Spectral Element Method in Structural Dynamics

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SPECTRAL ELEMENT METHOD IN STRUCTURAL DYNAMICS

Spectral Element Method in Structural Dynamics Usik Lee © 2009 John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82374-3

SPECTRAL ELEMENT METHOD IN STRUCTURAL DYNAMICS Usik Lee Inha University, Republic of Korea

Copyright Ó 2009

John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop, # 02-01, Singapore 129809

Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted by law, without either the prior written permission of the Publisher, or authorization through payment of the appropriate photocopy fee to the Copyright Clearance Center. Requests for permission should be addressed to the Publisher, John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop, #02-01, Singapore 129809, tel: 65-64632400, fax: 65-64646912, email: [email protected]. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. All trademarks referred to in the text of this publication are the property of their respective owners. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstrasse 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons Canada Ltd, 5353 Dundas Street West, Suite 400, Toronto, ONT, M9B 6H8, Canada Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Lee, Usik. Spectral element method in structural dynamics / Usik Lee. p. cm. Includes index. ISBN 978-0-470-82374-3 (cloth) 1. Structural dynamics–Mathematics. 2. Structural frams–Mathematical models. 3. Spectral theory (Mathematics) I. Title. TA654.15.L44 2009 624.1’71–dc22 2009015525 ISBN 978-0-470- 82374-3(HB) Typeset in 10/12pt Times by Thomson Digital, Noida, India. Printed and bound in Singapore by Markono Print Media Pte Ltd, Singapore. This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Contents Preface Part One

xi Introduction to the Spectral Element Method and Spectral Analysis of Signals

1 Introduction 1.1 Theoretical Background 1.1.1 Finite Element Method 1.1.2 Dynamic Stiffness Method 1.1.3 Spectral Analysis Method 1.1.4 Spectral Element Method 1.1.5 Advantages and Disadvantages of SEM 1.2 Historical Background 2 Spectral Analysis of Signals 2.1 Fourier Series 2.2 Discrete Fourier Transform and the FFT 2.2.1 Discrete Fourier Transform (DFT) 2.2.2 Fast Fourier Transform (FFT) 2.3 Aliasing 2.3.1 Aliasing Error 2.3.2 Remedy for Aliasing 2.4 Leakage 2.4.1 Leakage Error 2.4.2 Artificial Damping 2.5 Picket-Fence Effect 2.6 Zero Padding 2.6.1 Improving Interpolation in the Transformed Domain 2.6.2 Remedy for Wraparound Error 2.7 Gibbs Phenomenon 2.8 General Procedure of DFT Processing 2.9 DFTs of Typical Functions 2.9.1 Product of Two Functions 2.9.2 Derivative of a Function 2.9.3 Other Typical Functions

1 3 3 3 4 4 5 6 8 11 11 12 12 16 17 17 20 20 20 23 25 25 26 27 29 30 34 34 36 36

Contents

vi

Part Two

Theory of Spectral Element Method

39

3 Methods of Spectral Element Formulation 3.1 Force-Displacement Relation Method 3.2 Variational Method 3.3 State-Vector Equation Method 3.4 Reduction from the Finite Models

41 41 58 68 75

4 Spectral Element Analysis Method 4.1 Formulation of Spectral Element Equation 4.1.1 Computation of Wavenumbers and Wavemodes 4.1.2 Computation of Spectral Nodal Forces 4.2 Assembly and the Imposition of Boundary Conditions 4.3 Eigenvalue Problem and Eigensolutions 4.4 Dynamic Responses with Null Initial Conditions 4.4.1 Frequency-Domain and Time-Domain Responses 4.4.2 Equivalence between Spectral Element Equation and Convolution Integral 4.5 Dynamic Responses with Arbitrary Initial Conditions 4.5.1 Discrete Systems with Arbitrary Initial Conditions 4.5.2 Continuous Systems with Arbitrary Initial Conditions 4.6 Dynamic Responses of Nonlinear Systems 4.6.1 Discrete Systems with Arbitrary Initial Conditions 4.6.2 Continuous Systems with Arbitrary Initial Conditions

77 77 79 81 82 83 86 86

Part Three

87 89 90 99 104 105 107

Applications of Spectral Element Method

111

5 Dynamics of Beams and Plates 5.1 Beams 5.1.1 Spectral Element Equation 5.1.2 Two-Element Method 5.2 Levy-Type Plates 5.2.1 Equation of Motion 5.2.2 Spectral Element Modeling 5.2.3 Equivalent 1-D Structure Representation 5.2.4 Computation of Dynamic Responses Appendix 5A: Finite Element Model of Bernoulli–Euler Beam

113 113 113 114 119 119 120 125 126 130

6 Flow-Induced Vibrations of Pipelines 6.1 Theory of Pipe Dynamics 6.1.1 Equations of Motion of the Pipeline 6.1.2 Fluid-Dynamics Equations 6.1.3 Governing Equations for Pipe Dynamics 6.2 Pipelines Conveying Internal Steady Fluid 6.2.1 Governing Equations

133 133 134 136 137 138 138

Contents

6.2.2 Spectral Element Modeling 6.2.3 Finite Element Model 6.3 Pipelines Conveying Internal Unsteady Fluid 6.3.1 Governing Equations 6.3.2 Spectral Element Modeling 6.3.3 Finite Element Model Appendix 6.A: Finite Element Matrices: Steady Fluid Appendix 6.B: Finite Element Matrices: Unsteady Fluid

vii

139 144 146 146 147 153 157 159

7 Dynamics of Axially Moving Structures 7.1 Axially Moving String 7.1.1 Equation of Motion 7.1.2 Spectral Element Modeling 7.1.3 Finite Element Model 7.2 Axially Moving Bernoulli–Euler Beam 7.2.1 Equation of Motion 7.2.2 Spectral Element Modeling 7.2.3 Finite Element Model 7.2.4 Stability Analysis 7.3 Axially Moving Timoshenko Beam 7.3.1 Equations of Motion 7.3.2 Spectral Element Modeling 7.3.3 Finite Element Model 7.3.4 Stability Analysis 7.4 Axially Moving Thin Plates 7.4.1 Equation of Motion 7.4.2 Spectral Element Modeling 7.4.3 Finite Element Model Appendix 7.A: Finite Element Matrices for Axially Moving String Appendix 7.B: Finite Element Matrices for Axially Moving Bernoulli–Euler Beam Appendix 7.C: Finite Element Matrices for Axially Moving Timoshenko Beam Appendix 7.D: Finite Element Matrices for Axially Moving Plate

