been presented for a sliding isolated structural system, with no consideration made for the resilient (or recentering) mechanism. Because of this, rather large residual displacements may occur on the sliding isolation system, as have been numerically illustrated. If the concept of sliding isolators is to be applied to a real structure, it is important that the residual base displacements be controlled within certain limits, since they are not tolerable for some


22 37

Equipment

Equipment Superstructure

Superstructure

Ground (a)

Base Mat

Elastomeric Bearing

Ground Sliding Bearing

(b)

Base Mat

FPS Sliding Bearing

FIGURE 22.39 Schematic for a structure equipment system isolated by a sliding bearing with resilience capability: (a) combined isolation system; (b) friction pendulum system (FPS).

engineering applications. For example, residual base displacements may distort the networking of water and power lines, change the space between the isolated structure and adjacent buildings, and widen the gaps at building entrances. Therefore, in practice, a sliding isolation system is usually enhanced through inclusion of mechanisms that can provide some resilient force. However, for certain ground motions, the added resilient force may also present some negative effects not readily transparent to structural designers, as will be illustrated in the numerical studies later on. As shown in Figure 22.39, there are at least two ways of implementing the resilient mechanism in a sliding isolation system. Figure 22.39a shows an isolation system that combines the elastomeric bearings with sliding bearings (Chalhoub and Kelly, 1990), in which the elastomeric bearings are used to provide the resilient force, and the sliding bearings to uncouple the structural system from the ground motion. On the other hand, the resilient mechanism can also be incorporated into each single sliding bearing, in a way similar to that in the RFBI described in Section 22.2.3 (Mostaghel and Khodaverdian, 1987) or the FPS shown in Figure 22.39b (Mokha et al., 1991). The FPS isolation system has been implemented in many existing buildings and bridges. A typical FPS bearing consists a spherical sliding surface and a slider, which usually has a smooth coating of very low friction. When an FPS device is implemented under a structure, the slider will slide on the spherical surface during an earthquake, and the gravitational load of the structure, together with the curved sliding surface, will provide the resilient force for the system to return to its original position. The resilient stiffness of an FPS bearing depends on the radius of curvature of the sliding surface and the structural weight carried by the bearing. This section is aimed at investigating the behavior of a structure equipment system isolated by a sliding system with resilient device. For convenience of discussion, a system with resilient device will be referred to as the resilient sliding isolation (RSI), and a sliding system without resilient device, as the one studied previously, as the pure sliding isolation (PSI).

22.5.1

Mathematical Modeling and Formulation

Both the RSI systems shown in Figure 22.39 can be represented by the mathematical model given in Figure 22.40, for which the symbols used have been defined in Table 22.1. The RSI model shown in Figure 22.40 differs from the PSI model shown Figure 22.16 in that a linear spring of stiffness, kb ; is added to simulate the resilient force of the isolator. Obviously, an RSI model can be considered as the composition of a friction element and a spring element in parallel. Owing to addition of resilient stiffness, the number of vibration frequencies of the system is increased by one. The newly introduced frequency, which depends on the resilient stiffness, is called the isolation frequency, which can be approximated by pffiffiffiffiffiffiffiffiffiffiffi ð22:61Þ vb ¼ kb =ðW=gÞ

22 38


where W is the total weight of the isolated structure equipment system. The isolation fre quency commonly used in design is between 0.33 and 0.5 Hz, which implies a period of 2 to 3 sec (Naeim and Kelly, 1999). For the type of combined sliding system shown in Figure 22.39(a), the actual value of resilient stiffness, kb ; is decided by the total horizontal stiffness of the elastomeric bearings implemented. On the other hand, for the FPS shown in Figure 22.39(b), the resilient stiffness, kb ; is approximated by the following equation for small isolator displacements: kb ¼

W R

ce

kb

me

ke

ks

cs

μ

xe

ms

mb

xs

xb

.. xg

FIGURE 22.40 Model for a structure equipment system isolated by a sliding bearing with resilience capability.

ð22:62Þ

where R denotes the radius of curvature of the sliding surface. By substituting Equation 22.62 into Equation 22.61, one can verify that the isolation period of an FPS is equal to the oscillation period of a pendulum; that is pffiffiffiffi 2p 2p Tb ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ 2p R=g ð22:63Þ vb kb =ðW=gÞ When the sliding isolated system of Figure 22.40 is excited by a ground motion, its equation of motion may be written in exactly the same form as that of Equation 22.27; that is M€xðtÞ þ C_xðtÞ þ KxðtÞ ¼

ML1 x€ g ðtÞ þ L2 f ðtÞ

ð22:64Þ

All the variables used in the preceding equation are the same as those defined in Equation 22.28 to Equation 22.30, except that the stiffness matrix, K; should be modified as 2 3 ke ke 0 6 7 K¼6 ks 7 ð22:65Þ 4 ke ke þ ks 5 0

ks

ks þ kb

Note that, in Equation 22.64, the frictional force, f ðtÞ; which does not remain constant, is placed on the right hand side, while the resilient stiffness, kb ; which remains constant, is absorbed by the stiffness matrix, K; as in Equation 22.65. However, if the total shear force, sðtÞ of the isolation system is of interest, it should be computed as the summation of the frictional force and resilient force (see Figure 22.40); that is sðtÞ ¼ kb xb ðtÞ þ f ðtÞ

ð22:66Þ

The equation of motion as given in Equation 22.64 can be recast in the following form of the first order state space equation: z_ ðtÞ ¼ AzðtÞ þ E€xg ðtÞ þ Bf ðtÞ

ð22:67Þ

The definitions of the matrices zðtÞ; E; A; and B are the same as those defined in Equation 22.32 and Equation 22.33, except that the system matrix, A; should be modified to account for the addition of the resilient stiffness kb in the stiffness matrix, K:

22.5.2

Methods for Numerical Analysis

If one compares the equation of motion for the RSI system in Equation 22.67 with that for the PSI system in Equation 22.31, one will conclude that the only source of nonlinearity in both equations comes from


22 39

the same term, namely, the frictional force, f ðtÞ: As a result, the two methods of solution mentioned in Section 22.4.2, the shear balance method (Wang et al., 1998) and fictitious spring method (Yang et al., 1990), remain valid for the analysis of the RSI systems, with no modification required. Moreover, owing to inclusion of the resilient stiffness, kb ; in the structural stiffness matrix, K; the system matrix, A; becomes nonsingular and invertible. This introduces some advantage in computation of relevant coefficient matrices, including the B0 and B1 matrices in Equation 22.38a and Equation 22.38b. In Section 22.5.3, the shear balance method will be employed to simulate the response of an RSI system.

22.5.3

Simulation Results for Sliding Isolation with Resilient Mechanism

22.5.3.1

Numerical Model and Ground Excitations

In this section, the dynamic behavior of a sliding system represented by the model shown in Figure 22.40 will be investigated. The data adopted in the analysis for the equipment, structure, and the isolator have been listed in Table 22.4. To facilitate comparison, some of the data are selected to be the same as those in Table 22.3. In particular, the isolation frequency chosen is vb ¼ 0:4 Hz, falling in the common range of 0.33 to 0.5 Hz. Again, two types of ground excitations are considered, namely, the harmonic and earthquake excitations. For the harmonic excitation, a waveform of ground acceleration identical to the one given in Equation 22.60 is used. And for the earthquake excitation, the 1940 El Centro earthquake with different levels of PGA will be used, of which the acceleration waveform has been given in Figure 22.10. The harmonic excitation is adopted mainly for studying the frequency response of the sliding isolated system, while the earthquake excitation is for studying the effect of earthquake intensity. The dynamic responses computed for the RSI system, including the structure and equipment, will be presented, with emphasis placed on comparison with the PSI system of the same parameters. Similar to what was done in Section 22.4.3, the symbol m will be used to denote the frictional coefficient of the sliding isolation system in all figures, and the word “fixed” denotes the fixed base structure. 22.5.3.2

Harmonic Response of Structure

Time history. Consider an RSI system subjected to a harmonic excitation of vg ¼ 1 Hz. The base displacement and structural acceleration of the RSI system have been plotted in Figure 22.41 and Figure 22.42, respectively. Clearly, both the base displacement and structural acceleration of the RSI system reach their steady state harmonic responses in the first few cycles. Moreover, a smaller sliding frictional coefficient ðm ¼ 0:1Þ is more effective for suppressing the structural acceleration, as indicated by Figure 22.42. However, this is achieved only at the expense of a larger base displacement, as indicated by Figure 22.41. By comparing the result for the RSI system in Figure 22.41 with those for the PSI system in Figure 22.23, the effect of resilient mechanism in eliminating the permanent base displacement for the case with a small frictional coefficient of m ¼ 0:1 can be clearly appreciated. In spite of the large difference in base displacement, the structural accelerations for the RSI and PSI systems shown in Figure 22.42 and Figure 22.24, respectively, appear to be quite similar, when interpreted in terms of the waveform and response amplitude. This implies that, for the harmonic excitation considered, the resilient mechanism in RSI has little influence on the isolation effectiveness. Hysteretic behavior. In Figure 22.43, the hysteresis loops for RSI systems with m ¼ 0:1 and 0.25 subjected to a harmonic excitation of vg ¼ 1 Hz have been plotted, in which the vertical axis represents TABLE 22.4

System Parameters Used in Simulation (Section 22.5.3) Equipment

Superstructure

Isolation System

Parameter

Value

Parameter

Value

Parameter

Value

Mass me Frequency ve Damping ratio ze

3 t ð¼ ms =100Þ 5vs or a variable 5%

Mass ms Frequency vs Damping ratio zs

300 t 1.67 Hz 5%

Mass mb Frictional coefficient m Isolation frequency vb

100 t ð¼ ms =3Þ 0.05, 0.1, 0.25 0.4 Hz

22 40


0.3 0.2

Displacement (m)

0.1 0 −0.1 −0.2 −0.3 −0.4

m = 0.1 m = 0.25 0

FIGURE 22.41

5

Time (s)

10

15

Comparison of base displacements ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

the total shear, sðtÞ; of the bearing computed from Equation 22.66. It is interesting to note that, for an RSI system, the hysteresis loop is a parallelogram, of which the slope of the inclined upper and lower sides exactly represents the total resilient stiffness, kb ; and the height and width, respectively, are decided by the maximum frictional force and maximum base displacement. As the frictional coefficient decreases, the height of the parallelogram decreases, but the width increases. The total area of the hysteresis loop represents the portion of energy dissipated by the RSI system. Noteworthy is the fact that, when the resilient stiffness, kb ; reduces to zero, the hysteresis parallelogram reduces to a square as well, identical to the one shown in Figure 22.25 for the PSI system. Frequency responses. The maximum accelerations of the steady state response of the structure for four different frictional coefficients, that is, m ¼ 0:05; 0.1, 0.25 and 1 (fixed base), have been plotted in Figure 22.44. From this figure, the following observations can be made: (1) Compared with the fixed base case, the resonant peak occurring around the structural frequency, vs ; of 1.67 Hz was effectively suppressed by the RSI system, but a resonance of higher amplitude was induced in the lower frequency range (with frequencies lower than 0.6 Hz for the case studied). A further investigation reveals that the newly induced resonance is associated with the isolation frequency, vb ; of 0.4 Hz. Such an observation remains valid for all values of frictional coefficients, m: (2) The use of a lower frictional coefficient, m; will result in a smaller response in the high frequency range for the RSI system, for example, with frequencies higher than 0.6 Hz, but a larger response for the low frequency range. (3) Although both the RSI and PSI systems can effectively remove the resonant peak around the structural frequency of 1.67 Hz, the RSI system has the side effect of creating a low frequency resonant peak at the isolation frequency, vb : This implies that the RSI system is more sensitive to the excitation frequency. The frequency responses of the maximum base displacement for the RSI system with various frictional coefficients have been plotted in Figure 22.45. When compared with the results for the PSI system in Figure 22.27, it is clear that the resilient mechanism of the RSI system considerably reduces the base displacement in the nonresonant excitation range, but it also amplifies the base displacement in the region when the excitation frequency is close to the isolation frequency, vb : From Figure 22.44 and Figure 22.45, we observe that both the structural acceleration and base displacement of an RSI system may resonate at the isolation frequency, which is usually designed to be less than 0.5 Hz. This implies that an RSI system may be ineffective or unsafe for a ground motion with enriched low frequency


22 41

10

Acceleration (m/s2)

5

0

−5

−10 m = 0.1 Fixed −15 0

5

Time (s)

10

15

10

Acceleration (m/s2)

5

0

−5

−10

−15

m = 0.25 Fixed 0

FIGURE 22.42

5

Time (s)

10

15

Comparison of structural accelerations ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

(long period) vibrations, such as the case with a near fault earthquake containing a long period, pulse like waveform (Jangid and Kelly, 2001; Lu et al., 2003). Structural designers should be aware of such a side effect when designing an RSI system. 22.5.3.3

Harmonic Response of Equipment

Time history. Consider an equipment item with a frequency of ve ¼ 5vs (¼ 8.34 Hz), attached to the RSI system. The harmonic acceleration responses of the equipment for the case with m ¼ 0:1 and 0.25 have been plotted in Figure 22.46a and b, respectively, together with those for the fixed base case. As can be seen, the equipment acceleration has been effectively suppressed by the RSI with the smaller frictional coefficient ðm ¼ 0:1Þ: Moreover, the acceleration waveforms shown in Figure 22.46 are similar to those of

22 42


1500

Total base shear (kN)

1000

m = 0.1 m = 0.25

500 0 −500 −1000 −1500 −0.4

FIGURE 22.43

−0.2

0 Displacement (m)

0.2

0.4

Hysteresis loops of a sliding bearing with resilience capability ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

104

m = 0.05 m = 0.1 m = 0.25 Fixed

Acceleration (m/s2)

103 102 101 100 10−1 −1 10

100

101

Excitation Freq. (Hz) FIGURE 22.44

Maximum structural acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

the PSI system shown in Figure 22.28. This implies that for the given excitation, the behavior of the equipment was not altered by introduction of the resilient mechanism in the RSI system. Frequency responses. Figure 22.47 shows the acceleration frequency response curve of the attached equipment with a frequency of ve ¼ 5vs (8.34 Hz). A comparison of Figure 22.47 with Figure 22.44 indicates that the frequency responses of the equipment and primary structure are generally similar, except that a resonant peak associated with the equipment frequency around 8.34 Hz appears in Figure 22.47. Owing to such a similarity, the observations made previously for Figure 22.44 are applicable to Figure 22.47 for the attached equipment. Effect of equipment tuning. Figure 22.48 shows the frequency response of the attached equipment for the case when the equipment frequency is tuned to the structural frequency, that is, with ve ¼ vs : Similar to


22 43

104

m = 0.05 m = 0.1 m = 0.25

Displacement (m)

102 100 10−2 10−4 10−6 10−8 −1 10

Maximum base displacement vs. ground excitation frequency ðve ¼ 5vs Þ:

10

10

5

5

Acceleration (m/s2)

Acceleration (m/s2)

FIGURE 22.45

0 −5 −10 −15

m = 0.1 Fixed 0

101


5

10

15

Time (s) FIGURE 22.46

0 −5 −10 −15

m = 0.25 Fixed 0

5

10

15

Time (s)

Comparison of equipment accelerations ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

the structural frequency response shown in Figure 22.44, the equipment attached to the RSI system also resonates at the isolation frequency vb of 0.4 Hz. Such a resonance does not occur for the equipment attached to the PSI system (see Figure 22.30). Through comparison of the tuned case in Figure 22.48 with the detuned case in Figure 22.47, the following observations can be made: (1) Even when the equipment tuning occurs, an RSI system mitigates the equipment’s resonant peak associated with the structural frequency at 1.67 Hz, although the effectiveness of isolation has been reduced. (2) The tuning effect has no influence on the resonant response associated with the isolation frequency of 0.4 Hz. 22.5.3.4

Earthquake Response of Structure

Time history. For an RSI system subjected to the El Centro earthquake with PGA ¼ 0:5g; the structural acceleration and base displacement have been shown in Figure 22.49 and Equation 22.50, respectively. By comparing Figure 22.49 with Figure 22.31 for the corresponding PSI system, one observes that the structural accelerations of the RSI and PSI systems are generally similar, in terms of the response waveform and the response magnitude. Both systems reduce the maximum structural acceleration quite effectively, for example, by about 80% for m ¼ 0:1: However, significant difference does exist between the

22 44


104

m = 0.05 m = 0.1 m = 0.25 Fixed

Acceleration (m/s)

103 102 101 100 10–1 10–1

FIGURE 22.47


Maximum equipment acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

104

m = 0.05 m = 0.1 m = 0.25 Fixed

103 Acceleration (m/s)

101

102 101 100 10–1 10–1

FIGURE 22.48 ðve ¼ 5vs Þ:


101

Maximum equipment acceleration vs. ground excitation frequency under tuning condition

base displacements for the RSI system in Figure 22.50 and those for the PSI system in Figure 22.32. For example, for m ¼ 0:1; the maximum base displacement experienced by the RSI system has been reduced by about 30%, while the residual base displacement has been reduced by about 70%, as can be seen by comparing Figure 22.50 with Figure 22.32. This implies that the resilient mechanism of the RSI system plays an important role in reducing the maximum and residual base displacements, especially the latter. In spite of the observations made above, one should not forget that the frequency content of one earthquake may be different from an other. As was demonstrated in Figure 22.44 and Figure 22.45, an RSI system is generally sensitive to low frequency excitations and may resonate at the isolation frequency.


22 45

15

m = 0.1 Fixed

Acceleration (m/s2)

10 5 0 –5 –10 –15

0

10

20

30 Time (s)

40

15

60

m = 0.25 Fixed


50

5 0 –5 –10 –15

FIGURE 22.49

0

10

20

30 Time (s)

40

50

60

Comparison of structural accelerations ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

Therefore, if the RSI system is subjected to an earthquake containing more low frequency components, unlike the El Centro earthquake, it is likely that the maximum structural responses induced exceed those of the PSI system. Effect of earthquake intensity. The maximum structural acceleration and base displacement of the RSI system have been plotted with respect to the PGA in Figure 22.51 and Figure 22.52, respectively. These figures indicate that as the earthquake intensity increases from 0.1 to 1g; the structural acceleration is reduced by an increasing amount by the RSI system, while the maximum base displacement also increases. By comparing Figure 22.51 and Figure 22.52 with Figure 22.33 and Figure 22.34 for the PSI system, one observes that both the RSI and PSI systems perform equally well for the El Centro earthquake, although the PSI system induces a slightly larger base displacement. On the other hand, unlike the response for the PSI system, the use of a smaller frictional coefficient for the RSI system does not always lead to a lower structural acceleration, as can be verified by comparing the responses for m ¼ 0:1 and 0.05 with a PGA greater than 0:8g in Figure 22.33 and Figure 22.51. This can be attributed to the large resilient force induced by the large base displacement under higher PGA levels.

22 46


0.1

m = 0.1 m = 0.25

0.08 Displacement (m)

0.06 0.04 0.02 0 –0.02 –0.04 –0.06

0

FIGURE 22.50

10

30 Time (s)

40

50

60

Comparison of base displacements ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

m = 0.05 m = 0.1 m = 0.25 Fixed

20 Acceleration (m/sec)

20

15

10

5

0.2

FIGURE 22.51


1

Maximum structural acceleration vs. PGA ðve ¼ 5vs Þ:

Residual base displacement. Figure 22.53 shows the residual base displacement of the RSI system vs. the PGA of the earthquake. For a given m; it is difficult to establish a relation between the earthquake intensity and residual displacement, because a larger PGA may result in a smaller residual base displacement in some cases. However, if one takes the average of residual displacements over the PGA range from 0.1 to 1g; the following can be computed: xres ¼ 0:0065; 0.011, and 0.014 m for m ¼ 0:05; 0.1, and 0.25, respectively. These values indicate that a smaller frictional coefficient leads to a smaller residual base displacement, which can be attributed to the fact that for a SRI system with a smaller coefficient of friction, it is easier for the resilient mechanism to return the structure to its initial position after an earthquake. On the other hand, a comparison of Figure 22.53 with Figure 22.35 for the PSI system indicates that for the same value of m; the residual displacement was reduced substantially by the RSI system. This is certainly an advantage offered by the resilient mechanism of the RSI system.


22 47

0.5

m = 0.05 m = 0.1 m = 0.25

0.45

Displacement (m)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.2 FIGURE 22.52


Maximum base displacement vs. PGA ðve ¼ 5vs Þ:

m = 0.05 m = 0.1 m = 0.25

0.03 Displacement (m)

1

0.025 0.02 0.015 0.01 0.005 0.2

FIGURE 22.53

22.5.3.5


1

Residual base displacement vs. PGA ðve ¼ 5vs Þ:

Earthquake Response of Equipment

Time history. Let us consider an equipment item of natural frequency equal to five times the structural frequency, that is, ve ¼ 5vs (¼ 8.34 Hz). The acceleration responses of the equipment mounted on the RSI system that were subjected to the El Centro earthquake with a PGA of 0:5g for m ¼ 0:1 and 0.25 have been plotted in Figure 22.54a and b, respectively, along with those for the fixed base cases. As can be seen, the main shock response of the equipment appearing during the first 10 sec for the fixed base system was effectively suppressed by the RSI system with m ¼ 0:1 or 0.25. The level of reduction is more pronounced for the case with a smaller frictional coefficient, that is, with m ¼ 0:1: By comparing Figure 22.54 with Figure 22.36 for the PSI system, one concludes that the effect of the resilient mechanism of the RSI system on the equipment response is insignificant for the earthquake and equipment frequency considered. Effect of earthquake intensity. Figure 22.55 shows the maximum equipment acceleration vs. the PGA of the earthquake. This figure illustrates that for all values of the frictional coefficient, m; considered, an

22 48


15

m = 0.1 Fixed

Acceleration (m/s2)

10 5 0 –5 –10 –15

0

10

20

30 Time (s)

40

15

60

m = 0.25 Fixed


50

5 0 –5

–10 –15

0

FIGURE 22.54

10

20

30 Time (s)

40

50

60

Comparison of equipment accelerations ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

increasing amount of reduction can be achieved by the RSI system as the earthquake intensity increases from 0.1 to 1g: Because relatively stiff equipment (i.e., with ve ¼ 5vs ) was assumed in the simulation, the curves shown in Figure 22.55 are similar to those of Figure 22.51 for the primary structure. Therefore, the observations made previously for Figure 22.51 apply here. The maximum response of equipment items with other natural frequencies will be discussed below. Moreover, a comparison of Figure 22.55 with Figure 22.37 (for the PSI system) reveals that the resilient mechanism can have some minor effect on the equipment response, but only when a smaller frictional coefficient (i.e., m ¼ 0:05 or 0.1) is used and when the PGA level is high. Effect of equipment tuning. In order to study the effect of equipment tuning, the maximum acceleration of the equipment has been plotted in Figure 22.56 for equipment frequencies ranging from 0.1 to 10 Hz.


25

m = 0.05 m = 0.1 m = 0.25 Fixed


22 49

15

10

5 0.2

FIGURE 22.55

Acceleration (m/s2)

102


1

Maximum equipment acceleration vs. PGA ðve ¼ 5vs Þ:

m = 0.05 m = 0.1 m = 0.25 Fixed

101

100

10–1 10–1

100

101

Equipment Freq. (Hz) FIGURE 22.56

Maximum equipment acceleration vs. equipment frequency ðPGA ¼ 0:5gÞ:

As can be seen, for all the values of m considered, the equipment response is amplified when the equipment frequency is close to the structural frequency, vs ; of 1.67 Hz, which means that the tuning effect tends to enlarge the equipment response. However, the use of a smaller m can help in reducing the amplification of the equipment response resulting from the tuning effect. Finally, a comparison between Figure 22.56 and Figure 22.38 (for the PSI system) shows that the two diagrams are quite similar for an equipment item with a frequency higher than 1 Hz, but are different for that with a lower frequency. This implies that for the earthquake considered, the resilient mechanism of the RSI system has little effect on the response of the equipment with a higher stiffness.

22 50

22.5.4


Concluding Remarks

In this section, the behavior of a structure equipment system isolated by an RSI system under both the harmonic and earthquake excitations has been investigated. Both the responses of the structure and equipment were studied, with special attention given to the effect of the resilient mechanism that characterizes an RSI system. The numerical results demonstrated that when subjected to a harmonic excitation, an RSI system is able to effectively suppress the resonant peaks associated with the structural frequency for both the structure and equipment, but it may also induce some resonant response near the isolation frequency due to the presence of resilient stiffness. Therefore, an RSI system is more sensitive to the frequency content of the ground excitation than a PSI system, especially to excitations of low frequency components. As for the earthquake responses, the numerical results showed that the resilient mechanism of an RSI system can considerably reduce the residual base displacement. The resilient mechanism has a minor effect on the acceleration response of the structure and equipment, as long as no resonance is induced by the RSI system at the isolation frequency. By and large, both the RSI and PSI systems can be used as effective devices for reducing the acceleration responses of a structure and equipment.

22.6 22.6.1

Issues Related to Seismic Isolation Design Design Methods

Having been developing for over 30 years, the technology of seismic isolation has matured. Many earthquake prone countries, including the U.S., Japan, New Zealand, Taiwan, China, and European countries, have developed their own design codes, regulations, or guidelines (Fujita, 1998; Kelly, 1998; Martelli and Forni, 1998). Although most of the codes were developed based on the theory of structural dynamics, the design details outlined in the codes vary from one country to another. While a comprehensive explanation of the various design codes is not the purpose of this section, a brief overview of the concept underlying the design codes will be given. For more details, interested readers should refer to each code or to the books by Naeim and Kelly (1999) or Skinner et al. (1993). The design concept introduced herein is based on the series of Uniform Building Code (UBC, 1994, 1997). Given the fact that base isolation devices are diverse, most design codes or regulations have been written in such a way as not to be specific with respect to the isolation systems. For instance, in the UBC (1997), no particular isolation system is identified as being acceptable; rather, it requires that every isolation system is stable for required displacement, has properties that do not degrade under repeated cyclic loadings, and provides increasing resistance with increasing displacement. The design methods for base isolation can be classified as static analysis and dynamic analysis. The static analysis is applicable for stiff and regular buildings (in vertical and horizontal directions) that are constructed on soil of a relatively stiff condition. On the other hand, dynamic analysis is usually required for isolation systems with an irregular or long period superstructure, or constructed on relatively soft soils. For a sophisticated design case, static analysis may be used in the preliminary design phase in order to draft or initiate the isolation design parameters, while dynamic analysis is employed in the final design phase for tuning or finalizing the design details of the isolation system. For simple design cases, static analysis alone is considered sufficient.

22.6.2

Static Analysis

For static analysis, a number of formulas have been specified in the design codes, so that engineers can easily calculate the following design parameters (shown in the design sequence): maximum isolator displacement, D; isolator total shear, Vb ; total base shear, Vs ; of superstructure; and seismic load, Fi ; applied on each floor. These formulas were usually derived based on a simplified isolation model,


22 51

assuming the isolation system can be linearized (even though most isolation systems are nonlinear) and the superstructure can be modeled as a rigid block. Such a simplified model is considered reasonable, since the displacements of an isolated structure are concentrated at the isolation level, which implies that the superstructure behaves as a rigid block. Based on such a model, only the first vibration mode with the superstructure treated as a rigid body has been considered in deriving the formulas. This explains why static analysis is suitable only for rigid and regular structures. 22.6.2.1

Computation of Maximum Isolator Displacement

An isolation design by static analysis usually starts with the calculation of the maximum isolator displacement, D; which depends on several factors: D ¼ DðZ; N; S; Tef ; zef Þ

Force F+

ð22:68Þ

where Z denotes the earthquake zone factor, N the near fault factor, S the soil condition factor, Tef the effective isolation period, and zef the effective isolation damping. For example, in the UBC (1994), the formula derived from the constant velocity spectra over the period range of 1.0 to 3.0 sec has been given in the following form: 0:25ZNSTef D¼ B

ð22:69Þ

Kef

−D

Displacement D

F

−

FIGURE 22.57 Typical force displacement diagram for an isolation system.

where B is the damping factor, given as B ¼ Bðzef Þ < 0:25ð1

ln zef Þ

ð22:70Þ

In the above equations, the factors Z; N; and S depend on conditions of the construction site of the isolated structure; however, the factors Tef and zef depend solely on the properties of the chosen isolation system. The factors Tef and zef are called the “effective” period and damping of the isolation system, because they are frequently obtained by linearizing a nonlinear isolation system. The way to linearize an isolation system will be explained below, along with the formulas for computing Tef and zef : Suppose that for a nonlinear isolation system, the force displacement relation (hysteresis loop) obtained from a component test is shown in Figure 22.57. The effective stiffness of this isolation system can be computed by Kef ¼

Fþ

F 2D

ð22:71Þ

where F þ and F ; respectively, denote the largest positive and negative forces in the test. After the linearized stiffness is obtained from Equation 22.71, the corresponding effective quantities Tef and zef can be computed from the dynamic theory for a single DoF oscillation system; that is sffiffiffiffiffiffiffi W Kef g

ð22:72Þ

1 A 2p Kef D2

ð22:73Þ

Tef ¼ 2p

zef ¼

where W is the structural weight, g the gravitational acceleration, and A the total area enclosed by the hysteresis loop in Figure 22.57.

22 52

22.6.2.2


Computation of Maximum Isolator Shear

After the maximum isolator displacement, D; is obtained, the maximum isolator shear, Vb ; can be estimated by the following formula: Vb ¼ Kef D

ð22:74Þ

Obviously, the above equation represents an equivalent static force exerted on the isolation system, when the system is displaced by an amount, D: In some design codes, Vb has also been referred to as the design force beneath the isolation system. 22.6.2.3

Computation of Total Base Shear

The total base shear, Vs ; of the superstructure can be given as Vs ¼

Kef D RI

ð22:75Þ

where RI is a reduction factor (ductility factor) to account for structural ductility, which will be developed when the structure is subjected to an earthquake with intensity above the design level. In some codes, Vs has also been referred to as the design force above the isolation system. 22.6.2.4

Computation of Shear Force for Each Floor

Having computed the above total base shear, Vs ; a formula is employed to distribute this total shear to each floor of the isolated structure. For instance, in the, UBC (1997), the shear force, Fi ; exerted on each floor is computed by hi wi Fi ¼ V s X n hj wj

ð22:76Þ

j¼1

where n denotes the number of floors, wi the weight of the ith floor, and hi the height of the ith floor above the isolation level. Note that the sum of Fi ði ¼ 1 to nÞ must be equal to Vs : The general procedure for static analysis was illustrated in Figure 22.58. Once the design parameters, D; Vb ; Vs ; and Fi ; are all determined according to the code, they can be used in the detailed design of structural elements as well as of isolator elements. Nevertheless, in most applications, because the test data of the isolation system may not be available in the beginning of design, the values of Kef ; Tef ; and zef ; which are required in computing D; are not known to the designer. If this is the case, the design can begin with assumed values of Kef ; Tef ; and zef ; which may be obtained from experience or previous test data on similar isolators. After the preliminary design is completed, prototype isolators will be fabricated and tested. The actual values of Kef ; Tef ; and zef ; obtained from the tests will be used in the aforementioned code formulas to update the design parameters D; Vb ; Vs : Moreover, one observes from Equation 22.71 that the linearized isolator stiffness, Kef ; is a function of the design parameter, D; itself, and so are Tef ; and zef ; obtained from Equation 22.72 and 22.73. In order to obtain Kef ; as well as Tef ; and zef ; an initial guess of D is required at the beginning of design. As a result, the design procedure may have to be repeated iteratively until the difference between the final value of D and the value D0 computed in the last iteration is less than a preset tolerance. Such an iterative process is illustrated in Figure 22.58.

22.6.3

Dynamic Analysis

The dynamic analysis may be carried out in one of the two forms: response spectrum analysis and time history analysis. Response spectrum analysis usually involves application of the concepts of response spectrum and modal superposition, and so on. Since these concepts primarily come from the dynamics of linear systems, the response spectrum analysis is only suitable for isolation systems with linear


22 53

Start Given structure properties and site condition: Z, N, S

Assume isolator parameters: Kef, Tef, zef Choose smaller Tef (stiffer system)

Compute D by design code

yes

D too large

Choose larger Tef (softer system)

no Compute Vb and Vs by design code

Vb or Vs too large

yes

no Order or fabricate and test prototype isolators Compute actual Kef, Tef, zef Computer D by design code using actual Kef, Tef, zef

|D−D0| < tolerance

(D0 = D computed in previous design iteration)

Design iteration

no

yes Finalize Kef, Tef, zef and D, Vb, Vs

End

FIGURE 22.58

Flow chart of static analysis.

properties. For the case when the isolation system or the superstructure appears to be highly nonlinear, a time history analysis is generally required. Because dynamic analysis depends generally on the usage of computer programs, relatively few formulas have been given in the dynamic analysis sections of design codes. Nevertheless, for a successful time history analysis, the designer must prepare the following three basic elements: (1) a set of representative input ground motions, (2) accurate mathematic models for isolators and superstructures, and (3) a computer program that is capable of performing the nonlinear time history analysis. These three elements are explained below. 22.6.3.1

Input Ground Motions

The response of an isolated system depends greatly on the chosen input ground motions, which are usually expressed in the form of ground accelerations. Each ground motion is called one record event.

22 54


The chosen events must be representative of the site conditions and soil characteristics. Design codes usually specify the minimum number of events required for analysis. Each ground motion event must be scaled so that all events are compatible with each other and also with the code specified target spectrum. In the UBC (1997), the scaling factor for each event is obtained in response spectra, and then applied to the time domain of the record data. In particular, site specific ground motions are required in the UBC for the following cases: (1) an isolated structure located on a soft soil, (2) an isolated structure located within certain distance (e.g., 10 km) of an active fault, (3) an isolated structure with very long period of vibration (e.g., greater than 3 sec).

22.6.3.2

Mathematic Models

Before any time history analysis can be carried out, a mathematic model that can accurately reflect the mechanical behavior of the isolation system and the superstructure must be constructed. If the isolation system is nonlinear, the nonlinear parameters must be identified so that the constructed mathematic model can correctly describe the hysteretic behavior of the isolation system. In many cases, the isolation system is assumed to be nonlinear, but the superstructure linear. Establishing an accurate mathematic model is curial for obtaining reliable results in a time history analysis.

22.6.3.3

Computer Programs

In practice, the task of time history analysis is executed through the use of a computer program. The mathematic model properties mentioned above will be input to the program for analysis. The computer program selected should be capable of simulating the three dimensional behavior of structures with selected nonlinear elements. To serve the purpose of isolation design and analysis, several structural analysis programs running on the platform of personal computers have been developed for easy access. Some of the widely used programs include (but are not limited to): ETABS (ETABS, 2004), SAP 2000 Nonlinear (SAP, 2000), and 3D BASIS (Nagarajaiah et al., 1993). Most of these programs provide a set of imbedded mathematic models for the widely used isolator elements with linear or nonlinear parameters. The designers using these programs can easily build up the mathematic model for the isolated structure considered, specify the parameters of the isolator elements selected, and execute a nonlinear time history analysis on a personal computer.

22.6.4

Concluding Remarks

In this section, the design concept of seismic isolation for structures was briefly reviewed. The design methods can be based either on static or dynamic analysis. The fundamental issues that should be considered in each design method were highlighted, along with some relevant formulas for computing the relevant parameters. It is believed that, with the concepts and procedures presented in this section, the readers should have a general knowledge of the procedure for base isolation design of structures and equipment.

Acknowledgments The authors are indebted to the graduate student, Cheng Yan Wu, at the Department of Construction Engineering, National Kaohsiung First University of Science and Technology, for preparing some of the graphs presented in this chapter.


22 55

Nomenclature Symbol

Quantity

Symbol

Quantity

c; cs ce cEDF D fr Fi k; ks kb ke kEDF kf

damping coefficients of superstructure damping coefficients of equipment damping of the EDF system maximum isolator displacement interfacial frictional force seismic load applied on the ith floor stiffness of superstructure stiffness of isolation system stiffness of equipment stiffness of the EDF system stiffness of the fictitious spring in sliding isolation effective stiffness of isolation system mass of superstructure mass of base mat mass of equipment number of building stories near fault factor ductility factor soil factor natural period of superstructure damped period of superstructure effective isolation period

Vb Vs wi W x; xs ðtÞ

total shear force of isolation system total base shear of superstructure weight of the ith floor total weight of superstructure relative to the ground displacements of superstructure relative to the ground displacements of base mat relative to the ground displacements of equipment ground acceleration zone factor frictional coefficient of sliding isolation system frequency of isolation system frequencies of equipment frequency of ground excitation damping ratios of equipment effective damping ratios of isolation system damping ratios of superstructure damped natural frequency frequencies of superstructure

Kef m; ms mb me n N RI S T Td Tef

xb ðtÞ xe ðtÞ x€ g ðtÞ Z m

vb ve vg ze zef z; zs Vd V; vs

References Agrawal, A.K., Behaviour of equipment mounted over a torsionally coupled structure with sliding support, Eng. Struct., 22, 72 84, 2000. Chalhoub, M.S. and Kelly, J.M. 1990. Earthquake simulator test of a combined sliding bearing and rubber bearing isolation system, Report No. UCB/EERC 87/04, Earthquake Engineering Research Center, University of California, Berkeley, CA. Chaudhuri, S.R. and Gupta, V.K., A response based decoupling criterion for multiply supported secondary systems, Earthquake Eng. Struct. Dyn., 31, 1541 1562, 2002. Chen, G. and Soong, T.T., Energy based dynamic analysis of secondary systems, J. Eng. Mech., ASCE, 120, 514 534, 1994. Chen, G. and Soong, T.T., Exact solutions to a class of structure equipment systems, J. Eng. Mech., ASCE, 122, 1093 1100, 1996. Dey, A. and Gupta, V.K., Response of multiply supported systems to earthquakes in frequency domain, Earthquake Eng. Struct. Dyn., 27, 187 201, 1998. Dey, A. and Gupta, V.K., Stochastic seismic response of multiply supported secondary systems in flexible base structures, Earthquake Eng. Struct. Dyn., 28, 351 369, 1999. ETABS, Integrated Analysis, Design and Drafting of Building Systems, Version 8, Software by Computers and Structures Inc., Berkeley CA, 2004. Fujita, T., Seismic isolation of civil buildings in Japan, Prog. Struct. Eng. Mater., 1, 295 300, 1998. Gueraud, R., Noel Leroux, J.P., Livolant, M., and Michalopoulos, A.P., Seismic isolation using sliding elastomer bearing pads, Nucl. Eng. Des., 84, 363 377, 1985.

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Huang, C.D., Zhu, W.Q., and Soong, T.T., Nonlinear stochastic response and reliability of secondary systems, J. Eng. Mech., ASCE, 120, 177 196, 1994. Igusa, T., Response characteristic of inelastic 2 DOF primary secondary system, J. Eng. Mech., ASCE, 116, 1160 1174, 1990. Igusa, T. and Der Kiureghian, A.D., Dynamic characteristic of two degree of freedom equipment structure systems, J. Eng. Mech., ASCE, 111, 1 19, 1985a. Igusa, T. and Der Kiureghian, A.D., Dynamic response of multiply supported secondary systems, J. Eng. Mech., ASCE, 111, 20 41, 1985b. Igusa, T. and Der Kiureghian, A.D., Generation of floor response spectra including oscillator structure interaction, Earthquake Eng. Struct. Dyn., 13, 661 676, 1985c. Inaudi, J.A. and Kelly, J.M., Minimum variance control of base isolation floors, J. Struct. Eng., ASCE, 119, 438 453, 1993. Jangid, R.S. and Kelly, J.M., Base isolation for near fault motion, Earthquake Eng. Struct. Dyn., 30, 691 707, 2001. Juhn, G., Manolis, G.D., Constantinou, M.C., and Reinhorn, A.M., Experimental study of secondary systems in base isolated structure, J. Struct. Eng., ASCE, 118, 2204 2221, 1992. Kelly, J.M., Seismic isolation of civil buildings in USA, Prog. Struct. Eng. Mater., 1, 279 285, 1998. Kelly, J.M. and Tsai, H.C., Seismic response of light internal equipment in base isolated structures, Earthquake Eng. Struct. Dyn., 13, 711 732, 1985. Lai, M.L. and Soong, T.T., Statistical energy analysis of primary secondary structural systems, J. Eng. Mech., ASCE, 116, 2400 2413, 1990. Lu, L.Y. and Yang, Y.B., Dynamic response of equipment in structures with sliding support, Earthquake Eng. Struct. Dyn., 26, 61 77, 1997. Lu, L.Y., Shih, M.H., Tzeng, S.W., and Chang, C.S. 2003. Experiment of a sliding isolated structure subjected to near fault ground motion, In Proceedings of the Seventh Pacific Conference on Earthquake Engineering, February 13 15, Christchurch. Martelli, A. and Forni, M., Seismic isolation of civil buildings in Europe, Prog. Struct. Eng. Mater., 1, 286 294, 1998. Meirovitch, L. 1990. Dynamics and Control of Structures, Wiley, New York. Mokha, A.S., Constantinous, M.C., Reinhorn, A.M., and Zayas, V.A., Experimental study of friction pendulum isolation system, J. Struct. Eng., ASCE, 117, 1201 1217, 1991. Mostaghel, N., 1984. Resilient friction Base Isolator, Report No. UTEC 84 097, University of Utah, Salt Lake City, UT. Mostaghel, N. and Khodaverdian, M., Dynamics of resilient friction base isolator (R FBI), Earthquake Eng. Struct. Dyn., 15, 379 390, 1987. Mostaghel, N., Hejazi, M., and Tanbakuchi, J., Response of sliding structures to harmonic support motion, Earthquake Eng. Struct. Dyn., 11, 355 366, 1983. Naeim, F. and Kelly, J.M. 1999. Design of Seismic Isolated Structures: From Theory to Practice, Wiley, New York. Nagarajaiah, S., Li, C., Reinhorn, A.M., and Constantinou, M.C. 1993. 3D BASIS TABS: Computer program for nonlinear dynamic analysis of three dimensional base isolated structures, Technical report NCEER 93 0011, National Center for Earthquake Engineering Research, Buffalo, NY. Newmark, N.M., A method of computation for structural dynamics, J. Eng. Mech. Div., ASCE, 85, 67 94, 1959. Park, K.S., Jung, H.J., and Lee, I.W., A comparative study on aseismic performances of base isolation systems for multi span continuous bridge, Eng. Struct., 24, 1001 1013, 2002. Rivin, E.I., Vibration isolation of precision equipment, Precision Eng., 17, 41 56, 1995. SAP 2000. Integrated Structural Analysis and Design Software, Software by Computers and Structures, Inc., Berkeley, CA. Skinner, R.I., Robinson, W.H., and Mcverry, G.H. 1993. An Introduction to Seismic Isolation, Wiley, New York.


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Soong, T.T. and Dargush, G.F. 1997. Passive Energy Dissipation Systems in Structural Engineering, Wiley, New York. Steinberg, D.S. 2000. Vibration Analysis for Electronic Equipment, 3rd ed., Wiley, New York. Suarez, L.E. and Singh, M.P., Floor spectra with equipment structure equipment interaction effects, J. Eng. Mech., ASCE, 115, 247 264, 1989. UBC 1994. Uniform building code, International Conference of Building Officials, Whittier, CA. UBC 1997. Uniform building code, International Conference of Building Officials, Whittier, CA. Veprik, A.M. and Babitsky, V.I., Vibration protection of sensitive electronic equipment from harsh harmonic vibration, J. Sound Vib., 238, 19 30, 2000. Villaverde, R., Simplified seismic analysis of secondary systems, J. Eng. Mech., ASCE, 112, 588 604, 1986. Wang, Y.P., Chung, L.L., and Liao, W.H., Seismic response analysis of bridges isolated with friction pendulum bearings, Earthquake Eng. Struct. Dyn., 27, 1069 1093, 1998. Westermo, B. and Udwadia, F., Period response of a sliding oscillator system to harmonic excitation, Earthquake Eng. Struct. Dyn., 11, 135 146, 1983. Xu, Y.L., Liu, H.J., and Yang, Z.C., Hybrid platform for vibration control of high tech equipment in buildings subject to ground motion. Part 1. Experiment, Earthquake Eng. Struct. Dyn., 32, 1185 1200, 2003. Yang, J.N. and Agrawal, A.K., Protective systems for high technology facilities against microvibration and earthquake, Struct. Eng. Mech., 10, 561 575, 2000. Yang, Y.B. and Chen, Y.C., Design of sliding type base isolators by the concept of equivalent damping, Struct. Eng. Mech., 8, 299 310, 1999. Yang, Y.B. and Huang, W.H., Seismic response of light equipment in torsional buildings, Earthquake Eng. Struct. Dyn., 22, 113 128, 1993. Yang, Y.B. and Huang, W.H., Equipment structure interaction considering the effect of torsion and base isolation, Earthquake Eng. Struct. Dyn., 27, 155 171, 1998. Yang, Y.B., Lee, T.Y., and Tsai, I.C., Response of multi degree of freedom structures with sliding supports, Earthquake Eng. Struct. Dyn., 19, 739 752, 1990. Yang, Y.B., Hung, H.H., and He, M.J., Sliding and rocking response of rigid blocks due to horizontal excitations, Struct. Eng. Mech., 9, 1 16, 2000. Yang, Y.B., Chang, K.C., and Yau, J.D. 2002. Base isolation. In Earthquake Engineering Handbook, W.F. Chen and C. Scawthorn, Eds., CRC Press, Boca Raton, FL, chap. 17. Yang, Z.C., Liu, H.J., and Xu, Y.L., Hybrid platform for vibration control of high tech equipment in buildings subject to ground motion. Part 2. Analysis, Earthquake Eng. Struct. Dyn., 32, 1201 1215, 2003.

23

Vibration Control 23.1 Introduction ........................................................................ 23 1 Vibration Isolation vs. Vibration Absorption † Vibration Absorption vs. Vibration Control † Classifications of Vibration Control Systems † Performance Characteristics of Vibration Control Systems

23.2 Vibration Control Systems Concept ................................. 23 4 Introduction † Passive Vibration Control † Active Vibration Control † Semiactive Vibration Control † Adjustable Vibration Control Elements

23.3 Vibration Control Systems Design and Implementation ................................................................... 23 12

Nader Jalili Clemson University

Ebrahim Esmailzadeh

University of Ontario

Introduction Systems

†

Vibration Absorbers

†

Vibration Control

23.4 Practical Considerations and Related Topics .................... 23 38 Summary of Vibration Control Design Steps and Procedures Future Trends and Developments

†

Summary The fundamental principles of vibration control systems are formulated in this chapter. There are many important areas directly or indirectly related to the main theme of the chapter. These include practical implementation of vibration control systems, nonlinear control schemes, actual hardware implementation, actuator bandwidth requirements, reliability, and cost. Furthermore, in the process of designing a vibration control system, in practice, several critical criteria must be considered. These include weight, size, shape, center of gravity, types of dynamic disturbances, allowable system response, ambient environment, and service life. Keeping these in mind, general design steps and procedures for vibration control systems are provided.

23.1

Introduction

The problem of reducing the level of vibration in constructions and structures arises in various branches of engineering, technology, and industry. In most of today’s mechatronic systems, a number of possible devices such as reaction or momentum wheels, rotating devices, and electric motors are essential to the system’s operation and performance. These devices, however, can also be sources of detrimental vibrations that may significantly influence the mission performance, effectiveness, and accuracy of operation. Therefore, there is a need for vibration control. Several techniques are utilized either to limit or alter the vibration response characteristics of such systems. During recent years, there has been considerable interest in the practical implementation of these vibration control systems. This chapter presents the basic theoretical concepts for vibration control systems design and implementation, followed by an overview of recent developments and control techniques in this subject. Some related practical developments in variable structure control (VSC), as well as piezoelectric vibration control of flexible structures, are also provided, followed by a summary of design steps and procedures for vibration control systems. A further treatment of the subject is found in Chapter 32. 23 1

23 2


23.1.1

Vibration Isolation vs. Vibration Absorption

In vibration isolation, either the source of vibration is isolated from the system of concern (also called “force transmissibility”; see Figure 23.1a), or the device is protected from vibration of its point of attachment (also called “displacement transmissibility”, see Figure 23.1b). Unlike the isolator, a vibration absorber consists of a secondary system (usually mass spring damper trio) added to the primary device to protect it from vibrating (see Figure 23.1c). By properly selecting absorber mass, stiffness, and damping, the vibration of the primary system can be minimized (Inman, 1994).

23.1.2

Vibration Absorption vs. Vibration Control

In vibration control schemes, the driving forces or torques applied to the system are altered in order to regulate or track a desired trajectory while simultaneously suppressing the vibrational transients in the system. This control problem is rather challenging since it must achieve the motion tracking objectives while stabilizing the transient vibrations in the system. Several control methods have been developed for such applications: optimal control (Sinha, 1998); finite element approach (Bayo, 1987); model reference adaptive control (Ge et al., 1997); adaptive nonlinear boundary control (Yuh, 1987); and several other techniques including VSC methods (Chalhoub and Ulsoy, 1987; de Querioz et al., 1999; de Querioz et al., 2000). As discussed before, in vibration absorber systems, a secondary system is added in order to mimic the vibratory energy from the point of interest (attachment) and transfer it into other components or dissipate it into heat. Figure 23.2 demonstrates a comparative schematic of vibration control (both single input control and multi input configurations) on translating and rotating flexible beams, which could represent many industrial robot manipulators as well as vibration absorber applications for automotive suspension systems.

23.1.3

Classifications of Vibration-Control Systems

Passive, active, and semiactive (SA) are referred to, in the literature, as the three most commonly used classifications of vibration control systems, either as isolators or absorbers (see Figure 23.3; Sun et al., 1995). A vibration control system is said to be active, passive, or SA depending on the amount of

Vibration isolator c (a)

m Source of vibration

x(t) = X sin(w t)

F(t) = F0 sin(w t)

c

k Fixed base

FT

Absorber subsection

(c)

Device m

Vibration isolator

(b) absorber ma Ca

ka Primary device

k

y(t) = Y sin(wdtt)

Moving base xa(t)

F(t) = F0 sin(w t) Source of vibration

FIGURE 23.1 Schematic of (a) force transmissibility for foundation isolation; (b) displacement transmissibility for protecting device from vibration of the base and (c) application of vibration absorber for suppressing primary system vibration.

Vibration Control

23 3

s(t)

t(t)

Ο

l1

A

(a)

z2(t)

k2 z0(t)

w(x,t)

h Sec. A-A

Y

ka

Unsprung mass m2

y(x,t) X'

x

ba

U

b Y'

l2

za(t)

Absorber mass ma

L

X

q (t) A

U

mb

f (t)

z1(t)

Sprung mass m1

mt (b)

Road surface irregularities (c)

FIGURE 23.2 A comparative schematic of vibration control systems: (a) single input simultaneous tracking and vibration control; (b) multi input tracking and vibration control and (c) a two DoF vehicle model with dynamic vibration absorber.

external power required for the vibration control system to perform its function. A passive vibration control consists of a resilient member (stiffness) and an energy dissipater (damper) either to absorb vibratory energy or to load the transmission path of the disturbing vibration (Korenev and Reznikov, 1993; Figure 23.3a). This type of vibration control system performs best within the frequency region of its highest sensitivity. For wideband excitation frequency, its performance can be improved considerably by optimizing the system parameters (Puksand, 1975; Warburton and Ayorinde, 1980; Esmailzadeh and Jalili, 1998a). However, this improvement is achieved at the cost of lowering narrowband suppression characteristics. The passive vibration control has significant limitations in structural applications where broadband disturbances of highly uncertain nature are encountered. In order to compensate for these limitations, active vibration control systems are utilized. With an additional active force introduced as a part of absorber subsection, uðtÞ (Figure 23.3b), the system is controlled using different algorithms to make it more responsive to source of disturbances (Soong and Constantinou, 1994; Olgac and Holm Hansen, 1995; Sun et al., 1995; Margolis, 1998). The SA vibration control system, a combination of active and passive treatment, is intended to reduce the amount of external power necessary to achieve the desired performance characteristics (Lee Glauser et al., 1997; Jalili, 2000; Jalili and Esmailzadeh, 2002), see Figure 23.3c. (a) x

(b) x

m c

k

(c) x

m c

k

m c(t)

k(t)

Suspension subsection

u(t)

Suspension Point of attachment

Primary or foundation system

FIGURE 23.3 A typical primary structure equipped with three versions of suspension systems: (a) passive; (b) active and (c) SA configurations.

23 4

23.1.4


Performance Characteristics of Vibration-Control Systems

In the design of a vibration control system, it often occurs that the system is required to operate over a wideband load and frequency range that is impossible to meet with a single choice of required stiffness and damping. If the desired response characteristics cannot be obtained, an active vibration control system may provide an attractive alternative vibration control for such broadband disturbances. However, active vibration control systems suffer from control induced instability in addition to the large control effort requirement. This is a serious concern, which prevents them from the common usage in most industrial applications. On the other hand, passive systems are often hampered by a phenomenon known as “detuning.” Detuning implies that the passive system is no longer effective in suppressing the vibration it was designed for. This occurs due to one of the following reasons: (1) the vibration control system may deteriorate and its structural parameters can be far from the original nominal design, (2) the structural parameters of the primary device itself may alter, or (3) the excitation frequency or the nature of disturbance may change over time. A semiactive (also known as adaptive passive) vibration control system addresses these limitations by effectively integrating a tuning control scheme with tunable passive devices. For this, active force generators are replaced by modulated variable compartments such as variable rate damper and stiffness (see Figure 23.3c; Hrovat et al., 1988; Nemir et al., 1994; Franchek et al., 1995). These variable components are referred to as “tunable parameters” of the suspension system, which are retailored via a tuning control, thus resulting in semiactively inducing optimal operation. Much attention is being paid to these systems because of their low energy requirement and cost. Recent advances in smart materials, and adjustable dampers and absorbers have significantly contributed to applicability of these systems (Garcia et al., 1992; Wang et al., 1996; Shaw, 1998).

23.2 23.2.1

Vibration-Control Systems Concept Introduction

With a history of almost a century (Frahm, 1911), the dynamic vibration absorber has proven to be a useful vibration suppression device, widely used in hundreds of diverse applications. It is elastically attached to the vibrating body to alleviate detrimental oscillations from its point of attachment (see Figure 23.3). This section overviews the conceptual design and theoretical background of three types of vibration control systems, namely the passive, active and SA configurations, along with some related practical implementations.

23.2.2

Passive Vibration Control

The underlying proposition in all vibration control or absorber systems is to adjust properly the absorber parameters such that the system becomes absorbent of the vibratory energy within the frequency interval of interest. In order to explain the underlying concept, a single degree of freedom (single DoF) primary system with a single DoF absorber attachment is considered (Figure 23.4). The governing dynamics is expressed as

xa

ma

ca

xp

ka mp

cp

kp

f(t)

ma x€ a ðtÞ þ ca x_ a ðtÞ þ ka xa ðtÞ ¼ ca x_ p ðtÞ þ ka xp ðtÞ

ð23:1Þ

FIGURE 23.4 Application of a passive absorber to single DoF primary system.

Vibration Control

23 5

mp x€ p ðtÞ þ ðcp þ ca Þ_xp ðtÞ þ ðkp þ ka Þxp ðtÞ

ca x_ a ðtÞ

ka xa ðtÞ ¼ f ðtÞ

ð23:2Þ

where xp ðtÞ and xa ðtÞ are the respective primary and absorber displacements, f ðtÞ is the external force, and the rest of the parameters including absorber stiffness, ka ; and damping, ca ; are defined as per Figure 23.4. The transfer function between the excitation force and primary system displacement in the Laplace domain is then written as ( ) Xp ðsÞ ma s2 þ ca s þ ka TFðsÞ ¼ ¼ ð23:3Þ FðsÞ HðsÞ where HðsÞ ¼ {mp s2 þ ðcp þ ca Þs þ kp þ ka }ðma s2 þ ca s þ ka Þ

ðca s þ ka Þ2

ð23:4Þ

and Xa ðsÞ; Xp ðsÞ; and FðsÞ are the Laplace transformations of xa ðtÞ; xp ðtÞ; and f ðtÞ; respectively. 23.2.2.1

Harmonic Excitation

When excitation is tonal, the absorber is generally tuned at the disturbance frequency. For this case, the steady state displacement of the system due to harmonic excitation can be expressed as Xp ðjvÞ k ma v2 þ jca v ¼ a ð23:5Þ HðjvÞ FðjvÞ pffiffiffiffi 1: An appropriate parameter tuning scheme can then where v is the disturbance frequency and j ¼ be selected to minimize the vibration of primary system subject to external disturbance, f ðtÞ: For complete vibration attenuation, the steady state, lXp ðjvÞl; must equal zero. Consequently, from Equation 23.5, the ideal stiffness and damping of absorber are selected as ka ¼ ma v2 ; ca ¼ 0

ð23:6Þ

Notice that this tuned condition is only a function of absorber elements ðma ; ka ; and ca Þ: That is, the absorber tuning does not need information from the primary system and hence its design is stand alone. For tonal application, theoretically, zero damping in the absorber subsection results in improved performance. In practice, however, the damping is incorporated in order to maintain a reasonable trade off between the absorber mass and its displacement. Hence, the design effort for this class of application is focused on having precise tuning of the absorber to the disturbance frequency and controlling the damping to an appropriate level. Referring to Snowdon (1968), it can be proven that the absorber, in the presence of damping, can be most favorably tuned and damped if adjustable stiffness and damping are selected as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ma m2p v2 3kopt kopt ¼ ; copt ¼ ma ð23:7Þ ðma þ mp Þ2 2ðma þ mp Þ

23.2.2.2

Broadband Excitation

In broadband vibration control, the absorber subsection is generally designed to add damping to and change the resonant characteristics of the primary structure in order to dissipate vibrational energy maximally over a range of frequencies. The objective of the absorber design is, therefore, to adjust the absorber parameters to minimize the peak magnitude of the frequency transfer function ðFTFðvÞ ¼ lTFðsÞls¼jv Þ over the absorber parameters vector p ¼ {ca ka }T : That is, we seek p to min { p

max

vmin #v#vmax

{lFTFðvÞl}}

ð23:8Þ

Alternatively, one may select the mean square displacement response (MSDR) of the primary system for vibration suppression performance. That is, the absorber parameters vector, p, is selected such that

23 6


the MSDR E{ðxp Þ2 } ¼

ð1 0

{FTFðvÞ}2 SðvÞdv

ð23:9Þ

is minimized over a desired wideband frequency range. SðvÞ is the power spectral density of the excitation force, f ðtÞ; and FTF was defined earlier. This optimization is subjected to some constraints in p space, where only positive elements are acceptable. Once the optimal absorber suspension properties, ca and ka ; are determined, they can be implemented using adjustment mechanisms on the spring and the damper elements. This is viewed as a SA adjustment procedure as it adds no energy to the dynamic structure. The conceptual devices for such adjustable suspension elements and SA treatment will be discussed later in Section 23.2.5. 23.2.2.3

Example Case Study

To better recognize the effectiveness of the dynamic vibration absorber over the passive and optimum passive absorber settings, a simple example case is presented. For the simple system shown in Figure 23.4, the following nominal structural parameters (marked by an overscore) are taken: mp ¼ 5:77 kg; ma ¼ 0:227 kg;

kp ¼ 251:132 £ 106 N=m; 6

ka ¼ 9:81 £ 10 N=m;

cp ¼ 197:92 kg=sec ca ¼ 355:6 kg=sec

ð23:10Þ

These are from an actual test setting, which is optimal by design (Olgac and Jalili, 1999). That is, the peak of the FTF is minimized (see thin lines in Figure 23.5). When the primary stiffness and damping increase 5% (for instance during the operation), the FTF of the primary system deteriorates considerably (the dashed line in Figure 23.5), and the absorber is no longer an optimum one for the present primary. When the absorber is optimized based on optimization problem 8, the retuned setting is reached as ka ¼ 10:29 £ 106 N=m; ca ¼ 364:2 kg=sec

ð23:11Þ

which yields a much better frequency response (see dark line in Figure 23.5). The vibration absorber effectiveness is better demonstrated at different frequencies by frequency sweep test. For this, the excitation amplitude is kept fixed at unity and its frequency changes every 0.15 sec from

nominal absorber

de-tuned absorber

re-tuned absorber

1.0

FTF

0.8 0.6 0.4 0.2 0.0 200

400

600

800

1000 1200 Frequency, Hz.

1400

1600

1800

FIGURE 23.5 Frequency transfer functions (FTFs) for nominal absorber (thin solid line), detuned absorber (thin dotted line), and retuned absorber (thick solid line) settings. (Source: From Jalili, N. and Olgac, N., AIAA J. Guidance Control Dyn., 23, 961 970, 2000a. With permission.)

Vibration Control

23 7

1.75 Dimensional disp.

1.25 0.75 0.25 −0.25 −0.75 −1.25 −1.75 (a)

0.10

0.20

0.30

0.40

0.50 0.60 Time, sec.

0.70

0.80

0.90

1.00

1.75 Dimensional disp.

1.25 0.75 0.25 −0.25 −0.75 −1.25 −1.75 0.00

0.10

0.20

0.30

0.40 0.50 0.60 Time, sec.

0.70

0.80

0.90

0.10

0.20

0.30

0.40 0.50 0.60 Time, sec.

0.70

0.80

0.90

(b) 1.75 Dimensional disp.

1.25 0.75 0.25 −0.25 −0.75 −1.25 −1.75 0.00 (c)

FIGURE 23.6 Frequency sweep each 0.15 with frequency change of 1860, 1880, 1900, 1920, 1930, 1950, and 1970 Hz: (a) nominally tuned absorber settings; (b) detuned absorber settings and (c) retuned absorber settings. (Source: From Jalili, N. and Olgac, N., AIAA J. Guidance Control Dyn., 23, 961 970, 2000a. With permission.)

1860 to 1970 Hz. The primary responses with nominally tuned, with detuned, and with retuned absorber settings are given in Figure 23.6a c, respectively.

23.2.3

Active Vibration Control

As discussed, passive absorption utilizes resistive or reactive devices either to absorb vibrational energy or load the transmission path of the disturbing vibration (Korenev and Reznikov, 1993; see Figure 23.7, top). Even with optimum absorber parameters (Warburton and Ayorinde, 1980;

23 8


xa

ma Absorber

ka

ca

Point of attachment

x1

Primary

Structure

Sensor (Acceleration, velocity, or displacement measurement)

xa

ma ca

ka u(t)

x1

Compensator Point of attachment

Structure FIGURE 23.7

Absorber

Primary

A general primary structure with passive (top) and active (bottom) absorber settings.

Esmailzadeh and Jalili, 1998a), the passive absorption has significant limitations in structural applications where broadband disturbances of highly uncertain nature are encountered. In order to compensate for these limitations, active vibration suppression schemes are utilized. With an additional active force, uðtÞ (Figure 23.7, bottom), the absorber is controlled using different algorithms to make it more responsive to primary disturbances (Sun et al., 1995; Margolis, 1998; Jalili and Olgac, 1999). One novel implementation of the tuned vibration absorbers is the active resonator absorber (ARA) (Knowles et al., 2001b). The concept of the ARA is closely related to the concept of the delayed resonator (Olgac and Holm Hansen, 1994; Olgac, 1995). Using a simple position (or velocity or acceleration) feedback control within the absorber subsection, the delayed resonator enforces that the dominant characteristic roots of the absorber subsection be on the imaginary axis, hence leading to resonance. Once the ARA becomes resonant, it creates perfect vibration absorption at this frequency. The conceptual design and implementation issues of such active vibration control systems, along with their practical applications, are discussed in Section 23.3.

23.2.4

Semiactive Vibration Control

Semiactive (SA) vibration control systems can achieve the majority of the performance characteristics of fully active systems, thus allowing for a wide class of applications. The idea of SA suspension is very simple: to replace active force generators with continually adjustable elements which can vary and/or shift the rate of the energy dissipation in response to instantaneous condition of motion (Jalili, 2002).

23.2.5

Adjustable Vibration-Control Elements

Adjustable vibration control elements are typically comprised of variable rate damper and stiffness. Significant efforts have been devoted to the development and implementation of such devices for a variety of applications. Examples of such devices include electro rheological (ER) (Petek, 1992; Wang et al., 1994; Choi, 1999), magneto rheological (MR) (Spencer et al., 1998; Kim and Jeon, 2000) fluid dampers, and variable orifice dampers (Sun and Parker, 1993), controllable friction braces (Dowell and Cherry, 1994), and variable stiffness and inertia devices (Walsh and Lamnacusa, 1992;

Vibration Control

23 9

Nemir et al., 1994; Franchek et al., 1995; Abe and Igusa, 1996). The conceptual devices for such adjustable properties are briefly reviewed in this section. 23.2.5.1

Variable Rate Dampers

A common and very effective way to reduce transient and steady state vibration is to change the amount of damping in the SA vibration control system. Considerable design work on SA damping was done in the 1960s to the 1980s (Crosby and Karnopp, 1973; Karnopp et al., 1974) for vibration control of civil structures such as buildings and bridges (Hrovat et al., 1983) and for reducing machine tool oscillations (Tanaka and Kikushima, 1992). Since then, SA dampers have been utilized in diverse applications ranging from trains (Stribersky et al., 1998) and other off road vehicles (Horton and Crolla, 1986) to military tanks (Miller and Nobles, 1988). During recent years, there has been considerable interest in the SA concept in the industry for improvement and refinements of the concept (Karnopp, 1990; Emura et al., 1994). Recent advances in smart materials have led to the development of new SA dampers, which are widely used in different applications. In view of these SA dampers, ER and MR fluids probably serve as the best potential hardware alternatives for the more conventional variable orifice hydraulic dampers (Sturk et al., 1995). From a practical standpoint, the MR concept appears more promising for suspension, since it can operate, for instance, on vehicle battery voltage, whereas the ER damper is based on high voltage electric fields. Owing to their importance in today’s SA damper technology, we briefly review the operation and fundamental principles of SA dampers here. 23.2.5.1.1 Electro-Rheological Fluid Dampers ER fluids are materials that undergo significant y. y instantaneous reversible changes in material Moving cylinder characteristics when subjected to electric potentials (Figure 23.8). The most significant change is assoc iated with complex shear moduli of the material, and hence ER fluids can be usefully exploited in SA Fixed absorbers where variable rate dampers are utilized. cup Originally, the idea of applying an ER damper to vibration control was initiated in automobile Aluminum foil suspensions, followed by other applications (Austin, 1993; Petek et al., 1995). Ld Lo The flow motions of an ER fluid based damper can be classified by shear mode, flow mode, and squeeze mode. However, the rheological property of ER fluid is evaluated in the shear mode (Choi, 1999). As a result, the ER fluid r h ER Fluid damper provides an adaptive viscous and frictional damping for use in SA system (Dimarogonas FIGURE 23.8 A schematic configuration of an ER Andrew and Kollias, 1993; Wang et al., 1994). damper. (Source: From Choi, S.B., ASME J. Dyn. Syst. Meas. Control, 121, 134 138, 1999. With permission.)

23.2.5.1.2 Magneto-Rheological Fluid Dampers MR fluids are the magnetic analogies of ER fluids and typically consist of micron sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicon oil. When a magnetic field is applied, particle chains form and the fluid becomes a semisolid, exhibiting plastic behavior similar to that of ER fluids (Figure 23.9). Transition to rheological equilibrium can be achieved in a few milliseconds, providing devices with high bandwidth (Spencer et al., 1998; Kim and Jeon, 2000).

23 10


FIGURE 23.9 A schematic configuration of an MR damper. (Source: From Spencer, B.F. et al., Proc. 2nd World Conf. on Structural Control, 1998. With permission.)

23.2.5.2

Variable-Rate Spring Elements

In contrast to variable dampers, studies of SA springs or time varying stiffness have also been geared for vibration isolation applications (Hubard and Marolis, 1976), for structural controls and for vibration attenuation (Sun et al., 1995 and references therein). The variable stiffness is a promising practical complement to SA damping, since, based on the discussion in Section 23.2, both the absorber damping and stiffness should change to adapt optimally to different conditions. Clearly, the absorber stiffness has a significant influence on optimum operation (and even more compared to the damping element; Jalili and Olgac, 2000b). Unlike the variable rate damper, changing the effective stiffness requires high energy (Walsh and Lamnacusa, 1992). Semiactive or low power implementation of variable stiffness techniques suffers from limited frequency range, complex implementation, high cost, and so on. (Nemir et al., 1994; Franchek et al., 1995). Therefore, in practice, both absorber damping and stiffness are concurrently adjusted to reduce the required energy. 23.2.5.2.1 Variable-Rate Stiffness (Direct Methods) The primary objective is to directly change the spring stiffness to optimize a vibration suppression characteristic such as the one given in Equation 23.8 or Equation 23.9. Different techniques can be utilized ranging from traditional variable leaf spring to smart spring utilizing magnetostrictive materials. A tunable stiffness vibration absorber was utilized for a four DoF building (Figure 23.10), where a spring is threaded through a collar plate and attached to the absorber mass from one side and to the driving gear from the other side (Franchek et al., 1995). Thus, the effective number of coils, N; can be changed resulting in a variable spring stiffness, ka : d4 G ka ¼ ð23:12Þ 8D3 N where d is the spring wire diameter, D is the spring diameter, and G is modulus of shear rigidity. 23.2.5.2.2 Variable-Rate Effective Stiffness (Indirect Methods) In most SA applications, directly changing the stiffness might not be always possible or may require large amount of pcontrol ffiffiffiffiffiffiffi effort. For such cases, alternatives methods are utilized to change the effective tuning ratio ðt ¼ ka =ma =vprimary Þ; thus resulting in a tunable resonant frequency. In Liu et al. (2000), a SA flutter suppression scheme was proposed using differential changes of external store stiffness. As shown in Figure 23.11, the motor drives the guide screw to rotate with slide block, G; moving along it, thus changing the restoring moment and resulting in a change of store

Vibration Control

23 11

Spring driving gear

Spring collar

Absorber mass

Absorber spring

Guide rod Absorber base

Motor and geartrain (Potentiometer not shown) FIGURE 23.10 The application of a variable stiffness vibration absorber to a four DoF building. (Source: From Franchek, M.A. et al., J. Sound Vib., 189, 565 585, 1995. With permission.)

b w

S G

Left wing tip

FIGURE 23.11 A SA flutter control using adjustable pitching stiffness. (Source: From Liu, H.J. et al., J. Sound Vib., 229, 199 205, 2000. With permission.)

pitching stiffness. Using a double ended cantilever beam carrying intermediate lumped masses, a SA vibration absorber was recently introduced (Jalili and Esmailzadeh, 2002), where positions of moving masses are adjustable (see Figure 23.12). Figure 23.13 shows an SA absorber with an adjustable effective inertia mechanism (Jalili et al., 2001; Jalili and Fallahi, 2002). The SA absorber consists of a rod carrying a moving block and a spring and damper, which are mounted on a casing. The position of the moving block, rv ; on the rod is adjustable which provides a tunable resonant frequency. 23.2.5.3

Other Variable-Rate Elements

C

f(t) = F0 sin(wet)

q(t)

K

M m

m a L

FIGURE 23.12 A typical primary system equipped with the double ended cantilever absorber with adjus table tuning ration through moving masses, m. (Source: From Jalili, N. and Esmailzadeh, E., J. Multi Body Dyn., 216, 223 235, 2002. With permission.)

Recent advances in smart materials have led to the development of new SA vibration control systems using indirect influence on the suspension elements. Wang et al. used a SA piezoelectric network (1996) for structural vibration control. The variable resistance and inductance in an external RL circuit are used as real time adaptable control parameters. Another class of adjustable suspensions is the so called hybrid treatment (Fujita et al., 1991). The hybrid design has two modes, an active mode and a passive mode. With its aim of lowering the

23 12


FIGURE 23.13 Schematic of the adjustable effective inertia vibration absorber. (Source: From Jalili, N. et al., Int. J. Model. Simulat., 21, 148 154, 2001. With permission.)

control effort, relatively small vibrations are reduced in active mode, while passive mode is used for large oscillations. Analogous to hybrid treatment, the semiautomated approach combines SA and active suspensions to benefit from the advantages of individual schemes while eliminating their shortfalls (Jalili, 2000). By altering the adjustable structural properties (in the SA unit) and control parameters (in the active unit), a search is conducted to minimize an objective function subject to certain constraints, which may reflect performance characteristics.

23.3 23.3.1

Vibration-Control Systems Design and Implementation Introduction

This section provides the basic fundamental concepts for vibration control systems design and implementation. These systems are classified into two categories: vibration absorbers and vibration control systems. Some related practical developments in ARAs and piezoelectric vibration control of flexible structures are also provided.

23.3.2

Vibration Absorbers

Undesirable vibrations of flexible structures have been effectively reduced using a variety of dynamic vibration absorbers. The active absorption concept offers a wideband of vibration attenuation

Vibration Control

23 13

frequencies as well as real time tunability as two Imag S-plane major advantages. It is clear that the active control could be a destabilizing factor for the combined system, and therefore, the stability of the com bined system (i.e., the primary and the absorber subsystems) must be assessed. Real An actively tuned vibration absorber scheme utilizing a resonator generation mechanism forms the underlying concept here. For this, a stable primary system (see Figure 23.7, top, for instance) is forced into a marginally stable one through the addition of a controlled force in the active unit (see Poles of passive absorber Poles of ARA Figure 23.7, bottom). The conceptual design for generating such resonance condition is demon FIGURE 23.14 Schematic of the active resonator strated in Figure 23.14, where the system’s absorber concept through placing the poles of the dominant characteristic roots (poles) are moved characteristic equation on the imaginary axis. and placed on the imaginary axis. The absorber then becomes a resonator capable of mimicking the vibratory energy from the primary system at the point of attachment. Although there seem to be many ways to generate such resonance, only two widely accepted practical vibration absorber resonators are discussed next. 23.3.2.1

Delayed-Resonator Vibration Absorbers

A recent active vibration absorption strategy, the delayed resonator (DR), is considered to be the first type of active vibration absorber when operated on a flexible beam (Olgac and Jalili, 1998; Olgac and Jalili, 1999). The DR vibration absorber offers some attractive features in eliminating tonal vibrations from the objects to which it is attached (Olgac and Holm Hansen, 1994; Olgac, 1995; Renzulli et al., 1999), some of which are real time tunability, the stand alone nature of the actively controlled absorber, and the simplicity of the implementation. Additionally, this single DoF absorber can also be tuned to handle multiple frequencies of vibration (Olgac et al., 1996). It is particularly important that the combined system, that is, the primary structure and the absorber together, is asymptotically stable when the DR is implemented on a flexible beam. 23.3.2.1.1 An Overview of the Delayed-Resonator Concept An overview of the DR is presented here to help the reader. The equation of motion governing the absorber dynamics alone is ma x€ a ðtÞ þ ca x_ a ðtÞ þ ka xa ðtÞ

uðtÞ ¼ 0;

uðtÞ ¼ g x€ a ðt

tÞ

ð23:13Þ

where uðtÞ represents the delayed acceleration feedback. The Laplace domain transformation of this equation yields the characteristics equation ma s2 þ ca s þ ka

gs2 e

ts

¼0

ð23:14Þ

Without feedback ð g ¼ 0Þ; this structure is dissipative with two characteristic roots (poles) on the left half of the complex plane. For g and t . 0; however, these two finite stable roots are supplemented by infinitely many additional finite roots. Note that these characteristic roots (poles) of Equation 23.14 are discretely located (say at s ¼ a þ jv), and the following relation holds: g¼

lma s2 þ ca s þ ka l ta e ls2 l

where l·l denotes the magnitude of the argument.

ð23:15Þ

23 14


Using Equation 23.15, the following observation can be made: *

*

For g ¼ 0: there are two finite stable poles and all the remaining poles are at a ¼ For g ¼ þ1: there are two poles at s ¼ 0; and the rest are at a ¼ þ1:

1:

Considering these and taking into account the continuity of the root loci for a given time delay, t; and as g varies from 0 to 1, it is obvious that the roots of Equation 23.14 move from the stable left half to the unstable right half of the complex plane. For a certain critical gain, gc ; one pair of poles reaches the imaginary axis. At this operating point, the DR becomes a perfect resonator pffiffiffiffiand the imaginary 1: The subscript “c” characteristic roots are s ¼ ^jvc ; where vc is the resonant frequency and j ¼ implies the crossing of the root loci on the imaginary axis. The control parameters of concern, gc and tc ; can be found by substituting the desired s ¼ ^jvc into Equation 23.14 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ca vc tan 1 gc ¼ 2 ðca vc Þ2 þ ðma vc ka Þ2 ; tc ¼ þ 2ð‘ 1Þp ; ‘ ¼ 1; 2; … ma v2c ka vc vc ð23:16Þ When these gc and tc are used, the DR structure mimics a resonator at frequency vc : In turn, this resonator forms an ideal absorber of the tonal vibration at vc : The objective of the control, therefore, is to maintain the DR absorber at this marginally stable point. On the DR stability, further discussions can be found in Olgac and Holm Hansen (1994) and Olgac et al. (1997). 23.3.2.1.2 Vibration-Absorber Application on Flexible Beams We consider a general beam as the primary system with absorber attached to it and subjected to a harmonic force excitation, as shown in Figure 23.15. The point excitation is located at b; and the absorber is placed at a: A uniform cross section is considered for the beam and Euler Bernoulli assumptions are made. The beam parameters are all assumed to be constant and uniform. The elastic deformation from the undeformed natural axis of the beam is denoted by yðx; tÞ and, in the derivations that follow, the dot (·) and prime (0 ) symbols indicate a partial derivative with respect to the time variable, t; and position variable x; respectively. Under these assumptions, the kinetic energy of the system can be written as 1 ðL › y 2 1 1 T¼ r ð23:17Þ dx þ ma q_ 2a þ me q_ 2e 2 2 2 0 ›t The potential energy of this system using linear strain is given by !2 1 ðL › 2 y 1 1 U ¼ EI dx þ ka {yða; tÞ qa }2 þ ke {yðb; tÞ ›x 2 2 2 2 0

qa ca

qe

ma ka

.. gqta(−τ) E, I, A, L

ce

qe }2

ð23:18Þ

me ke

f(t) x

a b y

FIGURE 23.15 Beam absorber exciter system configuration. (Source: From Olgac, N. and Jalili, N., J. Sound Vib., 218, 307 331, 1998. With permission.)

Vibration Control

23 15

The equations of motion may now be derived by applying Hamilton’s Principle. However, to facilitate the stability analysis, we resort to an assumed mode expansion and Lagrange’s equations. Specifically, y is written as a finite sum “Galerkin approximation”: yðx; tÞ ¼

n X i¼1

Fi ðxÞqbi ðtÞ

ð23:19Þ

The orthogonality conditions between these mode shapes can also be derived as (Meirovitch, 1986) ðL 0

ðL

rFi ðxÞFj ðxÞdx ¼ Ni dij ;

0

EIF 00i ðxÞF 00j ðxÞdx ¼ Si dij

ð23:20Þ

where i; j ¼ 1; 2; …; n; dij is the Kronecker delta, and Ni and Si are defined by setting i ¼ j in Equation 23.20. The feedback of the absorber, the actuator excitation force, and the damping dissipating forces in both the absorber and the exciter are considered as non conservative forces in Lagrange’s formulation. Consequently, the equations of motion are derived. Absorber dynamics is governed by ( ) ( ) n n X X g q€ a ðt tÞ ¼ 0 ma q€ a ðtÞ þ ca q_ a ðtÞ Fi ðaÞ_qbi ðtÞ þ ka qa ðtÞ Fi ðaÞqbi ðtÞ ð23:21Þ i¼1

i¼1

The exciter is given by ( me q€ e ðtÞ þ ce q_ e ðtÞ

n X i¼1

)

þ ka

(n X i¼1

Fi ðaÞqbi ðtÞ

¼ f ðtÞFi ðbÞ;

n X

Fi ðbÞ_qbi ðtÞ þ ke qe ðtÞ

Finally, the beam is represented by (n X Ni q€ bi ðtÞ þ Si qbi ðtÞ þ ca Fi ðaÞ_qbi ðtÞ i¼1

(

) q_ a ðtÞ Fi ðaÞ þ ce

)

qa ðtÞ Fi ðaÞ þ ke

(n X i¼1

i¼1

(n X

Fi ðbÞqbi ðtÞ

i¼1

)

Fi ðbÞqbi ðtÞ ¼

f ðtÞ

ð23:22Þ

)

Fi ðbÞ_qbi ðtÞ

q_ e ðtÞ Fi ðbÞ

)

qe ðtÞ Fi ðbÞ þ g Fi ðaÞ€qa ðt

i ¼ 1; 2; …; n

tÞ ð23:23Þ

Equation 23.21 to Equation 23.23 form a system of n þ 2 second order coupled differential equations. By proper selection of the feedback gain, the absorber can be tuned to the desired resonant frequency, vc : This condition, in turn, forces the beam to be motionless at a; when the beam is excited by a tonal force at frequency vc : This conclusion is reached by taking the Laplace transform of Equation 23.21 and using feedback control law for the absorber. In short, Yða; sÞ ¼

n X i¼1

Fi ðaÞQbi ðsÞ ¼ 0

ð23:24Þ

where Yða; sÞ ¼ I{yða; tÞ}; Qa ðsÞ ¼ I{qa ðtÞ} and Qbi ðsÞ ¼ I{qbi ðtÞ}: Equation 23.24 can be rewritten in time domain as yða; tÞ ¼

n X i¼1

Fi ðaÞqbi ðtÞ ¼ 0

ð23:25Þ

which indicates that the steady state vibration of the point of attachment of the absorber is eliminated. Hence, the absorber mimics a resonator at the frequency of excitation and absorbs all the vibratory energy at the point of attachment.

23 16


23.3.2.1.3 Stability of the Combined System In the preceding section, we have derived the equations of motion for the beam exciter absorber system in its most general form. As stated before, inclusion of the feedback control for active absorption is, indeed, an invitation to instability. This topic is treated next. The Laplace domain representation of the combined system takes the form (Olgac and Jalili, 1998) AðsÞQðsÞ ¼ FðsÞ where

9 8 0 > > > > > > > > > > > > > > FðsÞ > > > > > > = < 0 ; FðsÞ ¼ > > > > > > .. > > > > > . > > > > > > > > > ; : 0 ðnþ2Þ£1

9 8 Qa ðsÞ > > > > > > > > > > > > Q ðsÞ > > e > > > > > > = < ; QðsÞ ¼ Qb1 ðsÞ > > > > > > > . > > > > .. > > > > > > > > > ; : Qbn ðsÞ ðnþ2Þ£1 0

B B B B B B B AðsÞ ¼ B B B B B B @

ma s2 þ ca s þ ka

ð23:26Þ

gs2 e

ts

0

F1 ðaÞðca s þ ka Þ

· ··

0

me s þ ce s þ ke

F1 ðbÞðce s þ ke Þ

· ··

ma F1 ðaÞs2

me F1 ðbÞs2

.. .

.. .

.. .

..

ma Fn ðaÞs2

me Fn ðbÞs2

0

· ··

2

N1 s2 þ cs þ S1 ð1 þ jdÞ · ·· .

Fn ðaÞðca s þ ka Þ

1

C C Fn ðbÞðce s þ ke Þ C C C C C 0 C C C C .. C . C A Nn s2 þ cs þ Sn ð1 þ jdÞ ð23:27Þ

In order to assess the combined system stability, the roots of the characteristic equation, detðAðsÞÞ ¼ 0 are analyzed. The presence of feedback (transcendental delay term for this absorber) in the characteristic equations complicates this effort. The root locus plot observation can be applied to the entire system. It is typical that increasing feedback gain causes instability as the roots move from left to right in the complex plane. This picture also yields the frequency range for stable operation of the combined system (Olgac and Jalili, 1998). 23.3.2.1.4 Experimental Setting and Results The experimental setup used to verify the findings is shown in Figure 23.16. The primary structure is a 3=80 in: £ 10 in: £ 120 in: steel beam (2) clamped at both ends to a granite bed (1). A piezoelectric actuator with a reaction mass (3 and 4) is used to generate the periodic disturbance on the beam. A similar actuator mass setup constitutes the DR absorber (5 and 6). The two setups are located symmetrically at one quarter of the length along the beam from the center. The feedback signal used to implement the DR is obtained from the accelerometer (7) mounted on the reaction mass of the absorber structure. The other accelerometer (8) attached to the beam is present only to monitor the vibrations of the beam and to evaluate the performance of the DR absorber in suppressing them. The control is applied via a fast data acquisition card using a sampling of 10 kHz. The numerical values for this beam absorber exciter setup are taken as *

*

*

Beam: E ¼ 210 GPa, r ¼ 1:8895 kg/m Absorber: ma ¼ 0:183 kg, ka ¼ 10,130 kN/m, ca ¼ 62:25 N sec/m, a ¼ L=4 Exciter: me ¼ 0:173 kg, ke ¼ 6426 kN/m, ce ¼ 3:2 N sec/m, b ¼ 3L=4

Vibration Control

23 17

(a) 12" 4

7

3

1

6 5

2

8

3/8"

(b)

14"

FIGURE 23.16 (a) Experimental structure and (b) schematic depiction of the setup. (Source: From Olgac, N. and Jalili, N., J. Sound Vib., 218, 307 331, 1998. With permission.)

23.3.2.1.5 Dynamic Simulation and Comparison with Experiments For the experimental set up at hand, the natural frequencies are measured for the first two natural modes, v1 and v2 : These frequencies are obtained much more precisely than those of higher order natural modes. Table 23.1 offers a comparison between the experimental (real) and analytical (ideal) clamped clamped beam natural frequencies. The discrepancies arrive from two sources: first, the experimental frequencies are structurally damped natural frequencies, and second, they reflect the effect of partially clamped BCs. The theoretical frequencies, on the other hand, are evaluated for an undamped ideal clamped clamped beam. After observing the effect of the number of modes used on the beam deformation, a minimum of three natural modes are taken into account. We then compare the simulated time response vs. the experimental results of vibration suppression. Figure 23.17 shows a test with the excitation frequency vc ¼ 1270 Hz. The corresponding theoretical control parameters are gc theory ¼ 0:0252 kg and tc theory ¼ 0:8269 msec. The experimental control parameters for this frequency are found to be gc exp: ¼ 0:0273 kg and tc exp: ¼ 0:82 msec. The exciter disturbs the beam for 5 msec, then the DR tuning is triggered. The acceleration of the beam at the point of attachment decays exponentially. For all intents and purposes, the suppression takes effect in approximately 200 ms. These results match very closely with the experimental data, Figure 23.18. The only noticeable difference is in the frequency content of the exponential decay. This property is dictated by the dominant poles of the combined system. The imaginary part, however, is smaller in the analytical study. This difference is a nuance that does not affect the earlier observations.

23 18


Comparison between Experimental and Theoretical Beam Natural Frequencies (Hz)

Natural Modes

Peak Frequencies (Experimental)

First mode Second mode

466.4 1269.2

Natural Frequencies (Clamped Clamped) 545.5 1506.3

FIGURE 23.17 Beam and absorber response to 1270 Hz disturbance, analytical. (Source: From Olgac, N. and Jalili, N., J. Sound Vib., 218, 307 331, 1998. With permission.)

FIGURE 23.18 Beam and absorber response to 1270 Hz disturbance, experimental. (Source: From Olgac, N. and Jalili, N., J. Sound Vib., 218, 307 331, 1998. With permission.)

23.3.2.2

Active Resonator Vibration Absorbers

One novel implementation of the tuned vibration absorbers is the ARA (Knowles et al., 2001b). The concept of the ARA is closely related to the concept of the delayed resonator (Olgac and Holm Hansen, 1994; Olgac and Jalili, 1999). Using a simple position (or velocity or acceleration) feedback control within the absorber section, it enforces the dominant characteristic roots of the absorber subsection to be on the imaginary axis, and hence leading to resonance. Once the ARA becomes resonant, it creates perfect vibration absorption at this frequency. A very important component of any active vibration absorber is the actuator unit. Recent advances in smart materials have led to the development of advanced actuators using piezoelectric ceramics, shape memory alloys, and magnetostrictive materials (Garcia et al., 1992; Shaw, 1998). Over the past two decades, piezoelectric ceramics have been utilized as potential replacements for conventional transducers.

Vibration Control

23 19

ma

Inertial mass PCB Series 712 Actuator Structure

ka

ca

u(t)

Structure

FIGURE 23.19 A PCB series 712 PZT inertial actuator (left), schematic of operation (middle), and a simple single DoF mathematical model (right). (Active Vibration Control Instrumentation, A Division of PCB Piezotronics, Inc., www.pcb.com.)

These materials are compounds of lead zirconate titanate (PZT). The PZT properties can be optimized to suit specific applications by appropriate adjustment of the zirconate titanate ratio. Specifically, a piezoelectric inertial actuator is an efficient and inexpensive solution for active structural vibration control. As shown in Figure 23.19, it applies a point force to the structure to which it is attached. 23.3.2.2.1 An Overview of PZT Inertial Actuators PZT inertial actuators are most commonly made out of two parallel piezoelectric plates. If voltage is applied, one of the plates expands as the other one contracts, hence producing displacement that is proportional to the input voltage. The resonance of such an actuator can be adjusted by the size of the inertial mass (see Figure 23.19). Increasing the size of the inertial mass will lower the resonant frequency and decreasing the mass will increase it. The resonant frequency, fr ; can be expressed as sffiffiffiffiffi 1 ka fr ¼ ð23:28Þ 2p ma where ka is the effective stiffness of the actuator, ma ; is defined as ma ¼ mePZT þ minertial þ macc

ð23:29Þ

The PZT effective mass is mePZT ; minertial is the inertial mass, and macc is the accelerometer mass. Using a simple single DoF system (see Figure 23.19), the parameters of the PZT inertial actuators can be experimentally determined (Knowles et al., 2001a). This “parameter identification” problem is an inverse problem. We refer interested readers to Banks and Ito (1988) and Banks and Kunisch (1989) for a general introduction to parameter estimation or inverse problems governed by differential equations. 23.3.2.2.2 Active Resonator Absorber Concept The concept of ARA is closely related to that of the DR (Olgac, 1995; Olgac and Holm Hansen, 1995; Olgac and Jalili, 1998). Instead of a compensator, the DR uses a simple delayed position (or velocity, or acceleration) feedback control within the absorber subsection for the mentioned “sensitization.” In contrast to that of the DR absorber, the characteristic equation of the proposed control scheme is rational in nature and is hence easier to implement when closed loop stability of the system is concerned. Similar to the DR absorber, the proposed ARA requires only one signal from the absorber mass, absolute or relative to the point of attachment (see Figure 23.7 bottom). After the signal is processed through a compensator, an additional force is produced, for instance, by a PZT inertial actuator. If the compensator parameters are properly set, the absorber should behave as an ideal resonator at one or even more frequencies. As a result, the resonator will absorb vibratory energy from the primary mass at given frequencies. The frequency to be absorbed can be tuned in real time. Moreover, if the controller or the actuator fails, the ARA will still function as a passive absorber, and thus it is inherently fail safe. A similar vibration absorption methodology is given by Filipovic´ and Schro¨der (1999) for linear systems. The ARA, however, is not confined to the linear regime.

23 20


For the case of linear assumption for the PZT actuator, the dynamics of the ARA (Figure 23.7, bottom) can be expressed as ma x€ a ðtÞ þ ca x_ a ðtÞ þ ka xa ðtÞ

uðtÞ ¼ ca x_ 1 ðtÞ þ ka x1 ðtÞ

ð23:30Þ

where x1 ðtÞ and xa ðtÞ are the respective primary (at the absorber point of attachment) and absorber mass displacements. The mass, ma ; is given by Equation 23.29 and the control, uðtÞ; is designed to produce designated resonance frequencies within the ARA. The objective of the feedback control, uðtÞ; is to convert the dissipative structure (Figure 23.7, top) into a conservative or marginally stable one (Figure 23.7, bottom) with a designated resonant frequency, vc : In other words, pffiffiffiffi the control aims the placement of dominant poles at ^jvc for the combined system, where j ¼ 1 (see Figure 23.14). As a result, the ARA becomes marginally stable at particular frequencies in the determined frequency range. Using simple position (or velocity or acceleration) feedback within the absorber section (i.e., UðsÞ ¼ UðsÞXa ðsÞ), the corresponding dynamics of the ARA, given by Equation 23.30, in the Laplace domain become ðma s2 þ ca s þ ka ÞXa ðsÞ

UðsÞXa ðsÞ ¼ CðsÞXa ðsÞ ¼ ðca s þ ka ÞX1 ðsÞ

ð23:31Þ

The compensator transfer function, UðsÞ; is then selected such that the primary system displacement at the absorber point of attachment is forced to be zero; that is CðsÞ ¼ ðma s2 þ ca s þ ka Þ

UðsÞ ¼ 0

ð23:32Þ

The parameters of the compensator are determined through introducing resonance conditions to the absorber characteristic equation, CðsÞ; that is, the equations Re{Cðjvi Þ} ¼ 0 and Im{Cðjvi Þ} ¼ 0 are simultaneously solved, where i ¼ 1; 2; …; l and l is the number of frequencies to be absorbed. Using additional compensator parameters, the stable frequency range or other properties can be adjusted in real time. Consider the case where UðsÞ is taken as a proportional compensator with a single time constant based on the acceleration of the ARA, given by UðsÞ ¼ UðsÞXa ðsÞ;

where UðsÞ ¼

gs2 1 þ Ts

ð23:33Þ

Then, in the time domain, the control force, uðtÞ; can be obtained from uðtÞ ¼

g ðt e T 0

ðt tÞ=T

x€ a ðtÞdt

ð23:34Þ

To achieve ideal resonator behavior, two dominant roots of Equation 23.32 are placed on the imaginary axis at the desired crossing frequency, vc : Substituting s ¼ ^jvc into Equation 23.32 and solving for the control parameters, gc and Tc ; one can obtain 0 1 pffiffiffiffiffiffiffi B C ca2 ka ca ka =ma B C ð23:35Þ gc ¼ ma B þ 1 ; T ¼ ; for v ¼ vc C c pffiffiffiffiffiffi 2 @ 2 2 A ka ka ma v2 ma v ma ka v ma ma The control parameters, gc and Tc ; are based on the physical properties of the ARA (i.e., ca ; ka ; and ma Þ as well as the frequency of the disturbance, v; illustrating that the ARA does not require any information from the primary system to which it is attached. However, when the physical properties of the ARA are not known within a high degree of certainty, a method to autotune the control parameters must be considered. The stability assurance of such autotuning proposition will bring primary system parameters into the derivations, and hence the primary system cannot be totally decoupled. This issue will be discussed later in the chapter.

Vibration Control

23 21

23.3.2.2.3 Application of ARA to Structural-Vibration Control In order to demonstrate the effectiveness of the proposed ARA, a simple single DoF primary system subjected to tonal force excitations is considered. As shown in Figure 23.20, two PZT inertial actuators are used for both the primary (model 712 A01) and the absorber (model 712 A02) subsections. Each system consists of passive elements (spring stiffness and damping properties of the PZT materials) and active compartment with the physical parameters listed in Table 23.2. The top actuator acts as the ARA with the controlled force, uðtÞ; while the bottom one represents the primary system subjected to the force excitation, f ðtÞ: The governing dynamics for the combined system can be expressed as ma x€ a ðtÞ þ ca x_ a ðtÞ þ ka xa ðtÞ

uðtÞ ¼ ca x_ 1 ðtÞ þ ka x1 ðtÞ

m1 x€ 1 ðtÞ þ ðc1 þ ca Þ_x1 ðtÞ þ ðk1 þ ka Þx1 ðtÞ

{ca x_ a ðtÞ þ ka xa ðtÞ

ð23:36Þ

uðtÞ} ¼ f ðtÞ

ð23:37Þ

where x1 ðtÞ and xa ðtÞ are the respective primary and absorber displacements. 23.3.2.2.4 Stability Analysis and Parameter Sensitivity The sufficient and necessary condition for asymptotic stability is that all roots of the characteristic equation have negative real parts. For the linear system, Equation 23.36 and Equation 23.37, when utilizing controller (Equation 23.34), the characteristic equation of the combined system (Figure 23.20, right) can be determined and the stability region for compensator parameters, g and T; can be obtained using the Routh Hurwitz method. 23.3.2.2.5 Autotuning Proposition When using the proposed ARA configuration in real applications where the physical properties are not known or vary over time, the compensator parameters, g and T; only provide partial vibration suppression. In order to remedy this, a need exists for an autotuning method to adjust the compensator

xa ca x1

ka

u(t) m1

c1

FIGURE 23.20 model (right).

ma

k1

f (t)

Implementation of the ARA concept using two PZT actuators (left) and its mathematical

TABLE 23.2 Experimentally Determined Parameters of PCB Series 712 PZT Inertial Actuators PZT System Parameters Effective mass, mePZT (gr) Inertial mass, minertial (gr) Stiffness, ka (kN/m) Damping, ca (Ns/m)

PCB Model 712 A01 7.20 100.00 3814.9 79.49

PCB Model 712 A02 12.14 200.00 401.5 11.48

23 22


parameters, g and T; by some quantities, Dg and DT; respectively (Jalili and Olgac, 1998b; Jalili and Olgac, 2000a). For the case of the linear compensator with a single time constant, given by Equation 23.33, the transfer function between primary displacement, X1 ðsÞ; and absorber displacement, Xa ðsÞ; can be obtained as gs2 ma s2 þ ca s þ ka X1 ðsÞ 1 þ Ts GðsÞ ¼ ¼ Xa ðsÞ ca s þ ka

ð23:38Þ

The transfer function can be rewritten in the frequency domain for s ¼ jv as

GðjvÞ ¼

X1 ðjvÞ ¼ Xa ðjvÞ

ma v2 þ ca vj þ ka þ ca vj þ ka

g v2 1 þ T vj

ð23:39Þ

where GðjvÞ can be obtained in real time by convolution of accelerometer readings (Renzulli et al., 1999) or other methods (Jalili and Olgac, 2000a). Following a similar procedure as is utilized in Renzulli et al. (1999), the numerator of the transfer function (Equation 23.39) must approach zero in order to suppress primary system vibration. This is accomplished by setting GðjvÞ þ DGðjvÞ ¼ 0

ð23:40Þ

where GðjvÞ is the real time transfer function and DGðjvÞ can be written as a variational form of Equation 23.39 as DGðviÞ ¼

›G ›G Dg þ DT þ higher order terms ›g ›T

ð23:41Þ

Since the estimated physical parameters of the absorber (i.e., ca ; ka ; and ma ) are within the vicinity of the actual parameters, Dg and DT should be small quantities and the higher order terms of Equation 23.41 can be neglected. Using Equation 23.40 and Equation 23.41 and neglecting higher order terms, we have " Dg ¼ Re½GðjvÞ " DT ¼ Re½GðjvÞ

# " # ka þ ka T 2 v2 ca T 2 v2 ca þ 2ka T þ Im½GðjvÞ ; v2 v2 ð23:42Þ # " # T 2 v2 ca þ 2ka T ka 2Tca v2 ka T 2 v2 T þ Im½GðjvÞ þ Dg g g v2 g v3

2Tca v2 ca

In the above expressions, g and T are the current compensator parameters given by Equation 23.35, ca ; ka ; and ma are the estimated absorber parameters, v is the absorber base excitation frequency, and GðjvÞ is the transfer function obtained in real time. That is, the retuned control parameters, g and T; are determined as follows gnew ¼ gcurrent þ Dg

and

Tnew ¼ Tcurrent þ DT

ð23:43Þ

where Dg and DT are those given by Equation 23.42. After compensator parameters, g and T; are adjusted by Equation 23.43, the process can be repeated until lGð jvÞl falls within the desired level of tolerance. GðjvÞ can be determined in real time as shown in Liu et al. (1997) by Gð jvÞ ¼ lGðjvÞleð jfðjvÞÞ

ð23:44Þ

where the magnitude and phase are determined assuming that the absorber and primary displacements are harmonic functions of time given by xa ðtÞ ¼ Xa sinðvt þ fa Þ;

x1 ðtÞ ¼ X1 sinðvt þ f1 Þ

ð23:45Þ

Vibration Control

23 23

With the magnitudes and phase angles of Equation 23.44, the transfer function can be determined from Equation 23.44 and the following: lGð jvÞl ¼

X1 ; Xa

fðjvÞ ¼ f1

fa

23.3.2.2.6 Numerical Simulations and Discussions To illustrate the feasibility of the proposed absorption methodology, an example case study is presented. The ARA control law is the proportional compensator with a single time constant as given in Equation 23.34. The primary system is subjected to a harmonic excitation with unit amplitude and a frequency of 800 Hz. The ARA and primary system parameters are taken as those given in Table 23.2. The simulation was done using Matlab/Simulinkw and the results for the primary system and the absorber displacements are given in Figure 23.21. As seen, vibrations are completely suppressed in the primary subsection after approximately 0.05 sec, at which the absorber acts as a marginally stable resonator. For this case, all physical parameters are assumed to be known exactly. However, in practice these parameters are not known exactly and may vary with time, so the case with estimated system parameters must be considered. To demonstrate the feasibility of the proposed autotuning method, the nominal system parameters ðma ; m1 ; ka ; k1 ; ca ; c1 Þ were fictitiously perturbed by 10% (i.e., representing the actual values) in the simulation. However, the nominal values of ma ; m1 ; ka ; k1 ; ca , and c1 were used for calculation of the compensator parameters, g and T: The results of the simulation using nominal parameters are given in Figure 23.22. From Figure 23.22, top, the effect of parameter variation is shown as steady state oscillations of the primary structure. This undesirable response will undoubtedly be encountered when the experiment is implemented. Thus, an autotuning procedure is needed. The result of the first autotuning iteration is given in Figure 23.22, middle, where the control parameters, g and T; are adjusted based on Equation 23.43. One can see tremendous improvement in the primary system response with only one iteration (see Figure 23.22, middle). A second iteration is

6

× 10−7 [m]

Absorber Primary

4 2 0 −2 −4 −6 −8

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time [sec] FIGURE 23.21

Primary system and absorber displacements subjected to 800 Hz harmonic disturbance.

23 24

1


× 10−6

0.5 0 − 0.5 −1

5

Absorber Primary 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

× 10−7

0

−5

Absorber Primary

−10

5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

× 10−7

0

−5

Absorber Primary

−10

0

0.01

0.02

0.03

0.04

0.05 0.06 Time, sec.

0.07

0.08

0.09

0.1

FIGURE 23.22 System responses (displacement, m) for (a) nominal absorber parameters; (b) after first autotuning procedure and (c) after second autotuning procedure.

performed, as shown in Figure 23.22, bottom. The response closely resembles that from Figure 23.21, where all system parameters are assumed to be known exactly.

23.3.3

Vibration-Control Systems

As discussed, in vibration control schemes, the control inputs to the systems are altered in order to regulate or track a desired trajectory while simultaneously suppressing the vibrational transients in the system. This control problem is rather challenging since it must achieve the motion tracking objectives while stabilizing the transient vibrations in the system. This section provides two recent control methods developed for the regulation and tracking of flexible beams. The experimental implementations are also discussed. The first control method is a single input vibration control system discussed in

Vibration Control

23 25

Section 23.3.3.1, while the second application utilizes a secondary input control in addition to the primary control input to improve the vibrational characteristics of the system (see Section 23.3.3.2). 23.3.3.1

Variable Structure Control of Rotating Flexible Beams

The vibration control problem of a flexible manipulator consists in achieving the motion tracking objectives while stabilizing the transient vibrations in the arm. Several control methods have been developed for flexible arms (Skaar and Tucker, 1986; Bayo, 1987; Yuh, 1987; Sinha, 1988; Lou, 1993; Ge et al., 1997; de Querioz et al., 1999). Most of these methods concentrate on a model based controller design, and some of these may not be easy to implement due to the uncertainties in the design model, large variations of the loads, ignored high frequency dynamics, and high order of the designed controllers. In view of these methods, VSC is particularly attractive due to its simplicity of implementation and its robustness to parameter uncertainties (Yeung and Chen, 1989; Singh et al., 1994; Jalili et al., 1997). 23.3.3.1.1 Mathematical Modeling As shown in Figure 23.23, one end of the arm is free and the other end is rigidly attached to a vertical gear shaft, driven by a DC motor. A uniform cross section is considered for the arm, and we make the Euler Bernoulli assumptions. The control torque, t; acting on the output shaft, is normal to the plane of motion. Viscous frictions and the ever present unmodeled dynamics of the motor compartment are to be compensated via a perturbation estimation process, as explained later in the text. Since the dynamic system considered here has been utilized in literature quite often, we present only the resulting partial differential equation (PDE) of the system and refer interested readers to Junkins and Kim (1993) and Luo et al. (1999) for detailed derivations. The system is governed by ðL Ih u€ðtÞ þ r x€zðx; tÞdx ¼ t ð23:46Þ

Ο t (t)

A X q (t)

A

y(x,t)

x

X′

b

Y′

Y

h Sec. A-A

FIGURE 23.23 Flexible arm in the horizontal plane and kinematics of deformation. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 719, 2001. With permission.)

0

rz€ðx; tÞ þ EIz 0000 ðx; tÞ ¼ 0

ð23:47Þ

with the corresponding boundary conditions zð0; tÞ ¼ 0;

z 0 ð0; tÞ ¼ uðtÞ;

z 00 ðL; tÞ ¼ 0;

z 000 ðL; tÞ ¼ 0

ð23:48Þ

where r is the arm’s linear mass density, L is the arm length, E is Young’s modulus of elasticity, I is the cross sectional moment of inertia, Ih is the equivalent mass moment of inertia at the root end of the arm, It ¼ Ih þ rL3 =3 is the total inertia, and the global variable z is defined as zðx; tÞ ¼ xuðtÞ þ yðx; tÞ

ð23:49Þ

Clearly, the arm vibration equation (Equation 23.47) is a homogeneous PDE but the boundary conditions (Equation 23.48) are nonhomogeneous. Therefore, the closed form solution is very tedious to obtain, if not impossible. Using the application of VSC, these equations and their associated boundary conditions can be converted to a homogeneous boundary value problem, as discussed next. 23.3.3.1.2 Variable Structure Controller The controller objective is to track the arm angular displacement from an initial angle, ud ¼ uð0Þ; to zero position, uðt ! 1Þ ¼ 0; while minimizing the flexible arm oscillations. To achieve the control

23 26


insensitivity against modeling uncertainties, the nonlinear control routine of sliding mode control with an additional perturbation estimation (SMCPE) compartment is adopted here (Elmali and Olgac, 1992; Jalili and Olgac, 1998a). The SMCPE method, presented in Elmali and Olgac (1992), has many attractive features, but it suffers from the disadvantages associated with the truncated model base controllers. On the other hand, the infinite dimensional distributed (IDD) base controller design, proposed in Zhu et al. (1997), has practical limitations due to its measurement requirements in addition to the complex control law. Initiating from the idea of the IDD base controller, we present a new controller design approach in which an online perturbation estimation mechanism is introduced and integrated with the controller to relax the measurement requirements and simplify the control implementation. As utilized in Zhu et al. (1997), for the tip vibration suppression, it is further required that the sliding surface enable the transformation of nonhomogeneous boundary conditions (Equation 23.48) to homogeneous ones. To satisfy vibration suppression and robustness requirements simultaneously, the sliding hyperplane is selected as a combination of tracking (regulation) error and arm flexible vibration as s ¼ w_ þ s w

ð23:50Þ

where s . 0 is a control parameter and w ¼ uðtÞ þ

m zðL; tÞ L

ð23:51Þ

with the scalar, m; being selected later. When m ¼ 0; controller (Equation 23.50) reduces to a sliding variable for rigid link manipulators (Jalili and Olgac, 1998a; Yeung and Chen, 1988). The motivation for such sliding a variable is to provide a suitable boundary condition for solving the beam Equation 23.47, as will be discussed next and is detailed in Jalili (2001). For the system described by Equation 23.46 to Equation 23.48, if the variable structure controller is given by

t ¼ cest þ

It 1þm

k sgnðsÞ

m y€ ðL; tÞ L

Ps

where cest is an estimate of the beam flexibility effect ðL c ¼ r x€yðx; tÞdx 0

s ð1 þ mÞu_

sm y_ ðL; tÞ L

ð23:52Þ

ð23:53Þ

k and P are positive scalars k $ 1 þ m=It lc cest l, 1:2 , m , 0:45; m – 1 and sgnð Þ represents the standard signum function, then, the system’s motion will first reach the sliding mode s ¼ 0 in a finite time, and consequently converge to the equilibrium position wðx; tÞ ¼ 0 exponentially with a time constant 1=s (Jalili, 2001). 23.3.3.1.3 Controller Implementation In the preceding section, it was shown that by properly selecting control variable, m, the motion exponentially converges to w ¼ 0 with a time constant 1=s; while the arm stops in a finite time. Although the discontinuous nature of the controller introduces a robustifying mechanism, we have made the scheme more insensitive to parametric variations and unmodeled dynamics by reducing the required measurements and hence easier control implementation. The remaining measurements and ever present modeling imperfection effects have all been estimated through an online estimation process. As stated before, in order to simplify the control implementation and reduce the measurement effort, the effect of Ð all uncertainties, including flexibility effect ð L0 x€yðx; tÞdxÞ and the ever present unmodeled dynamics, is gathered into a single quantity named perturbation, c; as given by Equation 23.53. Noting Equation 23.46, the perturbation term can be expressed as

c¼t

It u€ðtÞ

ð23:54Þ

Vibration Control

23 27

which requires the yet unknown control feedback t: In order to resolve this dilemma of causality, the current value of control torque, t; is replaced by the most recent control, t ðt dÞ; where d is the small time step used for the loop closure. This replacement is justifiable in practice since such an algorithm is implemented on a digital computer and the sampling speed is high enough to claim this. Also, in the absence of measurement noise, u€ðtÞ ø u€cal ðtÞ ¼ ½u_ðtÞ u_ðt dÞ =d: In practice and in the presence of measurement noise, appropriate filtering may be considered and combined with these approximate derivatives. This technique is referred to as “switched derivatives”. This backward differences are shown to be effective when d is selected to be small enough and the controller is run on a fast DSP (Cannon and Schmitz, 1984). Also, y€ ðL; tÞ can be obtained by attaching an accelerometer at arm tip position. All the required signals are, therefore, measurable by currently available sensor facilities and the controller is thus realizable in practice. Although these signals may be quite inaccurate, it should be pointed out that the signals, either measurements or estimations, need not be known very accurately, since robust sliding control can be achieved if k is chosen to be large enough to cover the error existing in the measurement/signal estimation (Yeung and Chen, 1989). 23.3.3.1.4 Numerical Simulations In order to show the effectiveness of the proposed controller, a lightweight flexible arm is considered (h .. b in Figure 23.23). For numerical results, we consider ud ¼ uð0Þ ¼ p=2 for the initial arm base angle, with zero initial conditions for the rest of the state variables. The system parameters are listed in Table 23.3. Utilizing assumed mode model (AMM), the arm vibration, Equation 23.47, is truncated to three modes and used in the simulations. It should be noted that the controller law, Equation 23.52, is based on the original infinite dimensional equation, and this truncation is utilized only for simulation purposes. We take the controller parameter m ¼ 0:66; P ¼ 7:0; k ¼ 5; 1 ¼ 0:01 and s ¼ 0:8: In practice, s is selected for maximum tracking accuracy taking into account unmodeled dynamics and actuator hardware limitations (Moura et al., 1997). Although such restrictions do not exist in simulations (i.e., with ideal actuators, high sampling frequencies and perfect measurements), this selection of s was decided based on actual experiment conditions. The sampling rate for the simulations is d ¼ 0:0005 sec, while data are recorded at the rate of only 0.002 sec for plotting purposes. The system responses to the proposed control scheme are shown in Figure 23.24. The arm base angular position reaches the desired position, u ¼ 0, in approximately 4 to 5 sec, which is in agreement with the approximate settling time of ts ¼ 4=s (Figure 23.24a). As soon as the system reaches the sliding mode layer, lsl , 1 (Figure 23.24d), the tip vibrations stop (Figure 23.24b), which demonstrates the feasibility of the proposed control technique. The control torque exhibits some residual vibration, as shown in Figure 23.24c. This residual oscillation is expected since the system TABLE 23.3 System Parameters Used in Numerical Simulations and Experimental Setup for Rotating Arm Properties Arm Young’s modulus Arm thickness Arm height Arm length Arm linear mass density Total arm base inertia Gearbox ratio Light source mass Position sensor sensitivity Motor back EMF constant Motor torque constant Armature resistance Armature inductance Encoder resolution

Symbol E b h L r Ih N Kb Kt Ra La

Value

Unit 9

207 £ 10 0.0008 0.02 0.45 0:06=L 0.002 14:1 0.05 0.39 0.0077 0.0077 2.6 0.18 0.087

N/m2 m m m kg/m kg m2 kg V/cm V/rad/sec N m/A V mH Deg/count

23 28


FIGURE 23.24 Analytical system responses to controller with inclusion of arm flexibility, that is, m ¼ 0:66: (a) arm angular position; (b) arm tip deflection; (c) control torque and (d) sliding variable sec. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 719, 2001. With permission.)

motion is not forced to stay on s ¼ 0 surface (instead it is forced to stay on lsl , 1) when saturation function is used. The sliding variable s is also depicted in Figure 23.24d. To demonstrate better the feature of the controller, the system responses are displayed when m ¼ 0 (Figure 23.25). As discussed, m ¼ 0 corresponds to the sliding variable for the rigid link. The undesirable oscillations at the arm tip are evident (see Figure 23.25b and c). 23.3.3.1.5 Control Experiments In order to demonstrate better the effectiveness of the controller, an experimental setup is constructed and used to verify the numerical results and concepts discussed in the preceding sections. The experimental setup is shown in Figure 23.26. The arm is a slender beam made of stainless steel, with the same dimensions as used in the simulations. The experimental setup parameters are listed in Table 23.3. One end of the arm is clamped to a solid clamping fixture, which is driven by a high quality DC servomotor. The motor drives a built in gearbox ðN ¼ 14:1Þ whose output drives an antibacklash gear. The antibacklash gear, which is equipped with a precision encoder, is utilized to measure the arm base angle as well as to eliminate the backlash. For tip deflection, a light source is attached to the tip of the arm, which is detected by a camera mounted on the rotating base. The DC motor can be modeled as a standard armature circuit; that is, the applied voltage, v; to the DC motor is v ¼ Ra ia þ La dia =dt þ Kb u_m

ð23:55Þ

Vibration Control

23 29

FIGURE 23.25 Analytical system responses to controller without inclusion of arm flexibility, that is, m ¼ 0: (a) arm angular position; (b) arm tip deflection; (c) control torque and (d) sliding variable sec. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 719, 2001. With permission.)

FIGURE 23.26 The experimental device and setup configuration. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 719, 2001. With permission.)

23 30


where Ra is the armature resistance, La is the armature inductance, ia is the armature current, Kb is the back EMF (electro motive force) constant, and um is the motor shaft position. The motor torque, tm from the motor shaft with the torque constant, Kt ; can be written as

tm ¼ Kt ia

ð23:56Þ

Ie u€m þ Cv u_m þ ta ¼ tm ¼ Kt ia

ð23:57Þ

The motor dynamics thus become where Cv is the equivalent damping constant of the motor, and Ie ¼ Im þ IL =N 2 is the equivalent inertia load including motor inertia, Im ; and gearbox, clamping frame and camera inertia, IL : The available torque from the motor shaft for the arm is ta : Utilizing the gearbox from the motor shaft to the output shaft and ignoring the motor electric time constant, ðLa =Ra Þ; one can relate the servomotor input voltage to the applied torque (acting on the arm) as

t¼

NKt v Ra

Cv þ

Kt Kb N 2 u_ Ra

Ih u€

ð23:58Þ

where Ih ¼ N 2 Ie is the equivalent inertia of the arm base used in the derivation of governing equations. By substituting this torque into the control law, the reference input voltage, V; is obtained for experiment. The control torque is applied via a digital signal processor (DSP) with sampling rate of 10 kHz, while data are recorded at the rate 500 Hz (for plotting purposes only). The DSP runs the control routine in a single input single output mode as a free standing CPU. Most of the computations and hardware commands are done on the DSP card. For this setup, a dedicated 500 MHz Pentium III serves as the host PC, and a state of the art dSPACEw DS1103 PPC controller board equipped with a Motorola Power PC 604e at 333 MHz, 16 channels ADC, 12 channels DAC, as microprocessor. The experimental system responses are shown in Figure 23.27 and Figure 23.28 for similar cases discussed in the numerical simulation section. Figure 23.27 represents the system responses when controller (Equation 23.52) utilizes the flexible arm (i.e., m ¼ 0:66). As seen, the arm base reaches the desired position (Figure 23.27a), while tip deflection is simultaneously stopped (Figure 23.27b). The good correspondence between analytical results (Figure 23.24) and experimental findings (Figure 23.27) is noticeable from a vibration suppression characteristics point of view. It should be noted that the controller is based on the original governing equations, with arm base angular position and tip deflection measurements only. The unmodeled dynamics, such as payload effect (owing to the light source at the tip, see Table 23.3) and viscous friction (at the root end of the arm), are being compensated through the proposed online perturbation estimation routine. This, in turn, demonstrates the capability of the proposed control scheme when considerable deviations between model and plant are encountered. The only noticeable difference is the fast decaying response as shown in Figure 27b and c. This clearly indicates the high friction at the motor, which was not considered in the simulations (Figure 24b and c). Similar responses are obtained when the controller is designed based on the rigid link only, that is, m ¼ 0: The system responses are displayed in Figure 23.28. Similarly, the undesirable arm tip oscillations are obvious. The overall agreement between simulations (Figure 24 and Figure 25) and the experiment (Figure 27 and Figure 28) is one of the critical contributions of this work. 23.3.3.2

Observer-Based Piezoelectric Vibration Control of Translating Flexible Beams

Many industrial robots, especially those widely used in automatic manufacturing assembly lines, are Cartesian types (Ge et al., 1998). A flexible Cartesian robot can be modeled as a flexible cantilever beam with a translational base support. Traditionally, a PD control strategy is used to regulate the movement of the robot arm. In lightweight robots, the base movement will cause undesirable vibrations at the arm tip because of the flexibility distributed along the arm. In order to eliminate

Vibration Control

23 31

FIGURE 23.27 Experimental system responses to controller with inclusion of arm flexibility, that is, m ¼ 0:66: (a) arm angular position; (b) arm tip deflection; (c) control voltage applied to DC servomotor. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 719, 2001. With permission.)

such vibrations, the PD controller must be upgraded with additional compensating terms. In order to improve further the vibration suppression performance, which is a requirement for the high precision manufacturing market, a second controller, such as a piezoelectric (PZT) patch actuator attached on the surface of the arm, can be utilized (Oueini et al., 1998; Ge et al., 1999; Jalili et al., 2002). In this section, an observer based control strategy is presented for regulating the arm motion (Liu et al., 2002). The base motion is controlled utilizing an electrodynamic shaker, while a piezoelectric (PZT) patch actuator is bonded on the surface of the flexible beam for suppressing residual arm vibrations. The control objective here is to regulate the arm base movement, while simultaneously suppressing the vibration transients in the arm. To achieve this, a simple PD control strategy is selected for the regulation of the movement of the base, and a Lyapunov based controller is selected for the PZT voltage signal. The selection of the proposed energy based Lyapunov function naturally results in velocity related signals, which are not physically measurable (Dadfarnia et al., 2003). To remedy this, a reduced order observer is designed to estimate the velocity related signals. For this, the control structure is designed based on the truncated two mode beam model. 23.3.3.2.1 Mathematical Modeling For the purpose of model development, we consider a uniform flexible cantilever beam with a PZT actuator bonded on its top surface. As shown in Figure 23.29, one end of the beam is clamped into

23 32

100


10 8 6 4 2 0 −2 −4 −6 −8 − 10

q (t), deg

75 50 25 0 −25 (a)

0

1

2

3

4 5 Time, sec. 6

6

7

8

(b)

y(L,t), mm

0

1

2

3

4 5 Time, sec.

6

7

8

v, volts

4 2 0 −2 −4 −6 (c)

0

1

2

3

4

5

6

7

8

Time, sec.

FIGURE 23.28 Experimental system responses to controller without inclusion of arm flexibility, that is, m ¼ 0: (a) arm angular position; (b) arm tip deflection; (c) control voltage applied to DC servomotor. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 719, 2001. With permission.)

a moving base with the mass of mb ; and a tip mass, mt ; is attached to the free end of the beam. The beam has total thickness tb ; and length L; while the piezoelectric film possesses thickness and length tb and ðl2 l1 Þ; respectively. We assume that the PZT and the beam have the same width, b: The PZT actuator is perfectly bonded on the beam at distance l1 measured from the beam support. The force, f ðtÞ; acting on the base and the input voltage, vðtÞ; applied to the PZT actuator are the only external effects. To establish a coordinate system for the beam, the x axis is taken in the longitudinal direction and the z axis is specified in the transverse direction of the beam with midplane of the beam to be z ¼ 0; as shown in Figure 23.30. This coordinate is fixed to the base. The fundamental relations for the piezoelectric materials are given as (Ikeda, 1990) F ¼ cS E¼

hD

hT S þ bD

ð23:59Þ ð23:60Þ

where F [ R6 is the stress vector, S [ R6 is the strain vector, c [ R6£6 is the symmetric matrix of elastic stiffness coefficients, h [ R6£3 is the coupling coefficients matrix, D [ R3 is the electrical displacement vector, E [ R3 is the electrical field vector, and b [ R3£3 is the symmetric matrix of impermittivity coefficients.

Vibration Control

23 33

An energy method is used to derive the equations of motion. Neglecting the electrical kinetic energy, the total kinetic energy of the system is expressed as (Liu et al., 2002; Dadfarnia et al., 2004)

s(t)

mb

f (t)

l1

1 1 ðl1 _ tÞÞ2 dx mb _sðtÞ2 þ b rb tb ð_sðtÞ þ wðx; 2 2 0 1 ðl2 þ b ðrb tb þ rp tp Þð_sðtÞ þ wðx; _ tÞÞ2 dx 2 l1 1 ðL þ b rb tb ð_sðtÞ þ wðx; _ tÞÞ2 dx 2 l2 1 _ tÞÞ2 þ mt ð_sðtÞ þ wðL; 2 1 1 ðL ¼ mb _sðtÞ2 þ rðxÞ _sðtÞ þ wðx; _ tÞÞ2 dx 2 2 0 1 _ tÞÞ2 ð23:61Þ þ mt ð_sðtÞ þ wðL; 2

Ek ¼

w(x,t) mt

FIGURE 23.29 Schematic of the SCARA/Cartesian robot (last link).

where

rðxÞ ¼ ½rb tb þ GðxÞrp tp b GðxÞ ¼ Hðx

l1 Þ

Hðx

L l2

ð23:62Þ l2 Þ

and HðxÞ is the Heaviside function, rb and rp are the respective beam and PZT volumetric densities. Neglecting the effect of gravity due to planar motion and the higher order terms of quadratic in w 0 (Esmailzadeh and Jalili, 1998b), the total potential energy of the system can be expressed as

Ep ¼

1 ðl1 ðtb =2 T 1 ðl2 ðtb =2 T 1 ðl2 ððtb =2Þþtp T b F S dy dx þ b F S dy dx þ b ½F S þ ET D dy dx 2 0 2 l1 2 l1 tb =2 tb =2 tb =2

1 ðL ðtb =2 T b F S dy dx 2 l2 tb =2 " #2 ðl2 ›2 wðx; tÞ ›2 wðx; tÞ 1 ðL 1 dx þ h D ðtÞ dx þ bl ðl2 cðxÞ ¼ l y 2 ›x 2 › x 2 0 2 l1 þ

z

beam zn

neutral axis

FIGURE 23.30

Coordinate system.

ð23:63Þ

PZT patch

(l1 − l2)

geometric t center of the p beam

l1 ÞDy ðtÞ2

tb 2 tb 2

x

23 34


where b cðxÞ ¼ 3

(

b 3 c11 tb 4

! þ GðxÞ

b 3c11 tb zn2

þ

p c11

tp3

tb 2

þ 3tp

zn

2

þ3tp2

)

tb 2

zn

ð23:64Þ

2zn Þ=2; bl ¼ b22 btp

hl ¼ h12 tp bðtp þ tb and

p

zn ¼

c11 tp ðtp þ tb Þ p b c11 tb þ c11 tp

p

b The beam and PZT stiffnesses are c11 and c11 ; respectively. Using the AMM for the beam vibration analysis, the beam deflection can be written as

wðx; tÞ ¼

1 X i¼1

fi ðxÞqi ðtÞ; Pðx; tÞ ¼ sðtÞ þ wðx; tÞ

ð23:65Þ

The equations of motion can now be obtained using the Lagrangian approach mb þ mt þ

ðL 0

rðxÞdx €sðtÞ þ

1 X j¼1

mj q€ j ðtÞ ¼ f ðtÞ

mi €sðtÞ þ mdi q€ i ðtÞ þ v2i mdi qi ðtÞ þ hl ðf0i ðl2 Þ hl

1 X j¼1

{ðf0j ðl2 Þ

f0j ðl1 ÞÞqj ðtÞ} þ bl ðl2

ð23:66aÞ

f0i ðl1 ÞÞDy ðtÞ ¼ 0

l1 ÞDy ðtÞ ¼ bðl2

l1 ÞvðtÞ

ð23:66bÞ ð23:66cÞ

where mdj ¼

ðL 0

rðxÞf2j ðxÞdx þ mt f2j ðLÞ; mj ¼

ðL 0

rðxÞfj ðxÞdx þ mt fj ðLÞ

ð23:67Þ

Calculating Dy ðtÞ from Equation 23.66b and substituting into Equation 23.66c results in mi €sðtÞ þ mdi q€ i ðtÞ þ v2i mdi qi ðtÞ ¼

h2l ðf0i ðl2 Þ bl ðl2

1 f0i ðl1 ÞÞ X {ðf0j ðl2 Þ l1 Þ j¼1

f0j ðl1 ÞÞqj ðtÞ}

hl bðf0i ðl2 Þ f0i ðl1 ÞÞ vðtÞ; i ¼ 1; 2; … bl

ð23:68Þ

which will be used to derive the controller, as discussed next. 23.3.3.2.2 Derivation of the Controller Utilizing Equation 23.66a and Equation 23.68, the truncated two mode beam with PZT model reduces to mb þ mt þ

ðL 0

rðxÞdx €sðtÞ þ m1 q€ 1 ðtÞ þ m2 q€ 2 ðtÞ ¼ f ðtÞ

h2l ðf01 ðl2 Þ f01 ðl1 ÞÞ bl ðl2 l1 Þ f01 ðl1 ÞÞq1 ðtÞ þ ðf02 ðl2 Þ f02 ðl1 ÞÞq2 ðtÞ

ð23:69aÞ

m1 €sðtÞ þ md1 q€ 1 ðtÞ þ v21 md1 q1 ðtÞ ðf01 ðl2 Þ ¼

hl bðf01 ðl2 Þ

bl

f01 ðl1 ÞÞ

vðtÞ

ð23:69bÞ

Vibration Control

23 35

h2l ðf02 ðl2 Þ f02 ðl1 ÞÞ bl ðl2 l1 Þ f01 ðl1 ÞÞq1 ðtÞ þ ðf02 ðl2 Þ f02 ðl1 ÞÞq2 ðtÞ

m2 €sðtÞ þ md2 q€ 2 ðtÞ þ v22 md2 q2 ðtÞ ðf01 ðl2 Þ hl bðf02 ðl2 Þ

¼

f02 ðl1 ÞÞ

bl

ð23:69cÞ

vðtÞ

The equations in Equation 23.69 can be written in the following more compact form € þ KD ¼ Fe MD where

2

c

m1

6 M¼6 4 m1

m2

2

0

7 6 6 0 7 5; K ¼ 4 0

md1

m2

3

0

md2

0

0 k11 k12

ð23:70Þ

8 9 8 9 sðtÞ > f ðtÞ > > > > > > > < = < = 7 ; D ¼ q ðtÞ vðtÞ e k12 7 ; F ¼ 1 1 e 5 > > > > > > > > : ; : ; q2 ðtÞ e 2 vðtÞ k22 0

3

ð23:71Þ

and

c ¼ mb þ mt þ

bl ðl2 h2l

bl ðl2

k22 ¼ v22 md2

0

rðxÞdx; e 1 ¼ h2l

k11 ¼ v21 md1 k12 ¼

ðL

l1 Þ

l1 Þ

ðf01 ðl2 Þ h2l

bl ðl2

l1 Þ

ðf01 ðl2 Þ

hl b 0 ðf ðl Þ bl 1 2

f01 ðl1 ÞÞ; e 2 ¼

hl b 0 ðf ðl Þ bl 2 2

f02 ðl1 ÞÞ;

f01 ðl1 ÞÞ2 ; ð23:72Þ

f01 ðl1 ÞÞðf02 ðl2 Þ ðf02 ðl2 Þ

f02 ðl1 ÞÞ;

f02 ðl1 ÞÞ2

For the system described by Equation 23.70, if the control laws for the arm base force and PZT voltage generated moment are selected as f ðtÞ ¼ vðtÞ ¼

kp Ds

kd _sðtÞ

kv ðe 1 q_ 1 ðtÞ þ e 2 q_ 2 ðtÞÞ

ð23:73Þ ð23:74Þ

where kp and kd are positive control gains, Ds ¼ sðtÞ sd ; sd is the desired set point position, and kv . 0 is the voltage control gain, then the closed loop system will be stable, and in addition lim {q1 ðtÞ; q2 ðtÞ; Ds} ¼ 0

t!1

See Dadfarnia et al. (2004) for a detailed proof. 23.3.3.2.3 Controller Implementation The control input, vðtÞ; requires the information from the velocity related signals, q_ 1 ðtÞ and q_ 2 ðtÞ; which are usually not measurable. Sun and Mills (1999) solved the problem by integrating the acceleration signals measured by the accelerometers. However, such controller structure may result in unstable closed loop system in some cases. In this paper, a reduced order observer is designed to estimate the velocity signals, q_ 1 and q_ 2 : For this, we utilize three available signals: base displacement, sðtÞ; arm tip deflection, PðL; tÞ; and beam root strain, e ð0; tÞ; that is y1 ¼ sðtÞ ¼ x1

ð23:75aÞ

y2 ¼ PðL; tÞ ¼ x1 þ f1 ðLÞx2 þ f2 ðLÞx3 t y3 ¼ e ð0; tÞ ¼ b ðf001 ð0Þx2 þ f002 ð0Þx3 Þ 2

ð23:75bÞ ð23:75cÞ

23 36


It can be seen that the first three states can be obtained by 8 9 x1 > > > = < > x2 ¼ C1 1 y > > > ; : > x3

ð23:76Þ

Since this system is observable, we can design a reduced order observer to estimate the velocity related state signals. Defining X1 ¼ ½ x1 x2 x3 T and X2 ¼ ½ x4 x5 x6 T ; the estimated value for X2 can be designed as ^ 2 ¼ Lr y þ z^ X

ð23:77Þ

z_^ ¼ F^z þ Gy þ Hu

ð23:78Þ

where Lr [ R3£3 ; F [ R3£3 ; G [ R3£3 ; and H [ R3£2 will be determined by the observer pole placement. Defining the estimation error as e2 ¼ X2

^2 X

ð23:79Þ

_^ X 2

ð23:80Þ

the derivative of the estimation error becomes _2 e_ 2 ¼ X

Substituting the state space equations of the system (Equation 23.77 and Equation 23.78) into Equation 23.80 and simplifying, we obtain e_ 2 ¼ Fe2 þ ðA21

Lr C1 A11

GC1 þ FLr C1 ÞX1 þ ðA22

Lr C1 A12

FÞX2 þ ðB2

Lr C1 B1

HÞu

ð23:81Þ

In order to force the estimation error, e2 ; to go to zero, matrix F should be selected to be Hurwitz and the following relations must be satisfied (Liu et al., 2002): F ¼ A22

Lr C1 A12

ð23:82Þ

H ¼ B2

Lr C1 B1

ð23:83Þ

G ¼ ðA21

Lr C1 A11 þ FLr C1 ÞC1 1

ð23:84Þ

The matrix F can be chosen by the desired observer pole placement requirement. Once F is known, Lr ; H; and G can be determined utilizing Equation 23.82, to Equation 23.84, respectively. The velocity variables, ^ 2 ; can now be estimated by Equation 23.77 and Equation 23.78. X 23.3.3.2.4 Numerical Simulations In order to show the effectiveness of the controller, the flexible beam structure in Figure 23.29 is considered with the PZT actuator attached on the beam surface. The system parameters are listed in Table 23.4. First, we consider the beam without PZT control. We take the PD control gains to be kp ¼ 120 and kd ¼ 20: Figure 23.31 shows the results for the beam without PZT control (i.e., with only PD force control for the base movement). To investigate the effect of PZT controller on the beam vibration, we consider the voltage control gain to be kv ¼ 2 £ 107 : The system responses to the proposed controller with a piezoelectric actuator based on the two mode model are shown in Figure 23.32. The comparison between the tip displacement, from Figure 31 and Figure 32, shows that the beam vibration can be suppressed significantly utilizing the PZT actuator. 23.3.3.2.5 Control Experiments In order to demonstrate better the effectiveness of the controller, an experimental setup is constructed and used to verify the numerical results. The experimental apparatus consists of a flexible beam with a PZT actuator and strain sensor attachments, as well as data acquisition,

Vibration Control

23 37

TABLE 23.4 System Parameters Used in Numerical Simulations and Experimental Setup for Translational Beam Properties

Symbol

6

6

5

5 4 3 2

3 2 1 0

1

2

3

4 (b)

50

0.4

25

0.2 0 −0.2

(c)

1 0 −1

0.6

v(t), volt

f (t), N

0

0

1

2 3 Time, sec.

(d)

0

1

0

1

2

3

4

2 3 Time, sec.

4

0 −25 −50

4

N/m2 mm mm mm kg/m3 N/m2 V/m m/F mm mm mm kg/m3 kg kg

69 £ 10 0.8125 20 300 3960.0 66.47 £ 109 5 £ 108 4.55 £ 107 0.2032 33.655 44.64 7750.0 0.455 0

tb b L rb p c11 h12 b22 tp l2 2 l1 l1 rp mb mt

4

Unit

9

P(L,t), mm

s(t), mm

Beam Young’s modulus Beam thickness Beam and PZT width Beam length Beam volumetric density PZT Young’s modulus PZT coupling parameter PZT impermittivity PZT thickness PZT length PZT position on beam PZT volumetric density Base mass Tip mass

(a)

Value

b c11

FIGURE 23.31 Numerical simulations for the case without PZT control: (a) base motion; (b) tip displacement; (c) control force and (d) PZT voltage.

amplifier, signal conditioner and the control software. As shown in Figure 23.33, the plant consists of a flexible aluminum beam with a strain sensor and a PZT patch actuator bound on each side of the beam surface. One end of the beam is clamped to the base with a solid clamping fixture, which is driven by a shaker. The shaker is connected to the arm base by a connecting rod. The experimental setup parameters are listed in Table 23.4. Figure 23.34 shows the high level control block diagram of the experiment, where the shaker provides the input control force to the base and the PZT applies a controlled moment on the beam. Two laser sensors measure the position of the base and the beam tip displacement. A strain gauge sensor, which is attached near the base of the beam, is utilized for the dynamic strain measurement. These three signals

23 38


6 4

P(L,t), mm

s(t), mm

5 3 2 1 0

0

0.5

1

1.5

2

(b)

0.6

50

0.4

25 v (t), volt

f (t), N

(a)

0.2 0 −0.2

(c)

6 5 4 3 2 1 0 −1

0

0.5

0

0.5

1

1.5

2

1 1.5 Time, sec.

2

0 −25

0

0.5

1 1.5 Time, sec.

2

−50 (d)

FIGURE 23.32 Numerical simulations for the case with PZT control: (a) base motion; (b) tip displacement; (c) control force and (d) PZT voltage.

are fed back to the computer through the ISA MultiQ data acquisition card. The remaining required signals for the controller (Equation 23.66) are determined as explained in the preceding section. The data acquisition and control algorithms are implemented on an AMD Athlon 1100 MHz PC running under the RT Linux operating system. The Matlab/Simulink environment and Real Time Linux Target are used to implement the controller. The experimental results for both cases (i.e., without PZT and with PZT control) are depicted in Figure 35 and Figure 36, respectively. The results demonstrate that with PZT control, the arm vibration is eliminated in less than 1 sec, while the arm vibration lasts for more than 6 sec when PZT control is not used. The experimental results are in agreement with the simulation results except for some differences at the beginning of the motion. The slight overshoot and discrepancies at the beginning of the motion are due to the limitations of the experiment (e.g., the shaker saturation limitation) and unmodeled dynamics in the modeling (e.g., the friction modeling). However, it is still apparent that the PZT voltage control can substantially suppress the arm vibration despite such limitations and modeling imperfections.

23.4 23.4.1

Practical Considerations and Related Topics Summary of Vibration-Control Design Steps and Procedures

In order to select a suitable vibration control system, especially a vibration isolator, a number of factors must be considered. 23.4.1.1

Static Deflection

The static deflection of the vibration control system under the deadweight of the load determines to a considerable extent the type of the material to be used in the isolator. Organic materials, such as rubber

Vibration Control

23 39

FIGURE 23.33 The experimental setup: (a) the whole system; (b) PZT actuator, ACX model No. QP21B; (c) dynamic strain sensor (attached on the other side of the beam), model No. PCB 740B02.

and cork, are capable of sustaining very large strains provided they are applied momentarily. However, if large strains remain for an appreciable period of time, they tend to drift or creep. On the other hand, metal springs undergo permanent deformation if the stress exceeds the yield stress of the material, but show minimal drift or creep when the stress is maintained below the yield stress.

Shaker DAC

Amplifier PZT

ADC Tip Laser Sensor Base Laser Sensor Starin sensor

FIGURE 23.34

High level control block diagram.

Plant

23 40


6

7 6 5 4 3 2 1 0 −1

P(L,t), mm

s(t), mm

5 4 3 2 1 0

0

2

4

6

(b)

100

0.4

50

0.2

2

4

6

0

−50

0 0

(c)

0

0.6

v(t), volt

f (t), N

(a)

2 4 Time, sec.

6

−100 (d)

0

2 4 Time, sec.

6

FIGURE 23.35 Experimental results for the case without PZT control: (a) base motion; (b) tip displacement; (c) control force and (d) PZT voltage.

6 P(L,t), mm

s(t), mm

5 4 3 2 1 0

0

0.5

1

1.5

2 (b)

100

0.4

50

0.2

0.5

1

1.5

2

1 1.5 Time, sec.

2

0 −50

0 (c)

0

0.6 v(t), volt

f(t), N

(a)

6 5 4 3 2 1 0 −1

0

0.5

1 1.5 Time, sec.

2

−100 (d)

0

0.5

FIGURE 23.36 Experimental results for the case with PZT control: (a) base motion; (b) tip displacement; (c) control force and (d) PZT voltage.

23.4.1.2

Stiffness in Lateral Directions

Resilient materials strained in compression are most useful when the load is relatively large and the static deflection is small. Such applications are difficult to design for a small load, unless the required

Vibration Control

23 41

static deflection is small. Otherwise, the small area and great thickness tend to cause a condition of instability. To a considerable extent, this limitation can be overcome by using sponge rubber, a material of lower modulus. In general, when the load is small, it is preferable to use rubber springs that carry the load in shear. 23.4.1.3

Environmental Conditions

It is highly common for vibration control systems to be subjected to harsh environmental conditions. Especially in military applications, extreme ambient temperatures are encountered in addition to exposure to substances like ozone, rocket fuels, and so on. Organic materials are usually more susceptible to these environmental conditions than metal materials. However, owing to the superior mechanical properties of organic materials, such as lighter weight, smaller size, greater damping, and the ability to store large amounts of energy under shock, organic materials that are capable of withstanding the harsh conditions are being developed. 23.4.1.4

Damping Characteristics

In most of the vibration control applications, the excitations cover a wide range of frequencies and may have random properties requiring the vibration control systems to possess adequate damping. Elastomers possess very good damping properties when compared with metal springs, and they also eliminate the trouble of standing waves that occurs at high frequencies. If a metal spring is used in vibration control applications requiring isolation of vibration at high frequencies, it is common to employ rubber pads in series with the metal spring, which also results in the damping of vibrations due to the addition of damping material. 23.4.1.5

Weight and Space Limitations

The amount of load carrying resilient material is determined by the quantity of energy to be stored. In most of the cases, the vibration amplitude tends to be small relative to the static deflection, and the amount of material may be calculated by equating the energy stored in the material to the work done on the vibration control system. 23.4.1.6

Dynamic Stiffness

In the case of organic materials like rubber, the natural frequency calculated using the stiffness determined from a static force deflection test of the spring is almost invariably lower than that experienced during vibration; that is, the dynamic modulus is greater than static modulus. The ratio between the dynamic and static modulus is generally between one and two. In many vibration control applications, it is not feasible to mount the equipment directly upon the vibration control system (isolator). Instead, a heavy, rigid block, usually made of concrete or heavy steel, supported by the isolator is employed.

23.4.2

Future Trends and Developments

During recent years, there has been considerable interest in the design and implementation of a variety of vibration control systems. Recent developments in multivariable control design methodology and microprocessor implementation of modern control algorithms have opened a new era for the design of externally controlled passive systems for use in such systems: fuzzy reasoning (Yoshimura, 1998); adaptive algorithms (Venhovens, 1994); observer design (Hedrick et al., 1994); and many others. Observing these developments combined with the substantial ongoing theoretical advances in the areas of adaptive and nonlinear controls (Astrom and Wittenmark, 1989; Alleyne and Hedrick, 1995), it is expected that the future will bring applications of these techniques in advanced vibration control system design. For practical implementation, however, it is preferable to simplify these strategies, thus leading to simpler software implementations. Suboptimal policy neglecting some performance requirements can serve as an example of such simplifications.

23 42


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Yeung, K.S. and Chen, Y.P., Regulation of a one link flexible robot arm using sliding mode control technique, Int. J. Control, 49, 1965 1978, 1989. Yoshimura, T., A SA suspension of passenger cars using fuzzy reasoning and the filed testing, Int. J. Vehicle Des., 19, 150 166, 1998. Yuh, J., Application of discrete time model reference adaptive control to a flexible single link robot, J. Robotic Syst., 4, 621 630, 1987. Zhu, G., Ge, S.S., Lee, T.H. 1997. Variable structure regulation of a flexible arm with translational base, pp. 1361 1366. In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA.

24

Helicopter Rotor Tuning 24.1 24.2 24.3 24.4 24.5

Kourosh Danai University of Massachusetts

24.6

Introduction ....................................................................... Neural Network Based Tuning ......................................... Probability Based Tuning .................................................. Adaptive Tuning ................................................................. The Interval Model † Estimation of Feasible Region Selection of Blade Adjustments † Learning

24 24 24 24

1 4 5 8

†

Case Study .......................................................................... 24 12 Simulation Model Evaluation

†

Interval Modeling

†

Performance

Conclusion ......................................................................... 24 17

Summary Before a helicopter leaves the plant, its rotors need to be tuned so that the helicopter vibration meets the required specifications during different flight regimes. For this, three different adjustments can be made to each rotor blade in response to the magnitude and phase of vibration. In this chapter, the basic concepts for determining the blade adjustments are discussed, and three methods with fundamentally different approaches are described. A neural network based method is described, which trains a feedforward network as the inverse model of the effect of the blade adjustments on helicopter vibrations, and uses the inverse model to determine the blade adjustments. Another is a probability based method that maximizes the likelihood of success of the selected blade adjustments based on a stochastic model of the probability densities of the vibration components. The third method is an adaptive method that uses an interval model to represent the range of effect of blade adjustments on helicopter vibration, so as to cope with the nonlinear and stochastic nature of aircraft vibration. This method includes the a priori knowledge of the process by defining the initial coefficients of the interval model according to sensitivity coefficients between the blade adjustments and helicopter vibration, but then transforms these coefficients into intervals and updates them after each tuning iteration, to improve the model estimation accuracy. The details of rotor tuning are described through a case study, which demonstrates the application of the adaptive method.

24.1

Introduction

Helicopter rotor tuning (track and balance) is the process of adjusting the rotor blades so as to reduce the aircraft vibration and the spread of rotors. Rotor tuning as applied to Sikorsky’s Black Hawk (H 60) helicopters is performed as follows. For initial measurements, the aircraft is flown through six different regimes, during which measurements of rotor track and vibration (balance) are recorded. Rotor track is measured by optical sensors, which detect the vertical position of the blades (see Chapter 15). Vibration is measured at the frequency of once per blade revolution ( per rev) by two accelerometers, A and B, attached to the sides of the cockpit (see Figure 24.1, detail B). The vibration data are vectorially combined into two components: A þ B, representing the vertical vibration of the aircraft, and A B, representing its roll vibration. A sample of peak vibration levels for the six flight regimes, as well as the peak angular positions relative to a reference blade, are given in Table 24.1, along with a sample of track data. 24 1

24 2


FIGURE 24.1 Illustration of the position of accelerometers A and B on the aircraft, and the rotor blade adjustments (push rod, trim tab, and hub weights). TABLE 24.1

Typical Track and Balance Data Recorded during a Flight

Flight Regime

Vibration AþB Magnitude (ips)

fpm hov 80 120 145 vh

0.19 0.07 0.02 0.04 0.02 0.10

A2B Phase (8)

Magnitude (ips)

332 247 86 28 104 312

0.38 0.10 0.04 0.04 0.07 0.12

Phase (8) 272 217 236 333 162 211

Track (mm) Blade #

fpm hov 80 120 145 vh

1

2

3

4

22 21 1 2 5 2

3 3 11 13 18 13

1 0 1 21 23 21

22 22 213 214 220 214


24 3

The six flight regimes in Table 24.1 are: ground ( fpm), hover (hov), 80 knots (80), 120 knots (120), 145 knots (145), and maximum horizontal speed (vh). The track data indicate the vertical position of each blade relative to a mean position. In order to bring track and one per rev vibration within specification, three types of adjustments can be made to the rotor system: pitch control rod adjustments, trim tab adjustments, and balance weight adjustments (see Figure 24.1). Pitch control rods can be extended or contracted by a certain number of notches to alter the pitch of the rotor blades. Positive push rod adjustments indicate extension. Trim tabs, which are adjustable surfaces on the trailing edge of the rotor blades, affect the aerodynamic pitch moment of the air foils and consequently their vibration characteristics. Tab adjustments are measured in thousandths of an inch, with positive and negative changes representing upward and downward tabbing, respectively. Finally, balance weights can be either added to or removed from the rotor hub to tune vibrations through changes in the blade mass. Balance weights are measured in ounces, with positive adjustments representing the addition of weight. In the case of the Sikorsky H 60 helicopter, which has four main rotor blades, a total of 12 adjustments can be made to tune the rotors (i.e., three adjustments per blade). Among them, balance weights primarily affect the ground vibration, so they are not commonly used for in flight tuning. Furthermore, since the symmetry of rotor blades in four bladed aircraft produces identical effects for adjustments to opposite blades, the combined form of blade adjustments to opposite blade pairs can be used as inputs. Accordingly, the input vector can be defined as Dx ¼ ½Dx1 ; Dx2 ; Dx3 ; Dx4

T

ð24:1Þ

where Dx1 and Dx3 denote the combined (condensed) trim tab adjustments ðDT Þ to blade combinations one/three and two/four, respectively, and Dx2 and Dx4 represent the combined pitch control rod adjustments ðDPÞ to blade combinations one/three and two/four, respectively. The relationships between the combined and individual adjustments are in the form: Dx1 ¼ DT3

DT1

ð24:2Þ

Dx2 ¼ DP3

DP1

ð24:3Þ

Dx3 ¼ DT4

DT2

ð24:4Þ

Dx4 ¼ DP4

DP2

ð24:5Þ

Ideally, identical adjustments made to any two aircraft with different tail numbers should result in identical changes in vibration. In reality, however, significant inconsistencies in vibration changes may be present for identical adjustments to different tail numbers. This is perhaps due more to nonuniformity of flight conditions from weather or error in implementing the blade adjustments than factors such as dissimilarities between aircraft and rotor blades. Virtually all of the current systems of rotor track and balance rely on the strategy shown in Figure 24.2, whereby the measurements of the flight just completed are used as the basis of search for the new blade adjustments. The search for blade adjustments is guided by the “process model” (see Figure 24.2), blade which represents the relationship between modifications aircraft vibration vibration changes and blade adjustments. Helicopter A difficulty of rotor tuning is the excess of equations compared to degrees of freedom (four Process inputs to control 24 outputs), which translates Model into one to many mapping. Another difficulty is caused by the high level of noise present in the Search vibration measurements. The traditional approach to rotor tuning uses linear relationships to define the process model FIGURE 24.2 Tuning strategy of the current methods.

24 4


and uses model inversion to streamline the search. The drawback of the traditional approach, therefore, is its neglect of the potential nonlinearity of track and balance, and the vibration noise, as well as its limited capacity to produce comprehensive solutions to facilitate model inversion due to its consideration of the most extreme vibration components. In an attempt to include the potential nonlinearity of the process, Taitel et al. (1995) trained a set of neural networks with actual track and balance data to map vibration measurements to blade adjustments as well as to evaluate the goodness of the solution. In effect, they developed an inverse model based on the solutions available in the historical track and balance data, and provided a forward model to evaluate the solution. The potential advantage of this method is that it can interpolate among the historical solutions to address potential nonlinearity and vibration noise. Its disadvantages are that it is only applicable to helicopters with extensive track and balance history, and that its solutions are constrained by those contained in the historical data. Another deviation from the traditional approach is introduced by Ventres and Hayden (2000), who define the relationships between blade adjustments and vibration in frequency domain, and provide an extension of these relationships to higher order vibrations. They use an optimization method to search for the adjustments to reduce per rev vibration as well as higher order vibrations. Accordingly, this approach has the capacity to provide a comprehensive solution, but it too neglects the potential nonlinearity between the blade adjustments and aircraft vibration as well as the noise in the measurements. The most recent solutions to rotor tuning are those by Wang et al. (2005a, 2005b), which are designed to address both the stochastics of vibration and the potential nonlinearity of the tuning process. In the first solution, which is a probability based method, the underlying model comprises two components: a deterministic component and a probability component. The method relies on the probability model to estimate the likelihood of the measured vibration satisfying the specifications and to search for blade adjustments that will maximize this likelihood. The likelihood measures in the probability model are computed according to the probability distribution of vibration derived from historical track and balance data. The second solution is an adaptive method that uses an interval model to cope with the potential nonlinearity of the process and to account for vibration noise. This method, which also incorporates learning to provide adaptation to the rotor tuning process, initializes the coefficients of the interval model according to the sensitivity coefficients between the blade adjustments and helicopter vibration. However, it modifies these coefficients after the first iteration to better represent the vibration measurements acquired. This method takes into account vibration data from all of the flight regimes during the search for the appropriate blade adjustments; therefore, it has the capacity to provide comprehensive solutions. The remainder of this chapter describes three of the methods discussed above to provide a representation of various solutions proposed for rotor tuning, followed by a case study to demonstrate the application of the adaptive method.

24.2

Neural Network-Based Tuning

As mentioned earlier, rotor tuning in four bladed aircraft is performed by first specifying a condensed set of adjustments to reduce vibrations, and then expanding these adjustments into a detailed set to satisfy the track requirements. This same strategy is implemented in the system of neural networks shown in Figure 24.3 (Taitel et al., 1995). The first network in this system, called the selection net, determines the condensed blade adjustments (output) that will bring about a given change in vibration (input). To eliminate vibration, the negatives of the vibration measurements from the flight are utilized as inputs to this network. The validity of the condensed adjustments is then checked by predicting their effect on vibration via the condensed simulation net. Theoretically, these simulated vibration changes should be the negative of the vibration measurements from the aircraft so that their summation will be zero. However, owing to the inexactness of the neural network models and noise, the resultant vibration will most likely not equal zero. In cases where the resultant vibration is not within specifications (usually less than 0.20 inches per second [ips]), the condensed adjustments may be refined by feeding the resultant


DESIRED ∆ VIBS. (24)

24 5

SELECTION NET

COND. MODS. (6)

EXPANDED MODS.

TRACK NET

(12) POSSIBLE FEEDBACK PREDICTED ∆ VIBS. (24)

PREDICTED ∆ VIBS. (24)

CONDENSED SIMULATION NET

VIBRATION NET

∆ TRACK (24)

BLADE MODS. (12)

SELECTION PACKAGE

FIGURE 24.3 Schematic of the rotor tuning system. The numbers inside parentheses represent the number of inputs or outputs of individual nets.

vibration back into the selection net. This feedback is depicted by the dashed feedback line in Figure 24.3. It should be noted that the condensed simulation net may also serve as a diagnostic tool by indicating behavior out of the norm. For example, an aircraft with vibrations significantly different from those predicted by this network may suffer from defective components. Just as with the traditional approach, once the condensed solution has been specified, it needs to be expanded into a detailed form to satisfy the rotor track requirements. As previously mentioned, the condensed set of adjustments may be viewed as the constraint on detailed adjustments so as to ensure that the vibration solution is not compromised for track. Each one of these detailed sets of adjustments is a candidate for the final rotor tuning solution, and it is left to the track net and the selection package to determine which set of detailed adjustments provides the best tracking performance. For selection purposes, the track net simulates the changes in track due to a candidate set of detailed adjustments, and then adds these changes to the initial track measurements from the flight to estimate the resultant track. The set of detailed adjustments that yields the smallest estimated track (i.e., smallest maximum blade spread) is selected as the solution to the rotor tuning problem. The selected set of detailed adjustments is then checked via the vibration net, which, similar to the condensed simulation net, serves as an independent evaluator of the selected adjustments.

24.3

Probability-Based Tuning

The noted contribution of this method is its introduction of the likelihood of success as a criterion in the search for the blade adjustments (Wang et al., 2005). This method speculates the effectiveness of various adjustment sets in reducing the vibration and selects the set with the maximum probability of producing acceptable vibration (see Chapter 5 for useful concepts of random or stochastic vibration). The concept of this method is explained in the context of a simple example. If the measured vibration from the current


flight is denoted by Vj ðk 1Þ and the estimated vibration change according to the model is represented by DV^ j ðkÞ ¼ f ðDxÞ as a function of the blade adjustments, Dx; then the predicted vibration of the next flight, V^ j ðkÞ; can be defined as V^ j ðkÞ ¼ Vj ðk

1Þ þ DV^ j ðkÞ

Vj ðkÞ ¼ V^ j ðkÞ þ e^j ðkÞ

ð24:6Þ ð24:7Þ

Probability Density

24 6

Specification Region

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0.4

where Vj ðkÞ denotes the measured vibration for the 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 next flight. In rotor tuning, the adjustments are Vibration Magnitude selected according to the predicted vibration, V^ j ðkÞ; whereas the objective is defined in terms of FIGURE 24.4 Illustration of improved placement of the measured vibration. The inclusion of the the predicted vibration within the specification range. probability model here is to account for the inevitable uncertainty in the actual position of the measured vibration. According to Equation 24.7, the mean value of the measured vibration is equal to the value of the predicted vibration plus the mean value of the prediction error. However, since the predicted vibration is a deterministic entity, the probability distribution of the measured vibration is the same as that of the prediction error. Accordingly, whereas the nominal value of the measured vibration can be controlled by the blade adjustment, its optimal position within the specification region should be determined according to its probability distribution. For a case where the prediction error, e^j ðkÞ; is zero mean and normally distributed, as illustrated in Figure 24.4, placing the predicted vibration at the center of the specification range will be synonymous with maximizing the probability that the measured vibration will be within the range. The likelihood of success of blade adjustments can therefore be measured by the area under the probability density function of prediction error located within the specification region. The blade adjustment set that produces the highest likelihood will be the preferred adjustment. The main difficulty with rotor tuning, however, is the limited number of DoFs, which precludes perfect positioning of the predicted vibration. This point is illustrated in Figure 24.5 for a case where two vibration components are to be positioned at the center of the specification region with only one adjustment. If one assumes that the effect of adjustment, Dx; on the change in the two vibration components, DV^ j ðkÞ; can be represented by a linear ^ model, as V1 L DV^ ðkÞ ¼ a Dx j

ij

then the position of the predicted vibration components will be constrained to the line L in Figure 24.5. As illustrated in this figure, since it will be impossible to place the predicted vibration components at the center, a compromised position needs to be selected. In this method, the best compromised position for the predicted vibration is that which renders the largest probability of satisfying the specifications for the measured vibration. This position, for the two component vibration example, Ð is one that maximizes Pr ½ðV1 ;V2 Þ [ S ¼ ðV1 ;V2 Þ[S pðV1 ;V2 ÞdV1 dV2 : The above formulation indicates that the placement of the predicted vibration requires knowledge of

Q P −s

s

^ V2

FIGURE 24.5 Restricted placement of vibration com ponents within the specification region for a two dimensional case.


24 7

the joint probability density function, pðV1 ;V2 Þ; of the vibration components. In the ideal case of independent vibration components with equal probability distributions, the loci of the points with equal probabilities Pr ½ðV1 ;…;Vn Þ [ S are surfaces of hyperspheres. Such ideal loci for the two component vibration example of Figure 24.5 are circles centered at the origin (see Figure 24.5), which lead to point P as the best compromised position closest on line L to the center of the specification circle. Point P, however, does not represent the best position if the two vibration components are dependent or have unequal distributions. The loci of equal probabilities for this more general case are elliptical, as also shown in Figure 24.5, indicating point Q as the best position on line L for placing the predicted vibration. The inadequacy of the DoFs illustrated here is exacerbated in rotor tuning, where 24 correlated vibration components need to be positioned within the specification region using only four condensed blade adjustments. For the 24 component vector of measured vibration VðkÞ ¼ ½Vc1 ðkÞ;Vs1 ðkÞ;…; Vc12 ðkÞ;Vs12 ðkÞ T ; where Vc and Vs represent the cosine and sine components of each vibration measurement, respectively, the joint probability density function of measured vibration for the kth flight, V(k), can be characterized as an N dimensional Gaussian function:

pðVðkÞÞ ¼

1 ð2pÞN=2 lFl1=2

e^ ðkÞ ¼ VðkÞ

1 e^ ðkÞT F 1 e^ ðkÞ 2

exp

Vðk

1Þ

CDxðkÞ

ð24:8Þ

ð24:9Þ

qffiffiffiffiffiffiffiffiffiffi where F represents the covariance matrix of the prediction error. Now, if G ¼ {lVj l ¼ Vcj2 þ Vsj2 # a;j ¼ 1;…;12} denotes the specification region in 24 dimensional Euclidean space, the blade adjustments, Dxp ; can be selected such that the probability that the measured vibration is within the acceptable range is maximized (see also Table 24.2). Formally, Dxp ¼ argDx max PrðVðkÞ [ G Þ ¼

TABLE 24.2

ð G

pðVðkÞÞdVðkÞ

ð24:10Þ

Summary of Probability Based Tuning

For the input vector: Dx ¼ ½Dx1 ; Dx2 ; Dx3 ; Dx4

T

where Dx1 and Dx3 denote the combined trim tab adjustments to blade combinations one to three and two to four, respectively, and Dx2 and Dx4 represent the combined pitch control rod adjustments to blade combinations one to three and two to four, respectively, the blade adjustments, Dx; can be selected such that the probability that the measured vibration is within the acceptable range is maximized. Formally, Ð Dxp ¼ argDx max PrðVðkÞ [ G Þ ¼ G pðVðkÞÞdVðkÞ where PrðVðkÞÞ denotes the probability of the measured vibration, G denotes the specification region in 24 dimensional Euclidean space, and pðVðkÞÞ represents the joint probability density of the measured vibration for the kth flight characterized as an N dimensional Gaussian function: 1 1 exp 2 e^ ðkÞT F 1 e^ ðkÞ pðVðkÞÞ ¼ 2 ð2pÞN=2 lFl1=2 with e^ ðkÞ ¼ VðkÞ 2 Vðk 2 1Þ 2 CDxðkÞ representing the predicted error in vibration.

24 8

24.4


Adaptive Tuning

The schematic of this method is shown in blade aircraft vibration modifications Figure 24.6 (Wang et al., 2005). As in the other Helicopter methods, it uses a process model as the basis of search for the appropriate blade adjustments, but Learning Algorithm instead of using a linear model, it uses an interval Interval model to accommodate process nonlinearity and Model estimated aircraft measurement noise. According to this model, the vibration feasible region of the process is estimated first, to include the adjustments that will result in Search acceptable vibration estimates. This feasible region is then used to search for the blade adjustments that will minimize the modeled vibration. If the FIGURE 24.6 The strategy of the proposed tuning method. application of these adjustments does not result in satisfactory vibration, the interval model will be updated to better estimate the feasible region and improve the choice of blade adjustments for the next flight. Important parameters of adaptive tuning are summarized in Table 24.3.

24.4.1

The Interval Model

In order to account for the stochastics and nonlinearity of vibration, an interval model (Moore, 1979) is defined to represent the range of aircraft vibration caused by blade adjustments. The interval model used here has the form: Dy yj ¼

n X i¼1

yji Dxi ; C

j ¼ 1; …; m

ð24:11Þ

where each coefficient is defined as an interval: yji ¼ ½CLji ; CUji C In the above model, the variables with the two sided arrow, $; denote intervalled variables, CLji and CUji represent, respectively, the current values of the lower and upper bounds of the sensitivity coefficients between each input, Dxi ; and output, Dy yj : The interval Dy yj denotes the estimated range of change of the jth output caused by the change to the current inputs, Dx1 ; …; Dxn :

TABLE 24.3

Summary of Adaptive Tuning

In adaptive tuning, each vibration component is defined as P yji Dxi ; Dy yj ¼ ni 1 C

j ¼ 1; …; m

where each coefficient is defined as an interval: yji ¼ ½CLji ; CUji C with CLji and CUji representing, respectively, the current values of the lower and upper bounds of the sensitivity coefficients between each input, Dxi ; and output, Dy yj : The blade adjustments are then sought by minimizing the objective function: PNe Distanceðxc ; xe Þ S ¼ QNse 1 1=Ns s 1 Distanceðxc ; xs Þ where xc represents a candidate set of blade adjustments within the feasible region, xe represents any set of blade adjustments within the selection region, xs denotes each of the previously selected blade adjustments, and Ne and Ns represent the number of the estimated feasible blade adjustments and the previously selected blade adjustments, respectively.


24 9

2

Actual input-output relationship Explored input-output pair Estimated range of output

Output

1.5 1 0.5 0 −4

FIGURE 24.7

−3

−2

−1

0 Input

1

2

3

4

Estimated range of output by the interval model using one reference input.

The fit provided by the interval model for a mildly nonlinear input/output relationship is illustrated in Figure 24.7, where the output range is estimated relative to one explored input.1 According to Equation 24.11, the estimated range of the output becomes larger, and therefore less accurate, as the potential input is selected farther from the current input (producing a large Dxi ). This potential drawback of the interval model is considerably reduced when multiple inputs have been explored so that the interval model can take advantage of several inputs for estimating the output range. The estimated output, yj ; at a potential input, xi ; may be computed relative to any set of previously explored inputs, yielding different estimates of yj (due to different values of Dxi ). In order to cope with the multiplicity of estimates, yj is defined as the common range among all of the yj estimates (Yang, 2000). The estimation of yj using this commonality rule is illustrated in Figure 24.8, which indicates that using this estimation approach enables representation of the system nonlinearities in a piecewise fashion. It can be shown that the lack of commonality between the estimated ranges of output will cause a part of the input output relationship to not be represented by the interval model. In such cases, however, the lack of compliance between the interval model and the input output relationship can be corrected by adaptation of the coefficient intervals through learning.

24.4.2

Estimation of Feasible Region

The feasible region comprises all sets of blade adjustments that will reduce the aircraft vibration within specifications. The feasible region is estimated here by comparing the individually estimated yj values with their corresponding constraints, so as to decide whether the corresponding blade adjustments belong to the feasible region. In this method, even when the interval yj partly overlaps the vibration constraint, the corresponding blade adjustments are included in the estimated feasible region. The above procedure of estimating the feasible region based on individual outputs is then extended to multiple outputs by forming the conjunction of the estimated feasible regions from each output.

24.4.3

Selection of Blade Adjustments

The blade adjustments provide the coordinates of the feasible region, therefore, they need to provide a balanced coverage of the input space. As such, blade adjustment selection becomes synonymous with maximizing the distance of the selected blade adjustments from the previous blade adjustments, as well as 1 An explored input represents an input for which the exact value of the output is available. In rotor tuning, an explored input would denote a blade adjustment that has been applied to the helicopter, and for which the corresponding vibration changes have been measured.

24 10


1

Actual input-output relationship Explored input-output pairs Estimated range of output

Output

0.8 0.6 0.4 0.2 0 −5

FIGURE 24.8

0 Input

5

Estimated range of output by the interval model using seven reference inputs.

bringing them closer to the center of the feasible region. This objective can be pursued by minimizing the following objective function: XNe Distanceðxc ; xe Þ S ¼ Y e¼1 ð24:12Þ 1=Ns Ns Distanceðxc ; xs Þ s¼1 where xc represents a candidate set of blade adjustments within the feasible region, xe represents any set of blade adjustments within the selection region, xs denotes each of the previously selected blade adjustments, and Ne and Ns represent, respectively, the numbers of the estimated feasible blade adjustments and the previously selected blade adjustments. Note that when the candidate set, xc ; is close QNs 1=Ns becomes small, and when the to the previously selected blade adjustments, s¼1 Distanceðxc ; xs Þ P e candidate set of blade adjustments, xc ; is far from the center of the feasible region, the value of N e¼1 Distanceðxc ; xe Þ becomes large. By minimizing S, the candidate blade adjustments are selected such that the above extremes are avoided.

24.4.4

Learning

Although an interval model defined according to the sensitivity coefficients may provide a suitable initial basis for tuning, it may not be the most representative of the rotor tuning process. As such, it may not be able to carry the search process to the end. A noted feature of the proposed method is its learning capability, which enables it to refine its knowledge base. To this end, the coefficients of the model are updated by considering new values for each of the upper and lower limits of individual coefficients. The objective is to make the range of the coefficients as small as possible while making sure that the interval model envelopes the acquired input output data. The learning problem can be defined as Minimize E ¼ subject to

KX1

K X

m¼1 k.m

{½yL ðm; kÞ

yðkÞ 2 þ ½yU ðm; kÞ

yðkÞ 2 }

ð24:13Þ

yU ðm; kÞ $ yðkÞ

ð24:14Þ

yL ðm; kÞ # yðkÞ

ð24:15Þ

g $ CLi

ð24:16Þ

CUi

where K represents the total number of sample points collected so far, yL ðm; kÞ and yU ðm; kÞ represent, respectively, the lower and upper limits of the estimated output range at the kth sample point relative to


24 11

the mth sample point, yðkÞ denotes the actual output value at the kth sample point, and CUi and CLi represent the upper and lower limits of the ith coefficient interval, respectively. The parameter g is a small positive number to control the range of the coefficients. Most of the approaches that can be potentially used for adapting the coefficient intervals, such as gradient descent (Ishibuchi et al., 1993) or nonlinear programming, cannot be applied to rotor tuning due to their demand for rich training data and their impartiality to the initial value of the coefficients representing the a priori knowledge of the process. As an alternative, a learning algorithm is devised here to cope with the scarcity of track and balance data while staying true to the initial values of the coefficients. In this algorithm, the coefficients of the interval model, initially set pointwise at the sensitivity coefficients, are adapted after each flight in two steps: enlargement and shrinkage. First, the vibration measurements from all of the flights completed for the present tail number are matched against the estimated output ranges from the current interval model. If any of the measurements do not fit the upper or lower limits of the estimates, the coefficient intervals are enlarged in small steps, iteratively, and the output ranges are re estimated at each iteration using the updated interval model. The enlargement of the coefficient intervals stops when the estimated output ranges include all of the measurements. At this point, even though the updated interval model provides a fit for the input output data, it may be overcompensated. In order to rectify this situation, the coefficient intervals are shrunk individually by selecting new candidates for their upper and lower limits. The shrinkage enlargement learning algorithm has the form: DCLi ¼

hdL Dxi ðm; kÞ

ð24:17Þ

DCUi ¼

hdU Dxi ðm; kÞ

ð24:18Þ

where, during the enlargement phase, dL and dU are defined as 8 DyL If Dxi ðm; kÞ . 0 and DyL . 0 > > < dL ¼ DyU If Dxi ðm; kÞ , 0 and DyU , 0 > > : 0 otherwise 8 DyU > > < dU ¼ DyL > > : 0

If Dxi ðm; kÞ . 0 and DyU , 0 If Dxi ðm; kÞ , 0 and DyL . 0

ð24:20Þ

otherwise

and during the shrinkage phase, they are defined as 8 DyL If Dxi ðm; kÞ . 0 and DyL , 0 > > < dL ¼ DyU If Dxi ðm; kÞ , 0 and DyU . 0 > > : 0 otherwise 8 DyU > > < dU ¼ DyL > > : 0

ð24:19Þ

ð24:21Þ

If Dxi ðm; kÞ . 0 and DyU . 0 If Dxi ðm; kÞ , 0 and DyL , 0

ð24:22Þ

otherwise

with Dxi ðm; kÞ ¼ xi ðkÞ

xi ðmÞ

ð24:23Þ

DyL ¼ yL ðm; kÞ

yðkÞ

ð24:24Þ

DyU ¼ yU ðm; kÞ

yðkÞ

ð24:25Þ

24 12


This procedure is repeated for each coefficient interval in an iterative fashion until the objective function E (Equation 24.13) is minimized. The minimization of E ensures limited adaptation of the coefficient intervals within the smallest possible range. At the beginning of tuning, the limited number of input output data available for learning will not provide a comprehensive representation of the process. Therefore, the coefficient intervals should not be shrunk drastically until enough input output data have become available. For this, the length of each coefficient interval ½CLi ; CUi is constrained by the minimal interval length for each tuning iteration as min L ¼ {CUi ð0Þ

CLi ð0Þ}ð1

bÞn

ð24:26Þ

where b [ ½0; 1 controls the shrinkage rate of the coefficient interval, and n denotes the number of tuning iterations. The coefficient interval cannot be shrunk when b ¼ 0 and can be shrunk without limit when b ¼ 1: Usually, b is selected closer to 0.

24.5

Case Study

The utility of the Interval Model (IM) method is demonstrated in application to Black Hawks. Ideally, the performance of the proposed method should be evaluated side by side against that of the traditional method. However, such an evaluation would require tuning the aircraft with one method, undoing changes, and tuning the aircraft with another. Since such testing is prohibitively costly and infeasible, a compromised approach of evaluating the method in simulation is utilized. A process simulation model is therefore used to represent the block “helicopter” in Figure 24.6, with the block “forward model” represented by an interval model.

24.5.1

Simulation Model

Considering the potential nonlinearity of the effect of blade adjustments on the helicopter vibration and the high level of noise present in vibration measurements, multilayer neural networks offer the most suitable framework for modeling. A series of neural networks were trained with historical balance data to represent the relationships between vibration changes and blade adjustments, and the stochastic aspects of vibration (see Chapter 5) were represented by the addition of random numbers to the outputs of the networks. A total of 102 sets of vibration data were used to train and test the neural networks. The inputs to these networks were the combined blade adjustments of push rods and trim tabs to opposite blade pairs, and their outputs were the resulting vibration changes between two consecutive flights. Since the vibration data are vector quantities that are represented by both magnitude and phase components (see Table 24.1), the vibration data were transformed into Cartesian coordinates, so that each vector element would denote the change in the cosine or sine component of the A þ B or A B vibration of each of the six flight regimes (see Table 24.1). In this study, each neural network model consisted of four inputs and one output, so a total of 24 networks were trained to represent all of the vibration components. Alternatively, all of the vibration measurements may be represented by one neural network, but such a network is more difficult to train. Formally, the outputs of the neural networks, which represent the cosine and sine components of the vibration at different regimes, vcj ðkÞ and vsj ðkÞ; respectively, are defined as V^ sj ðkÞ ¼ Vsj ðk

1Þ þ DVsj ðkÞ þ Rsj ðkÞ

ð24:27Þ

V^ cj ðkÞ ¼ Vcj ðk

1Þ þ DVcj ðkÞ þ Rcj ðkÞ

ð24:28Þ

DVsj ðkÞ ¼ Fsj ðDxÞ

ð24:29Þ

DVcj ðkÞ ¼ Fcj ðDxÞ

ð24:30Þ


24 13

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V^ j ðkÞ ¼ V^ sj ðkÞ2 þ V^ cj ðkÞ2

ð24:31Þ

where the input vector Dx ¼ {Dx1 ; Dx2 ; Dx3 ; Dx4 } denotes the set of combined blade adjustments, each of the functionals, Fsj and Fcj ; represent the change in vibration between two consecutive flights as represented by a neural network, and Rsj ðkÞ and Rcj ðkÞ denote random numbers added to the outputs of the networks to account for measurement noise. Each of the networks consisted of two hidden layers, with four and eight processing elements in the first and second layers, respectively. To avoid overtraining, the 102 sets of data were divided into two equal subsets, one set to train the network and the other to test its performance. The random numbers, Rcj and Rsj ; were generated according to the Gaussian distribution Nðm; s2 Þ; with the mean m and variance s 2 defined as

m^ ¼ s^ 2 ¼

M 1 X e M i¼1 i M X

1 M

1

i¼1

ðei

ð24:32Þ

m^Þ2

ð24:33Þ

In the above formulation, M represents the total number of data sets and ei denotes the difference between the measured and expected value of vibration, defined as ej ðkÞ ¼ Vj ðkÞ

Vj ðk

1Þ

DVj ðkÞ

ð24:34Þ

A sample of estimated vibration changes generated by the neural network model is compared side by side with the actual vibration changes in Figure 24.9. The results indicate close agreement between the predicted and actual vibration changes.

24.5.2

Interval Modeling

In application to the Black Hawks, a total of 24 interval models need to be constructed to approximate the changes in the cosine and sine components of the A þ B and A B vibrations at each of the six flight regimes. The interval models have the form ycj ðkÞ ¼ Vcj ðk V

1Þ þ

ysj ðkÞ ¼ Vsj ðk V

1Þ þ

4 X i¼1 4 X i¼1

ycji ðk C

1ÞDxi ðkÞ

ð24:35Þ

ysji ðk C

1ÞDxi ðkÞ

ð24:36Þ

0.25 Value of Vibration

0.2

A+B COS

0.15

A+B SIN

A-B COS

A-B SIN

0.1 0.05 0 −0.05 −0.1 −0.15

FIGURE 24.9

0

5

10 15 Ordinal Number of Vibration

20

25

A sample set of simulated vibration changes shown side by side with the actual vibration changes.

24 14


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yj ðkÞ ¼ V y 2 ðkÞ þ V y 2 ðkÞ; V sj cj

i ¼ 1; …; 4 and j ¼ 1; …; 12

ð24:37Þ

ycj ðkÞ and V ysj ðkÞ represent, respectively, the estimated cosine and sine components of A þ B or where the V yj ðkÞ denotes the magnitude of the vibration, and Dxi A B vibration at each of the six flight regimes, V are the same as those in Equation 24.2 to Equation 24.5. For this study, the feasible region was defined to include all of the blade adjustments associated with vibration estimates that satisfied the specification: y1 Þ; …; minðV y12 Þ} # 0:2: The above specification ensures that the lower limit of the estimated max{minðV vibration range of the largest vibration component will be less than 0.2 ips (an industry standard). The selection of the lower limit here is to ensure that the feasible region is as large as possible, so as not to eliminate any potentially good candidate blade adjustments. The computation of the feasible region was based on the range [ 0.015, 0.015] for push rods and [ 0.035, 0.035] for trim tabs, within which 20,000 random sets of blade adjustments were evaluated for their feasibility. The blade adjustments associated with vibration ranges satisfying the specification were included in the feasible region. As noted earlier, the proposed method uses the feasible region as the basis of search for the blade adjustments. For this study, the blade adjustments set that produced the smallest value for the objective function S (Equation 24.12) was selected to be applied to the helicopter. It should be noted that, given the stringent constraints on the vibration components, there were cases where the search algorithm could not find any feasible blade adjustments that would satisfy all of the constraints. In such cases, the set of blade adjustments that produced the smallest lower limit of the maximum estimated vibration was used as a compromised solution. The interval model was updated after each tuning iteration. For shrinkage enlargement learning, the parameter b in Equation 24.26 was set to 1 and g to 0, so that the coefficient intervals could be shrunk without limits. Learning was performed separately for each tail number to customize the interval model to individual tail numbers; that is, the interval model was set to the sensitivity coefficients for each tail number and was adapted after the first tuning iteration. Accordingly, the interval model was actually a pointwise model for the first iteration and took the form of an interval model thereafter.

24.5.3

Performance Evaluation

The interval model (IM) method was tested on 39 tail numbers, for which actual track and balance data were available from the field. For each tail number, the IM method was applied iteratively until either the simulated vibrations were within their specifications, or an upper limit of five process iterations had been reached. Since the stochastic aspects of vibration measurements impose randomness on the rotor tuning process, rotor tuning solutions cannot be evaluated by deterministic measures. This calls for the creation of performance measures that account for uncertainty. One such measure that assesses tuning efficiency is the average tuning iteration number (ATIN) which represents the average number of iterations taken for tuning each tail number. The number of flights used by the IM method for the 39 tail numbers is included in Table 24.4 along with those actually performed in the field. The results indicate that the IM method requires a smaller ATIN relative to that actually performed. Another potentially significant aspect of the IM method is its adaptation capability, which enables it to transform a pointwise model into an IM, and to subsequently update it after the first iteration. Adaptation capability, however, may not be as significant in rotor tuning, which offers limited possibility for training. In order to evaluate the significance of learning in the performance of the IM method, the results in Table 24.4 were reproduced in Table 24.5 with the learning feature turned off. The ATINs indicate that with learning, the IM method requires fewer iterations for tuning each tail number, despite the small number of iterations taken to tune each tail number. This, in turn, indicates that the interval model enhances the performance of the IM method, since without learning, the model remains pointwise at the sensitivity coefficients. However, perhaps an equally interesting set of results in Table 24.5 are those indicating that even without learning, the IM method requires fewer iterations than actually


24 15

TABLE 24.4 The Number of Tuning Iterations Required by the Interval Model Method and Those Applied in the Field Tail # (39)

Number of Tuning Iterations Actual

176 178 179 180 184 .. . 260 .. . 861

IM Method

1 1 3 1 2. .. 4. .. 1

Total ATIN

1 1 2 1 1. .. 1. .. 1

71 1.82

48 1.23

TABLE 24.5 The Number of Tuning Iterations Required by the IM Method (with and without Learning) along with Those Actually Applied in the Field Tail # (39)

Tuning Iteration Number Actual

IM Method With Learning

185 186 208 245 260 802 822 .. . Total ATIN

3 3 2 3 4 3 3. .. 71 1.82

2 2 2 2 2 2 2. .. 48 1.23

Without Learning 3 3 3 3 3 3 3. .. 62 1.59

performed in the field. Given that the adjustments associated with both sets of results were selected from the same model (i.e., sensitivity coefficients), the better performance of the IM method can only be attributed to its more effective search strategy that leads to more comprehensive solutions. A preferred aspect of a system of rotor tuning is its ability to tune the aircraft within one iteration. This aspect of the method was evaluated by checking the number of tail numbers tuned within one iteration. For these results, in order to eliminate the difference between the simulation model and the helicopter, only the vibration estimates from simulation were used to evaluate the suitability of the adjustments. The results of this study are shown in Table 24.6, where the tail numbers tuned within one iteration are p shown by a and those requiring more than one iteration are denoted by £ . The results indicate that the IM method satisfies this more stringent criterion better than the actual adjustments, further validating the claim that the IM method benefits from a more effective search engine. Owing to the randomness of the vibration measurements, repeated applications of an adjustment set may lead to slightly different vibration measurements. This, in turn, may cause a variance in the number of iterations produced by adjustments when the resulting vibration is close to the specified threshold. It would be beneficial, therefore, to devise a measure for the probability of success of adjustments.

24 16

Vibration and Shock Handbook TABLE 24.6 Tally of the Tail Numbers Tuned within One Iteration According to Simulated Vibration Tail # (39)

176 178 179 .. . 822 858 859 861 Total

Tuned within One Iteration Actual p £ p .. . £ £ p p

IM Method p p p .. . p p p p

19

30

The empirical measure, the acceptability index (AI), is defined here as AI ¼

N 1 X s N l¼1 l

ð24:38Þ

to denote the percentage of times an adjustment set will result in the vibration satisfying the specification. In the above equation, N represents the total number of flights simulated to represent the repeated applications of the same adjustment set, and ( 1 if vibration of the lth simulation flight is acceptable sl ¼ 0 if vibration of the lth simulation flight is unacceptable The AIs computed for both the actual and selected adjustments at the first iteration are included in Table 24.7. The results indicate that the IM method provides adjustments with a higher probability of success as judged by the acceptability of vibration estimates from the simulation model. These results, which indicate that the selected adjustments from the IM method can more consistently tune the rotors within one iteration, imply the better positioning of the adjustments within the feasible region. TABLE 24.7 The Values of Acceptability Index (Trial Mode) Computed for Both the Actual and Selected Adjustments at the First Flight Tail # (39)

Acceptability Index Actual

IM Method

176 178 179 .. . 260 261 263 .. . 822 857 858 859 861

0.92 0 0.61 .. . 0.40 0.09 0.18 .. . 0.00 0.62 0.64 0.94 0.96

0.87 0.54 0.52 .. . 0.89 0.95 0.12 .. . 0.64 0.55 0.93 0.67 0.74

Average

0.581

0.724


24 17

TABLE 24.8 Comparison of the First Iteration Solutions of IM Method and Actual Solutions from Sikorsky’s Production Line with the Cumulative Acceptable Adjustments Tail #

Modifications Actual Iteration 1 Modifications

801 802 822 858

6, 24, 210, 11 5, 2, 0, 0 6, 0, 220, 0 7, 0, 214, 0

CAM

IM Iteration 1 Modifications

2, 24, 24, 14 9, 0, 210, 10 10, 24, 223, 13 9, 22, 210, 3

3, 25, 26, 12 8, 22, 25, 10 8, 24, 222, 6 9, 22, 215, 4

Another evaluation basis for the adjustments can be established by comparing them to the actual P cumulative adjustments performed in the field. The cumulative adjustment set, x; can be defined as N X X x¼ Dxk

ð24:39Þ

k¼1

where N represents the total number of tuning iterations performed in the field for the tail number and Dxk denotes the adjustments applied at the kth iteration. A sample of actual first iteration adjustments, actual cumulative adjustments, and first iteration adjustments from the IM method is shown in Table 24.8. The results indicate that the adjustments from the IM method are closer to the actual cumulative adjustments than are the actual first iteration adjustments. Although the cumulative adjustments may not be the most desirable ones for the aircraft, they represent an acceptable set that has been proven in the field. The closeness of the IM method’s solutions to the actual cumulative adjustments further validates its effectiveness.

24.6

Conclusion

A logical feature for future rotor tuning systems will be the capability to adjust the blades during the flight. For this, these systems will need to have the capability to learn from their mistakes. They will also need to be able to monitor the condition of the rotor system in flight, so they will stop modifying the blade parameters when more drastic actions are necessary for saving the aircraft. As such, these systems will need to be used with strong operator interaction to prevent implementation of inappropriate adjustments, and must have the ability to explain the recommended adjustments to the operator.