163 163 163 165 170 172 172 174 178 178 181 181 183 188 189 192 192 195 204

210 212

8 Dynamics of Rotor Systems 8.1 Governing Equations 8.1.1 Equations of Motion of the Spinning Shaft 8.1.2 Equations of Motion of Disks with Mass Unbalance 8.2 Spectral Element Modeling 8.2.1 Spectral Element for the Spinning Shaft 8.2.2 Spectral Element for the Disk 8.2.3 Assembly of Spectral Elements 8.3 Finite Element Model 8.3.1 Finite Element for the Spinning Shaft

219 219 220 223 228 228 237 239 242 243

209 210

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8.3.2 Finite Element for the Disk 8.3.3 Assembly of Finite Elements 8.4 Numerical Examples Appendix 8.A: Finite Element Matrices for the Transverse Bending Vibration 9 Dynamics of Multi-Layered Structures 9.1 Elastic–Elastic Two-Layer Beams 9.1.1 Equations of Motion 9.1.2 Spectral Element Modeling 9.1.3 Spectral Modal Analysis 9.1.4 Finite Element Model 9.2 Elastic–Viscoelastic–elastic–Three-Layer (PCLD) Beams 9.2.1 Equations of Motion 9.2.2 Spectral Element Modeling 9.2.3 Spectral Modal Analysis 9.2.4 Finite Element Model Appendix 9.A: Finite Element Matrices for the Elastic–Elastic Two-Layer Beam Appendix 9.B: Finite Element Matrices for the Elastic–VEM–Elastic Three-Layer Beam

246 247 249 253 255 255 255 258 263 266 269 269 272 279 283 288 289

10 Dynamics of Smart Structures 10.1 Elastic–Piezoelectric Two-Layer Beams 10.1.1 Equations of Motion 10.1.2 Spectral Element Modeling 10.1.3 Spectral Element with Active Control 10.1.4 Spectral Modal Analysis 10.1.5 Finite Element Model 10.2 Elastic–Viscoelastic–Piezoelctric Three-Layer (ACLD) Beams 10.2.1 Equations of Motion 10.2.2 Spectral Element Modeling 10.2.3 Spectral Element with Active Control 10.2.4 Spectral Modal Analysis 10.2.5 Finite Element Model

293 293 293 297 300 301 303 305 305 308 312 313 315

11 Dynamics of Composite Laminated Structures 11.1 Theory of Composite Mechanics 11.1.1 Three-Dimensional Stress–Strain Relationships 11.1.2 Stress–Strain Relationships for an Orthotropic Lamina 11.1.3 Strain–Displacement Relationships 11.1.4 Resultant Forces and Moments 11.2 Equations of Motion for Composite Laminated Beams 11.2.1 Axial–Bending–Shear Coupled Vibration 11.2.2 Bending–Torsion–Shear Coupled Vibration 11.3 Dynamics of Axial–Bending–Shear Coupled Composite Beams 11.3.1 Equations of Motion

319 319 319 320 322 323 324 325 327 330 330

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11.3.2 Spectral Element Modeling 11.3.3 Finite Element Model 11.4 Dynamics of Bending–Torsion–Shear Coupled Composite Beams 11.4.1 Equations of Motion 11.4.2 Spectral Element Modeling 11.4.3 Finite Element Model Appendix 11.A: Finite Element Matrices for Axial–Bending–Shear Coupled Composite Beams Appendix 11.B: Finite Element Matrices for Bending–Torsion–Shear Coupled Composite Beams

330 336 339 339 339 346

12 Dynamics of Periodic Lattice Structures 12.1 Continuum Modeling Method 12.1.1 Transfer Matrix for the Representative Lattice Cell (RLC) 12.1.2 Transfer Matrix for an ET-Beam Element 12.1.3 Determination of Equivalent Continuum Structural Properties 12.2 Spectral Transfer Matrix Method 12.2.1 Transfer Matrix for a Lattice Cell 12.2.2 Transfer Matrix for a 1-D Lattice Substructure 12.2.3 Spectral Element Model for a 1-D Lattice Substructure 12.2.4 Spectral Element Model for the Whole Lattice Structure

355 355 356 361 362 365 366 367 368 369

13 Biomechanics: Blood Flow Analysis 13.1 Governing Equations 13.1.1 One-Dimensional Blood Flow Theory 13.1.2 Simplified Governing Equations 13.2 Spectral Element Modeling: I. Finite Element 13.2.1 Governing Equations in the Frequency Domain 13.2.2 Weak Form of Governing Equations 13.2.3 Spectral Nodal DOFs 13.2.4 Dynamic Shape Functions 13.2.5 Spectral Element Equation 13.3 Spectral Element Modeling: II. Semi-Infinite Element 13.4 Assembly of Spectral Elements 13.5 Finite Element Model 13.6 Numerical Examples Appendix 13.A: Finite Element Model for the 1-D Blood Flow

373 373 373 375 376 377 378 379 380 381 384 385 386 388 391

14 Identification of Structural Boundaries and Joints 14.1 Identification of Non-Ideal Boundary Conditions 14.1.1 One-End Supported Beam 14.1.2 Two-Ends Supported Beam 14.2 Identification of Joints 14.2.1 Spectral T-Beam Element Model for Uniform Beam Parts 14.2.2 Equivalent Spectral Element Model of the Joint Part 14.2.3 Determination of Joint Parameters

393 393 394 397 404 404 405 407

349 351

x

Contents

15 Identification of Structural Damage 15.1 Spectral Element Modeling of a Damaged Structure 15.1.1 Assembly of Spectral Elements 15.1.2 Imposition of Boundary Conditions 15.1.3 Reordering of Spectral Nodal DOFs 15.2 Theory of Damage Identification 15.2.1 Uniform Damage Representation 15.2.2 Damage Identification Algorithms 15.3 Domain-Reduction Method 15.3.1 Domain-Reduction Method 15.3.2 Three-Step Process

413 413 413 414 415 416 416 417 425 425 427

16 Other Applications 16.1 SEM–FEM Hybrid Method 16.2 Identification of Impact Forces 16.2.1 Force-History Identification 16.2.2 Force-Location Identification 16.3 Other Applications