References Ishibuchi, H., Tanaka, H., and Okada, H., An architecture of neural networks with interval weights and its applications to fuzzy regression analysis, Fuzzy Sets Syst., 57, 27 59, 1993. Moore, R.E. 1979. Methods and Applications of Interval Analysis, Society for Industrial and Applied Mathematics, Philadelphia. Taitel, H., Danai, K., and Gauthier, D.G., Helicopter track and balance with artificial neural nets, ASME J. Dyn. Syst. Meas. Contr., 117, 226 231, 1995. Ventres, S. and Hayden, R.E. 2000. Rotor tuning using vibration data only, American Helicopter Society 56th Annual Forum, Virginia Beach, VA, May 2 4, 2000. Wang, S., Danai, K., and Wilson, M., A probability based approach to helicopter track and balance, J. Am. Helicopter Soc., 50, 1, 56 64, 2005a. Wang, S., Danai, K., and Wilson, M., An adaptive method of helicopter track and balance, ASME J. Dyn. Syst. Meas. Contr., 2005b, March, in press. Yang, D.Z. 2000. Knowledge based interval modeling method for efficient global optimization and process tuning, Ph.D. Thesis. Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst.

Monitoring and Diagnosis

VI

VI 1

25

Machine Condition Monitoring and Fault Diagnostics 25.1 Introduction ....................................................................... 25.2 Machinery Failure ............................................................. Causes of Failure of Failure

†

Types of Failure

†

Frequency

25 2 25 2

25.3 Basic Maintenance Strategies ............................................

25 4

25.4 Factors which Influence Maintenance Strategy .............. 25.5 Machine Condition Monitoring ......................................

25 7 25 8

Run to Failure (Breakdown) Maintenance † Scheduled (Preventative) Maintenance † Condition Based (Predictive, Proactive, Reliability Centered, On Condition) Maintenance

Periodic Monitoring

†

Continuous Monitoring

25.6 Transducer Selection ......................................................... 25 10 Noncontact Displacement Transducers † Velocity Transducers † Acceleration Transducers

25.7 Transducer Location .......................................................... 25 14 25.8 Recording and Analysis Instrumentation ........................ 25 14 Vibration Meters † Data Collectors † Frequency Domain Analyzers † Time Domain Instruments † Tracking Analyzers

25.9 Display Formats and Analysis Tools ................................ 25 16 Time Domain † Frequency Domain Quefrency Domain

†

Modal Domain

†

25.10 Fault Detection .................................................................. 25 21 Standards Limits

†

Acceptance Limits

†

Frequency Domain

25.11 Fault Diagnostics ............................................................... 25 25

Chris K. Mechefske Queen’s University

Forcing Functions † Specific Machine Components † Specific Machine Types † Advanced Fault Diagnostic Techniques

Summary The focus of this chapter is on the definition and description of machine condition monitoring and fault diagnosis. Included are the reasons and justification behind the adoption of any of the techniques presented. The motivation behind the decision making in regard to various applications is both financial and technical. Both of these aspects are discussed, with the emphasis being on the technical side. The chapter defines machinery failure (causes, types, and frequency), and describes basic maintenance strategies and the factors that should be considered when deciding 25 1

25 2


which to apply in a given situation. Topics considered in detail include transducer selection and mounting location, recording and analysis instrumentation, and display formats and analysis tools (specifically, time domain, frequency domain, modal domain, and quefrency domain based strategies). The discussion of fault detection is based primarily on standards and acceptance limits in the time and frequency domains. The discussion of fault diagnostics is divided into sections that focus on different forcing functions, specific machine components, specific machine types, and advanced diagnostic techniques. Further considerations on this topic are found in Chapter 26 and Chapter 27.

25.1

Introduction

Approximately half of all operating costs in most processing and manufacturing operations can be attributed to maintenance. This is ample motivation for studying any activity that can potentially lower these costs. Machine condition monitoring and fault diagnostics is one of these activities. Machine condition monitoring and fault diagnostics can be defined as the field of technical activity in which selected physical parameters, associated with machinery operation, are observed for the purpose of determining machinery integrity. Once the integrity of a machine has been estimated, this information can be used for many different purposes. Loading and maintenance activities are the two main tasks that link directly to the information provided. The ultimate goal in regard to maintenance activities is to schedule only what is needed at a time, which results in optimum use of resources. Having said this, it should also be noted that condition monitoring and fault diagnostic practices are also applied to improve end product quality control and as such can also be considered as process monitoring tools. This definition implies that, while machine condition monitoring and fault diagnostics is being treated as the focus of this chapter, it must also be considered in the broader context of plant operations. With this in mind, this chapter will begin with a description of what is meant by machinery failure and a brief overview of different maintenance strategies and the various tasks associated with each. A short description of different vibration sensors, their modes of operation, selection criteria, and placement for the purposes of measuring accurate vibration signals will then follow. Data collection and display formats will be discussed with the specific focus being on standards common in condition monitoring and fault diagnostics. Machine fault detection and diagnostic practices will make up the remainder of the chapter. The progression of information provided will be from general to specific. The hope is that this will allow a broad range of individuals to make effective use of the information provided.

25.2

Machinery Failure

Most machinery is required to operate within a relatively close set of limits. These limits, or operating conditions, are designed to allow for safe operation of the equipment and to ensure equipment or system design specifications are not exceeded. They are usually set to optimize product quality and throughput (load) without overstressing the equipment. Generally speaking, this means that the equipment will operate within a particular range of operating speeds. This definition includes both steady state operation (constant speed) and variable speed machines, which may move within a broader range of operation but still have fixed limits based on design constraints. Occasionally, machinery is required to operate outside these limits for short times (during start up, shutdown, and planned overloads). The main reason for employing machine condition monitoring and fault diagnostics is to generate accurate, quantitative information on the present condition of the machinery. This enables more confident and realistic expectations regarding machine performance. Having at hand this type of reliable information allows for the following questions to be answered with confidence: *

*

Will a machine stand a required overload? Should equipment be removed from service for maintenance now or later?

Machine Condition Monitoring and Fault Diagnostics *

*

*

25 3

What maintenance activities (if any) are required? What is the expected time to failure? What is the expected failure mode?

Machinery failure can be defined as the inability of a machine to perform its required function. Failure is always machinery specific. For example, the bearings in a conveyor belt support pulley may be severely damaged or worn, but as long as the bearings are not seized, it has not failed. Other machinery may not tolerate these operating conditions. A computer disk drive may have only a very slight amount of wear or misalignment resulting in noisy operation, which constitutes a failure. There are also other considerations that may dictate that a machine no longer performs adequately. Economic considerations may result in a machine being classified as obsolete and it may then be scheduled for replacement before it has “worn out.” Safety considerations may also require the replacement of parts in order to ensure the risk of failure is minimized.

25.2.1

Causes of Failure

When we disregard the gradual wear on machinery as a cause of failure, there are still many specific causes of failure. These are perhaps as numerous as the different types of machines. There are, however, some generic categories that can be listed. Deficiencies in the original design, material or processing, improper assembly, inappropriate maintenance, and excessive operational demands may all cause premature failure.

25.2.2

Types of Failure

As with the causes of failure, there are many different types of failure. Here, these types will be subdivided into only two categories. Catastrophic failures are sudden and complete. Incipient failures are partial and usually gradual. In all but a few instances, there is some advanced warning as to the onset of failure; that is, the vast majority of failures pass through a distinct incipient phase. The goal of machine condition monitoring and fault diagnostics is to detect this onset, diagnose the condition, and trend its progression over time. The time until ultimate failure can then hopefully be better estimated, and this will allow plans to be made to avoid undue catastrophic repercussions. This, of course, excludes failures caused by unforeseen and uncontrollable outside forces.

25.2.3

Frequency of Failure

Anecdotal and statistical data describing the Wear In Normal Wear Wear Out frequency of failures can be summarized in what is called a “bathtub curve.” Figure 25.1 shows a Failure typical bathtub curve, which is applicable to an Rate individual machine or population of machines of the same type. The beginning of a machine’s useful life is Time In Service usually characterized by a relatively high rate of failure. These failures are referred to as “wear in” FIGURE 25.1 Typical bathtub curve. failures. They are typically due to such things as design errors, manufacturing defects, assembly mistakes, installation problems and commission ing errors. As the causes of these failures are found and corrected, the frequency of failure decreases. The machine then passes into a relatively long period of operation, during which the frequency of failures occurring is relatively low. The failures that do occur mainly happen on a random basis. This period of a machine’s life is called the “normal wear” period and usually makes up most of the life of a machine. There should be a relatively low failure rate during the normal wear period when operating within design specifications.

25 4


As a machine gradually reaches the end of its designed life, the frequency of failures again increases. These failures are called “wearout” failures. This gradually increasing failure rate at the expected end of a machine’s useful life is primarily due to metal fatigue, wear mechanisms between moving parts, corrosion, and obsolescence. The slope of the wearout part of the bathtub curve is machine dependent. The rate at which the frequency of failures increases is largely dependent on the design of the machine and its operational history. If the machine design is such that the operational life ends abruptly, the machine is underdesigned to meet the load expected, or the machine has endured a severe operational life (experienced numerous overloads), the slope of the curve in the wearout section will increase sharply with time. If the machinery is overdesigned or experiences a relatively light loading history, the slope of this part of the bathtub curve will increase only gradually with time.

*

*

*

*

*

25.3

Generally (outside of start up and shutdown) machinery is required to operate at constant speed and load. Machinery failure is defined based on performance, operating condition, and system specifications. Machinery failure can be defined as the inability of a machine to perform its required function. Causes of machinery failure can be generally defined as being due to deficiencies in the original design, material or processing, improper assembly, inappropriate maintenance, or excessive operation demands. The frequency of failure for an individual machine or a population of similar machines can be summarized using a “bathtub curve.”

Basic Maintenance Strategies

Maintenance strategies can be divided into three main types: (1) run to failure, (2) scheduled, and (3) condition based maintenance. Each of these different strategies has distinct advantages and disadvantages, which will be described below. Specific situations within any large facility may require the application of a different strategy. Therefore, no one strategy should be considered as always superior or inferior to another.

25.3.1

Run-to-Failure (Breakdown) Maintenance

Run to failure, or breakdown maintenance, is a strategy where maintenance, in the form of repair work or replacement, is only performed when machinery has failed. In general, run to failure maintenance is appropriate when the following situations exist: *

*

*

*

*

*

*

The equipment is redundant. Low cost spares are available. The process is interruptible or there is stockpiled product. All known failure modes are safe. There is a known long mean time to failure (MTTF) or a long mean time between failure (MTBF). There is a low cost associated with secondary damage. Quick repair or replacement is possible.

An example of the application of run to failure maintenance can be found when one considers the standard household light bulb. This device satisfies all the requirements above and therefore the most cost effective maintenance strategy is to replace burnt out light bulbs as needed.

Machine Condition Monitoring and Fault Diagnostics

Machine Capacity (Est.) Estimated Capacity and Load

25 5

Failures

Maintenance Activities

Machine Duty (Load) Time In Service FIGURE 25.2

Time vs. estimated capacity and actual load (run to failure maintenance).

Figure 25.2 shows a schematic demonstrating the relationship between a machine’s time in service, the load (or duty) placed on the machine, and the estimated remaining capacity of the machine. Whenever the estimated capacity curve intersects with (or drops below) the load curve, a failure will occur. At these times, repair work must be carried out. If the situation that exists fits within the “rules” outlined above, all related costs (repair work and downtime) will be minimized when using run to failure maintenance.

25.3.2

Scheduled (Preventative) Maintenance

When specific maintenance tasks are performed at set time intervals (or duty cycles) in order to maintain a significant margin between machine capacity and actual duty, the type of maintenance is called scheduled or preventative maintenance. Scheduled maintenance is most effective under the following circumstances: *

*

*

*

*

*

*

*

Data describing the statistical failure rate for the machinery is available. The failure distribution is narrow, meaning that the MTBF is accurately predictable. Maintenance restores close to full integrity of the machine. A single, known failure mode dominates. There is low cost associated with regular overhaul/replacement of the equipment. Unexpected interruptions to production are expensive and scheduled interruptions are not so bad. Low cost spares are available. Costly secondary damage from failure is likely to occur.

An example of scheduled maintenance practices can be found under the hood of your car. Oil and oil filter changes on a regular basis are part of the scheduled maintenance program that most car owners practice. A relatively small investment in time and money on a regular basis acts to reduce (but not eliminate) the likelihood of a major failure taking place. Again, this example shows how when all, or most, of the criteria listed above are satisfied, overall maintenance costs are minimized. Figure 25.3 shows a schematic demonstrating the relationship between a machine’s time in service, the load (or duty) placed on the machine and the estimated remaining capacity of the machine when scheduled maintenance is being practiced. In this case, maintenance activities are scheduled at regular intervals in order to restore machine capacity before a failure occurs. In this way, there is always a margin between the estimated capacity and the actual load on the machine. If this margin is always present, there should theoretically never be an unexpected failure, which is the ultimate goal of scheduled maintenance.

25 6



Margin Machine Duty (Load)

Maintenance Activities Time In Service

FIGURE 25.3

Time vs. estimated capacity and actual load (scheduled maintenance).

25.3.3 Condition-Based (Predictive, Proactive, Reliability Centered, On-Condition) Maintenance Condition based maintenance (which is also known by many other names) requires that some means of assessing the actual condition of the machinery is used in order to optimally schedule maintenance, in order to achieve maximum production, and still avoid unexpected catastrophic failures. Condition based maintenance should be employed when the following conditions apply: *

*

*

*

*

*

*

*

*

*

Expensive or critical machinery is under consideration. There is a long lead time for replacement parts (no spares are readily available). The process is uninterruptible (both scheduled and unexpected interruptions are excessively costly). Equipment overhaul is expensive and requires highly trained people. Reduced numbers of highly skilled maintenance people are available. The costs of the monitoring program are acceptable. Failures may be dangerous. The equipment is remote or mobile. Failures are not indicated by degeneration of normal operating response. Secondary damage may be costly.

An example of condition based maintenance practices can again be found when considering your car, but this time we consider the tires. Regular inspections of the tires (air pressure checks, looking for cracks and scratches, measuring the remaining tread, listening for slippage during cornering) can all be used to make an assessment of the remaining life of the tires and also the risk of catastrophic failure. In order to minimize costs and risk, the tires are replaced before they are worn out completely, but not before they have given up the majority of their useful life. A measure of the actual condition of equipment is used to utilize maintenance resources optimally. Figure 25.4 shows a schematic drawing that demonstrates the relationship between a machine’s time in service, the load (or duty) placed on the machine, and the estimated remaining capacity of the machine when condition based maintenance is being practiced. Note that the margin between duty and capacity is allowed to become quite small (smaller than in scheduled maintenance), but the two lines never touch (as in run to failure maintenance). This results in a longer time between maintenance activities than for scheduled maintenance. Maintenance tasks are scheduled just before a failure is expected to occur, thereby optimizing the use of resources. This requires that there exists a set of accurate measures that can be used to assess the machine integrity. Each of these maintenance strategies has its advantages and disadvantages and situations exist where one or the other is appropriate. It is the maintenance engineer’s role to decide on and justify the use of any one of these procedures for a given machine. There are also instances where a given machine will require more than one maintenance strategy during its operational life, or perhaps even at one


25 7


Minimum Margin

Machine Duty (Load)

Maintenance Activity

Time In Service FIGURE 25.4

Time versus estimated capacity and actual load (condition based maintenance).

time, and situations where more than one strategy is appropriate within a particular plant. Examples of these situations include the need for an increased frequency of monitoring as the age of a machine increases and the likelihood of failure increases, and the scheduling of maximum time between overhauls during the early stages of a machine’s useful life, with monitoring in between looking for unexpected failures.

*

*

*

*

*

25.4

Maintenance strategies can be divided into three main types: (1) run to failure, (2) scheduled, and (3) condition based maintenance. No one strategy should be considered as always superior or inferior to another. Run to failure, or breakdown maintenance, is a strategy where maintenance, in the form of repair work or replacement, is only performed when machinery has failed. When specific maintenance tasks are performed at set time intervals (or duty cycles) in order to maintain a significant margin between machine capacity and actual duty, the type of maintenance is called scheduled or preventative maintenance. Condition based maintenance requires that some means of assessing the actual condition of the machinery is used in order to optimally schedule maintenance, in order to achieve maximum production and still avoid unexpected catastrophic failures.

Factors which Influence Maintenance Strategy

While there are some general guidelines for choosing the most appropriate maintenance strategy, each case must be evaluated individually. Principal considerations will always be defined in economic terms. Sometimes, a specific company policy (such as safety) will outweigh all other considerations. Below is a list of factors (in no particular order) that should be taken into account when deciding which maintenance strategy is most appropriate for a given situation or machine: *

*

*

*

*

*

*

*

Classification (size, type) of the machine Critical nature of the machine relative to production Cost of replacement of the entire machine Lead time for replacement of the entire machine Manufacturers’ recommendations Failure data (history), MTTF, MTBF, failure modes Redundancy Safety (plant personnel, community, environment)

25 8

Vibration and Shock Handbook *

*

*

25.5

Cost and availability of spare parts Personnel costs, administrative costs, monitoring equipment costs Running costs for a monitoring program (if used)

Machine Condition Monitoring

With the understanding that condition based maintenance may not be appropriate in all situations, let us say that some preliminary analysis has been carried out and a decision made to apply machine condition monitoring and fault diagnostics in a selected part of a plant or on a specific machine. The following is a list of potential advantages that should be realized: *

*

*

*

*

*

*

*

*

*

*

Increased machine availability and reliability Improved operating efficiency Improved risk management (less downtime) Reduced maintenance costs (better planning) Reduced spare parts inventories Improved safety Improved knowledge of the machine condition (safe short term overloading of machine possible) Extended operational life of the machine Improved customer relations (less planned/unplanned downtime) Elimination of chronic failures (root cause analysis and redesign) Reduction of postoverhaul failures due to improperly performed maintenance or reassembly

There are, of course, also some disadvantages that must be weighed in the decision to use machine condition monitoring and fault diagnostics. These disadvantages are listed below: *

*

*

*

*

*

Monitoring equipment costs (usually significant). Operational costs (running the program). Skilled personnel needed. Strong management commitment needed. A significant run in time to collect machine histories and trends is usually needed. Reduced costs are usually harder to sell to management as benefits when compared with increased profits.

The ultimate goal of machine condition monitoring and fault diagnostics is to get useful information on the condition of equipment to the people who need it in a timely manner. The people who need this information include operators, maintenance engineers and technicians, managers, vendors, and suppliers. These groups will need different information at different times. The task of the person or group in charge of condition monitoring and diagnostics must ensure that useful data is collected, that data is changed into information in a form required by and useful to others, and that the information is provided to the people who need it when they need it. Further general reading can be found in these references: Mitchell (1981), Lyon (1987), Mobley (1990), Rao (1996), and Moubray (1997). The focus of this chapter will be on vibration based data, but there are several different types of data that can be useful for assessing machine condition and these should not be ignored. These include physical parameters related to lubrication analysis (oil/grease quality, contamination), wear particle monitoring and analysis, force, sound, temperature, output (machine performance), product quality, odor, and visual inspections. All of these factors may contribute to a complete picture of machine integrity. The types of information that can be gleaned from the data include existing condition, trends, expected time to failure at a given load, type of fault existing or developing, and the type of fault that caused failure. The specific tasks which must be carried out to complete a successful machine condition monitoring and fault diagnostics program include detection, diagnosis, prognosis, postmortem, and


25 9

prescription. Detection requires data gathering, comparison to standards, comparison to limits set in plant for specific equipment, and trending over time. Diagnosis involves recognizing the types of fault developing (different fault types may be more or less serious and require different action) and determining the severity of given faults once detected and diagnosed. Prognosis, which is a very challenging task, involves estimating (forecasting) the expected time to failure, trending the condition of the equipment being monitored, and planning the appropriate maintenance timing. Postmortem is the investigation of root cause failure analysis, and usually involves some research type investigation in the laboratory and/or in the field, as well as modeling of the system. Prescription is an activity that is dictated by the information collected and may be applied at any stage of the condition monitoring and diagnostic work. It may involve recommendations for altering the operating conditions, altering the monitoring strategy (frequency, type), or redesigning the process or equipment. The tasks listed above have relatively crisp definitions, but there is still considerable room for adjustment within any condition monitoring and diagnostic program. There are always questions, concerning such things as how much data to collect and how much time to spend on data analysis, that need to be considered before the final program is put in place. As mentioned above, things such as equipment class, size, importance within the process, replacement cost, availability, and safety need to be carefully considered. Different pieces of equipment or processes may require different monitoring strategies.

25.5.1

Periodic Monitoring

Periodic monitoring involves intermittent data gathering and analysis with portable, removable monitoring equipment. On occasion, permanent monitoring hardware may be used for this type of monitoring strategy, but data is only collected at specific times. This type of monitoring is usually applied to noncritical equipment where failure modes are well known (historically dependable equipment). Trending of condition and severity level checks are the main focus, with problems triggering more rigorous investigations.

25.5.2

Continuous Monitoring

Constant or very frequent data collection and analysis is referred to as continuous monitoring. Permanently installed monitoring systems are typically used, with samples and analysis of data done automatically. This type of monitoring is carried out on critical equipment (expensive to replace, with downtime and lost production also being expensive). Changes in condition trigger more detailed investigation or possibly an automatic shutdown of the equipment.

*

*

*

*

Potential advantages of machine condition monitoring include increased machine availability and reliability, improved efficiency, reduced costs, extended operational life, and improved safety. Some of the disadvantages of condition monitoring include monitoring equipment costs, operational costs, and training costs. The ultimate goal of machine condition monitoring and fault diagnostics is to get useful information on the condition of equipment to the people who need it, in a timely manner. The specific tasks which must be carried out to complete a successful machine condition monitoring and fault diagnostics program include detection, diagnosis, prognosis, postmortem, and prescription.

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25.6

Transducer Selection Relative Amplitude Response

A transducer is a device that senses a physical quantity (vibration in this case, but it can also be Displacement 10 temperature, pressure, etc.) and converts it into an electrical output signal, which is proportional to Velocity 1.0 the measured variable (see Chapter 15). As such, Acceleration the transducer is a vital link in the measurement 0.1 chain. Accurate analysis results depend on an accurate electrical reproduction of the measured 0.1 1.0 10 100 1,000 10,000 parameters. If information is missed or distorted Frequency (Hz) during measurement, it cannot be recovered later. Hence, the selection, placement, and proper use of FIGURE 25.5 Frequency versus response amplitude the correct transducer are important steps in the for various sensor types. implementation of a condition monitoring and fault diagnostics program. Considerable research and development work has gone into the design, testing, and calibration of sensors (transducers) for a wide range of applications. The transducer must be: *

*

*

*

Correct for the task Properly mounted In good working order (properly calibrated) Fully understood in terms of operational characteristics

Transducers usually require amplification and conversion electronics to produce a useful output signal. These circuits may be located within the sealed sensor unit or in a separate box. There are advantages and disadvantages to both of these configurations but they will not be detailed here. Traditional vibration sensors fall into three main classes: *

*

*

Noncontact displacement transducers (also known as proximity probes or eddy current probes) Velocity transducers (electro mechanical, piezoelectric) Accelerometers (piezoelectric)

Force and frequency considerations dictate the type of measurements and applications that are best suited for each transducer. Recently, laser based noncontact velocity/displacement transducers have become more commonplace. These are still relatively expensive because of their extreme sensitivity, and hence are still predominantly used in the laboratory setting. Figure 25.5 shows the relationship between the different transducer types in terms of response amplitude and frequency. For constant velocity vibration amplitude across all frequencies, a displacement transducer is more sensitive in the lower frequency range, while an accelerometer is more sensitive at higher frequencies. While it may appear as if the velocity transducer is the best compromise, transducers are selected to optimize sensitivity over the frequency range that is expected to be recorded. The type of motion sensed by displacement transducers is the relative motion between the point of attachment and the observed surface. Velocity transducers and accelerometers measure the absolute motion of the structure to which they are attached.

25.6.1

Noncontact Displacement Transducers

These types of sensors find application primarily in fluid film (journal) radial or thrust bearings. With the rotor resting on a fluid film there is no way to easily attach a sensor. A noncontact approach is then the best alternative. Noncontact measurements indicate shaft motion and position relative to the bearing. Radial shaft displacements and seal clearances can be conveniently measured. Another advantage of using


25 11

Vibration Amplitude

Vibration Direction

Output

(a)

Time

Vibration

Direction

X Output

Y Direction Displacement

Y Output

(b)

X Direction Displacement

FIGURE 25.6 (a) Shaft displacement with one sensor; (b) shaft orbit with two sensors (x direction versus y direction displacement assumes shaft is circular).

noncontact displacement probes is that when they are used in pairs, set 908 apart, the signals can be used to show shaft dynamic motion (orbit) within the bearing. Figure 25.6 shows a single channel and dual channel measurement result. When the two channels are plotted against one another, they clearly show what are known as shaft orbits. These orbits define the dynamic motion of the shaft in the bearing, and are valuable fault detection and diagnostic tools. The linearity and sensitivity of the proximity probe depends on the target conductivity and porosity. Calibration of the probe on the specific material in use is recommended. This type of sensor is capable of both static and dynamic measurements, but temperature and pressure extremes will affect the transducer output. The probe will detect small defects in the shaft (cracks or pits), and these may seem like vibrations in the output signal. Installation of these sensors requires a rigid mounting. Adaptors for quick removal and replacement without machine disassembly can be useful. The minimum tip clearance from all adjacent surfaces should be two times the tip diameter. Probe extensions must be checked to ensure that the resonant frequency of the extension is not excited during data gathering. As with all sensors, care must be taken when handling the cables and the connections must be kept clean.

25.6.2

Velocity Transducers

There are two general types of velocity transducers. They can be distinguished by considering the mode of operation. The two types are electro mechanical and piezoelectric crystal based. Piezoelectric crystal based transducers will be discussed in the next section, so the focus here will be on electro mechanical (see Chapter 15). Electro mechanical velocity transducers function with a permanent magnet (supported by springs) moving within a coil of wire. As the sensor experiences changes in velocity, as when attached to a vibrating surface, the movement of the magnet within the coil is proportional to force acting on the

25 12


Amplitude Response

sensor. The current in the coil, induced by the moving magnet, is proportional to velocity, which in turn is proportional to the force. This type of device is known as “self generating” and produces a low impedance signal; therefore, no additional signal conditioning is generally needed. Electro mechanical velocity sensors require the spring suspension system to be designed with a relatively low natural frequency. These devices have good sensitivity, typically above 10 Hz, but Frequency Natural ≈10 Hz their high frequency response is limited (usually Frequency around 1500 Hz) by the inertia of the system. Some devices may obtain a portion, or all, of FIGURE 25.7 Output sensitivity vs. frequency for an their damping electrically. This type must be electro mechanical velocity transducer. loaded with resistance of a specific value to meet design constraints. These are usually designed for use with a specific data collection instrument and must be checked and modified if they are to be used with other instruments. Figure 25.7 shows a plot of the sensitivity vs. frequency for an electro mechanical velocity transducer. While electro mechanical velocity transducers can be designed to have good dynamic range within a specific frequency range, there are several functional limitations. Because a damping fluid is typically used to provide most of the damping, this type of transducer is limited to a relatively narrow temperature band, below the boiling point of the damping liquid. The mechanical reliability of these sensors is also limited by the moving parts within the transducer, which may become worn or fail over time. This has resulted in this type of transducer being replaced by piezoelectric sensors in machine condition monitoring applications. The orientation of the sensor is also limited to only the vertical or horizontal direction, depending on the type of mounting used. Finally, as a damped system, such as an electro mechanical velocity transducer, approaches its natural frequency, a shift in phase relationships may occur (below 50 Hz). This phase shift at low frequencies will affect analysis work.

25.6.3

Acceleration Transducers Amplitude Response

By far the most commonly used transducers for measuring vibration are accelerometers (see Chapter 15). These devices contain one or more Natural Frequency piezoelectric crystal elements (natural quartz or 3 man made ceramics), which produce voltage when stressed in tension, compression or shear. This is the piezoelectric effect. The voltage generated across the crystal pole faces is proportional to the 0.1 0.2 0.4 0.6 0.8 1.0 applied force. Forcing Frequency Accelerometers have a linear response over a Natural Frequency wide frequency range (0.5 Hz to 20 kHz), with specialty sensors linear up to 50 kHz. This wide FIGURE 25.8 Typical accelerometer sensitivity vs. linear frequency range and the broad dynamic frequency. amplitude range make accelerometers extremely versatile sensors. Figure 25.8 shows the sensitivity vs. frequency relationship for a typical accelerometer. In addition, the signal can be electronically integrated to give velocity and displacement measurements. This type of transducer is relatively resistant to temperature changes, reliable (having no moving parts), produces a self generating output signal meaning no external power supply is needed unless there are onboard electronics, is available in a variety of sizes, is usually relatively insensitive to nonaxial vibration


25 13

Accelerometer Output



(,3% of main axis sensitivity), and can function well in any orientation. Signals from this type of transducer contain significantly more vibration components than other types. This means that there is a large amount of information available in the raw vibration signal. Installation of accelerometers requires as rigid ≈28kHz (a) Frequency a mount as possible. Permanent installation with studs or bolts is usually best for high speed machinery where high frequency measurements are required. The close coupling between the machine and the sensor allows for direct transmission of the vibration to the sensor. Stud mounting requires a flat surface to give the best amplitude linearity and frequency (b) Frequency ≈ 7kHz response. This type of mounting is expensive and may not be practical if large numbers of measurements are being recorded with a portable instrument. Magnetic mounts have the advantage of being easily movable and provide good repeatability in the lower frequency range, but have limited high frequency sensitivity (4 to 5 kHz). Hand held measurements are useful (c) ≈2kHz Frequency when conducting general vibration surveys, but usually result in significant variation between measurements. The hand held mount is least FIGURE 25.9 Accelerometer response vs. frequency expensive but only offers frequency response for various types of mounts (a stud; b magnet; c below 1 kHz. Figure 25.9 shows a plot of the hand held). response curves for the same accelerometer with different mounts. For machine condition monitoring and fault diagnostics applications, there will typically be a combination of all three mounting methods used, depending on the equipment being monitored and the monitoring strategy employed. As with the other types of vibration sensor, accelerometers have certain limitations. Because of their sensitivity and wide dynamic range, accelerometers are also sensitive to environmental input not related to the vibration signal of interest. Temperature (ambient and fluctuations) may cause distortion in the recorded signal. General purpose accelerometers are relatively insensitive to temperatures up to 2508C. At higher temperatures, the piezoelectric material may depolarize and the sensitivity may be permanently altered. Temperature transients also affect accelerometer output. Shear type accelerometers have the lowest temperature transient sensitivity. A heat sink or mica washer between the accelerometer and a hot surface may help reduce the effects of temperature. Piezoelectric crystals are sensitive to changes in humidity. Most accelerometers are epoxy bonded or welded together to provide a humidity barrier. Moisture migration through cables and into connections must be guarded against. Large electro magnetic fields can also induce noise into cables that are not double shielded. If an accelerometer is mounted on a surface that is being strained (bent), the output will be altered. This is known as base strain, and thick accelerometer bases will minimize this effect. Shear type accelerometers are less sensitive because the piezoelectric crystal is mounted to a center post not the base. Accelerometers are designed to remain constant for long periods of time; however, they may need calibrating if damaged by dropping or high temperatures. A known amplitude and frequency source (or another accelerometer that has a known calibration) should be used to check the calibration of accelerometers from time to time.

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25.7

Transducer Location

The placement (location) of a vibration transducer is a critical factor in machine condition monitoring and fault diagnostics. Using several locations and directions when recording vibration information is recommended. As always, it depends on the application and whether or not the expense is warranted. If the vibration component which relates to a given fault condition is not recorded, no amount of analysis will extract it from the signal. When selecting sensor locations consider the following: *

*

*

Mechanical independence The vibration transmission path Locations where natural frequency vibrations may be excited (flexible components or attachments)

*

*

*

*

*

*

*

25.8 25.8.1

A transducer is a device that senses a physical quantity and converts it into an electrical output signal, which is proportional to the measured variable. The selection, placement, and proper use of the correct transducer are important steps in the implementation of a condition monitoring and fault diagnostics program. The transducer must be correct for the task, properly mounted, in good working order (properly calibrated), and fully understood in terms of operational characteristics. Traditional vibration sensors fall into three main classes: noncontact displacement transducers, velocity transducers, and accelerometers. Transducers are selected to optimize sensitivity over the frequency range that is expected to be recorded. Using several locations and directions when recording vibration information is recommended. When selecting sensor locations, one must consider mechanical independence, the vibration transmission path, and locations where natural frequency vibrations may be excited.

Recording and Analysis Instrumentation Vibration Meters

Vibration meters are generally small, hand held (portable), inexpensive, simple to use, self contained devices that give an overall vibration level reading (see Chapter 15). They are used for walk around surveys and measure velocity and/or acceleration. Generally, these devices have no built in diagnostics capability, but the natural frequency of an accelerometer can be exploited to look for specific machinery faults. As an example, rolling element bearings generally emit “spike” energy during the early stages of deterioration. These are sharp impacts as rollers strike defects (pits, cracks) in the races. A spike energy meter is an accelerometer that has been tuned to have its resonant frequency excited by these impacts, thus giving a very early warning of deteriorating bearings.

25.8.2

Data Collectors

Most vibration data collectors available today for use in machine condition monitoring and fault diagnostics are microcomputer based. They are used together with vibration sensors to measures vibration, to store and transfer data, and for frequency domain analysis. Considerably more data can be recorded in a digital form, but the cost of these devices can also be considerable. Another


25 15

advantage of most data collectors is the ability to use these devices to conduct on the spot diagnostics or balancing. They are usually used with a PC to provide permanent data storage and a platform for more detailed analysis software. Data collectors are usually used on general purpose equipment.

25.8.3

Frequency-Domain Analyzers

The frequency domain analyzer is perhaps the key instrument for diagnostic work. Different machine conditions (unbalance, misalignment, looseness, bearing flaws) all generate characteristic patterns that are usually visible in the frequency domain. While data collectors do provide some frequency domain analysis capability, their main purpose is data collection. Frequency domain analyzers are specialized instruments that emphasize the analysis of vibration signals. As such, they are often treated as a laboratory instrument. Generally, analyzers will have superior frequency resolution, filtering ability (including antialiasing), weighting functions for the elimination of leakage, averaging capabilities (both in the time and frequency domains), envelope detection (demodulation), transient capture, large memory, order tracking, cascade/waterfall display, and zoom features. Dual channel analysis is also common.

25.8.4

Time-Domain Instruments

Time domain instruments are generally only able to provide a time domain display of the vibration waveform. Some devices have limited frequency domain capabilities. While this restriction may seem limiting, the low cost of these devices and the fact that some vibration characteristics and trends show up well in the time domain make them valuable tools. Oscilloscopes are the most common form. Shaft displacements (orbits), transients and synchronous time averaging (and negative averaging) are some of the analysis strategies that can be employed with this type of device.

25.8.5

Tracking Analyzers

Tracking analyzers are typically used to record and analyze data from machines that are changing speed. This usually occurs during run up and coast down of large machinery or turbo machinery. These measurements are typically used to locate machine resonances and unbalance conditions. The tracking rate is dependent on filter bandwidth, and there is a need for a reference signal to track speed (tachometer input). These devices usually have variable input sensitivity and a large dynamic range.

*

*

*

*

*

Vibration meters are generally small, hand held (portable), inexpensive, simple to use, self contained devices that give an overall vibration level reading. Most vibration data collectors available today for use in machine condition monitoring and fault diagnostics are microcomputer based. They are used together with vibration sensors to measures vibration, to store and transfer data, and for frequency domain analysis. Frequency domain analyzers are specialized instruments that emphasize the analysis of vibration signals, and as such they are perhaps the key instrument for diagnostic work. Time domain instruments are generally only able to provide a time domain display of the vibration waveform. Tracking analyzers are typically used to record and analyze data (locate machine resonances and unbalance conditions) from machines that are changing speed. This usually occurs during run up and coast down of large machinery or turbo machinery.

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25.9


Display Formats and Analysis Tools

Vibration signals can be displayed in a variety of different formats. Each format has advantages and disadvantages, but generally the more processing that is done on the dynamic signal, the more specific information is highlighted and the more extraneous information is discarded. The broad display formats that will be discussed here are the time domain, the frequency domain, the modal domain, and the quefrency domain. Within each of these display formats, several different analysis tools (some specific to that display format) will be described.

25.9.1

Time Domain

The time domain refers to a display or analysis of the vibration data as a function of time. The principal advantage of this format is that little or no data are lost prior to inspection. This allows for a great deal of detailed analysis. However, the disadvantage is that there is often too much data for easy and clear fault diagnosis. Time domain analysis of vibration signals can be subdivided into the following sections. 25.9.1.1

Time-Waveform Analysis

Time waveform analysis involves the visual inspection of the time history of the vibration signal. The general nature of the vibration signal can be clearly seen and distinctions made between sinusoidal, random, repetitive, and transient events. Nonsteady state conditions, such as run up and coast down, are most easily captured and analyzed using time waveforms. High speed sampling can reveal such defects as broken gear teeth and cracked bearing races, but can also result in extremely large amounts of data being collected much of which is likely to be redundant and of little use. 25.9.1.2

Time-Waveform Indices

A time waveform index is a single number calculated in some way based on the raw vibration signal and used for trending and comparisons. These indices significantly reduce the amount of data that is presented for inspection, but highlight differences between samples. Examples of time waveform based indices include the peak level (maximum vibration amplitude within a given pffiffitime signal), mean level (average vibration amplitude), root mean square (RMS) level (peak level= 2; reduces the effect of spurious peaks caused by noise or transient events), and peak to peak amplitude (maximum positive to maximum negative signal amplitudes). All of these measures are affected adversely when more than one machinery component contributes to the measured signal. The crest factor is the ratio of the peak level to the RMS level ðpeak level=RMS levelÞ; and indicates the early stages of rolling element bearing failure. However, the crest factor decreases with progressive failure because the RMS level generally increases with progressive failure. 25.9.1.3

Time-Synchronous Averaging

Averaging of the vibration signal synchronous with the running speed of the machinery being monitored is called time synchronous averaging. When taken over many machine cycles, this technique removes background noise and nonsynchronous events (random transients) from the vibration signal. This technique is extremely useful where multiple shafts that are operating at only slightly different speeds and in close proximity to one another are being monitored. A reference signal (usually from a tachometer) is always needed. 25.9.1.4

Negative Averaging

Negative averaging works in the opposite way to time synchronous averaging. Rather than averaging all the collected data, a baseline signal is recorded and then subtracted from all subsequent signals to reveal changes and transients only. This type of signal processing is useful on equipment or components that are isolated from other sources of vibrations.


25.9.1.5

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Orbits

As described above, orbits are plots of the X direction displacement vs. the Y direction displacement (phase shifted by 908). This display format shows journal bearing relative motion (bearing wear, shaft misalignment, shaft unbalance, lubrication instabilities [whirl, whip], and seal rubs) extremely well, and hence is a powerful monitoring and diagnostic tool, especially on relatively low speed machinery. 25.9.1.6

Probability Density Functions Probability Density(dB)

The probability of finding the instantaneous Faulty Bearing amplitude value from a vibration signal within a Normal Bearing certain amplitude range can be represented as a probability density function. Typically, the shape of the probability density function in these cases will be similar to a Gaussian (or normal) probability distribution. Fault conditions will have different characteristic shapes. Figure 25.10 shows two probability density functions. One is Normalized Vibration Amplitude characteristic of normal machine operating con ditions, and the other represents a fault condition. FIGURE 25.10 Normalized vibration amplitude vs. A high probability at the mean value with a wide probability density (normal and faulty bearings). spread of low probabilities is characteristic of the impulsive time domain waveforms that are typical for rolling element bearing faults. This type of display format can be used for condition trending and fault diagnostics. 25.9.1.7

Probability Density Moments

Probability density moments are single number indices (descriptors), similar to the time waveform indices except they are based on the probability density function. Odd moments (first and third, mean and skewness) reflect the probability density function peak position relative to the mean. Even moments (second and fourth, standard deviation and kurtosis) are proportional to the spread of the distribution. Perhaps the most useful of these indices is the kurtosis, which is sensitive to the impulsiveness in the vibration signal and therefore sensitive to the type of vibration signal generated in the early stages of a rolling element bearing fault. Because of this characteristic sensitivity, the kurtosis index is a useful fault detection tool. However, it is not good for trending. As a rolling element bearing fault worsens, the vibration signal becomes more random, the impulsiveness disappears, and the noise floor increases in amplitude. The kurtosis then increases in value during the early stages of a fault, and decreases in value as the fault worsens.

25.9.2

Frequency Domain

The frequency domain refers to a display or analysis of the vibration data as a function of frequency. The time domain vibration signal is typically processed into the frequency domain by applying a Fourier transform, usually in the form of a fast Fourier transform (FFT) algorithm. The principal advantage of this format is that the repetitive nature of the vibration signal is clearly displayed as peaks in the frequency spectrum at the frequencies where the repetition takes place. This allows for faults, which usually generate specific characteristic frequency responses, to be detected early, diagnosed accurately, and trended over time as the condition deteriorates. However, the disadvantage of frequency domain analysis is that a significant amount of information (transients, nonrepetitive signal components) may be lost during the transformation process. This information is nonretrievable unless a permanent record of the raw vibration signal has been made.

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25.9.2.1


Band-Pass Analysis

Band pass analysis is perhaps the most basic of all frequency domain analysis techniques, and involves filtering the vibration signal above and/or below specific frequencies in order to reduce the amount of information presented in the spectrum to a set band of frequencies. These frequencies are typically where fault characteristic responses are anticipated. Changes in the vibration signal outside the frequency band of interest are not displayed. 25.9.2.2

Shock Pulse (Spike Energy)

The shock pulse index (also known as spike energy; Boto, 1979) is derived when an accelerometer is tuned such that the resonant frequency of the device is close to the characteristic responses frequency caused by a specific type of machine fault. Typically, accelerometers are designed so that their natural frequency is significantly above the expected response signals that will be measured. If higher frequencies are expected, they are filtered out of the vibration signal. High speed rolling element bearings that are experiencing the earlier stages of failure (pitting on interacting surfaces) emit vibration energy in a relatively high, but closely defined, frequency band. An accelerometer that is tuned to 32 kHz will be a sensitive detection device. This type of device is simple, effective, and inexpensive tool for fault detection in high speed rolling element bearings. The response from this type of device is load dependent and may be prone to false alarms if measurement conditions are not constant. 25.9.2.3

Enveloped Spectrum

Another powerful analysis tool that is available in the frequency domain and can be effectively applied to detecting and diagnosing rolling element bearing faults is the enveloped spectrum (Courrech, 1985). When the vibration signal time waveform is demodulated (high pass filtered, rectified, then low pass filtered) the frequency spectrum that results is said to be enveloped. This process effectively filters out the impulsive components in signals that have high noise levels and other strong transient signal components, leaving only the components that are related to the bearing characteristic defect frequencies. This method of analysis is useful for detecting bearing damage in complex machinery where the vibration signal may be contaminated by signals from other sources. However, the filtering bands must be chosen with good judgment. Recall also, the impulsive nature of the fault signal at the characteristic defect frequency leaves as the fault deteriorates. 25.9.2.4

Signature Spectrum

The signature spectrum (Braun, 1986) is a baseline frequency spectrum taken from new or recently overhauled machinery. It is then later compared with spectra taken from the same machinery that represent current conditions. The unique nature of each machine and installation is automatically taken into account. Characteristic component and fault frequencies can be clearly seen and comparisons made manually (by eye), using indices, or using automated pattern recognition techniques. 25.9.2.5

Cascades (Waterfall Plots)

Cascade plots (also known as waterfall plots) are successive spectra plotted with respect to time and displayed in a three dimensional manner. Changing trends can be seen easily, which makes this type of display a useful fault detection and trending tool. This type of display is also used when a transient event, such as a coast down, is known to be about to occur. Cascade plots can also be linked to the speed of a machine. In this case, the horizontal axis is labeled in multiples of the rotational speed of the machine. Each multiple of the rotational speed is referred to as an “order.” “Order tracking” is the name commonly used to refer to cascade plots that are synchronously linked to the machine rotational speed via a tachometer. As the speed of the machine changes, the responses at specific frequencies change relative to the speed, but are still tracked in each time stamped spectra by the changing horizontal axis scale.


25.9.2.6

25 19

Masks

Like negative averaging in the time domain, masks are baseline spectra that are used with an allowable tolerance limit to “filter out,” or block, specific frequencies. This technique is similar to band pass analysis and requires a good knowledge of the full range of each machine’s operating limits (varying load or speed). 25.9.2.7

Frequency-Domain Indices

It has been noted that frequency spectra are more sensitive to changes related to machine condition (Mathew, 1987). Because of this sensitivity, several single number indices based on the frequency spectra have been proposed. Like the time waveform indices, frequency domain indices reduce the amount of information in frequency spectra to a single number. Because they are based on the frequency spectra, they are generally more sensitive to changes in machine condition than time domain indices. They are used as a means of comparing original spectra or previous spectra to the current spectra. Several frequency domain indices are listed below: *

Arithmetic mean (Grove, 1979):

( 20 log

*

*

*

*

N 1 X A N i¼1 i

!

Ai ¼ amplitude of ith frequency spectrum component N ¼ total number of frequency spectrum components Geometric mean (Grove, 1979): (N 1 X A 20 log pffiffii N i¼1 2

) 10

5

10

5

)

Matched filter RMS (Mathew and Alfredson, 1984): ( N 1 X Ai 10 log N i¼1 Ai ðref Þ

2

)

Ai ðref Þ ¼ amplitude of ith component in the reference spectrum RMS of spectral difference (Alfredson, 1982): ( )1=2 N 1 X ðLci Loi Þ2 N i¼1 Lci ¼ amplitude (dB) of ith component Loi ¼ amplitude (dB) of ith reference component Sum of squares of difference (Mathew and Alfredson, 1984): ( N 1 X ðL þ Loi Þ £ lLci Loi l N i¼1 ci

25.9.3

) 1=2

Modal Domain

Modal analysis is not traditionally listed as a machine condition monitoring and fault diagnostics tool, but is included here because of the ever increasing complexity of modern machinery. Often, unless the

25 20


natural (free and forced response) frequencies of machinery, their support structure, and the surrounding buildings are fully understood, a complete and accurate assessment of existing machinery condition is not possible. A complete overview of modal analysis will not be provided here, but a specific approach to modal analysis (operational deflection shape [ODS] analysis) will be described. ODS analysis is like other types of modal analysis in that a force input is provided to a structure or machine and then the response is measured. The response at different frequencies defines the natural frequencies of the structure or machine. Typically, an impact or constant frequency force is used to excite the structure. In the case of ODSs, the regular operation of the machinery provides the excitation input. With vibration sensors placed at critical locations and a reference signal linking together all the recorded signals, a simple animation showing how the machine or structure deflects under normal operation can be generated. These animations, along with the frequency information contained in each individual signal, can provide significant insights into how a machine or structure deforms under a dynamic load. This information, in turn, can be a useful addition to other data when attempting to diagnose problems.

25.9.4

Quefrency Domain

A quefrency domain (Randall, 1981, 1987) plot results when a Fourier transform of a frequency spectra (log scale) is generated. As the frequency spectra highlight periodicities in the time waveform, so the quefrency “cepstra” highlights periodicities in a frequency spectra. This analysis procedure is particularly useful when analyzing gearbox vibration signals where modulation components in spectrum (sidebands) are easily detected and diagnosed in the cepstrum.

*

*

*

*

*

*

*

*

Generally, the more processing that is done on the dynamic signal, the more specific useful information is highlighted and the more extraneous information is discarded. The primary display formats used in machine condition monitoring are the time domain, the frequency domain, the modal domain, and the quefrency domain. The time domain refers to a display or analysis of the vibration data as a function of time, allowing for little or no data to be lost prior to inspection. Time domain analysis includes: waveform analysis, time waveform indices, time synchronous averaging, negative averaging, orbit analysis, probability density functions, and probability density moments. The frequency domain refers to a display or analysis of the vibration data as a function of frequency, where the time domain vibration signal is typically processed into the frequency domain by applying a Fourier transform, usually in the form of a FFT algorithm. The principal advantage of frequency domain analysis is that the repetitive nature of the vibration signal is clearly displayed as peaks in the frequency spectrum at the frequencies where the repetition takes place. This allows for faults, which usually generate specific characteristic frequency responses, to be detected early, diagnosed accurately, and trended over time as the condition deteriorates. Frequency domain analysis includes the use of band pass analysis, shock pulse (spike energy), envelope spectrum, signature spectrum, cascades (waterfall plots), masks, and frequency domain indices. Quefrency domain analysis involves a Fourier transform of a frequency spectra (log scale). As the frequency spectra highlight periodicities in the time waveform, so the quefrency “cepstra” highlights periodicities in a frequency spectra.


25.10

25 21

Fault Detection

In many discussions of machine condition monitoring and fault diagnostics, the distinction between fault detection and fault diagnosis is not made. Here, they have been divided into separate sections in order to highlight the differences and clarify why they should be treated as separate tasks. Fault detection can be defined as the departure of a measurement parameter from a range that is known to represent normal operation. Such a departure then signals the existence of a faulty condition. Given that measurement parameters are being recorded, what is needed for fault detection is a definition of an acceptable range for the measurement parameters to fall within. There are two methods for setting suitable ranges: (1) comparison of recorded signals to known standards and (2) comparison of the recorded signals to acceptance limits.

25.10.1

Standards

One of the best known sources of standards is the International Organization for Standardization (ISO). These standards are technology oriented and are set by teams of international experts. ISO Technical Committee 108, Sub Committee 5 is responsible for standards for condition monitoring and diagnostics of machines. This group is further divided into a number of working groups who review data and draft preliminary standards. Each working group has a particular focus such as terminology, data interpretation, performance monitoring, or tribology based machine condition monitoring. While ISO is perhaps the most widely known standardization organization, there are several others that are focused on specific industries. Examples of these include the International Electrical Commission, which is primarily product oriented, and the American National Standards Institute (ANSI), which is a nongovernment agency. There are also different domestic government agencies that vary from country to country. National defense departments also tend to set their own standards. Standards Based on Machinery Type

Because different machines that are designed to perform approximately the same task tend to behave in a similar manner, it is not surprising that many standards are set based on machinery type. Figure 25.11 shows a generic plot separating vibration amplitude vs. rotating speed into different zones. For a specific type, size, or class of machine, a plot like this can be used to distinguish gross vibration limits relative to the speed of operation. Machines are usually divided into four basic categories:

ZONE D Vibration Amplitude

25.10.1.1

ZONE C

ZONE B

ZONE A

1. Reciprocating machinery: These machines may contain both rotating and reciprocat Rotational Speed ing components (e.g., engines, compres sors, pumps). FIGURE 25.11 Normalized vibration amplitude vs. 2. Rotating machinery (rigid rotors): These probability density (zone A new machine; zone machines have rotors that are supported B acceptable; zone C monitor closely; zone D on rolling element bearings (usually). The damage occurring). vibration signal can be measured from the bearing housing because the vibration signal is transmitted well through the bearings to the housing (e.g., electric motors, single stage pumps, slow speed pumps).

25 22


3. Rotating machinery (flexible rotors): These machines have rotors that are supported on journal (fluid film) bearings. The movement of the rotor must be measured using proximity probes (e.g., large steam turbines, multistage pumps, compressors). These machines are subject to critical speeds (high vibration levels when the speed of rotation excites a natural frequency). Different modes of vibration may occur at different speeds. 4. Rotating machinery (quasi rigid rotors): These are usually specialty machines in which some vibration gets through the bearings, but it is not always trustworthy data (e.g., low pressure steam turbines, axial flow compressors, fans). 25.10.1.2

Standards Based on Vibration Severity

It is an oversimplification to say that vibration levels must always be kept low. Standards depend on many things, including the speed of the machinery, the type and size of the machine, the service (load) expected, the mounting system, and the effect of machinery vibration on the surrounding environment. Standards that are based on vibration severity can be divided into two basic categories: 1. Small to medium sized machines: These machines usually operate with shaft speeds of between 600 and 12,000 rpm. The highest broadband RMS value usually occurs in the frequency range of 10 to 1000 Hz. 2. Large machines: These machines usually operate with shaft speeds of 600 to 1200 rpm. If the machine is rigidly supported, the machine’s fundamental resonant frequency will be above the main excitation frequency. If the machine is mounted on a flexible support, the machine’s fundamental resonant frequency will be below the main excitation frequency. While general standards do exist, there are also a large number of standards that have been developed for specific machines. Figure 25.12 shows a table with generic acceptance limits based on vibration severity.

25.10.2

Acceptance Limits

Standards developed by dedicated organizations are a useful starting point for judging machine condition. They give a good indication of the current condition of a machine and whether or not a fault exists. However, judging the overall condition of machinery is often more involved. Recognizing the changing machinery condition requires the trending of condition indicators over time. The development and use of acceptance limits that are close to the normal operating values for specific machinery will detect even slight changes in condition. While these acceptance limits must be tight enough to allow even small changes in condition to be detected, they must also tolerate normal operating variations without Vibration Amplitude Increasing

Vibration Severity for Separate Classes of Machines Class I Class II Class III Class IV A B

A B

C

C

D

D

FIGURE 25.12 Acceptance limits based on vibration severity levels (zone A zone C monitor closely; zone D damage occurring).

A B C D

A B C D

new machine; zone B

acceptable;


25 23

generating false alarms. There are two types of limits: 1. Absolute limits represent conditions that could result in catastrophic failure. These limits are usually physical constraints such as the allowable movement of a rotating part before contact is made with stationary parts. 2. Change limits are essentially warning levels that provide warning well in advance of the absolute limit. These vibration limits are set based on standards and experience with a particular class of machinery or a particular machine. Change limits are usually based on overall vibration levels. It is important to note that the early discovery of faulty conditions is a key to optimizing maintenance effort by allowing the longest possible lead time for decision making. As well as the overall vibration levels being monitored, the rates of change are also important. The rate of change of a vibration level will often provide a strong indication of the expected time until absolute limits are exceeded. In general, relatively high but stable vibration levels are of less concern than relatively low but rapidly increasing levels. An example of how acceptance limits may be used to detect faults and trend condition is provided when the gradual deterioration of rolling element bearings is considered. Rolling element bearings generate distinctive defect characteristic frequencies in the frequency spectrum during a slow, progressive failure. Vibration levels can be monitored to achieve maximum useful life and failure avoidance. Typically, the vibration levels increase as a fault is initiated in the early stages of deterioration, but then decrease in the later stages as the deterioration becomes more advanced. Appropriately, set acceptance levels will detect the early onset of the fault and allow subsequent monitoring to take place even after the overall vibration level has dropped. However, rapid bearing deterioration may still occur due to a sudden loss of lubrication, lubrication contamination, or a sudden overload. The possibility of these situations emphasizes the need for carefully selected acceptance limits. It should also be noted that changes in operating conditions, such as speed or load changes, could invalidate time trends. Comparisons must take this into consideration. 25.10.2.1

Statistical Limits

Statistical acceptance limits are set using statistical information calculated from the vibration signals measured from the equipment that the limits will ultimately be used with. As many vibration signals as possible are recorded, and the average of the overall vibration level is calculated. An alert or warning level can then be set at 2.5 standard deviations above or below the average reading (Mechefske, 1998). This level has been found to provide optimum sensitivity to small changes in machine condition and maximum immunity to false alarms. A distinct advantage to using this method to set alarm levels is the fact that the settings are based on actual conditions being experienced by the machine that is being monitored. This process accommodates normal variations that exist between machines and takes into account the initial condition of the machine.

25.10.3

Frequency-Domain Limits

Judging vibration characteristics within the frequency spectra is sometimes a more accurate method of detecting and trending fault conditions. It can also provide earlier detection of specific faults because, as mentioned previously, the frequency domain is generally more sensitive to changes in the vibration signal that result from changes in machine condition. The different specific methods are listed and described below. 25.10.3.1

Limited Band Monitoring

In limited band monitoring, the frequency spectrum is divided into frequency bands. The total energy or highest amplitude frequency is then trended within each band. Each band has its own limits based on experience. Generally, ten or fewer bands are used. Small changes in component specific frequency

25 24


ranges are more clearly shown using this strategy. Bandwidths and limits must be specific to the machine, sensor type, and location. Narrowband monitoring is the same as limited band monitoring, except it has finer definition of the bands. 25.10.3.2

Constant Bandwidth Limits

When limited band monitoring is practiced and the bands have same width at high and low frequencies, the procedure is called constant bandwidth monitoring (see Figure 25.13). This technique is useful for constant speed machines where the frequency peaks in the spectra remain relatively fixed. 25.10.3.3

Constant Percentage Bandwidth Limits

Amplitude

Constant percentage bandwidth monitoring involves using bandwidths that remain a constant percentage of the frequency being monitored (see Figure 25.14). This results in the higher frequency bands being proportionally wider than the lower frequency bands. This allows for small variations

Frequency Constant bandwidth acceptance limits.

Amplitude

FIGURE 25.13

Frequency FIGURE 25.14

Constant percentage bandwidth acceptance limits.


25 25

in speed without the frequency peaks moving between bands, which may have different acceptance limits.

*

*

*

*

*

*

*

25.11

Fault detection can be defined as the departure of a measurement parameter from a range that is known to represent normal operation. Such a departure signals the existence of a faulty condition. ISO Technical Committee 108, Sub Committee 5 is responsible for standards for Condition Monitoring and Diagnostics of Machines. Standards are based on machinery type or vibration severity. The development and use of acceptance limits that are close to normal operating values for specific machinery will detect even slight changes in condition. Statistical acceptance limits are set using statistical information calculated from the vibration signals measured from the equipment that the limits will ultimately be used with. Judging vibration characteristics within the frequency spectra is sometimes a more accurate method of the early detecting and trending of fault conditions because the frequency domain is generally more sensitive to changes in the vibration signal that result from changes in machine condition. Frequency domain limits include limited band monitoring, constant bandwidth limits, and constant percentage bandwidth limits.

Fault Diagnostics

Depending on the type of equipment being monitored and the maintenance strategy being followed, once a faulty condition has been detected and the severity of the fault assessed, repair work or replacement will be scheduled. However, in many situations, the maintenance strategy involves further analysis of the vibration signal to determine the actual type of fault present. This information then allows for a more accurate estimation of the remaining life, the replacement parts that are needed, and the maintenance tools, personnel, and time required to repair the machinery. For these reasons, and many more, it is often advantageous to have some idea of the fault type that exists before decisions regarding maintenance actions are made. There are obviously a large number of potential different fault types. The description of these faults can be systemized somewhat by considering the type of characteristic defect frequencies generated (synchronous to rotating speed, subsynchronous, harmonics related to rotating speed, nonsynchronous harmonics, etc.). Such a systemization requires a focus on frequency domain analysis tools (primarily frequency spectra). While this organization strategy is effective, it inherently leaves out potentially valuable information from other display formats. For this reason, the various faults that usually develop in machinery are listed here in terms of the forcing functions that cause them and specific machine types. In this way, a diagnostic template can be developed for the different types of faults that are common in a given facility or plant. Further reading on machinery diagnostics can be found in these references: Wowk (1991), Taylor (1994), Eisenmann and Eisenmann (1998), Goldman (1999), and Reeves (1999).

25.11.1

Forcing Functions

Listed and described below are a variety of forcing functions that can result in accelerated deterioration of machinery or are the result of damaged or worn mechanical components. The list is not meant to be exhaustive and is in no particular order.

25 26

25.11.1.1


Unbalance

Unbalance (also referred to as imbalance) exists when the center of mass of a rotating component is not coincident with the center of rotation. It is practically impossible to fabricate a component that is perfectly balanced; hence, unbalance is a relatively common condition in a rotor or other rotating component (flywheel, fan, gear, etc.). The degree to which an unbalance affects the operation of machinery dictates whether or not it is a problem. The causes of unbalance include excess mass on one side of the rotor, low tolerances during fabrication (casting, machining, assembly), variation within materials (voids, porosity, inclusions, etc.), nonsym metry of design, aerodynamic forces, and temperature changes. The vector sum of all the different sources of unbalance can be combined into a single vector. This vector then represents an imaginary heavy spot on the rotor. If this heavy spot can be located and the unbalance force quantified, then placing an appropriate weight 1808 from the heavy spot will counteract the original unbalance. If left uncorrected, unbalance can result in excessive bearing wear, fatigue in support structures, decreased product quality, power losses, and disturbed adjacent machinery. Unbalance results in a periodic vibration signal with the same amplitude each shaft rotation (3608). A strong radial vibration at the fundamental frequency, 1X, (1 £ rotational speed) is the characteristic diagnostic symptom. If the rotor is overhung, there will also be a strong axial vibration at 1X. The amplitude of the response is related to the square of the rotational speed, making unbalance a dangerous condition in machinery that runs at high rotational speeds. In variable speed machines (or machines that must be run up to speed gradually), the effects of unbalance will vary with the shaft rotational speed. At low speeds, the high spot (location of maximum displacement of the shaft) will be at the same location as the unbalance. At increased speeds, the high spot will lag behind the unbalance location. At the shaft first critical speed (the first resonance), the lag reaches 908, and at the second critical and above, the lag reaches 1808. A special form of unbalance is caused by a bent shaft or bowed rotor. These two conditions are essentially the same; only the location distinguishes them. A bent shaft is located outside the machine housing, while a bowed rotor is inside the machine housing. This condition is seen on large machines (with heavy rotors) that have been allowed to sit idle for a long time. Gravity and time cause the natural sag in the rotor to become permanent. The vibration spectrum from a machine with a bent shaft or bowed rotor is identical to unbalance, largely because it is an unbalanced condition. Bent shafts and bowed rotors are difficult to correct (straighten), so they need to be balanced by adding counterweights as described above. The best way to avoid this condition is to keep the shaft/rotor rotating slowly when the machine is not in use. 25.11.1.2

Misalignment

While misalignment can occur in several different places (between shafts and bearings, between gears, etc.), the most common form is when two machines are coupled together. In this case, there are two main categories of misalignment: (1) parallel misalignment (also known as offset) and (2) angular misalignment. Parallel misalignment occurs when shaft centerlines are parallel but offset from one another in the horizontal or vertical direction, or a combination of both. Angular misalignment occurs when the shaft centerlines meet at an angle. The intersection may be at the driver or driven end, between the coupled units or behind one of the coupled units. Most misalignment is a combination of these two types. Misalignment is another major cause of excessive machinery vibration. It is usually caused by improper machine installation. Flexible couplings can tolerate some shaft misalignment, but misalignment should always be minimized. The vibration caused by misalignment results in excessive radial loads on bearings, which in turn causes premature bearing failure. Elevated 1X vibrations with harmonics (usually up to the third, but sometimes up to the sixth) in the frequency spectrum are the usual diagnostic signatures. The harmonics


25 27

TABLE 25.1 Characteristics that Can Help Distinguish between Unbalance and Misalignment Unbalance High 1X response in frequency spectra Low axial vibration levels Measurements at different locations are in phase Vibration levels are independent of temperature Vibration level at 1X increases with rotational speed. Centrifugal force increases as the square of the shaft rotational speed

Misalignment High harmonics of 1X relative to 1X High axial vibration levels Measurements at different locations are 1808 out of phase Vibration levels are dependent on temperature (change during warm up) Vibration level does not change with rotational speed. Forces due to misalignment remain relatively constant with changes in shaft rotational speed

allow misalignment to be distinguished from unbalance. High horizontal relative to vertical vibration amplitude ratios (greater than 3:1) may also indicate misalignment. One final note regarding misalignment is that the heat of operation causes metal to expand resulting in thermal growth. Vibration readings should be taken when the equipment is cold and again after normal operating temperature has been reached. The changes in alignment due to thermal growth may be minimal, but should always be measured since they can lead to significant vibration levels. Because unbalance and misalignment are perhaps the two most common causes of excessive machinery vibrations and they have similar characteristic indicators, Table 25.1 has been included to help distinguish between them. 25.11.1.3

Mechanical Looseness

While there are many ways in which mechanical looseness may appear, there are two main types: (1) a bearing loose on a shaft and (2) a bearing loose in a housing. A bearing that is loose on a shaft will display a modulated time signal with many harmonics. The time period of modulation will vary and the time signal will also be truncated (clipped). A bearing that is loose in its housing will display a strong fourth harmonic, which can sometimes be mistaken for the blade pass frequency on a four blade fan. These faults may also look like rolling element bearing characteristic defect frequencies, but always contain a significant amount of wideband noise. Another way to diagnose mechanical looseness is by tracking the changes in the vibrations signal as the condition worsens. In the early stages, mechanical looseness generates a strong 1X response in the frequency spectrum along with some harmonics. At this stage, the condition could be mistaken for unbalance. As the looseness worsens, the amplitude of the harmonics will increase relative to the 1X response (which may actually decrease). The overall RMS value of the time waveform may also decrease. Further deterioration of the condition results in fractional harmonics 12 ; 13 ; 1 12 ; 2 12 increasing in amplitude. These harmonics are most visible in signals taken when the machine is only lightly loaded. These harmonics show up because of the clipping described above. 25.11.1.4

Soft Foot

Another condition that is in fact a type of mechanical looseness, but often masquerades as misalignment, unbalance, or a bent shaft, is soft foot. Soft foot occurs when one of a machine’s hold down bolts is not tight enough to resist the dynamic forces exerted by the machine. That part of the machine will lift off and set back down as a function of the cyclical forces acting on it. All the diagnostic signs associated with mechanical looseness will be present in the vibration signal. If the foundation (hold down points) of a machine does not form a plane, then tightening the hold down bolts will cause the casing and/or rotor to be distorted. This distortion is what leads to the misalignment, unbalance, and bent shaft vibration signatures. In order to check for a soft foot,

25 28


the vibration level must be monitored while each hold down bolt is loosened and then retightened. The appearance and/or disappearance of the diagnostic indicators mentioned above will determine if soft foot is the problem. When a machine’s vibration levels cannot be reduced by realignment or balancing, soft foot could well be the cause. 25.11.1.5

Rubs

Rubs are caused by excessive mechanical looseness or oil whirl. The result is that moving parts come into contact with stationary parts. The vibration signal generated may be similar to that of looseness, but is usually clouded with high levels of wideband noise. This noise is due to the impacts. If the impacts are repetitive, such as occurring each time a fan blade passes, there may be strong spectral responses at the striking frequency. In many cases, rubs are the result of a rotor pressing too hard against a seal. In these cases, the rotor will heat up unsymmetrically and develop a bowed shape. Subsequently, a vibration signal will be generated that shows unbalance. To diagnose this condition, it will be noted that the unbalance is absent until the machine comes up to normal operating temperature. 25.11.1.6

Resonances

The analysis of resonance problems is beyond the scope of this chapter. However, some basic description is provided here because of the high likelihood that at some time a resonance will be excited by repetitive or cyclic forces acting on or nearby a machine. A resonance is the so called “natural frequency” at which all things tend to vibrate. A machine’s natural resonant frequency is dictated by the relationship vn ¼ ðk=mÞ1=2 ; where vn is the natural frequency, k is the spring stiffness, and m is the mass. Most systems will have more than one resonance frequency. These resonances (also called modes) can be excited by any forcing function that is at or close to that frequency. The response amplitude can be 10 to 100 times that of the forcing function. The term “critical speed” is also used to refer to resonances when the machine rotating speed equals the natural frequency. The amount of response amplification depends on the damping in the system. A highly damped system will not show signs of resonance excitation, while a lightly damped system will be prone to resonance excitations. Resonances can be diagnosed by monitoring the vibration level while the speed of rotation of the machine is changed. A resonance will cause a dramatic increase in the 1X vibration levels as the speed is slowly changed. Most machines are designed to operate well away from known resonance frequencies, but changes to the machine (support structure, piping connections, etc.) and proximity to other machines may excite a resonance. 25.11.1.7

Oil Whirl

Oil whirl occurs when the fluid in a lightly loaded journal bearing does not exert a constant force on the shaft that is being supported and a stable operating position is not maintained. In most journal bearing designs, this situation is prevented by using pressure dams or tilt pads to insure that the shaft rides on an oil pressure gradient that is sufficient to support it. During oil whirl, the shaft pushes a wedge of oil in front of itself and the shaft then migrates in a circular fashion within the bearing clearance at just less than one half the shaft rotational speed. The rotor is actually revolving around inside the bearing in the opposite direction from shaft rotation. Because of the inherent instability of oil whirl, in many situations where oil whirl occurs, the time waveform will show intermittent whirl events. The shaft makes a few revolutions while whirl is present and then a few revolutions where the whirl is not present. This “beating” effect is often evident in the time waveform and can be used as a diagnostic indicator. Persistent oil whirl usually requires a replacement of the bearing. However, temporary measures to mitigate the detrimental effects include changing the oil viscosity (changing the operating temperature or the oil), running the machine in a more heavily loaded manner, or introducing a misalignment that will load the bearing asymmetrically. This last course of action is of course not recommended for more than relatively short term relief.


25.11.1.8

25 29

Oil Whip

Oil whip occurs when a subsynchronous instability (oil whirl) excites a critical speed (resonance), which then remains at a constant frequency regardless of speed changes. Oil whip often occurs at two times the critical speed because, at that speed, oil whirl matches the critical speed. Figure 25.15 shows a waterfall (cascade) plot of a mass unbalance that excites oil whirl and oil whip. Note how the oil whip “locks on” to the critical speed resonance. 25.11.1.9

Structural Vibrations

Structural vibrations can range dramatically in amplitude and frequency. Large amplitude, low frequency vibrations can be excited in multistory buildings during an earthquake or by the wind. These vibrations are usually the result of a building resonance being excited. While these sources of structural vibration are important, the source that we are concerned with here is that of machinery operating as part of a building’s utility system, as part of the production plant, or construction equipment close by. Fans, blowers, compressors, piping systems, elevators, and other building service machines all produce vibrations in a building and, if they are not properly isolated they can cause disruption and/or damage to other machines or processes operating close by. The same is true of heavy machinery operating within a plant (stamping machines, presses, forges, etc.) and construction equipment. High impact and repetitive vibrations can excite resonances large distances from the source of the excitation. 25.11.1.10

Foundation Problems

Machine foundations provide rigidity and inertia so that the machine stays in alignment. The energy generated by a machine in the form of vibrations is transmitted, reflected, or absorbed by the foundation. Especially on larger machines, the foundation is paramount to successful dynamic behavior. Maximum energy is transmitted through the foundation to the earth when the mechanical impedance of the foundation is well matched to that of the source of vibration. That is, the source of vibration and the foundation should have the same natural frequency. If this is the case, all frequencies of vibration below Rotor Speed Oil Whirl

OIL WHIP

Critical Speed FIGURE 25.15

Mass Unbalance

Frequency

Waterfall (cascade) plot of a mass unbalance that excites oil whirl and oil whip.

25 30


the natural frequency will be transmitted by the foundation to earth. A poor match will mean that more energy is reflected or absorbed by the foundation, which could effect the operation of the machine attached. Changing foundations can grossly affect amplitude and phase measurements, which means that vibration measurements can be used to easily detect a changing foundation or hold down system.

25.11.2

Specific Machine Components

25.11.2.1

Damaged or Worn Rolling-Element Bearings

Rolling element bearings produce very little vibration (low level random signal) when they are fault free, and have very distinctive characteristic defect frequency responses (see Eschman, 1985, for the equations for calculation of defect frequencies) when faults develop. This, and the fact that most damage in rolling element bearings occurs and worsens gradually, makes fault detection and diagnosis on this component relatively straightforward. Faults due to normal use usually begin as a single defect caused by metal fatigue in one of the raceways or on a rolling element. The vibration signature of a damaged bearing is dominated by impulsive events at the ball or roller passing frequency. Figure 25.16 shows the characteristic time waveform and frequency spectra at various stages of damage. As the damage worsens, there is a gradual increase in the characteristic defect frequencies followed by a drop in these amplitudes and an increase in the broadband noise. In machines where there is little other vibration that would contaminate or mask the bearing vibration signal, the gradual deterioration of rolling element bearings can be monitored by using the crest factor or the kurtosis measure (see above for definitions). A key factor in being able to accurately detect and diagnose rolling element bearing defects is the placement of the vibration sensor. Because of the relatively high frequencies involved, accelerometers should be used and placed on the bearing housing as close as possible to, or within, the load zone of the stationary outer race. Specific applications can also pose significant challenges to fault diagnosis. Very low speed machines have bearings that generate low energy signals and require special processing to extract useful bearing condition indications (Mechefske and Mathew, 1992a). Machines that operate at varying speeds also pose a problem because the characteristic defect frequencies are continuously changing (Mechefske and Liu, 2001). Bearings located close to, or within, gearboxes are also difficult to monitor because the high energy at the gear meshing frequencies masks the bearing defect frequencies (Randall, 2001).

Amplitude

Amplitude

Time Domain

Frequency Domain FIGURE 25.16 bearing.

Characteristic time waveform and frequency spectra at various stages of damage in a rolling element


25.11.2.2

25 31

Damaged or Worn Gears

Amplitude

Amplitude

Because gears transmit power from one rotating Sidebands Harmonics shaft to another, significant forces are present within the mating teeth. While gears are designed for robustness, the teeth do deflect under load and then rebound when unloaded. The local stresses are high at the tooth interface and root, which leads to fatigue damage. Proper design and perfect fabrica tion of gears (with perfect form and no defects) would result in relatively low vibration levels and a Frequency long life. However, the presence of nonperfect gears Rahmonics gives rise to excessive vibration (Smith, 1983). The time waveform, the frequency spectral, and the cepstral patterns generated by gear vibrations all contain critical information needed to diagnose defects (see Figure 25.17). In relatively simple gearboxes, the time waveform can be used to distinguish impacts due to cracked, chipped, or Quefrency missing teeth (McFadden and Smith, 1984, 1985). The frequency spectra and cepstra are powerful FIGURE 25.17 Frequency and quefrency plots tools when the gearbox contains several sets of (damaged gear). mating gears, which is most often the case. Even a significant defect on one tooth (or even a missing tooth) often does not produce an abnormally strong frequency spectral response at 1X. However, the defect will modulate the gear mesh frequency (number of teeth times the shaft rotational speed) and appear as 1X sidebands of the gear mesh frequency. That is, smaller spectral responses that appear a distance of 1X (and multiples of 1X for more severe gear faults) above and below the gear mesh frequency. Because these sidebands occur at multiples of 1X and a spectral plot can become quite cluttered with response lines, cepstral analysis is well suited to distinguish the frequency components that are strong fault indicators. Often, a change in the response at two times the gear mesh frequency is a good indicator of developing gear problems. The amplitude of the gear mesh frequency, and its multiples, vary with load. This makes it important to sample the vibration signal at the same load conditions. When unloaded, excessive gear backlash may also cause an increase in the amplitude of the gear mesh frequency. Because each gear tooth meshes with an impact, structural resonances may be excited in the gears, shafts, and housing. Proper design of a gearbox will minimize this effect, but resonances in gearboxes cause accelerated gear wear and should be monitored. Gears provide an excellent example of how machines must wear in during early use. New gears will have defects that are quickly worn away in the machine’s early life. Vibration levels will become steady and only increase gradually later in the machines life as the gears wearout. These gradual increases in vibration level are normal. Sudden changes in vibration levels (at gear mesh frequency, two times gear mesh frequency, or sidebands), especially decreases, are very significant. A drop in the vibration level usually means a decrease in stiffness, and that more of the transmission forces are being absorbed due to bending of the gear teeth. Catastrophic failure is imminent. Premature gear failures are usually a symptom of other problems such as unbalance, misalignment, bent shaft, looseness, improper lubrication, or contaminated lubrication.

25.11.3

Specific Machine Types

25.11.3.1

Pumps

There are two principal types of pumps: (1) centrifugal pumps and (2) reciprocating pumps. Reciprocating pumps will be discussed in a later section. The sources of vibration in pumps are widely

25 32


varied. In addition to the standard mechanical problems (unbalance, misalignment, worn bearings, etc.), problems that are particular to pumps include vane pass frequency generating conditions (starvation, impeller loose on the shaft, impeller hitting something) and cavitation. Starvation occurs when not enough liquid is not present to fill each vane on the impeller every revolution of the shaft. Pump starvation can be confused with unbalance (see Chapter 34). However, it can be distinguished by the varying amplitude 1X vibration at constant speed and the reduced load on the driving motor. When the vanes on the impeller are striking something, the vane pass frequency (the number of vanes times the rotational speed) is excited. Because the striking causes a force on the shaft, an unbalance is also present. The frequency spectrum will show a response at 1X and vane pass frequency. The time waveform will show a high frequency response (vane pass) riding on a frequency response at 1X. The vane pass frequency is in phase with the shaft speed. If the impeller is loose on the shaft, the vane pass frequency will be modulated by the shaft speed. Cavitation occurs when there is sufficient negative pressure (suction) acting on the liquid in the system that it becomes a gas (it boils). This usually takes place in localized parts of the system. Cavitation usually occurs in a pump when the suction intake is restricted and the liquid vaporizes when coming off the impeller. As the fluid moves past the low pressure region, the gas bubble collapses. If the collapsing bubble is close to a solid surface, it will aggressively erode the surface. Cavitation may be caused by a local decrease in atmospheric pressure, an increase in fluid temperature, an increase in fluid velocity, a pipe obstruction, or abrupt change in direction. The vibration signal that results will have significant vibration levels at 1X with harmonics and strong spectral responses at vane pass frequency. High frequency broadband noise is also common. An increase in the system pressure can reduce cavitation. Hydraulic unbalance will result if there has been poor design of suction piping (elbow close to inlet) or poor impeller design (unsymmetrical). The vibration signal will contain high 1X axial vibration components. Impeller unbalance is a specific form of mechanical unbalance as discussed above. High 1X vibration levels will result. Pipe stresses result from inadequate pipe support and cause stress on the pump casing. This may also cause misalignment. Pipe resonances can also be excited by vane pass frequency pressure pulsations. Diagnosis of pump problems can be improved by installing a pressure transducer in the discharge line of the pump. The measured pressure fluctuations can be processed in the same way as vibration signals. The frequencies measured represent the pressure fluctuations and the amplitude is the zero to peak pressure change. 25.11.3.2

Fans

Fans account for a significant number of field vibration problems due to their function and construction. Fans move air or exhaust gases that are often laden with grease, dust, sand, ash, and other corrosive and erosive particles (also see Chapter 34). Under these conditions, fans blades gain and lose material resulting in the need to regularly rebalance. The level of balance must also be relatively fine because fans often have large fan blade diameters and operate at relatively high speeds. Fans are usually mounted on spring/damper systems to help isolate vibrations, but they are also constructed in a relatively flexible manner, which adds to the demands for fine balancing. Along with fine balancing requirements, typical problems include looseness, misalignment, bent shaft, and defective bearings. Fans also generate a strong response at blade pass frequency (number of blades times the shaft rotational speed). This frequency response is present during normal operation, but it can become elevated if the blades are hitting something, the fan housing is excessively flexible, or an acoustical resonance is present. Acoustical resonances are relatively common where large volumes of air are being moved through large flexible ducts and/or the fan blades are of an air foil design. 25.11.3.3

Electric Motors

Electric motors can be divided into two groups: (1) induction motors and (2) synchronous motors. A full description will not be given here as to the differences. Like any machine, electric motors are subject to a


25 33

full range of mechanical problems, and vibrations signals can be used to detect and diagnose these problems. Apart from the conditions described elsewhere in this section, there are some problems that occur only in electric motors. For the sake of brevity, these problems and the vibration signals that typically accompany them are summarized in Table 25.2. 25.11.3.4

Steam and Gas Turbines

Steam and gas turbines (and high speed compressors) require special mention because of the high speeds and temperatures involved. Problems on steam turbines are usually limited to looseness, unbalance, misalignment, soft foot, resonance, and rubs. As discussed above, each of these conditions has a set of characteristic vibration responses that allow for relatively straightforward diagnosis. However, because of the high speeds, this type of machinery is usually designed to be lighter and less rigid than other rotating machines. Excessive vibration can therefore lead to catastrophic failure very quickly. Because of this, high speed turbines and compressors are designed to closer tolerances than other types of machines, and extra care is taken when balancing rotors. These machines also frequently operate above their first critical speed and sometimes between their second and third critical speeds. At these speeds, the rotor becomes quite flexible and the support bearings become very important in that they must provide the appropriate amount of damping. Because steam and gas turbines are supported on journal bearings, most monitoring and diagnostics work will be based solely on proximity probe signals. While this is not a problem in and of itself, accelerometer signals should also be taken in order to cover the higher frequencies, which are excited by conditions such as looseness and rubs. 25.11.3.5

Compressors

Compressors act in much the same way as pumps, except that they are compressing some type of gas. They come in many different sizes, but only two principal types: (1) screw type and (2) reciprocating compressors. Reciprocating compressors will be discussed in a later section. Screw type compressors have a given number of lobes or vanes on a rotor and generate a vane passing frequency. Screw compressors with multiple rotors can also generate strong 1X and harmonics up to vane pass frequency. The close tolerances involved result in relatively high vibration levels, even when the machine is in good condition. As with pumps, signals taken from pressure transducers in the discharge line can be useful for diagnostics. 25.11.3.6

Reciprocating Machines

Reciprocating machines (gas and diesel engines, steam engines, compressors, and pumps) all have one thing in common a piston that moves in a reciprocating manner. These machines generally have high overall vibration levels and particularly strong responses at 1X and harmonics, even when in good condition. The vibrations are caused by compressed gas pressure forces and unbalance. Vibrations at 12 X may be present in four stroke engines because the camshaft rotates at one half the crankshaft speed. TABLE 25.2

Mechanical Problems Particular to Electric Motors Condition

Motor out of magnetic center Motors with broken rotor bars Motor with turn to turn shorts in the windings Motor out of magnetic center with broken rotor bars or turn to turn shorts in the windings

Vibration Indicator High spectral response at 60 Hz High spectral response at motor running speed and/or second harmonic Motor runs at a slower than expected speed (high slit frequency) Side bands of slip frequency times the number of poles centered around the motor running speed and harmonics of the running speed

25 34


Many engines operate at variable speeds, which will allow the strong forcing functions to excite resonances of the components and the mounting structure, if it is not designed in a robust manner. Excessive vibrations in reciprocating machines usually occur due to operational problems such as misfiring, piston slap, compression leaks, and faulty fuel injection. These problems result in elevated 12 X vibrations, if only one cylinder is affected, and a decrease in efficiency and power output. Gear and bearing problems may also occur in reciprocating machines, but the characteristic defect frequencies for these faults are significantly higher.

25.11.4

Advanced Fault Diagnostic Techniques

Much of the discussion in the previous sections has highlighted the fact that many machine defects generate distinctive vibration signals. This fact has been exploited recently with the development of a variety of different automatic fault diagnostics techniques (Mechefske and Mathew, 1992b; Mechefske, 1995). The details of these systems will not be provided here, but the goal of automatic diagnostics is to augment and assist, rather than replace, the vibration signal analyst. If charac teristic defect indicators can be detected and extracted from a vibration signal without the intervention of a signal analyst, the analyst will have more time for other duties and will also have access to information that may not have been uncovered through normal signal processing and analysis. There are, however, still many situations where machine defects do not generate distinctive vibration signals or when the vibration signals are masked by large amounts of noise or vibrations from other machinery. In such cases, advanced diagnostic algorithms incorporating new signal processing techniques are currently being developed and implemented. Artificial neural networks (Timusk and Mechefske, 2002) have been found to provide an excellent basis for detecting and diagnosing faults. Wavelet analysis (Lin et al., 2004) and short time Fourier transforms (STFTs) have also been shown to effectively allow both time domain and frequency domain information to be displayed on the same plot. This provides an opportunity to clearly see short duration transient events as well as detect faults in machinery that is operating in nonsteady state conditions.

*

*

*

*

*

Analysis of the vibration signal to determine the actual type of fault present will allow for more accurate estimation of the remaining life, the replacement parts that are needed, and the maintenance tools, personnel, and time required to repair the machinery. A diagnostic template can be developed for the different types of faults that are common in a given facility or plant by listing various faults that usually develop in machinery in terms of the forcing functions that cause them and specific machine types. Common forcing functions include unbalance, misalignment, mechanical looseness, soft foot, rubs, resonances, oil whirl, oil whip, structural vibrations, and foundation problems. Specific machine components that need to be monitored include damaged or worn rolling element bearings and gears. Specific machine types that can be treated as common groups include pumps, fans, electric motors, steam and gas turbines, compressors, and reciprocating machines.

References Alfredson, R.J. 1982. A computer based system for condition monitoring, Symposium on Reliability of Large Machines, The Institute of Engineers Australia, Sydney, pp. 39 46. Boto, P.A., Detection of bearing damage by shock pulse measurement, Ball Bearing J., 5, 167 176, 1979. Braun, S. 1986. Mechanical Signature Analysis: Theory and Applications, Academic Press, London.


25 35

Courrech, J. 1985. New techniques for fault diagnostics in rolling element bearings, In Proceedings of the 40th Meeting of the Mechanical Failure Prevention Group, National Bureau of Standards, Gaithersburg, MD, pp. 47 54. Eisenmann, R.C. Sr. and Eisenmann, R.C. Jr. 1998. Machinery Malfunction: Diagnosis and Correction, Prentice Hall, Newark. Eschman, P. 1985. Ball and Roller Bearings, 2nd ed., Wiley, New York. Goldman, S. 1999. Vibration Spectrum Analysis, Industrial Press, New York. Grove, R.C. 1979. An investigation of advanced prognostic analysis techniques, Paper USARTL TR 79 10. Northrup Research and Technology Center, Palos Verdes, CA. Lin, J., Zuo, M.J., and Fyfe, K.R., Mechanical fault detection based on the wavelet de noising technique, ASME J. Vib. Acoust., 126, 9 16, 2004. Lyon, R.H. 1987. Machinery Noise and Diagnostics, Butterworth, Boston. Mathew, J., Machine condition monitoring using vibration analysis, J. Aust. Acoust. Soc., 15, 7 21, 1987. Mathew, J. and Alfredson, R.J., The condition monitoring of rolling element bearings using vibration analysis, Trans. ASME J. Vib. Acoust. Stress Reliab. Des., 106, 447 457, 1984. McFadden, P.D. and Smith, J.D., A model for the vibration produced by a single point defect in a rolling element bearing, J. Sound Vib., 96, 69 77, 1984. McFadden, P.D. and Smith, J.D., A signal processing technique for detecting local defects in a gear from the average of the vibration, Proc. IMechE, 199, 99 112, 1985. Mechefske, C.K., Fault detection and diagnosis in low speed rolling element bearings using inductive inference classification, Mech. Syst. Signal Process., 9, 275 282, 1995. Mechefske, C.K., Objective machinery fault diagnosis using fuzzy logic, Mech. Syst. Signal Process., 12, 855 864, 1998. Mechefske, C.K. and Liu, L., Fault detection and diagnosis in variable speed machines, Int. J. Condition Monit. Diagn. Eng. Manage., 5, 29 42, 2001. Mechefske, C.K. and Mathew, J., Fault detection in low speed rolling element bearings, part I: the use of parametric spectra, Mech. Syst. Signal Process., 6, 297 308, 1992a. Mechefske, C.K. and Mathew, J., Fault detection in low speed rolling element bearings, part II: the use of nearest neighbour classification, Mech. Syst. Signal Process., 6, 309 317, 1992b. Mitchell, J.S. 1981. An Introduction to Machinery Analysis and Monitoring, Penwell, Los Angeles. Mobley, R.K. 1990. An Introduction to Predictive Maintenance, Van Nostrand Reinhold, New York. Moubray, J. 1997. Reliability Centered Maintenance, Industrial Press, New York. Randall, R.B. 1981. A new method of modeling gear faults, ASME Annual Congress. Paper No. 81 Set 10. Randall, R.B. 1987. Frequency Analysis, Bruel & Kjaer, Copenhagen. Randall, R.B. 2001. Bearing diagnostics in helicopter gearboxes, Condition Monitoring and Diagnostic Engineering Management Conference, Manchester, U.K., August 2001. Rao, B.K.N. 1996. The Handbook of Condition Monitoring, Elsevier, Oxford. Reeves, C.W. 1999. The Vibration Monitoring Handbook, Coxmoor, Oxford. Smith, J.D. 1983. Gears and Their Vibrations, Marcel Dekker, New York. Taylor, J.I. 1994. The Vibration Analysis Handbook, Vibration Consultants, Tampa. Timusk, M. and Mechefske, C.K. 2002. Applying neural network based novelty detection to industrial machinery, 6th International Conference on Knowledge Based Intelligent Information Engineering Systems, Crema, Italy, November 2002. Wowk, V. 1991. Machinery Vibration: Measurements and Analysis, McGraw Hill, New York.

26

Vibration-Based Tool Condition Monitoring Systems 26.1 Introduction ........................................................................ 26 1 26.2 Mechanics of Turning ......................................................... 26 2 General Terms

†

Chatter Vibrations

†

Tool Wear

26.3 Vibration Signal Recording ................................................ 26 7 Direct and Indirect Systems † Sensor Requirements for Tool Wear Monitoring † Force Measurement † Acceleration Measurement † Acoustic Emission Measurement † Sensor Comparisons

26.4 Signal Processing for Sensor Based Tool Condition Monitoring ........................................................ 26 11

C. Scheffer University of Stellenbosch

P.S. Heyns

University of Pretoria

Feature Extraction

†

Feature Selection

26.5 Wear Model/Decision Making for Sensor Based Tool Condition Monitoring ............................................... 26 15 Trending, Threshold Other Methods

†

Neural Networks

†

Fuzzy Logic

†

26.6 Conclusion ........................................................................... 26 20

Summary Despite the high level of technology built into every aspect of modern metal cutting operations, the phenomenon of tool wear still hampers the reliability and complete automation of machining processes. Tool wear is the loss of material on the edge of the cutting tool. This chapter concerns sensor based tool condition monitoring (TCM), and specifically those methods that are based on vibration related properties such as force, acceleration, and acoustic emission (AE). References are made to systems proposed in the literature and also to commercially available hardware. The chapter focuses on turning operations. The mechanics of turning are briefly discussed. Various methods of obtaining vibration signals from turning operations are described. The vibration signal has to be processed in order to estimate the level of wear in the cutting edge of the tool, and several state of the art approaches are discussed. Effective methods of constructing a model relating sensor data and the tool wear, using processed vibration signals, are described. The chapter concludes by indicating some important points that should be considered when using vibration based systems for TCM, and some interesting topics for future research in this field of study. Chapters 25 and Chapter 27 present further information on the present subject.

26.1

Introduction

Millions of products are manufactured daily by a variety of processes. A basic method to form bulk metal into a desired final shape is through the process of metal cutting, also referred to as machining. Metal cutting is essentially the removal of excess material from a workpiece by moving a working tool over the 26 1

26 2


surface of the workpiece at a certain depth, speed, and feed rate. Conventional machining operations are turning, milling, and drilling. Despite the high level of technology built into every aspect of modern metal cutting operations, the phenomenon of tool wear still hampers the reliability and complete automation of machining processes. Tool wear is the loss of material on the edge of the cutting tool. Although tool wear can be minimized, it cannot be eliminated. Unfortunately, excessive or even a small quantity of tool wear may cause a defect in a machined component, and therefore it is always necessary to be aware of the extent of the current tool wear before machining can commence. Economic losses due to tool wear occur as a result of the scrapping of expensive parts and the nonoptimal use of tool inserts. A conservative approach is often taken, and the insert is recycled long before it should have been. Furthermore, secondary damage due to tool wear can be extreme and even catastrophic. For this reason, many approaches to tool condition monitoring (TCM) have been proposed through the years. There exist sensorless and sensor based TCM approaches. Sensorless approaches are generally tool life equations and not monitoring methods. Thus, sensorless approaches attempt to determine the optimal tool life under certain machining conditions. These are often extended versions of the famous Taylor equation, which is described by vT n ¼ C ð26:1Þ where v is the cutting speed, T is the tool life, and n and C are constants that must be determined experimentally for a given tool and workpiece combination. This chapter is focused on sensor based TCM, and specifically those methods that are based on vibration related properties such as force, acceleration, and acoustic emission (AE). These sensor types are known to be most effective for TCM. Furthermore, discussions will be focused on the application of TCM in turning operations, though reference will be made to other machining operations as well. Besides vibration based approaches, other sensor based TCM methods are: *

*

*

*

*

*

*

*

Use of noncontact capacitive sensors Vision systems Measurement of the motor current Surface roughness monitoring Ultrasonic monitoring Temperature monitoring Laser scatter methods Audible emission monitoring

The reader is also referred to other excellent overviews of sensor assisted TCM, published by Dan and Mathew (1990), Byrne et al. (1995), Scheffer and Heyns (2001a), and Dimla (2000). A TCM database was also published by the CIRP, supervised by Teti (1995), which includes more than 500 research papers focusing only on TCM.

26.2

Mechanics of Turning

26.2.1

General Terms

A typical turning operation is schematically shown in Figure 26.1. The cutting tool moves parallel to the workpiece and spindle, and hence reduces the diameter of the shaft. The most important machining parameters are: *

*

*

Cutting speed (usually expressed in m/min) Feed rate (usually expressed in mm/rev) Depth of cut (usually expressed in mm)

The force response on the tool tip due to the turning operation consists of three components: Fx, Fy, and Fz. These forces consist of a static and a dynamic part, as shown in Figure 26.2. The static forces are governed by the static pressure between the tool and workpiece, and are a function of the machining

Vibration Based Tool Condition Monitoring Systems

FIGURE 26.1

26 3

Turning operation.

parameters. The dynamic forces are governed by forced and free vibrations due to excitation from the cutting operation. Analytical models exist that can describe the static forces for basic machining operations (Merchant, 1945). The dynamic behavior is more difficult to model theoretically, although there is also continuous research in this area (Kapoor et al., 1998). One of the main difficulties of monitoring tool wear with vibration is to identify the frequency range that is influenced by tool wear, since machining processes entail various mechanisms that produce vibrations that are not related to tool wear. The frequency range of vibrations produced during ordinary machining operations usually falls between 0 and 10 kHz. From the literature, it can be concluded that the frequency range sensitive to tool wear depends entirely on the type of machining operation, and must be determined experimentally for each individual case. There are two important vibration frequencies present during cutting: *

*

The natural frequencies of the tool holder and its components The frequency of chip formation

force [N]

Dynamic tests should be conducted to identify the dynamic properties of tool holders (Scheffer and Heyns, 2002a). However, the interaction of the working tool engaged into the rotating workpiece complicates the situation, and as a result the dynamic behavior during cutting could be differ dynamic cutting ent from the expected behavior obtained from off force line tests. Scheffer and Heyns (2004) compared continuous cantilever models with modal hammer tests for different tool holder overhang lengths. static cutting force The natural frequency of the first mode as a func tion of overhang length is plotted in Figure 26.3 (for a specific tool holder). It can be seen that a 0 time [s] continuous fixed free cantilever beam model corresponds well with the results obtained with hammer tests. FIGURE 26.2 Static and dynamic forces.

26 4


26.2.2

Chatter Vibrations

7 1st mode frequency [kHz]

The chip formation frequency can be calculated with simple equations that take the machining conditions into account (Lee et al., 1989). The tool holder natural frequencies and chip formation frequency are independent. Generally, tool wear has a larger effect on the vibration amplitudes at the tool holder natural frequencies but can influence chip formation frequencies as well.

6 5 4 3 2 1

fixed-free cantilever beam hammer tests

Another phenomenon important to machining 0 operations is tool chatter vibrations. These are 40 45 50 55 60 overhang length [mm] self excited vibrations resulting from the gener ation of different chip thicknesses during machin ing. Initially, cutting forces excite a structural FIGURE 26.3 Frequency of first tool holder mode. mode of the machine workpiece system. This (Source: Scheffer, C. and Heyns, P.S., Mech. Syst. Signal leaves a wavy surface finish on the workpiece. Process, Elsevier, 2004. With permission.) During the next revolution, another wavy surface is produced in the same way. Depending on the phase shift between these two waves, the maximum chip thickness can grow and oscillate at a particular frequency that is close to that of a structural mode. This is called the regenerative chatter frequency. Chatter cause a poor surface finish and can also lead to tool breakage. The analysis and prediction of chatter has been the subject of research for many years. Morimoto et al. (2000) developed a piezoelectric shaker/actuator to regenerate the vibrations of the cutting process. In this way, unwanted vibrations such as chatter can be attenuated. The system is also helpful to determine the dynamic properties of the machine tool. Koizumi et al. (2000) used a very interesting approach called the correlation integral in the time domain to identify chatter onset. Lago et al. (2002) designed a sensor and actuator integrated tool for turning and boring to control chatter. The tool holder shank vibrations are sent to the actuator via a digital controller. An adaptive feedback control system is used to perform broadband vibration attenuation up to 40 dB at different frequencies simultaneously.

26.2.3

Tool Wear

26.2.3.1

Tool Failure Mechanisms

Tool wear is caused by mechanical loads, thermal loads, chemical reactions, and abrasive loads. The load conditions are in turn influenced by the cutting conditions and materials. The different loads can cause certain wear mechanisms that may occur in combination. These mechanisms have either a physical or chemical characteristic that causes loss or deformation of tool material. Tool wear mechanisms can be classified into several types, summarized as follows (Du, 1999): *

*

*

*

Abrasive wear resulting from hard particles cutting action Adhesive wear associated with shear plane deformation Diffusion wear occurring at high temperatures Fracture wear due to fatigue

Other wear mechanisms are plastic deformation and oxidation, which are not very common in industry. It is estimated that 50% of all tool wear is caused by abrasion, 20% by adhesion, 10% by chemical reactions and the remaining 20% by the other mechanisms (Kopac, 1998). Abrasion is basically the grinding of the cutting tool material. The volume of abrasive wear increases linearly with the cutting forces. Higher hardness of the tool material can reduce the amount of abrasive wear. During adhesion, the high pressures and temperatures on the roughness peaks on the tool and the workpiece cause


26 5

welding. These welding points are broken many times every second due to the workpiece movement and as a result cause removal of the tool material (Kopac, 1998). Diffusion wear occurs at even higher cutting speeds, where very high temperatures are present (especially when using hard metal tools). 26.2.3.2

Tool Failure Modes

Tool wear will generally occur as a combination of a number of wear modes, with one mode predominant. The dominant mode will depend on the dominant wear mechanism, which in its turn is influenced by the machining conditions and the choice of tool and workpiece material. For a given tool and workpiece combination, the dominant wear mode can be determined at different cutting speeds using the product of the cutting speed and the undeformed chip thickness (Dimla, 2000). The common wear modes are: *

*

*

*

*

*

*

*

Nose wear Flank wear Crater wear Notch wear Chipping Cracking Breakage Plastic deformation

Figure 26.4 is a graphical representation of the different tool failure modes. The consequences of tool wear are deviations in shape and roughness of the machined part that cause the part to be discarded because it is out of tolerance. Most wear modes cause an increase in cutting forces, although this is not always the case for all tool and workpiece combinations. The most widely researched tool failure modes for turning with single point tools are flank wear, breakage (fracture), and crater wear. Flank and crater wear are accepted as normal tool failure modes, because the other failure modes can be avoided by selecting the proper machining parameters. The growth of flank and crater wear is directly related to the total cutting time, unlike some of the other failure modes, which can occur unexpectedly even with a new tool.

FIGURE 26.4 Tool failure modes. (Source: Scheffer, C. and Heyns, P.S. 2001a. COMA ’01, University of Stellenbosch. With permission.)

26 6


26.2.3.3

Tool Wear Measurement

Wear measurements of tool inserts are done through the implementation of an appropriate international standard, ISO 3685. Flank wear is quantified in terms of VB, which is the mean of the wear height on the tool flank. The length of flank wear is also measured in terms of b. Crater wear is quantified in terms of the crater depth, K. The parameters are depicted in Figure 26.5, which is a scanning electron microscope (SEM) picture of a worn turning insert. 26.2.3.4

Tool Wear Stages

It is assumed by most authors that tool wear consists of an initial, a regular, and a fast wear stage (Zhou et al., 1995). Some authors divide tool wear into five distinct stages (Bonifacio and Diniz, 1994):

It has been established by various researchers that the initial and fast (before tool breakage) stages occur more rapidly than the regular stage. Bonifacio and Diniz (1994) explain that, during the fast wear stage with coated carbide tools, the tool loses its coating and the tool substrate (which has less resistance) begins to perform the cut and wears faster. During the initial stage, the tool edge loses its sharp edge rapidly, after this the process stabilizes for a given time. Flank wear in relation to total cutting time will typically appear as depicted in Figure 26.6. The geometrical growth and rate of wear is unique for every tool insert, even those used with the same machining parameters. Wear measure ments conducted on the shop floor of a piston manufacturer by Scheffer and Heyns (2004) are shown in Figure 26.7. It was found that the tools last between 1000 and 6000 components, which makes the optimal use of the tool extremely problematic if the wear is not monitored on line. The reason for this behavior is mainly attributed to fluctuating conditions on the shop floor, for example, the rate at which components are manufactured. If the time allowed for the tool to cool down between workpieces is not constant, large variations in the tool life can be expected.

flank wear

Initial stage of wear Regular stage of wear Microbreakage stage of wear Fast wear stage Tool breakage

Tool wear parameters.

initial

regular

1

2

fast 3

5

4

cutting time

FIGURE 26.6

Flank wear in relation to cutting time.

0.25 flank wear VB [mm]

1. 2. 3. 4. 5.

FIGURE 26.5

0.2 0.15 0.1 0.05 0

2000

4000

6000

number of workpieces

FIGURE 26.7 Typical variations in tool life. (Source: Scheffer, C. and Heyns, P.S., Mech. Syst. Signal Process, Elsevier, 2004. With permission.)


26.3

26 7

Vibration Signal Recording

The information from vibration sensors can be treated in numerous ways. The overall aim of a tool condition monitoring system (TCMS) is to utilize the best processing techniques to extract the relevant information from sensor signals. Generally, a TCMS consists of the steps depicted in Figure 26.8. Various methods that could be used in each step will be discussed in more detail.

26.3.1

Direct and Indirect Systems

TCMSs can be divided in two categories, namely, direct and indirect. Direct methods are concerned with a measurement of volumetric loss at the tool tip, while indirect methods use a pattern in sensor data to detect a failure mode (Byrne et al., 1995). Direct methods do not utilize vibration and will not be discussed here. In general, direct methods are sensitive to dirt and cutting chips, and consequently they are not commonly accepted in industry. Indirect methods have found more acceptance in industry due to the fact that most indirect methods are easily interpreted, cost effective, and often more reliable than direct methods. Also, for some applications, it might not be possible to use a direct monitoring method due to the nature of the process.

26.3.2

Sensor Requirements for Tool Wear Monitoring

Machine tools represent very hostile environments for sensors. Sensors used for TCM (also see Chapter 15) must meet certain requirements, such as (Byrne et al., 1995) the following: *

*

*

*

*

*

*

Must measure as close as possible to the point of metal removal Must not cause a reduction in the stiffness of the machine tool Must not cause a restriction of the working space of the machine Should be wear and maintenance free, easy to replace, and of low cost Must have resistance to dirt, chips, and electromagnetic and thermal influences Should function independent of tool and workpiece Must provide reliable signal transmission, e.g., from rotating to fixed machine components

26.3.3

Force Measurement

Worn tools cause an increase in the cutting force components. It is also known that both the dynamic and static components generally increase with tool wear due to frictional effects. The three components of the cutting force each responds uniquely to varying machining parameters and the different wear modes. Depending on the type of process that is investigated and the specific experimental setup, results among researchers vary. This can be attributed to dynamic effects of the machine tool and the measurement equipment. There are a number of different sensor configurations to collect forces from machining operations and these are described below. 26.3.3.1

Direct Measurement Dynamometers

Tool holder dynamometers are by far the most popular method for collecting cutting forces. These sensors utilize the piezoelectric effect and can measure quasistatic and dynamic cutting forces very accurately. However, dynamometers are very expensive and bulky instruments and are not practical for a sensor selection and deployment

signal recording and conditioning

generate signal features

FIGURE 26.8

select wear sensitive features

TCMS steps.

model features and wear relationship

26 8


typical shop floor. Furthermore, their usable frequency range is limited to approximately 1 kHz. An example of a tool holder dynamometer is shown in Figure 26.9. Tarmal and Opavsky (2000) investigated the dynamics of a conventional force dynamometer for machining operations. It was found that the dynamometer has significant amplitude distortion in the frequency range that is quoted as the operating range by the manufacturer. The authors suggest that the dynamic characteristics of the dynamometer (while clamped as it would be during measurements) be identified with a modal test and the effect of dynamometer dynamics be compensated for after measurements are made to obtain the true cutting force. 26.3.3.2

Indirect Force Sensors

There are numerous small force sensors available for the purpose of force measurement on machine tools. These measure forces in load carrying components of the machine tool and are thus not direct force measurement devices. The advan tages of these sensors are their size, low cost, and significantly higher operational frequency range. A disadvantage is that a suitable position for the sensor can only be determined experimentally. These sensors are suitable for tool breakage monitoring in rough machining or detection of other catastrophic events such as collisions. An example of a three component force sensor is shown in Figure 26.10. 26.3.3.3

FIGURE 26.9 KISTLER force dynamometer type 9121. (Source: KISTLER Brochure 2002. Courtesy of Kistler Instrumente AG.)

FIGURE 26.10 KISTLER three component force sen sor type 9251A. (Source: KISTLER Brochure 2003. Courtesy of Kistler Instrumente AG.)

Piezoelectric Strain Sensors

The use of piezoelectric strain sensors for wear monitoring of synthetic diamond tool inserts was reported by Scheffer and Heyns (2000a). These sensors are ultrasensitive to changes in cutting forces if they are installed in an appropriate location. The best location for the sensor must once again be determined experimentally, but generally it should be installed on a load carrying FIGURE 26.11 KISTLER strain sensor type 9232A. component of the machine as close as possible to (Source: KISTLER Brochure 2004. Courtesy of Kistler the tool tip, for example, on the tool holder itself Instrumente AG.) (Scheffer and Heyns, 2001b). An example of a piezoelectric strain sensor that can be used on machine tools is shown in Figure 26.11. 26.3.3.4

Resistance Strain Gauges

A quite simple method to estimate both the static and dynamic components of cutting forces without any distortion is to use resistance strain gauges (see Chapter 15). These comply with most of the requirements for TCM sensors, and they can accurately follow the static and dynamic response of a system up to


26 9

FIGURE 26.12 Application of resistance strain gauges. (Source: Scheffer, C. and Heyns, P. S., Mech. Syst. Signal Process., Elsevier, 2004. With permission.)

50 kHz. Scheffer and Heyns (2002a) developed a sensor integrated tool holder using strain gauges. It was shown that the system is robust, cost effective, and fit for an industrial TCMS. The physical layout of the strain gauges on a boring bar is shown in Figure 26.12. The system was calibrated with a special device to directly obtain the three cutting forces from the strain gauge signals. 26.3.3.5

Customized Force Sensors

There are a number of customized force sensors available that can be used with specific machining operations. These are: *

*

*

Force measuring plates, pins, and bearings Special force measuring bolts Force and torque measuring rings that fit on spindles

26.3.4

Acceleration Measurement

Piezoelectric accelerometers can measure the machine vibration caused by oscillations of cutting forces. It is well known that high frequency vibrations (higher than 1 kHz) yield large acceleration levels, giving accelerometers an advantage over force based monitoring. Accelerometers fulfill the environmental requirements for tool wear monitoring because they are resistant to the aggressive media present during machining. Accelerometers are also less expensive than force dynamometers and can measure vibration levels within a very wide frequency range, typically 5 Hz to 10 kHz. Various authors have shown that acceleration levels change with tool wear. Li et al. (1997) found that the coherence function of two crossed accelerations can be used as an easy and effective way to identify tool wear and chatter. They found that with progressive tool wear, the autospectra of the two accelerations and their coherence function increase gradually in magnitude around the first natural frequencies of the cross bending vibration of the tool shank. As the tool approaches a severe wear stage, the peaks of the coherence function increase to values close to unity. Scheffer et al. (2003) reported on the use of an accelerometer for wear monitoring during hard turning. It was found that certain frequencies show repeatable amplitude increase with increasing tool wear. These frequencies corresponded to the tool holder natural frequencies. Some authors, for example, Bonifacio and Diniz (1994), also found that a wear sensitive frequency will increase with increasing tool wear and then suddenly decrease near the end of tool life. This can be attributed to an increased damping effect due to plastic deformation and microbreakage of the cutting edge.

26.3.5

Acoustic Emission Measurement

Cutting processes produce elastic stress waves that propagate through the machine structure. Different sources in the cutting process generate these stress waves known as acoustic emission (AE). Sources of AE

26 10


in metal cutting are: *

*

*

*

*

Friction on the tool face and flank Plastic deformation in the shear zone Crack formation and propagation Impact of the chip at the workpiece Chip breakage

A typical AE sensor for use on machine tools is shown in Figure 26.13. FIGURE 26.13 Kistler AE sensor type 8152B. (Source: The fact that crack formation generates AE PCB Website 2002. Courtesy of Kistler Instrumente AG.) makes AE ideal for tool breakage detection. Collection of the AE requires special hardware that can bandpass filter the signals to the AE range (between approximately 50 and 250 kHz). Furthermore, amplification is required and an analogue root mean square (RMS) circuit with a short time constant is generally also included to collect the AERMS. The different steps required to collect AE are depicted in Figure 26.14 (adapted from Jemielniak, 2000). Araujo et al. (2000) investigated sliding friction as a possible source of AE during metal cutting. The AERMS values in different frequency ranges were collected for different widths of cut and also with the tool rubbing against the workpiece without cutting. It was found that the level of AE remains almost constant for all width of cut conditions, and hence it was concluded that the main mechanism for AE during metal cutting is the sliding friction between the tool and workpiece. Consequently, an increase or decrease of AE can be expected with tool wear depending on the effect on the sliding friction due to that tool wear. Furthermore, it is believed that the cutting temperatures will affect the AE due to thermal expansion effects. Chiou and Liang (2000) investigated AE with tool wear and chatter effects in turning. A model is presented that can predict the chatter AERMS amplitude with certain levels of flank wear. Good correlation was found between the model and the experimental results. Kim et al. (1999) reported on the use of AE to monitor the tool life during a gear shaping process. The AERMS is collected and used in a software program to predict the remaining tool life. Li (2002) presented an overview of using AE for TCM in turning operations. It is stated the AE is heavily dependent on cutting conditions and, as a result, methods should be employed to handle this problem effectively. Some methods are proposed that include advanced signal processing, sensor fusion and modeling techniques. Many other AE based tool wear and breakage monitoring systems have been implemented successfully in research. One problem still lies with an appropriate interpretation of the AE frequency spectrum. In most studies, an explanation for the choice of certain frequencies and their advantages are not given or not investigated. In fact, Jemielniak (2000) found that using the average value of AE (or AERMS) is the most suitable. A similar conclusion was made by Scheffer et al. (2003), who compared different processing methods of the AE signal during hard turning.

FIGURE 26.14

Steps for collecting AE during turning. (Adapted from Jemielniak, K., Ultrasonics, 2000.)


26.3.6

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Sensor Comparisons

Future research should be directed towards directly comparing different sensors for tool wear monitoring. Choi et al. (1999) developed a single sensor for parallel measurement of force and AE. A finite element analysis was carried out to determine the optimal position for the sensor away from the tool holder because the sensor obstructed the working space of the machine. The approach was successful for breakage detection but no wear estimations are reported. Barrios et al. (1993) compared AE, vibration, and spindle current for TCM during milling. It was found that the spindle current is the most sensitive sensor for detecting tool wear, with AE the least sensitive. However, contradictory results are reported in other publications, and hence more research would be required to determine ultimately which sensor is the best for which machining operation. Govekar et al. (2000) compared force and AE methods for TCM, and concluded that the best result is achieved when sensory information is combined. Dimla and Lister (2000a) compared the use of force and vibration signals for TCM and also combined the information in a single decision making technique (Dimla and Lister, 2000b). Similar comparative studies were reported by Scheffer et al. (2003).

26.4 Signal Processing for Sensor-Based Tool Condition Monitoring Using the sensor information from the different sensor systems described in the previous section, a decision must be made with respect to the tool condition. This decision is generally referred to as the data classification. It is often better to combine sensory information to solve a complex problem such as TCM. Such a combined approach is referred to as sensor fusion. Sick (2002) proposed a generic sensor fusion architecture for TCM, which summarizes the various sensor fusion levels of a TCMS. These are: *

*

*

*

*

Analogue preprocessing Digital preprocessing Feature extraction Wear model Decision making

Fusion of sensor information can occur at any of these levels. Analogue and digital preprocessing are activities such as signal amplification, conditioning, filtering, calibration, and temperature compen sation. The feature extraction step is probably the most important step, because here the sensor signals must be condensed and reduced to only a few appropriate wear sensitive values. Many different methods are available to achieve this. The wear model level establishes a relationship between the chosen features and the tool condition. In many cases, neural networks (NNs) are used in this step, and sensor fusion takes place within the NN. A decision level can also be included where a final decision is made with respect to the tool condition, for instance a “competing experts” formulation if a TCMS is used in conjunction with a tool life equation. Discussions on the various techniques follow.

26.4.1

Feature Extraction

Most decision making techniques for process monitoring are based on signal features. Through appropriate signal processing, features can be extracted from signals that show consistent trends with respect to tool wear. Features are mainly derived through processing in the time, frequency, or joint time frequency domain or statistical analysis. 26.4.1.1

Time Domain

Features extracted from the time domain are usually fundamental values such as the signal mean or RMS. Other techniques include the shape of enveloping signals, threshold crossings, ratios between

26 12


time domain signals, peak values, and polynomial approximations of time domain signals. Examples of time domain features from an interrupted cutting operation are shown in Figure 26.15. It has been found that some of the time domain features show good correlation with tool wear and are easy to implement (Scheffer, 1999). Bayramo glu and Du¨ngel (1998) investigated the use of several different force ratios (calculated from the static cutting forces). It was found that certain force ratios can be used to monitor tool wear under a wide range of cutting conditions. Most commercial TCMS rely on time domain infor mation. However, time domain features are known to be sensitive to disturbances and should be complemented with features from another domain. 26.4.1.2

FIGURE 26.15

Simple time domain features.

Frequency Domain

The power or energy of certain frequency bands in the fast Fourier transform (FFT) is often suggested as a feature for TCM. It is very challenging to identify spectral bands that are sensitive to tool wear. It is even more difficult to determine exactly why these frequencies are influenced by tool wear. Power in certain bands will often increase due to higher excitation forces because of the increase in friction when the tool starts to wear. Sometimes a peak in the FFT will also shift due to changing process dynamics as a result of tool wear. An early frequency domain approach is reported by Jiang et al. (1987), in which frequency band energy is determined from FIGURE 26.16 Frequency domain features. the power spectral density (PSD) function as a feature for tool wear. Some authors suggest that two frequency ranges be identified from the original signal (Bonifacio and Diniz, 1994). The one range must be sensitive to tool wear, the other must be insensitive. For instance, if the measurement was made from 0 to 8 kHz, it must be split (using appropriate filters) into a 0 to 4 kHz signal and a 4 to 8 kHz signal. If the lower range is more sensitive to tool wear, a ratio between the two ranges can be calculated. If this ratio exceeds a certain pre established value, it can be concluded that the end of the tool life has been reached. This can also apply for a ratio between the signal recorded from a fresh tool to that compared with a worn tool. Examples of frequency domain features from cutting forces are shown in Figure 26.16. One difficulty with frequency domain approaches is that the dynamics of the operation and measurement hardware is not always fully understood. The fact that measurement hardware dynamics instead of process dynamics are often measured was also recently identified by Warnecke and Siems (2002). The response of a force dynamometer is influenced by its clamping condition, which may cause it to experience nonlinearities at relatively low frequencies. There are also some uncertainties when using these instruments, relating to their calibration and other varying parameters. A model for expressing the uncertainties when collecting cutting forces with a dynamometer was proposed by Axinte et al. (2001). These uncertainties might be responsible for the scatter of force components often reported in the


26 13

literature. An interesting study is also reported by Ba¨hre et al. (1997), concerning determination of the natural frequencies of the machine tool components using the finite element method (FEM). These are taken into account for interpretation of the vibration/AE signal. 26.4.1.3

Statistical Processing

In the case of statistical features, signals are assumed to have a probabilistic distribution, and consequently, useful information can be extracted from the statistics of the distribution. Hence, the signal is regarded as a random process. Generally, machining processes are nonstationary but are assumed to be stationary for the short periods during which features are calculated. Several statistical features have been investigated for TCM and can be applied to several machining operations. The main features are those that describe the probability distribution of a random process (variance, standard deviation, skewness, kurtosis, etc.) and coefficients of time series models. There are also miscellaneous other statistical features, such as cross correlations, the coherence function, and the harmonic mean. One useful approach is the use of autoregressive (AR) and autoregressive moving average (ARMA) coefficients. AR coefficients computed for a signal represent its characteristic behavior. When the signal changes during the cutting operation due to tool wear, the model coefficients also change and can then be utilized to monitor the progressive tool wear. Baek et al. (2000) report on the use of an eighth order AR model for tool breakage detection during end milling. It was found that the AR approach is somewhat more accurate than the frequency band energy method. Yao et al. (1990) used the ARMA method to decompose the dynamic cutting force signals, and wear sensitive frequencies were identified. This assisted in identifying the importance of certain vibration modes with respect to TCM. The use of statistical process control (SPC) methods is also reported by some authors. Jun and Suh (1999) considered the X bar and exponentially weighted moving average (EWMA) for tool breakage detection in milling. Jennings and Drake (1997) used statistical quality control charts for TCM. Different statistical parameters are calculated and examples of one , two and three variable control charts are given. 26.4.1.4

Time–Frequency Domain

The most common time frequency domain processing method in TCM applications is wavelet analysis. A comprehensive discussion on the advantages and disadvantages of wavelet analysis for TCM is described by Sick (2002). It is often stated that wavelets are used because they provide information about the localization of an event in the time as well as in the frequency domain. However, locating discrete frequency related events in the time domain is rarely of importance with respect to tool wear (which is a gradually increasing phenomenon). In contrast, tool breakage will have a large localized effect in the time domain, but this can be monitored more effectively using time domain techniques. Furthermore, wavelets are time variant and the exact contribution of a particular frequency at any given time can never be determined accurately due to Heisenberg’s uncertainty principle. Despite the above arguments, the use of wavelet analysis for TCM is reported in several publications. Lee and Tarng (1999) use the discrete wavelet transform for cutter breakage detection in milling and find that the technique is reliable even under changing machining conditions. Luo et al. (2002) published results of a TCMS using wavelet analysis of vibration signals. In this case, the wavelet is used as a filter to enhance wear sensitive features in the signals. However, the results are not compared with conventional digital filtering. A comparative study between wavelets and digital filtering for tool wear monitoring was carried out by Scheffer (2002). It was found that, although the wavelet packets act as automated filters, a very similar (if not better) result could be achieved with appropriate digital filtering. The use of wavelets increase the complexity of the TCMS, which is a disadvantage for shop floor implementations. Furthermore, the results from digital filtering can be physically related to the machining operation and tool wear, whereas the behavior of wavelet packets is more difficult to interpret. Another method of time frequency analysis that can be applied for TCM is spectograms (e.g., the Gabor distribution). Spectograms are very useful to identify stationarity in dynamic signals, and for detection of disturbances that may be time localized in signals. The use of the Choi Williams time frequency distribution for TCM during multimilling is described by James and Tzeng (2000).

26 14


Wear sensitive regions on the time frequency distribution are calculated and used as inputs to a NN for wear classification. An example of a change in the dominant chip curl frequency during hard turning is shown in Figure 26.17. It is obvious that, due to some disturbance (perhaps tool wear), the dominant dynamic force frequency “jumps” from 27 to 9 Hz.

26.4.2

Feature Selection

Various authors attempt to generate features that are sensitive to tool wear but insensitive to changing machining parameters. For most oper ations, the machining parameters can be included in the wear model and hence the sensitivity of the features is not such an important issue. There are FIGURE 26.17 Time frequency distribution of cutting also other techniques to normalize sensor data force signal. (Source: Scheffer, et al., Int. J. Mach. Tools Manuf., Elsevier, 2003. With permission.) with respect to machining parameters, for instance the use of a theoretical model (Sick, 1998). This is very useful if the machining conditions change so often that not enough data can be collected for training or calibrating a model. Numerous techniques exist to select the most wear sensitive features or to reduce the input feature matrix to a lower dimension. The main techniques for feature selection and reduction are listed below: *

*

*

*

*

*

*

*

Principal component analysis (PCA) Statistical overlap factor (SOF) Genetic algorithm (GA) Partial least squares (PLS) Automatic relevance determination (ARD) Analysis of variance (ANOVA) Correlation coefficient Simulation error calculations

Al Habaibeh et al. (2000) presented a TCMS for a parallel kinematics machine tool for high speed milling of titanium. An interesting approach to feature selection is employed, called self learning automated sensors and signal processing selection (ASPS). This approach is based on an on line self learning methodology, whereby a certain feature will be selected automatically based on a correlation with tool wear. A linear regression is performed on each feature in the sensory feature matrix to detect the sensitivity of each feature with respect to tool wear. A very interesting cost analysis is then preformed to determine if the installation of a sensor justifies its costs. Ruiz et al. (1993) proposed the use of a discrimination power for feature selection in a TCM application. The method is similar to that of the SOF. An automated version is proposed that also checks for linear correlation between features. It is difficult to assess the success rate of the automated procedure because the experiments/simulations are not described in enough detail. Lee et al. (1998) describe the use of ANOVA to determine the best force ratio for TCM statistically. Several ratios between the three main cutting forces are computed and the influence of controllable parameters (e.g., machining conditions) on these ratios are investigated by means of ANOVA. Du (1999) describes the use of a blackboard system, which is a knowledge based approach for feature selection and decision making. An advantage is the fact that a physical interpretation of a feature can be linked to phenomena in the machining operation. The method is also flexible, but suffers from the disadvantage of requiring a large quantity of data and expertise to establish the knowledge based rules.


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FIGURE 26.18 Comparison of correlation coefficient and SOF for feature selection. (Source: Scheffer, C. and Heyns, P.S., Mech. Syst. Signal Process, Elsevier, 2004. With permission.)

Mdlazi et al. (2003) compared the performance of ARD and PCA for feature selection for two damage detection case studies. It was found that the performance of the methods is similar, but one might perform better on a particular data set. Generally speaking, the PCA yields better results for damage detection problems. Scheffer and Heyns (2002b) compared several feature selection methods for TCM, such as SOF, PCA, GA, ANOVA, and the linear correlation coefficient. It was found that the correlation coefficient approach and the SOF should be preferred for TCM applications. PCA could also be of assistance, but the feasibility of PCA for on line applications is still questionable. The correlation coefficient and SOF is expressed as percentages in Figure 26.18 (from Scheffer and Heyns, 2004) for 30 different wear monitoring features in a turning tool wear case study. Ideally, a feature with a high level of correlation and SOF should be selected. As a last step, engineering judgment is required for proper feature selection because automated methods will often select features that are dependant on one another, thus not achieving the goals of sensor fusion. The following rules can be used as a guideline for selecting features for TCM: *

*

*

*

*

Select features from the static and dynamic parts of force signals. Select features measured in different directions. Use time and frequency domain features. Features based on simple signal processing methods are preferred. There should be a reasonable physical explanation for the behavior of a feature with respect to tool wear.

26.5 Wear Model/Decision-Making for Sensor-Based Tool Condition Monitoring Trending, Threshold

A very simple decision making technique is to trend features and to establish threshold values. When a certain feature or set of features crosses a threshold value, an estimation of the tool con dition can be made. Unfortunately, these threshold values can only be determined experimentally. The difficulty with this method is to determine the correct threshold value, especially under diverse cutting conditions. Furthermore, the method is extremely sensitive to disturbances. The trend of the mean feed force with increasing flank wear is shown in Figure 26.19, and two thresholds are shown as examples. It is clear that this technique is not very reliable due to the large variance in the trend.

1

feature value [normalised]

26.5.1

0.5 0

threshold-replace threshold-warning

−0.5 −1 0.04

FIGURE 26.19

0.06 0.08 flank wear VB [mm]

Example of trend and thresholds.

26 16

26.5.2


Neural Networks

The use of NNs as a secondary, more sophisticated signal processing and decision making technique is often found in TCM applications. The simultaneous utilization of many features and the robustness towards distorted sensor signals are two of the most attractive properties of NNs. Neural networks also assist in the fusion of sensor information for TCM. In other words, combining features from acceleration, AE, and force signals in a NN can result in a method that can predict the tool condition with increased accuracy (Silva et al., 1998). The successful implementation of NNs is dependent on the proper selection of the network structure, as well as the use of the correct training and testing methods. It is important to make a distinction between supervised and unsupervised NN paradigms. Unsupervised NNs are trained with input data only, and are usually used for discrete classification of different stages of tool wear. Supervised NNs are trained with input and output data, and these are used for continuous estimations of tool wear. Furthermore, a distinction should be made between dynamic and static NNs. In the case of dynamic NNs, temporal (time) information is included in the network with the aim to model a time series. This can be done explicitly by using a time based feature as an input to the network, or implicitly by using recurrent networks or networks with tapped delay lines (TDLs). Dynamic networks are preferred for TCM because tool wear is time dependent (tool wear is a monotonically increasing parameter that is partly a function of machining time). 26.5.2.1

Unsupervised Networks

There are two basic network paradigms for unsu pervised classifications, namely adaptive resonance theory (ART) and the self organizing map (SOM). ART is based on competitive learning, addressing the stability plasticity dilemma of NNs. The main advantage is its ability to adapt to changing conditions. ART networks also have self stability and self organization capabilities. The SOM is actually a data mining method used to cluster multidimensional data automatically. A high dimensional feature matrix can be displayed on a FIGURE 26.20 Schematic representation of the SOM. two dimensional grid of neurons that are arranged in clusters with similar feature values. Clusters for new and worn tools can be formed, and these are used for automatic classification of the tool condition. A SOM is depicted schematically in Figure 26.20. There are many practical advantages for using unsupervised networks. One is the fact that the machining operation is not interrupted for tool wear measurements during the training phase. There is also the advantage of practical implementation if machining conditions change very often and appropriate training samples for supervised learning cannot be collected. Furthermore, the numerous different combinations of tool and workpiece materials and geometries can make supervised learning impossible. Normally, unsupervised NNs are used to identify discrete wear classes and cannot be used for a continuous estimation of tool wear. Silva et al. (2000) investigated the adaptability of the SOM and ART for tool wear monitoring during turning with changing machining conditions. It was found that, with appropriate training, the methods have enough adaptive capabilities to be employed in industrial applications. Govekar and Grabec (1994) use the SOM for drill wear classification, where the SOM is used as a kind of empirical modeler. It was found that the adaptability of the SOM and its ability to handle noisy data makes the technique viable for on line TCM. Scheffer and Heyns (2000b, 2001b) showed how a TCMS can be adaptable using SOMs. Different network sizes were compared with define discrete classes of new and worn tools. Larger networks yielded more continuous results. The TCMS using SOMs was applied to monitoring synthetic diamond tools for an industrial turning operation. It was found that the SOM can be used for industrial applications, especially if tool wear measurements are not available.


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FIGURE 26.21 Unsupervised approach to wear monitoring with the SOM. (Source: Scheffer, C. and Heyns, P.S., South African Inst. Tribol. 2002. With permission.)

Different NN paradigms were compared on a wear monitoring application for aluminum turning by Scheffer and Heyns (2002b). It was shown that the SOM is useful to identify discrete wear classes, as shown in Figure 26.21. If an exact value of the tool wear is required, supervised networks will yield better results but will require proper training samples. 26.5.2.2

Supervised Networks

Common supervised NNs used for TCM are the multilayer perceptron (MLP), multilayer feedforward (FF) network, recurrent neural network (RNN), supervised neuro fuzzy system (NFS S), time delay neural network (TDNN), single layer perceptron (SLP) and the radial basis function (RBF) network. The use of an SLP for TCM is described by Dimla et al. (1996), using the perceptron learning rule for training. The SLP is useful to identify discrete classes of the tool condition. FF networks are usually trained with the backpropagation algorithm. However, backpropagation should not always be the preferred choice because other methods are known that outperform this technique in terms of training time and generalization. The size of the hidden layers in multilayer networks should be optimized for performance. Many contradictory statements about the use of MLP networks can be found in the literature. One of the main problems is the selection of the number of input features, size of the network, and the number of training examples that should be used. A multilayer feedforward (FF) network is shown schematically in Figure 26.22. Normally, a nonlinear activation function should be used in the first layer, and linear neurons in the subsequent layers. In the case of the FF networks, the backpropagation algorithm is often used for training. Backpropagation is an optimization algorithm based on steepest gradient descent. The use of FF networks with the backpropaga tion training rule is reported by authors such as Zhou et al. (1995), Das et al. (1996), and Zawada Tomkiewicz (2001). Cutting conditions can also be included in such networks. Lou and Lin (1997) describe the use of a FF network using a Kalman filter to avoid the training problems encountered with backpropagation for a TCM application. The proposed method is less sensitive to the network initializations that often cause convergence problems with backpropagation. Monitoring a dynamic system such as a cutting process should be done with a dynamic modeling technique such as dynamic NN paradigms, for example, recurrent networks, TDNNs, or explicit FIGURE 26.22 Multilayer FF network.

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FIGURE 26.23 Recurrent networks: feedback connection (left) and Elman network (right). (Source: Scheffer, C. and Heyns, P.S., South African Inst. Tribol. 2002. With permission.)

inclusion of temporal information in static networks. Recurrent NNs have feedback connections from their output to their input. There are various types of recurrent NNs that are useful for specific applications. Elman networks are quite interesting. Generally, they are two layer networks with feedbacks from the first layer output to the first layer input. This type of network can be used to learn and model temporal patterns. A recurrent network and an Elman network are shown schematically in Figure 26.23. Liu and Altintas (1999) report on the use of a FF network using a combination of TDLs and recurrent connections. Machining conditions are also included. It is stated that the system was integrated into an industrial TCMS, but was never put to use due to lack of “… robust, practical cutting force sensors …” (Liu and Altintas 1999). Scheffer and Heyns (2002b) report on the use of an Elman NN for TCM. It was found that the Elman network has a very smooth response and yielded better results than static NN paradigms. It should be mentioned that the Elman network requires more time for training, but because this is done off line, training time should not be a criterion for evaluating NNs. Neuro fuzzy systems (NFS S) attempts to combine the learning ability of NNs with the interpretation ability of fuzzy logic. A TCMS using an NFS S can be generated almost automatically because the fuzzy rules can be learned by the NN. A combination of supervised and unsupervised training is used for NFS S. An in process NFS S system to monitor tool breakage was designed and implemented successfully by Chen and Black (1997), concentrating on end milling operations. Xiaoli et al. (1997) as well as Chungchoo and Saini (2002) also propose some of the advantages of using an NFS S for TCM. RBF networks are often preferred because of the convergence properties of the training algorithm. In essence, convergence can be guaranteed and is often achieved much faster than in MLPs. The accuracy of RBFs depends on the choice of the centers for the basis functions, and should be treated with care. Pai et al. (2001) reported on the use of a resource allocation network (RAN) for TCM. The RAN is a RBF network utilizing sequential learning. The RAN is compared with the MLP for wear estimation during face milling. It was found that the RAN has faster learning ability but the MLP is more robust. TDNNs have delay elements in the feedforward connections, called TDLs. One advantage of TDNNs over RNNs is that stability problems are avoided. An investigation towards the inclusion of one and two phase delays for a TCM application was reported by Venkatesh et al. (1997). Different network sizes were also investigated, and it was found that the NNs with temporal memory generally perform better than those without memory. It is also stated that new algorithms should be investigated for training. Sick and Sicheneder (1997) also describe the use of TDNNs for TCM in turning. The TDNN is compared with the MLP and a significant improvement was found when using


26 19

TDNNs. In another instance, Sick et al. (1998) compare the SOM, NFS S, and MLP networks for wear estimation. The following critical questions are used to evaluate the different NN paradigms (Sick et al., 1998): *

*

*

Are the generalization capabilities of the NN sufficient (tested on previously unseen data)? What rate of correct classification can be achieved for different wear stages? Are the results repeatable (e.g., with a new initialization)?

In the case study presented by Sick et al. (1998), the best results were found with MLPs. It is stated, however, that the results can be improved when using TDNNs, and such results are reported in Sick (1998). A novel combined approach is suggested by Sick (1998) to handle the effect of machining parameters. An empirical model is used to normalize the data with respect to machining parameters before the data are entered into the NN. Thus, machining parameters are not included in the NN itself. This approach solves the extrapolation limitations encountered when an NN is tested with data recorded with machining parameters it was not trained with. Although many authors test their NNs’ paradigms in such a way, NNs cannot be expected to extrapolate. NNs should instead be tested with previously unseen data recorded with same machining parameters it was trained with (hence an interpolation effect). This is a problem because training and testing patterns for each condition must be supplied. However, if data can be normalized with respect to machining parameters, training is only required for the normalized condition. This was in effect achieved by Sick (1998). A difficulty still lies with establishing an appropriate model, and in many cases it will also require a large number of experimental tests. A possible solution lies in the incorporation of numerical models, for example, finite element models. Scheffer (2002) presented another approach to tool wear monitoring of turning operations, using a combination of static and dynamic NNs. Static networks are trained off line to model selected features from cutting forces. A dynamic NN that uses explicit temporal information is then trained on line with the particle swarming optimization algorithm (PSOA). The training goal of the dynamic NN is to minimize the errors between the outputs of the static NNs and the on line measurements. The method was tested on various turning operations and was also tested on an industrial shop floor. It was found that the method is more accurate and reliable than other NN paradigms and can be used with cost effective hardware (Scheffer and Heyns 2002a, 2004). The method is depicted schematically in Figure 26.24.

FIGURE 26.24 Combined static and dynamic NN approach for turning. (Source: Scheffer, C. and Heyns, P. S., Mech. Syst. Signal Process, Elsevier, 2004. With permission.)

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26.5.3

Fuzzy Logic

Many authors have investigated the use of fuzzy logic to classify tool wear. It has been shown that fuzzy logic systems demonstrate great potential for use in intelligent manufacturing applications. While NN models cannot directly encode structured knowledge, it is often stated that fuzzy systems can directly encode structured knowledge in a numerical framework. Additionally, fuzzy systems are capable of estimating functions of a system with only a partial description of the system’s behavior. Du et al. (2002) propose a very interesting method called transition fuzzy probability, which was applied to a boring operation. This formulation can deal with the uncertainty of process conditions. The method performs well because TCM has two uncertainties: that of occurrence and that of appearance. The transition fuzzy probability solves this issue through the use of temporal information, similar to dynamic NNs. The method was shown to outperform a backpropagation NN, although very few details are given. It would be interesting to compare this method with dynamic NNs such as TDNNs. Fu et al. (1997) combined force, vibration, and AE in a fuzzy classifier for TCM during milling. Time and frequency domain features were used, and it was found that combining the sensory information achieved the best result. This is done within the fuzzy classifier. Li and Elbestawi (1996) and Kuo and Cohen (1998) combine fuzzy modeling steps with NNs at different levels for TCM. The latter combined force, vibration and AE in a multisensor approach with satisfactory results.

26.5.4

Other Methods

There are also a number of other decision making and modeling methods that have been applied to TCM, and these include: *

*

*

*

Knowledge based expert systems (Du, 1999) Pattern recognition algorithms (Kumar et al., 1997) Dempster Shafer theory of evidence (Beynon et al., 2000) Hidden Markov models (Ertunc and Loparo, 2001; Ertunc et al., 2001)

Of these four approaches, only hidden Markov models have the potential possibly to outperform NNs and fuzzy systems. However, not enough comparable research has been conducted in this area, and is it certainly a worthwhile topic for future research.

26.6

Conclusion

Techniques for achieving TCM with vibration based properties were presented in this chapter. The sensing methods that have proved to be effective for TCM are force, acceleration, and AE. The sensors employed must comply with certain requirements such as robustness and cost effectiveness. Sensors must be installed as close as possible to the point of metal removal in order to avoid signal to noise ratio problems. Various techniques exist to condition and process the signals in analogue and digital formats. The aim of signal processing is to generate wear sensitive features from the vibration signals. This could be done by time, frequency, joint time frequency, and statistical analysis. Feature selection can be automated with a variety of procedures, but care must be taken when using these to avoid selection of linearly dependent data. The selected features can be used to establish a model of tool wear. Numerous research papers have shown that NNs should be used due to the many advantages of NN modeling. The training and testing procedures of NNs are of utmost importance if the system is considered for industrial implementation. Care must be taken not to overtrain the networks because they will lose their ability to generalize. Furthermore, NNs cannot be expected to perform well if they are tested with previously unseen machining parameters. They should also be trained with the minimum and maximum tool wear that is


26 21

expected. Future work should be directed towards incorporating numerical machining models into the wear monitoring system to normalize the data with respect to machining parameters. If this can be achieved, the amount of training data required for an effective TCMS will be reduced, which in turn will provide a better solution to TCM for the manufacturing industry.

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26 23

Li, S. and Elbestawi, M.A., Fuzzy clustering for automated tool condition monitoring in machining, Mech. Syst. Signal Process., 10, 533 550, 1996. Li, X.Q., Wong, Y.S., and Nee, A.Y.C., Tool wear and chatter detection using the coherence function of two crossed accelerations, Int. J. Mach. Tools Manuf., 37, 425 435, 1997. Liu, Q. and Altintas, Y., On line monitoring of flank wear in turning with multilayerd feed forward neural network, Int. J. Mach. Tools Manuf., 39, 1945 1959, 1999. Lou, K. and Lin, C., An intelligent sensor fusion system for tool monitoring on a machining centre, Int. J. Adv. Manuf. Technol., 13, 556 565, 1997. Luo, G., Osypiw, D., and Irle, M. 2002. Tool wear monitoring by on line vibration analysis with wavelet algorithm. In Metal Cutting and High Speed Machining, pp. 393 405, Kluwer Academic/ Plenum Publishers, New York. Mdlazi, L., Marwala, T., Stander, C.J., Scheffer, C., and Heyns, P.S. 2003. The principal component analysis and automatic relevance determination for fault identification in structures. In Proceedings of the 21st International Modal Analysis Conference (IMAC), Kissimmee, FL, Paper 37. Merchant, M.E., Mechanics of the cutting process, J. Appl. Phys., 16, 318 324, 1945. Morimoto, Y., Ichida, Y., and Sata, R. 2000. Excitation technique by 2 axes shaker of an CNC lathe, pp. 1643 1648. In Proceedings of the 18th International Conference on Modal Analysis, San Antonio, TX. Pai, P.S., Nagabhushana, T.N., and Rao, P.K.R., Tool wear estimation using resource allocation network, Int. J. Mach. Tools Manuf., 41, 673 685, 2001. Ruiz, A., Guinea, D., Barrios, L.J., and Betancourt, F., An empirical multi sensor estimimation of tool wear, Mech. Syst. Signal Process., 7, 105 199, 1993. Scheffer, C. 1999. Monitoring of tool wear in turning operations using vibration measurements, Masters dissertation (MEng), Department of Mechanical and Aeronautical Engineering, University of Pretoria, South Africa. Scheffer, C. 2002. Development of a tool wear monitoring system for turning using artificial intelligence, Ph.D. thesis, Department of Mechanical and Aeronautical Engineering, University of Pretoria, South Africa. Scheffer, C. and Heyns, P.S. 2000a. Synthetic diamond tool wear monitoring using vibration measurements, pp. 245 251. In Proceedings of the 18th International Modal Analysis Conference, San Antonio, TX, 7 10 February. Scheffer, C. and Heyns, P.S. 2000b. Development of an adaptable tool condition monitoring system, pp. 361 370. In Proceedings of the 13th International Congress on Condition Monitoring and Diagnostic Engineering Management (COMADEM 2000), Houston, TX, 3 8 December. Scheffer, C. and Heyns, P.S. 2001a. Tool condition monitoring systems an overview, pp. 316 323. International Conference on Competitive Manufacturing (COMA ’01), Stellenbosch, South Africa, 31 January 2 February. Scheffer, C. and Heyns, P.S., Wear monitoring in turning operations using vibration and strain measurements, Mech. Syst. Signal Process., 15, 1185 1202, 2001b. Scheffer, C. and Heyns, P.S. 2002a. A robust and cost effective system for conducting cutting experiments in a production environment, pp. 329 334. In Proceedings of 3rd CIRP International Conference on Intelligent Computation in Manufacturing Engineering (ICME 2002), Ischia (Naples), Italy, 3 5 July. Scheffer, C. and Heyns, P. S. 2002b. Neural Network approaches for sensor based tool wear monitoring, In Proceedings of Metalworking Tools & Fluids, South African Institute of Tribology, Johannesburg, South Africa, 7 November. Scheffer, C. and Heyns, P. S., An industrial tool wear monitoring system for interrupted turning, Mech. Syst. Signal Process., 18, 1219 1242, 2004. Scheffer, C., Kratz, H., Heyns, P.S., and Klocke, F., Development of a tool wear monitoring system for hard turning, Int. J. Mach. Tools Manuf., 43, 973 985, 2003.

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Sick, B., On line tool wear monitoring in turning using neural networks, Neural Comput. Appl., 7, 356 366, 1998. Sick, B., Online and indirect tool wear monitoring in turning with artificial neural networks: a review of more than a decade of research, Mech. Syst. Signal Process., 16, 487 546, 2002. Sick, B. and Sicheneder, A. 1997. Time delay neural networks for on line tool wear classification and estimation in turning, pp. 461 466. In Proceedings of the Third Conference on Neural Networks and Their Applications, Kule, Poland, 14 18 October. Sick, B., Sicheneder, A., and Lindinger, H. 1998. A comparative evaluation of different neural network paradigms for tool wear classification in turning, pp. 139 146. In Proceedings of the 3rd International Workshop Neural Networks in Applications (NN ’98), University of Magdeburg, Germany, 12 13 February. Silva, R.J., Rueben, R.L., Baker, K.J., and Wilcox, S.J., Tool wear monitoring of turning operations by neural network and expert system classification of a feature set generated form multiple sensors, Mech. Syst. Signal Process., 12, 319 332, 1998. Silva, R.J., Baker, K.J., Wilcox, S.J., and Reuben, R.L., The adaptability of a tool wear monitoring system under changing cutting conditions, Mech. Syst. Signal Process., 14, 287 298, 2000. Tarmal, G.J. and Opavsky, P. 2000. Signal processing in measurement of milling forces, pp. 389 397. In Proceedings of the 13th International Congress on Condition Monitoring and Diagnostic Engineering Management (COMADEM 2000), Houston, TX. Teti, R., A review of tool condition monitoring literature database, Ann. CIRP, 44, 659 667, 1995. Venkatesh, K., Zhou, M., and Caudill, R.J., Design of artificial neural networks for tool wear monitoring, J. Intell. Manuf., 8, 215 226, 1997. Warnecke, G. and Siems, S. 2002. Dynamics in high speed machining. Metal Cutting and High Speed Machining, pp. 21 30, Kluwer Academic/Plenum Publishers, New York. Xiaoli, L., Yingxue, Y., and Zhejun, Y., On line tool condition neural network with improved fuzzy neural network, High Technol. Lett., 3, 30 38, 1997. Yao, Y., Fang, X.D., and Arndt, G., Comprehensive tool wear estimation in finish machining via multivariate time series analysis of 3 D cutting forces, Ann. CIRP, 39, 57 60, 1990. Zawada Tomkiewicz, A., Classifying the wear of turning tools with neural networks, J. Mater. Process. Technol., 109, 300 304, 2001. Zhou, Q., Hong, G.S., and Rahman, M., A new tool life criterion for tool condition monitoring using a neural network, Eng. Appl. Artif. Intell., 8, 579 588, 1995.

27

Fault Diagnosis of Helicopter Gearboxes 27.1 27.2 27.3 27.4

Introduction ........................................................................ Abnormality Scaling ........................................................... The Structure Based Connectionist Network .................. Sensor Location Selection ..................................................

Coverage Index Effectiveness

†

Overlap Index

†

Monitoring

27 1 27 5 27 8 27 11

27.5 A Case Study ....................................................................... 27 14

Kourosh Danai University of Massachusetts

Structural and Featural Influences † Evaluation of Influences Fault Detection Results † Fault Diagnostic Results † Sensor Location Evaluation † Sensor Location Validation

†

27.6 Conclusion .......................................................................... 27 23

Summary An overview of fault detection and diagnosis of helicopter gearboxes is presented in this chapter. Between oil analysis and vibration monitoring, the two predominant methods of gearbox monitoring, vibration monitoring is much more studied and relied upon because of its ability to reflect a wider variety of faults. In order to cover the concepts used in vibration monitoring, a method of diagnosis is discussed that relies on the knowledge of the gearbox structure and characteristics of the features of vibration for component fault isolation. Since a necessary part of vibration monitoring is selection of accelerometer locations on the housing, a method is also described whereby the suitability of accelerometer locations is quantified based on the structure of the gearbox and the monitoring effectiveness of the locations. Finally, a case study is included to illustrate the application of the concepts presented in the chapter to fault detection and diagnosis, as well as the ranking of suites of accelerometer locations. Some topics related to this chapter are covered in Chapter 15, Chapter 25, and Chapter 26.

27.1

Introduction

Present helicopter power trains are significant contributors to both flight safety incidents and maintenance costs. For example, for large and medium civil transport helicopters in the period 1956 to 1986, gearboxes were the principal cause of 22% of the accidents with potential loss of life and aircraft (Astridge, 1989). To prevent such incidents, routine maintenance is scheduled at a significant ratio of the total maintenance cost for the helicopter. Rapid and reliable detection and diagnosis (isolation) of faults1 in helicopter gearboxes is therefore necessary to prevent major breakdowns due to progression of undetected faults, and for enhancing personnel safety by 1

Determining whether the overall machinery is healthy is referred to as fault detection, whereas diagnosis is analogous to isolating the source of failure.

27 1

27 2


preventing catastrophic failures. Fault detection and diagnosis is also necessary for reducing maintenance costs by eliminating the need for routine disassembly of the gearbox, and for saving time during inspection. Detection and diagnosis of helicopter gearbox faults, like most rotating machinery, is based on oil analysis and vibration monitoring. Oil analysis, which is used to detect the presence of metallic debris, is performed by: (1) magnetic plugs (chip detectors), (2) oil filters, and (3) spectrometric oil analysis (SOA) (Lurton, 1994). Among these, magnetic plugs are the most popular because of their in flight utility and their ability to quantify the severity of wear by measuring the rate of detected debris. However, magnetic plugs can only collect ferromagnetic particles, and their capture efficiency may be poor. Oil filters used in helicopter gearboxes range between 3 and 150 mm, with finer meshes of 3 mm more common in the recent years. SOA, which is a ground based technique, is useful for detecting fine debris, typically below 10 mm in size, caused by wear conditions such as rubbing, cutting, and corrosion wear, and fine surface fatigue such as micropitting. Therefore, development of a reliable method of identifying common forms of fatigue such as spalling or wear that generate particles greater than 10 mm in size is of particular interest. The effectiveness of SOA is also affected by the level of filtering performed on the used oil, which often leaves the oil free of particles (Lurton, 1994). The need for improving wear detection has motivated development and use of other analysis methods such as image processing of oil filters, ferrography, thermography, and ultrasonic analysis (Thornton, 1994). The other method of gearbox fault detection and diagnosis, and by far the more popular one, is based on vibration monitoring. The basic principle behind vibration monitoring is that, under normal operating conditions, each component in the gearbox produces vibrations at specific frequencies related to the component’s rotational frequency. In the case of a component fault, the vibration generated by the faulty component is different from the normal vibration, and will be reflected at the component’s rotational frequency and its harmonics. As such, monitoring the changes of vibration should theoretically give an indication of the fault. In practice, however, changes in the measured vibration as a Sun Planet Gear(SG) Bearing(PB)

Ring Gear(RG)

Gear Roller Bearing(GRB)

Mast Ball Bearing(MBB)

Planet Gear(PG)

Spiral Bevel Gear(SBG) Spiral Bevel Pinion(SBP) Triplex Bearing(TB)

Duplex Bearing(DB)

FIGURE 27.1

Mast Roller Bearing(MRB)

Pinion Roller Bearing(PRB)

Layout of various components in a typical helicopter (OH 58A) gearbox.

Fault Diagnosis of Helicopter Gearboxes

27 3

result of component faults are not always distinct due to the attenuation of vibration by the housing and other components it travels through, as well as the noise in the signal. To provide a more tangible framework for the concepts in vibration monitoring, the layout of a typical helicopter gearbox is shown in Figure 27.1, with the location of the accelerometers on its test stand shown in Figure 27.2. In order to enhance identification of vibration changes by component faults, the raw vibration is generally processed to obtain “features” that characterize the vibration at the frequencies associated with the gearbox components. Accord ingly, the main focus of research in vibration monitoring has been the identification of individ ual features that consistently reflect specific gear box faults (Dyer and Stewart, 1978; McFadden and Smith, 1986; Mertaugh, 1986; Zakrajsek et al., 1995). A typical set of vibration features obtained from each vibration measurement is shown in Figure 27.3 (Stewart Hughes Ltd., 1986). Among them, envelope band and tone energies, cepstra, and synchronous time averaged signals, are associ FIGURE 27.2 Location and orientation of the accel ated with various bearing frequencies and gear erometers on the OH 58A test stand. mesh tones. Envelope band and tone energies, which are the sum of the harmonics of the Tape Recorder bearings’ fundamental rolling element frequencies Digitization / within a filtered bandwidth, could be used for early Processing detection of failure in rolling element bearings (Barkov and Barkova, 1995). Similarly, a cepstrum, STAT BBPS BRGA SGAV which is the power spectrum of the logarithm of a power spectrum, is often used because of its (16) BE For each of the 5 Gears (5) RMS (1) Skewness (17) BKV (6) WHT (1) MF1 (2) Kurtosis insensitivity to transmission path effects for (18) EB (2) FM4a (3) Crest Factor (7) RFR identification of families of uniformly spaced (8) TEO-G (19) ET (3) FM4B (4) Peak-to-Peak (9) TEO-P (4) FM4-B sidebands in gearbox vibration spectra (Randall, (10) TM1-G (5) ACH 1982). Synchronous time averaging is a signal (11) TM1-P (6) WCH (12) CEP(1911) (7) SCH processing technique that isolates the fundamental (13) CEP(572) (14) TON(1911) and harmonics of the gear meshing frequency, and (15) TON(572) is a primary analysis technique for detection of gear and shaft faults (McFadden and Smith, 1986). FIGURE 27.3 A typical set of vibration features The traditional approach to fault diagnosis has extracted from each accelerometer. relied on human expertise to relate vibration features to faults. In this approach, a diagnostician first identifies abnormalities in vibration features, then relates them to component faults, considering both the proximity of the accelerometer producing the feature to various components and the information about the type of fault characterized by the feature. Using this information, the diagnostician hypothesizes faults in specific components and then verifies or discards the hypothesis by examining features from other accelerometers in the proximity of the suspect component. The disadvantages of this approach arise from: (1) the difficulty in identifying abnormality in features which are contaminated with noise, and (2) the tediousness of examining the numerous features obtained from all of the accelerometers. Owing to the large number of features

27 4


Raw Vibration

associated with a gearbox2, the diagnostician Fault Detection cannot often pay equal attention to all the features and is likely to ignore information that contradicts Signal Abnormality Diagnostic the hypothesis. Processing Scaling features The most efficient method for integration of features is pattern classification. To this end, FIGURE 27.4 Generic structure of an unsupervised artificial neural networks have been widely inves diagnostic system. tigated and shown to provide excellent results (Solorzano et al., 1991; Chin et al., 1993; Kazlas et al., 1993; Chin et al., 1995). Neural networks offer the following advantages in diagnosis: (1) as pattern classifiers, they can efficiently cope with noise in features, (2) through training, they can form the signatures of individual faults in the multidimensional space of features, and (3) they can process vibration features in parallel, so diagnosis is not hindered by the enormity of feature space. The main disadvantage of supervised neural networks, however, is their need for prior training, which requires a comprehensive set of features during normal operation and at various fault instances. While training data can be obtained experimentally through accelerated fatigue tests (Lewicki et al., 1992) or seeded fault studies (Naval Command, Control, and Ocean Surveillance Center, 1995), their cost is considered too high, limiting the utility of supervised neural networks in fault diagnosis. The lack of training data poses a similar restriction for statistical pattern classifiers which need a priori statistics of the features. In the absence of training data required by supervised neural networks, the monitoring system can be formatted as in Figure 27.4 to take advantage of unsupervised pattern classification. The strategy depicted in Figure 27.4 considers fault detection independent of abnormality scaling of features, so it can be performed before or in parallel to this stage, based on the overall deviation of all of the features from their normal state. An example of abnormality scaling based on unsupervised pattern classification is described in the Section 27.2. For fault diagnosis, the proposed system should have detailed information about the relation between the individual features and component faults. One format for providing that information is expert diagnostic software, which can be developed at two different levels. At one level, “shallow expert systems” can be developed to compile a human diagnostician’s knowledge relating measurements to faults, often as “if …then” rules, similar to those already developed for simpler rotating machinery (Liddle and Reilly, 1993). The main obstacle in such a development will be the design of a robust inference engine that can resolve conflicting conclusions from the large pool of suspect features. At another level, “deep expert systems” can be developed to represent the diagnostic knowledge derived from the physics of the process. In deep expert systems, measurements need to be related to component faults by a model of the energy flow via the structural connections between components and sensors. A suitable format for defining the relation between components and sensors is the fuzzy set theory, which can address the approximate nature of vibration modeling. An example of a deep expert diagnostic system for helicopter gearboxes is the structure based connectionist network (SBCN) (Jammu et al., 1998) described in Section 27.3, which takes advantage of the integration capability of neural networks while avoiding the need for supervised training. The salient feature of this connectionist network is that its weights can be determined a priori, based on the proximity of components to various accelerometers and the type of faults characterized by individual features. Although this system has been developed on a very approximate model of vibration flow between the gearbox components and accelerometers, it has produced promising results when used for fault diagnosis of helicopter gearboxes (Jammu et al., 1998). A side benefit of a model of the structural connections between gearbox components and sensors is its use in sensor location selection (Wang et al., 1999). An important issue in helicopter gearbox diagnostics is the determination of the number of accelerometers to be used for monitoring and their location on the gearbox housing. Accelerometers are generally located by experts, based on their 2

The large number of frequencies (tones) associated with the various components of the gearbox necessitates a huge number of features (often in excess of a hundred) to be obtained for vibration readings from each of the several accelerometers.


27 5

proximity to gearbox components, orientation, and ease of mounting on the housing. However, this approach often leads to too many accelerometers and too high a demand for on line monitoring on the on board computer. Another problem with the extra accelerometers is the cost of extra mountings, cabling, and signal conditioning equipment. Sensor location selection based on a gearbox model is discussed in Section 27.4.

27.2

Abnormality Scaling

Vibration Features

Abnormality-Scaled Features

The abnormality scaling module described here Weight relies solely on the value of each feature during normal operation. Accordingly, it is referred to as the single category based classifier (SCBC) to signify its independence from feature values associated Matching with faulty conditions (Jammu et al., 1996). In order to perform abnormality scaling, the SCBC compares features with their normal values, and if they are sufficiently different, assigns values between zero and one to characterize their degree of deviation from their normal values. The Abnormality-Scaling schematic of the SCBC, which is implemented by a connectionist network, is shown in Figure 27.5. FIGURE 27.5 Schematic of the SCBC network. The inputs to SCBC are the raw features from signal processing, si ðtÞ; i ¼ 1; …; n; and its outputs are abnormality scaled features, fi ðtÞ; with values between zero and one. The value of zero indicates normality, and the other extreme of one denotes complete abnormality. The individual weights of the SCBC network, wi ; represent the normal values of the features, which are initially set equal to the first corresponding feature value supplied to the SCBC. Classification in the SCBC is performed by first measuring the Euclidean distance of each feature, si ðtÞ; from its weight value, wi ; and normalizing it into the range ½0; 1 using a matching factor, fi ; defined as (Figure 27.6)

fi ðtÞ ¼ 1

exp

ðsi ðtÞ wi

wi Þ

2

ð27:1Þ

A fi value of zero indicates that the feature value matches the weight value precisely, and a value of one denotes that it deviates from it considerably. Note that the exponential function used here is not unique, and that other functions that can map the Euclidean distance into the range ½0; 1 can also be used for the matching factor. Since during normal operation of the gearbox, noise in the features usually causes them Matching

Abnormality-Scaling

1

1 Normal Region FIGURE 27.6

Matching and abnormality scaling in SCBC.

fmin

27 6


to deviate from their normal values, a threshold, u, is considered to account for deviations by noise. The threshold, u, is used to hard limit fi ðtÞ in SCBC as ( 0 ðnormal Þ if fi ðtÞ , u ð27:2Þ fi ðtÞ ¼ otherwise fi ðtÞ ðhard limited Þ In the above relationship, the threshold u is obtained as

u¼

n 1 X 1 n i¼1

exp

½maxðsi Þ m2i

mi

2

! ð27:3Þ

where maxðsi Þ denotes the maximum value of the ith feature in a set of k samples of this feature recorded during normal operation, and mi represents its mean, estimated as

mi ¼

k 1 X s ðtÞ k t¼1 i

ð27:4Þ

The matching factor, defined by Equation 27.1, suppresses any positive value in ½0; 1 into the range ½0; 1 : As such, only very large deviations in the feature values will be scaled to the value of one. Since large deviations in feature values are uncommon for gearboxes, the value of matching factor is further scaled to yield abnormality scaled feature values fi ðtÞ as (Figure 27.6) fi ðtÞ ¼ fmin þ expða p fi ðtÞÞ

ð27:5Þ

where fmin represents the minimum abnormality value assigned to any feature that violates the threshold, u, and a controls the slope of the exponential curve. Since fi ðtÞ is defined to have a value between zero and one, it is set to one when fi ðtÞ in Equation 27.5 exceeds the value of one. After each round of classification of the vibration features, the weight values in the SCBC are updated so as to cope with noise and small variations in the operating conditions. Adaptation is carried out in two stages. In the first stage, called primary adaptation, a network weight is adapted if the feature associated with it is classified as normal. In the second stage, referred to as contrast enhancement (CE) (Carpenter and Grossberg, 1987), the rest of the weights are adapted to achieve homogeneity in the abnormality scaled values, thus increasing the likelihood of all of them being classified as normal or abnormal. Homogeneity, however, needs to exist only within specific feature groups, because gearbox faults do not necessarily cause abnormality in all of the features. For example, a gear fault will be reflected only by the features related to the gear, and is not expected to cause abnormality in bearing features. In order to preserve the functionality of individual feature groups (i.e., general features, gear features, and bearing features), CE is performed exclusively for each feature group. Adaptation in SCBC is performed as follows. Let wI represent the weight which is presently being updated and wi the remaining weights in the group. In primary adaptation, the weight value wI is modified according to the relationship wI ¼ wI þ dwI

ð27:6Þ

where ( dwI ¼

h½sI ðtÞ 0

wI

if fI ðtÞ ¼ 0 otherwise

ð27:7Þ

with the parameter h denoting the learning rate. For CE, if the majority of features are classified as normal, then the weight values associated with the features classified as abnormal will be adjusted such that the likelihood of all of the features being


27 7

classified as normal is increased for the same feature. CE is performed as wi ¼ wi þ dwi

for all i – I

ð27:8Þ

where (

hL½si ðtÞ

dwi ¼

wi

hL½si ðtÞ

wi

if fI ðtÞ ¼ 0 otherwise

ð27:9Þ

In CE, the amount by which the weight values are adjusted is controlled by a neighborhood function, L (Kohonen, 1989), which is assigned a value between zero and one. A value of zero is used for inputs with no noise, and a value at the other extreme of one is used for unreliable features with large amounts of noise. Usually, in practice, the value of L is set less than 0.5. For each round of primary adaptation (Equation 27.6 and Equation 27.7), I is varied to include all the features in the group. If the jth group of features contains mj features, then primary adaptation is applied by varying I from one to mj ; to cover all the weight values wI in the jth feature group. For each I, the remaining weight values, wi ; in the group (i ¼ 1 to mj and i – I) are adapted using CE according to Equation 27.8 and Equation 27.9. The adaptation algorithm presented in Equation 27.6 to Equation 27.9 is biased towards the most recent feature vector if only this vector were used for adaptation, whereas adaptation should be ideally performed using all of the feature vectors that pertain to the current operating conditions. However, as the number of available feature vectors for the operating condition progressively increases, adaptation based on all of the features becomes computationally demanding. As a compromise, in SCBC only the b most recent feature vectors are utilized for each adaptation sweep, such that adaptation is performed iteratively over the b most recent feature vectors. The learning rate, h, is progressively reduced for each adaptation iteration (Equation 27.7 and Equation 27.9). Abnormality scaling formulae are summarized in Table 27.1. TABLE 27.1

Summary of the Abnormality Scaling Formulae

Classification in the single category based classifier is performed by first measuring the Euclidean distance of each vibration feature, si ðtÞ; from its weight value wi ; and normalizing it into the range ½0; 1 using a matching factor, fi ; defined as

fi ðtÞ ¼ 1 2 exp

2ðsi ðtÞ 2 wi Þ wi

2

The matching factor is then hard limited by a threshold, u, as (

fi ðtÞ ¼

0

if fi ðtÞ , u

ðnormalÞ

fi ðtÞ ðhard limitedÞ otherwise

In the above relationship, the threshold, u, is obtained as

u¼

n 1 X 2½maxðsi Þ 2 mi 1 2 exp n i 1 m2i

2

!

where maxðsi Þ denotes the maximum value of the ith feature in a set of k samples of this feature, recorded during normal operation, and mi represents its mean, estimated as

mi ¼

k 1 X s ðtÞ kt 1 i

The vibration feature is then abnormality scaled as fi ðtÞ ¼ fmin þ expða p fi ðtÞÞ where fmin represents the minimum abnormality value assigned to any feature that violates the threshold, u, and a controls the slope of the exponential curve.

27 8

27.3


The Structure-Based Connectionist Network

The SBCN (Jammu et al., 1998) is developed to take advantage of the integration capability of neural networks, but to avoid the need for supervised training. It defines the weights of the network according to the structural knowledge of the gearbox and the type of fault represented by various vibration features. In the SBCN, the structural influences, which represent the proxi mity effect of faults on accelerometers, are determined based on the root mean square (RMS) value of the frequency response of a simplified lumped mass model of the gearbox. Fault diagnosis in SBCN is performed by propagating the abnormality scaled vibration fea tures from SCBC to produce outputs representing the fault possibility values for each gearbox component as (Figure 27.7) n X pk ðtÞ ¼ fi ðtÞvik ð27:10Þ

Abnormality-Scaled Features

Fuzzy Influence Weights

lik

uik

Faulty Components FIGURE 27.7 The structure based connectionist net work (SBCN) with its fuzzy weights for isolating faulty components.

i¼1

In the above equation, pk ðtÞ represents the fault possibility value for the kth component of the gearbox, fi ðtÞ denotes the abnormality scaled value of a feature and vik represents the weighting factor determined based on the lower and upper bounds of fuzzy influence weights ðlik ; uik Þ as vik ¼ lik þ ðuik

lik Þfi ðtÞ

ð27:11Þ

Note that Equation 27.10 represents propagation of inputs through the weights of the SBCN, similar to a regular connectionist network with vik as weights (Hertz et al., 1991). The difference is that the weights of the SBCN vary within the range ðlik ; uik Þ according to the magnitude of the corresponding input, fi ðtÞ: According to Equation 27.11, a higher abnormality scaled feature value produces a higher weight value, vik ; emulating the reasoning by the human expert who pays more attention to the features that exhibit higher abnormality values. In SBCN, in order to make uniform the interpretation of the fault possibility values pk ðtÞ; they are normalized to have values between zero and one as pk ðtÞ ck ðtÞ ¼ X n uik

ð27:12Þ

i¼1

Accordingly, a ck ðtÞ equal to one denotes a definite fault, whereas a value of zero represents normality. In this system, fault diagnosis is performed hierarchically. First, the faulty subsystem within the gearbox is identified by using the structural influences as vik : Then, the faulty components within the suspect subsystem(s) are isolated using the product of structural and featural influences as vik : The connection weights of the SBCN are defined based on the influences between the component faults and vibration features. Ideally, the structural influences should represent the strength of component vibration at the particular frequency (frequencies) represented by the feature. For this, the attenuation property of the “travel path” between each component and accelerometer needs to be modeled as a function of the moment of inertia, stiffness, and damping of the components in the path (Badgley and Hartman, 1974; Smith, 1983; Lyon, 1995). In order to appreciate the difficulties associated with vibration modeling of gearboxes, the vibration model of a simple gearbox is considered


(Choy and Qian, 1993): _ v ½X_ þ ½GA ½X þ ½Cb ½X_ ½M ½X€ þ ½G

27 9

X_ c þ ½Kb ½X

Xc þ ½Ks ½X

Xr ¼ ½FðtÞ þ ½FG ðtÞ ð27:13Þ

where X represents the generalized displacement vectors in the lateral x, y, and z, and rotational ux ; uy ; and uz directions, M denotes the inertia matrix, Gv represents gyroscopic forces, GA denotes the rotor angular acceleration, Cb and Kb represent the damping and stiffness matrices of bearings, respectively, ½X Xc denotes the casing vibration, ½X Xr represents the shaft residual bow, and Ks denotes the shaft bow stiffness matrix. The excitation force [F(t)] is due to mass imbalance, and [FG(t)] represents the nonlinear gear mesh force, which in the x direction has the form (Choy and Qian, 1993) FGxk ¼

n X i¼1;i–k

þ ðYci

Ktki ½ Rci uci

Rck uck þ ðXci

Xck Þ cosðaki Þ

Yck Þ sinðaki Þ ½cosðaki Þ þ SIGNðmÞ sinðaki Þ

ð27:14Þ

where FGxk denotes the gear mesh force in the x direction on the (n 1)th gear due to its mesh with (n 1) other gears, Ktki represents the nonlinear gear mesh stiffness between the kth gear and ith gear, Rci denotes the radius of the ith gear, aki represents the orientation angle between the kth and ith gears, and m denotes the coefficient of friction. Mathematical relationships similar to Equation 27.14 can be defined to represent the force in the y direction as well as torsional gear mesh forces. Furthermore, the vibration of the casing due to the vibration of the gear shaft system needs to be represented by a separate set of coupled equations of motion similar to Equation 27.13. The above equations need to be integrated numerically in order to estimate the vibration signal recorded on the housing, but the following difficulties exist. (1) The values of stiffness and damping coefficients for the components are not readily available. (2) The cross coupling terms in the stiffness matrices in the x, y, and z directions cannot be easily defined (Mitchell and Davis, 1985; Choy and Qian, 1993). (3) It is difficult to take into account the multitude of travel paths and the associated models of attenuation for the many component accelerometer pairs in the gearbox. For example, Ktki, the gear mesh stiffness, which is obtained by considering the gear tooth as a nonuniform cantilever beam (Lin et al., 1988; Boyd and Pike, 1989), is a function of the cross section of the tooth at the point of loading as well as load variation due to changes in the direction of load application (Mark, 1987; Lin et al., 1988; Choy and Qian, 1993), friction between the meshing teeth (Rebbechi et al., 1991), contact ratio (Cornell and Westervelt, 1978), the type of gears (spur, helical, etc.) (Mark, 1987; Lin et al., 1988; Boyd and Pike, 1989), and gear errors such as profile, transmission, and manufacturing errors (Smith, 1983; Mark, 1987). Similarly, the stiffness of bearings is a time varying, nonlinear function of bearing displacement and the number of rolling elements in the load zone, as well as the bearing type (roller, ball, etc.), axial preload, clearance, and race waviness (Harris, 1966; While, 1979; Walford and Stone, 1983). All these factors make it very difficult to obtain an accurate and computationally inexpensive vibration attenuation model for gearboxes. In order to avoid the difficulties associated with accurate modeling of vibration transfer, a simplified method is described here that accounts separately for the two main aspects of vibration change by faulty components: (1) the proximity effect of the faulty component on the accelerometer generating the feature (structural influence), and (2) the frequency related information represented by the feature (featural influence). The main simplification in this method is in the definition of structural influences as the average strength of the vibration signal across all frequencies measured by an accelerometer due to a component fault. To compute this average vibration, several simplifications have been adopted: 1. A lumped mass model of the gearbox is used to model vibration. 2. Only the average static values for the stiffness coefficients are used, to cope with the absence of accurate values for stiffness coefficients. 3. Damping ratios of bearings and shafts are neglected (Lin et al., 1988). 4. A damping ratio of 0.1 is used for all the gears as an approximation to actual gear ratios estimated between 0.03 and 0.17 (Kasuba and Evans, 1981).

27 10


5. The cross coupling terms in the stiffness matrix are neglected. 6. Only the shortest vibration travel path between each component accelerometer pair is considered. Using the above simplifications, the average vibration registered by an accelerometer due to a faulty ith component can be simulated by applying an excitation source y at the ith component in the lumped mass model (Figure 27.8). In order to represent all frequencies in the excitation source, y can be selected to consist of unit amplitude sine waves of all frequencies. The displacement of various components in the travel path due to an excitation exerted at the ith component can be obtained for a typical N mass path as (James et al., 1994) 2 32 3 2 3 a11 a12 0 ··· 0 x1 y 6 76 7 6 7 6a 7 6 7 6 ··· 0 76 x2 7 6 0 7 6 21 a22 a23 6 76 7 6 7 ð27:15Þ 6 . 6 7¼ .7 .. .. .. 7 6 . 76 .. 7 6 .. 7 7 6 . 76 . 7 6 . . · · · . 4 54 5 4 5 0

···

0

aN

0

xN

aNN

N 1

where ½x1 ; x2 ; · · ·; xN T represent the displacements of the N components in the path, y denotes the magnitude of excitation at the first component, and the coefficients aij are defined as ¼

jvcn

annþ1 ¼

jvcn

ann

1

kn

1

1

ann ¼

mn v2 þ jvðcn

1

þ cn Þ þ ðkn

1

þ kn Þ

kn

ð27:16Þ

In the above equation, mn denotes the mass of the nth component, kn and cn represent the stiffness and damping coefficients between the nth and (n þ 1)th components, respectively, and v denotes frequency. In SBCN, the average vibration representing the overall vibration transferred from the component to the accelerometer is characterized by the RMS value of vibration across all frequencies. RMS values of vibration are readily obtained from Equation 27.15 by numerical integration of the square of displacements across all frequencies. In these calculations, to avoid unnecessary numerical problems at the natural frequencies of the components with negligible damping, the integration is carried out excluding the natural frequencies. For the purpose of assigning structural influences, the RMS values are scaled so that the component directly adjacent to the accelerometer has the highest influence. Different functions can be used for defining the influences. For example, Ii can be defined as Ii ¼

logðri Þ logðrN Þ

ð27:17Þ

where ri represents the RMS value of vibration with the excitation source at the ith component, and rN denotes the RMS value when the excitation is at the Nth component. In both cases, the accelerometer is considered at the Nth component. The influence for the other components is obtained in a similar fashion, by moving the excitation source to them in the travel path. The RMS values, ri, are only approximate estimates due to the simplifications made for their computation. Such approximate RMS values would, in turn, result in approximate influences. To yi k0

k1 m2

m1 c0 FIGURE 27.8

k2

c1

Sensor k3

kN-1

mi c2

mN c3

cN-1

Illustration of the lumped mass model of a vibration travel path consisting of N masses.


27 11

characterize the approximate nature of influences, they can be defined as fuzzy variables (Zadeh, 1975) by mapping the influences into the range associated with fuzzy variables, such as nil: (0, 0.1), low: (0.1, 0.4), medium: (0.4, 0.6), high: (0.6, 0.9) and definite: (0.9, 1). Another body of knowledge commonly used by diagnosticians is that of the type of fault represented by a feature. To incorporate this knowledge, fuzzy featural influences can be defined according to the relation between the frequency content of each feature and the rotational frequencies of various components (McFadden and Smith, 1986; Stewart Hughes Ltd., 1986). For example, a feature such as envelope band energy (BE), which represents the energy at the bearings rotational frequencies and harmonics, is assigned a featural influence of “high” in relation to bearing faults. The SBCN is designed to provide fault possibility values for gearbox components without any prior training. However, its design does not preclude the possibility of training when confronted with misclassifications, which are in the form of undetected faults, false alarms, and misdiagnoses. Among these, undetected faults are safety hazards that should be avoided at all costs, and false alarms and misdiagnoses, although not as crucial as undetected faults, should be minimized so as to improve the reliability of the diagnostic system. One of the features of the SBCN is its ability to benefit from connectionist learning (Hertz et al., 1991) to improve diagnostic performance after each misdiagnosis. For this purpose, an error minimizing adaptation algorithm can be considered for adapting the fuzzy influence weights of SBCN so as to avoid reoccurrence of misdiagnosis. This algorithm reduces the error between the outputs of the SBCN ck(t) and the binary target Tk(t) obtained after inspection. The binary target takes the value of zero for all the normal components and one for the faulty components. Sequential update rules for adapting the fuzzy influences in SBCN have the form ( lik þ h3 ðTk ðtÞ ck ðtÞÞð1 fi ðtÞÞfi ðtÞ if 0 , lik , 1 lik ¼ ð27:18Þ lik otherwise ( uik ¼

uik þ h3 ðTk ðtÞ

ck ðtÞfi ðtÞÞ2

uik

if 0 , uik , 1 otherwise

ð27:19Þ

where h3 represents the learning rate. In this method, in order to allow uniform interpretation of the trained fuzzy influences with respect to their original values, adaptation is stopped when the weight values reach the bound of zero or one.

27.4

Sensor Location Selection

Suitable mounting locations for accelerometers on the housing are generally those that provide both close proximity to gearbox components and ease of mounting (also see Chapter 15). However, considering that only a limited number of accelerometers can be used, the candidate locations need to be selected so as to provide a comprehensive coverage of the components. The method presented here relies on indices to quantify the suitability of various mounting locations. It uses a coverage index to define the reach of each accelerometer, and an overlap index to represent the overlap of the locations in coverage of the gearbox components. The monitoring effectiveness of various combinations of accelerometer locations can then be estimated as a function of the coverage and overlap indices of the accelerometer locations (Wang et al., 1999).

27.4.1

Coverage Index

The coverage index denotes the reach of each accelerometer in monitoring various components of the gearbox. It can be computed based on an influence matrix such as that defined in the SBCN. It can be defined, for example, as the sum of the corresponding row of the influence matrix, as Cj ¼

N uij þ lij 1 X N i¼1 2

ð27:20Þ

27 12


where uij and lij represent the upper and lower limits, respectively, of the fuzzy influence coefficients between the gearbox component, i, and accelerometer, j, and N denotes the total number of gearbox components. According to this index, an accelerometer location with a higher coverage index should be considered a better candidate.

27.4.2

Overlap Index

The coverage index can be used to rank accelerometers individually. However, for a complex gearbox where more than one accelerometer is needed for monitoring, the sum of the coverage indices associated with all of the accelerometers in the suite will not provide a correct representation of their overall effectiveness because it will ignore the overlap among them. Although the overlap between accelerometer pairs may be defined according to the influence coefficients, that definition will only consider the distance of the accelerometers from the components, and will ignore factors such as accelerometer orientation and gearbox size. In order to consider these other factors, the overlap index may be defined independent of the influence coefficients, based on an overlap matrix that represents the level of cocoverage between all of the accelerometer pairs. The individual components Ojk in this matrix can be defined as Ojk ¼

Ejk ð1 þ Sjk Þ eaDjk

ð27:21Þ

where Ojk denotes the overlap between the accelerometers, j and k, Ejk represents the similarity of orientation between the two accelerometers, a denotes a constant to account for the size of the gearbox, Djk accounts for the physical distance between the two accelerometers, and Sjk represents a symmetry factor between the two accelerometers. The formulation of Equation 27.21 is based on the following understanding. The overlap in coverage of two accelerometers depends mainly on their orientation and location, that is, the more identical their orientation is and the closer they are mounted to each other, the higher is their expected level of overlap. In the above equation, the orientation factor, Ejk [ ½0; 1 ; is set to one when the accelerometers, j and k, have identical orientation. The distance factor, Djk [ ½0; 1 ; is defined as the normalized shortest geometrical distance between the two accelerometers along the casing, where the term eaDjk approximates the attenuation of vibration due to distance (Lindsay, 1960) and the constant a, with values ranging between zero and two, accounts for the effect of gearbox size on vibration attenuation. Although the main factors in the coverage overlap of two accelerometers are orientation and distance, the accelerometers that are symmetrically positioned are believed to have a larger overlap with each other. In order to account for this factor, a symmetry factor, Sjk [ ½0; 1 ; is also included in Equation 27.21, which is set to one when two accelerometers are perfectly symmetrical with respect to the housing. Based on Equation 27.21, the values of Ojk would ordinarily fall in the range ½0; 1 ; where the value of one indicates that the two accelerometers, j and k, have 100% overlap with each other and that one of them can be removed. Similarly, when Ojk is zero then the two accelerometers are assumed to be covering completely different components. It should be noted, of course, that some overlap between accelerometers is considered useful and is often factored in when selecting accelerometer locations. The overlap coefficients only represent the level of overlap between pairs of accelerometers and not the specific components covered jointly by each accelerometer pair. As such, defining the level of overlap between several accelerometers is not as straightforward. At one extreme, one can assume that the coverage overlap of each accelerometer pair in the suite does not coincide with the coverage overlap of the other accelerometer pairs, so the total overlap, Os, for the accelerometer suite can be computed as 0 1 m m X X Cj þ Ck @ min Ojk ; Cj A ð27:22Þ Os ¼ 2 j¼1 k¼jþ1 where m denotes the number of accelerometers in the suite, and the min function ensures that the computed overlap will not exceed the total coverage of accelerometer j. At the other extreme, it can


27 13

be assumed that the overlaps of all of the accelerometer pairs coincide, and that the total overlap of the accelerometer suite can be defined as Os ¼

m X j¼1

max Ojk

Cj þ Ck for all k . j 2

ð27:23Þ

The latter formulation only considers the largest overlap between accelerometer j and the other accele rometers in the suite. It is also possible to use the average of the above two extremes as a compromise: 0 1 20 13 m m X X Cj þ Ck Cj þ Ck 1 4@ Os ¼ min@ Ojk ; Cj A þ max Ojk for all k . j A5 ð27:24Þ 2 2 2 j¼1 k¼jþ1 It should be noted that, unlike the coverage index, the overlap index is suite related because the overlap of an accelerometer depends on the other accelerometers in the suite.

27.4.3

Monitoring Effectiveness

As mentioned earlier, the coverage index can be used to evaluate the effectiveness of individual accelerometers in monitoring, but its sum cannot be a sole measure of effectiveness of suites of accelerometers. For instance, a suite consisting of the top three accelerometer locations in terms of coverage may be inferior to another suite of three accelerometer locations with a lower total level of coverage but less overlap among the accelerometers. The best set of accelerometer locations is, therefore, one which provides the highest coverage of the gearbox components and the least overlap among the accelerometers. As such, the monitoring effectiveness (ME) of a suite can be expressed as 0 1 m X @ MEs ¼ Cj A Os ð27:25Þ j¼1

where m represents the number of accelerometers in the suite, Ci denotes the coverage of individual accelerometers in the suite, and Os represents the total overlap among the accelerometers in the suite (Equation 27.24), estimated from Equation 27.22, Equation 27.23, or Equation 27.24. The indices used in sensor location are summarized in Table 27.2. TABLE 27.2

Summary of Indices Used for Sensor Location Selection

Sensor location is performed according to the following indices: Coverage index Cj ¼

N uij þ lij 1 X N i 1 2

where uij and lij represent the upper and lower limits of the fuzzy influence coefficients between the gearbox component, I, and accelerometer, j, and N denotes the total number of gearbox components. Overlap index Ojk ¼

Ejk ð1 þ Sjk Þ eaDjk

where Ojk denotes the overlap between the accelerometers, j and k, Ejk represents the similarity of orientation between the two accelerometers, a denotes a constant to account for the size of the gearbox, Djk accounts for the physical distance between the two accelerometers, and Sjk represents a symmetry factor between the two accelerometers. According to the above indices, the monitoring effectiveness of an accelerometer suite is defined as 0 1 m X @ Cj A 2 O s MEs ¼ j 1

where m represents the number of accelerometers in the suite, Cj denotes the coverage of individual accelerometers in the suite and Os represents the total overlap among the accelerometers in the suite.

27 14

27.5


A Case Study

This case study illustrates the application of the SBCN to an OH 58A main rotor gearbox (Figure 27.1; Jammu et al., 1996). Experimental vibration data for the OH 58A gearbox were collected at the NASA Lewis Research Center as part of a joint NASA/Navy/Army advanced lubricants program (Lewicki et al., 1992). Various component failures in the OH 58A transmission were produced during accelerated fatigue tests. The vibration signals were recorded by eight piezoelectric accelerometers (frequency range of up to 10 kHz) using an FM tape recorder. The signals were recorded once every hour, for about 1 to 2 min per recording (using a bandwidth of 20 kHz). Two magnetic chip detectors were also used to detect the debris caused by component failures. The location and orientation of the accelerometers are shown in Figure 27.2. The OH 58A gearbox was run under a constant load, and was disassembled and inspected periodically or when one of the chip detectors indicated a failure. A total of eleven failures occurred during these tests. They consisted of three cases of planet bearing pitting fatigue, three cases of sun gear pitting fatigue, two cases of top housing cover cracking, and one case each of spiral bevel pinion (SBP) pitting fatigue, mast bearing micropitting, and planet gear pitting fatigue. In order to extract the vibration features, the vibration signals from the gearbox were digitized and processed by a commercially available signal analyzer (Stewart Hughes Ltd., 1986). Overall, 54 vibration features were extracted from each accelerometer for the OH 58A gearbox. Out of these features, 35 features were gear related features (7 for each of the five gears). The remaining 19 features were indicators of either general faults (e.g., wear and out of balance), or general gear and bearing faults.

27.5.1

Structural and Featural Influences

The structural influences for the OH 58A gearbox were obtained through five primary vibration travel paths: (1) duplex bearing spiral bevel mesh triplex bearing, (2) duplex bearing sun planet mesh ring gear, (3) mast roller bearing main shaft mast ball bearing, (4) ring gear planet bearing mast ball bearing, and (5) duplex bearing sun planet mesh mast ball bearing. The first travel path was in connection to accelerometers 4, 5, and 6, whereas all the other paths were connected to accelerometers 1, 2, 3, 6, 7, and 8. The RMS values of vibration were then computed using the lumped mass model of these paths with excitation sources at each of the gearbox components. These RMS values were then used as the basis for defining the fuzzy structural influences between each component accelerometer pair (Table 27.3). The structural influences in Table 27.3 indicate that all of the components in the gearbox are covered by the accelerometers, and that some accelerometers have identical influences with respect to the components. Although these influences are not completely accurate due to their neglect of the orientation of accelerometers and various approximations (Jammu et al., 1998), they can still be used for TABLE 27.3

Structural Influences between the Components of the OH 58A Gearbox and the Eight Accelerometers

Component/Accelerator Triplex bearing Spiral bevel pinion Pinion roller bearing Spiral bevel gear Duplex bearing Gear roller bearing Mast roller bearing Main shaft Mast ball bearing Sun gear Planet bearing Planet gear Ring gear The influences shown are: “

1

M M M M M H H H H

2

M M M M M H H H H

3

M M M M M H H H H

4

5

H H H H H H

H H H H H H

L L L L

M M M M

” nil, “L” low, “M” medium, and “H” high.

6

7

8

M M M M M H H H H

M M M M M H H H H

M M M M M H H H H


27 15

an overall assessment of the effectiveness of various accelerometers and their redundancy. For example, the influences in Table 27.3 are identical for accelerometers 1, 2, 3, 6, 7, and 8. This would indicate that one or more of these accelerometers can be discarded without any drastic effect on fault diagnostic effectiveness. However, it should be noted that the strategy used in SBCN relies on the averaging effect of accelerometers, and that it will not function without a certain level of overlap in accelerometer coverage. The featural influences were defined for individual vibration features according to the type of fault they are supposed to represent. For example, bearing related features such as envelope band and tone energy were assigned high in association with gearbox bearings. Similarly, the signal averaged features for the five gears in the OH 58A gearbox were assigned high with respect to these gears. Features that are indicators of faults in all rotating elements in the gearbox were assigned an influence of medium for gears as well as bearings. The structural and featural influences constitute the basis of the SBCN connection weights. As discussed in Jammu et al. (1998), diagnosis in SBCN is performed hierarchically, first isolating the faulty subsystems and then faulty components. As such, only the average of the structural influences associated with each subsystem is used as the connection weights, vik, of SBCN to yield the fault possibility values, pk(t), associated with each subsystem, according to Jammu et al. (1998): pk ðtÞ ¼

n X i¼1

fi ðtÞvik

ð27:26Þ

In subsystem diagnosis, fi(t) denotes the average abnormality scaled value of general features. In the second stage of fault diagnosis, at the component level, the combination of structural and featural influences are used as vik, and fi(t) consist of individual abnormality scaled features.

27.5.2

Evaluation of Influences

As explained in Jammu et al. (1998), the structural influences representing the proximity effect of component faults on accelerometer readings were obtained from the RMS value of the frequency response of the lumped mass model of the gearbox. In this case, however, the actual RMS values of the vibration were available at several fault instances for the OH 58A gearbox, which could also be used to yield a set of experimentally obtained structural influences. In order to evaluate the modeled influences, a comparison between these two sets of structural influences was conducted. The experimental influences were obtained by normalizing the experimental RMS values when a faulty component had been detected and using them as the basis for assigning the level of fuzzy influences (Table 27.4). The results in Table 27.4 indicate mixed agreement between experimental and structural influences. For example, the structural influences of sun gear, planet bearing, and planet gear on accelerometers 1, 2, and 3 are close to the experimentally obtained influences, but the influence of mast ball bearing on accelerometers 2 and 3 TABLE 27.4 Influences between Accelerometers and Gearbox Components Obtained from Experimental RMS Values of Vibration for the OH 58A Gearbox Influences from RMS Values of Vibration Accelerator/Parts

Spiral Bevel Pinion

1 2 3 4 5 6 7 8

( ) M( ) M( ) L (H) H (H) L( ) ( ) ( )

Mast Ball Bearing ( ) (H) (H) ( ) ( ) ( ) ( ) ( )

Sun Gear

Planet Bearing

H (H) M (H) M (H) H (L) H (M) L (H) H (H) M (H)

M (H) M (H) H (H) ( ) H (M) M (H) M (H) M (H)

For comparison, the influences from the lumped mass model are shown inside parentheses.

Planet Gear M (H) M (M) M (M) M( ) (M) L (H) L (H) M (H)

27 16

Vibration and Shock Handbook TABLE 27.5 Normalized Weight Values of the Supervised Connectionist Network Accelerator

1 2 3 4 5 6 7 8

Subsystem 1

2

3

p ( ) p ( ) p ( ) H (H) M (H) M (M) M( ) p ( )

p (M) H (M) p (M) p ( ) p ( ) p (M) L (M) H (M)

p (H) H (H) L (H) p (L) H (M) H (H) H (H) M (H)

For comparison, the subsystem influences of the SBCN are included inside parentheses. A “p” indicates a negative weight value.

do not match. In this case, the mismatch between the two sets of influences may be due to (1) the limitation of the RMS value in reflecting the change in vibration as a result of various component faults (e.g., mast ball bearing micropitting), (2) variation in the level of change of the RMS values as a function of the type and size of the fault, and (3) lack of faults in every component of the OH 58A gearbox, which limits the ability to determine the influences for every gearbox component. Defining influences based on an approximate model of the gearbox was motivated by the need to avoid supervised training of the SBCN. However, given that experimental data were available for the OH 58A gearbox, an evaluation of the structural influences could be performed by comparing them with the weights of a connectionist network structurally similar to SBCN, but trained by supervised learning. For this purpose, the OH 58A gearbox was divided into three subsystems (Figure 27.1), and the supervised network, having three output units for the three subsystems and eight input units for the eight accelerometers, was trained using least mean square (LMS) learning. The weights of this network were trained until the number of false alarms and misdiagnoses were reduced to zero. The trained weights were then normalized and converted into fuzzy variables for comparison with the structural influences obtained from the lumped mass model of the gearbox. Table 27.5 includes the influences of the two networks, where the modeled influences (inside parentheses) represent the average of component influences within each subsystem (Table 27.3). The results in Table 27.3 indicate general agreement between the trained weights and modeled influences. Some of the trained weights had negative values (indicated by “ p ”), which is inevitable due to the use of LMS learning. However, all of these negative weights were quite small in magnitude, which makes them consistent with their modeled nil counterparts (denoted by “ ”). As in the case of influences from the RMS values (Table 27.4), some of the mismatches in Table 27.5 are expected to be due to the limited number of faults represented in the experimental data. For example, the considerably different influence of subsystem 2 on accelerometer 1, or subsystem 3 on accelerometer 1, is attributed to the lack of specific faults in these subsystems that would lead to a more accurate influence on accelerometer 1. The comparison between the modeled structural influences and those obtained from the supervised neural network indicates that the modeled influences are in good agreement with the trained influences, and that the lumped mass modeling used here provides an acceptable set of structural influences for the SBCN.

27.5.3

Fault Detection Results

The fault detection network (FDN) in the proposed diagnostic system (Figure 27.4) is used first to identify the presence of faults in the gearbox. Fault diagnosis is then performed when the presence of a fault is prompted. A total of eight FDNs, one for each of the eight OH 58A accelerometers, were used. The inputs to each FDN were the 19 general features not specific to any particular gear or bearing. The initial weight

Fault Diagnosis of Helicopter Gearboxes TABLE 27.6

Fault Detection Results for the OH 58A Gearbox

Day

Fault Detection Network: Predicted and Actual Failures Test 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

27 17

( ) ( ) ( ) ( ) 1 (1) (1) 1 (1) 1 (1) (1p)

Test 2 ( ( ( ( ( ( ( ( (

) ) ) ) ) ) ) ) )

Test 3

1 1

1 1

( ) ( ) (1) (1p) ( ) ( ) ( ) ( ) (1p) ( ) (1) (1) (1p)

Test 4

1 1 1 1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1) (1) (1p) ( ) (1) (1p)

Test 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1) 1 (1) (1) (1p)

A “ ” indicates normality and a “1” represents the presence of a fault. For reference, the expected faults determined by an expert are included inside the parentheses, with “ p ” indicating actual observation of the fault.

values of the FDNs were set as the values of the first set of features for each of the five tests, and were subsequently adapted using 50 adaptation sweeps for each training batch. The occurrence of a fault was prompted when any of the FDNs indicated a fault. The detection results for individual test sets obtained from the FDNs are shown in Table 27.6. A “ ” in this table indicates normal conditions, whereas a “1” indicates the presence of a fault. The expected detection results are indicated inside parentheses. The results for test 1 indicate that the presence of faults was detected on days 5, 7, and 8, while faults were expected to be present from days 5 to 9. Of course, it should be noted that the gearbox was not inspected on a daily basis, so the actual condition of the gearbox is unknown for each day of the tests. In test 1, which was run for nine days, a fault was actually observed only on day 9 during a routine inspection of the gearbox (indicated by 1p). However, based on an inspection of the vibration features, it was estimated that the fault could have been present as early as day 5. For the other tests, the days when the faults were actually observed are also indicated by 1p. While it is discouraging to note that day 6 of test 1 was classified as normal though a fault was present on day 5, the results are in agreement with observations by experts who believe that sometimes increased noise levels immediately after the occurrence of faults mask the effect of faults on vibration features. For the other tests, the results indicate that except for an undetected fault in test 3 and a false alarm in test 4, excellent fault detection was obtained. It should also be noted that the fault on day 9 of test 3 was a hairline crack, which was perhaps undetectable through vibration monitoring. The quality of fault detection in the proposed system is particularly important to the overall diagnostic results, since it is only after a fault is detected that SBCN is engaged in diagnosis. In summary, the detection results obtained (Table 27.6) indicate that the occurrences of most of the faults were identified. This provides assurance that the later stage of diagnostics would not be hampered by the detection phase.

27.5.4

Fault Diagnostic Results

In the proposed system, fault diagnosis is performed by the SBCN only after the presence of a fault is detected. In this system, fault diagnosis is performed in two hierarchies so as to take full advantage of the separation of the structural and featural influences. In the first hierarchy the gearbox is divided into subsystems (Figure 27.1) and the faults in individual subsystems are isolated by the SBCN based on the structural influences alone. For each subsystem, the weights of the SBCN are set equal to the average of the structural influences of the components within that subsystem (Table 27.5). The inputs to the SBCN

27 18


for this stage of the diagnosis consist of the averaged values of abnormality scaled features from each accelerometer, and its outputs are the fault possibility values for each subsystem. In the second hierarchy, the faulty components within each subsystem are isolated. The inputs to the SBCN for this stage of the diagnosis are the abnormality scaled vibration features, and the weights are the product of featural influences and structural influences of the subsystem containing each component. The averaging of fuzzy influences here was done by taking the average of upper bounds and lower bounds of individual influences separately, and then defining the fuzzy variable that would match the range. The product of fuzzy influences was determined by multiplying the upper bounds and lower bounds of fuzzy variables separately, and then defining the fuzzy variable for the resultant range. 27.5.4.1

Faulty Subsystem Isolation

The fault possibility values for the three subsystems of the OH 58A gearbox are shown in Table 27.7. The results in this table represent the hard limited fault possibility values (threshold of 0.5) and include, for comparison, the actual condition of the gearbox reported from routine inspection inside parentheses. As before, a “ p ” indicates actual observation of the fault during inspection of the gearbox. The results in Table 27.7 indicate that in test 1, faults in subsystems 1 and 3 were correctly identified on days 5, 7, and 8. In test 3, the faults in subsystem 3 on days 3 and 4 were correctly identified, along with a possible fault in subsystem 1. The housing crack on day 9 of this test was left unidentified because it was never prompted during the detection phase. In any case, this particular fault (a housing crack) could not be isolated by the current SBCN due to absence of features that reflect this fault. Also for this test, faults in subsystems 2 and 3 were correctly identified on days 11 and 12. In Test 4, the fault in subsystem 3 was correctly diagnosed on days 10, 11, 12, 14, and 15. Moreover, on day 13 of test 4, even though the gearbox was supposed to be normal, the SBCN indicated faults in subsystem 3. This was due to the replacement of the three planet assembly with a four planet assembly, which changed the vibration characteristic of subsystem 3. In test 5, the fault in subsystem 3 was correctly identified on day 9. There was also a misdiagnosis in subsystem 1. In summary, the diagnostic results from the gearbox subsystems indicate that all of the eight subsystem faults were correctly identified in the OH 58A gearbox and that four faults were misdiagnosed. Considering that these results were obtained by using structural influences alone as the connection weights of the SBCN, the results validate the utility of these influences and of lumped mass modeling as a means of representing the vibration travel path of gearboxes for their model based fault diagnosis. TABLE 27.7

Faulty Subsystem Isolation Results for the OH 58A Gearbox

Day

Faulty Subsystems Isolation for OH 58A Test 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

( ) ( ) ( ) ( ) 1, 3 (1, 3) (1, 3) 1, 3 (1, 3) 1, 3 (1, 3) (1p, 3p)

Test 2 ( ( ( ( ( ( ( ( (

) ) ) ) ) ) ) ) )

Test 3

1, 3 1, 3

2, 3 2, 3

( ) ( ) (3) (3p) ( ) ( ) ( ) ( ) (3) ( ) (2, 3) (2, 3) (2p, 3p)

Test 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 (3) 3 (3) 3 (3p) 3( ) 3 (3) 3 (3p)

Test 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3) 1, 3 (3) (3) (3p)

The three subsystems in the table are the input subsystem (1), the output subsystem (2), and the transmission subsystem (3). For comparison, the actual faults are included inside parentheses with “ p ” indicating the observed faults.


27.5.4.2

27 19

Faulty Component Isolation

Fault possibility values associated with the components of the OH 58A gearbox obtained from the SBCN are included in Table 27.8. The results indicate that the diagnostics associated with individual components are not as accurate as those obtained for the subsystems. Briefly, for test 1, the SBP fault in subsystem 1 and the sun gear (SG) fault in subsystem 3 were correctly identified only on days 5 and 8, respectively, while other components were assigned higher fault possibility values on other days. For test 3, the three bearing faults in subsystems 2 and 3 (BRG2 and BRG3, respectively) were correctly identified on days 3, 4, and 12, but other components were also given high fault possibility values. In test 4, the bearing fault in subsystem 3 (BRG3) was correctly identified only on day 10, while the SG fault remained misdiagnosed. In test 5, the SG fault was correctly identified on day 9, while the planet gear (PG) fault was misdiagnosed. In view of the promising results obtained at the subsystem level, which confirm the validity of the structural influences, the cause of diagnostic inaccuracies at the component level should be attributed mainly to the deficiency of gear and bearing specific features used in this study. The strong cross coupling TABLE 27.8

Faulty Component Isolation by SBCN for the OH 58A Gearbox

Days

Faulty Component Isolation for OH 58A SS1

SS2

SBP

SBG

BRG1

0.90p

0.62

0.68p 0.65p

0.43 0.38

BRG2

SS3 SG

PG

RG

BRG3

0.89

0.52p

0.73

0.12

0.86

0.43 0.74

0.79 0.18

0.67p 0.98p

1.00 0.70

0.23 0.70

0.72 0.33

0.77 0.60

0.80 0.78

0.65 0.56

0.56 0.47

0.71 0.04

0.72p 0.79p

0.67 0.67

0.79 0.71

0.52 0.55

p

0.34 0.54 0.59 0.72 0.81p 0.79p

0.41 0.53 0.50 0.85 0.90 0.90

0.75 0.79 0.91 0.83 0.88 0.93

0.64p 1.00 0.68 0.48

0.60p

0.54p

0.50

0.58

Test 1 1 4 5 6 7 8 9

p

p

Test 2 1 9 Test 3 1 and 2 3 4 5 10 11 12 13

p

0.74 p

1.00p p

Test 4 1 9 10 11 12 13 14 15

0.79p p

Test 5 1 8 9 10 and 11

0.58

0.24

0.68

p

p

The components listed are SBP: spiral bevel pinion; SBG: spiral bevel gear; BRG1: bearings in subsystem (SS) 1; BRG2: bearings in SS2; SG: sun gear; PG: planet gear; RG: ring gear; and BRG3: bearings in SS3. As before, “ p ” indicates observation of the faulty component.

27 20


Maximum Values of Correlation Coefficients between Features and Faults Fault Feature Correlation Coefficients

Features/Faults

Gear Faults

Bearing Faults

Gear features Bearing features

0.49 (1) 0.38 (0)

0.57 (0) 0.44 (1)

The values inside parentheses are the expected ideal values.

between these features is illustrated by the maximum values of the correlation coefficients between the gear and bearing features and gear and bearing faults in Table 27.9. The results indicate reasonable correlation values of 0.49 between the gear features and gear faults, and 0.44 between the bearing features and bearing faults. These numbers, however, are not as impressive when they are compared with the cross correlation values of 0.57 between gear features and bearing faults, and 0.38 between bearing features and gear faults. The high cross correlation values in Table 27.9 indicate that the gear and bearing features do not provide the resolution necessary for faulty component isolation. The manifestation of the resolution problem caused by the coupling between gear and bearing features is observed in the similar fault possibility values of 0.9 and 0.89 on day 5 of test 1, in Table 27.8, for the SBP and bearings in subsystem 1 (BRG1). Based on the results in Table 27.8, one can conclude that the unsupervised pattern classification scheme incorporated in this research cannot be a substitute for well defined features, and that a more effective set of features with smaller cross correlation values are needed for diagnosis at the component level.

27.5.5

Sensor Location Evaluation

The eight candidate accelerometer locations (see Figure 27.2 for their locations and orientations), which were actually used for vibration measurement during accelerated fatigue tests of the gearbox, were analyzed and ranked for their significance in monitoring. In order to obtain the overlap coefficients for the OH 58A gearbox, the values of the orientation factors, Ejk, distance factors, Djk, and symmetry factors, Sjk, for the eight accelerometer locations were defined as 2

1

6 6 0:5 6 6 6 0:3 6 6 6 6 1 6 Ejk ¼ 6 6 0:4 6 6 6 6 0:8 6 6 6 1 4

0:5 0:3 1 0:3 0:5 0:4 0:7 0:5

1

0:4 0:8

1

0:8

3

2

0

0

0

7 6 6 0 0:3 0:5 0:4 0:7 0:5 0:7 7 0 0 7 6 7 6 7 6 1 0:4 1 0:3 0:3 0:3 7 0 0 6 0 7 6 7 6 6 1:5 1:5 1:5 0:4 1 0:4 0:5 1 0:8 7 7 6 7; aDjk ¼ 6 6 1:5 1:5 1:5 1 0:4 1 0:4 0:4 0:4 7 7 6 7 6 7 6 6 0:5 0:5 0:5 0:3 0:5 0:4 1 0:8 0:6 7 7 6 7 6 6 2:0 2:0 2:0 0:3 1 0:4 0:8 1 0:8 7 5 4

0:8 0:7 0:3 0:8 0:4 0:6 0:8 3 0 0 0 0 0 0 1 0 6 7 60 0 0 0 0 0 1 0 7 6 7 6 7 60 0 0 0 0 0 1 0 7 6 7 6 7 6 7 60 0 0 0 0 0 1 7 0 6 7 Sjk ¼ 6 7 60 0 0 0 0 0 0 7 0 6 7 6 7 6 7 60 0 0 0 0 0 0 7 1 6 7 6 7 6 1 1 1 1 0 0 0 0:5 7 4 5 2

0 0 0 0 0 1 0:5

0

1

1:5 1:5 0:5 2:0 0:5

3

7 1:5 1:5 0:5 2:0 0:5 7 7 7 1:5 1:5 0:5 2:0 0:5 7 7 7 7 0 0:8 1:0 0:5 2:0 7 7 7 0:8 0 1:0 0:5 2:0 7 7 7 7 1:0 1:0 0 1:5 1:0 7 7 7 1:5 1:5 1:5 0 1:5 7 5

0:5 0:5 0:5 2:0 2:0 1:0 1:5

0

Fault Diagnosis of Helicopter Gearboxes TABLE 27.10

Best and Worst Accelerometer Suites within Each Suite Size

Suite Size 1 accelerometer 2 accelerometers 3 accelerometers 4 accelerometers 5 accelerometers 6 accelerometers 7 accelerometers

27 21

Best Suites 5 (5, (2, (3, (2, (1, (1,

Worst Suites 3 (4, 7); (1, 6) (1, 6, 8); (1, 2, 6); (1, 2, 8) (1, 2, 7, 8); (1, 4, 7, 8); (1, 6, 7, 8) (1, 4, 6, 7, 8); (1, 2, 6, 7, 8); (1, 2, 4, 7, 8) (1, 2, 4, 6, 7, 8); (1, 3, 4, 6, 7, 8) (1, 2, 3, 4, 6, 7, 8)

8); (2, 5) 5, 7); (4, 5, 8); (2, 4, 5) 4, 5, 8); (2, 4, 5, 8); (2, 4, 5, 7) 3, 4, 5, 8); (2, 3, 5, 7, 8); (2, 3, 5, 6, 7) 2, 3, 4, 5, 7); (2, 3, 4, 5, 6, 8) 2, 3, 4, 5, 6, 8)

Using the above values, the overlap coefficients for the eight locations were determined from Equation 27.21 as 2

1

6 6 0:500 6 6 6 0:300 6 6 6 6 0:223 6 Ojk ¼ 6 6 0:089 6 6 6 6 0:485 6 6 6 0:271 4

0:50

0:3

1

0:3

0:300

1

0:112 0:089 0:089 0:223 0:425 0:182 0:135 0:081

0:2

0:089 0:485 0:271 0:485

3

7 0:112 0:089 0:425 0:135 0:425 7 7 7 0:089 0:223 0:182 0:081 0:182 7 7 7 7 1 0:180 0:184 0:446 0:108 7 7 7 0:180 1 0:147 0:089 0:054 7 7 7 7 0:184 0:147 1 0:179 0:441 7 7 7 0:446 0:089 0:179 1 0:268 7 5

0:485 0:425 0:182 0:108 0:054 0:441 0:268

ð27:27Þ

1

Maximum monitoring effectiveness

Given the above values of coverage and overlap, the ME values were determined from Equation 27.24 for various suites from the eight candidate accelerometer locations of the OH 58A main rotor gearbox. The ME values can be used, for example, to select the best set of accelerometer locations given a number of accelerometers to be used. A sample set of accelerometer locations with the highest and lowest rankings (from the 254 possible suites of the eight OH 58A locations) are shown in Table 27.10. Note that, for some suite sizes, several combinations of accelerometer locations are selected as the “best” and “worst” suites, because in these cases the ME values were too close to render a suite better or worse than the others. The ME values can also be used to assess the benefit of additional accelerometers, leading to larger suite sizes. To demonstrate this utility of monitoring effectiveness, the maximum ME value for each suite size was obtained (Figure 27.9). While additional accelerometers are expected to add to monitoring effectiveness, the added coverage they provide will 1 diminish for larger suite sizes due to the increase in 0.9 overlap between the accelerometers. This is clearly 0.8 reflected in Figure 27.9 where the maximum ME 0.7 value of the eight strong accelerometer suite is the 0.6 same as that of the seven strong accelerometer 0.5 suite, and the suite with five accelerometers has a 0.4 0.3 maximum ME value of 0.90, only 10% less than 0.2 that of the suite with eight accelerometers.

27.5.6

Sensor Location Validation

The validity of the ME values was evaluated by comparing the rankings they provided for various accelerometer suites to the rankings obtained

0.1

0

1

2 5 6 3 4 7 Number of accelerometers in the suite

8

FIGURE 27.9 Maximum values of monitoring effec tiveness for different accelerometer suite sizes.

27 22


empirically from a series of diagnostic tests. The suites were ranked into four evenly spaced categories. The suites with the highest monitoring effectiveness were ranked as one, and those with the lowest values were ranked into the fourth category. The empirical rankings were obtained from the SBCN according to the fault possibility values, pi [ ð0; 1Þ; of the individual gearbox subsystems (Jammu et al., 1998). For quantification purposes, a performance index, ps, was obtained for each suite of accelerometers to represent the accuracy of the diagnostic results, as X ps ¼ pi ð27:28Þ faults

where the summation was carried out over all the available faults that had occurred during the OH 58A experiments. Under ideal conditions and with a perfect accelerometer suite, the values of pi should be all equal to one (i.e., the value of ps should be equal to the number of faults). However, in practice, the fault possibility values are smaller than one due to imperfect signal conditioning, presence of noise, and so on. The understanding used here is that, everything else being the same, the value of ps is only affected by the quality of the accelerometer suite when the vibration features from different suites are used as inputs to the diagnostic system. The value of ps was computed for various accelerometer suites, and then normalized against the largest value of ps obtained for the same suite size. These normalized ps values were then used as the basis for ranking the suites. As with the monitoring effectiveness values, the suite with the highest normalized ps value was assigned to the first category. It should also be noted that, in both cases, the rankings provide only a relative measure of effectiveness among suites of the same size, and that they should not be perceived as global measures. The normalized ps values and the associated rankings for suites of seven accelerometers are included in Table 27.11, along with the ME values and their corresponding rankings. The results indicate remarkably close agreement between the estimated and empirical rankings and that, except for one mismatch, the rankings are identical. Similar analyses were performed for suites of other sizes. A summary of matches and mismatches for all the suites is given in Table 27.12. The results indicate that, out of the 254 possible suites, the estimated rankings of 174 suites match exactly the empirical rankings, and 103 mismatch by only one rank. The results summarized in Table 27.12 indicate that the proposed selection method is effective in assessing the monitoring effectiveness of suites of accelerometers. The experimental data set, although one of the most complete sets available in the industry, is not comprehensive enough to render a complete evaluation of the method. The main limitation is the absence of faults in all of the components of the gearbox. This could result in an overestimation of the significance of accelerometers that cover faulty components during the experiments. Similarly, it could lead to devaluation of accelerometers which cover healthy components during the experiments. For example, there was only a single fault in subsystem 2 (mast bearing micropitting), therefore, accelerometer locations that covered this subsystem were given a lower empirical ranking than they actually deserved. TABLE 27.11 Rankings Obtained from the Monitoring Effectiveness Values and from the Diagnostic Results for Suites of Seven Accelerometers Accelerometers Included

2, 3, 4, 5, 6, 7, 1, 3, 4, 5, 6, 7, 1, 2, 4, 5, 6, 7, 1, 2, 3, 5, 6, 7, 1, 2, 3, 4, 6, 7, 1, 2, 3, 4, 5, 7, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6,

8 8 8 8 8 8 8 7

Monitoring Effectiveness

Empirical

MEs

Rank

ps

Rank

0.983 0.791 0.705 0.908 0.200 0.990 1.000 0.940

1 2 2p 1 4 1 1 1

0.888 0.702 0.844 0.926 0.200 1.000 0.938 0.843

1 2 1 1 4 1 1 1

“ p ” indicates a mismatch between the rankings.


27 23

TABLE 27.12 Summary Comparison of Rankings Obtained from the Monitoring Effectiveness Values and from the Diagnostic Results for Suites of All sizes Suites of

7 6 5 4 3 2 1 Total

27.6

Match Exactly

7 17 39 48 38 17 8 174

Mismatch by 1

2

3

1 11 17 22 18 11 0 80

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Conclusion

Commercial fault detection systems developed for helicopter gearboxes, generally as part of a Health and Usage Monitoring System (HUMS)3, follow the traditional approach to fault detection and diagnosis. In these systems, the individual vibration signals obtained from each accelerometer are processed first to extract various features, and are then evaluated statistically or in comparison with established thresholds to identify possible abnormalities. The abnormal features are then scaled and weighted to provide a weighted average “figure of merit” that can be used for fault detection. Once a gearbox is identified as abnormal, diagnosis is performed by evaluating the synchronized time averaged features associated with individual shafts and gears the signal average of a particular shaft contains the mounted gear meshing characteristics which can be used to determine the gear condition. In HUMS, statistical analysis is usually based on the mean and standard deviation of individual features, and thresholds are established based either on experience or according to base lines determined from initial data points at the beginning of aircraft operation. It should be noted that the simplistic approach used by major HUMS producers for abnormality identification and fault detection and diagnosis is often due to the stringent restrictions imposed by the certification process4; otherwise, much development work has been conducted in the recent years that can be included in commercial systems. A truly integrated mechanical diagnostic system should have the following characteristics: (1) be able to automatically correlate the information from various sensor suites including accelerometers, oil debris sensors, acoustic emission (AE) transducers, and so on, and to consolidate the results into a reliable solution; (2) be able to “predict” (prognose) a degrading mechanical condition based on trending of the “features” or condition indices; (3) be compatible with the maintenance policies and operational situations, (4) account for the fatigue life of the monitored components; (5) account for flight conditions (such as transmission torque) that would affect the diagnostic results; (6) be based on open systems architecture (both hardware and software) for future technology insertion; and (7) be lightweight (including cabling and sensors) such that no significant extra weight will be added to the aircraft. Oil analysis will undoubtedly play a more critical role in fault detection and diagnosis. One logical step will be development of more reliable in flight methods of chip detection based on optical technology and image processing (Lukas and Yurko, 1996). The chip detectors of the future are expected to benefit from the micro electro mechanical systems (MEMS) technology, making available microdetectors that can be 3

A HUMS generally consists of structural usage monitoring, gearbox, engine, and rotor system fault detection and diagnosis, rotor track and balance, and the associated data processing and maintenance logistics. 4 No HUMS has been certified by the FAA yet because of the relative newness of the technology and lack of data to verify it. Some HUM systems have been flying in the North Sea area, mostly manufactured by Teledyne Stewart Hughes and GEC Bristows, but they need to be developed further before they can be totally reliable.

27 24


located practically anywhere within the gearbox. Continued emphasis is also expected on postflight oil analysis such as ferrography, thermography, ultrasonic analysis, and passive electric current analysis (Saba 1996). As for vibration monitoring, there will be continued need for algorithmic development to produce new features for reliable identification of component faults. In this domain, there will be more focus on acoustic monitoring and stress wave analysis. Accelerometers normally have a frequency response range of a few Hz to over 50 kHz. In acoustic monitoring, a much higher bandwidth is considered microphones are currently available that have a flat response to over 100 kHz. This provides the possibility of measuring a cleaner signal in higher frequency ranges, which may prove useful in detecting certain bearing faults. Another advantage of microphones is that they do not need to be mounted on the surface of the housing, therefore, unlike accelerometers, they are not affected by mounting resonance. For stress wave monitoring, AE sensors are available that detect the stress waves generated by strain (elastic) energy spontaneously released by materials when they undergo deformation. This type of energy, which has a very broad bandwidth (DC to several MHz) due to the impulse nature of the strain release, propagates away from a crack tip and throughout the structure body. For crack growth monitoring, analysis is usually focused on the high frequency region (. 100 kHz) of the AE signal so that the effect of low frequency noise caused by airframe, gearbox, and/or drive shaft vibration is minimized. Stress wave monitoring through AE signals may, therefore, reveal the presence of cracks, as well as their location and severity (Teller and Kwun, 1994). Another important area of research and development in fault diagnosis is sensor technology. A generic restriction in vibration monitoring is the limited number of mounting locations for accelerometers on the housing. This restriction, however, is expected to be alleviated with the advancement of the MEMS technology which will eventually make available miniaturized sensors that can be mounted practically anywhere on a housing, and perhaps inside it. These sensors are expected to have the added capability of processing the vibration signal and producing the features locally, so that the central processor can be dedicated solely to integration of features for fault detection and diagnosis.

References Astridge, D.G., Helicopter transmission design for safety and reliability, Proc. Inst. Mech. Engrs., 203, 123 138, 1989. Badgley, R.H. and Hartman, R.M., Gearbox noise reduction: prediction and measurement of mesh frequency vibrations within an operating helicopter rotor drive gearbox, J. Engng Ind., May, 567 577, 1974. Barkov, A. and Barkova, N., Condition assessment and life prediction of rolling element bearings parts 1 and 2, Sound Vibr., June, 14 20, 1995. Boyd, L.S. and Pike, J., Epicyclic gear dynamics, AIAA J., 27, 5, 603 609, 1989. Carpenter, G.A. and Grossberg, S., ART2: self organization of stable category recognition codes for analog input patterns, Appl. Optics, 1987. Chin, H., Danai, K., and Lewicki, D.G., Pattern classifier for fault diagnosis of helicopter gearboxes, IFAC J. Control Eng. Practice, 1, 5, 771 778, 1993. Chin, H., Danai, K., and Lewicki, D.G., Fault detection of helicopter gearboxes using the multivalued influence matrix method, ASME J. Mech. Des., 117, 2, 248 253, 1995. Choy, F.K. and Qian, W. 1993. Global Dynamic Modeling of a Transmission System. NASA Contractor Report 191117, ARL CR 11. Lewis Research Center, Cleveland, OH. Cornell, R.W. and Westervelt, W.W., Dynamic tooth loads and stressing for high contact ratio spur gears, J. Mech. Des., 100, 69 76, 1978. Dyer, D. and Stewart, R.M., Detection of rolling element bearing damage by statistical vibration analysis, ASME J. Mech. Des., 100, 229 235, 1978. Harris, T.A. 1966. Rolling Bearing Analysis, Wiley, New York.


27 25

Hertz, J., Krogh, A., Palmer, R.G., and Palmer, R.G. 1991. Introduction to the Theory of Neural Computation, Addison Wesley, Redwood City, CA. James, M.L., Smith, G.M., Wolford, J.C., and Whaley, P.W. 1994. Vibration of mechanical and structural systems, Harper Collins College Publishers, New York. Jammu, V.B., Danai, K., and Lewicki, D.G., Unsupervised pattern classifier for abnormality scaling of vibration features for helicopter gearbox diagnosis, Machine Vibr., 5, 3, 154 162, 1996. Jammu, V.B., Danai, K., and Lewicki, D.G., Structure based connectionist network for fault diagnosis of helicopter gearboxes, ASME J. Mech. Des., 120, 100 112, 1998. Kasuba, R. and Evans, J.W., An extended model for determining dynamic loads in spur gearing, J. Mech. Des., 103, 398 409, 1981. Kazlas, P.T., Monsen, P.T., and LeBlanc, M.J. 1993. Neural Network Based Helicopter Gearbox Health Monitoring System, presented at IEEE SP Workshop on Neural Networks for Signal Processing, Linthicum, MD, September. Kohonen, T. 1989. Self Organization and Associative Memory, Springer, Berlin. Lewicki, D.G., Decker, H.J., and Shimski, J.T. 1992. Full Scale Transmission Testing to Evaluate Advanced Lubricants. Technical Report, NASA TM 105668, AVSCOM TR 91 C 035. Liddle, I. and Reilly, S., Automatic analysis of rotating machinery using an expert system, Sound Vibr., February, 7 11, 1993. Lin, H. H., Huston, R.L., and Coy, J.J., On dynamic loads in parallel shaft trans mission: Part I modelling and analysis, J. Mech. Transm. Automation Des., 110, 221 229, 1988. Lindsay, R.B. 1960. Mechanical Radiation, McGraw Hill, New York. Lukas, M. and Yurko, R.J. 1996. Current technology in oil analysis spectrometers and what we may expect in the future, pp. 161 171. Proceedings of Integrated Monitoring Diagnostics and Failure Prevention, Mobile, AL. Lurton, E.H. 1994. Navy oil analysis program overview, pp. 4 6. In Proc. JOAP International Condition Monitoring Conference. Lyon, R.H., Structural diagnostics using vibration transfer functions, Sound Vibr., January, 28 31, 1995. Mark, W.D., Use of the generalized transmission error in equations of motion of gear systems, J. Mech. Transm. Automation Des., 109, 283 291, 1987. McFadden, P.D. and Smith, J.D., Effect of transmission path on measured gear vibration, J. Vibr. Acoust. Stress Reliabil. Des., 108, 377 378, 1986. Mertaugh, L.J. 1986. Evaluation of vibration analysis techniques for the detection of gear and bearing faults in helicopter gearboxes, pp. 28 30. In Proc. of the Mechanical Failure Prevention Group 41th Meeting, October, 1986. Mitchell, L.D. and Davis, J.W., Proposed solution methodology for the dynamically coupled, nonlinear geared rotor mechanics equations, J. Vibr. Acoust. Stress Reliabil. Des., 107, 112 116, 1985. Naval Command, Control, and Ocean Surveillance Center. 1995. CH 53E Aft Main Transmission Test Stand Data Set, San Diego, CA. Randall, R.B., A new method of modeling gear faults, ASME J. Mech. Des., 104, 259 267, 1982. Rebbechi, B., Oswald, F.B., and Townsend, D.P. 1991. Dynamic measurements of gear tooth friction and load, NASA Tech. Memorandum 103281, TR 90 C 0023. Lewis Research Center, Cleveland, OH. Saba, C.S. 1996. Alternate techniques for wear metal analysis, pp. 151 160. In Proceedings of Integrated Monitoring Diagnostics and Failure Prevention, Mobile, AL. Smith, J.D. 1983. Gears and Their Vibrations, Marcel Dekker/The Macmillan Press Ltd, Netowork, New York. Solorzano, M.R., Ishii, D.K., Nickolaisen, N.R., and Huang, W.Y. 1991. Detection and Classification of Faults from Helicopter Vibration Data Using Recently Developed Signal Processing and Neural Network Techniques, Code 535. Advanced Technology Development Branch, Naval Ocean Systems Center, San Diego, CA. Stewart Hughes Ltd. 1986. MSDA User’s Guide, Technical Report, Southamton, U.K.

27 26


Teller, C.M. and Kwun, H. 1994. Magnetostrictive sensors for structural health monitoring systems, pp. 297 305. In Proceedings of Mechanical Failure Prevention Group 48th Meeting, April. Thornton, M.G. 1994. Filter debris analysis, a viable alternative to existing spectrometric oil analysis techniques, pp. 108 119. In Proceedings of JOAP International Condition Monitoring Conference, 1994. Walford, T.L.H. and Stone, B.J. 1983. The sources of damping in rolling element bearings under oscillating conditions, pp. 225 232. In Proceedings of the Institute of Mechanical Engineers, Vol. 197c, December. Wang, K., Yang, D., Danai, K., and Lewicki, D.G., Model based selection of accelerometer locations for helicopter gearbox monitoring, J. Am. Helicopter Soc., 44, 4, 269 275, 1999. While, M.F., Rolling element bearing vibration transfer characteristics: effect of stiffness, J. Appl. Mech., 46, 677 684, 1979. Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning 1, Inf. Sci., 8, 199 249, 1975. Zakrajsek, J.J., Hanschuh, R.F., Lewicki, D.G., and Decker, H.J. 1995. Detecting gear tooth fracture in a high contact ratio face gear mesh, pp. 91 102. In Proceedings of 49th Meeting of the Society of Machinery Failure Prevention Technology, April.

28

Vibration Suppression and Monitoring in Precision Motion Systems 28.1 Introduction ........................................................................ 28 1 28.2 Mechanical Design to Minimize Vibration ...................... 28 2

K.K. Tan, T.H. Lee, K.Z. Tang, and S. Huang National University of Singapore

S.Y. Lim, W. Lin, and Y.P. Leow Singapore Institute of Manufacturing Technology

Stability and Static Determinacy of Machine Structures † Two Dimensional Structures † Three Dimensional Structures

28.3 Adaptive Notch Filter ......................................................... 28 10 Fast Fourier Transform

†

Simulation and Experiments

28.4 Real Time Vibration Analyzer ........................................... 28 17 Learning Mode † Monitoring Mode Mode † Experiments

†

Diagnostic

28.5 Practical Insights and Case Study ...................................... 28 29 28.6 Conclusions ......................................................................... 28 35

Summary Much research and development effort is going on to develop precision motion systems that are used in most production equipment. Market demands for better products (products with much higher performance, higher reliability, longer life, lower cost, and increasing miniaturization) are some of the main driving forces behind these efforts. There are several challenges ahead in order to meet these stringent requirements for precision motion systems. One of the main challenges is to suppress the mechanical vibrations in these motion systems. This chapter provides several possible approaches to this objective. The first approach will focus on a proper mechanical design, based on the determinacy of machine structure, to reduce the mechanical vibration in the motion systems to a minimum. In addition to a good design, a monitoring and suppression mechanism is necessary to cope with additional and usually unpredictable sources of vibration seeping in during the course of operations. An approach, utilizing an adaptive notch filter in the control system, is presented in the chapter, to continuously identify resonant frequencies present and suppress signal transmission into the system at these frequencies. Finally, the development of a low cost real time vibration analyzer for precision motion systems is presented. A case study is provided at the end of the chapter to illustrate the effectiveness of a remote vibration monitoring and control system for precision motion systems.

28.1

Introduction

Industrialization, propelled by advances in science and technology in the last millennium, has led to the development of mass production equipment to satisfy man’s relentless pursuit for products such as 28 1

28 2


automobiles, air conditioners, audio and video products, and so on. Today, apart from the consumer’s demand for more functionality in new product lines, a miniaturization phenomenon is also clearly evident where the physical size of the product can be smaller than its predecessor, even when it has expanded performance. In addition, the ever shortening time to market of products and productivity cost factors in today’s highly competitive world together pose tough and challenging requirements on the manufacturing and automation systems that produce them. Thus, while high accuracy and precision in production systems are necessary to produce highly delicate parts and products, a high production speed must still be maintained even though speed and precision adversely affect each other. It is not difficult to appreciate this dilemma since a high production speed can lead to excessive mechanical vibration (Fertis, 1995; Uicker et al., 2003), which means inevitably a loss in precision (Tan et al., 2001). This chapter will seek to address the vibration issues in precision motion systems. Different and numerous sources of vibration can be present in an industrial operating environment (Adams, 2000; Kelly, 2000). Vibration may be generated by rotating or reciprocating machineries such as: engines, motors, and compressors; impact processes such as drilling and PCB printing; the flow of fluid; and many others (Raichel, 2000). The vibration level can reach an excessive level when abnormal operating conditions occur due to unbalanced inertia, bearing failures in rotating systems such as turbines, motors, generators, pumps, drives, and turbofans, component failure and operation outside prescribed load ratings, and poor kinematical design (resulting in a nonrigid and nonisolating support structure). Vibration, whether naturally occurring or induced under abnormal conditions, is undesirable as it usually leads to dynamic stresses that, in turn, causes fatigue and deterioration of the machinery. Other adverse consequences include unnecessary energy losses, deterioration in performance, and an unduly high level of noise produced. These effects can be even more severe for high precision motion systems with stringent requirements on precision and accuracy, since vibration will lead directly to poor repeatability properties, impeding any effort for systematic error compensation. Thus, it is even more essential that the vibration level in such systems be suppressed as far as possible with an efficient mechanical design and that functionality be included in the control to monitor and possibly adaptively reduce excessive vibration when it occurs. This chapter provides several possible approaches to this objective. The first approach will focus on a proper mechanical design, based on the determinacy of the machine structure, to reduce the mechanical vibration to a minimum. While the system design approach is certainly a first and key step to minimizing vibration in mechanical systems, a parallel monitoring and suppression mechanism is necessary to cope with additional and usually unpredictable sources of vibration seeping in during the course of operations. An approach, utilizing an adaptive notch filter (narrow bandstop filter) in the control system, will be presented as a means of continuously identifying resonant frequencies present and suppress signal transmission into the system at these frequencies. Finally, the development of a low cost real time vibration analyzer will be presented. This analyzer can be implemented independently of the control system, and as such can be applied to existing equipment without much retrofitting being necessary. A vibration signature is derived from the vibration signal acquired using an accelerometer that is attached to the machine under normal operating conditions. A pattern recognition template is used to compare real time vibration signals against the normal condition signature and an alarm can be activated when the difference deviates beyond an acceptable threshold. Actions of rectification can then be invoked before damage is done to the machine. A case study will be provided at the end of the chapter to illustrate the effectiveness of a remote vibration monitoring and control system for precision motion systems.

28.2

Mechanical Design to Minimize Vibration

In the development of high speed and high precision motion systems, the notion of determinism is a key consideration (Evan, 1989), which implies that a physical system complies with the law of cause and effect, and this behavior allows the physical system to be modeled mathematically.

Vibration Suppression and Monitoring in Precision Motion Systems

28 3

The governing equations describing the model can then be used to predict the behavior of the system and thus allow for the compensation for possible errors to meet the demand of a tight error budget. A mechatronic approach, in which the structural design and the control design are to be seamlessly integrated (Rankers, 1997), is one of the possible approaches for machine design. This approach has been adopted efficiently by many scientists and engineers, and the benefits are clearly evident in the end products, such as the wafer scanner and stepper. In this section, the key issues to address in a sound mechanical design to keep mechanical vibration to a minimum will be highlighted. The issue of mechanical design represents a very large area in precision motion systems. In this section, only key points will be highlighted in general, to enable designers to design “rigid” structures during the initial phase, even before the physical modeling stage. Design, being an iterative process, always requires the designer to revisit the drawing board frequently, until an optimum design is achieved. The section will give qualitative ideas with abundant figures to illustrate key ideas, rather than using a purely quantitative approach. The reason for this is that, during the initial phase of a design, intensive quantification is normally not necessary for decision making. Iteration and optimization, which are normally mathematically intensive, should be addressed during the next stage of the design process.

28.2.1

Stability and Static Determinacy of Machine Structures

A structure can be a supporting framework that houses all the subassemblies that make up a machine, or it can be a collection of many smaller structures, or even a single component. The reaction forces of the high speed moving parts will excite the structural dynamics resulting in mechanical vibrations. These vibrations can be attenuated by reducing either the excitation or the response of the structure to that excitation (Beards, 1983). The first factor can be overcome by relocating the source within the structure or by isolating it from the structure so that the generated vibration is not transmitted to the structure via the supports. As for the second factor, changing the mass, the stiffness, or the damping can alter the structural response. In order to understand the dynamic responses of the structure, the real structure can be transformed into a physical model, which is usually a simplified model representative of the real structure. For example, a real machine can be modeled as a number of coupled spring mass systems. A derived physical model of the real system can be translated to a mathematical model which can be solved via software or by hand, thereby allowing engineers from different disciplines to communicate and refine their portion of the design. Every designer working on the structure for a machine needs to answer a very important question. Is the designed structure rigid and stable? A structure is rigid if its shape cannot be changed without deforming the members in the structure (Fleming, 1997), and a structure is stable if rigid body translation or rotation cannot occur. A good way to tell whether a structure is stable or not is the degree of indeterminacy. A structure is considered statically determinate if all the support reactions and internal forces in the members can be determined solely by the equations of static equilibrium. Otherwise, the structure is considered statically indeterminate. Statically indeterminate structures arise owing to the presence of extra supports, members, reaction forces, or reaction moments. For a structure to be statically determinate, it must first be constructed correctly and then supported correctly.

28.2.2

Two-Dimensional Structures

Most machine structures, in practice, are three dimensional. However, it is useful to look at a two dimensional problem first, before extending the problem to a three dimensional one. Generally, machine structures are stationary. Therefore, the sum of the forces and moments acting on it must be zero, which is in accordance with Newton’s Second Law. In mathematical form, X

Fx ¼ 0

ð28:1Þ

28 4


X X

Fy ¼ 0

ð28:2Þ

Mz ¼ 0

ð28:3Þ

where Fx and Fy are the forces in the x axis and y axis, respectively, while Mz is the moment about the z axis, where the z axis is pointing out of the page. For a plane structure, we will make use of the sign conventions as depicted in Figure 28.1. Since the static determinacy of a structure is a two fold issue, it is possible to proceed without considering the support first. Each structural configuration can be tested to verify if the plane structure satisfies the equation:

y

x θz FIGURE 28.1

Sign convention for plane structure.

2j ¼ m þ 3

ð28:4Þ

where j denotes the number of joints and m denotes the number of members. There are three possible cases, as follows: 1. If 2j ¼ m þ 3; then the structure is statically determinate. 2. If 2j . m þ 3; then the structure is unstable. 3. If 2j , m þ 3; then the structure is statically indeterminate. The three conditions are depicted in Figure 28.2. When a structure is unstable due to member deficiency, it appears that the structure becomes a four bar mechanism with one degree of freedom (DoF), as shown in Figure 28.2c. This DoF is undesirable, since it is the structure that is being designed and not the mechanism. If, however, there are too many members present, the structure becomes statically indeterminate. Under such a condition, it will be difficult to assemble the fifth bar of the structure shown in Figure 28.2d, if the

(a)

(c)

j=3 m=3 2j=m+3 Statically Determinate

j=4 m=4 2j>m+3 Unstable

(b)

(d)

j=4 m=5 2j=m+3 Statically Determinate

j=4 m=6 2j<m+3 Statically Indeterminate

FIGURE 28.2 Plane structure: (a) and (b) statically determinate structure; (c) unstable structure; (d) statically indeterminate structure.


extra member

new members

(a)

new joint FIGURE 28.3

28 5

(b)

ground

(a) Unstable structure; (b) extra member.

dimension of the fifth bar is not exact. Assembly is probably possible using brute force, but internal stresses will be built into the structure even without any external loading. When a structure is statically determinate, it will be stress free when it is not loaded externally other than by its own weight. In the event of thermal expansion of its member, owing to an increase in temperature, statically determinate structures allow expansion of their members, without inducing any stress resulting from an overconstrained condition due to the redundant members. The triangle is the basic shape for a plane structure as shown in Figure 28.2a. Statically determinate plane structure can be expanded from this basic structure, simply by linking two new members to two different existing joints for every new joint added, as shown in Figure 28.2b. However, the axis of the two new members must not form a line; in other words, the three joints must not be on the same line, as shown in Figure 28.3a. It is also noteworthy that the ground constitutes one member as well, and all joints are pin joints, as shown in Figure 28.3b. The second part of structure design lies in its supports. In this aspect, the whole structure can be treated as a rigid body. A plane structure has three DoF; that is, the plane structure is capable of motion in the x and y directions, and rotation about the z axis. Therefore, three members are needed providing three reactive forces to exactly constrain the plane structure in the plane.

(a)

(c) FIGURE 28.4

Stable

Stable

(b)

(d)

Stable

Unstable

(a), (b), and (c) Stable and exactly constraint supports; (d) unstable support.

28 6


Figure 28.4a c show some possible support for plane structure, while Figure 28.4d shows an unstable support scenario. It should be highlighted that the condition of having two support members at the same location can be replaced by a single pin joint, as shown in Figure 28.5. It is apparent that, in both cases, they constitute two reaction forces and do not constrain FIGURE 28.5 Equivalent of a two member support. the angular motion about the z axis present in the plane structure. The correct number of members in a structure, as well as the correct number of supports, must be in place for a stable and statically determinate structure. At this juncture, the issue of where the loads are to be applied on the structure must be addressed. To this end, it is necessary to examine the members that make up the structure. The stiffness of a bar member is affected by the way the load is applied with respect to its axial axis, its cross sectional geometry (e.g., the diameter, for a round bar) and the modulus of elasticity, E, of its material. In most cases, the bar is either loaded in tension, compression, or bending, as shown in Table 28.1. It is apparent from examining Table 28.1 that the stiffness of a bar is much better in axial loading than in bending loading. For a value of d ¼ 0:05 m and L ¼ 1:2 m, the ratio of kt =kb is 192. That is, a bar is 192 times stiffer when loaded axially than in bending. Therefore, when designing a rigid and stiff structure, the members must be loaded in tension or compression, never in bending. At times, redesigning the way an external load is applied on a structure can greatly improve the stiffness of the structure. Various configurations are shown in Table 28.2. As a general rule to observe, the loading point should be located at the joints.

≈

28.2.3

Three-Dimensional Structures

Next, space structures or three dimensional structures will be considered. These are structures that are of interest in most applications. In a very general sense, space structures can be perceived as a combination of many plane structures, arranged in a manner that all the planes are not coplanar. Therefore, for a space structure to be rigid, every plane structure that makes up the space structure

TABLE 28.1

Comparison of Stiffness for Axial Loading versus Bending Loading Configurations

(a)

F

Stiffness (N/m) 2

Normalized Stiffness

Tension

kt ¼ 0:25pEd =L

1

Compression

kc ¼ 0:25pEd2 =L

1

Bending

kb ¼ 0:75pEd4 =L3

3ðd=LÞ2

F L

(b)

F

F 0.5L (c)

Loading Condition

L

F

Units: E (N/m2), d (m), L (m), and L .. d:

Vibration Suppression and Monitoring in Precision Motion Systems TABLE 28.2

Comparison of Stiffness for Various Loading Configurations Configurations

(a)

28 7

L

F

Stiffness (N/m)

F

Normalized Stiffness

Compare

kt ¼ 0:25pEd2 =L

1

1

kb ¼ 0:75pEd4 =L3

3ðd=LÞ2

1/192

k^ ¼ 0:5pEd2 sin2 b=L

sin2 b

1/2

kcl ¼ 0:047pEd4 =L3

0:1875ðd=LÞ2

1/3072

k. ¼ 0:25pEd2 =L

1

1

0.5L F

F (c)

L b

b F

(d)

L

L (e)

b

b

F

L

(f)

b ¼ 458; d ¼ 0:05 m, L ¼ 1:2 m.

must be rigid in its own right. This is one reason to have a good understanding of plane structural rigidity. Since machine structures are stationary, the sum of the forces and moments acting on the machine must be zero, which is in accordance with Newton’s Second Law. Mathematically, this implies X F¼0 ð28:5Þ X M¼0

ð28:6Þ

where F and M are three dimensional force and moment vectors, respectively. The sign conventions as depicted in Figure 28.1 will be used. As before, each structural configuration can be tested to verify if the plane structure satisfies the equation: 3j ¼ m þ 6

ð28:7Þ

28 8


where j denotes the number of joints and m denotes the number of members. There are three possible cases, as follows: 1. If 3j ¼ m þ 6; then the structure is statically determinate. 2. If 3j . m þ 6; then the structure is unstable. 3. If 3j , m þ 6; then the structure is statically indeterminate. In the plane structure, the triangle is the basic shape, and is rigid and statically deter minate. In a space structure, the basic form for being rigid and statically determinant is the tetrahedron, which is depicted in Figure 28.6. Adding a new noncoplanar joint to the three existing joints of a triangular plane structure derives the tetrahedron struc ture. This new joint is connected to the existing joints with three new members. By following this procedure, a rigid and statically determinate space structure can be derived. Other space structures are shown in Figure 28.7 and Figure 28.8. It is also note worthy that the members are connected with ball joints. Thus far, the approach to obtain the tetrahedron space structure from the triangle plane structure, the pyramid from the tetra hedron, and the box from the tetrahedron has been illustrated. The next aspect of the design is to combine some of these structures. The structures can be treated as being coupled together as rigid bodies, and a rigid body in space has six DoF, i.e., the structure is capable of translations in the x, y, and z directions, and rotation about the x, y, and z axes. Therefore, six members are needed, providing six reactive forces to exactly constrain the structure in space. Figure 28.9 shows a typical gantry configuration, which is used extensively in many coordinate measuring machines (CMM). However, one of the mem bers is bearing a bending load, which has been shown earlier to be very detrimental to the stiffness of the structure. There are alternative structure configurations as shown in Figure 28.10 and Figure 28.11, although some redesign may be needed if such a configuration is to be utilized.

FIGURE 28.6 structure.

Basic space structure

the tetrahedron

FIGURE 28.7 Pyramid structure derived from a tetrahedron structure.

FIGURE 28.8 Box structure derived from a tetra hedron structure.


28 9

If the ground is perceived as another rigid body to which the space structure is to be coupled, then the design of the supports for a space structure is similar to those of coupling two space structures together; that is, six reactive forces are needed to exactly constrain the space structure. Some ways to arrange the six supporting members constrain ing a space structure are suggested in Figure 28.12. Examples of physical supports offering one, two, or three reactive forces are shown in Figure 28.13. This method of design, known as kinematical design, requires the use of point contact at the interfaces. Unfortunately, this method has some disadvantages: *

*

*

Load carrying limitation. Stiffness may be too low for application. Low damping.

There are, however, ways to overcome the disadvantages, which are via the semikinematical approach. This approach is a modification of the kinematical approach, and it aims to overcome the limitations of a pure kinematical design. The direct way is to replace all point contact with a small area, as shown in Figure 28.14; doing so decreases the contact stress, but increases the stiffness and load carrying capacity. However, the area contact should be kept reasonably small. This section has only illustrated some fundamental concepts in designing rigid and statically determinate machine structures. Interested readers may refer to Blanding (1999) for more details on designing a machine using the exact constraints principles.

FIGURE 28.9

Gantry space structure.

FIGURE 28.10 Coupling of a tetrahedron structure to a box structure with six members.

SUMMARY The approach taken to reduce the mechanical vibration in precision motion systems is to focus on a proper mechanical design, based on the determinacy of machine structures in this chapter. The aim of this approach is to design systems with stable and statistically determinate structures. Two dimensional structures and three dimensional structures are considered.

28 10

28.3


Adaptive Notch Filter

The task of eliminating or suppressing undesirable narrowband frequencies can be efficiently accom plished using a notch filter (also known as a narrowband stop filter), if the frequencies are known. The filter highly attenuates a particular frequency component and leaves the rest of the spectrum relatively unaffected. An ideal notch filter has a unity gain at all frequencies except in the so called null frequency band, where the gain is zero. A single notch filter is effective in removing single frequency or narrowband interference; a multiple notch filter is useful for the removal of multiple narrowbands, which is necessary in applications requiring the cancellation of harmo nics. Digital notch filters are widely used to retrieve sinusoids from noisy signals, eliminate sinusoidal disturbances, and track and enhance FIGURE 28.11 Coupling of a triangular plane struc time varying narrowband signals with wideband ture to a tetrahedron space structure with six members. noise. They have found extensive use in the areas of radar, signal processing, communications, biomedical engineering, and control and instru mentation systems. To create a null band in the frequency response of a digital filter at a normalized frequency, b0 ; a pair of complex conjugate zeros can be introduced on the unit circle at phase angles ^ b0, respectively. The zeros are defined as z1;2 ¼ e^jb0 ¼ cos b0 ^ j sin b0

ð28:8Þ

where the normalized null frequency, b0, is defined as

b0 ¼ 2p

f0 fs

ð28:9Þ

FIGURE 28.12 Examples for applying six constraints to a rigid body: (a) three sets of twin reactive forces; (b) 3 2 1 reactive forces.


FIGURE 28.13

28 11

Examples of support with (a) one reactive force; (b) two reactive forces; and (c) three reactive forces.

Note that fs is the sampling frequency in Hz (or rad/sec) and f0 is the notch frequency in Hz (or rad/sec). This yields a finite impulse response (FIR) filter given by the following transfer function: HðzÞ ¼ 1

2 cos b0 z

1

þz

2

ð28:10Þ

A FIR notch filter has a relatively large notch bandwidth, which means that the frequency components in the neighborhood of the desired null frequency are also severely attenuated as a consequence. The frequency response can be improved by introdu cing a pair of complex conjugate poles. The poles are placed inside the circle with a radius of a at phase angles ^ b0. The poles are defined as p1;2 ¼ a e^jb0 ¼ aðcos b0 ^ j sin b0 Þ ð28:11Þ where a # 1 for filter stability, and ð1 aÞ is the distance between the poles and the zeros. The poles introduce a resonance in the vicinity of the null frequency, thus reducing the bandwidth of the notch. The transfer function of the filter is given by HðzÞ ¼

ðz ðz

z1 Þðz p1 Þðz

z2 Þ p2 Þ

ð28:12Þ

Substituting the expression for zi and pi ; and dividing throughout by z 2 ; the resulting filter has

FIGURE 28.14 Kinematical vs. semikinematical design: (a) ideal condition point contact; (b) line contact; (c) area contact.

28 12


the following transfer function: a0 þ a1 z 1 þ b1 z

HðzÞ ¼ HðzÞ ¼

1 1

1 1

þ a2 z þ b2 z

2 cos b0 z 2a cos b0 z

2

ð28:13Þ

2

1

þz 2 1 þ a2 z

ð28:14Þ

2

Digitally, the filtered signal, y, is thus obtained from the raw signal, u, via the recursive formula in the discrete time domain as follows: yðnÞ ¼ a0 uðnÞ þ a1 uðn

1Þ þ a2 uðn

2Þ

b1 yðn

1Þ

b2 yðn

where the coefficients ai and bi are the same as those in Equation 28.13 because z time shift (delay through sampling period) operator. The bandwidth and the Q factor of the notch filter are, respectively pffiffi 2 2ð1 a2 Þ BW ¼ ½16 2að1 þ aÞ2 1=2 Q ¼ w0

½16

2að1 þ aÞ2 pffiffi 2 2ð1 a2 Þ

1=2

2Þ 1

ð28:15Þ

corresponds to the

ð28:16Þ ð28:17Þ

The filter transfer function, HðzÞ; has its zeros on the unit circle. This implies a zero transmission gain at the normalized null frequency, b0. It is interesting to note that the filter structure, Equation 28.14, allows independent tuning of the null frequency and the 3 dB attenuation bandwidth by adjusting b0 and a, respectively. The performance of the notch filter depends on the choice of the constant, a, which controls the bandwidth, BW, according to Equation 28.16. The bandwidth, which is a function of the distance of the poles and zeros ð1 aÞ; narrows when a approaches unity. Clearly, when a is close to 1, the corresponding transfer function behaves virtually like an ideal notch filter. Complete narrowband disturbance suppression requires an exact adjustment of the filter parameters to align the notches with the resonant frequencies. If the true frequency of the narrowband interference that is to be rejected is stable and known a priori, a notch filter with fixed null frequency and fixed bandwidth can be used. However, if no information is available a priori, or when the resonant frequencies drift with time, the fixed notch may not coincide exactly with the desired null frequency, particularly if the bandwidth is too narrow (i.e., a < 1). In this case, a tunable or adaptive notch filter is highly recommended. In Ahlstrom and Tompkins (1985) and Glover (1987), it is proposed to adapt the null bandwidth of the filter to accommodate the drift in frequency. In Bertran and Montoro (1998), it is suggested that an active compensator be used to suppress the vibration signals. Kwan and Martin (1989) adapt the null frequency, b0, while keeping the pole radii, a, constant. In other words, the parameters ai and bi of Equation 28.13 are adjusted such that the notch will center at the unwanted frequency while retaining the null bandwidth of the notch filter.

28.3.1

Fast Fourier Transform

The discrete Fourier transform (DFT) is a tool that links the discrete time domain to the discrete frequency domain (see Chapter 10, Chapter 21, and Appendix 2A). It is a popular off line approach, widely used to obtain the information about the frequency distribution required for the filter design. However, the direct computation of the DFT is prohibitively expensive in terms of required computation effort. Fortunately, the fast Fourier transform (FFT) is mathematically equivalent to the DFT, but it is a more efficient alternative for implementation purposes (with a computational speed that is exponentially faster) and can be used when the number of samples, n, is a power of two (which is not a serious constraint). For vibration signals where the concerned frequencies drift with time, the FFT can be continuously applied to the latest n samples to update the signal spectrum. Based on the updated spectrum, the filter characteristics can be continuously adjusted for notch alignment. The block diagram


. .. xd, xd, xd

e

Controller

Control signal

Adaptive Notch Filter

28 13

PMLM

x

Adjusting mechanism based on FFT

FIGURE 28.15 Block diagram of the adaptive notch filter with adjusting mechanism. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

of the adaptive notch filter that has been developed in the present work, with its adjusting mechanism, is shown in Figure 28.15.

28.3.2

Simulation and Experiments

A simulation study is carried out to explore the application of the adaptive notch filter in suppressing undesirable frequency transmission in the control system for a precision positioning system that uses permanent magnet linear motors (PMLM). In the simulation, a sinusoidal trajectory profile is to be closely followed and an undesirable vibration signal is simulated that drifts from a frequency of 500 Hz in the first cycle to a frequency of 1 to 5 Hz in the second cycle of the trajectory. Figure 28.16 shows

FIGURE 28.16 Simulation results without a notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

28 14


FIGURE 28.17 Simulation results using a fixednotch filter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.18 Simulation results using an adaptive notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)


28 15

the tracking performance of the precision machine without a notch filter. Figure 28.17 shows the performance when a fixed notch filter is used, and Figure 28.18 shows the performance with an adaptive notch filter. It is clearly evident that a time invariant narrowband vibration signal can be effectively eliminated using just a fixed notch filter. However, when the vibration frequencies drift, an adaptive notch filter is able to detect the drift and FIGURE 28.19 Hardware setup for the experimental align the notch to remove the undesirable study of the notch filter. (Source: Tan, K.K., Tang, K.Z., frequencies, with only a short transient period. de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy The notch filter is subsequently implemented in Syst., 2001, IOS Press. With permission.) the control system of a linear drive tubular linear motor (LD3810) equipped with a Renishaw optical encoder having an effective resolution of 1 mm. The hardware setup for this experimental study is shown in Figure 28.19 whereas Figure 28.20 shows the linear motor in more detail. The components of the linear motor consist of the thrust rod, the thrust block, the motor cable, and the optical encoder. The thrust rod is made of a thin walled stainless steel tube housing high energy permanent magnets. To enable the smooth FIGURE 28.20 Experimental platform (LD 3810) translation of the thrust block along the length of linear drive tubular motor with optical encoder the thrust rod, the thrust block is made from an attached. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., aluminum housing that contains cylindrical coils Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS arranged in a three phase star pattern. An Press. With permission.) electromagnetic field is produced by energizing these coils. The interactions between the permanent magnetic field of the thrust rod and the changing magnetic field of the thrust block provide the induced force for the translation of the block. Usually, a sinusoidal or trapezoidal motor commutation is utilized to smoothen the translation of the block. The popular proportional integral derivative (PID) control is used in this experimental study. The dSPACE DS1102 (dSPACE User’s Guide, 1996) digital signal processing (DSP) board is used as the data acquisition and control card. It is a single board system, which is specifically designed for the development of high speed multivariable digital controllers and real time simulations in various fields. The DS1102 is based on the Texas Instruments TMS320C31 third generation floating point DSP, which builds the main processing unit, providing fast instruction cycle time for numeric intensive algorithms. It contains 128K words memory that is fast enough to allow zero wait state operation. Besides these, the DS1102 DSP board supports a total memory space of 16M 32 bit words, including program, data, and I/O space. All off chip memory and I/O can be accessed by the host, even while the host is running, thus allowing easy system setup and monitoring. The TMS320C31 is object code compatible to the TMS320C30. The DSP is fully supplemented by a set of on board peripherals, frequently used in digital control systems. Analog to digital and digital to analog converters, a DSP microcontroller based digital input/output (I/O) subsystem, and incremental sensor interfaces make the DS1102 an ideal single board solution for a broad range of digital control tasks. The DS1102 DSP board is well supported by popular software design and simulation tools, including MATLAB and SIMULINK, which offer a rich set of standard and modular design functions for both classical and modern control algorithms. The SIMULINK model developed for the system (Figure 28.15), with the notch filter, is shown in Figure 28.21. This model is then downloaded to the DS1102 DSP board for real time implementation using one of the options available from the pull down menu.

28 16


FIGURE 28.21 SIMULINK model created for the system with the notch filter incorporated. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.22 Experimental results without a notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)


28 17

FIGURE 28.23 Experimental results using a notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

It is now in order to present the experimental results utilizing the notch filter in the system. Figure 28.22 shows the performance of the PMLM when no filter is used. Figure 28.23 shows the improvement in the control performance when the notch filter is incorporated into the control system.

SUMMARY An approach utilizing an adaptive notch filter in the control system is elaborated upon in this section. The notch filter is able to adaptively identify resonant frequencies present in the motion system and suppress vibration signal transmission into the system at these frequencies. The FFT is used to obtain the frequency distribution of the vibration signals. Experimental and simulation results are provided to illustrate the effectiveness of the adaptive notch filter.

28.4

Real-Time Vibration Analyzer

The development of an alternative, low cost approach towards real time monitoring and analysis of machine vibration (Vierck, 1979; de Silva, 2000) is described in this section. The main idea behind this approach is to construct a vibration signature based on pattern recognition of “acceptable” or “healthy” vibration patterns. The vibration analyzer can operate in three modes: learning, monitoring, or diagnostic. The learning mode, to be initiated first, will yield a set of vibration signatures based on which the monitoring and diagnostic modes will operate. In the monitoring mode, with the machine

28 18


Machine

Signal conditioning

Accelerometer

Vibration signals

Vibration monitoring program

Activate alarm or corrective action

DSP module

FIGURE 28.24 Schematic diagram of the real time vibration analyzer. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

under normal closed loop control, the analyzer only uses a naturally occurring vibration signal to deduce the condition of the machine. No test excitation is deliberately added to the input signal of the machine. More than one criterion may be used in the evaluation of the condition of the machine, in which case a fusion approach will generate a combined output (machine condition) based on the multiple inputs. In the diagnostic mode, explicit input signals are applied to the machine and the output signal (vibration) is logged for analysis with respect to the associated vibration signature. In what FIGURE 28.25 Hardware setup for the standalone follows, the details of the various components DSP module. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, and functions of the analyzer will be described IOS Press. With permission.) systematically. The block diagram of the real time vibration analyzer system that has been developed is shown in Figure 28.24. It consists of an accelerometer, which is mounted on the machine to be monitored. The accelerometer measures a multifrequency vibration signal and transmits it to an intelligent DSP module after performing appropriate signal conditioning. This module can be a standalone device (Figure 28.25), or one integrated to a personal computer (PC) host. The vibration analysis algorithm is downloaded to this DSP module. With this algorithm, it can be established as to whether the condition of the machine is within a predetermined acceptable threshold. If the condition is determined to be poor, the DSP module will trigger an alarm to the operator who would enable a corrective action, or automatically activate a corrective action (e.g., change the operating conditions of the machine, modify the parameters of the controller or shutdown the machine). The construction of the real time vibration analyzer is inexpensive and requires only commercially available, low cost components. The installation can be hassle free, as the accelerometer is able to gather vibration signals independent of the machine’s own control system. Thus, there is no need to disrupt the operation of the machine. In the prototype reported here, a DSP emulator board TMS320C24x model (TMSS320C24x DSP Controllers Evaluation Module Technical References, 1997), from Texas Instruments, is used as the standalone DSP module (Figure 28.25). This C24x series emulator board is built around the F240 DSP controller, operating at


28 19

20MIPS with an instruction cycle time of 50 ns. It is optimized for digital motor control and conversion applications. Other key components supported on this DSP module are analog to digital converters (ADCs), dual access RAM (DARAM), on chip flash memory, and an RS 232 compatible serial port. This DSP module and the accelerometer unit (with signal conditioning) constitute the only hardware requirements of the real time vibration analyzer. The vibration analysis algorithm, which is described in the sequel, will be downloaded to the flash ROM on the DSP module after satisfactory evaluation and tests on the PC. The algorithm is programmed using a Visual Cþþ based compatible environment, that is, a Code Composer integrated environment (IDE). The C based source code is then compiled into assembly code, using the built in compiler available in the IDE.

28.4.1

Learning Mode

In the learning mode, with the machine operating under normal conditions, the vibration signals are acquired by the accelerometer and stored in the DSP module. A suitable vibration signature (Ramirez, 1985) is then extracted from the vibration signals. There are many types of vibration signatures that are adequate for the purpose of machine monitoring. For example, one form of vibration signature may be based on the amplitude of the vibration; another form may use a time series analysis of the vibration; yet another form may employ the spectrum of the vibration, which can be efficiently obtained using the FFT algorithm. Regardless of the type, these vibration signatures are dependent on the nature of the input signals driving the machine. For example, a square wave input will produce a vibration spectrum that can be quite different from that resulting from an input of a chirp signal (i.e., a repeating sine wave of increasing frequency) or a pure sinusoid. Thus, a particular input signal will produce a unique spectrum based on which a unique vibration signature can be derived. Multiple vibration signatures corresponding to the natural vibrations of the machine (useful for the monitoring mode) or corresponding to different input signals (useful for the diagnostic mode) can thus be captured for subsequent diagnosis and monitoring of the machine.

28.4.2

Monitoring Mode

In the monitoring mode, the vibration signals are sampled periodically from the machine to monitor the condition of the machine. No deliberate or additional input signal is required, so the machine operation is not disrupted. The updated spectra are analyzed against the relevant vibration signatures. The analysis and comparison may be done in terms of the shift in the frequency or the amplitude of the spectrum, or a combination of the two. For example, one evaluation criterion (EV) may be based on the mean square (ms) value of the error between the current real time vibration spectrum and the vibration signature: N X

EV1 ¼

q¼1

ðSq

Spq Þ2 ð28:18Þ

M

where Sq is the discretized current real time vibration spectrum, the superscript p represents the vibration signature of the “healthy” machine, subscript q is the index for the data points, N is the total number of frequency points, and M is the total number of data points. Another EV may be formulated based on the difference in the amplitude of the current time series vibration pattern and its corresponding vibration signature: EV2 ¼

maxðTqp Þ

maxðTq Þ M

ð28:19Þ

28 20


where maxðTq Þ represents the highest amplitude of the current time series vibration pattern Tq ; and maxðTqp Þ is the highest amplitude of its corresponding vibration signature. More than one evaluation criterion may be used in the determination of machine condition. In this case, a fusion technique is necessary. The key idea of fusion is to associate the machine with a HEALTH attribute, which is computed from multiple evaluation criteria. These criteria are expected to influence, to a varying degree, the HEALTH of the machine. The HEALTH attribute is thus an appropriate function, I; of the various criteria (EVis); that is HEALTH ¼ IðEV1 ; EV2 ; …; EVn Þ

ð28:20Þ

where n refers to the number of criteria being evaluated. A fuzzy weighted approach may be used to realize the I function as follows. The HEALTH attribute is treated as a fuzzy variable (i.e., HEALTH [ ½0; 1 ). HEALTH ¼ 0 will represent absolute machine failure, while HEALTH ¼ 1 represents a perfectly normal machine condition. This attribute may be computed from a fuzzy operation on a combination of the evaluation criteria (EVis) obtained via an analysis of the vibration signals against their signatures. The final decision on the condition of the machine will be derived from the HEALTH attribute. Zadeh (1973) provides a comprehensive review on fuzzy logic. A Takagi and Sugeno (1985) type of fuzzy inference is used in this chapter. Consider the following p rules governing the computation of an attribute: IF EVi1 IS F1i ^ · · · ÊVin IS Fni ; THEN ui ¼ ai p X

ai ¼ 1

ð28:21Þ ð28:22Þ

i¼1

where ui [ ½0; 1 is a crisp variable output representing the extent to which the ith evaluation rule affects the final outcome. Thus, ai represents the weight of the ith rule, Fji represents the fuzzy sets in which the input linguistic variables (EVis) are evaluated, and ^ is a fuzzy operator that combines the antecedents into premises. The value of the attribute is then evaluated as a weighted average of the ui s: p X

HEALTH ¼

w i ui

i¼1 p X

ð28:23Þ w

i

i¼1

where the weight wi implies the overall truth value of the premise of rule, i, for the input. It is computed as wi ¼

n Y j¼1

mFji ðEVij Þ

mHIGH(MAX _ ERR)

ð28:24Þ

where mFji ðEVij Þ is the membership function for the fuzzy set, Fji ; related to the input linguistic variable, EVij ; (for the ith rule). For example, in this application, EVj may be the maximum error, MAX ERR, and Fji may be the fuzzy set HIGH. The membership function is represented as mHIGH ðMAX ERRÞ: It may have the characteristic shown in Figure 28.26. The decision as to whether any corrective action might be

MAX _ ERR

FIGURE 28.26 Membership function for the fuzzy input MAX ERR. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)


28 21

necessary can then be based on a simple IF THEN ELSE formulation as follows: IF HEALTH # g, THEN STRATEGY ¼ TRIGGER ALARM ELSE STRATEGY ¼ CONTINUE TO MONITOR Here, g is interpreted as a threshold value. Suitable values for g may be in the range 0.6 # g # 0.9. Here, STRATEGY is stated to trigger an alarm to the operator who will enable a corrective action, or automatically activate a corrective action (e.g., change the operating conditions of the machine, modify the parameters of the controller, or shutdown the machine). Under this framework, it is relatively easy to include additional criteria for analysis and decision making to the system. The procedure will involve setting up the membership functions for the criterion, formulating the additional fuzzy rules required, and adjusting the scaling parameters (the a terms in Equation 28.21) to reflect the relative weight of the new criterion as compared with the existing ones. In this manner, in the monitoring mode, foreboding trends can often be spotted long before the vibration condition reaches a level that is seriously detrimental to the machine.

28.4.3

Diagnostic Mode

In the diagnostic mode, the current vibration signal corresponding to each input signal (with standardized amplitude and frequency) is analyzed against the associated signature obtained earlier in the learning mode, depending on the type of machine (also see Chapter 25). Similar to the monitoring mode, there can be multiple evaluation criteria used in the diagnostic mode, so that the fusion technique described earlier is also applicable. The input signals applied to the machine must be designed carefully so as to yield as much information of the machine condition as possible in the operational regime of interest. Two important considerations are the choices of amplitude and frequency. Machines may have constraints in relation to the amount of travel that is possible. Too large an amplitude for the input signal may be not be viable for the machine owing to the limit of travel, or may even damage the machine. Also, the frequency range of the input should be chosen so that it has most of its energy in the frequency bands that are important for the system. Where input signals cannot be applied to the system in the open loop, the setpoint signal will serve as the input for the closed loop system since it may not be possible to directly access the system under closed loop control. Careful considerations of the mentioned issues will ensure that significant information can be obtained from the machine. The input signals considered here are square wave input (Figure 28.27), chirp input (Figure 28.29), and sine wave input (Figure 28.31), standardized in amplitude to 1 V and in frequency to 5 Hz. The corresponding vibration signatures are shown in Figure 28.28, Figure 28.30, and Figure 28.32, respectively.

28.4.4

Experiments

A shaker table (Figure 28.33) is used as the test platform for the experiments presented here (also see Chapter 15). The shaker table can be used to simulate machine vibrations and evaluate the performance, for example, of active inertial dampers. The shaker table is driven by a high torque direct drive motor (which has a maximum torque of 1.11 N m, a maximum design load of 11 kg and generates a maximum force of 175 N). The maximum linear travel of the table is ^ 2 cm. The learning mode is first initiated to obtain the vibration signals with the shaker table operating under normal conditions. It is assumed in the experiments that the normal condition corresponding to the input is a square wave signal (with a standardized amplitude of 1 V and frequency of 5 Hz). For the purpose of implementing the diagnostic mode, the vibration signals are also obtained for the input signals of the sinusoidal and chirp type, with standardized amplitudes of 1 V and

28 22


FIGURE 28.27 Square wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

frequencies of 5 Hz (see Figure 28.27 to Figure 28.32 for the inputs and their corresponding vibration signatures). 28.4.4.1

Input Variables — the Evaluation Criteria

Different types of EV can be used as input variables for the determination of the machine condition. For the present vibration analysis application, the input variables chosen for the computation of the HEALTH attribute are given below. 28.4.4.1.1

Monitoring Mode N X

EV1 ¼ EV2 ¼

q¼1

ðSsq;q

ð28:25Þ

M

ðmaxðTsq;q Þ M X

EV3 ¼

Spsq;q Þ2

q¼1

maxðTsq;q Þp Þ2 M

ðTsq;q M

ð28:26Þ

p Þ2 Tsq;q

ð28:27Þ

where Ssq;q represents the vibration spectrum for a square wave input (driving the machine) at the qth frequency point and Tsq;q represents the time domain signal of a square wave input at the qth time instant. N refers to the total number of points in the FFT spectrum and M is the number

FIGURE 28.28 Vibration signature of the square wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)


28 23

Time domain signal

1 0.8 0.6 0.4

Voltage (V)

0.2 0 –0.2 –0.4 –0.6 –0.8 –1

0

0.5

1

1.5

2

2.5 3 Time (secs)

3.5

4

4.5

5

FIGURE 28.29 Chirp wave input, with a standardized amplitude of 1 V and initial frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

of time series data points over an operational cycle. Hence, EV1 refers to the ms deviation between the vibration spectrum and its signature; EV2 refers to the square of the difference between the amplitude of the vibration signal over one operational cycle compared with its signature; EV3 refers to the ms deviation between the vibration signal and its signature (time domain) over one operational cycle. The superscript p represents the signature of the healthy machine. 28.4.4.1.2

Diagnostic Mode N X

EV4 ¼

q¼1

q¼1

ð28:28Þ

ðScp;q

Spcp;q Þ2 ð28:29Þ

M N X

EV6 ¼

Spsq;q Þ2

M N X

EV5 ¼

ðSsq;q

q¼1

ðSsn;q M

Spsn;q Þ2 ð28:30Þ

Here, cp denotes a chirp input signal and sn denotes a sine input signal. For the monitoring mode, the input attributes are related only to the square input owing to the assumption that the input signal, under normal operating conditions, is the square wave signal (with a standardized amplitude of 1 V and frequency of 5 Hz).

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FIGURE 28.30 Vibration signature of the chirp wave input, with a standardized amplitude of 1 V and starting frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

28.4.4.2

Evaluation Rules

The three rules for the computation of the HEALTH attribute are given below. 28.4.4.2.1

Monitoring Mode

IF EV1 IS LOW, THEN u ¼ a1 IF EV2 IS SHORT, THEN u ¼ a2 IF EV3 IS LOW, THEN u ¼ a3 The values of the scaling parameters, that is, a terms in Equation 28.21, reflect the relative importance of the fuzzy rules in the determination of the HEALTH of the machine. The scaling values of a1 ; a2 ; and a3 are set at 0.7, 0.2, and 0.1, respectively. The respective membership functions are

mi ðEVi Þ ¼ e

nðEVi Þb

ð28:31Þ

where n and b are scaling factors for normalization of EVi. Here, they are selected to be n ¼ 10 and b ¼ 0:5: 28.4.4.2.2 Diagnostic Mode The three evaluation rules for the computation of the HEALTH attribute in the diagnostic mode are IF EV4 IS LOW, THEN u ¼ a4 IF EV5 IS LOW, THEN u ¼ a5 IF EV6 IS LOW, THEN u ¼ a6


28 25

FIGURE 28.31 Sine wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

The scaling values of a4 ; a5 ; and a6 are set at 0.4, 0.2, and 0.4, respectively. Similar membership functions as for the monitoring mode are used here. The machine condition attribute HEALTH is then computed as in Equation 28.23. 28.4.4.3

Tests

28.4.4.3.1 Monitoring Mode In the monitoring mode, the normal input signal (i.e., the square wave with standardized amplitude of 1 V and frequency of 5 Hz) is applied to the shaker table system. At t ¼ 5 sec, a sinusoidal signal (with amplitude 0.4 V and frequency f ¼ 5 Hz) is also applied to the system to simulate a fault arising in the machine. The time domain signal of the machine (corresponding to the square input) is shown in Figure 28.34. The spectra of the machine before and after t ¼ 5 sec are shown in Figure 28.35. The vibration analysis algorithm is able to detect the fault in the machine. Before the introduction of the fault, the HEALTH attribute of the shaker table is found to be 0.98. After the introduction of the fault, the HEALTH attribute falls to 0.63, which is below the threshold value, set at 0.7. As a result, the alarm is triggered.

FIGURE 28.32 Vibration signature of the sine wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

28 26


FIGURE 28.33 Test platform: the shaker table. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

28.4.4.3.2 Diagnostic Mode In the diagnostic mode, three input signals (i.e., sine, square, and chirp wave with standardized amplitude and frequency) are selected to be applied to the shaker table system in turn. To simulate a fault arising at t ¼ 5 sec, the input gain is increased by a factor of 1.4 times at t ¼ 5 sec. The time domain

FIGURE 28.34 Time domain vibration signal corresponding to the square input, with a standardized amplitude of 1 V and frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)


28 27

FIGURE 28.35 (a) Vibration signature corresponding to the square input, with a standardized amplitude of 1 V and frequency of 5 Hz; (b) spectrum of the machine corresponding to the square input after fault occurs. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

vibration signal of the machine (corresponding to the chirp signal, with a standardized amplitude of 1 V and starting frequency of 5 Hz) is shown in Figure 28.36. The spectra (corresponding to the chirp signal) of the machine before and after t ¼ 5 sec are shown in Figure 28.37. The time domain vibration signal of the machine (corresponding to the sinusoidal wave input, with a standardized amplitude of 1 V and

FIGURE 28.36 Time domain vibration signal corresponding to the chirp input, with a standardized amplitude of 1 V and starting frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated).

28 28


FIGURE 28.37 (a) Vibration signature corresponding to the chirp input, with a standardized amplitude of 1 V and starting frequency of 5 Hz; (b) spectrum of the machine corresponding to the chirp input after a fault occurs.

FIGURE 28.38 Time domain vibration signal corresponding to the sinusoidal input, with a standardized amplitude of 1 V and frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated).

frequency of 5 Hz) is shown in Figure 28.38. The spectra (corresponding to the sinusoidal input) of the machine before and after t ¼ 5 sec are shown in Figure 28.39. The vibration analysis algorithm is able to detect the fault in the machine. Before the introduction of the fault, the HEALTH attribute of the shaker table is found to be about 0.97. After the introduction of the fault, the HEALTH attribute falls to 0.58, which is below the threshold value, set at 0.7. The alarm is triggered as a result.


28 29

FIGURE 28.39 (a) Vibration signature corresponding to the sinusoidal input, with a standardized amplitude of 1 V and frequency of 5 Hz; (b) spectrum of the machine corresponding to the sinusoidal input after fault occurs.

SUMMARY In this section, the development of a low cost approach towards real time monitoring and analysis of machine vibration is described. A real time analyzer, based on a fuzzy fusion technique, is used to monitor continuously and compare the actual vibration pattern against a vibration signature. This intelligent knowledge based real time analyzer is able to detect excessive vibration conditions much sooner than a resulting fault could be detected by an operator. Subsequently, appropriate actions can be taken. This approach may be implemented independently of the control system and as such can be applied to existing equipment without modification of the normal mode of operation. Experimental and simulation results are provided to illustrate the effectiveness of this real time vibration analyzer.

28.5

Practical Insights and Case Study

There are currently many companies in the industry that deal with machinery vibrations. Broadly, these companies can be classified into those that produce vibration related testing products (e.g., FFT analyzers [Steinberg, 2000], vibration data collectors, and end of line production test equipment), and those that provide solutions in resolving noise and vibration related problems for different industries (e.g., automotive, aerospace, manufacturing, and engineering companies). For instance, if one of the machines in operation on the shop floor is experiencing unexplained machinery noise or a high level of product failure, vibration analysis (Farrar et al., 2001) may provide some answers. The case study explores the feasibility of remote vibration monitoring and control of a number of machines or mechanical systems located at different sites. For the sake of generality, “machines” is used as a general term to represent vibrating machines, mechanical systems, and other systems that exhibit vibration behavior. There are strong motivations for such setups. Since the middle of the 1960s, the concept of the distributed system has been widely adopted in the process industries, such

28 30


as the chemical industry, manufacturing industry, food processing industry, primary industry, and many others. These distributed systems are also characterized by smart devices. As the costs of manufacturing, that is, labor costs, the cost of raw materials, fixed costs, rental costs, and other production costs, are a major concern for the location of plants, many processes are currently widely distributed geometrically. The layout of an entire plant can be rather extensive, spreading across continents in certain cases. The extensive distribution of an entire plant requires close coordination and synchronization of the distributed operations, as well as an efficient remote monitoring and control facility in place. The utilization of the Internet for this monitoring and control purpose perfectly complements the trend of distributed intelligence in the field controllers and devices on the shop floor. As a result, superior control decisions can be made with these readily available resources and information, that is, historical data, knowledge databases, and so on. Internet working has become essential for plant automation. A case study which implements remote vibration monitoring and control of various machines will be illustrated in this section. This vibration monitoring and control system possesses the structure as shown in Figure 28.40. It is able to monitor continuously the real time health conditions of multiple machines at different sites, connected via the Internet. The system operates in two modes: the learning mode and the monitoring mode, as mentioned earlier. In the learning mode, vibration signatures, representative of the health of the machines to be monitored, are stored in the supervisory controller. Accelerometers mounted on the machine directly provide measurements of the vibration signals. In the monitoring mode, real time vibration signals are streamed from the front end controllers to the supervisory controller. Based on these real time vibration signals from the various machines, the supervisory controller is able to generate decisions on the health condition of the machines, taking into consideration various criteria based on a fuzzy fusion technique, as mentioned earlier. Alarms will be activated when the health condition of the machines falls below an acceptable threshold level. Subsequently, rectification actions, to provide a warning or automatic corrective action (e.g., changing the operating conditions of the machine,

FIGURE 28.40

Remote vibration monitoring and control system.


28 31

modifying the parameters of the controller, or shutting down the machine), may be invoked before extensive damage is done to the machines. Referring to the architecture of the vibration monitoring and control system as shown in Figure 28.40, a data transfer and communication protocol needs to be adopted for such an application. There are several ways of connecting the supervisory controller and the remote machines. One of the possible ways is to apply DataSocket technology; yet another is to use low level protocols such as TCP/IP and User Datagram Protocol. As can be seen, this vibration monitoring and control system may be applied to existing setups at the shop floor level, without much modification necessary. The front end field devices (i.e., actuators, sensors, and other I/O modules) are linked up via the field level data highway. Two popular types of field level data highway are fieldbus and twisted pair. Access security is a main issue of such remote applications. There is a need to prevent unauthorized users from accessing and modifying the system so as to maintain the integrity and proper functioning of the system. This can be achieved by

FIGURE 28.41

Learning mode

snapshot of the control panel at the supervisory controller end.

28 32


imposing some access authentication scheme. One commonly used basic access authentication scheme is the challenge response authentication mechanism whereby the operator or user must provide some user logon information (e.g., user identification and a password) before accessing the supervisory controller terminal. The control panel at the supervisory controller end (Figure 28.40) is as shown in Figure 28.41. At the supervisory controller end, the operator is able to monitor and control the health condition of various machines at different remote sites. As shown here, two machines (i.e., machine A and machine B) are being monitored. The learning mode is first initiated to obtain the vibration signatures of the two machines in the normal operational condition. The vibration signature of machine A resembles a chirp like signal input whereas that of machine B resembles a sinusoidal like signal input. At the end of the learning mode, a message is shown in the message box. Figure 28.42 shows a snapshot of the control panel in the monitoring mode. Here, the two machines are in good health, as the health indices

FIGURE 28.42 Monitoring mode snapshot of the control panel at the supervisory controller end before any fault is invoked on machine A and machine B.


28 33

(i.e., current values are at 0.9) of the two machines are above the threshold value of 0.6. To simulate a fault arising in machine A, the input signal to machine A is changed to a sinusoidal input. This causes a simultaneous change in the vibration pattern of Machine A. A snapshot of the control panel at the supervisory controller end is as shown in Figure 28.43 after the fault is invoked to machine A. As can be clearly seen, the vibration pattern of machine A has changed. The message box displays that the health index of machine A (current value is at 0.49) has fallen below the threshold value of 0.6. To signal the deterioration of the health condition of machine A, the alarm LED is lit up. The health condition of machine B remains in the satisfactory region. To simulate a fault arising in machine B, the amplitude of the input signal to Machine B is increased slightly, by 20%. Consequently, this will also cause simultaneous change in the vibration pattern of machine B. A snapshot of the control panel at the supervisory controller end, after the fault is invoked to machine B, is as shown in Figure 28.44.

FIGURE 28.43 Monitoring mode snapshot of the control panel at the supervisory controller end when a fault is invoked on machine A (at t ¼ 3 sec).

28 34


FIGURE 28.44 Monitoring mode snapshot of the control panel at the supervisory controller end when a fault is invoked on machine B (at t ¼ 5 sec).

The vibration pattern of machine B has changed. The message box displays the health index of machine B (current value is at 0.53) has fallen below the threshold value of 0.6. In the same way, the deterioration of the health condition of machine B is highlighted by activating the alarm LED.

SUMMARY This section gives the reader practical insights into vibration monitoring and control applications in the industry. A case study is provided to explore the feasibility of remote vibration monitoring and control of a number of machines or mechanical systems located at different sites.


28.6

28 35

Conclusions

Approaches to address undue vibration arising in mechanical systems have been presented in the chapter. The first approach presented is concerned with a proper mechanical design with a view to reduce the sources of vibration. The second approach is the possibility of including functionalities in the control system to continuously monitor and possibly suppress vibration by terminating transmission at the resonant frequencies. The third approach uses a low cost monitoring device, which can trigger alarm when excessive vibration is detected to enable corrective actions. Precision motion systems, where the effects due to vibration will directly affect performance, will usually adopt a number of such strategies to keep vibration level to a minimum and acceptable level.

References Adams, M.L. Jr. 2000. Rotating Machinery Vibration: From Analysis to Troubleshooting, Marcel Dekker, New York. Ahlstrom, M.L. and Tompkins, W.J., Digital filters for real time ECG signal processing using microprocessors, IEEE Trans. Biomed. Eng., 32, 708 713, 1985. Beards, C.F. 1983. Structural Vibration Analysis: Modelling, Analysis and Damping of Vibrating Structures, Ellis Horwood, Chichester, U.K. Bertran, E. and Montoro, G. 1998. Adaptive suppression of narrow band vibrations, 5th International Workshop on Advanced Motion Control, Coimbra, Portugal, June 1998, pp. 288 292. Blanding, D.L. 1999. Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York. de Silva, C.W. 2000. Vibration: Fundamentals and Practice, CRC Press, Boca Raton, FL. dSPACE DS1102 User’s Guide, Document Version 3, dSPACE GmbH, Paderborn, Germany, 1996. Evan, C. 1989. Precision Engineering: An Evolutionary View, Cranfield Press, Bedford, U.K. Farrar, C.R., Doebling, S.W. and Duffey, T.A. 2001. Vibration based Damage Detection, Structural Dynamics @ 2000: Current Status and Future Directions, Research Studies Press, Baldock, U.K., pp. 145 174. Fertis, D.G. 1995. Mechanical and Structural Vibrations, Wiley, New York. Fleming, J.F. 1997. Analysis of Structural Systems, Prentice Hall, Englewood Cliffs, NJ. Glover, J.R. Jr., Comments on digital filters for real time ECG signal processing using microprocessors, IEEE Trans. Biomed. Eng., 34, 1, 962 963, 1987. Kelly, S.G. 2000. Fundamentals of Mechanical Vibrations, 2nd ed., McGraw Hill, Boston, MA. Kwan, T. and Martin, K., Adaptive detection and enhancement of multiple sinusoids using a cascade IIR filter, IEEE Trans. Circuits Syst., 36, 7, 937 947, 1989. Raichel, D.R. 2000. The Science and Applications of Acoustics, Springer Verlag, New York. Ramirez, R.W. 1985. The FFT Fundamentals and Concepts, Prentice Hall, Englewood Cliffs, NJ. Rankers, A.M. 1997. Machine Dynamics in Mechatronic Systems An Engineering Approach, In house Report, Philips, Amsterdam. Steinberg, D.S. 2000. Vibration Analysis for Electronic Equipment, 3rd ed., Wiley, New York. Takagi, T. and Sugeno, M., Fuzzy identification of systems and its applications to modelling and control, IEEE Trans. Syst. Man Cybern., 15, 116 132, 1985. Tan, K.K., Lee, T.H., Dou, H.F., Huang, S. 2001. Precision Motion Control: Design and Implementation, Springer Verlag, London. TMSS320C24x DSP Controllers Evaluation Module Technical References, Tarrant Dallas Printing, Dallas, TX, 1997. Uicker, J.J., Pennock, G.R. and Shigley, J.E. 2003. Theory of Machines and Mechanisms, 3rd ed., Oxford University Press, New York. Vierck, R.K. 1979. Vibration Analysis, Crowell, New York. Zadeh, L.A., Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. Syst. Man Cybern., 3, 28 44, 1973.

Seismic Vibration

VII

VII 1

29

Seismic Base Isolation and Vibration Control 29.1 Introduction ........................................................................ 29 1 From Ductility Design to Base Isolation and Control Design The Importance of Reducing Seismic Input and Response

†

29.2 Seismic Base Isolation ......................................................... 29 4

Hirokazu Iemura Kyoto University

Sarvesh Kumar Jain

Madhav Institute of Technology and Science

Mulyo Harris Pradono Kyoto University

Historical Development of Base Isolation † Basic Principle † Issues in Seismic Base Isolation † Seismic Isolation Devices † Design of Isolation Devices † Verification of Properties of Isolation Systems † Analysis of Base Isolated Structures † Experimental Methods for Isolated Structures † Implementation of Seismic Isolation † Performance during Past Earthquakes

29.3 Seismic Vibration Control ................................................. 29 33 Historical Development of Seismic Vibration Control † Basic Principles † Important Issues in Vibration Control † Vibration Control Devices † Control Algorithm † Experimental Performance Verification † Implementations

Summary This chapter presents seismic vibration control of civil engineering structures. It is divided into two main sections. The first part of the chapter deals with vibration control by seismic base isolation, whereas the second part covers methods of response control that use passive energy dissipation, active control, semiactive control, and hybrid control, respectively. Each part starts with a brief description of the historical development of these methods, followed by basic principles and important issues in their implementation. Thereafter, devices used for structural response control, their design methods, and recommended experimental procedures for verification of their properties and analytical modeling are discussed. Various methods generally used for analysis of such structures are then discussed in detail followed by a brief description of the implementation of these methods for various types of structures. In addition, performance of existing structures during past earthquakes is also included, to highlight the effectiveness of these methods during real earthquakes. Further information on the general topic of this chapter is found in Chapter 22, Chapter 30, and Chapter 31.

29.1

Introduction

Seismic isolation and vibration control systems are relatively new and sophisticated concepts that require more extensive design and detailed analysis than do most conventional seismic designs of structures. In general, these systems will be most applicable to structures whose designers seek superior earthquake performance. Seismic base isolation and passive energy dissipation systems are viable design strategies that have already been used for seismic protection of a number of structures. Other special seismic protective system techniques such as active control, semiactive control, hybrid combinations of active and passive devices, and tuned mass and liquid dampers may also provide practical solutions in the 29 1

29 2


near future. An innovative challenge is highly expected in this field for the seismic safety enhancement of civil structures.

29.1.1

From Ductility Design to Base Isolation and Control Design

The conventional method of seismic design mainly deals with increasing capacity. The approach is based on designing a strong and ductile structure Plastic hinges in a ductile structure, (see Figure 29.1), which can take care of the inertial enabling the whole forces generated by the earthquake shaking. The structural system to approach results in increasing the size of structural absorb seismic energy members and connections, and providing additional bracing members and shear walls, or other stiffening members. The increase in stiffness Ground excitation then attracts more seismic forces and in turn requires further strengthening, which becomes FIGURE 29.1 Schematic of a structure with ductile uneconomical. Therefore, the conventional prac members. tice permits safe design of a structure on the premise that inelastic action in a ductility based designed structure will dissipate significant energy and enable it to survive a severe earthquake without collapse. The conventional designs may permit some structural damage because of inelastic deformation in the members and also in nonstructural elements. Contents of structure can get damaged due to large interstory drift and high floor accelerations. It is difficult to control structural damage and it may be dangerous in unexpected strong seismic events. It has been observed that, in the event of major seismic events, structures based on the conventional design methods suffered damage, experienced high floor accelerations, and resulted in disruption of essential services such as transportation, communication, and so on. Thus, for the class of structures like nuclear power plants, museums, hospital buildings, buildings with artifacts, important bridges, and such structures located in high seismicity regions, this ductility based design is not suitable. The need to minimize earthquake damage in critical and important structures prompted civil engineers to search for other methods of earthquake resistant designs, which can not only protect structures from earthquake motions but also keep them functional during and after strong earthquakes. To this end, base isolation and structural control methods are found to be a solution. Base isolation has the capability to reduce the seismic response of a structure by isolating it from the ground shaking (Figure 29.2a). An isolation system reduces the transmission of ground vibration, thus enabling the structure to experience less shaking from the ground. Therefore, structural damage and occupants’ inconvenience can be minimized using this technique. However, at the expense of safety and the convenience of structure, the bearings undergo significant drift during large earthquakes that may disrupt the function of the bearings themselves and supply lines of services such as water and gas. Another way of reducing seismic response is by using the structural control method. It has the capability of modifying the structural properties, such as stiffness, mass, and damping, and providing passive or active counterforces. Figure 29.2b shows the schematic diagram of the structural control method in a civil structure. It shows some examples of devices generally used for applying control forces. The seismic safety enhancement of structures using the structural control method can be categorized as active and passive systems. There are also hybrid systems that represent combinations of active and passive, and semiactive systems to represent active controller that employs controllable passive devices.

Seismic Base Isolation and Vibration Control

29 3

Moving Mass Control

Joint Damper

Variable Damping Hysteretic Type Damping Active Varying Stiffness

Structure response

Base Isolation Bearings Ground shaking

(a) FIGURE 29.2

(b)

Schematic of a civil structure with (a) isolation bearings and (b) the structural control method.

Owing to changes in code provisions or upgradation of seismic zones, many structures come into the category of “unsafe” and require retrofitting. Response control strategies are found to be easier than other options, economical, and are often the only alternative for such cases.

29.1.2

The Importance of Reducing Seismic Input and Response

As mentioned above, by using the conventional method of seismic design, the design may permit some structural damage because of inelastic deformation in the members, and also in nonstructural elements, during large earthquakes. The ductility enables the structure as a whole to absorb the seismic energy. Once the structural response goes deeply into the plastic range during a large earthquake, structures may not be operational or repairable. If the seismic input to the structure and structural response can be reduced, then the structural damage can be minimized. For higher reliability of structures even under very severe earthquake motion, structural control techniques that effectively reduce seismic force to structures are developed. The fast development of technology, particularly in the fields of electronics and computer science, has provoked the researchers in some centers worldwide to intensify development of a new concept with the new philosophy of seismic design. Generally, this concept is known as a design of intelligent structures or smart structures. Owing to the experience of severe damage due to the Kobe earthquake, public demand for seismic performances of civil infrastructures became relatively clear in Japan. Civil infrastructures are constructed with the tax paid by the public, so a collapse or near collapse with unrepairable damage cannot be accepted, even under a very rare earthquake loading. Infrastructures are also expected to serve as public tools to help rehabilitate the affected society. For this purpose, infrastructures have to be repaired in a relatively short time, even though their functions are temporarily terminated due to severe earthquake loading.

29 4


Earthquake Performance Level Fully Operational Near collapse Operational Life safe Frequent (T = 43 years) Earthquake Occasional Design (T = 72 years) Level Rare (T = 475 years)

Unacceptable Performance (for New Construction)

Very rare (T = 970 years)

FIGURE 29.3

Public demand for seismic performance of infrastructure.

The public demand for seismic performance objectives of infrastructures shows that structural damage has to be limited even against very rare earthquake loading (Figure 29.3). The figure shows that civil infrastructures must be fully operational during and after frequent, weak earthquakes. They also expected to be operational even after very rare, strong earthquakes. To achieve the objectives, new technologies are to be developed that can result in the desired performance.

29.2 29.2.1

Seismic Base Isolation Historical Development of Base Isolation

Seismic base isolation is not a very new idea. More than a century ago, John Milne, a professor of engineering in Japan, built a small house of wood and placed it on ball bearings to demonstrate that a structure could be isolated from earthquake shaking (Housner et al., 1997). In 1891, after the Narobi earthquake ðM ¼ 8:0Þ; Kawai, a Japanese person, proposed a base isolated structure with timber logs placed in several layers in the longitudinal and transverse direction (Izumi, 1988; see Figure 29.4). In 1906, Jacob Bechtold of Germany applied for a U.S. patent in which he proposed to place building on rigid plate, supported on spherical bodies of hard material (Buckle and Mayes, 1990). In 1909, a medical doctor from England, Calentarients, applied for patents for his invention comprising FIGURE 29.4 Base isolation by timber logs. (Source: isolation system for earthquake proof building JSSI, Introduction of Base Isolated Structures, Japan (see Figure 29.5). He proposed separating a Society of Seismic Isolation, Ohmsa, Tokyo, 1995. With building from its foundation with a layer of sand permission.) or talc (Kelly, 1986). The Imperial Hotel in Tokyo, constructed in 1921, was intended to float on an underlying layer of mud. The building was founded on an 8 ft thick layer of firm soil under which exists a 60 to 70 ft thick layer of soft mud. The building was


29 5

FIGURE 29.5 Calentarients’ idea of seismic base isolation. (Source: JSSI, Introduction of Base Isolated Structures, Japan Society of Seismic Isolation, Ohmsa, Tokyo, 1995. With permission.)

highly decorative with many appendages. The soft mud acted as isolation system and the building survived the devastating 1923 Tokyo earthquake (Kelly, 1986; Buckle and Mayes, 1990). Attempts were made in the 1930s to protect the upper floors of multistory buildings by designing very flexible first story columns. In a later modification of the flexible first story columns approach, it was proposed that the first story columns should be designed to yield during an earthquake to produce an energy absorbing action. However, to produce enough damping, several inches of displacement is required, and a yielded column has greatly reduced buckling load, proving the concept to be impractical. It was then proposed that, if the soft first story is constructed underground, then energy dissipaters can be installed at the top of this story (which approximates ground level) that prevent the superstructure from moving too far, and dissipate the energy of ground motion before it enters the building. The superstructure, from the first floor up, can be an economically braced, nonductile concrete frame requiring no internal shear walls (Arnold, 2001). To overcome the inherent dangers of soft supports at the base, many types of roller bearing systems have been proposed. The rollers and the spherical bearings are very low in damping and have no inherent resistance to lateral load, and therefore some other mechanism that provides wind restraint and energy absorbing capacity is needed. A long duration between two successive earthquakes can result in the cold welding of bearings and plates, thus causing the system to become rigid after a time. Therefore, the application of rolling supports was restricted to the isolation of special components of low or moderate weight (Caspe, 1984). Parallel to the development of the soft first story approach, the flexibility of natural rubber was also seen as another solution for increasing the flexibility of the system. In 1968, large blocks of hard rubber, 54 in number, were used to isolate the three story Heinrich Pestalozzi School in Skopje, Republic of Macedonia. The building is constructed of reinforced concrete shear walls. This is the first building for which rubber bearings were used as base isolation against strong earthquakes. These rubber blocks are un reinforced and bulge sideways under the weight of this concrete structure (see Figure 29.6). Owing to having the same stiffness of the isolation system in all the directions, the building bounces and rocks FIGURE 29.6 Unreinforced rubber blocks. (Source: Ohashi, U.G. Earthquakes and Base Isolation, Pub. backwards and forwards (Jurukovski and Rakice Asakura, Tokyo, 1995. With permission.) vic, 1995). These types of bearings are unsuitable for the earthquake protection of structures. The subsequent development of laminated rubber bearings has made base isolation a practical reality (Figure 29.7). Later, a large number of isolation devices were developed, and now base isolation has reached the stage of gaining acceptance and replacing the conventional construction, at least for important structures. FIGURE 29.7 Rubber bearings with steel shims.

29 6

29.2.2


Basic Principle

Seismic base isolation is basically a lengthening of the fundamental time period of the structure with the help of a specially designed system that is placed between its superstructure and substructure (see Figure 29.8). Besides other advantages, the concept gained widespread acceptance due to the fact that most of the earthquake motions around the world have dominating frequencies in the range of 1.0 to 10 Hz, and the majority of conventionally designed structures also has their fundamental frequency of vibration lying in this range. Owing to this unwanted matching of the frequencies, these structures are subjected to high forces during earthquakes. The application of seismic base isolation shifts the fundamental time period away from the dominating frequencies of earthquake motions and thus detunes the frequencies. In other words, base isolation consists in filtering out high frequency waves from the ground motion, thereby preventing the transmission of high energy in the structure. The effect of base isolation on reduction in force is shown schematically in Figure 29.9. Under favorable conditions, seismic base isolation can reduce drift to 0.2 to 0.5 of that which would occur if the building were fixed base (Figure 29.10). Reduction in acceleration has more influence on the force deflection characteristics of the isolation system and may not be as significant as the reduction of drift (FEMA 356, 2000). However, the additional flexibility required for this period shift give rise to excessive relative displacement at the isolation level. Additional damping is introduced in the isolation system to limit this displacement response to within feasible limits. Still, it is necessary to provide an adequate seismic gap that can accommodate displacements at the isolation level. Most of the isolators have inherent damping, although sometimes supplemental energy dissipation devices may also be required at the isolation level. Various types of energy dissipation devices like metallic dampers and hydraulic dampers have been developed and can be used for this purpose (Skinner et al., 1993). The isolation damping also suppresses the resonance

FIGURE 29.8

Application of seismic isolation for different structures.

Normalized acceleration response

4.0 3.0

Conventionally designed structure

2.0

V = 2% Seismically isolated structure

1.0 0.0

V =15% 0.0

1.0

2.0

3.0

4.0

Natural period (seconds) FIGURE 29.9

Conceptual diagram for seismic isolation.

5.0


resulting due to higher period contents of the earthquake motion. Although damping is useful in reducing the required seismic gap, excess damping may result in an increase in acceleration that may affect the performance of nonstructural elements and contents. Thus, an isolation system should essentially be able to (1) support a structure, (2) provide horizontal flexibility, and (3) dissipate energy. These three functions can be incorporated in a single device or can be provided by means of different components. In addition, it may be necessary to provide buffers, which can limit the isolator displacements during extreme earthquakes.

29.2.3

29 7

FIGURE 29.10 Behavior of (a) fixed base and (b) base isolated building.

Issues in Seismic Base Isolation

A number of issues for seismic isolation design have been identified based on experiences of their behavior. Some of the issues that should be considered before choosing the base isolation approach for a project are touched in the following sections. 29.2.3.1

Performance Criteria

The performance criteria for the structure needs to be established in order to evaluate alternative seismic resisting systems; for example, it must be established whether the structure is required to be functional during and after major earthquakes, or if it is to be preserved for its historical importance. Whether seismic base isolation is a suitable design strategy for a particular project will depend primarily on the performance required. To achieve the fully operational or operational performance level, one can consider seismic base isolation as a possible design strategy, but if life safe is the required structural performance level, it may not be practical to choose seismic base isolation. 29.2.3.2

Type of Structure

Significant benefits obtained from isolation exist in structures for which the fundamental period of vibration without base isolation is short, that is, less than 1 sec. Certain structures may not be suitable for base isolation because of their shape; for example, this is true for slender high rise buildings that have a natural period long enough to attract low earthquake forces without isolation. Therefore, seismic isolation is mostly used for low rise buildings. Historical buildings, which generally are stiff masonry structures, can be appropriate structures for seismic base isolation. Bridges are the structures for which application of seismic isolation is very convenient. The provision of bearings at the tops of piers adds flexibility to stiffer piers and in turn avoids yielding of piers. It is easy to examine these bearings after a seismic event and replace them if needed. 29.2.3.3

Site Characteristics

In base isolation design, the basic objective is to filter out the high frequencies of the ground motion by lengthening the time period of vibration to approximately 2 sec. Thus, conventional base isolation is not suitable for structures on soft soils where the ground motions are dominated by low frequencies. Therefore, a detailed investigation of the site must be carried out before possible isolation can be considered. Another important aspect is near fault ground motions. Waves from such motions usually have long period velocity pulses, which impart lot of momentum to the structure. This is particularly damaging to base isolated structures because it may cause large horizontal base displacements. The displacement can result in instability of the structure, or it

29 8


can result in impact with moat walls, which can affect the sensitive equipment housed in the building. An isolator with bilinear force displacement behavior and a large ratio of yield force to supported weight can substantially reduce the displacement. The provision of high damping in an isolation system can also work. However, the degree of isolation during relatively frequent earthquakes without near fault pulses is much reduced due to these provisions (Skinner and McVerry, 1996). 29.2.3.4

Retrofit Issues

In selecting a suitable retrofit system and properties of seismic isolation system, consideration should be given to the characteristics of the existing building, such as foundation capacity and strength of the superstructure. For retrofit of buildings, successful implementation of a seismic isolation system requires that the sequences of temporary bracing, shoring, cutting of existing columns and walls, and installation of isolators be well planned. Base isolation for the retrofit of bridges is simpler as they usually have thermal bearings, which can easily be replaced by seismic isolation bearings. The retrofitting of monuments or buildings of historical importance requires special efforts to cope with the need of minimum alterations. Provisions must be made to protect them from any seismic event during the retrofitting. Also it must be identified whether workable spaces and access to the work area is available. 29.2.3.5

Design of Building Services

Depending on the base isolation system, base isolated structures under earthquake motions can exhibit significant base slab displacements due to the low horizontal stiffness of the isolation elements. This may create problems on the supply lines transitioning between the parts of structure below and above isolation level. Therefore, special attention is to be given to installations of building services such as water supply, sewerage, gas, air conditioning, and so on in order to prevent any damage to these supply lines, which might cause secondary effects. In the case of isolation of two or more structural units founded on a common foundation and connected by expansion joints, special care is needed regarding the proportions of the expansion joint in order to prevent the pounding of buildings during earthquakes. 29.2.3.6

Expected Life of Isolator

The isolation system should remain operational for the expected lifetime of the isolated structure. It should not require frequent maintenance during this period. Although the functioning of an isolator may be required few times during the lifetime of structure, it must perform well at such times. If life of the isolator is less than the life of structure, then it may be necessary to provide a mechanism for the inspection and replacement of the isolation system. Another related aspect is the protection of isolation elements against fire, and measures should to be taken for this. Furthermore, as an isolation system is provided mostly at the base, its resistance to chemical and biological reactions is also important (Jurukovski and Rakicevic, 1995).

29.2.4

Seismic Isolation Devices

The successful seismic isolation of a particular structure is strongly dependent on the appropriate choice of the isolation system. In addition to providing adequate horizontal flexibility and appropriate damping, the isolation system should essentially have the capability of self centering after deformation, high vertical stiffness to avoid rocking, and enough initial stiffness to avoid frequent vibration from wind and minor seismic events. Different types of isolators have been developed and proposed to achieve these properties, and some of them are discussed below.


29.2.4.1

29 9

Laminated Rubber Bearings

Load (ton)

Rubber bearings offer the simplest method of seismic base isolation and are relatively easy to manufacture. The bearings are made by vulcaniza tion bonding of sheets of rubber to thin steel reinforcing plates. Initially, the main function of the laminated rubber bearings was to provide flexibility for thermal displacements in bridges. Later, similar bearings found application in the isolation of buildings from vibration due to underground railways, and these bearings have FIGURE 29.11 Laminated rubber bearing. performed well over a substantial period of time (Kelly, 1990). The bearings are very stiff in the 20 vertical direction and flexible in the horizontal direction. High vertical stiffness of these bearings is achieved through the laminated construction of the bearing using steel plates. The cross section of a typical rubber bearing is shown in Figure 29.11. The most common elastomers used in elastomeric bearings are natural rubber, neoprene rubber, 0 butyl rubber, and nitrile rubber. The mechanical (tear strength, high strain fatigue resistance, creep resistance) and low temperature properties of natural rubber are superior to those of most synthetic elastomers used for seismic isolation bearings. Therefore, natural rubber is the most −20 frequently recommended material for use in elastomeric bearings, followed by neoprene. Butyl 0 100 −100 rubbers are suitable for low temperature appli Displacement (mm) cations and nitrile rubber has limited application in offshore oil structures (Taylor et al., 1992). FIGURE 29.12 Load displacement loops for high The damping ratio (i.e., the fraction of critical damping rubber bearing. (Source: Tanzo, W. et al. Res. damping) achieved from natural rubber is low, in Rpt. 92 ST 01, Kyoto Univ., 1992. With permission.) the order of 0.02 to 0.04, and therefore it is unusual to use it without some other element that is able to provide increased damping. In order to achieve better performance in a single unit, rubber used in the bearing is a compound formed with some filler agents. This compounding results in desired properties, such as (1) high damping and (2) high horizontal stiffness at low values of shear strain. The damping ratio (i.e., the fraction of critical damping) achieved is in the order of 0.10 to 0.20. These high damping rubber bearings, originally developed in England, found several applications in Japan and United States. A number of fillers are employed, such as metal oxides, clay, and cellulose, but the filler that is most commonly used in seismic isolation bearings is carbon black (Taylor et al., 1992). Force displacement behavior of these bearings depends upon the type of compounding. Figure 29.12 shows the results of cyclic loading test conducted on a four layer high damping rubber bearing specimen (Tanzo et al., 1992). In the experiment, the tests were carried out up to 200% (96 mm) shear strain. Vertical load for the tests was kept as 40 tonf (64 kgf/cm2). The application of steel shims in laminated rubber bearings provides necessary vertical stiffness, but at the same time makes these isolators heavy and expensive. Recently, Kelly (2001) proposed a seismic isolation system for developing countries, in which steel plates are replaced by fiber mesh. The fiber reinforced isolator is expected to be significantly lighter and could lead to a much less labor intensive manufacturing process.

29 10

Lead Rubber Bearing

This isolation system consists of a cylinder of lead enclosed in a rubber bearing. The system is also known as the NZ bearing system, and its components are shown in Figure 29.13. The lead plug produces a substantial increase in damping, from about 3% of critical damping in the natural rubber to 10 to 15%, and also increases the resistance to frequently occurring loads such as minor earthquakes or wind. The reason for choosing lead is that it is a crystalline material, and under normal conditions it changes its crystal structure under deformation but almost instantly regains its original crystal structure when the deformation ceases. Thus, repeated yielding due to cyclic loading does not cause fatigue. Lead yields in shear at the relatively low stress of about 8 to 10 N/ mm2, and therefore produces stable hysteretic behavior and dissipates significant energy in strong ground motion. Load displacement beha vior of a lead rubber bearing is shown in Figure 29.14 (JSSI, 1995). The hysteretic behavior of lead rubber bearings can be treated as bilinear, with initial stiffness in the range of 9 to 16 times the postyield stiffness. These bearings provide an economical and effective solution, incorporating period shifting, increased damping, and high stiffness at low strains, and providing vertical support in a single device (Skinner et al., 1993). It has found several applications in new con structions as well as for retrofitting of buildings and bridges in different parts of the world.

29.2.4.3

Friction-Based Systems

FIGURE 29.13 Lead rubber bearing. (Source: Skinner, R.I. et al. An Introduction to Seismic Isolation, Wiley, Chichester, UK, 1993. With permission.)

15

Stress (kg/cm2)

29.2.4.2


0

−15 −300

0 Shear strain (%)

300

FIGURE 29.14 Behavior of lead rubber bearing under cyclic loading. (Source: JSSI, Introduction of Base Isolated Structures, Japan Society of Seismic Isolation, Ohmsa, Tokyo, 1995. With permission.)

In this class of isolators, the superstructure is allowed to slide during major seismic events. The structure slides whenever the lateral force exceeds the friction force at the sliding interface. The horizontal friction force at the sliding surface offers resistance to motion and dissipates energy. Pure sliding systems have no inherent natural period and therefore are insensitive to variations in the frequency content of ground excitation. The acceleration at the base of the structure is limited to the coefficient of friction at the sliding interface. Thus, by keeping this coefficient of friction low, the acceleration felt by the structure can be reduced. Systems based on a Teflon (PTFE) and stainless steel interface have the potential to provide high levels of protection against floor acceleration because of the low friction of the materials. However, the friction coefficient cannot be reduced arbitrarily, as the sliding displacement may exceed the acceptable value. Another feature of this isolation system is that the frictional force is proportional to the vertical load coming on the bearing, and therefore the center of mass and the center of resistance of the sliding support coincide. As a result, the torsional effects produced by the asymmetry of building are diminished.


29 11

The main drawback of the system is the absence of restoring force due to which a large residual displacement from the original position of the structure may be left after a major earthquake event. There are other practical problems that can affect the efficiency of sliding isolation, such as cold welding, freezing, and deterioration of the sliding surfaces. Changes may occur in the friction coefficient due to FIGURE 29.15 Schematic diagram of friction pendu ageing, environmental attack, temperature vari ation, or wear during use (Skinner et al., 1993). lum system. As the sliding depends upon the coefficient of friction, the system may require regular inspection to maintain its coefficient of friction. The unsatisfactory predictability of friction coefficient and the absence of any centering force suggest that Teflon bearings should be used as seismic isolators only in combination with some other centering devices like steel dampers or rubber bearings (Priestley et al., 1996). Based on this concept another system called the resilient friction bearing (R FBI) system was developed (Mostaghel and Khodaverdian, 1987). The system consists of several concentric layers of Teflon coated friction plates that are in friction contact with each other and a central core of rubber. The rubber provides the resilient force while the friction forces dissipate the energy. An effective mechanism to provide recentering force by gravity has been utilized in a friction pendulum system (FPS). In this system, the sliding surface takes a concave spherical shape so that the sliding and recentering mechanisms are integrated in one unit. As the name indicates, there are two mechanisms that are employed to achieve isolation, namely, sliding friction and pendulum motion. The internal components consist of a stainless steel concave surface upon which slides a stainless steel articulated slider surfaced with a high load capacity and a low friction bearing material composite (see Figure 29.15). The radius of curvature determines the sliding or isolation period of the system. The period of the structure supported on the FPS is independent of the structure mass and therefore the period does not change if the structure weight changes or is different than assumed. This results in better control over the response of the systems, like liquid storage tanks in which weight varies in time because of variable liquid storage level. However, since the restoring force is linearly proportional to the sliding displacement, in case of high intensity earthquakes or a low coefficient of friction, the additional sliding introduces additional energy in the structure. To overcome this problem, Pranesh and Sinha (2002) proposed the variable frequency pendulum isolator, in which the geometry of the concave surface is designed such that its frequency decreases with the increase in sliding displacement and asymptotically approaches zero at very large displacement. 29.2.4.4

Other Systems

A base isolation system, which has found many applications in Japan, comprises of low damping rubber bearings with yielding metal devices. The yielding of metals provides the necessary dissipation of energy. The commonly used dampers are shown in Figure 29.16 (Aoyagi et al., 1988; Takayama et al., 1988; Yasaka et al., 1988). To standardize the designs of the sensitive structures of nuclear power plants for regions of different seismicity, an isolation system called the EDF (Electricite de France) system was developed. This is a laminated rubber bearing with friction plate at the top. An attractive feature of the EDF isolator is that, for the lower amplitude ground excitations, the lateral flexibility of the rubber provides the base isolation. At a high level of excitation, however, sliding will occur assuring maximum acceleration transmissibility of mg, where m is the coefficient of friction between materials of sliding surface (Constantinou, 1994).

29 12


FIGURE 29.16

(a) Elasto plastic steel damper; (b) lead damper; (c) steel rod damper.

FIGURE 29.17 Elements of a TASS system. (Source: Kawamura, S. et al. 1998. Proc. IX World Conf. on Earthquake Engineering, pp. 735 740. Tokyo, Japan. With permission.)

Taisei Corporation, Japan, developed the TASS (TAISEI Shake Suppression) system (Figure 29.17), which is composed of PTFE elastomeric bearings and neoprene springs (Kawamura et al., 1988). PTFE elastomeric bearings support the vertical load of the superstructure and reduce the horizontal seismic forces by sliding against severe earthquake motion. Horizontal springs provide weak lateral stiffness and restrain displacement. The TASS system never resonates to any type of excitation, stably supports superstructure, and limits the horizontal force transmitted to the superstructure to that equal to friction force. In situations where circumstances may result in substantial tension forces on the bearings, it has generally been accepted that elastomeric, sliding, or roller isolators alone are not suitable. For this, a system that controls uplift was developed. The system consists of rubber bearings with a central hole to accommodate a tension device (Kelly and Chalhoub, 1990). Logiadis et al. (1996) proposed a prestressed bearing isolation system in which vertical prestressed tendons are located in pairs, symmetrical on both sides of each bearing to avoid tension in the rubber bearing.


29 13

Tarics (1996) developed a composite isolator to provide equal protection in minor, moderate, or major earthquakes (see Figure 29.18). The composite seismic isolator has two distinctly different stiffnesses, which are activated by the displacement demand. During minor or moder ate earthquakes, the upper isolator provides flexibility. In the event of major earthquake, the higher stiffness of the lower isolator prevents excessive displacement. This isolator seems to be a promising alternative for near fault ground FIGURE 29.18 Composite isolator and its force motions. With similar objectives, Shimoda et al. displacement diagram. (Source: Tarics, A.G. Proc. XI (1992) proposed lead rubber bearings with a World Conf. on Earthquake Engineering, 1996. With stepped lead plug. Several other systems, for permission.) example, the sleeved pile system, GERB, alexisis mon, and so on, were developed by different researchers (Kelly, 1986; Buckle and Mayes, 1990) and the search for new isolation systems, which can give better performance under different conditions, is still continuing.

29.2.5

Design of Isolation Devices

As discussed above, several types of isolators have been proposed so far. Only a few bearings, namely, laminated rubber bearings, lead rubber bearing, FPS bearings, and Teflon steel sliding bearings with a restoring device, have been developed to the stage of practical application. These isolators are in general observed to be effective in reducing seismic response; however, their effectiveness depends on the situation and design constraints. For example, rubber bearings are most suitable for high frequency motions, while friction type isolators may also be applied at the site where ground motion can have low frequency content. Buildings with sophisticated electronic equipment and loose contents should preferably be isolated with rubber bearings to avoid transmission of high frequencies to the superstructure. In the case where, for some reason, it is not possible to provide large seismic gaps or for the conditions where light weight structure is to be isolated, the FPS type isolators are more suitable. The main requirements for the design of a base isolation system are (1) the ability to sustain gravity loads, (2) low horizontal stiffness that can lengthen the fundamental time period to the desired value, (3) large vertical stiffness to minimize amplification in vertical direction and complications due to rocking, (4) energy dissipation capacity to keep displacements at the isolation level within acceptable limits, and (5) sufficient initial stiffness to avoid unwanted vibrations due to wind loads and frequent minor seismic events. The design procedure of most widely used isolation systems, namely, high damping rubber bearings, lead rubber bearings, and the FPS, is discussed in the following sections.

29.2.5.1

High-Damping Rubber Bearing

The design of high damping laminated rubber bearings involves determining the plan size, the number of rubber layers, the thickness of each rubber layer, and steel shim. The steps involved in the design of high damping rubber bearings are as follows: 1. Specify the design vertical load on the bearing ðWÞ; the design period for the isolated structure ðTI Þ; and site conditions. The load, W; is computed for the dead load and live load combination. The design period, TI ; depends on site conditions. Its minimum value is usually taken as three times the fundamental time period of the structure when it is fixed base.

29 14


2. Find corresponding horizontal stiffness of the bearing, Kh ; using Kh ¼

W ð2p=TI Þ2 g

ð29:1Þ

3. Assign the values to quantities such as the maximum permissible shear strain ðgÞ; Young’s modulus ðEÞ; and the shear modulus ðGÞ for the rubber compound. Generally, g is considered to be in the range of 100 to 150%. G can be taken from design specifications provided by the manufacturer. The value of G for this range of shear strains varies from 0.69 to 0.86 MPa, depending on the rubber composition (Kelly, 1997). 4. Specify the effective equivalent damping in the isolation system and allowable pressure on the bearing ðpÞ for vertical loads. The allowable pressure ðpÞ for bearings with metal reinforcement can be taken as about 6.9 N/mm2 (Kelly, 1997). Damping can be taken as 15 to 20%, subject to verification. 5. Using a response spectrum or code formulae, find the maximum horizontal displacement, that is, the design displacement ðDÞ: This depends on the spectral coefficient (i.e., the site condition), design period, and damping. 6. The plan area of the rubber ðAÞ in bearing is primarily controlled by the design vertical load ðWÞ and allowable pressure ðpÞ; and is determined by the expression A ¼ W=p

ð29:2Þ

7. The total thickness of rubber, tr ; in the bearing is determined from the following equation: tr ¼ GA=Kh

ð29:3Þ

Total thickness, tr ; should not be less than D=g: 8. Evaluate the shape factor ðSÞ for the desired value of ratio of vertical stiffness ðKv Þ and horizontal stiffness ðKh Þ using following expressions: 6S2 ¼ Kv =Kh

for circular bearings

6:73S2 ¼ Kv =Kh

for square bearings

ð29:4aÞ ð29:4bÞ

The minimum recommended value for the stiffness ratio is 400. These expressions are fairly accurate for shape factors of ten or less. However, for higher values of S; the effect of compressibility should also be taken into account (Kelly, 1997). 9. Using a basic definition of shape factor as the ratio of the cross sectional area of the bearing to the force free area of a single layer of rubber, the thickness of each rubber layer can now be determined by t ¼ f=4S for circular bearings ð29:5aÞ t ¼ b=4S

for square bearings

ð29:5bÞ

where f is the diameter for circular bearing and b is the side for square bearing. Knowing the values of tr and t; the number of rubber layers ðnÞ can be computed. 10. The thickness of steel shims provided for laminated construction should be in the range of one tenth to one eighth of an inch (Kelly, 1997). The material properties for rubber compounds used for isolation devices can be related to the rubber hardness. Properties for the range of rubber hardness normally used in bearing are listed in Table 29.1 (Bridgestone, 1990). The designed bearing should be checked for the following criteria also. 29.2.5.1.1 Stability against Buckling Buckling load may become critical in situations where bearings with a high rubber thickness relative to the plan dimension are to be provided. For example, in the case of base isolation for low vertical loads,

Seismic Base Isolation and Vibration Control TABLE 29.1

29 15

Relationship between Rubber Hardness and Other Properties

Rubber Hardness (IRHD ^2)

Young’s Modulus, E (MPa)

Shear Modulus, G (MPa)

Material Constant, k

Minimum Elongation at Break (%)

35 40 45 50 55 60

1.18 1.50 1.80 2.20 3.25 4.45

0.37 0.45 0.54 0.64 0.81 1.06

0.89 0.85 0.80 0.73 0.64 0.57

650 600 600 500 500 400

rubber bearings are generally slender. Average compressive stress in the bearing ðW=AÞ should not exceed the critical stress ðpcr Þ given by Kelly (1997): pffiffi 2pGS pcr ¼ r ð29:6Þ tr pffiffi where r is radius of gyration (b=2 3 for a square bearing and f/4 for a circular bearing). Maximum shear strain is described as follows. Under vertical loading only, as per AASHTO recommendations for service loads (i.e., for the dead and live load), a factor of safety of three should be adopted for maximum shear strain values. Thus, maximum shear strain ðgs Þ should satisfy the following condition: W 1 gs i:e: 6S # b Ec A 3 where 1b is the minimum tensile strain at which rubber breaks and Ec is compression modulus for bearing and is given by Ec ¼ Eð1 þ kS2 Þ: E and k, for a given value of rubber hardness, can be taken from Table 29.1. Under earthquake loading, the following applies. In the case of ultimate load that includes effect of earthquake loads, AASHTO recommends a factor of safety of 1.33. Thus, maximum shear strain (gu) due to combined effect of compression (gc), torsion (gt), and lateral load (geq) should not exceed 1b =1:33; that is, gc þ geq þ gt # 1b =1:33: geq is given by D=tr ; and gt can be evaluated by

gt ¼

b2 12De 2ttr ðB2 þ L2 Þ

ð29:7Þ

Here, B and L are dimensions of a structure with rectangular plan and e is the eccentricity. gc ð¼ 6SW 0 =Ec Ar Þ is computed for vertical load (W 0 ), which also includes the effect of earthquake load. In this case, the effective area (Ar) is reduced due to lateral displacement (D) and it is the overlapping area of top and bottom in displaced condition. Ar can be determined using the following expressions: Ar ¼ A 1 ( 1 Ar ¼ f2 sin 2

1

pffiffiffiffiffiffiffiffiffiffiffi f2 D2 f

D b

for square bearings qffiffiffiffiffiffiffiffiffiffiffi!) for circular bearings D f2 D2

ð29:8aÞ ð29:8bÞ

29.2.5.1.2 Stability against Roll Out Maximum horizontal displacement should not exceed the roll out displacement (d) given by

d¼

WE b W E þ Kh h

ð29:9Þ

where vertical load (WE) also includes the effect of earthquake load, b is the side for square and the diameter for the circular bearing, and h is the overall height of the bearing (Figure 29.19).

29 16


The check should be made for bearings with bolted connections also, so as to avoid the development of tensile stresses. 29.2.5.2

Lead Rubber Bearings

The design of lead rubber bearings can be split into two parts, namely, (a) the design of the lead plug and (b) the design of the laminated rubber bearing. 29.2.5.2.1 Design of Lead Plug In fact, the lead rubber bearing can be designed effectively by assuming its force displacement behavior comprises two distinct stiffnesses, namely, the preyielding and postyielding stiff nesses, as shown in Figure 29.20. The characteristic strength, Qd ; for the desired energy dissipation (WD) or the known effective damping (jeff ) can be computed using Qd ¼ WD =4ðD

FIGURE 29.19 Stability against roll out. (Source: Kelly, J.M. Earthquake Resistant Design with Rubber, Springer, London, 1997. With permission.)

F

Fy

Qd Ke

dy Þ

¼ 2pKeff D2 jeff =4ðD

dy Þ

dy

ð29:10Þ

Qd and yield displacement (dy) are the unknowns. So for a first trial, considering dy is very small as compared with design displacement (D), we have, as a first approximation Qd ¼ WD =4D

Kp

ð29:11Þ

FIGURE 29.20 bearing.

Keff D Displacement

Design parameters for lead rubber

Using this approximate value for Qd ; the postyield stiffness (Kp) can be evaluated by Keff D ¼ Qd þ Kp D

ð29:12Þ

This may lead to modified value of yield displacement, dy ; given by dy ¼ Qd =ðKe

Kp Þ

ð29:13Þ

where, depending upon the physical parameters of the laminated rubber bearing, Ke can be assumed to be about ten times the value of Kp : This value of dy can be used in the next trial to compute revised values of Qd ; Kp ; and dy : Trials are to be repeated until dy converges. Now, the yield force can be evaluated by Fy ¼ Ke dy

ð29:14Þ

Thus, the area of lead plug (Ap) can be determined using Ap ¼ Fy =f y ; in which f y is the yield strength of the lead in shear and has a value of 10.5 MPa (Skinner et al., 1993). In design, owing to considerations for the confinement of the plug, a reduced value of 7 to 8 MPa is generally considered.


29 17

29.2.5.2.2 Design of Laminated Rubber Bearing Postyield stiffness of the bearing (Kp) is typically higher than the corresponding stiffness of the bearing without the lead plug (Kh): Kp ¼ f L Kh where f L is a factor that takes into account the effect of lead plug and is always larger than unity. Typically, f L is 1.15 (FEMA 356, 2000). Another expression for computing Kp is given by (Yang et al., 2003) Kp ¼ Kh 1 þ 12

Ap A0

ð29:15Þ

where A0 is the area of the bearing based on allowable normal stress under the vertical load. The final value of Kp obtained during design of the lead plug is used in these expressions to find Kh : This contribution (Kh) towards combined stiffness (Kp) controls the design of the laminated rubber bearing. The steps from Step 3 onwards, described earlier in the design of high damping rubber bearings, can be followed to design this laminated rubber bearing. One additional check for the lead plug should be applied: 1:25 #

hp # 5:0 fp

where hp is effective height of the lead plug and fp is its diameter (Yang et al., 2003). 29.2.5.3

Design of Friction Pendulum System

The basic design variables in FPS are (1) the radius of curvature (R), (2) the material friction coefficient ðmÞ; and (3) the plan dimension. The desired isolation period governs the radius of curvature, whereas the plan dimension is controlled by the design displacement. R for a known value of the design period of the isolated structure (TI) can be computed using R ¼ gðTI =2pÞ2

ð29:16Þ

where g is acceleration due to gravity. The coefficient of friction ðmÞ; along with the displacement, controls the energy dissipation. The effective damping (b) for this system is given by expression

b¼

m 2 p m þ D=R

ð29:17Þ

The desired damping, depending upon the properties of the system, may vary from 10% to 20% of critical damping. As the design displacement (D) depends upon damping (b), it becomes a trial and error process and requires few iterations to get the values of b, m, and D for desired response. The effective stiffness of the isolation system at design displacement can be evaluated by Keff ¼

29.2.6

mW W þ D R

ð29:18Þ

Verification of Properties of Isolation Systems

As they are the most critical part of the base isolated structure, the quality of isolation bearings is of paramount importance. Looking into uncertainties in the manufacture, the deformation characteristics and damping values of the isolation system must be verified by tests. Therefore, the selected units of the isolation system have to undergo testing for examining quality control and determining their actual load deformation behavior. Force deformation characteristics of the isolation system obtained from tests are used for analyzing the base isolated structures. These tests also serve as a means to verify the

29 18


ability of the isolation bearings to withstand long term vertical loading and numerous cycles of shear displacement during design basis and maximum capable earthquake. The primary function of an isolation system is called on during major earthquakes only and, for most of its design life, it is subjected to nonseismic loads. Therefore its behavior for such loads should also be verified by tests. 29.2.6.1

Tests for Isolation Systems

System characterization tests, prototype tests, and quality control tests are the three levels of tests that are recommended for an isolation system. The first tests provide the fundamental characteristics of the isolation system; the second allow one to know the design parameters of the isolation system fabricated for the project; and the third verify the quality control and consistency achieved in manufacturing. These tests are briefly discussed in the following sections (Taylor et al., 1995). 29.2.6.1.1 System Characterization Tests These tests are performed to evaluate the basic characteristics of the isolation systems, especially its dependence on temperature, frequency, bilateral loading, wear and fatigue, and so on. The tests are also intended to provide the ultimate and reserve capacity of a device for various loading conditions. These tests are conducted only once for a system of a given design, material, and construction. However, they should be repeated if there are major changes. These tests are formally not required by codes; however, in practice, they are conducted for a new isolation system and the test results provide some idea about the suitability of the isolation system for a particular project. 29.2.6.1.2 Prototype Tests These tests are conducted to verify the design properties of an isolation system prior to its construction. Effective stiffness, energy dissipation capacity, stability of hysteretic behavior (in a check for system degradation), and stability at maximum seismic displacement are the properties generally verified through these tests. The stability of an isolation system is verified for maximum and minimum vertical load conditions. Tests are also carried out for nonseismic loads. In the case of isolation systems for buildings, the major nonseismic force is wind load, and for bridges it may be thermal displacements and braking or centrifugal forces. If the isolation system is frequency dependent, then prototype tests should be performed dynamically for a range of frequency that represents full scale prototype loading rates. Similarly, if the behavior of the isolation system is significantly different for unilateral and bilateral loadings, then the isolation system should be tested for different combinations of displacements in two orthogonal directions. Specimens in these tests are subjected to extreme conditions and therefore are not to be used for construction. 29.2.6.1.3 Quality Control Tests These tests are carried out to verify the quality and consistency of the manufacturing process, and to measure the as built properties of the isolation system prior to installation. The quality control tests are divided into production tests and completed unit tests. Production tests are carried out on materials or components that are used in the making of the isolation unit. In the case of elastomeric bearings, the elastomer is tested for hardness, tensile strength and elongation at break, bond strength, compression set, low temperature properties, high temperature aging, and ozone resistance. Elements used for a sliding system are supposed to be tested for surface roughness, trueness of surface, interface material properties, bearing pad attachment, and sliding interface attachment. All completed units are tested for (1) sustained compression, and (2) combined compression and shear. The purpose of the sustained compression test is to verify (1) the quality of the bond between the elastomer and the steel for laminated elastomeric bearings and (2) the bond between the bearing liner and the metal backing plate for sliding bearings. A combined compression and shear test is conducted to verify that actual values of effective stiffness and damping are in close agreement with design values. After quality control testing, laminated elastomeric bearings are visually inspected for rubber to steel bond, surface cracks, laminate placement, and permanent deformation, while sliding bearings should be inspected for bearing liner to metal bond,


29 19

scoring of the stainless steel plate, leakage, and permanent deformation. In the case that the bearings do not meet the requirements, they should be rejected. 29.2.6.2

Uniform Building Code Recommendations

Uniform Building Code (UBC, 1997) recommends that the deformation characteristics and damping values of the isolation system used in the design and analysis of seismic isolated structures is based on the sequence of cyclic loading tests of a selected sample of the components (Figure 29.21). First, the specimen is tested under a cyclic load corresponding to the design wind load. Sub FIGURE 29.21 Schematic diagram for cyclic loading sequently, it is tested, for a prescribed number of test of bearing. cycles, with cyclic displacements varying from one fifth of the design displacement to the total maximum design displacement. Vertical load (dead load plus 50% of the live load) for all the above tests is prescribed as the average load on all isolators of common type and size. If an isolator unit is also a vertical load carrying element, the cyclic load tests should be performed for maximum and minimum vertical loads, which includes the effects of an earthquake overturning evaluated corresponding to the test displacement. 29.2.6.2.1 Units Dependent on Loading Rates If there is greater than 10% difference in effective stiffness values at the design displacement (i) obtained by performing test on an isolator unit at a frequency equal to the inverse of the effective period TI of the isolated structure, and (ii) obtained by performing test on the same unit at any frequency in the range of 0.1 to 2.0 times the inverse of TI, then the force deflection properties of the isolator unit are considered to be dependent on rate of loading. In such cases, the tests are prescribed to be performed at a frequency equal to the inverse of TI. 29.2.6.2.2 Units Dependent on Bilateral Load If the bilateral and unilateral force deflection properties have greater than a ^10% difference in effective stiffness at the design displacement, the force deflection properties of an isolator unit are be considered to be dependent on bilateral load. In such cases, the tests specified above shall be augmented to include bilateral load increments of the total design displacement 0.25 and 1.0, 0.50 and 1.0, 0.75 and 1.0, and 1.0 and 1.0. 29.2.6.2.3 System Adequacy The performance of the test specimens is assessed as adequate if the following conditions are satisfied: 1. The load displacement curves for all tests have a positive incremental load carrying capacity. 2. The difference between the average values of effective stiffness of the two test specimens of a common type and size does not exceed the prescribed value of 10%. 3. The effective stiffness of each test specimen for each cycle of test is within ^10% of the average value of effective stiffness for the specimen. 4. There is no greater than a 20% change in the initial effective stiffness of each test specimen over the prescribed number of cycles. 5. There is no greater than a 20% decrease in the initial effective damping of each test specimen over the prescribed number of cycles. 6. All specimens of vertical load carrying elements remain stable at the total maximum displacement for the prescribed vertical load.

29 20

29.2.7


Analysis of Base-Isolated Structures

Depending upon the code requirements, one can choose the static or dynamic analysis procedure for a particular project. Regarding the static analysis method, codes generally have some bindings, whereas time history analysis can be used for all types of situations. In UBC (1997), minimum values of some design parameters to be used in dynamic analysis are defined in terms of percentage of values obtained by static analysis and thus most of the steps of static analysis are necessary for a designer, even if the dynamic analysis procedure is followed. These analysis methods require the modeling of the base isolation system and the superstructure. In most of the cases, the linear model for the superstructure of a base isolated building is enough. However, the isolation systems generally require nonlinear model. 29.2.7.1

Modeling of a Superstructure

The superstructure of a base isolated building can be modeled as rigid blocks, a two dimensional and a three dimensional linear model, or nonlinear models. The rigid block assumption is the simplest one, which is used in the design of isolation system, for preliminary analysis and feasibility studies. Moreover, it can also be used as a tool to crosscheck the results obtained from other rigor ous analysis methods. Assuming the superstruc ture as a single degree of freedom (single DoF) model and the isolation system as linear spring plus linear viscous damper, Kelly (1990) developed FIGURE 29.22 A two DoF model of base isolated linear theory for a base isolated structure (Figure building. (Source: Kelly, J.M. Earthquake Spectra, 6(2), 29.22). In general, an isolation system aims at 223 244, 1990. With permission.) providing significant protection not only to the building but also to nonstructural components and contents. Therefore, a three dimensional linear model for the superstructure is generally sufficient for analyzing a base isolated structure. However, for the situations when the response to extreme earthquakes must be investigated, nonlinear models for superstructure should be used. In case of the analysis of seismically isolated bridges, it may be important to take into account the pier masses and their own modes of vibration. The pier can be modeled using a linear beam element with mass, and the deck can be modeled with linear beams with mass. However, for very squat piers, the shear flexibility should also be considered. In the case of a regular structure, the coupling effect of the deck can be neglected, and a bend can be considered independent of the others. However, if a stiff deck is supported on piers of different flexibility, the structure should be modeled as a linear multi DoF system. To ascertain the response for extreme loading conditions, the piers should be modeled with nonlinear models; however, the deck can still be simulated with linear elements. In such cases, proper account should be made for change in the effective stiffness of piers due to cracks (Priestley et al., 1996). In the case of equipment, the modeling depends upon the level of protection; however, the rigid block assumption for equipment generally suffices. Liquid storage containers should be modeled with proper consideration to the sloshing affect of liquid during base motion. 29.2.7.2

Modeling of Isolation System

The isolation system can be modeled as a linear or nonlinear element. The linear model of an isolation system consists of effective stiffness and effective damping at the design displacement. These values are derived from force displacement test data for the isolation system. However, except in preliminary designs, the isolation system is generally modeled as nonlinear. Commonly used isolation systems, namely, high damping rubber bearings, lead rubber bearings, friction bearings, and FPSs, can be modeled by bilinear model (Figure 29.23). However, for more precise results, the isolation elements should be modeled by their actual force displacement behavior, for example, in the modeling of


29 21

high damping rubber bearings, which in extreme conditions follow a trilinear model with stiffening at larger strains. The model used for analysis should be verified with force displacement characteristics obtained by prescribed tests, dis cussed in Section 29.2.6. Moreover, the isolation system should be modeled with considerations for the following requirements (UBC, 1997):

F

F

msW

msW d

d

(a)

(b)

1. Account for the spatial distribution of FIGURE 29.23 Analytical models for isolators: (a) pure isolator units. friction system; (b) friction pendulum system. 2. Calculate translation, in both horizontal directions, and the torsion of the structure above the isolation interface, considering the most disadvantageous location of mass eccentricity. 3. Assess overturning or uplift forces on individual isolator units. 4. Account for the effects of vertical load, bilateral load, and/or the rate of loading if the force deflection properties of the isolation system are dependent on one or more of these attributes. 29.2.7.3

Static Analysis Procedure

This procedure may be sufficient for analyzing a base isolated structure that satisfies certain code requirements related to the height of a building, soil conditions, geometry of a structure, effective period of base isolated building, force deflection, and restoring force characteristics of the isolation system, and so on. The procedure involves the use of simple expressions for computing design displacements (D) and lateral forces. Using the response spectrum or code formula, the design displacement is computed for a target period of the isolation system, its effective damping, and site conditions. Since time period and effective damping itself depends on design displacement, it becomes an iterative procedure. Appropriate values of damping and the target time period can be assumed for the first iteration. Force displacement data for the isolation system, obtained from tests, is used to determine effective stiffness ðkeff Þ and damping at design displacement. The design lateral seismic force for the isolation system and structural elements at or below the isolation system is simply evaluated as Vb ¼ keff D

ð29:19Þ

Total lateral force for the structure above the isolation system is given by Vs ¼

keff D RI

ð29:20Þ

The factor RI is based on the type of lateral force resisting system used for the structure above the isolation level. The value of Vs is not to be taken as less than the following (UBC, 1997): 1. The lateral seismic force required for a fixed base structure of the same weight, W, and a period equal to the isolated period 2. The base shear corresponding to the design wind load 3. One and a half times the lateral seismic force required to fully activate the isolation system The total lateral force shall be distributed over the height of the structure in accordance with the formula W x hx Fx ¼ X Vs ð29:21Þ n WI hI i¼1

where Wx is weight at level x and hx is the distance of level x from the base. Fx ; the lateral force at level x, is applied over the area of the building at that level in accordance with the mass distribution. The maximum

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displacement of the isolation system is calculated in the most critical direction and for maximum capable earthquake. The total design and maximum displacement of the elements of the isolation system should include the additional displacement due to actual and accidental torsion, calculated by taking into account the spatial distribution of the lateral stiffness of the isolation system and the most disadvantageous location of mass eccentricity. 29.2.7.4

Dynamic Analysis Procedure

The method involves response spectrum or time history analysis, and is essential when, owing to code requirements, it is not possible to use the static analysis procedure alone. However, for the situations when (1) the structure above isolation system may reach the nonlinear range for the design earthquake motion, (2) the soil is soft or site conditions require site specific evaluation, or (3) the isolation system does not meet certain prescribed code criteria, even the use of the response spectrum method is not sufficient and the time history analysis procedure becomes essential. 29.2.7.4.1 Response Spectrum Method This method of analysis is very convenient for the laminated rubber bearing type isolation system. A design spectrum is constructed for the design basis and the maximum capable earthquake. Total design displacement of the isolation system and the lateral forces and displacements of the isolated structure are computed for the design basis earthquake. Maximum displacement of the isolation system should be obtained for the maximum capable earthquake. In this method of analysis, damping for fundamental mode in the direction of interest is generally restricted to the effective damping of the isolation system or 30% of the critical damping, whichever is less. Damping for higher modes is consistent with those appropriate for the response spectrum analysis of similar fixed base structure. To take into account the effect of bidirectional excitation, the total maximum displacement includes simultaneous excitation of the model by 100% of the most critical direction of the ground, and not less than 30% of the ground in the orthogonal direction (UBC, 1997). The maximum displacement of the isolation system is calculated as the vector sum of the two orthogonal displacements. The total lateral force shall be distributed over the height of the structure in accordance with Equation 29.21. 29.2.7.4.2 Time-History Analysis This method of analysis can be used for base isolated structures under all types of situation. The method is necessary for conditions where (1) isolation systems have a sliding system, (2) the structure is founded on very soft soil, (3) the isolation system allows low displacement, or (4) the force displacement behavior of the isolation system depends on (a) the rate of loading, (b) the vertical load, or (c) the bilateral load. Appropriate horizontal ground motion time histories are selected, and their magnitudes, fault distances, and source mechanism should be consistent with those that control the design basis (or the maximum capable) earthquake. Where appropriate recorded ground motion time history pairs are not available, appropriate simulated ground motion time history pairs may be used. Each pair of time histories is applied simultaneously to the model, considering the most disadvantageous location of mass eccentricity. The maximum displacement of the isolation system is calculated from the vectorial sum of the two orthogonal displacements at each time step. 29.2.7.5

Software for Analysis

The need for an isolation system that is stiff under low levels of frequently occurring loads but flexible under higher levels necessarily leads to a nonlinear system. The force displacement behavior of most of the isolators can be modeled as bilinear. However, in the case of high damping rubber bearings, in extreme conditions, a trilinear model with stiffening at large shear strains better represents the behavior. These nonlinear force displacement relationships of the bearings need the use of specialized computer programs to analyze base isolated structures. The program should be capable of analyzing base isolated


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buildings with isolator models made of combination of discrete nonlinear elements and superstructure models that are fully elastic or that permit some localized nonlinear behavior. Several programs are available for this purpose. The computer code NPAD has plasticity based nonlinear elements that can be used to model certain types of elastomeric bearings. The program simulates the superstructure using a three DoF per floor model and can analyze base isolated structures with linear superstructure (Tospelas et al., 1994). The justification behind the approach of considering superstructure as linear is that seismic base isolation is generally provided to reduce the force transmitted to the superstructure to the point where it can be assumed to remain elastic. The computer program SADSAP has the capability to analyze three dimensional structures with localized nonlinearities. The nonlinear elements available in the software can be combined to simulate the behavior of various types of isolation systems. The program may also be useful in analyzing base isolated buildings with nonlinear bracing elements provided for supplemental damping (Kelly, 1997). Some general purpose programs such as the DRAIN series and ANSR have the capability of nonlinear analysis of two and three dimensional structures and have provisions that can be used to simulate seismic isolators, which exhibit bilinear behavior (Tospelas et al., 1994). Some commonly used finite element analysis software such as ABAQUS, ANSYS, and COSMOS also have the capability for analyzing complex superstructures on nonlinear isolators (Kelly, 1997). However, sliding bearings cannot be modeled precisely with these programs. SAP2000 (2002) can be used to analyze these structures and also has the facility to model friction bearings. The 3D BASIS group of programs is a group of special purpose programs developed for nonlinear dynamic analysis of three dimensional base isolated structures, and has been used for the analysis of many base isolated structures (Constantinou, 1994). The programs assume that the superstructure is elastic and the isolation system is nonlinear. Models used for modeling the isolation elements can also capture precisely frictional behavior. It is not possible to account for stiffening of laminated rubber bearings at higher strain levels using the programs discussed above. The program LPM (lumped parameter model), originally developed for nonlinear three dimensional analysis of masonry structures, has the capability to model this stiffening and at the same time can also incorporate a nonlinear element in the superstructure (Kelly, 1997).

29.2.8

Experimental Methods for Isolated Structures

Although analytical simulation has progressed tremendously through the use of powerful computers utilizing accumulated theoretical and experimental data on structures, it is not yet possible to analyze and predict the inelastic response of base isolated structures with total confidence. The responses of base isolated structures are very much dependent on the characteristics of the input earthquake motion, isolation system, and the supported structures. Experimental tests have been used to verify and calibrate mathematical models, as well as to provide data for the design of structures. However, extensive tests are needed to study the complex behavior of base isolated structures under realistic conditions that are likely to occur during earthquakes. Different test methods such as quasistatic testing, shaking table testing, online hybrid testing, and substructured online testing have been proposed. A typical problem that appears when performing quasistatic tests on an isolated structure is the alteration of the restoring forces due to strain rate effect existing for this material. This problem does not exist for shaking table tests performed at the real size and real time scale. Shaking table tests of base isolated specimens have been used successfully for investigating the behavior. However, it requires a large capacity table for testing a full scale specimen. The involvement of high cost for large size shaking tables and the limited capacity of available shaking tables necessitate the structure to be drastically scaled down. This scaling of a structure presents some limitations for the extrapolation of the results to the real scale. As a promising option, the online hybrid test method seems to be well suited for obtaining the seismic response of isolated structures thanks to available quasistatic loading equipment and high speed computers. However, this technique involves the fabrication of a complete structural prototype

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FIGURE 29.24 Substructured hybrid testing for seismically isolated structures. (Source: Tanzo, W. et al. Res. Rpt. 92 ST 01, Kyoto Univ., 1992. With permission.)

for proper modeling. Thus, test models must be simplified, for example, into a single DoF model, and/or specimens have to be reduced in scale. By incorporating substructure concepts into the online hybrid test technique, a substructured online hybrid test method can be used, in which a complete system is considered to be divided into analytical substructures and experimental substructures (Tanzo et al., 1992). In the case of seismically isolated structures, inelastic deformations are designed to occur only in isolation system. The isolation system, which is generally difficult to model mathematically, is taken as experimental substructures (Figure 29.24). The remaining part of the structure is taken as analytical substructures in which presently available analytical models are used to describe their restoring force characteristics. For the experimental substructure, restoring force information is directly measured from a specimen loaded according to its current deformation state. The substructured online hybrid test procedure for the earthquake response of base isolated structure is explained in Figure 29.25.

29.2.9

Implementation of Seismic Isolation

29.2.9.1

New Construction

29.2.9.1.1 Buildings In the early stages of the development of seismic isolation, the main target of the earthquake resistant design of the structure was the prevention of collapse. However, later, other additional considerations, such as the value of structure or its content, have exerted their importance on the seismic design of structures. For example, facilities of postearthquake importance such as fire stations, police stations, hospitals, communication centers, and so on, have a critical role and are required to remain operational immediately after a major seismic event. Similar reasoning applies to museums housing unique artifacts or buildings that are architecturally important. The required low level of structural and nonstructural damage may be achieved by using an isolation system, which limits structural deformations and ductility demands to low values. The isolation system can be provided at different levels. However, while deciding the isolation level, considerations should be given for (1) the seismic gap, (2) the continuity of supply lines, stairways, and elevators, and (3) details of cladding below the isolation level. In addition, provisions for (1) access to bearings for inspection and replacement, (2) backup system for vertical loads, and (3) full diaphragm for the uniform distribution of lateral load to individual bearings also need to be considered. Figure 29.26 shows typical positions for bearings in buildings (Mayes and Naeim, 2001). The first building constructed using modern isolation bearings was a government facility, the William Clayton Building in Wellington, New Zealand, which was completed in 1981. After that, a number of seismically isolated buildings have been constructed there, and in most of them the isolation system used is lead rubber bearings. The most widespread use of seismic isolation systems is in Japan. The first seismically isolated structure to be completed in Japan was the Yachiyodai Residential Dwelling, a two story building completed in 1982. Better performance of seismically isolated buildings during the


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FIGURE 29.25 Procedure for substructured online test method for a base isolated structure. (Source: Tanzo, W. et al. Res. Rpt. 92 ST 01, Kyoto Univ., 1992. With permission.)

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FIGURE 29.26 Typical positions of isolation level for buildings. (Source: Mayes, R.L. and Naeim, F. 2001. The Seismic Design Handbook, 2nd ed., Kluwer, Boston. With permission.)

Kobe earthquake of 1995 triggered the rapid increase in number of seismic isolation projects. Before the Kobe earthquake, the number of such buildings in Japan was 82; this suddenly increased to about 650 within 4 years of the earthquake. Besides rubber bearings, there have been some applications of frictional sliding systems also for seismic isolation of buildings (Nishitani, 2000). The first newly constructed base isolated building in the United States was the Foothill Communities Law and Justice Center in California, which was completed in 1986. Until 1990 there were four isolated buildings. After the good performance of base isolated buildings during the Northridge earthquake, the number increased and, at the end of 1998, there were approximately 40 isolated buildings completed or in construction in the United States (Clark et al., 2000). Some buildings in other countries, including Italy, France, Canada, Mexico, China, England, Russia, Armenia, Indonesia, Iran, Chile, and India, now use these systems. Until now, the emphasis on base isolation applications has mostly been for structures with sensitive and expensive contents or high risk structures like nuclear power plants and computer centers. However, there is an increasing interest in applying this technology to public housing and schools in developing countries (Fuller and Muhr, 1995). Most of the base isolation projects have made use of laminated rubber bearings and lead rubber bearings. In Japan, laminated rubber bearings are often used with energy dissipating devices, both with and without provision for recentering. 29.2.9.1.2 Bridges In most cases, bridges are strategic structures and require a higher degree of protection to ensure their functionality after a seismic event. For many simple bridges, it has been found that seismic isolation of the superstructure gives improved seismic resistance, often at a reduced cost, while it is also effective for thermal expansion of the superstructure (Skinner et al., 1993). The aim of seismically isolating bridge superstructures is usually to protect the piers and their foundations, and sometimes to protect the abutments also. With this approach, bearings are placed between the superstructure and the top of the substructure (the piers and abutments). Under normal conditions, they behave like regular thermal bearings. However, in the event of a strong earthquake, they add flexibility to the structure by elongating its period and dissipating energy. In addition, most of the mass of a bridge is concentrated at the deck level, which is inherently strong and can be assumed to be rigid. This permits the superstructure to oscillate at a lower frequency than its piers, which results in the reduction or elimination of deformation of the substructure components beyond their elastic range, particularly at locations that are difficult to inspect or repair (e.g., the piles). Superstructure isolation systems are designed, as far as is practical, to provide moderate flexibility, high damping, torsional balance, and an appropriate distribution of seismic loads between the superstructure supports. Isolated bridges in the United States are generally designed for the effects of a design basis earthquake; however, the components of the isolation system are provided with increased displacement capacity and are tested for the effects of maximum capable earthquake. This design philosophy is based on observations of bridge failures in past earthquakes, which were primarily due to excessive bearing displacement and the loss of bearing support rather than the collapse of substructure. Providing for strong restoring force in this design philosophy prevents the accumulation of significant permanent displacements and allows for a relatively reliable estimation of peak bearing displacement and substructure force.


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Isolation systems adopted by Italian engineers restrict the transmission of force to elements of the substructure to a predetermined level, which is independent of seismic action. Thus, the Italian engineers chose to limit the force transmitted to the substructure to a desired level, at the expense, however, of large variation in peak bearing displacements and the development of permanent displacements. In Japan, menshin (reduction of response) design is followed for the design of bridges. Although menshin design is closely related to the seismic isolation design, the natural period of a bridge is not forcibly elongated in this method, because there are various restrictions in increasing the natural period. Instead of elongating the period, the emphasis in the menshin design is on increasing the energy dissipating capability and distribution of lateral forces to as many substructures as possible in order to decrease the lateral forces for the design of substructures (Iemura, 1994a, 1994b). It is common to restrain the transverse displacement of bridge isolators, and therefore many bridges in Japan are isolated in longitudinal direction only. The application of seismic isolation for bridges started in the early 1970s. Bridges isolated in New Zealand before 1978 used metallic dampers for seismic isolation. Later, lead rubber bearings replaced the metallic dampers for their seismic isolation. Italian engineers also started the application of modern technology of seismic isolation for bridges in the mid 1970s and now have a long list of isolated bridges. The first seismically isolated bridge in Japan was the Hokuso line viaduct in Chiba Prefecture. The viaduct was completed in 1990 and used lead rubber bearings for its isolation system. The United States also obtained its first seismically isolated new bridge in 1990, when two bridges, namely, Sexton Creek Bridge in Illinois and Toll Plaza Road Bridge in Pennsylvania, were completed. Lead rubber bearings were used as the isolation system for both the bridges (Skinner et al., 1993). 29.2.9.2

Seismic Retrofit

Postearthquake analysis of recent earthquakes around the world shows that the damage was especially concentrated in the buildings and bridge structures that were not designed for earthquake loads or were designed by following the old seismic design guidelines. Moreover, a significant proportion of the damage was due to nonstructural elements and the contents housed in the buildings (Kelly, 1997). In order to prevent the recurrence of damage in such seismically deficient structures, it is vital to understand their vulnerability and retrofit them for expected future seismic events. In the case of important structures, retrofit techniques are required to provide a solution that can also lead to continued functioning during and after major earthquakes, protection of nonstructural components and contents, and, sometimes, to preserve their historic character. 29.2.9.2.1 Buildings Conventional retrofitting has been the most common way of enhancing the performance of existing, seismically deficient structures. However, to maintain the usability of important structures after a big earthquake and to protect the valuable contents housed in the buildings, conventional methods can help a little. Simple strengthening or stiffening of a structure through reducing inelastic response displace ment, however, may increase the response acceleration and the forces. Some of the other demerits of this approach are (i) the large amount of labor required to increase the strength and ductility of critical sections distributed throughout the complex structural system and (ii) damage may occur due to inelastic behavior of structural elements during a major seismic event (Iemura and Adachi, 2003). Besides this, conventional methods of retrofitting often require restricting the use of the building during the retrofit. Some buildings, housing important activities, cannot afford to have their routine usage stopped and therefore require retrofitting through an earthquake resistance system that enables continued building use even during the retrofit operation. Seismic isolation has proved to be very effective not only in enhancing the safety of existing seismically deficient buildings, but also to preserve the function and protect the contents. Unlike the conventional approach, this retrofit method aims to reduce the earthquake force to a level lower than the strength of the existing structure (Figure 29.27). Seismic isolation is the method that can mitigate both interstory drift and high floor accelerations simultaneously, and it has been used in retrofitting of important buildings, buildings housing valuable contents, and structures of postearthquake importance.

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Safety margin

Required strength

Required strength

Seismic isolation Safety margin Lack of strength

Acceleration (Strength) Lack of strength

Fundamental Period

Fundamental Period Before retrofit Retrofit by seismic isolation

Before retrofit After retrofit (a)

(b) Note: Region above the required strength curve indicates safe zone.

FIGURE 29.27 Concept of retrofitting methods: (a) conventional; (b) seismic isolation. (Source: Sugano, S., Proc. XII World Conf. on Earthquake Engineering, 2000. With permission.)

Moreover, as the modification or demolition of building features is minimized, the method has found application in many monuments of historical importance, protecting them from future earthquakes (Buckle, 1995). Recently, the old masonry Chapel of Rikkyo (St Paul’s) University, Tokyo, constructed in 1920, was retrofitted with seismic isolation to preserve this historical asset to the university (Seki et al., 2000). Retrofitting using seismic isolation of the 72 year old Clock Tower, a symbol of Kyoto University, Japan, is presently underway (Ogura et al., 2003). The seismic isolation method is found to be more appropriate for the retrofitting of seismically vulnerable buildings that are required to function even during retrofit operation. To meet such a demand, the retrofitting method should affect minimum alterations to the existing structures and should produce a low level of noise and vibration. The technique has successfully been applied to a number of such buildings. In the retrofitting of Rankine Brown Building of Victoria University, Wellington, New Zealand, which houses the main library and administration of the university, one of the main requirements was the building’s uninterrupted usage during the retrofit operation (Robinson, 2003). Meeting a similar challenge, the government office of Toshima Ward, in Tokyo, has been retrofitted with rubber and sliding isolators below the existing foundation and was used as usual by the officials and citizens even during the retrofit works. Similarly, a building of Nihon University in Tokyo, Japan, in which seismic isolation was applied at the top of the columns of basement floor, was available for research and education during the retrofit operation (Kawamura et al., 2000). Although seismic isolation reduces earthquake forces, it does not eliminate them. Consequently, the strength and ductility of an existing structure must at least be sufficient to resist the reduced earthquake forces that remain even after isolation. Further, the requirement of space for the seismic gap and provisions for services, elevators, and so on to follow large displacements at isolation level are some issues that need prior attention. Although not applicable to all structures and all site conditions, seismic isolation still has great potential to reduce the risk in existing buildings and bridges that are not capable of resisting expected future earthquakes. This method, in the beginning, was used generally for preservation of cultural assets, but now has many applications in the retrofitting of other buildings also. 29.2.9.2.2 Bridges A large number of old bridges were constructed with little or no consideration for seismic requirements. Some of them have been retrofitted, mostly by conventional methods. However, the remaining bridges are waiting for the enhancement of their strength to meet the latest code requirements. In existing bridges, bearings have generally been provided to accommodate longitudinal thermal movements between the superstructure and the supports. Therefore, little structural modification is required to use seismic isolation as a retrofit method. A common structural configuration, a continuous deck supported


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on bearings at the tops of piers, attracts the option of seismic upgrading of existing bridge structures by seismic isolation and is often practical and relatively inexpensive. Seismic isolation has been used for the retrofitting of number of bridges in New Zealand. During the early period of its application in the United States, seismic isolation was mainly used for the retrofitting of bridges and the isolation system used was mainly lead rubber bearings (Skinner et al., 1993). However, to replace an existing bearing with an isolation bearing, enough space is required between the deck and pier top. All the jacks used for replacing the bearings should be driven by the same hydraulic pump and raised at the same time to prevent stressing any cross members between girders. The dampers can be connected to the substructure using heavy duty mechanical or chemical anchorages, with the other end bolted to the girders. Since an earthquake can shift a bridge in any direction, universal joints are recommended on each end of the damping device to prevent the shaft from being bent during movements other than axial ones. The damper should be set to the bridge in its midstroke position or a position that takes full advantage of the free movements (Hipley, 1997). In situations where rigid connection of footing and piers has made the bearing capacity of existing piles inadequate for the new seismic requirements, seismic isolation bearings can be used at the base of columns. Sometimes, it may be costly to retrofit a seismically deficient pier and foundation by conventional methods due to submerged conditions that may require construction of cofferdams. In such cases, seismic isolation can reduce the demand on the seismically deficient piers by redistributing the force and transferring it to the abutments, which are significantly less expensive to retrofit. Seismic retrofits are intended to bring bridges up to reasonable earthquake standards; however, in some old bridges, the retrofit method is also required to have the least effect possible on the historic appearance and seismic isolation is the appropriate alternative for this requirement. 29.2.9.3

Other Implementations

Apart from in buildings and bridges, seismic isolation has found application in other structures. It has been used for a number of reactor units in France for which the site safe shutdown earthquake acceleration was 0.2g. An isolation system of neoprene pads with topping of lead bronze alloy limits the acceleration to a value of 0.2g. A similar system was used for reactor units in Koeberg, South Africa, where the site safe shutdown earthquake acceleration was 0.3g (Tajiran, 1998). Seismic isolation has a lot of potential for the protection of sensitive equipment that can malfunction when the acceleration reaches beyond its allowable limit. Communication centers, hospitals, computer centers, and other facilities housing such sensitive equipment can use seismic isolation to remain functional even after a major seismic event. Proposed systems for the seismic isolation of sensitive equipment are to (1) base isolate the entire building or a portion of the building, (2) isolate essential flooring systems only (raised floors), and (3) isolate individual pieces of equipment by placing them on separate isolation bearing systems (see Figure 29.28). In Japan, there are many buildings that are using the raised floor system to protect computers and other sensitive equipment. Seismic isolation is found to be very useful for semiconductor facilities, in which cost of the delicate items is as high as 75% of the total investment (Amick et al., 1998). Liquid storage tanks are important components of chemical factories and are also used in increasing

FIGURE 29.28 Seismic isolation for protection of equipment. Unreinforced rubber blocks. (Source: Ohashi, U.G. Earthquakes and Base Isolation, Pub. Asakura, Tokyo, 1995. With permission.)

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FIGURE 29.29

Three dimensional base isolation system for Moroku Bosatsu statue.

TABLE 29.2 Details of Isolation System and Dynamic Characteristics of the Structure

Elastic support Damping mechanism Natural frequency (Hz)

Horizontal Direction

Vertical Direction

Coil spring Friction damper 0.25 0.35

Coil spring Air damper 2.0 2.5

number to store liquefied natural gas. The FPS was applied for seismic isolation of a large size tank in Greece, and steel rubber bearings were used for another tank in Korea, storing liquefied natural gas (Tajiran, 1998). Seismic base isolation can also be used for the seismic protection of antique pieces. Iemura and coworkers designed a special seismic isolation system to protect a wooden carved statue of Moroku Bosatsu (Maitreya). It is an antique cultural artifact, considered to be the first wooden carved statue in Japan. The isolation system provides protection in all three directions FIGURE 29.30 Overall size of the three dimensional (Figure 29.29). The details of the isolation system base isolation system. are provided in Table 29.2. The overall size of the isolation system is shown in Figure 29.30. The shaking table test was also carried out to verify the effectiveness of the isolation system. Results of the experiment are presented in Table 29.3 and clearly show that the isolation system is very effective in reducing the accelerations.

29.2.10

Performance during Past Earthquakes

Analytical models and experimental tools have been developed by researchers to simulate the behavior of a structure under earthquake motion. However, none of these methods can validate the behavior of a structure as does a real earthquake. In fact, many of the concepts of earthquake resistant design are

Seismic Base Isolation and Vibration Control TABLE 29.3 System

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Results of Shaking Table Tests of a Three Dimensional Base Isolation

Direction

Level Acceleration (gal) Shaking Table

NS EW Vertical

795 505 314

Above Base Isolation System 166 136 250

Relative Displacement of Base Isolation (cm) 15.3 10.0 1.1

the outcome of the analysis of observations made or recorded during past earthquakes. In the case of base isolation, as it is a relatively newer approach, it can help to assess the modeling techniques, existing analysis and design procedures, and methods followed in construction. Such analyses may help in evaluating the effectiveness of base isolation and also exposing weak aspects of designs. Strong motion recording instrument networks have been installed on some of the base isolated structures in different parts of the world in order to record the response of those structures during earthquakes. During the Northridge, Kobe, and Turkey earthquakes, a few base isolated structures had their isolation system reaching nonlinear range, and the recorded performance of these structures can give a fair idea of the behavior of base isolated structures during strong earthquake motions.

29.2.10.1

Performance of Buildings

The Northridge earthquake of January 1994 saw thousands of conventionally designed buildings and their contents suffer extensive damage and be closed for several days. However, two base isolated buildings, namely, University of Southern California Hospital building and Fire Command and Control Facility, performed well during the earthquake. During the event, the seven story base isolated hospital building experienced a peak ground acceleration of 0.37g. The recorded peak acceleration at the roof was 0.21g. Analytical study showed that presence of the isolation system reduced the peak story shear at the base to about one third of that if the building had its base fixed (Nagarajaiah et al., 1996). “Workers in a large supply room within the hospital reported no disruption of supplies, or material falling off shelves, as a result of the earthquake. In contrast a pharmacy at the ground level in an adjacent medical building with fixed base reported substantial disruption of the supplies” (EERI, 1996). The two story base isolated Fire Command and Control Facility could not perform as per expectation due to minor negligence in detailing the seismic gap. The building is base isolated by high damping rubber bearings and experienced peak ground acceleration of 0.19g. The recorded response shows the presence of sharp acceleration spikes that were due to pounding. However, the analytical study by Nagarajaiah et al. (2001) shows that, even with pounding, maximum base story shear of the base isolated building was half that of the fixed base case and without pounding it would have been one fourth. In January 1995, the unexpected earthquake of magnitude 7.2 in Kobe shocked the engineering community. The 7 sec of strong shaking caused a heavy loss of life (over 5000 people died) and a catastrophic loss to infrastructure. The strong earthquake tested two existing base isolated buildings in the region, namely, the six story building of West Japan Postal Savings Computer Center (WJPSCC) and the three story building of Technical Research Institute of Matsumura Gumi Corporation (TRIMGS). The six story WJPSCC building that is base isolated with elastomeric isolators, lead rubber bearings, and steel coil dampers experienced peak ground acceleration of 0.306g. A nearby six story fixed base building experienced peak ground acceleration of 0.27g. However, during the event, the maximum acceleration recorded at the roof of the fixed base building was about nine times of that recorded at the roof of the WJPSCC building. Similarly, a

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comparison of the roof acceleration of the three story base isolated TRIMGS building with that of a nearby three story fixed base building reveals that, due to base isolation, the peak accelerations at the roof were reduced by a factor of approximately 4.9 in the north south direction and 2.5 in the east west direction. It was later realized that the performance of this base isolated building might be adversely affected due to freezing cold, when the rubber has a tendency to become stiffer (Izawa et al., 1996). 29.2.10.2

Performance of Bridges

In the year 1999, Turkey faced two major earthquakes, the first one on 17th August and the second on 12th November. At the time of the events, there were two base isolated structures in the region, namely, Bolu Viaduct and Bolu Bridge. The structures were designed for a peak ground acceleration of 0.4g with isolation system comprising of a friction system and crescent moon type hysteretic elements. During the earthquake of August 17, 1999, when the estimated ground acceleration at the Bolu viaduct site was 0.39g in the longitudinal direction and 0.31g in the transverse direction, the viaduct survived without any damage. The residual displacement was of the order of 1 mm only. However, the second event produced near fault, or rather on fault, pulse type motion with a peak ground acceleration at the viaduct site in the order of 1.0g. The unexpected level of ground acceleration caused heavy damage to the viaduct. It was reported that the sliding bearings fell down from most of the piers before any cyclic movement. This performance again raised the issue of feasibility of base isolation for such near fault pulse type motions. During the same event, Bolu Bridge experienced the estimated peak ground motion of 0.65g, much higher than the design value. However, the only significant damage suffered by the bridge was the failure of some elements of the expansion joints because they were designed for a movement of 100 mm only (corresponding to the earthquake with a return period of 50 years). After the earthquake, the bridge showed a residual displacement in longitudinal direction of 45 mm and in transversal direction varying from 60 to 100 mm (Marioni). The Eel River Bridge (U.S.), after being retrofitted with lead laminated rubber bearings, was hit by the magnitude 7.0 Cape Mendocino (Petrolia) earthquake in the year 1992. The peak ground acceleration recorded at a nearby recording station was 0.55g. The bridge suffered structural damage limited to the spalling of concrete at joints. Another base isolated bridge, Sierra Point overpass (U.S.), was subjected to the 1989 Loma Prieta earthquake when peak ground acceleration at the base of bridge columns was 0.090g. The bridge, first constructed in 1956, was later retrofitted with lead rubber bearings in 1985. However, the abutments were not modified for the larger clearance required in isolation. This design weakness was exposed during the Loma Prieta earthquake and caused significant amplification in the steel superstructure of the bridge. However, no damage was sustained (Lee et al., 2001). Rangitaiki River Bridge at Te Teko, New Zealand, with lead rubber bearings on each pier and plain rubber bearings on abutments, suffered minor damage and a small permanent displacement during the 1987 Bay of Plenty earthquake. The peak acceleration, recorded at about 11 km from the bridge site was 0.33g. It was observed that owing to the inadequate fastening details for the isolators, one of the two bearings on abutment was dislocated (McKay et al., 1990). Bai Ho bridge (Taiwan), a three span, continuous nonprismatic prestressed concrete girder bridge, was subjected to the magnitude 6.0 Gia I earthquake in year 1999. The bridge is seismically isolated in its longitudinal direction with lead rubber bearings; however, shear keys and specially designed steel rods are installed on both abutments to restrict the transverse movement of the superstructure. The peak acceleration recorded on the deck in longitudinal direction was slightly higher than that in the foundation while, owing to restraints, the peak acceleration in the transverse direction was 2.5 times than that of foundation (Lee et al., 2001). During the Kobe earthquake, the base isolated Matsunohama Bridge had good recorded performance; however, the ground acceleration was not profound enough to test the effectiveness of the isolation system (Izawa et al., 1996).


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In summary, analysis of the records obtained from a variety of base isolated structures during real earthquakes presents some very interesting facts. In general, it was observed that, during strong earthquakes, numerous conventionally designed buildings suffered damage and remained closed for several days; however, the base isolated buildings in the same region not only survived but continued to operate. On the other hand, in some cases, strong earthquakes exposed the lack of attention to minor construction details and the effectiveness of base isolation for near fault earthquakes. Better performance of base isolated structures during the Northridge and Kobe earthquakes have now persuaded the structural engineers and owners and triggered a rapid increase in the number of base isolated buildings. In Japan alone, the number of buildings of this kind constructed during 1995 was almost the same as that of those constructed during the period from 1985 to 1994.

29.3 29.3.1

Seismic Vibration Control Historical Development of Seismic Vibration Control

It is worth noting that a structural vibration control system for reducing seismic responses was developed several centuries ago in Japan, by an unknown engineer. This was applied to the construction of a Gojunoto (pagoda). The Goju noto was five stories tall and was constructed of closely fitting mortised wooden beams and Wooden pagoda columns (Figure 29.31). During an earthquake, the vibrations of such a vertical cantilever structure would produce bending moments that could not be resisted by tension at the mortised joints. To Suspended overcome this weakness, a long wooden pole was pole suspended freely from the upper part of the pagoda so that it could undergo pendular vibrations if the pagoda was excited into motion by an earthquake. The weight of the pole exerted a compressive prestress on the pagoda, thus increas ing the bending resistance. The bottom of the pole extended into a cylindrical hole in the ground that was of a larger diameter than the pole. Thus, when the pagoda was excited into vibrations by an earthquake, some of the vibrational energy would be transferred into oscillations of the pole and the impact of the pole on the sides of the hole would dissipate energy. In 1969, Gupta and Chandrasekaran (1969) FIGURE 29.31 Sketch of a wooden pagoda with studied the effectiveness of a number of tuned suspended pole that is free to undergo pendular vibration. (Source: Tanabashi, R., Proc. II World Conf. mass dampers (TMDs) put in a single DoF on Earthquake Engineering, 1960. With permission.) system to reduce seismic responses. The TMD system has both linear and elasto plastic restor ing force characteristics and viscous type damping. The study was shown in the Fourth World Conference on Earthquake Engineering (WCEE) in Santiago, Chile. In the Fifth WCEE in 1973, Skinner et al. (1973) presented their studies on energy absorption devices for earthquake resistant structures. The devices were based on the plastic deformation of mild steel. In 1977, Ohno et al. (1977) introduced optimum tuning of the dynamic damper for civil structures under earthquake

29 34


excitations. Skinner et al. (1977) showed their Floor Force works on studying hysteretic dampers for Fy increasing structural earthquake resistance, and Friction Tyler (1978) demonstrated the usefulness of damper Deflection flexural yielding of steel plate energy dissipation Column devices in New Zealand. In the seventh WCEE in 1980, Sladek and Klingner (1980) reported their studies on tuned (b) mass dampers for seismic response reduction in (a) civil structures, whereas Keightley (1980) presented his work on a dry friction damper for multistory FIGURE 29.32 (a) Friction dampers in a framed structures. Figure 29.32 shows the schematic of a structure; (b) hysteretic loop of the damper. (Source: framed structure with additional friction dampers Keightley, W.O. Proc. VII World Conf. on Earthquake Engineering, 1980. With permission.) and the hysteretic loop of the damper. In 1984, Pall (1984) numerically studied the performance of a framed building equipped with friction devices. The performance was found to be superior to that of a conventional building. Scholl (1984) introduced brace dampers. The stiffness and damping of the damper can be set to alter the stiffness and damping in a structure. It was typically installed at the cross points in braced frame structure systems. Liu et al. (1984) investigated the effectiveness of an active mass damper control system for controlling coupled lateral torsional motions of tall buildings subjected to strong earthquakes. A special themed session named “Seismic response control of structural systems” was held at the Ninth WCEE in 1988. Some studies on passive systems were reported such as: a hysteretic damper that is put between two adjacent structures (Kobori et al., 1988); a multiple passive TMD (Clark, 1988); viscoelastic materials (Bergman and Hanson, 1988); viscous damping walls (Arima et al., 1988); and magnetic damping (Shimosaka et al., 1988). Also, some studies on active control were reported, such as an experimental study on active mass dampers (Aizawa et al., 1988) and air injection for suppressing sloshing in a liquid tank (Hara and Saito, 1988; Sogabe et al., 1988). Since then, the active vibration control of buildings and other civil structures has attracted a worldwide growing interest as an innovative technology in earthquake engineering. During the Tenth WCEE in 1992, more thorough studies on structural control were reported such as: predictive control of structures with reduced number of sensors and actuators (Lopez Almansa et al., 1992); applicability of vibration control to nonlinear structure (Shimada et al., 1992); full scale implementation of active control (Soong et al., 1992); effects of soil structure interaction (SSI) on actively controlled structure (Wong and Luco, 1992); the time delay problem (Hou and Iwan, 1992); and fuzzy logic control (Tani et al., 1992). In this conference, the decision to form an international association and to hold a world conference dedicated to structural control was made. The International Association for Structural Control (IASC) was formed in 1993. The efforts led to the successful First World Conference on Structural Control in 1994 in Pasadena, California (IASC, 1994). It was followed by the Second World Conference on Structural Control in 1998, which was held in Kyoto, Japan (IASC, 1999). The Third World Conference on Structural Control was held in April 2002, in Como, Italy (IASC, 2003). The conferences showed increasing interest among researchers and an increase of innovations and applications of structural control technologies in civil structures.