429 429 434 435 436 439

References

441

Index

449

Preface Owing to the rapid developments in computer technology, impressive progress in the computational methods used in engineering and science has been made over recent decades. The classical finite element method (FEM) has probably been the most popular in many areas of engineering and science, being one of the most convenient and easy-to-use computational methods. Though the FEM is applicable to most geometries, boundary conditions and material variations, it can be extremely expensive and it is often impossible to work out solutions to the large scale finite element models using a desktop computer. Thus, an alternative method that can provide accurate solutions while reducing the computational burden, but retaining the key advantage features of FEM, is mandatory, even today. The FEM is a time-domain solution method in which the finite element equation is formulated in the time domain and solved by using a numerical integration method. On the other hand, the spectral element method (SEM) is a frequency-domain solution method in which the spectral element equation is formulated in the frequency domain and solved by using the fast Fourier transform (FFT) based spectral analysis method. In SEM, the exact dynamic stiffness matrix, known as the spectral element matrix, is formulated in the frequency domain by using exact wave solutions for the governing differential equations. Accordingly, in theory, the SEM will provide exact frequency-domain solutions while using only a minimum number of degrees-of-freedom. Although there have been a huge number of journal publications since the basic concept of SEM was initially introduced more than two decades ago, very few books exist on the subject. Thus, the present book presents a concise introduction to the theory of SEM and its applications to various problems in structural dynamics and other related areas. It is intended as a valuable reference book for graduate students, professors, and professional researchers in the areas of mechanical engineering, civil engineering, aerospace engineering, naval architecture, structural engineering, applied mechanics, biomechanical engineering, and other related areas including computational methods. This book could be also used as a textbook for graduate students. There are three parts to the book. The first part (Chapters 1 and 2) addresses the background and history of the SEM and the fundamentals of the spectral analysis of signals. In the second part (Chapters 3 and 4), the methods of spectral element formulation and the spectral element analysis method are addressed. The last part, from Chapter 5 to Chapter 16, presents the applications of SEM to various engineering problems in structural dynamics and related areas. The following overview summarizes the key features of each chapter. Chapter 1 addresses key features of the SEM, together with a brief historical perspective on the development of SEM and its applications.

xii

Preface

Chapter 2 introduces the fundamentals of the DFT (discrete Fourier transform) and FFT theories necessary for the spectral analysis of signals. Chapter 3 is devoted to various methods of spectral element formulation: the forcedisplacement relation method, the variational method, and the state-vector equation method. Chapter 4 addresses the general procedure of spectral element analysis: formulation and assembly of spectral elements; imposition of boundary conditions; and the computation of eigensolutions and time responses. Chapter 5 deals with the dynamics of beams and Levy-type plates. For distributed dynamic forces, the two-element method is introduced as an approximation approach. Chapter 6 is devoted to the spectral element modeling and analysis of the flow-induced vibrations of pipelines conveying two types of internal fluids: steady fluid and unsteady fluid. Chapter 7 addresses the spectral element modeling and analysis of the dynamics and stability of axially moving structures, such as the string, the Bernoulli–Euler beam, the Timoshenko beam, and the thin plates. Chapter 8 is devoted to the spectral element modeling and analysis of the dynamics of rotor systems, which consist of the flexible shafts, thin and thick rigid disks and bearing supports. Chapter 9 discusses the spectral element modeling and analysis of the dynamics of multi-layered beam structures, such as the elastic–elastic two-layer beams and the elastic– viscoelastic–elastic passive constrained layer damping (PCLD) beams. Chapter 10 discusses the spectral element modeling and analysis of the dynamics of adaptive structures, such as the elastic–piezoelectric two-layer beams and the active constrained layered damping (ACLD) beams with and without active control. Chapter 11 is devoted to the spectral element modeling and analysis of the dynamics of composite laminate beams subjected to axial-bending-shear coupled vibration and to bendingtorsion-shear coupled vibration. Chapter 12 addresses the SEM-based continuum modeling method and the spectral transfer matrix method for the efficient dynamic analysis of various types of periodic lattice structures. Chapter 13 extends the SEM application to a biomechanics problem: blood flow analysis. The pseudo-force based non-linear spectral element modeling is introduced. Chapter 14 discusses the SEM-based methods to identify non-ideal structural boundaries and the joints in a beam structure from experimentally measured frequency response function (FRF) data. Chapter 15 addresses the SEM-based linear and non-linear structural damage identification methods to identify multiple local damage generated in a structure from experimentally measured FRF data. Chapter 16 discusses other promising SEM applications, such as the SEM–FEM hybrid method and the impact force identification, while providing a brief list of applications from the existing literature. In most chapters the derivations of the associated governing equations are provided. Furthermore, the conventional finite element models are also provided in the appendices of each chapter for the use in validating or comparing with the corresponding spectral element models. Though no source codes are included in this book, some code examples (M files) are available for download from the book’s companion website http://www.wiley.com/go/ulee. This book is the product not only of my knowledge, research and teaching experience, but of numerous discussions with my past and present graduate students over a period of more than 15 years. I wish to acknowledge the contributions of my graduate students during the various

Preface

xiii

stages of the manuscript, without which this book would not have been completed. All support and cooperation from the staff of John Wiley & Sons, senior commissioning editor Mr. James Murphy, project editor Mr. Roger Bullen, and production editor Ms. Sarah Karim are accordingly acknowledged. The author also acknowledges the financial support from Inha University (Inha University Research Grant) during the course of manuscript preparation. Finally, I dedicate this book to my mother Chungkyung Koh and to the memory of my father Seokbong Lee. The author will be pleased to hear from readers who find misprints and errors, or who can provide hints to other ways of improving the book in any future editions. Usik Lee Inha University, Incheon, Republic of Korea

Part One Introduction to the Spectral Element Method and Spectral Analysis of Signals