29.3.2

Basic Principles

Structural vibration control basically involves the regulation of structural properties in order to achieve structural a desirable response to a given external load and modification of the excitation. The regulation of structural properties includes the modification of mass, damping, and stiffness of the structure so that


29 35

it can respond more favorably to the external excitation. Moreover, it is also possible to reduce the level of excitation transmitted to the structure. Structural control can be defined as a mechanical system that is installed in a structure to reduce structural vibrations during loadings such as strong earthquakes. 29.3.2.1

Passive Control System

The basic function of a passive control system is to absorb or consume a portion of the input energy, thereby reducing energy dissipation demand on primary structural members and minimizing the possible structural damage. In the passive system, controlling forces develop at the locations of installation of the mechanism itself. The energy necessary for generation of these forces is provided through the motion of the mechanism during the dynamic excitation. The relative motion of the mechanism defines the amplitude and the direction of the controlling force. Passive control has four main advantages: (1) it is usually relatively inexpensive; (2) it consumes no external energy (energy may not be available during a major earthquake); (3) it is inherently stable; and (4) it works even during a major earthquake. In the passive system area, structural response control can be divided into two groups. The first group uses supplemental damping to reduce structural responses by conversion of kinetic energy to heat. The second group uses supplemental oscillators to reduce the structural response by transferring energy among vibration modes. The first group includes devices that operate on principles such as frictional sliding, the yielding of metals, phase transformation in metals, deformation of viscoelastic solids or fluids, and fluid orificing. The second group includes dynamic vibration absorbers such as TMDs and tuned liquid dampers. The major difference between viscous/viscoelastic devices and friction/yielding devices is the maxi mum force that each device will develop during an earthquake. The maximum earthquake forces developed in viscous/viscoelastic devices are determined by the maximum displacements and velocities across these devices. The maximum earthquake forces in a friction/yielding device equals the design friction force/design yield force plus strain hardening. Thus, the maximum earthquake forces are more easily controlled in the friction/yielding devices. The effect of increasing damping in a structure can be appreciated by studying the earthquake response spectrum. The earthquake response spectrum of a quantity is a plot of the peak value of the response quantity as a function of the natural vibration period of the system. Each such plot is for single DoF system having a fixed damping ratio, and several such plots for different values of damping ratio are included to cover the range of damping values. The most commonly used spectra in earthquake engineering are the absolute acceleration response spectrum, the velocity response spectrum, and the displacement response spectrum. For more information on how to develop earthquake response spectrum, readers are referred to Chopra (1995). Figure 29.33 shows the response spectra of measured ground acceleration during the Kobe, Japan, earthquake on January 17, 1995. The ground acceleration north south (NS) component was recorded at the Kobe Marine Meteorological Observatory in Chuo ku. As is shown in the figure, increasing the damping ratio from 5 to 30% decreases the response spectra. Therefore, on a level of ordinary structure that has 5% of damping, the response can be reduced further by increasing the structural damping. In the area of passive systems, a variety of mechanical energy dissipaters to increase structural damping has been developed and tested in the laboratory and, in some cases, in actual structural applications, such as bracing systems, friction dampers, viscoelastic dampers, and other mechanical dampers. The principle of the TMD dates back to the 1940s when Den Hartog (1947) reintroduced the dynamic absorber invented by Frahm in 1909. Figure 29.34 shows the principle of the TMD. A large system under alternating force P0 sin vt is represented by a K M system where K and M are the stiffness and the mass of the system, respectively. The vibration absorberpconsists of a comparatively small vibratory system ffiffiffiffiffi with stiffness k and mass m. The natural frequency, k=m; of the attached absorber is chosen to be equal to the frequency, v, of the disturbing force. It has been shown (Den Hartog, 1947) that the main mass, M, does not vibrate at all and that the small system, k m, vibrates in such a way that its spring force is at all instants equal and opposite to P0 sin vt: Thus, there is no net force acting on M and therefore that mass

29 36


FIGURE 29.33

Effect of damping on earthquake response spectra.

does not vibrate sffiffiffiffi rffiffiffiffi K k m ; m¼ ð29:22Þ vM ¼ ; vm ¼ M m M

x1 K

M

P0 sinwt k

x2 m

It is useful to introduce the new notations, above. Assume that m ¼ 0:2; and vM ¼ vm : For this FIGURE 29.34 Additional vibratory k m system to a condition, the resonance curve of the system is larger K M system to reduce vibration of the larger illustrated in Figure 29.35. At v ¼ vm or v=vm ¼ system. 1; the motion of the main mass ceases altogether (Chopra, 1995). Since the system has two DoF, two resonant frequencies exist and the response is unbounded at those frequencies. If the exciting frequency, v, is close to the natural frequency, vm, of the attached absorber, and the operating restrictions make it possible to vary either one, the vibration absorber can be used to reduce the response amplitude of the main system to near zero. It is worth noting that the larger the attached absorber mass, m, the smaller the absorber displacement, ðx2 x1 Þ; relative to the displacement of main mass, M. However, a large absorber mass presents a practical problem. At the same time, the smaller is the mass, m, the narrower is the operating frequency range of the absorber (Chopra, 1995). 29.3.2.2

Active Control System

Active control systems are used to control the response of structures to internal or external excitation, such as machinery or traffic noise, wind, or earthquakes, where the safety or comfort level of the


29 37

occupants is of concern. Active control makes use 8 of a wide variety of actuators, which may employ x1 hydraulic, pneumatic, electromagnetic, or motor Po /K driven ball screw actuation. An essential feature of active control is that external power is used to effect the control action. This makes the system vulnerable to power failure, which is always a 1 0 possibility during a strong earthquake. 01 0.8 1 1.25 ω/ωm For the actuators properly to apply the desired forces, sensors are placed in the structure FIGURE 29.35 Resonance curve for the system in in order to measure structural response. The Figure 29.34 that shows response amplitude vs. exciting sensors convey response information to a central frequency (the dashed curve indicates negative x or 2 computer that then uses this information to phase opposite to excitation) for m ¼ 0:2 and vM ¼ vm . calculate the desired actuator forces. The advan (Source: Chopra, A.K. 1995. Dynamics of Structures, tage of an active control system is that it attains Theory and Applications to Earthquake Engineering, excellent control results. However, there are many Prentice Hall, New York. With permission.) drawbacks to using the active control system. It is relatively expensive to design and to operate due to Excitation Structure Response the large amount of power needed. It has the potential to destabilize the structural system. Control Furthermore, it tends to take up more space than passive control devices. Sensors Controller Sensors A fully active structural control system has the basic configuration as shown schematically in Figure 29.36. The structural control system in the FIGURE 29.36 Schematic diagram of active control figure basically consists of sensors, controllers, system. and actuators. Sensors are used to measure either external excitations or structural responses, Desired Control or both. Controllers process the measured infor Output output input Controller Plant mation and compute necessary control forces needed based on a given control algorithm. Actua tors are used to produce the required forces and are FIGURE 29.37 An open loop control system; the usually powered by external energy sources. controller applies control input without knowing the According to the characteristics of the control plant output. ling effects, active control can be classified into two categories: open loop control and closed loop control. A control system in which the control input is applied without the knowledge of the plant output is called an open loop control system. Figure 29.37 shows a block diagram of an open loop control system, where the subsystems (controller and plant) are shown as rectangular blocks. Open loop control will be successful only if the controller has a reasonably good prior knowledge of the behavior of the plant, which can be defined as the relationship between the control input and the plant output. Mathematically, the relationship between the output of a linear plant and the control input can be described by a transfer function. However, in actuality, the presence of plant behavior uncertainty is unavoidable. Therefore, it is clear that an open loop control system is unlikely to be successful. In the case that the controller adjusts the control input according to the actual observed output, the system is called a closed loop system. In this system, the control input is a function of the plant’s output. Since in a closed loop system the controller is constantly in touch with the actual output, it is likely to succeed in achieving the desired output even in the presence of uncertainty in the linear plant’s behavior (the transfer function). The mechanism by which the information about the actual output is conveyed to the controller is called feedback. On a block diagram, the path from the plant output to the controller is called a feedback loop. A block diagram of a possible closed loop system is given in Figure 29.38.

29 38


Desired output +−

Error

Controller

Control input

Plant

Output

Feedback loop FIGURE 29.38 Example of a closed loop control system with feedback; the controller applies a control input based on the plant output.

When the desired output is a constant, the resulting controller is called a regulator. If the desired output changes with time, the corresponding control system is called a tracking system. In any case, the principal task of a closed loop controller is to minimize the “error” as quickly as possible. 29.3.2.3

Semiactive Control System

Semiactive control falls between passive and active on the control spectrum. A semiactive control system is similar to an active system in that the system uses sensors and controllers, and operates on external power. However, the source of external energy is used only for adjustment of the mechanical characteristics of the system (Rakicevic and Jurukovski, 2001). The inherent benefit of a semiactive control device is that the mechanism used does not require large amounts of external power. Many semiactive devices can be powered by batteries protecting them from sudden power loss during earthquakes. Semiactive control systems are basically derived from passive systems; they are modified in such a way that they enable adjustment or correction of their mechanical characteristics. A typical strategy for a semiactive control system is that an “ideal” actively controlled device is first assumed and appropriate primary controller designs for this device are designed. Then, a secondary controller is used that clips the optimal control force so it is dissipative in a manner consistent with the physical nature of the device. This strategy has been widely used (e.g., Dyke et al., 1996a, 1996b, Jung et al., 2001). Spencer and Sain (1997) found that many active control systems for civil engineering applications operate primarily to modify structural damping. They claimed that preliminary studies indicate that appropriately implemented semiactive systems perform significantly better than passive devices and have the potential to achieve the majority of the performance of fully active systems, thus allowing for the possibility of effective response reduction during a wide array of dynamic loading conditions. In other words, semiactive control devices offer the adaptability of active control devices without requiring the associated large power sources (Spencer and Sain, 1997). Moreover, according to presently accepted definitions (Housner et al., 1997), a semiactive control device is one that cannot inject mechanical energy into the controlled structural system (including the structure and the control device), but has properties that can be controlled optimally to reduce the responses of the system. Therefore, in contrast to active control devices, semiactive control devices do not have the potential to destabilize (in the bounded input/bounded output sense) the structural system. Semiactive control devices are often viewed as controllable passive devices. Semiactive control technologies have recently been widely investigated in terms of the reduction of the dynamic response of structures subjected to earthquake and wind excitations (Housner et al., 1997; Spencer et al., 1997; Patten, 1998; Kurata et al., 1999; He et al., 2001; Iemura et al., 2001; Jung et al., 2001). Various semiactive devices have been proposed that utilize forces generated by surface friction or viscous fluids to dissipate vibratory energy in a structural system. 29.3.2.4

Hybrid Control System

The common usage of the term “hybrid control” implies the combined use of active and passive control systems. For example, this may be a structure equipped with distributed viscoelastic damping


29 39

supplemented with an active mass damper on the top of the structure, or a base isolated structure with actuators actively controlled to enhance performance. The method, which consists of using both passive and active devices, should utilize the merits of both passive and active methods and avoid the demerits of these methods. Thus, higher levels of performance should be achievable. Additionally, the resulting hybrid control system can be more reliable than a fully active system because, should the active control malfunction, the minimum seismic protection of the structure can be done by the passive control. 29.3.2.5

Categorization of Basic Principles

From the above group of structural control systems, there are basically five principles concer ning earthquake induced structural response control that are important to consider. They are listed below: 1. Cutting off the input energy from the earthquake ground motion; examples: (a) Floating structures (b) Frictional structures 2. Isolating the natural frequencies of the structures from the predominant seismic power components; examples: (a) Base isolated structures (b) Long period structures 3. Providing nonlinear structural characteristics and establishing a nonstationary state nonresonant system; examples: (a) Inelastic structures (b) Varying stiffness and damping structures 4. Utilizing energy absorption mechanism; examples: (a) Viscous damper (b) Viscoelastic damper (c) Inelastic behavior 5. Supplying control force to suppress the structural response; examples: (a) Active mass damper (b) Active tendon (c) Joint damper

29.3.3

Important Issues in Vibration Control

29.3.3.1

Soil–Structure Interaction

Seismic vibration control of a civil structure deals with methods to suppress the response in a structure subjected to earthquake excitation. To control the structural dynamic response, the structural system model must be known. Modeling of a structure is relatively straightforward, using a finite number of DoF, because the dimensions of the structure are finite. However, in general, civil structures are supported on surrounding soil from which the tremor excites the structure. This makes the structure interact with the surrounding soil. Therefore, it is very important to include SSI for controlling the seismic response. SSI will result in a structural response that may be quite different from the structural response computed for a fixed base building. The frequency of vibration of the structure may be lower because of the interaction. The change in frequency may also affect the response of the overall structure or its substructures or components. Moreover, soils are notoriously nonlinear when subjected to strong ground motions at the level of engineering interest (Marshall, 2001). Damping of the final system increases because of the radiation of energy of the propagating waves away from the structure (Wolf, 1985).

29 40

29.3.3.2


Device –Structure Interaction and Time Delay

Other important effects come from the device structure dynamic interaction. Dyke et al. (1995) studied the role of control structure interaction in mitigating dynamic response. Their findings show that accounting for control structure interaction and actuator dynamics in the design process can improve the performance and robustness of a control system. Time delay between control command and actual control force also causes problems. A study by Hou and Iwan (1992) showed the problems of time delay in the vibration control of structures. Also, Agrawal and Yang (2000) claimed that applications of unsynchronized control forces due to time delay may result in a degradation of the control performance, and it may even render the controlled structure unstable. Therefore, they provide a state of the art review for available methods of time delay compensation. 29.3.3.3

Design Guidelines

Since structural control is a new concept in civil structures, design guidelines are now being developed to provide the design engineers with tools for the safe design or seismic rehabilitation of structures. For a passive control system, the American Society of Civil Engineers has prepared for the Federal Emergency Management Agency a “Prestandard and Commentary for the Seismic Rehabilitation of Buildings” (ASCE, 2000), in which requirements for the systematic rehabilitation of buildings using energy dissipation systems is set forth. In the standard, analysis and design criteria for passive energy dissipation systems are provided. In this prestandard, two sections deal with structural control technology. First, is “Passive Energy Dissipation Systems,” which contains (1) General Requirements, (2) Implementation of Energy Dissipation Devices, (2) Modeling of Energy Dissipation Devices, (3) Linear Procedures, (4) Nonlinear Procedures, (5) Detailed Systems Requirements, (G) Design Review, and (H) Required Test of Energy Dissipation Devices. Each subsection is explained in detail in the prestandard for the users. However, in the second section, “Other Response Control Systems,” it is mentioned that analysis and design of response control other than in the passive systems above shall be reviewed by an independent engineering review panel. This is because the technology of active, semiactive, and hybrid control is not sufficiently mature and the necessary hardware is not sufficiently robust to warrant the preparation of general guidelines for the implementation of the technology.

29.3.4

Vibration-Control Devices

29.3.4.1

Passive Control Systems

29.3.4.1.1 Metallic Yielding Dampers One of the effective mechanisms useful for the dissipation of energy input to a structure from an earthquake is through the inelastic deformation of metals. The idea of utilizing added metallic energy dissipaters within a structure to absorb a large portion of the seismic energy began with the conceptual and experimental work of Kelly et al. (1972) and Skinner et al. (1975). Devices considered by them included torsional beams, flexural beams, and U strip energy dissipaters as shown schematically in Figure 29.39. During the following years, considerable progress has been made in the development of metallic dampers. For example, many new designs have been proposed, including the hourglass or X shaped and triangular plate dampers as shown in Figure 29.40. The X shaped steel plates form devices called ADAS (Added Damping and Stiffness; see Perry et al., 1993), because the devices essentially add stiffness as well as damping to the element where they are installed. For example, if they are installed between two adjacent floors, then it will increase the stiffness and damping between those two floors. Increasing the stiffness will generally attract more seismic forces. However, since the devices have much lower yielding forces than the elements where they are installed, then the postyield stiffness of the devices is dominant during a strong earthquake.


29 41

Force

Force Force Force

Force (a)

(b)

Force

FIGURE 29.39 Schematic of metallic dampers: (a) torsional beam and (b) U strip, and the directions of alternating forces. (Source: Skinner, R.I. et al. Earthquake Eng. Struct. Dyn., 3, 287 296, 1975. With permission.)

Force Beam of Upper Floor

Force

Braced to Beam of Lower Floor

Force (a)

(b)

Force

FIGURE 29.40 Schematic of (a) X shaped (Source: Tsai, K. C. et al. Earthquake Spectra, 9, 505 528, 1993. With permission.) and (b) triangular plate dampers, and the directions of alternating forces. (Source: Perry, C.L. et al. Earthquake Spectra, 9, 559 579, 1993. With permission.)

The devices dissipate energy through the flexural yielding deformation of mild steel plates. Each device consists of a series of steel plates arranged in parallel, with boundary elements at the top and bottom to establish end fixity, which bend in double curvature flexure when subjected to lateral loading. The direction of the lateral loadings is shown in Figure 29.41. The plates are cut in an hour glass shape to match the moment diagram and thus maximize the uniformity of plastification over the height of the steel plates. Cyclic loading tests to determine the reliability of the ADAS were performed by Bergman and Goel (1987). The tests demonstrated that the devices maintained stable hysteretic properties, dissipated significant energy with no pinching or slip zones, and continued to have these properties until plate fracture at high strain or high numbers of cycles. The schematic hysteretic loop of ADAS

29 42


Force Moment diagram

Force FIGURE 29.41 ADAS element: (a) lateral view; (b) longitudinal view with lateral forces; and (c) moment diagram over the height of the steel plates.

devices tested by Bergman and Goel (1987) is shown in Figure 29.42. It is worth noting that the apparent stiffening of the device in the figure is due to finite deformation, not strain hardening of the material; therefore, the effects of finite deformation should be included in the assessment of a design. Tsai et al. (1993) have performed cyclic loading tests on TADAS (Triangular plate ADAS; see Figure 29.40b). The hysteretic loops of the tested TADAS are similar to those of ADAS, such as the apparent stiffening at large displacement, since the mechanism is the same. Although the slotted pin connection at the apex reduces the axial tension force, some friction at the pin location contributes to the stiffening of hysteretic loops (Soong and Dargush, 1997). 29.3.4.1.2 Friction Dampers Another excellent mechanism for energy dissipation is friction. The mechanism has been used for many years in automotive brakes to dissipate kinetic energy of motion. In friction dampers, stick slip phenomena must be minimized to avoid introducing high frequency excitation. 10

8 Force (kips)

(a) −0.5

Displacement (inch)

−8 0.5

(b)

−10 −1.1

Displacement (inch)

1.1

12 Force (kips) (c)

−12 −1.6

Displacement (inch)

1.6

FIGURE 29.42 Schematic hysteresis loop of the ADAS device with maximum displacements of (a) 0.42 in. (10.7 mm); (b) 1.04 in. (26.4 mm); and (c) 1.56 in. (39.6 mm). (Source: Bergman, D. and Goel, S., Rpt. UMCE 87 10, Univ. Michigan, 1987. With permission.)


brace

29 43

A

C

column

beam

links

slip lap joint with friction pad damper

(a)

B (b)

FIGURE 29.43

D brace

Pall friction damper: (a) set up in a structure; (b) schematic of the damper.

Furthermore, friction materials should have a Force (kN) (lbs) consistent coefficient of friction over the intended 2000 8.9 life of the device (Housner et al., 1997). 0.7 (in) −0.7 The Pall device (Pall and Marsh, 1982) is one of Displacement the friction dampers that can be installed in a −17.78 17.78 (mm) structure in an X braced frame as illustrated in −2000 −8.9 Figure 29.43. The mechanism of the damper is that, when there is axial tension force in the bracing system, as shown in the figure by outward arrows, FIGURE 29.44 Schematic of force displacement then plates A and B are moving outward from each hysteresis loop of the Pall friction damper. (Source: other. This movement is resisted by some friction Filiatrault, A. and Cherry, S. Earthquake Spectra, 3, force at the slip lap joint between plates A and B. 57 78, 1987.) This movement also causes plates C and D to move toward each other because of the existence of the links. This movement is also resisted by some frictional force between plates C and D. Without the existence of the links, the bracing may buckle because of axial compression force. The friction level between plates is designed so that the plates will not slip to each other during wind storms or moderate earthquakes. Under severe loading conditions, the devices slip in order to dissipate energy so that structural response can be reduced. The force displacement relationship of Pall dampers has been studied extensively. A plot of its typical cyclic response is illustrated in Figure 29.44 (Filiatrault and Cherry, 1987). 29.3.4.1.3 Viscoelastic Dampers The application of viscoelastic dampers to civil Viscoelastic Force/2 material engineering structures seems to have begun in 1969, when approximately 10,000 viscoelastic Force dampers were installed in each of the twin towers Force/2 of the late World Trade Center in New York to Center reduce wind induced vibrations. An example of a plate Steel flange viscoelastic damper is illustrated in Figure 29.45. The damper consists of a viscoelastic material FIGURE 29.45 Typical viscoelastic damper con bonded to steel plates. The viscoelastic materials figuration. used in structural application are typically copoly mers or glassy substances, which dissipate energy when subjected to shear deformation (Soong and Dargush, 1997). With induced structural vibration, the damper will absorb and dissipate the vibrational

29 44


1.00 Elastic stiffness

Force (kips) 0.00

Energy dissipated in one cycle of oscillation −1.00 −0.05

Displacement (in)

0.05

FIGURE 29.46 Typical schematic hysteresis loop of viscoelastic damper. (Source: Shen, K.L. and Soong, T.T. J. Eng. Mech. ASCE, 121, 694 701, 1995. With permission.)

energy by shearing deformation within the viscoelastic material. Heat will be generated in the viscoelastic material and released through the steel members of the damper. A typical hysteresis loop produced by viscoelastic dampers is shown in Figure 29.46. By using viscoelastic dampers, although the structural response is elastic, hysteresis loops are formed because of the viscoelastic material. The area enclosed in the hysteresis loops is the energy dissipated by the viscoelastic dampers during one cycle of oscillation. The behavior of viscoelastic materials under dynamic loading depends on vibrational frequency, strain, and ambient temperature. In general, the relationship between shear strain, gðtÞ; and shear stress, t ðtÞ; under harmonic shear strain with frequency, v; can be expressed as (Zhang et al., 1989) G00 ðvÞ g_ðtÞ ð29:23Þ v G0 ðvÞ and G00 ðvÞ are shear storage modulus and shear loss modulus of the viscoelastic material, respectively. The loss factor is defined by h ¼ G00 ðvÞ=G0 ðvÞ: In general, as the vibrational frequency increases, the values of G0 ðvÞ and G00 ðvÞ become larger. However, if the ambient temperature increases, those values become smaller. Test results of a typical viscoelastic damper averaged over the first 20 cycles are shown in Table 29.4. For a viscoelastic damper with shear area, A; and thickness, d; the corresponding force displacement relationship is ð29:24Þ FðtÞ ¼ kd ðvÞX þ cd ðvÞX_

tðtÞ ¼ G0 ðvÞgðtÞ þ

in which X and X_ are the relative displacement and velocity of the damper, respectively, and kd ðvÞ ¼

TABLE 29.4

AG0 ðvÞ AG00 ðvÞ cd ðvÞ ¼ d vd

ð29:25Þ

Typical Viscoelastic Damper Properties

Temperature (8C)

Frequency (Hz)

Strain (%)

kd a (lb/in.)

G 0 (psi)

G 00 (psi)

h

24 24 24 24 36 36 36 36

1 1 3 3 1 1 3 3

5 20 5 20 5 20 5 20

2124 2082 4084 3840 880 873 1626 1542

142 139 272 256 59 58 108 103

193 192 324 306 67 65 119 112

1.36 1.38 1.19 1.20 1.13 1.12 1.10 1.09

a

The definition of kd is shown in the text. Source: Data from Chang, K.C. et al. Earthquake Spectra, 9, 371 387, 1993.


29 45

From the above equations, it is clear that a linear system with added viscoelastic dampers remains linear with the dampers contributing to increased viscous damping as well as stiffness. However, it needs to be pointed out that, for viscoelastic material at large strains, there is considerable self heating due to the large amount of energy dissipated. The heat generated changes the mechanical properties of the material, and the overall behavior becomes nonlinear. This means that a linear analysis utilizing the above equations can only be for approximation of the response. 29.3.4.1.4 Viscous Fluid Dampers The most convenient and common functional output equation for a damper comes from classical system theory, and is that of the so called “linear” or “viscous” damping element _ FðtÞ ¼ cd XðtÞ

ð29:26Þ

_ are the damping force, damping coefficient, and relative velocity across the in which FðtÞ; cd, and XðtÞ damper, respectively. In mechanical engineering, it is difficult to manufacture a useable fluid filled component having a purely viscous output, because even moderate pressure hydraulic flows through a simple orifice follow a very different output equation, in which differential pressure varies with the fluid velocity squared. Therefore, the output of the basic hydraulic damping element is _ _ 2 FðtÞ ¼ sgnðXðtÞÞc d lXðtÞl

ð29:27Þ

As a result of long research in this area, which was started in the 1960s, the most useful dampers being used in buildings today are the so called “low exponent” type, with an output equation of the form: _ _ a FðtÞ ¼ sgnðXðtÞÞc d lXðtÞl

ð29:28Þ

In most cases, a is an exponent having a specified value in the range of 0.3 to 1.0. Values of a that have proven to be the most popular are in the range of 0.4 to 0.5 for building designs with seismic input (Taylor, 2002). The design elements of a fluid damper are relatively few. However, the detailing of these elements varies greatly. Typical elements for a fluid damper are shown in Figure 29.47. Typical experimentally measured force displacement loops are shown in Figure 29.48. 29.3.4.1.5 Tuned Mass Dampers Much of the early development of dynamic vibration absorbers, as mentioned in Section 29.3.2.1, has been limited to the use of dynamic absorbers in mechanical engineering systems in which one operating frequency is in resonance with a machine’s fundamental frequency. However, building structures are subjected to earthquakes which posses many frequency components. The performance of a Piston rod

Seal retainer

Chamber 1

Cylinder

Piston head with orifices

Compressible silicon fluid

Chamber 2

Control valve

Accumulator housing

Rod make-up accumulator

High strength acetal resin seal

FIGURE 29.47 Typical schematic of a fluid damper. (Source: Taylor, D.P. Passive Structural Control Symp., 2002. With permission.)

29 46


dynamic vibration absorber, referred to as the TMD herein, in complex multi DoF systems is expected to be different. Consider the resonance curve in Figure 29.49 of a system shown in Figure 29.34, but now with different parameters: m ¼ 0:05; but still vM ¼ vm (see Equation 29.22 for the definition of these parameters). An additional parameter, the damping ratio of TMD, is shown in Equation 29.29. Where c is damping coefficient of a linear viscous damper installed between structural mass, M; and TMD mass, m; c ð29:29Þ 6TMD ¼ 2mvm

1000 Force (lb)

f = 4 Hz f = 1 Hz

0

−1000 −1.5

0.0

1.5 Displacement (in)

FIGURE 29.48 Typical schematic of fluid damper hysteretic loops. (Source: Constantinou, M.C. and Symans, M.D. Struct. Des. Tall Build., 2, 93 132, 1993. With permission.)

When the damping ratio in the TMD equals zero, the response amplitude is infinite at the two resonant frequencies. When the damping ratio becomes infinite, the two masses are virtually stuck to each other; the result is a single DoF system with mass, 1.05 ðMÞ with the amplitude becoming infinite again at a resonant frequency (see Figure 29.49). Therefore, somewhere between these extremes, there must be a value of the TMD damping ratio for which the peak becomes a minimum. Therefore, the objective in adding the TMD herein is to bring the resonant peak of the amplitude to its lowest possible value so that smaller amplifications over a wider frequency bandwidth can be achieved. There are many types of TMD for implementation and the following are some examples (Figure 29.50). An innovative challenge is highly expected in this field. There is another type of TMD that uses liquid as the mass; this damper is called the tuned liquid damper (TLD). The TLD has been used in ships for controlling vibrations because of water waves. The TLD uses water or other liquid as the moving mass and the restoring force is generated by gravity. Energy absorption comes from boundaries between liquid and containers and turbulence in the liquid flow. The basic principle of the TLD in absorbing kinetic energy of the main structure is the same as the TMD. Figure 29.51 shows types of TLD. Favorable properties of TLD compared with TMD are as follows: *

*

*

Smooth movement in small vibration is possible because of no mechanical friction. It is reasonable in cost and maintenance because of no complex mechanism. It can be applied easily in two horizontal directions with a single TLD. It can be compact and portable if large numbers are used. 16

8

*

ζ TMD = 0

x1 Po/K 8

0

0.32 0.1

4 0 0.6

1

ω/ωm

1.3

FIGURE 29.49 Resonance curve for the system in Figure 29.34 that shows response amplitude vs. exciting frequency for m ¼ 0:05 and vM ¼ vm :


29 47

(b)

(a)

(c)

(e)

(d)

(f)

FIGURE 29.50 Examples of different types of TMD: (a) pendulum with damper; (b) inverted pendulum with spring and damper; (c) pendulum of which hangers are winded to save space; (d) swinging mass on rotational bearings; (e) sliding mass with spring and damper; (f) mass on rubber bearings. (Source: Iemura, H. Passive and Active Vibration Control in Civil Engineering, Springer, New York, 1994a,b. With permission.) p

(a)

Vibration

(b)

Vibration

FIGURE 29.51 Types of TLD: (a) tuned sloshing damper with meshes and rods; (b) tuned liquid column damper with orifice. (Source: Iemura, H. Passive and Active Vibration Control in Civil Engineering, Springer, New York, 1994a,b. With permission.)

The TLD can be divided into two categories. First, is the sloshing damper as shown in Figure 29.51a. The vibration period is adjusted by the size of the container and the depth of the liquid. The damping capacity is increased by placing meshes or rods in the liquid. The second category is the tuned liquid column damper as shown in Figure 29.51b. The vibration period is adjusted by the shape of the column or the air pressure in the column. The damping capacity is increased by adjusting the orifice in the column, which generates a high turbulence. 29.3.4.2


29.3.4.2.1 Electromagnetic Actuator An electromagnetic actuator is frequently employed where it is necessary to provide a mechanical force depending on an electric current. Figure 29.52 shows the elements of such a device. A magnetic structure supports a circuit of magnetic flux driven by the coil of N turns carrying current i, in ampere. Part of the magnetic circuit is a movable armature that slides smoothly on the support member. An air gap (or possibly a vacuum gap) of length d m is also in the magnetic circuit. Assume that d is small enough for the magnetic flux density to be essentially constant at B Wb/m2 across the face of the armature. The cross sectional area of the armature face is A m2. A tractive force, F, in Newtons, developed on the face of the armature, is related to the area and field strength as F¼

AB2 2m0

ð29:30Þ

29 48


where m0 is the magnetic permeability of the air gap having a value of 4p £ 10 7 H/m (Henries/ meter). The magnetic field intensity is pro portional to the magnetomotive force N £ i ampere turns. For an efficient magnetic structure, the magnetic flux density is

m B¼ 0N £i d

ð29:31Þ

Magnet flux density = B Webers/meter2 z (t) Armature

"

Am0 N 2 2

#

i2 d2

! Newton

ð29:32Þ

Note that F is positive when i is either positive or negative. This means that the force, F, is always tractive if the current, I, is either positive or negative. The maximum value for magnetic flux density, B, which can be realized using modern magnetic materials in normal environments, is approximately 1 Wb/m2 (Clark, 1996). In most cases, it is necessary to apply force in both the positive and negative directions. In this case, the device is constructed with two air gaps having the corresponding traction forces opposed to one another, as shown in Figure 29.53. It is interesting to observe how the net force, F, on the armature depends on z and Di (differential current ¼ iL i0 ¼ iR i0 ; see Figure 29.53 for the parameters) when these are constants. The schematic plot of F vs. z, with Di as a parameter, is shown in Figure 29.54. It is worth noting that the plot is nonlinear. It is also noted from the figure that, if z is positive, F is also positive. It means that the force developed on the armature due to a small displacement from the equilibrium position is in the direction to increase that small displacement. This character izes a condition of static instability. Feedback control can be used to stabilize this inherently unstable device when such a device is used as an actuator.

d

Sliding armature

N turns

which, combined with Equation 29.30, gives a useful relationship: F¼

A

F

i (Amperes)

FIGURE 29.52 Elements of an electromagnet. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.) z(t) Armature

N turns

N turns iL

2i0

iR

FIGURE 29.53 Push pull electromagnet. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.) 30

Di = 2A

F (Newton)

Di = 1A Di = 0A

0 Di = −1A −30 −3

i0 = 2 A d0 = 5 mm

Di = −2A 0

z (mm)

3

FIGURE 29.54 Force vs. displacement of push pull electromagnet with differential current as a parameter. (Source: Clark, R.N. Control System Dynamics, Cam bridge Univ. Press, U.K., 1996. With permission.)

29.3.4.2.2 Hydraulic Actuators Hydraulic actuators can produce large forces even at large displacements, which is useful for seismic response control applications. The actuator is usually driven using electrical signals. In other words, an electro hydraulic device is used to convert the low powered signals into high powered hydraulic fluid flow. One example of such a device is an electro hydraulic servo valve. The schematic of such a device is shown in Figure 29.55.


29 49 Hydraulic Return

Single-state spool valve

To push-pull current amplifier

To push-pull current amplifier

Hydraulic Pressure

Actuator Force Low hydraulic pressure

High hydraulic pressure

FIGURE 29.55 Schematic of servo valve actuator. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.)

The valve shown in the figure is a single stage spool valve. The simple design shown in the figure exhibits the basic principles, inherent in most types of servo valves, of conversion from electromagnetic to hydraulic energy; in other words, actuating force. By changing the electric signals going to the spool valve, the actuating force can be generated because the hydraulic pressure acting on the piston in one side is greater than the other side (see Figure 29.55). The actuating force can also be altered in real time, enabling real time control of seismically excited structures. 29.3.4.3


29.3.4.3.1 Variable-Orifice Hydraulic Dampers A variable damping device used for the semiactive control method can be achieved by using a Controllable Valve controllable, electro mechanical, variable orifice valve to alter the resistance to flow of a conven Load tional hydraulic fluid damper. Figure 29.56 shows a schematic of such a device. The effectiveness of variable orifice dampers in controlling seismically excited buildings has been demonstrated through both simulation and small scale experimental FIGURE 29.56 Schematic diagram of variable orifice studies (Hrovat et al., 1983; Mizuno et al., 1992; valve oil damper. Kurata et al., 1994; Patten et al., 1994; Sack et al., 1994; Liang et al., 1995; Niwa et al., 2000). 29.3.4.3.2 Controllable Fluid Dampers Another class of semiactive devices use controllable fluids. The advantage of controllable fluid devices over controllable valve devices is that they contain no moving parts other than the piston. The essential characteristic of controllable fluids is their ability to change reversibly from a free flowing, linear viscous fluid to a semisolid with a controllable yield strength in milliseconds, when exposed to an electric (for electro rheological fluid) or magnetic (for magneto rheological [MR] fluid) field (Housner et al., 1997).

29 50


MR fluids typically consist of micron sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil. When a magnetic field is applied to the fluid, particle chains form and the fluid becomes a semisolid and exhibits viscoplastic characteristics. Transition to rheological equilibrium can be achieved in a few milliseconds. A schematic diagram of the controllable fluid damper is shown in Figure 29.57.

Wires to MR fluid electromagnet

Magnetic Choke

Accumulator Load

Load Rod

Piston

FIGURE 29.57 Schematic diagram of magneto rheological damper. (Source: Spencer, B.F. et al. J. Eng. Mech., 123, 1997. With permission.)

29.3.4.3.3 Controllable Friction Dampers Various controllable friction devices have been Upper Plate proposed to dissipate vibratory energy in a Slotted Bolts structural system. Akbay and Aktan (1990, 1991) and Kannan et al. (1995) proposed a variable N(t) N(t) friction device in which the force at the frictional + + interface was adjusted by allowing slippage in − − controlled amounts. A similar device was also studied by Cherry (1994) and Dowdell and Cherry Lower Plate Friction Pad (1994a, 1994b). A recent work by He et al. (2003) studied a semiactive electromagnetic friction damper FIGURE 29.58 Schematic drawing of semiactive (SAEMFD) for controlling seismic responses. electro magnetic friction damper. (Source: He, W.L. et al. J. Struct. Eng., ASCE, 129, 941 950, 2003. With Figure 29.58 shows a schematic diagram of the permission.) SAEMFD. The device consists of a friction pad sandwiched between two steel plates. These three layers are slot bolted together so that sliding takes place between the steel plates and the friction pad. The friction force between steel plates and the friction pad depends on the coefficient of friction ðmÞ and the normal force NðtÞ: Two insulated solenoids are installed on the outer surfaces of the steel plates and the electric current in these solenoids is regulated such that an electromagnetic attractive force exists between the two solenoids. Hence, the normal force NðtÞ between the steel plates is directly proportional to the square of the current in the solenoids. 29.3.4.4

Hybrid Control System

The hybrid control methods which consist of both passive and active devices have been proposed and implemented, utilizing the merits of both passive and active methods and avoiding the demerits of these methods. Thus, higher levels of performance may be achievable. Additionally, the resulting hybrid control system can be more reliable than a fully active system, although it is also often somewhat more complicated. One example of a hybrid system is a TMD with actuators that is put between the TMD mass and its support so that the effectiveness of the TMD is increased by this technique. Figure 29.59 shows a schematic diagram of a passive TMD, an active AMD (active mass damper), and a hybrid ATMD (active TMD). Another example of hybrid control is a combination of base isolation with some form of active control to limit excessive displacement (Fujii et al., 1992; Kageyama, M. and Yasui, 1992; Feng et al., 1993; Reinhorn and Riley, 1994).

29.3.5

Control Algorithm

29.3.5.1


The most important part of active control is the algorithm, because the control forces are based on this. Research efforts in active structural control have been focused on a variety of control algorithms based

Seismic Base Isolation and Vibration Control Auxiliary Mass

29 51

Auxiliary Mass

m2

Auxiliary Mass

m2

m2 Controller

Spring

Damper

m1

Actuator

Controller

G

m1

G

m1 Sensor

Sensor

Passive

FIGURE 29.59

Active

Hybrid

Schematic diagram of a passive, an active, and a hybrid mass damper system.

on several control design criteria. An example of a famous control algorithm is linear optimal control because all of the control design parameters can be determined for multiinput and multioutput systems. Also, the control allows us to formulate directly the performance objectives of a control system. The adjective optimal above means that a control system can be designed to meet the desired performance objectives with the smallest control energy, which is the energy associated with generating the control inputs. Such a control system that minimizes the cost associated with generating control inputs is called an optimal control system. The optimal control system directly addresses the desired performance objectives, while minimizing the control energy, by formulating an objective function that must be minimized in the design process. If the transient energy of a system is the total energy of the system when it is undergoing the transient response, then the successful control system must have the capability to decay quickly the transient energy to zero. By including the transient energy and the control energy in the objective function, both parameters can be minimized. The objective function for the optimal control problem is a time integral of the sum of transient energy and control energy expressed as a function of time. The general, optimal control formulation for regulators can be explained as follows. Consider a structure (Figure 29.60) under dynamic loading that is represented by Equation 29.33: M€xðtÞ þ C_xðtÞ þ KxðtÞ ¼ DuðtÞ þ EfðtÞ

ð29:33Þ

where M, C, K are, respectively, the n £ n mass, damping, and stiffness matrices, and xðtÞ is the n dimensional displacement vector, fðtÞis an r vector representing applied load or external excitation, and uðtÞ is the m dimensional control force vector. The n £ m matrix, D, and n £ r matrix, E, are location matrices that define locations of the control force and the excitation, respectively. The equation can be rewritten using the state space representation in the form z_ ðtÞ ¼ AzðtÞ þ BuðtÞ þ HfðtÞ; zð0Þ ¼ z0

ð29:34Þ

where " zðtÞ ¼

xðtÞ x_ ðtÞ

# ð29:35Þ

29 52


xn mn

un

kn

cn

x2 m2 x1 m1

k2

c2 c1

FIGURE 29.60

f2

u2 u1

k

fn

f1

m1    m 2  M=   ...   m  n  c1 + c2  −c 2 C=    

−c2

k1 + k2  −k K= 2   

−k2 ... ...

...

...

...

...

   −cn   cn 

−cn

... ... −kn

   −kn   kn 

Typical civil structure with its dynamic properties.

is the 2n dimensional state vector

"

0

A¼ is the 2n £ 2n system matrix, and

" B¼

M 1K

M 1D

ð29:36Þ

M 1C "

#

0

#

I

and H ¼

I M 1E

# ð29:37Þ

are 2n £ m and 2n £ r location matrices specifying, respectively, the locations of controllers and external excitation in the state space. In Equation 29.36, 0 and I denote the null matrix and the identity matrix of appropriate dimensions, respectively. For simplicity, assume that we have a linear time invarying plant, as shown in Equation 29.34 above, and suppose we would like to design a full state feedback regulator for the plant such that the control input vector is given by uðtÞ ¼

GxðtÞ

ð29:38Þ

where G is a feedback gain matrix. The control law given by the above equation is linear. Since the plant is also linear, the closed loop control system is linear. The control energy, CE, can be expressed as CE ¼ uT ðtÞRuðtÞ

ð29:39Þ

where R is a square, symmetric matrix called the control cost matrix. Such an expression for control energy is called a quadratic form, because the scalar function in Equation 29.39 contains quadratic functions of the elements of uðtÞ: The transient energy, TE, can also be expressed in a quadratic form as TE ¼ xT ðtÞQxðtÞ

ð29:40Þ

where Q is a square, symmetric matrix called the state weighting matrix. The objective function can then be written as follows: ðtf Jðt; tf Þ ¼ ½xT ðtÞQxðtÞ þ uT RuðtÞ dt ð29:41Þ t

where t and tf are the initial and final times, respectively; that is, the control begins at t ¼ t and ends at t ¼ tf ; where t is the variable of integration.


29 53

The optimal control problem consists of solving for the feedback gain matrix, G, such that the scalar objective function, Jðt; tf Þ; given by Equation 29.41, is minimized. Hence, the optimal control problem solves a regulator gain matrix, G, which minimizes Jðt; tf Þ; subject to the constraint given by Equation 29.34. MATLABw Control System Toolbox (MathWorks, 1998) provides a function “lqr” for the solution of the linear optimal control problem shown above (also see Appendix 32A). By using this command, the gain matrix, G, can easily be obtained. For a numerical simulation, the matrix, G, is used for obtaining optimal forces, uðtÞ; as shown in Equation 29.38. The MATLABw Control System Toolbox user manual is a good reference (MathWorks, 1998) for applying this command. For a real application in a structure, the optimal forces, uðtÞ; must be converted to real forces by actuating devices. Actuating devices are covered in Section 29.3.4. In the above explanation, the linear optimal controller has been derived with full state feedback that minimizes a quadratic objective function. The controller robustness to process and measurement noise can only be indirectly ensured by iterative techniques. There are more advanced topics in modern control that directly address the problem of robustness by deriving controllers that maintain system response and error signals to within prescribed tolerances. One example is the H1 (pronounced H infinity) optimal control design technique. The reader may refer to control design textbooks, such as that written by Tewari (2002). 29.3.5.2


Because of the intrinsically nonlinear nature of semiactive control devices, development of control strategies that are practically implementable and can fully utilize the capabilities of these unique devices is an important and challenging task. Various nonlinear control strategies have been developed to take advantage of the particular characteristics of semiactive devices, including bang bang control (Mukai et al., 1994; McClamroch and Gavin, 1995), clipped optimal control (Dyke et al., 1996a), bistate control (Patten et al., 1994), fuzzy control methods (Sun and Goto, 1994), and adaptive nonlinear control (Kamagata and Kobori, 1994). Caughey (1993) proposed a variable stiffness system that employed a semiactive implementation of the Reid (1956) spring as a structural element. He et al. (2001) proposed a resetting semiactive stiffness damper used for controlling seismically excited cable stayed bridges. Iemura and Pradono (2003) introduced a pseudonegative stiffness control algorithm used for producing artificially rigid perfectly plastic force deformation hysteretic loops by using controllable damper. 29.3.5.2.1 Common Control Schemes for Controllable Dampers An example will be given here for common control schemes for a controllable damper. Examples of a controllable damper are variable orifice damper and MR fluid dampers. Both types of dampers are covered in Section 29.3.4. The strategy of a clipped optimal control algorithm (Dyke et al., 1996a, 1996b) for seismic protection using MR fluid dampers is as follows. First, an “ideal” active control device is assumed, and an appropriate primary controller for this active device is designed. Then a secondary bang bang type controller causes the smart damper to generate the desired active control force, so long as this force is dissipative. The primary controller can be one of active control algorithms shown above. For the general smart damping device, the secondary control strategy is given by ( fsa;i ¼

fa;i ; fa;i £ x_ dev , 0 0; otherwise

ð29:42Þ

where fsa,i is the control force of the ith MR fluid damper, fa,i is the desired control force for the ith device, and x_ dev is the velocity across the ith damper. Since the controllable damper is an energy dissipative device that cannot add mechanical energy to the structural system, special care must be taken in the design of the primary controller so that the desired control force, fa,i, is dissipative during the majority of the seismic event.

29 54


29.3.5.2.2 Special Control Schemes for Controllable Damper The term “special” here refers to control schemes that have certain objectives for the hysteretic loops of the controllable dampers. The control schemes are not centralized, so that each damper is controlled separately in one structure or, in other words, one controller is for one damper. The advantage is that should one controller malfunction, this controller will not affect the other controller. One example is given by He et al. (2001). A resetting semiactive stiffness damper (RSASD) is used to control the peak seismic response of a structure. The RSASD consists of a cylinder piston system with an on off valve in the bypass pipes connecting two sides of the cylinder. Basically, the damper is similar to a variable orifice oil damper as shown in Figure 29.58; however, the orifice valve is replaced by an on off valve. This damper serves as a stiffness element in which the stiffness is provided by the bulk modulus of the fluid in the cylinder when the valve is closed. When the valve is open, the piston is free to move, and the hydraulic damper provides only a small damping, without stiffness. Such a stiffness damper can be operated in the resetting mode. During the operation in this mode, the valve is always closed. The energy is then stored in the hydraulic oil of the damper in the form of potential energy. At an appropriate time, the valve is pulsed to open and close quickly. At that moment, the piston is at the resetting position, and the energy stored in the hydraulic damper is released and converted into the head loss across the damper. Hence, by pulsing the valve at appropriate times, structural response can be reduced by drawing energy from the system (He et al., 2001). Another example is that of Kurino et al. (2003). They presented a device developed for an actual application whose system employs a decentralized control algorithm that uses information only from built in sensors. The pseudonegative stiffness control algorithm (Iemura and Pradono, 2003) is also intended for controlling a variable damper based only on the displacement and velocity sensors located within the damper. The purpose is to control the variable damper’s hysteresis loop. An example of the application of this control algorithm to a bridge model is shown in Section 29.3.7.2.

29.3.6

Experimental Performance Verification

29.3.6.1 Shaking Table Tests of a Flexible Structural Model with TMD, AMD, and ATMD For practical implementation of TMD, AMD, and ATMD for structures, it is important to find the efficiency of each control system for random excitations. The author has made an analytical and numerical studies on the efficiency of different control methods. To verify the results, a three DoF structural frame model (Figure 29.61) with and without control devices is tested a on shaking table at Kyoto University, Japan (Iemura et al., 1992). Natural periods for modes 1, 2, and 3 are 0.6578, 0.2580, and 0.1568 sec, respectively. The relevant participation factors for each mode are 1.2204, 0.3493, and 0.1341, respectively. The moving mass and other mass of the TMD are 3.5 and 5.5 kg, respectively. The spring constant is 0.581 kgf/cm. The damping ratio is 25.06%. These properties are used for the experiment. The masses of the TMD consist of the AC servo motor, moving mass, driving guides, and velocity meter. At the time of the experiment of the TMD, the moving mass was fixed and the TMD

Velocity Meter Velocity Meter

Motor Moving Mass

Velocity Meter

Velocity Meter

Velocity Meter Shaking Table

FIGURE 29.61 TMD/ATMD.

Three DoF experimental model with


29 55

was hung from the third floor. In order to work as the hybrid type ATMD, the moving mass was driven by the motor. For the pure active control experiment, the motor and the moving mass are set directly on the third floor. It was clearly found that the second mode response is not reduced by the TMD but is effectively reduced by the AMD and ATMD. The TMD is effective only in the first mode frequency range, while active control force can cover a wide frequency range. It was also found that the control force of the ATMD is much lower than that of AMD, especially in the first mode frequency range. The first mode response is reduced mainly by the TMD and the second mode response is reduced by the AMD. This verifies the energy efficiency of ATMD. This is the reason that the ATMD concept is now popularly used for practical application. 29.3.6.2

Substructure Hybrid Test

Before introducing additional dampers to a structure for seismic response reduction, the precise frequency dependent properties of the damper should be obtained from the damper. These properties will be used for numerical simulation on the effectiveness of the damper in reducing seismic responses. It is quite difficult to work out the appropriate model for a specified device because of the existence of strong nonlinearity. For a more reliable experimental test of structures with structural control devices, so called “substructure hybrid experiment” techniques have been developed. The term “substructure hybrid” implies a technique that combines the device loading experiment and numerical simulation of structural response. Why should it be separated? It is because civil structures are relatively large and expensive to construct in a laboratory. Therefore, only the inelastic part is tested experimentally. The elastic part, which is easier to model, is numerically simulated on a computer. Both results are combined in real time at every time step of the simulation. Up until now, various kinds of test methods have been proposed. Most of them can be classified into three categories from the viewpoint of the loading equipment. 29.3.6.2.1 Hybrid Tests Using Hydraulic Actuator Hydraulic actuators are commonly used for loading experiments. They are advantageous for testing specimens that needs large excitation force and displacements (Tanzo et al., 1992; Igarashi et al., 1993; Igarashi, 1994; Williams and Blakeborough, 2001). Various algorithms and techniques have been proposed in order to conduct precise, real time experiments such as the “operator splitting numerical integration scheme,” which is suitable for online controlled experiments (Nakashima, 1993). A compensation method based on extrapolation is proposed by Horiuchi and Konno (2001) for the response delay of the actuator. Similar feedforward based compensation methods are widely used for the numerical algorithm’s development and real time testing (Nakashima and Masaoka, 1999; Nakashima et al., 1999; Blakeborough et al., 2001). 29.3.6.2.2 Hybrid Tests Using Shaking Table Substructure hybrid loading test systems have been developed for shaking table equipment. Since most shaking tables are driven by hydraulic actuators, algorithms as well as technologies for the hydraulic actuators system are directly applicable to the shaking table test systems. Iemura et al. (2002) introduced the inverted digital filter of the shaking table for compensating its dynamics, and conducted a real time hybrid experiment using the electromagnetic mass damper installed in the nonlinear structure. The shaking table test is applicable to a test specimen such as the TMD. 29.3.6.2.3 Hybrid Tests Using Inertia-Force-Driven Loading System This is a newly developed method for hybrid loading test (Toyooka, 2002; Iemura et al., 2003). The system consists of a large size mass, rubber and roller supports, and active mass driver. The schematic figure of the system is shown in Figure 29.62 and the property of the system is in Table 29.5. The active mass driver is attached to the mass. The test specimen is attached to the mass and the ground. The mass

29 56


FIGURE 29.62 TABLE 29.5

Test set up of the IFDL system.

Properties of the IFDL System

Weight of slab mass Total stiffness Total damping Natural frequency Equivalent damping Stroke limit

23.853 tonf 344.43 kN/m 6.32 kN/m/sec 0.55 Hz 3.86% ^10 cm

(i.e., the concrete slab) can be excited with large displacement, velocity, and acceleration by making use of the shaker (Iemura et al., 2003). The Inertia Force Driven Loading (IFDL) system was developed to allow an economical and accurate loading environment for energy dissipation devices to characterize the dynamic properties and to comprehend the performances of these devices under the realistic loading conditions.

29.3.7

Implementations

29.3.7.1 Semiactive Control of Full-Scale Structures Using Variable Joint Damper System 29.3.7.1.1 Background of Study In order to verify the effectiveness of the application of the semiactive control technique to the joint damper system (JDS), seismic response control tests using full scale multistory steel frame structures, excitation devices, and a variable damper are performed at the Disaster Prevention Research Institute, Kyoto University, Japan. The variable damper allows external control of damping force by the electric servo valve that regulates the oil flow through the cylinder/piston mechanism. The test results show that the variable damper was successfully controlled with high accuracy, as well as having the advantage of JDS application of the semiactive control in reducing the dynamic response of structures over the


29 57

conventional passive control. A comparison of the semiactive control algorithms to JDSs in terms of the feasibility and the advantage in the engineering application is also based on the test results. Extensive research on the semiactive control approach has been conducted in order to reduce the seismic response of structures, induced especially by strong earthquake ground motions. The JDS, which aims to achieve the dynamic response reduction of adjacent structures through the use of connection devices with energy absorbing capability, has been considered a promising approach to establish effective semiactive structural systems for earthquakes. The purpose of this study is to verify experimentally the effectiveness of the application of semiactive control to the JDS. Seismic response control tests using full scale multistory frame structures, excitation devices, and a variable damper are performed at the Disaster Prevention Research Institute, Kyoto University. The variable damper allows for external control of the damping force using the electric servo valve that regulates the oil flow through the cylinder/piston mechanism. Two types of semiactive control algorithms were employed, namely the linear quadratic regulator (LQR) control theory and the newly proposed pseudonegative stiffness control for JDSs. Parameter setting strategies for the algorithms are studied prior to the tests through numerical simulations based on the modeling of the full scale steel frame structures and the control device used in the tests. 29.3.7.1.2 Test System Set Up As shown in Figure 29.63, the test structure of the JDS consists of two full scale structural steel frames: a five story frame (1 £ 2 span) and a three story frame (1 £ 1 span). The dimensions of both frames are shown in Table 29.6. Natural frequencies are also shown in the table. In this test system, mass driver devices are used to reproduce the vibration conditions under both sinusoidal and real earthquake inputs. One directional horizontal earthquake excitation is applied. Although three mass driver systems can be seen in the figure, two of them are used at the same time. The accuracy of the simulated response using the mass driver devices has been verified by a series of research Variable Damper

Velocity Sensors

Shaker

Data Acq. PC FIGURE 29.63

Shaker Control PC

Damper Control PC

Schematic diagram of joint damper system.

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Test Frames and Mass Drivers

Height (m) Weight (tonf) First mode natural frequency (Hz) Mass driver

Five Story

Three Story

17.22 163.1 1.78 5 ton mass at the fourth floor

10.65 62.1 2.41 2 ton mass at third floor

conducted prior to this study. Velocities, relative displacements, and absolute accelerations of all floors are measured through instrumentation. All measured responses are sent to a digital signal processing based system for the feedback control. As the control device in the JDS, a variable damping device (variable damper) is used in the test system. The variable damper is installed at the third story of the five story frame so as to connect the two frames. The mechanism of the variable damper is similar to that shown in Figure 29.56. It is a semiactive hydraulic damper consisting of a cylinder/piston mechanism filled with oil, double rods that connect the frames, a by pass pipe that contains a flow control valve, and an accumulator that keeps the by pass line pressure constant. The opening ratio of the flow control valve can be changed by a servo controller using an external signal. The flow volume through the valve can be regulated to control the pressure loss. The delay time for the opening ratio control is sufficiently short to allow real time control. 29.3.7.1.3 Control Algorithm In this study, three types of control algorithms are used: linear viscous damper control, LQR control (discussed in Section 29.3.5), and pseudonegative stiffness (PNS) control. The linear viscous damper control algorithm is intended to be the reference response in the case of passive control, and the effectiveness of the semiactive control is demonstrated by comparing the other two cases with the linear viscous damper control case. In linear viscous control algorithm, the damping force demanded of the variable damper, Fd ðtÞ; is Fd ðtÞ ¼ Cc vr ðtÞ

ð29:43Þ

where Cc is the connecting damping coefficient and vr is the relative velocity of the damper position. Real time control of the valve opening ratio is required to generate the demanded control force, Fd, with the variable damper, even for this simplest control algorithm. The LQR control theory is used as a semiactive control algorithm in this study, as extensively used in past studies. In the LQR control algorithm, the optimal control force, Fd, is regarded as the demanded force and the variable damper is controlled to track the demanded force as close as possible within the constraint, depending on the piston velocity. The control gain parameters are chosen on the basis of numerical simulation in consideration of the capacity of the variable damper. The control force is calculated in the following manner: Fd ðtÞ ¼

GxðtÞ

ð29:44Þ

where G is the optimal gain matrix given by the LQR theory and xðtÞ is the state vector for the structural frames. If the main purpose of a JDS is the response reduction of the upper floors in the adjacent structures, the most interesting feature of the system is obtained by connecting them at lower stories with a negative stiffness element. Although this characteristic has been reported theoretically and analytically in many studies, there are many problems left in applying the active control device at the present time, mainly owing to the difficulty in realizing the negative stiffness with a passive control device. However, semiactive devices such as the variable damper can generate an apparent negative stiffness by controlling the damping. Therefore, taking into account JDS and negative stiffness, a new, simple control algorithm to realize pseudonegative stiffness with a damping element is proposed in this study. To generate the


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negative stiffness, the demanded force of variable damper Fd is defined as follows: Fd ðtÞ ¼ Kc xr ðtÞ þ Cc vr ðtÞ

ð29:45Þ

where Kc and Cc are the connecting stiffness (negative value) and the damping coefficient, respectively. The relative displacement and velocity of the damper piston, respectively, are xr and vr. It follows that this algorithm simulates the state in which the two frames are connected by a negative stiffness and a positive damping elements. The eigenvalue analysis is used to determine the value of Kc and Cc. This pseudonegative stiffness algorithm has a great advantage in practical application. Most of the previously proposed control algorithms require direct measurement of the structural system to produce feedback and to calculate the demanded control force. That means that a considerable number of sensors should be installed in the object structure. Considering practical application, it is difficult to install many sensors because of the economical disadvantage. On the other hand, for the pseudonegative stiffness algorithm, only the relative displacement and velocity are used for feedback and a sensor is required only at the damper. In addition to the simplicity in installation, the required parameters are limited to the connecting stiffness and damping. 29.3.7.1.4 Test Results The response of variable damper to sinusoidal input with 1.8 Hz (max 10 gal), which is approximately the first resonance frequency in the connected state, is shown in Figure 29.64. The five story top floor velocity response in the LQR control theory and pseudonegative stiffness are smaller than that in the viscous damper; especially; the peak value in pseudo negative stiffness is improved at 25% compared with that in the viscous damper, though the velocities at the top story of the three story frame are almost equal for all control algorithms. On the other hand, the LQR control theory can reduce the peak response of both frames as compared with the viscous damper. When the object is to moderate the response of total system, the LQR theory is the most effective algorithm of the three being compared. Based on the test, semiactive controls based on the LQR control theory and pseudonegative stiffness can reduce the peak responses of the total system more effectively than viscous damper type passive control. For the earthquake excitation (El Centro 1940 N S and Kobe 1995 N S, scaled to 20 gal max), the influence of the friction of the variable damper appears in the dynamic characteristics of the variable damper in every control algorithm because of the relatively small responses. The relative velocity and displacement responses of the variable damper in both semiactive controls are larger than that in the viscous damper. Judging from the test result, it is confirmed that the variable damper is controlled effectively in the different control algorithm for real earthquake inputs. With respect to the velocity response of the top story of the three story frame, the responses are not very different when dif ferent control algorithms are used. On the other hand, for the five story top floor response, both semiactive controls can reduce the response more effectively than the passive control to both real

FIGURE 29.64 Variable damper responses in the sinusoidal excitation test: (a) at the top story of five story fame; (b) at the top story of three story fame.

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FIGURE 29.65 (a) Viscous type control, (b) LQR type control, and (c) pseudonegative stiffness type control under El Centro 1940 N S scaled to 20 gal max.

earthquake excitations. Semiactive controls are more effective than passive control to reduce the response of the top story of the five story frame in the earthquake excitation cases. Figure 29.65 shows the hysteretic loops resulting from viscous type, LQR type, and pseudonegative stiffness type controls under El Centro 1940 N S, scaled to 20 gal max. It is obvious that the pseudonegative hysteretic loop can be achieved experimentally by using a variable damper. 29.3.7.2 Application of Structural Control Technologies to Seismic Retrofit of a Cable-Stayed Bridge 29.3.7.2.1 Background of Study Owing to severe damage to bridges caused by the Hyogo ken Nanbu earthquake in 1995, very high ground accelerations (level II design) are now required in the new bridge design specification set in 1996, in addition to the relatively frequent earthquake motion (level I design) by which old structures were designed and constructed. Hence, the seismic safety of cable stayed bridges that were built before the present specification has to be reviewed and seismic retrofit has to be done, if it is found necessary. In order to study the effectiveness of passive and semiactive control on the seismic retrofit of a cable stayed bridge, numerical analyses on a model is carried out. An existing cable stayed bridge that has fixed hinge connections between the deck and towers is modeled and its connections are replaced with isolation bearings and dampers. The isolation bearings are assumed to be elastic or hysteretic type. The dampers are linear and variable type. The objective is to increase the damping ratio of the bridge by using passive and semiactive control technologies. The calculation of the structural damping ratio at the main mode is feasible, as the passive or semiactive control method produces certain hysteretic loops under


FIGURE 29.66

29 61

Side view of the Tempozan Bridge.

harmonic motion, and the main mode has an effective modal mass that is larger than 90% of the total mass. SSI effects on the structural damping ratio are also studied. The Tempozan Bridge (Hanshin Highway Public Corporation, 1992), built in 1988, is a three span continuous steel cable stayed bridge that is situated on reclaimed land and crosses the mouth of the Aji River, Osaka, Japan. The total length of the bridge is 640 m with a center span of 350 m, and the lengths of the side spans are 170 and 120 m (Figure 29.66). The main towers are A shaped to improve the torsional rigidity. The cable in the superstructure is a two plane fan pattern multicable system with nine stay cables each plane. The bridge is supported on a 35 m thick soft layer and the foundation consists of cast in place RC piles of 2 m in diameter. The main deck is fixed at both towers to resist horizontal seismic forces. The bridge is relatively flexible, with a predominant period of 3.7 sec. As to the seismic design in the transverse direction, the main deck is fixed at the towers and the end piers. Figure 29.67 shows the original design spectrum used for designing the bridge and the new design spectrum specified in the bridge design specification set in 1996 for level I and level II earthquakes (Japan Roadway Association, 1996). A level II earthquake has type I (interplate type)

Absolute Acceleration (gal)

10000

1000

100 New Design Spectrum Level II (Type I) New Design Spectrum Level II (Type II) New Design Spectrum Level I Original Design Spectrum

10

0.1

1 Natural Period (sec.) FIGURE 29.67

Design spectra for bridges.

10

29 62


and type II (intraplate type). As can be seen in the figure, the new design spectrum shows higher acceleration response in all period ranges than the original one. 29.3.7.2.2 Basic Concept of Seismic Retrofit If the deck is connected with very flexible bearings to the towers, the induced seismic forces will be kept to minimum values, but the deck may have a large displacement response. On the other hand, a very stiff connection between the deck and the towers will result in a lower deck displacement response but will attract much higher seismic forces during an earthquake. This is the case in the original bridge structure, the Tempozan Bridge. Therefore, it is important to replace the existing fixed hinge bearings with special bearings or devices at the deck tower connection both to reduce seismic forces and to absorb large seismic energy and reduce the response amplitudes. Additionally, energy absorbing devices may also be put between the deck ends and piers; however, this will attract relatively large lateral force of the piers, and therefore this kind of method has been avoided for this bridge at this time. The bridge model that represents the existing Tempozan Bridge is termed the “original bridge model.” The bridge model with the spring and damper (viscous, hysteretic, and semiactive) between the deck and the towers is termed the “retrofitted bridge model.” The original and retrofitted bridge models are shown in Figure 29.68. The original structure system has fixed hinge connections between the towers and the deck and rollers connection between the deck ends and piers, so that the deck longitudinal movement is constrained by the towers (Figure 29.68a). For the retrofitted bridge, the isolation bearings and dampers connect the deck to the towers (Figure 29.68b). The cables are modeled by truss elements. The towers and deck are modeled by beam elements, and the isolation bearings are modeled by spring elements. The models were analyzed by a commercial finite element program (Prakash and Powell, 1993). The moment curvature relationship of the members is calculated based on the sectional properties of members and material used. 29.3.7.2.3 Modal Shape Analysis The first modes of the structures are interesting here because these modes have the largest contribution to the longitudinal movement of the bridge (also see Chapters 3 and Chapter 4). The mode shapes of the original bridge and the retrofitted bridge are shown in Figure 29.69. The first mode shape of the original structure is shown in Figure 29.69a. The natural period ðTÞ of this mode is 3.75 sec (frequency ¼ 0.266 Hz), which is close to the design value for the bridge (3.7 sec, frequency ¼ 0.270 Hz; Hanshin Highway Public Corporation, 1992). This first mode shape has the effective modal mass as a percentage of the total mass of 84%. For the retrofitted structure, the stiffness of the bearings is an important issue, as large stiffness pro duces a large bearing force. However, very flexible connections produce a large displacement response. Therefore, based on a study on a simplified model of the bridge under seismic motion, the bearing stiffness

(a)

Fixed Hinge

(b) FIGURE 29.68

Isolation Bearings + Passive or Semi Active Dampers

Cable stayed bridge models: (a) original structure system; (b) retrofitted structure system.


29 63

1st mode shape, frequency = 0.266 Hz

Fixed deck-tower connections

(a)

1st mode shape, frequency = 0.158 Hz

Flexible deck-tower connections (b) FIGURE 29.69

First mode shapes of (a) original structure and (b) retrofitted structure.

that produces retrofitted main period ðT 0 Þ 1.7 times the original main period ðTÞ was chosen. This bearing stiffness makes the energy absorbing devices work well in reducing seismic induced force and displacement. The main natural period ðT 0 Þ of the retrofitted bridge then becomes 6.31 sec and the effective modal mass as a percentage of total mass is 92%. It is clear from the figures that smaller curvatures are found at the towers and the decks of the retrofitted structure than in the original structure. This shows that the retrofitted structure is expected to produce smaller moments at the towers and the decks than the original structure during a seismic attack. 29.3.7.2.4 Time-History Analysis The models were analyzed by a commercial finite element program (Prakash and Powell, 1993), which produces a piece wise dynamic time history using Newmark’s constant average acceleration ðb ¼ 1=4Þ integration of the equations of motion, governing the response of a nonlinear structure to a chosen base excitation. The input earthquake motions were type I III 3, I III 2, and I III 1 earthquakes, which are artificial acceleration data used for design in Japan for soft soil condition. Those data are intended to be type I (interplate type). With numerical comparison (Figure 29.67), type I earthquake motion gives higher effect to the bridge than type II motion, in longer period range. Table 29.7 shows the seismic response effects because of different kinds of bearings and dampers: fixed hinge bearings for the original bridge model; elastic bearings, elastic bearings plus viscous dampers, and hysteretic bearings for the retrofitted bridge models. The input earthquake was type I III 3 earthquake data and was in the longitudinal direction. From the table, it is clearly seen that if only elastic bearings are used for seismic retrofit, then the sectional forces are reduced to about 40% of the original ones. However, the displacement response is increased to 176% of the original one. By adding viscous dampers to the elastic bearings, the sectional forces can be reduced to about 25% of the original ones, and the displacement response is reduced to 63% of the original. Thus, the viscous dampers together with bearings work to reduce the seismic response of retrofitted bridge. The structural damping ratio is calculated as 35%. If hysteretic bearings are used for seismic retrofit, the sectional forces are reduced to about 29% of the original ones and the displacement response is reduced to 67% of the original one. The equivalent structural damping ratio is calculated as 13.1% by using pushover analysis to obtain a hysteretic loop at the main mode. The hysteretic bearings are modeled by a bilinear model, and the second stiffness of the hysteretic bearings is 0.03 times the initial stiffness and produces a first mode natural period of 6.31 sec.

29 64


TABLE 29.7

Maximum Earthquake Responses and Damping Ratios in Longitudinal Direction

Items

Deck displacement (m) Tower momenta (MN m) Tower axial forcea (kN) Cable force (kN) Bearing forceb (kN) Deck moment (MN m) Deck axial force (kN) Damping ratio (%) Natural period (sec) a b c

Original Structure

Retrofitted Structure Elastic Bearings

2.37 3,100 48,000 24,000 94,000 370 56,000 2 3.75

Elastic Bearings þ Viscous Damping

4.17 2,000 15,000 3,440 44,000 95 21,000 2 6.31

1.50 900 15,000 4,000 17,000 75 11,000 35 6.73

Hysteretic Bearings 1.58 900 21,000 5,000 25,000 95 15,000 13.1 3.86 and 6.31c

Base of tower AP3. At connection between deck and tower AP3. Initial and postyield stiffness.

29.3.7.2.5 Soil–Structure Interaction Effect One method to study the SSI effects is to take into Kx account the effects of flexible foundations and the Cr K C r x radiation of energy from foundations. In this method, the cable stayed bridge is idealized as in Figure 29.70 (Kawashima and Unjoh, 1991). The a a elastic half space subsoil supporting the foundation was assumed to be an elastic half space. The subsoil was assumed to FIGURE 29.70 Cable stayed bridge model with flex be elastic with no energy dissipation. The foun dation was idealized as a rigid massless circular ible foundation and energy radiating from foundation. plate. The radius of the rigid circular plate was simply assumed so that it gives the same surface area as the foundation. Dynamic stiffness of the foundation was assumed in a frequency independent form: pGs a2 Vs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3 8Gs a 0:25p 2ð1 y Þ=ð1 2y ÞGs a Cr ¼ Kr ¼ Vs 3ð1 y Þ Kx ¼

8Gs a 2 y

Cx ¼

ð29:46Þ ð29:47Þ

in which Kx and Cx represent the spring and damping coefficient for sway motion, and Kr and Cr represent those for rocking motion. Vs and a represent shear wave velocity of subsoils and the radius of foundation, respectively. The result shows that SSI increases the natural period and the damping ratio of the original structure. However, the damping ratio of the retrofitted structure is reduced and the effectiveness of the bearings and dampers in reducing seismic responses is also reduced (Table 29.8). This is mainly because the SSI model introduces flexibility at the base. A flexible base will reduce the frequency of the structure. A smaller frequency will reduce the effectiveness of viscous damping devices in absorbing earthquake energy. Moreover, a flexible base will increase the elastic strain energy of the structure that reduces the damping ratio. If the SSI model possesses an elemental damping ratio, as is a usual case for the soil, the structural damping ratio will also be influenced by the SSI model damping characteristics. 29.3.7.2.6 Semiactive Control The semiactive control herein uses the pseudonegative stiffness control algorithm (Iemura and Pradono, 2003) so that the sum of the damper force and bearing force (plus other connecting stiffness forces) are expected to produce a hysteresis loop that is as close as possible to that of rigid perfectly plastic


TABLE 29.8

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Maximum Earthquake Responses and Damping Ratios (SSI Included)

Items

Original Structure

Retrofitted Structure Elastic Bearings þ Viscous Damping

Deck displacement (m) Tower momenta (MN m) Tower axial forcea (kN) Cable force (kN) Deck moment (MN m) Deck axial force (kN) Foundation displacement (m) Damping ratio (%) Natural period (sec) a b

2.78 1,500 36,200 12,300 228 31,900 0.171 3.1 5.04

Hysteretic Bearings

2.57 800 12,500 3,010 58 9,100 0.079 23 7.66

2.77 882 19,800 4,470 86 12,300 0.093 9.3 5.13 and 7.46b

Base of tower AP3. Initial and postyielding stiffness.

force deformation characteristics (Figure 29.71a). Moreover, no residual displacement is expected at the bearings after an earthquake attack, because the hysteresis loop is velocity dependent. Figure 29.71 shows ideal and realistic force deformation characteristics of the variable damper that can produce artificial rigid perfectly plastic force deformation characteristics by using variable damper. One algorithm that can approach the hysteretic loop in Figure 29.71b requires the following variable damper force (Iemura et al., 2001): Fd;t ¼ Kd ut þ Cd u_ t

F u F F

u u

(a)

F

Connecting Stiffness + Variable Damper = Total

F + F

u u u

(b)

FIGURE 29.71 (a) Ideal and (b) realistic hysteretic loops produced by variable damper.

ð29:48Þ

where Kd is connecting stiffness (negative value) and Cd is damping coefficient (positive value). The algorithm is practical because only displacement and velocity sensors are placed in the dampers. Therefore, each damper can have its own controller. Should a malfunction happen in one damper or controller, it will not affect the other dampers or controllers. This algorithm produces the hysteretic loop shown in Figure 29.72b under harmonic motion. It is clear from the figure that the variable damper is superior to the linear viscous damper, because the maximum variable damper plus the connecting stiffness force can be set to be equal to the maximum connecting stiffness force (Figure 29.72b). One can calculate that the damping ratio of the hysteresis loop in Figure 29.72b is 53.4%. For the same damping ratio, the hysteresis loop in Figure 29.72a produces a total force 1.46 times larger than the connecting stiffness force (Iemura and Pradono, 2003). The connecting stiffness between the deck and the tower of the retrofitted cable stayed bridge comes from the contribution of cable stiffness, upper tower stiffness, and bearing stiffness. The cable stayed bridge model with isolation bearings and variable dampers controlled with the pseudonegative stiffness algorithm was analyzed by a program developed by the authors under the MATLAB (MathWorks, 2000) and SIMULINK (MathWorks, 1999) environments. The program produces a piece wise dynamic time history, using Newmark’s constant average acceleration ðb ¼ 1=4Þ integration of the equations of motion, governing the response of a nonlinear structure to a chosen base excitation. The input motions were type I III 1, I III 2, and I III 3 earthquakes, which are artificial acceleration data used for design in Japan (Japan Roadway Association, 1996). The results show that the application of the pseudonegative stiffness control algorithm is effective in reducing seismic response of the bridge model. Figure 29.73 shows the base shear deck displacement

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Linear damping + connecting stiffness Linear damping

Variable damping + connecting stiffness

Variable damping

F

F

u

u

Connecting stiffness (a) FIGURE 29.72

(b) Hysteresis loops for (a) linear viscous damping and (b) pseudonegative stiffness damping.

80000

40000

−3.0 −2 .0 −1.0

0

1.0

2 .0

3.0

Base Shear (kN)

Base Shear (kN)

80000

40000

−3.0 −2.0 −1.0

10

20

3.0

−40000

− 40000

−80000

−80000 (a)

00

Deck Displacement (m)

Deck Displacement (m)

(b)

(a)

40000

40000

20000

20000

0 −1.5 −1.0 −0.5

0.5

1.0

−20000 − 40000 Bearing Displacement (m)

15

Damping Force (kN)

Damping Force (kN)

FIGURE 29.73 Base shear vs. deck displacement relationship of a cable stayed bridge model with (a) linear dampers (b) pseudonegative stiffness dampers (type I III 1 earthquake).

(b)

−1.5 −1.0

−0.5 0 0

05

1.0

1.5

−20000 −40000 Bearing Displacement (m)

FIGURE 29.74 Damping force vs. damping displacement relationship of a cable stayed bridge model with (a) linear dampers, (b) pseudonegative stiffness dampers (type I III 1 earthquake).

40000 20000

0 −1.5 −1.0 − 0 .5

0

0.5

1.0

− 20000

15

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Damping + Bearing Force (kN)

Damping + Bearing Force (kN)


Bearing Displacement (m)

20000 0

−1.5 −1.0 −0.5

− 40000 (a)

40000

00

05

10

1.5

−20000 − 40000

(b)

Bearing Displacement (m)

FIGURE 29.75 Damping plus bearing force vs. damping displacement relationship of a cable stayed bridge model with (a) linear dampers, (b) pseudonegative stiffness dampers (type I III 1 earthquake).

relationship for both bridges, with a linear damper and pseudonegative stiffness damper, respectively, under type I III 1 earthquake input. The bridge model with pseudonegative stiffness dampers shows lower base shear than that of the bridge with linear damper. Figure 29.74 shows the hysteretic loops produced by both linear dampers and pseudonegative stiffness dampers (at tower AP3). The damping force produced by the pseudonegative stiffness damper is larger than that of the linear damper. However, the total force of damping plus the isolation bearing is lower for the pseudonegative stiffness damper (Figure 29.75). Therefore, the base shear of the cable stayed bridge model is lower for the pseudonegative stiffness dampers.

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Williams, M.S. and Blakeborough, A., Laboratory testing of structures under dynamic loads: an introductory review, R. Soc. Lond., 359, 1651 1669, 2001. Wolf, J.P. 1985. Dynamic Soil Structure Interaction, Prentice Hall, New York. Wong, H.L. and Luco, J.E. 1992. Effects of soil structure interaction on the seismic response of structures subjected to active control, Vol. 4, pp. 2137 2142. In Proceedings Tenth World Conference on Earthquake Engineering, Madrid, Spain. Yang, Y. B., Chang, K. C., and Yau, J. D. 2003. Base isolation. In Earthquake Engineering Handbook, 1st ed., W.F. Chen and C. Scawthorn, eds., pp. 17.1 17.31. CRC Press, Boca Raton, FL. Yasaka, A., Koshida, H., and Iizuka, M. 1988. Base isolation system for earthquake protection and vibration isolation of structures, Vol. 5, pp. 699 704. In Proceedings of IX World Conference on Earthquake Engineering, Tokyo, Japan. Zhang, R.H., Soong, T.T., and Mahmoodi, P., Seismic response of steel frame structures with added viscoelastic dampers, Earthquake Eng. Struct. Dyn., 8, 389 396, 1989.

30

Seismic Random Vibration of Long-Span Structures 30.1 Introduction ........................................................................ 30 2 Basic Concepts of Random Vibration Structural Seismic Analysis

†

Three Methods for

30.2 Seismic Random Excitation Fields .................................... 30 11 Power Spectral Density of Spatially Varying Ground Acceleration † Several Coherence Models † Generation of Ground Acceleration Power Spectral Density Curves from Acceleration Response Spectrum Curves † Seismic Equations of Motion of Long Span Structures † Seismic Waves and Their Geometrical Expressions

30.3 Pseudoexcitation Method for Structural Random Vibration Analysis ............................................................... 30 16 Structures Subjected to Stationary Random Excitations Structures Subjected to Nonstationary Random Vibration † Precise Integration Method

†

30.4 Long Span Structures Subjected to Stationary Random Ground Excitations ............................................. 30 27 The Solution of Equations of Motion Using the Pseudoexcitation Method † Numerical Comparisons with Other Methods

30.5 Long Span Structures Subjected to Nonstationary Random Ground Excitations ............................................. 30 34

Jiahao Lin Dalian University of Technology

Yahui Zhang

Dalian University of Technology

Modulation Functions † The Formulas for Nonstationary Multiexcitation Analysis † Expected Extreme Values of Nonstationary Random Processes † Numerical Comparisons with the Corresponding Stationary Analysis

30.6 Conclusions ......................................................................... 30 39

Summary Particular considerations must be made during the design of long span bridges with regard to safety during earthquakes. These include: the wave passage effect caused by the different times at which seismic waves arrive at different supports; the incoherence effect due to loss of coherency of the motion caused by either reflections and refractions of the waves in the inhomogeneous ground medium or the difference in the manner of superposition of waves from an extended source arriving at various supports; and the local effect because of the differences in soil conditions at different supports and the manner in which these influence the amplitude and frequency content of the bedrock motion. This chapter deals with the random vibration approach to analyzing these structures, which is based on a statistical characterization of the set of motions at the supports. This approach is particularly suitable for dealing with the above spatially varying input motions. The computational problems may be largely overcome by the pseudo excitation method. This approach is presented here. Numerical comparisons are given to show the 30 1

30 2


accuracy of the method and its capability of dealing with the spatial effects and nonstationary effects. Further topics related to this chapter are discussed in Chapter 5.

30.1

Introduction

Seismic computations of long span structures have long been an issue of great concern. Such computations are usually executed numerically using schemes in the time domain (i.e., time history). For short span bridges, all supports can be assumed to move uniformly and the response spectrum method (RSM) is a suitable computation tool. For long span bridges, however, various spatial effects such as the wave passage effect, the incoherence effect, the local site effect, and so on, may be important. Such spatial effects cannot be dealt with directly by the conventional RSM. Instead, the time history method (THM) is the most widely used method for these systems. The time history scheme requires solving the dynamic equations for a number of seismic acceleration samples. The results are then processed statistically to produce the quantities required by the designs. This process is rather complex and requires a considerable computational effort. As a result, more efficient and effective methods are under investigation. Seismic motions are random (stochastic) in nature (Housner, 1947). Spatial effects of long span bridges can be analyzed using the random vibration approach (see also Chapter 5). In the last two decades, many scholars and experts (Lee and Penzien, 1983; Dumanoglu and Severn, 1990; Lin et al., 1990; Berrah and Kausel, 1992; Kiureghian and Neuenhofer, 1992; Ernesto and Vanmarcke, 1994) have made great progress in promoting the seismic random analysis of long span structures and its engineering applications. Although available computational methods still need further improvement both in precision and efficiency, the random vibration approach, as a theoretically advanced tool, has been gradually accepted by the earthquake engineering community. For example, it has been adopted by the European Bridge Code (European Committee for Standardization, 1995). Developed recently (Lin, 1992; Lin et al., 1994a, 1995a, 1995b, 1997a, 1997b; Lin and Zhan, 2003), the pseudoexcitation method (PEM) is an accurate and highly efficient approach to the stationary and nonstationary random seismic analysis of long span structures. For typical three dimensional finite element models of long span bridges with thousands of degrees of freedom (DoF) and dozens of supports, when using 100 to 300 modes for mode superposition analysis, the seismic responses can be implemented quickly and accurately on a standard personal computer. Numerical results show that the wave passage effect is of particular importance for the seismic analysis of long span bridges, and the incoherence effect is of comparatively less importance. The details will be given in this chapter. The PEM has been successfully applied to some practical engineering analyses (Wang et al., 1999; Liu and Liu, 2000; Xue et al., 2000; Fan et al., 2001), and has been proven to be quite effective.

30.1.1

Basic Concepts of Random Vibration

30.1.1.1

Stationary Random Process

The probabilistic properties of stationary random processes are independent of time. A random process is said to be strictly stationary (or strongly stationary) if its probability density function does not change with time. However, such a condition is very difficult to satisfy in practical engineering problems. Therefore, a wide sense stationary (or weakly stationary) process is defined for which only the mean value and autocorrelation function of the process are not permitted to vary with time. A random variable x is said to be Gaussian distributed if its probability density can be written in the form ! 1 ðx xÞ2 pðxÞ ¼ pffiffiffiffi exp ð30:1Þ 2s 2 s 2p in which s is the standard deviation of x; and the variance is given by ð1 s2 ¼ ðx xÞ2 pðxÞdx 1

ð30:2Þ

Seismic Random Vibration of Long Span Structures

30 3

p(x)

0.5

s = 0.5

0.4 0.3 0.2

s = 1.0 s = 2.0

0.1 −4

−3

−2

FIGURE 30.1

0.0 −1 0

x 1

2

3

4

5

6

7

8

Probability density functions of Gaussian random variables.

Probability density functions of typical Gaussian random processes are shown in Figure 30.1, in which x is the mean value given by the abscissa (horizontal coordinate) of the peak value. A smaller s corresponds to a narrower and higher peak. For a random process xðtÞ; if the joint probability density function of its values xðt1 Þ; xðt2 Þ; …; xðtn Þ at n arbitrary time instants is Gaussian, then xðtÞ is said to be a Gaussian random process. Since the joint probability density function depends only on the mean values and covariances of the n values, a weakly stationary Gaussian random process is also strongly stationary. If a stationary random process has statistical properties that can be computed by taking the time average of an arbitrary sample over a sufficiently long period, the process is said to be ergodic. A typical seismic ground motion record is usually assumed to be a Gaussian and ergodic stationary random process. Its expected value (i.e., mean value) can be computed by ðþ1 E½xðtÞ ¼ xðtÞ ¼ xðtÞpðx; tÞdx ð30:3Þ 1

in which pðx; tÞ is the probability density function of xðtÞ: In order to investigate the relation between the values of a random process xðtÞ at two different times, the autocorrelation function of xðtÞ is defined as Rxx ðtÞ ¼ E½xðtÞxðt þ tÞ ¼ lim

T!1

1 ðT=2 xðtÞxðt þ tÞdt T T=2

ð30:4Þ

A stationary random process, denoted as xðtÞ; is not absolutely integrable in a region of t [ ð 1; 1Þ: Therefore, a subsidiary function xT ðtÞ is defined: ( xðtÞ when T=2 # t # T=2 xT ðtÞ ¼ ð30:5Þ 0 elsewhere Obviously, xT ðtÞ is absolutely integrable within t [ ð 1; 1Þ: Therefore, its Fourier transformation (see Appendix 2A and Chapter 10) can be computed by 1 ð1 XT ðvÞ ¼ ð30:6Þ xT ðtÞexpðjvtÞdt 2p 1 Let Sxx ðvÞ ¼ lim

T!1

1 lX ðvÞl2 T T

Equation 30.7 is the definition of the auto PSD (power spectral density) function of xðtÞ:

ð30:7Þ

30 4


Note that the repeated subscripts in Rxx or Sxx can be represented by just one; that is, they can be denoted as Rx or Sx : When xðtÞ is a zero mean stationary random process, its variance is given by ð1 s2x ¼ ð30:8Þ Sxx ðvÞdv 1

Figure 30.2 gives the auto PSD curves of four typical stationary random processes, which show the energy distribution with frequency for each kind of random process. The energy of a narrowband random process is concentrated within a narrow frequency band (see Figure 30.2b) whereas the energy of a wideband random process is distributed over a rather wide frequency range, as shown in Figure 30.2c. The energy of a white noise process is distributed uniformly over an infinite region, v [ ð 1; 1Þ; as shown in Figure 30.2d. Using such a random process model results in mathematical convenience. However, the white noise process does not physically exist. A single harmonic random wave has nonzero values only at two isolate frequencies ^v0 (see Figure 30.2a), and its initial phase angle w is usually regarded as uniformly distributed over ½0; 2pÞ:

x(t) = x0 sin(w 0t+f)

Sxx(w)

t

−w 0

w0

ω

(a) x(t)

Narrow-banded

Sxx(w)

t

ω

(b) x(t)

Wide-banded

Sxx(w)

ω

t (c) x(t)

White noise Sxx(w) t

(d) FIGURE 30.2

Auto PSD curves of four typical stationary random processes.

ω


30 5

The auto PSD Sxx ðvÞ of a stationary random process xðtÞ has the following properties: 1. Sxx ðvÞ is a nonnegative real number; that is Sxx ðvÞ $ 0

ð30:9Þ

This can be judged from the definition of auto PSD functions, that is, Equation 30.7. 2. Sxx ðvÞ is an even function; that is Sxx ðvÞ ¼ Sxx ð vÞ

ð30:10Þ

This can also be deduced from Equation 30.7. 3. The auto PSDs of the derivatives of xðtÞ can be computed from Sxx ðvÞ directly by Sx_ x_ ðvÞ ¼ v2 Sxx ðvÞ;

Sx€ x€ ðvÞ ¼ v4 Sxx ðvÞ

ð30:11Þ

30.1.1.1.1 Wiener–Khintchine Theorem Wiener and Khintchine proved that for an arbitrary stationary random process xðtÞ; its auto PSD Sxx ðvÞ and autocorrelation function Rxx ðtÞ are a Fourier transform pair; that is 1 ð1 ð30:12Þ Sxx ðvÞ ¼ Rxx ðtÞexpð jvtÞdt 2p 1 Rxx ðtÞ ¼

ð1 1

Sxx ðvÞexpðjvtÞdv

ð30:13Þ

According to this theorem, if either of Sxx ðvÞ or Rxx ðtÞ has been found, the other can be directly obtained. If stationary random processes xðtÞ and yðtÞ are both ergodic, then their cross correlation function can be computed in terms of their sample functions x^ ðtÞ and y^ ðtÞ: 1 ðT=2 x^ ðtÞ^yðt þ tÞdt Rxy ðtÞ ¼ lim ð30:14Þ T!1 T T=2 Ryx ðtÞ ¼ lim

T!1

1 ðT=2 y^ ðtÞ^xðt þ tÞdt T T=2

ð30:15Þ

Also, their cross PSD functions can be defined by means of the Fourier transforms of the corresponding cross correlation functions: 1 ð1 ð30:16Þ Rxy ðtÞexpð ivtÞdt Sxy ðvÞ ¼ 2p 1 Syx ðvÞ ¼

1 ð1 Ryx ðtÞexpð ivtÞdt 2p 1

ð30:17Þ

For more details, see Lin (1967). 30.1.1.2

Nonstationary Random Process

Nonstationary random processes are generally short in duration. Their basic characteristic is that the statistical properties vary significantly with time. An example is the process of a typical earthquake record, for which the medium flat segment is often regarded as a stationary random process in order to simplify the structural analysis. However, such simplification sometimes causes significant errors. For instance, some long span bridges are very flexible, with fundamental periods of approximately 15 to 20 sec. The period of the strong earthquake portion of a typical earthquake record is only approximately 20 to 30 sec. For such slender long span bridges, the seismic excitations exhibit clear nonstationary characteristics. In order to avoid computational complexities in the structural analyses, such excitations are usually assumed to be stationary random processes. This chapter shows that the analysis of such nonstationary random responses is made very simple by using PEM.

30 6


Nonstationary random processes are not ergodic because their statistical properties vary with time. In earthquake engineering, the evolutionary random process defined by Priestly (1967) has been investigated extensively. It is expressed in terms of the Riemann Stieltjes integration as ð1 Aðv; tÞexpðivtÞdaðvÞ f ðtÞ ¼ ð30:18Þ 1

in which aðvÞ satisfies the relations xðtÞ ¼

ð1 1

expðivtÞdaðvÞ

E ½dap ðv1 Þdaðv2 Þ ¼ Sxx ðv1 Þdðv2

ð30:19Þ

v1 Þdv1 dv2

ð30:20Þ

Here, xðtÞ is a zero mean stationary random process, with auto PSD Sxx ðvÞ; Aðv; tÞ is a deterministic slowly varying nonuniform modulation function, and d is a Dirac delta function. The variance of f ðtÞ is ð1 ð1 s2f ðtÞ ¼ Sff ðvÞdv ¼ lAðv; tÞl2 Sxx ðvÞdv ð30:21Þ 1

1

The PSD of f ðtÞ as given by Sff ðv; tÞ ¼ lAðv; tÞl2 Sxx ðvÞ

ð30:22Þ

is known as an evolutionary power spectral density function. Responses of structures subjected to nonstationary random excitations expressed by Equation 30.18 are not easy to compute. Therefore, the nonuniform modulation assumption is often replaced by a uniform modulation assumption; that is, the nonuniform modulation function Aðv; tÞ is replaced by a uniform modulation function gðtÞ: Thus, Equation 30.18 reduces to ð1 f ðtÞ ¼ gðtÞexpðivtÞdZðvÞ ¼ gðtÞxðtÞ ð30:23Þ 1

x(t)

(a)

g(t) (b)

f(t) = g(t)x(t)

(c) FIGURE 30.3

A uniformly modulated evolutionary random excitation f ðtÞ:


30 7

Equation 30.18 and Equation 30.23 are known as the nonuniformly modulated and uniformly modulated evolutionary random processes, respectively. Figure 30.3 shows a stationary random process xðtÞ and the corresponding uniformly modulated evolutionary random excitation f ðtÞ with a given modulation function gðtÞ:

30.1.2

Three Methods for Structural Seismic Analysis

30.1.2.1

Response Spectrum Method

The equations of motion of a linear multi DoF structure subjected to a ground acceleration excitation x€ g ðtÞ can be written as (see Chapter 3 and Chapter 8) M€y þ C_y þ Ky ¼

Me€xg ðtÞ

ð30:24Þ

in which M, C, and K are the n £ n mass, damping, and stiffness matrices of the structure, and e is the index vector of inertia forces. For short span structures, all supports can be assumed to move uniformly with the same ground acceleration x€ g ðtÞ: If the structure under consideration has a very large number of DoF, Equation 30.24 can be solved by using the mode superposition scheme (see Chapter 3). First, the lowest q natural angular frequencies vj ð j ¼ 1; 2; …; q; q ,, nÞ and the corresponding n £ q mass normalized mode matrix F should be extracted. Then, yðtÞ can be decomposed in terms of these modes: yðtÞ ¼ FuðtÞ ¼

q X j¼1

uj wj

ð30:25Þ

With proportional damping assumed (see Chapter 3 and Chapter 19), Equation 30.24 can be decoupled into q single DoF equations u€ j þ 26j vj u_ j þ v2j uj ¼

gj x€ g ðtÞ

ð30:26Þ

in which 6j is the jth damping ratio and gj is the jth modal participation factor

gj ¼ wTj Me

ð30:27Þ

According to the response spectrum theory, the solution of Equation 30.26 is uj ¼ gj aj g=v2j

ð30:28Þ

in which g is the gravity acceleration and aj is the value of the ground acceleration response spectrum (ARS) at frequency vj : If the kth element of y, denoted as yk ; is required, then the kth elements of all yj ð j ¼ 1; 2; …; qÞ are taken to compose a vector yk ; which is then used in the computation of the response (or demand) yk : qffiffiffiffiffiffiffiffiffi yk ¼ yTk Rc yk ð30:29Þ Here, Rc is the correlation matrix representing the degree of correlation between all participating modes. Based on random vibration theory, Wilson and Kiureghian (1981) have derived the expression for its elements as pffiffiffiffiffi 8 zi zj ðzi þ rzj Þr3=2 ð30:30Þ Rij ¼ ð1 r2 Þ2 þ 4zi zj rð1 þ r2 Þ þ 4ðz2i þ z2j Þr 2 in which r ¼ vj =vi : This is the widely used Complete Quadratic Combination (CQC) algorithm in the RSM. If the correlation coefficients between all modes are neglected, that is, if Rij ¼ dij (Dirac delta function), then Rc becomes a unity (identity) matrix and Equation 30.29 reduces to the square root of the sum of squares (SRSS) algorithm. The RSM, as outlined above, is very popular in the seismic analysis of short span structures. Some extensions have been published (Lee and Penzien, 1983; Dumanoglu and Severn, 1990; Lin et al., 1990;

30 8


Berrah and Kausel, 1992; Kiureghian and Neuenhofer, 1992; Ernesto and Vanmarcke, 1994) in order to deal with the seismic analysis of long span structures. However, the efficiency and accuracy still need further improvement before they can be widely accepted in engineering practice. 30.1.2.2

Time-History Method

Assume that all supports move uniformly with the same acceleration x€ g ðtÞ; which is now given in a discrete numerical form. Equation 30.24 can now be solved using the Newmark method, the Wilson u method (Clough and Penzien, 1993), or the precise integration method (Zhong and Williams, 1995). In these THMs, the structural parameters can be modified at any time. Therefore, this method is good for nonlinear problems for which structural parameters often vary with time, for example in seismic elasto plastic analysis. A major disadvantage of THMs is that the computational results rely heavily on the selected ground acceleration records. In general, a number of records must be selected for structural analyses, and statistical results are then used in the designs. In order to reduce the computational effort, usually only about three to ten records are used for statistical purposes. When the wave passage effect needs to be taken into account, the same ground acceleration record is applied to different supports with time lags and this generates x€ b on the right hand side of Equation 30.78. If the incoherence effect between the supports must also be considered, then the process for generating x€ b becomes rather complicated (Deodatis, 1990). In fact, real records of this type are difficult to find. 30.1.2.3

Random Vibration Method

The random vibration approach is appealing for seismic random analysis of long span structures. Previously, because of its high complexity and low efficiency, it was not accepted as a method of analysis by practicing engineers. However, this situation has changed considerably in recent years. Let us still begin with Equation 30.24, which we can also apply to structures subjected to uniform stationary random ground excitations. Now x€ g ðtÞ is a zero mean Gaussian stationary random process with a known auto PSD Sa ðvÞ representing acceleration excitations uniformly applied to all supports of the structure. By means of the modal superposition scheme, that is, Equation 30.25 to Equation 30.27, the traditional CQC method can be established (Clough and Penzien, 1993): Syy ðvÞ ¼

q X q X j¼1 k¼1

gj gk wj wTk Hjp ðvÞHk ðvÞSa ðvÞ

ð30:31Þ

in which wj and gj are the jth mode and the jth modal participation factor, and Hj ¼ ðv2j

v2 þ 2i6j vvj Þ

1

ð30:32Þ

is the jth frequency response function. For a real long span bridge, the number of structural DoF n usually ranges from 103 to 104, and the numbers of v and q typically range from 102 to 103. Equation 30.31 includes all quadratic terms of the participating modes, and it must be repeatedly computed for dozens or hundreds of frequencies. Although it is a simple form of excitation, the computational effort is still considerable. Therefore, in engineering practice, the following SRSS method obtained by neglecting all j – k terms in Equation 30.31, is generally used in place of the above CQC method: Syy ðvÞ ¼

q X j¼1

g2j wj wTj lHj ðvÞl2 Sa ðvÞ

ð30:33Þ

This is frequently recommended in academic literature. The SRSS formula is an approximation of Equation 30.31 that neglects the cross correlation terms between participating modes, thereby reducing the computational effort to about 1/q of that required by Equation 30.31. However, this approximation can be used only for lightly damped structures for which the participating frequencies must be sparsely spaced. For most structures (in particular their three dimensional structural models), some participating frequencies are often closely spaced. Hence, the applicability of the SRSS approximation is


30 9

somewhat questionable. It will be seen in the next section that PEM will produce results identical to those from Equation 30.31 with much less computational effort. The random vibration analysis outlined above is executed in terms of power spectral densities in the frequency domain and therefore it is also referred to as the power spectrum method. A diagonal element Sjj in the PSD matrix represents the auto PSD of a random response j: Assume that this response is significant only within the frequency domain v [ ½vL ; vU : Thus, the ith spectral moment of j can be computed by ð1 ðvU li ¼ 2 vi Sjj ðvÞdv < 2 vi Sjj ðvÞdv ð30:34Þ vL

0

The PSD values at the negative frequencies do not have any intuitive physical significance and so the single sided PSD Gxx ðvÞ is defined for applications in many engineering fields: ( 2Sxx ðvÞ v $ 0 ð30:35Þ Gxx ðvÞ ¼ v,0 0 Thus, Equation 30.34 becomes

li ¼

ð1 0

vi Gjj ðvÞdv
< ne Ts . 2:1 1=2 13 þ ð2 ln ne Ts Þ3:2 T Þ n ð2 ln ð30:48Þ sh ¼ e s > : 0:65 ne Ts # 2:1 in which

(

ne ¼

ð1:63q0:45

n0

0:38Þn0

d0 , 0:69 d0 $ 0:69

ð30:49Þ

Gupta and Trifunac (1998) made numerical experiments to compare the above two models using 1000 simulated time history excitations. Their research shows that for most practical purposes in earthquake engineering studies, the effect of the dependence among level crossings is not significant. 30.1.2.4

Comparisons of the Three Seismic-Analysis Methods

The RSM is the most popular method for the seismic analysis of short span structures. Some extensions have been made to allow this method to be used in the seismic analysis of long span structures.


30 11

However, the accuracy and efficiency still need further improvement for practical applications. The THM can be used in the seismic analysis of long span structures without theoretical difficulties. However, its major disadvantages are that it requires a good selection of the ground acceleration record samples and its computational cost is very high. The random vibration method is appealing because of its statistical nature. In the past, the random vibration computation of complex structures has been very costly. This issue has received much attention in the last two decades. As will be described below, PEM has remarkably improved this situation. Now, long span structures with thousands of DoF and dozens of supports can be computed far more quickly and accurately on a personal computer.

30.2 30.2.1

Seismic Random-Excitation Fields Power Spectral Density of Spatially Varying Ground Acceleration

For multiple excitation problems, the PSD matrices of ground acceleration excitations have the form 2 3 SX€ 1 X€ 1 ðivÞ SX€ 1 X€ 2 ðivÞ · · · SX€ 1 X€ N ðivÞ 6 7 6 S € € ðivÞ S € € ðivÞ · · · S € € ðivÞ 7 6 X2 X1 7 X2 X2 X2 XN 7 Sxx ðivÞ ¼ 6 ð30:50Þ 6 7 6 7 ··· ··· ··· ··· 4 5 SX€ N X€ 1 ðivÞ SX€ N X€ 2 ðivÞ · · · SX€ N X€ N ðivÞ in which

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SX€ k X€ l ðivÞ ¼ rkl ðivÞ SX€ k ðvÞSX€ l ðvÞ

ð30:51Þ

rkl ðivÞ ¼ lrkl ðivÞlexp½iukl ðvÞ

ð30:52Þ

and rkl ðivÞ is the acceleration coherence function between the kth and lth supports. Its norm must satisfy the relation lrkl ðivÞl # 1: The values of SX€ k ðvÞ and SX€ l ðvÞ in Equation 30.51 can be different due to local effects. However, earthquake records show that the structural responses due to the difference between SX€ k ðvÞ and SX€ l ðvÞ are in general of relatively low significance. The factor exp½iukl ðvÞ expresses the wave passage effect, which can be further expressed as exp½iukl ðvÞ ¼ exp½ ivdklL =vapp

ð30:53Þ

in which dkl is the horizontal distance between the two supports; dklL is the projection of dkl along the propagation direction of the seismic waves; and vapp is the apparent velocity of seismic waves along the surface. Assume that the time lags between the supports and the origin of the reference coordinate system are t1 ; t2 ; …; tN ; respectively. Without losing generality, let tl $ tk (when l . k). Then dklL =vapp ¼ tl tk ; and so Equation 30.53 becomes exp½iukl ðvÞ ¼ exp½ivðtk

tl Þ

ð30:54Þ

The factor lrkl ðivÞl reflects the incoherence effect (Kiureghian and Neuenhofer, 1992). Some mathematical models of rkl ðivÞ have been established based on practical earthquake records, which will be shown in the next section. Using Equation 30.51 to Equation 30.54, Equation 30.50 can be written as SðivÞ ¼ Bp JRJB

ð30:55Þ

B ¼ diag½expð ivt1 Þ; expð ivt2 Þ; …; expð ivtN Þ

ð30:56Þ

in which

J ¼ diag

hqffiffiffiffi pffiffiffiffi qffiffiffiffiffi i SX€ 1 ; SX2 € ; …; SX€ N

ð30:57Þ

30 12


2

lr12 l

1

lr1N l

···

6 6 lr l 1 ··· 6 21 R¼6 6 6 ··· ··· ··· 4 lrN1 l lrN2 l · · ·

30.2.2

3

7 lr2N l 7 7 7 7 ··· 7 5 1

ð30:58Þ

Several Coherence Models

A number of coherence models have been established based on practical earthquake records. Some of them are outlined below. 30.2.2.1

Feng–Hu Model lrkl ðv; dkl Þl ¼ exp½ ðr1 v þ r2 Þdkl

ð30:59Þ

in which r1 and r2 are the coherence parameters. According to the Hai Cheng earthquake records (China) and the Niigata earthquake records (Japan), the values of these parameters are (Feng and Hu, 1981): Hai Cheng: r1 ¼ 2 £ 10 5 sec/m, r2 ¼ 88 £ 10 4 m Niigata: r1 ¼ 4 £ 10 4 sec/m, r2 ¼ 19 £ 10 4 m 1 30.2.2.2

1

Harichandran –Vanmarcke Model

lrkl ðv; dkl Þl ¼ A exp

2d ð1 auðvÞ

A þ aAÞ þ ð1

2d ð1 auðvÞ

AÞexp

A þ aAÞ

ð30:60Þ

in which

uðvÞ ¼ K 1 þ ðv=v0 Þb

1=2

ð30:61Þ

According to the acceleration records of the SMART 1 array (Harichandran and Vanmarcke, 1986), the parameters in Equations 30.60 and Equations 30.61 are: A ¼ 0:736; a ¼ 0:147; K ¼ 5210; v0 ¼ 6:85 rad/sec, b ¼ 2:78: 30.2.2.3

Loh –Yeh Model

" lrkl ðv; dkl Þl ¼ exp

vdkl a 2pvapp

# ð30:62Þ

in which a is the wave number of the seismic waves. According to the 40 acceleration records of the SMART 1 array (Loh and Yeh, 1988), a ¼ 0:125 is proposed. Parameters vapp and dkl have been explained above. 30.2.2.4

Oliveira –Hao–Penzien Model lrkl ðv; dkl Þl ¼ expð b1 dklL

b2 dklT Þexpð

qffiffiffiffi a1 dklL

qffiffiffiffi a2 dklT ðv=2pÞ2 Þ

ð30:63Þ

in which ai ¼ ð2pai =vÞ þ ðvbi =2pÞ þ ci : dklL and dklT are the projections of dkl along the propagation direction of the seismic waves and along its normal direction, respectively. Based on 17 acceleration records of the SMART 1 array (Oliveira et al., 1991), parameters b1 ; b2 ; a1 ; a2 ; b1 ; b2 ; c1 , and c2 are given for each of these records.


30.2.2.5

Luco–Wong Model

" lrkl l ¼ exp

avdkl vs

30 13

2

# ð30:64Þ

in which a is the coherence factor, dkl is the horizontal distance between the kth and lth supports and vs is the shear wave velocity (Luco and Wong, 1986). 30.2.2.6

Qu–Wang–Wang Model

T. Qu, J. Wang, and Q. Wang (QWW) (Qu et al., 1996) proposed the following model h i lrkl l ¼ exp aðvÞdklbðvÞ

ð30:65Þ

in which aðvÞ ¼ a1 v2 þ a2 ;

bðvÞ ¼ b1 v þ b2

ð30:66Þ

a1 ¼ 0:00001678; a2 ¼ 0:001219; b1 ¼ 0:0055; b2 ¼ 0:7674: This model was established based on the statistics of dozens of records from four closely located arrays with SMART 1 as the leading one. This model has given reasonable results in some applications in China.

30.2.3 Generation of Ground Acceleration Power Spectral Density Curves from Acceleration Response Spectrum Curves In order to carry out the seismic random vibration analysis of important long span bridges, the local seismic motion PSD (usually the acceleration PSD) must be established by specialists. For bridges of less importance, however, such seismic acceleration PSD can be derived from the local ARS curve. Two methods to perform the transformation are given below. 30.2.3.1

Kaul Method

Kaul (1978) proposed an approximate transformation method that consists of the two equations Sðv0 Þ ¼ r2 ¼ 2 ln

4jR2a ðj; v0 Þ pv0 r2 p ln p v0 Ts

ð30:67Þ 1

ð30:68Þ

in which Ra ðj; v0 Þ is the ARS curve of the seismic absolute acceleration when the damping ratio is j and the natural angular frequency is v0 (usually j takes 0.05); Sðv0 Þ is the equivalent PSD curve corresponding to the given ARS curve; Ts is the seismic duration; and p is the probability of the peaks that do not cross with the given positive or negative barriers usually p ¼ 0:85 is assumed. 30.2.3.2

Iteration Scheme

An iteration based scheme for the transformation has been proposed (Sun and Jiang, 1990), which produces more accurate equivalent PSD curve than the Kaul method. For a given ARS curve Ra ðj; v0 Þ; in order to transform it into an equivalent PSD curve SðvÞ; specify its initial values S0 ðvi Þ ¼ Si ; for i ¼ 1; 2; …; N: Then the standard deviation of the acceleration response for a SDOF system can be computed from " #1=2 ð1 1 þ 4j2 ðv=v0 Þ2 dv SðvÞ ð30:69Þ s0 ðj; v0 Þ ¼ ð1 ðv=v0 Þ2 Þ2 þ 4j2 ðv=v0 Þ2 0 The peak factor r is

.pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi r ¼ 2 lnðvTs Þ þ 0:5772 2 lnðvTs Þ

ð30:70Þ

30 14


in which Ts is the period of the strong earthquake portion. The average zero crossing rate is pffiffiffiffiffiffiffi approximately v ¼ l2 =l0 p < v0 =p: The acceleration responses computed by means of Equation 30.69 and Equation 30.70 are Am ðj; v0 Þ ¼ r s0 ðj; v0 Þ

ð30:71Þ

The percentage errors between Ra ðj; v0 Þ and Am ðj; v0 Þ can be computed from Eðv0 Þ ¼

lRa ðj; v0 Þ Am ðj; v0 Þl £ 100% Ra ðj; v0 Þ

ð30:72Þ

Compute Eðv0 Þ for each frequency. If Eðv0 Þ is found to be greater than the given tolerance 1 for at least one frequency, modify all PSD values according to the following equation Skþ1 ðvi Þ ¼ Sk ðvi ÞR2a ðj; vi Þ A2m ðj; vi Þ;

i ¼ 1; 2; …; N

ð30:73Þ

and then repeat the computations of Equation 30.69 to Equation 30.72. The above process is continued until Equation 30.72 is satisfied at all frequencies.

30.2.4

Seismic Equations of Motion of Long-Span Structures

For long span structures subjected to differential ground motion, the equations of motion in the global coordinate system (assumed to be fixed to the center of the Earth) can be written in partitioned form as #( ) ( ) #( ) " " #( ) " xs 0 Ks Ksb x_ s Cs Csb x€ s Ms Msb ð30:74Þ ¼ þ þ T T T xb pb x_ b Ksb Kb Csb Cb x€ b Msb Mb in which the subscript m represents the master DoF, that is, the support displacements, while the subscript s represents the slave DoF. The absolute displacement vector xs can be decomposed into the two parts xs ¼ y s þ y r

ð30:75Þ

where ys is the quasi static displacement vector (Clough and Penzien, 1993), which satisfies ys ¼

Ks 1 Ksb xb

ð30:76Þ

Substituting Equation 30.75 and Equation 30.76 into Equation 30.74 gives Ms y€ r þ Cs y_ r þ Ks y r ¼ Ms Ks 1 Ksb x€ b þ ðCs Ks 1 Ksb

Csb Þ_xb

ð30:77Þ

It should be pointed out that Equation 30.77 cannot be reduced to the conventional Equation 30.24 when xb represents uniform ground displacements (Clough and Penzien, 1993). This is because Equation 30.74 assumes the damping forces to be proportional to the absolute velocity vector {_xTs ; x_ Tb }T : In order to avoid this inconsistency, the damping forces should be assumed to be proportional to the relative velocity vector {_yTr ; 0}T in Equation 30.74. This leads to the equations Ms y€ r þ Cs y_ r þ Ks yr ¼ Ms Ks 1 Ksb x€ b

ð30:78Þ

x€ b ¼ eb x€ g

ð30:79Þ

for uniform ground motion

Note that the following rigid displacement condition is satisfied: " #( ) ( ) es 0 Ks Ksb ¼ T Ksb Kb eb 0

ð30:80Þ


30 15

Its second half gives Ksb eb ¼

K s es

ð30:81Þ

Substituting Equation 30.79 into Equation 30.78 and using Equation 30.81 gives Equation 30.24.

30.2.5

Seismic Waves and Their Geometrical Expressions

Seismic waves can be divided into body waves and surface waves. Body waves include longitudinal waves (or pressure waves, primary waves or P waves) and transverse waves (or shear waves, secondary waves or S waves). Surface waves include Rayleigh waves and Love waves. For P waves, the soil particles move parallel to the traveling direction of waves; for S waves, however, their motion is normal to the wave traveling direction (see Figure 30.4). For horizontal shear waves (SH waves), all particles move horizontally. For vertical shear waves (SV waves), all particles move vertically. Assume that both x and y axes lie in the horizontal plane. The angle between axis x and the horizontal traveling direction of these waves is b; as shown in Figure 30.5. Thus, the acceleration components along the coordinate axes can be expressed by the components parallel, or normal, to the wave traveling direction, that is, for P waves x€ i ¼ u€ i cos b; z€i ¼ 0

y€ i ¼ u€ i sin b;

z

y SH

FIGURE 30.4 waves.

u€ j sin b;

.. yi P

z€i ¼ 0

ð30:83Þ

and for SV waves x€ i ¼ 0;

x Particle motion directions for P and S

üi .. xi

.. yi

üj

ð30:82Þ b

y€ i ¼ u€ j cos b;

SV

y

for SH waves x€ i ¼

P

β

FIGURE 30.5 components.

y€ i ¼ 0;

z€i ¼ u€ k

.. xi

SH x

Transform of ground acceleration

ð30:84Þ

In the equations of motion of an multi DoF system under uniform ground excitations, that is, Equation 30.24, e is the index vector of inertia forces. Its mathematical expressions for the different waves are: for P waves e ¼ ex cos b þ ey sin b

ð30:85Þ

for SH waves e¼

ex sin b þ ey cos b

ð30:86Þ

for SV waves e ¼ ez Clearly, for P waves, e ¼ ex when b ¼ 0 and e ¼ ey when b ¼ 908:

ð30:87Þ

30 16


If a structure has N supports, its ground acceleration excitations along the wave traveling direction can be expressed by the N dimensional vector u€ b ¼ {€u1 ; u€ 2 ; …; u€ N }T

ð30:88Þ

The m dimensional ground acceleration vector in Equation 30.78 is x€ b ¼ {€x1 ; x€ 2 ; …; x€ m }T

ð30:89Þ

x€ b ¼ EmN u€ b

ð30:90Þ

The relation between these two vectors is in which EmN is a m £ N block diagonal matrix EmN ¼ diag½eb ; eb ; …; eb

ð30:91Þ

m£N

If only three translations are considered for each support, then m ¼ 3N and each submatrix eb would be 8 9 8 9 8 9 sin b > 0> cos b > > > > > > > > > < = < = < > = sin b ; cos b and 0 > > > > > > > > > > : ; > : ; : > ; 0 0 1 for the P, SH and SV waves, respectively. Using Equation 30.90, Equation 30.76 and Equation 30.78 can be rewritten as ys ¼

Ks 1 Ksb EmN ub

ð30:92Þ

Ms y€ r þ Cs y_ r þ Ks yr ¼ Ms Ks 1 Ksb EmN u€ b

ð30:93Þ

30.3 Pseudoexcitation Method for Structural Random Vibration Analysis 30.3.1

Structures Subjected to Stationary Random Excitations

30.3.1.1

Single Stationary Random Excitations

The basic principle of the PEM for structural stationary random vibration analysis can be explained by Figure 30.6. Consider a linear system subjected to a zero mean stationary random excitation xðtÞ

(a)

(b)

(c)

x(t)

Linear Structure

Sxx

Linear Structure

∼ x(t)= Sxx(w)exp(iwt)

FIGURE 30.6

Linear Structure

y(t) z(t) Syy(w) = Hy*(w)Sxx(w)Hy(w) Syz(w) = Hy*(w)Sxx(w)Hz(w) ∼ y = Sxx(w)Hy(w)exp(iwt) ∼z = Sxx(w)Hz(w)exp(iwt)

Basic principle of pseudoexcitation method (stationary analysis).


30 17

(see Figure 30.6a) and with a given PSD Sxx ðvÞ: Suppose that for two arbitrarily selected responses yðtÞ and zðtÞ; the auto PSD Syy ðvÞ and cross PSD Syz ðvÞ are desired. Figure 30.6b gives the conventional formulas for computing these PSD functions. Hy ðvÞ and Hz ðvÞ are the frequency response functions, that is, if xðtÞ is replaced by a sinusoidal excitation expðivtÞ; the harmonic responses of yðtÞ and zðtÞ would be Hy ðvÞexpðivtÞ and Hz ðvÞexpðivtÞ; respectively. Thus, if xðtÞ is replaced by a sinusoidal excitation (Lin et al., 1994b) pffiffiffiffiffiffiffiffi ð30:94Þ x~ ¼ Sxx ðvÞexpðivtÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi the responses of yðtÞ and zðtÞ would be y~ ¼ Sxx ðvÞHy ðvÞexpðivtÞ and z~ ¼ Sxx ðvÞHz ðvÞexpðivtÞ (see Figure 30.6c). It can be readily verified that pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ð30:95Þ y~ p y~ ¼ Sxx ðvÞHyp ðvÞexpð ivtÞ Sxx ðvÞHy ðvÞexpðivtÞ ¼ lHy ðvÞl2 Sxx ðvÞ ¼ Syy ðvÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi y~ p z~ ¼ Sxx ðvÞHyp ðvÞexpð ivtÞ Sxx ðvÞHz ðvÞexpðivtÞ ¼ Hyp ðvÞSxx ðvÞHz ðvÞ ¼ Syz ðvÞ

ð30:96Þ

If yðtÞ and zðtÞ are two arbitrarily selected random response vectors of the structure, and y~ ¼ ay expðivtÞ and z~ ¼ az expðivtÞ are the corresponding harmonic response vectors due to the pseudo excitation (30.94), it can also be proven that the PSD matrices of yðtÞ and zðtÞ are Syy ðvÞ ¼ y~ p y~ T ¼ apy aTy

ð30:97Þ

Syz ðvÞ ¼ y~ p z~ T ¼ apy aTz

ð30:98Þ

This means that the auto and cross PSD functions of two arbitrarily selected random responses can be computed using the corresponding pseudoharmonic responses. Now, consider a structure subjected to a single seismic random excitation. Its equations of motion are M€y þ C_y þ Ky ¼

Me€xg ðtÞ

ð30:99Þ

in which the ground acceleration x€ g ðtÞ is a stationary random process. Its PSD Sx€ g ðvÞ is known and e is a given constant vector, indicating the distribution of inertia forces. In order to solve Equation 30.99, let the pseudoground acceleration be qffiffiffiffiffiffiffiffi ð30:100Þ x€~ g ðtÞ ¼ Sx€ g ðvÞexpðivtÞ then Equation 30.99 becomes My€~ þ Cy_~ þ K~y ¼

qffiffiffiffiffiffiffiffi Me Sx€ g ðvÞexpðivtÞ

ð30:101Þ

and its stationary solution is y~ ðtÞ ¼ ay ðvÞexpðivtÞ

ð30:102Þ

Using its first q normalized modes for mode superposition, then (Clough and Penzien, 1993) ay ðvÞ ¼

q X j¼1

pffiffiffiffiffiffiffiffi gj Hj wj Sxx ðvÞ

ð30:103Þ

in which vj ; wj ; 6j ; Hj and gj are the jth natural angular frequency, mass normalized mode, damping ratio, frequency response function, and mode participation factor, respectively. According to PEM, the PSD matrix of y is Syy ðvÞ ¼ y~ p y~ T ¼ apy ðvÞaTy ðvÞ

ð30:104Þ

Substituting Equation 30.102 into Equation 30.104 and expanding it also gives Equation 30.31. This means these two equations are mathematically identical. However, the computational effort required by Equation 30.104 is approximately only 1=q2 of that required by Equation 30.31. Therefore, Equation 30.104 is also known as the fast CQC algorithm (Lin, 1992).

30 18


Example 30.1 Derivation of the Kanai – Tajimi PSD Formula Consider the system shown in Figure 30.7(a). The single layered homogeneous soil and the super single DoF structure can be modeled by the system of Figure 30.7(b). Assume that the horizontal acceleration of the bedrock x€ 0 is a stationary random process with white noise spectrum S0 : The ground displacement relative to the bedrock is y, and the displacement of the superstructure (assumed to be an single DoF system) relative to the ground is x. The effective horizontal shear resistant stiffness of the soil layer is kg ; while the corresponding damping coefficient is cg : Assuming that m is very small in comparison with the equivalent ground mass mg ; then the equation of motion of the effective ground mass is mg y€ þ cg y_ þ kg y ¼

mg x€ 0

ðiÞ

x€ 0

ðiiÞ

or y€ þ 26g vg y_ þ v2g y ¼

in which v2g ¼ kg =mg ; 26g vg ¼ cg =mg : Now, form the horizontal pseudoacceleration for the bedrock pffiffiffi x€~ 0 ¼ S0 expðivtÞ Substituting it into Equation ii gives y€~ þ 26g vg y_~ þ v2g y~ ¼

ðiiiÞ

pffiffiffi S0 expðivtÞ

ðivÞ

Its right hand side is harmonic. Therefore, the stationary solution of y~ is pffiffiffi S0 y~ ¼ 2 expðivtÞ vg v2 þ 2i6g vg v

ðvÞ

The pseudoabsolute displacement of the ground is x~ g ¼ x~ 0 þ y~

ðviÞ

The corresponding pseudoabsolute acceleration is " # pffiffiffi pffiffiffi v2g þ 2i6g vg v v2 €x~ g ¼ x€~ 0 þ y€~ ¼ S0 expðivtÞ 1 þ ¼ S expði v tÞ 0 v2g v2 þ 2i6g vg v v2g v2 þ 2i6g vg v

ðviiÞ

Using PEM, the PSD of x€~ g is Sx€ g ðvÞ ¼ x€~ pg x€~ g ¼ S0

v4g þ 462g v2g v2 ðv2g v2 Þ2 þ 462g v2g v2

ðviiiÞ

This is exactly the Kanai Tajimi filtered white noise spectrum formula (Clough and Penzien, 1993).

x

m k

c

wg (a) FIGURE 30.7

m k

y

x c

y

mg

Vg

kg cg ..

x0

(b)

..

x0

Structural modeling for Kanai Tajimi PSD formula.


30.3.1.2

30 19

Multiple Stationary Random Excitations

Consider a linear structure subjected to a number of stationary random excitations, which are denoted as an m dimensional stationary random process vector xðtÞ with known PSD matrix Sxx ðvÞ: It is a Hermitian matrix and so it can be decomposed, for example, by using its eigenpairs cj and dj ð j ¼ 1; 2; …; rÞ; into Sxx ðvÞ ¼

r X j¼1

dj cpj cTj ðr # mÞ

ð30:105Þ

in which r is the rank of Sxx ðvÞ: Next, form the r pseudoharmonic excitations qffiffiffi x~ j ðtÞ ¼ dj cj expðivtÞ ð j ¼ 1; 2; …; rÞ

ð30:106Þ

By applying each of these pseudoharmonic excitations, two arbitrarily selected response vectors yj ðtÞ and zj ðtÞ of the structure, which can be displacements, internal forces, or other linear responses, may be easily obtained and expressed as y~ j ðtÞ ¼ ayj ðvÞexpðivtÞ

ð30:107Þ

z~ j ðtÞ ¼ azj ðvÞexpðivtÞ

ð30:108Þ

The corresponding PSD matrices can be computed by means of the following formulas (Lin et al., 1994a; Zhong, 2004): Syy ðvÞ ¼ Syz ðvÞ ¼

r X j¼1 r X j¼1

y~ pj ðtÞ~yTj ðtÞ ¼ y~ pj ðtÞ~zTj ðtÞ ¼

r X j¼1 r X j¼1

apyj ðvÞaTyj ðvÞ

ð30:109Þ

apyj ðvÞaTzj ðvÞ

ð30:110Þ

The method used to decompose Sxx ðvÞ into the form of Equation 30.105 is not unique. In fact, the Cholesky scheme is perhaps the most efficient and convenient way to do it; that is, Sxx ðvÞ is decomposed into Sxx ðvÞ ¼ Lp DLT ¼

r X j¼1

dj lpj lTj

ðr # mÞ

ð30:111Þ

in which L is a lower triangular matrix with all its diagonal elements equal to unity and D is a real diagonal matrix with r nonzero diagonal elements dj : The implementation of Cholesky decomposition for a Hermitian matrix is very similar to that for a real symmetric matrix (Wilkinson and Reinsch, 1971).

Example 30.2

A Massless Trolley Subjected to Excitations with Phase-Lags

The massless trolley shown in Figure 30.8 is connected to two abutments by springs and viscous dashpots (linear viscous dampers) as shown. The abutments move distances x1 ðtÞ and x2 ðtÞ with spectral densities S0 (constant), but x2 ðtÞ ¼ x1 ðt TÞ; in which T is a fixed time difference. The response spectral density of the trolley displacement Syy ðvÞ is to be determined. The equation of motion of the trolley is (Newland, 1975) ðc1 þ c2 Þ_y þ ðk1 þ k2 Þy ¼ k1 x1 þ c1 x_ 1 þ k2 x2 þ c2 x_ 2

ðiÞ

Since the PSD of x1 ðtÞ is S0 ; the pseudoexcitation corresponding to x1 ðtÞ is pffiffiffi x~ 1 ðtÞ ¼ S0 expðivtÞ

ðiiÞ

30 20


Because x2 ðtÞ ¼ x1 ðt

TÞ; we have pffiffiffi x~ 2 ðtÞ ¼ S0 exp½ivðt TÞ

Clearly

pffiffiffi x_~ 1 ðtÞ ¼ iv S0 expðivtÞ; pffiffiffi x_~ 2 ðtÞ ¼ iv S0 exp½ivðt

TÞ

y(t)

ðiiiÞ

x2(t)

x1(t)

ðivÞ

k1

k2

c1

c2

Substituting the above equations into Equation i gives the harmonic equation ðc1 þ c2 Þy_~ þ ðk1 þ k2 Þ~y ¼ ½ðk1 þ ivc1 Þ þ ðk2 þ ivc2 Þ expð ivTÞ expðivtÞ

FIGURE 30.8

Two phase input trolley.

ðvÞ

Its solution can be readily obtained as k þ ivc1 þ ðk2 þ ivc2 Þexpð ivTÞ expðivtÞ y~ ¼ 1 k1 þ k2 þ ivðc1 þ c2 Þ

ðviÞ

Hence, Syy ¼ y~ p y~ ¼

k21 þ k22 þ c12 v2 þ c22 v2 þ 2ðk1 k2 þ c1 c2 v2 Þcos vT þ 2ðk1 c2 v ðk1 þ k2 Þ2 þ ðc1 þ c2 Þ2 v2

k2 c1 vÞsin vT

S0

ðviiÞ

This result is identical to that given by Newland (1975). However, the process given here is quite simple.

30.3.2

Structures Subjected to Nonstationary Random Vibration

30.3.2.1

Structures Subjected to Uniformly Modulated Evolutionary Random Excitations

30.3.2.1.1 Single Excitation Problems The basic principle of the PEM for nonstationary random vibration analyses can be described by Figure 30.9. Consider a linear system subjected to an evolutionary random excitation (see Figure 30.9a) f ðtÞ ¼ gðtÞxðtÞ

ð30:112Þ

in which gðtÞ is a slowly varying modulation function, while xðtÞ is a zero mean stationary random process with auto PSD Sxx ðvÞ: The deterministic functions gðtÞ and Sxx ðvÞ are both assumed to be given.

(a)

(b)

(c)

f(t) = g(t)x(t)

Linear Structure

∼ f(t) = Sxx(w)g(t)exp(iwt)

Linear Structure

Sff (w,t) =

FIGURE 30.9

g2(t)S

xx(w)

Linear Structure

y(t) z(t) ~ y(w, t) ~z(w, t) Syy(w, t) = ~y* ~ yT Syz(w, t) = ~ y*~ zT

Basic principle of pseudoexcitation method (nonstationary analysis).


30 21

The auto PSD of f ðtÞ (see Figure 30.9b) is Sff ðv; tÞ ¼ g 2 ðtÞSxx ðvÞ

ð30:113Þ

In order to compute the PSD functions of various linear responses due to the action of f ðtÞ; we note that the pseudoexcitation has the form pffiffiffiffiffiffiffiffi f~ðv; tÞ ¼ gðtÞ Sxx ðvÞexpðivtÞ ð30:114Þ Now suppose that yðtÞ and zðtÞ are two arbitrarily selected response vectors (see Figure 30.9a) and y~ ðv; tÞ ~ v; tÞ with the and z~ ðv; tÞ are the corresponding transient responses due to the pseudoexcitation fð structure initially at rest (see Figure 30.9b). It has been proven (Lin et al., 1994a; Lin et al., 2004) that the desired PSD matrices of yðtÞ and zðtÞ are Syy ðv; tÞ ¼ y~ p ðv; tÞ~yT ðv; tÞ

ð30:115Þ

Syz ðv; tÞ ¼ y~ p ðv; tÞ~zT ðv; tÞ

ð30:116Þ

as shown in Figure 30.9c. Now, consider the Equation 30.99 of a linear structure subjected to the evolutionary random excitation pffiffiffiffiffiffiffiffi x€ g ðtÞ ¼ gðtÞxðtÞ: The pseudoground acceleration is now x€~ g ðtÞ ¼ gðtÞ Sxx ðvÞexpðivtÞ: Substituting this into the right hand side of Equation 30.99 gives the deterministic equations pffiffiffiffiffiffiffiffi My€~ þ Cy_~ þ K~y ¼ MegðtÞ Sxx ðvÞexpðivtÞ ð30:117Þ For seismic problems, the structure is initially at rest, that is, y~ ¼ y_~ ¼ 0 when t ¼ 0: Thus, the time history of y~ ðv; tÞ can be computed using the Newmark or Wilson u schemes. Furthermore, any other linear pseudo response vectors, denoted as uð ~ v; tÞ and v~ ðv; tÞ; can be computed from y~ ðv; tÞ: The response PSD matrices of uðtÞ and vðtÞ can also be accurately computed by using their pseudoresponses uð ~ v; tÞ and v~ ðv; tÞ Suu ðv; tÞ ¼ u~ p ðv; tÞu~ T ðv; tÞ

ð30:118Þ

Suv ðv; tÞ ¼ u~ p ðv; tÞ~vT ðv; tÞ

ð30:119Þ

Example 30.3 Single-Degree of Freedom System Subjected to Suddenly Applied Stationary Random Excitation Consider the following single DoF example, which was first given by Caughey and Stumpf (1961) and has been widely used to compare the efficiency and precision of various methods: y€ þ 26v0 y_ þ v20 y ¼ f ðtÞ ¼ gðtÞxðtÞ; in which

( gðtÞ ¼

1:0

t$0

0

t,0

yð0Þ ¼ y_ ð0Þ ¼ 0

ðiÞ

ðiiÞ

and xðtÞ is a zero mean valued stationary random process with its PSD Sxx ðvÞ given. Now, constitute a pseudo excitation pffiffiffiffiffiffiffiffi f~ ðtÞ ¼ Sxx ðvÞexpðivtÞ ðt $ 0Þ ðiiiÞ Thus, the motion equation that should be solved is pffiffiffiffiffiffiffiffi y€~ þ 26v0 y_~ þ v20 y~ ¼ Sxx ðvÞexpðivtÞ;

y~ð0Þ ¼ y_~ ð0Þ ¼ 0

ðivÞ

30 22


The solution of this simple equation is y~ ðv; tÞ ¼ H expðivtÞ

iv þ 1 sin v1 t v1

expð 1tÞ cos v1 t þ

pffiffiffiffiffiffiffiffi Sxx ðvÞ

ðvÞ

in which H ¼ ðv20

v2 þ 2i6v0 vÞ 1 ;

pffiffiffiffiffiffiffiffi v1 ¼ v0 1 62

1 ¼ 6v0 ;

ðviÞ

Therefore, the auto PSD of y is Syy ðv; tÞ ¼ y~ p y~ ¼ l~yl2 ¼ lHl2

expð 1tÞ

þ expð 1tÞ

1 sin v1 t þ cos v1 t v1

v sin v1 t v1

sin vt

2

cos vt

2

ðviiÞ

Sxx ðvÞ

This result is identical to the exact solution given by Caughey and Stumpf (1961), who used a more complicated process and expressed it less concisely as " 2

Syy ðv; tÞ ¼ lHl ½1 þ expð 21tÞ 1 þ

2v 0 6 sin v1 t cos v1 t v1

expð1tÞ 2 cos v1 t þ

# 2v ðv0 6Þ2 v21 þ v2 2 sin v1 t sin vt þ expð1tÞ sin v1 t Sxx ðvÞ v1 v21

2v0 6 sin v1 t cos vt: v1

ðviiiÞ 30.3.2.1.2 Multiple Excitation Problems Fully coherent excitations. In order to include the phase lags between ground excitations, that is, the wave passage effect, the evolutionary random excitation vector fðtÞ to which the structure is subjected should be 9 9 8 8 a1 gðt t1 ÞFðt t1 Þ > > > F1 ðtÞ > > > > > > > > > > > > > > > > > > > > > a gðt t ÞFðt t Þ F ðtÞ < 2 = < 2 2 2 = ¼ GðtÞfðtÞ ð30:120Þ ¼ fðtÞ ¼ .. > .. > > > > > > > . > > > > . > > > > > > > > > > > ; : ; > : Fn ðtÞ an gðt tn ÞFðt tn Þ in which 2 6 6 6 6 GðtÞ ¼ 6 6 6 4

a1 gðt

3

t1 Þ a2 gðt

7 7 7 7 7; 7 7 5

t2 Þ ..

. an gðt

tn Þ

fðtÞ ¼

9 t1 Þ > > > > > t2 Þ > =

8 > Fðt > > > > > < Fðt > > > > > > :

.. . Fðt

tn Þ

> > > > > > ;

ð30:121Þ

Here, all components of fðtÞ clearly have the same form, although there are time lags tj ðj ¼ 1; 2; …; nÞ between them; gðtÞ is the modulation function; aj ðj ¼ 1; 2; …; nÞ are given real numbers; and tj are given constants. FðtÞ is a stationary random process, of which the auto PSD SFF ðvÞ is known. The pseudo excitations corresponding to {fðtÞ} are (Lin and Zhang, 2004) pffiffiffiffiffiffiffiffi ~ ¼ GðtÞVq0 SFF ðvÞexpðivtÞ ð30:122Þ fðtÞ


30 23

in which 2

1

6 61 6 R0 ¼ q0 qT0 ¼ 6 6 .. 6. 4 1

1

3

···

1

1 .. .

7 ··· 17 7 7 .. .. 7 . .7 5

1

···

1

ð30:123Þ

V ¼ diag½expð ivt1 Þ; expð ivt2 Þ; …; expð ivtn Þ qT0 ¼ {1; 1; …; 1} The excitation PSD matrix can be written as Sff ðv;tÞ ¼ f~p ðtÞf~T ðtÞ ¼ SFF ðvÞGðtÞVp q0 qT0 VT GT ðtÞ ¼ SFF ðvÞ 2 g 2 ðt t1 Þ gðt t1 Þgðt t2 Þexpðivðt1 t2 ÞÞ 6 6 6 gðt t2 Þgðt t1 Þexpðivðt2 t1 ÞÞ g 2 ðt t2 Þ 6 6 6 6 .. .. 6 . . 6 4 gðt tn Þgðt t1 Þexpðivðtn t1 ÞÞ gðt tn Þgðt t2 Þexpðivðtn t2 ÞÞ

·· · gðt ·· · gðt ..

.

·· ·

3 tn Þexpðivðt1 tn ÞÞ 7 7 t2 Þgðt tn Þexpðivðt2 tn ÞÞ7 7 7 7 7 .. 7 . 7 5 2 g ðt tn Þ ð30:124Þ t1 Þgðt

~ can be computed using a numerical integration An arbitrarily chosen response y~ k ðv;tk Þ excited by fðtÞ scheme and the results can be expressed as pffiffiffiffiffiffiffiffi y~ k ðv; tk Þ ¼ ayk ðv;tÞ SFF ðvÞ ð30:125Þ in which ayk ðv;tk Þ ¼

ðtk 0

Hðtk

tk ÞGðtk ÞVq0 expðivtk Þdtk

ð30:126Þ

Here, ayk ðv;tÞ is the response excited by GðtÞVq0 expðivtÞ and HðtÞ is the impulse response function matrix. The cross PSD matrix between yk ðtÞ and yl ðtÞ is Syk yl ðv;tÞ ¼ y~ pk ðv; tÞ~yTl ðv;tÞ

ð30:127Þ

By letting k ¼ l; the auto PSD matrix of yk ðtÞ can be directly computed by Equation 30.127. Partially coherent excitations. In addition to the time lags between excitations, if the arbitrary coherence between such nonstationary excitations is also taken into account, then the problem is more difficult. According to PEM, however, it is only required that the matrix R0 in Equation 30.123 is changed into 2

1

6 6r 6 21 6 R¼6 . 6 . 6 . 4 rn1

r12

···

1

···

.. .

rn2

r1n

3

7 r2n 7 7 7 .. 7 .. 7 . 7 . 5

···

ð30:128Þ

1

in which rij reflects the coherence between the excitations at points i and j, and matrix R is usually symmetric and positive definite or semipositive definite. Denoting its rank as rðr $ 1Þ; R can be

30 24


expressed as R¼

r X j¼1

aj cpj cTj

ð30:129Þ

This means that the global excitation PSD matrix is decomposed into r matrices with rank unity where the jth of them corresponds to the pseudoexcitation qffiffiffiffiffiffiffiffiffiffiffi f~j ðtÞ ¼ GðtÞVcj aj SFF ðvÞexpðivtÞ ð30:130Þ If y~ kj ðtÞ and y~ lj ðtÞ are two arbitrary responses due to f~j ðtÞ; then Syk yl ðv; tÞ ¼

r X j¼1

y~ pkj ðtÞ~yTlj ðtÞ

ð30:131Þ

By letting k ¼ l; the auto PSD matrix of y k ðtÞ can be directly computed by Equation 30.131. 30.3.2.2 Structures Subjected to Nonuniformly Modulated Evolutionary Random Excitations 30.3.2.2.1 Single Excitation Problems Consider a nonuniformly modulated evolutionary random excitation f ðtÞ (Priestly, 1967) ð1 f ðtÞ ¼ Aðv; tÞexpðivtÞdaðvÞ 1

ð30:132Þ

in which Aðv; tÞ is a given nonuniform modulation function and a satisfies the equation E ½dap ðv1 Þdaðv2 Þ ¼ Sxx ðvÞdðv2

v1 Þdv1 dv2

ð30:133Þ

Here, Sxx ðv1 Þ is the auto PSD of the stationary random process xðtÞ: The Riemann Stieltjes integration in Equation 30.132 causes difficulties in conventional computations. However, this problem can be conveniently solved using PEM as follows. First, constitute the following pseudoexcitation (Lin et al., 1997a, 1997b): pffiffiffiffiffiffiffiffi f~ ðv; tÞ ¼ Aðv; tÞ Sxx ðvÞexpðivtÞ ð30:134Þ Second, replace gðtÞ in Equation 30.117 by Aðv; tÞ to yield My€~ þ Cy_~ þ K~y ¼

pffiffiffiffiffiffiffiffi MeAðv; tÞ Sxx ðvÞexpðivtÞ

ð30:135Þ

Third, with the structure initially at rest, compute the time history, y~ ðv; tÞ; for any arbitrary frequency v; and the required time histories uð ~ v; tÞ and v~ ðv; tÞ: Finally, the PSD matrices of uðtÞ and vðtÞ can be computed from Suu ðv; tÞ ¼ u~ p ðv; tÞu~ T ðv; tÞ

ð30:136Þ

Suv ðv; tÞ ¼ u~ p ðv; tÞ~vT ðv; tÞ

ð30:137Þ

Evidently, the PEM is nearly identical for uniformly or nonuniformly modulated evolutionary random excitations. The unique difference is the use of either Aðv; tÞ or gðtÞ in the pseudoexcitation expressions, see Equation 30.114 and Equation 30.134. 30.3.2.2.2 Multiple Excitation Problems For multiple nonstationary random excitation problems, the PEM based analysis process for uniformly modulated evolutionary random excitations can be immediately extended to that for nonuniformly modulated evolutionary random excitations if the modulation function gðtÞ is replaced by Aðv; tÞ:


30 25

All other formulas remain exactly the same. The difference in computational effort from using different modulation functions is negligible.

30.3.3

Precise Integration Method

30.3.3.1

Precise Integration of Exponential Matrices

Structural equations of motion, for example, Equation 30.99, can be written as M€y þ G_y þ Ky ¼ fðtÞ

ð30:138Þ

in which M, G, and K are given time invariant n £ n matrices, respectively, and fðtÞ is the given external force vector. The initial displacement yðtÞ and the initial velocity y_ ðtÞ of the system are specified. The equation of motion, Equation 30.138, combined with the identity y_ ¼ y_ leads to the first order equations of motion in the state space being v_ ¼ Hv þ r in which

"

H¼

0 B

ð30:139Þ

#

( ) ( ) 0 y ; B¼ ; r¼ ; v¼ M 1 fðtÞ D y_ I

M 1 K;

D¼

M 1G

ð30:140Þ

The homogeneous solution of Equation 30.139 is vh ðtÞ ¼ TðtÞc

ð30:141Þ

TðtÞ ¼ exp ðHtÞ

ð30:142Þ

in which Consider the current integration interval t [ ½tk ; tkþ1 ; t ¼ t tk : When t ¼ 0 or t ¼ tk ; TðtÞ ¼ I and, therefore, c is a constant vector. If the particular solution to Equation 30.139, vp ðtÞ; is temporarily assumed to have been found, then the general solution of Equation 30.139 is vðtÞ ¼ TðtÞðvðtk Þ

vp ðtk ÞÞ þ vp ðtÞ

ð30:143Þ N

In order to compute TðtÞ accurately, it is desirable to subdivide the step t into m ¼ 2 equal intervals, that is, Dt ¼ t=m ¼ 2

N

t

ð30:144Þ

For application purposes, the use of N ¼ 20 is sufficient, because it leads to Dt < 10 6 t: Such a small Dt is, in general, much smaller than the highest natural period of any practical discretized system. Using the Taylor series expansion, we have expðH £ DtÞ < I þ Ta0

ð30:145Þ

Ta0 ¼ ðH £ DtÞ þ ðH £ DtÞ2 2! þ ðH £ DtÞ3 3! þ ðH £ DtÞ4 4!

ð30:146Þ

in which

Substituting Equation 30.145 into Equation 30.142 gives TðtÞ ¼ ðexpðH £ DtÞÞm ¼ ðI þ Ta0 Þm

ð30:147Þ

Note that I þ Tai ¼ ðI þ Ta;i 1 Þ2 ¼ ðI þ 2 £ Ta;i

1

þ Ta;i

1

£ Ta;i 1 Þ;

ði ¼ 1; 2; …; NÞ

ð30:148Þ

so that I þ TaN ¼ ðI þ Ta;N

1Þ

2

¼ ðI þ Ta;N

4 2Þ

¼ · · · ¼ ðI þ Ta0 Þm ¼ TðtÞ

ð30:149Þ

30 26


Equation 30.148 and Equation 30.149 suggest the following computing strategy. In order to avoid the loss of significant digits in the matrix TðtÞ; it is necessary to compute Ta1 directly from Ta0 ; Ta2 directly from Ta1 ; and so on, by using Tai ¼ 2 £ Ta;i

1

Then TðtÞ should be computed from

þ Ta;i

1

£ Ta;i 1 ;

ði ¼ 1; 2; …; NÞ

TðtÞ < I þ TaN

ð30:150Þ ð30:151Þ

In Equation 30.151, the approximation is caused by the truncation of the Taylor expansion of Equation 30.146. It is generally negligibly small because when N ¼ 20; the first term ignored by the truncation is of the order OðDt 5 Þ ¼ 10 30 Oðt5 Þ; which is on the order of the round off errors of a typical computer. 30.3.3.2 Particular Solutions and Precise Integration Formulas for Various Forms of Loading 30.3.3.2.1 Linear Loading Form (HPD-L Form) Assume that the loading varies linearly within the time step ðtk ; tkþ1 Þ; that is, r ¼ r0 þ r1 £ ðt

t0 Þ

ð30:152Þ

in which r0 and r1 are time invariant vectors. The particular solution of Equation 30.139 is then (Lin et al., 1995a, 1995b; Zhong, 2004) vp ðtÞ ¼ ðH

1

þ ItÞð H 1 r1 Þ

H 1 ðr0

r1 t k Þ

ð30:153Þ

Substituting Equation 30.152 into Equation 30.143 gives the HPD L (High Precision Direct integration Linear) integration formula vðtkþ1 Þ ¼ TðtÞðvðtk Þ þ H 1 ðr0 þ H 1 r1 ÞÞ

H 1 ðr0 þ H 1 r1 þ r1 tÞ

ð30:154Þ

30.3.3.2.2 Sinusoidal Loading Form (HPD-S Form) If the applied loading is sinusoidal within the time region t [ ðtk ; tkþ1 Þ; then rðtÞ ¼ r1 sin vt þ r2 cos vt

ð30:155Þ

in which r1 and r2 are time invariant vectors. Substituting Equation 30.155 into Equation 30.139 enables the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004) v p ðtÞ ¼ v1 sin vt þ v2 cos vt

ð30:156Þ

in which v1 ¼ ðvI þ H2 =vÞ 1 ðr2

Hr1 =vÞ

v2 ¼ ðvI þ H2 =vÞ 1 ð r1

Hr2 =vÞ

ð30:157Þ

Substituting Equation 30.156 into Equation 30.143 gives the general solution of Equation 30.139, that is, the HPD S direct integration formula vðtkþ1 Þ ¼ TðtÞðvðtk Þ

v1 sin vtk

v2 cos vtk Þ þ v1 sin vtkþ1 þ v 2 cos vtkþ1

ð30:158Þ

The time interval t ¼ tkþ1 tk can cover an arbitrary segment, or even many periods, of a sinusoidal wave because, no matter how large the step size may be, exact responses will be obtained provided the matrix TðtÞ has been generated accurately, without any instability occurring. 30.3.3.2.3 Exponentially Decaying Sinusoidal Loading Form (HPD-E Form) Suppose that the applied loading varies according to the following exponentially decaying sinusoidal law within the time region t [ ðtk ; tkþ1 Þ: rðtÞ ¼ expðatÞðr1 sin vt þ r2 cos vtÞ

ð30:159Þ


30 27

in which r1 and r2 are time invariant vectors. Substituting Equation 30.159 into Equation 30.139 enables the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004) as vp ðtÞ ¼ expðatÞðv1 sin vt þ v2 cos vtÞ

ð30:160Þ

in which v 1 ¼ ððaI

H2 Þ þ v2 IÞ 1 ððaI

HÞr1 þ vr2 Þ

v 2 ¼ ððaI

H2 Þ þ v2 IÞ 1 ððaI

HÞr2

vr1 Þ

ð30:161Þ

Thus, substituting Equation 30.161 into Equation 30.143 gives the general solution of Equation 30.139, that is, the HPD E direct integration formula vðtkþ1 Þ ¼ TðtÞðvðtk Þ

expðatk Þðv1 sin vtk þ v 2 cos vtk ÞÞ þ expðatkþ1 Þðv 1 sin vtkþ1 þ v2 cos vtkþ1 Þ ð30:162Þ

The time interval is t ¼ tkþ1

tk :

30.4 Long-Span Structures Subjected to Stationary Random Ground Excitations 30.4.1 The Solution of Equations of Motion Using the Pseudoexcitation Method In Equation 30.93, the PSD matrix of u€ b ; that is, SðivÞ in Equation 30.55, is an N dimensional Hermitian matrix, while R is an N dimensional real symmetric matrix. Both matrices are usually positive definite or semipositive definite. If the rank of R is rðr # NÞ; then by using Equation 30.111 it can be readily decomposed into the product of an N £ r matrix Q and its transposition; that is R ¼ QQT

ð30:163Þ

SðivÞ ¼ Bp JQQT JB ¼ Pp PT

ð30:164Þ

P ¼ BJQ

ð30:165Þ

Thus, Equation 30.55 can be written as in which To solve Equation 30.93, the right hand side u€ b can be replaced by the pseudoground acceleration €~ ¼ P expðivtÞ U b

ð30:166Þ

Thus, Equation 30.93 becomes the following sinusoidal equations of motion: Ms Y€~ r þ Cs Y_~ r þ Ks Y~ r ¼ Ms Ks 1 Ksb EmN P expðivtÞ

ð30:167Þ

The stable solution of Equation 30.167 is the pseudorelative displacement vector Y~ r ; whilst the pseudostatic displacement vector Y~ s can be computed by 1 Y~ s ¼ 2 Ks 1 Ksb EmN P expðivtÞ v

ð30:168Þ

Thus, the pseudoabsolute displacement vector is ~ s ¼ Y~ r þ Y~ s X

ð30:169Þ

~ s can be computed from X ~ s by means of a quasi If necessary, any arbitrary pseudointernal force vector N static analysis. Then, the corresponding PSD matrix is ~ ps N ~ Ts SNs Ns ðvÞ ¼ N

ð30:170Þ

30 28


If it is assumed that SX€ 1 ¼ SX€ 2 ¼ · · · ¼ SX€ N ; denoted as Sa ; then Equation 30.165 becomes pffiffiffi P ¼ Sa BQ

ð30:171Þ

If only the wave passage effect is considered, that is, all lrij l ¼ 1 in Equation 30.58, then matrix Q will reduce to a vector q0 with all its elements unity; that is Q ¼ q0 ¼ {1; 1; …; 1}T

ð30:172Þ

pffiffiffi P ¼ Sa e 0

ð30:173Þ

e0 ¼ {expð ivt1 Þ; expð ivt2 Þ; …; expð ivtN Þ}T

ð30:174Þ

Thus, Equation 30.171 reduces to in which e0 is a complex vector

Therefore, when only the wave passage effect is considered, Equation 30.167 reduces to pffiffiffi Ms y€~ r þ Cs y_~ r þ Ks y~ r ¼ Ms Ks 1 Ksb EmN e0 Sa expðivtÞ

ð30:175Þ

Furthermore, if the structure is subjected to a uniform ground motion, then the vector e0 in Equation 30.174 should be replaced by q0 ; and so Equation 30.175 can be further reduced to pffiffiffi Ms y€~ r þ Cs y_~ r þ Ks y~ r ¼ Ms Ks 1 Ksb EmN q0 Sa expðivtÞ ð30:176Þ

30.4.2

Numerical Comparisons with Other Methods

30.4.2.1

Song-Hua-Jiang Suspension Bridge

The Song Hua Jiang suspension bridge (see Figure 30.10) is located in Jilin Province of China. Its overall length is 450 m, with a main span of 240 m and a width of 28 m. The finite element model had 2076 DoF, 445 nodes (includ ing 12 supports) and 574 elements. The static equilibrium position of the bridge included the effects of the initial tensions of the cables. The earthquake action was determined based on the Chinese National Standard (Code for Seismic Design of Buildings GB 50011 2001), which FIGURE 30.10 Song Hua Jiang suspension bridge. directly gives the ground ARS curve for the bridge. The PSD curve was obtained in terms of the Kaul method from which the samples of the ground acceleration time history can be produced (Kaul, 1978). For the analyses associated with the SH and SV waves, 100 modes were used for mode superposition, whereas for the P waves, only 30 modes were used. The apparent wave speeds used were 3 km/s for P waves and 2 km/s for SH or SV waves. 30.4.2.1.1 All Supports Move Uniformly Figure 30.11(a) gives the axial force distribution of this bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck. All supports of the bridges are assumed to move uniformly. The following four computational models were used: 1. 2. 3. 4.

Response spectrum method Pseudoexcitation method Time history method using three samples Time history method using ten samples


kN

RSM

PEM

30 29

THM (3 samples)

THM (10 samples)

4.0E+2

3.0E+2

2.0E+2

1.0E+2

0.0E+0

(a)

0

50

100 RSM

kN

150 PEM

200

250

300

THM (3 samples)

350

400

m

THM (10 samples)

4.0E+3

3.0E+3

2.0E+3

1.0E+3

0.0E+0

(b)

0

50

100 RSM

kN

150 PEM

200

250

300

THM (3 samples)

350

400

m

THM (10 samples)

5.0E+2

4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0

(c)

0

50

100

150

200

250

300

350

400

m

FIGURE 30.11 Deck force distribution of Song Hua Jiang bridge due to uniform ground motion: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

30 30


TABLE 30.1

Central Processing Unit (CPU) Times Required by Different Methods for Stationary Analysis Method Used

RSM (sec)

PEM (sec)

THM (for One Sample) (sec)

Uniform ground motion Wave passage effect Wave passage effect and incoherence effect

80.6

15.1 24.2 209.1

29.9 36.7

Note: the CPU time for mode extraction is not included; extracting 100 modes needs 180.9 sec.

Figure 30.11(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck. All supports move uniformly. The above four computational models were used. Figure 30.11(c) gives the vertical shear force distribution along the deck due to the seismic SV waves traveling along the deck. All supports move uniformly. The above four computational models were also used here. All computations were executed on a P3 750 personal computer. The computation times for different methods are listed in Table 30.1. Figure 30.11 and Table 30.1 show that, when ground motion is assumed uniform, that is, the earthquake spatial effects are not taken into account, the RSM, PEM, and THM (using ten samples) give very close results if the excitations are properly produced. The RSM is the most popular method, but the newly developed PEM may be the more efficient one. The THM needs to be executed for a number of ground acceleration samples and so was inefficient.

30.4.2.1.2 Wave Passage Effect Is Taken into Account Figure 30.12(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck. All supports of the bridges are assumed to move with certain time lags; that is, the wave passage effect is taken into account. The apparent P wave speed is 3 km/sec. The following four computational models were used: 1. 2. 3. 4.

RSM (uniform ground motion is assumed for comparison only) PEM (wave passage effect is considered) THM (wave passage effect is considered using three ground acceleration samples) THM (wave passage effect is considered using ten ground acceleration samples)

Figure 30.12(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck and Figure 30.12(c) gives the corresponding vertical shear force distribution. The above four computational models were used. Figure 30.12 and Table 30.1 show that when the seismic wave passage effect is taken into account, that is, the earthquake spatial effects are partly taken into account, the PEM and THM (using ten samples) give very close results. The RSM, which does not consider the wave passage effect, may give quite different results; these may appear larger or smaller than, or very close to, the results by the more reasonable PEM or THM analyses. Therefore, such computations are necessary for evaluating the seismic spatial effects of long span structures. The PEM gives the most reliable results with the least computational effort and, therefore, this method is strongly recommended.

30.4.2.1.3 Wave-Passage Effect and Incoherence Effect Are Jointly Taken into Account Figure 30.13(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck. All supports of the bridge are assumed to move with certain time lags; that is, the wave passage effect is taken into account. In addition, the incoherence effects are also taken into account. Two coherence models, that is, the Loh Yeh model


30 31

FIGURE 30.12 Deck force distribution of Song Hua Jiang bridge due to wave passage effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

30 32


RSM (uniform) PEM (Loh model)

kN

PEM (uniform) PEM (QWW model)

PEM (v=3km/s)

5.0E+2 4.0E+2

3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a)

0

50

100

150


kN

200

250

300


350

400

m

PEM (v=2km/s)

4.0E+3

3.0E+3

2.0E+3

1.0E+3

0.0E+0 (b)

0

50

100

150


kN

200

250

300


350

400

m

PEM (v=2km/s)

5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0

0

50

100

150

200

250

300

350

400

m

(c)

FIGURE 30.13 Deck force distribution of Song Hua Jiang bridge due to incoherence effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.


30 33

and the QWW model were used for evaluating such effects. The following five computational models were used: 1. 2. 3. 4. 5.

RSM (uniform ground motion is assumed) PEM (uniform ground motion is assumed) PEM (wave passage effect is considered) PEM (wave passage effect is considered also using the Loh Yeh coherence model) PEM (wave passage effect is considered also using the QWW coherence model)

Figure 30.13(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck, and Figure 30.13(c) gives the corresponding vertical shear force distribution due to the seismic SV waves. For both analyses the above five computational models were used. The above computations as well were executed on a P3 750 personal computer, and the computation times required by different methods are listed in Table 30.1. From Figure 30.13 and Table 30.1, we can conclude that: 1. When ground motion is assumed to be uniform, the RSM and PEM give very close internal force responses (i.e., demands), with the PEM being more efficient. 2. The wave passage effect is an important factor that affects the seismic responses of long span structures. To execute such seismic analyses, the PEM is not only theoretically quite reasonable, but also very efficient. 3. The incoherence effect appears to diverge when using different coherence models. Herein, the influence caused by the QWW model is more evident than that caused by the Loh Yeh model. However, compared with the wave passage effect, the influence of the incoherence effect is of less importance. 30.4.2.2

San Joaquin Concrete Bridge

San Joaquin Bridge, located in California (see Figure 30.14), is a reinforced concrete bridge built in 2001. Its length is 36 þ 50 þ 50 þ 50 þ 36 ¼ 222 m, its width is 12 m, and the height of all piers is 16.76 m. The finite element model had 367 nodes (including 10 ground z y nodes) and 366 elements. Its basic natural period x is 0.811s. Twenty modes were used in the mode superposition analysis with all damping ratios being 0.05. The seismic analysis was carried out using the RSM and PEM, respectively. The RSM 36 m 50 m 50 m 50 m 36 m analysis was conducted according to the CAL z TRANS Code (1999) with ARS ¼ 0.2 g, Type D x soil profile and magnitude Mw ¼ 7.0. The equivalent ground acceleration power spectral density curve was produced by means of the FIGURE 30.14 Structural model of San Joaquin Kaul method. All seismic waves were assumed to bridge. travel along the longitudinal direction of the bridge. The apparent P and S wave speeds were 3000 and 2000 m/s, respectively. The internal forces in the deck (i.e., the axial forces due to P waves, the transverse shear forces due to SH waves, and the vertical shear forces due to SV waves) were all computed using the following computational models: 1. RSM (uniform ground motion is assumed) 2. PEM (uniform ground motion is assumed)

30 34


3. PEM (wave passage effect is considered) 4. PEM (wave passage effect is considered also using the Loh Yeh coherence model) 5. PEM (wave passage effect is considered also using the QWW coherence model) The computational results are shown in Figure 30.15(a) (c). This bridge is not very long. However, similar phenomena to those found for the bridge of Example 30.1 are still found. Clearly, when the ground motion is assumed to be uniform, the RSM and PEM still give very close results. If the wave passage effect is taken into account, then the internal force distribution with the PEM will change considerably, particularly at the midpoint of the deck. It is known that, for symmetric bridges, the antisymmetric modes will not participate in the symmetric motions under the assumption of uniform ground motion. However, when the wave passage effect is taken into account, this conclusion does not hold. It is obvious that, even for this shorter bridge, the wave passage effect seems to be quite significant. The incoherence effect is comparatively not so important.

30.5 Long-Span Structures Subjected to Nonstationary Random Ground Excitations A typical strong motion earthquake record consists of three stages. In the first stage, the intensity of the ground motion increases, which mainly reflects the motion of P waves. The intensity of the ground motion remains the strongest in the second stage, which mainly reflects the motion of S waves. The ground motion will die down in the last stage. Such a complete seismic motion is usually regarded as a nonstationary random process. If the nonstationary property is assumed to takes place only for the intensity of the motion, then this random process is regarded as a uniformly modulated evolutionary random process. However, if the shape of the ground motion PSD curve also varies with time (in other words, the intensity and the distribution with frequency of the ground motion energy both depend on time), then the ground motion is regarded as a nonuniformly modulated evolutionary random process. It is usually accepted that when the intensity of the seismic motion in the second stage appears quite stationary while the time interval of this stage is much longer (e.g., three times or over) than the basic period of the structure under consideration, a simplified, stationary based random analysis may be acceptable as a substitute of the nonstationary analysis. In fact, the basic periods of many long span bridges range from 10 to 20 sec, and the stationary portion of a typical strong earthquake is usually less than 1 min, being only 20 to 30 sec in most cases. Therefore, nonstationary analyses are appropriate for such problems. Previously, such nonstationary random analyses have been considered very difficult. However, by using the recently developed PEM, combined with the precise integration method, such analyses have become relatively easy.

30.5.1

Modulation Functions

Some popular uniform modulation functions are listed below: 8 I ðt=t Þ2 0 # t # t1 > > < 0 1 gðtÞ ¼ I0 t1 # t # t2 > > : I0 exp{cðt t2 Þ} t $ t2 ( gðtÞ ¼ gðtÞ ¼ a½expð a1 tÞ

1

t$0

0

t,0

expð a2 tÞ ;

gðtÞ ¼ sin bt

ð30:177Þ

ð30:178Þ 0 # a1 , a2

ð30:179Þ ð30:180Þ


30 35

3

2

2

FIGURE 30.15 Deck force distribution of San Joaquin bridge due to incoherence effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

30 36


Nonuniform modulation models have rarely been investigated. Lin et al. (1997a, 1997b) suggested the following nonuniform modulation model: Aðv; tÞ ¼ bðv; tÞgðtÞ ¼ exp

h0

vt gðtÞ v a ta

ð30:181Þ

in which gðtÞ is an amplitude modulation function; bðv; tÞ is a frequency modulation function; and va and ta are the reference frequency and time, which are introduced to transform v and t into dimensionless parameters. In principle, va and ta can be arbitrarily selected. Once they have been selected, the factor h0 ðh0 . 0Þ can be adjusted accordingly to make the high frequency components of the nonstationary random process decay more quickly than the low frequency components and, thus, simulate the seismic motion more accurately. When h0 ¼ 0; that is, bðv; tÞ ¼ 1; Aðv; tÞ reduces to the uniform modulation function gðtÞ:

30.5.2

The Formulas for Nonstationary Multiexcitation Analysis

For nonuniformly modulated multiexcitation problems, the pseudoexcitation for the corresponding stationary problems, that is, Equation 30.166, is extended to €~ ðv; tÞ ¼ Aðv; tÞPexpðivtÞ U b

ð30:182Þ

in which the kth diagonal element of the N £ N diagonal matrix Aðv; tÞ is the modulation function Ak ðv; tÞ of the excitation which is applied to the kth support of the structure. In the case of uniformly modulated excitations, it is only necessary to replace all the nonuniform modulation functions Ak ðv; tÞ by the uniform modulation functions gk ðtÞ: Other formulae remain entirely unchanged. The N £ r matrix P can be generated by means of Equation 30.165 to Equation 30.174. Each column of €~ ðv; tÞ can be regarded as a deterministic acceleration excitation vector. By substituting it into the U b right hand side of Equation 30.93 and solving the equations of motion, a column of the matrix Y~ r ðv; tÞ can be produced. Because Aj ðv; tÞ is a time dependent and slowly varying function, the pseudoground displacement matrix can be computed approximately from ~ b ðv; tÞ ¼ U

1 €~ U ðv; tÞ v2 b

ð30:183Þ

The pseudoquasi static displacement matrix Y~ s ðv; tÞ can then be computed from Equation 30.92. Then, the PSD matrix of the absolute displacement vector Xs ðv; tÞ is SXs Xs ðv; tÞ ¼ ðY~ r ðv; tÞ þ Y~ s ðv; tÞÞp ðY~ r ðv; tÞ þ Y~ s ðv; tÞÞT

ð30:184Þ

~ e ; has been computed, then the PSD matrix of the If a group of pseudointernal forces, denoted as N corresponding internal forces Ne can be computed from ~ pe N ~ Te SNe Ne ðv; tÞ ¼ N

ð30:185Þ

When the ground acceleration PSD matrix is known, the corresponding pseudoacceleration vector u€~ b is easy to generate according to Equation 30.163 to Equation 30.166. If instead, the ground displacement PSD matrix or velocity PSD matrix is known, then the acceleration PSD matrix can be obtained by multiplying the displacement or velocity PSD matrices by v4 or v2 ; respectively.


30.5.3

30 37

Expected Extreme Values of Nonstationary Random Processes

The evaluation of the peak amplitude responses of structures subjected to nonstationary seismic excitations has also received much attention (Shrikhande and Gupta, 1997; Zhao and Liu, 2001). Previously, only very simple structures could be computed. However, by using the PEM, complicated structures can be analyzed, as is briefly described below. To evaluate the expected extreme value responses of a structure subjected to nonstationary Gaussian excitations, the duration of which the intensity of the excitation peaks exceeds 50% of the maximum peak intensity denoted by ½t0 ; t0 þ t is taken as the equivalent stationary duration in order to use Equation 30.40 to Equation 30.49 to evaluate the desired expected extreme values. Provided that the time dependent PSD of any arbitrary response yðtÞ; that is Syy ðv; tÞ, has been computed over that equivalent duration using the PEM, then the equivalent stationary PSD over that duration is

S0yy ðvÞ ¼

1 ðt0 þt Syy ðv; tÞdt t t0

ð30:186Þ

To compute the extreme value responses based on Equation 30.177, the parameters t0 and t are chosen as t0 ¼ t1

pffiffi 2;

t ¼ t2 þ ln 2=c

pffiffi t1 = 2

ð30:187Þ

Thus, the equivalent stationary random responses are obtained and the subsequent processing can still use Equation 30.40 to Equation 30.49.

30.5.4

Numerical Comparisons with the Corresponding Stationary Analysis

The example of the Song Hua Jiang suspension bridge of the last section is used here for the seismic nonstationary random vibration analysis. The results are compared with those from the corresponding stationary random vibration analyses with the ground assumed to move uniformly (i.e., at an apparent wave speed vapp ¼ 1), or to move at a limited apparent wave speed vapp (with the wave passage effect is taken into account), which is 3 km/sec for P waves and 2 km/sec for S waves. The nonstationary random excitation model zðtÞ ¼ gðtÞxðtÞ was used in which the auto PSD of xðtÞ is assumed to be identical to that used for the stationary excitation in the preceding section. The frequency domain parameters also remained the same. The modulation function had the form of Equation 30.177 with t1 ¼ 8:0; t2 ¼ 20:0; and c ¼ 0:20: The duration of the earthquake was t [ ½0; 25 ; and the time step size was Dt ¼ 0:5: The nonstationary analysis results are shown in Figure 30.16(a) to (c), and are compared with the results of the corresponding stationary random vibration analyses. Clearly, for such a long span bridge, the wave passage effect is quite significant in its seismic analysis, as seen in Figure 30.11 to Figure 30.13. In addition, whether for uniform ground motion or for differential ground motion (i.e., the wave passage effect is considered), the nonstationary responses are always smaller than the corresponding stationary responses. The maximum difference between their corresponding peak values may reach up to 23.1% for the present problem, as shown in Table 30.2. For very slender bridges, this nonstationary property will be even stronger. By means of the PEM combined with the precise integration method (its HPD E form for the modulation function used in this example), such modification can be fulfilled quickly and conveniently. The computational effort required by the nonstationary analysis is only about 25 min (see Table 30.3).

30 38

Vibration and Shock Handbook kN

Uniform-Nonstationary v=3km/s-Nonstatinary

Uniform-Stationary v=3km/s-Stationary

5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a)

0

50

kN

100

150

200

250

300


350

m

400


4.0E+3

3.0E+3

2.0E+3

1.0E+3

0.0E+0 (b)

0

50

kN

100

150

200

250

300


350

400

m


5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0

0

50

100

150

200

250

300

350

400

m

(c)

FIGURE 30.16 Deck force distribution of Song Hua Jiang bridge due to uniform and differential, and stationary and nonstationary random ground motion: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

Seismic Random Vibration of Long Span Structures TABLE 30.2

30 39

Comparisons of the Expected Extreme Values of Deck Internal Forces

Ground Motion

Internal Forces in the Deck (kN)

Uniform ground motion

Axial force N due to P waves Transverse shear force Fy due to SH waves Vertical shear force Fz due to SV waves

Wave passage effect

Axial force N due to P waves Transverse shear force Fy due to SH waves Vertical shear force Fz due to SV waves

Distance from the Left End (m)

Nonstationary Responses

Stationary Responses

Increases (%)

6 72 258

274.95 712.19 146.00

337.60 832.47 165.81

22.8 16.9 13.6

420 270 246

264.51 1292.17 134.00

325.61 1515.75 152.89

23.1 17.3 14.1

TABLE 30.3 CPU Times Required by PEM for Nonstationary Analyses (Units: Seconds)

30.6

Modes Used

30

100

Uniform ground motion Wave passage effect Extracting modes

450 543 143

1142 1479 181

Conclusions

For short span structures, the ground spatial effects are negligible, and the seismic analyses using RSM, PEM, or THM (with a sufficient number of samples) are relatively close to one another provided that the ground accelerations have been produced properly, and so are almost equivalent. Although they have almost the same accuracy level, their efficiencies are quite different. Of the three methods, if the structural models are rather complex (e.g., the FEM models have thousands or more DoF and need dozens or hundreds of modes for mode superposition), then the PEM will have the highest computational efficiency. For long span structures, the wave passage effect is an important factor for structural seismic responses. The influence may produce more conservative, or more dangerous, designs, which is difficult to predict by intuitive experience. Thus, computer based analysis is a preferable choice. The PEM is comparatively efficient and accurate, and is recommended. If the apparent seismic wave speed is not available, then a few possible speeds can be taken for computation, with the most unfavorable results being used in the practical design. The reasonable selection of an incoherence model for a special region needs further study, but its influence seems to be much less than that caused by the wave passage effect. Therefore, for less important long span structures, the incoherence effect can be neglected. The influence of nonstationarity is also significant for very flexible long span structures, and should be taken into account. The PEM, combined with the precise integration method, provides a powerful tool for such nonstationary analysis.

References Berrah, M. and Kausel, E., Response spectrum analysis of structures subjected to spatially varying motions, Earthquake Eng. Struct. Dyn., 21, 461 470, 1992. Caughey, T.K. and Stumpf, H.J., Transient response of a dynamic system under random excitation, Trans. ASME, JAM, 28, 563 566, 1961. Clough, R.W. and Penzien, J. 1993. Dynamics of Structures, McGraw Hill, New York. Davenport, A.G., Note on the distribution of the largest value of a random function with application to gust loading, Proc. Inst. Civil Eng., 28, 187 196, 1961.

30 40


Deodatis, G., Simulation of ergotic multivariate stochastic processes, J. Eng. Mech., 122, 778 787, 1996. Dumanoglu, A. and Severn, R.T., Stochastic response of suspension bridges to earthquake forces, Earthquake Eng. Struct. Dyn., 19, 133 152, 1990. Ernesto, H.Z. and Vanmarcke, E.H., Seismic random vibration analysis of multi support structural systems, J. Eng. Mech., ASCE, 120, 1107 1128, 1994. European Committee for Standardization, 1995. Eurocode 8, Structures in Seismic Regions Design Part 2: Bridge, Brussels, Belgium. Fan, L.C., Wang, J.J., and Chen, W., Response characteristics of long span cable stayed bridges under non uniform seismic action, Chin. J. Comput. Mech., 18, 358 363, 2001 (in Chinese). Feng, Q.M. and Hu, Y.X., Mathematical model of spatial correlative ground motion, Earthquake Eng. Eng. Vib., 1, 1 8, 1981 (in Chinese). Gupta, I.D. and Trifunac, M.D., A note on statistic of level crossing and peak amplitude in stationary stochastic process, Eur. Earthquake Eng., 12, 52 58, 1998. Harichandran, R.S. and Vanmarcke, E.H., Stochastic variation of earthquake ground motion in space and time, J. Eng. Mech., ASCE, 112, 154 175, 1986. Housner, G.W., Characteristic of strong motion earthquakes, Bull. Seism. Soc. Am., 37, 17 31, 1947. Kaul, M.K., Stochastic characterization of earthquake through their response spectrum, Earthquake Eng. Struct. Dyn., 6, 497 510, 1978. Kiureghian, A.D., Structural response to stationary random excitation, J. Eng. Mech., Div. ASCE, 106, 1195 1213, 1980. Kiureghian, A.D. and Neuenhofer, A., Response spectrum method for multi support seismic excitations, Earthquake Eng. Struct. Dyn., 21, 713 740, 1992. Lee, M. and Penzien, J., Stochastic analysis of structures and piping systems subjected to stationary multiple support excitations, Earthquake Eng. Struct. Dyn., 11, 91 110, 1983. Lin, Y.K. 1967. Probabilistic Theory of Structural Dynamics, McGraw Hill, New York. Lin, J.H., A fast CQC algorithm of PSD matrices for random seismic responses, Comput. Struct., 44, 683 687, 1992. Lin, J.H., Li, J.J., Zhang, W.S., and Williams, F.W., Non stationary random seismic responses of multi support structures in evolutionary inhomogeneous random fields, Earthquake Eng. Struct. Dyn., 26, 135 145, 1997a. Lin, J.H., Shen, W.P., and Williams, F.W., A high precision direct integration scheme for non stationary random seismic responses of non classically damped structures, Struct. Eng. Mech., 3, 215 228, 1995a. Lin, J.H., Shen, W.P., and Williams, F.W., A high precision direct integration scheme for structures subjected to transient dynamic loading, Comput. Struct., 56, 113 120, 1995b. Lin, J.H., Sun, D.K., Sun, Y., and Williams, F.W., Structural responses to non uniformly modulated evolutionary random seismic excitations, Commun. Numer. Meth. Eng., 13, 605 616, 1997b. Lin, J.H., Zhang, Y.H. 2004. Pseudo Excitation Method of Random Vibration, Science Press, Beijing (in Chinese). Lin, J.H., Zhang, W.S., and Li, J.J., Structural responses to arbitrarily coherent stationary random excitations, Comput. Struct., 50, 629 633, 1994a. Lin, J.H., Zhang, W.S., and Williams, F.W., Pseudo excitation algorithm for non stationary random seismic responses, Eng. Struct., 16, 270 276, 1994b. Lin, Y.K., Zhang, R., and Yong, Y., Multiply supported pipeline under seismic wave excitations, J. Eng. Mech., 116, 1094 1108, 1990. Liu, T.Y. and Liu, G.T., Seismic random response analysis of Arch Dams, Eng. Mech., 17, 20 25, 2000, in Chinese. Loh, C.H. and Yeh, Y.T., Spatial variation and stochastic modeling of seismic differential ground movement, Earthquake Eng. Struct. Dyn., 16, 583 596, 1988. Luco, J.E. and Wong, H.L., Response of a rigid foundation to a spatially random ground motion, Earthquake Eng. Struct. Dyn., 14, 891 908, 1986.


30 41

National Standard of The People’s Republic of China, 2001. Code for Seismic Design of Buildings GB50011 2001, Chinese Architectural Industry Press, Beijing. Newland, D.E. 1975. An Introduction to Random Vibration and Spectral Analysis, Longmans, London. Oliveira, C.S., Hao, H., and Penjien, J., Ground motion modeling for multiple input structural analysis, Struct. Saf., 10, 79 93, 1991. Priestly, M.B., Power spectral of non stationary random responses, J. Sound Vib., 6, 86 97, 1967. Qu, T.J., Wang, J.J., and Wang, Q.X., A practical power spectrum model for spatially varying seismic motion, J. Seismol., 18, 55 62, 1996. Shrikhande, M. and Gupta, V.K., A generalized approach for the seismic response of structural systems, Eur. Earthquake Eng., 11, 3 12, 1997. Sun, J.J. and Jiang, J.R., Parameters of Kanai Tajimi power spectrum according to the response spectrum of Chinese code, Chin. J. World Earthquake Eng., 8, 42 48, 1990 (in Chinese). Vanmarcke, E.H., Properties of spectral moments with applications to random vibration, J. Eng. Mech., Div. Proc. ASCE, 98, 425 446, 1972. Wang, G.W., Cheng, G.D., Shao, Z.M., Chen, H.Q. 1999. Optimal Fortification Intensity and Reliability of Aseismic Structures, Part 4, Science Press, Beijing, in Chinese. Wilkinson, J.H., Reinsch, G. 1971. Linear Algebra, Oxford University Press, London. Wilson, E.L. and Kiureghian, A.D., A replacement for the SRSS method in seismic analysis, Earthquake Eng. Struct. Dyn., 9, 187 192, 1981. Xue, S.D., Li, M.H., Cao, Z., and Zhang, Y.G. 2000. Random vibration analysis of lattice shells subjected to multi dimensional earthquake inputs. In Proceedings of the International Conference on Advances in Structural Dynamics, December 2000, pp. 777 784. Elsevier, Hong Kong. Zhao, F.X. and Liu, A.W., Relationship between power spectral density functions and response spectra of earthquake ground motions, Earthquake Eng. Eng. Vib., 21, 30 35, 2001 (in Chinese). Zhong, W.X. 2004. The Dual System in Applied Mechanics and Optimal Control, Kluwer Academic Publishers, Boston. Zhong, W.X. and Williams, F.W., A precise time step integration method, Proc. Inst. Mech. Engrs, 208C, 6, 427 430, 1995.

31

Seismic Qualification of Equipment 31.1 31.2 31.3


Introduction ......................................................................... 31 1 Distribution Qualification ................................................... 31 1 Drive Signal Generation Procedures

†

Distribution Spectra

†

Test

Seismic Qualification ........................................................... 31 6 Stages of Seismic Qualification † Test Preliminaries Single Frequency Testing † Multifrequency Testing Generation of RRS Specifications

† †

Summary Product qualification is intended for determining the adequacy of a product of good quality for a specific application or a range of applications. A good example is the seismic qualification of a nuclear power plant. This chapter presents test procedures used in product qualification, with emphasis on seismic qualification. Both single frequency testing and multifrequency testing are described. Generation of test signal specifications is outlined.

31.1

Introduction

Vibration testing is used in product qualification. Here, the objective is to test the adequacy of a product of good quality for a specific use in a typical operating environment. Clearly, the nature of testing and the test requirements, including test specifications, will depend on the type of application and the class of product. In this chapter, we will consider just two types of product qualification: distribution qualification and seismic qualification. Procedures of vibration testing for other types of qualification will be similar.

31.2

Distribution Qualification

The term “distribution qualification” denotes the process by which the ability of a product to withstand a clearly defined distribution environment is established. Dynamic effects on the product due to handling loads, characteristics of packaging, and excitations under various modes of transportation (truck, rail, air, and ocean) must be properly represented in the test specifications used for distribution qualification. If a product fails a qualification test, then corrective measures and subsequent requalification are necessary prior to commercial distribution. Product redesign, packaging redesign, and modification of existing shipping procedures might be required to meet qualification requirements. Often, the necessary improvements can be determined by analyzing data from prior tests. Proper distribution qualification will result in improved product quality (and associated reliability and performance), reduced wastage and inventory problems, cost effective packaging, reduced shipping and handling costs, and reduced warranty and service costs. 31 1

31 2


TABLE 31.1

Comparison of Test Types

Simultaneous multimodal (multiresonant) excitation possible? Test duration Power requirements Represents a random environment? Test system cost Overtesting possibility

Sine Testing

Random Testing

Narrowband Random Sweep

Broadband Random Sweep

No

Yes

No

Yes

Long Low No Low High

Short High Yes High Low

Long Low Yes Moderate to high High

Moderate High Yes High Low

Random testing can more accurately represent vibrations in distribution environments. Several characteristics make it superior to sine testing. A sine test is a single frequency test; thus, only one frequency is applied to a test object at a given instant. As a result, failure modes caused by the simultaneous excitation of two or more modes of vibration cannot be realized by sine testing, at least under steady excitations. On the other hand, in random testing, many frequencies are simultaneously applied to the test object. Therefore, conditions are more conducive to multiple mode excitations and associated complex failures. A comparison of testing with four types of excitation signals is given in Table 31.1.

31.2.1

Drive-Signal Generation

The first step in signal synthesis for driving the exciter is to assign independent and identically distributed, random phase angles to the digitized spectral magnitude (spectral lines) of the drive spectrum. The number of lines chosen is consistent with the fast Fourier transform (FFT) algorithm that is employed and the desired numerical accuracy. The inverse Fourier transform is obtained from the resulting discrete, complex Fourier spectrum. In general, the signal thus obtained would not possess ergodicity and Gaussianity. Stationarity can be attained by randomly shifting the signal with respect to time and summing the results. The resulting signal would also be weakly ergodic. Ergodicity is improved by increasing the duration of the signal. To obtain Gaussianity, a sufficiently large number of time shifted signals must be summed as dictated by the central limit theorem. Furthermore, because the magnitude of a Gaussian signal almost always remains within three times its standard deviation (99.7% of the time), Gaussianity can be imposed simply by windowing the time shifted signal. The amplitude of the window function is governed by the required standard deviation of the drive signal. Unwanted frequency components introduced as a result of sharp end transitions in each time shifted signal component can be suppressed by properly shaping the window. This process introduces a certain degree of nonstationarity into the synthesized signal, particularly if the windowed signal segments are joined end to end to generate the drive signal. A satisfactory way to overcome this problem is to introduce a high overlap from one segment to the next. Because the processing time increases in proportion to the degree of overlap, however, a compromise must be reached. In summary, for a given drive spectral magnitude the drive signal can be synthesized as follows: 1. 2. 3. 4. 5. 6.

Assign independent, identically distributed random phase values to the drive spectral lines. Perform an inverse Fourier transform of the resulting spectrum using FFT. Generate a set of independent and identically distributed time shift values. Perform a time shift of the signal obtained in Step 2 using the values from Step 3. Window the time shifted signals. Join the windowed signals with a fixed overlap.

The resulting digital drive signal is converted into an analog signal using a digital to analog converter (DAC) and passed through a low pass filter to remove any unwanted frequency components before it is used to drive the shaker. This procedure is illustrated in Figure 31.1.

Seismic Qualification of Equipment

FIGURE 31.1

31.2.2

31 3

The synthesis of a random drive signal.

Distribution Spectra

The distribution environment to which a product is subjected depends on several factors. In particular, one must consider (1) the nature and severity of handling prior to and during shipment, (2) the mode of transportation (truck, rail, air cargo, ship), (3) the geographic factors, (4) the environmental conditions, (5) the characteristics of the protective packaging used, and (6) the dynamic characteristics of the product itself. These factors are complex and essentially random in nature. Laboratory simulation of such an environment is difficult even if a combination of several types of tests e.g., vibration, shock, drop, and thermal cycling is employed. A primary difficulty arises from the requirement that test specifications should be simp1e, yet accurately represent the true environment. The test must also be repeatable to allow standardization of the test procedure and to facilitate evaluation and comparison of test data. Finally, testing must be cost effective. During transportation, a package is subjected to multi degree of freedom (multi DoF) excitations that can include rectilinear and rotational excitations at more than one location simultaneously. However, test machines are predominantly single axis devices that generate excitations along a single direction. Thus, any attempt to duplicate a realistic distribution environment in a laboratory setting can prove futile. An alternative might be to use trial shipments. However, because of the random nature of the distribution environment, many such trials would be necessary before the data would be meaningful. Therefore, trial shipments are not appealing from a cost benefit point of view and also because test control and data acquisition would be difficult. Data from trial shipments are extremely useful, however, in developing qualification test specifications and in improving existing laboratory test procedures. A more realistic goal of testing would be to duplicate possible failure and malfunction modes without actually reproducing the distribution environment. This, in fact, is the underlying principle of testing for distribution qualification. For instance, sine tests can reproduce some types of failure caused during shipment even though the test signal does not resemble the actual dynamic environment; however, random testing is generally superior. Test specifications are expressed in terms of distribution spectra in distribution qualification where random testing is used. Specification development begins with a sufficient collection of realistic data.

31 4


Sources of data include field measurements during trial shipments, computer simulations (e.g., Monte Carlo simulations), and previous specifications for similar products and environments. For best results, all possible modes of transportation, excitation levels, and handling severities should be included. The data, expressed as power spectral density (PSD), must be reduced to a common scale particularly with respect to the duration of excitation for comparison purposes. Scaling can be accomplished by applying a similarity law based on a realistic damage criterion. For example, a similarity law might relate the excitation duration and the PSD level such that the value of a suitable damage function would remain constant. Time dependent damage criteria are developed primarily on the basis of fatigue strength characteristics of a test product. Owing to nonlinearities of the environment, spectral characteristics (frequency content) will change with the excitation level. If such changes are significant, then they should be properly accounted for. The influence of environmental conditions, such as temperature and humidity, must be considered as well. The PSD curves conditioned in this manner are plotted on a log log plane to establish an envelope curve. This curve represents the worst composite environment that is typically expected. The envelope is then fitted with a small number of straight line segments. At this point, the PSD curve should be scaled so that the root mean square (RMS) value is equal to that before the straight line segments were fitted. The resulting PSD curve can be used as the test specification. Test duration can be established from the time scaling criterion. If the corresponding test duration is excessively long, thereby making the test impractical, then the test duration should be shortened by increasing the test level according to a realistic similarity criterion. Product overtesting can be significant only if one reference spectrum is used to represent all possible distribution environments. Shipping procedures should thus be classified into several groups depending on the dynamic characteristics of the shipping environment and a representative reference spectrum should be determined for each group. If a range of products with significantly diverse dynamic characteristics is being qualified, then reference spectra should be modified and classified according to product type. At the testing stage, a reference spectrum must be chosen from a spectral database depending on the product type and applicable shipping procedures. Alternatively, a general composite spectrum can be developed by assigning weights to a chosen set of reference spectra and computing the weighted sum. Vibration levels in land vehicles and aircraft can range up to several kilohertz (kHz). Ships are known to have lower levels of excitation. In general, the energy content in vibrations experienced during the distribution of computer products is known to remain within 20 Hz. Consequently, the test specification spectra (reference spectra) used in distribution qualification are usually limited to this bandwidth. The typical specification curve shown in Figure 31.2 can be specified simply from the co ordinates of the break points of the PSD curve. Intermediate values can be determined easily because the break points are joined by straight line segments on a log log plane. The area beneath the PSD curve gives the required mean square value of the test excitation. The square root of this value is the RMS value. It is specified along with the PSD curve, even though it can be determined directly from the PSD curve. An acceptable tolerance band for the control spectrum usually ^ 3 db is also specified. Test duration should be supplied with the test specification.