1 Introduction 1.1 Theoretical Background 1.1.1 Finite Element Method As the dynamic behavior and characteristics of a structure are of great importance in engineering, it is necessary to predict them accurately in an efficient and economic manner. The finite element method (FEM) is probably one of the powerful and popular computation methods to have been used in many areas of engineering and science. As an illustrative example, the vibration pattern of a structure certainly varies depending on the vibration frequency, and its wavelength at high frequency is very low. Because a sufficiently accurate dynamic response can only be obtained by capturing all necessary high frequency wave modes, the mesh (finite element) size used in the finite element modeling must be sufficiently small, comparable to the lowest wavelength of the vibrating structure. However, as the conventional finite element models are formulated by using frequencyindependent (static or fixed) polynomial shape (interpolation) functions, the FEM cannot capture all necessary high frequency wave modes of interest. Thus the FEM solutions become significantly inaccurate, especially at high frequencies, where associated wavelengths are very short. The so-called h-method is one of well-known approaches to improving the FEM accuracy by refining the meshes. Unfortunately this approach will make the size of the system extremely large, and hence from the computational aspect, the conventional FEM often becomes prohibitive for most complex, large flexible structures. It is known that, as a rough guide, the mesh size must be 10–20 times smaller than the wavelength of the highest frequency wave mode of interest [6]. An alternative approach to improve the solution accuracy is to use the shape functions, which can vary depending on vibration frequency. Accordingly, the candidate shape functions will be frequency dependent and they are known as dynamic shape functions in the literature. As the dynamic shape functions can readily capture all necessary high frequency wave modes of interest, extremely highly accurate solutions can be obtained, and the need to refine the meshes is no longer necessary. This elegant concept has led to the so-called dynamic stiffness method (DSM) [11, 108].


4

Spectral Element Method in Structural Dynamics

1.1.2 Dynamic Stiffness Method The exact dynamic stiffness matrix is used in the DSM. The exact dynamic stiffness matrix is formulated in the frequency domain by using exact dynamic shape functions that are derived from exact wave solutions to the governing differential equations. To obtain the exact wave solutions in the frequency domain, the time-domain governing differential equations are transformed into the frequency domain by assuming harmonic solutions of a single frequency. Accordingly the exact dynamic stiffness matrix is also frequency dependent and it can be considered as a mixture of the inertia, stiffness and damping properties of a structure element. As the exact dynamic stiffness matrix is formulated by using exact frequency dependent dynamic shape functions derived from the exact wave solutions, it automatically deals with the continuous mass distribution in a structure element exactly. Thus the structure element represented by an exact dynamic stiffness matrix is often called a continuum element. Consequently, the DSM guarantees exact frequency-domain solutions to the governing differential equations (or the mathematical model) adopted for the problem under consideration: this is why the DSM is referred to as an exact solution method in the literature. Of course the absolute accuracy of the DSM will be limited to the accuracy level of the governing differential equations adopted to formulate the exact dynamic stiffness matrix. For instance, the DSM based on the Timoshenko-beam model will provide more accurate frequency-domain solutions when compared with that based on the Bernoulli–Euler beam model. However, as the assumptions made for the DSM will be less severe than those made for the conventional FEM and other approximate solution methods, the DSM will still provide better solutions. As the exact dynamic stiffness matrix is formulated by using exact dynamic shape functions, it treats the mass distribution in a structure member exactly. Thus only one single element is sufficient for a regular part of a structure (without any structural or material discontinuities inside), regardless of its length between any two successive structural or material discontinuities, to acquire exact solutions. That is, we no longer need to refine a regular part of a structure into multiple fine meshes. This will significantly reduce the size of the problem, in other words, the total number of meshes and degrees-of-freedom (DOFs). In due course, this will significantly reduce the computation cost and time, together with improving the solution accuracy by reducing the computer round-off errors or numerical errors that are inevitable for most large size problems. In addition, the DSM provides an infinite number of eigensolutions from the exact dynamic stiffness matrix represented in terms of a minimum number of DOFs. Because the exact dynamic stiffness matrices are stiffness formulated (as the conventional finite element stiffness matrices are), they can be assembled in a completely analogous way to that used in the FEM. Thus, the meshing and assembly features of the FEM can be applied equally to the DSM. The only difference from FEM is that the assembly must be repeated at all discrete frequencies by using the Do-Loop routine. By combining the advantage features of the DSM with those of the spectral analysis method (SAM), Beskos [124] introduced the fundamental concept of the spectral element method (SEM) for the first time in his 1978 work.

1.1.3 Spectral Analysis Method The solution methods for the governing differential equations formulated in the time domain can be categorized into two major groups. The first group consists of the time-domain methods, such as the numerical integration methods and the modal analysis method, which is commonly

Introduction

5

used for the vibration analysis. The second group consists of the frequency-domain methods. The spectral analysis method (SAM) is one of the frequency-domain methods most popularly reported in the literature. It is worth remembering that throughout this book, the word “frequency” is used to mean the Fourier transform of the “time” (that is, “time frequency”) rather than that of the “spatial” coordinate (that is, “space frequency”). In SAM, the solutions to the governing differential equations are represented by the superposition of an infinite number of wave modes of different frequencies (or periods). This corresponds to the continuous Fourier transform of the solutions. This approach involves determining an infinite set of spectral components (or Fourier coefficients) in the frequency domain and performing the inverse Fourier transform to reconstruct the time histories of the solutions. The continuous Fourier transform is feasible only when the function to be transformed is mathematically simple and the inverse transform is the biggest impediment to most practical cases, especially when dealing with digitized experimental data measured through a modern data acquisition system. Thus, instead of using the continuous Fourier transform, the discrete Fourier transform (DFT) is widely used in practice. The DFT is an approximation of the continuous Fourier transform. In contrast to the continuous Fourier transform, the solutions are represented by a finite number of wave modes of discrete frequencies and thus, as an enormous advantage, one can use the fast Fourier transform (FFT) algorithm to compute the DFT and its inversion economically and quickly. Thus, the use of the FFT algorithm makes it possible to efficiently take into account as many spectral components as are needed up to the highest frequency of interest. Accordingly, the DFT/FFT-based SAM can provide very accurate solutions, while reducing the computation cost and time significantly. It is worth mentioning that firstly, although the DFT is the approximation of the continuous Fourier transform, the DFT processing itself is exact in the sense that it does allow the time signal to be regained exactly at discrete times [127], and secondly, although one uses a computer to accomplish the DFT/FFT-based spectral analysis, it is certainly not a numerical method in the sense that the analytical descriptions of the Fourier transforms are used in the computation.

1.1.4 Spectral Element Method As shown in Figure 1.1, the spectral element method (SEM) can be considered as the combination of the key features of the conventional FEM, DSM and SAM. The key features

Figure 1.1

Key features of the spectral element method

6


of each method can be summarized as follows: 1. Key features of FEM. Meshing (spatial discretization) and the assembly of finite elements. 2. Key features of DSM. Exactness of the dynamic stiffness matrix formulated in terms of a minimum number of DOFs. 3. Key features of SAM. Superposition of wave modes via DFT theory and FFT algorithm. It is worth mentioning that the above key features of the both FEM and SAM have been of little interest in most of the existing literature associated with the classical DSM. In SEM, exact dynamic stiffness matrices are used as the element stiffness matrices for the finite elements in a structure. To formulate an exact dynamic stiffness matrix for the classical DSM, the dynamic responses of a structure are usually assumed to be the harmonic solutions of a single frequency. However, for the SEM, the dynamic responses are assumed to be the superposition of a finite number of wave modes of different discrete frequencies based on the DFT theory. Accordingly, the computation of the exact dynamic stiffness matrix must be repeated at all discrete frequencies up to the highest frequency of interest. By virtue of the exactness of the spectrally formulated dynamic stiffness matrix, one-element modeling will suffice for a regular structure member. The SEM is an element method, just like the conventional FEM. Thus, the mesh refining can also be applied in the SEM when any geometric or material discontinuities in the spatial domain of concern exist, and also when any externally applied forces exist. The details of the spectral element formulation procedure and some examples are provided in Chapter 3. The spectrally formulated exact dynamic stiffness matrix to be used in the SEM in conjunction with the SAM is known as the “spectral element matrix” in this book and in much of the existing literature. In some of the literature, it is often called a “spectral finite element matrix” or “spectral dynamic stiffness matrix.” Accordingly, the spectral element method is also often called the “spectral finite element method” and, sometimes the “continuum method.” The finite length of the structure element represented by the spectral element matrix is called the “spectral element,” as the terminology “finite element” is commonly used in the FEM. The terminologies “spectral element” and finite element” will be used interchangeably throughout this book without introducing any unnecessary confusion. The frequency-domain nodal DOFs specified on a spectral element are known as “spectral nodal DOFs.” It should be pointed out that unfortunately the same terminology “spectral element method” or “SEM” has also been used for a different class of finite element methods developed in 1984 by Anthony Patera at MIT [135]. In fact, the word “spectral” for Patera’s SEM is not time-wise, but space-wise. The SEM is stiffness formulated. Thus, the spectral elements can be assembled to form a global system matrix equation for the whole problem domain by using exactly the same assembly techniques as used in the conventional FEM. The global system matrix equation is then solved for the global spectral nodal DOFs, of course, repeatedly at all discrete frequencies. Finally, we use the inverse-FFT (IFFT) algorithm to compute the time histories of dynamic responses (time-domain solutions).

1.1.5 Advantages and Disadvantages of SEM By virtue of the apparent advantages of the exact dynamic stiffness matrix used in conjunction with the DFT/FFT-based SAM, the major advantages of the SEM (although not all can be listed

Introduction

7

here) may be summarized as follow: 1. Extremely high accuracy. In theory, the SEM provides exact frequency-domain solutions, such as the eigensolutions (natural frequencies and modes) and the frequency response functions (system transfer functions). The SEM may also provide extremely accurate timedomain solutions (the time histories of dynamic responses) by efficiently taking into account as many high frequency wave modes as are required by using the FFT algorithm. Accordingly, the problems solved by using the spectral element model can be utilized as benchmark problems for evaluating the accuracy and performance of a newly developed solution method. 2. Smallness of the problem size and DOFs. This is true because one-element modeling suffices for the representation of a regular structure member of any size, without any structural and material discontinuities inside. 3. Low computation cost. For computing time-domain responses, this is certainly true due to the smallness of problem size as well as due to the use of the FFT algorithm. The quick computation of the time-domain responses also enables us to treat the wave propagation simulation more realistically. However, it is worth mentioning that the most significant part of time consumption in SEM is for computing eigenfrequencies by using an iteration method of root finding. 4. Effective to deal with frequency-domain problems. This is the situation because the SEM itself is a frequency-domain solution method. Accordingly the SEM can be successfully applied to the following situations: (i) when the characteristics of a system (e.g., constitutive equation or internal damping) are dependent on frequency (e.g., viscoelastic materials), and (ii) when the boundary conditions of a system are specified in the frequency domain (e.g., the impedance boundary conditions for the fluid-structure interaction problems). 5. Effective to deal with the non-reflecting boundary conditions of the infinite- or semi-infinite-domain problems. This is achieved by using the semi-infinite spectral element (often called through-off element), which can be formulated by simply removing the wave modes reflected from the boundary at the infinite from the dynamic shape functions. 6. Locking-free method. The SEM does not exhibit the shear locking problems often raised in the conventional FEM, because the exact wave solutions to the governing equations are used as the dynamic shape functions in the SEM. 7. Effective to deal with digitized data. This is true because the FFTalgorithm used in SEM is an efficient tool to deal with digitized data as follows: (i) the experimental data measured digitally through the analogue-to-digital converters, and (ii) the excitation forces measured or specified in the forms of numerical values at regular intervals of time rather than in the forms of analytical functions. 8. The system transfer functions (the inverse of global dynamic stiffness matrix) are the byproducts of the spectral element analysis. Thus, it is very straightforward to perform the inverse problems, such as the system identification (e.g., identification of system parameters, boundary conditions, structural joints, or structural damages) and the force identification. Despite the aforementioned advantages of SEM, there are also some disadvantages as follows: 1. The exact spectral element formulation is possible for the problems where the exact wave solutions to the governing equations are available. However, unfortunately the exact wave solutions are not always available for most complex and multi-dimensional problems. Thus,

8


the exact spectral element models developed to-date have been mostly for one-dimensional (1-D) problems (e.g., rods and beams) or some multi-dimensional problems that can be transformed into the equivalent 1-D problems (e.g., Levy-type plates). If the exact spectral element model is not available for a problem, the approximate spectral element modeling approach can be applied by adopting approximate dynamic shape functions obtained by using the wavenumbers computed from the approximate dispersion relation or by using another appropriate approximation method. As the assumptions possibly made for the approximate spectral element modeling will be less severe than those usually made for the conventional FEM, the approximate spectral element models may still provide very accurate solutions. 2. As the SEM is a frequency-domain analysis method based on DFT/FFT techniques, it cannot be directly applied to time-variant, non-linear systems for which the principle of superposition does not hold. Thus, for the case of non-linear systems, for instance, one may need to use an iteration method by treating non-linear terms as the pseudo-forces. 3. In the SEM, the time-domain solutions are post-processed by convolving the transfer functions with external loadings in the frequency domain via the IFFTalgorithm. Thus, even though the SEM guarantees exact frequency-domain solutions, it is not true for the timedomain solutions, because errors due to aliasing or leakage are inevitable in practice. As the aliasing or leakage-induced errors depend on the FFT conditions, special care must be taken in determining the successful FFT conditions.