31.2.3

Test Procedures

Dynamic test systems with digital control are easy to operate. In menu driven systems, a routine or mode is activated by picking the appropriate item from a menu that is displayed on the cathode ray tube screen. The system asks for necessary data, and then necessary parameter values are entered into the system. Typically, the user supplies lower and upper RMS limits for test abort levels, break point co ordinates of the reference spectrum, and test duration. The tolerance bands for test spectrum equalization and the accelerometer sensitivities are also entered. More than one test setup can be stored, a number being assigned to denote each test. Preprogrammed tests can be modified using a similar procedure in the edit mode. Any preprogrammed test can be carried out simply by entering the corresponding test number. Computed


FIGURE 31.2

31 5

A reference spectrum for the distribution qualification of personal computers.

results, such as PSD curves and transmissibility functions, are stored for future evaluation. If desired, these results can be displayed, printed, or plotted with proper annotations and scales while the test is in progress. Main steps of a typical test procedure are as follows: 1. Carefully examine the test object and record obvious structural defects, abnormalities, and hazardous or unsafe conditions. 2. Perform a functional test (i.e., operate the product) according to specifications and record any malfunctions and safety hazards. Note: The test may be abandoned at this stage if the test object is defective. 3. Mount the test object rigidly on the shaker table so that the loading points and the excitation axis are consistent with standard shipping conditions and the specified test sequence. 4. Perform an exploratory test at half the specified RMS level (one fourth the specified PSD level). Monitor the response of the test package at critical locations including the control sensor location. 5. Perform the full level test for the specified duration. Record the response data. 6. Change the orientation in accordance with the specified test sequence and repeat the test. 7. After the test sequence is completed, carefully inspect the test object and record any structural defects, abnormalities, and safety hazards. 8. Conduct a functional test and record any malfunctions, failures, and safety problems. An exploratory test at a fraction of the specified test level is required for new product models that are being tested for the first time, or for older models that have been subjected to major design modifications. Three mutually perpendicular axes are usually tested, including the primary orientation (vertical axis) that is used for shipping. If product handling during distribution is automated, then testing only the primary axis is adequate. When multiple tests are required, the test sequence is normally stipulated. If the test sequence is not specified, then it can be chosen such that the least severe orientation (orientation least likely to fail) is tested first. The test is repeated for the remaining orientations, ending with the most severe one. The rationale is that with this choice of test sequence, the aging of the most severe direction would be maximized, thereby making the test more reliable.

31 6


The test report should contain the following: 1. Description of the test object: Serial number, size (dimensions and weight), product function (e.g., system unit, hard drive, power supply, printer, keyboard, mouse, monitor, and floppy disk drive of a personal computer), and packaging particulars. Descriptive photos are useful. 2. Test plan: Usually standard and attached to the report as an appendix. 3. Test setup: Test orientations, sensor (accelerometer) locations, details of mounting fixtures, and a brief description of the test apparatus. Photos may be included. 4. Test procedure: A standard attachment that is usually given according to corporate specifications. 5. Test results: Ambient conditions in the laboratory (e.g., temperature, humidity), pretest observations (e.g., defects, abnormalities, malfunctions), test data (e.g., reference spectrum, equalized control spectrum, drive spectrum, response time histories and corresponding spectra, transmissibility plots, coherence plots) and posttest observations. 6. Comments and recommendations: General comments regarding the test procedure and test item and recommendations for improving the test, product or packaging. Names and titles of the personnel who conducted the test should be given in the test report, with appropriate signatures, dates, and location of the test facility. Tests for distribution qualification can be conducted on both packaged products and those without any protective packaging, even though it is the packaged product that is shipped. However, the reference spectra used in the two cases are usually not the same. The spectrum used for testing a product without protective packaging is generally less severe. Response spectra that are used for testing an unpackaged product should reflect the excitations experienced by the product during packaging.

31.3

Seismic Qualification

Frequently, it is necessary to determine whether a given piece of equipment is capable of withstanding a pre established seismic environment in a specific application. This process is known as seismic qualification. For example, electric utility companies should qualify their equipment for seismic capability before installing it in earthquake prone geographic localities. Safety related equipment in nuclear power plants also requires seismic qualification. Regulatory agencies usually specify the general procedures to follow in seismic qualification. Seismic qualification by testing is appropriate for complex equipment, but in such cases, the equipment size is a limiting factor. For large systems that are relatively simple to model, qualification by analysis is suitable. Often, however, both testing and analysis are needed in the qualification of a given piece of equipment. Seismic qualification of equipment by testing is accomplished by applying a dynamic excitation by means of a shaker to the equipment, which is suitably mounted on a test table, and then monitoring structural integrity and functional operability of the equipment. Special attention should be given to the development of the dynamic test environment, mounting features, the operability variables that should be monitored, the method of monitoring functional operability and structural integrity, and the acceptance criteria used to decide qualification. In monitoring functional operability, the test facility would normally require auxiliary systems to load the test object or to simulate in service operating conditions. Such systems include actuators, dynamometers, electrical load and control signal circuitry, fluid flow and pressure loads, and thermal loads. In seismic qualification by analysis, a suitable model is first developed for the equipment, and then static or dynamic analysis (or computer analysis) is performed under an analytically defined dynamic environment. The analytical dynamic environment is developed on the basis of the specified dynamic environment for seismic qualification. By analytically or computationally determining system response at various locations, and by checking for such crucial parameters as relative deflections, stresses, and strains, qualification can be established.


31.3.1

31 7

Stages of Seismic Qualification

Consider the construction of a nuclear power plant. In this context, the plant owner is the customer. Actual construction of the plant is done by the plant builder, who is directly responsible to the customer concerning all equipment purchased from the equipment supplier or vendor. Often, the vendor is also the equipment manufacturer. The equipment may be purchased by the customer and handed over to the plant builder or directly purchased by the plant builder. Accordingly, the purchaser could be the plant builder or the plant owner. A regulatory agency might stipulate seismic excitation capability requirements for the equipment used in the plant, or might specify the qualification requirements for various categories of equipment. The customer is directly responsible to the regulatory agency for adherence to these stipulations. The vendor, however, is responsible to the plant builder and the customer for the seismic capability of the equipment. The vendor may perform seismic qualification on the equipment according to required specifications. More often, however, the vendor hires the services of a test laboratory, which is the contractor, for seismic qualification of equipment in the plant. A reviewer, who is hired by the plant builder or the customer, may review the qualification procedure and the report, which are usually developed by the test laboratory by adhering to the qualification requirements. Figure 31.3 gives a flowchart for test object movement and for associated information interactions between various groups in the qualification program. A basic step in any qualification program is the preparation of a qualification procedure. This is a document that describes in sufficient detail such particulars as the tests that will be conducted on the test object, pretest procedures, the nature of test input excitations and the method of generating these signals, inspection and response monitoring procedures during testing, definitions of equipment malfunction, and qualification criteria. Where analysis is also used in the qualification, the analytical methods and computer programs that will be used should be described adequately in the qualification procedure. The qualification procedure is prepared by the test laboratory (contractor), equipment particulars are obtained from the vendor or the purchaser, and the purchaser usually supplies the test environment for which the equipment will be qualified. Before the qualification tests are conducted, the test procedure is submitted to the purchaser for approval. The purchaser normally hires a reviewer to determine whether the qualification procedure satisfies the requirements of both the regulatory agency and the purchaser. There could be several stages of revision of the test procedure until it is accepted by the purchaser upon the recommendation of the reviewer. The approved qualification procedure is sent to the test laboratory and qualification is performed accordingly. The test laboratory prepares a qualification report that also includes the details of static or dynamic analysis when incorporated. The qualification report is sent to the purchaser for evaluation. The purchaser might obtain the services of an authority to review the qualification report. The report might have to be revised and analysis and tests might have to be repeated before a final decision is made on the qualification of the equipment. Information flow in a typical qualification program is shown in Figure 31.4. Regulatory Agency Equipment To Be Equipment Qualified Test Supplier Laboratory (vendor) (Contractor) (Manufacturer)

Customer (Plant Owner)

Plant Builder Qualified Equipment

FIGURE 31.3

Test object movement and information interactions in seismic qualification.

31 8


Equipment Supplier Purchaser

Equipment Data

Test Environment

Qualification Procedure Test Laboratory (Contractor) Standards Guides Specification

Qualification Procedure Reviewer Test Report

Approved Test Report

31.3.2

Test Laboratory

Purchaser

Regulatory Agency

FIGURE 31.4

Approved Procedure

Purchaser

Test Report

Information flow in a seismic qualification program.

Test Preliminaries Acceleration Amplitude

Seismic qualification tests are usually conducted by one of two methods depending on whether single frequency or multifrequency excitation inputs are employed in the main tests. The two test categories are (1) single frequency tests and (2) multifre quency tests. Although the second test method is more common in seismic qualification by testing, the first method is used under some conditions depending on the nature of the test object and its Frequency mounting features (for example, line mounted vs. floor mounted equipment). Typically, multifre FIGURE 31.5 A typical required input motion curve. quency excitations are preferred in qualification tests, and single frequency excitations are favored in design development and quality assurance tests. In single frequency testing, amplitude of the excitation input is specified by a required input motion (RIM) curve, similar to that shown in Figure 31.5. If single frequency dwells (e.g., sine dwell, sine beat) are employed, then the excitation input is applied to the test object at a series of selected frequency values in the frequency range of interest for that particular test environment. In such cases, dwell times (and number of beats per cycle where sine beats are employed) at each frequency point should be specified. If a single frequency sweep (such as a sine sweep) is employed as the excitation signal, then the sweep rate should be specified. When the single frequency test excitation is specified in this manner, the tests are conducted very much like multifrequency tests. Multifrequency test are normally conducted by employing the response spectra method to represent the test input environment. Basically, the test object is excited using a signal whose response spectrum, known as the test response spectrum (TRS), envelops a specified response spectrum, known as the required response spectrum (RRS). Ideally, the TRS should equal the RRS but it is practically impossible to achieve this condition. Hence, multifrequency tests are conducted using a TRS that envelops the RRS so that in significant frequency ranges, the two response spectra are nearly equal (see Figure 31.6). However, excessive conservatism, which would result in overtesting, should be avoided. It is usually acceptable to have TRS values below the RRS at a few frequency points. The RRS is part of the data supplied to the test laboratory prior to the qualification tests being conducted. Two types of RRS are provided, representing (1) the operating basis earthquake (OBE), and (2) the safe shutdown earthquake (SSE). The response spectrum of the OBE represents the most severe motions produced by an earthquake under which the equipment being tested would remain functional without undue risk of malfunction or safety hazard. However, if the equipment is allowed to operate at a


31 9

disturbance level higher than the OBE level for a prolonged period then there would be a significant risk of malfunction. The response spectrum of the SSE represents the most severe motions produced by an earthquake that the equipment being tested could safely withstand while the entire nuclear power plant is being shutdown. However, prolonged operation (i.e., more than the duration of one earthquake) could result in equipment malfunction. In other words, equipment is designed to withstand only FIGURE 31.6 The TRS enveloping the RRS in a multifrequency test. one SSE in addition to several OBEs. A typical seismic qualification test would first subject the equipment to several OBE level excitations, primarily for aging the equipment mechanically to its end of design life condition, and then would subject it to one SSE level excitation. When providing RRS test specifications, it is customary to supply only the SSE requirement. The OBE requirement is then taken as a fraction (typically, 0.5 or 0.7) of the SSE requirement. Test response spectra corresponding to the excitation signals are generated by the test laboratory during testing. The purchaser usually supplies the test laboratory with an FM tape containing frequency components that should be combined in some ratio to generate the test input signal. Qualification tests are conducted according to the test procedure approved and accepted by the purchaser. The main steps of seismic qualification testing are outlined in the following subsections.

31.3.3

Single-Frequency Testing

Seismic ground motions usually pass through various support structures before they eventually are transmitted to equipment. For seismic quali fication of that equipment by testing, in theory we should apply the actual excitations felt by it, and not the seismic ground motions. In an ideal case, the shaker table motion should be equivalent to the seismic response of the supporting structure at the point of attachment of the equipment. The supporting structure would have a particu lar frequency response function between the ground location and the equipment support location (see Figure 31.7). Consequently, it could FIGURE 31.7 Schematic representation of the filtering be considered a filter that modifies seismic ground of seismic ground motions by a supporting structure. motions before they reach the equipment mounts. In particular, the components of the ground motion that have frequencies close to a resonant frequency of the supporting structure will be felt by the equipment at a relatively higher intensity. Furthermore, the ground motion components at very high frequencies will be almost entirely filtered out by the structure. If the frequency response of the supporting structure is approximated by a lightly damped simple oscillator, then the response felt by the equipment will be almost sinusoidal, with a frequency equal to the resonant frequency of the structure. When the equipment supporting structure has a very sharp resonance in the significant frequency range of the dynamic environment (for example, 1 to 35 Hz for seismic ground motions), it follows from the previous discussion that it is desirable to use a short duration single frequency test in seismic qualification of the equipment. Equipment that is supported on pipelines (valves, valve actuators, gauges, and so forth) falls into this category and is termed line mounted equipment.

31 10


Resonant frequency of the supporting structure is usually not known at the time of the seismic qualification test. Consequently, single frequency testing must be performed over the entire frequency range of interest for that particular dynamic environment. Another situation in which single frequency testing is appropriate is when the test object (equipment) itself does not have more than one sharp resonance in the frequency range of interest. In this case, the most prominent response of the test object occurs at its resonant frequency, even when the dynamic environment is an arbitrary excitation. Consequently, a single frequency excitation would yield conservative test results. Equipment that has more than one predominant resonance may employ single frequency testing provided that each resonance corresponds to a dynamic DoF (e.g., one resonance along each dynamic principal axis), and that cross coupling between these DoF is negligible. In summary, single frequency testing may be used where one or more of the following conditions are satisfied: 1. The supporting structure has one sharp resonance in the frequency range of interest (line mounted equipment is included). 2. The test object does not have more than one sharp resonance in the frequency range of interest. 3. The test object has a resonance in each DoF, but the DoF are uncoupled (for which adequate verification should be provided in the test procedure). 4. The test object can be modeled as a simple dynamic system (such as a simple oscillator), for which adequate justification or verification should be provided. Usually, the required SSE excitation level for a single frequency test over a frequency range is specified by a curve such as the one shown in Figure 31.5. This curve is known as the RIM magnitude curve. The OBE excitation level is usually taken as a fraction (typically, 0.5 or 0.7) of the RIM values given for the SSE. For a sine sweep test, the sweep rate and the number of sweeps in the test should also be specified. Typically, the sweep rate for seismic qualification tests is less than 1 octave/min. One sweep, from the state of rest to the maximum frequency in the range and back to the state of rest, is normally carried out in an SSE test (for example, 1 35 to 1 Hz). Several sweeps (typically, five) are performed in an OBE test. In an SSE sine dwell test, the dwell time for each dwell frequency should be specified. The dwell frequency intervals should not be high (typically, a half octave or less). For an OBE test, the dwell times are longer (typically, five times longer) than those specified for an SSE test. For an SSE test using sine beats, the minimum number of beats and the minimum duration of excitation (with or without pauses) at each test frequency should be specified. In addition, the pause time for each test frequency should be specified when sine beats with pauses are employed. For an OBE test, the duration of excitation should be increased (as in a sine dwell test). The dwell time at each test frequency should be adequate to perform at least one functional operability test. Furthermore, a dwell should be carried out at each resonant frequency of the test object as well as at those frequencies that are specified. Total duration of an SSE test should be representative of the duration of the strong motion part of a standard safe shutdown earthquake. Sometimes, narrow band random excitations may be used in situations where single frequency testing is recommended. Narrowband random signals are those that have their power concen trated over a narrow frequency band. Such a signal can be generated for test excitation purposes by passing a random signal through a narrow band pass filter. By tuning the filter to different center frequencies in narrowbands, the test excitation frequency can be varied during testing. This center frequency of the filter should be swept up and FIGURE 31.8 A typical RRS for a narrow band down over the desired frequency range at a excitation test.


31 11

reasonably slow rate (e.g., 1.0 octave/min) during the test. Thus, a multifrequency test with a sharp frequency response spectrum (the RRS), as illustrated in Figure 31.8, is adequate in cases where single frequency testing is recommended. In this case, a requirement that has to be satisfied by the test excitation signal is that its amplitude should be equal to or greater than the zero period acceleration of the RRS for the test.

31.3.4

Multifrequency Testing

When equipment is mounted very close to the ground under its normal operating conditions, or if its supporting structure and mounting can be considered rigid, then seismic ground motions will not be significantly filtered before they reach the equipment mounts. In this case, the seismic excitations that are felt by the equipment will retain broadband characteristics. Multifrequency testing is recommended for seismic qualification of such equipment. Whereas single frequency tests are specified by means of an RIM curve along with the test duration at each frequency (or sweep rates), multifrequency tests are specified by means of an RRS curve. The test requirement in multifrequency testing is that the response spectrum of the test excitation (the TRS) felt by the equipment mounts should envelop the RRS. Note that all frequency components of the test excitation are applied simultaneously to the test object. This is in contrast to single frequency testing, in which only one significant frequency component is applied at a given instant. When random excitations are employed in multifrequency testing, enveloping of the RRS by the TRS may be achieved by passing the random signal produced by a signal generator through a spectrum shaper. As the analyzing frequency bandwidth (e.g., one third octave bands, one sixth octave bands) decreases, the flexibility of shaping the TRS improves. A real time spectrum analyzer (or a personal computer) may be used to compute and display the TRS curve corresponding to the control accelerometer signal (see Figure 31.9). By monitoring the displayed TRS, it is possible to adjust the gains of the spectrum shaper filter to obtain the desired TRS that would envelop the RRS. Most test laboratories generate their multifrequency excitation signals by combining a series of sine beats that have different peak amplitudes and frequencies. Using the same method, many other signal types (such as decaying sinusoids) may be superimposed to generate a required multifrequency excitation signal. A combination of signals of different types could also be employed to produce a desired test input. A commonly used combination is a broadband random signal and a series of sine beats. In this combination, the random signal is adjusted to have a response spectrum that will envelop the broadband portion of the RRS without much conservatism. The narrow band peaks of the RRS, which generally will not be enveloped by such a broadband response spectrum, will be covered by a suitable combination of sine beats.

Random Signal Generator

Random Signal

Shaped Signal

Spectrum Shaper

Power Amplifier

Spectrum Control TRS

Spectrum Analyzer

Control Accelerometer Output

FIGURE 31.9

Test Table

Test Input Excitation

Matching of the TRS with the RRS in multifrequency testing.

31 12


By employing such mixed composite signals, it is possible to envelop the entire RRS without having to increase the amplitude of the test excitation to a value that is substantially higher than the ZPA of the RRS. One important requirement in multifrequency testing is that the amplitudes of the test excitation be equal to or greater than the ZPA of the RRS.

31.3.5

Generation of RRS Specifications

Seismic qualification of an object is usually specified in terms of an RRS. The excitation input that is used in seismic qualification analysis and testing should conservatively satisfy the RRS; that is, the response spectrum of the actual excitation input should envelop the RRS (without excessive conservatism, of course). For equipment to be installed in a building or on some other supporting structure, the RRS generally cannot be obtained as the response spectrum of a modified seismic ground motion time history. The supporting structure usually introduces an amplification effect and a filtering effect on seismic ground motions. This amplification factor alone could be as high as three. Some of the major factors that determine the RRS for a particular seismic qualification test are as follows: 1. Nature of the building that will be qualified 2. Dynamic characteristics of the building or structure and the location (elevation and the like) where the object is expected to be installed 3. In service mounting orientation and support characteristics of the object 4. Nature of the seismic ground motions in the geographic region where the object would be installed 5. Test severity and conservatism that is required by the purchaser or the regulator agency The basic steps in developing the RRS for a specific seismic qualification application include the following: 1. Development of representative safe shutdown earthquake (SSE) ground motion time histories for the building (or support structure) location 2. Development of a suitable building (or support structure) model 3. Response analysis of the building model, using the time histories that are obtained in Step 1 4. Development of response spectra for various critical locations in the building (or support structure), using the response time histories obtained in Step 3 5. Normalization of the response spectra obtained in Step 4 to unity ZPA (that is, dividing by their individual ZPA values) 6. Identification of the similarities in the set of normalized response spectra that are obtained in Step 5 and grouping them into a small number of groups 7. Representation of each similar group by a response spectrum consisting of straight line segments that envelop all members in the group, giving a normalized RRS for each group 8. Determination of scale factors for various locations in the building for use in conjunction with the corresponding normalized RRS curves Representative strong motion earthquake time histories (SSEs) are developed by suitably modifying actual seismic ground motion time histories that have been observed in that geographic location (or a similar one), or by using a random signal generation (simulation) technique or any other appropriate method. These time histories may be available as either digital or analog records, depending on the way in which they are generated. If computer simulation is used in their development, then a statistical representation of the expected seismic disturbances in the particular geographic region (using geological features in the region, seismic activity data, and the like) should be incorporated in the algorithm. The intensity of the time histories can be adjusted, depending on the required test severity and conservatism. The normalized response spectra are grouped so that the spectra that have roughly the same shape are put in the same group. In this manner, relatively few groups of normal response spectra (normalized) are


31 13

obtained. Then, the response spectra that belong to each group are plotted on the same graph paper. Next, straight line segments are drawn to envelop each group of response spectra. This procedure results in a normalized RRS for each group of analytical response spectra. The RRS that is used for a particular seismic qualification scheme is obtained as follows. First, the normalized RRS corresponding to the location in the building where the object would be installed is selected. The normalized RRS curve is then multiplied by the appropriate scaling factor. The scaling factor normally consists of the product of the actual ZPA value under SSE conditions at that location (as obtained from the analytical response spectrum at that location, for example) and a factor of safety that depends on the required test severity and conservatism. In fact, three RRS curves corresponding to the vertical, east west and north south directions might be needed, even for single degree of freedom (single DoF) seismic qualification tests because, by mounting three control accelerometers in these three directions, triaxial monitoring could be accomplished. If only one control accelerometer is used in the test, then only one RRS curve is used. In this case, the resultant of the three orthogonal RRS curves should be used. One way to obtain the resultant RRS curve is to apply the square root of the sum of squares (SRSS) method to the three orthogonal components. Alternatively, the envelope of the three orthogonal RRS curves is obtained and multiplied by a safety factor (greater than unity). Note that more than one building (or even many different geographic locations) could be included in the described procedure for developing RRS curves. The resulting RRS curves are then valid for the collection of buildings or geographic locations considered. When the generality of an RRS curve is extended in this manner, the test conservatism increases. This will also result in an RRS curve with a much broader band. In practice, in a particular seismic qualification project, only a few normalized RRS curves are employed. In conjunction with these RRS curves, a table of data is provided that identifies the proper RRS curves and the scaling factors that should be used for different physical locations (for example, elevations) in various buildings that are situated at several geographic locations.

Bibliography Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, Wiley Interscience, New York. Brigham, E.O. 1974. The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ. de Silva, C.W. 1983. Dynamic Testing and Seismic Qualification Practice, D.C. Heath and Co., Lexington, MA. de Silva, C.W., A dynamic test procedure for improving seismic qualification guidelines, J. Dyn. Syst. Meas. Control, Trans. ASME, 106, 143 148, 1984. de Silva, C.W., Hardware and software selection for experimental modal analysis, Shock Vibration Digest, 16, 3 10, 1984. de Silva, C.W., Matrix eigenvalue problem of multiple shaker testing, J. Eng. Mech. Div., Trans. ASCE, 108, 457 461, 1982. de Silva, C.W., Optimal input design for the dynamic testing of mechanical systems, J. Dyn. Syst. Meas. Control, Trans. ASME, 109, 111 119, 1987. de Silva, C.W., Seismic qualification of electrical equipment using a uniaxial test, Earthquake Eng. Struct. Dyn., 8, 337 348, 1980. de Silva, C.W., Selection of shaker specifications in seismic qualification tests, J. Sound Vibration, 91, 21 26, 1983. de Silva, C.W., Shaker test fixture design, Meas. Control, 17, 152 155, 1983. de Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control application in the distribution qualification of microcomputers, Shock Vibration Digest, 18, 3 13, 1986. de Silva, C.W., Loceff, F., and Vashi, K.M., Consideration of an optimal procedure for testing the operability of equipment under seismic disturbances, Shock Vibration Bull., 50, 149 158, 1980.

31 14


de Silva, C.W. 2000. Vibration Fundamentals and Practice, CRC Press, Boca Raton, FL. Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth, England. McConnell, K.G. 1995. Vibration Testing, Wiley, New York. Randall, R.B. 1977. Application of B & K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark.

Design and Applications

VIII

VIII 1

32

Vibration Design and Control 32.1 32.2 32.3 32.4

32.5

32.6 32.7 32.8 32.9

Introduction .....................................................................

32 2

Specification of Vibration Limits ...................................

32 3

Vibration Isolation ...........................................................

32 5

Shock and Vibration

Peak Level Specification † Root Mean Square Value Specification † Frequency Domain Specification Design Considerations Systems

†

Vibration Isolation of Flexible

Balancing of Rotating Machinery ................................... 32 15

Static Balancing † Complex Number/Vector Approach † Dynamic (Two Plane) Balancing Experimental Procedure of Balancing

†

Balancing of Reciprocating Machines ............................ 32 26

Single Cylinder Engine † Balancing the Inertia Load of the Piston † Multicylinder Engines † Combustion/ Pressure Load

Whirling of Shafts ............................................................ 32 33

Equations of Motion † Steady State Whirling Self Excited Vibrations

†

Design through Modal Testing ....................................... 32 39

Component Modification

†

Substructuring

Passive Control of Vibration ........................................... 32 45

Undamped Vibration Absorber † Damped Vibration Absorber † Vibration Dampers

Active Control of Vibration ............................................ 32 61


†

Control Techniques

32.10 Control of Beam Vibrations ............................................ 32 67 State Space Model of Beam Dynamics Problem † Use of Linear Dampers


†

Control

Appendix 32A MATLAB Control Systems Toolbox ............................................................... 32 73

Summary There are desirable and undesirable types and situations of mechanical vibration. Undesirable vibrations are those that cause human discomfort and hazards, structural degradation and failure, performance deterioration and malfunction of machinery and processes, and various other problems. This chapter discusses ways of either eliminating or reducing the undesirable effects of vibration. Specifically, some useful topics on design for vibration suppression and the control of vibration are addressed. General approaches to vibration mitigation may be identified from the dynamic systems point of view. Typically, a set of vibration specifications is given as simple threshold values (bounds) or frequency spectra, and the goal is to either design or control the system to meet these 32 1

32 2


specifications. Frequency domain techniques based on transfer functions such as transmissibility and time domain techniques using the state space representation, optimal control, and modal control are presented. Applications considered here include vibration isolation, balancing of rotating and reciprocating machinery, whirling suppression, and passive and active control of vibration.

32.1

Introduction

Consider the schematic diagram of a vibratory system shown in Figure 32.1. Forcing excitations f(t) to the mechanical system S cause the vibration responses y. Our objective is to suppress y to a level that is acceptable. Clearly, there are three general ways of doing this:

Vibration Response y Vibration Excitation f(t)

b m k

1. Isolation. Suppress the excitations of Mechanical Vibrating System (S) vibration. This method deals with f. 2. Design modification. Modify or redesign the mechanical system so that for the same FIGURE 32.1 A vibrating mechanical system. levels of excitation, the resulting vibrations are acceptable. This method deals with S: 3. Control. Absorb or dissipate the vibrations using external devices, through implicit or explicit sensing and control. This method deals with y. Within each of these three categories, several approaches can be used to achieve the objective of vibration mitigation. Essentially, each of these approaches involves designing (either complete through redesign or incremental design modification) of the system on the one hand, and controlling the vibration through external means (passive or active devices) on the other. Note that removal of faults (e.g., misalignments and malfunctions by repair or parts replacement) can also remove vibrations. This approach may fall into any of the three categories listed above. The category of vibration isolation involves “isolating” a mechanical system (S) from vibration excitations (f) so that the excitation signals are “filtered” out or dissipated prior to reaching the system. The use of properly designed suspension systems, mounts, and damping layers falls within this category. The category of design modification will involve making changes to the components and the structure of a mechanical system according to a set of specifications and design guidelines. Balancing of rotating machinery and structural modification through modal analysis and design techniques fall into this category. The category of control will involve either passive devices (which do not use external power), such as dynamic absorbers and dampers, or active control devices (which need external power for operation). In the passive case, the control device implicitly senses the vibration response and dissipates it (as in the case of a damper) or absorbs and stores its energy where it is slowly dissipated (as in the case of a dynamic absorber). In the active case, the vibrations y are explicitly sensed through sensors and transducers. The forces that should be acted on the system to counteract and suppress vibrations are determined by a controller, and the corresponding forces or torques are applied to the system through one or more actuators. Note that there may be some overlap in the three general categories of vibration mitigation that were mentioned above. For example, the addition of a mount (category 1) may also be interpreted as a design modification (category 2) or as incorporating a passive damper (category 3). It should also be noted that the general approach, commonly known as source alteration, may fall into either category 1 or category 2. In this case, the purpose is to alter or remove the source of vibration. The source could either be external (e.g., road irregularities that result in vehicle vibrations) a category 1 problem, or internal (imbalance or misalignment in rotating devices that results in periodic forces, moments, and vibrations) a category 2 problem. It can be more difficult to alter external vibration sources (e.g., resurfacing the roadways) than to modify the internal sources (e.g., balancing of rotating machinery).

Vibration Design and Control

32 3

Furthermore, the external source of vibration may be quite random and may not be accessible for alteration at all (e.g., aerodynamic forces on an aircraft).

32.1.1

Shock and Vibration

Sometimes, response to shock loads is considered separately from response to vibration excitations for the purpose of design and control of mechanical systems. For example, shock isolation and vibration isolation are treated under different headings in some literature. This is actually unnecessary. Even though vibration analysis predominantly involves periodic excitations and responses, transient and random oscillations (vibrations) are also commonly found in practice. The frequency band of the latter two types of signals is much broader than that of a simple periodic signal. A shock signal is transient by definition, and has a very short duration (in comparison to the predominant time constants of the mechanical system to which the shock load is applied). Hence, it will possess a wide band of frequencies. Consequently, frequency domain techniques are still applicable. Furthermore, time domain techniques are particularly suited to dealing with transient signals in general and shock signals in particular. In that context, a shock excitation may be treated as an impulse whose effect is to instantaneously change the velocity of an inertia element. Then, in the time domain, a shock load may also be treated as an initial velocity excitation of an otherwise free (unforced) system.

32.2

Specification of Vibration Limits

Design and control procedures of vibration have the primary objective of ensuring that under normal operating conditions, the system of interest does not encounter vibration levels that exceed the specified values. In this context, then, the ways of specifying vibration limits become important. This section will present some common ways of vibration specification.

32.2.1

Peak Level Specification

Vibration limits for a mechanical system may be specified in either the time domain or the frequency domain. In the time domain, the simplest specification is the peak level of vibration (typically, acceleration in units of g the acceleration due to gravity). Here, the techniques of isolation, design, or control should ensure that the peak vibration response of the system do not exceed the specified level. In this case, the entire time interval of operation of the system is monitored, and the peak values are checked against the specifications. Note that, in this case, it is the instantaneous peak value at a particular time instant that is of interest, and what is used in representing vibration is an instantaneous amplitude measure rather than an average amplitude or an energy measure.

32.2.2

Root-Mean-Square Value Specification

The root mean square (RMS) value of a vibration signal yðtÞ is given by the square root of the average of the squared signal as yRMS ¼

1 ðT 2 y dt T 0

1=2

ð32:1Þ

Note that, by squaring the signal, its sign is eliminated and, essentially, the energy level of the signal is used. The period T; over which the squared signal is averaged, will depend on the problem and the nature of the signal. For a periodic signal, one period is adequate for averaging. For transient signals, several time constants (typically four times the larger time constant) of the vibrating system would be sufficient. For random signals, a value that is as large as feasible should be used.

32 4


In the method of RMS value specification, the RMS value of the acceleration response (typically, acceleration in gs) is computed using Equation 32.1 and is compared with the specified value. In this method, instantaneous bursts of vibration do not have a significant effect as they are filtered out because of the integration. It is the average energy or power of the response signal that is considered. The duration of exposure enters into the picture indirectly and in an undesirable manner. For instance, a highly transient vibration signal can initially have a damaging effect. However, the larger the T that is used in Equation 32.1, the smaller the computed RMS value. Hence, in this case, the use of a large value for T would lead to diluting or masking the damage potential. In practice, the longer the exposure to a vibration signal, the greater the harm caused. Hence, when using specifications such as peak and RMS values, they have to be adjusted according to the period of exposure. Specifically, a larger specification should be used for longer periods of exposure.

32.2.3

Frequency-Domain Specification

It is not realistic to specify the limitation to vibration exposure of a complex dynamic system by just a single threshold value. Usually, the effect of vibration on a system depends on at least the following three parameters: 1. Level of vibration (peak, RMS, power, etc.). 2. Frequency content (range) of excitation. 3. Duration of exposure to vibration.

Velocity (m/s) 0.01 0.001 m

0.001 1.0

10.0

1g

100.0

1000.0

This is particularly true because the excitations Frequency (Hz) that generate the vibration environment may not necessarily be a single frequency (sinusoidal) signal and may be broadband and random. FIGURE 32.2 Operating vibration specification Furthermore, the response of the system to the (nomograph) for a machine. vibration excitations will depend on its frequency transfer function, which determines its resonances and damping characteristics. Under these circumstances, it is desirable to provide specifications in a nomograph where the horizontal axis gives frequency (Hz) and the vertical axis could represent a motion variable such as displacement (m), velocity (m/s), or acceleration (m/s2 or g). It is not important which of these motion variables represents the vertical axis of the nomograph. This is true because in the frequency domain Velocity ¼ jv £ displacement Acceleration ¼ jv £ velocity and one form of motion may be easily converted into one of the remaining two motion representations. In each of the forms, assuming that the two axes of the nomograph are graduated in a logarithmic scale, the constant displacement, constant velocity, and constant acceleration lines are straight lines. Consider a simple specification of machinery vibration limits as given by the following values: Displacement limit ðpeakÞ ¼ 0:001 m Velocity limit ¼ 0:01 m=sec Acceleration limit ¼ 1:0g This specification may be represented in a velocity vs. frequency nomograph (log log) as in Figure 32.2. Usually, such simple specifications in the frequency domain are not adequate. As noted above, the system behavior will vary depending on the excitation frequency range. For example, motion sickness in humans may be predominant in low frequencies in the range of 0.1 to 0.6 Hz and passenger discomfort in


RMS Acceleration (m/s2)

32 5

8 4

Motion Sickness Region

2

1.0 0.8 0.6 0.4

30 min. Trips 2 hr Trips 8 hr Trips

0 .2 0.1 0 .2 0 .4 Frequency (Hz) FIGURE 32.3

0 .6

0 .8 1.0

A severe discomfort vibration specification for ground transit vehicles.

ground transit vehicles may be most serious in the frequency range of 4 to 8 Hz for vertical motion and 1 to 2 Hz for lateral motion. In addition, for any dynamic system, particularly at low damping levels, the neighborhoods of resonant frequencies should be avoided and, hence, should be specified by low vibration limits in the resonant regions. Furthermore, the duration of vibration exposure should be explicitly accounted for in specifications. For example, Figure 32.3 presents a ride comfort specification for a ground transit vehicle, where lower vibration levels are specified for longer trips. Finally, it should be noted that the specifications we are concerned with in the present context of design and control are upper bounds of vibration. The system should perform below (within) these specifications under normal operating conditions. Test specifications are lower bounds. The test should be conducted at or above these vibration levels so that the system would meet the test specifications. Some considerations of vibration engineering are summarized in Box 32.1.

32.3

Vibration Isolation

The purpose of vibration isolation is to “isolate” the system of interest from vibration excitations by introducing an isolator in between them. Examples of isolators are machine mounts and vehicle suspension systems. Two general types of isolation can be identified: 1. Force isolation (related to force transmissibility) 2. Motion isolation (related to motion transmissibility) In force isolation, vibration forces that would be ordinarily transmitted directly from a source to a supporting structure (isolated system) are filtered out by an isolator through its flexibility (spring) and dissipation (damping) so that part of the force is routed through an inertial path. Clearly, the concepts of force transmissibility are applicable here. In motion isolation, vibration motions that are applied at a moving platform of a mechanical system (isolated system) are absorbed by an isolator through its flexibility and dissipation so that the motion that is transmitted to the system of interest is weakened.

32 6


Box 32.1 VIBRATION ENGINEERING Vibration mitigation approaches: *

*

*

Isolation (buffers system from excitation) Design modification (modifies the system) Control (senses vibration and applies a counteracting force: passive/active)

Vibration specification: *

*

Peak and RMS values Frequency domain specs on a nomograph p Vibration levels p Frequency content p Exposure duration

Note: lVelocityl ¼ v £ lDisplacementl lAccelerationl ¼ v £ lVelocityl Limiting specifications: Operation (design) specifications: specify upper bounds Testing specifications: specify lower bounds

The concepts of motion transmissibility are applicable in this case. The design problem in both cases is to select applicable parameters for the isolator so that the vibrations entering the system are below specified values within a frequency band of interest (the operating frequency range). Let us revisit the main concepts of force transmissibility and motion transmissibility. Figure 32.4(a) gives a schematic model of force transmissibility through an isolator. Vibration force at the source is f ðtÞ: In view of the isolator, the source system (with impedance Zm ) is made to move at the same speed as the isolator (with impedance Zs ). This is a parallel connection of impedances. Hence, the force f ðtÞ is split so that part of it is taken up by the inertial path (broken line) of Zm : Only the remainder ð fs Þ is transmitted through Zs to the supporting structure, which is the isolated system. Force transmissibility is Tf ¼

fs Zs ¼ f Zm þ Zs

ð32:2Þ

Figure 32.4(b) gives a schematic model of motion transmissibility through an isolator. Vibration motion vðtÞ of the source is applied through an isolator (with impedance Zs and mobility Ms ) to the isolated system (with impedance Zm and mobility Mm ). The resulting force is assumed to transmit directly from the isolator to the isolated system and hence, these two units are connected in series. Consequently, we have the motion transmissibility: Tm ¼

vm Mm Zs ¼ ¼ v Mm þ Ms Zs þ Zm

ð32:3Þ

It can be seen that, according to these two models, we have Tf ¼ Tm

ð32:4Þ


32 7

Generated Vibration Force f(t) Inertial System (Source)

Zm

Mm = Zs

Isolator fs

(a)

Transmitted Vibration Force

Ms =

Fixed Supporting Structure (Isolated System)

Inertial System (Isolated System)

1 Zm

1 Zs

(b)

Received Vibration Motion vm Applied Vibration Motion v(t)

Isolator Moving Platform

f (t) m

vm m b

k

k (c)

FIGURE 32.4

fs

b

v(t)

(d)

(a) Force isolation; (b) motion isolation; (c) force isolation example; (d) motion isolation example.

As a result, the concepts of force transmissibility and motion transmissibility may be studied using just one common transmissibility function T: Simple examples of force isolation and motion isolation are shown in Figure 32.4(c) and (d). For both cases, the transmissibility function is given by T¼

ðk

k þ bjv mv2 þ bjvÞ

ð32:5Þ

where v is the frequency of vibration excitation. Note that the model (Equation 32.5) is not restricted to sinusoidal vibrations. Any general vibration excitation may be represented by a Fourier spectrum, which is a function of frequency v: Then, the response vibration spectrum is obtained by multiplying the excitation spectrum by the transmissibility function T: The associated design problem is to select the isolator parameters k and b to meet the specifications of isolation. Equation 32.5 may be expressed as T¼

ðv2n

v2n þ 2zvn vj v2 þ 2zvn vjÞ

where pffiffiffiffiffi vn ¼ k=m ¼ undamped natural frequency of the system b z ¼ pffiffiffiffi ¼ damping ratio of the system 2 km

ð32:6Þ

32 8


Equation 32.6 may be written in the nondimensional form: T¼

1

1 þ 2zrj r2 þ 2zrj

ð32:7Þ

where the nondimensional excitation frequency is defined as r ¼ v=vn The transmissibility function has a phase angle as well as magnitude. In practical applications, the level of attenuation of the vibration excitation (rather than the phase difference between the vibration excitation and the response) is of primary importance. Accordingly, the transmissibility magnitude sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4z2 r2 lTl ¼ ð32:8Þ ð1 r2 Þ2 þ 4z2 r2 pffiffi is of interest. It can be shown that lTl , 1 for r . 2; which corresponds to the isolation region. Hence, pffiffi the isolator should be designed such that the operative frequencies v are greater than 2vn : Furthermore, a threshold value for lTl would be specified, and the parameters k and b of the isolator should be chosen so that lTl is less than the specified threshold in the operating frequency range (which should be given). This procedure may be illustrated using an example.

Example 32.1 A machine tool and its supporting structure are modeled as the simple mass spring damper system shown in Figure 32.5.

f(t) Machine Tool Inertia

1. Draw a mechanical impedance circuit for m this system in terms of the impedances of the three elements: mass (m), spring (k), and viscous damper (b). 2. Determine the exact value of the frequency ratio r in terms of the damping ratio z; at b k Flexibility and which the force transmissibility magnitude Dissipation will peak. Show that for small z; this value is r ¼ 1: 3. Plot lTf l vs. r for the interval r ¼ ½0; 5 ; with one curve for each of the five z values 0.0, 0.3, 0.7, 1.0, and 2.0, on the same plane. Discuss the behavior of these transmissi Supporting bility curves. Structure 4. From part (3), determine for each of the five fs z values and the excitation frequency range with respect to vn ; for which the transmis FIGURE 32.5 A simplified model of a machine tool sibility magnitude is: and its supporting structure. Less than 1.05 Less than 0.5 5. Suppose that the device in Figure 32.5 has a primary, undamped natural frequency of 6 Hz and a damping ratio of 0.2. It is necessary that the system has a force transmissibility magnitude of less than 0.5 for operating frequency values greater than 12 Hz. Does the existing system meet this requirement? If not, explain how you should modify the system to meet the requirement. *

*


32 9

Solution 1. Here, the elements m; b, and f are in parallel with a common velocity v across them, as shown in Figure 32.6. In the circuit, Zm ¼ mjv; Zb ¼ b; and Zk ¼ k=jv: Force transmissibility Fs F =V Zs ¼ s ¼ F Zs þ Z0 F=V Zb þ Zk ¼ Zm þ Zb þ Zk

v fs

f−fs Zm

Zb

f(t) Zk 0

Tf ¼

ðiÞ

FIGURE 32.6 The mechanical impedance circuit of the force isolation problem.

Substitute the element impedances. We obtain bþ Tf ¼

k jv

k mjv þ b þ jv

bjv þ k ¼ v2 m þ bjv þ k

¼

jvb=m þ k=m v2 þ jvb=m þ k=m

ðiiÞ

The last expression is obtained by dividing the numerator and the denominator by m: Now, use the fact that k ¼ v2n m

and

b ¼ 2zvn m

and divide Equation ii throughout by v2n : We obtain Tf ¼ The transmissibility magnitude is

1 þ 2zrj v2n þ 2zvn jv ¼ v2n v2 þ 2zvn jv 1 r2 þ 2zrj

ðiiiÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4z 2 r 2 lTf l ¼ ð1 r 2 Þ2 þ 4z2 r2

ð32:9Þ

where r ¼ v=vn is the normalized frequency. 2. To determine the peak point of lTf l, differentiate the expression within the square root sign in Equation iv and equate to zero: ½ð1 Hence,

r2 Þ2 þ 2z2 r2 8z2 r ½ð1

4r{½ð1

½1 þ 4z2 r2 ½2ð1 r 2 Þ 2 þ 4z 2 r 2 2

r2 Þð 2rÞ þ 8z2 r

r 2 Þ2 þ 2z2 r2 2z2 þ ½1 þ 4z2 r2 ½ð1

r2 Þ

2z 2

¼0

}¼0

which simplifies to rð2z2 r4 þ r2 The roots are

1Þ ¼ 0

pffiffiffiffiffiffiffiffiffi 1 ^ 1 þ 8z 2 r ¼ 0 and r ¼ 4z 2 The root r ¼ 0 corresponds to the initial stationary point at zero frequency. That does not represent a peak. Taking only the positive root for r 2 and then its positive square root, the peak point of the transmissibility magnitude is given by hpffiffiffiffiffiffiffiffiffi i1=2 1 þ 8z2 1 r¼ ðivÞ 2z 2

32 10


For small z; Taylor series expansion gives qffiffiffiffiffiffiffiffiffi 1 1 þ 8z2 < 1 þ £ 8z2 ¼ 1 þ 4z2 2 With this approximation, Equation iv equates to 1. Hence, for small damping, the transmissibility magnitude will have a peak at r ¼ 1; and from Equation 32.9, its value is 1 pffiffiffiffiffiffiffiffiffi 1 þ £ 4z2 1 þ 4z 2 2 lTf l < < 2z 2z or lTf l
>” as illustrated next. 32A.2.2

Arithmetic

An example of a simple computation using MATLAB is given below: >> x ¼ 2; y ¼ 3; >> z ¼ x^2 x p y þ 4 z¼ 14 In the first line, we have assigned values 2 and 3 to two variables x and y: In the next line, the value of an algebraic function of these two variables is indicated. Then, MATLAB provides the answer as 14. Note that if you place a “;” at the end of the line, the answer will not be printed/displayed. Table 32A.1 gives the symbols for common arithmetic operations used in MATLAB. Following example shows the solution of the quadratic equation ax2 þ bx þ c ¼ 0: >> a ¼ 2; b ¼ 3; c ¼ 4; >> x ¼ ( b þ sqrt(b^2 x¼ 0.7500 þ 1.1990i

4 p a p c)) (2p a)

pffiffiffiffi The answer is complex, where i denotes 1: Note that the function sqrt( ) is used, which provides the positive root only. Some useful mathematical functions are given in Table 32A.2. 32A.2.3

Arrays

An array may be specified by giving the start value, increment, and the end value limit. An example is given below. >> x ¼ (0.9: x¼

0.1:0.42)

0:9000 0:8000

0:7000

0:6000 0:5000

TABLE 32A.1

MATLAB Arithmetic Operations

Symbol

Operation

þ 2 p / ^

Addition Subtraction Multiplication Division Power

Vibration Design and Control TABLE 32A.2

32 75 Useful Mathematical Functions in MATLAB

Function

Description

abs( ) acos( ) acosh( ) asin( ) atan( ) cos( ) cosh( ) exp( ) imag( ) log( ) log10( ) real( ) sign( ) sin( ) sqrt( ) tan( )

Absolute value/magnitude Arc cosine (inverse cosine) Arc hyperbolic cosine Arc sine Arc tan Cosine Hyperbolic cosine Exponential function Imaginary part of a complex number Natural logarithm Log to base 10 (common log) Real part of a complex number Signum function Sine Positive square root Tan function

Note that MATLAB is case sensitive.

The entire array may be manipulated. For example, all the elements are multiplied by p as below: >> x ¼ x p pi x¼ 2:8274 2:5133 2:1991

1:8850 1:5708

The second and the fifth elements are obtained by >> x([2 5]) ans ¼ 2:5133 1:5708 Next, we form a new array y using x; and then plot the two arrays, as shown in Figure 32A.1: >> y ¼ sin(x); >> plot(x,y) A polynomial may be represented as an array of its coefficients. For example, the quadratic equation ax2 þ bx þ c ¼ 0 as given before, with a ¼ 2; b ¼ 3; and c ¼ 4; may be solved using the function 1 “roots” as below: 0.9 >> p ¼ ½ 2 3 4 ; >> roots(p) ans ¼

0:7500 þ 1:1990i 0:7500 1:1990i The answer is the same as we obtained before. 32A.2.4

Relational and Logical Operations

Useful relational operations in MATLAB are given in Table 32A.3. Basic logical operations are given in Table 32A.4.

0.8 0.7 0.6 0.5 0.4 0.3 1.5

2

FIGURE 32A.1

2.5 A plot using MATLAB.

3

32 76

Vibration and Shock Handbook TABLE 32A.3

Some Relational Operations

Operator

Description

, ,¼ . .¼ ¼¼ ,¼

Less than Less than or equal to Greater than Greater than or equal to Equal to Not equal to

TABLE 32A.4

Basic Logical Operations

Operator

Description

& l ,

AND OR NOT

Consider the following example: >> x ¼ (0:0.25:1) p pi x¼ 0 0:7854 1:5708 2:3562 >> cos(x) . 0 ans ¼ 1 1 1 0 0 >> (cos(x) . 0)&(sin(x) . 0) ans ¼ 0 1 1 0 0

3:1416

In this example, first an array is computed. Then the cosine of each element is computed. Next it is checked whether the elements are positive. (A truth value of 1 is sent out if true and a truth value of 0 if false.) Finally, the “AND” operation is used to check whether both corresponding elements of two arrays are positive. 32A.2.5

Linear Algebra

MATLAB can perform various computations with vectors and matrices (see Appendix 3A and Appendix 6A). Some basic illustrations are given here. A vector or a matrix may be specified by assigning values to its elements. Consider the following example: >> b ¼ ½ 1:5 2 ; >> A ¼ ½ 2 1; 1 1 ; >> b ¼ b0 b¼ 1:5000 2:0000 >> x ¼ inv(A) p b x¼ 1:1667 0:8333 In this example, first a second order row vector and 2 £ 2 matrix are defined. The row vector is transposed to get a column vector. Finally the matrix vector equation Ax ¼ b is solved according to x ¼ A 1 b: The determinant and the eigenvalues of A are determined by


32 77

TABLE 32A.5

Some Matrix Operations in MATLAB

Operation

Description

þ 2 p / ^ ‘

Addition Subtraction Multiplication Division Power Transpose

TABLE 32A.6

Useful Matrix Functions in MATLAB

Function

Description

det( ) inv( ) eig( ) [,] ¼ eig( )

Determinant Inverse Eigenvalues Eigenvectors and eigenvalues

>> det(A) ans ¼ 3 >> eig(A) ans ¼ 1:5000 þ 0:8660i 1:5000 0:8660i Both eigenvectors and eigenvalues of A computed as >>[V,P] ¼ eig(A) V¼ 0:7071 0:3536 þ 0:6124i P¼

0:7071 0:3536 0:6124i

1:5000 þ 0:8660i 0

0 1:5000

0:8660i

Here, the symbol V is used to denote the matrix of eigenvectors. The symbol P is used to denote the diagonal matrix whose diagonal elements are the eigenvalues. Useful matrix operations in MATLAB are given in Table 32A.5 and several matrix functions are given in Table 32A.6. 32A.2.6

M-Files

The MATLAB commands have to be keyed in on the command window, one by one. When several commands are needed to carry out a task, the required effort can be tedious. Instead, the necessary commands can be placed in a text file, edited as appropriate (using text editor), which MATLAB can use to execute the complete task. Such a file is called an M file. The file name must have the extension “m” in the form filename.m. A toolbox is a collection of such files, for use in a particular application area (e.g., control systems, fuzzy logic). Then, by keying in the M file name at the MATLAB command prompt, the file will be executed. The necessary data values for executing the file have to be assigned beforehand.

32A.3

Control Systems Toolbox

There are several toolboxes with MATLAB, which can be used to analyze, compute, simulate, and design control problems. Both time domain representations and frequency domain representations can be

32 78


used. Also, both classical and modern control problems can be handled. The application is illustrated here through several control problems. 32A.3.1

MATLAB Modern Control Examples

Several examples in modern control engineering are given now to illustrate the use of MATLAB in control. 32A.3.1.1 Pole Placement of a Third-Order Plant A mechanical plant is given by the input output differential equation ffl x þ x€ ¼ u; where u is the input and x is the output. Determine a feedback law that will yield approximately a simple oscillator with a pffiffi damped natural frequency of 1 unit and a damping ratio of 1= 2: To solve this problem, first we define the state variables as x1 ¼ x; x2 ¼ x_ 1 ; and x3 ¼ x_ 2 : The corresponding state space model is 2 3 2 3 x_ 1 x1 0 1 0 0 6 7 6 7 7 6 7 x_ ¼ 6 4 x_ 2 5 ¼ 00 00 11 4 x2 5 þ 01 u |fflfflfflffl{zfflfflffl} |{z} x_ 3 x3 B A y ¼ ½1 0 0 |fflffl{zfflffl}

Cx

The open loop poles and zeros are obtained using the following MATLAB commands: >> A ¼ ½ 0 1 0; 0 0 1; 0 0 1 ; >> B ¼ ½ 0; 0; 1 ; >> C ¼ ½1 0 0 ; >> D ¼ [0]; >> sys open ¼ ss(A,B,C,D); >> [nat freq open,damping open,poles open] ¼ damp(sys open) >> pzmap(sys open) The open loop poles are: ½0 0 1 T: The step response of the open loop system is obtained using the command: >> step(sys open) Step Response

45 35

0.05

30

0.04

Amplitude

Amplitude

40

25 20 15 10

0.03 0.02 0.01

5 0

Step Response

0.06

0

2

4

6

8

10

0

0

2

4

6

FIGURE 32A.2 (a) Step response of the open loop system; (b) step response of the third order system with pole placement control.


32 79

The result is shown in Figure 32A.2(a). Clearly, the system is unstable. pffiffi With the desired damped natural pffiffi frequency vd ¼ 1 and damping ratio z ¼ 1= 2; we get the undamped natural frequency vn ¼ 2 and, hence, zvn ¼ 1: It follows that we need to place two poles at 1 ^ j: Also the third pole has to be far from these two on the left half plane (LHP); say, at 10: The corresponding control gain K can be computed using the “place” command in MATLAB: >> p ¼ [ 1 þ j 1 >> K ¼ place(A,B,p) place:ndigits ¼ 15 K¼ 20:0000 22:0000

j

10];

11:0000

The corresponding step response of the closed loop system is shown in Figure 32A.2(b). 32A.3.1.2 Linear Quadratic Regulator for a Third-Order Plant For the third order plant in the previous example, we design a linear quadratic regulator (LQR), which has a state feedback controller, using MATLAB Control Systems Toolbox. The MATLAB command K ¼ lqr(A,B,Q,R) computes the optimal gain matrix K such that the state feedback law u ¼ Kx minimizes the quadratic cost function ð1 J¼ ðxT Qx þ uT RuÞdt 0

The weighting matrices Q and R are chosen to apply the desired weights to the various states and inputs. The MATLAB commands for designing the controller are

The step response of the system with the designed LQR controller is shown in Figure 32A.3. 32A.3.1.3 Modal Analysis Example Consider the two DoF mechanical system shown in Figure 32A.4. We now solve the modal analysis problem using MATLAB, for the numeri cal values

a ¼ 0:5; b ¼ 0:5; m ¼ 1 kg; k ¼ 1 N=m For the given mass matrix M and the stiffness matrix K, the solution steps for the alternative approach of modal analysis are 1. Determine M1=2 and M 1=2 : 2. Solve for the eigenvalues l and the eigenvectors f of M 1=2 KM 1=2 : These eigenvalues are the squares of the natural frequencies of the original system.

Step Response

1.4 1.2 1 Amplitude

>> A ¼ ½ 0 1 0; 0 0 1; 0 0 1 ; >> B ¼ ½ 0; 0; 1 ; >> C ¼ ½ 1 0 0 ; >> D ¼ [0]; >> Q ¼ ½ 2 0 0; 0 2 0; 0 0 2 ; >> R ¼ 2; >> Klqr ¼ lqr(A,B,Q,R) >> lqr closed ¼ ss(A BpKlqr,B,C,D); >> step(lqr closed)

0.8 0.6 0.4 0.2 0

0

2

4

6

8

FIGURE 32A.3 Step response of the third order system with LQR control.

k

Mass 1 m

FIGURE 32A.4

βk

Mass 2 am

A two DoF system.

32 80


3. Determine the modal vectors c of the original system using the transformation c ¼ M The program code is given below: %Modal Analysis Example clear; m ¼ 1.0; k ¼ 1.0; M ¼ ½ m 0; 0 m=2 ; K ¼ ½3=2 p k k=2; k=2 k=2 ; M sqrt ¼ M^0.5; M s inv ¼ inv(M sqrt); lemda ¼ eig(M s inv p K p M s inv); [U,D] ¼ eig(M s inv p K p M s inv); V ¼ M s inv p U; disp(‘Natural frequencies’) fprintf(‘omega1 ¼ %10.3f %14.3f \n’ ,sqrt(lemda(2,1))); fprintf(‘\n omega2 ¼ %10.3f%14.3f \n’ ,sqrt(lemda(1,1))); fprintf(‘\n’) fprintf(‘\nMode shapes \n’) fprintf(‘First mode Second Mode \n’) for i ¼ 1:2 ,V(i,2)/V(1,2),V(i,1)/V(1,1)); fprintf(‘%10.3f %14.3f \n’ end The necessary results are obtained as shown below: >> Natural frequencies omega1 ¼ 1:414 omega2 ¼ 0:707 Mode Shapes First Mode 1.000 21.000

Second Mode 1.000 2.000

1=2

f:

33

Structural Dynamic Modification and Sensitivity Analysis 33.1 Introduction ........................................................................ 33 2 33.2 Structural Dynamic Modification of Finite Element Model .................................................................... 33 2 33.3 Perturbation Method of Vibration Modes ....................... 33 4 First Order Perturbation of Distinct Modes † Second Order Perturbation of Distinct Modes † Numerical Examples

33.4 Design Sensitivity Analysis of Structural Vibration Modes ................................................................. 33 8 Direct Differential Method for Sensitivity Analysis Perturbation Sensitivity Analysis † Numerical Example † Concluding Remarks

†

33.5 High Accuracy Modal Superposition for Sensitivity Analysis of Modes ............................................................... 33 11 Method One

†

Method Two

33.6 Sensitivity of Eigenvectors for Free Free Structures ............................................................................. 33 13 33.7 Matrix Perturbation Theory for Repeated Modes ........... 33 14 Basic Equations † The First Order Perturbation of Eigensolutions † High Accuracy Modal Superposition for the First Order Perturbation of Repeated Modes

33.8 Matrix Perturbation Method for Closely Spaced Eigenvalues .......................................................................... 33 16 Method One of Perturbation Analysis for Close Eigenvalues Method Two of Perturbation Analysis for Close Eigenvalues Concluding Remarks

† †

33.9 Matrix Perturbation Theory for Complex Modes ........... 33 22

Su Huan Chen Jilin University

Basic Equations † Matrix Perturbation Method for Distinct Modes † High Accuracy Modal Superposition for Eigenvector Derivatives † Matrix Perturbation for Repeated Eigenvalues of Nondefective Systems † Matrix Perturbation for Close Eigenvalues of Unsymmetric Matrices

Summary The matrix perturbation theory for structural dynamic modification and sensitivity analysis is presented in this chapter. The theory covers a broad spectrum of subjects, specifically, matrix perturbation of real modes of complex structures and matrix perturbation of complex modes. The contents include nine sections. Section 33.2 provides the preliminaries to matrix perturbation and sensitivity analysis. Section 33.3 presents the matrix perturbation method including first order and second order perturbation. Section 33.4 presents methods for design sensitivity 33 1

33 2


analysis. In Section 33.5, high accuracy modal superposition for sensitivity analysis of modes is given. Section 33.6 presents the sensitivity analysis of eigenvectors for free free structures. In Section 33.7 and Section 33.8, matrix perturbations for repeated modes and closely spaced modes are discussed. In Section 33.9, the matrix perturbation approach for complex modes is presented.

33.1

Introduction

In modern engineering problems, the dynamic design of structures is becoming increasingly important. In order to achieve an optimal design, we repeatedly have to modify the structural parameters and solve the generalized eigenvalue problem. The iterative vibration analysis can be very tedious for large and complex structures. Therefore, it is necessary to seek a fast computation method for sensitivity analysis and reanalysis. The matrix perturbation method is an extremely useful tool for this purpose. The matrix perturbation method is concerned with how the natural frequencies and modal vectors change if small modifications are imposed on the parameters of structures. Engineering problems often involve many small modifications in the structural parameters, such as material property variations, manufacturing errors, iterative design of structural parameters, design sensitivity analysis, random eigenvalue analysis, robustness analysis of control systems, and so on. In this chapter, it is assumed that the reader has an undergraduate knowledge in vibration theory (see Chapter 1 to Chapter 5, and Chapter 14) and a working knowledge in the finite element method (see Chapter 9). The contents of the chapter include the basic preliminaries: vibration equations of the finite element model, eigenvalue problem, modal vectors, orthogonality conditions, modal expansion theorem, and the power series expansion of eigensolutions. The chapter also covers such topics as: the perturbation method for distinct eigenvalues and corresponding eigenvectors; sensitivities of eigenvalues and eigenvectors; the high accuracy modal superposition method for eigenvector derivatives; eigenvector derivatives for free free structures; perturbation method for systems with repeated eigenvalues and close eigenvalues; and perturbation method of the complex modes of systems with real unsymmetric matrices.

33.2

Structural Dynamic Modification of Finite Element Model

The finite element method is an important tool to obtain numerical and computational solutions to problems in structural vibration analysis. By applying the finite element method to a structure, a discrete analysis model to idealize the continuum can be obtained. The finite equation of vibrations of a structure in the global coordinate system is Mq€ þ C_q þ Kq ¼ Q

ð33:1Þ

where M; K; and C are the mass, stiffness, and damping matrices, respectively, q€ ; q_ ; and q are the acceleration, velocity, and displacement vectors, respectively, and Q is the external load vector. Neglecting the damping force and external load vector, Equation 33.1 becomes Mq€ þ Kq ¼ 0

ð33:2Þ

This is the natural vibration equation for the structure. Its solution (the natural vibration) is harmonic (see Chapter 30), and is given by q ¼ u cosðvt

wÞ

ð33:3Þ

where u is modal vector, and v the natural frequency of the system. Substituting Equation 33.3 into Equation 33.2, the eigenproblem of structural vibration can be obtained as Ku ¼ lMu where l ðl ¼ v2 Þ denote the eigenvalues of the system.

ð33:4Þ

Structural Dynamic Modification and Sensitivity Analysis

33 3

In structural vibration analysis, the natural frequencies and the corresponding modal vectors can be obtained by solving the eigenproblem (Equation 33.4). The solutions for n eigenvalues and corresponding eigenvectors satisfy KU ¼ MUL

ð33:5Þ

where U; which is called the modal matrix, is an ðn £ nÞ matrix with its columns equal to the n eigenvectors, and L is an ðn £ nÞ diagonal matrix consisting of the corresponding eigenvalues as the diagonal elements; specifically U ¼ ½u1 ; u2 ; …; un

ð33:6Þ

L ¼ diagðli Þ; i ¼ 1; 2; …; n

ð33:7Þ

An important relation for eigenvectors is that of M orthogonality and K orthogonality; that is, we have uTi Muj ¼ dij

ð33:8Þ

uTi Kuj ¼ li dij

ð33:9Þ

where dij is the Kronecker delta. For n eigenpairs, Equation 33.8 and Equation 33.9 can be written as UT MU ¼ I

ð33:10Þ

UT KU ¼ L

ð33:11Þ

Since the modal vectors are independent, an arbitrary displacement vector, u; can be expressed as a linear combination of ui ; i ¼ 1; 2; …; n; that is u¼

n X r¼1

cr ur ¼ UC

ð33:12Þ

where cr is a constant. Each constant cr can be determined by cr ¼ uTr Mu; r ¼ 1; 2; …; n

ð13:13Þ

This is known as the expansion theorem. Suppose the physical parameter of a given structure is given a small modification. This will cause a small change in the matrices K0 and M0 ; that is M ¼ M0 þ 1M1 ; K ¼ K0 þ 1K1

ð33:14Þ

where 1 is a small parameter, K0 and M0 are the original mass and stiffness matrices, respectively, and 1M1 and 1K1 are the corresponding modifications. It is obvious that if M0 and K0 are symmetric, the matrices M1 and K1 are also symmetric. If 1M1 and 1K1 are small, the changes of eigenvalues and eigenvectors of the structure are also small. According to the matrix perturbation theory, the eigensolutions of Equation 33.4 can be expressed in the form of a power series in 1; thus ui ¼ u0i þ 1u1i þ 12 u2i þ · · ·

ð33:15Þ

li ¼ l0i þ 1l1i þ 12 l2i þ · · ·

ð33:16Þ

where u0i and l0i are the eigensolutions of the original structure, l1i and l2i are the first and the second order perturbations of the eigenvalues, and u1i and u2i are the first and the second order perturbation of the eigenvectors. Since the eigensolutions of the original structure, u0i and l0i ; are known, only the first and the second order perturbations of the eigensolutions are required without solving Equation 33.4.

33 4

33.3


Perturbation Method of Vibration Modes

The perturbation methods of vibration modes are well developed (Fox and Kapoor, 1968; Rogers, 1977; Chen and Wada, 1979; Hu, 1987; Chen, 1993). In this section, it is assumed that all eigenvalues of the original structure are distinct.

33.3.1

First-Order Perturbation of Distinct Modes

According to the expansion theorem, the first order perturbation, u1i ; can be expanded by the modal vectors, u0s ; of the original structure as u1i ¼

n X s¼1

c1s u0s

ð33:17Þ

where c1s ¼

1

l0i

l0s

ðuT0s K1 u0i

c1i ¼

1 2

l0i uT0s M1 u0i Þ; s – i

uT0i M1 u0i

ð33:18Þ ð33:19Þ

The first order perturbation of the eigenvalues is

l1i ¼ uT0i K1 u0i

33.3.2

l0i uT0i M1 u0i

ð33:20Þ

Second-Order Perturbation of Distinct Modes

If the parameter modification is fairly large, in order to obtain high computing accuracy, the second order perturbation must be used. According to the expansion theorem, the second order perturbation, u2i ; can be expanded by the modal vectors, u0s ; of the original structure as u2i ¼

n X s¼1

c2j u0s

ð33:21Þ

where c2s ¼

1

l0i

l0s

ðuT0s K1 u1i

l0i uT0s M1 u1i

c2s ¼

l1i uT0s M0 u1i

l1i uT0s M1 u0i Þ; s – i

1 T ðu M u þ uT0i M1 u1i þ uT1i M1 u0i Þ 2 1i 0 1i

ð33:22Þ ð33:23Þ

The second perturbation of the eigenvalues is

l2i ¼ uT0i K1 u1i

33.3.3

l0i uT0i M1 u1i

l1i uT0i M0 u1i

l1i uT0i M1 u0i

ð33:24Þ

Numerical Examples

As illustrations of the matrix perturbation method, several numerical examples are given now.

Example 33.1 Consider the five degree of freedom (five DoF) system shown in Figure 33.1. The physical parameters are given as m1 ¼ m2 ¼ m3 ¼ m4 ¼ 1:0; m5 ¼ 0:5; k1 ¼ k2 ¼ k3 ¼ k4 ¼ k5 ¼ 1:0


K1

K2

K3

m1

K4

m2

FIGURE 33.1

33 5

K5

m3

m4

m5

Mass spring system for Example 33.1.

In order to study the computing accuracy of first and second order perturbations, let us assume that the fifth mass undergoes a decrement of 5 to 30%, and the stiffness of the first spring undergoes a decrement of 5 to 30%. The computed results for the natural frequencies are presented in Table 33.1, in which the initial solutions mean the eigensolutions of the original structure. TABLE 33.1 Mode Number

Comparison of Natural Frequencies Changes of Structural Parameter (%) 5

10

15

20

25

30

1

A B C D C E C

0.3022 0.3128 3.52 0.3017 0.14 0.3022 0.0033

0.2922 0.3128 7.14 0.2903 0.58 0.2922 0.068

0.2821 0.3128 10.9 0.2783 1.34 0.2827 0.24

0.2724 0.3128 14.38 0.2658 2.43 0.2740 0.58

0.2629 0.312 18.97 0.2527 3.9 0.2660 0.93

0.2536 0.3128 23.3 0.2388 5.81 0.2588 2.08

2

A B C D C E C

0.8788 0.9079 3.31 0.8775 0.15 0.3789 0.0076

0.8512 0.98079 6.66 0.8460 0.61 0.8518 0.062

0.8249 0.9079 10.1 0.8133 1.41 0.8267 0.21

0.7998 0.98079 13.52 0.7792 2.58 0.8089 0.52

0.7756 0.9079 17.06 0.7435 4.14 0.7835 0.91

0.7523 0.98079 20.68 0.7060 6.15 0.7659 1.80

3

A B C D C E C

1.3732 1.1421 2.99 1.371 0.15 1.3733 0.0074

1.3348 1.1421 5.94 1.3266 0.62 1.3356 0.060

1.2989 1.1421 8.88 1.2806 1.41 1.3015 0.20

1.2650 1.1421 11.79 1.2328 2.54 1.2712 0.49

1.2332 1.1421 14.7 1.1832 4.05 1.2449 0.74

1.2031 1.1421 17.5 1.1313 5.96 1.2231 1.66

4

A B C D C E C

1.7355 1.7820 2.68 1.7331 0.14 1.7356 0.0070

1.6923 1.7820 5.30 1.6828 0.56 1.6933 0.057

1.6520 1.7820 7.78 1.6319 1.28 1.6552 0.19

1.6143 1.7820 10.38 1.5773 2.29 1.6216 0.45

1.5790 1.7820 12.85 1.5209 3.62 1.5830 0.68

1.5457 15.28 1.5209 5.27 1.5695 1.53

A B C D C E C

1.9273 1.9753 2.49 1.9248 0.1300 1.9274 0.0064

1.8825 1.9753 4.93 1.8929 0.51 1.8835 0.051

1.8408 1.9753 7.31 1.896 1.15 1.8439 0.17

1.8017 1.9753 9.63 1.7696 2.06 1.8090 0.41

1.7649 1.9753 11.92 1.7079 3.23 1.7790 0.83

1.7304 1.9753 14.16 1.6492 4.69 1.7541 1.37

5

33 6


TABLE 33.2

Comparison of Natural Frequencies

Mode No.

A (Hz)

B (Hz)

F (%)

D (Hz)

G (%)

E (Hz)

H (%)

1 2 3 4 5 6

27.78 109.1 157.4 230.5 320.7 391.1

25.45 107.7 153.2 233.2 325.6 393.7

8.39 1.28 2.67 1.17 1.53 0.66

29.27 110.39 159.53 231.24 320.58 392.06

5.36 1.18 1.35 0.32 0.04 0.24

28.14 109.35 158.79 231.02 320.89 319.35

1.29 0.28 0.88 0.022 0.04 0.06

As can be seen from the results, if the change of the structural parameter is 15%, the average change of the natural frequencies is 9.0%. Using the first order perturbation, the average error of the frequencies is reduced to 1.32%. If the change of the structural parameter is 30%, the average error of the natural frequencies is 18%. Using the first order perturbation, the average error of natural frequencies is reduced to 1.6%. The notation used in Table 33.1 and Table 33.2 is as follows: *

*

*

*

*

*

*

*

A: the exact solutions of the modified structure B: the initial solutions of the original structure C: percent error D: the first order perturbation solutions E: the second order perturbation solutions F: the percent errors of the initial solutions G: the percent errors of the first order perturbation H: the percent errors of the second perturbation

Example 33.2 Consider a truss structure (as shown in Figure 33.2) with 20 rods. The cross section area of the second rod is changed from 1:0 to 2:0 cm2. The results calculated are listed in Table 33.2.

5

2 1

6

4 3

Example 33.3 Consider a torsional vibration system with five disks, as shown in Figure 33.3. The physical parameters of the system are as follows: *

*

*

*

*

*

*

*

*

I1 ¼ 10:78 kg cm sec2 I2 ¼ 82:82 kg cm sec2 I3 ¼ 14:27 kg cm sec2 I4 ¼ 29:56 kg cm sec2 I5 ¼ 21:66 kg cm sec2 K1 ¼ 10:48 £ 104 kg cm=rad K2 ¼ 34:30 £ 104 kg cm=rad K3 ¼ 24:40 £ 104 kg cm=rad K4 ¼ 40:60 £ 104 kg cm=rad

The corresponding constrained system is shown in Figure 33.3b, in which the hung stiffness is Ks ¼ 4060 kg cm=rad: The exact eigensolutions of the constrained system are taken as the initial results. Using matrix

7

FIGURE 33.2

Truss structure for Example 33.2.

I1

I2 K1

I3 K2

I4 K3

I5 K4

(a) K5

(b) FIGURE 33.3 Example 33.3.

Torsional

vibration

system

for

Structural Dynamic Modification and Sensitivity Analysis TABLE 33.3

33 7

Comparison for Natural Frequencies No. 1

v0 vx d 1v1 v 00 d0

2

0.000000 1.595950

62.554934 62.604815 0.079700 20.049895 62.554920 2.24 £ 10

21.525707 0.080243

TABLE 33.4

3 105.668169 105.669360 0.001070 20.001132 105.448168 9.46 £ 10

5

7

177.680405 177.704224 0.013406 20.023834 177.680390 8.22 £ 10

5

8

224.361444 224.386235 0.011000 20.002508 224.383727 9.93 £ 10

5

Comparison for Eigenvectors ui0

No.

4

uix

1ui1

u0 i0

1

0.079283 0.079283 0.079283 0.079283 0.079283

0.097363 0.079342 0.079287 0.079198 0.079189

20.000080 20.000059 20.000004 0.000085 0.000153

0.079823 0.079283 0.079283 0.079283 0.079342

2

20.095647 20.057148 0.008612 0.099081 0.125223

20.095611 20.057064 0.008719 0.099192 0.125259

0.000036 20.000084 20.000107 20.000111 0.000036

20.095647 20.057148 0.008612 0.099081 0.125223

3

0.277894 20.041278 20.027509 0.009810 0.024263

0.277884 20.041284 20.027497 0.009840 0.024279

0.000010 0.000005 20.000013 20.000030 20.000016

0.277894 20.41279 20.027510 0.009810 0.024263

4

20.009038 0.020312 20.125560 20.098791 0.144375

20.009033 20.020309 20.125580 0.098735 0.144413

20.000005 0.000002 0.000024 20.000053 20.000038

20.009038 0.020312 20.125556 20.098788 0.144375

5

0.004823 20.020156 0.217244 20.088717 0.052618

0.004823 20.020154 0.217227 20.088725 0.052652

0.000001 20.0000002 0.000018 0.000008 20.000034

0.004824 20.020156 0.217245 20.088717 0.052618

perturbation, the eigensolutions of the free free system can be obtained. The results are listed in Table 33.3 and Table 33.4. The notation used in Table 33.3 is as follows: *

*

*

v0: exact solution of the natural frequency of the free free system (l/sec) v0 0: perturbation solutions of the natural frequency of the free free system (l/sec) vx: natural frequency of the constrained system (l/sec)

1v1 ¼ v0 0

vx the perturbation of the natural frequency d¼ d0 ¼

lv0 lv0

v0 v0

vx l v0 0 l

ð%Þ ð%Þ

33 8


The notation used in Table 33.4 is as follows: *

*

*

*

ui0: exact solution of eigenvectors of the free free system u0 i0: perturbation solution of eigenvectors of the free free system uix: eigenvector of the constrained system 1ui1: first order perturbation of eigenvectors

As can be seen from Table 33.3, the natural frequencies of the free free system are increased by the hung elastic elements. For example, the frequency of the rigid mode is increased to 1.595950 (l/sec), and the frequency of the first elastic mode is increased by 0.8124%. By modifying the eigensolutions with the perturbation method, the frequency of the rigid mode is reduced to 0.079700 (l/sec), which is nearly equal to zero, and all the frequencies of the elastic modes become almost exact solutions. The results in Table 33.4 show that the mode shapes of the free free system, u0 i0 ; are close to the exact solution, ui0 :

33.4

Design Sensitivity Analysis of Structural Vibration Modes

In the optimization of structural analysis, the design sensitivity analysis of eigenvalues and eigenvectors plays an essential role. The designer can use this information directly in an interactive computer aided design procedure as a valuable guide. Significant work has been done in this area (Haug et al., 1985; Adelmen and Haftka, 1986; Chen and Pan, 1986; Wang, 1991).

33.4.1

Direct Differential Method for Sensitivity Analysis

Design sensitivity analysis of eigenvalues and eigenvectors will reveal how the changes in some design parameters in the system affect the dynamic characteristics of the structure. Let li;j and ui;j denote the sensitivity of the eigenvalue, li ; and the eigenvector, ui ; respectively, with respect to the design variables bj ð j ¼ 1; 2; …; LÞ; and let K;j and M;j denote the derivative of the stiffness and mass matrices, respectively, with respect to bj : The design sensitivity of the eigenvalue is

li;j ¼ uTi ðK;j

li M;j Þui

ð33:25Þ

The sensitivity of the eigenvector, ui;j ; can be expressed as the following series: ui;j ¼

n X s¼1

cijs us

ð33:26Þ

where cijs ¼

1

li

ls

uTs ðK;j

li M;j Þui ; i – s; i; s ¼ 1; 2; …; n; i – s ciji ¼

33.4.2

1 T u M u 2 i ;j i

ð33:27Þ ð33:28Þ

Perturbation Sensitivity Analysis

Let DK and DM denote the increments of the stiffness and the mass matrices resulting from an incremental change of the design variable, Dbj ; and let Dli and Dui denote the corresponding perturbations of the eigenvalue and eigenvector, respectively. The direct differential method of design sensitivity analysis of vibration modes can now be put into perturbation form, approximately as

li;j ¼

Dli Dbj

ð33:29Þ

ui;j ¼

Dui Dbj

ð33:30Þ


33 9

where Dli and Dui can be evaluated by the perturbation formulas presented in this chapter. In practical analysis, the design variables could be the cross sectional area of the truss members, bending moment of inertia, equivalent torsional moment of inertia of a beam, the thickness of a plate, or other variable. In some complex structures, a mass, mr ; may be placed at a node point and moving in the direction of the rth DoF, or an elastic support with spring stiffness, Kr ; may be placed at a certain node point. It is also possible that an elastic connector of stiffness, Kj ; might exist between two components. They can also be considered as design variables. In finite element analysis, DK and DM are known to be the sum of the element increments, DKe and DMe ; thus X DK ¼ DKe ð33:31Þ e

DM ¼

X

DMe

e

ð33:32Þ

Hence, the sensitivity formulas of vibration modes as given above can be transformed into the finite element perturbation form (Chen and Pan, 1986)

li;j ¼ and

1 X T u ðDKe Dbj e i

li DMe Þui

ð33:33Þ

1

0 ui;j ¼

n 1 XB 1 BX uT ðDKe B Dbj e @s¼1 li ls s

li DMe Þui us

C 1 T C ui DMe ui ui C A 2

ð33:34Þ

s–i

In these formulas, the overbar signifies that the eigenvector concerned contains only the components needed for the eth finite element. It is important to observe that, in Equation 33.33 and Equation 33.34, calculations are done on the element basis, and as a result, the calculations are greatly simplified. Using the shorthand notations

lei;j ¼ 0 uei;j ¼

1 T e u DK Dbj i

n BX

1 B 1 uT DKe B @ Dbj s¼1 li ls s

li DMe ui

ð33:35Þ 1

li DMe ui us

C 1 T C ui DMe ui ui C A 2

ð33:36Þ

s–i

Equation 33.33 and Equation 33.34 can be written as X e li;j ¼ li;j

ð33:37Þ

e

and ui;j ¼

X e

uei;j

ð33:38Þ

where lei;j and uei;j are the design sensitivity of the eth element for the eigenvalue li and the eigenvector ui ; respectively. Let us consider the following important cases. For a concentrated mass, mr ; placed at a node point and moved in the direction of the rth DoF, Equation 33.33 and Equation 33.34 become

li;r ¼

Dli ¼ Dmr

li u2ir

ð33:39Þ

33 10


and ui;r ¼

n X Dui li u u u ¼ l ks sr ir s Dmr i s¼1

1 2 u u 2 ir i

ð33:40Þ

s–i

where uir is the rth element of the ith eigenvector ui : For an elastic connector with stiffness Kj between the rth and the lth DoF of two components, Equation 33.33 and Equation 33.34 become

li;j ¼

Dli ¼ ðuir Dkj

uil Þ2

ð33:41Þ

and ui;j ¼

n X Dui 1 ¼ ðu u Dkj l ls sr ir i s¼1

usl uir

usr uil þ usl uil Þus

ð33:42Þ

s–i

For an elastic support with spring stiffness Kr placed in the direction of the rth DoF, Equations 33.33 and Equation 33.34 become Dli ¼ u2ir ð33:43Þ li;r ¼ Dkr and ui;r ¼

n X Dui 1 u u u ¼ l ls sr ir s Dkr i s¼1

ð33:44Þ

s–i

33.4.3

Numerical Example

The design sensitivity analysis of an automotive chassis is presented here as an illustration of the method.

Example 33.4 The finite element model of an automobile chassis consists of 39 beam elements involving 30 nodal points and 180 DoF (Figure 33.4). The design variables for the sensitivity analysis of eigenvalues in this example are the equivalent torsional moment of inertia, J; and the bending moment of inertia, Iy ; of the beam element of

FIGURE 33.4

Finite element model of the automotive chassis for Example 33.4.


33 11

Sensitivities of the First Four Chassis Frequencies

1

NE 15 le1J 9.97 NE 24 le1Iy 0.015

11 19 24 6 37 34 36 5.23 5.23 3.64 3.64 3.34 3.32 3.31 6 23 7 22 8 33 21 0.15 £ 1022 0.33 £ 1023 0.33 £ 1023 0.28 £ 1023 0.19 £ 1023 0.11 £ 1023 0.11 £ 1023

2

NE 16 14 11 19 18 12 13 17 le2J 6.67 £ 1022 0.67 £ 1022 0.45 £ 1022 0.45 £ 1022 0.11 £ 1022 0.11 £ 1022 0.53 £ 1023 0.53 £ 1023 NE 38 10 21 9 22 8 24 6 2.92 1.78 1.78 1.49 1.49 1.44 1.44 le2Iy 2.92

3

NE 15 le3J 95.9 NE 10 le3Iy 2.96

19 47.2 38 2.96

11 47.2 9 1.97

37 26.5 21 1.97

31 25.4 8 1.74

30 25.2 22 1.74

32 23.8 6 1.70

1 12.5 24 1.70

4

NE 14 le4J 0.19 NE 10 le4Iy 22.4

16 0.19 38 22.4

19 0.12 39 15.7

11 0.12 20 15.7

12 0.03 5 11.8

18 0.03 25 11.8

17 0.015 6 10.9

13 0.015 24 10.9

the structure. Results for the sensitivity of eigenvalues with respect to J and Iy are given in Table 33.5, in which NE denotes the number of the element. Only the highest eight values are given, and they are listed in descending order. From Table 33.5, it is seen that for this particular chassis, the sensitivities of the first natural frequency, le1Iy ; are much smaller than le1J : This indicates that there is very little effect of the change of bending moment of inertia, Iy ; of the beams on the vibration of the chassis at its first natural frequency. Thus, we can conclude that the first mode is a torsional mode. Similarly, the results indicate that the third mode is also a torsional mode. On the other hand, the second and the fourth modes are recognized to be bending modes. This information is very useful to the designer when deciding on a change in the design. For example, if he wants to increase the first torsional frequency, the efficient way is for him to increase the equivalent torsional moments of inertia of beam elements 15, 11, 19, 24, 6, and so on. It should be noted that only the first low frequency modes are available and can be used as basis vectors of eigenvector derivatives in Equation 33.26. However, modal truncation induces errors, and the errors become significant if more high frequency modes are truncated. An improvement to truncated modal superposition representation of eigenvector derivatives is presented in the next section.

33.4.4

Concluding Remarks

As can be seen from the numerical examples given above, the matrix perturbation method is an extremely useful tool for fast reanalysis of a modified structure. It is widely used in a range of structural modifications, such as the modification of various types of elements, local modification of structures, sensitivity analysis of vibration modes, and so on. Therefore, matrix perturbation plays an important role in dynamic analysis and optimization of structures.

33.5 High-Accuracy Modal Superposition for Sensitivity Analysis of Modes The modal superposition method is often used to compute the derivatives of modal vectors. Because of the cost of generating computer solutions for a dynamic analysis, it is impractical to obtain all modes.

33 12


Therefore, only the first L low frequency modes are computed and are used as basis vectors of eigenvector derivatives. However, as noted above, modal truncation induces errors, which can be significant if more high frequency modes are truncated. An explicit method to improve the truncated modal superposition representation of eigenvector derivatives is presented (Wang, 1991), in which a residual static mode is used to approximate the contribution due to unavailable high frequency modes (method one). In this section a more accurate modal superposition method (method two; Chen, 1993a; Liu and Chen, 1994a) than method one is given. In this method, the contribution of the truncated modes to the eigenvector derivatives is expressed exactly, as a convergent series that can be evaluated by a simple iterative procedure.

33.5.1

Method One

The modal sensitivity can be expressed as ui;j ¼

N X s¼1

cs us ¼

L X j¼1

cj uj þ SR

ð33:45Þ

where SR ¼

N X j¼Lþ1

cj uj

ð33:46Þ

Since li ,, lLþ1 ; Equation 33.46 can be approximated as SR < SRA ¼ H0

W0

ð33:47Þ

where H0 ¼ K 1 ð K;j þ li;j M þ li M;j Þ W0 ¼

33.5.2

L X 1 T uj ð K;j þ li;j M þ li M;j Þuj l j¼1 j

ð33:48Þ ð33:49Þ

Method Two

The contribution of ui;j ; SR due to truncated high frequencies modes is as follows: SR ¼

1 X j¼0

j

li ðHj

Wj Þ

ð33:50Þ

where Wj ¼ U L L L

j 1

UTL ð K;j þ li;j M þ li M;j Þ

UL ¼ ½u1 ; u2 ; …; uL

ð33:51Þ ð33:52Þ

Hj can be obtained with the following iterative procedure:

9 H0 ¼ K 1 ð K;j þ Ki;j M þ li M;j Þ > > > = 0 F j 1 ¼ MHj 1 ; j $ 1 > > > ; Hj ¼ K 1 F0 j 1

ð33:53Þ


33 13

Define SR ðkÞ as SR ðkÞ ¼

k X j¼0

j

li ðHj

Wj Þ

ð33:54Þ

Using this definition, the given iterative process can be terminated if the following inequality kSR ðkÞ

SR ðk

1Þk2 # 1

ð33:55Þ

is satisfied, where 1 is a specified accuracy requirement. It should be noted that, if only the first term in the series (Equation 33.50) is retained with all the other terms neglected, then method two is reduced to method one. In addition, the series (Equation 33.50) can be used to estimate the errors induced by the modal truncation.

33.6

Sensitivity of Eigenvectors for Free–Free Structures

As can be seen from Equation 33.48 and Equation 33.53, both method one and method two fail to deal with the free free structures with rigid body modes because they involve the inversion of the stiffness matrix. However, we can transform the eigenproblem with a singular stiffness matrix into its equivalent eigenproblem with a nonsingular stiffness matrix, in the sense that these two eigenproblems have the same derivatives of eigenvalues and eigenvectors (Liu and Chen, 1994b). Consider the eigenvalue problem Kui ¼ li Mui

ð33:56Þ

mM

ð33:57Þ

where K;K

Here, m is a nonzero scalar parameter and K is nonsingular if m – li ði ¼ 1; 2; …; nÞ It can be shown that

li ¼ li

m

ð33:58Þ

ui ¼ ui ; i ¼ 1; 2; …; N

ð33:59Þ

dli dli ¼ db db

ð33:60Þ

dui dui ¼ db db

ð33:61Þ

and

The derivatives dui =db can be obtained from the derivatives dui =db of the eigenproblem of Equation 33.56, in which K is nonsingular. In this context, both method one and method two, discussed in Section 33.5.1 and Section 33.5.2, can be applied to deal with the free free structures with rigid body modes. To achieve a faster average convergent speed for all the first m eigenvector derivatives, m can be determined as 0 1 8 m > X > > @ lj A > < j¼1 ð33:62Þ m¼ ; j ¼ 1; 2; …; m > > m > > : m – lj ;

33 14

33.7 33.7.1


Matrix Perturbation Theory for Repeated Modes Basic Equations

In this section, let us consider the case of repeated eigenvalues, namely, l0i ¼ l0iþ1 ¼ · · · ¼ l0iþm 1 : The system is known as a degenerate system. In engineering, many complex and large structures, such as airplanes, rockets, tall towers, bridges, and ocean platforms, often have multiple or cluster eigenvalues. The matrix perturbation for the repeated modes is presented in Haug, et al. (1980), Chen and Pan (1986), Hu (1987), Mills Curran (1988), Ojalvo (1988), Dailey (1989), Lim et al. (1989) and Shaw and Jayasuriya (1992). Assume that l0 ¼ l01 ¼ l02 ¼ · · · ¼ l0m ; that is, l0 is a repeated eigenvalue with multiplicity equal to m; and u01 ; u02 ; · · ·; u0m are the eigenvectors associated with l0 : Then, a linear combination of u0j ðj ¼ 1; 2; …; mÞ; denoted as U0 ; will also be the eigenvector associated with l0 : U0 ¼ U0m a

ð33:63Þ

U0m ¼ ½u01 ; u02 ; …; u0m

ð33:64Þ

aT a ¼ I

ð33:65Þ

where

and a ¼ ½a1 ; a2 ; …; am

T

ð33:66Þ

Note that a is a constant matrix to be determined. According to the matrix perturbation method, the eigenvalues and eigenvectors of the structure with repeated eigenvalues for the perturbed structure can be expressed as Lm ¼ L0 þ 1L1

ð33:67Þ

Um ¼ U0m a þ 1ðU0 Cm þ UA CA Þ ¼ U0m a þ 1ðU0m aCm þ UA CA Þ

ð33:68Þ

where UA is the n £ ðn mÞ modal matrix containing all the eigenvectors except U0m ; Lm is the m £ m eigenvalue diagonal matrix of the perturbed structure, L1 is the m £ m diagonal matrix with its diagonal elements equal to the first order perturbations of eigenvalues, Cm is an m £ m matrix to be determined, and CA is an ðn mÞ £ ðn mÞ matrix to be determined.

33.7.2

The First-Order Perturbation of Eigensolutions

L1 and a can be computed from the following ðm £ mÞ eigenproblem: Wa ¼ aL1 ; aT a ¼ I

ð33:69Þ

where W ¼ UT0m ðK1

l0 M1 ÞU0m

ð33:70Þ

Solving the m £ m eigenproblem of Equation 33.69 can produce L1 and a: If matrix W has no repeated eigenvalues, a can be uniquely determined; if matrix W has repeated eigenvalues, a can be determined using the higher order perturbation equations. Here, we assume that matrix W has no repeated eigenvalues; that is, l1i – l1j ; ði – jÞ; where l1k ð0 , k # mÞ are the elements of the diagonal matrix L1 : The matrix CA is CA ¼ ðLA

l0 IÞ 1 UTA ðl0 M1

K1 ÞU0m a

ð33:71Þ


33 15

The elements of Cm are Cm ij ¼

Rij

lð1Þ jm

lð1Þ im

; i – j i; j ¼ 1; 2; …; m

ð33:72Þ

where Rij are the elements of R given by R¼

aT UT0m M1 U0m aL1 þ aT U0m K1 UA CA

l0 aT U0m M1 UA CA

aT UT0m M0 UA CA L1

ð33:73Þ

and 1 Q 2 ii

ð33:74Þ

aT UT0m M1 U0m a

ð33:75Þ

Cm ii ¼ where Q ii is the diagonal elements of Q; given by Q¼

33.7.3 High-Accuracy Modal Superposition for the First-Order Perturbation of Repeated Modes In Section 33.5, the high accuracy modal superposition for the first order perturbation of eigenvectors of distinct eigenvalues is given. In this section, we extend these methods to the situation with repeated modes. 33.7.3.1

Method One for Computing U1

Assuming UAL and LAL are the first L modes and eigenvalues excluding the repeated modes, the first order perturbation of eigenvectors is U1 ¼ U0m aCm þ UAL CAL þ SR SR ¼ US

ð33:76Þ

. . ½U0m ..UAL diagðl0 1 ; LAL1 Þ½U0m ..UAL T T

ð33:77Þ

where US is the static displacement obtained by KUS ¼ T

ð33:78Þ

and T ¼ M0 U0m aL1 þ l0 M1 U0m a

K1 U0m a

ð33:79Þ

In Equation 33.79, L1 and a can be obtained from Equation 33.69. The matrix CAL is given by CAL ¼ ðLAL

l0 IÞ 1 UTAL ðl0 M1

K1 ÞU0m a

ð33:80Þ

; i – j; i; j ¼ 1; 2; …; m

ð33:81Þ

and the elements of matrix Cm are Cm ij ¼

Rij

lð1Þ jm

lð1Þ im

where R is given by R ¼ aT UT0m M1 U0m L1

aT UT0m ðl0 M1

K1 ÞðUAL CAL þ SR Þ

aT UT0m M0 SR L1

ð33:82Þ

The diagonal elements of Cm are Cm ii ¼

1 Q 2 ii

ð33:83Þ

33 16


where Q¼

33.7.3.2

aT UT0m M1 U0m a

aT UT0m M0 SR

STR M0 U0m a

ð33:84Þ

Method Two for Computing U1

The first order perturbation of eigenvectors can be expressed as U1 ¼ U0m aCm þ UAL CAL þ SR

ð33:85Þ

where CAL can also be calculated using Equation 33.80; that is

l0 IÞ 1 UTAL ðl0 M1

CAL ¼ ðLAL

K1 ÞU0m a

and SR is given by SR ¼

1 X j¼0

j

l0 ðHj

Wj Þ

ð33:86Þ

where . . j 1 Wj ¼ ½U0m ..UAL L0 ½U0m ..UAL T T; j $ 0

ð33:87Þ

T ¼ M0 U0m aL1 þ l0 M1 U0m a

ð33:88Þ

K1 U0m a

The iterative method for computing Hj is as follows: H0 ¼ K 1 T; F0 j

1

¼ MHj 1 ; j $ 1

ð33:89Þ

Hj ¼ K 1 F0 j 1 ; This iterative process can be terminated according to the accuracy requirement. If we define SR ðkÞ as SR ðkÞ ¼

k X j¼0

lj0 ðHj

Wj Þ

ð33:90Þ

the termination condition can be stated as kSR ðkÞ

SR ðk

1Þk2 # 1; j ¼ 1; 2; …; m

ð33:91Þ

where 1 is a specified accuracy requirement. The computation method for Cm in Equation 33.85 is similar to that of Equation 33.81 to Equation 33.84. The only difference is that SR in Equation 33.82 and Equation 33.84 can be replaced with SR ðkÞ in Equation 33.90.

33.8

Matrix Perturbation Method for Closely Spaced Eigenvalues

The vibration modes with close frequencies, that is, with clusters of frequencies, often occur in certain structural systems including large space structures, multispan beams, and in some nearly periodic structures and symmetric structures. Therefore, it is important here to present the perturbation method for vibration modes with close eigenvalues (Liu, 2000). The perturbation analysis of close eigenvalues can be transformed into a problem with a repeated eigenvalue, which is equal to the average value of the close eigenvalues (Chen, 1993).


33.8.1

33 17

Method One of Perturbation Analysis for Close Eigenvalues

Consider vibration eigenproblem . . K0 ½U0 ..UA ¼ M0 ½U0 ..UA diagðL0 ; LA Þ

ð33:92Þ

. . ½U0 ..UA T M0 ½U0 ..UA ¼ I

ð33:93Þ

where K0 and M0 are n £ n real symmetric matrices, and L0 and U0 are the m £ m diagonal matrix of close eigenvalues and the corresponding n £ m modal matrix. Using the spectral decomposition of K0 ; the problem can be expressed as K0 ¼ K0 þ 1 dK0

ð33:94Þ

K0 ¼ M0 ðl0 U0 UT0 ÞM0 þ M0 ðUA LA UTA ÞM0

ð33:95Þ

1 dK0 ¼ M0 ðU0 ð1 dL0 ÞUT0 ÞM0

ð33:96Þ

where

0X m 1 dL0 ¼ L0

l0 I ¼ L0

1 l 0i C B B i¼1 C B C B m CI @ A

ð33:97Þ

It can be seen that K0 given by Equation 33.95 satisfies . . K0 ½U0 ..UA ¼ M0 ½U0 ..UA diagðl0 I; LA Þ

ð33:98Þ

. . ½U0 ..UA T M0 ½U0 ..UA ¼ I

ð33:99Þ

This indicates that l0 and U0 are the repeated eigenvalues and the corresponding eigenvector subspace with multiplicity m of the eigenproblem (Equation 33.92), and LA and UA are also the eigensolution of eigenproblem (Equation 33.92). If L0 ! l0 I; 1 dL0 ! 0; and K0 ! K0 ; and if the small parameter modifications 1K1 and 1M1 are introduced to the matrices K0 and M0 ; the eigenproblem with close eigenvalues becomes ðK0 þ 1K1 ÞU ¼ ðM0 þ 1M1 ÞUL

ð33:100Þ

UðM0 þ 1M1 ÞUT ¼ I

ð33:101Þ

Substituting Equation 33.94 into Equation 33.100, we obtain ðK0 þ 1K1 ÞU ¼ ðM0 þ 1M1 ÞUL

ð33:102Þ


ð33:103Þ

1K1 ¼ 1 dK0 þ 1K1

ð33:104Þ

L ¼ l0 I þ 1L1 þ 12 L2 þ · · ·

ð33:105Þ

U ¼ U0 a þ 1U1 þ 12 U2 þ · · ·

ð33:106Þ

where

33 18


Therefore, the eigenproblem of Equation 33.102 and Equation 33.103 can be considered to be a perturbed eigenproblem with the perturbation matrices equal to ðdK0 þ K1 Þ and M1 ; respectively. The eigensolutions, L and U; can be obtained from Equation 33.102 and Equation 33.103 by using the perturbation method for repeated eigenvalues as discussed in Section 33.7. Accordingly, the perturbation problem of modes with close eigenvalues is transformed into one of the repeated eigenvalues. The complete algorithm for L and U is given below. (1) Compute m X

l0 ¼

i¼1

l0i

m

(2) Compute W ¼ UT0 ðdK0 þ K1

l0 M1 ÞU0

(3) Solve the eigenvalue problem Wa ¼ aL1 aT a ¼ I (4) Compute the perturbed eigenvalues of the close eigenvalues L ¼ l0 I þ L1 (5) Compute the new eigenvectors U0 a corresponding to l0 . (6) Compute the matrix CA CA ¼ ðLA

l0 IÞ 1 UTA ðl0 M1

K1

dK0 ÞU0 a

(7) Compute R ¼ ½Rij R¼

aT UT0 M1 U0 aL1

l0 aT UT0 M1 UA CA

aT UT0 ðdK0 þ K1 ÞUA CA

(8) Compute Cm ¼ ½C m ij Cm ij ¼

Rij

l1j

l1i

; i – j; i; j ¼ 1; 2; …; m

Cm ii ¼ Q¼

1 Q 2 ii

aT UT0 M1 U0 a

aUT0 M0 UA CA L1


33 19

(9) Compute the perturbed eigenvectors U U ¼ U0 a þ U0 aCm þ UA CA

33.8.2

Method Two of Perturbation Analysis for Close Eigenvalues

Because of the importance of the problem in both theory and practice, we now present method two of perturbation analysis for close eigenvalues, which is equivalent to method one given above. Using the spectral decomposition of M0 ; the problem can be expressed as M0 ¼ M0 þ 1 dM0

ð33:107Þ

. . K0 ½U0 ..UA ¼ M0 ½U0 ..UA diagðl0 I; LA Þ

ð33:108Þ

. . ½U0 ..UA T M0 ½U0 ..UA ¼ I

ð33:109Þ

M ¼ l0 2 K0 U0 UT0 K0 þ K0 ½UA ðLA 1 Þ2 UTA K0

ð33:110Þ

1 dM0 ¼ K0 ½U0 1 dðL0 1 Þ2 UT0 K0

ð33:111Þ

Then, the following equations hold:

where

2

1 dL0 ¼ L0

2

2

l0 I

ð33:112Þ

and m X

l0 ¼

i¼1

l0i ð33:113Þ

m

It can be seen that M0 and 1 dM0 given by Equation 33.110 and Equation 33.111 satisfy Equation 33.108 and Equation 33.109; that is, l0 and U0 are the repeated eigenvalues and corresponding modal matrix of Equation 33.108 and Equation 33.109. LA and UA are the eigenvalue diagonal matrix and the corresponding modal matrix excluding L0 and U0 ; respectively. If K0 and M0 are modified to K0 þ 1K1 and M0 þ 1M1 ; the eigenvalue problem becomes ðK0 þ 1K1 ÞU ¼ ðM0 þ 1M1 ÞUL

ð33:114Þ


ð33:115Þ

Substituting Equation 33.107 into Equation 33.114 yields ðK0 þ 1K1 ÞU ¼ ðM0 þ 1M1 ÞUL T

ð33:116Þ

UðM0 þ 1M1 ÞU ¼ I

ð33:117Þ

1M1 ¼ 1 dM0 þ 1M1

ð33:118Þ

where Thus, Equation 33.116 and Equation 33.117 can be considered to be a perturbed eigenproblem with repeated eigenvalues, and the perturbation method for repeated eigenvalues can be used to obtain the perturbed eigensolutions of Equation 33.116 and Equation 33.117: L ¼ l0 I þ 1L1 þ · · ·

ð33:119Þ

U ¼ U0 a þ 1U1 þ · · ·

ð33:120Þ

33 20


Example 33.5 For the six DoF mass spring system shown in Figure 33.5, the stiffness and mass matrices K0 and M0 are given by 2 3 1500 1000 6 7 6 1000 1200 7 200 6 7 6 7 6 200 15; 200 5000 5000 5000 7 6 7 7 ðN=mÞ K0 ¼ 6 6 7 6 7 5000 5000 6 7 6 7 6 7 5000 5000 4 5 5000

5000

M0 ¼ diagð200; 300; 50; 20; 20; 20:004Þ (kg) The perturbation eigensolutions are computed for the following three cases: Case 1

2

0

6 60 6 6 60 6 1K1 ¼ 6 6 60 6 6 60 4

0

0

0

0

0

0

0

0

5

0

0

0

0

0

0

5

7 07 7 7 07 7 7 ðN=mÞ 7 07 7 7 07 5

0

0

0

0

0

5

0

0

5

0

3

0

1M1 ¼ diagð0; …; 0Þ (kg)

Case 2

2 6 6 6 6 6 6 1K1 ¼ 6 6 6 6 6 6 4

0

3

5:00

0

0 0

0

0

0

0 0

0

0

0

0 0

0

0

0

0 0

0

0

0

0 0

0

7 07 7 7 07 7 7 ðN=mÞ 7 07 7 7 07 5

0

0

0 0

0

0

1M1 ¼ diagð0; 0; 0; 0; 0; 0:5Þ (kg) Case 3

1K1 ¼ 0; 1M1 ¼ diagð0; 0; 0; 0; 2:0; 0Þ (kg)

The unperturbed eigensolutions have a single pair of close eigenvalues given by L0 ¼ diagð249:966642; 250:000000Þ " UT0

¼

0:00000

0:00000

0:00000

0:00000

0:00012 0:00000

0:091283 0:158114

0:091283 0:158114

The other unperturbed eigensolutions are as follows: LA ¼ diagð0:594885; 2:478725;10:234656;552:175102Þ

0:182560 0:00000

#


UA

m4

2

6 6 6 6 6 6 ¼6 6 6 6 6 6 4

0:058595

0:027542 0:028427

0:000001

m5

3

7 0:027659 0:039259 0:000127 7 7 7 0:006732 0:074593 0:058387 0:104793 7 7 7 7 0:007020 0:075340 0:058527 0:086699 7 7 7 0:007020 0:075340 0:058527 0:086699 7 5

0:032047

33 21

K4

m6

K6

K5

m3 K3

0:007020 0:075340 0:058527 0:086667

m2

K2 The perturbed eigensolutions associated with the single pair of close eigenvalues for the three m1 cases are summarized in Table 33.6. These results show that the perturbation analysis of K1 distinct eigenvalues is not only inaccurate but also misleading when applied to close eigen values, and that the perturbed eigensolutions given by the present method are in good FIGURE 33.5 Six DoF mass spring system for agreement with the exact solutions. Example 33.5. For example, in Case 3, the eigenvalue errors induced by the present method are reduced to 1.047950 and 0.000000, while the errors induced by the perturbation of distinct eigenvalues are 4.174100 and 3.025595. The eigenvectors obtained by the perturbation method of distinct eigenvalues are not only

TABLE 33.6

Comparison of Eigensolutions with Close Eigenvalues Exact

Perturbation Method of Distinct Eigenvalues

Eigenvalues Eigenvectors

249.973872 0.000000 0.000000 0.000018 0.150492 0.014254 20.164753

250.159351 0.000000 0.000000 20.000066 0.103411 20.181965 0.078701


243.878599 0.000000 0.000000 20.000015 20.000622 20.155859 0.156472

247.932109 0.000000 20.000004 0.001506 0.182090 20.091163 20.090082


234.474405 0.000000 0.000016 20.005558 20.089507 0.175420 20.089778

249.975015 0.000000 0.000000 20.000015 20.158189 0.000158 0.158022

Case 1 250.008324 0.000000 0.000000 20.000043 20.433574 0.251058 0.182585 Case 2 244.759777 0.000000 0.000002 20.000772 8.467433 28.648272 0.180521 Case 3 245.800915 0.000000 20.000008 0.003030 34.141638 2 34.321586 0.181586

Perturbation Method of Close Eigenvalues

250.125000 0.000000 0.000000 0.000079 0.039471 0.355699 20.395285

249.973872 0.000000 0.000000 0.000011 0.150501 0.014251 20.164745

250.159451 0.000000 0.000000 0.000005 20.103354 0.182015 20.078658

246.875000 0.000000 0.000005 20.001976 25.098718 24.784467 9.882118

243.725997 0.000000 0.000000 0.000011 0.001103 20.157555 0.158052

247.908461 0.000000 0.000000 0.000006 0.183060 20.092069 20.090988

237.500000 0.000000 0.000021 20.007905 219.920531 219.612210 39.528471

233.326455 0.000000 0.000000 0.000006 20.091153 0.182572 20.091415

249.975015 0.000000 20.000000 0.000010 0.158189 20.000158 20.158025

33 22


inaccurate but also misleading, while good agreement with the exact eigenvectors has been obtained by the present method.

33.8.3

Concluding Remarks

Perturbation analysis of vibration modes with close frequencies is presented in this section. It can be regarded as a general treatment of perturbation analysis, because the perturbation analysis of both distinct eigenvalues and repeated eigenvalues is contained in the present method. The results obtained by this method allow one to analyze the influence of parameter changes in a system on the dynamic characteristics of the system, which is very important for effective structural design.

33.9

Matrix Perturbation Theory for Complex Modes

In Section 33.2 to Section 33.8, the matrix perturbation for real modes of systems with real symmetric mass and stiffness matrices, M and K; was given. However, in many engineering problems such as systems with nonproportional damping (see Chapter 3 and Chapter 19), dynamic systems under nonconservative forces, analysis of aero elastic flutter, and structural vibration control systems, the system matrices are not symmetric and may not be diagonalizable. In this case, the matrix perturbation for real modes cannot be used, and we must use the matrix perturbation for complex modes (Murthy and Haftka, 1988; Chen, 1993; Liu, 1999; Adhikari and Friswell, 2001). In the following, we assume that the system is not defective; that is, the system has a complete eigenvector set to span the eigenspace. The discussion in this chapter is limited to the nondefective systems.

33.9.1

Basic Equations

The vibration equation of a linear system with n DoFs is given by Mq€ þ C_q þ Kq ¼ QðtÞ

ð33:121Þ

where the matrices M; C; and K; are assumed to be real and unsymmetric. The free vibration equation of the system is Mq€ þ C_q þ Kq ¼ 0

ð33:122Þ

The corresponding right eigenvalues problem is ðMs2 þ Cs þ KÞx ¼ 0

ð33:123Þ

and its adjoint eigenvalue problem is ðMs2 þ Cs þ KÞT y ¼ 0 yT ðMs2 þ Cs þ KÞ ¼ 0

ð33:124Þ

It is common in literature to call y the left eigenvector, while x in the original system, a column vector, is called the right eigenvector. Let us introduce a state vector ( ) sx u¼ ¼ Tx ð33:125Þ x where T is the state transformation matrix

( T¼

sI I

) ð33:126Þ


Similarly, we introduce the state vector

( v¼

sy

33 23

) ¼ Ty

y

ð33:127Þ

Hence, Equations 33.123 and Equation 33.124 become ðAs þ BÞu ¼ 0

ð33:128Þ

T

ðAs þ BÞ v ¼ 0

ð33:129Þ

or v T ðAs þ BÞ ¼ 0 where

" A¼ B¼

C

K

#

I 0 " # M 0 0

I

It is well known that the eigenvalues of the adjoint eigenproblem (Equation 33.129) are identical to that of the original eigenproblem (Equation 33.128). The characteristic equation is detðA þ sBÞ ¼ 0 This characteristic determinant is a polynomial of 2n order in s; and 2n eigenvalues si ði ¼ 1; 2; …; 2nÞ can be found in the complex domain. The left and right modal vectors, vi and ui ; corresponding to si satisfy Aui ¼ si Bui

ð33:130Þ

A T v i ¼ si B T v i

ð33:131Þ

vTj Bui ¼ 0

ð33:132Þ

vTj Aui

ð33:133Þ

and The orthogonality conditions are ¼0

The normalization conditions are v Ti Bui ¼ 1 uTi Bui ¼ 1

ð33:134Þ

Therefore, the orthogonality conditions can be written as vTj Bui ¼ dij vTj Aui ¼ si dij

33.9.2

ð33:135Þ

Matrix Perturbation Method for Distinct Modes

If small changes are made on the structural parameters, the mass, damping, and stiffness matrices of the system also have small changes given by M ¼ M0 þ 1M1 C ¼ C0 þ 1C1 K ¼ K0 þ 1K1

ð33:136Þ

33 24


and we have A ¼ A0 þ 1A1

ð33:137Þ

B ¼ B0 þ 1B1

ð33:138Þ

where 1 is a small parameter. In the following, we first consider the case of distinct eigenvalues, s0i ; of the original system. According to the matrix perturbation theory, the eigenvalues and eigenvectors can be expressed as a power series in 1; that is S ¼ S0 þ 1S1 þ 12 S2 þ · · ·

ð33:139Þ

U ¼ U0 þ 1U1 þ 12 U2 þ · · ·

ð33:140Þ

2

V ¼ V0 þ 1V1 þ 1 V2 þ · · ·

ð33:141Þ

where S0 ; U0 ; and V0 are the eigensolutions of the original system; S1 ; U1 ; and V1 are the first order perturbations of eigensolutions; and S2 ; U2 ; and V2 the second order perturbations. U1 can be expressed as a linear combination of the right eigenvectors of the original system as U1 ¼ U0 C1

ð33:142Þ

1

where C is to be the determined matrix given by C 1ij ¼

1 S0j

S0i

Pij1 ; j – i; i; j ¼ 1; 2; …

ð33:143Þ

Also where

Pij1

1 1 S1 ¼ diagðP11 ; P22 ; …Þ

ð33:144Þ

P1 ¼ VT0 ð A1 U0 þ B1 U0 S0 Þ

ð33:145Þ

1

are the elements of P given by

The V1 can be expressed as the expansion of V0 V1 ¼ V0 D1

ð33:146Þ

1

where D is to be the determined coefficient matrix given by D1ij ¼

1 S0j

S0i

R1ij ; j – i; i; j ¼ 1; 2; …

ð33:147Þ

and R1ij are the nondiagonal elements of R1 R1 ¼ UT0 ðBT1 V0 S0

AT1 V0 Þ

ð33:148Þ

If the modification of the parameter is fairly large, the second order perturbation must be used to obtain high accuracy. According to the expansion theorem, the second order perturbation of eigenvectors, U2 ; can be expressed as U2 ¼ U0 C2

ð33:149Þ

In a similar manner, S2 and the elements C 2ij can be obtained as S2 ¼ diagðP 211 ; P 222 ; …Þ C 2ij ¼

1 S0j

S0i

ð33:150Þ

P 2ij ; j – i; i; j ¼ 1; 2; …

ð33:151Þ

where P 2ij are the nondiagonal elements of P2 P2 ¼ VT0 B0 U1 S1 þ VT0 B1 U0 S1 þ VT0 B1 U1 S0

VT0 A1 U1

ð33:152Þ


33 25

V2 can be expressed as V2 ¼ V0 D2

ð33:153Þ

where D2 is to be the determined coefficient matrix given by D2ij ¼

1 S0j

S0i

R2ij ; j – i; i; j ¼ 1; 2; …

ð33:154Þ

and R2ij are the nondiagonal elements of R2 R2 ¼ UT0 ðBT0 V1 S1 þ BT1 V0 S1 þ BT1 V1 S0

AT1 V1 Þ

If j ¼ i; the coefficients C 1ii ; D1ii ; C 2ii ; and D2ii can be computed as 0 C 1ii ¼

uT0i ðB0

where

¼

Q 1ii

C 1ii

ð33:157Þ

is the diagonal element of Q1 VT0 B1 U0

uT1i B0 u1i þ uT0i B1 u1i þ uT1i B1 u0i C 2ii ¼

ð33:158Þ n X j¼1 j–i

Cij uT0j ðB0 þ BT0 Þu0i

uT0i ðB0 þ BT0 Þu0i D2ii ¼ Q 2ii

and

ð33:156Þ

j–i

Q1 ¼

Q 2ii

1

n B C X 1 B T C Cij uT0j ðB0 þ BT0 Þu0i C Bu0i B1 u0i þ T A þ B0 Þu0i @ j¼1

D1ii Q 1ii

ð33:155Þ

C 2ii

ð33:159Þ ð33:160Þ

2

is the diagonal element of Q

Q2 ¼

33.9.3

VT0 B0 U1

VT1 B0 U1

VT1 B1 U0

ð33:161Þ

High-Accuracy Modal Superposition for Eigenvector Derivatives

For a large scale structure, only a small number of the first lower L modes are extracted, and the higher modes are truncated in order to reduce the computational cost. The modal superposition method may not only give inaccurate result, but also may be misleading if the truncation is considerable. In this section, we give a high accuracy modal superposition method for derivatives of the complex mode of nonsymmetric matrices. 33.9.3.1

Improved Modal Superposition

An improved modal superposition (IMS) to reduce the computation errors by modal truncation was proposed (Lim et al., 1989). The derivatives of modes can be expressed as

›ui ¼ aii ui þ zi ›b zi ¼

ð33:162Þ

L L vT F X X vTj Fi j i uj þ A 1 Fi þ u S S Sj j j j¼1 j¼1 i

ð33:163Þ

j–i

Fi ¼

›A ›b

›Si B ›b

Si

›B u ›b i

ð33:164Þ

33 26


where A 1 Fi is the contribution of the truncated higher modes to the derivatives of modes, as given by

aii ¼ 33.9.3.2

1 T ›B u u þ uTi Bzi þ zTi Bui 2 i ›b i

ð33:165Þ

High-Accuracy Modal Superposition

Assume that the eigenvalues are ordered according to their modular magnitude, and satisfy the following condition: lSi l , lSj l; j . L

ð33:166Þ

The derivatives of modes can be expressed as

›ui ¼ aii ui þ zi ¼ aii ui þ ziL þ ziH ›b

ð33:167Þ

zi ¼ ziL þ ziH

ð33:168Þ

where

ziL ¼

L X v Tj Fi u S Sj j j¼1 i

ð33:169Þ

j–i

ziH ¼

K X

A 1 ðBA 1 Þ j

1

UL ððSL Þ 1 Þ j VTL Fi

j¼1

ð33:170Þ

Also, aii can be obtained from Equation 33.165, and K denotes the number of terms used in series (Equation 33.170). It can be shown that for K ¼ 1; Equation 33.167 is equivalent to Equation 33.162. 33.9.3.3

Numerical Example

Example 33.6 Consider a 20 DoF system, as shown in Figure 33.6, with the parameters given by m1 ¼ m2 ¼ · · · ¼ m19 ¼ 2m; m20 ¼ m ¼ 1:0 kg k1 ¼ k2 ¼ · · · ¼ k21 ¼ 1:0 £ 103 N=m c1 ¼ c2 ¼ · · · ¼ c7 ¼ 3c; c8 ¼ c9 ¼ · · · ¼ c14 ¼ 2c c15 ¼ c16 ¼ · · · ¼ c21 ¼ c ¼ 0:1 N sec=m

C1

C2 m1

K1

Ci+1

Ci m2

K2

FIGURE 33.6

mi Ki

C21 m5

Ki+1

The 20 DoF system for Example 33.6.

K21


33 27

Errors of Eigenvector Derivatives (%) Modes used

TMS

IMS

HAMS 2

3

4

›u1 =›b1

4 8 12

60.18 26.66 16.76

13.14 1.01 0.66

1.41 0.03 0.01

0.01 0.00 0.00

0.00 0.00 0.00

›u1 =›b1

4 8 12

101.00 48.65 23.98

50.94 10.32 3.37

27.72 1.29 0.35

7.64 0.38 0.06

1.51 0.03 0.01

›u1 =›b1

4 8 12

69.74 26.60 25.92

14.82 1.39 0.85

3.07 0.23 0.12

0.02 0.00 0.00

0.00 0.00 0.00

›u1 =›b1

4 8 12

90.63 44.86 23.76

43.79 6.65 3.23

21.33 1.56 0.37

9.01 0.35 0.11

2.00 0.04 0.02

›u1 =›b1

4 8 12

72.52 37.95 28.54

15.07 1.22 0.52

8.46 0.57 0.03

1.11 0.03 0.00

0.03 0.00 0.00

›u1 =›b1

4 8 12

63.33 27.56 25.00

9.19 0.75 0.47

1.92 0.49 0.15

0.42 0.21 0.01

0.00 0.00 0.00

For the purpose of comparison, the errors of the truncation modal superposition (TMS), the IMS, and the high accuracy superposition (HAMS) in computing the derivatives of eigenvectors are listed in Table 33.7. For the sake of simplicity, the errors of eigenvector derivatives are represented by ! ! ›ui ›ui ›bj 1 ›bj a where ð›ui =›bj Þ1 denotes the exact solution and ð›ui =›bj Þa denotes those obtained by the three methods presented above. In the computation of the eigenvector derivatives, the parameters m1 and m10 are functions of the design variable b1 ; the parameters c8 and c15 are functions of design variable b2 ; and K denotes the number of terms used in series (Equation 33.170). The results in Table 33.7 confirm that the solution accuracy of the high accuracy modal superposition is much higher than that of the TMS and the IMS. For example, if only the first four modes are used, the errors of ›u2 =›b1 are 101.00 and 50.94% for the truncated modal superposition and the IMS, and the errors are reduced to about 27.27, 7.64, and 1.51% for the high accuracy modal superposition. If the first 12 modes are used, the error of ›u1 =›b1 is 16.76% for the truncated modal superposition, and the errors are reduced to 0.01, 0.00 and 0.00% for the case of K ¼ 2; 3, 4 in the series (Equation 33.170), where the first four modes are used, respectively.

33.9.4 Matrix Perturbation for Repeated Eigenvalues of Nondefective Systems 33.9.4.1

Basic Equation

Consider a system having repeated eigenvalues, S0 ¼ S1 ¼ · · · ¼ S0m ; with multiplicity m, and the corresponding right and left modal matrices U0m ¼ ½ u01

u02

···

u0m

ð33:171Þ

33 28


V0m ¼ ½ v 01

v02

···

v0m

ð33:172Þ

the remaining eigenvalues being distinct. The repeated eigenvalues satisfy the following equations: A0 U0m ¼ B0 U0m S0

ð33:173Þ

AT0 V0m ¼ BT0 V0m S0

ð33:174Þ

VT0m B0 U0m

¼I

uT0i B0 u0i ¼ 1

ð33:175Þ ð33:176Þ

If small changes are made to the parameters, we have A ¼ A0 þ 1A1

ð33:177Þ

B ¼ B0 þ 1B1

ð33:178Þ

The eigenvalues and eigenvectors of the perturbed system can be expressed as power series expansions in 1: Sm ¼ S0 þ 1S1 þ 12 S2 þ · · ·

ð33:179Þ

Um ¼ U0 þ 1U1 þ 12 U2 þ · · ·

ð33:180Þ

2

Vm ¼ V0 þ 1V1 þ 1 V2 þ · · ·

ð33:181Þ

U0 ¼ U0m a

ð33:182Þ

V0 ¼ V0m b

ð33:183Þ

where

and am£m and bm£m are to be determined coefficient matrices. 33.9.4.2

The First-Order Perturbation of Eigenvalues

The first order perturbation diagonal matrix, S1 ; of the repeated eigenvalues and the coefficient matrix, a; can be obtained from the equations: Wa ¼ aS1 T

W b ¼ bS1 W ¼ VT0m ðA1

S0 B1 ÞU0m

ð33:184Þ ð33:185Þ ð33:186Þ

and the normalization conditions aT a ¼ I T

b a¼I

ð33:187Þ ð33:188Þ

If matrix W has no repeated eigenvalues, a and b can be uniquely determined. If W has repeated eigenvalues, we must consider the higher order perturbation equations for determining a and b: Here, we assume that S1i are distinct eigenvalues, that is, S1i – S1j ði – j Þ; where S1i are the diagonal elements of S1 : 33.9.4.3

The First-Order Perturbation of Eigenvectors

According to the modal expansion theorem, the first order perturbation of the right and left eigenvectors, U1 and V1 ; can be expressed as U1 ¼ U0m aC1m þ UA C1A

ð33:189Þ

V1 ¼ V0m bD1m þ VA D1A

ð33:190Þ


33 29

where C1m and C1A are coefficient matrices which are to be determined, and UA and VA are the right and left modal matrices corresponding to the distinct eigenvalues: S0 Þ 1 VTA ðS0 B1

C1A ¼ ðSA D1A

¼ ðSA

S0 Þ

1

UTA ðS0 BT1

A1 ÞU0m a

ð33:191Þ

AT1 ÞV0m b

ð33:192Þ

The elements of matrix C1m can be computed by 1 Cmij ¼

R1ij

l1j

l1i

; i – j; i; j ¼ 1; 2; …; m

ð33:193Þ

where R1ij are the nondiagonal elements of R1: R1 ¼ bT VT0m B0 UA C1A S1 þ bT VT0m B1 U0m aS1 þ bT VT0m B1 UA C1A S0 0 1 Cmii ¼

bT VT0m A1 UA C1A

1

ð33:194Þ

m n B C X X 1 B T C 1 T T 1 u B u þ c u ðB þ B Þu þ cAij uT0j ðB0 þ BT0 Þu0i C ð33:195Þ B 0i 1 0i mij 0j 0 0 0i T T @ A u0i ðB0 þ B0 Þu0i j¼mþ1 j¼1 j–i

D1mij ¼

R2ij S1j

S1i

; i – j; i; j ¼ 1; 2; …; m

ð33:196Þ

where R2ij are the nondiagonal elements of R2: R2 ¼ aT UT0m B0 VA D1A S1 þ aT UT0m BT1 V0m bS1 þ aT UT0m BT1 VA D1A S0 1 Cmii

D1mii ¼ Q 2ii where

Q 2ii

ð33:197Þ ð33:198Þ

2

are the diagonal elements of Q : Q2 ¼

33.9.5

aT UT0m AT1 VA D1A

bT VT0m B1 U0m a

ð33:199Þ

Matrix Perturbation for Close Eigenvalues of Unsymmetric Matrices

Assume that S0 is a diagonal matrix with m close eigenvalues; U0n£m and V0n£m are the corresponding right and left eigenvectors matrices; SA is the remaining distinct eigenvalue diagonal matrix; UAn£ðn mÞ and VAn£ðn mÞ are the corresponding right and left eigenvector matrices. They satisfy the following equations: . . A0 ½U0 ..UA ¼ B0 ½U0 ..UA diagðS0 ; SA Þ

ð33:200Þ

. . AT0 ½V0 ..VA ¼ BT0 ½V0 ..VA diagðS0 ; SA Þ

ð33:201Þ

. . ½V0 ..VA T B0 ½U0 ..UA ¼ I

ð33:202Þ

uT0i B0 u0i

¼1

ð33:203Þ

Construct the matrix A0 as A0 ¼ A0 þ 1A0

ð33:204Þ

. . A0 ¼ B0 ½U0 ..UA diagðS0 I; SA Þ½V0 ..VA T B0

ð33:205Þ

where

1A0 ¼

B0 U0 ð1½dS0 ÞVT0 B0

1½dS0 ¼ S0

S0 I

ð33:206Þ ð33:207Þ

33 30

Vibration and Shock Handbook n 1 X S0 ¼ S m k¼1 0i

! ð33:208Þ

Here, S0i are the close eigenvalues, and S0 is the average of S0i ði ¼ 1; 2; …; mÞ: It can be shown that the following equations hold: A0 U0 ¼ B0 U0 S0 I

ð33:209Þ

BT0 V0 S0 I

ð33:210Þ

AT0 V0

¼

These equations indicate that S0 is the repeated eigenvalue with multiplicity, m; for the eigenproblem defined by Equation 33.209 and Equation 33.210, and U0 and V0 are the corresponding right and left modal matrices, respectively. If small modifications 1A1 and 1B1 are imposed on the matrices A0 and B0 ; then the eigenproblems of the perturbed system become ðA0 þ 1A1 ÞU ¼ ðB0 þ 1B1 ÞUS T

T

ð33:211Þ

ðA0 þ 1A1 Þ V ¼ ðB0 þ 1B1 Þ VS

ð33:212Þ

1A1 ¼ 1A0 þ 1A1

ð33:213Þ

where

The eigensolutions of Equation 33.211 and Equation 33.212 are given by U ¼ U0 a þ 1U1

ð33:214Þ

V ¼ V0 a þ 1V1

ð33:215Þ

S ¼ S0 þ 1S1

ð33:216Þ

It should be noted that Equation 33.211 and Equation 33.212 are the eigenproblem for repeated eigenvalues. That is, the perturbation analysis for close eigenvalues has been transferred into that of repeated eigenvalues. Hence, the methods given by Section 33.9.4 can be used to compute S1 ; a; b; U1 ; and V1 in Equation 33.214 to Equation 33.216.

References Adelmen, H.M. and Haftka, R.T., Sensitivity analysis of discrete structural systems, AIAA J., 24, 823, 1986. Adhikari, S. and Friswell, M.I., Eigenderivative analysis of asymmetric non conservative systems, Int. J. Numer. Methods Eng., 39, 1813, 2001. Chen, S.H. 1993. Matrix Perturbation Theory in Structural Dynamics, International Academic Publishers, Beijing. Chen, S.H. and Liu, Z.S., High accuracy modal superposition for eigenvector derivatives, Chin. J. Mech., 25, 432, 1993a. Chen, S.H. and Liu, Z.S., Perturbation analysis of vibration modes with close frequencies, Commun. Numer. Methods Eng., 9, 427, 1993b. Chen, S.H. and Pan, H.H., Design sensitivity analysis of vibration modes by finite element perturbation, Proc. of the Fourth IMAC, 38, 1986. Chen, J.C. and Wada, B.K., Matrix perturbation for structural dynamics, AIAA J., 15, 1095, 1979. Dailey, R.L., Eigenvector derivatives with repeated eigenvalues, AIAA J., 27, 486, 1989. Fox, R.L. and Kapoor, M.P., Rates of change of eigenvalues and eigenvectors, AIAA J., 12, 2426, 1968. Haug, E.J., Komkov, V., and Choi, K.K. 1985. Design Sensitivity Analysis of Structural Systems, Academic Press, Orlando, FL.


33 31

Haug, E.J. and Rousselet, B., Design sensitivity analysis in structural dynamics, eigenvalue variations, J. Struct. Mech., 8, 161, 1980. Hu, H. 1987. Natural Vibration Theory for Multi DoF Structures, Science Press, Beijing (in Chinese). Lim, K.B., Juang, J.N., and Ghaemmaghani, P., Eigenvector derivatives of repeated eigenvalues using singular value decomposition, J. Guidance Control Dyn., 12, 282, 1989. Liu, J.K., Perturbation technique for non self adjoint systems with repeated eigenvalues, AIAA J., 37, 222, 1999. Liu, X.L., Derivation of formulas for perturbation analysis with modes of close eigenvalues, Struct. Eng. Mech., 10, 427, 2000. Liu, Z.S. and Chen, S.H., Contribution of the truncated modes to eigenvector derivatives, AIAA J., 32, 1551, 1994a. Liu, Z.S. and Chen, S.H., An accurate method for computing eigenvector derivatives for free free structures, Int. J. Comput. Struct., 52, 1135, 1994b. Mills Curran, W.C., Calculation of eigenvector derivatives for structures with repeated eigenvalues, AIAA J., 26, 867, 1988. Murthy, D.V. and Haftka, R.T., Derivatives of eigenvalues and eigenvectors of a general complex matrix, Int. J. Numer. Methods Eng., 26, 293, 1988. Ojalvo, E.U., Efficient computation of modal sensitivities for systems with repeated frequencies, AIAA J., 26, 361, 1988. Rogers, L.C., Derivatives of eigenvalues and eigenvectors, AIAA J., 8, 943, 1977. Shaw, J. and Jayasuriya, S., Modal sensitivities for repeated eigenvalues and eigenvalues derivatives, AIAA J., 30, 850, 1992. Wang, B.P., Improved approximate methods for computing eigenvector derivatives in structural dynamics, AIAA J., 29, 1018, 1991.

34

Vibration in Rotating Machinery 34.1 Introduction ....................................................................

34 1

34.2 Vibration Basics ..............................................................

34 6

History of Vibration in Rotating Machinery Forced Vibration † Self Excited Vibration Instability † Torsional Vibration

†

Parametric

34.3 Rotordynamic Analysis .................................................. Analysis Methods

†

Modeling

†

Design

34.4 Vibration Measurement and Techniques ..................... Units of Measurement Methods

H. Sam Samarasekera Sulzer Pumps (Canada), Inc.

†

Measured Parameters and

34.5 Vibration Control and Diagnostics .............................. Standards and Guidelines † Vibration Cause Identification † Vibration Analysis Case Study

34 18 34 39 34 39

Summary This chapter concerns vibration in rotating machinery. Although it is impractical to totally eliminate such vibrations, it is essential that they be controlled to within acceptable limits for safe and reliable operation of such machines. The two major categories of vibration phenomena that occur in rotating machinery are forced vibration and self excited instability. Monitoring, diagnosis and control of these vibrations requires a sound understanding of rotor dynamics in machinery. Predicting the vibration behavior of a rotating machine by analytical means has become customary in many industries. With the advent of computer technology, several computer based programs have been developed to accurately predict the behavior of rotating machinery. Significant strides in modeling techniques have also been made over the past century to accurately represent components such as shaft sections, disks, impellers, bearings, seals, rotor dampers, and rotor stator interactions. This has enhanced the accuracy and reliability of both analytical and computational procedures. The chapter presents useful techniques of analysis, measurement, diagnosis, and control of vibration in rotating machinery.

34.1

Introduction

Vibrations are an inherent part of all rotating machinery. Residual mass imbalance and dynamic interaction forces between the stationary and rotating components, which are practically impossible to eliminate, cause these vibrations. The challenge is to identify the source of vibration and control it to within reasonable limits. Because of economic advantages, the trend in industry has been to move towards high speed, high power, lighter and more compact machinery. This has resulted in machines operating above their first critical speeds, which was unheard of in the past. The new operating parameters have required concurrent development of vibration technology without which it is not 34 1

34 2


possible to safely and reliably operate such machinery. Industry has also come to realize that vibration is an essential phenomenon, which could be used to assess the performance, durability, and reliability of rotating machinery. Engineers at different levels approach the subject of vibration in rotating machinery differently. The machinery designer has to recognize the potential sources of vibration and control them to within acceptable levels. In the past few decades, owing to the advancement in computers and modeling techniques, better understanding of the dynamics of rotating machinery, including the identification of potential sources of vibration, has been realized. This has enabled designers to accurately predict the rotordynamic behavior of machinery, allowing it to reach higher operating speeds and larger energy capacities safely and reliably. Approaching vibration from a different perspective, the maintenance engineer uses vibration standards and guidelines to monitor the health of equipment for their timely repair and refurbishment. Reliable vibration monitoring and diagnostics techniques have moved industry into predictive rather than preventive maintenance practices, which considerably reduce plant downtimes that rely on key rotating machinery. Premature replacement of machinery components has also been minimized. The resulting financial and economic benefits provide an added incentive for the study and understanding of vibration in rotating machinery. The vibration specialist or troubleshooter has to use his knowledge of rotordynamics and his diagnostic capabilities to solve vibration problems in rotating machinery. In most cases, it is also important to have an understanding of the interfacial dynamics of the rotating machinery with the surrounding system in order to solve a vibration problem. From a safety and reliability standpoint, the public must be concerned with vibration in rotating machinery. Their concerns are addressed through vibration standards and guidelines. These procedures have been developed for rotating machinery by numerous organizations, both at the national and international levels. Some of these standards are industry specific and some are equipment type specific, while a number of them try to cover a wide range of rotating machinery. The objective of most of these standards is to establish and control quality, safety, durability and reliable performance of rotating machinery for the benefit of those who use or operate it.

34.1.1

History of Vibration in Rotating Machinery

Although various types of rotating machinery have been in use for many centuries, understanding of their rotordynamic behavior did not begin until 1869 (Rankine, 1869). Since that time, there has been steady growth in the development and understanding of the vibration behavior of rotating machinery. A tabulation of major historical events that have contributed to this growth is presented in Table 34.1.

*

*

*

*

*

All rotating machinery vibrates to some degree. For public safety and machine reliability, the vibrations have to be controlled to within acceptable limits. Modern trends towards more sophisticated, higher speed compact rotating machinery have contributed to the rapid development in vibration technology through a better understanding of their rotordynamics. Vibration technology is integrated into the areas of design, maintenance, and trouble shooting of rotating machinery. From a safety and reliability standpoint, the public is protected by the implementation of vibration standards and guidelines. The first publicly reported rotordynamic study was made in 1869.


34 3

TABLE 34.1 A Chronological Listing of Major Contributions that Have Led to the Development and Understanding of Vibration in Rotating Machinery Year 1869

Contributor

Description

Rankine, W.J.M.

He examined the equilibrium of a frictionless, uniform shaft disturbed from its initial position. The resulting recorded article is recognized to be the first on the subject of rotor dynamics. He proposed that motion is stable below the first critical speed, is neutral or indifferent at the critical speed, and unstable above the critical speed He also developed numerical formulae for critical speeds for the cases of a shaft resting freely on a bearing at each end and for an overhanging shaft fixed in direction at one end He studied the effect of end thrust and torque on the stability of a long shaft and concluded that they were both unimportant. He also obtained formulae for the cases of an unloaded shaft resting on bearings at each end and fixed in direction at each end He extended the theory developed by Rankine and Greenhill for the case of a shaft loaded with pulleys He developed formulae for critical speeds for loaded shafts in terms of the diameter of the shaft, weights of pulleys, the manner in which the shaft is supported, and so on, and verified them by experiment He postulated that any degree of unbalance will excite the shaft at the critical speed to very high amplitudes and that it is possible to operate above the first critical speed. The dependence of critical speed on the moment of inertia of the rotating pulley was identified He developed an approximate method to calculate the natural frequency of a continuous beam with distributed mass and flexibility using the energy method He was responsible for the first experimental demonstration that a steam turbine is capable of sustained operation above the first critical speed He discovered the effects of transverse shear deflection on the natural frequency of a continuous beam and applied the principle to the case of the rotating shaft He examined the effect of unbalance on the whirl amplitudes and the forces transmitted to the bearings. The case of a light uniform shaft supported freely on bearings at its ends and carrying a thin pulley of mass m at the center of the span was studied. He assumed the moment of inertia of the pulley to be negligible. Using this model, later known as the Jeffcott model, a comprehensive theory was developed to explain the behavior of the rotor as it passed through the critical speed The effect of damping on the whirl amplitude, a phase change of angle p as it passes through the critical speed, and the concept of synchronous rotor whirling (precession) were introduced and explained. He also recognized that with a separation margin of 10% on either side of a critical speed, the amplitude of vibration would not be excessive. He demonstrated that it is better from the vibration point of view to design the shaft with its critical speed below the working speed rather than to have a critical speed the same proportion above the working speed. Accordingly, he explained the behavior of the De Laval steam turbine and the economic advantages of operation above the critical speed They found that a torque and an end thrust of constant magnitude lowers the critical speed of a rotating shaft, disproving Greenhill’s earlier (1883) conclusions

1883

Greenhill, A.G.

Circa 1890

Reynolds, O.

1893

Dunkerley, S.

1894

Rayleigh, J.W.S.

1895

DeLaval, G.

1916

Timoshenko, T.

1919

Jeffcott, H.H.

1921

Southwell, R.V. and Gough, B.S.


34 4 TABLE 34.1 Year 1921 1924

Vibration and Shock Handbook (continued) Contributor Holzer, H. Newkirk, B.L.

1924

Kimball, A.T.

1924

Newkirk, B.L.

1925

Newkirk, B.L.

1925

Stodola, A.

1927

Stodola, A.

Description He developed a numerical method to calculate torsional critical speeds and mode shapes for a multidisk rotor system He observed that a rotor operating at a speed above the first critical speed can enter into high, violent whirling and the center of the rotor will precess in the forward direction at a rate equal to that of the critical speed. Unlike in the case of synchronous whirling, if the speed is increased beyond the initial whirl speed, the whirl amplitude will continue to increase, eventually leading to failure. This was the first time that it was realized that nonsynchronous unstable motion can exist in a high speed rotor Based on experiments, he made the following key observations on nonsynchronous whirling. The amplitude and the onset speed of whirling are independent of the rotor balance. Whirling always occurs at speeds above the critical speed, and the whirl speed is always constant at the critical speed, regardless of the rotor speed. The whirl threshold speed can vary even for machines of similar construction. Whirling occurs only in built up rotors, and not in single piece constructions. Increasing the foundation flexibility, distortion or misalignment of the bearing housings, or introducing damping to the foundation or increasing the axial thrust bearing load, increased the threshold speed of whirling Suggested that internal friction or viscous action due to bending may cause a shaft to whirl when rotating at any speed above the first critical speed. He postulated that the nonsynchronous whirling observed by Newkirk was due to this phenomenon Based on Mr Kimball’s theory, he concluded that similar frictional forces are generated at the mating face between the shrunk on disk and the shaft of a built up rotor, and the nonsynchronous whirling observed by him was due to this effect. However, he was unable to explain some of his experimental findings, in particular, the effects of bearing or foundation flexibility, damping, and misalignment He experienced another form of nonsynchronous whirling, similar but different to that caused by the frictional effects of a shrink fit disk. It occurred at rotor speeds just exceeding twice the first critical speed on shafts mounted on journal bearings. He recognized that the oil in the journal bearing was responsible for the violent motion and called it oil whip. The whirl speed and direction of whirling were the same as that for friction induced whirling, that is, the first critical speed in the forward direction. A theory to explain how the oil film can produce the whirling motion of a journal and to account for why it took the same direction as rotation of the shaft was proposed. However, the theory does not explain why whirling does not commence until the rotor speed reaches twice the critical speed value. The influence of foundation flexibility on the rotor stability was also found to be confusing to Newkirk. In the case of friction induced whirl, he was able to totally eliminate the rotor instability by means of a flexibly mounted bearing. When this was tried with the journal bearings, the whirl amplitudes magnified. External damping at the bearing was found to have a favorable influence on whirl amplitudes He developed an iterative procedure to calculate the fundamental frequency of a vibrating system based on an assumed mode shape He provided an explanation and formulae for the gyroscopic moment effect on the critical speed of a rotor. He also introduced the notion of synchronous and nonsynchronous reverse precession of a rotor under specific conditions

Vibration in Rotating Machinery TABLE 34.1 Year

34 5

(continued) Contributor

Description

1933

Robertson, D.

1933

Smith, D.M.

1944

Myklestad, N.

1945

Prohl, M.

1953

Poritsky, H.

1953

Miller, D.F.

1955

Pinkus, O.

1958

Lomakin, A.

1958

Thomas, H.

1966

Gunter, E.J. Jr.

1969

Black, H.F.

1970

Ruhl, R.

1974

Lund, J.

1976

Nelson, H. and McVaugh, J.

In order to understand oil whip, he studied the stability of the ideal 3608 infinitely long journal bearing, and erroneously concluded that the rotor will be unstable at all speeds and not only at speeds above twice the critical speed value He studied the case of unsymmetrical rotors on unsymmetrical supports and obtained four different critical speed values in comparison to the single value for a symmetrical system. He also discussed the presence of additional critical speeds due to gyroscopic effects of large disks A lumped parameter transfer matrix method to calculate natural frequencies for airplane wings was developed by him He developed a lumped parameter transfer matrix method for calculating critical speeds of flexible rotors Using the small displacement theory, he derived a radial stiffness coefficient for the journal bearings and analyzed the rotor behavior under oil whirl conditions. He concluded that the rotor was stable below twice the critical speed and indicated that increasing the rotor or bearing flexibility will reduce the threshold speed of instability. He also proposed a stability criterion for a rotor based on the bearing and rotor stiffness He introduced a solution to the steady state forced vibration problem, for a beam or rotating shaft on damped, flexible end supports. The response of the rotor to an unbalance force and the damped resonance frequencies are calculated by this method He investigated oil whirl in various journal bearing types and made the following major conclusions. The unbalance of the rotor has minimal effect on stability. The threshold of instability occurs at approximately twice the first critical speed of the rotor. In the unstable region, the whirl frequency remained constant at the first critical speed, irrespective of the shaft rotating speed. At speeds nearly equal to three times the first critical speed, whipping motion stops with a heavy shaft rotor, whereas with a light shaft rotor it does not cease. High loads, high viscosity, flexible mountings, and bearing asymmetry favor stability The influence of the dynamic characteristics of seals on the critical speeds and stability of pump rotors were introduced by him He proposed that an eccentric turbine rotor would generate a destabilizing force due to the circumferential variation in clearance He combined the different theories on whirling developed by the rotor dynamist and the bearing specialist, and elegantly explained some of the conflicting experimental evidence gathered thus far. He emphasized the importance of considering the combined effects of rotor parameters and the bearing and foundation characteristics on rotor stability, and developed more comprehensive criteria for self excited whirl instability He provided a comprehensive analysis of annular pressure seals on the vibrations of pump rotors He introduced finite element models for flexible rotors for calculating rotor critical speeds and mode shapes. These models did not take into account gyroscopic effects and axial loading A transfer matrix method to calculate damped critical speeds of a rotor taking into account the cross coupling terms as well were introduced by him They extended the finite element model of a rotor to account for rotary inertia, gyroscopic effect, and axial loads (continued on next page)

34 6


TABLE 34.1

(continued)

Year

Contributor

1978 1980

Benckert, H. and Wachter, J.

1980

Nelson, H.

1980

Brennen, C. et al.

1986

Muszynska, A.

34.2

Description A method to calculate flow induced spring constants for labyrinth gas seals and the use of swirl breaks to reduce the destabilizing force caused by tangential velocity in labyrinth seals was introduced by them He further developed a finite element model of a rotor to include shear deflection and axial torque effects They recognized the presence of substantial shroud forces, which influences the rotor dynamics of a pump She demonstrated that oil whirl occurs at about one half the running speed in a vertical rotor. With further increase in speed, oil whip will commence when the whirl frequency approaches the critical speed of the rotor

Vibration Basics

The vibration phenomena that manifest in rotating machinery can be divided into two major categories: forced vibration and self excited instability. A stimulus or a source of excitation is required to initiate and sustain vibratory motion in a rotor. When the stimulus is a forcing phenomenon such as mass unbalance, it will produce forced flexural vibration in the rotor analogous to linear forced vibration response in a simple spring mass system. On the other hand, self excited vibration (instability) does not require a forcing phenomenon for its initiation or sustenance. A description of these phenomena is given next.

34.2.1

Forced Vibration

A rotating force vector (unbalance), a steady directional force (gravity), or a periodic force (pump impeller/diffuser interface action), will cause forced vibration in a rotating machine. The response of the rotor will depend on the nature of the forcing function and how it relates to rotor characteristics. The rotor responses to the most common excitation phenomena are examined below.

Z

C

M

34.2.1.1 Unbalance Response — Synchronous Whirling As an introduction to the theory on rotating machinery vibration and understanding unbalance response, it is most appropriate to examine Jeffcott’s (1919) rotor, which is a simple model that has many of the basic characteristics of more complex rotating machinery. The Jeffcott rotor represents a massless elastic shaft supported freely in bearings at its ends and carrying a disk of mass m at the center of its span. The mass center of the disk is eccentric to its geometric center by a distance e. Refer to Figure 34.1. C ¼ geometric center of the disk b ¼ phase angle

O

X

Y M e β C

r wt O

q FIGURE 34.1

X Jeffcott rotor.


34 7

M ¼ mass center of the disk c ¼ viscous damping coefficient of on the rotor O ¼ bearing center r ¼ deflection of rotor from origin u ¼ angle of precession k ¼ shaft stiffness v ¼ angular velocity of the rotor ¼ u_ þ b_ Whirling is defined as the angular velocity of rotation of the rotor geometric center ðcÞ or the time derivative ðu_Þ of the angle of precession ðuÞ (also see Chapter 32). Synchronous whirling is when the rate of whirling, u_; is equal to the total angular velocity, v; of the system. Applying Newton’s Laws of motion to the rotor, the differential equations of motion in polar coordinates ðr; uÞ are obtained as c k r€ þ r_ þ u_ 2 r ¼ ev2 cos b ð34:1Þ m m c ru€ þ ð34:2Þ r þ 2_r u_ ¼ ev2 sin b m For a steady state condition, the values of r; b; u_; and v are constant. For synchronous whirling, Equation 34.1 and Equation 34.2 reduce to k v2 r ¼ ev2 cos b ð34:3Þ m c vr ¼ ev2 sin b ð34:4Þ m From Equation 34.3 and Equation 34.4 ev2 r ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k cv 2 v2 þ m m c v b ¼ tan 1 k m v2 m F¼

kr kev2 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k cv 2 v2 þ 2 m m

ð34:5Þ

ð34:6Þ

ð34:7Þ

Using the following relationships: sffiffiffiffi k Natural frequency of rotor without damping vN ¼ m pffiffiffiffi ccr ¼ 2 km Critical damping coefficient c Damping ratio z¼ ccr Equation 34.6 and Equation 34.7 are reduced to the following nondimensional form: r 2F ðv=vN Þ2 ffi ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ke ð1 ðv=vN Þ2 Þ2 þ ð2zv=vN Þ2

b ¼ tan

1

ð1

2zðv=vN Þ ðv=vN Þ2 Þ

ð34:8Þ

ð34:9Þ

34 8


6.0 Damping Ratio = 0

Amplification Factor r/e

5.0

0.1

4.0

0.15 3.0 0.25 2.0

0.4

0.5

0.707

1.0

1.0 0.0

2.0 4.0 0

0.5

FIGURE 34.2

1

1.5 Speed Ratio w/w N

2

Jeffcott rotor response with mass eccentricity

2.5

3

amplification vs. speed.

Figure 34.2 is a graphical representation of the unbalance response of the rotor as a function of rotating speed, v: Upon examination of the phase relationship, it is important to note that the phase angle, b; changes from approximately 08 at low speed to values approaching 1808 at the higher speed. At vN ; b ¼ 908: A pictorial illustration of this phenomenon is given in Figure 34.3.

200.00 180.00 Damping Ratio = 0.01

160.00

0.15

Phase Angle Deg.

140.00 120.00

0.25

100.00

0.5

0.707

80.00

1.0

4.0

2.0

60.00 40.00 20.00 0.00

0

FIGURE 34.3

0.5

1

1.5 Speed Ratio w /wN

Jeffcott rotor response to unbalance

2

2.5

phase angle vs. speed.

3


34 9

For the case of zero damping, when v ¼ vN ; the rotor deflection and the bearing forces are unbounded. For all other cases, the rotor deflection and the bearing forces are bounded, and their amplitude depends on the damping ratio. If a shaft is quickly accelerated through its critical speed to a higher working speed, then there may not be enough time for large rotor deflection to take place. At high speeds, v .. vN ; the amplitude of the rotor deflection decreases and approaches the value e; the eccentricity of the rotor. The critical speed, vcr ; of a rotor in the general case, is the speed at which the rotor deflection amplitude or the force amplitude transmitted to the bearings is a maximum. This implies that, at v ¼ vcr dr dF ¼ ¼0 dv dv Using Equation 34.8, the following relationship between the natural frequency of the rotor and its critical speed is derived:

vN vcr ¼ pffiffiffiffiffiffiffiffiffiffi 1 2z 2

ð34:10Þ

From Equation 34.10, it is evident that the critical pffiffi speed of a rotor is not a fixed value and is dependent on the degree of rotor damping. When z ¼ 1= 2; the system is said to be critically damped. It is important to note that rotor response to unbalance (or imbalance) is recognizable and controllable. The amplitude of the force transmitted to the bearing can be reduced by operation at speeds above the critical speed, reducing unbalance, increasing viscous damping, and avoiding operation close to critical speeds. 34.2.1.2

Shaft Bow

A rotor with a bent shaft will behave in a similar manner to a rotor with an eccentric mass (Ehrich, 1999). At high rotor speeds ðv .. vcr Þ; the shaft will tend to correct the bow as illustrated in Figure 34.4. When shaft bow is combined with mass eccentricity, unique behavior patterns are produced depending on the phase angle between the bow and the eccentric mass (Childs, 1993). 6.0 Damping Ratio = 0

Amplification Factor r/s

5.0

0.1

4.0

0.15

3.0

0.25

2.0

0.4

1.0 0.707

0.0

0.5

4.0 2.0 1.0

0

0.5

FIGURE 34.4

1 1.5 Speed Ratio w/w N

2

Jeffcott rotor response with shaft bow

2.5

3

amplification vs. speed.

34 10

34.2.1.3


Gravity Critical

A special case of synchronous whirling may occur in certain types of horizontal rotors due to the gravitational force. It is a secondary critical speed commonly called the gravity critical, which can occur in a very heavy lightly damped rotor. The critical speed will occur at approximately half the natural frequency of the rotor and its amplitudes of deflection at the critical speed are bounded and approximately twice the static deflection of the rotor (Gunter, 1966). 34.2.1.4

The Influence of Rotor Inertia and Gyroscopic Action

The effect of rotor inertia is ignored in the Jeffcott model. However, in practice, it is recognized that rotor inertia and gyroscopic action has an influence on the natural frequencies, critical speeds, and unbalance response of the rotor, including reverse whirling. In the case of the natural frequency of the rotor (zero speed), the diametral or rotary inertia provides an additional natural frequency associated with the rotational degree of freedom (DoF). Also, the inertia effect lowers the first natural frequency (Childs, 1993). In the rotating case, the effect of inertia generates both forward and reverse whirling critical speeds (Childs, 1993). These forward whirling critical speeds tend to be higher (stiffening effect) and the reverse whirling critical speed lower than the natural frequency of the rotor. At the forward critical speeds, large amplitude whirling motion due to imbalance occurs, whereas the reverse critical speeds are insensitive to imbalance of the rotor. 34.2.1.5

Rotor Housing Response across an Annular Clearance

If the rotor deflection due to imbalance exceeds the uniform annular gap, continuous contact would occur between the rotor and stator resulting in coupled motion between the rotor and stator (Childs, 1993). For low contact frictional forces, synchronous forward whirling driven by the imbalance forces will occur. If the contact friction force is large enough to prevent slipping between the rotor and stator, reverse whirling will take place. For the case of synchronous forward whirling in a certain range of running speeds, instability will occur due to engagement between the rotor and stator (Black, 1968). The zones of instability depend on the coupled natural frequency of the rotor and stator and the degree of rotor deflection with respect to the annular gap. 34.2.1.6

Effect of Nonlinearity and Asymmetry on Forced Vibration Response

The foregoing analysis has assumed that stiffness and damping are linear and symmetric and the resulting forces are proportional to the deflection and velocity of the rotor. However, in reality, rotating machinery components have inherent nonlinearities and asymmetries that can have a profound influence on their rotordynamic behavior. At large amplitudes of motion, stiffness and damping coefficients become nonlinear and result in modifying the response amplitude and critical speeds of the rotor. Nonlinearity in the support stiffness will introduce considerable distortion to the otherwise simple harmonic vibration behavior of a purely linear system. The stiffness and damping coefficients of the bearings and their supports are asymmetric in most cases, in particular in horizontal machines. As a result, the forced vibratory responses in the two principal directions are different and can behave independent of each other. Each principal direction will display a critical speed unique to itself. Ehrich (1999) has presented a discussion on how nonlinearity and asymmetry of stator systems influence forced vibration response. The influence of rotor stiffness asymmetry and inertia asymmetry on rotor stability is discussed in Section 34.2.3.

34.2.2

Self-Excited Vibration

Instability (nonsynchronous whirling) is a self induced excitation phenomenon, sometimes described as sustained transient motion, that can occur in rotating machinery. At the inception of instability, the rotor deflection will continue to build up with increase in speed, whereas in the case of a critical speed resonance, the amplitude of the deflection reaches a maximum value and then decreases. If the rotor speed is increased above the instability threshold speed, the large amplitudes of motion will normally


34 11

Rotat

ion

mw2r cwr

Destabilizing force Fq

r

mr cr

FIGURE 34.5

l hir on W ecti dir

2mwr kr

Bearing axis

Rotor instability

general case.

result in damage to the machine. Unlike forced flexural vibration, rotor instability is self induced and does not require a sustained forcing phenomenon to initiate or maintain the motion. It is known to occur only in machines operating at speeds well above the critical speeds of the rotor. Furthermore, the rotor whirling speed is different to the rotor speed (nonsynchronous whirling), and it is identical to the critical speed irrespective of the rotor speed. In general, the rotor instability is associated with the existence of a tangential force vector, Fu ; acting at right angles to the deflection vector and directly opposing the damping force vector, as illustrated in Figure 34.5. The nature of Fu is such that its magnitude increases proportionately with the rotor deflection. At the point where Fu equals the external damping force, rotor instability will commence due to the nullification of the stabilizing force. This will produce a whirling motion of ever increasing amplitude. Several phenomena inherent in the rotor system that generates such tangential force vectors have been identified and are discussed below. Rotordynamists believe that there still remain more such phenomena to be discovered. 34.2.2.1

Internal Friction Damping

This type of instability was first experienced in the early 1920s in blast furnace compressors made by the General Electric Company. These machines were subject to occasional fits of violent vibration. Newkirk (1924) carried out a series of experiments to understand the unusual behavior of these machines. Based on the internal friction theory of Kimball (1924). Newkirk (1924) concluded that the interfacial friction damping forces at the disk shaft interface caused the subsynchronous whirling. In order to understand the internal friction damping phenomena let us examine the shaft stresses in the whirling Jeffcott rotor (Figure 34.1). Figure 34.6 is a cross section of the shaft disk interface. Owing to its deflection, all of the fibers in the right half of the cross section are in tension, Te ; and those in the left half are in compression, Ce : These fiber stresses tend to straighten the shaft and produce a restoring force, Fr ; which opposes the centrifugal force, mu_r: Furthermore, a set of frictional forces are generated at the shaft disk interface due to stretching and compression of the fibers. The fibers in the bottom lower half will be stretched and are under frictional tension, Tf ; and those in the upper half are being compressed under frictional compression Cf : Similarly to the reaction force, Fr ; produced from right to left by Te and Ce ; a reaction force, Fu ; from bottom to top will be produced by the frictional stresses Tf and Cf : The disturbing force, Fu ; is in the same direction as the whirling motion and as a result will increase as

34 12


Fq

Whirl path

Cf

Shaft cross section

Wh

irl

FIGURE 34.6

Ce

E

Te

Rotation

Fr

O rad

ius

Tf

Internal friction damping forces acting on a rotor.

the whirling motion increases. The force, Fu ; will oppose the external damping force, cu_r: At the threshold of instability, the two forces nullify each other. It is also know that the frequency of whirling, u_; at the threshold of instability equals the natural frequency, vn ; of the rotor. Mathematically it can be expressed as follows: Fu ¼ ci rðv

u_Þ

ð34:11Þ

u_Þ

ð34:12Þ

where ci is the rotor internal damping coefficient: cu_r ¼ ci rðv

u_ ¼ vN

ð34:13Þ

Equation 34.12 and Equation 34.13 yield the following relationship between the threshold speed of instability, the first critical speed, and the damping factors (both internal and external): c v ¼1þ ð34:14Þ ci vN 34.2.2.2

Tip Clearance Excitation (Alford’s Force, Steam Whirl)

Thomas (1958) investigated the instability of steam turbines and suggested that nonsymmetric radial clearances caused by an eccentric rotor could result in destabilizing forces, and called them clearance excitation forces. Subsequently Alford (1965) discovered a similar phenomenon in aircraft gas turbines and, as a result, the destabilizing forces are sometimes referred to as Alford forces in North America. The destabilizing force is created as a result of the variation in the gap between the blade tip and the stator. When the gap decreases, the leakage decreases and consequently the efficiency increases, resulting in a torque higher than the average torque produced by a uniform gap. When the gap increases, there is a corresponding decrease in the torque relative to the average. The variation in torque produced by the eccentricity results in a tangential force, which is normal to the radial deflection and is in the direction of the whirling motion as shown in Figure 34.7. Furthermore, it has been illustrated that the magnitude of the resulting force increases proportionately with the increase in rotor deflection, that is, the decrease


34 13

Rotat

ion

Increase force due to low blade clearance

mw 2r Destabilizing force Fq

cwr

Whirl

direction

Bearing axis kr

Decreased force due to large blade clearance

FIGURE 34.7

Tip clearance excitation (Alford forces).

in the gap. The resulting force is a destabilizing force that will oppose the external damping force, and at some point when they balance each other, rotor instability will occur. A detailed analysis of the tip clearance forces has been made by Urlichs (1977). 34.2.2.3

Impeller Diffuser Excitation Forces

Instability experienced in centrifugal compressors provides a strong suspicion that impeller diffuser interaction phenomena are involved in the development of destabilizing forces. However, to date, no satisfactory destabilizing mechanism or source involving impellers has been identified. In the last two decades there have been several studies related to rotordynamic forces arising from shrouded centrifugal pump impellers, but very little work has been done on compressor impellers. The destabilizing force arising from the impeller diffuser/volute interaction of a pump has been determined to be relatively small (Jery et al., 1984; Bolleter et al., 1985; Adkins and Brennen, 1986; Ohashi et al., 1986). The major portion of the destabilizing force is known to be generated in the narrow gap region between the casing and shroud of the impeller (Childs, 1986; Bolleter et al., 1989; Baskharone et al., 1994; Moore and Palazzolo, 2001). In the case of centrifugal compressors, an empirical method to determine stability of multistage machines has been proposed (Kirk and Donald, 1983). The stability maps proposed by them for flow through and back to back centrifugal compressors are shown in Figure 34.8 and Figure 34.9. 34.2.2.4

Propeller Whirl

Propeller whirl (Taylor and Browne, 1938; Houbolt and Reed, 1961) is another form of instability which occurs in aircraft rotors when there is a mismatch in the angular velocity vector of the propeller and the linear velocity vector of the aircraft. This angular mismatch results in the generation of a moment whose vector has a component of significant magnitude, which contributes to the instability of the

34 14


104

PRESSURE PARAMETER, P2∆P/1000 (LB2/IN.4)

5

2 EXPERIENCE LIMIT FOR AUTHORS' COMPANY (STABLE DESIGNS)

103 5

2

102

PRESSURE PARAMETER VERSUS SPEED RATIO P2∆P N 1000 VS NCR

5

2

CENTRIFUGAL COMPRESSORS NO PROBLEMS FROM AERODYNAMIC EXCITED INSTABILITIES • FOR UNITS WITH N/NCR ≥ 2.0 • •

101 1.0

2.0

3.0

4.0

CRITICAL SPEED RATIO, N/NCR (DIM.) FIGURE 34.8 Proposed stability map for flow through centrifugal compressors. (Source: Rotor Dynamical Instability, 1983. With permission.)

propeller (Vance, 1988). Its magnitude is proportional to both the angular mismatch and the linear speed of the aircraft. With increasing speed, the magnitude of the destabilizing moment will exceed the rotor viscous damping moment and result in propeller instability (refer to Figure 34.5). Since the propeller is supported only from one end, the whirling motion is conical and is found to be in the reverse direction to propeller rotation. The instability is sensitive to the velocity and density of the air and not a function of the torque of the machine. 34.2.2.5

Fluid Trapped in a Hollow Rotor

Wolf (1968) has demonstrated that trapped fluid inside a hollow rotor can produce a force component tangential to the whirl orbit due to viscous drag forces. Under subsynchronous whirling speeds, this force component acts in the same direction of whirling motion and its magnitude is proportional to the rotor deflection. With reference to Figure 34.5, this force has all the markings of a destabilizing force, which can produce instability in the rotor. The threshold speed of instability is reached when the whirling speed equals the first critical speed of the rotor. It has been shown (Ehrich, 1999) that, at the threshold of instability, the rotor speed is less than twice the first critical speed. This results in a ratio of whirl speed to rotor speed in the range of 0.5 to 1.0. 34.2.2.6

Dry Friction Rubs

In Section 34.2.1.5, a dry friction rub situation was identified, where slipping was prevented between the rotor and stator under contact conditions. The contact was made possible by the deflection of the rotor


34 15

105

PRESSURE PARAMETER, P2∆P/1000 (LB2 / IN.4)

5

EKOFISK ORIGINAL (UNSTABLE)

EKOFISK FINAL (STABLE)

2 UNCCEPTABLE (BACK TO BACK DESIGN)

104 KAYBOB FINAL (STABLE)

5

KAYBOB ORIGINAL (UNSTABLE)

2 ACCEPTABLE

103

PRESSURE PARAMETR VERSUS SPEED RATIO

5

2 102

P2 = ∆P/1000 VS N/NCR P2 = ∆DISCHARGE PRESSURE (PSIA) ∆P = PRESSURE RISE (PSI) N = COMPRESSOR OPERATING SPEED (RPM) NCR = COMPRESSOR RIGID BEARING CRITICAL (PRM)

1.0

2.0

3.0

4.0

CRITICAL SPEED RATIO. N/NCR (DIM.) FIGURE 34.9 Proposed stability map for back to back centrifugal compressors. (Source: Rotor Dynamical Instability, 1983. With permission.)

due to unbalance forces. When contact is made between the rotor and stator, Coulomb friction produces a tangential force in the direction opposite to shaft rotation. Since the frictional force prevents slipping, whirling in the reverse direction to rotation occurs. The whirling speed is equal to rv=C; where r is the radius of the rotor, C the radial gap, and v the speed of the rotor. Since the frictional force is in the same direction of whirling, it will cause the magnitude of whirling to increase, resulting in further increase of the frictional force. When the magnitude of this force exceeds the viscous damping force, rotor instability will occur. Another possibility is for the dry friction whirling speed to approach the coupled natural frequency of the rotor and stator, in which case unstable motion termed dry friction whipping takes place (Ehrich, 1999). In addition to the case described above, dry friction rubs can occur in journal bearings, seals, wear rings, or any situation where a small clearance between a rotor and stator exists. The inadvertent closure of the clearance due to unbalance or lack of proper lubrication can initiate dry friction rub induced instability in these cases as well. 34.2.2.7

Torque Whirl/Load Torque

When the rotor disk axis is not aligned with the bearing axis, as in the case with an overhung rotor, Vance (1988) has shown that nonsynchronous whirling (torque whirl) can occur as a result of the misalignment between the load torque and the driving torque. His findings are based on the analysis of a simple

34 16


rotor model. It appears that torque whirl instability can occur only in the case of long slender shafts with high load torque values. The practical implications of this theory are still to be fully explored. 34.2.2.8

Oil Whirl/Whip

Newkirk and Taylor (1925) first experienced shaft whipping due to oil action in journal bearings during their investigation into internal friction induced whirling of rotors. They found that under certain conditions, a rotor mounted on journal bearings whipped when the rotor was running at any speed above double the critical speed; the whirling motion was in the forward direction and its speed matched the critical speed of the rotor. He provided a qualitative explanation of the phenomenon based on the fact that the oil film rotates at half the velocity of the shaft due to friction drag. Hence, for rotational speeds near twice the critical speed, the oil film provides the stimulus as its speed matches the critical speed value resulting in large displacements and whipping. Others have also drawn similar conclusions based on the suggestion of an oil wedge rotating at half speed, or rotating fluid force fields at half shaft speed. However, the foregoing fails to explain why oil whip persists at speeds greater than twice the critical speed. Ehrich (1999) has also provided a qualitative explanation for oil whirl based on the general theory of rotor instability. Although a comprehensive explanation of the physical phenomena of oil whirl is still outstanding, numerous analytical models to identify where it could be encountered have been suggested. Gunter (1966) has analytically demonstrated that the instability in a rotor supported on journal bearings can be attributed to the cross coupling bearing coefficients. As a result, most of the research on oil whirl instability has narrowed to accurate estimation of bearing cross coupling coefficients. 34.2.2.9

Influence of Bearings and Supports on Rotor Instability

The results of the Jeffcott model can be easily adopted to include bearing stiffness and bearing support stiffness effects, provided they are both linear and circumferentially symmetric (isotropic). For this particular case, the rotor stiffness, k; is the equivalent stiffness resulting from the series connection of the shaft, bearings, and support stiffness. The resulting values for v N and vcr will be less than those for the simply supported Jeffcott model. This will result in lowering the threshold speed of instability. If the bearing stiffness or the bearing support stiffness is not symmetric (orthotropic) then it can be shown (Childs, 1993) that the threshold speed of instability is increased and the maximum amplitude of deflection of the rotor is reduced in comparison to the case with symmetric bearings. The effect of damping at the bearings or at the bearing support is very similar to the influence of stiffness. It reduces the amplitude of the synchronous rotor response at the critical speed, and elevates the threshold speed of instability. However, there is a limit to the amount of damping that can be applied. Excessive damping causes a reduction in stability (Childs, 1993). The mass of the bearings plays a significant role on rotor stability. If the bearing mass is significantly larger than the rotor mass, the threshold speed of instability is lowered.

34.2.3

Parametric Instability

The instability phenomena described in Section 34.2.2 can be represented by linear differential equations where the system parameters such as mass, inertia, stiffness, damping, and natural frequency are assumed to be constants. There is another subcategory of self excited motion, referred to as parametric instability, since it is induced by the periodic variation of the system parameters such as inertia, mass, and stiffness. A discussion of the more common forms of this phenomenon follows. 34.2.3.1

Shaft Stiffness Asymmetry

If the shaft of a rotor contains a sufficient level of stiffness asymmetry in the two principal axis of flexure, rotor instability could occur. Smith (1933) investigated the rotor behavior under unsymmetrical flexibility of the bearing supports and unsymmetrical transverse flexibility of the shaft, taking into consideration the damping effects as well. The following conclusions were derived based on his investigation:


34 17

In the presence of stiffness asymmetry, the onset speed of internal friction induced instability is lowered. When there is no external damping, the rotor becomes unstable at all speeds between the two undamped natural frequencies in the two orthogonal directions. However, if external damping is significant, parametric instability may be eliminated. Within the unstable range, the whirling motion is in the forward direction and is synchronous with the shaft speed. Further, unlike in the case of internal friction induced instability; it is theoretically possible to run through parametric instability. This makes parametric instability quite similar to the case of unbalance response. When the asymmetric rotor is acted upon by a transverse disturbing steady force such as gravitational force, the rotor whirls at twice the speed of the shaft. This motion exhibits a resonant increase in amplitude at a speed that is approximately half the mean of the two natural frequencies. 34.2.3.2

Rotor Inertia Asymmetry

Crandall and Brosens (1961) analyzed the parametric excitation of a rotor with nonsymmetrical principal moments of inertia. Their results indicate that the rotor behavior is very similar to the case of rotors with stiffness asymmetry described in Section 34.2.3.1, and parametric instability over similar speed ranges occurs. 34.2.3.3

Pulsating Torque

Constant torque acting on a rotor is known to lower its critical speeds because they effectively reduce the rotor’s lateral stiffness. A pulsating torque introduces lateral vibrations and instabilities into a rotor. When a combination of a pulsating and constant torque is applied to a rotor, it will induce unstable lateral vibrations in a specific range of rotor speeds and certain combinations of torque amplitudes. In the region of unstable lateral motion, the whirling speed of the rotor will coincide with the first critical speed of the rotor regardless of the rotor speed or the frequency of the pulsating torque. At rotor speeds outside the unstable region, the whirling speed of the rotor will be coincident with the pulsating torque frequency. 34.2.3.4

Pulsating Longitudinal Loads

Pulsating axial forces on a shaft that are in the order of magnitude of the buckling load will effectively cause a periodic variation in its lateral stiffness. This will result in a proportionate reduction of the lateral natural frequency of the shaft. Therefore, pulsating axial loads are capable of inducing parametric instability in a shaft for both the rotating and the stationary cases. 34.2.3.5

Nonsymmetric Clearance Effects

Bentley (1974) recognized that large subsynchronous whirling can occur in rotating machinery due to certain types of nonsymmetric clearance conditions. One such condition is when a rotor’s whirling motion causes rubbing with a stationary surface over a portion of the rotor orbit. This effectively results in an increase in the rotor stiffness during the contact portion of the orbit, producing a periodic variation in rotor stiffness during each cycle. Another situation that produces cyclic variation in rotor stiffness can occur in the case of a rotor supported on antifriction bearings mounted with a clearance fit to the housing. The cyclic variation of the effective rotor stiffness produces a subsynchronous whirling motion at exactly half the rotational speed. Occasionally, whirling at one third and one fourth the running speed has also been observed by Childs (1993). Instability will occur when the whirling speed corresponds to a critical speed of the rotor.

34.2.4

Torsional Vibration

Rotating machinery with rotors that have relatively large moments of inertia are susceptible to torsional vibration problems. Torsional vibration is an oscillatory angular motion that is superimposed on the steady rotational motion of the rotor. In practice it can easily go undetected, as standard vibration

34 18


monitoring equipment is not geared to measure torsional vibration. Special equipment must be used to detect torsional vibration in rotating machinery. Since the introduction of electric motor variable frequency drives (VFD), the incidence of field torsional vibration problems has increased. This has been attributed to the inherent torsional excitation forces present in current designs of VFDs. An added complication when using VFDs, as compared with using fixed speed drives, is the requirement of eliminating torsional natural frequencies over a wider speed range. Large synchronous electric motors are known to produce a large pulsating torque at a frequency that changes from twice the line frequency at the start, down to zero at the synchronous operating speed. In this case, any torsional natural frequency between zero and twice the line frequency may be subject to excitation. Most industries have come to recognize that torsional vibration is a potential hazard, and therefore, needs to be investigated at the design stage of rotating machinery. Several standard design specifications now require that torsional analysis is part of the design procedure. The standard design practice in modeling the system for torsional analysis has been to calculate the undamped eigenvalues of the rotor as a free body in space. This practice is acceptable for most types of rotating machinery since the torsional stiffness and damping of bearings is insignificant. Also, in most cases, the torsional damping of the rotors itself is extremely low. Although the absence of damping is favorable from an analysis point of view, it makes it extremely difficult to eliminate a torsional vibration problem when encountered. This deficiency has been partly addressed with the introduction of several new lines of couplings that have a significant degree of torsional damping. Although not commonly used in rotating machinery, several torsional dampers such as the Lanchester damper have been developed for use on reciprocating machines. Dampers of similar design could be developed for use in rotating machinery to solve torsional vibration problems. The torsional critical speeds of a simple rotor with one or two DoFs can be calculated by analytical methods. Numerical methods are used to calculate critical speeds and mode shapes of more complex systems with higher DoF. The Holzer numerical method (Holzer, 1921) described in Section 34.3.1.11 is one of the common methods used for these analyses.

*

*

*

34.3

The two major categories of vibration phenomena that occur in rotating machinery are forced vibration and self excited instability. Parametric instability is a special case of self excited instability where some of the normally constant parameters vary, influencing rotor motion. Torsional vibrations are similar to lateral vibrations but occur in the planes perpendicular to the shaft axis.

Rotordynamic Analysis

Rotordynamic analysis is a part of current design procedures for rotating machinery that is carried out to predict the vibration behavior during operation of the machine. Potential problems are identified and eliminated by analytical means well before the manufacturing of components is begun. Furthermore, when a machine in operation displays unusual vibration behavior, analytical means are employed to study, identify, and help resolve the problem. In order for the analysis to be useful, it must be accurate and cost effective. During the last 100 years, several analytical procedures have been developed to understand the vibration behavior of rotating machinery. Some of these techniques are of historical interest only, and their usefulness in practical systems is very limited. With the advent of computer technology and advanced modeling techniques, several computer based procedures have been developed to predict the vibration behavior of rotating machinery quite precisely. Of the most commonly used procedures, two are based on the lumped parameter model where the distributed elastic and inertial properties are


34 19

represented as a collection of rigid bodies connected by massless elastic beam elements. These two procedures are the transfer matrix formulations introduced by Myklestad (1944) and Prohl (1945), and the direct stiffness matrix formulations proposed by Biezeno and Grammel (1954). Ruhl and Booker (1972) introduced the third commonly used method, based on the finite element analysis (FEA) model in which the rotor is represented as an assemblage of elements with distributed elastic and inertial properties. Several of the well known procedures are discussed next, some of them for historical significance.

34.3.1

Analysis Methods

34.3.1.1

Rankine’s Numerical Method

Rankine (1869) proposed that, for a shaft of a given length, diameter, and material, there is a limit of speed, and for a shaft of a given diameter and material, turning at a given speed, there is a limit of length, below which centrifugal whirling is impossible. The limits of length and speed depend on the way the shaft is supported. The critical speed of the shaft is given by the following equation:

v¼

kðHgÞ1=2 b2

ð34:15Þ

where

v ¼ critical speed in rad/sec k ¼ radius of gyration of the cross section of the shaft g ¼ acceleration due to gravity H ¼ modulus of elasticity expressed in units of height of itself ðH ¼ E=rÞ E ¼ Young’s modulus r ¼ density of the material l ¼ shaft length b ¼ l=p for a simply supported shaft b ¼ l=0:595p for an overhanging shaft 34.3.1.2

Greenhill’s Formulae

Greenhill (1883) introduced the following differential equation of motion for a uniform shaft slightly deformed from straightness by centrifugal whirling: d4 y dx4

mv2 y¼0 gEak2

ð34:16Þ

The general solution to Equation 34.16 is given by y ¼ B cosh mx þ A cos mx

where

ð34:17Þ

m4 ¼ mv2 =gEak2 m ¼ weight of the shaft per unit length v ¼ rotational speed a ¼ cross sectional area of the shaft The constants A and B depend on the boundary conditions at the support locations. 34.3.1.3

Reynolds’ Equations

Reynolds extended the differential equation of motion for a uniform rotating shaft (Equation 34.16) to include shafts loaded with pulleys (disks) and for multispan rotors (Dunkerly, 1894).

34 20


At a bearing support, the difference in shear force must equal the bearing load P: dMr dx

dMl ¼P dx

ð34:18Þ

where Mr and Ml are bending moments in the right (r) and left (l) sides of the load. At a load consisting of a revolving weight, W; the above equation becomes dMr dx

dMl W 2 vy ¼ dx g

ð34:19Þ

A further equation may be obtained by considering the centrifugal couple (gyroscopic moment) as given by Mr

Ml ¼ v2 I 0

dy dx

ð34:20Þ

where I 0 ¼ moment of inertia of the pulley. The solution to Equation 34.20 is given by y ¼ A cosh mx þ B sinh mx þ C cos mx þ D sin mx

ð34:21Þ

The values of A; B; C; and D will depend on the boundary conditions between any two singular points. 34.3.1.4

Dunkerley Method

When considering the effects of the pulleys and the shaft together, the formulae derived by Reynolds were found to be limited for practical purposes. Dunkerly (1894) proposed an empirical method to consider the effects of the shaft and each of the pulleys separately, and then combine them using the following formula to obtain the critical speed of the rotor:

where

n X 1 1 1 ¼ 2 þ 2 2 vc vs v i i¼1

ð34:22Þ

vc ¼ critical speed of the rotor vs ¼ critical speed of the shaft alone vi ¼ critical speed of the ith disk on a weightless shaft In the case of the unloaded shaft, the critical speed, vs ; is given by the following formula !1=4 mv2s l¼a gEI

ð34:23Þ

where I ¼ sectional inertia of shaft l ¼ length of the span a ¼ a coefficient dependent on the manner of support of the shaft The critical speed of the rotor vi with a single disk of weight Wi on a weightless shaft is given by

vi ¼ u

gEI Wi c3

1=2

where c ¼ distance of disk from nearest support u ¼ a coefficient dependant on the manner in which the shaft is supported, the position of the disk within the span and the dimensions of the disk

ð34:24Þ


34.3.1.5

34 21

Rayleigh Method

The Rayleigh method is based on the premise that, when a system vibrates at its natural frequency, the maximum potential energy stored in the elastic components is equal to the maximum kinetic energy stored in the masses (Rayleigh, 1945). The first natural frequency of a vibrating uniform beam is given by the following equation: ! ðl d2 y 2 EI dx dx2 0 2 ð34:25Þ v ¼ ðl m y 2 dx 0

The Rayleigh formula for a lumped mass system is n X 2

v ¼

i¼1 n X i¼1

mi yi mi yi2

ð34:26Þ

where

v ¼ first natural frequency y ¼ deflection of the beam x ¼ distance along x axis l ¼ length of the beam mi ¼ ith lumped mass yi ¼ static deflection of ith mass The accuracy of the Rayleigh method depends upon the selection of a suitable deflection curve that approximates the fundamental mode shape. If the assumed curve represents the true mode shape, then the correct fundamental natural frequency will result. All deviations from the true mode shape will yield frequencies that are higher than the correct value. 34.3.1.6

Ritz Method

The Ritz method is an improvement on the Rayleigh method (Timoshenko et al., 1974) where the mode shape is represented by several orthogonal functions with unknown coefficients that satisfy the boundary conditions. The orthogonal functions are represented by a series of functions, Fi ðxÞ; where i varies from 1 to n: The mode shape is represented by the following expression: y¼

n X i¼1

ai Fi ðxÞ

ð34:27Þ

In order for the coefficients ai in the above equation to yield minimum values when substituted in the energy balance equation proposed by Rayleigh, the following expression needs to be satisfied: ! ðl d2 y 2 dx › 0 dx2 ¼0 ð34:28Þ ðl ›ai y2 dx 0

From Equation 34.25 and Equation 34.28, we find

› ðl › ai 0

"

d2 y dx2

!2

# v2 m 2 y dx ¼ 0 EI

ð34:29Þ

34 22


Substituting Equation 34.27 for y in Equation 34.29 and performing the mathematical operations, a system of linear equations in ai is obtained. The number of such equations will be equal to n: These equations will yield solutions different from zero only if the determinant of the coefficients of ai is equal to zero. This condition yields the frequency equation, from which the frequency of each mode can be derived. 34.3.1.7

Stodola –Vianello Method

The Stodola Vianello method is a numerical iterative process (Timoshenko et al., 1974) that can be used to calculate the natural frequencies and mode shapes of vibrating systems. An approximate mode shape is first assumed and by successive iterations it is refined until convergence is obtained to the desired level of accuracy. This method can be used to refine the assumed mode shape when using the Rayleigh formulae or in the more general case of the matrix iteration process illustrated below. Using Newton’s Second law, the equations of motion for a multi DoF system in matrix notation are {Y} ¼ v2i ½A ½m {Y} ½A ¼ ½K

where

1

ð34:30Þ ð34:31Þ

½A is the flexibility matrix ½m is the mass matrix ½K is the stiffness matrix To start the iterative process a trial vector, {Y}1 ; representing the mode shape is substituted to both sides of Equation 34.30 and solve for the natural frequency, vi : For this reason, let the product of ½A ; ½m ; and {Y}1 be {Y}02 : The first approximation for vi may be obtained by dividing any one of the elements on {Y}1 by {Y}02 (Note that, if {Y}1 was the true mode shape, then the ratio for all such elements will be equal.) The vector {Y}02 is then normalized by dividing all the elements by the first element to produce {Y}2 : The vector {Y}2 is premultiplied by ½m and ½A to produce {Y}03 : Once again the ratio of corresponding elements of {Y}03 and {Y}2 are compared for equality. This procedure is repeated until the mode shape and the associated frequency is determined to the desired level of accuracy. In the above iteration procedure, the mode shape converges to the one corresponding to the lowest natural frequency. If the stiffness matrix had been used instead of the flexibility matrix, then convergence at the highest natural frequency is obtained. After the first mode of vibration is determined, it is removed from the system matrices by the use of a sweeping matrix so that higher modes can be obtained. This procedure is repeated until all the desired mode shapes and natural frequencies are determined. 34.3.1.8

Myklestad –Prohl Method (Transfer Matrix Method)

The Myklestad Prohl transfer matrix formulation (Myklestad, 1944; Prohl, 1945) is commonly used to analyze lumped parameter models of rotating machinery. The distributed elastic and inertial properties of the rotor are represented as a collection of rigid bodies connected by massless elastic beam elements as illustrated in Figure 34.10. This method is best suited to calculate critical speeds and mode shapes of rotors neglecting the effects of viscous damping. The Myklestad Prohl procedure can also be adopted to perform synchronous response and stability analysis, including for the effects of damping. In order to demonstrate the transfer matrix procedure, an axisymmetric rotor is analyzed to determine its undamped critical speeds and mode shapes. Refering to Figure 34.10, the rotor is divided into n nodes, and each node is connected to the adjacent node by a massless elastic beam with uniform cross sectional properties. The mass of components such as disks, impellers, and so on, together with the mass of the adjacent portion of the shaft, is lumped at the nodes. The Myklestad Prohl method is based on the solution of the Bernoulli Euler equation and the variables of interest are displacement ðyÞ; slope ðuÞ; moment ðMÞ; and shear ðVÞ: The development of the following procedure follows Childs (1993).


34 23

Rotors (rigid bodies)

Massless elastic beam elements

1

N

2

1

N-1

4

2 3

n-2

n

n-1

FIGURE 34.10

Lumped parameter model of rotor.

At a typical nodal point ðnÞ; the variables on the left hand side (l) are related to the variables on the right hand side (r) by the following relationship: 8 r 9 2 38 l 9 1 0 0 0 > yn > > > > > > yn > > > 6 7> > > > > l > > > r 6 7 = < un = 6 0 < un > 1 0 0 7> 7 ¼6 ð34:32Þ 6 7> l > 2 r > > 6 0 > > Jn v 1 0 7 Mn > Mn > > > > > 4 5 > > > > > > ; ; : r> : l > 0 0 1 mn v 2 Vn Vn For the purpose of abbreviation: ðQÞTn ¼ ð yn

un

Mn

Vn Þ

ð34:33Þ

and Equation 34.32 can be written in a more compact form as follows: ðQÞrn ¼ ½Tmn ðQÞln

ð34:34Þ

where ½Tmn represents the transfer mass matrix at node n: At a massless beam section, connecting node n to node n þ 1 the transfer matrix is given by 2 3 l2n l3n 8 9 6 1 ln 8 l 7> y r 9 > 6 ynþ1 > 2EIn 6EIn 7 n > > > > 6 7> > > > > 6 > r > > 7> > > 2 7> = 6 < < u lnþ1 > ln ln 7 u n = 60 1 ð34:35Þ ¼6 7 6 > > EIn 2EIn 7 > > > > Mnr > M lnþ1 > 6 7 > > > > > > 6 > 7> > > > ; 6 : l > 1 ln 7 40 0 5: Vnr ; V nþ1 0 0 0 1 Equation 34.35 may be written in a more abbreviated form as ðQÞlnþ1 ¼ ½Tbn ðQÞrn

ð34:36Þ

where ½Tbn represents the beam element transfer matrix connecting node n to node n þ 1: From Equation 34.34 and Equation 34.36, we obtain the combined transfer matrix for nodes n and n þ 1: ðQÞlnþ1 ¼ ½Tbn ½Tmn ðQÞln ¼ ½Tn ðQÞln

ð34:37Þ

34 24


Starting with node one, successive matrix multiplications are carried out until node n þ 1 is reached. The last node ðn þ 1Þ is a dummy node with the beam length, l; equal to zero, and the mass and inertias also equal to zero. This makes the nodal parameters on the left hand side of node n þ 1 equal to those on the right hand side of node n: The result is as follows: ðQÞrn ¼ ½Tn ½Tn

1

· · ·½T1 ðQÞl1 or ðQÞrn ¼ ½T ðQÞl1

ð34:38Þ

The matrix ½T is a function of the rotational speed; v: The Myklestad Prohl method uses a trial and error solution to determine the values of v which satisfy the boundary conditions and Equation 34.38 simultaneously. It is not necessary to store and multiply all the matrices together. The transfer matrix procedure is used to proceed from one end to the other without having to store all the nodal matrices. In all cases, two boundary conditions each are known at the two ends of the shaft, and the frequencies that satisfy these boundary conditions are the critical speeds of the rotor. Once the critical speeds are calculated, the corresponding mode shapes can also be determined using the transfer matrix procedure. It should be noted that other types of elements, such as elastic supports, flexible couplings, and so on, could also be introduced very conveniently. 34.3.1.9

Direct Stiffness Method

The direct stiffness method uses a lumped parameter formulation to evaluate the dynamic characteristics of a flexible rotor. The general differential equation of motion that characterizes its behavior (less the damping and gyroscopic forces) is as follows: " #( ( ) ( ) ) ½m 0 ðFÞ ðYÞ ðY€ Þ ¼ ð34:39Þ þ ½K 0 ½J ðTÞ ðuÞ ðu€Þ where ½m and ½ J are diagonal matrices which contains the nodal masses, mi ; and nodal moments of inertia, Ji ; respectively. The stiffness matrix, ½K ; contains the internal stiffness terms of the beam elements as well as any external spring stiffness at the supports. The vectors ðFÞ and ðTÞ represent external forces and moments acting on the system, respectively. The stiffness matrix for a typical beam element based on the Bernoulli Euler equations is as follows (Childs, 1993): 2 3 6 3li 6 3li 6 7 6 3l 3li l2i 7 2l2i 7 i 2EIi 6 i 7 ð34:40Þ ½K ¼ 3 6 7 li 6 6 6 7 3l 6 3l i i 4 5 3li 2l2i 3li l2i The overall stiffness matrix, ½K ; has to be assembled by combining the individual component matrices in a systematic manner. The following procedure illustrates the process. The stiffness matrix of the ith beam element in matrix notation is ½K i ¼ ½kij;k

ð34:41Þ

where j and k vary from ð2i 1Þ to ð2i þ 2Þ: To form the overall stiffness matrix, the elements with the same subscripts of adjacent beam elements are added over n beam elements as given by the following equation: ½K ¼ ½Kj;k ¼

n 2iþ2 X X

2iþ2 X

i¼1 j¼2i 1 k¼2i 1

kij;k

ð34:42Þ

Once the inertia matrix and the stiffness matrix for the entire system are assembled, the eigenvalues and eigenvectors can be evaluated by solving the following homogeneous equation derived


34 25

from Equation 34.39: ½M ðY€ Þ þ ½K ðYÞ ¼ 0

ð34:43Þ

There are numerous analysis procedures (Meirovitch, 1986) for the solution of Equation 34.43 that yield the eigenvalues and eigenvectors of the system. The method of choice will depend on the complexity and nature of the inertia and stiffness matrices. Perhaps the most widely known is the matrix iteration using the power method in conjunction with the sweeping technique. However, this method is not necessarily the most efficient, particularly for higher order systems. The Jacobi’s method, which uses matrix iteration to diagonalize a matrix by successive rotations, is more commonly used owing to its higher efficiency. Details of these techniques are given in the text by Meirovitch (1986). When the damping matrix and the gyroscopic matrix is also included in Equation 34.39, the direct stiffness method can be used to calculate damped critical speeds, forced rotor response, and instability of the rotor in addition to the eigenvalues using similar methods of solution. 34.3.1.10

The Finite Element Analysis Method

The basis of the FEA method is to provide formulation for complex and irregular systems that can utilize the automation capabilities of computers (also see Chapter 9). The FEA method considers a rotordynamic system as an assemblage of discreet elements, where every such element has distributed and continuous properties, namely, the consistent representation of both mass and stiffness as distributed parameters. As illustrated in Section 34.3.1.9, the lumped parameter method uses a consistent stiffness matrix equation (Equation 34.40) in its formulation, and therefore, the identical procedure can be adopted for the finite element method as well. For the distributed mass representation of an element, Archer (1963) procedure, which is based on the assumption that the mass distribution is proportional to the elastic distribution similar to the Rayleigh Ritz formulation, is utilized. The resulting mass matrix is as follows: 2 3 156 22li 54 13li 6 7 6 22l 13li 3l2i 7 4l2i 7 i m i li 6 i 6 7 ½m ¼ ð34:44Þ 7 420 6 6 54 7 13l 156 22l i i 4 5 22li 4l2i 13li 3l2i The overall stiffness matrix, ½K ; for the entire system is assembled by combining the individual component matrices in a systematic manner according to Equation 34.41 and Equation 34.42. The overall mass matrix can also be assembled in precisely the same manner, as given by Equation 34.45 and Equation 34.46. The mass matrix of the ith beam element in matrix notation can be represented as ½mi ¼ ½mij;k where j and k varies from ð2i

ð34:45Þ

1Þ to ð2i þ 2Þ : ½M ¼ ½Mj;k ¼

n 2iþ2 X X

2iþ2 X

i¼1 j¼2i 1 k¼2i 1

mij;k

ð34:46Þ

Once the mass matrix and the stiffness matrix for the entire system are assembled, Equation 34.43 that describes the free vibration of the complete system can be solved. The solution methods of the eigenvalue problem, which can be utilized, are the same as those used for the direct stiffness method illustrated in Section 34.3.1.9 above. Details of the FEA methods are given in Ruhl and Booker (1972).

34 26

34.3.1.11


Torsional Analysis (Holzer Method)

The development of torsional analysis methods have gone through a similar evolutionary process to lateral vibration methods. Holzer (1921) first introduced the lumped parameter numerical method to calculate torsional natural frequencies of a multi DoF system. Even to date, this is the most commonly used method because of its simplicity and reasonable degree of accuracy. The Holzer method is a transfer matrix formulation that uses a lumped parameter model similar to that used in the Myklestad Phrol method described in Section 34.3.1.8. The only difference is that the transfer matrices represented by Equation 34.32 and Equation 34.35 are replaced by the equations # ( )l ( )r " u u 1 0 ¼ ð34:47Þ 2 T n vJ 1 n T n 2 3 ( )l ( )r 1 u u 1 6 7 k5 ¼4 ð34:48Þ T nþ1 0 1 nþ1 T n Starting with node one, successive matrix multiplications are carried out until node n þ 1 is reached. The result can be represented by Equation 34.38. The matrix ½T is a function of the rotational speed, v: In all cases, one boundary condition at each end of the rotor is known. A trial and error solution to determine the values of v which satisfy the boundary conditions and Equation 34.38 are simultaneously determined. These values are the torsional critical speeds of the rotor. Once the critical speeds are calculated, the corresponding torsional mode shapes can also be determined using the transfer matrix procedure. In the case of branched systems and geared systems, particular attention has to be paid to the relative rotational speeds of the components. The rule is quite simple: multiply all stiffness and inertias of the geared shaft by N 2 ; where N is the speed ratio of the geared shaft to the reference shaft. Other methods such as the distributed mass matrix method, direct stiffness method, and finite element method can also be used to determine torsional critical speeds of rotors. These procedures are very similar to those for lateral critical speed analysis.

34.3.2

Modeling

The design and analysis of rotordynamic systems require the development of models that simulate the behavior of the physical system. In the past, the critical speed of the rotor was considered to be the main criterion for stable operation. Today, stable, well damped rotordynamic response to the exciting forces within a machine is considered to be a necessary condition for high reliability. The accuracy and reliability of the results greatly depends on the credibility of the system model and its adaptability to the analytical procedure. Even the most accurate and efficient analytical method cannot produce good results from a bad model. The methods that are commonly used to model shaft sections and disks and other such elements attached to shafts have been discussed in the previous sections. Useful formulae for calculating critical speeds of simple systems are given in Table 34.2. Models to represent bearings, rotor dampers, seals, and rotor stator interactions are discussed in the following sections. 34.3.2.1

Journal Bearings

Journal bearings were used in rotating machinery for a long time before their dynamic characteristics were fully understood. Considerable effort has been expended in the last few decades to understand and develop techniques for their accurate representation in rotordynamic analysis. A variety of bearing types with improved characteristics have been developed over the years. Figure 34.11 shows the most commonly used types in rotating machinery. Hagg and Sankey (1958) were amongst the first to provide dynamic stiffness and damping coefficients for a number of these bearing types. However, these coefficients are considered incomplete as cross coupling terms were not considered. Soon after, there was a flurry of activity related to the analysis of journal bearings; Sternlicht (1959), Warner (1963),

Vibration in Rotating Machinery TABLE 34.2

34 27

Useful Formulas in Vibration Analysis and Design

Rankine formula

v¼

kðHgÞ1=2 b2

Note: This formula is of historical interest only and has limited practical value

Greenhill formula

d4 y mv2 2 y¼0 4 dx gEak2

Dunkerly equation

n X 1 1 1 ¼ 2 þ v2c vs v2i i 1

The above equation reduces to g v2c ¼ P ystat Formulas for natural frequency calculation (Blevins, 2001; Gorman, 1975)

a

a W

vc ¼

1 ab

a

vc ¼

1 a

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6EI Wð3L 2 4aÞ

a

vc ¼

1 a

rffiffiffiffiffiffi 3EI WL

W

L

W

L

W

L a

b

vc ¼

12EIL3 3 Wa b2 ð3L þ bÞ

b

vc ¼

3L3 EI Wa3 b3

vc ¼

3EI WL3

1=2

vc ¼

98EI mL4

1=2

vc ¼

64a 9L

W

L a

W

L

W

L

m

rffiffiffiffiffiffiffi 3EIL W

b

!1=2

!1=2

L m L

a

þ

16 3

2

Þ

2

2

6a L

EI mL4

1=2

;

a $ 0:25 L


34 28 TABLE 34.2

Vibration and Shock Handbook (continued) m

vc ¼

L

237EI mL4

1=2

vc ¼ 274:7

Þ

2

þ 3:14 a , 0:25 L

m

vc ¼

502EI mL4

m

vc ¼

12:4EI mL4

vc ¼

k J

2

a L

a L

þ22:1

EI mL4

1=2

;

1=2

L 1=2

L Formulas for torsional natural frequency calculation

k

k1 J 1

k1

J

k2

1 k þ k2 k þ 2 vc ¼ pffiffi 1 J1 J2 2 ( k1 þ k2 k þ 2 7 J1 J2

J2

k1 þ k2 J

vc ¼

k2

J

2

k

J1

k1 J 2

Rayleigh equations

EI

v2 ¼

J2

k2

0

m

vc ¼

ðl 0

2

1=2

d2 y dx2

4ðk1 k2 þ k2 k3 þ k1 k3 Þ J1 J2

k k þ J1 J2

4k1 k2 J1 J2

# 1=2 1=2

1=2

1 k k þ k2 k þ 2 vc ¼ pffiffi 1 þ 1 J2 J3 2 J1 ( k1 k þ k2 k 2 7 þ 1 þ 2 J1 J2 J3 # 1=2 1=2 4k k ðJ þ J2 þ J3 Þ 2 1 2 1 J1 J2 J3

J3

ðl

2

1 k þ k2 k þ k3 þ 2 vc ¼ pffiffi 1 J1 J2 2 k1 þ k2 k2 þ k3 2 7 þ J1 J2

k1 J k2 J k2 1 2

J1

1=2

!2

y2 dx

dx

# 1=2 1=2


34 29

(continued) n X i 1 v2 ¼ X n i 1

Ritz method

y¼

Pn

ai Fi ðxÞ !2 d2 y dx dx2

0

ðl

y2 dx

0

› ›ai Stodola Vianello method Transfer matrix Myklestad Phrol method

Stiffness matrix for a beam element

Mass matrix for a beam element

ðl " 0

d2 y dx2

!2 2

¼0

# v2 m 2 y dx ¼ 0 EI

{Y} ¼ v2i ½A ½m {Y}; ½A ¼ ½K 2 9 61 8 l > 6 ynþ1 > > > > > > > 6 6 > < u lnþ1 > = 6 6 ¼ 60 6 > > M lnþ1 > > 6 > > > 6 > l > > : ; 6 40 V nþ1 0

ln

l2n 2EIn

1

ln EIn

0

1

0

0

2 ½K i ¼

6

6 6 3l 2EIi 6 6 i 3 li 6 6 26 4 3li 2

156 6 i 6 22l 6 i m li 6 ½mi ¼ 420 6 6 54 4 213li

Squeeze film damper coefficients

k¼ c¼ k¼

c¼

Unbalance sensitivity Rolling element bearing defect frequencies

mi yi2

i 1

ðl

› ›ai

mi yi

fbor

1

3 2l3n 8 7> y r 9 6EIn 7 n > > 7> > > > 7> > < u rn > = l2n 7 7 7 r > > 2EIn 7 Mn> 7> > > > 7> > > 2ln 7 5: V rn ; 1 26

2l2i

23li

23li

6

l2i

23li

7 l2i 7 7 7 7 23li 7 5 2l2i

22li

54

213li

4l2i

13li

13li

156

23l2i

222li

3li

3

3li

3

7 23l2i 7 7 7 7 222li 7 5 4l2i

24R3 Lmv1 Cr3 ð2 þ 12 Þð1 2 12 Þ Cr3 ð2

12pR3 Lm þ 12 Þð1 2 12 Þ1=2

2RL3 mv1 Cr3 ð1 2 12 Þ2 pRL3 m 3 2Cr3 ð1 2 12 Þ 2

a SF ¼ M U " # 2 ND d ¼ 12 cos u 60d D

N¼ rotational speed (rpm), D ¼ rolling element pitch diameter (continued on next page)

34 30 TABLE 34.2

Vibration and Shock Handbook (continued)

Lomakin formula for radial stiffness for a close clearance bushing

fir ¼

Nn d 1 þ cos u 120 D

d ¼ rolling element diameter, N ¼ number of rolling elements

for ¼

Nn d 1 2 cos u 120 D

u ¼ contact angle with respect to axis, bor ¼ ball or roller defect

fc ¼

N d 1 2 cos u 120 D

k¼

p l ð1 þ 6Þlm4 8 bm

m2 ¼

2

ir ¼ inner race defect, or ¼ outer race defect, c ¼ cage defect DpD

1 1 þ 6 þ ðll=2bm Þ

bm ¼ radial clearance, l ¼ length of bushing,

z ¼ inlet loss coefficient,

D ¼ diameter, Dp ¼ differential pressure across bushing, l ¼ friction coefficient

Lund (1964), Lund (1965), Glienicke (1966), Orcutt (1967), Lund (1968), Someya et al. (1988), and several others provided complete bearing coefficients, including cross coupling terms, for several bearing types. This information is considered to be a valuable resource for those engaged in rotordynamic analysis. The general form of the rotordynamic model for a journal bearing resulting from the above contributions is given by the following equation: #( ) " #( ) " ( ) c11 c12 k11 k12 X FX X_ ð34:49Þ ¼ c21 c22 k21 k22 Y FY Y_ Since the dawn of the digital computer era, several computer codes have been developed to analyze all aspects of journal bearings, including stiffness and damping coefficients. Many of these codes have been developed by equipment manufactures and research centers for their exclusive use. Several commercially available software codes popularized in North America are given in Table 34.3. Although bearing coefficients given in the form of charts and tables from the earlier studies are still in use, computer based codes are growing in popularity. 34.3.2.2

Rolling Element Bearings

Rolling element bearings are used in numerous types of rotating machinery which are required to be compact, manage high loads, and have low heat rejection and simple lubrication systems. Unlike journal bearings, their load carrying capacity is not speed dependent and as a result is capable of full load capacity down to zero speed. Some of these salient features make rolling element bearings very attractive to many industries. From a rotordynamic standpoint, rolling element bearings are modeled as linear spring elements with direct spring coefficients only. The damping terms are insignificant and as a result do not attenuate rotor deflections at critical speeds. A typical rolling element bearing is represented by the following equation: " #( ) ( ) FX k 0 X ¼ ð34:50Þ 0 k Y FY The absence of cross coupling stiffness and damping terms signifies that bearing induced rotor instability will not occur. Although, for convenience of analysis, the spring stiffness is considered linear, its


34 31

20∞

C

C e R+ R

W

(a) Plain Cylindrical

C e R+ R

W

(e) Two groove Elliptical

(c) Partial-arc Cylindrical

W

+C

R 2d

R+

C

C

C R+ R

2d

(d) Offset Cylindrical

10 W

C

R+

C R+

W

W

R+

C R+ 2d

C e R+ R

(b) Two groove Cylindrical

W 20

W

R+

C

R+

2d

R

2d

20

(f) Three lobe

FIGURE 34.11

(g) Four lobe

(h) Tilting pad

Common types of journal bearings.

true behavior can be quite the opposite, leading to some calculation inaccuracies. The nonlinearities are most significant where the bearings have no preload and some internal clearance in the bearings exists. Preloaded bearings with little or no internal clearance behave quite linearly. Jones (1960), Harris (1991), and Kramer (1993) have analyzed the bearing stiffness coefficients for the common types of rolling element bearings, and this data can be utilized for rotordynamic study of rotating machinery. 34.3.2.3

Squeeze-Film Dampers

Squeeze film dampers are used to introduce damping capacity to a rolling element bearing or, in the case of journal bearings, to provide additional damping and stiffness to eliminate rotor instability problems. Squeeze film dampers have come into prominence through the modern aircraft gas turbine industry where the bearing of choice is the rolling element bearing. In the mid 1970s, several designs were introduced to add damping capacity and predictable stiffness to the rolling element bearings. In its basic form, a squeeze film damper is very similar to a nonrotating cylindrical journal bearing where the outer race of the rolling element bearing forms the journal as illustrated in Figure 34.12. The addition of end seals (to control leakage) and centering springs are modifications that have been introduced to enhance its performance. The interactive force at a bearing using a squeeze film damper can be represented by the following equation: " #( ) " #( ) ( ) FX k 0 X c 0 X_ ¼ ð34:51Þ FY 0 k Y 0 c Y_ The stiffness and damping coefficients for the squeeze film dampers have been derived (Ehrich, 1999), from the solution of the Reynolds’ equation for the case of a nonrotating journal bearing. For dampers with end seals, the long journal bearing theory is used to generate the following stiffness and damping

34 32 TABLE 34.3

Vibration and Shock Handbook Rotordynamic Analysis Software

Name of Software CAD20 CAD21 CAD21a CAD22 CAD24 CAD25 CAD25a CAD26 CAD27 CAD30 CAD30a CAD31 CAD32 CAD34a CAD34b CAD36 CAD38 CAD40 CAD41 CAD42 CAD42i FEATURE COJOUR

Type of Analysis Lateral critical speeds of flexible rotors Unbalance response of flexible rotors Response of flexible rotors to nonsynchronous sinusoidal excitation Torsional critical speeds and response of geared systems Transient torsional critical speeds of geared system Dynamic stability of flexible rotors Transient response of flexible rotors Lateral critical speeds of multilevel rotors Unbalance response of multilevel rotors Dynamic coefficients of liquid lubricated journal bearings Dynamic coefficients of ball bearings Dynamic coefficients of liquid lubricated tilting pad journal bearings

Supplier CADENSE Programs, Foster Miller Technologies Inc., Albany, NY, USA

Dynamic coefficients of liquid lubricated axial groove and single pad journal bearings Performance of tilting pad thrust bearings Performance of tapered land thrust bearings Dynamic coefficients of liquid lubricated pressure dam journal bearings Dynamic coefficients of liquid lubricated deep pocket hydrostatic journal bearings Dynamic coefficients of gas lubricated journal bearings Dynamic coefficients of gas lubricated tilting pad journal bearings Dynamic coefficients of gas lubricated spiral groove journal bearings Dynamic coefficients of liquid lubricated spiral groove journal bearings Rotor bearing system analysis Analysis of journal bearings

DYNROT

A program designed to perform a complete study of the rotordynamic behavior of rotors. It is capable of linear, nonlinear and torsional analysis of rotors

Dipartimento di Meccanica, Politecnico di Torino, Torino, Italy

DyRoBeS

Comprehensive rotordynamic analysis software for lateral and torsional analysis, including bearing analysis of rotor bearing systems A software package for agile modeling of rotor systems, bearings, and seals. It combines the tasks of design, modeling, analysis, post processing, and data management into a consistent user interface

AGILE SOFTWARE CONCEPTS NREC White River Junction, VT, USA

RotorLab

DAMBRG2

HYDROB PDAM2D SQFDAMP THBRG

Coefficients and rigid rotor stability information for ROMAC Rotating Machinery two lobe isoviscous bearings with a pressure dam in and Controls Laboratory, only one pad University of Virginia, Charlottesville, VA, USA Predicts the steady state and dynamic operating characteristics of hybrid journal bearings This program can analyze stiffness and damping coefficients, and the rigid rotor stability threshold of multipad pressure dam bearings Determines stiffness and damping coefficients for short and long squeeze film bearings with and without fluid film cavitation Dynamic coefficients of multilobe journal bearings with incompressible fluid


(continued)

Name of Software THPAD THRUST CRTSP2 MODFR2 TWIST2 FRESP2 RESP2V3 HCOMB LABY3 SEAL2 SEAL3 TURSEAL FSTB3 ROTSTB COTRAN TORTRAN3 hydrosealt

hydroflext hydrotran hydrojet hydroTRC hseal2p fembear sfdfem sfdflexs hsealm

34 33

Type of Analysis

Supplier

Dynamic coefficients of tilting pad journal bearings with incompressible fluid Predicts the steady state operating characteristics of tilting pad and fixed geometry fluid film thrust bearings Undamped lateral critical speeds of dual level rotor systems Undamped lateral critical speeds of single or dual level rotor systems Undamped torsional critical speeds and mode shapes of rotor systems Predicts the modal frequency forced response of dual rotor systems with a flexible substructure Nonplanar synchronous unbalance response of dual level multimass flexible rotors Dynamic coefficients of straight through honeycomb seals with a compressible gas Dynamic coefficients for straight through and uniform interlocking type labyrinth seals with a compressible fluid Stiffness and damping coefficients for plain and grooved seals with incompressible turbulent axial flow Stiffness, damping and mass coefficients of both plain and circumferentially grooved seals Stiffness and damping coefficients of turbulent flow annular seals or water lubricated bearings Stability, damped critical speeds, and whirl mode shapes of multispool rotor systems Stability, damped critical speeds, and whirl mode shapes of single spool rotor systems Nonlinear time transient analysis of multilevel rotors with substructure Transient torsional rotor response Stiffness and damping coefficients, and threshold speed of instability of cylindrical pad journal bearings and pad hydrostatic bearings of arbitrary arc lengths and preloads Stiffness and damping coefficients, and threshold speed of instability of a variety of bearing and seal types Predicts the transient force response of a rigid rotor supported on fluid film bearings Force coefficients for a variety of hybrid bearing and seal types handling process fluids Stiffness and damping coefficients for a variety of bearing and seal types and for different types of fluids Stiffness and damping coefficients of seals that operate under two phase flow conditions Stiffness and damping coefficients of cylindrical and fixed arc pad hydrostatic and hydrodynamic bearings for laminar and isothermal flow conditions Damping force coefficients of finite length squeeze film dampers executing circular centered motion Instantaneous fluid film forces for arbitrary journal motions and circular centered orbits in multiple pad integral squeeze film dampers Stiffness and damping coefficients of cylindrical annular pressure seals

Rotordynamics Laboratory, Texas A&M University, College Station, TX, USA


34 34 TABLE 34.3

Vibration and Shock Handbook (continued)

Name of Software

Type of Analysis

lubsealn

Stiffness and damping coefficients of single land and multiple land high pressure oil seal rings and cylindrical journal bearings

ROTECH

Lateral rotordynamic analysis for critical speeds; unbalance response, linear stability and nonlinear transient response of rotors. Also includes a torsional rotordynamic analysis program A comprehensive software package for lateral rotordynamic analysis of rotating equipment

ROTOR E

Supplier

ROTECH Engineering Services, Delmont, PA, USA Engineering Dynamics Inc., San Antonio, TX, USA

ROTORINSA

A software package devoted to the prediction of the steady state lateral dynamic behavior of rotors

Laboratoire de Mecanique des Structures, LMST INSA Lyon, Lyon, France

TURBINE PAK TURBINE PAK NONLINEAR XLrotor

A software package for rotordynamic analysis of nonlinear multibearing rotor bearing foundation systems Designed to study transient responses of rotor bearing foundation systems, including the loss of stability of the system A complete suite of analysis tools for rotating machinery dynamics. Handles both lateral and torsional analysis of rotors. Also includes codes for calculating coefficients for fluid film and antifriction bearings

Scientific Engineering Research, Mt Best, Vic., Australia.

XLTRC

A suite of codes for executing a complete lateral The Turbomachinery rotordynamic analysis of rotating machinery Laboratory, Texas A&M Force and moment coefficients for annular turbulent seals University, College Station, in the laminar, turbulent, and transition flow regimes TX, USA Coefficients for centered grooved stator, turbulent flow, annular pump seals Stiffness and damping coefficients for tooth on rotor or tooth on stator gas labyrinth seals Coefficients for smooth rotor/honeycomb stator annular seals Coefficients for high pressure oil bushing seals of compressors or smooth pump seals in the laminar flow regime Stiffness and damping coefficients for fixed arc and tilting pad bearings Stiffness and damping coefficients for hydrostatic and hybrid journal pad bearings in the laminar flow regime Stiffness and damping coefficients for fixed arc, tilting pad and flexure pivot hydrostatic bearings Stiffness and damping coefficients for multilobed, rigid pad arc bearings with preload and pressure dam bearings with relief tracks Stiffness coefficients for ball bearings Damping and mass coefficients for locally sealed squeeze film dampers Damping and mass coefficients for open ended squeeze film dampers Damping coefficients for squeeze film dampers with various types of end seals Stiffness, damping, and mass matrices for centrifugal pump impellers Destabilizing cross coupled force coefficients for impellers of centrifugal compressors Destabilizing cross coupled stiffness coefficients for unshrouded turbines

XLAnSeal XLCGrv XLLaby XLIsotSL XLLubGT XLJrnl HLHydPad XLTFPBrg XLPresDm XLBalBrg XLLSFD XLOSFD XLSFDFEM XLPIMPLR XLWachel XLClrEx

Rotating Machinery Analysis Inc., Austin, TX, USA


coefficients: k¼ c¼

24R3 Lmv1 Cr3 ð2 þ 12 Þð1 12 Þ

12pR3 Lm Cr3 ð2 þ 12 Þð1 12 Þ1=2

34 35

Oil Inlet

ð34:52Þ ð34:53Þ

where

'O' Rings

R ¼ the damper radius v ¼ whirl speed L ¼ length of damper m ¼ viscosity of oil Cr ¼ the radial clearance 1 ¼ eccentricity ratio (orbit radius/Cr )

Rolling element bearing

Similarly, for a damper without end seals, the short journal bearing theory yields the following stiffness and damping coefficients: k¼ c¼

2RL3 mv1 Cr3 ð1 12 Þ2 pRL3 m 2Cr3 ð1

3

12 Þ 2

ð34:54Þ ð34:55Þ

Although the above equations, based on the Reynolds’ equation, have been proposed to predict damper characteristics, the experimental evidence does not validate these equations. Therefore, these equations should be used with caution for practical purposes. 34.3.2.4

Spring element

FIGURE 34.12

A squeeze film damper.

Annular Seals

Annular seals are primarily used in pumps, compressors, gas turbines, and steam turbines to minimize leakage and thereby improve the volumetric efficiency of the machine. In addition to their basic function, they also play a vital role in the rotordynamics of the machine, especially in multistage machines, providing stiffness and damping and thereby enhancing high speed operational capability. In fact, in the last few decades, most of the development work on seals has focused on understanding and improving their dynamic vibration characteristics rather than improving their efficiency in sealing. Lomakin (1958) was the first to publish on the restoring forces in smooth annular clearances in pumps. However, it was more than a decade later that Black (1968) provided the major initial impetus for the understanding and development of seals. Childs (1993) provided an excellent compendium of the research work conducted in the area of seals. His book also provides the most comprehensive coverage of the subject of seal dynamics. In the present context, seals are handled in the same manner as the stiffness and damping characteristics of journal bearings with some degree of modifications. In particular, fluid inertia effects are included, and it is assumed that the center of the shaft orbit is the same as the center of the stationary seal ring. Assuming rotational symmetry the reaction force seal motion model can be represented by the following equation: " #( ) " #( ) " #( ) ( ) FX k kc X c cc m 0 X_ X€ ¼ ð34:56Þ FY kc k Y cc c 0 m Y_ Y€

34 36


An added complexity is the predominance of turbulent flow in annular seals. This invalidates the use of Reynolds’ equation for the derivation of seal coefficients. The highest degree of accuracy can be obtained by the direct solution of the Navier Stokes and continuity equations. However, at the present moment, such methods are considered to be excessively costly and impractical. As a result, two practical semiempirical methods have been developed to derive seal coefficients. In the first approach, the semiempirical turbulent model is directly substituted in the Navier Stokes equation and a numerical technique is used for its solution. The second, most commonly used technique uses a bulk flow model together with control volume formulations, namely, the continuity equation and momentum equation, to obtain the desired results. For a detailed discussion of these methods, solution techniques, the influence of various physical parameters on the coefficients, and an excellent compilation of computational and experimental results, the publication by Childs (1993) is recommended. 34.3.2.5

Impeller –Diffuser/Volute Interface

It is widely known that the flow fields within certain types of rotating machinery can significantly influence its vibration behavior. Thomas (1958) recognized and explained the presence of destabilizing clearance excitation forces in axial flow steam turbines. Black (1974) was the first to suggest that centrifugal pump impellers could also develop destabilizing forces. The nature of these forces and their influence on rotor instability has been explained in Section 34.2.2.2 and Section 34.2.2.3 of this chapter. The impeller diffuser/volute forces assuming rotational symmetry can generally be modeled by an equation of the following form: #( ) #( ) " " #( ) " ( ) m mc c cc k kc X FX X€ X_ ð34:57Þ ¼ mc m cc c kc k Y FY Y€ Y_ For analytical procedures for the derivation of impeller interaction coefficients and a comparison of experimental data, the work by Childs (1993) is recommended. It is well recognized that a considerable amount of work still needs to be done towards understanding the complex nature of impeller diffuser/ volute interactive forces, especially at off design conditions.

34.3.3

Design

Since the real machine is not available for tests, at the preliminary design stage it is a common practice to develop an accurate mathematical model of the machine to predict its dynamic behavior in operation. It is also prudent to understand and estimate how the machine will interact with its operating environment and how the environment could influence the operation of the machine. A suitable model of the rotor can be developed using the techniques described in Section 34.3.2, and the rotordynamic characteristics of the machine can be analyzed using one of the methods described in Section 34.3.1, above. Based on these methods, numerous computer based rotordynamic analysis programs have been developed. A listing of the most widely known computer programs in North America is given in Table 34.3. The objectives of the analysis are to predict the critical speeds, excitation frequencies, the amplitudes of deflection, and the magnitude of the forces of the rotor within its full operating range. In certain situations, evaluation of the energy content of the excitation may also be required. Once the mathematical model is developed, the eigenvalues of the rotor and the mode shapes can be determined. The results can then be presented in the form of a Campbell diagram, where the eigenvalues along with the excitation frequencies are plotted as a function of rotor speed. Critical speeds occur at the speeds corresponding to the points of intersection of the excitation frequency lines and the eigenvalue lines. The Campbell diagram presentation (Figure 34.13) of the results is very useful since the influence of key parameters such as stiffness, damping, clearances (new and worn conditions), and so on can all be shown on the same diagram. A critical speed, although present, may be of little consequence if it is associated with sufficient damping. As illustrated in Figure 34.4, when the damping ratio z $ 0:707; the system is critically damped and above this level of damping there is no amplification of the rotor deflection. At or near a critical speed the amplification factor is < 1=2z: Using this estimated value,

160

200

34 37

3080 3600 Rotor Speed [rpm]

2560

3080 3600 Rotor Speed [rpm]

60

75

2560

0

15

30

45

Damping [%]

1000 1520 2040 ---- Synchronous Excitation

n max

0

nn

nmin

40

80

120

Frequency [Hz]


1000

1520

2040

List of Symbols: Mode 1. Mode 2. Mode 3. Pump Stat: New Analysis...... 807281012. FIGURE 34.13

FEED PUMP

Campbell diagram for a multistage pump.

depending on the internal clearances of the machine, it is possible to assess if the rotor can pass through a critical speed without causing damage to the components. An amplification factor of 2.5 or below is a typical acceptance limit for centrifugal pumps, even for continuous operation at or near it. However, if the amplification factor exceeds the acceptable limits or the critical speeds are too close to the continuous operating speed, then design modifications have to be made to change the critical speed values. At the design stage, it is considered good practice to ensure that the critical speeds are not within ^10% of the continuous operating speed; these limits are sometimes referred to as separation margins. The mode shapes of the rotor are important from the standpoint of identifying where the maximum deflections occur. It also provides a good guide for assessing design modifications to improve damping or reduce sensitivity to unbalance forces.

34 38


The eigen analysis only provides relative deflections of the rotor. In order to estimate true deflections, a forced response analysis has to be made. Forcing functions of estimated magnitude are applied at selected locations to determine the resulting deflections at specific points on the rotor. This type of analysis is typically carried out for synchronous excitation forces only. The nature of the forcing function depends on the type of the machine; mechanical unbalance is common to all types of machines, whereas hydraulic unbalance is relevant to centrifugal pumps and electrical unbalance to electric motors. The challenge, of course, is to determine the magnitudes, directions, and locations of the forces to apply and how the resulting rotor response should be judged. Of course, these criteria are machine type dependent and not necessarily applicable to all types of rotating machinery. An example of how forced response analysis on centrifugal pumps is evaluated is given below (Bolleter et al., 1992): 1. Maximum amplification factors and required separation margins are defined by specifications; example as shown in API 610, 8th edn., 1995. 2. Excitation forces are defined and the response is judged relative to admissible shaft vibration limits, and relative to clearances. 3. Apply unbalance forces of such a magnitude that maximum permissible vibration limits at the vibration probe locations are reached, and then evaluate if the deflections exceed the minimum clearances in the machine. 4. Apply an unbalance force of arbitrary magnitude and determine the resulting response at the same or another location, and calculate the sensitivity factor (SF) using the following formula: SF ¼ where a ¼ rotor deflection U ¼ unbalance force M ¼ rotor mass

a M U

ð34:58Þ

The sensitivity factor should then be compared with experimental base values of similar machines for acceptance. The rotor responses to the applied forces can be further analyzed to extract other parameters of interest, such as phase angles and force magnitudes at the bearings, in order to evaluate the design. In order to optimize the rotating machine design in terms of placement of critical speeds and control of deflections and forces, a parameter sensitivity coefficients analysis (Lund, 1979; Rajan et al., 1986; Rajan et al. 1987) may be carried out. For speed and convenience of analysis, the optimization routine can be automated.

*

*

*

*

*

Rotordynamic analysis is a part of the current rotating machinery design practice used to predict their vibration behavior. The most current rotordynamic analytical procedures are computer based and are derived from the lump parameter model or the transfer matrix method. In the lumped parameter model method, the distributed elastic and inertial properties of the rotor are represented as a collection of rigid bodies connected by massless elastic beams. In the transfer matrix method, commonly called the FEA method, the rotor is represented as an assembly of elements with distributed elastic and inertial properties. Accurate modeling and representation of rotor components is vital to the accuracy and reliability of analysis results. As a result, significant advancement in modeling shaft sections, disks, impellers, bearings, seals, rotor dampers, and rotor stator interactions have been made.


34.4

34 39

Vibration Measurement and Techniques

34.4.1

Units of Measurement

The most commonly used units of measurement of vibration are as follows: *

*

*

*

*

Displacement (peak to peak): millimeter (mm) or micron (10 6 m) in metric units, and thou or mil (0.001 in) in imperial units Velocity (peak, RMS or “true peak”): mm/sec or m/sec in metric units, and in./sec in imperial units Acceleration (RMS): g or m/sec2 in metric units, and g or in./sec2 in imperial units Frequency: hertz (Hz) or cycles/min (cpm) in both systems of units Phase angle: degrees in both systems of units

34.4.2

Measured Parameters and Methods

Under steady state conditions, the vibration from a rotating machine is a periodic signal of a complex waveform. During unstable operation or upset conditions, the signal may become random in nature. In certain types of machines, transient signals that are nonperiodic could also be present due to internal impacts and damping. Based on the simple spring mass system and the mathematical Fourier analysis procedure, all periodic complex waveforms can be reduced to the sum of a series of sinusoidal functions. In the case of random signals, averaging techniques are used to reduce them to periodic signals for convenience of analysis. A quantitative assessment of the vibration can be made in terms of the amplitude, velocity, acceleration, or the magnitude of force of the motion. Other key parameters such as frequency, phase angle, and the time varying nature of the signal are important in fully characterizing it. Because the true signal is not purely sinusoidal, it is important to identify its magnitude as, peak, peak to peak or root mean square (RMS). The preferred parameter of measurement varies throughout the industry and depends on the nature, complexity, and type of machine, and the purpose for which it is measured. A general classification of measured parameters and techniques based on industry and rotating machine type is given in Table 34.4.

*

*

34.5 34.5.1

The displacement amplitude, velocity, acceleration, frequency, and phase angle of the vibration signal are some of the parameters that can be used to assess the condition of a rotating machine. The vibration signals generated by a typical rotating machine is complex in nature and, therefore, requires various mathematical analysis procedures and signal averaging techniques to reduce them to simple and interpretable forms.

Vibration Control and Diagnostics Standards and Guidelines

Given the fact that some degree of vibration is always present in rotating machinery, some means of judging “how much is too much” has to be established so that vibrations can be controlled within reasonable limits. When such a judgment is not based on a scientific method, there is room for speculation and it will depend on the one making the decision, the manufacturer, end user, the governing authority, and so on. Since vibrations are a key indication parameter of the performance of rotating machinery, different interest groups monitor it for a variety of purposes. It can be used as a measure of

34 40 TABLE 34.4

Vibration and Shock Handbook Measurement Parameters and Techniques

Measurement/Technique Acceleration, RMS

Bode/Nyquist plot Cepstrum analysis Condition monitoring

Displacement peak to peak (a) Absolute (b) Relative (c) External Modal analysis

Orbit analysis Polar plots

Phase angle

Description When high frequency or force of the vibration is of interest

Gear boxes, rolling element bearings, gas/steam turbines. Mainly used by defense and aerospace industry Plot of displacement amplitude and phase Observe critical speeds and instability in angle vs. speed machines using journal bearings Inverse Fourier transform of logarithmic To detect families of harmonics and power spectrum sidebands in gearboxes, rolling element bearings, and electric motors On critical equipment where predictive Analysis of signals generated by the maintenance programs are used. Can machine to determine its condition on reduce equipment redundancy a continuous or periodic basis

Absolute displacement amplitude of rotor vibration

Where the rotor mass is very much larger than the stator mass. Large motors, generators, and fans Relative displacement amplitude of Commonly used on machines with journal rotor vibration bearings or in close clearance seals Absolute displacement amplitude of stator Low speed machines (less than 1000 rpm) component vibration To measure the vibration response of To determine the modal mass, stiffness a structure to an applied force. and damping properties of a structure. The force can be periodic or Also used to measure structural natural an impact force frequencies The path of the shaft centerline Diagnostics of machines using journal motion during rotation bearings. Provides a picture of the motion of the journal in the bearing A polar graph of amplitude versus Similar to Bode plots, can be used to phase at various machine speeds detect critical speeds and instability. Modal properties can also be extracted from polar plots Phase angle of vibration signal Useful in balancing, diagnosing critical speeds, and misalignment problems

Rolling element bearing analysis (a) Acceleration Measure the amplitude of all pass and discreet frequency accelerations (b) Shock pulse method (c) Envelope technique

(d) Spike energy method (e) Kurtosis method Run up, run down analysis (waterfall/ cascade plots)

When/Where Used

A high frequency resonance technique tuned to the detector natural frequency Bearing defects cause periodic impacts, which make bearing components resonate. Demodulation and enveloping techniques are used to detect the impact (fault) frequencies Measure the broadband acceleration over the 5 45 kHz range The normalized fourth moment of the probability distribution of acceleration over the 2 80 kHz range Three dimensional plot of frequency or time spectrum vs. time or speed

When damage has progressed to generate audible noise amplitudes increase. Useful from 5 5 kHz Early detection of failure, measures ultrasonic noise. Proprietary technique Used for early failure detection (ultrasonic noise) as well as for advanced stages of damage (audible noise) Used for early failure detection (ultrasonic noise) as well as for advanced stages of damage (audible noise) Used for early failure detection (ultrasonic noise) as well as for advanced stages of damage (audible noise) Used for diagnosing a variety of vibration problems. Helpful in analyzing transient signals


34 41

(continued)

Measurement/Technique

Description

Spectrum analysis

Plot of amplitude vs. frequency of vibration

Trend analysis

Time domain analysis

Vibration data collected periodically over an extended time domain Averaging of time records using triggering at the same point of the waveform of a repetitive signal Plot of amplitude versus time

Velocity peak or RMS

Velocity amplitude of vibration signal

Time averaging

When/Where Used Used for diagnostics, to determine frequency, harmonics, side bands, beats, transfer functions, etc., and to control of vibrations levels at discreet frequencies Useful in predictive maintenance programs in assessing machine conditions Used in the analysis of faulty gearboxes. Can reduce asynchronous components in the signal and improve signal to noise ratio To observe amplitude modulation, beats, impacts, transients, and phase angle. Very useful in diagnostics Parameter most commonly used in many industries to monitor vibrations. Peak readings relate to peak stress levels and rms to energy of vibrations

quality and workmanship or the common basis for acceptance between the user and manufacturer. From a safety point of view, the operator can establish normal, alarm, and shutdown levels based on vibration limits. Vibration levels are also used in making maintenance decisions in rotating machinery. Rathbone (1939) was the first to publish guidelines for vibration limits for machinery, based on his experience as an insurance agent. Since that time, numerous individuals, organizations, and governing bodies have developed a variety of guidelines and standards for vibration levels in rotating machinery. A listing of the more commonly used guidelines and standards is given in Table 34.5. It should be recognized that these are experience based standards, and therefore, will grow and develop with technology. ISO 10816 Part 1 to Part 5 is a comprehensive set of standards that has been developed for the evaluation of vibration of rotating machinery by measuring the vibration response on nonrotating, structural components such as bearing housings. Vibration measuring points as specified by these standards are shown in Figure 34.14. In a similar vein, ISO 7919 Part 1 to Part 5 has been developed for the evaluation of vibration by measuring the vibration on rotating shafts. These standards cover the most widely used types of rotating machinery and they relate to both operational monitoring and acceptance testing of equipment. Table 34.6 and Table 34.7 are derived from these standards and are presented as a general guideline for vibration limits of rotating machinery. For specific details, including the limitations of the standards, the reader is advised to refer to the relevant sections of ISO 10816 and ISO 7919 Standards.

34.5.2

Vibration Cause Identification

Vibrations are an inherent part of all rotating machinery. Vibration can be due to many causes: improper design, practical manufacturing limits, poor installation, the effect of system environment, component deterioration, operation outside of design limits, or a combination of the above. At times, finding the exact cause of vibration can be quite a challenge, as several of the causes have similar symptoms. Table 34.8 is a list of the more commonly known causes of vibration in rotating machinery and their symptoms.

34.5.3

Vibration Analysis — Case Study

In the past, it was common practice to operate centrifugal pumps at a fixed speed and attain required flow changes by means of throttling. This forces the pump to operate at low efficiency conditions

34 42 TABLE 34.5 Year

Vibration and Shock Handbook Vibration Guidelines and Standards Author/ Organization

Reference Number

2002

AGMA

ANSI/AGMA 6000 B96

2003

API

ANSI/API std 541 2003

1997 2004

API API

1997

API

API STD 546, second edition API STD 610/ISO 13709, ninth edition API STD 611, fourth edition

2005

API

2003

API

API STD 612/ISO 10437, sixth edition API STD 613, fifth edition

1998

API

API STD 616, fourth edition

2002

API

API STD 617, seventh edition

2000 2004

API API

API STD 670, fourth edition API STD 672, fourth edition

2001 1997

API API

API STD 673, second edition API STD 677, second edition

1996

API

API STD 681, first edition

2000

API

API STD 685, first edition

1965 1964

BDS Blake, M.P.

1963 1975 1971 1968 1966

CAGI CAGI CSN Dresser Industrial Gosstandart

BDS 5626 65 Hydrocarbon Processing, January 1964

2002

HI

ANSI/HI 9.6.4

1996

IEC

IEC 60034 14

1964 IRD 1995 2001 ISO

CSN 011410 GOST 12379 66

IRD #305D ISO 10816 1:1995 ISO 10816 2:2001 ISO 10816 3:1998 ISO 10816 4:1998

Title/Description Specification for Measurement of Linear Vibration on Gear Units Form Wound Squirrel Cage Induction Motors 500 hp and Larger Brushless Synchronous Machines, 500 kVA and Larger Centrifugal Pumps for Petroleum, Petrochemical and Natural Gas Industries General Purpose Steam Turbines for Petroleum, Chemical and Gas Industry Services Petroleum, Petrochemical and Natural Gas Industries Steam Turbines Special Purpose Applications Special Purpose Gear Units for Petroleum, Chemical and Gas Industry Services Gas Turbines for the Petroleum, Chemical, and Gas Industry Services Axial and Centrifugal Compressors and Expander compressors for Petroleum, Chemical and Gas Industry Services Mechanical Protection Systems Packaged Integrally Geared, Centrifugal Air Compressors for Petroleum, Chemical, and Gas Industry Services Special Purpose Fans General Purpose Gear Units for Petroleum, Chemical, and Gas Industry Services Liquid Ring Vacuum Pumps for Petroleum, Chemical, and Gas Industry Services Sealless Centrifugal Pumps for Petroleum, Heavy Duty Chemical, and Gas Industry Services Measurement of Vibration on Electrical Rotating Machines New Vibration Standards for Maintenance In Service Standards for Centrifugal Compressors Standard for Centrifugal Air Compressors Permitted Limits for Unbalanced Solid Machine Elements General Guidelines for Vibration on Clark Centrifugal Compressors Measurement of Vibration on Electrical Rotating Machines Centrifugal and Vertical Pumps Vibration Measurement and Allowable Values Rotating Electrical Machines, Part 14: Mechanical Vibrations of Certain Machines with Shaft Heights 56 mm and Higher Measurement, Evaluation and Limits of Vibration General Machinery Vibration Severity Chart Mechanical Vibration Evaluation of Machine Vibration by Measurements on Nonrotating Parts: Part 1: General Guidelines Part 2: Land Based Steam Turbines and Generators in Excess of 50 MW with Normal Operating Speeds of 1500, 1800, 3000 and 3600 rpm Part 3: Industrial Machines with Nominal Power above 15 kW and Nominal Speeds between 120 and 15,000 rpm when Measured In Situ Part 4: Gas Turbine Driven Sets Excluding Aircraft Derivations

Vibration in Rotating Machinery TABLE 34.5 Year

(continued) Author/ Organization

Reference Number ISO 10816 5: 2000

2002

34 43

ISO

1996 2001 ISO

ISO 14839 1: 2002 ISO/CD 14839 2:2004 ISO 7919 1: 1996 ISO 7919 2: 2001 ISO 7919 3: 1996 ISO 7919 4: 1996 ISO 7919 5: 1997

1993

ISO

2004

ISO

ISO 8579 2

ISO 15242 1:2004 ISO 15242 2:2004 ISO/CD 15242 3

1959 1967

Kruglov, N.V. Maten, S

1983

McHugh, J.D.

1974

MIL

Teplonerg, 8 (85), 1959 Hydrocarbon Processing, January 1967 J. Lub. Tech., Trans. ASME, 1983, 105 MIL STD 167 1

2003

NEMA

NEMA MG 1 2003

1991 1991 1965

NEMA NEMA PKN

NEMA SM 23 1991 NEMA SM 24 1991 PN 65/E 04255

1939 1964

Rathbone, T.C. Power Plant Engineering, November 1939 VDI VDI 2056

1982

VDI

VDI 2059 P1

1990

VDI

VDI 2059 P2

1985

VDI

VDI 2059 P3

1981

VDI

VDI 2059 P4

1982

VDI

VDI 2059 P5

1949

Yates, H.G.

Trans. N.E. Coast Inst. Engrs Ship Builders, Vol. 65, 1949

Title/Description Part 5: Machine Sets in Hydraulic Power Generating and Pumping Plants Mechanical Vibration Vibration of Active Magnetic Bearing Equipped Rotating Machinery Part 1: Vocabulary Part 2: Evaluation of Vibration Mechanical Vibrations of Nonreciprocating Machines Measurement on Rotating Shafts and Evaluation Criteria Part 1 (1996): General Guidelines Part 2 (2001): Land Based Steam Turbines and Generators in Excess of 50 MW with Normal Operating Speeds of 1500, 1800, 3000 and 3600 rpm Part 3 (1996): Coupled Industrial Machines Part 4 (1996): Gas Turbine Sets Part 5 (1997): Machine Sets in Hydraulic Power Generating and Pumping Plants Acceptance Code for Gears, Part 2: Determination of Mechanical Vibration of Gear Units During Acceptance Testing Rolling Bearings Measuring Methods for Vibration Part 1: Fundamentals Part 2: Radial Ball Bearings with Cylindrical Bore and Outside surface Part 3: Spherical and Taper Radial Roller Bearings with Cylindrical Bore and Outside Diameter Turbomachine Vibration Standards New Vibration Velocity Standards Estimating the Severity of Shaft Vibration within Fluid Film Journal Bearings Mechanical Vibration of Shipboard Equipment, Type I: Environmental, Type II: Internally Excited Motors and Generators, Part 7 Mechanical Vibration Measurement, Evaluation and Limits Steam Turbines for Mechanical Drive Service Land Based Steam Turbine Generator Sets 0 to 33,000 kW Measurement of Vibration of Electrical Rotating Machines Vibration Tolerances Evaluation Criteria for Mechanical Vibrations in Machines Shaft Vibrations of Turbosets Principles for Measurement and Evaluation Shaft Vibrations of Steam Turbosets for Power Station Measurement and Evaluation Shaft Vibrations of Industrial Turbosets Measurement and Evaluation Shaft Vibrations of Gas Turbosets Measurement and Evaluation Shaft Vibrations of Hydraulic Machinesets Measurement and Evaluation Vibration Diagnosis of Marine Geared Turbines

34 44


FIGURE 34.14 (a) Measuring points; (b) measuring points for vertical machine sets. (Source: ISO 10816 3, 1998 05 15. With permission.)


34 45

FIGURE 34.14 (continued )

34 46 TABLE 34.6A

Vibration and Shock Handbook Acceptable Vibration Levels for Rotating Machinery Measured on Nonrotating Parts

Machinery Type

Power Level

Speed Range (rpm)

Applicable Vibration Level Rigid Support Flexible Support

Steam turbines

15 # P # 300 kW 300 kW # P # 50 MW P . 50 MW P . 50 MW P . 50 MW

120 # N # 15,000 120 # N # 15,000 N , 1,500 or N .3,600 N ¼ 1,500 or 1,800 N ¼ 3,000 or 3,600

V1 and D3 V3 and D5 V3 and D5 V5 V7


Gas turbines

15 # P # 300 kW 300 kW # P # 3 MW P . 3 MW

120 # N # 15,000 120 # N # 15,000 3,000 # N # 20,000

V1 and D3 V3 and D5 V8

V3 and D7 V6 and D8 V8

Hydraulic turbines and pump turbine

Horizontal machines P . 1 MW P . 1 MW

60 # N # 300 300 , N # 1,800

N/A V2 and D6

V4 N/A

P . 1 MW P . 1 MW

60 , N # 1,800 60 , N # 1,000

V2 and D6 V2 and D6

N/A V4 and D9

P . 15 kW P . 15 kW

120 # N # 15,000 120 # N # 15,000

V3 and D2 V1 and D1

V6 and D4 V3 and D2

Electric motors Shaft height H $ 315 mm P . 15 kW Shaft height 160 # H , 315 mm P . 15 kW

120 # N # 15,000 120 # N # 15,000

V3 and D5 V1 and D3

V6 and D8 V3 and D7

Vertical machines Centrifugal pumps Separate driver Integral driver

Generators, excluding those used in hydraulic power generation

15 # P # 300 kW 300 kW # P # 50 MW P . 50 MW P . 50 MW P . 50 MW

120 # N # 15,000 120 # N # 15,000 N , 1,500 or N . 3,600 N ¼ 1,500 or 1,800 N ¼ 3,000 or 3,600



Generators and motors used in hydraulic power generation

Horizontal machines P . 1 MW P . 1 MW

60 # N # 300 300 , N # 1,800

N/A V2 and D6

V4 N/A

P . 1 MW P . 1 MW 15 # P # 300 kW 300 kW # P # 50 MW

60 , N # 1,800 60 , N # 1,000 120 # N # 15,000 120 # N # 15,000

V2 and D6 V2 and D6 V1 and D3 V3 and D5

N/A V4 and D9 V3 and D7 V6 and D8

Vertical machines Compressors, rotary, blowers, and fans

resulting in wasted energy and premature failure of components due to high vibration. The current practice to obtain flow changes in the pump is by means of speed change. This eliminates flow throttling and allows the pump to operate close to its best efficiency point, where energy is not wasted and vibrations are a minimum. However, as illustrated below, variable speed operation of a pump motor set over a wide speed range could pose several challenging problems. TABLE 34.6B

Maximum Vibration Velocity Limits for Different Levels (mm/sec, RMS)

Vibration Level

Zone A

Zone B

Zone C

Alarm

Trip

V1 V2 V3 V4 V5 V6 V7 V8

1.4 1.6 2.3 2.5 2.8 3.5 3.8 4.5

2.8 2.5 4.5 4.0 5.3 7.1 7.5 9.3

4.5 4.0 7.1 6.4 8.5 11.0 11.8 14.7

3.5 3.1 5.6 5.0 6.6 8.9 9.4 11.6

5.6 5.0 8.9 8.0 10.6 13.8 14.8 18.4

Vibration in Rotating Machinery TABLE 34.6C

34 47

Maximum Vibration Displacement Limits for Different Levels (mm, RMS)

Vibration Level

Zone A

Zone B

Zone C

Alarm

Trip

D1 D2 D3 D4 D5 D6 D7 D8 D9

11 18 22 28 29 30 37 45 65

22 36 45 56 57 50 71 90 100

36 56 71 90 90 80 113 140 160

28 45 56 70 71 63 89 113 125

45 70 89 113 113 100 141 175 200

Zone A: Newly commissioned machines should fall within this zone. Zone B: Machines with vibrations within this zone are considered acceptable for long term operation. Zone C: Machines with vibrations within this zone are normally considered unsatisfactory for long term operation. Such a machine may be operated for a short period in this condition. Alarm: The values chosen will normally be set relative to a baseline value determined from experience. However, it is recommended that the alarm value shall not exceed those given herein. Trip: The values will generally relate to the mechanical integrity of the machine. They will generally be the same for all machines with similar design. It is recommended that the trip value shall not exceed those given herein. Notes: (1) The measured vibration is broadband, and the frequency range will depend on the type of machine being considered. A range from 2 to 1000 Hz is typical except for in high speed machines, .10,000 rpm, where the upper limit should at least be six times the rotational frequency. (2) It is common practice to evaluate rotating machinery based on the broadband RMS vibration velocity, since it can be related to the vibration energy levels. However, other quantities such as vibration displacement or acceleration may be preferred. Especially low speed machines can have unacceptably large vibration displacements when the 1 £ rpm component is dominant. Therefore, where specified, both the velocity and displacement criteria are met. (3) Since typical vibration waveforms measured on rotating machinery are complex in nature, there is no simple relationship between broadband velocity, displacement, and acceleration. (4) Vibration measurements shall be taken on bearing support housings, or other structural components, which adequately respond to the dynamic forces of the machine. Recommended locations for bearing housings are shown in Figure 34.14. (5) For certain types of machines, the axial vibration limits may differ from those for radial directions. Also, within the same machine set, in particular hydraulic power generating sets, the applicable level may differ from bearing to bearing depending on its classification as a rigid or flexible support. (6) Above vibration limits apply to steady state/normal operating conditions of the machine. If the vibration levels are sensitive to the operational conditions, then evaluation of the machine for operating conditions outside steady state conditions will have to be based on different criteria. (7) The vibration limits specified herein should not be used to assess the condition of rolling element type bearings although it encompasses machines that may have these types of bearings. (8) It must be recognized that the vibration measurement on nonrotating parts alone does not form the only basis for judging the condition of a machine. In certain types of machines, it is common practice also to judge the vibration based on measurements taken on rotating shafts. (9) A support may be considered as rigid in a specific direction only if its natural frequency in that direction exceeds the main excitation frequency by at least 25%, otherwise it is considered to be flexible. In some cases, a support may be rigid in one direction and flexible in another. (10) In the case of hydraulic machine sets, major differences in radial bearing support arrangement can occur. For evaluation of the support type it is recommended that the reader refer to ISO 10816 5.

Description. The following case study is taken from a petroleum pipeline pump application where pump motor sets with VFDs were installed in a new pipeline starting in Alberta, Canada and terminating in Minnesota, USA. VFDs are frequently used in the pipeline industry to power high horsepower pumps to eliminate power wasted by throttling, reduce inrush current at motor startup, and to provide greater operating flexibility. However, variable speed operation can cause vibration problems in the pump, motor, and the couplings that are not normally experienced with fixed speed pumps. Unexpected high torsional and lateral vibrations were experienced with these pumps and motors at certain operating speeds. A rotor torsional resonance, motor housing resonance, acoustic resonance in the internals of the pump, and discharge piping were identified to be the causes of the high vibration in the pump motor set. Details on diagnosing the problems and the corrective measures taken to resolve them are given below: Pump type: The pump was a centrifugal, two stage, double volute horizontal pump, with six vane impellers, normally designed to operate at a fixed speed. Generation of pressure pulsations at the vane passing frequencies of 6 £ and 12 £ rotational speed is normally expected.

34 48


TABLE 34.7A

Acceptable Vibration Levels for Rotating Machinery, Measured on Rotating Shafts

Machinery Type

Power Level

Speed Range (RPM)

Applicable Vibration Level Relative Displacement

Steam turbines

Gas turbines Hydraulic turbines and pumps used in hydraulic power generation and pumping plants Centrifugal pumps Electric motors Generators, excluding those used in hydraulic power generation

Generators and motors used in hydraulic power generation Compressors, rotary, blowers, and fans

P P P P P P P P

# 50 MW . 50 MW . 50 MW . 50 MW . 50 MW . 3 MW # 3 MW . 1 MW

1,000 # N # 30,000 N ¼ 1,500 N ¼ 1; 800 N ¼ 3,000 N ¼ 3,600 3,000 # N # 30,000 1,000 # N # 30,000 60 # N # 1,800

D8 D5 D4 D2 D1 D8 D8 D9

All All P # 50 MW P . 50 MW P . 50 MW P . 50 MW P . 50 MW P . 1 MW P . 1 MW All

1,000 # N # 30,000 1,000 # N # 30,000 1,000 # N # 30,000 N ¼ 1,500 N ¼ 1,800 N ¼ 3,000 N ¼ 3,600 60 # N # 1,000 1,000 , N # 1,800 1,000 # N # 30,000

D8 D8 D8 D5 D4 D2 D1 D9 D8 D8

Absolute Displacement D7 D6 D5 D3 D9

D7 D6 D5 D3 D9

Motor: The motor was a 3000 hp, two pole horizontal induction motor, designed to operate at 3600 rpm. Supply: The supply was a VFD of the current source inverter type. These drives are known to generate an oscillatory torque at 6 £ and 12 £ the operating frequency. Coupling: Flexible disc type coupling with a spacer was used. These couplings have very little torsional damping capacity. Speed range: The speed range was from 1440 rpm (24 Hz) to 3900 rpm (65 Hz). Reference: Refer to Figure 34.15 to Figure 34.18. As for the resonance at second and third torsional critical speeds (Figure 34.15) the second and third torsional modes are excited when the 6 £ component of rotational speed corresponds to the critical speeds of 92 and 268 Hz, respectively. The 6 £ rpm torsional excitation is caused by the pressure pulsations in the pump. The waterfall plot (Figure 34.15a) was taken during a run down of the set with the power to the motor turned off. A similar plot taken during run up of the motor (Figure 34.15b) shows excitations at the same frequencies but having different amplitude. Since both the pump and motor generate 6 £ excitation, it suggests a phase difference between the excitation torques. It is important to note that the conventional vibration monitoring devices cannot detect the torsional resonance problem. The only indication of a problem was the unusual chattering noise emitted by the coupling. Special techniques to measure dynamic torque using strain gauges had to be used to detect the torsional vibrations. As for the motor housing resonance (Figure 34.16), the 2 £ rotational speed vibration of the motor is dominant and peaks at 118 Hz corresponding to a natural frequency of the motor frame. In the waterfall plot, the natural frequencies corresponds to excitations that are parallel to the axis. Excitation at harmonics, including the 6 £ component, is present but is not dominant. As for the pump vibrations at the vane passing frequency (Figure 34.17), the 6 £ vane passing frequency is dominant at all operating speeds. It peaks at 238 Hz, possibly due to an acoustic resonance in

Vibration in Rotating Machinery TABLE 34.7B

Maximum Vibration Displacement Sp Zone A

D1 D2 D3 D4 D5 D6 D7 D8 D9

10ð2:3381

75 80 90 90 100 110 120 pffiffi 4800= n

0:0704 log nÞ

34 49

p

Limits (mm) Peak to Peak Limits for Different Levels Zone B

10ð2:5599

150 165 180 185 200 220 240 pffiffi 9000= n

0:0704 log nÞ

Zone C

10ð2:8609

240 260 290 290 320 350 385 pffiffi 13,200/ n 0:0704 log nÞ

Zone A: Newly commissioned machines should fall within this zone. Zone B: Machines with vibrations within this zone are considered acceptable for long term operation. Zone C: Machines with vibrations within this zone are normally considered unsatisfactory for long term operation. Such a machine may be operated for a short period in this condition. Alarm: The values chosen will normally be set relative to a baseline value determined from experience. However, it is recommended that the alarm value shall not exceed those given herein. Trip: The values will generally relate to the mechanical integrity of the machine. They will generally be the same for all machines with similar design. It is recommended that the trip value shall not exceed those given herein. Notes: (1) The measured vibration is broadband and is shaft vibration displacement peak to peak. Where applicable, vibration limits for both absolute and relative radial shaft vibrations are given in certain cases. (2) Relative displacement is the vibratory displacement between the shaft and an appropriate structural component such as the bearing housing. Absolute displacement is the vibratory displacement of the shaft with reference to an inertial frame of reference. (3) Relative measurements are carried out with a noncontacting transducer. Absolute readings are obtained by one of the following methods: by a shaft riding probe on which a seismic transducer is mounted so that it measures absolute shaft displacement directly, or with the combination of a noncontacting transducer which measures relative shaft displacement and a seismic transducer which measures support vibration. Their conditioned outputs are vectorially added to provide a measure of the absolute shaft motion. (4) The vibration evaluation criteria are dependent upon a variety of factors and the criteria adopted will vary for different types of machines. Some of these factors are the bearing type, clearance, and diameter. The adopted criteria have to be compared with the bearing diametral clearance ðCÞ and adjusted to suit. Typical values are: Zone A #0.4C; Zone B #0.6C; and Zone C #0.7C. (5) Above vibration limits apply to steady state/normal operating conditions of the machine. If the vibration levels are sensitive to the operational conditions then evaluation of the machine for operating conditions outside steady state conditions will have to be based on different criteria. (6) It is recommended that vibration readings at each location be made with a pair of transducers and that the transducers are mounted perpendicular to the shaft axis and they are at an angle of 908 to one another. The vibration limits apply to each measured direction. (7) The mechanical and electrical run out at each measurement location must be assessed and should be ,25% of the allowable limit or 6 mm, whichever is greater. (8) It must be recognized that the vibration measurement on rotating shafts does not form the only basis for judging the condition of a machine. In certain types of machine, it is common practice also to judge the vibration based on measurements taken on nonrotating parts. (9) ALARM levels should be set relative to a baseline value determined from experience for the measurement position, direction and type of machine. It must provide a warning that a defined value, which is significantly above the baseline value, has been reached. The maximum ALARM setting should be #0.75C. (10) The TRIP values should be based on protecting the mechanical integrity of the machine. Consideration of damage to bearings is typical; therefore, maximum TRIP setting should be #0.9C.

the discharge pipe. It does not seem to correspond to a structural natural frequency due to the absence of excitations at 238 Hz at all speeds. As for the acoustic resonance in the pump cross over pipe (Figure 34.18), dynamic pressure pulsation measurements made on the pump cross over pipe from the first stage discharge to the second stage suction show an acoustic resonance at 540 Hz. The consistent presence of some excitation at 540 Hz at all speeds confirms that it is an acoustic natural frequency of the cross over pipe. When the 6 £ rpm pressure pulsation frequency coincides with the acoustic natural frequency, a resonance condition occurs and the magnitude of the pressure pulsation increases by almost a factor of 30. As for corrective action, for a pump that has to operate over a wide speed range, totally eliminating the coincidence of all the frequencies of exciting forces with the system natural frequencies is impractical. Therefore, the system has to be designed such that the resulting magnitudes of the forces are controlled to within tolerable levels so that safe and reliable operation can take place. This can be accomplished by a

34 50


TABLE 34.8

Vibration Cause Identification

Cause

Dominant Frequency

Mass unbalance

1£

Shaft bow

1£

Misalignment

1 £ and 2 £

Worn journal bearings 1 £ , 1/2 £ Gravity critical 2£ Asymmetric shaft

Shaft crack

Loose components

Coupling lockup Thermal instability Oil whirl

Internal rubs

Trapped fluids in rotor Defective rolling element bearings Damaged gears

2£

Spectrum, Time Domain, Orbit Shape High 1 £ with much lower harmonics; circular or elliptic orbits Run down plot shows decrease of vibration at critical speed Equally high 1 £ and 2 £ , figure 8 orbits

Equally high 1 £ and 1/2 £ Run down plot will show excitation at 1/2 critical speed Run down plot will show excitation at 1/2 critical speed

Characteristics, Corrections, Comments Corrected by shop or field balancing The shaft has to be straightened using an acceptable method Realign at operating conditions; loads causing misalignment, such as nozzle loads, may have to be reduced Difficult to balance Can be corrected by balancing

Typically occurs on multistage machines when all the keyways lie in the same plane; correct by staggering them 1 £ and 2 £ High 1 £ and run down plots Confirmation and detection may show excitation at of location of the crack 1/2 critical speed may require NDE techniques 1 £ and higher orders plus High 1 £ with lower level Shimming and peening may fractional subharmonics orders and fractional be used as temporary subharmonics methods to fix the problem 1 £ and 2 £ Equally high 1 £ and 2 £ , Stop starts may change figure 8 orbits vibration pattern 1£ High 1 £ varies with temperature. Proper prewarming or Phase angle may change compromise balancing can correct the problem ,1/2 £ , typically 0.35 £ Run up plot will show Temporary problem may be to 0.47 £ 1/2 £ increasing and locking caused by excess clearances, into fixed value ,1/2 £ oil viscosity, or unloading of the bearing; if it is a design problem, correct by changing to tilting pad bearings 1/4 £ , 1/3 £ , 1/2 £ , 2 £ , Run down plots may May get progressively 3 £ , 4 £ , etc. show decreasing worse; galling between amplitudes and contact surfaces or heat disappearance; build up may cause seizure loops in orbits and shaft failure 0.8 £ to 0.9 £ Time domain signal will Balancing the rotor may show beating reduce the vibration At bearing defect frequency Peaks at defect frequencies Shock pulse measurements in spectrum can also be used to detect problem Gear mesh frequency High peaks at gear mesh To determine exact nature frequency with side bands. of damage further analysis Time domain may may be required also show pulses


34 51

(continued)

Cause

Dominant Frequency

Spectrum, Time Domain, Orbit Shape

Characteristics, Corrections, Comments In the case of two pole motors, it can be confused with mechanical causes as the rotational speed is the same as line frequency Caused by high nozzle loads, casing not free to expand, soft foot or foundation distortion Causes misalignment between bearings or between coupled equipment More common in machines originally designed for fixed speed operation, later converted to variable speed operation Increase or decrease stiffness of structure or add or remove mass to change natural frequency Occurs in built up rotors with transitional fits

Electric motor problems

1 £ (line frequency), 2 £ (line frequency)

High peaks at 1 £ and 2 £ line frequency with side bands; disappears when power to motor is turned off

Casing distortion

1£

High 1 £ , may change with time

Piping forces

1£,2£

Equally high 1 £ and 2 £

Rotor and bearing critical

1£

High 1 £ , on rundown plot 1 £ decreases rapidly, may also show a large phase angle change

Structural resonance

1£,2£

High 1 £ and some 2 £;can be easily identified on run down plot

Rotor hysteresis

0.65 £ to 0.85 £

Hydraulic causes

1 £ (vane pass frequency), 2 £ (vane pass frequency)

Spectrum will show high magnitudes at 0.65 £ to 0.85 £ High 1 £ and 2 £ vane pass frequency

Common in centrifugal pumps due to flow recirculation or inadequate gap between impeller and casing

direct reduction of the exciting force or by means of increased damping. Based on these guidelines, the following modifications were proposed to correct the problem: 1. Torsional resonance Use an electrometric type coupling that has a high degree of torsional damping to reduce the magnitude of the torsional excitation forces such that the torsional stresses within the rotors are within acceptable limits. Since both the pump and VFD generate excitation at 6 £ rpm, their effects could be compounding one another. Introducing either five vane or seven vane impellers into the pump will eliminate this possibility. Additional filters could be introduced into the VFD to reduce the 6 £ and 12 £ component periodic torsional excitation. Consider not operating (lock out) the pump within ^10% of the frequency at which torsional resonance occurs. 2. Motor housing resonance at 2 £ Although the 2 £ vibration is dominant, its magnitude is within tolerable levels. The fact that some 2 £ vibration is also present in the pump indicates that the 2 £ vibration is perhaps caused by misalignment between the pump and the motor. This can be corrected by proper *

*

*

*

*

34 52


FIGURE 34.15 (a) Torsional resonance run up and run down plots; (b) torsional resonance run down plot. (Source: Private communique, Insight Engineering Services Ltd. Alta., Canada. With permission.)


34 53

FIGURE 34.16 Motor frame resonance. (Source: Private communique, Insight Engineering Services Ltd., Alta., Canada. With permission.)

FIGURE 34.17 Pump bearing housing resonance. (Source: Private communique, Insight Engineering Services Ltd., Alta., Canada. With permission.)

34 54


FIGURE 34.18 Pump cross over pipe acoustic resonance. (Source: Private communique, Insight Engineering Services Ltd., Alta., Canada. With permission.)

alignment and thus reducing the 2 £ excitation forces. In some cases, due to an unequal air gap between the rotor and stator of the motor, the motor could generate the 2 £ vibration. Under such conditions, accurate centering of the motor bearings will generally correct the problem. 3. Pump vibrations at vane passing frequency Generally, high vibrations at vane passing frequency are caused by pressure pulsations generated at the discharge of the impeller. There are several hydraulic modifications that can be made to the pump to reduce the amplitude of these pulsations that occur at vane passing frequency. The most common method is to increase the gap between the impeller discharge vanes and diffuser/volute. Also, changing the ratio of the number of impeller vanes to diffuser/volute vanes can help in reducing vane passing frequency pressure pulsations and the resulting vibration. 4. Acoustic resonance in the pump cross over pipe Once the pump is constructed, it is not possible to change the acoustic natural frequency of the cross over pipe. However, the excitation force, pressure pulsations generated at the impeller discharge, can be reduced by the methods outlined above. *

*

*

*

*

The root cause of a vibration problem in a rotating machine can be determined by careful study and analysis of the vibration signals. Industrial and international vibration standards and guidelines have been developed to ensure safe and reliable operation of rotating machinery. Equipment manufacturers, users, insurance companies, and public interest groups use vibration standards to control vibration to within acceptable levels.


34 55

References Adkins, D. and Brennen, C. 1986. Origins of hydrodynamic forces on centrifugal pump impellers, NASA CP No 2443, p. 467, In Proceedings of a Workshop held at Texas A&M University, Dallas, TX. Alford, J., Protecting turbomachinery from self excited rotor whirl, Trans. ASME, J. Eng. Power, 87, 333, 1965. API Standard 610, 8th ed., Centrifugal Pumps for General Refinery Service, 1995. Archer, J.S., Consistent mass matrix for distributed mass systems, J. Struct. Div., Proc. ASCE, 89, ST4, 161, 1963. Baskharone, E.A., Daniel, A.S., and Hensel, S.J., Rotordynamic effects of the shroud to housing leakage flow in centrifugal pumps, Trans. ASME, J. Fluid Eng., 116, 558, 1994. Benckert, H., Wachter, J. 1980. Flow induced spring coefficients of labyrinth seals for application in turbomachinery, NASA CP No. 2133, p. 189, In Proceedings of a Workshop held at Texas A&M University, Dallas, TX. Bentley, D. 1974. Forced Subrotative Speed Dynamic Action of Rotating Machinery, ASME, Dallas, TX, 74 PET 16. Biezeno, C.B. and Grammel, R. 1954. Engineering Dynamics, Steam Turbines, Vol. III. D. Van Nostrand Co. Inc., New York (originally published in German in 1939 as Technische Dynamik by Julius Springer, Berlin, Germany). Black, H.F., Interaction of a whirling rotor with a vibrating stator across a clearance annulus, J. Mech. Eng. Sci., 10, 1, 1968. Black, H.F. 1974. Lateral stability and vibrations of high speed centrifugal pump rotors, p. 56, In Proceedings IUTAM Symposium on Dynamics of Rotors, Lyngby, Denmark. Blevins, Robert, D. 2001. Formulas for Natural Frequency and Mode Shapes, Krieger Publishing Co. Inc., Melbourne, FL. Bolleter, U., Frei, A., Florjancic, S., Leibundgut, E., and Stu¨rchler, R., Rotordynamic Modeling and Testing of Boiler Feedpumps, EPRI TR 100980, 1992. Bolleter, U., Leibundgut, E., Stu¨rchler, R., and McCloskey, T. 1989. Hydraulic interaction and excitation forces of high head pump impellers, p. 187, In Proceedings of the Third ASCE/ASME Mechanical Conference, La Jolla, CA. Bolleter, U., Wyss, A., Welte, I., and Stu¨rchler, R., Measurement of hydrodynamic interaction matrices of boiler feed pump impellers, Trans. ASME, J. Vib. Stress Reliab. Des., 1985, SME, 85 DET 147, New York. Brennen, C., Acosta, A., and Caughey, T. 1980. A test program to measure cross coupling forces in centrifugal pumps and compressors, NASA CP No. 2133, p. 229, In Proceedings of a Workshop held at Texas A&M University, Dallas, TX. Childs, D.W. 1986. Force and moment rotordynamic coefficients for pump impeller shroud surfaces, NASA CP No. 2443, p. 467, In Proceedings of a workshop held at Texas A&M University Dallas, TX. Childs, D. 1993. Turbomachinery Rotordynamics, Wiley, New York. Chree, C., The whirling and transverse vibration of rotating shafts, Phil. Mag., 7, 504, 1904. COJOUR, User’s Guide: Dynamic Coefficients for Fluid Film Journal Bearings, EPRI CS 4093. Crandall, S.H. and Brosens, P.J., Whirling of unsymmetrical rotors, J. Appl. Mech., 28, 567, 1961. Dunkerly, S., On the whirling and vibration of shafts, Phil. Trans. R. Soc., London A, 185, 279, 1894. Ehrich, F.F. 1999. Handbook of Rotordynamics, Revised ed., Krieger Publishing Co. Inc., Melbourne, FL. Foppl, A., Das Problem der Laval’schen Turbinewelle, Civilingenieur, 41, 333, 1885. Glienicke, J. 1966. Experimental investigation of the stiffness and damping coefficients of turbine bearings and their application to instability prediction, p. 122, In Proceedings of the Journal Bearings for Reciprocating and Turbo Machinery Symposium, Nottingham, UK. Gorman and Daniel, J. 1975. Free Vibration Analysis of Beams and Shafts, Wiley, New York. Greenhill, A.G., On the strength of shafting when exposed both to torsion and to end thrust, Proc. I. Mech. Eng (London), 182, 1883.

34 56


Gunter, E.J. Jr., Dynamic stability of rotor bearing systems, NASA SP 113, 1966. Hagg, A.C. and Sankey, G.O., Elastic and damping properties of oil film journal bearings for application to unbalance vibration calculations, Trans. ASME, J. Appl. Mech., 25, 141, 1958. Harris, T. 1991. Rolling Bearing Analysis, 3rd ed., Wiley, New York. Holzer, H. 1921. Die Berechnung der Drehschwingungen, Springer, Berlin. Houbolt, J.C. and Reed, W.H., Propeller nacelle whirl flutter, Inst. Aerospace Sci., 1, 61, 1961. ISO 10816 1. Mechanical vibration evaluation of machine vibration by measurements on non rotating parts. Part 1. General Guidelines, ISO, Geneva, Switzerland, 1995. ISO 10816 2. Mechanical vibration evaluation of machine vibration by measurements on non rotating parts. Part 2. Land based Steam Turbines and Generators in excess of 50 MW with normal operating speeds of 1500 r/min, 1800 r/min, 3000 r/min and 3600 r/min, ISO, Geneva, Switzerland, 2001. ISO 10816 3. Mechanical vibration evaluation of machine vibration by measurements on non rotating parts. Part 3. Industrial machines with nominal power above 15 kW and nominal speeds between 120 r/min and 15 000 r/min when measured in situ, ISO, Geneva, Switzerland, 1998. ISO 10816 4. Mechanical vibration evaluation of machine vibration by measurements on non rotating parts. Part 4. Gas Turbine Driven Sets Excluding Aircraft Derivations, ISO, Geneva, Switzerland, 1998. ISO 10816 5. Mechanical vibration evaluation of machine vibration by measurements on non rotating parts. Part 5. Machine Sets in Hydraulic Power Generating and Pumping Plants, ISO, Geneva, Switzerland, 2000. ISO 7919 1. Mechanical vibrations of non reciprocating machines measurement on Rotating Shafts and Evaluation Criteria. Part 1. General Guidelines, ISO, Geneva, Switzerland, 1996. ISO 7919 2. Mechanical vibrations of non reciprocating machines Measurement on Rotating Shafts and Evaluation Criteria. Part 2. Land Based Steam Turbines and Generators in Excess of 50 MW with Normal Operating Speeds of 1500 r/min, 1800 r/min, 3000 r/min and 3600 r/min, ISO, Geneva, Switzerland, 2001. ISO 7919 3. Mechanical vibrations of non reciprocating machines Measurement on Rotating Shafts and Evaluation Criteria. Part 3. Coupled Industrial Machines, ISO, Geneva, Switzerland, 1996. ISO 7919 4. Mechanical vibrations of non reciprocating machines Measurement on Rotating Shafts and Evaluation Criteria. Part 4. Gas Turbine Sets, ISO, Geneva, Switzerland, 1996. ISO 7919 5. Mechanical vibrations of non reciprocating machines Measurement on Rotating Shafts and Evaluation Criteria. Part 5 Machine Sets in Hydraulic Power Generating and Pumping Plants, ISO, Geneva, Switzerland, 1997. Jeffcott, H.H., The lateral vibration of loaded shafts in the neighbourhood of a whirling speed the effect of want of balance, Phil. Mag., 37, 304, 1919. Jery, B., Acosta, A., Brennen, C., and Caughey, T. 1984. Hydrodynamic impeller stiffness, damping, and inertia in the rotordynamics of centrifugal flow pumps, NASA CP No. 2338, p. 137, In Proceedings of a workshop held at Texas A&M University, Dallas, TX. Jones, A., A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions, Trans. ASME J. Basic Eng., 82, 309, 1960. Kimball, A.L. Jr., Internal friction theory of shaft whirling, Gen. Electr. Rev., 27, 244, 1924. Kirk, R.G., Donald, G.N. 1983. Design Criteria of Improved Stability of Centrifugal Compressors, AMD Vol. 55, Rotor Dynamical Instability. ASME, New York, p. 59. Kramer, E. 1993. Dynamics of Rotors and Foundations, Springer, Berlin. Lomakin, A.A., Calculating the critical speed and the conditions to ensure dynamic stability of the rotors in high pressure hydraulic machines, taking account of the forces in the seals, Energomashinos troenie, 4, 1, 1958. Lund, J.W., Spring and damping coefficients for the tilting pad journal bearing, Trans. ASLE, 7, 342, 1964.


34 57

Lund, J.W. 1965. Rotor bearing Dynamics Design Technology. Part III. Design Handbook for Fluid film Bearings, AFAPL TR 64 45. Wright Patterson Air Force Base, Dayton, OH. Lund, J.W. 1968. Rotor bearing Dynamics Design Technology. Part VII. The Three Lobe Bearing and Floating Ring Bearing, AFAPL TR 65 45. Wright Patterson Air Force Base, Dayton, OH. Lund, J.W., Sensitivity of the critical speeds of a rotor to changes in the design, Trans. ASME, J. Mech. Des., 102, 115, 1979. Lund, J.W. and Orcutt, F.K., Calculation and experiments on the unbalance response of a flexible rotor, Trans. ASME, J. Eng. Ind., 89, 785, 1967. Meirovitch, L. 1986. Elements of Vibration Analysis, 2nd ed., McGraw Hill, New York. Miller, D.F., Forced lateral vibration of beams on damped flexible end supports, Trans. ASME, J. Appl. Mech., 20, 167, 1953. Moore, J.J. and Palazzolo, A.B., Rotordynamic force prediction of Whirling Centrifugal Impeller Shroud passages using Computational Fluid Dynamic techniques, Trans. ASME, J. Eng. Gas Turbine Power, 123, 910, 2001. Muszynska, A., Whirl and whip rotor/bearing stability problems, J. Sound Vib., 110, 443, 1986. Myklestad, N.O., A new method for calculating natural modes of uncoupled bending vibrations of airplane wings and other types of beams, J. Aeronaut. Sci., 11, 153, 1944. Nelson, H., A Finite rotating shaft element using Timoshenko beam theory, Trans. ASME, J. Mech. Des., 102, 793, 1980. Nelson, H. and McVaugh, J., The dynamics of rotor bearing systems using finite elements, Trans. ASME, J. Eng. Ind., 98, 593, 1976. Newkirk, B.L., Shaft Whipping Gen. Electr. Rev., 27, 169, 1924. Newkirk, B.L. and Taylor, H.D., Shaft whipping due to oil action in journal bearings, Gen. Electr. Rev., 28, 559, 1925. Ohashi, H., Hatanaka, R., and Sakurai, A. 1986. Fluid force testing machine for whirling centrifugal impeller, In Proceedings of the International Federation for Theory of Machines and Mechanisms, International Conference on Rotordynamics, JSME, Tokyo, Japan. Orcutt, F.K., The steady state and dynamic characteristics of the tilting pad journal bearing in laminar and turbulent flow regimes, Trans. ASME, J. Lubricat. Technol., 89, 392, 1967. Perera, L. 2002. Private communique´, Insight Engineering Services Ltd, Alta., Canada. Pinkus, O. and Sternlicht, B. 1961. Theory of Hydrodynamic Lubrication, McGraw Hill, New York. Poritsky, H., Contribution to the theory of oil whip, Trans. ASME, 75, 1153, 1953. Prohl, M.A., A general method for calculating Critical Speeds of flexible rotors, Trans. ASME, J. Appl. Mech., 12, A 142, 1945. Rajan, M., Nelson, H.D., and Chen, W.J., Parameter sensitivity in the dynamics of rotor bearing systems, Trans. ASME, J. Vib. Acoust. Stress Reliab. Des., 108, 197, 1986. Rajan, M., Rajan, S.D., and Nelson, H.D., and Chen, W.J., Optimal placement of critical speeds in rotor bearing systems, Trans. ASME, J. Vib. Acoust. Stress Reliab. Des., 109, 152, 1987. Rankine, W.J.M., On the centrifugal force of rotating shafts, Engineer, 249, 9, 1869. Rathbone, T.C., Vibration tolerances, Power Plant Eng., November, 1939. Rayleigh, J.W.S. 1945. Theory of Sound, Dover Publications, New York. Robertson, D., Whirling of a journal in a sleeve bearing, Phil. Mag., 15, 96, 113, 1933. Robertson, D., Transient whirling of a rotor, Phil. Mag., 20, 793, 1935. Ruhl, R.L. and Booker, J.F., A finite element model for distributed parameter turborotor systems, Trans. ASME, J. Eng. Ind., 94, 126, 1972. Smith, D.M., The motion of a rotor carried by a flexible shaft in flexible bearings, Proc. R. Soc. London A, 142, 92, 1933. Someya, T. 1989. Journal Bearing Databook, Springer, New York. Southwell, R.V. and Gough B.S., 1921, Complex Stress Distributions in Engineering Materials, British Association For Advancement of Science Reports, 345.

34 58


Sternlicht, B., Elastic and damping properties of cylindrical journal bearings, Trans. ASME, J. Basic Eng., 81, 101, 1959. Stodola, A. 1927. Steam and Gas Turbines, Vol. I, McGraw Hill, New York. Taylor, E.S. and Browne, K.A., Vibration isolation of aircraft power plants, J. Aeronaut. Sci., 6, 43, 1938. Thomas, H.J., Unstable natural vibration of turbine rotors excited by the axial flow in stuffing boxes and blading, Bull. AIM, 71, 1039, 1958. Timoshenko, S., Young, D.H., and Weaver, W. Jr. 1974. Vibration Problems in Engineering, 4th ed., Wiley, New York. Urlichs, K., Leakage flow in thermal turbo machines as the origin of vibration exciting lateral forces, NASA, TT 17409, 1977. Vance, J.M. 1988. Rotordynamics of Turbomachinery, Wiley, New York. Warner, P.C., Static and dynamic properties of partial journal bearings, Trans. ASME, J. Basic Eng., 85, 247, 1963. Wolf, J.A. 1968. Whirl Dynamics of a Rotor Partially Filled with Liquids, ASME, New York, 68 WA/ APM 25.

35

Regenerative Chatter in Machine Tools 35.1 Introduction .......................................................................... 35 1 35.2 Chatter in Turning Operations ............................................ 35 3 Example 1

35.3 Chatter in Face Milling Operations .................................... 35 9 Example 2

35.4 Time Domain Simulation .................................................. 35 14 Example 3

†

Example 4

35.5 Chatter Detection ............................................................... 35 18 Example 5

35.6 Chatter Suppression ............................................................ 35 20

Robert G. Landers University of Missouri at Rolla

Spindle Speed Selection † Example 6 † Feed and Depth of Cut Selection † Spindle Speed Variation Example 7

†

35.7 Case Study ........................................................................... 35 24

Summary Regenerative chatter, a result of unstable interactions between machining forces and structural deflections, is a great limitation in machining operations. This chapter describes the modeling, analysis, simulation, detection, and control of regenerative chatter in machining operations, and, in particular, turning and face milling. An analytical method is applied to calculate the limiting stable depth of cut and corresponding spindle speeds to generate stability lobe diagrams. The method is applied to both turning and face milling operations. Time domain simulation is described and applied to turning and face milling operations. Methods for chatter detection are presented and experimental results from a face milling operation are given. Chatter suppression techniques, namely spindle speed selection, feed selection, depth of cut selection, and spindle speed variation, are presented and two simulations of a turning operation are used to illustrate the spindle speed selection and spindle speed variation techniques. Finally, a case study of a face milling operation is presented. The nomenclature used in the presentation is listed at the end of the chapter.

35.1

Introduction

Regenerative chatter is a major limitation in machining operations. This phenomenon is a result of an unstable interaction between the machining forces and the structural deflections. The forces generated when the cutting tool and part come into contact produce significant structural deflections. These structural deflections modulate the chip thickness that, in turn, changes the machining forces. For certain cutting conditions, this closed loop, self excited system becomes unstable and regenerative chatter occurs. Regenerative chatter may result in excessive machining forces and tool wear, tool failure, and scrap parts due to unacceptable surface finish, thus severely decreasing operation productivity and part quality. 35 1

35 2


Depth-of-cut

A typical chatter stability chart, the so called stability borderline stability lobe diagram, is shown in Figure 35.1. asymptotic borderline If the process parameters are above the stability borderline, chatter will occur, and if the process parameters are below the stability borderline, chatter will not occur. The asymptotic stability unstable borderline is the depth of cut below which stable machining is guaranteed regardless of the spindle speed. The lobed nature of the stability stable borderline allows stable pockets to form; thus, at specific ranges of spindle speeds, the depth of Spindle speed cut may be substantially increased beyond the asymptotic stability limit. These pockets become FIGURE 35.1 Stability lobe diagram. smaller as the spindle speed decreases. The stability borderline is “pulled up” for low spindle speeds due to process damping (i.e., the back side of the tool rubbing on the part surface). If accurate models of the structural components and the cutting process are available, the stability lobe diagram may be used to plan chatter free machining operations. The analysis of regenerative chatter as the interaction between the cutting forces and structural vibrations was established by Tobias (1965) and Koenigsberger and Tlusty (1971). Merritt (1965) used systems theory to determine stability and construct the stability lobe diagram by generating specialized plots from the harmonic solutions of the system’s characteristic equation. Chatter analysis reveals a natural delay in the system leading many researchers to use Nyquist techniques to generate stability lobe diagrams (Minis et al., 1990a, 1990b; Lee and Liu, 1991a, 1991b; Minis and Yanushevsky, 1993). A set of process parameters is selected and the characteristic equation is formed. The Nyquist criterion is applied to determine if the system for this process parameter set is stable. The depth of cut is adjusted and the procedure is repeated until the critical depth of cut is determined. Another chatter analysis technique capable of generating stability lobe diagrams analytically for linear systems has recently been introduced (Altintas and Budak, 1995; Budak and Altintas, 1998a, 1998b). This technique is utilized in this chapter. The theoretical analysis of regenerative chatter laid the foundation for developing techniques to automatically detect its occurrence and to automatically suppress it. Since there is a dominant chatter frequency, which is near a structural frequency, that occurs when chatter develops, most monitoring techniques analyze the frequency of a process variable, and chatter is detected when significant energy is present near a structural frequency. Most automatic chatter suppression routines either adjust the spindle speed to be in a pocket of the stability lobe diagram or vary the spindle speed to bring the current and previous tooth passes into phase. While automatic monitoring and control of regenerative chatter shows great promise, it has been mostly limited to laboratory applications. Therefore, commercial tools are not currently available. While regenerative chatter in turning and face milling operations is discussed in this chapter, this phenomenon is not limited to these specific manufacturing operations. Other machining operations for which chatter has been analyzed include end milling (Budak and Altintas, 1998a, 1998b), grinding (Inasaki et al., 2001), drilling (Tarng and Li, 1994), and so on. Also, the regenerative chatter phenomenon occurs in other manufacturing operations, most notably in rolling (Yun et al., 1998; Tlusty, 2000). Section 35.2 and Section 35.3 present an analytical method to examine regenerative chatter in turning and face milling operations, respectively. Section 35.4 discusses a numerical technique known as time domain simulation that may be used to analyze regenerative chatter for nonlinear systems. The subject of chatter detection is presented in Section 35.5, then methods to perform chatter suppression are discussed and illustrated in Section 35.6. Section 35.7 presents a case study of a face milling operation.


35.2

35 3

Chatter in Turning Operations

A schematic of a turning operation is shown in Figure 35.2. The part structure is assumed to be perfectly rigid, while the cutting tool structure is capable of vibrations in the longitudinal (i.e., the z) direction only. The machining force in the longitudinal direction is FðtÞ ¼ Pdf ðtÞ

structural stiffness

x direction of tool motion

z

tool

y structural damping

ð35:1Þ

part

Ns

fnom

The depth of cut is assumed to be constant; d however, the feed, and hence the machining force, is time varying due to structural vibrations. It is assumed here that the machining force does FIGURE 35.2 Turning operation schematic: current not explicitly depend upon the cutting speed. pass (solid line) and previous pass (dotted line). The feed is the chip thickness in the longitudinal direction. The nominal feed is the distance the tool advances relative to the part each spindle revolution and is constant once the tool fully engages the part. However, the cutting tool vibrates, leaving an undulated surface on the part and, thus, modulates the feed. The instantaneous feed is f ðtÞ ¼ fnom þ DzðtÞ ¼ fnom þ zðtÞ

zðt

TÞ

ð35:2Þ

The term fnom is the nominal feed, also known as the static feed. The term DzðtÞ is the feed due to the cutting tool vibrations and is known as the dynamic feed. The parameter T is the spindle rotation period. The structural vibration, zðtÞ; known as the inner modulation, is the cutting tool vibration at the current time. The delayed structural vibration zðt TÞ; known as the outer modulation, is the cutting tool vibration as of when the part was at the current angular during the previous spindle rotation. The modulation in feed due to structural vibrations is illustrated in Figure 35.3. Inserting Equation 35.2 into Equation 35.1: FðtÞ ¼ Pdfnom þ PdDzðtÞ ¼ Fnom þ DFðtÞ

ð35:3Þ

The force Fnom ¼ Pdfnom is due to the nominal chip thickness and does not vary since the depth of cut and nominal feed are constant. The force DFðtÞ ¼ PdDzðtÞ is due to changes in the nominal feed caused by structural vibrations. The structural vibrations are related to the machining force by zðsÞ ¼

gðsÞFðsÞ

ð35:4Þ

where gðsÞ is the transfer function relating the structural vibrations to the machining forces. Since FðtÞ Fðt TÞ ¼ DFðtÞ DFðt TÞ; the structural vibrations are related to the machining

f = fnom

No vibration

f

Vibrations in phase

f

Vibrations out of phase

FIGURE 35.3 Modulation in feed due to structural vibrations in a turning operation: current pass (solid line) and previous pass (dotted line).

35 4


forces by DzðsÞ ¼

ð1

e

Substituting for Dz in Equation 35.5 and rearranging: n DFðsÞ 1 þ Pd 1 e

sT

ÞgðsÞDFðsÞ

sT

o gðsÞ ¼ 0

ð35:5Þ

ð35:6Þ

Equation 35.6 is now solved, based on the method presented by Budak and Altintas (1998a, 1998b), to determine the stability lobe diagram. Assuming the steady state solution is a harmonic function at a single chatter frequency vc ; Equation 35.6 becomes n o DFðjvc Þe jvc t 1 þ Pd 1 e jvc T gð jvc Þ ¼ 0 ð35:7Þ where j2 ¼

1: For nontrivial solutions of Equation 35.7, the following eigenvalue problem is derived: n o ð35:8Þ det 1 þ Pd 1 e jvc T gðjvc Þ ¼ 0

Since the structural dynamics are one dimensional, Equation 35.8 reduces to 1 þ Pd 1

e

jvc T

gð jvc Þ ¼ 0

ð35:9Þ

L ¼ Pd 1

e

jvc T

¼ LR þ j LI

ð35:10Þ

The parameter L is defined as

Using the Euler identity 1

e

jvc T

¼1

dlim ¼

cosðvc TÞ þ j sinðvc TÞ; the limiting stable depth of cut is 1 P 1

LR þ j LI cosðvc TÞ þ j sinðvc TÞ

Equation 35.11 is rewritten as ( ) LR ½1 cosðvc TÞ þ LI sinðvc TÞ 1 LR sinðvc TÞ þ LI ½1 cosðvc TÞ dlim ¼ þj 1 cosðvc TÞ 2P 1 cosðvc TÞ

ð35:11Þ

ð35:12Þ

Since the limiting depth of cut must be a real number:

LR sinðvc TÞ þ LI ½1

cosðvc TÞ ¼ 0

ð35:13Þ

The parameter k is defined as

k¼

sinðvc TÞ LI ¼ 1 cosðvc TÞ LR

ð35:14Þ

The limiting stable depth of cut is solved explicitly as dlim ¼

LR ð1 þ k2 Þ 2P

ð35:15Þ

Note that LR must be positive for dlim to be positive. From Equation 35.9, the parameter L is

L¼

1 gðjvc Þ

ð35:16Þ

Equation 35.16 is used to determine LR and LI ; and these values are used to solve for dlim : Next, the spindle speed at which the limiting depth of cut occurs is determined. The trivial solution to Equation 35.14 is

vc T ¼ 0 þ 2lp; l ¼ 0; 1; 2; …

ð35:17Þ


35 5

The quantity vc T may be interpreted as the number of vibration cycles during a spindle rotation. The trivial solution indicates that the successive vibrations are in phase (i.e., there is no regeneration). The nontrivial solution to Equation 35.14 is cosðvc TÞ ¼

k2 1 k2 þ 1

ð35:18Þ

l ¼ 0; 1; 2; …

ð35:19Þ

and may be rewritten as

vc T ¼ 1 þ 2lp; where 1 ¼ cos

1

k2 1 k2 þ 1

! ð35:20Þ

The parameter 1 is the fraction of the vibration cycles during a spindle rotation. The angle of L in the complex plane is

w ¼ tan

1

LI LR

¼ tan 1 ðkÞ

ð35:21Þ

cosð2wÞ

ð35:22Þ

Substituting k ¼ tanðwÞ into Equation 35.18 yields cosðvc TÞ ¼ A solution to Equation 35.22 is

vc T ¼ p

2w þ 2lp;

l ¼ 0; 1; 2; …

ð35:23Þ

Comparing Equation 35.19 and Equation 35.23, it is seen that the fraction of vibration cycles is 1 ¼ p 2w: Since 0 # 1 # 2p; one must ensure that p=2 # w # p=2 when computing w: For example, if Equation 35.21 is solved using a four quadrant inverse tangent function whose solution is bounded between p and p, then p=2 # w # p=2 since LR is positive. For milling applications, it will be seen that LR must be negative; therefore, the following conditions must be enforced to ensure 0 # 1 # 2p: if LI , 0 then w ! w þ p if LI . 0 then w ! w

p

ð35:24Þ

The spindle speed is Ns ¼

60 60vc ¼ ; 1 þ 2lp T

l ¼ 0; 1; 2; …

ð35:25Þ

To construct a stability lobe diagram, the following steps are implemented: Select a chatter frequency ðvc Þ near a dominant structural frequency. Calculate LR and LI using Equation 35.16. Calculate dlim using Equation 35.15. Select a stability lobe number ðlÞ and calculate Ns using Equation 35.25. The point ðNs ; dlim Þ is the point on the stability lobe diagram corresponding to the chatter frequency, vc ; and the stability lobe number, l: 5. Repeat Step 4 for the desired number of stability lobes. The result is a vector of spindle speeds, ~ s ¼ {Ns Ns · · · Ns }: Each point {ðNs ; dlim Þ ðNs ; dlim Þ · · · ðNs ; dlim Þ}: corresponds N 1 2 n 1 2 n to a different stability lobe, and all of the points correspond to the chatter frequency vc : 6. Select another chatter frequency and repeat Steps 2 to 5. In this manner, the stability lobe diagram is constructed. The smaller the difference between successive chatter frequencies, the greater the resolution of the stability lobe diagram. In general, the lobes will overlap. In this case, the

1. 2. 3. 4.

35 6


minimum limiting depth of cut is the smallest depth of cut. If the lobes do not overlap, then the range of chatter frequencies must be increased.

35.2.1

Example 1

The feed force for a turning operation is given by Equation 35.1 and the structural dynamics are given by Equation 35.26. An analytical expression for the limiting depth of cut and corresponding spindle speed for a given chatter frequency and stability lobe number is developed. The stability lobe diagram is plotted for P ¼ 0:6 kN/mm2, vn ¼ 600 Hz, z ¼ 0:2; and k ¼ 12 kN/mm. The stability lobe diagram is compared to stability lobe diagrams for z ¼ 0:1; 0.3, and 0.4. The stability lobe diagram is then compared with stability lobe diagrams for vn ¼ 500; 700, and 800 Hz. The first ten lobes are included for all stability lobe diagrams: v2n z€ðtÞ þ 2zvn z_ðtÞ þ v2n zðtÞ ¼ FðtÞ ð35:26Þ k The parameter L is

L¼

1 k ¼ LR þ jLI ¼ 2 ðv2c gðjvc Þ vn

The limiting depth of cut is dlim

v2n Þ

j

k ð2zvc vn Þ v2n

ð35:27Þ

" # kðv2c v2n Þ 4z2 v2c v2n ¼ 1þ 2 2K v2n ðvc v2n Þ2

ð35:28Þ

The spindle speed is Ns ¼

p

2 tan

1

60vc 2zvc vn v2c v2n

þ 2lp

;

l ¼ 0; 1; 2; …

ð35:29Þ

where l is the stability lobe number. Note that the chatter frequency must be greater than the structural natural frequency for the limiting depth of cut to be positive. The first ten lobes of the

30

Depth-of-cut (mm)

25 20 15 10 5

lobe 1 0

20

FIGURE 35.4

lobe 0 40 60 80 Spindle speed (krpm)

100

Unprocessed stability lobe diagram for Example 1.

120


35 7

25

Depth-of-cut (mm)

20

15

10

5

0

10

20

FIGURE 35.5

30

40 50 60 70 Spindle speed (krpm)

80

90

100

Processed stability lobe diagram for Example 1.

stability lobe diagram are plotted in Figure 35.4 and Figure 35.5. In Figure 35.4, the entire solution for each of the ten stability lobes is shown. The largest stability lobe is the zeroth lobe on the right. The lobe number increases from right to left on the stability lobe diagram and successive lobes become closer together. The stability lobe diagram is processed in Figure 35.5 such that the minimum depth of cut is selected at each spindle speed showing the true stability borderline.

25

15

Depth-of-cut (mm)

Depth-of-cut (mm)

20 z = 0.1

10 5 0

10

20

30

40

15 10 5 0

50

z = 0.2

20

10

Spindle speed (krpm)

40

50

40

30

Depth-of-cut (mm)

Depth-of-cut (mm)

30


35 z = 0.3

25 20 15 0

20

10

20

30


FIGURE 35.6

40

50

z = 0.4

35 30 25 20 0

10

20

30

40

50


Stability lobe diagrams for Example 1 with z ¼ 0.1, 0.2, 0.3, and 0.4.

60

35 8


25

wn = 500 Hz Depth-of-cut (mm)

Depth-of-cut (mm)

25 20 15 10 5

0

20

20 15 10 5

40

wn = 600 Hz

0

20


Spindle speed (krpm) 25

wn = 700 Hz Depth-of-cut (mm)

Depth-of-cut (mm)

25 20 15 10 5

0

20

40

20 15 10 5

60

wn = 800 Hz

0

20

Spindle speed (krpm) FIGURE 35.7

40

40

60


Stability lobe diagrams for Example 1 with vn ¼ 500, 600, 700, and 800 Hz.

In Figure 35.6, the effect of the structural damping ratio is illustrated: as the structural damping ratio increases, the lobes shift slightly to the left and the asymptotic stability boundary shifts up dramatically. The effect of the structural natural frequency is illustrated in Figure 35.7: as the structural natural frequency increases, the lobes shift to the right but the magnitude remains the same.

This section presented an analytical method to generate stability lobe diagrams for turning operations. The limiting depth of cut in a turning operation is given by L dlim ¼ R ð1 þ k2 Þ 2P where LR is the real part of 1=gðjvc Þ; gðjvc Þ is the structural transfer function evaluated at the chatter frequency, vc ; k ¼ LI =LR ; and LI is the imaginary part of 1=gðjvc Þ: The corresponding spindle speed is 60vc Ns ¼ 1 þ 2lp where l ¼ 0; 1; 2; … is the stability lobe number and 1 ¼ cos

1

k2 1 k2 þ 1

!


35.3

35 9

Chatter in Face-Milling Operations

A schematic of a face milling operation is shown in Figure 35.8. In milling operations, multiple teeth may be in contact with the part simultaneously, the feed naturally varies as a function of the tooth angle even when structural vibrations are not present, and each tooth enters and leaves contact with the part every spindle revolution. The depth of cut is the chip thickness in the z direction and is assumed to be constant, since the machine tool and part structures are typically much stiffer in the z direction than in the x and y directions. The instantaneous feed of the ith tooth, illustrated in Figure 35.9, is fi ðtÞ ¼ ft cos½ui ðtÞ þ DxðtÞ cos½ui ðtÞ þ DyðtÞ sin½ui ðtÞ

ð35:30Þ

where DxðtÞ ¼ {xt ðtÞ

xt ðt

Tt Þ}

{xp ðtÞ

xp ðt

Tt Þ}

ð35:31Þ

DyðtÞ ¼ {yt ðtÞ

yt ðt

Tt Þ}

{yp ðtÞ

yp ðt

Tt Þ}

ð35:32Þ

The term ft cos½ui ðtÞ in Equation 35.30 represents the feed due to the distance the part advances relative to the cutting tool each tooth rotation and is known as the static feed. The terms DxðtÞ cos½ui ðtÞ y structural stiffness (y)

x

structural damping (y)

z

part

qex

ft

qi fi

direction of part motion

structural stiffness (x)

qen

ith tooth

structural damping (x)

Ns

FIGURE 35.8 Face milling operation schematic: current pass (solid line), previous pass (dotted line), and depth of cut in z direction.

qi

qi

ith tooth

fi

ith tooth fi

fi = ft cos (qi) No vibration

qi

Vibrations in phase

ith tooth

Vibrations out of phase

FIGURE 35.9 Modulation in feed due to structural vibrations in a face milling operation: current pass (solid line) and previous pass (dotted line).

35 10


and DyðtÞ sin½ui ðtÞ in Equation 35.30 represent the feed due to tool and part vibrations in the x and y directions, respectively, at the tooth angle ui ðtÞ; and are known as the dynamic feed. The machining forces in the x and y directions, respectively, are Fx ðtÞ ¼ dft

o PT cosðcr Þ cos2 ½ui ðtÞ þ PC cos½ui ðtÞ sin½ui ðtÞ s½ui ðtÞ

Nt n X i¼1


Nt n X

þ dDxðtÞ

i¼1

o PT cosðcr Þ sin½ui ðtÞ cos½ui ðtÞ þ PC sin2 ½ui ðtÞ s½ui ðtÞ

Nt n X

þ dDyðtÞ

i¼1

Fy ðtÞ ¼ dft

Nt n X

o PC cos2 ½ui ðtÞ s½ui ðtÞ

PT cosðcr Þ cos½ui ðtÞ sin½ui ðtÞ

i¼1

Nt n X

þ dDxðtÞ

PT cosðcr Þ cos½ui ðtÞ sin½ui ðtÞ

i¼1

þ dDyðtÞ

Nt n X

PT cosðcr Þ sin2 ½ui ðtÞ

i¼1

where

(

s½ui ðtÞ ¼

ð35:33Þ


o PC cos½ui ðtÞ sin½ui ðtÞ s½ui ðtÞ

1

if uen # ui ðtÞ # uex

0

if uen . ui ðtÞ . uex

ð35:34Þ

ð35:35Þ

The function s½ui ðtÞ determines if the ith tooth is in contact with the part at the tooth angle, ui ðtÞ: The first terms in Equation 35.33 and Equation 35.34 are the machining forces acting on the tool in the x and y directions, respectively, due to the static feed. The second terms in Equation 35.33 and Equation 35.34 are the machining forces acting on the tool in the x and y directions, respectively, due to the dynamic feed resulting from structural vibrations in the x direction. The third terms in Equation 35.33 and Equation 35.34 are the machining forces acting on the tool in the x and y directions, respectively, due to the dynamic feed resulting from structural vibrations in the y direction. The dynamic portion of the face milling force process model may be written compactly as "

DFx ðtÞ

#

DFy ðtÞ

" ¼ dAðtÞ

DxðtÞ DyðtÞ

#

" ¼d

A11 ðtÞ A12 ðtÞ A21 ðtÞ A22 ðtÞ

#"

DxðtÞ

#

DyðtÞ

ð35:36Þ

where A11 ðtÞ ¼ A12 ðtÞ ¼ A21 ðtÞ ¼ A22 ðtÞ ¼

Nt n X i¼1 Nt n X i¼1 Nt n X i¼1 Nt n X i¼1


ð35:37Þ

o PT cosðcr Þ sin½ui ðtÞ cos½ui ðtÞ þ PC sin2 ½ui ðtÞ s½ui ðtÞ

ð35:38Þ


ð35:39Þ

o PC cos½ui ðtÞ sin½ui ðtÞ s½ui ðtÞ

ð35:40Þ

PT cosðcr Þ cos½ui ðtÞ sin½ui ðtÞ PT cosðcr Þ sin2 ½ui ðtÞ


35 11

These coefficients modulate the instantaneous feed as the tooth angular displacement changes. The summation from i ¼ 1 to Nt represents the contribution to this modulation for each of the Nt teeth. Note the matrix AðtÞ is time varying and periodic with the tooth passing period, Tt : For chatter analysis, the matrix AðtÞ is typically expanded in a Fourier series using the zeroth term (Minis and Yanushevsky, 1993; Budak and Altintas, 1998a). The zeroth term of the Fourier expansion of the force process matrix AðtÞ is " 0 0 # Nt A11 A12 0 ð35:41Þ A ¼ 2p A021 A022 where 1 sinð2uÞ þ PC sin2 ðuÞ 2

u¼uex

A011 ¼

1 2

PT cosðcr Þ u þ

A012 ¼

1 2

PT cosðcr Þ sin2 ðuÞ þ PC u

1 sinð2uÞ 2

u¼uex

A021 ¼

1 2

PT cosðcr Þ sin2 ðuÞ

1 sinð2uÞ 2

u¼uex

A022 ¼

1 2

PT cosðcr Þ u

PC u þ

1 sinð2uÞ 2

PC sin2 ðuÞ

ð35:42Þ

u¼uen

ð35:43Þ

u¼uen

ð35:44Þ

u¼uen u¼uex

ð35:45Þ

u¼uen

The dynamic force process is now approximated by the linear, time invariant relationship: " # " # DFx ðtÞ DxðtÞ 0 ¼ dA DFy ðtÞ DyðtÞ The tool and part vibrations, respectively, are related to the machining forces by #" # " # " # " Fx ðsÞ Fx ðsÞ Gt11 ðsÞ Gt12 ðsÞ xt ðsÞ ¼ Gt ðsÞ ¼ Fy ðsÞ Fy ðsÞ Gt21 ðsÞ Gt22 ðsÞ yt ðsÞ "

xp ðsÞ yp ðsÞ

#

" ¼

Gp ðsÞ

Fx ðsÞ Fy ðsÞ

#

" ¼

Gp11 ðsÞ

Gp12 ðsÞ

Gp21 ðsÞ

Gp22 ðsÞ

#"

Fx ðsÞ Fy ðsÞ

ð35:46Þ

ð35:47Þ # ð35:48Þ

where Gt ðsÞ and Gp ðsÞ are the transfer functions relating the tool structural and part structural vibrations, respectively, to the machining forces. The negative sign in Equation 35.48 is due to the fact that the forces acting on the part are equal in magnitude and opposite in direction to the machining forces given in Equation 35.33 and Equation 35.34. Since " # " # " # " # Fx ðtÞ Fx ðt Tt Þ DFx ðtÞ DFx ðt Tt Þ ¼ Fy ðtÞ Fy ðt Tt Þ DFy ðtÞ DFy ðt Tt Þ the structural vibrations can be related to the machining forces by " # " # DFx ðsÞ Dx ¼ ð1 e sTt Þ½Gt ðsÞ þ Gp ðsÞ DFy ðsÞ Dy

ð35:49Þ

The machine tool and part vibrations are assumed to occur at a chatter frequency, vc ; when a marginally stable depth of cut is taken. Assuming the steady state solution is a harmonic function at a chatter

35 12


frequency, vc ; and substituting for the structural vibrations, Equation 35.49 becomes " # " # DFx DFx dNt 0 jvc t jvc Tt e jvc t ÞG ðjvc Þ e ¼ ð1 e 2p DFy DFy

ð35:50Þ

where the matrix G0 is G0 ðjvc Þ ¼

2p 0 A ½Gt ðjvc Þ þ Gp ðjvc Þ Nt

ð35:51Þ

Equation 35.50 is now solved based on the method presented by Budak and Altintas (1998a, 1998b) to determine the stability lobe diagram. The characteristic equation of Equation 35.50 is dNt det I2 ð35:52Þ ð1 ejvc Tt ÞG0 ðjvc Þ ¼ 0 2p where I2 is the 2 £ 2 identity matrix. The solution of Equation 35.52 yields the limiting stable depth of cut. The inverse of the eigenvalue of G0 is defined as dNt ð35:53Þ Lðjvc Þ ¼ LR ðjvc Þ þ jLI ðjvc Þ ¼ ð1 e jvc Tt Þ 2p Expanding the exponential term in Equation 35.53 and noting that the depth of cut must be a real number, the limiting stable depth of cut may be written as pLR dlim ¼ ð1 þ k2 Þ ð35:54Þ Nt where the parameter k is defined by the transcendental equation L sinðvc Tt Þ k¼ I ¼ 1 cosðvc Tt Þ LR

ð35:55Þ

Equation 35.55 is solved for the tooth passing period of the lth stability lobe and the tooth passing period is related to the spindle speed to yield 60vc Ns ¼ ; l ¼ 0; 1; 2; … ð35:56Þ Nt ½p 2w þ 2lp where, again, w ¼ tan 1 ðkÞ: A chatter frequency is selected and the limiting stable depth of cut is calculated from Equation 35.54 corresponding to the spindle speed on the lth lobe as given by Equation 35.56.

35.3.1

Example 2

The cutting and thrust pressures in a face milling operation are given by PC ¼ 2:0 kN/mm2 and PT ¼ 0:8 kN/mm2, respectively, and the lead angle is 458. The part is assumed to be perfectly rigid and the tool structural dynamics for the x and y directions are given by Equation 35.57 and Equation 35.58, respectively. The nominal parameters are uen ¼ 458; uex ¼ 458; Nt ¼ 4; kx ¼ 14 kN/mm, ky ¼ 17 kN/mm, zx ¼ 0:15; zy ¼ 0:1; vx ¼ 3000 rad/sec, and vy ¼ 4000 rad/sec. Stability lobe diagrams are generated for the nominal parameters and Nt ¼ 1; 2, and 8 teeth. Next, stability lobe diagrams are generated for the nominal parameters and uex ¼ 308; 608, and 758. The first 15 lobes are included for all stability lobe diagrams. x€ t ðtÞ þ 2zx vx x_ t ðtÞ þ v2x xt ðtÞ ¼

v2x F ðtÞ kx x

ð35:57Þ

y€ t ðtÞ þ 2zy vy y_ t ðtÞ þ v2y yt ðtÞ ¼

v2y F ðtÞ ky y

ð35:58Þ

The effect of the number of teeth is illustrated in Figure 35.10: as the number of teeth increases, the lobes shift to the left and the asymptotic stability borderline decreases. In Figure 35.11, the effect of the exit angle is illustrated: as the exit angle increases, the asymptotic stability borderline decreases.

Regenerative Chatter in Machine Tools 15

Nt= 1 Depth-of-cut (mm)

Depth-of-cut (mm)

30

35 13

25 20 15 10

0

10

20

30

40

10

5

50

Nt= 2

0

Spindle speed (krpm) 4

Nt= 4

7 6 5 4 3

0

2

4 6 8 10 Spindle speed (krpm)

FIGURE 35.10

Depth-of-cut (mm)

Depth-of-cut (mm)

2

0

2 4 Spindle speed (krpm)

6

Stability lobe diagrams for Example 2, with Nt ¼ 1, 2, 4, and 8.

qex = 30°

6 4

0

5

qex = 45° 6

4

2

10

0

5

10


8

6 Depth-of-cut (mm)

qex = 60° 6

4

0

5 Spindle speed (krpm)

FIGURE 35.11

25

Nt = 8


Depth-of-cut (mm)

20

8

8

2

15

3

1

12

10

2

10


Depth-of-cut (mm)

Depth-of-cut (mm)

8

5

10

qex = 75°

5 4 3 2

0

5

10


Stability lobe diagrams for Example 2, with uex ¼ 308, 458, 608, and 758.

35 14


This section presented an analytical method to generate stability lobe diagrams for face milling operations. The limiting depth of cut in a face milling operation is given by pLR ð1 þ k2 Þ dlim ¼ Nt where Nt is the number of teeth, LR is the inverse eigenvalue of ð2p=Nt ÞA0 ½Gt ðjvc Þ þ Gp ðjvc Þ ; A0 is the zeroth term of the Fourier expansion of the force process matrix, Gt ðjvc Þ and Gp ðjvc Þ are the transfer functions relating the tool structural and part structural vibrations, respectively, to the machining forces evaluated at the chatter frequency vc ; k ¼ LI =LR ; and LI is the imaginary part of ð2p=Nt ÞA0 ½Gt ðjvc Þ þ Gp ðjvc Þ : The corresponding spindle speed is 60vc Ns ¼ Nt ½p 2 tan 1 ðkÞ þ 2lp where l ¼ 0; 1; 2; … is the stability lobe number.

35.4

Time-Domain Simulation

Time domain simulation (Tlusty and Ismail, 1981, 1983; Tlusty, 1986; Tsai et al., 1990; Lee and Liu, 1991a, 1991b; Smith and Tlusty, 1993; Elbestawi et al., 1994; Tarng and Li, 1994; Weck et al., 1994) is an alternative method for determining regenerative chatter. In a time domain simulation, the machining forces and structural vibrations are simulated in the time domain for a specific set of process parameters and the resulting signals (i.e., forces and displacements) are examined to determine if chatter is present. The analyses presented above for the turning and face milling operations assume that the tool always maintains contact with the part and that the cutting and thrust pressures are independent of the process parameters. Further, the face milling analysis approximated the time varying force process matrix, AðtÞ; by the zeroth term of its Fourier expansion. With time domain simulations, nonlinear effects may be directly incorporated into the simulation; thus, more accurate stability prediction is possible. The disadvantage of time domain simulations is the extreme computational cost that is required. For a specific spindle speed, several simulations must be conducted at different depths of cut; thus, the stability boundary for that spindle speed is determined iteratively. This procedure is repeated for a range of spindle speeds to construct a complete stability lobe diagram. For turning operations, the machining force is calculated using Equation 35.1, the feed is calculated using Equation 35.2, and the tool displacement is calculated using Equation 35.4. For face milling operations, the feed is calculated using Equation 35.30, the machining forces in the x and y directions are calculated using Equation 35.33 to Equation 35.35, and the tool and part displacements, respectively, are calculated using Equation 35.47 and Equation 35.48. To calculate the machining forces in the face milling operation, the angular displacement of each tooth is required. The angular displacement of the ith tooth is

ui ðtÞ ¼

2p 2p Ntþ ði 60 s Nt

1Þ

ð35:59Þ

The feed and force equations are static, while the structural displacement equations are dynamic and must be solved via a numerical integration technique. A sufficiently small time step must be utilized in the numerical integrations to account for the small system time constants associated with the large structural frequencies.

35.4.1

Example 3

The feed force for a turning operation is given by FðtÞ ¼ 0:6df 0:7 ðtÞ: The structural dynamics are given by Equation 35.26 with the following parameters: vn ¼ 600 Hz, z ¼ 0:2; and k ¼ 12 kN/mm.

Feed force (kN)


1.5 1 0.5 0

Toll displacement (mm)

35 15

0

0.02

0.04

0.06 Time (s)

0.08

0.1

0

0.02

0.04

0.06 Time (s)

0.08

0.1

0

−0.05 −0.1

−0.15

FIGURE 35.12

Time domain simulations for Example 3 with fnom ¼ 0.1 mm.

A time domain simulation of the system including the effect of the tool disengaging from the part is constructed, and simulations for Ns ¼ 10,000 rpm, d ¼ 8 mm, and fnom ¼ 0.1 mm are conducted. The simulation is repeated for fnom ¼ 0.2 mm. For both simulations, the time history of the feed force and the tool displacement are plotted. To account for the phenomenon of the tool disengaging from the part, the feed in Equation 35.2 must be modified as follows: ( fnom þ zðtÞ fp ðt TÞ if fnom þ zðtÞ fp ðt TÞ $ 0 f ðtÞ ¼ ð35:60Þ 0 if fnom þ zðtÞ fp ðt TÞ , 0 where ( fp ðtÞ ¼

zðtÞ fnom þ fp ðt

if fnom þ zðtÞ

fp ðt

TÞ $ 0

TÞ if fnom þ zðtÞ

fp ðt

TÞ , 0

ð35:61Þ

If the feed at the current time is calculated to be negative, then the cutting tool has disengaged from the part and the feed is zero. The term fp ðtÞ accounts for feed due to structural vibrations at the previous spindle rotation, even when the cutting tool disengages from the part. The results for fnom ¼ 0:1 mm and fnom ¼ 0:2 mm are shown in Figure 35.12 and Figure 35.13, respectively. As the nominal feed is increased, chatter is suppressed.

35.4.2

Example 4

The cutting and thrust forces in a face milling operation are given by FC ðtÞ ¼ 1:4df 0:6 ðtÞ and FT ðtÞ ¼ 0:4df 0:8 ðtÞ; respectively. The lead angle is 458, the entry angle is 608; the exit angle is 608, the number of teeth is four, and the feed per tooth is ft ¼ 0:15 mm. The part is assumed to be perfectly rigid, and tool structural dynamics for the x and y directions are given by Equation 35.58 and Equation 35.59, respectively. A time domain simulation is developed to determine the limiting stable depth of cut for spindle speeds of 1000 and 32,000 rpm. For both spindle speeds, the system is simulated for a depth of cut 10% below the limiting stable depth of cut and for a depth of cut 10% above the limiting stable depth of cut. The cutting force, thrust force, x tool displacement, and y tool displacement are plotted.


Feed force (kN)

35 16

1.6 1.4 1.2 1 0.8 0

0.02

0.04

0.06

0.08

0.1

0.08

0.1

Tool displacement (mm)

Time (s) 0 −0.05 −0.1 −0.15

0

0.02

0.04

0.06 Time (s)

FIGURE 35.13

Time domain simulations for Example 3 with fnom ¼ 0.2 mm.

The nonlinear effect of tooth disengagement is included: x€ t ðtÞ þ 2ð0:15Þð3000Þ_xt ðtÞ þ 30002 xt ðtÞ ¼

30002 F ðtÞ 15 x

ð35:62Þ

40002 F ðtÞ ð35:63Þ 17 y To account for the phenomenon of the tool disengaging from the part, the feed in Equation 35.30 must be modified as follows: ( fci ðtÞ fpi ðt Tt Þ if fci ðtÞ $ 0 fi ðtÞ ¼ ð35:64Þ 0 if fci ðtÞ , 0 0.5

0

0

−0.5

Fy (kN)

Fx (kN)

y€ t ðtÞ þ 2ð0:1Þð4000Þ_yt ðtÞ þ 40002 yt ðtÞ ¼

−0.5 −1 0

0.5 Time (s)

−1 −1.5 0

1

0.5 Time (s)

1

0.5 Time (s)

1

0

0.1 y (mm)

x (mm)

0.05 0

−0.05

−0.05 −0.1 0

FIGURE 35.14

0.5 Time (s)

1

−0.1 0

Time domain simulation for Example 4 with Ns ¼ 1000 rpm and d ¼ 2.025 mm.


35 17 0

0

Fy (kN)

Fx (kN)

2

−2 −4

0

0.5 Time (s)

−4

1

0.2

0.2 y (mm)

0.4

x (mm)

0.4

0

−0.4

0

0.5 Time (s)

{xt ðt

0.5 Time (s)

1

0

0.5 Time (s)

1

0

−0.4

1

Time domain simulation for Example 4 with Ns ¼ 1000 rpm and d ¼ 2.475 mm.

fci ðtÞ ¼ ft cos½ui ðtÞ þ {xt ðtÞ

fpi ðtÞ ¼

0

−0.2

FIGURE 35.15

(

−2 −3

−0.2

where

−1

Tt Þ

xp ðtÞ} cos½ui ðtÞ þ {yt ðtÞ

Tt Þ} cos½ui ðtÞ þ {yt ðt

xp ðt

ft cos½ui ðtÞ þ fpi ðt

Tt Þ

yp ðt

yp ðtÞ} sin½ui ðtÞ

Tt Þ} sin½ui ðtÞ

Tt Þ

if fci ðtÞ $ 0 if fci ðtÞ , 0

ð35:65Þ ð35:66Þ

Note that ui ðtÞ ¼ uiþ1 ðt Tt Þ and i 1 ! Nt if i ¼ 1: The term fpi ðtÞ represents the contribution to the instantaneous feed when the previous tooth was at the same angular location as the ith tooth. If the tooth and part are in contact, this contribution is due to the tool and part vibrations. If the tooth and part are not in contact, this contribution is the previous contribution added to the static portion and the instantaneous feed is set to zero. Through time domain simulations, the limiting stable depth of cut for

−0.5

0 −1 −2

−1 −1.5

0

0.01 0.02 Time (s)

−2

0.03

0.02

0

0

−0.05 y (mm)

x (mm)

0 Fy (kN)

Fx (kN)

1

−0.02 −0.04 −0.06

FIGURE 35.16

0

0.01 0.02 Time (s)

0.03

0

0.01 0.02 Time (s)

0.03

−0.1 −0.15

0

0.01 0.02 Time (s)

0.03

−0.2

Time domain simulation for Example 4 with Ns ¼ 32,000 rpm and d ¼ 3.105 mm.

Vibration and Shock Handbook 4

0

2

−1

Fy (kN)

Fx (kN)

35 18

0 −2 −4

−2 −3

0

0.01 0.02 Time (s)

−4

0.03

0

0.05

0 y (mm)

0.1

x (mm)

0.1

0

−0.05 −0.1

FIGURE 35.17

0.01 0.02 Time (s)

0.03

0.01 0.02 Time (s)

0.03

−0.1 −0.2

0

0.01 0.02 Time (s)

0.03

−0.3

0

Time domain simulation for Example 4 with Ns ¼ 32,000 rpm and d ¼ 3.795 mm.

Ns ¼ 1000 rpm is found to be 2.25 mm and the limiting depth of cut for Ns ¼ 32,000 rpm is found to be 3.45 mm. The results are shown in Figure 35.14 to Figure 35.17. The system is stable in Figure 35.14 and Figure 35.16, while instability is evidenced in Figure 35.15 and Figure 35.17 by the force in the y direction saturating at 0 kN.

This section presented the technique of time domain simulation as an alternative means to analyze regenerative chatter. Time domain simulations are the direct numerical simulations of the force process and structural vibrations. A process parameter is changed iteratively from simulation to simulation to determine the critical value at which chatter occurs. A sufficiently small time step must be utilized in the numerical integrations to account for the small system time constants associated with the large structural frequencies.

35.5

Chatter Detection

Regenerative chatter is easily detected by an operator due to the loud, high pitched noise it produces and the distinctive “chatter marks” it leaves on the part surface. However, automatic detection is required for intelligent manufacturing (Cho and Ehmann, 1988; Delio et al., 1992). At the onset of chatter, process signals (e.g., force, vibration) contain significant energy at the chatter frequency. It is a well known fact that the chatter frequency will be close to a dominant structural frequency. The most common method to detect the presence of chatter is to threshold the frequency signal of a process signal. To analyze the frequency content of a signal, a Fourier transform, or fast Fourier transform, is performed. If the frequency content of the resulting signal near a dominant chatter frequency is above a threshold value, then chatter is determined to be present. It should be noted that machining process signals also contain significant energy at the tooth passing frequency. If the dominant structural frequencies and tooth passing frequency are sufficiently separated, then the tooth passing frequency may be ignored when determining the presence of chatter. If the dominant structural frequencies and tooth passing frequency are close, then the signal must be filtered at the tooth passing frequency using a notch filter. Also, forced vibrations, such as those resulting from the impact between the cutting tool and part, must not be allowed to falsely trigger the chatter


35 19

detection algorithm. These thresholding algorithms all suffer from the lack of an analytical method of selecting a threshold value. This value is typically selected empirically and will not be valid over a wide range of cutting conditions and machining operations.

35.5.1

Example 5

Power spectral density (N2S2)

Power spectral density (N2S2)

An experimental face milling operation, a com × 10−5 6 plete description of which is given in Landers (1997), is conducted with a spindle speed of 5 1500 rpm and a tool with four teeth. The 4 dominant structural frequencies are 334, 414, 653, and 716 Hz. The machining force Fz is 3 sampled at a frequency of 2000 Hz, and the time 2 domain signal is transformed into the frequency domain via a Fourier transform using 80 points 1 (i.e., one spindle revolution). The power spectral 0 density of the force signal is shown for depths of 0 100 200 300 400 500 600 700 800 900 1000 cut of 1.0 and 1.5 mm in Figure 35.18 and Frequency (Hz) Figure 35.19, respectively. In Figure 35.18, there is significant energy at 100 Hz, which is the tooth FIGURE 35.18 Power spectral density of Fz in a face passing frequency. There is also significant energy milling operation with d ¼ 1.0 mm. at 750 Hz due to structural vibrations; however, the system did not chatter, as evidenced by the lack of chatter marks on the part and a high pitched × 10−4 4 sound during machining. In Figure 35.19, there is significant energy at 665 Hz as well as 100 Hz. 3.5 Chatter was evidenced by the chatter marks left on 3 the part surface and the high pitched sound 2.5 during machining. The results demonstrate that 2 the chatter frequency is 665 Hz, which is near the 1.5 dominant structural frequency of 653 Hz. Note 1 that the power spectral density at the frequency of 0.5 0 Hz is ignored in Figure 35.18 and Figure 35.19. 0 This component is stronger than the components 0 100 200 300 400 500 600 700 800 900 1000 at all other frequencies since the machining force Frequency (Hz) Fz fluctuates about a static, nonzero value. In this application, a thresholding algorithm may ignore FIGURE 35.19 Power spectral density of Fz in a face the low frequencies where the tooth passing milling operation with d ¼ 1.5 mm. frequency is strong; however, if the operation were to be performed at a higher spindle speed, say 7500 rpm, or the number of teeth were increased from 4 to 20, the tooth passing frequency would be 500 Hz, close to the structural frequencies. In this case, the force signal would have to be filtered at the tooth passing frequency.

This section presented techniques to detect the occurrence of regenerative chatter. The phenomenon of regenerative chatter is easily detected by an operator due to the loud, high pitched noise it produces and the distinctive “chatter marks” it leaves on the part surface. The most common method to detect the presence of chatter is to threshold the frequency signal of a process signal. In this case, one must be careful to separate out the spindle rotation and tooth passing frequencies.

35 20

35.6


Chatter Suppression

Most machining process plans are derived from handbooks or from a database. Since these plans do not consider the physical machine that will be used, chatter free operations cannot be guaranteed. Thus, multiple iterations, where the feed or spindle speed are adjusted using the operator’s experience, are typically required. The tool position may also be adjusted (e.g., the depth of cut may be decreased) to suppress chatter and, while this is guaranteed to be effective due to the presence of the asymptotic stability borderline, this approach is typically not employed since part program must be rewritten to add multiple passes, thereby drastically decreasing productivity. The stability lobe diagram can be used as a tool to plan chatter free machining operations and productivity can be greatly increased by selecting the process parameters to lie in a pocket between two lobes. A cutting tool design methodology (Altintas et al., 1999) has also been proposed for milling tools where the pitch is slightly adjusted such that the teeth are not evenly spaced. The variable pitch has the effect of changing the phase difference between successive teeth vibrations and, if designed properly, will suppress chatter. These design techniques are very sensitive to parameter variations and model uncertainty, and may not be used reliably for a large range of operating conditions. This section will describe methods for automatic chatter suppression.

35.6.1

Spindle-Speed Selection

For the stability lobe diagram generated from a system modeled as having a one dimensional structure, it is seen that the maximum depths of cut are located at the tooth passing frequencies (i.e., the number of teeth multiplied by the spindle speed) corresponding to the dominant structural frequency and integer fractions thereof. If the dominant structural frequency is known, it may be used as an aid in selecting spindle speeds; however, the structural dynamics are often unknown and may be determined only through costly testing. Further, structural dynamics change drastically over time. It is known, however, that during chatter, the dominant frequency seen in the cutting process output is close to a dominant structural frequency. This fact is used in Smith and Delio (1992) to suppress chatter automatically. The following steps are taken: 1. Implement a chatter detection routine to determine the presence of chatter. 2. If chatter is detected, determine the chatter frequency, vc : This will be the frequency at which the process signal has the greatest energy. 3. Set the new spindle speed to be Ns ¼ vc =½Nt ðN þ 1Þ ; where N is the smallest positive integer such that the new spindle speed does not violate the maximum spindle speed constraint. 4. Repeat Steps 1 to 3 until the chatter has been suppressed. The equation Ns ¼ vc = ½Nt ðN þ 1Þ may be interpreted as selecting the tooth passing frequency, or an integer fraction thereof, corresponding to the approximate dominant structural frequency. Note that if the depth of cut is too large and the maximum spindle speed is too small, this technique will not be effective and the feed or depth of cut must be adjusted, or the spindle speed must be continuously varied.

35.6.2

Example 6

The feed force for a turning operation is given by Equation 35.1, and the structural dynamics are given by Equation 35.26. The system parameters are P ¼ 0.75 kN/mm2, fnom ¼ 0.1 mm, vn ¼ 750 Hz, z ¼ 0.1, and k ¼ 15 kN/mm. The depth of cut is 5 mm. The spindle speed that should be selected to suppress chatter if the chatter frequency is 725 Hz, when the spindle speed is not constrained, is determined. The spindle speed that should be selected to suppress chatter if the maximum spindle speed is 15,000 rpm is also determined. The system is simulated for a spindle speed of 10,000 rpm for ten spindle revolutions and then for ten spindle revolutions for the spindle


35 21


Feed force (kN)

0.5 0.4 0.3 0.01

0.02

0.03

0.04 0.05 Time (s)

0.06

0.07

0.01

0.02

0.03

0.04 0.05 Time (s)

0.06

0.07

0 −0.01 −0.02 −0.03 −0.04

Time domain simulations using spindle speed selection with Ns ¼ 43,500 rpm.

FIGURE 35.20


Feed force (kN)

speed calculated when the spindle speed is not constrained. The simulation is then repeated for the spindle speed calculated when the spindle speed is constrained. Feed force and tool displacement are plotted for both cases. For a chatter frequency of 725 Hz, the optimal spindle speed is 60(725) ¼ 43,500 rpm. Other possible spindle speeds are 43,500/2 ¼ 21,750 rpm, 43,500/3 ¼ 14,500 rpm, 43,500/4 ¼ 10,875 rpm, and so on. Therefore, when the maximum spindle speed is 15,000 rpm, a spindle speed of 14,500 rpm is used. The time domain simulations are in Figure 35.20 and Figure 35.21. The results illustrate that a depth of cut of 5.3 mm is stable at 43,500 rpm, but not at 14,500 rpm. Therefore, if the spindle speed is limited to 15,000 rpm, spindle speed selection may not be used to suppress the chatter present in the machining operation.

0.5 0.4 0.3 0.01

0.02

0.03

0.04

0.05 0.06 Time (s)

0.07

0.08

0.09

0.1

0.01

0.02

0.03

0.04

0.05 0.06 Time (s)

0.07

0.08

0.09

0.1

0 −0.01 −0.02 −0.03 −0.04

FIGURE 35.21

Time domain simulations using spindle speed selection with Ns ¼ 14,500 rpm.

35 22

35.6.3


Feed and Depth-of-Cut Selection

When chatter occurs, operators will sometimes increase the feedrate via the feedrate override button on the machine tool control panel. This has the effect of increasing the feed, assuming the spindle speed remains constant. When linear chatter analysis techniques are employed, the force process gains are linearized about the nominal feed, and stability does not appear to be affected by the nominal feed. However, the stability results are only valid for a small region about the nominal feed. It is well known that there is a nonlinear relationship between the machining forces and the feed of the form F ¼ Pðf Þdf : The pressure can be expressed in the form Pðf Þ ¼ Kf a where a , 0; thus, the pressure decreases as the feed increases. Since the stable depth of cut is inversely proportional to the pressure, the stability limit will increase as the feed increases, assuming the spindle speed remains constant. An illustration of this phenomenon was shown in Example 3: when the feed was increased from 0.1 to 0.2 mm, chatter was suppressed. While increasing the feed can suppress chatter, the sensitivity of chatter to feed is limited and other adverse phenomenon, such as tooth chippage, may occur. Another method to suppress chatter is to decrease the depth of cut (Weck et al., 1975). This method is guaranteed to work as evidenced by stability lobe diagrams. However, this method is typically not preferred as it dramatically decreases operation productivity by increasing the total number of tool passes that are required to complete the operation.

35.6.4

Spindle-Speed Variation

tool displacement (mm)

feed force (kN)

Spindle speed variation (SSV) is another technique that has shown the ability to suppress chatter (Inamura and Sata, 1974; Lin et al., 1990). The spindle speed is varied about some nominal value, typically in a sinusoidal manner. Although SSV is a promising technique, the theory required to guide the designer in the selection of suitable amplitudes and frequencies is in its infancy (Radulescu et al., 1997a, 1997b; Sastry et al., 2002). Also, in some cases, SSV may create chatter that would not occur when using a constant spindle speed.

FIGURE 35.22

0.5 0.4 0.3 0.2 0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

0.02 0 −0.02 −0.04 −0.06

Time domain simulations using spindle speed variation with A ¼ 0.1 and V ¼ 20 Hz.

Feed force (kN)


0.45 0.4 0.35 0.3


0.25

FIGURE 35.23

35.6.5

35 23

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

0 −0.01 −0.02 −0.03 −0.04


Example 7

The feed force for a turning operation is given by Equation 35.1 and the structural dynamics are given by Equation 35.26. The system parameters are P ¼ 0.75 kN/mm2, fnom ¼ 0.1 mm, vn ¼ 750 Hz, z ¼ 0.1, and k ¼ 15 kN/mm. The depth of cut is 5 mm. The system is simulated for a nominal spindle speed of Nnom ¼ 10,000 rpm for 10 spindle revolutions and then for 30 spindle revolutions for the spindle speed calculated from Equation 35.67 for the following three cases: A ¼ 0.1 and V ¼ 20 Hz, A ¼ 0.25 and V ¼ 20 Hz, and A ¼ 0.25 and V ¼ 160 Hz. Feed force and tool displacement are plotted for all three cases.

Feed force (kN)

Ns ðtÞ ¼ Nnom ½1 þ A sinðVtÞ

0.45 0.4 0.35 0.3


0.25

FIGURE 35.24

ð35:67Þ

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

0 −0.01 −0.02 −0.03 −0.04


35 24


The time domain simulations are in Figure 35.22 to Figure 35.24 for the respective cases. The results illustrate that SSV may be utilized to suppress chatter; however, the amplitude and frequency of the spindle speed vibration must be carefully chosen.

This section presented several techniques to suppress regenerative chatter. The three major techniques to suppress chatter are spindle speed selection, feed selection, and SSV. In spindle speed selection, the spindle speed is adjusted to be a multiple of the chatter frequency to place the spindle speed in a pocket of the stability lobe diagram. In feed selection, the feed is increased to suppress chatter. In SSV, the spindle speed is varied in a sinusoidal manner to decrease the phase difference between the current and previous tooth passes.

35.7

Case Study

A case study of regenerative chatter for a face milling operation is now presented. Further details are presented in Landers (1997). The machine tool is a three axis vertical milling machine (Figure 35.25). Each axis has a linear encoder with a resolution of 10 mm mounted on it. The axis motors (186 W) drive pulleys that rotate leadscrews and provide motion to the linear axes. The spindle (2240 W) drives the face mill (Carboloy R/L220.13 02.00 12, 50 mm diameter). The tool holds four carbide inserts (Carboloy SEAN 42AFTN M14 HX, 458 lead angle). The part is 6061 aluminum. The spindle is run open loop. A Kistler 9293 piezoelectric three component dynamometer was utilized for force process modeling and chatter detection. The x and y channels have a natural frequency of 4.5 kHz, rigidity of 0.7 kN/mm, and range of 20 to 20 kN. The z channel has a natural frequency of 5 kHz, rigidity of 7 kN/mm, and range of 100 to 200 kN. A Bently Nevada 3000 Series Type 190 proximity transducer was utilized to measure the static stiffnesses of the structural components. The sensor gain is 8 V/mm, the response is flat to 10 kHz, and the range is 1.02 mm. A Kistler Quartz Model #802A accelerometer (resonant frequency 36.7 kHz) was utilized to measure the dynamic characteristics of the structural components. The cutting and thrust pressures, respectively, are PC ¼ 0:29f

0:25

d

0:13

PT ¼ 0:16f

0:40

d

0:41

V 1000

0:72

V 1000

0:58

ð35:68Þ ð35:69Þ

The transfer function matrices, respectively, between the tool structure and machining forces, and the part structure and machining forces are modeled as 2 3 ð45002 =14Þ " # 6 2 0 7" F ðsÞ # xt ðsÞ 6 s þ 2ð0:07Þð4500Þs þ 45002 7 x 7 ¼6 ð35:70Þ 6 7 2 4 5 Fy ðsÞ yt ðsÞ ð4100 =14Þ 0 s2 þ 2ð0:11Þð4100Þs þ 41002 2 "

xp ðsÞ yp ðsÞ

# ¼

ð26002 =9:5Þ 6 2 6 s þ 2ð0:09Þð2600Þs þ 26002 6 6 4 0

3 0 ð21002 =9:5Þ s2 þ 2ð0:22Þð2100Þs þ 21002

7" F ðsÞ # 7 x 7 7 5 Fy ðsÞ

ð35:71Þ


FIGURE 35.25

35 25

Three axis vertical machine tool schematic.

Using the machining force and structural models, Depth-of-cut (mm) a stability lobe diagram was constructed using time 2.0 domain simulations. Experimental data were 1.5 collected by adjusting the depth of cut in incre ments of 0.1 mm until chatter occurred. The time 1.0 domain simulations and the experimental data are plotted in Figure 35.26. The cutting conditions 0.5 were ft ¼ 0.10 mm/tooth, N t ¼ 4 teeth, uen ¼ 908, and uex ¼ 908. The chatter detection 0.0 1000 1500 2000 2500 methodology for this system was described in Spindle speed (rpm) Example 5. Spindle speed adjustment is typically a more productive chatter suppression option. However, FIGURE 35.26 Stability lobe diagram for a face time domain simulations (empty the machine tool in this case study is not equipped milling operation boxes) and experimental points (filled circles). with automatic spindle speed control and, thus, the depth of cut is adjusted to suppress chatter. When chatter is detected, the chatter suppressor rewrites the part program to accommodate one additional tool pass (Figure 35.27). Therefore, the new operation depth of cut is dn ¼

dp 1 þ Nc

ð35:72Þ

35 26

Vibration and Shock Handbook 8b 4b 4a 3b

TOOTH

3a

5b

2b

7b

1a

1b

2a 6b

chatter detected

WORKPIECE

original tool path (a)

FIGURE 35.27 tool pass.

new tool path (b)

Original tool path (a) is rewritten when chatter occurs. New tool path (b) contains an additional

where dp is the previous operation depth of cut and Nc is the number of times the chatter suppression routine has been invoked. The new value may be well below the stability limit; however, making all passes an equal depth of cut provides a good balance between productivity and the search for a stable depth of cut. Results of this controller are presented in Landers and Ulsoy (1998, 2001).

Nomenclature Symbol Quantity

Symbol Quantity

d dlim f ft F FC FT Fx

T Tt xp

Fy Fz k Ns Nt P PC PT t

depth of cut (mm) limiting stable depth of cut (mm) feed (mm) feed per tooth (mm) machining force (kN) cutting force (kN) thrust force (kN) force acting on cutting tool in x direction (kN) force acting on cutting tool in y direction (kN) force acting on cutting tool in z direction (kN) structural stiffness (kN/mm) spindle speed (rpm) number of teeth machining pressure (kN/mm2) cutting pressure (kN/mm2) thrust pressure (kN/mm2) time (sec)

xt yp yt z

u uen uex vc vn cr z

spindle rotational period (sec) tooth rotational period (sec) part structural displacement in x direction (mm) cutting tool structural displacement in x direction (mm) part structural displacement in y direction (mm) cutting tool structural displacement in y direction (mm) cutting tool structural displacement in z direction (mm) tooth angle (rad) tooth entry angle (rad) tooth exit angle (rad) chatter frequency (rad/sec) structural natural frequency (rad/sec) lead angle (rad) structural damping ratio


35 27

References Altintas, Y. and Budak, E., Analytical prediction of stability lobes in milling, Ann. CIRP, 44/1, 357 362, 1995. Altintas, Y., Engin, S., and Budak, E., Analytical stability prediction and design of variable pitch cutters, ASME J. Manuf. Sci. Eng., 121, 173 178, 1999. Budak, E. and Altintas, Y., Analytical prediction of chatter stability in milling, part I: general formulation, ASME J. Dyn. Syst. Meas. Control, 120, 22 30, 1998a. Budak, E. and Altintas, Y., Analytical prediction of chatter stability in milling, part II: application of the general formulation to common milling systems, ASME J. Dyn. Syst. Meas. Control, 120, 31 36, 1998b. Cho, D.W. and Ehmann, K.F., Pattern recognition for on line chatter detection, Mech. Syst. Signal Process., 2, 279 290, 1988. Delio, T., Tlusty, J., and Smith, S., Use of audio signals for chatter detection and control, ASME J. Eng. Ind., 114, 146 157, 1992. Elbestawi, M.A., Ismail, F., Du, R., and Ullagaddi, B.C., Modeling machining dynamics including damping in the tool workpiece interface, ASME J. Eng. Ind., 116, 435 439, 1994. Inamura, T. and Sata, T., Stability analysis of cutting under varying spindle speed, Ann. CIRP, 23/1, 119 120, 1974. Inasaki, I., Karpuschewshi, B., and Lee, H. S., Grinding chatter origin and suppression, Ann. CIRP, 50/2, 515 534, 2001. Koenigsberger, I., Tlusty, J. 1971. Structures of Machine Tools, Pergamon Press, New York. Landers, R.G. 1997. Supervisory machining control: a design approach plus force control and chatter analysis components, Ph.D. Dissertation. Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor. Landers, R.G. and Ulsoy, A.G., Supervisory machining control: design and experiments, Ann. CIRP, 47/1, 301 306, 1998. Landers, R.G. and Ulsoy, A.G., Supervisory control of a face milling operation in different manufacturing environments, Trans. Control Automat. Syst. Eng., 3, 1 9, 2001. Lee, A. C. and Liu, C. S., Analysis of chatter vibration in a cutter workpiece system, Int. J. Mach. Tools Manuf., 31, 221 234, 1991a. Lee, A. C. and Liu, C. S., Analysis of chatter vibration in the end milling process, Int. J. Mach. Tools Manuf., 31, 471 479, 1991b. Lin, S.C., DeVor, R.E., and Kapoor, S.G., The effects of variable speed cutting on vibration control in face milling, ASME J. Eng. Ind., 112, 1 11, 1990. Merritt, H.E., Theory of self excited machine tool chatter: contribution to machine tool chatter research 1, ASME J. Eng. Ind., 87, 447 454, 1965. Minis, I., Magrab, E., and Pandelidis, I., Improved methods for the prediction of chatter in turning, Part III: a generalized linear theory, ASME J. Eng. Ind., 112, 28 35, 1990a. Minis, I. and Yanushevsky, R., A new theoretical approach for the prediction of machine tool chatter in milling, ASME J. Eng. Ind., 115, 1 8, 1993. Minis, I., Yanushevsky, R., and Tembo, A., Analysis of linear and nonlinear chatter in milling, Ann. CIRP, 39/1, 459 462, 1990b. Radulescu, R., Kapor, S.G., and DeVor, R.E., An investigation of variable spindle speed face milling for tool work structures with complex dynamics, part 1: simulation results, ASME J. Manuf. Sci. Eng., 119, 266 272, 1997a. Radulescu, R., Kapor, S.G., and DeVor, R.E., An investigation of variable spindle speed face milling for tool work structures with complex dynamics, part 2: physical explanation, ASME J. Manuf. Sci. Eng., 119, 273 280, 1997b. Sastry, S., Kapor, S.G., and DeVor, R.E., Floquet theory based approach for stability analysis of the variable speed face milling process, ASME J. Manuf. Sci. Eng., 124, 10 17, 2002.

35 28


Smith, S. and Delio, T., Sensor based chatter detection and avoidance by spindle speed selection, ASME J. Dyn. Syst. Meas. Control, 114, 486 492, 1992. Smith, S. and Tlusty, J., Efficient simulation programs for chatter in milling, Ann. CIRP, 42/1, 463 466, 1993. Tarng, Y.S. and Li, T.C., Detection and suppression of drilling chatter, ASME J. Dyn. Syst. Meas. Control, 116, 729 734, 1994. Tlusty, J., Dynamics of high speed milling, ASME J. Eng. Ind., 108, 59 67, 1986. Tlusty, J. 2000. Manufacturing Processes and Equipment, Prentice Hall, Upper Saddle River, NJ. Tlusty, J. and Ismail, F., Basic nonlinearity in machining chatter, Ann. CIRP, 30/1, 299 304, 1981. Tlusty, J. and Ismail, F., Special aspects of chatter in milling, ASME J. Vib. Acoust. Stress Reliab. Des., 105, 24 32, 1983. Tobias, S.A. 1965. Machine Tool Vibration, Wiley, New York. Tsai, M.D., Takata, S., Inui, M., Kimura, F., and Sata, T., Prediction of chatter vibration by means of a model based cutting simulation system, Ann. CIRP, 39/1, 447 450, 1990. Weck, M., Altintas, Y., and Beer, C., CAD assisted chatter free NC tool path generation in milling, Int. J. Mach. Tools Manuf., 34, 879 891, 1994. Weck, M., Verhagg, E., and Gather, M., Adaptive control of face milling operations with strategies for avoiding chatter vibrations and for automatic cut distribution, Ann. CIRP, 24/1, 405 409, 1975. Yun, I.S., Wilson, W.R.D., and Ehmann, K.F., Review of chatter studies in cold rolling, Int. J. Mach. Tools Manuf., 38, 1499 1530, 1998.

36

Fluid-Induced Vibration

36.1 Description of the Ocean Environment ............................ 36 1 Spectral Density † Ocean Wave Spectral Densities † Approximation of Spectral Density from Time Series † Generation of Time Series from a Spectral Density † Short Term Statistics † Long Term Statistics † Summary

36.2 Fluid Forces ......................................................................... 36 16 Wave Force Regime † Wave Forces on Small Structures Morison Equation † Vortex Induced Vibration † Summary

36.3 Examples .............................................................................. 36 23

Seon M. Han Texas Tech University

Static Configuration of a Towing Cable † Fluid Forces on an Articulated Tower † Distribution of Significant Wave Heights Weibull and Gumbel Distributions † Reconstructing Time Series for a Given Significant Wave Height † Available Numerical Codes

Summary This chapter gives an overview on the subject of fluid induced vibration in an ocean environment. The main objective is to show how the fluid forces on an offshore structure due to current and random waves are modeled. The chapter is divided into three sections. The first section describes the ocean environment, especially the currents and random waves. The second section is dedicated to obtaining fluid forces utilizing the results from the first section and the third section gives some examples to show how the results from the first two sections can be used in practice. In the first section, the concept of spectral density is introduced. For a given spectrum, methods to obtain a sample time series are given. In the second section, the forces that the fluid can exert on a body are discussed. The regimes in which inertia, drag, or diffraction forces are dominant are shown in terms of the ratio of the wave height to the structural diameter and the ratio of the structural diameter to the wavelength. The Morison equation is extended to the case of a moving inclined cylinder. The Morison equation requires the use of experimentally determined fluid coefficients such as added mass, inertia, and drag coefficients. Plots of these fluid coefficients for various values of the fluid parameters are reproduced here. The vortex shedding force is discussed briefly. In the third section, four examples are given to show how fluid forces affect the static and dynamics of ocean structures, how the significant wave height can be chosen to represent the condition in a certain area for a long time, and how the time series can be constructed from a given spectrum. Finally, the available numerical codes for modeling slender flexible bodies in fluids are listed.

36.1

Description of the Ocean Environment

In modeling offshore structures, one needs to account for the forces exerted by the surrounding fluid. In depth studies are given in Kinsman (1965), Sarpkaya and Isaacson (1981), Wilson (1984), Chakrabarti (1987), and Faltinsen (1993). The vibration characteristics of a structure can be significantly altered when it is surrounded by water. For example, damping by the fluid (or the added mass) lowers the natural 36 1

36 2


frequency of vibration. When considering the dynamics of an offshore structure, one must also consider the forces due to the surrounding fluid. The two important sources of fluid motion are ocean waves and ocean currents. Most steady large currents are generated by the drag of the wind passing over the surface of the water, and they are confined to a region near the ocean surface. Tidal currents are generated by the gravitational attraction of the sun and the moon, and they are most significant near coasts. The ultimate source of the ocean circulation is the uneven radiation heating of the Earth by the Sun. Isaacson (1988) suggested an empirical formula for the current velocity in the horizontal direction as a function of depth: Uc ðxÞ ¼ ðUtide ðdÞ þ Ucirculation ðdÞÞ

x d

1=7

þ Udrift ðdÞ

x

d þ d0 d0

ð36:1Þ

where Udrift is the wind induced drift current, Utide is the tidal current, Ucirculation is the low frequency long term circulation, x is the vertical distance measured from the ocean bottom, d is the depth of the water, and d0 is the smaller of the depth of the thermocline and 50 m. The value of Utide is obtained from tide tables, and Udrift is about 3% of the 10 min mean wind velocity at 10 m above the sea level. It should be noted that these currents evolve slowly compared with the time scales of engineering interests. Therefore, they can be treated as a quasisteady phenomenon. Waves, on the other hand, cannot be treated as a steady phenomenon. The underlying physics that govern wave dynamics are too complex and, therefore, waves must be modeled stochastically. The subsequent section discusses the concept of the spectral density, available ocean wave spectral densities, a method to obtain the spectral density from wave time histories, methods to obtain a sample time history from a spectral density, the short term and long term statistics, and a method to obtain fluid velocities and accelerations from wave elevation using linear wave theory.

36.1.1

Spectral Density

Wave Elevation h(t)

Wave Elevation h(x)

Here, we will consider only surface gravity waves. l Let us first consider a regular wave in order to familiarize ourselves with the terms that are used A H to describe a wave. The wave surface elevation is denoted as hðx; tÞ and can be written as hðx; tÞ ¼ Distance t=0 A cosðkx vtÞ; where k is the wave number, and v t=t is the angular frequency. Figure 36.1 shows the surface elevation at two time instances (t ¼ 0 and t ¼ t) and the surface elevation at a fixed location T ðx ¼ 0Þ: A is the amplitude, H is the wave height or the distance between the maximum and minimum A H wave elevation or twice the amplitude, and T is the period given by T ¼ 2p=v: Time x=0 In practice, waves are not regular. Figure 36.2 shows a schematic time history of an irregular wave surface elevation. The wave height and frequency are not easy to find. Therefore, we rely FIGURE 36.1 Regular wave. on a statistical description for the wave elevation such as the wave spectral density. The spectral density tells us how the energy of the system is distributed among frequencies. The random surface elevation hðtÞ can be thought of as a summation of regular waves with different frequencies. The surface elevation hðtÞ is related to its Fourier transform XðvÞ by

hðtÞ ¼

1 ð1 XðvÞ expð ivtÞdv 2p 1

36 3

Suppose that the energy of the system is pro portional to h2 ðtÞ so that we can write the energy as E¼

1 C h2 ðtÞ 2

where C is the proportionality constant. Let us assume that the expected value of the energy is given by E{E } ¼

Wave Elevation h (t)

Fluid Induced Vibration

H

FIGURE 36.2

1 CE{h2 ðtÞ} 2

Time

Time history of random wave.

where E{h2 ðtÞ} is the mean square of hðtÞ: If hðtÞ is an ergodic process (see Chapter 5 and Chapter 30), then the mean square of hðtÞ can be approximated by the time average over a long period of time: E{h2 ðtÞ} ¼ lim

Ts !þ1

1 ðTs =2 2 1 1 ð1 h ðtÞdt ¼ lim lXðvÞl2 dv Ts !þ1 Ts 2p Ts 1 Ts =2

where we have used Parseval’s theorem ð1 1

h2 ðtÞdt ¼

where lXðvÞl2 ¼ XðvÞX p ðvÞ; XðvÞ ¼

ð1 1

1 ð1 lXðvÞl2 dv 2p 1

hðtÞ expð ivtÞdt; X p ðvÞ ¼

ð36:2Þ

ð36:3Þ ð1 1

hðtÞ expðivtÞdt

We define the power spectral density (or simply the spectrum) as (see Chapter 5 and Chapter 30) Shh ðvÞ ;

1 lXðvÞl2 2pTs

ð36:4Þ

so that E{h2 ðtÞ} is given by E{h2 ðtÞ} ¼

ð1 1

Shh ðvÞdv

ð36:5Þ

For a zero mean process, E{h2 ðtÞ} is also the variance s2h : The spectral density has units of h2 t: Where h is the wave elevation, the spectral density has a unit of m2 sec. It can also be shown that Shh ðvÞ is related to the autocorrelation function, RðtÞ; by the Wiener Khinchine relations (Wiener, 1930; Khinchine, 1934): ð1 1 ð1 Shh ðvÞ ¼ Rhh ðtÞ expð ivtÞdt; Rhh ðtÞ ¼ Shh ðvÞ expðivtÞdv ð36:6Þ 2p 1 1 It should be noted that, in some textbooks, the factor 1=2p appears in the second equation instead of the first. Figure 36.3 shows some important pairs of Shh ðvÞ and Rhh ðtÞ: There are a few properties of the spectral density that readers should become familiar with. The first property is that the spectral density function of a real valued stationary process is both real and symmetric. That is, Shh ðvÞ ¼ Shh ð vÞ (Equation 36.4). Secondly, the area under the spectral density is equal to E{h2 ðtÞ} (Equation 36.5) and is also equal to Rhh ð0Þ ¼ s2h m2h ; where s2h is the variance and m2h is the mean of hðtÞ: In most cases, we only consider a zero mean process so that the area under the spectral density is just s2h : If the process does not have a zero mean, the mean can be subtracted from it so that the process has a zero mean. For ocean applications, a one sided spectrum in terms of cycles per second (cps) or hertz is often used. We will denote the one sided spectrum with a superscript “o”. The one sided spectrum can be obtained from the two sided spectrum by o ðvÞ ¼ 2Shh ðvÞ; v $ 0 Shh

36 4


R(t) =

•

S(w) = 2p

1

R(t)e-iwtdw

-•

1 t

0

t

cos w0t

1 2

sin w 0t t

1 2

d(w +w 0)

w0

0

t

T t

4 sin2 (wT/2) Tw 2 0 2p /T

-aΩtΩcos w0t 2p 2pe

0

w

0

2p

T

w

2a a 2+w 2

2p e-aΩtΩ

0

d(w -w 0)

t

0

0

FIGURE 36.3

w

0

-w 0

-T

w

1

0

2p

d(w)

0

2pd (w)

1

0

•

1

S(w)e iw tdw

-•

w

1/a t

-w 0

0

w0

w

Relationship between the autocorrelation function and the power spectral density.

The two sided spectrum in terms of v can be transformed to the spectrum in terms of f (where v ¼ 2pf ) by Shh ðf Þ ¼ 2pShh ðvÞ; f ; v $ 0 Then, the two sided spectrum in terms of v can be transformed to the one sided spectrum in terms of cps (or hertz) by Sohh ðf Þ ¼ 4pShh ðvÞ; f ; v $ 0


36 5

It should be noted that the spectral density that we have defined here is the amplitude half spectrum. The amplitude, height, and height double spectra are related to the amplitude half spectrum by SA ðvÞ ¼ 2SðvÞ; SH ðvÞ ¼ 8SðvÞ; S2H ðvÞ ¼ 16SðvÞ

36.1.2

Ocean Wave Spectral Densities

In this section, we will discuss spectral density models to describe a random sea. An excellent review of existing spectral density models is given in Chapter 4 of Chakrabarti (1987). The ocean wave spectrum models are semiempirical formulas. That is, they are derived mathematically but the formulation requires one or more experimentally determined parameters. The accuracy of the spectrum depends significantly on the choice of these parameters. In formulating spectral densities, the parameters that influence the spectrum are fetch limitations, decaying vs. developing seas, water depth, current, and swell. The fetch is the distance over which a wind blows in a wave generating phase. Fetch limitation refers to the limitation on the distance due to some physical boundaries so that full wave development is prohibited. In a developing sea, the sea has not yet reached its stationary state under a stationary wind. In contrast, a wind has blown for a sufficient time in a fully developed sea, and the sea has reached its stationary state. In a decaying sea, the wind has dropped off from its stationary value. Swell is the wave motion caused by a distant storm and persists even after the storm has died down or moved away. The Pierson Moskowitz (P M) spectrum (Pierson and Moskowitz, 1964) is the most extensively used spectrum for representing a fully developed sea. It is a one parameter model in which the sea severity can be specified in terms of the wind velocity. The P M spectrum is given by ! !4 8:1 £ 10 3 g 2 g o 4 Shh ðf Þ ¼ exp 0:74 v Uw;19:5 m v5 where g is the gravitational constant and Uw;19:5 m is the wind speed at a height of 19.5 m above the still water. The P M spectrum is also called the wind speed spectrum because it requires wind data. It can also be written in terms of the modal frequency vm as Sohh ðf Þ ¼

8:1 £ 10 3 g 2 exp v5

1:25

vm v

4

ð36:7Þ

Note that the modal frequency is the frequency at which the spectrum is the maximum. In some cases, it may be more convenient to express the spectrum in terms of significant wave height rather than the wind speed or modal frequency. For a narrowband Gaussian process1, the significant wave height is related to the standard deviation by Hs ¼ 4sh : The standard deviation is the square root of the Ð area under the spectral density, 11 Shh ðvÞdv ¼ s2h : Then, the spectrum can be written as ! 8:1 £ 10 3 g 2 0:0324g 2 o 4 ð36:8Þ exp Shh ðf Þ ¼ v v5 Hs2 and the peak frequency and the significant wave height are related by pffiffiffiffiffiffi vm ¼ 0:4 g=Hs

ð36:9Þ

The P M spectrum is applicable for deep water, unidirectional seas, fully developed and local wind generated seas with unlimited fetch, and was developed for the North Atlantic. The effect of swell is not accounted for. Although it was developed for the North Atlantic, the spectrum is valid for other locations. However, the limitation that the sea is fully developed may be too restrictive because it cannot model the 1

See Section 36.1.5 for details.

36 6


o Shh ðf Þ ¼ 0:169

v4s 2 H exp v5 s

0:675

vs v

2.5 Spectral Density (m2 s)

effect of waves generated at a distance. Therefore, we consider a two parameter spectrum, such as the Bretschneider spectrum, in order to model a sea that is not fully developed as well as a fully developed sea. The Bretschneider spectrum (Bretschneider, 1959, 1969) is a two parameter spectrum in which both the sea severity and the state of development can be specified. The Bretschneider spectrum is given by

spectrum { Pierson-Moskowitz w = 0.731

Hs = 4m

s

2

ws = 0.877 ws = 1.023 ws = 1.169 ws = 1.315

1.5 1 0.5

4

0

0

0.5

1 1.5 2 2.5 Frequency w (rad/s)

3

where vs ¼ 2p=Ts and Ts is the significant FIGURE 36.4 Bretschneider spectrum with various period. The sea severity can be specified by Hs values of vs : and the state of development can be specified by pffiffiffiffi vs : It can be shown that the relationship vs ¼ 1:167vm (equivalent to vs ¼ 1:46= Hs ) renders the Bretschneider spectrum and the P M spectrum equivalent. Figure 36.4 shows the Bretschneider spectra for Hs ¼ 4 m. When vs ¼ 0:731 rad/sec, the P M and the Bretschneider spectra are identical. It should be noted that the developing sea will have a slightly higher modal frequency than the fully pffiffiffi developed sea, and can be described by vs greater than 1:46= Hs : Other two parameter spectral densities that are often used are the International Ship Structures Congress (ISSC) and the International Towing Tank Conference (ITTC) spectra. The ISSC spectrum is written in terms of the significant wave height and the mean frequency, where the mean frequency is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ð1 u vSðvÞdv u u 0 v¼u ¼ 1:30vm ð 1 t SðvÞdv 0

Thus, the ISSC spectrum is given by o ðf Þ ¼ 0:111 Shh

v4 2 H exp v5 s

0:444

v v

4

The ITTC spectrum is based on the significant wave height and the zero crossing frequency and is given by o ðf Þ ¼ 0:0795 Shh

v4z 2 H exp v5 s

0:318

vz v

4

where the zero crossing frequency, vz , is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ð1 u v2 SðvÞdv u u 0 vz ¼ u ¼ 1:41vm t ð1 SðvÞdv 0

The Bretschneider, ITTC, and ISSC spectra are called two parameter spectra, and they can be written as Sohh ðf Þ ¼ with A and v given in Table 36.1.

A v~4 2 H exp 4 v5 s

A

v~ v

4


36 7

TABLE 36.1 Two Parameter Spectrum Models o ðvÞ ¼ ðA=4ÞHs2 v4 =v5 expð Aðv=vÞ 4 Þ Shh Model Bretschneider ITTC ISSC

A

v~

0.675 0.318 0.4427

vs vz v

The spectra that we have discussed so far do not allow us to generate spectra with two peaks to represent local or distant storms or to specify the sharpness of the peaks. The Ochi Hubble (O H) spectrum (Ochi and Hubble, 1976) is a six parameter spectrum with the form: o ðvÞ ¼ Shh

2 1X ðð4li þ 1Þv4mi =4Þli Hsi2 exp Gðli Þ 4 i¼1 v4li þ1

4li þ 1 4

vmi v

4

where Gðli Þ is the Gamma function, Hs1 ; vm1 ; and l1 are the significant wave height, modal frequency, and shape factor for the lower frequency components, respectively, and Hs2 ; vm2 ; and l2 are those for the higher frequency component. Assuming that the entire spectrum is that of a narrow band, the equivalent significant wave height is given by qffiffiffiffiffiffiffiffiffiffiffiffi 2 þ H2 Hs ¼ Hs1 s2 For l1 ¼ 1 and l2 ¼ 0; the spectrum reduces to the P M spectrum. With the assumption that the entire spectrum is narrowband, the value of l1 is much higher than l2 : The O H spectrum represents unidirectional seas with unlimited fetch. The sea severity and the state of development can be specified by Hsi and vmi ; respectively. In addition, li can be selected to control the frequency width of the spectrum. For example, a small li (wider frequency range) describes a developing sea, and a large li (narrower frequency range) describes a swell condition. Figure 36.5 shows the O H spectrum with l1 ¼ 2:72; vm1 ¼ 0:626 rad/ sec, Hs1 ¼ 3:35 m, l2 ¼ 2:72; vm2 ¼ 1:25 rad/sec, and Hs2 ¼ 2:19 m. Finally, another spectrum that is commonly used is the Joint North Sea Wave Project (JONSWAP) spectrum developed by Hasselmann et al. (1973). It is a fetch limited spectrum because the growth over a limited fetch is taken into account. The attenuation in shallow water is also taken into account. The JONSWAP spectrum is written as o Shh ðvÞ ¼

ag 2 exp v5

1:25

vm v

4

gexpð

ðv vm Þ=2t 2 v2m Þ

a ¼ 0:076ðXÞ X¼

0:22

or 0:0081 if fetch independent

gX=Uw2

higher frequency spectrum

1 0

X ¼ fetch length ðnautical milesÞ Uw ¼ wind speed ðknotsÞ

vm ¼ 2p £ 3:5ðg=Uw ÞX

Spectrum Density (m2 s)

where g is the peakedness parameter and t is the shape parameter. The peakedness parameter g is the ratio of the maximum spectral energy to the maximum spectral energy of the corresponding P M spectrum. That is, when g ¼ 7; the peak spectral energy is seven times that of the P M spectrum. 8 > 7:0 for very peaked data > 5 < Hs = 4 m g ¼ 3:3 for mean of selected JONSWAP data > lower frequency > 4 : spectrum 1:0 for P M spectrum ( 3 0:07 for v # vm t¼ 2 0:09 for v . vm

0:33

0

0.5

1 1.5 2 Frequency w (rad/s)

FIGURE 36.5

2.5

Ochi Hubble spectrum.

3

36 8


Figure 36.6 shows the JONSWAP spectrum when a ¼ 0:0081 and vm ¼ 0:626 rad/sec for three peakedness parameters.

18 Spectral Density (m2s)

16

36.1.3 Approximation of Spectral Density from Time Series

14 12

g = 7.0

a = 0.0081 w m = 0.626

10 8

From the time history of the wave elevation, the 6 g = 3.3 spectral density function can be obtained by two 4 g =1 methods. 2 The first method is to use the autocorrelation 0 function Rhh ðtÞ; which is related to the spectral 0 0.5 1 1.5 2 2.5 3 density function Shh ðvÞ by the Wiener Khinchine Frequency w (rad/s) relations (Equation 36.6). The autocorrelation Rhh ðtÞ is the expected value FIGURE 36.6 JONSWAP spectrum for g ¼ 1.0, 3.3, of hðtÞhðt þ tÞ or Rhh ðtÞ ¼ E{hðtÞhðt þ tÞ}; and 7.0. where t is an arbitrary time and t is the time lag. For a weakly stationary process, the autocorrela tion is a function of the time lag only. Assuming that the process is ergodic, the autocorrelation function for a given time history of length Ts can be approximated as R^ hh ðtÞ ¼ lim

Ts !1

ðTs

1 Ts

t

t

0

hðtÞhðt þ tÞdt for 0 , t , Ts

Note that the superscript ‘ is used to emphasize that the variable is an approximation based on a sample time history of length Ts : The spectral density is then obtained by taking the Fourier cosine transform of R^ hh ðtÞ; 1 ðT s ^ R ðtÞ cos vt dt S^ hh ðvÞ ¼ p 0 hh

ð36:10Þ

The second method for obtaining the spectral density function is to use the relationship between the spectral density and the Fourier transform of the time series. They are related by S^ hh ðvÞ ¼ lim

Ts !1

1 ^ lXðvÞX^ p ðvÞl 2pTs

ð36:11Þ

^ vÞ is given by where Xð ^ vÞ ¼ Xð

ðTs 0

hðtÞ expð ivtÞdt

and X^ p ðvÞ is the complex conjugate given by X^ p ðvÞ ¼

ðT s 0

hðtÞ expðivtÞdt

In order to obtain the Fourier transforms of the time series (see Chapter 2, Chapter 10, Chapter 21, and Appendix 2A), the discrete Fourier transform (DFT) or the fast Fourier transform (FFT) procedure can be used. For detailed descriptions of how this is done, see Appendix 1 in Tucker (1991). Nowadays, spectral analysis is almost always carried out via FFTs because it is easier to use and faster than the formal method via correlation function. It should be noted that the length of the sample time history only needs to be long enough so that the limits converge. Taking a longer sample will not improve the accuracy of the estimate. Instead, one should take many samples or break one long sample into many parts. For n samples, the spectral densities


36 9

are obtained for each sample time history using either Equation 36.10 or Equation 36.11, and they are averaged to give the estimate. The determination of the spectral density from wave records depends on the details of the procedure such as the length of the record, sampling interval, degree and type of filtering and smoothing, and time discretization.

36.1.4

Generation of Time Series from a Spectral Density

In a nonlinear analysis, the structural response is found by a numerical integration in time. Therefore, one needs to convert the wave elevation spectrum into an equivalent time history. The wave elevation can be represented as a sum of many sinusoidal functions with different angular frequencies and random phase angles. That is, we write hðtÞ as

hðtÞ ¼

N X i¼1

cosðvi t

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wi Þ 2Shh ðvi ÞDvi

ð36:12Þ

where wi is a uniform random number between 0 and 2p, vi are discrete sampling frequencies, Dvi ¼ vi vi 1 ; and N is the number of partitions. Recall that the area under the spectrum is equal to the variance, sh2 : The incremental area under the spectrum, Shh ðvi ÞDvi ; can be denoted as si2 such that P 2 the sum of all the incremental area equals the variance of the wave elevation or sh2 ¼ N i¼1 si : The time history can be written as

hðtÞ ¼

N X i¼1

cosðvi t

pffiffi wi Þ 2si

The sampling frequencies, vi ; can be chosen at equal intervals such that vi ¼ iv1 : However, the time history will then have the lowest frequency of v1 and will have a period of T ¼ 2p=v1 : In order to avoid this unwanted periodicity, Borgman (1969) suggested that the frequencies are chosen so that the area under the spectrum curve for each interval is equal or si2 ¼ s 2 ¼ sh2 =N: The time history is written as rffiffiffiffi N X 2 ð36:13Þ hðtÞ ¼ sh cosðvi t wi Þ N i¼1 where vi ¼ ðvi þ vi 1 Þ=2: The discrete frequencies, vi , are chosen such that the area between the interval 0 , v , vi is equal to i=N of the total area under the curve between the interval 0 , v , vN or ðvi 0

Shh ðvÞdv ¼

i ðvN S ðvÞdv for i ¼ 1; …; N N 0 hh

where it is assumed that the area under the spectrum beyond vN is negligible. If hðtÞ is a narrowband Gaussian process, the standard deviation can be replaced by sh ¼ Hs =4; and the time history can be written as rffiffiffiffi N Hs 2 X hðtÞ ¼ cosðvi t wi Þ 4 N i¼1 Shinozuka (1972) proposed that the sampling frequencies, vi , in Equation 36.13 should be randomly chosen according to the density function, f ðvÞ ; Sohh ðvÞ=s2h : This is equivalent to performing an integration using the Monte Carlo method. The random frequencies v distributed according to f ðvÞ can be obtained from uniformly distributed random numbers, x, by v ¼ F 1 ðxÞ; where FðvÞ is the cumulative distribution of f ðvÞ: The random frequencies obtained this way are used in Equation 36.13 to generate a sample time series. It should be noted that many sample time histories should be obtained and averaged to synthesize a time history for use in numerical simulations.

36 10

36.1.5


Short-Term Statistics Random process

In discussing wave statistics, we often use the positive maxima negative maxima term significant wave to describe an irregular sea surface. The significant wave is not a physical Z wave that can be seen but rather a statistical description of random waves. The concept of significant wave height was first introduced by Time Sverdrup and Munk (1947) as the average height of the highest one third of all waves. Usually, ships co operate in programs to find sea statistics FIGURE 36.7 A sample time history. by reporting a rough estimate of the storm severity in terms of an observed wave height. This observed wave height is consistently very close to the significant wave height. Stationarity and ergodicity are two assumptions that are made in describing short term waves statistics. These assumptions are valid only for “short” time intervals approximately two hours or the duration of a storm but not for weeks or years. The wave elevation is assumed to be weakly stationary so that its autocorrelation is a function of time lag only. As a result, the mean and the variance are constant, and the spectral density is invariant with time. Therefore, the significant wave height and the significant wave period are constant when we consider short term statistics. In this case, the individual wave height and wave period are the stochastic variables. We then need to determine certain statistics for the analysis and design of offshore structures when we consider short time intervals. Consider a sample time history of a zero mean random process, as shown in Figure 36.7. The questions that we ask are how often is a certain level (e.g., z in the figure) exceeded, and how are the maxima distributed? Likewise, we can ask when we can expect to see that a certain level is exceeded for the first time, and what are the values of the peaks of a random process? The first question is important when a structure may fail due to a one time excessive load, and the second question is important when a structure may fail due to cyclic loads. It is found that the rate at which a random process XðtÞ crosses Z with a positive slope (zero up crossing) may be calculated from ð1 nzþ ¼ vfX X_ ðz; vÞdv 0

_ where fX X_ ðx; x_ Þ is the joint probability density function of X and XðtÞ: The expected time of the first up crossing is then the inverse of the crossing rate or E{T} ¼ 1=nzþ The probability density function of the maxima, A, can be calculated from ð0 vfX X_ X€ ða; 0; vÞdv fA ðaÞ ¼ ð10 vfX_ X€ ð0; vÞdv 1

_ and X: € where fX X_ X€ ðx; x_ ; x€ Þ is the joint probability density function of X, X; If XðtÞ is a Gaussian process, then we can write the joint probability density functions as " !2 # 1 1 x 2 1 x_ exp ; 1 , x , 1; 1 , x_ , 1 fX X_ ðx; x_ Þ ¼ 2psX sX_ 2 sX 2 sX_ and fX X_ X€ ðx; x_ ; x€ Þ ¼

1 3=2

1=2

ð2pÞ lMl

exp

1 ð{x} 2

{mX }ÞT ½M

1

ð{x}

{mX }Þ


where

36 11

2

s2X

0

6 6 ½M ¼ 6 0 4 s2X_

s2X_ 0

s2X_

3

2

7 7 0 7 and {x} 5 s2X€

x 6 {mX } ¼ 6 4 x_

mX

x€

mX€

7 mX_ 7 5

Then, for a stationary Gaussian process, the up crossing rate is given by " ð1 1 1 Z 2 ð1 1 þ exp exp nz ¼ fX_ X€ ðZ; x_ Þ_x d_x ¼ 2psX sX_ 2 sX 2 0 0 ¼

sX_ exp 2psX

1 Z 2 sX

3

x_ sX_

!2 # x_ d_x

2

ð36:14Þ

and the probability density function of maxima is given by the Rice density function (Rice, 1954) " # ! ! pffiffiffiffiffiffiffiffi 1 aa 1 a2 a2 a 1 a2 pffiffiffiffiffiffiffiffi exp fA ðaÞ ¼ pffiffiffiffi þa 2 F for 1 , a , 1 exp 2 sh2 ð1 a2 Þ 2 sh2 sh 2psh sh a2 1 where FðxÞ is the cumulative distribution function of standard normal random variable 1 ðx FðxÞ ¼ pffiffiffiffi expð z 2 =2Þdz 1 2p and a is the irregularity factor equivalent to the ratio of the number of zero up crossings (number of times that h½t crosses zero with a positive slope) to the number of peaks. a ranges from 0 to 1, and it is also equal to

a¼

sh_2 sh2 sh2€

If XðtÞ is a broadband process, a ¼ 0 and the Rice distribution is reduced to the Gaussian probability density function given by " # 1 1 a2 fA ðaÞ ¼ pffiffiffiffi for 1 , a , 1 exp 2 s2h 2psh If XðtÞ is a narrowband process, it is guaranteed that it will have a peak whenever hðtÞ crosses its mean. In this case, the irregularity factor is close to unity, and the Rice distribution is reduced to the Rayleigh probability density function given by " # 0.7 a 1 a2 fA ðaÞ ¼ 2 exp for 0 , a , 1 a = 1 (Rayleigh) 2 2 sh sh 0.6 0.5 fA (a)

In other words, the amplitudes of a narrowband stationary Gaussian process are distributed according to the Rayleigh distribution. Figure 36.8 shows the Rice distribution for various values of a. Note that the Rice distribution includes both positive and negative maxima except when a ¼ 1; in which case all the maxima are positive. The positive maxima are the local maxima that occur above the mean of XðtÞ; and the negative maxima are the local maxima that occur below the mean, as shown in Figure 36.7. In some cases, the negative maxima may not mean

0.4

a = 0 (Gaussian)

0.3 0.2

a = 0.8 a = 0.6 a = 0.4 a = 0.2

0.1 0 -4

-3

-2

FIGURE 36.8

-1 0 1 Maxima, A

2

3

Rice distribution for maxima.

4

36 12


much physically. In those cases, we can use the truncated Rice distribution, where only the positive portion of fA ðaÞ is used. fA ðaÞ is normal ized by the area under the probability density for positive maxima (Longuet Higgins, 1952; Ochi, 1973):

0.8 0.7

a = 1 (Rayleigh)

fAtrunc (a)

0.6

f ðaÞ fAtrunc ðaÞ ¼ ð1 A ; a$0 fA ðaÞda

0.5

a = 0.8 a = 0.6 a = 0.4 a = 0.2

0.4 0.3 0.2

a=0

0.1

0

The truncated Rice distribution is shown in Figure 36.9. If XðtÞ is the wave elevation, its maxima, A, are the amplitudes of the wave elevation. The wave height, H ¼ 2A; is then distributed according to

0

0.5

1

FIGURE 36.9

" dA h exp ¼ fH ðhÞ ¼ fA ðH=2Þ dH 4s2h

1 h2 2 4s2h

1.5 2 2.5 Maxima, A

3

3.5

4

Truncated Rice distribution.

# for 0 , h , 1

For any given wave, the probability that the height is less than h (the cumulative distribution) is " # 1 h2 FH ðhÞ ¼ 1 exp for 0 , h , 1 2 4s2h If hðtÞ is a stationary narrowband process so that the peaks pare distributed according to the Rayleigh ffiffiffiffiffiffiffiffi distribution, we find that the root mean square wave height, E{H 2 }; is given by qffiffiffiffiffiffiffiffi ð1 pffiffi E{H 2 } ¼ h2 fH ðhÞdh ¼ 2 2sh 0

In addition, it can be shown that the average and the significant wave heights are given by pffiffiffiffi H0 ; E{H} ¼ 2psh ; Hs ; E{H1=3 } ¼ 4sh

ð36:15Þ

where E{H1=3 } means that it is the expectation of the highest one third of the waves.

36.1.6

Long-Term Statistics

Because offshore structures are designed for long life spans, we must also consider long term wave statistics. Previously, when we considered the short term statistics, the significant wave height and spectrum were assumed to be invariant with time. This assumption is valid only over time periods of days at most. For longer time periods, the significant wave height has its own statistics and is a random variable. When one uses short term statistics to describe long term events, improbable events seem unjustifiably probable. For example, let us consider the probability that the wave height, distributed according to the Rayleigh distribution, exceeds a certain extreme value. Let us assume that the mean period of this wave is 10 sec and the probability that the height of any given wave is greater than 300 ft is 10 10. The value is small and the occurrence of a 300 ft wave seems improbable. However, the probability that the height will exceed 300 ft at least once in 10 years (3 £ 108 sec) is given by 1

ð1

10

10 3£108 =10

Þ

¼ 0:997


36 13

Thus, the statistical description states that it is almost certain that the wave height will exceed 300 ft at least once in 10 years. This prediction is a shortcoming of the short term statistics since waves of this magnitude do not arrive at this probability. In order to compute the probability that a wave height will exceed a certain extreme value, we require statistics for these extreme events. The actual maximum amplitude in a sequence of random amplitudes is a random variable itself. It has a probability distribution with mean value, standard deviation and other statistical properties. In fact, the distributions of these maximum values are called the extreme value distributions (EVDs). Gumbel (1958) obtained three methods of extrapolation known as three asymptotes. They are the Gumbel, Fretchet, and Weibull distributions. We will discuss the Gumbel and Weibull distributions in the next section. For the moment, we will discuss the concept of the N year storm. In long term statistics, we often speak of an N year storm. It means that, for any given year, the probability that we will have an N year storm is p¼

1 N

It follows that the probability that we will have m storms in n years is given by 1 N

Pr{mN year storms in n years} ¼n Pm

m

n m

1 N

1

where n Pm is the permutation given by n

Pm ¼

ðn

n! mÞ!