1.2 Historical Background The history of spectral analysis, which is also known as the Fourier analysis or frequencydomain analysis, began with the pioneering work “Theorie analytique de la chaleur (The analytical theory of heat)” published in 1822 by Joseph Fourier. He showed how an infinite series of sine and cosine functions can be used to analyse heat conduction in solids. Owing to the distrust in the use of series then, Fourier’s method did not gain acceptability during his lifetime. Since then, works by Dirichlet, Riemann and other mathematicians have resolved any doubts about the validity of the Fourier series, and spectral analysis has followed two major pathways: the continuous Fourier transform and the discrete Fourier transform (DFT). The continuous Fourier transform has the drawback that it involves the integration of a signal (function) specified analytically over the time period, which is not always easy to carry out. This is especially true when the signal is experimentally measured in a digitized data form. Thus, the continuous Fourier transform is usually transformed in a discrete form by replacing the integration by the summation of a finite number of signal values sampled at regular intervals of time. The discrete version of the continuous Fourier transform is the DFT and it is a very effective means for transforming a periodic signal sampled in the time domain into an expression of the data in the frequency domain and vice versa. The major difficulty in using the DFT is the requirement of extensive computation time. Although some techniques and ideas to reduce the computation time appeared early in the twentieth century, it was not until 1965 when James W. Cooley (IBM researcher) and John W. Tukey (Princeton faculty member) developed a computational algorithm that is now known as the fast Fourier transform (FFT). The FFT is simply a remarkably efficient algorithm for computing the DFT and its inversion. The FFT algorithm reduces the number of arithmetical

Introduction

9

operations for computing the DFT from the order N2 to the order Nlog2N, where N is the number of samples. This is a dramatic reduction for large values of N: the reduction ratio is over 100 : 1 when N ¼ 1024 and 400 : 1 when N ¼ 4096. Historically it has been recognized that Carl Friedrich Gauss developed the same FFT algorithm around 1805, but without getting any attention until Cooley and Tukey rediscovered it, independently, 160 years later. The FFT rediscovery has made spectral analysis highly efficient, with widespread applications to digital signal processing and certain areas of engineering analysis. However, the applications of DFT/ FFT-based spectral analysis to the dynamic analysis of structures are fairly recent and mostly limited to 1-DOF or multi-DOFs discrete systems. Extensive reviews on the subject and related applications can be found in the books by Ginsberg [49], Humar [58] and Newland [127]. In classical DSM, most researchers have been predominantly interested in formulating exact dynamic stiffness matrices to obtain more accurate natural frequencies and modes, rather than in developing it further in the form of the SEM by combination with the DFT/FFT-based SAM. As mentioned in the previous section, the exact dynamic stiffness matrix for the classical DSM is usually formulated from exact wave solutions obtained by assuming the harmonic solutions of a single frequency to the governing differential equations. Over the last seven decades, there have been a large number of publications related to the formulation and applications of exact dynamic stiffness matrices. The 1941 work by Kolousek [70] is probably the first to derive the dynamic stiffness matrix for the Bernoulli–Euler beam. Przemieniecki [142] introduced the formulation of the frequency-dependent mass and stiffness matrices for both bar and beam elements in his book. In contrast to the conventional finite element mass and stiffness matrices, which result in the linear eigenproblems, the exact dynamic stiffness matrices result in transcendental eigenproblems, the coefficients of which are the transcendental functions of frequency. Thus, a drawback of the DSM, is that it is not an easy task to compute all natural frequencies (without missing any one within a specified frequency range of interest) accurately by solving the transcendental eigenvalue problems. In 1971, this difficulty was resolved by Wittrick and Williams [169] by developing the well-known Wittrick–Williams algorithm for automatic calculation of undamped natural frequencies [169]. The Wittrick–Williams algorithm has certainly enhanced the applicability of the DSM. Since then numerous exact dynamic stiffness matrices have been developed for various structures. The extensive literature up to 1976 and 1983 can be found in Ref. [4] and Ref. [115], respectively. In addition, the extensive literature up to 1993, together with elegant theories for the dynamic stiffness and substructures, can be found in a book by Leung [108]. Although not all published literature can be cited herein, exact dynamic stiffness matrices have been developed mostly for the 1-D structures including the Timoshenko beams with or without axial force [26, 30, 31, 54, 159], extension-bending-torsion coupled beams [10, 16, 18, 19, 47, 52], non-uniform beams [17, 56], curved beams and springs [61, 78, 107, 136], Rayleigh– Timoshenko beams [4, 115], composite beams [20, 44], sandwich beams [14], aircraft wing [13], the beams on elastic foundation [167], axially moving string [77], and other structures [106, 110]. Although the history of the DSM spans about seven decades, it was not until 1978 when Beskos [124] introduced the fundamental concept of the SEM for the first time. He derived an exact dynamic stiffness matrix for the beam element and employed FFT for the dynamic analysis of plane frame-works. His work was further improved and generalized in his 1982 work [148], while claiming that his 1978 work [124] was the first to employ FFT for the dynamic analysis of plane frame-works in conjunction with the DSM. It was in 1988 that Doyle

10


at Purdue University [38] published his first work on the formulation of the spectral element for the longitudinal wave propagation in rods. He seemed to be the first to use the terminology “spectral element method” in his 1992 work [146] for the DFT/FFT-based spectral element analysis approach. Doyle and his colleagues have applied the SEM mostly to wave propagations in structures. A comprehensive list of the works by Doyle’s research group and other researchers up to 1997 can be found in a book by Doyle [40]. On the other hand, the present author and his students have extensively applied the SEM to various problems in structural dynamics and the outcome of this research [33, 67, 72, 80–82, 84–101, 103, 104, 130] is the major source of the material presented in this book.