The probability that we will have at least one N year storm in n years is Pr{at least one N year storms in n years} ¼ 1

1 N

1

n

For a large N, the probability can be approximated as 1 expðn=NÞ: It should be noted that the probability that we will have exactly one N year storm in N years is not one, but Pr{one N year storm in N years} ¼ 1

M 1

1 N

As N ! 1; we find that P ¼ 1=e < 0:3679 The probability that we will have at least one N year storm in N years is Pr{at least one N year storms in n years} ¼ 1

1 N

1

N

As N ! 1; we find that P¼1 36.1.6.1

1=e < 0:6321

ð36:16Þ

Weibull Distribution

The Weibull distribution fits probabilities of extremes quite satisfactorily. In long term statistics, the significant wave height follows the Weibull distribution closely. The probability density and the cumulative distribution are given by f ðhÞ ¼

m h g b b

m 1

exp

g

h

b

m

; FðhÞ ¼ 1

exp

g

h

b

m

for g , h

ð36:17Þ

36 14


where m is called the shape parameter. Manipulating the cumulative distribution, we can write lnð ln{1

FðhÞ}Þ ¼ m{lnðh

gÞ

lnðbÞ}

where the left hand side is known from data. If we let y ¼ lnð ln{1 straight line with slope of m and a y intercept of m ln b: y ¼ mx

FðhÞ}Þ and x ¼ lnðh

gÞ; y is a

m ln b

Suppose we have significant wave height data over a long period of time, and our goal is to find the Weibull parameters, g, b, and m that best fit the distribution of the significant wave heights. These parameters can be determined by the least squares method or using the Weibull paper. Using the latter method, we first guess g so that the discrete points ðx; yÞ or ðlnðh gÞ; lnð ln{1 FðhÞ}ÞÞ form a straight line. The slope of this line is m, and the value of y when the line intersects the y axis is m ln b. This method will be illustrated in Section 36.3.3. The Gumbel distribution is given by f ðhÞ ¼ a expð aðh FðhÞ ¼ exp{

bÞÞexp{exp½ aðh bÞ } 1,h,1 exp½ aðh bÞ } for

ð36:18Þ

When ln½ ln FðhÞ is plotted against h, the result is a line with a slope of a and y intercept of ab. Another distribution that may be used is the lognormal distribution given by (Jasper, 1954) " # 1 1 ðln h mÞ2 ln h m f ðhÞ ¼ pffiffiffiffi ; FðhÞ ¼ F for 0 # h ð36:19Þ exp 2 s s 2 2psh 36.1.6.2

Wave Velocities via Linear Wave Theory

The wave velocities that correspond to the wave 2p elevation given in Equation 36.12 can be obtained A,H/2 k by linear wave theory. Linear wave theory, also h called airy wave theory, sinusoidal wave theory, t1 and small amplitude theory, is the simplest wave t2 theory. It is also the most important wave theory d because it forms the basis for the probabilistic x z spectral description of waves. y Linear wave theory assumes that the wave height is small compared with the wavelength and wave depth. In addition, fluid particles are assumed to follow a circular orbit. The readers should refer to FIGURE 36.10 A schematic of a simple sinusoidal Kinsman (1965) and LeMehaute (1976) for wave shown at two different times. detailed descriptions. In linear wave theory, the surface elevation is given by

hðy; tÞ ¼ A cosðvt

kyÞ

ð36:20Þ

which is a plane wave traveling to the right in Figure 36.10. Linear wave theory relates this sinusoidal surface elevation to the wave velocities given by wy ðx; y; tÞ ¼ Av

cosh kx cosðvt sinh kd

kyÞ;

wy ðx; y; tÞ ¼ Av

sinh kx sinðvt sinh kd

kyÞ

ð36:21Þ

where k, v, and A are wave number, angular frequency, and amplitude of a surface wave, respectively. The velocities vary with time, horizontal coordinate y, and depth x measured from the ocean floor. The wave velocities are sinusoidal in y and t, but exponentially decrease with the distance from the surface.


36 15

The frequency v is related to the wave number k by the dispersion relation given by

v2 ¼ gk tanh kd where d is the water depth. For deep water, tanh kd approaches unity and the frequency is given by lim v2 ¼ gk

d!1

For the surface elevation given in Equation 36.12, the surface elevation and the wave velocities are given by

hðy; tÞ ¼

N X i¼1

wy ðx; y; tÞ ¼ wy ðx; y; tÞ ¼

N X i¼1 N X i¼1

cosðvi t

ki y

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wi Þ 2Shh ðvi ÞDvi

vi

cosh ki x cosðvi t sinh ki d

ki y


vi

sinh ki x sinðvi t sinh ki d

ki y


ð36:22Þ

The wave accelerations can be obtained by differentiating the wave velocities with respect to time. Sample time histories of the wave velocity and acceleration can be obtained using either Borgman’s or Shinozuka’s method.

36.1.7

Summary

In this section, the concept of spectral density is introduced. It is then shown how the concept is used to describe the ocean wave heights. The spectral density or spectrum is related to the autocorrelation function by the Wiener Khinchine relations: 1 ð1 Shh ðvÞ ¼ Rhh ðtÞexpð ivtÞdt 2p 1 ð1 Rhh ðtÞ ¼ Shh ðvÞexpð ivtÞdv 1

In addition, the spectral density function of a real valued stationary process is also real and symmetric, or Shh ðvÞ ¼ Shh ð vÞ and the area under the spectral density is given by ð1 Shh ðvÞdv ¼ Rhh ð0Þ ¼ s2h 1

ð36:23Þ

m2h

ð36:24Þ

If the spectral density is given only for v $ 0; then this one sided spectrum is related to the two sided spectrum by Sohh ðvÞ ¼ 2Shh ðvÞ;

v$0

When the frequency is given in Hertz instead of in rad/sec, the spectra are related by Shh ðf Þ ¼ 2pShh ðvÞ;

v$0

The spectra that are often used to describe wave heights are the P M, Bretschneider, ITTC, ISSC, O H, and JONSWAP spectra. The most widely used spectrum is the P M spectrum, which is a single parameter spectrum. The P M spectrum is applicable for deep water, unidirectional seas, fully developed and local wind generated sea with unlimited fetch, and was originally developed for the North Atlantic. The single parameter for this spectrum can be expressed as the wind velocity at 19.5 m above sea level or the significant wave height that specifies the sea severity. When it is written

36 16


in terms of the significant wave height, it is given by Sohh ðf Þ

8:1 £ 10 3 g 2 exp ¼ v5

0:0324g 2 v Hs2

! 4

Bretschneider, ITTC, and ISSC spectra are two parameter spectra in which the state of development as well as the sea severity can be specified. The O H spectrum is a six parameter spectrum that allows us to represent local and distant storm effects and to specify sharpness of the peaks as well as to specify the sea severity and the state of development. The JONSWAP spectrum allows us to account for growth over a limited fetch. For a given ocean spectrum, a sample time history can be obtained by rffiffiffiffi N X 2 hðtÞ ¼ sh cosðvi t wi Þ N i¼1 In Borgman’s method, the sampling frequencies vi ¼ ðvi þ vi 1 Þ=2 are chosen so that the area between vi 1 and vi are equal. In Shinozuka’s method, the sampling frequencies are chosen randomly. The traditional method of choosing the sampling frequency at even intervals is not recommended. When a relatively short interval of time is considered, for example, about two hours or the duration of a storm, it can be assumed that the spectrum and its statistics are invariant with time. In this case, the distribution of local maxima or peaks of a stationary Gaussian process is described by the Rice distribution. The Rice distribution can be reduced to the Rayleigh distribution when the process is narrowband and to the Gaussian distribution when the process is broadband. In the long term statistics, the spectrum and its statistics may vary with time. In this case, the term “N year storm” is often used to indicate the sea severity, and the significant wave heights closely follow the Weibull distribution. Finally, the wave velocities and accelerations are related to the wave velocities using linear wave theory.

36.2

Fluid Forces

The following is a list of several types of forces that the fluid can exert on a body: 1. Drag force. This is due to the pressure difference between the downstream and upstream flow region. It can be thought of as the force required to hold a body stationary in a fluid of constant velocity. The drag force is proportional to the square of the velocity of the fluid relative to the structure. 2. Inertia force. This is the force exerted by the fluid while it accelerates and decelerates as it passes the structure. It is also the force required to hold a rigid structure in a uniformly accelerating flow, and it is proportional to the fluid acceleration. The concept of the inertia force in an inviscid flow was first formulated by Lamb (1945). 3. Added mass. As the body accelerates or decelerates in a stationary fluid, the body carries a certain amount of the surrounding fluid along with it. This entrained fluid is called the added, apparent, or virtual mass. In order to accelerate the body, additional force is required to accelerate or decelerate the added mass. 4. Diffraction force. This is due to the scattering of an incident wave on the surface of the structure. It is important when the body is large compared with the wavelength of the incident wave. 5. Froude Kryloff force. This is the pressure force on the structure due to the incident wave, assuming that the structure does not exist and does not interfere with the incident wave. 6. Lift force. This is due to nonsymmetrical separation of the fluid or due to vortices that are shed in a nonsymmetrical way. The component of the force perpendicular to the flow direction is the lift force.


36 17

7. Wave slamming force. This is due to a single occasional wave with a particularly high amplitude and energy, and it may be important at the free surface. Sarpkaya and Isaacson (1981) reviewed the research on slamming of water against circular cylinders. Miller (1977, 1980) found that the peak wave slamming force on a rigidly held horizontal circular cylinder is proportional to the square of the horizontal water particle velocity.

36.2.1

Wave Force Regime

Previously, we discussed various types of forces caused by waves and currents. In some cases, one type of force may be dominant. Hogben (1976) gave a literature review of the fluid force in various regimes. The load regime of importance can be demonstrated for the case of a vertical cylinder in Figure 36.11 in terms of H=D and pD=l; where H is the wave height, D is the cylinder diameter, and l is the wavelength. When linear wave theory is used, H=D is related to the Keulegan Carpenter number by K ¼ pH=D The Keulegan Carpenter number gives a measure of the importance of drag force relative to the inertia force. The term pD=l is called the diffraction parameter, and it determines the importance of the diffraction effect. As H=D increases, the drag force becomes more important and the inertia force becomes less important. As pD=l increases, the diffraction force becomes important. Using linear wave theory, the maximum drag force to the maximum inertia force can be written as fdrag 1 H K ¼ ¼ 2p D 2p2 finertia From the last relation, we find that the drag force is 5% of the inertial force when H=D ¼ 0:314: The Morison equation may be used for D=l , 0:2 and fdrag =finertia . 0:1 or thereabouts. It should be noted that Figure 36.11 is valid only near the surface. The drag force is predominant for a cylinder that extends from the bottom to the near surface, so that the Morison equation may be used. For example, consider a fixed jacket platform with legs with a diameter of 10 m and bracings with a diameter of 0.8 m. For a 10 year storm with l ¼ 100 m and H ¼ 8 m, the ratios H=D and D=l for the leg are 0.8 and 0.1, respectively. Similarly, the ratios H=D and D=l for the bracings are 10 and 0.08, respectively. Figure 36.11 shows that the inertia force is dominant for the legs, and both inertia and the drag forces are important for the bracings. 100

10

large drag

no

wa ves

inertia and drag

l H D

25% drag

1

large inertia

5% drag

0.1

0.01 0.01

D/l = 0.2

H/D

inertia CD = 0 0.1

diffraction 1.0

pD/l FIGURE 36.11

Load regimes near surface.

10

36 18

36.2.2


Wave Forces on Small Structures — Morison Equation

The added mass, MA ; can be written as MA ¼ CA Mdisp where CA is called the added mass coefficient and Mdisp is the mass of the fluid displaced by the structure. For a cylinder with a diameter, D, and height, h, the displaced fluid mass is pD2 h=4: It should be noted that the added mass is a tensor quantity. That is, we can speak of the added mass force in the xi direction due to the acceleration of the body in the xj direction, denoted as MijA : MijA is symmetric so that the added mass force in the xi direction due to the acceleration in the xj direction is equal to the added mass force in the xj direction due to the acceleration in the xi direction. The off diagonal terms are not zero if the cross section is not symmetric. Similarly, the inertia force can be written as FM ¼ CM Mdisp w_

ð36:25Þ

where the proportionality constant, CM ; is called the inertia coefficient. It should be noted that the added mass and the inertia effects are often neglected for a body vibrating in air since the displaced air mass is negligible. The drag force is proportional to the square of the fluid velocity, w, the density of the fluid, r, and the area of the body projected onto the plane perpendicular to the flow direction, Af ; FD ¼

1 C rA wlwl 2 D f

where CD is the drag coefficient. The absolute value sign is used to ensure that the drag force always acts in the direction of the flow. For a cylinder with a diameter D and height h, the projected area Af is Dh. For a body with nonzero velocity, the drag force is given by FD ¼

1 C rA ðw 2 D f

vÞlw

vl

ð36:26Þ

where w v is the velocity of the fluid relative to the body. Morison et al. (1950) combined the inertia and drag terms (Equation 36.25 and Equation 36.26) so that the fluid force on a body is given by f ¼

1 C rA wlwl þ CM Mdisp w_ 2 D f

For a cylinder, the fluid force per unit length can be written as f ¼

1 D2 CD rDwlwl þ CM rp w_ 2 4

For a moving cylinder with velocity v, the Morison force is given by f ¼ 36.2.2.1

1 C rDðw 2 D

vÞlw

vl þ CM rp

D2 w_ 4

Inclined Cylinder

Let us now consider the inclined cylinder shown in Figure 36.13. The direction of the flow makes an angle of u with the cylinder. Often, only the fluid force in the normal direction is considered. The normal component is given by fn ¼

1 C rDðwn 2 D

vn Þlwn

vn l þ CM rp

D2 n w_ 4

ð36:27Þ

where the superscript is used for the normal component. The term, wn vn ; is the normal component of the relative velocity of the fluid with respect to the structure. Suppose that fluid is flowing to the right,


36 19

and the cylinder is also moving to the right, as shown in Figure 36.12. The normal components of the fluid and cylinder velocities are wn ¼ lwl cos u; vn ¼ lvl cos u

Direction of flow

In three dimensions, it may be difficult to picture what the normal component should be. Here, we can find the normal component using the formula ðwn

~ vn Þ~n ¼ ~t £ ðw

~vÞ £ ~t

Cylinder velocity, v q t Fluid velocity, w

ð36:28Þ

n

where ~t is the unit vector tangent to the cylinder FIGURE 36.12 Inclined cylinder. and n~ is the unit vector normal to the cylinder. Note that the normal direction depends on the direction of the flow as well as the inclination of the cylinder. In some cases, the tangential drag force may be included, and it can be written as 1 f t ¼ CT rDðwt vt Þlwt vt l ð36:29Þ 2 where CT is the tangential drag coefficient. Note that CT is usually a very small number. The normal component of the fluid force is more dominant than the tangential component. It may seem strange that the fluid force does not act in the direction of the fluid motion. Instead, the force is predominantly in the normal direction defined by Equation 36.28. In Section 36.3.1, we will demonstrate what this means by considering a towing cable. 36.2.2.1.1 Determination of Fluid Coefficients The drag, inertia, and added mass coefficients must be obtained by experiment. However, for a long cylinder, CM approaches its theoretical limiting value (uniformly accelerated inviscid flow) of 2, and CA approaches unity (Lamb, 1945; Wilson, 1984). In reality, the inertia and drag coefficients are functions of at least three parameters (Wilson, 1984): CM ¼ CM ðRe; K; cylinder roughnessÞ CD ¼ CD ðRe; K; cylinder roughnessÞ where Re is the Reynolds number and K is the Keulegan Carpenter number given by UT r UD Re ; f ; K; D m

ð36:30Þ

where rf is the density of the fluid, U is the free stream velocity, D is the diameter of the structure, m is the dynamic or absolute viscosity, and T is the wave period. Sarpkaya looked at the variation of these hydrodynamic coefficients extensively and obtained the plots shown in Figure 36.13 to Figure 36.15 (Sarpkaya, 1976; Sarpkaya et al., 1977). Figure 36.13 shows the inertia and drag coefficients for a smooth cylinder as a function of K for various values of Re and the reduced frequency b, defined by b ¼ Re=K: From this figure, we find that for low Re and b, the inertial coefficient decreases and the drag coefficient increases at about 10 , K , 15: It is found that the drop and the increase in these coefficients are due to shedding vortices, which also exert forces perpendicular to the structure and the flow. Figure 36.14 and Figure 36.15 show the inertia and drag coefficients for a rough cylinder, whose roughness is measured by k=D: Figure 36.14a shows a drop in the drag coefficient for Re between 104 and 105, and this is called the “drag crisis.” For a larger Re, the drag coefficient stays constant. As the surface becomes rougher, the drop occurs at lower Re and the drag coefficients for the larger Re increases. Figure 36.14 to Figure 36.16 can be used to obtain proper values of the drag and inertia coefficients for fluid with known Re, Keulegan Carpenter number, and cylinder roughness.

36 20

Vibration and Shock Handbook 3.0 2.0

1985

1.5

Re × 10−3 = 10 15 20

b = 497 1107 784

30 40 50

CD 1.0

80 5260

8370 150

0.5 0.4

(a)

60

3123 4480

0.3 2.5 3

5

10

50

100

150 200

3.0 Re × 10−3 = 60

2.0

150 200

5260 4480

1.5

CM

1.0 b = 497

0.5 0.4

(b)

3

5

10

20

50

30

40

31231985 1107 784

60

100

80 100

150 200

K

FIGURE 36.13 Drag and inertia coefficients as functions of K for various values of Re and b. (Source: Sarpkaya, 1976, Proceedings of the Eighth Offshore Technology Conference. With permission.)

36.2.3

Vortex-Induced Vibration

When the flow passes around a fixed cylinder, for a very low Re ð0 , Re , 4Þ; the flow separates and reunites smoothly. When the Re is between 4 and 40, eddies are formed and are attached to the downstream side of cylinder. They are stable and there is no oscillation in the flow. For a flow with a Reynolds number greater than about 40, the fluid near the cylinder starts to oscillate due to shedding vortices. These shedding vortices exert an oscillatory force on the cylinder in the direction perpendicular to both the flow and the structure. The frequency of oscillation is related to the nondimensionalized parameter, the Strouhal number, defined by St ¼

fv D U

ð36:31Þ

where fv is the frequency of oscillation, U is the steady velocity of the flow, and D is the diameter of the cylinder. For circular cylinders, the Strouhal number stays roughly at 0.22 for laminar flow ð103 , Re , 2 £ 105 Þ and 0.3 for turbulent flow (Patel, 1989). The lift force due to these shedding vortices can be written as fL ¼

1 C rA U 2 cos 2pfv t 2 L f

ð36:32Þ

where CL is the lift coefficient, which is also a function of Re, K, and the surface roughness. The experimental data of the lift coefficients show considerable scatter with typical values ranging from 0.25 to 1. For smooth cylinders, the lift coefficient approaches about 0.25 as Re and K increase.


36 21

1.9 1.8

CD

k /D = 1/50

1/100

1.6

1/200 1/400

1.4

1/800

1.2 1.0 0.8 0.6

smooth 0.5

0.1

1 Re ×

(a)

5

7

5

7

10 −5

2.0

CM

1.8 1.6 1.4

1/100 k /D = 1/50

1/200

1/400

1/ 800

smooth

1.2 1.0 0.1 (b)

0.5

1 Re × 10 −5

FIGURE 36.14 Drag and inertia coefficients for a rough cylinder as functions of Re for various values of cylinder roughness (as measured by k=D) for K ¼ 20: (Source: Sarpkaya et al., 1977, Proceedings of the Ninth Offshore Technology Conference. With permission.)

It should be noted that the vortex forces are not generally correlated on the entire cylinder length. That is, the phase of the vortex shedding forces varies over the length. The correlation length the length over which vortex shedding is synchronized for a stationary cylinder is about three to seven diameters for laminar flow. If sectional forces are randomly phased, the net effect will be small. The total force on a cylinder of length L will be only a fraction of LfL : This fraction is called the joint acceptance and depends on the ratio of the correlation length to the total length. When the flow passes by a cylinder that is free to vibrate, the shedding frequency is also controlled by the movement of the cylinder. When the shedding frequency is close to the first natural frequency of the cylinder (^ 25 to 30% of the natural frequency [Sarpkaya and Isaacson, 1981]), the cylinder takes control of the vortex shedding. The vortices will shed at the natural frequency instead of at the frequency determined by the Strouhal number. This is called lock in or synchronization, which is a result of nonlinear interaction between the oscillation of the body and the action of the fluid. Figure 36.16 shows the shedding frequency, as a function of flow velocity in the presence of a structure. f1 and f2 are the natural frequencies of the structure. The amplitude of the structural response and the range of the fluid velocity over which the lock in phenomenon persists are functions of a reduced damping parameter the ratio of the damping force to the exciting force (Vandiver, 1985, 1993). If the reduced damping parameter is small, the lock in can persist over a greater range of flow velocity.

36 22


1.8

k/D = 1/50

1/100

1.6 1.4

1/400

1.2 CD

1/200

1/800

1.0 0.8

smooth

0.6 0.4 (a)

0.2

0.5

1

2.0 1/100

5

10

15

10

15

1/800

1/400

1.8

Re × 10−5

smooth

1/ 200

1.6 CM

1.4

k /D = 1/50

1.2 1.0

0.2

0.5

1

(b)

Re × 10−5

5

FIGURE 36.15 Drag and inertia coefficients for a rough cylinder as functions of Re for various values of cylinder roughness (as measured by k=D) for K ¼ 60: (Source: Sarpkaya et al., 1977, Proceedings of the Ninth Offshore Technology Conference. With permission.)

The existing models for vortex induced oscillation for a rigid cylinder include single degree of freedom models and coupled models. The single DoF models assume that the effect of vortex shedding is an external forcing function, which is not affected by the motion of the body. The coupled models assume that the equations that govern the motion of the structure and the lift coefficients are coupled so that the fluid and the structure affect each other (Billah, 1989).

36.2.4

Summary

Some of the fluid forces are discussed briefly, and the regimes where inertia, drag, and diffraction forces are important are shown as functions of the ratio of the structural diameter to the wave

fv

fv = St U D f2

f1

U FIGURE 36.16

An example of fluid elastic resonance.


36 23

length, D=l; and the ratio of the wave height to the structural diameter, H=D: The wave forces on small structures are modeled by the Morison equation, and it is valid for D=l , 0:2 and H=D . 0:63 or thereabouts. The Morison equation includes the effects of added mass, inertia, and drag. The added mass is simply MA ¼ CA Mdisp For a cylinder with transverse velocity, v, the normal and the tangential components of the drag and the inertia forces are given by 1 C rDðwn 2 D 1 f t ¼ CT rDðwt 2

fn ¼

vn Þlwn vt Þlwt

vn l þ CM rp

D2 n w_ 4

vt l

The fluid coefficients are at least functions of three parameters: the Reynolds number, the Keulegan Carpenter number, and the cylinder roughness. The plots of these coefficients are reproduced in Figure 36.13 to Figure 36.15. The frequency of the lift force that is exerted by shedding vortices is closely related to the Strouhal number given by St ¼

fv D U

The lift force due to these shedding vortices can be written as fL ¼

1 C rA U 2 cos 2pfv t 2 L f

If the structure is free to vibrate, then lock in or synchronization may occur when the shedding frequency is close to the structure’s natural frequency. The structure takes control of the vortex shedding. Many nonlinear models are available to capture this phenomenon.

36.3

Examples

Four examples are given in this section. The first example illustrates the roles of the normal and the tangential components of the drag force in the static configuration of a towing cable. The second example shows how the equation of motion of an articulated tower can be formulated in the presence of surrounding fluid. The third example shows how to choose a single significant wave height to represent a certain condition from significant wave height data over a long period of time. The final example shows how to reconstruct time series data from a given spectrum.

36.3.1

Static Configuration of a Towing Cable

For the purpose of ocean surveillance, oceanographic or geographic measurements, or ocean exploration, marine cables with instrument packages or Remotely Operated Vehicles are often towed behind ships or submarines. For example, the goal of the VENTS program by the National Oceanic and Atmospheric Administration (NOAA) is to conduct research on the impacts and consequences of submarine volcanoes and hydrothermal venting on the global ocean. In attempts to locate and map the distributions of hydrothermal plumes in the Mid Ocean Ridge system, an instrument package called a CTD (Conductivity, Temperature and Depth Sensors) is towed behind a ship.

36 24


Let us consider a cable and a body towed behind a ship at a constant velocity with no current as shown in Figure 36.17. What kind of shape will the cable take? What will be the distance between the ship and the towed body? We immediately recognize that this is equivalent to having a stationary ship with a steady current in the opposite direction. The equation of motion is given by

Ship

q t

n s

Cable Tension Drag

y x

X › F~ ¼ m~aðs; tÞ ¼ 0~ ¼ ðT~t Þ þ f n n~ þ f t~t þ mg ~k ›s

Weight

Instrument package where m is the mass of the cable per unit length, a~ðs; tÞ is the acceleration of the cable, s is the FIGURE 36.17 Towed system in equilibrium and the coordinate along the cable, T is the tension which forces acting on the towed body. ~ is the set of unit vectors of is a function of s, ð~t; n~ ; bÞ the curvilinear coordinate system, ~k is the unit vector downward in the direction of gravity, g is the gravitational acceleration, f n is the normal drag force, and f t is the tangential drag force. The added mass and the inertial terms are zero because the fluid acceleration and the cable acceleration are zero. The normal and tangential drag forces are given in Equation 36.27 and Equation 36.29. In our case, they are given by

f n ¼ CD r

D 2 U cos2 u; f n ¼ 2

CT r

D 2 2 U sin u 2

The corresponding scalar equations are given by dT ds

D 2 2 U sin u mg cos u ¼ 0 2 du D þ CD r U 2 cos2 u mg sin u ¼ 0 T ds 2 CT r

ð36:33Þ

where u is the angle that the tangential vector makes with the vertical and measured positive clockwise. Note that we have used ›~t=›s ¼ ð ›u=›sÞ~n and ~k ¼ cos u~t sin un~ : Equation 36.33 shows that the tangential components of the external forces act to increase the tension, while the normal components cause the towline to bend. Because the normal component of the drag force is much larger than the tangential component, most of the fluid force is used to turn the cable. From the force diagram (in Figure 36.17), the angle that the cable makes with the vertical where it is connected to the towed body is given by Tð0Þcos uð0Þ ¼ W;

Tð0Þsin uð0Þ ¼ Drag

Once we know the weight and the drag force on the towed body, the tension and the angle at s ¼ 0 can be found. If the drag is negligible compared with the weight, then the cable must be near vertical and the tension must be equal to the weight of the towed body at s ¼ 0 : Tð0Þ < W and uð0Þ < 0 For now, let us assume that this is the case. Then, with these initial conditions, the system of ordinary differential equations (Equation 36.33) can be solved numerically for TðsÞ and uðsÞ: For example,


36 25

even very simple finite difference equations will work. A set of equations Tiþ1 ¼ Ti þ mg cos ui

uiþ1 ¼ ui

D 2 2 U sin ui Ds 2

ð36:34Þ

D 2 U cos2 ui Ds=Ti 2

ð36:35Þ

CT r

mg sin ui þ CD r

where Ti ¼ TðiDsÞ; are used here, and it works very well for Ds ¼ 0:05: The Cartesian coordinates, x and y, are related to u by dx dy ¼ sin u and ¼ cos u ds ds and can also be obtained by integrating them numerically. Figure 36.18 shows the results when mg ¼ 1:5 N/m, CD rDU 2 =2 ¼ 10 N/m, CT rDU 2 =2 ¼ 0:1 N/m, W ¼ 100 N, and the cable is 100 m long. Care is taken so that the ship is located at x ¼ 0 and y ¼ 0: It is interesting to note that u approaches a critical value, and the shape gradually becomes linear toward the ship. Mathematically, du=ds becomes zero. This is when the drag force is completely balanced by the normal component of the cable weight. The angle at which this occurs, ucr ; can be obtained from the second governing equation and sin ucr D 1 ¼ CD r U 2 cos ucr 2 mg

f n;

mg sin ucr ¼

Water depth, y (m)

In our case, ucr ¼ 1:184 rad, and this value agrees with Figure 36.18.

0 −10 −20 −30 −40 −50 −90

−80

−70

−60

−50

−40

−30

−20

−10

0

Horizontal coordinate, x (m)

q (rad)

1.5 1 0.5 0

0

10

20

30

40

50

60

70

80

90

100

Coordinate along the cable, s (m) FIGURE 36.18 The equilibrium configuration of a towed cable and the angle that the cable makes with the vertical when mg ¼ 1:5 N/m, CD rDU 2 =2 ¼ 10 N/m, CT rDU 2 =2 ¼ 0:1 N/m, and W ¼ 100 N.

36 26

36.3.2


Fluid Forces on an Articulated Tower

Offshore structures are used in the oil industry as exploratory, production, oil storage, and oil landing facilities. They are designed to be self supporting and sufficiently stable for offshore activities such as drilling and production of oil. An articulated tower as seen in Figure 36.19 is an example of an offshore platform that consists of a base, shaft, universal joint that connects the base and the shaft, ballast chamber, buoyancy chamber, and deck. The ballast chambers provide the extra weight so that the tower’s bottom stays on the ocean floor, and the buoyancy chamber adds the necessary buoyancy so that the tower does not fall. An articulated tower can be effectively modeled as a rigid inverted pendulum, where the deck is modeled as a point mass, the shaft as a uniform rigid bar, and the buoyancy chamber by a point buoyancy. In two dimensions, motion of the tower can be described with a single DoF (Chakrabarti and Cottor, 1979; Bar Avi, 1996). The equation of motion in terms of the tower’s deflection angle is obtained by summing the moment about the point O in Figure 36.20 and is given by I

X d2 u L MO ¼ mg sin u 2 ¼ 2 dt þ MgL sin u

Bl sin u þ

Deck

Buoyancy chamber Shaft Ballast chamber Universal joint Base

FIGURE 36.19

Schematic of an articulated tower.

q fn

B Mg

L l

ðL 0

f n x dx

L/2

Mg where I is the mass moment of inertia about the point O given by I ¼ mL2 =3 þ ML2 ; m is the mass of the shaft, g is the gravitational acceleration, L is O the length of the shaft, M is the point mass at the top, B is the buoyancy provided by the buoyancy FIGURE 36.20 Free body diagram. chamber, l is its moment arm, f n is the normal fluid force per unit length, and x is the coordinate along the shaft from O. The fluid force per unit length in the normal direction is given by

f n ¼ CD r

D n ðw 2

vn Þlwn

vn l þ CM rp

D2 n w_ 4

CA rp

D2 n a 4

where the last term is the force in the normal direction due to the added mass. vn and an are the velocity and the acceleration of the body in the normal direction and are given by

vn ¼ x

du d2 u and an ¼ x 2 dt dt


36 27

If we assume that the surrounding fluid is stationary, then the normal velocity and the acceleration of the fluid (w and w) _ are zeros. Thus, the moment due to the fluid force is given by ðL 0

n

f x dx ¼

¼

ðL

D du CD r x 2 dt 2

0

CD r

D L4 2 4

du dt

2

2

! du D2 d2 u sign þ CA rp x dx x dt 4 dt 2

sign

du D2 L3 d2 u þ CA rp dt 4 3 dt 2

and the equation of motion is given by L2 D2 L3 m þ ML2 þ CA rp 3 4 3

!

d2 u L ¼ mg þ MgL dt 2 2

Blb sin u

CD r

D L4 du sign dt 2 4

du dt

2

Note that the normal fluid drag force adds directly to the restoring moment in the case of a rigid bar. The equation of motion can be solved numerically once the initial conditions (u½0 and du=dt½0 ) are given. The equation of motion can be simplified if we assume that the angle of rotation u is small. More specifically, if we assume that u 2 is negligible when compared with 1, then we find that2 sin u < u The equation of motion can be simplified to ! L2 D2 L3 d2 u L 2 m þ ML þ CA rp mg þ MgL 2 3 4 3 dt 2

Blb u þ CD r

D L4 du sign 2 4 dt

du dt

2

¼0

which resembles the equation for a linear oscillator with a nonlinear damping term. Note that the system becomes unstable when the stiffness term (the coefficient of u) becomes negative. This occurs when the buoyancy is not sufficient or B,

1 L mg þ MgL lb 2

36.3.3 Distribution of Significant Wave Heights — Weibull and Gumbel Distributions The National Buoy Data Center (NBDC) run by NOAA collects ocean data such as wind, current, wave, pressure, and temperature data in various locations and the records are made public. Let us say that we are to design an articulated tower (in Section 36.3.2) in one of these locations where the data are available. The first task is to characterize the environment. Using all of the information that is collected is inefficient and impractical. Instead, we are interested in choosing a single number that can represent typical and extreme situations such as 10 and 50 year storms. For now, let us only consider random waves. We are then interested in finding the significant wave heights representing 10 and 50 year storms. From NBDC data for a buoy outside Monterey Bay, the number of occurrences for ranges of significant wave heights is constructed in Table 36.2. The measurements were taken every hour for about 12 years. We first construct the corresponding Weibull distribution using the method described in Section 36.1.6. We first guess g so that a pair of lnð ln{1 FðhÞ}Þ and lnðh gÞ form a 2

This is called the small angle assumption.

36 28


Number of Occurrences of Various Sea States

Significant Wave Height, h (m)

Number of Occurrences

,1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 Total

2,367 46,353 3,4285 1,3181 3,813 716 145 32 8 2 100,902

2,367 48,720 83,005 96,186 99,999 100,715 100,860 100,892 100,900 100,902

3 2 ln(−ln{1−F(h)})

straight line. Figure 36.21 shows that the pair yields nearly a straight line when g < 0:84: The slope and the y intercept of this line are 1.6 and 0.78, respectively. The Weibull parameters are then m ¼ 1:6 and b ¼ 1:6: Similarly, we can find the corresponding Gumbel probability density function by plotting pairs of ðh; lnð ln{FðhÞ}ÞÞ to form a line. For the data shown in Table 36.2, the line has a slope of 1.52 and y intercept of 2.84 so that a ¼ 1:52 and b ¼ 1:87: Figure 36.22 shows the Weibull probability density and the cumulative distribution (Equation 36.17) in solid lines, the Gumbel probability density and the cumulative distribution in dotted lines (Equation 36.18), and the discrete probability density and the cumulative distribution derived from Table 36.2 in symbols.

Sum

0

−2 −4 −5

0.99 γ = 0.2 0.5 0.84 0.95 −3

−1 ln(h−γ)

FIGURE 36.21 Plots of ðlnðh for various values of g.

1

gÞ; ln½ ln{1

3

FðhÞ} Þ

1 FGumbel (h)

Probability density and cumulative distribution

0.8

FWeilbull (h)

0.6 FGumbel (h) 0.4 0.2 0

fWeilbull (h) 0

2

4 6 8 Significant wave height, h (m)

10

FIGURE 36.22 Weibull approximations of the probability density and cumulative distribution of significant wave heights measured in the outer Monterey Bay area. The symbols are the values given in Table 36.2.


36 29

TABLE 36.3 Comparison of Representative Significant Wave Heights for Long Term Predictions from Gumbel and Weibull Distributions 5 Year (m) Weibull Gumbel

10 Year (m)

7.84 8.83

50 Year (m)

8.15 9.33

8.79 10.4

The next step is to find a significant wave height that can represent an N year storm, hN : The probability that we will not have an N year storm in any given year is 1 1=N and is equivalent to the probability that the significant wave height will not exceed hN in the same year. The probability that h , hN in a single measurement is FðhN Þ; and the probability that h , hN in every measurement taken in a year is FðhN Þ24£365 : Then, we have 1 ¼ FðhN Þ24£365 N

1

Table 36.3 shows significant wave heights that represent 5 , 10 , and 50 year storms obtained using the Weibull and Gumbel distributions. The Gumbel probability distribution gives higher significant wave heights. For this particular set of data, the Weibull distribution seems to fit the data better (Figure 36.22), and the Weibull distribution is the most often used distribution in the offshore industry.

36.3.4

Reconstructing Time Series for a Given Significant Wave Height

Previously, we found significant wave heights that could represent 5 , 10 , and 25 year storms for a given site. Recall that the significant wave height can entirely characterize the Pierson Moskowitz spectra. Once the spectral density is determined, a sample time history of the wave profile, hðtÞ; can be determined using either Borgman’s or Shinozuka’s method (Section 36.1.4). Here, Shinozuka’s method is used to generate the random wave elevations. o Let us first find the random frequencies distributed according to Shh ðvÞ=s2h : The P M spectrum in terms of the significant wave height is given by Equation 36.8. o Shh ðvÞ ¼ 0:7795v

5

3:118 v Hs2

exp

4

The variance is given by

s2h ¼

ð1 0

Hs2 16

Sohh ðvÞdv ¼

The probability density and the cumulative distribution functions are given by f ðvÞ ¼ Sohh ðvÞ=s2h ¼

12:472 v Hs2

5

exp

3:118 v Hs2

4

FðvÞ ¼ 1

;

The inverse of the cumulative distribution function is given by F

1

ðxÞ ¼

Hs2 lnð1 3:118

!

exp

3:118 v Hs2

4

1=4

xÞ

The random frequencies distributed according to f ðvÞ can be obtained from uniformly dis tributed random numbers x from 0 and 1. Table 36.4 shows uniform random numbers between

36 30

Vibration and Shock Handbook TABLE 36.4 Generation of Random Frequencies Distributed According to f ðvÞ from Uniform Random Numbers Uniform Random Numbers 0 , x , 1

Random Frequencies v Distributed According to f ðvÞ

Wave elevation (m)

0.950 0.231 0.606 .. .

Wave velocities (m/s)

1/4

¼ 0.360 ¼ 0.662 1/4 ¼ 0.483 1/4

20 10 0 −10 −20

Wave velocities (m/s)

(219.713 ln[1 2 0.950]) (219.713 ln[1 2 0.231]) (219.713 ln[1 2 0.606]) .. .

0

10

20

30

40

50 60 Time (s)

70

80

90

100

0

10

20

30

40

50 60 Time (s)

70

80

90

100

0

50

100

150

200

350

400

450

500

10 5 0 −5 −10

2 1 0 −1 −2 −3

250

300

Water Depth (m) FIGURE 36.23

Wave elevation and velocities.

0 and 1 and the random frequencies distributed according to f ðvÞ3. The significant wave height of 7.84 m is used. We can obtain 100 in this way, and the wave elevation is also obtained using Equation 36.13. The random phase wi is obtained by multiplying uniform random numbers (different from the ones used to generate the random frequencies) by 2p. Figure 36.23 shows the surface elevation as a function of time, the corresponding wave velocities at the water surface (Section 36.1.7) as functions of time, and the wave velocities at t ¼ 0 as functions of the water depth. Note that the wave velocities decay with depth. 3

The uniform random numbers can be generated by the MATLAB rand function.


36.3.5

36 31

Available Numerical Codes

Many numerical codes are available for modeling the dynamics of slender structures such as risers, tether, umbilicals, and mooring lines. The first example in this section was solved by a numerical code, WHOI Cable, developed at Woods Hole Oceanographic Institution. WHOI Cable is a time domain program that can be used for analyzing the dynamics of towed and moored cable systems in both two and three dimensions. It takes into account bending and torsion as well as extension. Comparative studies investigating flexible risers were carried out by ISSC Committee V7 from computer programs developed by 11 different institutions in the period between 1988 and 1991, and the results were reported by Larsen (1992). More recently, Brown and Mavrakos (1999) conducted a comparative study on the dynamic analysis of suspended wire and chain mooring lines and reported results from 15 different numerical codes. The participants included engineering consultancies, and academic and research institutions involved in marine technology. Some of the time domain programs that were included in the comparative study are MODEX by Chalmers University of Technology, FLEXAN C by Institute Francais du Petrole, DYWFLX95 by MARIN, R.FLEX by MARINTEK, CABLEDYN by National Technical University of Athens, DMOOR by Noble Denton Consultancy Services Ltd, V.ORCAFLEX by Orcina Ltd Consulting Engineers, ANFLEX by Petrobras SA, TDMOOR DYN by University College London, FLEXRISER by Zentech International. Some of these programs are available to academic institutions and government laboratories at no cost.

Acknowledgments The author wishes to express gratitude for the funding from the Woods Hole Oceanographic Institution and the Department of Mechanical Engineering at Texas Tech University.

References Bar Avi, P. 1996. Dynamic response of an offshore articulated tower, Ph.D. thesis, The State University of New Jersey, Rutgers, May 1996. Billah, K. 1989. A study of vortex induced vibration, Ph.D. thesis, Princeton University, May 1989. Borgman, L., Ocean wave simulation for engineering design, J. Waterway Harbors Div., 95, 557 583, 1969. Bretschneider, C. 1959. Wave variability and wave spectra for wind generated gravity waves. Technical Memorandum No.118, Beach Erosion Board, U.S. Army Corps of Engineers, Washington D.C. Bretschneider, C. 1969. Wave forecasting. In Handbook of Ocean and Underwater Engineering, Ed. J.J. Myers, McGraw Hill, New York. Brown, D. and Mavrakos, S. Comparative study on mooring line dynamic loading, Marine Structures, 12, 131 151, 1999. Chakrabarti, S.K. 1987. Hydrodynamics of Offshore Structures, Computational Mechanics Publications, Southampton, U.K. Chakrabarti, S.K. and Cottor, D., Motion analysis of articulated tower, J. Waterway Port Coast. Ocean Div., 105, 281 292, 1979. Faltinsen, O.M. 1993. Sea Loads on Ships and Offshore Structures, Cambridge University Press, Cambridge. Gumbel, E. 1958. Statistics of Extremes, Columbia University Press, New York. Hasselmann, K., Barnett, T., Bouws, E., Carlson, H., Cartwright, D., Enke, K., Ewing, J., Gienapp, H., Hasselmann, D., Kruseman, P., Meerburg, A., Muller, P., Olbers, D., Richter, K., Sell, W., and Walden, H. 1973. Measurement of wind wave growth and swell decay during the joint North Sea wave project (JONSWAP), Technical Report 13 A. Deutschen Hydrographischen Zeitschrift. Hogben, N. 1976. Wave loads on structures, Behavior of Offshore Structures (BOSS),Oslo, Norway.

36 32


Isaacson, M., Wave and current forces on fixed offshore structures, Can. J. Civil Eng., 15, 937 947, 1988. Jasper, N., Statistical distribution patterns of ocean waves and of wave induced ship stresses and motions with engineering applications, Trans. Soc. Nav. Arch. Mar. Engrs, 64, 375 432, 1954. Khinchine, A., Korrelations theorie der stationaren stochastischen prozesse, Math. Ann., 109, 604 615, 1934. Kinsman, B. 1965. Wind Waves, Prentice Hall, Englewood Cliffs, NJ. Lamb, H. 1945. Hydrodynamics, 6th Ed., Cambridge University Press, New York. Larsen, C. Flexible riser analysis comparison of results from computer programs, Marine Structure, 5, 103 119, 1992. LeMehaute, B. 1976. Introduction to Hydrodynamics and Water Waves, Springer Verlag, New York. Longuet Higgins, M., On the statistical distribution of the height of sea waves, J. Mar. Res., 11, 3, 245 266, 1952. Miller, B., Wave slamming loads on horizontal circular elements of offshore structures, Nav. Arch., 3, 81 98, 1977. Miller, B. 1980. Wave slamming on offshore structures, Technical Report No. NMI R81. National Maritime Institute. Morison, J., O’Brien, M., Johnson, J., and Schaaf, S., The force exerted by surface waves on piles, Pet. Trans., AIME, 189, 149 157, 1950. Ochi, M., On prediction of extreme values, J. Ship Res., 17, 29 37, 1973. Ochi, M. and Hubble, E. 1976. Six parameter wave spectra. ASCE pp. 301 328. Proceedings of the Fifteenth Coastal Engineering Conference, Honolulu, HI. Patel, M. 1989. Dynamics of Offshore Structures, Butterworths, London. Pierson, W. and Moskowitz, L., A proposed spectral form for fully developed wind seas based on the similarity theory of S.A. Kitaigorodskii, J. Geophys. Res., 69, 24, 5181 5203, 1964. Rice, S.O. 1954. Mathematical analysis of random noise. In Selected Papers on Noise and Stochastic Processes, N. Wax, Ed., Dover Publications, New York. Sarpkaya, T. 1976. In line and transverse forces on cylinders in oscillating flow at high Reynolds numbers, OTC 2533, pp. 95 108. Proceedings of the Eighth Offshore Technology Conference, Houston, TX. Sarpkaya, T., Collins, and N., Evans, S. 1977. Wave forces on rough walled cylinders at high Reynolds numbers, OTC 2901, pp. 175 184. In Proceedings of the Ninth Offshore Technology Conference, Houston, TX. Sarpkaya, T., and Isaacson, M. 1981. Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reihold, New York. Shinozuka, M., Monte Carlo Solution of structural dynamics, Comput. Struct., 2, 855 874, 1972. Sverdrup, H., and Munk, W. 1947. Wind, sea, and swell: theory of relations for forecasting, Technical Report 601. U.S. Navy Hydrographic Office. Tucker, M. 1991. Waves in Ocean Engineering: Measurements, Analysis, and Interpretation, Ellis Horwood, Chichester, U.K. Vandiver, J. 1985. Prediction of lockin vibration on flexible cylinders in sheared flow, May 1985. In Proceedings of the 1985 Offshore Technology Conference, Paper No. 5006, Houston, TX. Vandiver, J., Dimensionless parameters important to the prediction of vortex induced vibration of long, flexible cylinders in ocean currents, J. Fluids Struct., 7, 5, 423 455, 1993. Wiener, N., Generalized harmonic analysis, Acta Math., 55, 117 258, 1930. Wilson, J. 1984. Dynamics of Offshore Structures, Wiley, New York.

Acoustics

IX

IX 1

37

Sound Levels and Decibels 37.1 Introduction .......................................................................... 37 1 37.2 Sound Wave Characteristics ................................................. 37 1 Velocity of Sound

S. Akishita Ritsumeikan University

37.3 Levels and Decibels ............................................................... 37 3 Sound Power Level † Sound Pressure Level Sound Pressure Level

†

Overall

Summary In this chapter, the basic characteristics of sound and sound propagation are described. Levels and decibels, which represent the magnitude of sound waves, are defined and explained.

37.1

Introduction

Sound is related to vibration, and is described as a propagating perturbation through a fluid, which is air or water in most cases. A very wide variety of noise sources exists. Each source is peculiar to its generation mechanism, which may cover a wide range of phenomena including fluid mechanics and the vibration of structures. Sound is perceived by the ear of the listener as a pressure wave superimposed upon the ambient air pressure. The sound pressure is the incremental variation about the ambient atmospheric pressure. Generally, it is detected by a microphone and expressed as oscillatory electric signal output from an audio measurement instrument. We shall present a mathematical description of these pressure waves that are known as sound. The field of acoustics concerns sound and vibration, and is treated in Chapter 37 to Chapter 45 of this book.

37.2

Sound Wave Characteristics

The characteristics of a sound wave are described by a pressure oscillation of a pure tone. A “pure tone” is a sinusoidal pressure wave of a specific frequency and amplitude, propagating at a velocity determined by the temperature and pressure of the medium (air). Let us consider a hypothetical sound field in a duct with constant cross sectional area, as shown in Figure 37.1a. A reciprocating piston at the left end emits the sound wave and it propagates toward the right side end along the indicated axis. It is detected by a microphone at the right end. Figure 37.1b shows the instantaneous pressure distribution in a duct at time t ¼ t0 : Figure 37.1c shows the pressure variation of the time history detected by the microphone at x ¼ x0 : 37 1

37 2


The wavelength, l; is the distance between successive two peaks in the waveform in Figure 37.1b. Wavelength is related to the frequency, f ; and the velocity of wave propagation, c; by

l¼

c f

ðft or mÞ

ð37:1Þ

The period, T; of the sinusoidal wave is the time interval required for one complete cycle, as depicted in Figure 37.1b. The period, T; is related to the frequency, f ; by T¼

37.2.1

1 ðsecÞ f

ð37:2Þ

Velocity of Sound

The velocity of sound is identical to the velocity of wave propagation, c; and in air it is given by c¼

sffiffiffiffiffiffi gp0 ðft=sec or m=secÞ r

ð37:3Þ

where g denotes the ratio of specific heat, p0 denotes the ambient or equilibrium pressure, and r denotes the ambient or equilibrium density. For air, g is taken as 1.4. Equation 37.3 then becomes sffiffiffiffiffiffiffiffi 1:4p0 ðft=sec or m=secÞ ð37:4Þ c¼ r

FIGURE 37.1 (a) Propagating sound wave in a duct; (b) instantaneous pressure distribution; (c) pressure variation in time history detected by a microphone at x ¼ x0 :

which can be further simplified by the fact that the ratio p0 =r is related to the temperature of the gas. On assuming that the air behaves virtually as an ideal gas, the velocity, c; is related to the absolute temperature in degrees Kelvin (K) by pffiffi c ¼ 20:05 T ðm=secÞ

ð37:5Þ

where T; the temperature in degrees Kelvin, is T ¼ 273:28 þ ð8CÞ K

Example 37.1 Calculate the velocity of sound, c; giving the temperature of 158C. Solution T ¼ 273:28 þ 158 ¼ 288:2 K; then pffiffiffiffiffiffiffi c ¼ 20:05 288:2 ¼ 340:4 m=sec is obtained. This value means a typical velocity of sound in the air.

ð37:6Þ

Sound Levels and Decibels

37.3

37 3

Levels and Decibels

Sound pressure and power are commonly expressed in terms of decibel levels. This allows us to use a logarithmic rather than a linear scale. It provides the distinct advantage of allowing accurate computations using small numerical values, while accommodating a wide range of numerical values.

37.3.1

Sound Power Level

Sound power level describes the acoustical power radiated by a given source with respect to the international reference of 10 12 W. The sound power level, LW ; is defined as LW ¼ 10 log

W Wre

where W denotes sound power in question and Wre ¼ 10

ðdBÞ 12

ð37:7Þ

W (reference).

Example 37.2 Determine the sound power level of a small ventilation fan that generates 10 W of sound power. Solution LW ¼ 10 log

37.3.2

W Wre

¼ 10 log

10 10 12

¼ 130 dB

Sound Pressure Level

Sound pressure levels are expressed in decibels, as are sound power levels. The sound pressure level, Lp ; is defined as ! p2 p ðdBÞ ð37:8Þ Lp ¼ 10 log 2 ¼ 20 log pre pre where p denotes root mean square (RMS) sound pressure in question Pa or N/m2 and pre ¼ 20 £ 10 6 Pa ¼ 0.0002 mbar. The pressure of 20 £ 10 6 Pa has been chosen as a reference because it has been found that the average young adult can perceive a 103 Hz tone at this pressure. This reference is often referred to as the threshold of hearing at 103 Hz.

Example 37.3 Giving Lp ¼ 50 dB for the Aeolian tone of 200 Hz, determine the RMS pressure of the tone.

Solution Given Lp as 50 dB, then p is determined by using Equation 37.8. 50 ¼ 20 log

p pre

then p ¼ 1050=20 pre ¼ 316:2pre p ¼ 6:32 £ 10

3

Pa ¼ 0:0632 mbar

Note that this value is very small, contradicting the magnitude of the sensory impression of the human ear.

37 4


37.3.3

Overall Sound Pressure Level

The sound pressure level is defined assuming “pure tone” sound. However, practically any real sound contains various components of pure tone sound. Let us consider a set of n components of pure tone, denoted by 8 p1 ðtÞ ¼ a1 sinð2pf1 t þ f1 Þ > > > < .. ð37:9Þ . > > > : pn ðtÞ ¼ an sinð2pfn t þ fn Þ pðtÞ ¼ p1 ðtÞ þ · · · þ pn ðtÞ

ð37:10Þ

If Lp of pðtÞ is evaluated in RMS pressure, p; we have p ¼ lim

T!1

1 ðT 2 p ðtÞdt T 0

Since lim

T!1

is valid, p is obtained as

1=2

¼ lim

T!1

1 ðT ðp ðtÞ þ · · · þ pn ðtÞÞ2 dt T 0 1

1=2

ð37:11Þ

1 ðT p ðtÞpj ðtÞdt ¼ 0; i – j T 0 i

i1=2 i1=2 h h ¼ p21 þ · · · þ p2n p ¼ p1 ðtÞ2 þ · · · þ pn ðtÞ2

where p2i ; pi ðtÞ2 ; lim

T!1

1 ðT 2 1 p ðtÞdt ¼ a2i T 0 i 2

ð37:12Þ

ð37:13Þ

Let us define Lpi ; 10 logðp2i =p2re Þ ði ¼ 1; 2; …; nÞ: Then the overall sound pressure level, Lp ; of pðtÞ is expressed by Lp ; 20 log

1 p ¼ 10 log 2 ð p21 þ · · · þ p2n Þ pre pre

or Lp is expressed by Lpi ði ¼ 1; 2; …; nÞ as follows: Lp ¼ 10 logð10Lp1 =10 þ 10Lp2 =10 þ · · · þ 10Lpn =10 Þ

ð37:14Þ

Example 37.4 Determine the overall sound pressure level of the combination of three pure tones, the sound pressure levels of which are expressed by Lp1 ¼ 60 dB ðf1 ¼ 250 HzÞ;

Lp2 ¼ 65 dB ðf2 ¼ 500 HzÞ;

Lp3 ¼ 55 dB ð f3 ¼ 1000 HzÞ

Solution We have 10Lp1 =10 ¼ 106 ; 10Lp2 =10 ¼ 106:5 ; and 10Lp3 =10 ¼ 105:5 : Then the overall level, Lp ; is determined by using Equation 37.14 as follows: Lp ¼ 10 logð106 þ 100:5 £ 106 þ 10 ¼ 10 log106 ð1 þ 100:5 þ 10

0:5

0:5

£ 106 Þ

Þ ¼ 60 þ 10 log4:479 ¼ 66:5 dB

Note that the sum of 65, 60 and 55 dB is just 66.5 dB.

38

Hearing and Psychological Effects


38.1 38.2 38.3 38.4 38.5

Introduction .......................................................................... Structure and Function of the Ear ...................................... Frequency and Loudness Response ..................................... Hearing Loss ......................................................................... Psychological Effects of Noise .............................................

Loudness Interpretation

†

Noise Criteria Curves

†

Sound Level

38 38 38 38 38

1 1 2 4 4

Summary In this chapter, first the characteristics of human hearing are discussed, including a brief description of the anatomy and function of the hearing mechanisms. Next, the frequency and loudness responses of the human hearing are explained, and then hearing loss causing permanent damage is described. Finally, the psychological response to noise is discussed by defining the indices, loudness (sones), noise criteria curves, and sound level.

38.1

Introduction

This chapter considers the characteristics of human hearing. After a brief description of the anatomy and function of the hearing mechanism, those aspects of hearing that are important in noise control are discussed. The perception of sound by the human ear is a complicated process, dependent both on the frequency and pressure amplitude of the sound. We shall consider the structure of the ear and hearing mechanism. We will also briefly discuss various means of measuring the psychological effects of noise.

38.2

Structure and Function of the Ear [1]

The main components of the human ear are depicted in Figure 38.1(a). The ear is commonly divided into three main components: (1) the outer ear, (2) the middle ear, and (3) the inner ear. The visible portion of the ear is called the pinna. Because of its small size compared with the primary wavelengths that we hear, the pinna serves only to produce a small enhancement of the sounds that arrive from the front of the listener as compared to those which arrive from behind; that is, the human sound reception system has a small frontal directivity. The remainder of the outer ear, which consists of the ear canal terminated in the ear drum, forms a resonant cavity at about 3 kHz. This resonant or near resonant condition allows for a nearly reflection free termination of the ear canal and thus a good impedance match of the ear drum to the air in which the sound wave was propagated. The middle ear consists of three small ear bones, the hammer, anvil, and stirrup. The middle ear serves as an impedance transformer, which matches the low impedance of the air in which sound travels and in 38 1

38 2


FIGURE 38.1

(a) Main components of the human ear; (b) functional diagram of the ear.

which the ear drum is located to the high impedance of the lymphatic fluid of the cochlea beyond the oval window. Without this impedance matching transformation, a mismatch would occur, resulting in a loss of approximately 30 dB. In the inner ear, the cochlea is the main component where the actual reception of sound takes place. The schematic extended structure of the cochlea is depicted in Figure 38.1(b). The cochlea, which is located in extremely hard temporal bone, is divided almost its entire length by the basilar membrane. At the end of the cochlea, the two canals are connected by the helicotrema, which allows for the flow of the lymphatic fluid between the two sections. The basilar membrane, which is about 3 cm long and 0.02 cm wide, has about 24,000 nerve ends terminated in hair cells located on the membrane. The motion of the oval window is transmitted to the basilar membrane and its associated sensing cells. This motion is sensed as sound.

38.3

Frequency and Loudness Response

The threshold of hearing, defined for binaural listening, is that sound pressure in the free field which one can just still hear as the signal is decreased. The threshold of hearing, for what is considered normal hearing, is shown in Figure 38.2. As seen from the curve, human hearing is most sensitive in the range of 2000 to 5000 Hz; furthermore, we note that the response in this range is very close to 0 dB, or 20 £ 10 6 Pa. At the other end of the scale, there is the threshold of pain, which is usually taken as about 135 to 140 dB. Thus, there is a dynamic range of normal hearing of approximately 140 dB.


38 3

140

Phon

130

130

120

120

110

110

Sound pressure level (dB)

100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10 0

10

Normal binaural threshold of hearing

Phon −10 −20

30 40 50

80 100

200 300

500 800 1k

2k 3k

5k

8 k 10 k

20 k

Frequency [Hz] FIGURE 38.2

Equal loudness contour for free field binaural listening.

One also readily notes from the curve of Figure 38.2 that the threshold of hearing is a function of frequency. For example, with normal hearing, one would just be able to hear a 2000 Hz tone at a 0 dB level. However, one would require a pressure level of about 15 dB to be able to barely hear a 200 Hz tone. Thus, in describing the subjective loudness of sound, it is necessary to consider the characteristics of the human ear. This concept of loudness is quantized by the loudness level. The loudness level of a particular sound is determined by the subjective comparison of the loudness of the sound to that of a 1000 Hz pure tone. The level, measured in phons, is equal numerically to the sound pressure level, in dB, of the 1000 Hz tone, which was regarded to be of equal loudness. A set of internationally standardized equal loudness contours is plotted in Figure 38.2. In keeping with the definition of loudness level, note that at 1000 Hz, all the equal loudness contours are equal in phons to the sound pressure level in dB.

Example 38.1 Determine the sound pressure level of a 100 Hz tone with a loudness level of 30 phon. Solution From Figure 38.2, we find the sound pressure level to be 44 dB.

38 4

38.4


Hearing Loss

Excessive and prolonged noise exposure causes permanent hearing loss. Various theories have been put forth in an effort to characterize and predict the possible damage that might be caused by a given exposure. Absolute proof of any theory concerning such a complex biological phenomenon is virtually impossible to achieve. However, reliable data have been collected, which deal with situations where workers have been continuously exposed to more or less the same noise environment for many years. It is well established that excessive noise exposure causes permanent hearing damage by destroying the auditory sensor cells. These cells are hair cells located on the basilar membrane. Furthermore, other types of inner ear damage include harm to the auditory neurons, as well as damage to the structure of the organ of Corti. In all, the various theories and data have been taken advantage of establishing the noise exposure criteria set forth in the noise exposure regulations.

38.5

Psychological Effects of Noise

In this section, certain generally accepted aspects of the psychological effects of noise will be discussed and quantified. Various indexes have been proposed that quantify the psychological effects of noise. However, only a few of indices, loudness (sones), noise criteria (NC) curve, and sound level, are introduced in the following presentation.

38.5.1

Loudness Interpretation

As was discussed relating to Figure 38.2, loudness level is measured in phons, and the related quantity, loudness, is measured in sones. A sone is defined as the loudness of a 1000 Hz pure tone with a sound pressure level of 40 dB. On recalling the definition of loudness level, or by referring to Figure 38.2, one notes that 40 phon have a loudness equal to 1 sone. This relationship may be simply expressed as S ¼ 2ðLL

40Þ=10

sone

ð38:1Þ

where S ¼ loudness (sones), LL ¼ loudness level (phons), or conversely LL ¼ 33:2 log S þ 40 phon

ð38:2Þ

Example 38.2 Make the following two conversions using the appropriate equation (Equation 38.1 or Equation 38.2): (1) convert 80 phon to sone, (2) convert 100 sone to phon. Solution 1. To convert phons to sones, use Equation 38.2: S ¼ 2ðLL

40Þ=10

¼ 2ð80

40Þ=10

¼ 24 ¼ 16 sone

2. To convert sones to phons, use Equation 38.2: LL ¼ 33:2 log S þ 40 ¼ 33:2 log 100 þ 40 ¼ 66:4 þ 40 ¼ 106:4 phon How should we determine the “total loudness” (sones), when the sound is composed of multiple frequency components? Probably the most widely used method for establishing the loudness of a complex noise is that developed by Stevens [2]. The method is based on the measurement of the 1 octave, 1/3 octave, or 1/2 octave band pressure levels. The measured band pressure levels are used


38 5

120 120

Loudness Index

110

150 100 80 60 50 40 30

100

Band pressure [dB]

90

20

80

15 12 10 8

70

6 5 4 3 2.5 2 1.5

60 50 40

1.0 0.7 0.5 0.3 0.2 0.1

30 20 10

50

100

FIGURE 38.3

200

500 1 k 2 k Frequency [Hz]

5k

10 k

Equal loudness index contour [2].

in conjunction with the equal loudness index contours shown in Figure 38.3 to determine the loudness or loudness level by means of a simple calculation. A step by step outline of the procedure is as follows: 1. Measure the band pressure levels (1 octave, 1/2 octave, or 1/3 octave) over the frequency range of interest. Usually, the range chosen is from about 50 to 10,000 Hz. 2. Enter the center frequency and band pressure level for each band in the contour of Figure 38.3, and determine the loudness index for each band. 3. Calculate the total loudness, St ; in sones, by using St ¼ Im ð1

KÞ þ K

n X i¼1

Ii ðsoneÞ

ð38:3Þ

where St ¼ the total loudness (sones), Im ¼ the largest of the loudness indices, Ii ¼ the loudness indices, including Im, K ¼ weighting factor for the bands chosen. K ¼ 0:3 for 1 octave bands, K ¼ 0:2 for 1/2 octave bands, K ¼ 0:15 for 1/3 octave bands.

38 6


4. If so desired, one may calculate the loudness level in phons using Equation 38.2, or one may convert to loudness level by means of the conversion curve of Figure 38.3.

Example 38.3 A particular complex noise was measured to yield the one octave band pressure given in the following table Center Frequency (Hz)

Band Pressure Level (dB)

63 125 250 500 1000 2000 4000 8000

Loudness Index (sone)

66 63 65 70 73 76 81 79

2.5 3.2 4.8 7.5 10.6 15.2 25.1 29.0

Compute the loudness level using the procedure described before.

Solution As a first step, the loudness indices are determined from Figure 38.3 and recorded in tabular form with the band pressure levels. Next, we note that one octave bands have been used. Therefore K ¼ 0:3 in Equation 38.3, and St ¼ Im ð1

0:3Þ þ 0:3

8 X i¼1

Ii

From the table above, we find that Im ¼ 29.0 sone and, summing up, find St ¼ 29ð1

P

Ii ¼ 97:9 sone: Therefore,

0:3Þ þ 0:3ð97:9Þ ¼ 49:67 sone

We find that the loudness, St ø 50 sone: The loudness level may now be calculated by means of Equation 38.2 as LL ¼ 33:2 log St þ 40 ¼ 33:2 log 50 þ 40 ¼ 96:4 phon Therefore, LL ; the loudness level, is 96 phon.

38.5.2

Noise-Criteria Curves

Noise criteria curves, which are neglected here, were established in 1957 for rating indoor noise. The curves have been utilized as one method of rating background noise level in a room. Each curve specifies the maximum octave band sound pressure level for a given NC rating. If the octave band levels for a given noise spectrum are known, the rating of that noise in terms of the NC curves is given by plotting the noise spectrum on the set of NC curves to determine the point of highest penetration. In 1971, some objections to the NC curves led to their modification. The new curves, which are shown in Figure 38.4, are called the preferred noise criteria (PNC) curves. Although these curves differ from the NC curves, they are used in exactly the same manner.


38 7

80

70

Octave band pressure level (dB)

60

PNC-65 PNC-60

50

PNC-55 PNC-50

40

PNC-45 PNC-40

30

PNC-35 PNC-30

20

PNC-25

10

PNC-20

The approximate threshold of hearing for continuous noise

31.5

63

125

PNC-15

250

500

1000

2000

4000

8000

Octave band center frequencies (Hz) FIGURE 38.4

1971 preferred noise criteria curves.

Example 38.4 Determine the PNC rating for the octave band noise spectrum tabulated below. Center frequency (Hz) Band pressure level (dB)

63 65

125 60

250 60

500 63

1000 55

2000 50

4000 45

8000 40

Solution The highest penetration is found at 500 Hz on PNC 60. Hence, the answer is PNC 60.

38.5.3

Sound Level

Sound levels are sound pressure levels that have been weighted according to a particular weighting curve. Three weightings, A, B, and C, and associated sound levels, have been developed as a method to subjectively evaluate the impact of noise upon the human ear, in a proper manner. The frequency response and decibel conversions from a flat response for each of these weightings are given in Figure 38.5 and Table 38.1, respectively. The A weighting network is now used almost exclusively in measurements that relate directly to the human response to noise, both from the viewpoint of hearing damage and of annoyance. Such measurements are referred to as sound level measurements. Sound level is designated by L and the designated unit is the dBA. Similarly, dBB and dBC are used to designate sound level weighted by B weighting and C weighting networks, respectively.

38 8


FIGURE 38.5

Frequency response for the A, B, and C weighting networks.

TABLE 38.1 Sound Level Conversion Chart from Flat Response to A Weighting Frequency (Hz)

A Weighting (dB)

50 63 80 100 125 160 200 250 315 400 500 630 800 1,000 1,250 1,600 2,000 2,500 3,150 4,000 5,000 6,300 8,000 10,000 12,500 16,000

230.2 226.2 222.5 219.1 216.1 213.4 210.9 28.6 26.6 24.8 23.2 21.9 20.8 0 þ0.6 þ1.0 þ1.2 þ1.3 þ1.2 þ1.0 þ0.5 20.1 21.1 22.5 24.3 26.6

Hearing and Psychological Effects TABLE 38.2 fc (Hz)

Octave Band Sound Pressure Levels Lflat (dB)

63 125 250 500 1000 2000 4000 8000

38 9

74 71 61 60 62 60 62 69

DLA (dB) 226.2 216.1 28.6 23.2 0 1.2 1.0 21.1

LA ¼ Lflat þ DL (dB) 47.8 54.9 52.4 56.8 62.0 61.2 63.0 67.9

IiA 0.60 £ 105 3.09 1.74 6.31 1.585 £ 106 1.318 1.995 6.166 12.238 £ 106

Sum Note: *

*

*

*

fc: band center frequency Lflat: sound pressure level with flat weighting DLA: A weighting level IiA: sound pressure intensity with A weighting.

In noise abatement problems, it is often necessary to convert calculated a 1 octave band or 1/3 octave band sound pressure level to a total sound level in dBA. Table 38.1 gives sound level conversion by A weighting from flat response pressure.

Example 38.5 Determine the total A weight sound level, L; of the set of octave band sound pressure levels given in Table 38.2.

Solution Refer Table 38.1 for the dB conversion from a flat response level, Lflat ; for each of the octave bands to a sound pressure intensity with A weighting IiA ; and then the sum of IiA : Finally, the total sound level with A weighting Ltotal;A is given by Ltotal;A ¼ 10 log

n X i¼1

IiA ¼ 10 log 12:238 £ 106 ¼ 70:9 ðdBÞ

References 1. Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood Cliffs, NJ. 2. American National Standard USAS S3.4 1968. 1968. Procedure for the Computation of Loudness of Noise, America National Standards Institute, New York, NY.

39

Noise Control Criteria and Regulations 39.1 Introduction ..................................................................... 39.2 Basic Ideas behind Noise Policy ..................................... 39.3 Legislation ........................................................................ Action Plans Commission Action


† †

Publication of Data by the European Proposal for Further European Union

39.4 Regulation ........................................................................ 39.5 Measures of Noise Evaluation ........................................

39 1 39 1 39 2

39 4 39 5

Summary In this chapter, the basic ideas behind the development of noise control criteria and regulations are discussed, taking into consideration that the standards and criteria vary from country to country and depend on governments in power. Legislations in the European Union and regulations in Japan are introduced as typical examples. Some indexes as measures of noise evaluation are described.

39.1

Introduction

In order to protect people from being exposed to excessive noise, different communities have implemented different types of legislative control. While the controls vary in scope, control mechanisms, and technical requirements, and are based on different control philosophies, they are intended to achieve a balance between the demand for a tranquil environment and the need for maintaining economic and social activities. In general, the noise standards vary according to the time of day and the use of the land concerned, with the more stringent standards applied to rest periods and areas where the noise sensitivity is high, such as those with schools and hospitals, and exclusive residential areas. Different countries have adopted different noise standards and regulations to meet their local situations and requirements. This chapter cannot describe all major control criteria and regulations in the world, or even in the major industrialized countries. Only the main issues of legislation on noise emission and reception are briefly introduced in the chapter. More details in the on going noise control issues are found in Refs. [2,3].

39.2

Basic Ideas behind Noise Policy

Every noise policy originates from the idea of protecting the quality of life from noise pollution of all kinds. When establishing a noise policy, it is useful to consider the distinction between noise emission and immission (or reception) [6]. The former means literally emitting or radiating 39 1

39 2


sound energy or power from a noise source, whereas the latter means receiving, perceiving, or observing radiated noise, which leads to the extent of the noise exposure at a position near the noise source. Therefore, noise emission is controlled with noise regulation law by the government, whereas noise immission is legislated with environmental quality standards. The measure of the extent describing the former is the “sound power level,” and that describing the latter is the sound pressure level. The global professional organization on noise control, the International Institute of Noise Control Engineering (I INCE) recently started its activities to develop a global noise policy [5]. In response to the question “is noise policy a global issue, or is it a local issue?”, I INCE had a common theme presented in special session. It was felt that noise is primarily a global policy issue, but many noise problems can only be solved with the active participation of local authorities. The task of the technical study group is to take a global approach to noise in order to define the requirements for an international noise control policy to be effective, stated as follows: All vehicles, devices, machinery, and equipment that emit audible sound are manufactured products; most are entered into world trade and many are produced in two or more different countries by companies with worldwide operations. The noise emission of these products is an appropriate subject of international agreements and regulations. The noise immisions resulting from the operation of these products are growing in severity as traffic flow and the pace of industrialization continues to increase in many parts of the globe. The technical study group reports the classification of noise areas as follows: 1. OCCUPATION NOISE noise received at the workplace, indoors and outdoors, caused by all noise sources in the vicinity of the workplace. 2. ENVIRONMENTAL NOISE noise perceived by individuals in the domestic environment, indoors and outdoors, caused by sources controlled by others. 3. CONSUMER PRODUCT NOISE noise perceived by users and bystanders of noise generating products over which the individual has some control, including noise in the passenger compartment of vehicles, excluding occupational and environmental noise.

39.3

Legislation

The World Health Organization (WHO) published the historic “Guidelines for Community Noise” in 2000, which has been accepted as the most significant recommendations for noise exposure criteria. The bodies that are responsible for enacting the regulations as law include the Federal Government in the USA, the European Union (EU) in Europe, and the Japanese Government in Japan. In the following, the EU’s legislation on noise immission is shown, as an example of the flow of legislation process [8]. On July 18th, 2002, a European Directive on the assessment and management of environmental noise was published in the Official Journal of European Communities. It was required to be implemented in the national legislation of the EU Member States no later than July 18th, 2004. From then on, a program was to start, containing periodic noise mapping, the making of action plans, and information of the public. The directive also has strengthened the position of the European Commission regarding the reduction of noise emission. In 2002, the development of the European Directive on environmental noise resulted in an approved directive relating to the assessment and management of environmental noise, for which the acronym “DAMEN” is used. According to Article 1 of the DAMEN its objective is to “define a common approach to avoid, prevent or reduce harmful effects, including annoyance, due to exposure to environmental noise.” A rough description of actions in the DAMEN is shown in Figure 39.1. Brief notes are given next to supplement Figure 39.1.


HARMONIZATION OF NOISE INDICATORS AND ASSESSMENT METHODS

39 3

LIMIT VALUES (optional; set by the Member States)

NOISE MAPS for major agglomerations, roads, railways and airports

REPORT (European Commission)

PROPOSALS for: 1. Further EU actions, particularly on NOISE SOURCES 2. Protection of QUIET AREAS IN OPEN COUNTRY

ACTION PLANS

INFORMATION/CONSULTATION of the PUBLIC

European Council and European Parliament

FIGURE 39.1 Overview of the DAMEN. (Source: Wolde, T.T. 2003. The European Union’s legislation on noise immission, pp. 4367 4371. In Proceedings of Inter noise 2003 (N832). With permission.)

39.3.1

Action Plans

In a case where the mapping results are such that they violate the local or national limit value, or are found unsatisfactory for other reasons, action plans shall be developed for the improvement of the situation. These action plans shall be discussed with the citizens involved. A summary of the action plans shall be sent to the European Commission.

39.3.2

Publication of Data by the European Commission

Every five years, starting in 2009, the Commission shall publish a summary report from the noise maps and the action plans.

39.3.3

Proposal for Further European Union Action

In 2004, the European Commission was to submit a report to the European Parliament and the Council containing a review of existing EU measures relating to sources of environmental noise and present proposals for improvement, if appropriate. In 2009, the European Commission will submit to the European Parliament and the Council a report on the implementation of the directive. That report will in particular assess the need for further EU action and, if appropriate, propose implementing strategies on aspects such as: *

*

*

Long term and medium term goals for the reduction of the number of persons harmfully affected by environmental noise Additional measures on noise emission by specific sources The protection of quiet areas on the open country

39 4

39.4


Regulation

Noise regulation is executed by local governments once the central government enacts a noise regulation law. The law is considered the “national minimum.” For example, factory noise, construction work noise, and road traffic noise are under the purview of the Noise Control Act, which means the central government is responsible of regulating these kinds of noise. On the other hand, community noise and factory noise are under the purview of the original regulation of local governments. It can be said that local governments are responsible for a great part of the noise policy, although they may not always fully understand the situations concerned. In what follows, an outline of the legal system for environmental noise problem in Japan is given as an example of a typical legal system for noise regulation [6]. In Japan, the “Environmental Quality Standards for Noise” was revised in 1999 after 27 years with the old law. Figure 39.2 outlines of the legal system in Japan.

Basic Environment Law (1968, revised in 1993) General and roadside areas Environmental Quality Standards for Noise (1971, revised in 1998) Areas around large-scale airport (including airbases) Environmental Quality Standards for Aircraft Noise (1973) Areas along Shinkansen Super-express Railways Environmental Quality Standards for Shinkansen Super-express Railway Noise (1975)

Noise Regulation Law (1968, amended in 1999)

Environmental Impact Assessment Law (1997)

Guideline for the Preservation of Living Environment around Small Airfields (1990)

Guideline of Noise Measures for Conventional Railways (1995) FIGURE 39.2 Legal system for environmental noise in Japan. (Source: Tachibana, H. and Kaku, J. 2003. Acoustic measures for the environmental noise assessment in Japan, pp. 3317 3322. In Proceedings of Inter noise 2003 (N1007). With permission.)


39 5

Each of these laws and standards is legislated for a specific noise problem (a noise source), and therefore different noise indices are specified according to the respective noise problems. To review this situation from a historical viewpoint, it can be said that each law or standard responds to a specific noise problem promptly, through the use of available measurement technology at that time. However, some laws and standards have become outdated since their establishment, when considering the current situation, international dynamics, and the current acoustic measurement technology.

39.5

Measures of Noise Evaluation

Basically, the A weighting networks are applied to obtain a measure of noise evaluation. As the measures are legislated by governments, they are dependent on the legislative regulations and standards. In what follows, the measure of noise evaluation legislated in Japan is given to show the concepts behind the legislation [6]. In the regulations and standards for environmental noise problems, a variety of noise measures are used. In order to improve these legislative regulations and standards in the future, the present measures shall be reviewed considering the difference between noise emission and immission and the difference between noise measurement and monitoring, and impact assessment and prediction. These measures legislated in Japan are listed and classified in Table 39.1 by considering the difference between noise characteristics.

TABLE 39.1

Assessment Methods Specified in Laws and Standards for Environmental Noises in Japan

Noise Sources Roads Shinkansen superexpress railways Conventional railways Aircrafts

Construction works Factories Large scale retail stores

a

Law and Standards Environmental Quality Standards for Noise Environmental Quality Standards for Shinkansen Superexpress Railways Guideline of Noise Measures for Conventional Railways Environmental Quality Standards for Aircrafts Noise Guideline for the Preservation of Living Environment around Small Airfields Noise Regulation Law (specific noise sources) Law concerning the measures by large scale retail stores for preservation of living environment

Noise Indices LAeq,Ta LA,Smaxb LAeq,T WECPNLc

Assessment Time Daytime (6:00 22:00); nighttime (22:00 6:00) Every event Daytime (7:00 22:00); nighttime (22:00 7:00) Time weighting

Ldend According to time variation: LAe; LA,Fmaxf; LA5g; LA,Fmax,5h

Not specified; every event

LAeq,T, equivalent continuous A weighted sound pressure level. LA,Smax, SLOW maximum value of A weighted sound pressure level. c WECPNL, weighted equivalent continuous perceived noise level(calculated from LA,Smax). d Lden, day/evening/night equivalent continuous A weighted sound pressure level. e LA, FAST maximum value of A weighted sound pressure level. f LA,Fmax, A weighted sound pressure level. g LA5, upper value of the 90% range of A weighted sound pressure level. h LA,Fmax,5, upper value of the 90% range of the FAST maximum Aweighted sound pressure level. Source: Tachibana, H. and Kaku, J. 2003. Acoustic measures for the environmental noise assessment in Japan, pp. 3317 3322. In Proceedings of Inter noise 2003 (N1007). With permission. b

39 6


When considering the consistency between noise measurement and monitoring, and noise prediction for impact assessment, it is most reasonable to use energy based indices such as LAeq. Of course, LAeq is not a panacea and some secondary adjustment may be needed for the exact assessment of environmental noise with different characteristics. Nevertheless, the possibility of unification by LAeq should be considered in the near future in Japan. Although LAeq is now being widely used for the assessment of aircraft noise in almost all countries, WECPNL is still being used in Japan. WECPNL is very close to LAeq in concept and it is not difficult to change the assessment index from WECPNL to LAeq. The aim of the laws and standards shown in Figure 39.2 is to measure and assess the environmental noise for prevention or maintenance of the present situation. Therefore, any noise index should be appropriately used for each of noise problems, as shown in Table 39.1, which presents assessment methods specified in laws and standards for environmental noise in Japan. In particular, when predicting the future noise situation in environmental impact assessments, the indices should be suitable for theoretical calculation. The statistical noise indices such as the percentile level (LA5) and maximum level (LA,Fmax or LA,Smax) specified in the laws and standards have to be predicted statistically. It is difficult to predict these quantities by a simple physical calculation model, in principle. In this respect, the energy based noise indices such as LAeq can be easily treated in energy based calculation, and the prediction model becomes simple and clear in physical meaning. In an environmental impact assessment, the predicted results are to be compared with the related laws or standards. In the case of road traffic noise, LAeq has been adopted in the new environmental quality standards, and therefore prediction has become very simple in theory, founded on energy based indices. In the prediction of road traffic noise, a motor vehicle as the noise source can be treated as a stationary sound sources of a constant sound power for a limited path. On the other hand, in the case of predicting construction noise, there are many complicated problems because various kinds of machines and equipment with various temporal variations of characteristics must be treated. Therefore, in the construction noise prediction method given in the ”Acoustic Society of Japan CN model 2002” [7], various noise indices for describing the acoustic output of various types of noise sources are specified as given in Table 39.2, which presents classification of noise sources and indices for expressing their acoustic output. Finally, Table 39.3 presents definitions and indices of measurement for acoustical output of noise sources. TABLE 39.2 Output

Classification of Noise Sources and Indices for Expressing Their Acoustic

Temporal Variation

Stationary

Fluctuating randomly and widely

Indices for Expressing Acoustic Output Sign LWA

LA(r0) LWAeff LAeff(r0) LA,Fmax,5(r0)

Intermittent impulsive

LJA

LWAeff LAE ðr0 Þ LA;Fmax ðr0 Þ

Terms

A weighted sound power level A weighted sound power level at the reference distance (r0 ¼ 1 m) Effective A weighted sound power level Effective A weighted sound pressure level at the reference distance (r0 ¼ 1 m) 5% value of A weighted sound pressure level at the reference distance (r0 ¼ 1 m) A weighted sound energy level Effective A weighted sound power level Single event sound exposure level at the reference distance (r0 ¼ 1 m) FAST max. of A weighted sound pressure level at the reference distance (r0 ¼ 1 m)

Source: Tachibana, H. and Kaku, J. 2003. Acoustic measures for the environmental noise assessment in Japan, pp. 3317 3322. In Proceedings of Inter noise 2003 (N1007). With permission.

Noise Control Criteria and Regulations TABLE 39.3

Definitions and Measurements of Indices for Acoustical Output of Noise Sources

Indices LWA

Definition LWA ¼ 10 log

LWAeff

39 7

PA P0

ð1Þ

Measurement Method LWA ¼ LA ðrÞ þ 20 log

r þ8 r0

ð2Þ

Here, P0 ¼ 1 pW

Here, LA ðrÞ is the A weighted sound pressure level measured at a distance of r; r0 ¼ 1 m

Effective A weighted sound power level applied to fluctuating, intermittent and impulsive sounds

LWAeff ¼ LAeff ðrÞ þ 20 log

r þ8 r0

ð3Þ

Here, LAeff is the A weighted sound pressure level measured at a distance of r " # 1 ð2 p2A ðtÞ LAeff ¼ 10 log dt ð4Þ T 1 p20 Here, Tðt1 2 t2 Þ is averaging time (sec), p0 ¼ 20 mPa

LJA

LJA ¼ 10 log

EA E0

ð5Þ

Here, E0 ¼ 1 pJ

LJA ¼ LAE ðrÞ þ 20 log

r þ8 r0

ð6Þ

Here, LAE is the single event sound exposure level measured at a distance of r " # 1 ð2 p2A ðtÞ dt ð7Þ LAE ¼ 10 log 2 T 1 p0 Here, T0 ¼ 1 sec; t1 2 t2 is the time including the event (sec)

LA ðr0 Þ LA;Fmax ðr0 Þ

A weighted sound pressure level converted to the value at the reference distance ðr0 Þ ¼ 1 m

LA ðr0 Þ ¼ LA ðrÞ þ 20 log

r þ8 r0

ð8Þ

Here, LA ðrÞ is the A weighted sound pressure level measured at a distance of r

Source: Tachibana, H. and Kaku, J. 2003. Acoustic measures for the environmental noise assessment in Japan, pp. 3317 3322. In Proceedings of Inter noise 2003 (N1007). With permission.

References 1. 2. 3. 4. 5.

Fahy, F. 1985. Sound and Structural Vibration, Radiation, Transmission and Response, Academic Press, New York, chap. 2. Fields, J.M. and de Jong, R.G., Standardized general purpose noise reaction questions for community noise survey: research and a recommendation, J. Sound Vib., 242, 641 679, 2001. Harris, C.M., Ed. 1979. Handbook of Noise Control, 2nd ed., McGraw Hill, New York, chap. 37. Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, New York, chap. 5. Lang, W.W. and Wolde, T.T. 2003. Progress report for TSG#5 ‘Global Noise Policy’, pp. 98 101. In Proceedings of Inter noise 2003 (N872).

39 8

6. 7. 8.


Tachibana, H. and Kaku, J. 2003. Acoustic measures for the environmental noise assessment in Japan, pp. 3317 3322. In Proceedings of Inter noise 2003 (N1007). Tachibana, H. and Yamamoto, K. 2003. Construction Noise Prediction Model, ASJ CN Model 2002, proposed by the Acoustical Society of Japan, EURONOISE, in Naples (2003.5). Wolde, T.T. 2003. The European Union’s legislation on noise immission, pp. 4367 4371. In Proceedings of Inter noise 2003 (N832).

40

Instrumentation 40.1 40.2

Kiyoshi Nagakura

40.3

Railway Technical Research Institute

Sound Intensity Measurement ........................................... 40 1 Theoretical Background † Measurement Method Measurement of Sound Intensity † Applications

†

Errors in

Mirror Microphone System .............................................. 40 4 Principle of Measurement

†

Applications

Microphone Array .............................................................. 40 6 Principle of Microphone Array Pattern † Applications

†

Array’s Directivity

Summary This chapter describes some measuring methods for the identification and ranking of noise source that are of benefit in noise control projects. Sound intensity measurement and directional measuring devices such as the mirror microphone system and microphone array are introduced and their principles and applications are described.

40.1

Sound Intensity Measurement

Every noise control project starts with the identification and ranking of the noise sources. Several methods have been proposed for the purpose and have proved to be useful and widely utilized. In this chapter, sound intensity measurement and directional measuring devices such as the mirror microphone system and microphone array are introduced and their principles and applications are described. Other useful measurements, such as acoustic holography method [1,2] and spatial transformation of sound fields [3], are described in the literature.

40.1.1

Theoretical Background

Sound intensity is a measure of the magnitude and direction of the flow of sound energy. The instantaneous intensity vector, IðtÞ; is given by the product of the instantaneous sound pressure, pðtÞ; and the corresponding particle velocity, uðtÞ; that is, IðtÞ ¼ pðtÞuðtÞ: In practice, the time averaged intensity, I; is more important, and is given by the equation: I ¼ lim

T!1

1 ðT=2 pðtÞuðtÞdt T T=2

ð40:1Þ

The intensity vector denotes the net rate of flow of energy per unit area (watts/m2). Thus, the acoustic power, W; of the source located in a closed surface, S; is given by the integral of the intensity passing through the surface, S; as W¼

ðð s

I·dS

ð40:2Þ 40 1

40 2


Equation 40.2 indicates that the measurement of sound intensity over a surface enclosing a source enables the estimation of its sound power, which shows the usefulness of the sound intensity concept.

40.1.2

Measurement Method

The principle of intensity measurement systems in commercial production employs two closely spaced pressure microphones [4,5], as shown in Figure 40.1. The particle velocity, ur ðtÞ; in a particular direction, r; can be approximated by integrating over time the difference of sound pressures at two points separated by a distance Dr in that direction: ur ðtÞ ¼

1 ðt r0

p2 ðtÞ 1

Dr

p1 ðtÞ

dt

p1 ðtÞ þ p2 ðtÞ 2

θ

r

∆r

ð40:3Þ

where p1 and p2 are the sound pressure signals from the two microphones. The sound pressure at the center of two microphones is approximated by pðtÞ ¼

Sound

p1(t)

p2(t)

Microphones FIGURE 40.1 Microphone arrangement used to measure sound intensity.

ð40:4Þ

Thus, the intensity in the direction r can be calculated as Ir ðtÞ ¼

ðt 1 ½p1 ðtÞ þ p2 ðtÞ 2r0 Dr

1

½p2 ðtÞ

p1 ð t Þ d t

ð40:5Þ

Some commercial intensity analyzers use Equation 40.5 to measure the intensity. Another type of analyzer uses the equation in the frequency domain: Ir ðvÞ ¼

Im½G12 vr0 Dr

ð40:6Þ

where G12 is the cross spectrum between the two microphone signals. Equation 40.6 makes it possible to calculate sound intensity with a dual channel fast fourier transform (FFT) analyzer.

40.1.3

Errors in Measurement of Sound Intensity

The principal systematic error of the two microphone method is due to the approximation of the pressure gradient by a finite pressure difference. When the incident sound is a plane wave, the ratio of the measured intensity, I^r , and the true intensity, Ir , is given by sinðkDr cos uÞ I^r =Ir ¼ kDr cos u

ð40:7Þ

where the angle u is as defined in Figure 40.1 and k is the wave number. Equation 40.7 indicates that the upper frequency limit is inversely proportional to the distance between the microphones. Another serious error is caused by the phase mismatch between the two measurement channels. In the calculation of intensity from Equation 40.5, the phase difference, w; between the two microphone signals, p1 and p2 ; is very important. Hence, the phase mismatch between the two measurement channels, Dw; must be much smaller than w: Since w increases with frequency, this error is serious in lower frequencies. Other possible errors, such as in the sensitivity of microphones and random errors associated with a given finite averaging time, are usually less serious.

Instrumentation

40.1.4

40 3

Applications

One important application of sound intensity measurement is the determination of the sound power level using Equation 40.2. Furthermore, measurement of the intensity in the very near field of a source surface makes it possible to identify and rank the noise sources. Plots of the sound intensity measured on a surface near a sound source are useful for investigating noise source distributions. Figure 40.2 shows sound intensity of noise from a wheel of a railway car. An intensity probe is located in the vicinity of the wheel and the normal component of sound intensity is measured by traversing the probe on a plane 100 mm away from the side surface of the wheel. These figures show a free vibration behavior of the wheel at each frequency; the wheel vibrates with one nodal diameter at 700 Hz and with three nodal diameters at 1150 Hz. Visualization by intensity vectors also gives valuable information about a noise source. Figure 40.3 shows the sound intensity vectors at each octave band measured in the vicinity of a railway car running at 120 km/h. These results suggest that the main radiator of rolling noise is the rail at the 500 Hz to 1 kHz band and the wheels at the 2 to 4 kHz band.

FIGURE 40.2

FIGURE 40.3

Measurements of the sound intensity radiated by a wheel of a railway car (1 dB contour).

Sound intensity vectors measured in the vicinity of a railway car running at 120 km/h.

40 4


Mirror –Microphone System Principle of Measurement

A mirror microphone system consists of a reflec tor of elliptic or parabolic shape and an omnidir ective microphone located at its focus [6,7]. Figure 40.4 shows the layout of a reflector of elliptic shape, an omnidirective microphone, and a noise source. Here, S and S0 denote the front and back surfaces of the mirror, respectively; PðrÞ denotes the pressure field on this configuration; Pi ðrÞ denotes the pressure field of free space; rm is the position of the microphone; r is a point on the mirror surface. The normal, n0 ; directs toward the medium. Using Green’s theorem, the pressure at the microphone position Pðrm Þ is obtained by Pðrm Þ ¼ Pi ðrm Þ þ "

ðð ðsþs 0 Þ

# eikRm d2 r 4pRm

PðrÞ

› ›n0

Pðrm Þ ¼

cos uðrÞd2 r

r Rs

Rm θ D

rm

S′

S

Far focus

n0

Microphone = Near focus

rs Noise source

B L

FIGURE 40.4 noise source.

Layout of a reflector, microphone, and

ð40:8Þ

where k ¼ 2pf =c0 is the wave number, f is the frequency of sound, c0 is the speed of sound, and Rm ¼ lr rm l is the distance between the micro phone and the mirror surface. If the wavelength is sufficiently smaller than the diameter of the reflector, the pressure field PðrÞ is approximated by 2Pi ðrÞ on the front surface, S; and by zero on the back surface, S0 : In such a frequency range, the incident field term Pi ðrm Þ can be ignored. With these approximations, assuming that the noise source is a monopole type point source located at a position, rs ; Equation 40.8 reduces to mð f Þ ð ð eikðRm þRs Þ ik 8p 2 Rm Rs s

Reflector

1 Rm

G 3dB

40.2.1

SPL at microphone position (re SPL in freefild)

40.2

w

0

Displacement of the mirror

FIGURE 40.5 Directivity pattern of a mirror micro phone system.

ð40:9Þ

Here, mð f Þ is the amplitude of the mass flux rate of the source, Rs ¼ lr rs l is the distance between the sound source and the mirror surface and the angle uðrÞ is defined in Figure 40.4. When the noise source is located at the far focus of the mirror, the sound pass length Rm þ Rs is constant with respect to r; and a strong signal is obtained. As the noise source is moved away in the direction perpendicular to the mirror axis, the variance of the sound pass length, Rm þ Rs ; due to the position r increases, and thus the microphone signal drops off due to interference (see Figure 40.5, which we call the “directivity pattern”). The ratio of the peak level to the free field level at the microphone, G; is referred to as the “gain factor.” The spatial resolution of the mirror is characterized by the displacement of the mirror position, w; at which the microphone signal drops off by a given relative amount, such as 3 dB. The quantities G and w

Instrumentation

40 5

can be related to the mirror geometries in Figure 40.4 by G < 10 logðCD4 =l2 B2 Þ ðC ¼ const:Þ w / lL=D

ð40:10Þ ð40:11Þ

The gain factor, G; increases with frequency at the rate of 6 dB per octave, and the spatial resolution, w; is inversely proportional to the frequency. The lower frequency limit is decided by the size of the mirror. On the other hand, there is no higher frequency limit, except for the capacity of an omnidirectional microphone itself. Thus, measurements with the mirror microphone system are more suited to a scaled model test.

40.2.2

Applications

The mirror microphone system has proved useful for identification of a noise source because of its directional property [8 10]. A scan of the source region produces a noise source map. It has an advantage in that the measurement is possible at a far field and it needs only one sensor, but has a disadvantage in that the measuring process is a time consuming task. Figure 40.6 shows an example of source maps of aerodynamic noise generated by a one fifth scale high speed train model, obtained from measurements by a mirror microphone system, in a wind tunnel test. The surface of the car model is divided into several noise source areas and the noise source distribution in each area is measured by traversing the mirror microphone system over the surface. The diameter and focal distance of the reflector are 1.7 and 3 m, respectively. Detailed maps of noise source strength are obtained, which show that aerodynamic noise from high speed trains is generated in relatively localized areas, namely, the local surface structures. The mirror microphone system can be used for the measurement of the source distribution of a moving noise source. Figure 40.7 gives a time

FIGURE 40.6 Noise source distribution of a one fifth scale Shinkansen car model in a wind tunnel test measured with an elliptic mirror microphone system.

40 6


FIGURE 40.7 Time history of the A weighted one third octave band ( f0 ¼ 8 kHz) sound pressure level measured with a parabolic mirror microphone system (D ¼ 1 m, train speed ¼ 274 km/h).

history of noise from a high speed train measured with a parabolic mirror microphone system, the diameter of which is 1 m. Peaks of the time history correspond to pantographs, doors, gaps between cars and the step up of windows, which shows that they are main noise sources.

40.3 40.3.1

Microphone Array Principle of Microphone Array

A microphone array [11] consists of several microphones distributed spatially to measure an acoustic field. The time signals from each microphone are added, accounting for the time delay between sound sources and microphones, and a directional output signal can be obtained as a result. The algorithm is called “beamforming.” Now, consider M omnidirectional microphones distributed in a far field of noise sources. The output signal of the array focused to a particular location in the source region, r; and zðr; tÞ; is calculated as a sum of delayed and weighted signals of each microphone: zðr; tÞ ¼

M X m¼1

wm pm ðt

Dm Þ

ð40:12Þ

Here, pm ðtÞ is the signal from the mth microphone, wm is a weighting factor, and Dm is a time delay applied to signal of the mth microphone, as given by Dm ¼

ro

c0

rm

ð40:13Þ

where ro and rm are the distances from the focus point to the reference point o and the mth microphone, respectively. When the focus location coincides with the source location, a strong signal is obtained (see Figure 40.8). If this process is repeated for various focus locations, r, on the source surface, then a noise source map can be obtained.

Instrumentation

40 7

p1(t)

Delay w1 ∆1

p2(t)

Delay w2 ∆2

pM(t)

Delay wM ∆M

Σ

z(t)

FIGURE 40.8 Principle of a microphone array. Individual time delays are chosen such that signals arriving from a given point will be added up coherently.

40.3.2

Array’s Directivity Pattern

The performance of a microphone array is characterized by the spatial resolution and signal to noise ratio. For simplicity, consider a linear array of M ¼ 2N þ 1 microphones spaced equally by d: When a harmonic plane wave is propagating with an incident angle u; and weighting factors all equal 1=M; the ratio of the output signal of the array to that of the center microphone is computed using WðuÞ ¼

1 sinððM=2Þkd sin uÞ M sinðð1=2Þkd sin uÞ

ð40:14Þ

where k is the wave number. Figure 40.9 shows the directivity patterns for different values of the product kd based on Equation 40.14. The highest peak appears at u ¼ 0; which we call a “main lobe,” and lower peaks also appear at some locations that are separate from a true source direction, which we call “side lobes.” The width of the main lobe decides the performance of the array to separate two closely lying sources (which we call spatial resolution), and the ratio of main lobe to side lobe decides the signal to noise ratio of the array. The spatial resolution improves as kd increases, that is, in proportion to the ratio of the array length to the wavelength. However, when kd ¼ 2p; a peak of the same strength as the true source appears due to a spatial aliasing at u ¼ 908; which occurs when d . l=2; where l is the wavelength. Thus, the acoustic frequency, f ; is restricted by f , c0 =2d; to avoid aliasing.

W(q) 1.2

PN (t)

0.8

P2 (t)

0.6

P1 (t)

d

P0 (t)

0.4 θ

P–1 (t)

0.2 0

P–2 (t)

−0.2

P– N (t) FIGURE 40.9

kd=2π kd=π kd=π/2

1

−0.4

0

30 60 Incident angle q (degree)

90

Directivity patterns of a linear array for different values of the product kd ðM ¼ 2N þ 1 ¼ 9Þ:

40 8


Pantographs

10 dB

Sound pressure level [dB]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Leading car

0

1

2

3 4 Time (s)

5

6

7

FIGURE 40.10 Time history of sound pressure level for a passing train measured with a linear microphone array located at a point 25 m away from a track (train velocity ¼ 285 km/h).

In the above case of the linear array, the directivity exists only in the direction of the array (one dimensional). If microphones are arranged in a two dimensional plane, a two dimensional directivity can be obtained. Recently, many microphone arrangements have been proposed that obtain better spatial resolution and to reduce side lobes [12 15].

40.3.3

Applications

Microphone arrays have been used for identification of the noise source in various situations, for example, in wind tunnel tests. Many actual examples can be found in published literature [10,16 18]. The measurement with a microphone array has the advantage of much shorter measuring time than that of a mirror microphone system. Furthermore, the lower frequency limit is not so serious because the size of the apparatus can be easily extended. Another fundamental example is given now. Nine microphones are arranged equally spaced by d ¼ l=2 for each one third octave band, and their signals are summed without time delay. In this case, the array is focused to a fixed direction, perpendicular to the array axis. Figure 40.10 shows a time history of noise generated by a high speed train, measured with a linear microphone array located at a point 25 m away from the track. It is found that pantographs, the leading car, and gaps between cars are the main noise sources in this example.

References 1. 2. 3. 4.

Ferris, H.G., Computation of far field radiation patterns by use of a general integral solution to the time independent scalar wave equation, J. Acoust. Soc. Am., 41, 1967. Maynard, J.D., Nearfield acoustic holography: theory of generalized holography and the development of NAH, J. Acoust. Soc. Am., 78, 1985. Ginn, K. and Hald, J., The effect of bandwidth on spatial transformation of sound field measurements, Inter Noise, 87, 1987. Fahy, F.J., Measurement of acoustic intensity using the cross spectral density of two microphone signals, J. Acoust. Soc. Am., 62, 1977.

Instrumentation

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

40 9

Chung, J.Y., Cross spectral method of measuring acoustic intensity without error caused by instrument phase mismatch, J. Acoust. Soc. Am., 64, 1978. Grosche, F.R., Stiewitt, H., and Binder, B., On aero acoustic measurements in wind tunnels by means of a highly directional microphone system, Paper AIAA 76 535, 1976. Sen, R., Interpretation of acoustic source maps made with an elliptic mirror directional microphone system, Paper AIAA 96 1712, 1996. Blackner, A.M. and Davis, C.M., Airframe noise source identification using elliptical mirror measurement techniques, Inter Noise 95, 1995. Dobrzynski, W., Airframe noise studies on wings with deployed high lift devices, Paper AIAA 98 2337, 1998. Dobrzynski, W., Research into landing gear airframe noise reduction, Paper AIAA 2002 2409, 2002. Johnson, D.H. and Dudgeon, D.E. 1993. Array Signal Processing, Prentice Hall, Englewood Cliffs, NJ. Elias, G., Source localization with a two dimensional focused array: optimal signal processing for a cross shaped array, Inter Noise 95, 1995. Dougherty, R.P. and Stoker, R.W., Sidelobe suppression for phased array aeroacoustic measurements, Paper AIAA 98 2242, 1998. Nordborg, A., Optimum array microphone configuration, Inter Noise 2000, 2000. Hald, J. and Christensen, J.J., A class of optimal broad band phased array geometries designed for easy construction, Inter Noise 2002, 2002. Piet, J.F. and Elias, G., Airframe noise source localization using a microphone array, Paper AIAA 97 1643, 1997. Hayes, J.A., Airframe noise characteristics of a 4.7% scale DC 10 model, Paper AIAA 97 1594, 1997. Stoker, R.W., Underbrink, J.R., and Neubert, G.R., Investigation of airframe noise in pressurized wind tunnels, Paper AIAA 2001 2107, 2001.

41

Source of Noise S. Akishita

41.1 41.2

Ritsumeikan University

Introduction ......................................................................... 41 1 Radiation of Sound ............................................................. 41 1 Point Source † Sources of Finite Volume † Radiation from a Plane Surface † Estimation of Noise Source Sound Power

Summary In this chapter, a mathematical description of sound radiation is briefly presented accompanied by an introduction to sound sources, monopole, and dipole. The modeling of a simple source of noise is discussed, introducing Green’s function. As an example of simple sound radiation, the sound field generated by a source embedded in a plane surface is described. Finally, an estimation of noise source sound power is presented by introducing the power conversion factor of actual machinery.

41.1

Introduction

A careful examination of noise measurement data reveals that there exists a very wide variety of noise sources. Each source is peculiar to its generation mechanism, which can be any of a wide range of phenomena including fluid mechanics and the vibration of structures. However, in analysis, sources are normally simplified to rather simple and typical models in their generation mechanism. The vibration of a solid body, which may be in contact with the fluid medium, generates sound waves or vibratory forces acting directly on a fluid, will result in the emission of acoustic energy in the medium. In the next section, an expression for an idealized sound source is introduced. We will assume that the fluid medium outside the source region is initially uniform and at rest. Also, we will concentrate on wave propagation in an infinite medium. Generally, acoustic waves sensed as a sound represent a very small energy density in the medium. Only a very small fraction of the mechanical energy of a source body is converted into acoustic energy. The conversion factor, defined as the ratio of sound power to the mechanical power of the source, is in the order of 10 7 to 10 5. Some examples of estimated sound power conversion factor are given for typical common noise sources.

41.2

Radiation of Sound

41.2.1

Point Source

41.2.1.1

Simple Source: Spherical Wave by a Monopole

Propagation of sound pressure wave pðx; y; z; tÞ is described by the following partial differential equation for a medium where a field point is expressed by orthogonal coordinate system O xyz; as shown

41 1

41 2


in Figure 41.1(a):

7 2p 2

7 ;

1 ›2 p ¼ 0; c2 ›t 2

›2 ›2 ›2 þ 2 þ 2 2 ›x ›y ›z

!

ð41:1Þ

where c denotes the velocity of sound propagation. If the source region is compact and the generating motion has no preferred direction, it will produce a wave, which spreads spherically outwards. As the medium is assumed infinite in extent, the wave form will depend on the distance, r; from the center of the source. The wave equation in this case is 1 › 2 ›p r ›r r 2 ›r

1 ›2 p ¼0 c2 ›t 2

ð41:2Þ

When a monopole source of angular frequency v is assumed, the simplest solution for the outward propagating waveform is expressed as pðr; tÞ ¼ pv ðrÞe pv ðrÞ ¼

i vt

;

vr S eikr ; 4pr v

k¼

v 2p ¼ c l

ð41:3Þ

where r denotes the density of the medium and k denotes the wave number. Here, pv is used to denote the sinusoidal component of the sound pressure with angular frequency, v: The subscript FIGURE 41.1 Directivity of monopole and dipole, v on a variable typically indicates the sinusoidal (a) Spherical sound field by a monopole; (b) Axi component of a variable, but the variables related symmetric sound field by a dipole. to sound energy and sound power, such as w; I; W, do not have the subscript v even if they mean the sinusoidal component. In this case, the source of sound is taken as a “pulsating globe” of radius a and radial velocity Uv on the surface. Therefore, the flow outward from the origin, Sv ; is related to Uv as follows, and as shown in Figure 41.1(a): Sv ¼ 4pa2 Uv

ð41:4Þ

We should note that, while a pulsating globe with a finite radius a is assumed as the physical sound source, the sound field by a monopole with infinitesimal small size and finite magnitude, as expressed mathematically in Equation 41.3, is used. The other quantities related to the spherical wave are described next [1]: 8 1 > > ðikr 1ÞSv eikðr ctÞ ; radial velocity ur v ¼ > > 4 p r2 > > > > > 1 2 1 > > ; energy density w¼r lSv l2 ðkrÞ2 þ > < 4pr2 2 ð41:5Þ > k 2 lpl2 > 2 > I ¼ rc lSv l ¼ ; energy flux intensity > > > rc 4pr > > > > 2 > > : W ¼ ð4pr 2 ÞI ¼ rc p lSv l2 ¼ rv lSv l2 ; total power 2 4pc l

Source of Noise

41 3

We should note that the first term in the parentheses, kr; is negligible in the region where lkrl ,, 1 is valid. Hence, w¼

r lS l=4pr2 2 v

2

¼

r 2 u ¼ 2 rv

I l v 4pr 2

is reduced. Conversely, when lkrl .. 1 is valid, then w ¼ rðk=4prÞ2 lSv l2 ¼ I=c is valid. 41.2.1.2

Simple Source: Plane Wave by an Alternating Piston

Another example of simple sound wave is generated in the one dimensional field of fluid medium, as shown in Figure 41.1. Let us set the coordinate x along the axis of wave propagation, for example, the axis of duct with a constant cross sectional area. Then the wave equation is

›2 p ›x2

1 ›2 p ¼0 c 2 ›t 2

ð41:6Þ

The solution for a periodic source is given by pðx; tÞ ¼ pv ðxÞe

i vt

;

pv ðxÞ ¼ rcuv eikx

ð41:7Þ

This is known as a plane wave, which is generated by the piston motion at the origin, the velocity of which is expressed by uðtÞ ¼ uv ðxÞe

ivt

ð41:8Þ

The other quantities related with the plane wave are given below: 8 > u ðxÞ ¼ uv eikx ; > < v w ¼ I ¼ rcluv l2 ¼ lpv l2 =rc2 ; > > : W ¼ SI ðS; cross sectional areaÞ;

particle velocity energy density; sound intensity

ð41:9Þ

total power

A plane wave is generated in very limited situations, but its utility is rather wide since the sound wave propagating through a duct or duct like space with a gradually varying cross section is approximated as the plane wave. Network theory is applied to the sound wave propagating through a branch and junction by using the description of a plane wave. 41.2.1.3

Dipole and Multipoles and Their Sound Field

Let us return to the three dimensional sound field. The second simple solution to Equation 41.1 is the “dipole” sound field. Suppose that a pair of monopoles, close together, opposite in sign, and equal in magnitude, Sv ; are located along the x axis as shown in Figure 41.1(b). Since only a preferred direction is assigned along the x axis, the sound field is axisymmetric as represented by pðr; u; tÞ ¼

k2 Dv

rc cos u i 1þ e 4pr kr

iðvt krÞ

ð41:10Þ

Dv is defined by Dv ¼ Sv d

ð41:11Þ

where d denotes the separation of the monopoles as shown in Figure 41.1(b). Mathematically, d tends to zero, keeping Dv finite, and the preferred axis is the x axis. Physically, a sound field is commonly realized by a pair of monopoles with a finite separation that is short compared with the wavelength, l; as illustrated next with realistic examples.

41 4


The characteristic quantities relating with dipole sound field are described below [1]: 8 > k2 Dv cos u 2i 2 ivtþkr > > ur v ¼ ; radial velocity 1þ > 2 r2 e > kr k 4 p r > > > > > k2 Dv sin u i > > ¼ i 1þ e ivtþkr; peripheral velocity u > uv 2 > 4pr kr > > > > !2 > < k2 lDv l 1 1 2 1 1 4 w¼r cos2 u þ þ ð1 þ 3 cos2 uÞ ; energy density > 4p r 2 kr 2 kr > > > !2 > > > k2 lDv l > > sound intensity I ¼ rc cos2 u; Iu ¼ 0; > > > r 4pr > > > > > > rv4 > :W ¼ lD l2 ; total power 12pc3 v

ð41:12Þ

We should add the following notes on the dipole sound field. When lkrl .. 1 is assumed, the second term in parentheses in Equation 41.10 is negligible. Then the directivity for pðr; u; tÞ is expressed by cos u: A similar directivity is found on Ir and on urv ; uuv ; and w with the assumption lkrl .. 1: A pair of dipoles produces a quadrupole, a pair of quadrupoles produce an octopole, and so on. These are called multipole in general. Out of multipoles, the quadrupole is common in representing a sound field generated by mixing fluid flow, especially jet flow. More details on multipoles are found in Ref. [1].

41.2.2

Sources of Finite Volume

41.2.2.1

Description of Sound Field by Green’s Function

In order to describe the sound field from distributed sources, source terms are introduced to the right side of Equation 41.1. The partial differential equation with source term is derived from the equation system representing the dynamics of fluid flow with periodic motion at angular frequency v:

7 2 pv þ k 2 pv ¼

mv þ div Fv

ð41:13Þ

where pv denotes the acoustic pressure amplitude according to pðx; y; z; tÞ ¼ pv ðx; y; zÞe ivt ; mv denotes the effective monopole source density expressed by ivrsv ; sv denotes the generalization of the point source strength, Sv ; of Equation 41.4 for a distributed source, and Fv denotes the vector representation of point force density in the fluid. Introduction of Green’s function, gv ; of angular frequency, v; satisfying the following equation is useful in general:

7 2 g v þ k2 g v ¼

dðx

x0 Þdðy

y0 Þdðz

z0 Þeivt0

ð41:14Þ

Here, dðzÞ denotes the Dirac impulse (delta) function of the variable z; the coordinate ðx0 ; y0 ; z0 Þ denotes the position of unit source, r0 ; with periodic angular velocity, v; and t0 denotes the time pertaining to the source. The solution of Equation 41.14 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ikr e ; lr r0 l ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 ð41:15Þ gv ðr; r0 Þ ¼ 4pR where r denotes the position vector of sound field with coordinates ðx; y; zÞ: The solution of Equation 41.13 is described by using gv ðr; r0 Þ as follows: pv ðrÞ ¼

ððð

½mv ðr0 Þ

div Fv ðr0 Þ gv ðr; r0 Þdx0 dy0 dz0

ð41:16Þ

Source of Noise

41 5

FIGURE 41.2

41.2.2.2

Practical monopole sources.

Radiation from Vibrating Small Body

A pulsating globe, as the simplest sound source, is rarely realized in the real world. The approximated monopole source required in most measurements is an eight sided polyhedron, as depicted in Figure 41.2, where a loud speaker is installed on each of the surfaces. The sound field radiated by a thus approximated source is almost the same as that by a pulsating globe at the far field, where kr .. 1 is valid. In this case, the sound field is represented by Equation 41.16, assuming that Fv ðr0 Þ ¼ 0; mv ðr0 Þ ¼ ivrUv dðnÞ; and dx0 dy0 dz0 ¼ dndS; where the source mv ðr0 Þ is distributed on a thin layer of thickness, dn; on the spherical surface element, dS: Substitution of these relationships into Equation 41.16 yields the formula pv ðrÞ ¼ ¼

ð ð ivrU v ikR e dS S 4pR ivr ð4pa2 ÞUv eikR0 4pR0

A

Wavefronts

B

ð41:17Þ

ÐÐ FIGURE 41.3 Field of a pulsating spherical source. where S dS means integration on the approxi matelypspherical surface, and then R ; lr r l ø 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ¼ x2 þ y2 þ z 2 is applied. The final reduction of Equation 41.17 gives the same expression as is deduced from Equation 41.3 and Equation 41.4. The sound field generated by a monopole source is illustrated in Figure 41.1. This field is unchanged with the presence of a rigid plane AB in Figure 41.3. It is clear, on account of symmetry, that the presence of the rigid plane does not alter the sound field in any way, because only the tangential component of particle velocity is induced on the plane. This utilization of the symmetry and construction of the semi infinite field is applied to the expression of the sound field generated by the baffled structure, as shown in Figure 41.4. Let us imagine a sound field generated by an oscillating small body in an infinite medium, as shown in Figure 41.4(a). Since the oscillation is caused by an external force, the sound source is modeled by the force Fv ðr0 Þ: Therefore, the sound field is described by Equation 41.16, assuming mv ðr0 Þ ¼ 0: After applying a law of vector analysis, such as gv ðr; r0 Þdiv Fv ðr0 Þ·grad0 gv þ div0 ðgv Fv Þ; the sound field is

41 6


FIGURE 41.4 along x axis.

Dipole field generated by an oscillating small body: (a) oscillating small body; (b) oscillating sphere

represented by pv ðrÞ ¼

ððð

Fv ðr0 Þ·7 0 gv ðr; r0 Þdx0 dy0 dz0

ð41:18Þ

ÐÐÐ ÐÐ where div0 ðgv Fv Þdx0 dy0 dz0 ¼ S gv Fv dSn ðx0 ; y0 ; z0 Þ ¼ 0 is applied. Assume that a small sphere of radius a0 is oscillating along the x axis with angular frequency v: Instead of an oscillating sphere with velocity Ut ¼ Uv e ivt ; the sound field is generated by the concentrated body force, FðtÞ ¼ mðd=dtÞUðtÞ ¼ ivmUv e ivt ; (m is the mass of the sphere), at the origin, r0 ¼ 0: Then Fv ðr0 Þ is expressed by Fv ðr0 Þ ¼ ðFxv iÞdðx0 Þdðy0 Þdðz0 Þ; Fxv ¼ imvUv : This approximate reduction is appropriate when ka0 ,, 1 is valid. Substituting the approximation, Equation 41.18 is rewritten as pv ðrÞ ø Fxv

› g ðr; r0 ; uÞ ¼ ›x0 v

k2 Dv

Dv ¼

i F krc vx

rc i ikr cos u 1 þ e 4pr kr

ð41:19aÞ ð41:19bÞ

As the sound field is axisymmetric about x axis, the sound pressure, pv ðr; uÞ; depends only on r and u; where r and u are defined in Figure 41.4(b). The expression is applicable to the sound field generated by an oscillating small body in a free space.

41.2.3

Radiation from a Plane Surface

41.2.3.1

Radiation from a Small Body in Infinite Plane Surface

The introduction of an infinite rigid plane surface to the sound field, as shown in Figure 41.3, simplifies the formulation of the sound field generated by an oscillating body adjacent to a large plane.

Source of Noise

41 7

The configuration discussed in this section relates to a source in the presence of an infinite plane barrier, so that the medium is confined to one side of the plane. By taking the effect of the image caused by the rigid plane surface, the Green’s function, gv ðr; r0 Þ; in this case simplifies as given below, to what is called Rayleigh’s formula [2]: 1 ikr e ; R ¼ lr r0 l ð41:20Þ gv ðr; r0 Þ ¼ 2p R where r0 denotes the projection position of the source on the surface. Therefore, Equation 41.16 is rewritten as follows: ð ð ð eikR pv ðrÞ ¼ ½m ðr Þ div Fv ðr0 Þ dx0 dy0 dz0 ð41:21Þ 2pR v 0 41.2.3.2

Radiation from a Circular Piston

Let us consider the sound field generated by a rigid circular piston of radius a mounted flush with the surface of an infinite baffle and vibrating with simple harmonic motion of angular frequency, v: The solution of this example is applicable to a number of related problems, including the radiation from the open end of a flanged organ pipe. The coordinate system is shown in Figure 41.5, where the infinite baffle and the piston are placed in the ! Oxy plane. Note that the observation point, P, is denoted by r ¼ OP ; while the source point, Q, is ! denoted by r0 ¼ OQ: Since sound is observed only in the semi infinite plane where z . 0 is valid, only mv ðr0 Þ remains nonzero in Equation 41.21. As the velocity of the piston is denoted by Uv e ivt along the z axis, mv ðr0 Þ ¼ ivrUv dðz0 Þ; is distributed only on the circular piston. Finally, Equation 41.21 is rewritten as pv ðrÞ ¼

ivrUv ð ð eikR dx0 dy0 2p S R

ð41:22Þ

z

P (x,y,z) q r

Z R

S a

j

0

P′(x,y,0) x

r0 Q (x0,y0,0) dS y FIGURE 41.5

The rigid circular piston and the coordinate system.

41 8


Assuming a ,, r; the following approximation can be made: ðx0 cos w þ y0 sin wÞ sin u;

Rør 1 1 ø R r

8

ð41:23Þ

where w and u denote the angles defining the observation point, P, as shown in Figure 41.5. By changing from Cartesian coordinates, x0 ; y0 ; to polar coordinates, r; w0 ; such that x0 ¼ r cos w0 ; y0 ¼ r sin w0 in the integral above, we rewrite Equation 41.22 as ivrUv ikr ð2p pv ðrÞ ¼ dw0 e 2pR 0 ða exp½ kr cosðw0 wÞ sin u rdr 0

6

4

2

−2

−1

1

ð41:24Þ

λ/∆ = 10 λ /∆ = 2

The integration is performed by introducing the Bessel function of the first order, J1 ðzÞ as follows: pv ðrÞ ¼

FIGURE 41.6 Directivity of sound intensity generated by an oscillating piston.

ivr eikr f ðuÞ; 2pr v 2

fv ðuÞ ¼ pa Uv

2

2J1 ðka sin uÞ ka sin u

ð41:25Þ

Note that dependency on w has disappeared in the integration process, which follows from the axisymmetry of the sound field. The corresponding intensity at r is given by Ir ðuÞ ø

lpv l 2J1 ðka sin uÞ rcUv2 a2 ¼ ðkaÞ2 rc 4r2 ka sin u

2

ð41:26Þ

Figure 41.6 illustrates the dependency of Ir on u for two cases of ka: Note that, for the smaller ka ¼ 2pa=l; Ir is almost independent on u, which is similar to the dependence of the monopole. 41.2.3.3

Radiation from a Rectangular Plate

An normal velocity distribution, uv ðx0 ; y0 Þ; is prescribed over a baffled planar radiator located in the plane z0 ¼ 0 in the region Lx # x0 # Lx ; Ly # y0 # Ly ; as shown in Figure 41.7. In this case, the sound pressure field is represented by the following equation, similar to the previous section: pv ðrÞ ¼

ðLx U ðx ; y ÞeikR ivr ðLy v 0 0 dx0 dy0 2p R Ly Lx

ð41:27Þ

Assuming 2Lx ; 2Ly > > 2>
a =m pÞ2 1 ½ðb=npÞ2 ½ð > > > :

ð41:32Þ 9 > > > > = 1 > > > > ;

ð41:33Þ

41 10


where cosða=2Þ is used when m is an odd integer, and sinðb=2Þ is used when m is an even integer; cosðb=2Þ is used for even n and sinðb=2Þ for odd n:

41.2.4

Estimation of Noise-Source Sound Power

41.2.4.1

Power Conversion Factor of Machinery

It is often necessary to estimate the expected sound power that a particular machine might introduce into an environment [4]. One way where such an estimate may be approached for a particular class of machine is by means of the sound power conversion factor, hn : This factor is defined as P ð41:34Þ hn ¼ Pm where P ¼ sound power of the machine (W), and Pm ¼ power of the machine (W). This relationship is valid for both mechanical and electrical machinery. The conversion factors for some common noise sources are given in Table 41.1.

Example 41.1 Estimate the sound power level of a typical 1 kW electric motor that operates at 1200 rpm.

Solution From Table 41.1, we find that for typical electric motors, hn ¼ 1 £ 10 7 : Thus, using Equation 41.34, we obtain P ¼ hn Pm ¼ ð1 £ 10 7 Þ £ 1000 ¼ 10 4 ðWÞ as the total sound power of motor. Then using Equation 37.7, LW is given by ! 10 4 ¼ 80ðdBÞ LW ¼ 10 log 10 12 41.2.4.2

Fan Noise

We are familiar with noise nuisance caused by a domestic ventilating fan. The mechanical power of the fan is expressed by Pm ¼ pT Q ð41:35Þ where pT denotes the total pressure rise through the fan and Q denotes the volumetric flow rate. According to the law of sixth power of flow velocity deduced by the aeroacoustics theory, sound power of the fan is proportional to pT2 5 Q: Therefore, the specific ratio kT defined below is more useful than hn for the fan: kT ¼ TABLE 41.1

Pm pT 2 5 Q

ð41:36Þ

Estimated Sound Power Conversion Factors for Common Noise Sourcesa

Noise Source

Conversion Factor Low

Compressor, air (1 100 hp) Gear trains Loud speakers Motors, diesel Motors, electric (1200 rpm) Pumps, over 1600 rpm Pumps, under 1600 rpm Turbines, gas

3 £ 10 1.5 £ 10 3 £ 10 2 £ 10 1 £ 10 3.5 £ 10 1.1 £ 10 2 £ 10

Midrange 7 8 2 7 8 6 6 6

5.3 £ 10 5 £ 10 5 £ 10 5 £ 10 1 £ 10 1.4 £ 10 4.4 £ 10 5 £ 10

7 7 2 7 7 5 6 6

High 1 £ 10 1.5 £ 10 1 £ 10 2.5 £ 10 3 £ 10 5 £ 10 1.6 £ 10 5 £ 10

6 6 1 6 7 5 5 5

a Total sound power for the four octave bands from 500 to 4000 Hz. Source: Irwin, J. D. and Graf, E. R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood Cliffs, NJ.

Source of Noise

41 11 TABLE 41.2

Specific Fan Noise Level for Low Pressure Fans

Type

KT (dB)

Axial Centrifugal/cirroco Centrifugal/radial Centrifugal/turbo Cross flow

287.5 to 270.7 287.5 to 285.7 298.7 to 284.7 2104.7 to 289.7 276.7 to 268.7

Furthermore, usually the specific fan noise level, KT ; as given below is more useful than kT : KT ¼ LW

10 logðpT2 5 QÞdB

ð41:37Þ 3

where pT denotes the total pressure rise in mmAg, Q denotes the flow rate in m /min, LW denotes the total sound power level in dB, and KT represents the radiation efficiency of the fan noise. This efficiency varies with the type of the fan and with the flow rate when the model is assigned. In particular, KT will be the lowest at the flow rate at which the aerodynamic power efficiency is the highest. Therefore, we will have the best advantage on the sound environment when the fan is operated at the highest efficiency. For convenience of design, the LW is often evaluated by A weighted total sound level (dB A) not by linear total level (dB). Table 41.2 gives the specific fan noise level evaluated with dB A for five types of the low pressure fans.

Example 41.2 Consider a ventilating axial flow fan, the specifications of which are: pT ¼ 10 mmAq; Q ¼ 30 m3 =min; D ¼ 30 cm (diameter of the duct containing fan rotor). Estimate the directional distribution of intensity Ir ðuÞ; assuming r ¼ 3 m and f ¼ 150 Hz for the main component of the fan noise, and KT ¼ 79:0 dB A from Table 41.2. The fan noise is assumed to be radiated as a plane sound wave at the mouth of the duct.

Solution First, we modify the KT in dB A to that of linear scale. From the frequency response for A weighting network shown in Figure 2.5, for f ¼ 150 Hz; the modification is found as DKT ¼ 15 dB: Then, the modified specific fan noise level is KT ¼ 79:0 þ 15:0 ¼ 64:0 dB: By using Equation 41.37, LW ¼ KT þ 25 log PT þ 10 log Q ¼ 64 þ 25 þ 15 ¼ 24 dB is obtained. This means the emitted total sound power is W ¼ 10 24=10 ¼ 3:98 £ 10 3 W: Since we assume a plane sound wave at the mouth of the duct for the fan noise, we can use the sound radiation model of a circular piston with a ¼ 15 cm for the radiated sound wave. In our case ka ¼ 2pa=ðc=f Þ ¼ 0:415; or l=a ¼ 15:1: A monopole model will be valid for the directional distribution of intensity from Figure 41.6. Therefore, Ir ¼ W=2pr2 ¼ 0:704 £ 10 4 W=m2 or lpl2 ¼ Ir rc ¼ 2:92 £ 10 2 ðPa2 Þ: This is the same as the sound pressure level Lp ¼ 10 loglpl2 =pref ¼ 78:6 dB or Lp ðAÞ ¼ Lp 15 ¼ 63:6 dB A:

References 1. Morse, P.M. and Ingard, K.U. 1986. Theoretical Acoustics, Princeton University Press, Princeton, NJ, chap. 7. 2. Junger, M.C. and Feit, D. 1986. Sound, Structures, and Their Interaction, 2nd ed., MIT Press, Cambridge, chap. 4. 3. Fahy, F. 1985. Sound and Structural Vibration. Radiation, Transmission and Response, Academic Press, New York, chap. 2. 4. Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood Cliffs, NJ, chap. 5.

42

Design of Absorption 42.1 42.2

Introduction ....................................................................... 42 1 Fundamentals of Sound Absorption ................................ 42 2

42.3

Sound Absorbing Materials .............................................. 42 3

42.4

Acoustic Characteristic Computation of Compound Wall ................................................................. 42 6

Attenuation of Sound

Porous Material † Tubular Material † Membrane Material † Perforated Plate † Acoustic Resonator

Absorption Coefficient of Combined Plate with Porous Blanket † Transmission Loss through a Single Porous Board † Transmission Loss through a Sandwich Board

Teruo Obata Teikyo University

42.5

Attenuation of Lined Ducts .............................................. 42 10

42.6

Attenuation of Dissipative Mufflers ................................. 42 12

Computation of Attenuation in a Lined Duct † Attenuation in a Lined Bend † Attenuation in Splitter Lined Duct Transmission Loss of Lined Expansion Chamber Transmission Loss of a Plenum Chamber

†

42.7

General Considerations ..................................................... 42 15

42.8

Practical Example of Dissipative Muffler ......................... 42 17

Surface Treatment with Lining of Acoustic Material † Gas Flow Velocity † Gas Temperature † Dust and Water Exposure

Summary This chapter presents the basics of designing devices for sound absorption. The absorption coefficient and acoustic impedance are introduced. Characteristic properties and parameters of sound absorption material and basic elements are presented. Acoustic modeling, analysis, and design considerations of components, such as ducts, and noise attenuation devices, such as mufflers, are presented. A practical design example is presented for illustration of the concepts presented in the chapter.

42.1

Introduction

Sound absorption equipment is used for multiple purposes in architectural acoustics, mechanical noise countermeasures, and so on. In this context, the necessity for designing sound absorption equipment from the viewpoint of noise control is explained. Architectural acoustics is an important area of study, which involves architecture, sound absorption design, and sound measurement facility. In the area of noise reduction, the characteristics of sound are important, and proper sound absorbing material should be selected based on how much attenuation is necessary for each frequency of sound. In particular, acoustic characteristics of sound absorbing material such as the type of material and the

42 1

42 2


sound absorption mechanism are important. In this chapter, the basics of sound absorption are given, and the prediction and calculation methods for attenuation of lined or dissipative mufflers are outlined.

42.2

Fundamentals of Sound Absorption

42.2.1

Attenuation of Sound

When an acoustic wave propagates in a medium, the sound energy attenuates due to such reasons as viscosity, heat conduction, and the effects of molecular absorption. In a medium of small volume surrounded by a boundary surface, the attenuation is particularly considerable, for example, when the medium is a thin tube. This is because there is the dissipation of the energy controlled by the viscosity of the medium and heat conduction between the material and the medium of tube wall. A sound absorbing material may be utilized to adjust such dissipation of acoustic energy. 42.2.1.1

Absorption Coefficient and Normal Acoustic Impedance

Some amount of energy is lost when an acoustic wave hits the surface of a sound absorbing material. Figure 42.1 illustrates an infinite medium of absorbing material separated by air and the reflected wave (sound pressure pr) from the boundary surface with the air where a plane wave of sound pressure pi is emitted in the direction indicated by an arrow, at an angle u: When u ¼ 0; sound pressure p in air is given by p ¼ pi þ pr ¼ ðAe

jkx

þ Bejkx Þejvt

ð42:1Þ

where A; B ¼ the amplitude of sound pressure of inci dent and reflected waves (in Pa), pffiffiffiffi j¼ 1; k ¼ 2pf =c; wave number (1/m), v ¼ angular frequency (rad/sec). The sound pressure, pm ; in the absorbing material may be expressed using a complex propagation constant, by the equation:

FIGURE 42.1 Plane wave incidence on an infinite absorbing material.

pm ¼ px¼0 e

g x j vt

e

ð42:2Þ

where

g ¼ the propagation constant in the absorbing material (m 1). Note: g ¼ d þ jb: g is a property of the material itself and is not dependent on the mounting conditions when large areas of material are considered. d ¼ attenuation constant. Note: d tells us how much of the sound wave will be reduced as it travels through the material. b ¼ phase constant. Note: b is a measure of the velocity of propagation of the sound wave through the material. The relation for determining the velocity of sound in the material is given by cm ¼ v=b

ð42:3Þ

Design of Absorption

42 3

Boundary conditions must be satisfied on the boundary surface. The acoustic impedance of a unit area of air and of absorbing material are, respectively, denoted by z and za : The pressure and the particle velocity on both sides of the boundary are equal. We have 9 pi þ pr ¼ px¼0 > = ð42:4Þ pi pr px 0 > ¼ ; za z The amplitude of reflectance of sound pressure, r; is obtained from Equation 42.4, and is given by r¼

pr z z ¼ a pi za þ z

ð42:5Þ

The reflectivity is the energy reflection rate. The absorption coefficient, a; of an absorbing material is defined as

a¼1

lr2 l

ð42:6Þ

The impedance, zn ; through a surface is the quantity that represents the dissipation of energy of sound as well as the absorption coefficient. It is given as a ratio between sound pressure and particle velocity on boundary surface in the reflecting acoustic wave: zn ¼

p u

x¼0

pi þ pr rc cos u pi pr

¼

ð42:7Þ

Note that zn is a complex quantity and involves both amplitude and phase, both of which depend on the sound pressure at the boundary surface in the reflecting acoustic wave. In the case of oblique incidence, the surface impedance can be expressed by following equation: zn ¼ Z gz=q

ð42:8Þ

where z ¼ the acoustic impedance (Pa sec/m3). Here, Z¼

zl coshðqlÞ þ ðgz=qÞsinhðqlÞ zl sinhðqlÞ þ ðgz=qÞcoshðqlÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ g2 þ k2 sin2 u

The absorption coefficient, aðuÞ; for an oblique incidence with angle u may be expressed by

aðuÞ ¼ 1

42.3 42.3.1

zn cos u rc zn cos u þ rc

2

ð42:9Þ

Sound-Absorbing Materials Porous Material

Porous acoustical materials are a special category of a more general class of gas solid mixtures. They range from porous solids, for example, porous rocks, fibrous granular solids, expanded plastics, and form materials, to porous or turbid gases, for example, suspensions and emulsions. Sound is attenuated in a gas saturated porous solid due to the restriction on the gas movement within it. A convenient microstructure model for such materials is one of a rigid solid matrix through which run cylindrical, capillary pores (tubing) with constant radius, normal to its surface. This model enables the use of Kirchoff ’s theory of sound propagation in narrow tubes with rigid walls. Accordingly, this mechanism of dissipation may be identified as (1) a viscous loss in the boundary layer at the wall of each capillary tube

42 4


associated with the relative motion between the viscous gas and the solid wall, or (2) heat conduction between compressions and rarefactions of the gas and the conducting solid walls.

42.3.2

Tubular Material

Consider the absorption of low frequency sound using the tubular absorbing material. By itself, sound absorption is not satisfactory with the tubular absorbing material. The material produces bending vibration due to an acoustic wave through it, and sound absorption occurs by the internal friction of the material. For hard plywood and gypsum boards, there is a natural frequency in the range 100 to 200 Hz, and the absorption coefficient ranges from 0.3 to 0.5. It is possible to increase the absorption coefficient by coating the board surface with fibrous absorbing material.

42.3.3

Membrane Material

For membrane material, the sound absorption mechanism makes use of resonant vibration. Hence, resonant frequency is a governing parameter. The imaginary part (the reactance term) of the acoustic impedance of a membrane gives rise to a resonance. The associated natural frequency is given by ( ! ) 1 1 1:4 £ 105 fr ¼ þ Km ð42:10Þ 2 m L where fr ¼ natural frequency (Hz) m ¼ surface density (kg/m2) L ¼ thickness of air space (m) Km ¼ board rigidity (kg/m2 sec2) The Km values of some boards are shown in Figure 42.2. The absorption coefficient is approximately 0.3 to 0.4 in the frequency range of 300 to 1000 Hz, when the thickness of the air space between the membrane and the rigid wall behind it is 50 to 100 mm.

42.3.4

Perforated Plate

A perforated board of sound absorbing material (i.e., a board with holes) is placed over a rigid wall at a fixed clearance, as shown in Figure 42.3. The sound absorption characteristics depend on the board thickness, t; the hole diameter of the perforations, d; the clearance, L; between the perforated board and the rigid wall, and so on. The absorption coefficient becomes a maximum at resonant frequency. In the present case, the resonant frequency is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 fr ¼ ð42:11Þ 2p ðt þ 0:8dÞL where sound speed is c; the airspace thickness is L (typically, 300 mm or less), and the ratio of the total area of holes to the total area of the board is 1: The absorption coefficient is approximately 0.3 to 0.4.

42.3.5

Acoustic Resonator

Yet another method of achieving sound absorption is using an acoustic resonator of Helmholtz type, which consists of a vessel of any shape containing a volume air, as shown in Figure 42.4. The air volume is in direct communication with the ambient air in the room through an interconnecting tube, which may be long or short and of any cross sectional shape. An example of a resonator of Helmholtz type may be a 1 gal jar. When a sound wave impinges on the aperture of neck of the jar, the air in the neck will be set in oscillation, periodically expanding and compressing the air in the vessel.


42 5

(a) Vinyl sheet (1) Back air space 50 mm (2) back air space 105 mm

a

0.4

0.2

(2) (1)

0 (b)

102

FIGURE 42.2

103 Frequency (Hz)

104

Some Km values for membrane absorbing materials.

The resulting amplified motion of the air particles in the neck of the jar, due to phase cancellation between the air plug in the neck and the air volume in the vessel, causes energy dissipation due to friction in and around the neck. This type of absorber can be designed to produce maximum absorption over a very narrow frequency band or even a wide frequency band. The resonant frequency of a Helmholtz resonator may be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 fr ¼ ð42:12Þ 2m ðt þ 0:8dÞL where c ¼ speed of sound (m/sec) Sn ¼ cross sectional area of neck of jar (m2) dn ¼ diameter of neck of jar (m) V ¼ volume of vessel (m3)

42 6


FIGURE 42.3 (b) plan view.

42.4

Sound absorption characteristics of a perforated plate structure: (a) cross sectional view;

Acoustic Characteristic Computation of Compound Wall

42.4.1

Absorption Coefficient of Combined Plate with Porous Blanket

A common form of problem in noise control is the need to reduce the sound radiated from a duct or some other object. A way to achieve this is by lining the duct with several centimeters of porous acoustic material, and covering it with a solid plate of some type, as indicated in Figure 42.5. Consider the case of normal incidence with the sound absorbing structure of Figure 42.5. Assume that the boundary conditions for the sound pressure and the volume flow rate are identical. For plane wave incidence on the hard wall, the magnitude of reflection coefficient is 1 [1]. The following equation is obtained: 2

1 6 6 6 1 6 6 6 6 0 4 0 2 6 6 6 6 ¼6 6 6 4

e

1

1

m1

m1

gl1

egl1

m2 e 3 1 7 7 17 7 7 7 0 7 5 0

gl1

m2 egl1

32

3 B1 76 7 76 7 0 76 A1 7 76 7 76 7 7 2jkl2 76 ð1 þ e Þ 76 B 1 7 54 5 0

ð1

e

2jkl2

Þ

B2

ð42:13Þ

FIGURE 42.4 Geometry of a Helmholtz resonator. Volume, V; is connected to an infinitely open area by a neck tube of diameter d and length ln :


42 7

where pffiffiffiffi j¼ 1 m1 ¼ z0 =z1 ; m2 ¼ z1 =z2 z0 ; z1 ; z2 : acoustic impedance of each medium (Pa sec/m3) g ¼ complex propagation constant (1/m) The absorption coefficient for normal incidence is given by the following equation:

a0 ¼ 1

B0 2 ¼1 A0

lB0 l2

ð42:14Þ

The absorption coefficient for random incidence may be approximated by n 1 X a¼ aðuÞi n i¼1

ð42:15Þ

FIGURE 42.5 Structure for sound absorption using a blanket and an air space showing angles u1 in the air and u2 in the blanket.

where u ¼ the incident angle of sound, 0 , u , p=2: It is known that the propagation speed of the sound in fibrous materials changes with air, and the following equation holds on the boundary surface: sin u=sin u 0 ¼ c=cm

ð42:16Þ

Here, cm is the sound speed in fibrous materials, which is calculated from the imaginary part of Equation 42.18, given later. The angle of reflection, u 0 ; in the boundary surface of the back air space is obtained in a similar way. Hence, the following equation is substituted in Equation 42.13 instead of the thickness of the absorber, l1 ; and the thickness of the air space, l2 ; to obtain the absorption coefficient in oblique incidence: l 01 ¼ l1 =cos u 0 ; l 02 ¼ l2 =cos u 00

ð42:17Þ

The complex propagation constant, g; is an important physical quantity in absorbing material of propagated sound, which is given per unit length of acoustic attenuations, and phase changes. Between the aeroelasticity rate, Ka ; of absorbing material and the bulk modulus, Q; of absorbing material, g is given by the following equation, for Ka . 20 Q [2,3]: pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ jv Y=K kr1 l jkR1 l=v kR1 l ¼

kr1 l ¼ r0 k

ð42:18Þ

R1 ½1 r0 ð1 YÞ=rm " r0 ðk 1Þ 2 R21 1þ 2 2 1þ rm rm v ½1 þ r0 ðk R21 ðY=k þ rm =r0 kÞ 2 2 rm v ½1 þ r0 ðk 1Þ=rm 2 1þ

where

rm ¼ density of acoustical material (kg/m3) r0 ¼ density of air (kg/m3) c0 ¼ speed of sound in air (m/sec)

r2m v2 ½1

þ

R21 þ r0 ðk

# 1Þ=rm

1 þ r0 Yðk 1 þ r0 ðk 1Þ=rm

2

2

1Þ=rm k 1Þ=rm

42 8


K ¼ volume coefficient of elasticity of air (N/m2) R1 ¼ alternating flow resistance for unit thickness of material due to the difference between the velocity of the skeleton and the velocity of air in the interstices (Pa sec/m2). R1 values are given in Table 42.1 Y ¼ porosity ¼ the ratio of the volume of the voids in the material to the total volume; porosity equals the total volume minus the fiber volume, all divided by total volume k ¼ 5:5 4:5Y; the structure factor of the interstices in the skeleton v ¼ 2pf ; the angular frequency (radians/sec) The acoustic impedance, z1 ; of absorbing material is given by z1 ¼ R þ jX ¼ in which

42.4.2

jK g vY

ð42:19Þ

o n R ¼ r0 c0 1 þ 0:0571ðr0 f =Rf Þ 0:754 n o X ¼ r0 c0 0:0870ðr0 f =Rf Þ 0:732

Transmission Loss through a Single Porous Board

Assume that a sound wave impinges on the left side of a porous board at normal incidence and emerges with a reduced amplitude from the right side. The associated transmission loss of the porous board is obtained from 9 TL0 ¼ 10 log10 ðX þ YÞ > > ( )2 > > > 2 2 > v m PRf > = X ¼ 1þ 2 2 2 2 2r0 c0 ðv m P þ Rf Þ ð42:20Þ > > ( )2 > > 2 > vmRf > > Y¼ ; 2 2 2 2 2r0 c0 ðv m P þ Rf where m ¼ surface density of the blanket (kg/m2) P ¼ porosity of the blanket (porosity ¼ the total volume minus the fiber volume, all divided by the total volume) Rf ¼ specific flow resistance of material (Pa sec/m) TABLE 42.1

Flow Resistance Values of Glass Wool Board (Quality Regulation Range by JIS)

Board Type

K value

Gross Specific Gravity (kg/m3)

#1 Glass wool board

8 12 16 20 24 12 16 20 24 32 48 64 96 96

8^2 12 ^ 2 16 ^ 2 20 ^ 3 24 ^ 3 12 ^ 2 16 ^ 2 20 ^ 3 24 ^ 3 32 ^ 4 48 ^ 5 64 ^ 6 96 ^ 10 96 ^ 10

#2 Glass wool board

#3 Glass wool board

Specific Flow Resistance ( £ 10 3 N sec/m4) 1.5 , 7.0 2.5 , 12.0 4.7 , 17.0 5.0 , 22.0 6.5 , 27.0 1.5 , 7.0 2.5 , 10.0 3.0 , 13.0 4.0 , 16.0 6.0 , 22.0 11.0 , 38.0 18.0 , 60.0 27.0 , 95.0 15.0 , 40.0

Standard of JIS for Glass Wool JIS A 9505 A

JIS A 9505 B

JIS A 9505 C


42 9

r0 ¼ density of air (kg/m3) c0 ¼ sound speed in air (m/sec)

42.4.3

Transmission Loss through a Sandwich Board

Consider a wide wall formed by two panels (sheets) of infinite area separated with a homogeneous filling of fibrous acoustical material, as shown in Figure 42.6. Suppose that a plane wave impinges at an angle u: As the pressure of both sides of the wall is equal with regard to the amplitude of the progres sing wave and the reflected wave in each boundary surface, the following result may be established [4]: 9 > > > > > > ðA0 B0 Þ=z0 ¼ ðA1 B1 Þ=z1 > > > > jkl 01 jkl 01 > > þ B1 e ¼ A2 þ B2 A1 e > > > 0 0 > jkl 1 jkl 1 = B1 e Þ=z1 ¼ ðA2 B2 Þ=z2 > ðA1 e A0 þ B0 ¼ A1 þ B1

> > > > > 0 0 > gl 2 gl 2 > ðA2 e B2 e Þ=z2 ¼ ðA3 B3 Þ=z3 > > > > > gl 03 gl 03 > A3 e þ B3 e ¼ A4 þ B4 > > > > 0 0 ; gl 2 gl 2 ðA3 e B3 e Þ=z3 ¼ ðA4 B4 Þ=z0 A2 e

gl 02

0

þ B2 egl 2 ¼ A3 þ B3

ð42:21Þ

FIGURE 42.6 panel.

Cross sectional view of a sandwich

where A and B are the amplitude of sound pressures. From Equation 42.17, l 01 ¼ l1 =cos u1 ; l 02 ¼ l2 =cos u 02 ; and l 01 and l 02 may be calculated. The speed of sound in the walls is given by the following equation in terms of the modulus of longitudinal elasticity, Ei : pffiffiffiffiffiffi ci ¼ Ei =ri

ð42:22Þ

The real part of acoustic impedance, zi (i ¼ 1; 3), is given by Ri ¼ ri =cos u; and of the imaginary part is given at Xi ¼ mi v: The internal resistances, ri ; are functions of such factors as the material, frequency, temperature, and density. Some typical values are given in Table 42.2. If the space of the transmission side is infinite, B4 in Equation 42.21 becomes equal to zero. Then, the transmission loss is given is given by TLðuÞ ¼ 10 log10

TABLE 42.2

A4 A0

2

ð42:23Þ

Internal Resistance Values of Several Useful Materials

Material

Thickness (mm)

Aluminum Plywood Plaster board

0.4 3.0 7.0

Internal Resistance (Pa sec/m3) 3.0 7.5 15.0

42 10

42.5 42.5.1


Attenuation of Lined Ducts Computation of Attenuation in a Lined Duct

A lined duct is an air passage with one or more of the interior surfaces covered with an acoustical material such as a glass or mineral fiber blanket. The parallel baffles are merely a series of side by side ducts that generally have a rectangular or round cross section. If the walls are covered with absorptive material, attenuation will occur because of the viscous motion of the air in and out of the porous of blanket. Figure 42.7 shows an isometric illustration of a lined duct. The attenuation of sound for a lined duct is dependent primarily on the duct length, le ; the thickness of the lining, b; the density of the lining, r; the width of the air passage, l; and the wavelength of sound, l: At low frequencies ðl=l , 0:1Þ; the attenuation of sound in a lined duct may be calculated from the following empirical formula: ATT ¼ Kl P=S

ð42:24Þ

where K l ¼ the coefficient, which is determined from the random incidence absorption coef ficient of lined material, given in the chart of Figure 42.8 P ¼ acoustically lined perimeter of duct (m) S ¼ cross sectional open area of duct (m2)

FIGURE 42.7

If the absorbing material is lined in the rectangular cross section as shown in Figure 42.9 to Figure 42.11, the attenuation can be estimated using the formulas given in Table 42.3 [5].

2

Attenuation in a Lined Bend

A lined bend duct is shown in Figure 42.12. The insertion loss, IL, of a lined bend results from two mechanisms: the reflection of sound back toward the source side, and the scattering of sound energy into the high frequency region is rapidly attenuated by the lining beyond the bend. Higher frequency modes will be attenuated by even an unlined duct for frequencies below the ratio of the air passage between the linings to the wavelength of sound equal to 0.5. At frequencies well above this ratio, the insertion loss of a lined bend is expected to be comparable to the reverberant field

Kl

42.5.2

Illustration of a lined duct.

1

0

0

0.5 a

1

FIGURE 42.8 Kl value for sound absorption co efficient by reverberation room method.


42 11

FIGURE 42.9

Duct liner configurations corresponding to Table 42.3.

end correction derived for the duct. The insertion loss of a lined bend may be obtained as following equation [6]: IL ¼

Kl P þ ðl1 þ l2 Þ þ F S

ð42:25Þ

120

12

110

11

100

10

90

9

80

8

70

7

60

6

50

5

40

4

30

3

20

2

10

1

0 0

FIGURE 42.10

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Absorption Coefficient a

0.8

0.9

1

n

n

where F is obtained from Figure 42.13.

0

Relationship between absorption coefficient and stationary wave factor, n:

42 12


The total insertion loss for a lined bend is given in Figure 42.13 along with the attenuation of the lining beyond the bend.

42.5.3 Attenuation in Splitter Lined Duct The use of parallel or zigzag baffle type separators (splitters) to increase the perimeter area ratio results in more compact attenuators. In rock wool blankets, the attenuation of a parallel type splitter 2 duct may be obtained directly from Figure 42.14. FIGURE 42.11 Damping function KSL as a function The peak value of the attenuation is related to of dimensionless frequency, vl=nc: wavelength of sound and the splitter interval. With the zigzag arrangement of acoustic blankets, the attenuation of high frequencies is improved over that of the parallel splitter [7].

42.6

Attenuation of Dissipative Mufflers

42.6.1

Transmission Loss of Lined Expansion Chamber

The geometry and nomenclature for a dissipative muffler are given in Figure 42.15. For f , 1:2c=D; the assumption of plane wave is acceptable where D ¼ the diameter of the muffler. The transmission loss for the light lining in the chamber may be obtained using [8,9]: TL ¼ 10 log10

TABLE 42.3 See Figure 42.7

coshðde le =2Þ þ

2 2 mþ1 mþ1 sinhðde le =2Þ cos2 kle þ sinhðde le =2Þ þ coshðde le =2Þ sin2 kle 2m 2m ð42:26Þ

Formulas for Attenuation of Several Lined Ducts Low Frequencies vl Range: ,1 nc 4:34 nly

(A)

b¼

(B)

b ¼ 4:34

(C)

b¼

(D)

b ¼ 4:34

(E)

b ¼ 8:7

(F)

b¼

1 1 þ ny l y nx l x

8:7 nly

17:4 nl

2 1 þ ny ly nx lx 1 1 þ ny ly nx lx

High Frequencies vl Range: .5 nc

Middle Frequencies Range (KySyly; see Figure 42.7)

b¼

8:7c ðK S l 2 Þ l 2y v y y y

b ¼ 21:4

c2 n v2 l3y

b¼

8:7c Ky Sy l 2y K S l2 þ x 2x x 2 ly v lx

b ¼ 21:4

c2 v2

b¼

34:7c Ky Sy l 2y 4 l 2y v

b ¼ 171

b¼

8:7c Ky Sy l 2y K S l2 þ x 2x x v l 2y lx

b ¼ 21:4

b¼

34:7c Ky Sy l 2y K S l2 þ x x2 x 4l 2y v 4l x

b¼

171c2 v2

b¼

69:5c KSl 2 4l 2 v

b¼

341c2 n v2 l3

ny n þ 3x l 3y lx

c2 ny v2 l 3y c2 v2

8ny n þ 3x l 3y lx ny n þ 3x l 3y lx

b is attenuation (dB/m), n is absorbing factor plotted in Figure 42.7, KySyly is damping function, plotted in Figure 42.8, c is sound speed, l is the width of the duct, v ¼ 2pf : angular frequency, x; y: coordinates, see Figure 42.7. Source: Bruël, P.V. 1951. Sound Insulation and Room Acoustics, Chapman & Hall, London, p.159. With permission.


42 13

FIGURE 42.12 Sketch of a typical lined bend with plane wave incidence. (Source: Beranek, L.L. Noise Reduction, McGraw Hill, 1960. With permission.)

20

F db

Plane axial wave input

10 Random input 0

0.1

0.2

0.5

1 /l

2

5

10

Z

FIGURE 42.13 Insertion loss for lined bend. (The lining must extend two to four duct widths beyond the bend for this data to be valid.) (Source: Beranek, L.L. Noise Reduction, McGraw Hill, 1960. With permission.)

200

50

d = 75 mm

a = 2b

150

20

300

10

600

5 2 1 0.1

100

a=b

50 ATT (dB/m)

ATT (dB/m)

100

d=50mm

100 200

20 10

400

5 2

0.2

0.5

1 d /λ

2

5

1

0.05 0.1 0.2

0.5

1

2

5

d /λ

FIGURE 42.14 Sound attenuation for a splitter duct. Each baffle is constructed with two sheets of perforated metal filled with mineral wool, with about 100 to 140 kg/m3 gross density; a ¼ the width of the open space, b ¼ the width of the baffle, d ¼ the center to center distance of baffles, l ¼ the wavelength of the sound.

42 14


FIGURE 42.15 Filled up Factor of Glass Wool, ag

TABLE 42.4 Vg/V ag

A dissipative muffler.

0.05 0.106

0.10 0.124

0.15 0.288

0.20 0.365

0.30 0.529

0.40 0.677

0.50 0.794

0.60 0.885

0.70 0.935

0.80 0.960

0.90 0.987

1.00 1.0

Vg ¼ Filled up volume (factors of 100 kg/m3), V ¼ Volume of chamber

in which de ¼ the attenuation per unit length for the lined duct, which is given by the following equation: 20 log10 ðde le Þ ¼

Kl Ple S

ð42:27Þ

The Kl values are obtained from the absorption coefficient, as shown earlier (see Figure 42.8). In particular, de is given by 1 de ¼ 100:05Kl Ple =S le

ð42:28Þ

where m ¼ the ratio of the area of expanded or lined sections to the area of inlet or outlet sections of muffler; k ¼ 2pf =c; and le ¼ the length of the muffler. The transmission loss for the case of a thick lining of glass wool in the chamber is obtained using the empirical formula [10] TL ¼ 10 log10 1 þ

1 a mkle 2 g

2

ð42:29Þ

where

ag ¼ the coefficient, which is obtained from Table 42.4, using the filling volume and the density of glass wool m ¼ the ratio of the area of expanded or lined sections to the area of inlet or outlet sections of muffler k ¼ 2pf =c le ¼ the length of muffler

42.6.2

Transmission Loss of a Plenum Chamber

The geometry and nomenclature for a plenum chamber are given in Figure 42.16. A plenum chamber is similar in many ways to a lined expansion chamber. The main difference is that the inlet and outlet of a plenum chamber are not located in line. Generally, there is an offset to direct transmission of sound. Sound is reflected at the square cornered bend as the cross section dimension of the duct is


42 15

sufficiently large. Particularly at high frequencies, almost all of the sound energy may reflect many times off the lined sides when propagating from the inlet to the outlet. The transmission loss of a single plenum chamber can be obtained approxi mately from [11]: TL ¼ 10 log10 Sw

cos u 1 þ 2pd2 R

ð42:30Þ

where Sw ¼ lW ¼ area of the inlet and outlet d ¼ {ðL lÞ2 þ H 2 }1=2 ¼ the slant distance from inlet to outlet cos u ¼ H=d R ¼ a=ð1 am Þ a ¼ the total lined area in chamber times absorption coefficient am ¼ the statistical absorption coefficient of the lining

42.7

FIGURE 42.16 A single plenum chamber showing the nomenclature used in Equation 42.26.

General Considerations

In order to carry out the design of noise control measures for a particular problem, we must consider not only the fundamental acoustical properties of the material as discussed before, but also such practical aspects of the problem as (1) gas flow velocities, (2) temperature of gas, (3) moisture exposure, and (4) head losses for gas flow. The client depends heavily on the expertise of the designer to realize adequate protection of the noise control equipment under operating conditions.

42.7.1

Surface Treatment with Lining of Acoustic Material

Fibrous material in the market has some form of resin binder. Comparatively long fiber flocculent and comparatively short fiber are available. The packaging density of flocculent is about 60 to 100 kg/m3. It is necessary to cover with perforated thin metal or wire netting so that an arbitrary shape may be maintained in the absorbing material. The perforated metal does not take into account the numerical aperture, hole shape, hole diameter, and metal thickness. From the acoustic viewpoint, a suitable numerical aperture is given in Table 42.5.

42.7.2

Gas Flow Velocity

Noise control problems often involve the use of an acoustical material in high velocity gas flow such as those found in the exhaust of engines or ventilating systems. Deterioration of the acoustical TABLE 42.5 Perforated Metal for Treatment of Absorbing Material (Gas Flow Velocity is 25 m/sec or Less) Perforation rate: 30 to 50% Hole diameter: 5 to 10 mm Hole shape: round, plus, slit and interminglement Metal: iron, stainless steel (used in case of the corrosive gas)

42 16


material due to high velocity gas flowing past it can be a serious problem. In addition, turbulence in the gas flow subjects the materials to vibration and can cause further deterioration. One solution to this problem is to install the acoustical material behind some form of protective facing, which will vary in complexity depending on the gas velocity. A limited amount of information on this subject is available through field experience, as shown in Figure 42.17 [12]. However, the parameters of the treatment structure are not well established, for example, those concerning perforated metal, wire net, absorbing material, and gas flow. Multiple layers are used under conditions of flow velocity exceeding 25 m/sec, and the associated performance analysis can become rather complex.

42.7.3

FIGURE 42.17 Protective surface for absorbing material subjected to high velocity gas flow.

Gas Temperature

In many noise control problems, temperature is a very important factor. Sometimes high tempera ture ducts that are radiating noise, for example, in diesel engines, and induced draft fans, must be wrapped. With a proper choice, it is possible to combine thermal and acoustical insulations using one single material. Under extremely high temperatures, the tensile strength of materials tends to decrease, and the material may be subjected to thermal shock. Examples of absorbing materials that are currently available for use where temperature is an important consideration are given in Table 42.6.

42.7.4

Dust and Water Exposure

The holes of perforated metal can be blocked if a dust treatment is not carried out, and the sound absorption performance will deteriorate with adhesion to the surface of the absorbing material. Methods of dust accumulation and removal may be designed into cavity type mufflers used on the sound absorption equipment. A fan of a cooling tower, for example, experiences a considerable amount of moisture. Precautions must be taken so that water droplets are not deposited on the sound absorbing material. The underside of the equipment should be treated with rust prevention material. Figure 42.18 shows the degradation

TABLE 42.6

Fibrous Materials of Use in Hot Gas Flows

Materials

Maximum Allowable Temperature (8C)

Glass fibers with binder Glass wool Mineral wool felts Mineral wool Asbestos fibers Alumina silica

320 , 360 960 , 1060 1160 1660 760 1900


42 17

of the acoustic characteristic of absorbing material due to moisture, using the normal incidence absorption coefficient [13].

42.8 Practical Example of Dissipative Muffler An example is given on the design of a dissipative muffler for noise reduction in an axial flow fan for a ventilation system. 1. Specification of the axial flow fan *

*

*

*

*

*

Volume flow rate: Q ¼ 125 m3/min Wind pressure: p ¼ 80 mm Aq Rotor blade number: Z ¼ 10 Stationary blade number: Zs ¼ 5 Rotational speed: N ¼ 2580 rpm Shaft horsepower: P ¼ 3.75 kW

2. The desired values of attenuation and head loss with the muffler installation follow. The FIGURE 42.18 Degradation of the absorption coeffi noise of the fan propagates both intake and cient by water content. discharge sides. A performance level (noise reduction) of about 37.5 dB is required, when specific sound level Ks is obtained on the basis of the axial flow fan specification given by K s ¼ LA

10 log10 ðp2 QÞ

ð42:31Þ

3. Figure 42.19 gives the noise spectrum for the axial flow fan. The blade passing frequency, BPF, is a fundamental component of the velocity fluctuation as the flow passes the blades. It is seen in the noise spectrum in Figure 42.19 at 430 Hz ðQ £ Z ¼ ð2580=60Þ £ 10Þ: By adding the background noise spectrum to this spectrum, it is seen that a muffler that provides an attenuation over 20 dB near 430 Hz, and about 15 dB in the frequency range of 800 to 1000 Hz is necessary. The head loss value of the muffler is to be maintained within 4 mm Aq.

FIGURE 42.19

Noise spectrum of the axial flow fan of a factory ventilation system.

42 18


FIGURE 42.20

Half cross sectional view of dissipative muffler for the axial flow fan.

4. The structure of the dissipative muffler is shown in Figure 42.20. The maximum value of outer diameter of the muffler is 750 mm, and the length chosen to optimize the performance. 5. The packing density of glass wool is chosen as 65 kg/m3. The surface treatment of glass wool uses perforated metal with 1 mm thickness, 36% open area with 6 mm hole diameter. A sound absorption body of 200 mm diameter is supported in the center part, and it is welded to the outside cylinder by three props in the flow direction, and two in the circumferential direction. 6. The attenuation characteristics of the dissipative muffler may be calculated using Equation 42.27 TL ¼ 10 log10 1 þ

1 a mkle 2 g

2

The proportion of the volume of glass wool filled into the muffler is approximately (0.752 0.552 þ 0.22)/0.752 ¼ 0.53. For a packing density of 100 kg/m3, we have 0.53 £ 65/ 100 ¼ 0.347. The value of ag < 0.6 is obtained from Table 42.2. The required expansion ratio, m; length, le ; and wave number, k; are given by m ¼ ð750=500Þ2 ¼ 2:25;

le ¼ 1:2 m;

k ¼ 2pf =c

The speed of sound c depends on the environmental temperature. For a temperature of 258C, we get 346.5 m/sec (¼ 331.5 þ 0.6 £ 25). The TL values at 100 to 1000 Hz are calculated. We have f ¼ 100 Hz; TL ¼ 5:0 dB f ¼ 430 Hz; TL ¼ 16:1 dB f ¼ 1000 Hz; TL ¼ 23:4 dB The flow velocity satisfies desired value of head loss ploss (in mm Aq), and is calculated by following empirical equation [12]: ( ) 3=4 dm 1=3 u2 0:1 le ploss ¼ 0:142mf ð42:32Þ d1 d1 g Use the numerical values as follows: *

*

*

*

*

*

mf ¼ (550/500)2 ¼ 1.21, ratio of cross sectional area between air passage and muffler. d1 ¼ 500 mm, diameter of inlet. le ¼ 1.2 m, length of muffler. dm ¼ 200 mm, diameter of absorption body. u ¼ 10 m/sec or less, flow velocity at inlet. g ¼ 9.8 m/sec2, acceleration of gravity force.


FIGURE 42.21

42 19

Noise reduction characteristics of the designed dissipative muffler.

The corresponding head loss is 3.25 mm Aq, which corresponds to JIS B 833, and nearly agrees with the predicted value. Specifically, the condition of 4 mm Aq or less of the designed value is satisfied. The connection of axial flow fan and the muffler uses vibration isolation, using the thick synthetic rubber. 7. The result of the attenuation realized from the spectrum after the muffler installation is shown in Figure 42.21. It is proven that the attenuation characteristics almost parallel the designed value. The frequency range where the approximation is valid is given by f , c=d ¼ 346:5=0:175 ¼ 1980 Hz For frequencies below 250 Hz, the estimated result of the attenuation becomes slightly overestimated.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Obata, T., Hirata, M., Nishiwaki, N., Ohnaka, I., and Kato, K., Noise reduction characteristics of dissipative mufflers, 1st report, acoustical characteristics of fibrous materials, Trans. Japan Soc. Mech. Eng., 42, 363, 3500, 1976. Zwikker, C. and Kosten, C.W. 1949. Sound Absorbing Materials, Elsevier, New York. Beranek, L.L., Acoustical properties of homogeneous isotropic rigid tiles and flexible blankets, J. Acoust. Soc. Am., 19, 4, 556, 1947. Obata, T. and Hirata, M., Estimation of acoustical transmission loss for combined walls, Proc. Japan Soc. Mech. Eng. Annu. Meet., 780, 1, 42, 1978. Bruël, P.V. 1951. Sound Insulation and Room Acoustics, Chapman & Hall, London, p. 159. Beranek, L.L. 1971. Noise and Vibration Control, McGraw Hill, New York, chap. 17, p. 390. King, A.J., Attenuation of lined ducts, J. Acoust. Soc. Am., 30, 6, 505, 1958. Davis, D.D. Jr. and Stokes, G.M., 1954. Natl. Advisory Comm. Aeronaut. Ann. Rept., 1192. Davis, D.D. Jr. 1957. Acoustical Filters and Mufflers. Handbook of Noise Control, C.M. Harris, Ed., McGraw Hill, New York, chap. 21. Hagi, S., Studies on Silencer for Ventilating System, A Doctoral Thesis of University of Tokyo, 1961. Wells, R.J., Acoustical plenum chambers, Noise Control, 4, 4, 9, 1958. Obata, T. and Hirata, M., Estimation of acoustic power of flow generated noise within silencer and head losses, J. Acoust. Soc. Japan, 34, 9, 532, 1978. Koyasu, M., Acoustical properties of fibrous materials, personal letter, RC SC35, Japan Soc. Mech. Eng. Div. Meet., 15, 1975.

43

Design of Reactive Mufflers 43.1 Introduction ........................................................................ 43 1 43.2 Fundamental Equations ...................................................... 43 2 Analytical Model

†

Boundary Conditions

43.3 Effects of Reactive Mufflers ................................................ 43 3 Insertion Loss

†

Transmission Loss

43.4 Calculation Procedure ........................................................ 43 5 43.5 Application Range of Model .............................................. 43 6

Teruo Obata Teikyo University

Condition for Approximation of Plane Wave † Effect of Temperature † Effect of Gas Flow in Pipe † Effect of Friction Loss in Pipe

43.6 Practical Example ................................................................ 43 13 Expansion Type Muffler for Reciprocating Compressor

Summary This chapter concerns the design of noise suppression devices such as mufflers. In particular, reactive mufflers that are inserted into long ducts are considered in detail. Analytical and empirical equations and information that are useful in the modeling and analysis of mufflers are presented, with an indication of their application ranges and limitations. A design procedure, complete with the necessary computations, is given. Methodologies and parameters of the performance analysis of acoustic systems with mufflers are indicated. An illustrative example of a muffler for a double acting reciprocating compressor is presented.

43.1

Introduction

In noise reduction applications, the need for a reactive muffler usually arises when transporting gas through a duct. For sound transmission through the duct to be minimized, an acoustic suppression device must be incorporated into the duct system. For example, in internal combustion engines, it is required to reduce the intake and exhaust noise to acceptable levels. This may be accomplished by inserting a muffler in the intake and exhaust ducting to attenuate the pressure pulsations before they reach the environment. A successful muffler design must satisfy at least the following three criteria: (1) muffler performance as a function of frequency (the maximum permissible noise generated by the gas flow through the muffler may have to be specified as well); (2) the maximum permissible average pressure drop through the muffler at a given temperature and mass flow; (3) the maximum allowable volume and restrictions on space utilization. The customer may ask for a muffler with unrealistically high noise attenuation, virtually no backpressure, and very small size. In addition, it is important to the customer that the muffler is inexpensive and durable, and presents no maintenance problems. Needless to say, in practice, these 43 1

43 2


criteria for muffler design are unrealistic, and have to be modified to practical levels. In this chapter, we will present some of the analytical and empirical tools that are helpful in muffler design.

43.2 43.2.1

Fundamental Equations Analytical Model

The physical behavior of a reactive muffler may be adequately modeled by linear differential equations. The law of conservation of mass must hold, while three simultaneous equations, Newton’s, Boyle Charles, and that of conservation, must be satisfied. When these equations are combined, we obtain the wave equation for the plane, one dimensional sound pressure wave:

›2 j ›2 j ¼ c2 2 2 ›t ›x › j p ¼ rc2 ›x

where

j¼ c¼ p¼ r¼ t¼ x¼

ð43:1Þ ð43:2Þ

displacement of particle motion (m) velocity of sound (m/sec) sound pressure (Pa) density of air (kg/m3) time (sec) coordinate system along which wave travels (m)

The stationary solutions for angular frequency v of Equation 43.1 and Equation 43.2 are given by

j ¼ ðA e

jkx

B e jkx Þe jvt

ð43:3Þ

p ¼ rc2 kðA e jkx þ B e jkx Þe jvt ð43:4Þ where A; B ¼ amplitudes of sound pressure or particle motion for traveling and reflecting waves, pffiffiffiffi k ¼ 2pf =c; wave number, and j ¼ 1.

43.2.2

Boundary Conditions

The boundary conditions are given below. (1) Sound source. The sound source is assumed to be independent of the existence of the muffler, and the volume rate of the particles is assumed constant, as given by Sj_ ¼ const ð43:5Þ in which · denotes the time derivative. (2) Open end of duct. The reflection coefficient, R; at the open end of an unflanged circular pipe is available, and is given by B e jkx R¼ ð43:6Þ A e jkx The magnitude of the reflection coefficient, lRl; is shown in Figure 43.1a as a function of ka; where a is the pipe radius. The phase shift can be determined from Figure 43.1b, which is a plot of a=a as a function of ka: Also, the reflection coefficient is [1]: ð43:7Þ R ¼ lRle 2jka For the small values of ka that are most often encountered in reactive muffler design, lRl < 1 and a=a ¼ 0:613: (3) Closed end. The displacement of particle motion is zero at a rigid wall. Hence, we have

j¼0

ð43:8Þ

Design of Reactive Mufflers

43 3

1

0.6

0.8

|R|

α/a

0.6

0.4

0.4 0.2 0.2

(a)

0 0.0

1.0

2.0 ka

3.0

4.0

(b)

0 0.0

1.0

2.0 ka

3.0

4.0

FIGURE 43.1 (a) Magnitude of the reflection coefficient at open end of an unflanged circular pipe; (b) End correction for an unflanged circular pipe. (Source: Levine, H. and Schwinger, J., On the radiation of sound from an unflanged circular pipe, J. Phys. Rev., 73, 383, 1948. With permission.)

(4) Junction conditions. The following equations correspond to the continuity of volume flow rate of the particles and the continuity of pressure, even if the cross section changes suddenly:

43.3

Si j_i ¼ Siþ1 j_iþ1

ð43:9Þ

pi ¼ piþ1

ð43:10Þ

Effects of Reactive Mufflers

The acoustic behavior of a reactive muffler may be expressed in term of the insertion loss, the difference in the noise levels measured at some external point with and without the muffler in the system. The transmission loss is defined as the insertion loss for a nonreflecting source and the end of exhaust duct.

43.3.1

Insertion Loss

A single expansion type muffler installation is shown schematically in Figure 43.2. At the open end of a pipe, as in Figure 43.2, the traveled and reflected waves of the source become A0 e jkl ; B0 e jkl ; over a length l; where the amplitudes are denoted by A0 ; B0 : The reflective coefficient for length l0 is given by R0 ¼

B0 e jkl0 A0 e jkl0

ð43:11Þ

This is obtained from Equation 43.6 with x ¼ l0 : The energy, W0 ; of the acoustic wave escaping from the open end of the pipe is given by W0 /

S0 A20 ð1 R20 Þ r0 c0

ð43:12Þ

in which r0 ¼ density of air, and c0 ¼ speed of sound in air. The equation of the sound pressure level measured at an open point at some distance is given by p0 ¼ 10 log10

Qd 4 þ 4pr02 Rr

! þ PWLr0

ð43:13Þ

43 4


Sound source

A0e −jkl0

A0

p0(dB)

L

S B0e jkl0

B0 l0

Expansion chamber Inlet pipe A1

I A1e − jkl1

S

A2

B2e jkl2

B3

B1e jkl1 l2

l1

FIGURE 43.2

p0(dB)

L

A3

A2e −jkl2

B2 B1

Outlet (or tail) pipe

II

l3

Measurement of insertion loss.

where r0; ¼ distance Rr ¼ A=ð1 aÞ; room constant A ¼ the indoor sound absorbing power (indoor surface area times indoor average absorption coefficient) a ¼ the indoor average absorption coefficient Qd ¼ the directivity factor from the open end Therefore, the measured value of insertion loss can be obtained from Equation 43.14, when Qd values are equal. Power level is defined as PWL ¼ 10 log10

W 10 12

IL ¼ PWLr0

PWLr

Now, ð43:14Þ

Using Equation 43.14, it can be shown that IL can be expressed by IL ¼ 10 log10

S0 A0 S3 A3

2

1 1

R20 R23

ð43:15Þ


43 5

S1 A1

A2

S2 TL = 10 log10 A1 A2

Muffler

S1 S2

B2

B1 FIGURE 43.3

3.2

2

Definition of transmission loss for a muffler.

Transmission Loss

It is desirable to eliminate the source and radiation characteristics from the system in Figure 43.3, and to look only at some property of the muffler itself. This may be accomplished by defining a quantity called “transmission loss” (TL) as follows: TL ¼ 10 log10

A1 A2

2

S1 S2

ð43:16Þ

In the measurement of TL, it is difficult to separate the reflected wave. The theoretical calculation is easy and useful.

43.4

Calculation Procedure

For the reactive muffler shown in Figure 43.2, the following equations are obtained from Equation 43.5 to Equation 43.7 and Equation 43.9. The inlet pipe, cavity, and tail pipe are denoted by suffix in the figure [2]. (1) The sound source: S0 ðA0

B0 Þ ¼ S1 ðA1

(2) The open end of pipes:

ð43:17Þ

9 R0 = ¼ A3 e jkl3 R3 ;

B0 e jkl0 ¼ A0 e B3 e jkl3

B1 Þ

jkl0

ð43:18Þ

(3) The sudden expansion (junction I): A1 e

jkl1

S1 ðA1 e

þ B1 e jkl1 ¼ A2 þ B2 jkl1

B1 e jkl1 Þ ¼ S2 ðA2

9 =

(4) The sudden contraction (junction II): jkl2

A2 e S2 ðA2 e

þ B2 e jkl2 ¼ A3 þ B3

jkl2

B2 e jkl2 Þ ¼ S3 ðA3

9 =

R3 e

2jkl3

ð43:20Þ

B3 Þ ;

From these linear equations, the following equation can be obtained: n A0 m10 ¼ j 1 þ R3 e 2jkl3 sin kl1 cos kl2 þ m21 kl1 sin kl2 B0 1 R0 e 2jkl0 þm32 1

ð43:19Þ

B2 Þ ;

m21 cos kl1 cos kl2

sin kl1 sin kl2

o

ð43:21Þ

43 6


When the reflection coefficients are absent, R0 ¼ R3 ¼ 0; we have IL ¼ 10 log10 m03 {m210 ðsin kl1 cos kl2 þ m21 cos kl1 sin kl2 Þ2 þ m210 m232 ðm21 cos kl1 cos kl2

sin kl1 sin kl2 Þ2 }

ð43:22Þ

where m10 ¼ S0 =S1 ; m21 ¼ S2 =S1 ; m32 ¼ S3 =S2 ; m03 ¼ S0 =S3 ; which are ratios of cross sectional areas of the pipes. For the reflection factors, R0 ¼ R3 ¼ 1; S0 ¼ S1 ¼ S3 ; and m21 ¼ m; the equation of insertion loss becomes IL ¼ 20 log10

cos kl1 cos kl21 cos kl3 cos kl0

m sin kl2 sin kl3

tan kl1 cos kl2 sin kl3 þ

1 sin kl2 cos kl3 m ð43:23Þ

The ratio lA1 =A3 l may also be obtained from Equation 43.17 to Equation 43.20. When the magnitude of reflection coefficients R0 ¼ R3 ¼ 0; S0 ¼ S1 ¼ S3 ; and expansion ratio of the cross sectional open area of the pipes m21 ¼ m; the transmission loss is given by TL ¼ 10 log10 1 þ When R0 ¼ R3 ¼

1 m 4

1 m

2

sin2 kl2

ð43:24Þ

1; S0 ¼ S1 ¼ S3 ; and m21 ¼ m; TL is obtained from the following equation:

TL ¼ 10 log10 1 þ

1 ðm2 m2

1Þ{ðm2 þ 1Þsin2 kl3

1}sin2 kl2 þ

1 2

mþ

1 sin 2kl2 sin 2kl3 m

ð43:25Þ

Computation formulas of insertion loss and transmission loss for the case of an expansion chamber with insertion pipe and resonator are shown in Table 43.1. The principal structures of several reactive mufflers are shown in Figure 43.4.

43.5 43.5.1

Application Range of Model Condition for Approximation of Plane Wave

The frequency range where the approximation of a plane wave is valid is given by fc , 1:22

c D

ð43:26Þ

where fc ¼ critical frequency of plane wave (Hz) c ¼ speed of sound (m/sec) D ¼ diameter of muffler (m) It is seen that the expansion ratio of an open area of a pipe increases with IL or TL. However, the application range of the analytical model decreases with increasing diameter of chamber.

43.5.2

Effect of Temperature

Under conditions of high temperature and high speed gas flow, as in an engine exhaust system, the primary effect of a change in pipe temperature is the corresponding change in the speed of sound, which is proportional to the square root of the absolute temperature. In the design of a reactive muffler, it is necessary to use the actual speed of sound in the gas inside the pipe. The most accurate values available for density ðr0 Þ and the speed of sound ðc0 Þ at each element should be used in calculating the impedance

TABLE 43.1

Transmission Loss of Reactive Mufflers and Insertion Loss of Reactive Mufflers

Transmission loss of reactive mufflers (a)

TL (dB)

( TL ¼ 10 log10 1 þ

kS 4a

2

Application Limits and Comments

)

a=l , 0:1; R ¼ 0

a: radius of orifice (b)

f , 1:22c=D

ð1 þ R3 e 2jkl3 Þðcos kl2 þ jm21 sin kl2 Þ þ m32 ð1 2 R3 e 2jkl3 Þð j sin kl2 þ m21 cos kl2 Þ

TL ¼ 20 log10


Muffler (see Figure 43.4)

m10 ¼ S1 =S0 ; m21 ¼ S2 =S1 ; m32 ¼ S3 =S2 ; R0 ; R3 : the reflection coefficient of the open end, l3 ¼ length of the tail pipe (1) when R0 ¼ R3 ¼ 0; S0 ¼ S1 ¼ S3 TL ¼ 10 log10 1 þ

1 1 m21 2 4 m21

2

sin kl2

(2) when R0 ¼ R3 ¼ 21; S0 ¼ S1 ¼ S3 " ! # m221 2 1 1 1 2 2 2 TL ¼ 10 log10 1 þ 2 m21 sin 2kl2 sin 2kl3 {ðm21 þ 1Þsin kl3 2 1}sin kl2 þ m221 2 m21 (c)

TL ¼10 log10 þ

m21 sin kðl1 2 l11 2 l22 Þðtan kl11 þ tan kl22 Þ m

2

R >
> =

C 1B m C B TL ¼ 10 log10 1 þ B C A @ > 4 kS2 > : 2 cot kl C0

> > ;

43-7

C0 ¼ NCi ; N: number of holes, Ci ¼ 2pa2i =ðlb þ paiÞ ; lb ; lb : thickness of the pipe, ai : radius of a hole, m ¼ S12 =S

R >
tan klb 2 b C> = m2 B kV C B TL ¼ 10 log10 1 þ A> > 4 @ Sb > : ; tan klb þ 1 > kV

R < 0; lb ,, l Resonator size ,, l

m ¼ Sb =S (g)

R > > > > > > > = > > > > > > > > > ;

ð43:32Þ

The suffix of d gives the thickness of absorbing material in mm.

43.6

Practical Example

43.6.1

Expansion-Type Muffler for Reciprocating Compressor

Consider a double acting (i.e., fluid on both sides of the piston in the cylinder) reciprocating compressor for supplying high pressure air to a machine shop of a factory, for example. The specifications of the reciprocating compressor follow: *

*

*

*

Delivery pressure: 6.9 £ 105 Pa Rotational speed of driving shaft: 600 rpm Power of driving shaft: 450 kW Diameter of inlet pipe: 380 mm

Pressure pulsations of 10 and 20 Hz are produced by the compressor due to the rotational speed, as seen in Figure 43.5. The pressure wave from the inlet propagates the free space and can the damage nearby private houses. Wooden doors with glass paneling, wooden sliding doors, and leaves of plants and foliage, and have been found to vibrate due to low frequency audible sound. An attenuation of 15 to 20 dB was

43 14

FIGURE 43.5


Noise spectrum at inlet of reciprocating compressor and insertion loss of designed muffler.

necessary at the frequencies 10 and 20 Hz. A muffler using an expansion and tail pipe type was suggested to handle the problem. The reflection coefficient of the tail pipe is approximately R ¼ 1 and ka ¼ 2 pfa=c ¼ 2p £ 20 £ 0:19=345 ¼ 0:0692: IL ¼ 20 log10 lcos kl2 cos kl3

m sin kl2 sin kl3 l

where k ¼ 2pf =c; wave number (1/m) l2 ¼ the length of the chamber (m) l3 ¼ the length of the tail pipe (m) m ¼ the expansion ratio of the cross section between the chamber and inlet

Tail pipe

Air flow

φ380 × 6000 Filter

With kl2 ¼ kl3 ¼ p=2; we have IL ¼ 20 log10 lml We need m . 10 in order to satisfy the desired value of IL. For 20 Hz, we use kl2 ¼ kl3 ¼ p: When kl2 ¼ kl3 ¼ p=2 at f ¼ 10 Hz, we have IL ¼ 0. Then, using kl2 ¼ kl3 ¼ p=2 at frequency 15 Hz, we can satisfy the IL condition of 20 dB at both frequen cies. Hence, l2 ¼ l3 ¼ 345=ð4 £ 15Þ ¼ 5:75 (m) is chosen at a speed of sound c ¼ 345 m/s. The noise spectrum at the inlet of the recipro cating compressor under study and insertion loss of the muffler design in this example are shown in Figure 43.5. The diameters or the lengths of the chamber and the tail pipe are properly selected, as shown and in Figure 43.6. At 10 Hz, IL is determined as

Sound wave

Expansion chamber φ1600 × 4500

To compressor

FIGURE 43.6 The muffler designed for the noise control of a reciprocating compressor.


43 15

indicated below: IL ¼ 20 log10 lcosð2p £ 10 £ 4:5=345Þcosð2p £ 10 £ 6=345Þ £ sinð2p £ 10 £ 6=345Þl ¼ 20 log10 l0:6825 £ 0:4600

ð1:6=0:38Þ2 sinð2p £ 10 £ 4:5=345Þ

17:728 £ 0:7308 £ 0:8879l ¼ 21:0 ðdBÞ

Similarly, at 20 Hz, we have IL ¼ 20 log10 lcosð2p £ 20 £ 4:5=345Þcosð2p £ 20 £ 6=345Þ £ sinð2p £ 20 £ 6=345Þl ¼ 20 log10 lð 0:0682Þ £ ð 0:5767Þ

ð1:6=0:38Þ2 sinð2p £ 20 £ 4:5=345Þ

17:728 £ 0:9977 £ 0:8170l ¼ 23:2 ðdBÞ

Clearly, the attenuation at both frequencies satisfies the desired lower limit of 20 dB. Calculated values of IL at low frequencies are shown by a curved continuous line in Figure 43.5.

References 1. Levine, H. and Schwinger, J., On the radiation of sound from an unflanged circular pipe, J. Phys. Rev., 73, 383, 1948. 2. Ohnaka, I., Lecture for noise reduction of machines, no. 2, J. Marine Eng. Soc. Jpn., 4, 179, 1969. 3. Suyama, E. and Hirata, M., Attenuation constant of plane wave in a tube, J. Acoust. Soc. Jpn., 35, 152, 1979.

44

Design of Sound Insulation 44.1 Theory of Sound Insulation .............................................

Kiyoshi Okura Mitsuboshi Belting Ltd.

Expressions of Sound Insulation † Transmission Loss of a Single Wall † Transmission Loss of Multiple Panels † Transmission Loss of Double Wall with Sound Bridge

44 1

44.2 Application of Sound Insulation ..................................... 44 13 Acoustic Enclosure

†

Sound Insulation Lagging

Summary This chapter presents useful theory and design procedures for sound insulation. Related concepts and representations of transmission loss, the transmission coefficient, and impedance are given. Analysis and design procedures for sound insulation structures such as single and multiple panels and walls with sound absorption material are presented. Practical applications for the design of sound insulation components and systems are described.

44.1

Theory of Sound Insulation

44.1.1

Expressions of Sound Insulation [1]

44.1.1.1

Transmission Coefficient

Let us denote by Ii the acoustic energy incident on a wall per unit area and unit time. Some energy is dissipated in the wall, and, apart from the energy that is reflected by the wall, the rest is transmitted through the wall. Using It to denote the transmitted acoustic energy, the transmission coefficient of the wall is defined as

t¼ 44.1.1.2

It Ii

ð44:1Þ

Transmission Loss

As an expression for sound insulation performance, we may use transmission loss (TL), which is defined as (also see Chapter 42 and Chapter 43) TL ¼ 10 log

44.1.2

1 t

¼ 10 log

Ii It

ð44:2Þ

Transmission Loss of a Single Wall

Consider a plane sound wave incident on a impermeable infinite plate at angle u, which is placed in a uniform air space as shown in Figure 44.1. The sound pressure of the incident, reflected, and transmitted 44 1

44 2


waves, denoted by pi ; pr ; and pt ; respectively, are given by pi ¼ Pi e jvt pr ¼ Pr e jvt pt ¼ Pt e

y pr

jkðx cos u þ y sin uÞ jkð x cos u þ y sin uÞ

ð44:3Þ

pt

q

jvt jkðx cos u þ y sin uÞ

q 0

q

x

where Pi ; Pr ; and Pt are the sound pressure amplitudes of incident, reflected, and transmitted pi waves, respectively; v is angular frequency; k is the wave number of the sound wave; c is the speed of sound, respectively in the air. Assuming FIGURE 44.1 Plane sound wave incidence on an that the plate is sufficiently thin compared infinite plate. with the wavelength of the incident sound wave, the vibration velocities on the incident and transmitted surfaces of the plate are equal. Then vibration velocity, u, of the plate in the x direction is equal to the particle velocity of the incident and transmitted sound waves, and we obtain relations u¼

1 ›ðpi þ pr Þ ¼ jvr ›x pi þ pr u

pt

1 ›pt jvr ›x

ð44:4Þ

¼ Zm

ð44:5Þ

where r is the air density and Zm is the mechanical impedance of the plate per unit area. From these equations, the transmission coefficient, tu ; and then the transmission loss, TLu ; at the incident angle, u; are obtained according to 2

TLu ¼ 10 log 44.1.2.1

1 p Z cos u ¼ 10 log i ¼ 10 log 1 þ m pt tu 2r c

2

ð44:6Þ

Coincidence Effect

Consider the vibration of the plate in the x y plane shown in Figure 44.1. Denoting by m the surface density, and by B the bending stiffness per unit length of the plate, the equation of motion of the plate is given by m

›2 j ›4 j ¼ pi þ pr 2 þ Bð1 þ jhÞ ›t ›y4

pt ;

B¼

Eh3 Eh3 2 ø 12ð1 n Þ 12

ð44:7Þ

where

j ¼ displacement in the x direction E ¼ Young’s modulus of the plate h ¼ thickness of the plate h ¼ loss factor of the plate n ¼ Poisson’s ratio of the plate The plane sound wave of angular frequency, v; and of incidence angle, u; causes a bending wave in the plate where displacement is assumed to be j ¼ j0 e jðvt k1 yÞ ; as a solution of Equation 44.7. Hence, the mechanical impedance per unit area is obtained: ! p þ pr pt Bk4 Bk41 Zm ¼ i ð44:8Þ ¼ h 1 þ j vm ›j=›t v v where k1 ¼ k sin u ðk ¼ v=cÞ is the wave number of the bending wave in y direction caused by the incident sound wave. Propagation speed of the forced bending wave, c1 ; and a free bending wave of


44 3

the plate, cB ; are given by c1 ¼ v =k1 ;

cB ¼

v2 B m

!1=4 ð44:9Þ

Equation 44.9 reduces Equation 44.8 to Zm ¼ hv m

4

cB c1

þ jvm 1

cB c1

4

ð44:10Þ

When the speed of forced bending wave, c1 ; and the speed of free bending wave, cB ; are equal in Equation 44.10, the imaginary part of Zm becomes 0, and a form of “resonance” occurs. Then the transmission loss decreases rapidly. This phenomenon is called the coincidence effect, and the resonant frequency dependent on the incident angle is given by rffiffiffiffi c2 m f ¼ ð44:11Þ 2p sin2 u B The minimum of the resonant frequency is called coincidence critical frequency, or critical frequency for short, and it reduces to rffiffiffiffi c2 m c2 fc ¼ ø ð44:12Þ 2p B 1:8cL h pffiffiffiffiffiffi where cL ¼ E=rP is the speed of longitudinal wave in the plate, and rP denotes the density of the plate. Let us show the relations of the critical frequency and the plate thickness pffiffiffiffi of typical material of sound insulation shown in Figure 44.2. Using the relation cB =c1 ¼ f =fc sin u; Equation 44.10

104

104

8 6 4 3

5 (CL = 1000m/s)

2

Critical frequency fc (Hz)

8 6

4 (CL = 2000m/s) 3 (CL = 3000m/s)

103

4 3 2

103

8 6

8 6

4 3

4 3

2 (CL = 4000m/s)

2

2

1 (CL = 5000m/s)

102

102 8 6

8 6 4 3

4 3

2

2

10 0.1

2

3 4 5 6 7 81.0

2

3 4 5 6 7 8 10

2

10

3 4 5 6 7 8100

Thickness h (cm) FIGURE 44.2 Critical frequency vs. plate thickness of typical sound insulation materials: (1) aluminum, steel, or glass; (2) hardboard or copper; (3) dense concrete, plywood, or brick; (4) gypsum board; (5) lead or light weight concrete. (Source: Beranek, L.L. 1988. Noise and Vibration Control, INCE/USA. With permission.)

44 4


becomes f Zm ¼ hvm fc

44.1.2.2

2

"

f fc

4

sin u þ jvm 1

2

# 4

sin u

ð44:13Þ

Mass Law of Transmission Loss

When f ,, fc ; Equation 44.13 becomes Zm ø jvm: Then, the transmission loss depends on the incident angle, the frequency and the surface density of the plate. This is called the mass law of transmission loss. Mass law of normal incidence represents the transmission loss at the incident angle u ¼ 0; as given by TL0 ¼ 10 log 1 þ

vm 2r c

2

ð44:14Þ

For vm .. 2rc; it becomes TL0 ø 10 log

vm 2rc

2

¼ 20 log mf

42:5; for air

ð44:15Þ

Mass law of random incidence represents the transmission loss at the angle averaged over a range of u from 0 to 908, which is realized for perfectly diffused sound field. We have TLr ¼ 10 logð1=tr Þ ø TL0

10 logð0:23TL0 Þ

ð44:16Þ

where the random incident transmission coefficient, tr ; is defined as

tr ¼

ðp=2 0

tu cos u sin u du

ðp=2 0

cos u sin u du

ð44:17Þ

An approximation for Equation 44.16, as given below, is generally used for a practical use and this is often useful. TLr ¼ 18 log mf

44

ð44:18Þ

Mass law of field incidence represents the transmission loss at the angle averaged over a range of u from 0 to about 788, which is said to agree with actual sound field. We have TLf ¼ TL0

5

ð44:19Þ

The three types of transmission loss presented above are compared in Figure 44.3. 44.1.2.3

Stiffness Law of Transmission Loss [2]

The plate described above is assumed to be infinite. However, an actual plate is always supported by some structures at its boundaries and the plate size is finite. Transmission loss of a finite plate is considered to be related to the nature of excitation of vibration in the plate, for example, sound wave incidence, modes of vibration and characteristics of sound radiation. Therefore, the governing relationships become very complex. However, in the following frequency range, it is known that the transmission loss conforms to the mass law c , f ,, fc 2a

ð44:20Þ

where a is length of shorter edge for rectangular plate. When f , c=2a; the whole plate is excited in phase, and stiffness effects from the supports of its edges will appear. If we denote the equivalent stiffness of the plate as K and assume a loss factor of 0,


44 5

FIGURE 44.3

Theoretical transmission loss based on mass law.

the mechanical impedance of the plate is obtained by using Equation 44.8; thus Zm ¼ j vm

K v

ð44:21Þ

The frequency at Zm ¼ 0 corresponds to the first mode natural frequency, f11 ; of the plate (see Chapter 4), and consequently, the equivalent stiffness of rectangular plate with simple edge support is given by rffiffiffiffi rffiffiffiffi 1 K p B 1 2 1 2 f11 ¼ ; þ ð44:22Þ 2p m 2 m a b Then, K ¼ Bp4

1 1 þ 2 a2 b

2

¼ 10 log 1 þ ø 20 logðK=f Þ

K 2vrc 74:5

2

ð44:23Þ

This is called the stiffness law of the transmission loss, and it shows a 6 dB decay per octave. The characteristics mentioned above for single wall transmission loss are shown in Figure 44.4 and summarized below.

Transmission loss TL (dB)

where a and b are the length of the short and long edges for the rectangular plate, respectively. When f ,, f11 is assumed in Equation 44.21, the mass term can be neglected, and from Equation 44.6 the normal incidence transmission loss, TLS0, is given by Region II K 2 Region I Region III Region IV TLS0 ¼ 10 log 1 j 2vrc −6

dB

/oc

tav e

5.4

e tav

c B/o

d

f11

fc

Frequency (Hz) FIGURE 44.4 single wall.

Transmission loss characteristics of a

44 6


1. Region I ð f ,, f11 Þ: Transmission loss is controlled by the stiffness of the panel: TL ¼ TL0

40 log

f f11

ð44:24Þ

2. Region II ð f < f11 Þ: Transmission loss is controlled by the lower mode natural frequencies of the panel, and the estimation becomes very complex. 3. Region III ð f11 ,, f # fc =2Þ: Transmission loss is controlled by the mass (surface density) of the panel: TL ¼ 18 log mf

44

ð44:25Þ

4. Region IV ðf . fc =2Þ: Transmission loss is controlled by the mass and the damping of the panel, and it is reduced by coincidence effects. For fc =2 , f # fc: TL is represented by a straight line connecting the value at f ¼ fc =2 of Equation 44.25 and the value at f ¼ fc of Equation 44.26. For f . fc: TL ¼ TL0 þ 10 log

44.1.3

2h f p fc

ð44:26Þ

Transmission Loss of Multiple Panels

To realize sound insulation of high performance, we often use a double wall or a multiple panel composed of insulation materials like steel plates and absorbing materials like fiber glass. In this subsection, transmission loss of a multiple panel is described [3]. 44.1.3.1

Calculation Method

Consider a multiple panel of infinite lateral extent as shown in Figure 44.5, which is composed of n acoustic elements, each element consisting of three basic materials, an impermeable plate, air space, and an absorption layer. Furthermore, consider a plane wave incident on the left hand side surface of the nth element at angle u: Let the sound pressure of the incident wave be pi ; and of the reflected wave be pr ; and the wave be propagating through the structure, and then radiating from the right hand side of the first element as a plane wave of pressure pt into a free field at transmission angle u: In the analysis, we append the subscript kð¼ 1; 2; …; nÞ to the physical parameters of the kth element, and “2” and “1” to the left and right hand side values of these parameters, respectively, pressure at the incident surface, pn2 ; to the incident

y

pr

n

k

pt

1 q

q q

x

pi Zn2 pn2 FIGURE 44.5 tiple panel.

Zk2 Zk1 pk2 pk1

Z11 p11

Calculation model of n element mul

as shown in Figure 44.5. The ratio of the sound wave, pi ; is given by

pn2 p þ pr 2Zn2 ¼ i ¼ pi pi Zn2 þ rc=cos u

ð44:27Þ

where Zn2 is the acoustic impedance of the left hand side normal to the surface of the nth element and rc=cos u is the acoustic impedance normal to the surface, which is equal to the radiation impedance of the


44 7

first element, Z11 ; shown in Figure 44.5. Using the usual condition of pressure matching at each interface, we can write the expression for the oblique incidence transmission coefficient as 2

t ðuÞ ¼

2

2

pt p p p p p ; 11 ¼ n2 · n1 · · · k1 · · · 11 pi pi pi pn2 pk2 p12

2

ð44:28Þ

Hence, we obtain the following expression for the random incidence transmission loss: 0 ðu 1 l cos u sin u du C B B C TL ¼ 10 logB ðul 0 C @ A t ðuÞ cos u sin u du

ð44:29Þ

0

where u l is the limiting angle above which no sound is assumed to be received, and it varies between 788 and 858. If we know Zn2 in Equation 44.27 and the pressure ratio across each of the single elements in Equation 44.28, we can calculate the TL using Equation 44.29. We can obtain Zn2 by using the conditions of impedance matching at each interface from the rightmost to the leftmost element in order, if we know the impedance relations across each of the single elements. Now, we present the pressure ratios and the acoustic impedance relations across three basic elements. 44.1.3.2

Impermeable Plate

Consider the vibration of an infinite impermeable plate of thickness, h, induced by the sound pressure difference on each side of the plate, as illustrated in Figure 44.6. In this case, the particle velocity on both sides of the plate must be the same as the plate vibration velocity. Then, from Equation 44.8, the following expressions are obtained: Z2 ¼ Z1 þ Zm

ð44:30Þ

p2 Z ¼ 2 p1 Z1

ð44:31Þ

y Pt

Pr q q Pi

q 0 P2

Z2

h

x

P1

Z1

where p2 ; p1 are the sound pressure at the incident FIGURE 44.6 Excitation of infinite plate by a plane surface x ¼ 0 and at the transmitted surface x ¼ h; sound wave. respectively, Zm is the mechanical impedance of the plate, and Z2 ; Z1 are the acoustic impedance normal to the incident surface at x ¼ 0 and the transmitted one at x ¼ h; respectively. 44.1.3.3

Sound Absorbing Material

For a sound absorbing material layer of thickness d and infinite lateral extent, consider a plane wave incident at an angle u to the normal, as shown in Figure 44.7. Deriving the wave equation in the sound absorbing material and applying the continuity conditions of the sound pressure across the surface at x ¼ 0 and x ¼ d; with some mathematical manipulation we get the following results: Z2 ¼

gZ0 coth ðqd þ wÞ q

p2 cosh ðqd þ wÞ ¼ cosh w p1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 2 sin u; w ¼ coth q¼g 1þ g

ð44:32Þ ð44:33Þ 1

qZ1 gZ0

ð44:34Þ

44 8


where g is a propagation constant and Z0 is a characteristic impedance of a homogeneous, iso tropic absorbing material. If porous material is used as the absorbing material, the following relations are applicable for g and Z0 [4]:

y

y

Pi

Pi

q

0

Z0 ¼ R þ jX

z

R =rc ¼ 1 þ 0:0571ðr f =Rl Þ X=rc ¼

x

f

0:0870ðr f =Rl Þ

0:754

Z2

ð44:35Þ FIGURE 44.7 directions.

0:732

x

d Z1

Schematic relation of sound wave

g ¼ a þ jb 0:595

b=k ¼ 1 þ 0:0978ð r f =Rl Þ

ð44:36Þ 0:700

Flow resistivity R1 (MKS rayls/m)

a =k ¼ 0:189ðr f =Rl Þ

ð0:01 # r f =Rl # 1Þ where r is the air density, f is the frequency, and Rl is the flow resistivity, respectively. Specifically, note that Rl is defined as the flow resistance of the porous absorbing material per unit thickness. With data measured with a measuring tube of flow resistance, we can write Dp Rl ¼ l·u

ð44:37Þ

where Dp is pressure difference between the inlet and the outlet of the absorbing material in the tube, u is the mean flow velocity in the tube, and l is the thickness of the absorbing material. It is known that the flow resistivity of porous absorbing material such as fiber glass or rock wool is related to the bulk density, as shown in Figure 44.8. 44.1.3.4

105 5

104 Fiber glass Rock wool

5

10

5

102

Bulk density rm

5

(Kg/m2)

FIGURE 44.8 Flow resistivity vs. bulk density for porous, sound absorbing materials.

Air Space

For an air space, Z0 ¼ rc and g ¼ jk: Hence, Equation 44.32 to Equation 44.34 reduce to Z2 ¼

rc coth ð jkd cos u þ dÞ cos u

p2 cosh ð jkd cos u þ dÞ ¼ cosh d p1

d ¼ coth

44.1.3.5

1

Z1 cos u rc

ð44:38Þ ð44:39Þ ð44:40Þ

Double Wall [2]

Applying the theory formulated above, we can easily obtain the transmission loss of a double wall composed of the three elements: impermeable plate, air space, and impermeable plate, as shown in Figure 44.9. Assume that the two impermeable plates have the same surface density, m, and the mechanical impedance of the plates is jvm: Then, we can obtain following equations for element


44 9

one and element three:

rc Z12 ¼ Z11 þ Zm ¼ þ jvm; cos u p12 Z vm cos u ¼ 12 ¼ 1 þ j p11 Z11 rc

y 3

ð44:41Þ

2

1

pr

Z32 ¼ Z31 þ Zm ¼ Z22 þ jvm; ð44:42Þ

p32 Z vm ¼ 32 ¼ 1 þ j p31 Z22 Z22

pt q

q q

For element two: rc coth ð jkd cos u þ d 0 Þ; Z22 ¼ cos u 0

p22 cosh ð jkd cos u þ d Þ ¼ cosh d 0 p21

d

pi

m

ð44:43Þ FIGURE 44.9 air space.

x

m

Calculation model of a double wall with

where, by applying impedance matching conditions at the interface of element one and element two, the following definition is introduced: vm cos u Z21 cos u Z12 cos u d 0 ¼ coth 1 ¼ coth 1 ¼ coth 1 1 þ j ð44:44Þ rc rc rc In this case, Equation 44.27 reduces to pi Z þ rc=cos u Z þ jvm þ rc=cos u ¼ 22 ¼ 32 2Z32 2ðZ22 þ jvmÞ p32

ð44:45Þ

Substituting Equation 44.41 to Equation 44.45 into Equation 44.28, we obtain the transmission loss of the double wall: ð44:46Þ TLu ¼ 10 log½1=t ðuÞ ¼ 10 log½1 þ 4a2 cos2 u ðcos b a cos u sin bÞ2 a ¼ vm=2rc;

b ¼ kd cos u

In Equation 44.46, the transmission loss is zero, and full passage (i.e., “all pass” in the filter terminology) of sound occurs when the following equation holds: cos b

a cos u sin b ¼ 0

When b ,, 1ðkd ,, 1Þ; the frequency of full passage for normal incidence is given by sffiffiffiffiffiffiffi 1 2rc2 fr ¼ 2p md

ð44:47Þ

ð44:48Þ

This is the natural frequency of a vibrating system consisting of two masses, m, connected by a spring of spring constant, rc2 =d: When b .. 1ðkd .. 1Þ; the solution of Equation 44.47 for b is b ø np; and the frequency of all passage for normal incidence is given by nc ðn ¼ 1; 2; 3; …Þ ð44:49Þ fn ¼ 2d These are the acoustic resonant frequencies of the air space d. Characteristics of the transmission loss given by Equation 44.46, in case of normal incidence ðu ¼ 0Þ; are as follows: pffiffiffiffiffiffiffiffi 1. f , fr b , 2rd=m TL ø 10 logð4a2 Þ ¼ TL0 þ 6 This is equal to the transmission loss of a single wall of surface density 2m.

ð44:50Þ

44 10


2. fr # f , f1 =p

pffiffiffiffiffiffiffiffi 2rd=m # b , 1 TL ø 10 logð4a4 b2 Þ ¼ 2TL0 þ 20 logð2kdÞ

ð44:51Þ

This transmission loss indicates an 18 dB increase per octave. 3. f ¼ ð2n 1Þc =4dðb ¼ np p=2Þ TL ø 10 logð4a4 Þ ¼ 2TL0 þ 6

ð44:52Þ

A straight line connecting the transmission losses at these frequencies in Figure 44.10 indicates a 12 dB increase per octave. When the two impermeable plates have different surface densities, m1 and m2 ; Equation 44.41 to Equation 44.52 reduce to pffiffiffiffiffiffiffiffi 1. f , fr b , 2rd=m

2. fr # f , f1 =p

3. f ¼ ð2n

TL ¼ 20 log½vðm1 þ m2 Þ=2rc

ð44:53Þ

TL ¼ TL1 þ TL2 þ 20 logð2kdÞ

ð44:54Þ

TL ¼ TL1 þ TL2 þ 6

ð44:55Þ

pffiffiffiffiffiffiffiffi 2rd=m # b , 1

1Þc=4dðb ¼ np

p=2Þ

In these equations, TL1 and TL2 are the transmission losses of each plate, which are given by Equation 44.15. The transmission loss of a double wall, as mentioned above, is shown schematically in Figure 44.10. An actual double wall, however, is finite in size and the air space forms a closed acoustic field, which


TL = TL1 + TL2 + 6

TL = TL1 + TL2 + 20 log(2kd) f3 f2

TL = 20 log[(m1 + m2)f ] − 42.5 fr

f1/π

f1


Transmission loss of a double wall with air space.


44 11

m1 tav e

m1

18 dB /oc


Double wall with sandwiched porous material

12

ve cta

o B/

d

Double wall with air space

w TL s La

Mas

√2 fr

2 fr

1

3 fc

Frequency (Hz) FIGURE 44.11 or air space.

Design chart for estimating the transmission loss of a double wall with sandwiched porous material

makes the transmission loss deviate from the theoretical value. Figure 44.11 gives a design chart of an actual double wall, which is based on theory and experiments.

44.1.4

Transmission Loss of Double Wall with Sound Bridge [5]

In the previously presented theory, each plate of the multiple panel is considered to be structurally independent. In actual multiple panels, such as partitions of a building or sound insulation laggings of a duct, however, each plate is connected with steel sections, stud bolts, and the like, which are called sound bridges. This is illustrated in Figure 44.12. The sound pressure of the transmitted wave through a double wall with sound bridges is given by the summation of radiated sound pressure from the vibration of the transmitted side plate excited by the sound in the air space and that mechanically excited by the sound bridges.

FIGURE 44.12

Examples of actual double wall with sound bridges.

44 12


The acoustic power radiated from the area, S, of an infinite plate excited by sound pressure is given by WP ¼ rcSv22

ð44:56Þ

v22

where is the space averaged mean square vibration velocity over the plate. The acoustic power radiated from the plate mechanically excited by a point force or a line force is WB ¼ rcxv2 ð f ,, fc Þ 8 2 l p3 c x¼ 2 ll p c

ð44:57Þ

ðpoint force excitationÞ

ð44:58Þ

ðline force excitationÞ

where v2 is the mean square vibration velocity of the plate at the excitation point, lc ¼ c=fc is the wavelength of the bending wave at the critical frequency, and l is the length of the line force. By comparing Equation 44.56 and Equation 44.57, it is noted that x is the effective area of the acoustic power radiated from the infinite plate excited by the point or line force. Acoustic power, WB ; is the power radiated from a small area near the excitation point, because a free bending wave propagating in an infinite plate can radiate little sound when f , fc : From the equations given above, the total acoustic power radiated from the transmitted side plate is obtained as WT ¼ WP þ WB ¼ rcSv22 1 þ

nx v S v2

2

ð44:59Þ

where n is the number of excitation forces applied to the area, S. Then, transmission loss TL T of the double wall with sound bridges is given by TL T ¼ 10 log

WI WT

¼ 10 log

WI WP · WP WT

¼ TL

TLB

ð44:60Þ

where WI is the acoustic power incident on the double wall, TL is the transmission loss of the double wall without a sound bridge, and TLB denotes the transmission loss reduction by the sound bridges, and is given by TLB ¼ 10 log

WT WP

¼ 10 log 1 þ

nx v S v2

2

ð44:61Þ

We assume the following: 1. The vibration velocity of the incident side plate is not affected by the sound bridges. 2. The vibration velocity of the transmitted side plate at the excitation points (connecting points) is equal to the velocity v1 of the incident side plate, and consequently, the next equation holds v v ø 1 v2 v2 With these assumptions, we apply the method presented in section 44.1.3, to determine v=v2 as

v2 m2 d v v ø 1 ¼ rc 2 v2 v2

ð fr , f , f1 =pÞ ¼

vm2 rc

ð f . f1 =pÞ

ð44:62Þ

Using Equation 44.60 and Equation 44.53, we can obtain the increase in transmission loss, DTL; from the transmission loss of mass law based on the total mass of the double wall, as presented below.



44 13

Double wall with air space 6 dB/octave (12 dB/octave) Double wall with sandwiched porous material (18 dB/octave)

D TL

Mass law 18 log[(m1 + m2 ) f ]− 44

√ 2 fr 2 fr

FIGURE 44.13

1/3 min ( fc1, fc2) Frequency (Hz)

Design chart for estimating the transmission loss of a double wall with sound bridges.

1. Point connection DTL ¼ TL

TLB

20 log

vðm1 þ m2 Þ 2r c

m1 ¼ 20 logðefc Þ þ 20 log m1 þ m2

p3 þ 10 log 8c2

! ð44:63Þ

2. Line connection DTL ¼ TL

TLB

20 log

¼ 10 logðbfc Þ þ 20 log

vðm1 þ m2 Þ 2rc m1 m1 þ m2

þ 10 log

p 2c

ð44:64Þ

pffiffiffiffi where e ¼ S=n is the distance between point forces and b ¼ S=nl is the distance between line forces. Figure 44.13 presents a practical and useful design chart of the transmission loss for a double wall with sound bridges, which is based on Figure 44.11 and Equation 44.63 and 44.64.

44.2 44.2.1

Application of Sound Insulation Acoustic Enclosure

Performance of an enclosure may be represented by the insertion loss (IL), which is the difference of acoustic power level before and after installation of the enclosure. When we assume a noise source and also an enclosure with one dimensional model as shown in Figure 44.14a, the insertion loss through frequency is shown in Figure 44.14b. It is divided into the following four regions:

44 14


FIGURE 44.14 One dimensional model for calculating the insertion loss characteristics of an acoustic enclo sure: (a) One dimensional model; (b) insertion loss of an enclosure.

1. Region I ð f , fr Þ is controlled by the stiffness of the enclosure plate and air space. 2. Region II ðf ø fr Þ is the resonance region for a vibrating system consisting of the mass, the stiffness of the enclosure plate, and the capacitance of air space. 3. Region III ðfr , f , c=2dÞ is controlled by the mass of the enclosure plate. 4. Region IV ðf . c=2dÞ is controlled by the diffused sound field. This region cannot be represented by a one dimensional model. 44.2.1.1

Near Field Type [4]

When the distance between the noise source and the enclosure is less than half of a wavelength of the emitted sound from the source, insertion loss corresponds to the characteristics of the regions I to III, and we can represent them with a one dimensional acoustic model, as shown in Figure 44.14a. Consider an infinite flat enclosure plate with distance d from a noise source of plane sound wave, as shown in Figure 44.14a. The insertion loss of the plate is given by " # v0 2 2 sin uðX cos u R sin uÞ sin2 uðX 2 þ R2 Þ IL ¼ 10 log þ ¼ 10 log 1 v1 rc r2 c2 ð44:65Þ

u ¼ kd ¼ vd=c;

R ¼ hvm;

X ¼ ðvm

K=vÞ ¼ vm½1

ðv11 =vÞ2 ;

pffiffiffiffiffiffi v11 ¼ K=m

where m and K are the density and equivalent stiffness of the plate per unit area, respectively, and h is the loss factor of the plate. If the enclosure is a rectangular plate of size a £ b and is simply supported at its edges, the equivalent stiffness is given by Equation 44.22, and v11 is the natural (angular) frequency of the first mode. In Equation 44.65, the conditions in which the brackets of the right hand side are equal to zero or IL ¼ 1 are satisfied by following frequencies: (1) u ,, p sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi rc2 1 K þ rc2 =d 1 ¼ ð44:66Þ fr ¼ v211 þ m md 2p 2p This is the natural frequency of vibration of the one degree of freedom (one DoF) system determined by the stiffness of the plate, the spring constant of the air space, and the surface density of the plate, as shown in Figure 44.15.


44 15

(2) u ¼ np ðn ¼ 1; 2; …Þ nc fn ¼ ðn ¼ 1; 2; …Þ 2d

K+

ð44:67Þ

These are the resonant frequencies of the air space. The frequency characteristics of the IL given by Equation 44.65 are shown in Figure 44.16, where the normal incidence transmission losses are shown by broken lines, as a reference. Equation 44.65 is approximated by

m

V1 V0

1. f , fr 2Kd rc2

ð44:68Þ

2. fr # f , f1 4p2 IL ø 20 logðmdf Þ þ 20 log rc2

!

2

¼ 20 logðmdf 2 Þ

ð44:69Þ

71

44.2.1.2 Far Field-Type (Absorption Type) Enclosure [1] When the distance between the noise source and the enclosure is larger than half of a wavelength of the emitted sound, insertion loss may be rep resented by the characteristics of Region IV, and it can be analyzed using the theory of room or hall acoustics. Consider the enclosure shown in Figure 44.17, with a noise source of power level LW0 : From the theory of room acoustics, the average sound pressure level, LP0 ; on the inner surface of the enclosure plate is obtained as the sum of the direct and reverberant sound pressures: LP0 ¼ LW0 þ 10 log ¼ LW0

FIGURE 44.15 system.

Insertion loss (dB)

IL ø 10 log 1 þ

rc2 d

60 50 40 30 20 10 0 −10 −20 −30 10

One degree of freedom vibrating

Mass law TL

C B

Resonance in air space

A

100 1000 Frequency (Hz) m = 16 Kg/m2, d = 0.19 m ~ 0.033 (at 33 Hz) Curve A: f11 = 0 Hz, h = Curve B: f11 = 100 Hz, h ~ = 0.033 (at 105 Hz) ~ 0.033 (at 475 Hz) Curve C: f11 = 475 Hz, h =

FIGURE 44.16 Example of theoretical insertion loss of a near field type enclosure.

1 4 þ S R

10 log S þ 10 log 1 þ R¼

ð1

aS aÞ

4S R

ð44:70Þ ð44:71Þ FIGURE 44.17

Calculation model of a far field type

where S is the inner surface area of the enclosure enclosure. and a is the average absorption coefficient on the inner surface of the enclosure. In Equation 44.70, the first and the second terms of the right hand side represent the influence of the direct sound field, and the third term represents the buildup caused by the covering. When we use SP for the area of the enclosure plate and SO ð¼ S SP Þ for the area of the enclosure opening, and assume diffusing condition for the sound field in the enclosure, acoustic power levels

44 16


radiated from the plate and the opening are given, respectively, by LWP ¼ LP0

ðTL þ 6Þ þ 10 log SP ;

LW0 ¼ LP0

6 þ 10 log SO

ð44:72Þ

where TL is the random incident transmission loss of the enclosure plate. Then, the insertion loss of the enclosure is IL ¼ LW0

10 log 10LWP =10 þ 10LW0 =10 ð44:73Þ

In the design of an acoustic enclosure, special attention should be paid to the following points: 1. Buildup increases the sound pressure in the enclosure and also the power level radiated FIGURE 44.18 Example of a silencer at the opening. from the enclosure. We must treat the inner surface of the enclosure with sound absorbing materials to reduce the absorption coefficient and to decrease the buildup. 2. Opening radiates more acoustic power than the enclosure plate by TL, as is clear from Equation 44.72. It is desirable to make the opening as small as possible within the range given by SO # 10 SP

TL=10

¼t

ð44:74Þ

This equation means that the acoustic power from the opening is less than that from the enclosure plate. If the relation in Equation 44.74 cannot be satisfied because of ventilation requirements, and so on, some type of silencers should be provided at the opening, as shown in Figure 44.18. 3. Structure borne noise, which is caused by the vibration propagating from the base of the machine (noise source) to the enclosure plate, significantly decreases the insertion loss of the enclosure. In this case, some means of noise/vibration suppression should be provided, for example, the following: Place supporting structures of the enclosure at the points of lowest vibration level, and the vibrations of the machine should be prevented from propagating to the enclosure plate, using vibration isolation materials. Add damping materials to the enclosure plate so as to reduce the vibration level of the plate. *

*

44.2.2

Sound Insulation Lagging

In electric power plants and chemical plants, for example, piping for high pressure water or steam, and ducts for air or gas flow form major noise sources. For controlling these noise sources, we usually use sound insulation laggings, which cover the noise sources with heavy and impermeable plates or sheets with sound absorbing materials, as shown in Figure 44.19. 44.2.2.1

Pipe Lagging [2]

Approximate the cylindrical piping and pipe lagging with a one dimensional model as shown in Figure 44.20. The insertion loss of one layered lagging approximated by a one DoF system is given by Equation 44.69 in the frequency region fr # f , f1 ; as mentioned before. It is not practical, however, to directly apply Equation 44.69 to actual laggings, and we approximate the insertion loss of actual laggings by IL ¼ a logðmdf 2 Þ þ b where a and b are constants.


FIGURE 44.19

44 17

Examples of typical sound insulation laggings: (a) pipe lagging, (b) duct lagging.

By taking mdf 2 as the horizontal axis and plotting the insertion loss data from laboratory tests and field tests (the vertical coordinates), we obtain Figure 44.21. Apply regression analysis to the data in Figure 44.21 to obtain the insertion loss of one layered lagging as IL ¼ 11:7 logðmdf 2 Þ

43:3 ð5 £ 103 # mdf 2 # 108 Þ

ð44:75Þ

Applying the same method to double layered lagging, approximated by a two DoF vibrating system, we get Figure 44.22, and the insertion loss IL ¼ 6:9 logðm1 m2 d1 d2 f 4 Þ

40:3 ð106 # m1 m2 d1 d2 f 4 # 1015 Þ

ð44:76Þ

where the subscripts “1” and “2” denote the first layer and the second layer, respectively.

FIGURE 44.20 lagging.

Examples of pipe laggings and calculation model: (a) one layered lagging; (b) double layered

44 18


FIGURE 44.21 Measured insertion loss data obtained from laboratory and field tests for one layered laggings and regression analysis.

FIGURE 44.22 Measured data of insertion loss obtained from laboratory and field tests for double layered laggings and regression analysis.

44.2.2.2

Duct Lagging [4]

Various types of the duct laggings are used according to the need, for example, as shown in Figure 44.19. A simpler and more practical approach is to place a thin plate on the duct casing through absorbing materials, as shown in Figure 44.23. In this case, assuming that vibration of the duct casing is not affected by the placed plate, the insertion loss of the duct lagging is obtained by following equations, which can be deduced from the method used in the transmission loss of double wall with sound bridges.


44 19

FIGURE 44.23 Examples of duct laggings and connection types of thin plate to the duct casing: (a) point connection; (b) line connection.

1. Point connection

"

IL ¼

2. Line connection

8 c2 f 10 log b 3 nP 2 þ r p fc f

"

IL ¼

4

2

c f 10 log 0:64nL þ r fc f

4

# þ 10 log s

ð44:77Þ

# þ 10 log s

ð44:78Þ

where b ¼ vibration isolation factor of the flexible support (b ¼ 1 for rigid support) nP ¼ number of attachment points per unit area nL ¼ number of studs per unit length

FIGURE 44.24 Plateau height for point connection as a function of the number of bridges per unit area [4]. (Source: Beranek, L. L. 1988. Noise and Vibration Control, INCE/USA. With permission.)

44 20


FIGURE 44.25 Plateau height for line connection as a function of the number of studs per unit length [4]. (Source: Beranek, L. L. 1988. Noise and Vibration Control, INCE/USA. With permission.)

fc ¼ critical frequency of the thin plate given by Equation 44.12 fr ¼ resonant frequency given by Equation 44.66 s ¼ sound radiation efficiency of the thin plate Knowing the critical frequency, fc ; and the resonant frequency, fr ; we can obtain the insertion loss from the charts given in Figure 44.24 and Figure 44.25 instead of using Equation 44.77 and Equation 44.78. In Figure 44.24 and Figure 44.25, it is assumed that s ¼ 1: Note from Equation 44.77 and Equation 44.78 that we must consider the following measures to obtain a higher insertion loss. 1. Make the distance between attachment points or studs as large as possible (decrease nP and nL ). 2. Make the air space as large as possible (decrease fr ).

References 1. 2. 3. 4. 5.

Shiraki, K., ed. 1987. From Designing of Noise Reduction to Simulation (in Japanese), Ouyou gijutsu Shuppan, Chiyoda ku, Tokyo. Tokita, Y., ed. 2000. Sound Environment and Control Technology, Vol. I, Basic Engineering (in Japanese), Fuji techno system, Bunkyo ku, Tokyo. Okura, K. and Saito, Y., Transmission loss of multiple panels containing sound absorbing materials in a random incidence field, Inter noise, 78, 637, 1978. Beranek, L.L., ed. 1988. Noise and Vibration Control, INCE/USA, Ames, IA. Sharp, B.H. 1973. A study of techniques to increase the sound insulation of building elements, Wyle Laboratory Report WR 73 5, El Segundo, CA.

45

Statistical Energy Analysis 45.1 45.2

Introduction ........................................................................ 45 1 Power Flow Equations ........................................................ 45 2

45.3

Estimation of SEA Parameters ........................................... 45 4

45.4

Takayuki Koizumi Doshisha University

Power Flow Equations of a Two Subsystem Structure † Power Flow Equations of a Multiple Subsystem Structure

Modal Density † Internal Loss Factor Factor † Input Power

†

Coupling Loss

Application in Structures ................................................... 45 7 Application for Prediction of Noise in a Tractor Cabin Application for Prediction of Noise and Vibration in a Building

†

Summary This chapter presents the basics of statistical energy analysis (SEA) as applied to acoustic problems in structural systems. Power flow equations for structures consisting of two or more subsystems are described. The modeling and analysis procedures for the structural subsystems and acoustic subsystems are given. An estimation procedure of the necessary SEA parameters is given. The practical application of the SEA procedure in structures is illustrated using a two story building as an example.

45.1

Introduction

This chapter describes the basic concepts of the method of statistical energy analysis (SEA) and presents its application to structures. The analysis and computation techniques for vibration response and radiating sound in instruments and structures vary according to the characteristics of the physical object and the frequency range of interest. Here, we analyze vibration and noise in relation to a rather large scale structure over a wide frequency band. Extensive computations are usually required, when, for example, the finite element method (see Chapter 9) is used for the computations, with respect to a given oscillation mode. In particular, when the computations must be performed in the high frequency range and when many modes are included in the frequency band, the level of computation becomes considerable, generally resulting in reduced computational accuracy. To supplement the weak point of the traditional approach, it is necessary to redistribute statistically the energy equally from all modes in the analytical frequency band. This allows computed results to be compared with experimental results for a structure across a wide frequency band. This is the SEA method [1]. Early in its development, the objective of this analytical method was to predict the vibration response of artificial satellites and rockets that receive sound excitation when the jet discharges, and to predict the response of vibration stress in the boundary layer noise of an aircraft’s airframe. It also became a model that allows an 45 1

45 2


exciting force to be statistically (randomly) diffused (distributed) over a wide frequency band. This technique considers energy of excitation of a diffused (distributed) sound field and its variables that represent the sound pressure, acceleration, and force. Thus, it can be applied to problems of solid borne sound in which vibration propagates through each element [2] and problems of air borne sound in which multiple barriers exist [3], even when more excitation points than one are present.

45.2

Power Flow Equations

With the SEA method, we do not deal with specific characteristic modes (see Chapter 3 and Chapter 4) of the analyzed structure. Instead, we consider the structural components as a set of equivalent vibrating elements, and evaluate the vibration condition of the components as a macroscopic quantity averaged statistically over the frequency band and space (by describing the energy). We assume that the vibration modes within a given frequency band are distributed uniformly and are excited to the same degree. Using the SEA method, we can formalize the relationships of power flows between subsystems, and by solving these relationships, we can compute the energy stored in each subsystem. Next, the equations of such basic power flow [4] are explained.

45.2.1

Power Flow Equations of a Two-Subsystem Structure

The power flow relationships of a structure consisting of a two subsystem structure are shown in Figure 45.1. The equations for the power flows between subsystem 1 and subsystem 2 under typical conditions are expressed as Subsystem 1: Pi1 ¼ Pl1 þ Pl2

ð45:1Þ

Subsystem 2: Pi2 ¼ Pl2 þ P21

ð45:2Þ

where Pi1 is the input power to subsystem 1 from outside, Pl1 is the internal power loss of sub system 1, and P12 is the transmitted power from subsystem 1 to subsystem 2. The internal power loss, Pl1 ; is written as Pl1 ¼ vh1 E1

ð45:3Þ

Pi1

Pi2

Sub system 1 n1

h1

Pl1 FIGURE 45.1 subsystems.

E1

P12 h12 P21

Sub system 2 n2

h2

E2

Pl2 Power flow relationships between two

where v is the central angular frequency in the band, E1 is the energy in the bandwidth Dv of subsystem 1, and h1 is the internal loss factor (ILF). The average modal energy Em1 in subsystem 1, and Em2 in subsystem 2, are given by Em1 ¼

E1 E ; Em2 ¼ 2 N1 N2

ð45:4Þ

where N1 is number of modes in the bandwidth Dv of subsystem 1, and N2 is number of modes in the bandwidth Dv of subsystem 2. The transferred power, P12 ; between subsystems 1 and 2 is expressed as P12 ¼

P21 ¼ P 012

P 021

ð45:5Þ

P 012 ¼ vh12 E1 ¼ vh12 N1 Em1

ð45:6Þ

P 021 ¼ vh21 E2 ¼ vh21 N2 Em2

ð45:7Þ

Statistical Energy Analysis

45 3

where h12 and h21 are the coupling loss factors (CLFs) from subsystem 1 to subsystem 2, and from subsystem 2 to subsystem 1. They satisfy the reciprocity relationship h12 n1 ¼ h21 n2 : Therefore, transferred power, P12 ; becomes P12 ¼ vh12 N1 ðEm1

Em2 Þ ¼ vh12 N1

E1 N1

E2 N2

ð45:8Þ

Consequently, the power flow equations (Equation 45.1 and Equation 45.2) can be expressed as follows: Pi1 ¼ vh1 E1 þ vh12 N1

E1 N1

E2 N2

ð45:9Þ

Pi2 ¼ vh2 E2 þ vh21 N2

E2 N2

E1 N1

ð45:10Þ

If the SEA parameters (i.e., the modal density, intrinsic loss factor, CLF, and input power) are given, then each subsystem’s energy condition can be easily computed.

45.2.2

Power Flow Equations of a Multiple Subsystem Structure

By expanding the formulation in the previous section, it is possible to formalize the power flow relationships of a structure composed of multiple subsystems in the same way. The power flow equation for a structure composed of N subsystems is expressed by the following equation in the matrix form: ! 2 3 N X þ n n · · · n h h h h 6 1 7 1i 1 12 1 1N 1 6 7 2 3 2 3 i–1 6 7 E1 =n1 Pi1 6 7 ! 6 7 6 N 7 6 7 X 6 7 6 6 7 6 7 6 E2 =n2 7 h21 n2 h2 þ h2i n2 · · · h2N n2 7 6 Pi2 7 6 7 6 7 6 7 7£6 i–2 v6 ¼ 6 . 7 ð45:11Þ 6 7 6 . 7 6 . 7 6 7 6 .. 7 7 6 . 7 6 7 4 .. .. .. .. 5 4 5 6 7 . . . . 6 7 6 7 E =n P ! N N iN 6 7 NX1 4 5 ··· ··· hN1 nn hN þ hNi nN i–N

From Equation 45.11, if the SEA parameters are given in the same way as for the structure of two subsystems, then the energy equation of each subsystem can be obtained. The average energy of a subsystem is expressed by the following equations by using the vibration velocity and sound pressure: E ¼ Mkv2 l

ð45:12Þ

Mkp2 l Z02

ð45:13Þ

E¼

where M is the mass of the subsystem, kv2 l is the average spatial square of the vibration velocity, kp2 l is the average spatial square of the sound pressure, and Z0 is the specific acoustic impedance of air. Accordingly, if each condition of component’s energy is determined from Equation 45.11, it is possible to compute the vibration variable and the sound pressure with Equation 45.12 and Equation 45.13.

45 4

45.3


Estimation of SEA Parameters

To solve the power flow equations, it is necessary to determine the SEA parameters (i.e., the modal density, ILF, CLF, and input power). In this subsection, a method is given for computing the SEA parameters.

45.3.1

Modal Density

45.3.1.1

Structural Subsystem

Modal density is a key parameter for determining the dynamic characteristic of a structure. The number of modes, N; included in the frequency band (for estimation), is a factor denoting how easily energy, in transferring between subsystems, can be obtained. To determine N in the prescribed frequency band, it is first necessary to determine the modal density nð f Þ; that is, the gradient of N in the frequency band. The modal density of a structural subsystem is computed by using the following equation [4,5]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dN 1 A 12rð1 n 2 Þ nð f Þ ¼ ¼ ¼ ð45:14Þ df f0 2t E where A is the area of cross section, t is the thickness of the structural subsystem, r is the mass density, n is the Poisson’s ratio, E is the Young’s modulus, and f0 is the fundamental natural frequency of the structural subsystem. 45.3.1.2

Acoustic Subsystem

The modal density of an acoustic subsystem is determined by the following analytical equation [6]: nð f Þ ¼

dN 4pV ¼ 3 f2 c df

ð45:15Þ

where c is the speed of sound propagation within the acoustic subsystem, and V denotes the volume of the acoustic subsystem. Modal density of the cavity in the low frequency band is deduced in a similar manner to that in the two dimensional space. Define the depth of the cavity by d; and the frequency of the standing wave in the cavity by fd ¼ c=2d: If f , fd ; then the modal density is assumed to be uniformly distributed, and is estimated by nð f Þ ¼

2pS f c2

ð45:16Þ

where S is the area of the cavity. If f . fd ; modal density can be estimated using Equation 45.15, because the cavity is designated as acoustically three dimensional.

45.3.2

Internal Loss Factor

45.3.2.1

Structural Subsystem

The ILF, h1 ; of a subsystem gives the loss ratio when the input power to the subsystem from the outside is converted to kinetic energy of the subsystem. An excitation test to measure the damping ratio is employed to estimate the ILF of the structural subsystem. There are several methods for estimating the internal loss factor. The ILF applied in the SEA method is estimated by measuring input energy and output energy simultaneously, or by measuring the attenuation ratio within a given period of time. Both methods require the same setup to conduct an excitation test, and one is able to improve the measurement precision by conducting both methods.


45 5

With the energy measuring methods mentioned above, the ILF can be estimated by ðf 2

ReðYÞF 2 df * + h¼ ðf 2 2 v df v0 M f1

ð45:17Þ

f1

where Y is the complex mobility at the driving point in the range of f1 to f2 ; F 2 is the power spectrum of the input vibration force, and v2 is the power spectrum of the response speed. In addition, k l denotes the space average operator. 45.3.2.2

Acoustic Subsystem

The ILF of an acoustic subsystem is determined by [7]

h¼

cSa 4V v

ð45:18Þ

where a is the average acoustic absorption coefficient, V is the volume of the acoustic subsystem, and S denotes the surface area. The acoustic absorption coefficient can be estimated by measuring the reverberation time. Both the ILF of the cavity in the low frequency band and the modal density are deduced similar to that in two dimensional spaces. For f , fd ; the ILF is estimated using

h¼

cSp ap pvVc

ð45:19Þ

where ap is the acoustic absorption coefficient in the cavity, Sp is the peripheral area of the cavity, and Vc is the volume of the cavity. For f . fd ; the modal density can be estimated using Equation 45.18 because the cavity is taken as an acoustic subsystem.

45.3.3

Coupling Loss Factor

45.3.3.1

Between Structural Subsystems

The CLF hij gives the loss ratio when power transmits between two subsystems [4]. For example, the CLF between two flat plates can be estimated using

hij ¼

cgi Lc t pvSi

(a)

I-type

(b)

L-type

(c)

T-type

ð45:20Þ

where cgi is the group velocity of the bending waves, Lc is the coupled length, Si is the surface area, and tij is the energy transmission factor from subsystem i to subsystem j. The transmission factor varies with the type of coupling, for example, I type, L type, or T type shown in Figure 45.2. In this section, we use energy transmission efficiency of vertical incidence, reported by Cremer [7].

FIGURE 45.2

Coupled type.

45 6


45.3.3.2

Between a Structural Subsystem and an Acoustic Subsystem

Coupling power between a structural subsystem and an acoustic subsystem is the power flow based on resonance at transmission. The CLF between a structural subsystem and an acoustic subsystem is given by

hij ¼

Z 0 Sc s vMi

9 12 7

11

4

14

3

16

ð45:21Þ

where Z0 is the specific acoustic impedance of air, Sc is the surface area of coupling, s is the acoustical radiation efficiency, and Mi is the mass of the structural subsystem. 45.3.3.3 Between an Acoustic Subsystem and a Cavity

10

1

2

cs Scs tm 4vVs

Input Power

45.3.4.1

Vibration Input Power

13 8 6

FIGURE 45.3

Modeling of the tractor cabin.

Pi4

The vibration input power PiN is given by PiN ¼ vMi kvi2 l

4

Pl4

14

9

12

16

15

6

7

11

18

17

8

10

1

ð45:23Þ

where Mi is the equivalent mass, kvi2 l is the spatial average of square of the vibration velocity, and v is the central angular frequency. 45.3.4.2

Acoustical Input Power

13

The acoustical input power Ps is given by Ps ¼

kp2 lS2 nð f Þ c2 srad 4M 2pSf 2

17

ð45:22Þ

where cs is the sound velocity in the acoustic space, Vs is the volume in the acoustic space, tm is the transmission factor at random incidence depend ing on mass flow through the partition, and Scs is the coupling area.

45.3.4

18

5

Coupling power between an acoustic subsystem and a cavity is power flow based on resonance in transmission. The CLF between an acoustic subsystem and a cavity is given by [8]

hsc ¼

15

5

ð45:24Þ 19

2

2

where kp l is the square average of the input sound pressure, S is the surface area of the component, M is the mass of the component, nð f Þ is the modal density, and srad is the sound radiation factor [2].

3 FIGURE 45.4

Power flows in the tractor cabin.


Application in Structures

In this section, we present an example of modeling with significant analytical accuracy, and discuss the application of the SEA method for structures.

45.4.1 Application for Prediction of Noise in a Tractor Cabin Figure 45.3 shows a model of the tractor cabin. This figure shows that the cabin consists of a floor, a door, a ceiling, and other components. Figure 45.4 presents the power flow relationships within the cabin [9,10].

FIGURE 45.6

Sound Pressure Level [dB]

45.4

45 7

70 65 55 50

Experimental Value Analytical Value

45 40

125 160 200 250 315 400 500 630 800 1000 1250

Frequency [Hz] FIGURE 45.5 Results of estimating the sound pressure level in a cabin.

The configuration of the building.

42

39

29

36

24

22

9

28

11

38

8 10 12

1

7 4

23

25

13

14

2

35

19

33

41

37

6

34

5 30

21

15

20

3 17

16

18

26 31 40

27

32

FIGURE 45.7

The power flow relationships between structural subsystems in the entire building.

45 8


28

42

53

10 4 14 5

31 15

6

7

8

9

24

38

23 37

FIGURE 45.8

The power flow relationships in the acoustical laboratory room.

The results obtained for the cabin are shown in Figure 45.5. According to this figure, the disagreement between the computation and the measurement was found about 2 dB in the medium to high frequency band.

45.4.2

Application for Prediction of Noise and Vibration in a Building Sound Pressure Level [dB] Re : 2×10−5 [Pa]

Consider a two story laboratory reinforced with 70 Experimental Value concrete [11]. The building configuration is shown Analytical Value 60 in Figure 45.6. This building comprises a driving room, an acoustical laboratory room, a computer 50 room, a measurement room, an equipment room, 40 and others. We modeled the structural subsystem using 30 I , L , or T type connected points, and the acoustic subsystem as an element shown in 20 31.5 63 125 250 500 1000 Figure 45.6. Frequency [Hz] The SEA model constructed in this manner is composed of 61 elements, and has 244 connecting points. Subsystems 1 to 17 and subsystems 19 to 42 FIGURE 45.9 Estimated sound pressure level results are concrete components. Subsystem 18 and for other rooms. subsystems 43 to 48 are plasterboard components; subsystems 49 to 55 are room components; and subsystems 56 to 61 are cavity components. For example, Figure 45.7 shows the power flow relationships between structural subsystems in the entire building, while Figure 45.8 shows them in the acoustical laboratory room. Here, the thin dotted, dotted, and solid lines indicate the I , L , and T type combinations, respectively. Subsystem 53 is the room component, and it is connected with all structural components shown in Figure 45.8. The plasterboards located between the computer room and the measurement room are considered as a partition; therefore, connections between subsystem 49 and subsystems 56 to 59 (cavity components), and subsystem 50 and subsystems 60 and 61 (cavity components) are derived from nonresonant modes. The results obtained for some other rooms are shown in Figure 45.9. Computing accuracy in this building is worse than in the cabin because the structure of this building is complicated, although the differences between the computed values and the measured values were approximately 4 dB in the medium to high frequency band. The computations take approximately 10 sec, so the workload on the personal computer is quite light.


45 9

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Lyon, R.H. 1975. Statistical Energy Analysis of Dynamical Systems: Theory and Applications, MIT Press, Cambridge, MA. Irie, Y., Solid propagation sound analysis by SEA, Mitsubishi Heavy Ind. Tech. Rev., 21, 4, 571 578, 1984. Crocker, M.J., Sound transmission using statistical energy method, AIAA J., 15, 2, 75 83, 1977. Irie, Y., Solid propagation sound analysis by SEA, Nihon Onkyo Gakkai shi, 48, 6, 433 444, 1992. Lyon, R.H. and Dejong, R.G. 1995. Theory and Applications of Statistical Energy Analysis, 2nd ed., Butterworth Heinemann, Oxford. Craik, R.J.M. 1996. Sound Transmission Through the Buildings Using Statistical Energy Analysis, Gower Press, England. Cremer, L., Heckle, M., and Unger, E.E. 1973. Structure Borne Sound, Springer, New York, pp. 347 370. Mechanical Noise Handbook, JSME, Japan Society of Mechanical Engineers, pp. 179 181, 1991 (in Japanese). Koizumi, T., Tsujiuchi, N., Kubomoto, I., and Ishida, E., Estimation of the noise and vibration response in a tractor cabin using statistical energy analysis, JSAE Paper, 28, 4, 49 54, 1997. Koizumi, T., Tujiuchi, N., Kubomoto, I., and Ishida, E., Estimation of the noise and vibration response in a tractor cabin using statistical energy analysis, SAE Paper, 1999 01 2821, 1999. Koizumi, T., Tujiuchi, N., Tanaka, H., Okubo, M., and Shinomiya, M., Prediction of the vibration in building using statistical energy analysis, IMAC Paper, 7 13, 2002.

Glossary

A 1 B vibration Vertical vibration of an aircraft represented by the sum of the vibrations measured at the two sides of the cockpit. A 2 B vibration Roll vibration of an aircraft represented by the difference in the vibrations measured at the two sides of the cockpit. A-weighted sound level Sound pressure level that has been weighted according to A weighting networks, commonly used in measurements that relate to human response to noise (dBA). See Sound level. Abnormality scaling The process of scaling the level of abnormality in each vibration feature between 0 and 1 to signify the degree of deviation from its normal value. Absolute acceleration shock response spectrum This is an SRS for acceleration. See Response spectrum (RS) and Shock response spectrum (SRS). Absolute error The absolute value of the difference between the exact solution and the computed solution, or between the true value and the measured value. Absolute limit Conditions that could result in catastrophic failure. Absorber See Vibration absorber. Absorption coefficient Fraction of the incident energy absorbed by a sound surface. AC Alternating current. AC bridge A bridge circuit that uses an AC excitation. It is more stable than a DC bridge. G-1

G2


AC motor A motor that uses an AC excitation. Single phase and multiphase, induction, and synchronous varieties are available. Acceleration Rate of change of velocity with respect to time along a specified axis (m/sec2 or g) or about an axis (rad/sec2). Second derivative of displacement. Gives both magnitude and direction (vector). Accelerogram A recording of the acceleration of the ground during an earthquake. Accelerograph An instrument (accelerometer) that records the acceleration of the ground during an earthquake. Accelerometer Sensor or transducer measuring acceleration; output (typically voltage) is proportional to acceleration. A piezoelectric accelerometer contains one or more piezo electric crystal elements, which produce electric charges or voltages when stressed in tension, compression, or shear. Can be two or three axial. The output can be in digital in some cases, with proper transducers. Acceptability index The percentage of times an adjustment set will result in the vibration satisfying the specification. Accuracy How well a measurement (reading) or a computed result represents the true (correct) value. Accuracy of a value and accuracy of an instrument are related because the latter can determine the former. Acoustic absorption coefficient A ratio (fraction) expressing the ability of a material to absorb acoustic power. Acoustic emission (AE) Sound or ultrasound pulses generated during crack initiation or propagation in materials and coatings from being subjected to stress. Acoustic enclosure A rigid box surrounding noise sources to provide noise isolation for machinery or equipment. Acoustic holography A method using phase and amplitude information over a closed surface to reconstruct a sound field in the region outside the surface. Acoustic impedance Acoustic pressure acting on a surface divided by the volume (or particle) velocity at the surface. Assumes a plane wave is falling perpendicular to the surface. May assume acoustic material of infinite depth on which the wave is applied. Acoustic power The rate of acoustic energy flow emitted or radiated by a specified sound or noise source (W). Acoustic resonance Condition of peak (maximum) response of a fluid borne system to a periodic driving pressure oscillation. Typically, the frequency at which the system tends to oscillate naturally.

Glossary

G3

Across wind response Response perpendicular to the direction of wind. Action plan When measured values of noise or vibration violate the local or national regulatory limits, “action plans” shall be developed for correcting of the situation. Active bridge A bridge circuit (with four bridges; e.g., Wheatstone bridge) that uses an operational amplifier (an active device, which uses an external power source). Active control See Active vibration control. Active damping Controlled damping where the damping force is generated by an actuator (with an external power source) in response to sensing (e.g., active suspensions and magnetorheology). Active device A device (hardware) that needs an external power source to operate. Active filter Uses an operational amplifier (an active device), as opposed to a passive filter, which uses only passive devices. Active gauge The strain gauge mounted on the deforming member and forms a part of the bridge circuit. See Dummy gauge. The bridge output is proportional to the change in the active gauge (automatically compensated for ambient changes by the dummy gauges). Active mount An engine mount that uses actuators (with external power) for providing proper suspension and isolation (force and motion). Active resonator absorber See Vibration absorber and Resonator absorber. A resonator absorber that utilizes an active device to generate resonance (see Resonance) within the absorber subsection. Active suspension A suspension system (of a vehicle) that uses actuators (with external power) with feedback control for providing proper suspension. Active vibration control A vibration control method in which an external force (utilizing external power) is applied to control vibrations. ADC An analog to digital converter, which samples an analog signal at some frequency (sampling rate) and converts the sampled data into a digital form. Adiabatic process A process that takes place with no heat transfer with the environment.

G4


Adjustable suspension element Typically includes variable rate damper and stiffness. Aeroelastic model The model that simulates the dynamic properties of buildings to capture the motion dependent loads. Air-borne sound A sound that is transmitted from one structure to another. Aliasing distortion The error that results due to sampling (discretization) of an analog signal. Along-wind response Response in the direction of wind. Amplitude Peak magnitude of variation (displacement, velocity, acceleration, etc.) of each oscillation of vibration. The maximum value of a sinusoidal quantity. A measure of floor vibration. Amplitude-dependent damping Nonlinear form of damping where its value depends on the amplitude but not frequency of motion, e.g., fluid friction or aerodynamic damping or whose magnitude is proportional to the square of the velocity; nonhysteretic internal damping. Amplitude ratio The ratio of forced vibration amplitude to the static amplitude. Analog device A device with analog circuitry (hardware) but no digital components. It deals with analog or time continuous signals but not digital signals. Analog-to-digital conversion See ADC. Anelasticity Property of real solids responsible for internal friction (prefix “an” means “other than”). Anisotropic damping where there are differences depending on directions relative to crystalline axes. Angular frequency Frequency of a periodic quantity, measured in radians per unit time. Cyclic frequency multiplied by 2p: The usual symbol is v: Angular velocity The rate of change of angular position (i.e., orientation) of an object (radians per second [rad/sec], revolutions per minute [rpm], etc.). Annular clearance The uniform gap between the rotor and stator of a turbine or compressor motor or some rotating machine. Antialiasing filter A low pass filter with its cut off set near half the sampling rate of the filtered signal in order to minimize the aliasing error.

Glossary

G5

Appendages Additional secondary devices. Additional nonstructural elements to the main structure to increase the building’s function to the occupants. Asymptotic stability Stability representation where the response naturally decays to zero. Asymptotic stability borderline Maximum depth of cut below which chatter will not occur for all spindle speeds. Attenuate Decrease. To make small. Attenuation of muffler Decrease in sound power between two points in a muffler system (noise suppressor). Autocorrelation Correlation of a signal with itself, with a phase shift (reference time shift) expressed as a function of the shift. Inverse Fourier transform of the power spectrum. Autonomous system Self contained system with internal control (no external control). A system whose parameters are time invariant (i.e., do not vary with time). Autotuning Automatic tuning (see tuning) of controller parameters to achieve desired performance (e.g., in vibration control). Average tuning iteration number (ATIN) The average number of iterations taken for tuning an aircraft. Average value A representative value of a data set. The centriod of the data. The approximate point at which the sum of moments of all the data values is zero. A first order statistic of random data. Mean value. Expected value. Mathematical expectation. Axi-symmetric wave A sound wave generated by a point dipole. When propagation of sound wave is described by polar coordinates, its magnitude is independent of the angle about the given axis. Axial Along the shaft centerline. Along the axis. Back e.m.f. An electromotive force (e.m.f.) is a voltage generated in a conductor located in a magnetic field when there is a relative motion (in an orthogonal direction). The back e.m.f. is the voltage generated in a moving coil that is connected to a voltage source (like motor rotor windings) and located in a magnetic field, where the generated e.m.f. opposes the applied voltage. Backlash Mechanical looseness in between meshed gear wheels due to unavoidable clearances between meshing teeth.

G6


Baffled space The idealized boundary condition to model the sound field generated by a compact source embedded in a plane large in size compared to wavelength, e.g., a radiation field generated by a circular vibrating piston mounted flush with the surface. Balance A condition where the rotating centerline between bearings coincides with that which defines the center of mass distribution. See Static balancing and Dynamic balancing. Balance weight adjustments Weights added to or removed from the rotor hub to tune vibrations through changes in blade mass. Balance weights are measured in grams or ounces, with positive adjustments representing the addition of weight. Balancing A procedure for adjusting the mass distribution of a rotor by adding or removing weight, with the goal of achieving less vibration amplitude at rotational speed. See Static balancing and Dynamic balancing. Band-pass filter A filter that allows through the signal components within a given range of frequencies, stopping (filtering) those above and below that band. Bandstop filter A filter that suppresses a given range of frequencies, transmitting only those above and below that band. Also called band elimination filter. Also called a notch filter when the band is very small. Bandwidth Operating frequency range of a device. The difference between the upper and lower cut off frequencies at which the signal is attenuated by 3 dB. The difference between the limiting frequencies within which performance of a device falls within specified limits with respect to some characteristic. A measure of speed of response of a device. Barkhausen effect Jerkiness in the magnetization curve of a ferrous specimen. Base isolation See Isolation. An engineering technology that uncouples the motion of a structural system from the ground motion by inserting an isolation layer under the structure. Basis Countable set of linearly independent functions or vectors that can represent a general function or a vector space. Bathtub curve Curve (based on anecdotal and statistical data) describing the frequency of failures. Applicable to an individual machine or population of machines of the same type. Beam A structural (vibrating) component that is long and slender and has some flexibility (bending or flexure). Commonly modeled by Euler Bernoulli equation or Timoshenko equation.

Glossary

G7

Bearing A support for the ends of a shaft, a rotating member, etc. May be dry or lubricated. Will introduce some damping, but it should be minimized for proper operation. Linear bearing is one that provides a guideway for rectilinear motion (rather than rotary motion). Beats The alternating and significant rise and fall of vibration amplitude caused by two sources vibrating at nearly equal frequencies (or engines or rotating machines rotating at nearly equal speeds). Bernoulli Euler beam A common analytical model (partial differential equation with respect to time and length coordinate) used to analyze slender beams. Rotatory inertia and deformation due to shear are neglected in this model (holds for slender beams), unlike the Timoshenko beam model. Bicoherence Second order correlation measure between signals. Biorthogonal wavelets Symmetric wavelets with compact support created by two sets of wavelets; one set for decomposition and one set for reconstruction. Bisection method A nonlinear root finding method which repeatedly bisects the interval to find a root. For this method to be successful, the initial interval has to be such that the function is positive at one endpoint and negative at the other. Blast mitigation Physical defensive measures taken to reduce the damaging effects of explosions from devices introduced by hostile parties. Includes both exterior hardening (wall, etc.) and internal measures taken to improve the survival of the protected resources (e.g., using more stable chemicals in a protected process). Bode plot The pair of plots of amplitude versus frequency and phase versus frequency. Usually, the amplitude is given in dB (or in log scale) as a nondimensional magnification factor of a transfer function magnitude. The frequency is given in log scale. Boom A long, overhanging pole or beam like structure. As in a crane or a robot, a boom can be used to position and manipulate objects at a remote location. Boundary conditions (BCs) The conditions (force, moment, motion, deformation, etc.) at the boundary locations of an object (e.g., beam, plate, structure) used for solving the governing equations of the object. These are conditions expressed with respect to spatial coordinates (space) rather than time. Also require initial conditions (ICs) which are conditions at the initial time, for a complete solution of the system equations. Boundary layer A thin stationary layer of liquid with a hypothetical boundary next to a solid surface where much of the velocity changes (gradients) occur.

G8


Bracing Temporary supports of a structure using struts in vertical and lateral directions during seismic rehabilitation of the structure. Bridge circuit An electric circuit with four arms having resistances or impedances, and input (source) and an output. Where pure resistances are used, we have a Wheatstone bridge. Hydraulic circuits also may be represented as bridge circuits, and analyzed using similar techniques. Bridge balance The condition (the relation between bridge resistances or impedances) such that the output of the bridge circuit is zero (even when the source provides an excitation). Bridge constant The constant parameter relating a variation in a bridge arm (resistance or impedance) to the corresponding bridge output, starting with a balanced bridge. Broadband (wideband) process A random process where most of the energy is in a wideband of frequency. Broadband excitation An excitation with a significant range of frequency (almost a white noise spectrum). Buffer A device provided at an isolation bearing which will be hit by the moving plate of an isolation bearing to limit the bearing displacement during extreme earthquakes. Bulk modulus Change in volume divided by initial volume gives the bulk strain. The bulk stress (or pressure) divided by the bulk strain is the bulk modulus. Bump Simple shock that is generally repeated many times when testing, e.g., half sine, 10g; 16 msec, 3000 bumps (shock) per axis, 3 bumps per second. Bump test See Bump. An impact test. A test carried out on an object (equipment, structure, etc.) using a bump impact. Butterworth filter A low pass filter with many poles (i.e., high order) to provide good roll off (i.e., sharp cut off). Calibration Adjusting some parameters or conditions (using knobs, keys, computer interface, etc.) of a device so that its performance (output or reading) is accurate with known engineering units. Cantilever A flexible beam fixed at one end (zero displacement and zero slope) and free at the other (zero bending moment and zero shear force). Carrier A sufficiently high frequency signal that can carry another (transient) signal of lower frequency for maintaining the accuracy of the carried signal during signal transmission and analysis.

Glossary

G9

Carrier frequency Frequency of the carrier signal. Carrier signal See Carrier. Cartesian coordinates Orthogonal (perpendicular) coordinates in a rectangular coordinate frame with a common origin. Cascade connection Connection in series of two or more devices. Cascade plots Successive spectra plotted with respect to time and displayed in a three dimensional manner. Also known as Waterfall plots. Catastrophic failure Sudden and complete failure of a machine. Cannot be analyzed by conventional methods of nonlinear system analysis. Causality Cause effect behavior of a dynamic system. The numerator order (or the input order) cannot be greater than the denominator order (or the output order) of the system transfer function. This determines the physical realizability of the system. Center of gravity The point representing the average position of weight in a body. The point of intersection of orthogonal axes, relative to which the first moment of weight is zero. Typically the same as the center of mass. Denoted by COG or CG. Center of mass The point representing the average position of matter (mass) in a body. The point of intersection of orthogonal axes, relative to which the first moment of mass is zero. Typically the same as the center of gravity, denoted by COG or CG. Center of stiffness The point of intersection of orthogonal axes relative to which the first moment of stiffness is zero. Centered difference Finite difference approximation that uses the Taylor series expansions on either side of the point in question. Centrifugal acceleration Acceleration of a rotating mass (e.g., attached to a string or arm) directed towards the center rotation. Equal to v2 r where v is the angular speed of rotation and r is the radius of rotation. Centrifugal compressor A compressor (a machine in which air or gas is compressed) by the mechanical action of rotating vanes or impellers. The flow is in a radial direction through the impeller, which imparts velocity (kinetic energy) to the fluid. Passing through a diffuser section at exit, the kinetic energy is converted into pressure (potential energy).

G 10


Centrifugal pump See Centrifugal compressor. As the impeller of the pump (enclosed in a casing or volute with an inlet and a discharge connection), kinetic energy is imparted to the fluid by the rotating impeller and is converted into pressure at the outlet. Centripetal acceleration See Centrifugal acceleration (synonymous). Centroid Same as the center of mass or center of gravity for practical purposes. Cepstra Plural of cepstrum. Equivalent to spectra, but in the “quefrency domain” rather than the frequency domain. Cepstrum The inverse Fourier spectrum of the logarithm of the Fourier spectrum of a signal. Change limit Warning levels that provide early warning well in advance of the absolute limit. Channel A signal line in a piece of equipment or hardware. Both input channels and output channels are important. The capability and complexity increase as the number of channels increases. Chaos Deterministic form involving the “butterfly effect” (work of E. Lorenz). Characteristic equation See Characteristic polynomial. Characteristic polynomial The polynomial, when set to zero (characteristic equation) and solved, gives the eigenvalues (or poles or characteristic roots) of a dynamic system. The denominator polynomial of the transfer function of a dynamic system. The roots are complex in general. The imaginary parts give the natural frequencies. The real parts give the reciprocal of time constants or damping level. Charge amplifier An electronic amplifier used to convert the high output impedance of a device (e.g., piezoelectric accelerometer) into low output impedance so that it is compatible with common signal acquisition and readout devices (processors, controllers, displays, recorders, etc.). Chatter An instability in a machine tool, which occurs at high frequencies, causing poor machining quality, increasing tool wear, generating noise, and causing other problems to the machine. Chatter frequency Dominant frequency of vibration during the occurrence of chatter (unstable vibration in a machine tool).

Glossary

G 11

Chatter marks Distinctive undulating patterns left on a part after machining that are indicative of chatter. Cholesky decomposition A symmetric positive definite matrix may be written as the product of a lower triangular matrix and its transpose (an upper triangular matrix): A ¼ U t U: Circle fit Fitting vibration test data to a transfer function that approximates to a circle on the complex plane (horizontal real axis and vertical imaginary axis). True for a single degree of freedom (single DoF) in relation to some specific transfer functions (e.g., mobility) with some types of damping. Parameters of the system (damping, natural frequency) can be extracted from the fitted circle. Used in experimental modal analysis or modal testing. Cofferdam A temporary watertight enclosure that is pumped dry to expose the bottom of a body of water so that construction (e.g., of piers) may be undertaken. COG Centre of gravity (or center of mass). See their definitions. Coherence First order correlation measure (normalized) between different signals, expressed in the frequency domain (as a function of frequency). It has a magnitude and a phase. Coincidence effect A form of acoustic resonance phenomena where the transmission loss decreases rapidly when the acoustically forced bending wave speed is equal to the free bending wave speed (flexural natural frequency) in a structure (plate). Cold welding Bonding between two parts of seismic isolation bearings (which are supposed to move relatively to each other during earthquake attacks) because of long durations of contact. Color centers Defects introduced into a crystal such as lithium fluoride by ionizing radiation. Combined adjustments The net adjustments to opposite blade pairs in a four bladed aircraft. Common-mode rejection Improvements in signal to noise ratio (SNR) by the reduction of noise signals that are common to both inputs of a differential amplifier. Commutator A device that provides power to a set of coil segments (circuits) in a rotating electric machine (motor). Analogous methods may be used in linear actuators (which perform rectilinear motion rather than rotary motion). A commutator with a slip ring with split segments connected to the circuit segments, and brushes (carbon blocks) in contact to provide the power, is called a “slip ring brush commutator.” Alternatively, electronic commutation may be used where the power is switched into to different circuit segments by electronic means.

G 12


Compact source condition When the scale of the source region is small (compared with the wavelength of generated sound wave), the compact source condition is mathematically valid for modeling the sound field. See Monopole and Dipole. Compensating element A resistor or a coil in a circuit (e.g., eddy current sensor, bridge circuit, potentiometer circuit), which is used to compensate for environmental changes (e.g., change in temperature). Complex modes In some vibrating systems with some complex types of damping, the mode shapes (eigenvectors) obtained analytically (see mode shapes) can be complex (as opposed to real). Strictly then, these mode shapes are not realizable in practice. In fact, some distinct shapes will appear (similar to real modes) but with some changes (e.g., movements of nodes, small changes in phase of the moving elements) at the corresponding natural frequencies. Complex propagation constant This constant g has two parts; the real part, d; called the attenuation constant in decibels per meter (dB/m), and the imaginary part, b; called the phase constant in radians per meter (rad/m). Compound pendulum See Pendulum. A pendulum having more than one degree of freedom (e.g., two swinging degrees and one twisting degree). Compressor circuit In controlling a vibration shaker, this circuit ensures that the drive signal to the shaker satisfies the required spectrum of the test signal that needs to be applied to the test object. It compresses the frequency components that exceed the specification. Condition-based maintenance A strategy where maintenance is performed after assessing the actual condition of the machinery (rather than scheduling it at fixed intervals) in order to optimally schedule maintenance, thereby achieving maximum production and still avoiding unexpected catastrophic failures. Condition monitoring Monitoring the operation of a machine or other dynamic system, for the purpose of checking proper operation and possible faults or malfunction. Vibration signal analysis is one approach to condition monitoring. Confined concrete Concrete confined by transverse reinforcement, commonly using closely spaced steel spirals or hoops to increase the ductility. Consistent mass Representation of the mass of a system based on the virtual work principle and that results in a nondiagonal mass matrix. Constant percentage bandwidth Frequency bandwidths over the entire frequency range that remain a constant percentage of the frequency being monitored.

Glossary

G 13

Consumer product noise Noise perceived by a user or bystander of a noise generating product over which the individual has some control. Excludes occupational noise and environmental noise. Contrast enhancement Adaptation of the weights associated with features classified as normal or abnormal to enhance the contrast. Continuous monitoring Constant or very frequent data collection and analysis, usually with a permanently installed monitoring system. Continuous wavelet transform Wavelet transform on continuous ranges of scale shift parameters. Controllability The ability of a controller to drive the response of the dynamic system, which it controls, to any desired value in a finite time, using a finite input (excitation). Convolution integral Integral of the product of two functions (usually of time), where one function is the shifted complement of the other (i.e., one is a function of t and the other is a function of t t; and the integration is done over t). If one function is the system input and the other is the response of the system to a unit impulse, then the convolution integral gives the system response to the given input. The Fourier transform of a convolution integral is the product of the Fourier spectra of the individual time functions. In general, a product in the frequency (or Laplace) domain is a convolution integral in the time domain, and vice versa. Correlation function The correlation function is the mathematical expectation of a random function with another random function in the time domain. It is a measure of how two random functions correlate with each other over various time differences. Coulomb damping Friction force that is constant in magnitude, and acts on a body in the opposite direction to the velocity of the body relative to the surface in contact. Coupling loss factor The fraction of the original vibrating energy between two subsystems that is lost when interconnected. Coverage index The reach of each sensor (accelerometer) location in reflecting faulty components. It is defined in terms of the structural influences that represent the proximity of each sensor (accelerometer) to various system (gearbox) components. Creep Slow dimensional changes of a specimen due to structural defects. They progress through three distinctive phases called primary, secondary, and tertiary. Crest factor Peak Level The ratio RMS Level of a waveform. It can indicate early stages of rolling element (bearing) failure.

G 14


Critical damping The minimum value of damping that produces a nonoscillatory decaying motion of a system under free vibration. Value of damping that provides most rapid response to a step function without overshoot. When this occurs, damping ratio ¼ 1. Critical damping coefficient Damping coefficient corresponding to critical damping. See Damping coefficient and Critical damping. This occurs when ðc=2mÞ2 ¼ k=m; where k ¼ spring constant, m ¼ mass, and c ¼ damping coefficient or damping constant. At this value, damping ratio ¼ 1. Critical frequency The minimum frequency at which the coincidence effect occurs when the propagation speed of a bending wave in a plate equals the speed of sound in air. Critical speed The rotational speed that is equal to a natural frequency of bending of a rotor shaft bearing system (rotor dynamic resonant frequency). It can cause large amplitude vibration or whirling with only slight excitation. Cross sensitivity The sensitivity of a device (e.g., sensor) in a direction orthogonal (perpendicular) to the direction of direct sensitivity. Normally, the cross sensitivity of a sensor has to be minimized and the direct sensitivity maximized. Crust The outermost major layer of the Earth, ranging from about 10 to 65 km in thickness worldwide. The uppermost 15 to 35 km of crust is brittle enough to produce earthquakes. Cumulative modification set The total number of tuning iterations performed in the field on an aircraft. Curve-fitting Fitting a set of data to an analytical curve, as accurately as possible. See Circle fit, which is a single DoF fit. Multi DoF curve fitting is used for complex systems with closely spaced modes. Used in experimental modal analysis and modal testing. Cut-off frequency The frequency of a filter beyond which the signal entering it is cut off (filtered out). Cycle The full range of states or values through which a periodic (cyclic) phenomenon or function passes before completely repeating itself. The complete sequence of values of a periodic quantity that occur during one period. For a periodic waveform (signal), this corresponds to one period (or 1/frequency). Cyclic frequency Frequency of a periodic variation in cycles per unit time (Units: Hertz or Hz, or cycles or cycles). Typical symbol is f. See also Angular frequency. D’Alembert’s principle An equivalent version of Newton’s Second Law where the acceleration is multiplied by mass, and the resulting “inertia force” is applied in the direction opposite to that of the acceleration. The dynamic systems become an equivalent static system in this manner.

Glossary

G 15

DAC Digital to analog conversion. Digital data from a digital device (e.g., computer) are interpolated and converted by a DAC into an analog signal for use with analog instruments. Damage boundary curve For a given acceleration, curve giving the critical velocity change and, for a given velocity change, the critical maximum shock acceleration that leads to a damage on the material (deformation, fracture, faulty operation after the shock, etc.). Damped natural frequency The natural frequency of free vibration of a damped linear system. Damper A device that dissipates mechanical energy and progressively diminishes the response of a mechanical system (in free motion). In the case of linear viscous damping, this device is sometimes called a dashpot. The resulting damping force is the product of the damping coefficient and the relative velocity of the two components of the damper. Damping The dissipation of energy of a mechanical system, which progressively diminishes (dampens) the magnitude of response (or the amplitude of vibration) of the system in free motion. The damping process converts and dissipates mechanical energy into other forms of energy such as thermal energy (heat transfer) and noise. Damping capacity Energy loss per cycle of the motion. Damping coefficient The constant by which the relative velocity of a body is multiplied to yield the viscous damping force acting on the body. Damping constant The same as damping coefficient. Damping ratio Ratio of actual damping coefficient to the critical damping coefficient in a system with viscous damping. Expressed mathematically, the damping ratio z ¼ c=cc : If less than 1, the system will be underdamped and will exhibit oscillations when disturbed or excited. If larger than 1, the system will be overdamped and the system response will diminish and die out with no oscillations. Damping redshift Change in frequency of a harmonic oscillator due to damping. Predicted by the viscous damping model, it is virtually immeasurable because of the small size of the shift. Dashpot Same as “damper” but assumes a linear viscous damping model and generally has three orthogonal damping forces. Daubechies wavelets A family of orthogonal wavelets with compact support and a specified number of vanishing moments.

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DC Direct current. DC bridge A bridge circuit using a DC excitation. Compare with an AC bridge. DC motor An electric motor that uses a DC excitation. Requires a commutator for operation. Commonly used as a servomotor, where motion sensing and feedback control are employed for accurate generation of motion trajectories. DC tachometer A speed sensor that uses the principle of an electric generator (electromagnetic induction). The excitation is DC. The output DC voltage is proportional to the measured speed. Detuning Deviation of a passive vibration control system (see passive vibration control) from its original design due to structural parameters changes, change in the excitation frequency, or change in nature of disturbance. Deflection Movement (or deformation) of a structure or parts of a structure under applied loads. In the case of an elastic body or spring, the distance it moves when subjected to a static or dynamic force. Degrees of freedom Denoted by DoF or dof or df. Minimum number of independent coordinates necessary to determine completely the state (position and orientation) of the considered system. This definition assumes a holonomic system. For a general (nonholonomic) system, the definition must use “incremental” coordinates and corresponding “incremental” displacements. Often used to classify mechanical structures, such as robotic arms. In the finite element analysis, DoF may be associated with nodes or elements. Three dimensional shell and beam elements have six DoF per node (three translations and three rotations). Delayed resonator (DR) absorber A form of a resonator absorber (see resonator absorber) to optimally utilize the effect of time delay in the processed signal. Delta function Dirac delta function. A unit impulse function. A function whose value is infinity at one point and zero elsewhere, with the integral of the function equal to 1. Demodulation Recovering the original signal from a modulated signal by removing the carrier signal. A common method is to multiply (by digital or analog means) the modulated signal by the carrier signal and then low pass filter the result. Depth-of-cut selection Technique where the depth of cut is chosen to avoid or suppress chatter in a machine tool. See Chatter. Design (basis) earthquake Denoted by DBE. The earthquake (as defined by various parameters, such as peak ground acceleration or PGA, response spectra, etc.) for which the structure will be, or was, designed.

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Desorption Release of gas molecules from a solid undergoing evacuation. Determinacy of machine structure A degree at which a physical structure complies with the law of cause and effect (see causality). This allows the system dynamic behavior to be modeled and analyzed mathematically, and the modeled system to be physically realizable. Detonation An explosive reaction that consists of the propagation of a shock wave through the explosive, accompanied by a chemical reaction that furnishes energy to sustain the shock propagation in a stable manner, with gaseous formation and pressure expansion following shortly thereafter. DFT Discrete Fourier transform or digital Fourier transform. A discrete version of the Fourier integral for use in numerical computation. The fast Fourier transform (FFT) uses a much more efficient algorithm for this computation. Dielectric constant A parameter of a capacitive material indicating the degree of its ability to store electric charges. Also known as permittivity. Differential transformer See LVDT and RVDT. A transformer with a movable ferromagnetic element in its core. Serves as a noncontact displacement sensor. Difference equation A discrete version of a differential equation. Used in computer simulation of dynamic systems. In the nonhomogeneous case (i.e., when there is an input function) there is an equivalent z transform transfer function (just like a nonhomogeneous differential equation having an equivalent Laplace transfer function or a frequency response function). Diffraction parameter This indicates the importance of the diffraction force. A larger parameter value means that the diffraction force is more important than the drag or inertia forces. Diffuser A component in a centrifugal pump adjacent to the impeller exit, which has multiple passages of increasing area for converting velocity (kinetic energy) to pressure (potential energy). Digital filter In contrast to an analog filter, it uses a digital processor and a filter algorithm to perform filtering on digital data. Digital Fourier analysis See DFT and FFT. Fourier analysis using digital data. Converts time domain information into frequency domain (spectral) information. Can determine Fourier spectra, transfer functions, power spectra densities, coherence functions, etc. through this analysis. Digital oscilloscope An instrument that has the usual oscilloscope capabilities (acquisition and display of one or more signals) together with a variety of other capabilities such as extraction of signal parameters (mean, time constants, periods) and even FFT and spectral analysis.

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Digital signal analyzer Signal analyzer. Digital spectrum analyzer. A hardware digital device that carries out Fourier analysis and related operations on signals at very high speed (almost real time). Faster than a software based analyzer. Digital-to-analog conversion See DAC. Digital transducer A transducer that can sense a signal and present it as a pulse sequence (pulse count or pulse rate) or digital data. It is generally very accurate, but necessarily introduces a quantization error. Dilation The squeezing or stretching of the mother wavelet in the time domain in order to analyze the signal in different scales. Dipole See Monopole. The second basic sound source where a pair of monopoles, close together, opposite in sign, and equal in magnitude are located along one axis. The sound sources like a compact vibrating body, with a characteristic axis or a point force acting in the medium that may be modeled as a dipole. Dirac delta function See Delta function. Direct sensitivity See Cross sensitivity. Directivity The dependency of a sound field on the angular position. Although the monopole sound field has no preferred direction, the dipole and the multipole sound fields have a particular dependency on the angular position. The dependency varies with the condition of the compactness of the sound source. Discrete Fourier transform See DFT. A mathematical transformation used to transform a discrete time signal (sampled data) into a discrete set of spectral values equally spaced in the discrete frequency domain. It can be used to numerically (computationally) determine the spectrum of any general (e.g., nonperiodic) time signal. Discrete wavelet transform Wavelet transform on a discrete set of scale shift parameters. Dislocation Disruption to the order of an ideal crystal, e.g., by an extra plane of atoms inserted partway into the crystal to form an edge dislocation. Edge and screw dislocations are described by means of the Burgers vector. Displacement Change of position of a body, usually measured from the mean position or position of rest. Usually applies to the uniaxial case. In the case of angular motions, it may be termed a “rotation.” In a general mutidimensional case, this is a vector quantity that specifies the change of a body or particle, and is usually measured from the mean position or position of rest. In general, it can be represented as a translation vector, a rotation vector, or both.

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Dissipative muffler A muffler where the acoustic performance is determined mainly by the presence of sound absorbing (e.g., flow resistive) material. See Muffler and Reactive muffler. Distinct eigenvalues The case where there are no repeated eigenvalues. All the eigenvalues are unequal. Distributed-parameter system A system where at least one member is continuous (e.g., beam, plate), needing one or more spatial coordinates to analytically represent (model) it. A partial differential equation in time and space is needed. Compare with lumped parameter system. Doppler effect When the wave source moves towards the receiver, the received frequency increases. When it moves away from the receiver, the received frequency decreases. True for both sound and light. Drag force Force on a body in the direction of wind or any other fluid (liquid or gas). Drift The undesirable variation of the output of an instrument when the input conditions are kept constant. Ductile detailing Special requirements for civil engineering structures, such as close spacing of lateral reinforcement to attain confinement of a concrete core in reinforced concrete and masonry, appropriate relative dimensioning of beams and columns, hooks on main beam reinforcement within the column, etc. Ductile structure A structure that is able to undergo a repeated large displacement after yielding, without significant degradation of yield strength. Ductility factor The ratio of the total displacement (elastic plus inelastic) to the elastic (i.e., yield) displacement. Duffing oscillator Classic nonlinear elastic (mechanical) oscillator characterized by a double well potential. Dummy gauge A strain gage in a bridge that is not active. Used for bridge completion and compensation (for temperature changes, etc.). Active gauge is the strained gauge, which is mounted on the deforming member. Dunkerley’s formula A formula that provides a lower bound for the fundamental (first or lowest) natural frequency of a mechanical (vibrating) system. Duration of shock pulse The time required for the acceleration of a shock pulse to rise from some specified fraction of the maximum amplitude (peak) of the pulse and to decay to this value (see Shock pulse). Dwell frequency The frequency at which a “sine dwell” test is carried out.

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Dyadic Follows a geometric sequence of ratio 2. Dynamic absorber See Absorber or Vibration absorber. Dynamic balancing See Balancing. Balancing achieved on two parallel planes of vibration perpendicular to the axis of rotation. Dynamic magnification factor The ratio of the maximum displacement of a single DoF system excited by a harmonic force to the deflection that would result if a force of that magnitude was applied statically. Eccentric mass An equivalent point mass in a rotary device, located at a radial offset (eccentricity) from the axis of rotation such that the corresponding centrifugal force is equal to the imbalance force of the original device. The mass that needs to be placed on a rotor (eccentrically) in order to balance it. Eccentricity The distance between the geometric center and the center of mass of a rotor. Eddy current damping Damping due to eddy currents generated in a conducting surface and associated dissipation of the energy through electrical resistance to eddy currents. Eddy current sensor Eddy currents are generated on the surface of a conducting medium when a high frequency (e.g., radio frequency in the MHz level) magnetic field is applied to it. An eddy current sensor is a proximity sensor where the probe emits a high frequency magnetic field towards the monitored object, which acts as a thin conducting layer. The inductance change caused by the eddy currents in the probe coil is a measure of the intensity of eddy currents and the distance (proximity) of the object. The inductance change can be measured using an inductance bridge circuit. Effective isolation period A linearized equivalent isolation period of a nonlinear isolation system. Eigenfrequencies Natural frequencies determined by an eigenvalue analysis. Eigenfunctions Mode shape functions, in a general sense, including complex modes. Eigenvalue analysis Analysis carried out to obtain the roots of the characteristic equation of the matrix formulation of an engineering problem. In system dynamics and vibration problems, the imaginary parts of the eigenvalues are the natural frequencies of the system. The reciprocal of the imaginary parts are the time constants of the system. Imaginary parts divided by the corresponding undamped natural frequencies provide the modal damping ratios. The analysis also provides eigenvectors, which give the mode shape information. The analysis that provides both natural frequencies and mode shapes is called modal analysis. See Eigenvalue problem. In stress analysis, the eigenvalues and eigenvectors give the values and the directions

Glossary

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of the principal stresses or principal strains. In other engineering problems, there are eigenvalue problems with corresponding interpretations. Eigenvalues The equation of motion for undamped free vibrations of a system with n DoF can be represented in matrix notation as the product of the characteristic matrix and the vector representing the corresponding displacement coordinates. For nontrivial solutions, the determinant of the characteristic matrix is set equal to zero, yielding a characteristic equation. The roots of the characteristic equation of the system are called eigenvalues. See Eigenvalue analysis for the interpretation of eigenvalues. Eigenvalue problem The mathematical formulation of the dynamics of a system as a problem of determining eigenvalues and eigenvectors of a matrix. This matrix is M 1 K; where M is the mass matrix and K is the stiffness matrix in the undamped case, giving natural frequencies, time constants and modal damping ratios. See Eigenvalue analysis. In stress analysis, the eigenvalue analysis gives the values and directions of the principal stresses or principal strains. In other engineering problems, there are eigenvalue problems with corresponding interpretations. Eigenvectors Many engineering problems can be cast as the solution of eigenvalues and eigenvectors of a characteristic matrix. In vibrations problems, this matrix is M 1 K; where M is the mass matrix and K is the stiffness matrix, in the undamped case. The corresponding eigenvectors give mode shape vectors (see Eigenvalue analysis). In stress analysis, the eigenvectors give the direction of the principal stresses or strains. Elastomer A generic term that encompasses all types of rubber, natural or synthetic. Elastomeric bearing One type of base isolators, which are usually made of rubber or soft materials. Elastomeric coupling A flexible shaft coupling, which employs nonmetallic elastomeric elements for transmitting power, and accommodates shaft misalignment from one rotating shaft to another. Electricite de France (EDF) One type of seismic isolators, which consists of an elastomeric component and a sliding bearings in series. Electrical impedance The transfer function (voltage or current), in the frequency domain. Electro-rheological (ER) fluid Materials that undergo significant instantaneous reversible changes in characteristics when subjected to electric field. Element In finite element analysis, this is a small portion or a sub domain of a system having small but finite dimensions and bounded by straight or curved boundaries. Elements are connected to each other via a set of connection points called nodes. The minimum number of nodes for an element would be the corner nodes defining its shape.

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EMA See Experimental modal analysis. e.m.f. Electromotive force. See Back e.m.f. Emission Emitting or radiation of sound energy or power from a noise source. Noise emission is controlled through noise government legislation. Energy equivalence A simplified analysis method of a dynamic system where a simplified (e.g., lumped parameter) model (of reduced number of degrees of freedom) is established such that its energy (kinetic energy and potential energy) is equal to that of the original system. Information about the original system can be derived (approximately) by analyzing the equivalent system. Ensemble average The average of ensembles or samples (records) of possible outcomes. This is different from averaging a single record or signal (temporal average). Enveloped spectrum The envelope of a set of spectra obtained by different records of a vibration signal. Environmental noise Noise received by individuals indoors and outdoors, caused by noise sources that are controlled by others. Environmental quality standard World Health Organization (WHO) is the world body that recommends this regulatory standard of noise exposure. E.g., see Guideline for Community Noise published by WHO in 2000. Epicenter The projection on the surface of the earth directly above the hypocenter of an earthquake. Equalization In vibration testing, the spectrum of the actual excitation signal applied to the test object has to be (approximately) equal to the specified test spectrum. The process of achieving this by examining small frequency band of the actual spectrum and changing the spectral values (by adjusting the gains) is called equalization. Equalizer A device that achieves equalization in a dynamic test system. Equipment interaction Feedback effect between the motions of the structure and attached equipment. Equipment tuning When the natural frequency of a piece of equipment is such that undesirable resonances occur in vibratory response and equipment tuning may be necessary to avoid this condition. One option is to tune a vibration absorber (dynamic absorber) to this frequency.

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Equivalent static acceleration Steadily applied acceleration producing the same deformation of the structure as that resulting from the action of a shock. Also termed pseudoacceleration. Equivalent mass See Energy equivalence. The mass of a single DoF model, in the “energy equivalence” method. Equivalent stiffness See Energy equivalence. The stiffness of a single DoF model, in the “energy equivalence” method. Equivalent viscous damping The value of viscous damping assumed to analyze a vibratory motion, such that the dissipation of energy per cycle is the same for the viscous and the actual cases. Ergodic hypothesis For a stationary process, the assumption that the ensemble average can be approximated by the time (temporal) average. Ergodic process A stationary process where the ensemble average can be approximated by the time (temporal) average. This assumption is known as the ergodic hypothesis. Euclidean norm Square root of sum of squares (SRSS) of a set of independent values. Euclidean space See Cartesian coordinates. Space represented by a general Cartesian coordinate system with n coordinate axes. Cannot be represented in a physical space when n . 3: Euler angles A set of orthogonal rotation angles normally used to describe orientation of bodies in three dimensional (3 D) space. Euler Bernoulli beam See Bernoulli Euler beam. Euler’s method First order Runge Kutta method of numerical integration. Excel Microsoftw spreadsheet software. Excitation Force (or even a motion such as support motion) applied to a mechanical system. This is the input to the system. The resulting response is the output. Exciter Shaker. The device that generates and applies an excitation to the test object in dynamic (vibration) testing.

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Expansion joint A connection between two or more seismically isolated structures for the passage of occupants from one building to another, and also to let the adjacent buildings move relatively to each other during a seismic excitation. Expected value Also known as “mathematical expectation” of a random function. It is the integral of that function weighted by its probability density function. If the function is the random variable X; then the expected value gives the mean value ðX Þ; which is the first moment statistic. If it is X 2 ; then we get the second moment statistic. If it is ðX XÞ2 ; then we get the variance, the square root of which is the standard deviation of a random parameter. Other higher order statistics of X can be derived similarly. Experimental modal analysis (EMA) A method of determining a model of a dynamic (vibrating system) through testing and modal parameter extraction. A signal is applied to the system using an exciter. Various responses are acquired and analyzed (usually by Fourier analysis methods in the frequency domain) to determine the natural frequencies, damping ratios, and mode shapes of the system, and using these an analytical model (mass matrix, stiffness matrix, and a damping matrix) is computed for the system. Explosion interaction Two structures collide during an explosion. Extrinsic Opposite of intrinsic. A method where the device is used as the medium (or vehicle) in transmitting the objective (e.g., signal), rather than generating the objective. Some types of fiber optic and ultrasound sensors use the extrinsic approach as well as the intrinsic approach. Fast Fourier Transform (FFT) A fast and efficient computer algorithm to calculate the discrete Fourier transform (DFT). The input data are samples of a time waveform from (e.g., digitized voltage measurements). The result is a display of discrete spectral points giving amplitude versus frequency, and phase versus frequency. The frequency interval is so small that the spectrum appears continuous in the display or printout. Fatigue Failure of a structural member exposed to a repeated load whose amplitude is well under the member’s strength. Fault A zone of the Earth’s crust within which the two sides have moved. Faults may be hundreds of miles long, from 1 to over 100 miles deep, and not readily apparent on the ground surface. Fault detection The process of determining if the overall machinery is healthy or a fault has occurred. Fault detection may performed by such methods as oil analysis and vibration monitoring. Data gathering, data analysis (e.g., FFT, cepstrum, wavelet analysis), comparison to standards, comparison to limits set in plant for specific equipment, etc., may be involved when detecting whether machinery is developing a faulty condition. Fault diagnosis The process of identifying the type, source, and severity of a failure (e.g., a faulty gearbox) for a detected fault. Vibration based fault diagnosis may be performed by studying the abnormality of vibration features combined with the locations of the accelerometers generating those features.

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Fault prognosis Estimating the expected time to failure, and trends in the condition of the equipment being monitored, and planning and scheduling appropriate maintenance procedures. FEA Finite element analysis. See also Finite element method (FEM). A computational technique that starts by discretizing the continuum equations of the problem. Feasible region All sets of blade adjustment in jet engines that will reduce the aircraft vibration to within specifications. Featural influence The coefficients representing the weight of each vibration feature in the structure based connectionist network. Feed selection Technique where the feed is chosen to avoid or suppress chatter. Feedback control A common control procedure where a sensor measures or monitors the system response and feeds the information into the controller. The controller compares the measurement with the desired value. If there is an error, then a correcting signal is generated by the controller, which is then applied to the system by the drive unit (or actuator). Feedback resistor The resistance element in the feedback path of an operational amplifier. It makes the device stable. Feedforward action A control signal generated without monitoring the response. Using a model of the system or measurement of unknown inputs may be used to generate it. Feedforward control See Feedforward action. FEM Finite element method. See FEA. Ferromagnetic core A member (e.g., rod or disk made of laminations) made of ferromagnetic material for use in a magnetic circuit (e.g., windings) to concentrate the field and reduce losses. FFT See Fast Fourier transform. Fiber-optic Devices made using optical fibers (a bundle of thin and flexible fibers that can carry light or laser signals). Fictitious spring method An analysis method for sliding isolated structures, in which a fictitious spring is introduced between the base mat and ground to simulate nonlinear behavior of friction.

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Filter An analog hardware or software (programmed) device used to filter out some frequency bands of a signal (while allowing through the other bands). Final peak sawtooth shock (FPS) See Terminal peak sawtooth shock (TPS). Finite difference Computational methods involving a difference approximation for the derivatives. Finite element method A method for solving differential equations by approximating the solution by a linear combination of trial functions, which are typically polynomials. Finite element method (FEM) A numerical technique to solve the equations of an engineering (e.g., vibration) problem. A continuum is divided into a number of relatively small regions called elements that are inter connected at selected nodes. The kinetic energy or potential energy of the entire continuum is expressed as the sum of the element kinetic or potential energy. The solution is approximated by a linear combination of trial functions, which are typically polynomials. This leads to the assembled mass matrix and the assembled stiffness matrix, and finally to the finite element equations of the problem (vibration). Fixed end A boundary condition where the displacement and slope are zero (bending moment and shear force are not zero). Cantilever is obtained by fixing one end of a beam. Flexible mount An engine mount made of flexible material (e.g., elastomer or spring) to provide vibration or shock isolation. Flexibility matrix The aij element of this matrix is the displacement at i due to a unit force at j with all other forces equal to zero. Inverse of the stiffness matrix. Flexural vibration Bending vibration. The oscillatory motion of a flexible shaft (possibly with a rotor) or a beam or plate in any transverse plane of bending. Flow resistance Resistance due to friction between a fluid and some constriction (e.g., a pipe or duct) or between a gas and fibers in acoustic problems. The energy loss may take place in a very narrow layer of air adjacent to each fiber of acoustic material. Flow resistivity Unit area flow resistance per unit thickness of a porous material. Given by the applied air pressure difference across the two sides of the material divided by particle velocity through the material. Fluid damping Force of external friction whose magnitude is proportional to the square of the velocity, and whose direction is opposite the velocity.

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Foil-type strain gauge A strain gauge consisting of a thin metallic foil. As it is stretched (strained), its resistance increases, which can be measured using a bridge circuit to provide a strain reading. Force isolation See Isolation. Vibration isolation of a system from a force source. Force transmissibility See Transmissibility. Transmitted vibration force divided by an applied vibration force (or actuator force). Forced vibration The steady state vibration caused by an external excitation (force or input). Typically (in linear problems in particular), forced vibration occurs at the frequency of the exciting force. Forcing frequency The frequency of the excitation function (force). Fourier analysis Analysis carried out using the Fourier transform. This provides frequency domain information (magnitude and phase with respect to frequency) of a time signal. Fourier integral See Fourier transform. Fourier series expansion Expressing a general periodic signal into a series (sum) of harmonic (sinusoidal, real or complex) components of integer multiples of a fundamental frequency. Fourier spectrum Frequency spectrum of a signal. The spectrum of a signal, as obtained by Fourier transform. Fourier transform An integral transform (weighted by a complex exponential function). This converts time domain information (time signals) into frequency domain information (frequency spectra) and vice versa. Frahm absorber See Absorber. A dynamic (vibration) absorber. Frame Countable set of linearly dependent functions. Free-body diagram Diagram showing the basic components of an object and all the forces acting on them. Free vibration Vibration that occurs in a system after the removal of excitation. Typically, free vibrations occur at one or more natural frequencies. Free vibration The oscillatory motion (vibration) of a system that occurs in the absence of an external forcing function (excitation).

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Frequency The repetition rate or periodicity of a periodic event, usually expressed in cycles per second (hertz or Hz), or revolutions per minute (rpm). This is the cyclic frequency, which is the reciprocal of the period. The angular frequency (radians per second or rad/sec) is obtained by multiplying cyclic frequency by 2p: Frequency domain The representation of the system variables (including inputs and outputs) as functions of frequency. Related to the time domain through the Fourier transform. Vibration data or results are displayed or usually analyzed as a function of frequency. The independent variable is frequency ðvÞ: Frequency-domain analysis Response analysis of a dynamic system with respect to frequency. Typically, uses Fourier analysis, frequency spectra, and transfer functions. Frequency-domain index A single number calculated from the frequency spectrum and used for trend analysis and comparison. Frequency modulation (FM) A signal modulation method, where a carrier signal is used to transmit another signal (modulating signal, carried signal) over a long distance without distortion. In this method, the amplitude of the carrier is fixed, but its frequency is varied according the amplitude of the carried (modulating) signal. Typically, radio signals are transmitted by this FM method. Frequency response The amplitude and phase response characteristics of a system as functions of frequency under external excitation. Frequency response function (FRF) Output or input in the frequency domain, assuming zero initial conditions. Laplace transfer function can be converted into FRF by simply changing the Laplace variable into complex frequency variable jv: The FRF has amplification and phase change information of a response with respect to an input at various frequency values. It may be plotted as a Bode diagram or a Nyquist diagram. Frequency transfer function (FTF) Same as Frequency response function (FRF). Friction pendulum system (FPS) One type of sliding isolators, which has a spherical concave sliding surface to produce a recentering force for the isolated structure under excitations. Fringes A pattern of dark and light lines obtained when two light waves of nearly equal frequency are mixed. This method is used in some sensors, for example, for measuring small velocity changes (by Doppler effect). Full bridge A bridge circuit with four arms. It can be used to sense small changes in one of the arms (consisting of a resistor, inductor, capacitor, etc.). A half bridge contains two arms. Full-scale drift The drift (undesirable variation of the output when the input conditions are kept constant) of an instrument, when the input conditions correspond to its full scale (maximum) value.

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Fully differential capacitive sensor A bridge in which all four components change in symmetric manner. Fully operational The highest performance level at which a structure is expected to operate with all functions, after an earthquake attack, if there is no (or at least very little) structural and nonstructural damage. Fundamental frequency Fundamental natural frequency. The lowest (first) natural frequency of a (vibrating) system. Fundamental natural mode of vibration The mode of vibration of a system having the lowest natural frequency. Fuzzy logic An approach to computing (soft computing) and knowledge based decision making, based on the “degrees of membership” in a set, rather than the conventional binary membership (member or nonmember). Fuzzy data are matched with a fuzzy knowledge base to generate fuzzy inferences (decisions). A more general form of “true or false” boolean logic on which the modern computer is based, and incorporates such logic states as slightly true or very true. FVC Frequency to voltage converter. g units Units of acceleration due to the Earth’s gravity (by international agreement, g ¼ 9:80665 m=sec2 or 386.087 in/sec2 or 32.1739 ft/sec2). Divide an acceleration value by the value of g to obtain (as a ratio) the acceleration in g units. Gabor transform A sampled short time Fourier transform. Gaussian elimination The process by which a matrix is converted by elementary row operations to an upper triangular matrix. Generalized coordinates A coordinate system, which does not necessarily use physical coordinates to represent the dynamics of the system. An example is the modal coordinates. Generalized forces See Generalized coordinates. When multiplied by the increment of the corresponding generalized coordinate, the product is equal to the actual incremental work done by that force in moving through that incremental motion. Geometric modeling The process of transcribing the geometry of a system into a finite element or CAD data base. Geometric nonlinearity Nonlinearity caused by geometrical constraints, such as rubber mount casing, or by trigonometric functions in the kinematic equations of the system (e.g., robot). Gradient wind Wind velocity above the boundary layer.

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Green’s function The solution that represents the sound field generated by an impulsive source in space and in time. Useful in describing the sound field generated by a continuously distributed source, with convolution integration, both in free space and bounded space. Ground vibration A wave transmitted through the ground that causes the surface particles of ground to oscillate as it passes. Guyan reduction A procedure to reduce the model scale for vibration analysis by removing those DoF (called slave DoF) that can be approximately expressed by the rest of the DoF (called master DoF) through a static relation. Gyroscopic action A torque (or moment) is necessary to change the direction of angular momentum vector (the product of inertia and angular velocity vector) of a spinning member. Without such a torque, the member will remain oriented in a fixed direction. Gyroscopic motion Motion of bodies in 3 D space subjected to multiple angular rotations, causing gyroscopic effects. Gyroscopic sensor A sensor based on the gyroscopic principle, commonly used to measure angles from a fixed reference, when a physical surface representing such a reference is not available. h-conversion The process of achieving convergence or improving the accuracy of a finite element solution through increasing the number of elements, while keeping the element order the same. Haar wavelet The first known wavelet derived from a step function. Half bridge See Full bridge. Half-power bandwidth pffiffi The frequency interval in a spectrum where the magnitude drops to 1= 2 the peak, resonant value (i.e., half the power). Half-power points See Half power bandwidth. The two frequency points representing the half power bandwidth. Half-sine shock Simple shock for which the acceleration time curve has the form of a half period (positive or negative part) of a sinusoid. Hamilton’s principle Variation of the difference of the system kinetic energy and the system potential energy plus the line integral of the virtual work done by the nonconservative forces during any time interval must be equal to zero.

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Hammer test An impact test where the test object is impacted with an instrumented hammer and the corresponding excitation and response signals of the object are measured. It can be used to determine natural frequencies and damping ratios. Harmonic A sinusoidal quantity having a frequency that is an integral multiple of the frequency of a periodic quantity to which it is related (normally the fundamental frequency). Harmonic oscillator This oscillator produces periodic oscillation described mathematically by either of the harmonic trigonometric functions (sine or cosine), which has only one frequency component (undamped natural frequency of the oscillator). Similar to an undamped simple oscillator. Harmonic response analysis Harmonic response analysis is used to determine the response of a vibration system to a harmonic (i.e., sinusoidal) excitation force. A typical output is a graph showing response (usually displacement of a certain degree of freedom) vs. frequency. The response has an amplitude (amplification) and a phase change with respect to the harmonic excitation, which are characteristics of the system itself. Hearing loss Permanent hearing loss is caused by excessive and prolonged noise exposure, which destroys the auditory sensor cells. High-tech facility Production lines for printed circuit boards, high precision devices, semiconductor devices, electronic appliances, etc., employing robotics, automation, and other modern and advanced technologies. Holonomic system A mechanical dynamic system where all of the constraints can be represented by algebraic equations (not differential equations). In this case, the number of DoF is equal to the number of independent coordinates needed to represent an arbitrary position and orientation of the system. Homogeneous equation An unforced ordinary differential equation. For a dynamic system with zero input, the right hand side of the system differential equation will be zero, giving a homogeneous equation. Hooke’s Law Linear spring equation. Deflection is proportional to the applied force. Strain is proportional to stress in one dimension. Hydraulic actuator A piston cylinder (i.e., ram) device to which a controlled and pressurized fluid is sent, thereby moving the piston. This motion can be used to move another object. This described a linear (rectilinear) actuator, although rotary hydraulic actuators are also available. Hyperpatch or Volume A portion of tri cubic solid completely bounded by four areas (for tetrahedron volumes) or six areas (for brick volumes).

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Hypocenter The location of initial radiation of seismic waves (i.e., the first location of dynamic rupture). Hysteresis Greek word meaning “arrive late,” used to describe a specific nonlinear processes where the response curve when the input is increasing is different from that when the input is decreasing, resulting in a hysteresis loop. Examples include magnetic systems (due to residual magnetism) and plastic systems (due to residual stresses and strains). Hysteresis loop A diagram that describes the force deformation relation of a nonlinear device of the hysteresis type. See Hysteresis. Vibration isolator with hysteretic damping produces a hysteresis loop, the area of which gives the energy dissipation in one cycle of vibration. Hysteretic damping Internal friction type damping in material where the energy dissipation per cycle is independent of frequency. Imbalance The eccentricity that exists between the center of mass of a rotor and the axis of rotation that generates centrifugal forces, resulting in vibrations and undesirable forces at the bearings. IC engine Internal combustion engine. A mixture of fuel and air is ignited in a cylinder, causing the piston of the cylinder to move. Imission Receiving, perceiving or observing radiated noise, causing noise exposure at a location near the noise source. Noise imission is legislated using an Environmental Quality Standard. Impedance Resistance exhibited by a “flow” variable (e.g., current, velocity) to an applied “effort” variable (e.g., voltage, force), expressed as a transfer function in the frequency domain. See Mechanical impedance and Electrical impedance. Impedance bridge A bridge circuit with inductor or capacitor elements in addition to resistors in at least one of its four arms. Impedance transformer A device with a very high input impedance and very low output impedance. Typically, an operational amplifier with feedback. Impeller A bladed member of the rotor of a pump or compressor that imparts principal energy into the fluid that is being pumped or compressed. Impulse The product of the force on a body and the time during which that force acts, equal to the change in momentum.

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Impulse response function The response of a system to a unit impulse input. Inverse Fourier transform of the frequency response function (or inverse Laplace transform of the transfer function) of the system. Incipient failures Partial and usually gradual failure of a machine. Inelastic response A structure subjected to large excitations, resulting in yielding of longitudinal rebars. Inertia A property of matter where it resists a change in motion, including resistance to change from a position at rest. Inertia asymmetry The moments of inertia in two orthogonal planes are not equal. Inertia block A block or a heavy plate attached to the base of a machine in order to achieve good vibration isolation. Inertia matrix A matrix consisting of direct and cross moment of inertia terms in three dimension of a rigid body. Mass matrix. The matrix multiplied by the acceleration vector of a multi DoF system in its Newtonian equations of motion expressed in the vector matrix form. Inertial actuator A class of actuators in which the actuation force is generated through the addition of inertial masses and applying the resulting inertial forces on the system at the actuator point of attachment. Influence coefficient A coefficient representing the effect at a particular point of an action some other point. Initial conditions (ICs) The conditions of the response of a dynamic system at time t ¼ 0: These are temporal conditions, different from boundary conditions, which are special conditions. The required number of ICs is equal to the “order” of the system, or the number of state variables needed for the system, or the order of the highest derivative of the input output differential equation. Inner ear Cochlea is the main component of inner ear, where the actual reception of sound takes place. The mechanical motion caused by the sound wave is transmitted to basilar membrane and its associated sensing cells, where this motion is sensed as sound. Inner modulation Current structural vibration. Input The force or energy (or a motion such as support motion) applied to a mechanical system. Input output equation A single differential equation with the response as the dependent variable and the input as the independent variable.

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Insertion loss The insertion loss of a muffler is defined as the difference between the sound pressure or acoustic power level at the same point in space before and after the muffler is connected to the noise source. Instability A self induced excitation that can occur in rotating machinery, which does not require a sustained forcing phenomenon to initiate or maintain the motion. At the inception of instability, the rotor deflection will continue to build up with speed, and if further increased the large amplitudes of motion will normally result, causing damage to the machine. Intensity A metric of the effect, or the strength, of an earthquake hazard at a specific location, commonly measured on qualitative scales such as MMI, MSK, and JMA. Internal friction Damping that derives from structural defects internal to structural components of a vibrating system. Interplate Processes between the Earth’s crustal plates. Interval model A linear regression model with interval coefficients to represent the range of aircraft vibrations caused by blade adjustments. This model can account for the stochastic and nonlinear nature of aircraft vibration. Intraplate Processes within the plates. Intrinsic Opposite of extrinsic. A method where the device is used to directly generate the objective (e.g., signal) rather than simply to transmit it. Some types of fiber optic and ultrasound sensors use the intrinsic approach (as well as extrinsic approach). Intrinsic loss factor Factor in a subsystem to represent the loss of vibration power with respect to the total power. ISO International Organization for Standardization. Technical Committee 108, Subcommittee 5 is responsible for standards for Condition Monitoring and Diagnostics of Machines. Isolation Diminishing the transfer of vibration amplitude from a machine to another machine or a structure by the judicious selection of barrier materials and barrier configuration. Vibration isolation of a system from a vibration source. Inverse of “transmissibility.” Isolation period Vibration period of the isolation system. Isotropic Being the same in all directions. See Orthotropic. Jerk The rate of change of acceleration with time (third derivative of displacement with respect to time).

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Johnson noise White noise (approx.), whose value depends on both bandwidth and resistance. Journal bearings Journal bearings consist of a circular section of shaft, called the journal, rotating inside a cylindrical bush, called the bearing. The space between the two is partially filled by the lubricating fluid, which is drawn up around the journal due to the effect of elasto hydrodynamic lubrication. Jump phenomenon A nonlinearity where the frequency response of a system sharply shifts up or down (jumps) at a particular frequency, as the frequency of excitation to the system increases or decreases through this frequency. K-normal A modal vector z that is normalized with respect to the stiffness matrix K, such that z T Kz ¼ 0 Keulegan Carpenter number The Keulegan Carpenter number indicates the relative importance of the drag and inertia forces for small (small compared to the wavelength) diameter structures. A small Keulegan Carpenter number means that the inertia force is more important than the drag force. Key point A coordinate location in space. Kinematic design Design of a structure by its motion, independent of mass and force. Kinematics Motion geometry of bodies. Kinetic energy Energy associated with motion (velocity). Kronecker delta A function of two indices, which is equal to one when the two indices are identical, and zero otherwise. It may be used to represent a unit matrix (or identity matrix), a diagonal matrix with unity diagonal elements. Kurtosis Fourth probability density moments sensitive to the impulsiveness in the vibration signal and, therefore, sensitive to the type of vibration signal generated in the early stages bearing fault of a rolling element. LabVIEW Industry standard equipment, software, and hardware marketed by National Instruments. LaCoste zero-length spring Spring of a vertical seismometer, which would collapse to zero length if wire of the coil was of infinitesimal diameter. Lagrange’s equations Differential equations of motion of a dynamic system, obtained using the Lagrangian energy approach, in terms of a set of generalized coordinates. There are as many Lagrange’s equations as there are generalized coordinates. The forcing terms are generalized forces.

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Lagrangian Equal to total kinetic energy minus the total potential energy of the system. Laminated rubber bearing (LRB) A type of elastomeric isolators that is usually composed of alternating layers of steel and hard rubber ensuring it is stiff enough to sustain the vertical loads, yet flexible under the lateral forces. Laplace transform An integral transform weighted by an exponential function of the Laplace variable s. It converts a derivative into a power of s, of the same order. Hence, a differential equation is converted into an algebraicpequation. Related through the Fourier transform through s ¼ jv; where v ¼ angular frequency ffiffiffiffi and j ¼ 1: An input output equation is converted into a transfer function. Lateral force resisting system A structural system for resisting horizontal forces due, for example, to earthquake or wind (as opposed to the vertical force resisting system, which provides support against gravity). Learning Adaptation of the interval coefficients to represent the range of vibration caused by blade adjustments more effectively. Life safety The third performance level in which a structure is expected to survive an earthquake. The life safety of occupants is not endangered either by structural collapse or the presence of heavy falling debris, although substantial damage may occur to both structural and nonstructural building components. LIGO Laser Interferometer Gravitational Wave Observatories of the U.S. Limit cycle An instability in a nonlinear system that is exhibited as a steady oscillation of amplitude that is independent of the input amplitude. Limiting stable depth-of-cut Depth of cut below which chatter will not occur for a given spindle speed of a machine tool. Line noise Noise at the frequency of the voltage signal in a power line (60 or 50 Hz) caused by the power lines in the vicinity of a system (e.g., through ground loops). Line or line segment A portion of a cubic spline curve bounded on both ends by a key point. A line may be straight or curved. Linear quadratic regulator (LQR) A linear feedback controller with constant gains, which minimizes a cost function consisting of the time integral of the sum quadratic functions (positive definite and semidefinite) in the state variables and input variables. Linear system A system is linear if it can be represented by a set of linear differential equations. The principle of superposition holds. Its magnitude of response is directly proportional to its magnitude of excitation for every part of the system.

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Linear-variable differential transformer See LVDT. Load displacement relationship The relationship between the load on a structure and the corresponding displacement. Logarithmic Decrement Measure of damping. Equal to the natural logarithm of the ratio of successive amplitudes of free decay. It is inversely proportional to the magnification factor (quality factor), Q. Long-term statistics Statistics of the short term parameters such as the significant wave height. Lorentizian Resonant response of a driven harmonic oscillator with damping. Also, the shape of spectral lines in atomic physics. Lorenz force Force on a current carrying wire located orthogonally in a magnetic field. Loudness level The loudness of sound is determined by subjective comparison with the loudness of a pure tone of 1000 Hz. The level, measured in “phons,” is numerically equal to the sound pressure level, in decibels (dB), of the 1000 Hz tone. LPA Last point algorithm for performing numerical integration. Will not become unstable like Euler’s first point approximation does. Lumped mass A point presentation of the mass of a system. Results in a diagonal matrix. Lumped mass matrix The diagonal mass matrix obtained by lumping the inertia of a system by element masses at the nodes finite element mesh. Lumped-parameter model A model where the system components are represented by parameters (e.g., mass, stiffness, damping coefficient) concentrated at a finite number of points. An approximation of a distributed parameter system. LVDT Linear variable differential transformer. A popular noncontact sensor for measuring linear (rectilinear) displacements. Operates by the transformer effect, with a movable ferromagnetic core element (attached to the sensed object). M-normal A modal vector z that is normalized with respect to the mass matrix M, such that z T Mz ¼ 0: Machinery failure The inability of a machine to perform its required function due to some defect in the machine components.

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Machining Removal of excess material from a piece of material (workpiece) by moving a working tool over the surface of the workpiece to achieve a desired shape. Magneto-rheological (MR) fluid A magnetic fluid analogous to electro rheological (ER) fluid. Typically consists of micron sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicon oil. When a magnetic field is applied to the fluid, particle chains are formed, and the fluid becomes a semisolid and exhibits viscoplastic characteristics. Magnetorheological damping Damping associated with a magnetic fluid and a magnetic field. Magnitude A measure of an earthquake’s release of strain energy, measured on a variety of scales, of which the moment magnitude MW (derived from seismic moment) is preferred. The absolute value of a complex quantity (e.g., FRF). Masks Baseline spectra that are used with an allowable tolerance limit to “filter out” or block specific frequencies. Optical plates with windows for concentrating light from a source to generate better contrast in an optical pulse generating transducer (e.g., optical encoder). Masonry structure A structure built from bricks and cement mortar that has relatively small lateral and tensile strength, which is vulnerable to an earthquake attack. Mass law For a single plate, the transmission loss depends on frequency and the surface density of the plate in the frequency range from the fundamental (lowest) natural frequency to the critical frequency of the plate. Mass matrix The matrix multiplied by the acceleration vector of a multi DoF system, which, in its Newtonian equations of motion, is expressed in the vector matrix form. Typically, a diagonal matrix with its diagonal elements giving the masses corresponding to various DoF of the system. Mathematical expectation Also known as “expected value.” The mathematical expectation of a random function is the integral of that function weighted by its probability density function. If the function is the random variable X, the expected value gives the mean value ðXÞ; which is the first moment statistic. If it is X 2, we get the second moment statistic. If it is ðX X Þ2 we get the variance, the square root of which is the standard deviation of a random parameter. Other higher order statistics of X can be derived similarly. Matlab MATLABw is a commercial software package (tool) for numerical computation, visualization and symbolic manipulation. Various toolboxes are available for specialized computation on important subjects such as control and signal analysis. Maximax shock response spectrum Envelope of the absolute values of the positive and negative shock response spectra. See Response spectrum.

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Mechanical design Design of a structure or building utilizing the properties of electronic and mechanical subassemblies and parts. Mechanical impedance Force or velocity in the frequency domain. This transfer function is one of many used in the analysis and design of vibrating systems. Mechanical looseness A condition where two or more mechanical components, which are designed to be rigidly connected or in firm contact, can move relative to one another. See Backlash (in gears). Mechatronics Synergistic integration of electronic and electrical engineering, computer technology and control engineering with mechanical engineering, using concurrent and integrated design techniques. This area forms a crucial part in the design, manufacture, and maintenance of a wide range of engineering products and processes. Mean free path Distance a molecule travels on average in a gas between molecular collisions. Measure of noise evaluation Energy based indices like LAeq (equivalent A weighted sound pressure level) for noise evaluation. All of these measures are legislated by the government, and are dependent on legislative regulations or standards. Member Any individual component of a structural frame. An element is within a set (membership ¼ 1). MEMS Microelectromechanical systems. A relatively new and important field concerned with modeling, analysis, design, and development of very small mechanical devices with electrical or electronic features. Mesodynamics A new term to describe complex behavior of otherwise harmonic mechanical oscillators with energies approximating 10 11 J. Mesomechanics Focused on the complex behavior of granular materials. Mexican hat wavelets Obtained by calculating the second derivative of a Gaussian. Meyer wavelets Wavelets associated with enveloped trigonometric functions. Microdynamics Dynamics describing complex features of contact friction, largely advanced by NASA. Microphone A device for producing an electrical signal from an acoustic pressure (sound or voice). Standard equipment in acoustic or sound acquiring and generating systems.

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Microphone array A directional measuring device that consists of an array of miniature microphones distributed spatially in an acoustic field. Middle ear Consists of three small ear bones, the hammer, anvil and stirrup. Serves as an impedance transformer, which matches the low impedance of the air in ear canal with the high impedance of inner ear. MIMO Multi input multioutput system. A system with many inputs and many outputs. A multivariable system. Mirror microphone system A directional measuring device that consists of a reflector of elliptic or parabolic shape and an omnidirective microphone located at its focus. Misalignment A condition when two rotating machines (or components) are not coupled together along a single axis (straight line) of rotation. Mobility The transfer function (velocity or force) in the frequency domain. The inverse of mechanical impedance. Modal analysis The dynamics of a system may be formulated as a problem of determining eigenvalues and eigenvectors of a matrix. This matrix is M 1 K; where M is the mass matrix and K is the stiffness matrix in the undamped case giving natural frequencies, time constants and modal damping ratios. See Eigenvalue analysis and Eigenvalue problem. The imaginary parts of the eigenvalues are the natural frequencies of the system. The reciprocal of the imaginary parts are the time constants of the system. Imaginary parts divided by the corresponding undamped natural frequencies provide the modal damping ratios. The eigenvectors provide the mode shape information. The analysis that provides both natural frequencies and mode shapes is called modal analysis. This is the process of decomposing the vibrations (or response) of a complex system into its component modes of vibration (motion). One goal may be to find the locations and frequencies of maximum (undesirable) response for application of vibration control or suppression methods (e.g., stiffener and dampers). Modal density The number of mode shapes present in a unit spatial area or the number of natural frequencies in a unit frequency interval. Modal matrix The square matrix formed by placing all the independent mode shape vectors as column vectors. Modal vector A mode shape vector expressed as a column vector. Mode The condition where a mechanical system moves at one natural frequency while maintaining a characteristic shape (mode shape). Can also apply to nonmechanical systems.

Glossary

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Mode mixing Transfer of energy among modes of different frequencies. This assumes nonlinearity and is not the same as beating. Mode shape The deflected characteristic shape, usually normalized, that a system would take when it vibrates at one of its natural frequencies. All elements move in step at this frequency while maintaining a specific shape. Some points will remain stationary throughout, which and they are called nodes. Only the shape of the curve would be known from a modal analysis. If the mode shapes are real, they correspond to natural modes. Otherwise, we have complex modes, which is the case in the presence of some types of damping. Model identification Experimental modeling. Determining an analytical model of a dynamic system using input output test data. See Experimental modal analysis. Model reduction Reduction of the order or the number of DoF (of a vibrating system) by using some accuracy limits (error tolerance). Modal truncation is used, where modes above a specific frequency are neglected. Means of simplification and idealization based on the nature of the vibration problem. Modulation Embedding a signal within a carrier signal so that the original signal (carried signal or modulating signal) does not lose accuracy during transmission. Examples are amplitude modulation (AM), frequency modulation (FM), and pulse width modulation (PWM). Modulus of elasticity Young’s modulus. Direct stress divided by direct strain in one dimension. Mohr’s circle A circle for representing stresses, strains, or moments of inertia at a particular point of system in different directions. The horizontal axis indicates the direction of the principal values and the vertical axis the shear (or cross) values. The angle around the circle is double the physical angle in the system. Moment of inertia A mass parameter that is measure of the resistance (inertia) of a body to angular acceleration about a given axis. Equal to the sum of (or integral to) the products of each element of mass in the body and the square of the element’s distance from the axis. Momentum A property of a moving body that exists by virtue of its mass and motion, to resist any change to that motion. Equal to the product of the body’s mass and velocity. It is a vector (with a direction as well as a magnitude) in the same direction as the velocity. Monopole An acoustic source with energy density concentrated at a single point in space. The most basic sound source, where the source region is compact and the generating motion has no preferred direction. Monopole produces a wave that spreads spherically outwards. Mother wavelet The function used to generate all the basis functions of the space by dilation and translation.

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Motion isolation See Isolation. Vibration isolation of a system from a motion source. Motion transmissibiliy See Transmissibility. Transmitted vibration motion divided by an applied vibration motion (or support motion). Mounts Usually, vibration mounts used for vibration isolation. Muffler A structural device that suppresses noise when fluid (e.g., hot and high speed gas) flows through it. See Dissipative muffler and Reactive muffler. MTBF Mean time between failures. MTTF Mean time to failure. Multibody-spring-dashpot Collection of rigid bodies joined together by springs and dashpots. Multiresolution analysis The process of studying a signal at different scales by use of different time windows to capture characteristics varying from coarse to fine. Narrowband Corresponding to a narrow (small) frequency interval (band). Narrowband process A random process where most of the energy is concentrated in a narrow band (interval) of frequency. Natural frequency The frequency of free vibrations of a system without the influence of an external forcing, except for an initial excitation to start the motion. Also a self induced phenomenon. For a single DoF system, vn ¼ ðk=mÞ1=2 ; where vn is the undamped natural frequency, k is the spring stiffness, and m is the mass. When damping pisffiffiffiffiffiffiffiffi present, the natural frequency decreases to vd ; which is the damped natural frequency given by vd ¼ 1 z2 vn ; where z ¼ damping ratio. For a multi DoF system, the natural frequencies are the frequencies of the normal modes of vibration. Natural mode See Mode shape. Natural period The time period of one complete cycle of free vibration. The inverse of the natural frequency (cyclic). Natural vibration analysis See Modal analysis. The natural vibration analysis is to determine the natural vibration frequencies and the associated mode shapes of a vibration problem when the excitation forces are zero. They are intrinsic characteristics of the vibration problem and independent of the excitation forces.

Glossary

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Near-fault ground motions Ground motions, typically assumed to be within a distance of about 20 to 60 km from a ruptured fault, which usually have long period velocity pulses that impart considerable momentum to a long period structure. Negative averaging A baseline signal that is recorded and then subtracted from subsequent signals before analysis to reveal changes and transients. Negative definite A symmetric matrix is negative definite if xt Ax , 0 for every nonzero vector x. Negative shock response spectrum See Shock response spectrum (SRS). SRS plotted, considering the highest negative response due to the shock, without reference to the duration of the shock. It is the envelope of the negative primary and residual spectra. Neural Network (NN) A nonanalytical (computational) modeling technique, inspired by the human nervous system. Considered as a “learning” network. A network of massively parallel network is trained (by adjusting the network weights, etc.) to yield a known (correct) output for a given input. Once trained, it may be used as a computational model of the system. Further training and adaptations may be done subsequently. Several variations of this feed forward network architecture are available (e.g., feedback, recurrent, dynamic, reinforcement learning). Neutral axis The axis along a bending beam where the stress and strain are zero. Undeformed axis when a beam is bent. Nodal point In finite elements: a point of connection between elements. A node will have a specific number of degrees of freedom, indicating its ability to move or rotate. In vibrations: a point or a line in the mode shape of a structure that does not move (see Node). Node See Mode shape. Points that are stationary in a modal vibration. The mode shape curve passes through the relaxed configuration of the structure at these points. Noise (1/f ) Also known as pink or flicker noise. Common in complex systems of both electronic and mechanical type. Noise Criteria (NC) NC curves are utilized for rating the background noise level in a room. Each curve specifies the maximum octave band sound pressure level for a given NC rating. See Preferred noise criteria (PNC). Noise regulation law This regulatory standard defines the national minimum for noise suppression, for example, factory noise, construction work noise, and road traffic noise. The central government is the regulating agency. See Emission.

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Nomograph A logarithmic frequency velocity plotting graph sheet. The sheet gives constant deplacement, constant velocity, and constant acceleration lines, properly graduated. A period axis (period ¼ 1/cyclic frequency) is given for convenience of plotting. Nonclassical damping Damping that cannot lead to modal decoupling of a structural system. For this case, the mode shapes of a structure can only be expressed in terms of complex (not real valued) functions. Nonholonomic system A mechanical dynamic system where one or more of its constraints need differential relations (not algebraic equations) for mathematical representation. In this case, the number of DoF is equal to the number of independent “incremental” coordinates needed to represent an arbitrary increment (change) in position and orientation of the system, and is less than the number of independent coordinates needed to represent an arbitrary position and orientation of the system. Nonlinearity A characteristic that is not linear (e.g., hysteresis, saturation, deadband, jump phenomenon, nonlinear or nonproportional force motion relationship). Represented by nonlinear equations. The principle of superposition does not hold. See Linear system. Normal modes See Natural mode. In a multiple DoF system, the normal modes are the free natural configurations in which the system vibrates. A mode is characterized by its natural frequency and mode shape. The mode shape depends only on the mass and stiffness of the system and how they are distributed, and not on damping. Normal modes exist for all undamped systems and only for a selected class of damped systems (e.g., proportional damping). Normal wear period A relatively long period of operation during which the frequency of failure occurrence is relatively low. Notch filter A bandstop filter when the band is very small. Nyquist frequency Half the sampling rate. The frequency above which the spectrum of a digital signal has no significance and should be ignored. Nyquist plot Plot of frequency transfer function (FTF) on a complex plane where the horizontal axis gives the real part, and the vertical axis gives the imaginary part of the FTF. OBE Operating basis earthquake. An earthquake representation (e.g., response spectrum) under which a nuclear power plant can operate safely. Occupational Noise Noise received at the workplace, indoors and outdoors, caused by all noise sources in the vicinity of the workplace. ODS Operational deflection shape of a structure.

Glossary

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Offset Steady state error in response of a device. Oil whip A condition that occurs when a subsynchronous instability (oil whirl) excites a critical speed (resonance), which then remains at a constant frequency regardless of speed changes. Oil whirl A condition that occurs when the lubrication in a lightly loaded journal bearing does not exert a constant force on the supported shaft, and a stable operating position is not maintained. Oil whirl/whip See Whirling. The whirling motions that occur due to oil action in journal bearings when operated at or above double the critical speed of the rotor. The whirling motion is in the forward direction and its speed will match the critical speed of the rotor. The oil film is believed to rotate at half the velocity of the shaft due to friction drag. At rotational speeds near twice the critical speed, the oil film provides the stimulus as its speed matches the critical speed value resulting in large displacements and whipping. Omnidirective microphone A microphone whose response does not depend on the direction of the incident sound. Operating basis earthquake See OBE. Operational The second performance level in which a structure is expected to function in a limited manner after an earthquake because of some damage to structural or nonstructural members. Optical encoder A digital transducer that measures displacement. The encoder plate (disk) has light passing areas (windows) on one or more track, with a light source on one side and a light sensor on the other side. As the plate moves, a pulse signal is generated by the light sensor. The pulse count gives displacement and a pulse rate gives speed. Orbit A plot of the displacement Y direction (ordinate) versus the displacement in X direction (abscissa). Order The order of the highest derivative of a differential equation (DE) and the highest difference in a difference equation. The order of a dynamic system corresponds to the number of first order DEs needed to represent the system. Factor of rotational speed of a machine used to represent nondimensional frequency in tracking (monitoring) its spectral response (in the frequency domain). See also order tracking. Order tracking Cascade plots of machine response (vibration) spectra that are synchronously linked to the machine rotational speed via a tachometer. Orthogonality Generalized perpendicularity. If the vectors x and y are orthogonal, then xT y ¼ y T x ¼ 0: If the vectors x and y are orthogonal with respect to matrix P, then xT Py ¼ yT Px ¼ 0:

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Orthotropic Not uniform in all directions (not isotropic). Oscillation A periodically increasing and decreasing variation of a response. In forced oscillation, the force itself oscillates causing an oscillatory response. In free oscillations, the cause is the continuous interchange of energy between two forms (e.g., kinetic energy and potential energy), where the variation can be either constant amplitude (undamped) or decaying amplitude (damped). If the amplitude progressively increases, it is an unstable oscillation. No oscillations occur in overdamped systems because energy is more rapidly dissipated than it is interchanged, leaving no energy for a returning (reversed) response. Outer modulation Structural vibration delayed by one tooth passing period. Outer ear Composed of pinna, ear canal, and eardrum. Pinna produces only a small enhancement of the sounds arriving from the front of the listener. Ear canal works as a resonant cavity allowing for a nearly reflection free termination, and thus a good impedance match of eardrum. Eardrum transforms sound wave to vibration in the membrane. Overdamped response No oscillations occur in overdamped systems because energy is more rapidly dissipated than it is interchanged between two forms (e.g., kinetic and potential), leaving no energy for a returning (reversed) response. For a simple DoF mass (m) spring (k) damper (c) system, the condition for overdamped behavior is 4k=m , ðc=mÞ2 : Overall sound pressure level The pressure is evaluated as the root mean square (RMS) of all components of pure tones. Can be converted into decibels (dB) as for any sound pressure. Overlap index Denotes the overlap in coverage of more than one gearbox component by one accelerometer location. p-conversion The process of achieving convergence or improving the accuracy of a finite element solution through increasing the order of the elements, while keeping the number of elements the same. Parametric instability A self excited motion, representing an instability that is induced by the periodic variation of the system parameters such as inertia, mass, and stiffness. The system cannot be represented by linear differential equations with constant coefficients. Particle velocity The fluctuating velocity of a material particle associated with a sound field (units: m/s). Using Newton’s Second Law, we can solve for the particle velocity u in terms of sound pressure p. Particular integral A particular solution (just one solution) that is satisfied by a nonhomogeneous differential equation. Particular solution Same as particular integral.

Glossary

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Passive vibration control A vibration control method in which a resilient member (stiffness) and an energy dissipater (damper) are utilized to either absorb the vibratory energy or dissipate it inside the damper. By definition, no external power source is needed in a “passive” system. Patch or area A portion of a bicubic surface completely bounded by three line segments (for triangular patches) or four line segments (for quadrilateral ones). Peak Maximum value of a varying quantity (or a spectrum) measured from the zero or mean value. Peak ground acceleration (PGA) The maximum amplitude of a recorded acceleration (similar to the ZPA, or zero period acceleration, which is the acceleration at infinite frequency). Peak level The highest absolute value in a vibration signal. Peak-to-peak value For an oscillating quantity, this is the algebraic difference between the maximum and minimum extremes of the quantity. Pendulum A simple pendulum. A point mass suspended from a fixed point by a string. A mechanical swing oscillator whose motion is harmonic at small amplitudes, but nonlinear otherwise an archetype of deterministic chaos. These are assumed to have just one DoF (the angle of swing). More complex pendula such as compound pendulum and double pendulum, which have more than one DoF, are available. Perimeter The edge or boundary of an area. Period Duration of one complete cycle in a cyclic waveform (e.g., vibration signal, pressure or particle local velocity in an acoustic system). The inverse of the cyclic frequency (Hz). Period shift effect Reducing the structural response by increasing the structural period. Periodic monitoring Regular and intermittent data gathering and analysis with portable and removable, monitoring equipment. Periodic quantity An oscillating quantity whose value repeats itself in cyclic (periodic) manner. Phase The fractional part of a period through which a variable (signal) has advanced, measured from an arbitrary reference. A time difference, expressed in degrees, between two signals gives their phase difference. A shift of one period of the signal is considered a phase shift of 2p (or 3608). Phase lag corresponds to a time delay and phase lead corresponds to a time advance of the signal.

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Phase angle The fractional part of a period through which the quantity has advanced as a measure from a reference independent variable. Phase difference See Phase. The differences between the phase angles of two periodic quantities. In theory, only those components with the same frequency can be compared for phase. Phase noise See Phasor. In phasor representation, the noise vector that is perpendicular, rather than parallel, to the oscillator (primary signal) vector. Phasor A rotating vector of length (magnitude) a and angular speed v: Its projection on the vertical axis gives a sinusoidal wave of amplitude a and angular frequency v: The phase angle may be represented by the starting angle of the phasor from a reference line (e.g., vertical axis). Phons Unit of loudness level. See Loudness level. Photogate Infrared beam device used for time interval measurement. Physical pendulum See Pendulum. Contrasted with a simple pendulum, its motion at very small amplitudes is not harmonic an archetype of mesodynamic complexity. Physical realizability Whether an analytical model is physically realizable (whether the cause effect condition or causality is satisfied). Pierson Moskowitz spectrum A parameter ocean wave spectrum first derived for the North Atlantic, which is applicable for deep water, fully developed, and unidirectional seas. Piezoelectric A phenomenon exhibited by some materials in which application of a strain (or stress) causes the establishment of an electric field (or charge) and vice versa. The forward principle is used in piezoelectric accelerometers and the reverse principle is used in piezoelectric actuators. Pinned beam See Simply supported beam. Pitch control rod adjustments Extension or contraction of pitch control rods by a number of notches to alter the pitch of the rotor blades. Positive pitch rod adjustments indicate extension. Plane wave A sound wave that is uniform over any plane normal to a specified axis. This is applicable when propagation of sound wave is one dimensional and the amplitude of the sound pressure variation is constant. This is approximately valid locally for a sound field very far from the source region.

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Poisson’s ratio Lateral strain divided by the direct strain (absolute value, disregarding the sign) in a member in tension in one dimension. Poles Eigenvalues of a system. The roots of the characteristic equation. The characteristic polynomial is the denominator of the transfer function. Portevin LeChatelier Effect “Jerky” strain of a loaded specimen as a result of internal friction arising from mesoscale structural changes. Post mortem The investigation of root cause failure mechanisms. Postprocessing The process of calculating and graphically representing additional quantities and results (e.g., stresses, reactions) from the primary finite element solution results (e.g., displacements). Positive definite A symmetric matrix is positive definite if xt Ax . 0 for every nonzero vector x. Positive shock response spectrum SRS plotted, considering the highest positive response due to the shock, without reference to the duration of the shock. It is about the envelope of the positive primary and residual spectra. Postshock Shock pulses added after the specified shock test carried out on a shaker to cancel the velocity and displacement of the shaker table at the end of its movement (return to the rest position with a zero velocity). Potential energy (PE) Energy due to deformation of a spring or other flexible member (elastic PE) or due to elevated state (gravitational PE). Potentiometer (Pot) A displacement sensor consisting of a conductor with a constant voltage applied to it and a slider moving along it. The voltage of the slider is proportional to the displacement. Power conversion factor The ratio is sound power generated from a machine divided by the total power of the machine. This factor ranges from 10 8 to 10 5 for both mechanical and electrical machinery. Power flow Acoustic power translation from a subsystem to the next connected subsystem. Power spectral density The Fourier transform of the autocorrelation function of a signal. Also called power spectrum. It is a measure of how energy is distributed as a function of frequency for a random signal. A cross spectrum is similarly defined corresponding to the cross correlation function of two signals.

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Power supply A device that supplies DC power to instruments and circuits. In a regulated power supply, the output voltage does not drop when connected to an external instrument. Predictive maintenance An equipment maintenance strategy based on measuring the condition of equipment in order to assess whether it will fail during some future period, and then taking appropriate action to avoid the consequences of that failure. The performance of the equipment is assessed using such approaches as condition monitoring, statistical process control, or the human experience. See also Preventive maintenance. Preferred noise criteria (PNC) See Noise criteria (NC). PNC curves are used to modify NC curves and are utilized in exactly the same manner as NC curves. Preprocessing The process of building up the geometric and finite element model for a physical system. Preshock Shock pulses added before the specified shock test carried out on a shaker to cancel the velocity and displacement of the shaker table at the end of its movement (return to the rest position with a zero velocity). Pressure of sound Incremental variation of pressure due to a sound wave, about the ambient atmospheric pressure, with an accompanying variation of the air density. Pressure pulsations The oscillatory pressure variations that occur at a point in a fluid system. Preventive maintenance An equipment maintenance strategy based on replacing, overhauling or refurbishing machine parts and components at a fixed interval, regardless of its state or condition at that time. See also Predictive maintenance. Primary adaptation In feature classification: the process of adapting each weight of the single category based classifier (SCBC) closer to the current value of the feature if the corresponding feature is classified as normal. Primary (or initial) negative shock response spectrum Shock response spectrum (SRS) plotted using the highest negative response observed during the shock. Primary (or initial) positive shock response spectrum Shock response spectrum (SRS) plotted using the highest positive response observed during the shock. Principal axes of inertia Orthogonal axes relative to which no cross inertia exist. The inertia matrix will be diagonal in this coordinate frame. Principal axes of stiffness Axes along which any deflection will not create resultant loading in any other direction. The stiffness matrix will be diagonal in this coordinate frame.

Glossary

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Principal directions See Principal axes. Principal inertia Moment of inertia about a principal axis where the cross inertia are zero and the inertia matrix is diagonal. Principal stresses Stresses along the principal axes where the shear stresses are zero. Probability density function A function that defines the likeliness (or probability) that a random parameter will take a particular value of the function variable. Derivative of the “probability distribution function.” This function completely defines the random parameter. In vibration, this can be used to determine the probability of realizing a particular instantaneous amplitude value from a vibration signal within a certain amplitude range. Probability density moments See Expected value. Single number indices expressed as moments weighted by the probability density function. Probability distribution function A function that defines the probability that a random parameter will take a value less than or equal to a particular value of the function variable. Integral of the probability density function. This function completely defines the random parameter. Process damping In machining, damping due to the rubbing between the back side of tool and the surface of the machined part. This decreases chatter. Propeller Rotor with blades or a screw like device that imparts kinetic energy to a fluid (which can be converted into pressure and thrust). Used in compressors, pumps, jet engines. Provides thrust to push a boat through the water, or an aircraft through air. Proportional damping The damping matrix of a mechanical (vibrating) system is such that the modal vectors are real. The damping matrix is proportional to the mass matrix, stiffness matrix or both. Pseudoacceleration Product of the squared natural frequency ð2pfn Þ2 and the highest relative displacement response of a linear one degree of freedom system to a shock input. This product has the dimension of acceleration but does not represent the acceleration of the mass, except when damping is zero. In such cases, this term is then strictly equal to the absolute acceleration of the mass. Pseudovelocity Product of the natural frequency ð2pfn Þ2 and the highest relative displacement response of the mass of a linear one degree of freedom system to a shock input. This product has the dimension of velocity but does not represent the velocity of the mass. Pulse rise time Interval of time required for a shock pulse to rise from some specified small fraction to a large fraction (typically 10% to 90%) of its maximum value.

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Pyrotechnic shock (or pyroshock) Transient response of a structure to loading induced by the ignition of pyrotechnic (explosive or propellant activated) devices. Pyroshocks are characterized by high peak acceleration, short duration and high frequency content. Quadrature error The secondary signal that is at 908 phase to the primary signal, produced by an LVDT when the slider is symmetrically placed, while in theory it should be zero. Quadrature signals The two signals from a shaft encoder that are 908 out of phase, which are used to determine the direction of motion. Quality factor (Q-factor) A measure of the sharpness of a resonance or frequency selectivity of a vibratory linear one degree of freedom system. Assuming a lightly damped single DoF system, the Q factor is related to damping ratio by Q ¼ 1=ð2z Þ: Other approximate relations for Q are: 1. 2. 3. 4.

Transmissibility at resonance p/logarithmic decrement 2pW=DW where W is the stored energy and DW the energy dissipation per cycle fr =Df where fr is the resonant frequency and Df is the bandwidth between the half power points

Nonlinear damping causes it to vary with time. The concept is applicable to both mechanical and electrical systems. Quantization error The error created by the finite representation of numbers (i.e., finite number of digits or bits) within a computer (i.e., for a digital quantity). Quefrency See Cepstrum. Similar to frequency, but in the Quefrency domain. The independent variable of a cepstrum (similar to spectrum). Quefrency domain See Cepstrum. The domain of a cepstrum of a signal.

inverse Fourier transform of a log of the Fourier transform

Radiational sound Sound in an acoustic field generated from a radiating source. Random vibration A vibration whose magnitude cannot be precisely predicted for any given instant of time. Range of normal hearing Human hearing is most sensitive in the range 2000 to 5000 Hz, where the threshold response is very close to 0 dB. At the other end of the scale, we have the threshold of pain, which is usually about 135 to 140 dB. Reactive muffler A muffler where there are one or more chambers, resonators, or finite sections of pipe, which collectively provide impedance mismatch for the acoustic energy traveling along the duct, causing noise suppression. See Muffler and Dissipative muffler.

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Real-time Refers to the relative response time for a particular process under computer control to occur after an action is taken. Should be consistent with the response time of the physical system. Receptance The transfer function of displacement or force in the frequency domain. Rectangular shock Simple shock for which the acceleration time curve increases instantaneously up to a given value, remains constant throughout the signal and decreases instantaneously to zero. Practically, trapezoidal shocks are rather carried out. Rectilinear Along a straight line. Often (incorrectly) called “linear.” Recurrence relation Discrete equation (difference equation) that allows the computation of numbers in a series one by one. Regenerative chatter Unstable, self excited interaction between machining forces and structural vibrations. Relative displacement shock spectrum See Shock response spectrum. SRS plotted, taking its ordinate (y axis) the pseudoacceleration of the mass of a linear one DoF system. Relaxation Process characterized by an exponential variation, for example, sample strain response to a constant load. Reluctance Magnetic resistance. Resistance to magnetic field in a magnetic circuit (magnetic path). Repeatability The ability of a transducer to reproduce value is applied to it repeatedly, under Repeatability is expressed as the maximum scale. This concept is also applicable to the robots.

output readings accurately when the same measured the same conditions (in the same direction, etc.). difference between output readings as a percent of full operation of other instruments and machinery such as

Required response spectrum See RRS. Residual base displacement A permanent base displacement of an isolated structure after an earthquake. Resilient friction base isolator (RFBI) One type of sliding isolator, which is made of a central rubber core and Teflon coated steel plates. Resilient mechanism An automatic recentering mechanism of isolators by which the isolated structure can return to its original position after an earthquake.

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Resolution The smallest change (in a parameter, signal, instrument output, etc.) that can be meaningfully detected and used. Resonance A resonance occurs in a mechanical system excited by a harmonic force if the excitation frequency is equal to the resonant frequency of the system. A small change in the frequency of excitation in either direction will cause the system response to decrease rapidly if the system damping is low. At a resonance, very little energy input into a structure results in a very large displacement (for low damping levels). Resonant frequency is almost equal to (but less than) the undamped natural frequency when the damping is low. Although the resonant frequency is nearly equal to natural frequency, the two concepts are not the same. Resonant frequency is excited by a harmonic forcing function at that frequency. The natural frequency is excited by an initial condition (corresponding the particular mode shape) followed by a free (unforced) response. Resonance search A vibration testing procedure whose objective is to identify and observe various resonances in a test object. A harmonic excitation with variable frequency may be used for this. See Sine sweep. Resonator absorber A method in which a vibration absorber (see Vibration absorber) is tuned to have a natural frequency equal to the external excitation frequency to mimic optimally the vibratory energy from the primary system (system of concern). The absorber then “absorbs” the energy of vibration of the primary system, thereby suppressing those vibrations. Actually, the absorber generates a force in the opposite direction to the force of vibration of the primary system. Response The response of a device or system is the motion (or other output) resulting from an excitation (force, stimulus, or input) under specified conditions. Response spectrum (RS) A plot of maximum amplitudes (acceleration, velocity, or displacement) of a single DoF oscillator (Single DOF) as the natural period (or natural frequency) of the Single DOF is varied across a spectrum of engineering interest (typically, for natural periods from 0.03 to 3 or more seconds, or frequencies of 0.3 to 30 Hz; in seismic applications). One RS curve is obtained for a specified damping ratio of the Single DOF system. Response spectrum analysis Computation of the response spectrum (RS) of a given signal (see Response spectrum). This is quite different from spectral analysis (or spectrum analysis), which is to compute the Fourier spectrum of a signal. These are different from spectral density analysis, which determines the power spectral density function of a “random” signal. Reverse whirling Whirling is the rotation of a spinning rotor center of mass, which has an offset (due to shaft deflection) with respect to the axis through the bearings of its shaft. Reverse whirling occurs if the direction of this rotation about the bearing axis is in the opposite direction to the angular speed of spinning of the rotor with respect to a fixed frame of reference. Reynolds number A dimensionless parameter (that depends on viscosity and velocity), whose value determines if a fluid flow is laminar or turbulent.

Glossary

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Ridge Curve of local maxima in a mean square wavelet map. Rigid bodies Bodies exhibiting no deformation under acceleration or forces. Rigid-body mode A mode of motion that corresponds to zero natural frequency. Such a mode will exist in an unrestrained structure (no end fixtures) that is free to move as a rigid body without any deformation. Rigid-body motion A structural motion that results in no deformations or strains. By definition, these are zero frequency motions. Rigid perfectly plastic force deformation characteristics Characteristics of hysteretic loops produced by an ideal friction damping device. Rigid structure A structure that cannot be deformed and whose shape cannot be changed. RMS value Root mean square value. Square root of the arithmetical average of a set of squared instantaneous values of a response quantity. Rolling element bearings A rolling element bearing generally consists of two rings (races or raceways) with a set of rolling elements running in their tracks. The rolling elements take the form of balls or various types of rollers. The outer race is located in a housing and the inner race on the shaft. Root cause failure The basic cause for a given failure. Root Mean Square See RMS value. Rotating machinery The group of electrical or mechanical machinery composed of a rotating component that rides on some form of bearings encased in a stationary casing (housing). Rotor The rotating part (as opposed to the stator) of an electrical or mechanical machine. Rotor instability An excitation mode, which is self induced and does not require a sustained forcing phenomenon to initiate or maintain the motion. It occurs in machines operating at speeds well above the critical speeds of the rotor, and the rotor whirling speed is identical to the critical speed irrespective of the rotor (spinning) speed. Rotor internal damping A damping phenomenon, which occurs within the internal fibers of a material due to friction when there is relative movement between the fibers.

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Rotor track The vertical position of the blades of a rotor, usually measured by optical sensors. Rotor tuning The process of adjusting the rotor blades of a jet engine to reduce the system vibration (e.g., aircraft) and the spread of rotors. Rotordynamics The dynamics of rotating machinery. Round-off error The error created by the finite representation of numbers (i.e., finite number of digits) within a computer. Similar to Quantization error. RRS In a dynamic test, the response spectrum specification that has to be satisfied (enveloped) by the excitation signal. Rub A condition when a moving part comes into contact with, and slides against, a stationary part. Run-to-failure maintenance A strategy where maintenance is performed only when machinery has failed. See Scheduled maintenance, Predictive maintenance, Preventive maintenance, and Condition based maintenance. Runge Kutta Numerical methods (integration schemes) based on approximating the solution by a Taylor series, and weighted averages of corresponding differences. RVDT Rotary variable differential transformer. See LVDT. Similar to an LVDT, but for measuring angular displacements. Safe shutdown earthquake (SSE) An earthquake representation (e.g., by a response spectrum) for which a nuclear reactor can be safely shutdown without causing damage or fault in equipment due to the earthquake. Sampling The process of converting continuous signals into discrete values for digital processing. Sampling frequency The rate at which samples of a continuous time signal is recorded in discrete fixed time periods. Sampling theorem When a signal is sampled, its spectrum has no meaning beyond half the sampling frequency, called Nyquist frequency. Saturation A nonlinearity state in a device where a further increase in input no longer changes the output. Scaling To stretch or compress a function.

Glossary

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Scheduled maintenance A strategy where maintenance is performed at set time intervals (or duty cycles) in order to maintain a significant margin between machine capacity and actual duty. SEA Statistical energy analysis. Second moment of area Similar to moment of inertia, except uses area elements rather than mass elements. (The two are proportional for uniform material density and constant thickness along the axis.) Secondary (or Residual) negative shock response spectrum SRS plotted, considering the highest negative response observed after the end of the shock. Secondary (or Residual) positive shock response spectrum SRS plotted, considering the highest positive response observed after the end of the shock. Secondary system A general term for equipment attached to a primary structural system. Seismic Measurements and activity concerning earth motion, such as earthquakes. Seismic gap A gap provided between a seismically isolated building and the ground so that the ground level story can move in relation to the ground during earthquake excitations. Seismic hazards The phenomena or expectation of an earthquake related agent of damage, such as fault rupture, vibratory ground motion (i.e., shaking), inundation (e.g., tsunami, seiche, dam failure), various kinds of permanent ground failure (e.g., liquefaction), fire, or hazardous material release. Seismic isolation Application of base isolation technology to structures subjected to earthquake motions. Seismic risk The product of the hazard and the vulnerability (i.e., the expected damage or loss, or associated full probability distribution). Seismicity The geographic and historical distribution of earthquakes. Self excited instability Same as Rotor instability. Self-induced Created within itself and does not require an external influence. Self-organizing map (SOM) A type of unsupervised neural network that yields a graphical representation of input data where similar input vectors are placed in close proximity.

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Semiactive vibration control A vibration control system in which both active and passive (see Active vibration control and Passive vibration control) treatments are combined to reduce the amount of external power necessary to achieve the desired vibration control performance. Shaker See Exciter. Shape function Is a spatial function (i.e., function of position vectors) describing the change of a variable within an element domain. Shear balance method An analysis method for sliding isolated structures, which uses the force balance condition of total shear under the isolation system to compute the friction force. Shear force An internal force acting normal to the longitudinal axis given by the algebraic sum of all forces to one side of the section chosen. Shear wall A structural wall between stories of a building to increase shear strength but at the expense of increasing stiffness. Shifting The process of translating (moving) a function with respect to some reference (coordinate system). Shock Transient condition where the equilibrium of a system is disrupted by a sudden applied force or by a sudden change in the direction or magnitude of a velocity vector. The time duration of this change is shorter than about twice the natural period of the excited mechanical system. Shock duration Interval of time during which the shock pulse has a higher acceleration value than some specified small fraction of its maximum value (usually 10%). Shock machine (or shock test machine) Device for subjecting a system to controlled and reproducible mechanical shock. Two principal categories are usually distinguished. Impulse machines: increase the velocity of the test item during the shock. The initial velocity is generally zero. Impact machines: decrease the velocity of the test item throughout the shock or change its direction. Shock motion An excitation involving motion of a foundation. Shock programmer Same as Shock simulator. Device on a shock machine that is used to format the applied shock waveform according to a desired shape during impact (usually half sine, terminal peak sawtooth, and rectangular pulses).

Glossary

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Shock pulse A substantial disturbance characterized by a rise of acceleration from a constant value to a substantial peak value, and decay of acceleration to the constant value in a very short period of time. Shock pulses are normally displayed graphically as curves of acceleration as functions of time. Shock pulse index An accelerometer tuned to have its resonant frequency excited by a specific shock can provide early warning of fault (e.g., deteriorating bearings). The sensor output level is the shock pulse index (also known as spike energy). Shock response spectrum (SRS) The response of an array of single DoF systems to a shock input represented as the maximum response versus the natural frequency of single DoF systems for a specified damping ratio. This response may be expressed in units of acceleration, velocity, or displacement. Shock simulator Same as shock programmer. Device on a shock machine that is used to format the applied shock waveform during impact according to a desired shape (usually half sine, terminal peak sawtooth, and rectangular pulses). Shock spectrum See shock response spectrum. A plot (amplitude vs. natural frequency) of the response of a natural frequency variable single DoF system to an applied shock for a specified damping ratio. This response may be expressed in units of acceleration, velocity, or displacement. Shock wave A strong pressure wave produced by explosions or other phenomena that create violent changes in pressure. Shoring Temporary support at the base floors of a structure during replacement of its originally fixed support to seismically isolated support. Short-term statistics Statistics of a random process that are only good for a few hours or a few days at the most. It is assumed that the significant wave height and the wave spectrum are invariant with time. Short time FFT Time sequence of concatenated FFT of short period records of a signal. Shroud of impeller The circular section that forms a sidewall of a pump or compressor impeller. Signal-to-noise ratio (SNR) Determined by both electronic and mechanical noise (but not thermal noise due to the fluctuation dissipation theorem). Signature spectrum A base line frequency spectrum taken from new or recently overhauled machinery.

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Significant wave height A statistical description of random waves. Average of the highest one third of all waves or four times the variance. Consistently very close to the observed wave height. Simple beam See Simply supported beam. A beam restrained at its end only against vertical movement. Simple harmonic motion See Simple harmonic vibration. Simple harmonic vibration A motion such that the displacement, velocity and acceleration are pure sinusoidal functions of time. Sometimes called “harmonic motion.” Simple oscillator A single DoF mass spring damper system (or an LRC electric circuit), which can produce damped oscillations at one frequency (damped natural frequency). An undamped simple oscillator is a harmonic oscillator, which can produce a periodic oscillation described mathematically by either of the harmonic trigonometric functions (sine or cosine), and has only one frequency component (undamped natural frequency). Simple (or perfect) shock Shock whose signal can be represented exactly in simple mathematical terms. Standards specify generally one of the three following: half sine (approached by a versed sine waveform); terminal peak sawtooth; and rectangular shock (approached by a trapezoid waveform). Simply supported beam A beam pinned (or simply supported) at its ends so that the displacement and bending moment at these ends will be zero. Slopes and shear forces will not be zero. Sine beat A signal formed by adding two sine signals whose frequencies nearly equal, so as to generate beats. Such a signal is used for excitation in some types of vibration testing. Sine-dwell A vibration test where a shake applies a sinusoidal excitation of a specific frequency to the test object over a specific duration, changes the frequency, and repeats the dwell in specific frequency steps. Sine sweep A vibration test procedure where a harmonic excitation is applied to a test object (using an exciter or shaker) and the frequency of excitation is slowly and continuously varied at a specified rate. Single-category-based classifier (SCBC) The pattern classifier that performs abnormality scaling by comparing the value of its feature with its no fault value. Single-degree-of-freedom (single-DoF) system A system with only one coordinate (strictly, one incremental coordinate to include nonholonomic systems) to define completely the configuration of the system at any instant. Single plane balancing See Static balancing, which is synonymous.

Glossary

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SISO Single input single output system. A system with just one input and one output. Skeleton Wavelet transform values along a ridge. Sliding bearing A type of base isolator that protects a structure from ground excitations by using a sliding frictional interface at the base. SNR Signal to noise ratio, determined by both electronic and mechanical noise (but not thermal noise due to the fluctuation dissipation theorem). Soft foot A condition that occurs when one of a machine’s hold down bolts are not tight enough to resist the dynamic forces exerted by the machine. Soft story A story of a building that is more flexible than the other stories so that the stories above this story are isolated from the ground shaking. Softened compression stress strain relationship Viewing the shear action on a membrane element as a two dimensional problem, the compressive strength of concrete in one direction is reduced by cracking due to tension in the perpendicular direction. Softening parameter When a membrane element is subjected to shear, the compressive strength of concrete is reduced to approximately 40 to 60%. Soil structure interaction A dynamic interaction between a structure and its supporting soil, in which dynamic properties of one will affect the response of the other. Solenoid A linear (rectilinear) two position (on off) actuator. A ferromagnetic rod surrounded by a coil of many turns. An electric current through the coil will generate a flux that will move the rod, thereby turning on the solenoid. A version (linear motor) is able to provide continuous movements. Sone Unit of the loudness that is defined as the loudness of a 1000 jHz pure tone with a sound pressure level of 40 dB. This relationship is expressed as S ¼ 2ðLL 40Þ=10 ; S (sone) and L (phons). Sound absorbing material Porous materials such as glass fiber blankets used to attenuate sound propagating in ducts to improve sound transmission loss and to reduce standing waves in air spaces. Sound bridge Structures such as studs or stud bolts connecting the plates added to a noise source in insulation lagging or the plates of a double wall. They can considerably reduce insertion loss of the lagging or transmission loss of the double wall.

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Sound insulation lagging Thin plates separated by airspace or sound absorbing materials added to noise sources such as pipes, ducts and machines to isolate their noise. Sound intensity The time average rate of energy flow per unit area, denoted by the vector I (W/m2). Sound level Sound pressure level that has been weighted according to particular weighting curves: A B , C weighting networks and associated levels. The frequency response and decibel conversions from a flat response for each of these weightings are available. Sound power level The acoustic power radiated by a source with respect to the international reference of 10 expressed in decibels (dB).

12

(watts),

Sound pressure The difference between the instantaneous pressure at a fixed point in a sound field and the pressure at the same point without the sound (Pa). Alternatively, this may be defined as the ratio (dB) of the instantaneous pressure at a fixed point in a sound field with respect to a reference of 2 £ 10 6 (Pa), for which a pure tone of sound is assumed. Span The distance between two consecutive supports of a beam, truss, bridge, guideway, etc. Spatial coupling Effect of multiple support motions occurring when the attached equipment is mounted at multiple locations of the structure. Specific fan noise level Ratio (dB) of sound power generated from a fan due to its pressurized flow with respect to some standard power level. Specifications Conditions that must be satisfied by operating or design parameters during operation or design of a product in order to satisfy the objectives of the product. Spectral density See Power spectral density. Fourier transform of the autocorrelation function of a random signal (random process). It shows how the variance of the system is distributed over frequency. It may also be expressed as a “spectral energy density” since the unit of the variance often has the unit of energy. Spectrogram A graph of time vs. frequency, e.g., the Gabor distribution. Spectrum A description of a quantity as a function of frequency or wavelength. Fourier transform of a signal is called the Fourier spectrum (or simply a spectrum).

Glossary

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Spherical wave When propagation of sound wave is described by radial (polar) coordinates, the sound field is represented by a “spherical wave,” which is generated by a point monopole. Spike energy meter An accelerometer tuned to have its resonant frequency excited by the impacts in a deteriorating bearing, thus giving an early warning of an impending failure. Spindle speed selection In machining: a technique where the spindle speed is chosen to avoid or suppress chatter. Spindle speed variation In machining: a technique where the spindle speed is continuously adjusted to suppress chatter. Spring The basic flexibility element characterized by its stiffness, force, or deflection. Stores elastic potential energy. In three dimensional problems, a spring may assume three orthogonal stiffness parameters. SRSS Square root of sum of squares of a set of values. SSE See Safe shutdown earthquake. Stability Stability of a dynamic system: If the natural (unforced) response of a system decays to zero, then the system is asymptotically stable. If for a finite excitation, the system response is bounded, we have bounded input bounded output (BIBO) input. Stability of a numerical method: Numerical errors do not grow without bound. Stability lobe diagram Chatter stability chart in the process parameter space. Standardized shock response spectrum See Shock response spectrum (SRS). Standardized or reduced SRS plotted in dimensionless coordinates. The abscissa is the product of natural frequency and shock duration. The ordinate is the ratio spectrum value or shock pulse amplitude. Standoff distance Distance away from a specific structure that determines the amount of damage that would occur if an explosion of a specific size was detonated. For standoff threats, the distance away from a potential vantage point with direct line of sight to the asset. State space formulation A generalized analytical model for a dynamic system expressed as a minimum set of first order ordinary differential equations (state equations). The associate variables are the state variables that may include structural displacements and velocities, which will represent the dynamic state of the system. Static balancing See Balancing. Balancing achieved on a single plane of vibration perpendicular to the axis of rotation.

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Static loading Constant loading of a member. Stationary process A random process where the statistics can be considered independent of time. Stationary process is used synonymously with weakly stationary process where the first order statistics such as the mean and the variance are independent of time, and the second order statistics such as the autocorrelation function depend on the difference in time. Statistically determinate A structure is considered statistically determinate if all the support reactions and internal forces in the members can be determined solely by the equations of static equilibrium. Stator The stationary structure surrounding the rotating member (rotor) of a rotating electrical or mechanical machine. Steady-state error Offset. The error in the response of a system under steady state conditions. Steady-state vibration Steady state vibration exists in a system if the velocity of each particle is a continuous periodic quantity. Stiffness Characteristic parameter of a spring element. Ratio of change of force (or torque) to the corresponding change in translational (or rotational) deflection of an elastic element, i.e., force required to deflect an elastic element through a unit distance. Typical units are Newtons/meter (N/m). Stiffness asymmetry The stiffness (or flexibility) is not symmetrical in two or more orthogonal directions. Stiffness law Transmission loss characteristic of a single plate that the transmission loss depends on frequency and equivalent stiffness of the plate below the first order natural frequency of the plate. Stiffness matrix The matrix multiplied by the displacement vector in the Newtonian form of differential equations of motion of a mechanical (vibrating) system, expressed in the vector matrix form. Stochastic process An indexed collection of correlated (in general) random variables. Stochastic resonance Counterintuitive increase of SNR when the noise level is raised. Only possible in nonlinear systems. Stokes law Viscous friction force on a sphere of radius R, given by 6mRv, where m is the fluid’ viscosity. Strain Change in length divided by length. A nondimensional measure of the physical deformation or deflection resulting from stress (force per unit area).

Glossary

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Strain gauge A miniature resistive element (metal foil or semiconductor) whose resistance changes when stretched (deformed), which can be measured using a Wheatstone bridge circuit. When properly mounted on (i.e., bonded to) a structural member, it can measure the local direct strain in the member. Stress strain curve of a steel bar in concrete A steel bar in concrete is stiffened by tensile stress of concrete. It relates the average stress to the average strain of a large length of bar crossing several cracks. Stretched exponentials Discovered with Leyden jars (capacitors), a multiexponential model used by some to explain features of anelasticity. Stribeck damping A sophisticated friction model where the coefficient of friction drops rapidly and then increases gradually (somewhat linearly) as the relative velocity increases from zero. Stroke The maximum rectilinear movement that is possible in a device. Strouhal frequency Frequency at which the vortices are shed from a fixed cylinder subjected to a constant velocity flow. It is proportional to the diameter of the structure and inversely proportional to the stream velocity. Structural influence The coefficient that represents the proximity effect of the accelerometer generating a feature of a machine (gearbox) component. Structural wall An in plane wall to reduce the relative interstory distortions of a building caused by seismic induced motions. Structure-based connectionist network (SBCN) In machine monitoring for fault detection, this network takes advantage of the integration capability of neural networks, but avoids the need for supervised training. In the SBCN, the network weights are defined according to the “structural” knowledge of the monitored machine (e.g., gearbox) and the type of fault represented by various vibration features. The structural influences in the SBCN, which represent the proximity effect of faults on accelerometers, are determined according to the RMS value of the frequency response of a simplified lumped mass model of the monitored machine (e.g., gearbox). Structure-borne noise Type of noise that is based on the generation and propagation of timewise varying motions and forces in solid bodies, as well as on the associated sound radiation. Subharmonic Sinusoidal quantity of a frequency that is an integral submultiple of a fundamental frequency. Subharmonic resonancy A phenomenon observed in sliding isolation where the isolated system possesses extra harmonic resonant frequencies in the low frequency range in addition to the original ones as a result.

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Substructured online hybrid test A test in which only inelastic parts of a structure are experimentally tested, while the remaining elastic parts are modeled in a computer. Both parts are interconnected using online communications so that the complete structural responses are obtained. Substructuring Modeling, analysis and design development method where subsystems are developed first and assembled together (interconnected) through appropriate connecting elements. The analysis is simplified by using interconnection laws and making use of the results for the subsystems. Subsystem structure A structure that is regarded as a subsystem in the analyzing process of statistical energy analysis (SEA). Sweep rate The rate at which the frequency is changed in a sine sweep test. Symmetric matrix The corresponding elements of a symmetric matrix about its main diagonal are identical. Specifically, aij ¼ aji where a is an element, i is the row number, and j is the column number. The transpose does not change the matrix in this case. Synchronous averaging Averaging of the vibration signal synchronous with the running speed of the machinery being monitored. Synchronous whirling See Whirling. The angular velocity of whirling rotation of the rotor geometric center is equal to the rotational speed (spin) of the rotor shaft. Tachometer A speed sensor. See DC tachometer. AC tachometers have a primary coil with an AC signal (carrier signal), a secondary coil whose induced signal is demodulated to determine the speed, and a rotor that is either a permanent magnet of a closed conduction (as in induction motor). Thermal bearings Flexible bearings used in long span bridges for accommodating the changes of deck length due to varying temperature to protect bridge piers from receiving large lateral forces. Terminal peak sawtooth shock (TPS) Same as Final peak sawtooth shock (FPS). A simple shock for which the acceleration time curve has the shape of a triangle where acceleration increases linearly up to a maximum value and then instantly decreases to zero. Test factor Where the shock test specification is derived from measurements of the real environment, this is the coefficient applied to mean SRS of the measured shock pulses to take into account the variability of the resistance of the test item and the very small number of specimens tested (often only one specimen). Test response spectrum See TRS. Thermoelastic damping Energy dissipation (damping) due to the anharmonic, interatomic potential independent of defects.

Glossary

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Threshold The minimum value a stimulus may have to create a desired effect. Threshold of hearing This is defined for binaural listening as the lowest sound pressure in free field that one can just hear as the signal reduced. Threshold speed The speed at which a stimulus is perceptible or can produce an effect. Throttling Reducing the fluid flow in a system by a valve or regulator. Time constant The inverse of the real part (disregarding the negative sign) of an eigenvalue. Inversely proportional to the speed of response and the level of stability. Time-dependent effects Normally referred to material properties that are influenced by motion history. Time domain Domain of analysis of a dynamic (vibrating) system where the independent variable is time (t). In this domain, vibration data (and systems) are represented, displayed or analyzed as a function of time. Time-domain analysis Analysis of dynamic (vibration) systems and response (motion) results, which are described with respect to time. Time-domain simulation Computer simulation of the response of a dynamic system as a function of time. Technique to analyze chatter where delay differential equations defining the machining forces and structural vibrations are numerically simulated in the time domain. Time-history analysis The analysis of a response of a dynamic system (e.g., structure) with respect to time. Time waveform index A single number representative of a waveform, calculated using a raw vibration signal and used for trending and comparisons. TNT equivalent The weight of trinitrotoluene (TNT) that has an equivalent energy output to that of a specified weight of another explosive compound. Tool chatter Self excited vibration between the cutting tool and the workpiece due to a cutting tooth machining over a portion of the workpiece that was machined by a previous tooth of the cutter. This causes uneven chip thickness and consequently, excessive self excited vibrations. Tool condition monitoring (TCM) A method by which the condition or state of cutting tools can be monitored.

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Tool condition monitoring system (TCMS) A combination of sensors and algorithms to achieve TCM. Tool wear The loss of material on the edges of a cutting tool due to machining. Tooth passing frequency Frequency at which the cutting tool teeth pass a given point during a spindle revolution. Tooth passing period Time interval for two successive cutting tool teeth to pass a given point during a spindle revolution. Tornado A violently rotating column of air in contact with, and extending between, a convective cloud and the surface of the Earth. It is the most destructive of all storm scale atmospheric phenomena. They can occur anywhere in the world given the right conditions, especially after landfall of hurricanes. Torque whirl A whirling instability that can occur in overhung, long, slender, rotors with high load torque values. See Whirling. Torsional response Response causing a twisting motion. Torsional stiffness Torque applied to a member divided by the resulting angular displacement (twist) of the member. Torsional vibrations Oscillatory angular motion causing twisting of a shaft (carrying a rotor) in a rotating machine. Tracking analyzer Devise used to record and analyze data from machines with variable speed. Transducer A device that senses a physical quantity is called a sensor. A transducer is a device that converts the sensed signal (measurand) into a form (typically, an electrical output signal of sufficient value) proportional to the measured variable. Often, the terms sensor and transducer are used interchangeably. Transfer function The ratio of the output of a dynamic system to its input in the Laplace domain or frequency domain (see Frequency domain and FRF) assuming zero initial conditions. Transient-response analysis Sometimes called time history analysis, this determines the dynamic response of a dynamic system (e.g., structure, machine) under the action of time varying excitation. Excitation forces are explicitly defined in the time domain. Computed response usually includes the time varying displacements, velocities, accelerations, strains, and stresses. Transient vibration The vibratory motion of a mechanical system as it changes from one state to another as a function of time. This may be a decaying or growing signal but not a steady vibration. It may consist of forced or free vibration or both.

Glossary

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Transmissibiliy Transmitted vibration (force or motion) divided by an applied vibration (force or motion). Inverse of Isolation. Transmission loss The difference between incident and transmitted acoustic intensity levels. It is used to rate the sound insulating properties of a wall or some other structure or device. For a muffler, this is the ratio between the energy in incident sound wave at the muffler inlet and energy transmitted at the muffler outlet into perfectly absorbing termination. Transmitted power An acoustic power that is transmitted from a subsystem to another subsystem. Trapezoidal shock Simple shock for which the acceleration time curve grows linearly up to a given value, remains constant during a certain time, and then decreases linearly to zero. Trim tab adjustments Trim tabs are adjustable surfaces on the trailing edge of the rotor blades of a rotating machine (e.g., jet engine). They affect the aerodynamic pitch moment of their foils and consequently, their vibration characteristics. Tab adjustments are measured in thousandths of an inch (thou), with positive and negative changes representing upward and downward tabbing, respectively. TRS Test response spectrum. The response spectrum of the excitation signal that is applied to a test object during testing. Truncation error The error created by deleting a signal after a finite time period to obtain a finite segment before digitally analyzing it. This is equivalent to multiplying the signal by a rectangular window. The error can be reduced (or ripples in the frequency spectrum of the truncated signal can be suppressed) by using other types of window functions with smoothing at the two ends. Tuning The process of adjusting the controller parameters in order to achieve the required performance (e.g., vibration control). Unbalance (or imbalance) A condition when the center of mass of a rotating component is not coincident with the center of rotation. Unbalance response The displacement vibratory response of a rotor to its mass imbalance as a function of rotor speed. Uncertainty factor Where the shock test specification is derived from measurements of a real environment, this is the coefficient applied to the mean SRS of the measured shock pulses to take into account the variability of the shock environment and the variability of the resistance of the material.

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Uncertainty principle The product of uncertainties of conjugate variables is greater than or equal to a constant, which in quantum mechanics is Planck’s constant. Also known by the name Heisenberg. Undamped natural frequency The frequency of free vibration resulting from only elastic and inertial forces of the system while neglecting damping. Undamped system A system where there is no energy dissipation. It has energy storage capability (e.g., inertia elements to store kinetic energy and springs to store elastic potential energy). If there is no external force applied on the system, then the system will experience free steady vibrations. Motion of the system are established by an initial disturbance (i.e., initial conditions). Undetermined coefficient method Use a trial solution consisting of the sum of possible set of functions multiplied by unknown coefficients. Substitute in the equation to be solved and equate like terms to determine the unknown coefficients. Uplift force Force perpendicular to the flow. Van der Pol oscilaltor Classic nonlinear electronic oscillator. Variable frequency drives A controlling device to vary the frequency of AC supplied to an electric motor in order to vary proportionately the speed of rotation of the motor. Variable structure control (VSC) A class of nonlinear switching controllers whose structure consists of different phases. Typically associated with a discontinuity function (e.g., signum function). When the system response enters a different phase, a different controller (corresponding to the particular phase) is switched on. VCC Voltage to current converter. Velocity A vector quantity that specifies the magnitude and direction of the rate of change of displacement with time along a specified axis with respect to a reference frame (m/sec). First derivative of displacement with respect to time. It may refer to angular motion as well as to uniaxial (rectilinear) motion. Where the reference frame is not inertial, the velocity is often designated “relative velocity.” Velocity change Variation of the velocity during a shock, i.e., integral of the acceleration time history signal over the shock duration. Velocity of sound Velocity of sound wave propagation. In air, it is proportional to the square root of ambient pressure and of the inverse of ambient density, or it is proportional to the square root of the absolute temperature. For example, in stationary air, the velocity is 340.4 m/sec at 158C.

Glossary

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Velocity shock Shock motion characterized by a sudden velocity change of the support (base) of a system. Versed-sine (or Haversine) shock Simple shock for which the acceleration time curve has the form of one period of the curve representative of the function [1 cos( )], with this period starting from the zero value of the function. It is thus a signal ranging between two minima of a sine wave. VFC Voltage to frequency converter. Vibration The oscillating, reciprocating, or other periodic motion of a rigid or elastic body or medium (such as a floor) when its position or state of equilibrium has been changed. Vibration absorber A secondary system (usually a mass spring damper trio with zero or low damping) to protect the primary device from vibrating. Typically, the secondary system vibrates at its natural frequency (resonance) generating an opposing force to suppress vibration in the primary system. Vibration absorption Addition of a vibration absorber to protect the primary device from vibrating. Vibration analyzer The device that could monitor and analyze the machine conditions using the machine’s vibration signals. Vibration control Means of controlling vibration using external or internal forces and loads. Vibration features Aspects of vibration that can illuminate the effect of individual faulty components of a machine (e.g., gearbox) on vibration. Usually, vibration features are defined as functions of specific frequencies affected by the components. Vibration isolation See Isolation. Isolating the source of vibration from the concerned system protecting a device from vibration of its point of attachment. Motion isolation and force isolation are of interest. Vibration measurements For a bladed rotating machine (e.g., jet engine, turbine, compressor), obtained at the frequency of once per blade revolution (1 per rev) by two accelerometers, A and B, attached to the sides of the cockpit. The vibration data is vectorially combined into two components: A þ B, representing the vertical (heave) vibration of the system (e.g., aircraft), and A B, representing its roll vibration. Vibration meter Small, handheld, inexpensive, simple to use, self contained device that gives an overall vibration level reading. Vibration mode See Mode shape. For a flexible plate, the vibration represents the particular spatial configuration of vibration amplitude (i.e., mode shape) with phase information of the moving elements, and the associated natural frequency. The actual vibration of the plate is obtained by the superposition of the

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distinct vibration modes. For continuous (i.e., distributed parameter) systems (e.g., plate), there are infinite distinct modes, many of which at the high frequency end can be neglected depending on the frequency range of operation. Vibration modeling The process of converting an engineering vibration problem into a mathematical model, whereby the major vibration characteristics of the original problem can be accurately predicted. Vibration mounts Devices that are placed at the base (or legs) of a machine or equipment to reduce its vibration problems and specifically, to reduce the vibration forces transmitted from the machine or equipment to the support structure and its environment. A key purpose is vibration isolation. Inertia blocks may be attached to the machine base and the mounts. Vibration response The output of a vibrating system for an arbitrary input. Vibration signature A unique frequency response or pattern that corresponds to a particular input or excitation signal. Vibration standards National and international vibration standards and guidelines published by recognized professional organizations and regulatory agencies. The objective of these publications is to establish and control quality, safety, durability and reliable performance of rotating machinery. Viscoelasticity Model of anelasticity where an elastic solid also has fluid like (viscous) properties. Viscous damping The dissipation of energy that occurs when a particle in a vibrating system is resisted by a force that has a magnitude proportional to the magnitude of the relative velocity of the particle and a direction opposite to the direction of the particle. Volume coefficient of elasticity of air Generally, isothermal conditions prevail for frequencies below 100 Hz, whereas adiabatic conditions prevail above 1 kHz. In between these limits, there is a volume coefficient of elasticity (similar to bulk modulus) that varies between 1.0 £ 105 and 1.4 £ 105 N/m2. Volute One of the major components of a centrifugal pump forming the casing around the impeller, which channels the flow exiting the impeller into the outlet of the pump while converting the velocity energy generated by the impeller into pressure energy. Vortex Any circular or rotary flow in the atmosphere that possesses vorticity. Vulnerability The expected damage given a specified value of a hazard parameter.

Glossary

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Wake Region surrounded by the shear layers originating from separation points. Wake excitation Excitation caused by the vortices in the wake. Waterfall plots Successive spectra plotted with respect to time and displayed in a three dimensional manner. Also known as Cascade plots. Wavelength In acoustics, the distance between successive two peaks in the repetitive spatial variation of the pressure or the particle local velocity. Wavelet Functions having representations in the time and frequency domains of small support set. Wavelet analysis Similar to Fourier analysis, wavelet analysis is the breaking up of a time waveform into shifted and scaled versions of a wavelet. A wavelet is a waveform of limited duration with an average value of zero. Wavelet transform Representation of a function as a series of wavelets. Wear-in failures Failures that occur at the beginning of a machine’s useful life, usually characterized by a relatively high rate of failure. Wearout failures Failures that occur as a machine reaches the end of its design life. WECPNL Acronym of weighted equivalent continuous perceived noise level. It is very close to LAeq in concept and it would not be difficult to change the assessment index from WECPNL to LAeq. WECPNL is used for the assessment of aircraft noise. Whipping A term used to describe relatively large motion associated with whirling of a rotor. Typically, occurs when the whirling speed matches the critical speed of the rotor. Whirling Condition when the rotor shaft bends so that the center of mass of the rotor rotates about the fixed axis through the bearings while spinning about the shaft geometric axis. See Synchronous whirl. White noise process A random process where the energy is equally distributed over the entire frequency domain. It is a mathematical simplification rather than a real process. Wideband Corresponding to a wide (large) frequency band.

G-74


Wideband process A random process where most of the energy is in a wideband of frequency. Wiener–Khinchine relationship Relates the autocorrelation function to the spectral density of a random process. Young’s modulus See Modulus of elasticity. Zener model Classic description of damping that involves relaxation processes — named after the individual who also contributed to a well-known voltage controlling diode. Zero drift See Drift. The drift in a device under conditions of zero output. Zero period acceleration See ZPA. Zero shift Slow variation of the average of a signal due to the fact that high accelerations make the crystal of the accelerometer work in a temporarily nonlinear field. This shift can affect the calculation of the shock response spectrum. Zeros The roots of the numerator polynomial equation of a transfer function. Together with poles, they represent the entire system (transfer function) and govern the system behavior (stability, phase, speed of response, percentage overshot, rise time, etc.). ZPA Zero period acceleration. The acceleration shown on a RS or shock SRS at infinite frequency. This can be used as a representative parameter of a signal (e.g., earthquake ground motion) for testing and designing a system.

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