2 Spectral Analysis of Signals In mathematics, it is well known that an arbitrary periodic signal can be decomposed into many harmonic components and, in the reverse transform, the signal can be synthesized by superposing all its harmonic components. This is the basis of spectral (Fourier) analysis. For a signal given in functional form, we may evaluate its harmonic components by analytically evaluating the continuous Fourier transform integral. However, if the signal function is so complicated, we may decide to use numerical integration instead. In this case we need to know the signal values only at a discrete set of instances. A similar situation arises in most experimental measurements, because modern data acquisition systems based on analogueto-digital converters store digitized data sampled at discrete, uniformly spaced intervals. The discrete Fourier transform (DFT) enables us to deal with such discrete sampled data from a signal. The important advantage of DFT theory is that it allows us to use the fast Fourier transform (FFT), which is an extremely efficient algorithm for computing the DFT. To formulate spectral element models, the exact solutions to the governing equations are in general represented in the spectral forms by using the DFT. Thus this chapter provides a brief review of the theories and applications associated with DFT and FFT.

2.1 Fourier Series If x(t) is a continuous periodic function of time t, with period T, we can always represent it in the form of a Fourier series as 1 X xðtÞ ¼ a0 þ 2 ðan cos vn t þ bn sin vn tÞ ð2:1Þ n¼1

where a0, an and bn are constant Fourier coefficients given by ð 1 T a0 ¼ xðtÞdt T 0 ð 1 T xðtÞ cos vn t dt an ¼ T 0 ð 1 T xðtÞ sin vn t dt bn ¼ T 0 Spectral Element Method in Structural Dynamics Usik Lee © 2009 John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82374-3

ð2:2Þ

12


In the above equations, vn ¼ nDv is the frequency of the nth harmonic and Dv is the frequency spacing (fundamental frequency) defined by Dv ¼

2p T

ð2:3Þ

The cosine and sine functions can be decomposed by using the Euler identities as follows: 1 ivn t ðe þ eivn t Þ 2 i sin vn t ¼ ðeivn t eivn t Þ 2 cos vn t ¼

ð2:4Þ

pffiffiffiffiffiffiffiffi where i ¼ 1 is an imaginary unit. By substituting Equation 2.4 into Equation 2.1, we get an expression as follows xðtÞ ¼ a0 þ

1 X

ðan ibn Þeivn t þ

n¼1

1 X ðan þ ibn Þeivn t

ð2:5Þ

n¼1

Defining Xn ¼ an ibn

ð2:6Þ

we can rewrite Equation 2.5 into a compact form xðtÞ ¼

1 X n¼ 1

Xn eivn t

ð2:7Þ

where Xn is the nth Fourier coefficient for a continuous periodic function x(t). By substituting Equation 2.2 into Equation 2.6 and then by applying Equation 2.4, we obtain Xn ¼

1 T

ðT

xðtÞeivn t dtðn ¼ 0; 1; 2; . . . ; 1Þ

ð2:8Þ

0

Equations 2.7 and 2.8 constitute the Fourier transform pair for the continuous periodic function x(t). Equation 2.7 is referred to as the synthesis equation and Equation 2.8 as the analysis equation for a continuous periodic function x(t). In summary, the Fourier transform decomposes a periodic function into a discrete spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components.

2.2 Discrete Fourier Transform and the FFT 2.2.1 Discrete Fourier Transform (DFT) Although x(t) is a continuous periodic function of time, with period T, it is often the case that only sampled values of the function are available. This is true, for instance, when a time history x(t) is experimentally measured in the form of digitized data taken at equally spaced instances by using a modern data acquisition system. Let N be the number of samples in the

13

Spectral Analysis of Signals

x(t ) x x1 0

x2

xN/2 xN-1

Δt

0

t1 t2 ··· tN/2 ···

t

tN-1

T Figure 2.1

Sampling of a periodic function of period T at uniform time interval Dt

time domain and, as shown in Figure 2.1, assume that they are sampled at equally spaced time intervals given by Dt ¼

T N

ð2:9Þ

We express the N samples in a discrete time series {xr}, where xr ¼ x(tr), tr ¼ rDt and r ¼ 0, 1, 2, . . . , N 1. By using the discrete time series {xr}, the integral in Equation 2.8 can be approximately replaced by the summation as Xn ¼

1 1 NX xr eivn tr Dt T r¼0

ð2:10Þ

By using Equation 2.9, Equation. 2.10 can be rewritten as Xn ¼

1 1 NX xr eivn tr N r¼0

ð2:11Þ

Equation 2.11, the process of converting from discrete time series {xr} to frequency data is called the discrete Fourier transform (DFT). The frequency data Xn are called the DFT coefficients. As the discrete time series x(tr) are real, we can readily show from Equation 2.11 that X0 ¼ real

ðaÞ

XN=2 ¼ XN=2 ¼ real

ðbÞ

Xn ¼ Xn

ðcÞ

XN=2 þ n ¼ XN=2 þ n ¼ XN=2 n

ðdÞ

XN n ¼ Xn ¼ Xn N

ðeÞ

XN þ n ¼ X n

ðfÞ

ð2:12Þ

14


by using the following relationships eivp þ q tr ¼ eivp tr eivq tr

ðaÞ

eiv p tr ¼ eivp tr

ðbÞ

eivN=2 tr ¼ ð1Þr

ðcÞ

eivN tr ¼ 1

ðdÞ

ð2:13Þ

The asterisk symbol ( ) is commonly used in this book to denote the complex conjugate of a complex number. The synthesis Equation 2.7 can also be approximated by truncating the summation index n as N=2 1 X

xr ffi

Xn eivn tr

ð2:14Þ

n¼ N=2

We can rewrite the right-hand side of Equation 2.14 as xr ffi

1 X

Xn eivn tr þ

N=2 1 X

Xn eivn tr

ð2:15Þ

n¼0

n¼ N=2

By using Equations 2.12 and 2.13, the first summation of Equation 2.15 can be expressed as 1 X

Xn eivn tr ¼

n¼ N=2

¼ ¼

N=2 1 X k¼0 N=2 1 X k¼0 N=2 1 X

X N=2 þ k eiv N=2 þ k tr XN=2 þ k eiv N=2 þ k tr XN=2 þ k eivN=2 þ k tr ¼

k¼0

ð2:16Þ N 1 X

Xn eivn tr

n¼N=2

Substitution of Equation 2.16 into Equation 2.15 yields xr ffi

N 1 X

Xn eivn tr

ð2:17Þ

n¼0

This is the inverse discrete Fourier transform (IDFT), which is the process of converting DFT coefficients into discrete time series data. Equations 2.11 and 2.17 constitute a DFT pair. The range of the Fourier coefficients Xn is limited to n ¼ 0 to N 1 in order to maintain the symmetry of the DFT pair. Although the DFT pair, Equations 2.11 and 2.17, is the approximation of Equations 2.8 and 2.7, respectively, it does allow all discrete time series {x(tr)} to be regained exactly. In other words, the DFT processing is indeed exact. This important fact can be readily verified by showing that the discrete time series {x(tr)} are exactly regained by simply substituting Xn from Equation 2.11 into the right-hand side of Equation 2.17.

15


The 1/N factor before the integral in Equation 2.11 can be placed before the summation in Equation 2.17 ensuring that there is no net change in amplitude when one transforms from one domain to the other and back. Thus we can also use an alternative form of the DFT pair as Xn ¼

N 1 X

xr eivn tr

ðn ¼ 0; 1; 2; . . . ; N 1Þ

ð2:18Þ

1 1 NX Xn eivn tr N n¼0

ðr ¼ 0; 1; 2; . . . ; N 1Þ

ð2:19Þ

r¼0

xr ffi

The DFT pair defined by Equations 2.18 and 2.19 is adopted in MATLAB . Thus, the DFT pair defined by Equations 2.18 and 2.19 will be adopted in this book for readers who are familiar with MATLAB. Figure 2.2a shows the DFT coefficients arrangement for an example case of N ¼ 23 ¼ 8. The real and imaginary parts of the DFT coefficients are arranged in the order ½XN=2

X1

X0

X1

XN=2 1

ð2:20Þ

that is, for the example case, they are displayed in the order ½X4

X3

X2

Re[Xn ]

–4 –3 –2 –1 0

1 2

3

n

2 3

n

Xn

–4 –3 –2 –1 0

X2

X3

ð2:21Þ

0 1 2

3

4 5 6 7

3

4

3

4 5 6 7

n

5 6 7 0 1 2

n

Xn

1

2 3

(a) fftshift [Xn]

Figure 2.2

X1

Im[Xn ]

–4 –3 –2 –1 1

X0

Re[Xn ]

Im[Xn ]

0

X1

n

0 1 2

n

(b) fft [x(tr)]

Two types of arrangement of the DFT coefficients using the MATLAB functions

16


However, as XN/2 þ k ¼ XN/2 þ k (k ¼ 0, 1, . . ., N/2 1) is satisfied by Equation 2.12, [XN/2, XN/2 þ 1, . . ., X2, X1] is identical with [XN/2, XN/2 þ 1, . . ., XN2, XN1]. Thus the arrangement of Equation 2.20 can be changed into the form ½X0

X1

XN=2

XN=2 þ 1

XN=2 þ 2

XN 2

XN 1

ð2:22Þ

For the example case, the DFT coefficients are displayed in the order ½X0

X1

X2

X3

X4 ¼ X4

X5 ¼ X3

X6 ¼ X2

X7 ¼ X1

ð2:23Þ

which is shown in Figure 2.2b. The MATLAB fft function provides the DFT coefficients in the arrangement of Equation 2.22 and it also enables us to transform it into the arrangement of Equation 2.20. As the DFT coefficients arrangement of Equation 2.22 matches the IDFT formula of Equation 2.19, it will be adopted for use in this book. As illustrated in Figure 2.3, owing to the properties of Equation 2.12a to Equation 2.12f, the DFT replicates periodically on the frequency axis with a period of 2p/Dt rad/s (or 1/Dt Hz) if the discrete time data constitutes a set of N values sampled at uniform time interval Dt (seconds). The real part is symmetrical about the middle frequency, vN/2 ¼ p/Dt rad/s (or fN/2 ¼ 1/2Dt Hz), while the imaginary part is anti-symmetric, because the DFT coefficients in the frequency range vN/2 v vN are complex conjugates of those in the frequency range of 0 v vN/2. Thus only the DFT coefficients at the frequencies below vN/2 are unique, while those at the frequencies above vN/2 are spurious. This means that N real discrete time data are transformed into N/2 complex frequency data without gaining or losing any information from a signal. Accordingly, the highest frequency we can deal with by the use of DFT is vN/2 ¼ p/Dt rad/s or fN/2 ¼ 1/2Dt Hz, which it is called the Nyquist frequency vNq (rad/s) or fNq (Hz): vNq ¼

p ðrad=sÞ Dt

or fNq ¼

1 ðHzÞ 2Dt

ð2:24Þ

Figure 2.3 Periodicity of the DFT coefficients

2.2.2 Fast Fourier Transform (FFT) The fast Fourier transform (FFT) is an ingenious highly efficient computer algorithm to compute the DFT and its inverse, IDFT, originally discovered by Gauss around 1805 and

17


rediscovered by Cooley and Tukey at IBM in 1965. The FFT is not a different transform from the DFT, but rather just a means of computing the DFT with a considerable reduction in the computational time. The Cooley–Tukey algorithm makes the FFT extremely efficient by reducing the arithmetic operations (multiplications and additions) from something of the order of N2 to N/log2N. This is a dramatic reduction in computation time for large values of N. It is worth pointing out that although the FFT uses a computer, it is not a numerical method in the usual sense, because the analytical descriptions of DFT and IDFT, Equations 2.18 and 2.19, are still retained. An individual working with FFT does not necessarily need to know the details inside of the FFT algorithm, but must know how to prepare the data and how to interpret the results. More details of FFT algorithms can be found in Refs. [63, 127]. As a rule, data to be transformed by the FFT consists of N uniformly spaced discrete time series. For the maximum efficiency of computation, the optimal N is required to be an integer power of 2: that is, N ¼ 2p where p ¼ integer. Although some FFT algorithms do not require that N be an integer power of 2, N ¼ 2p is optimal for the maximum execution speed of the FFT algorithm. In most practical situations, the number of data sampled by using a fixed sampling time interval Dt for a fixed sampling time window Ts is unlikely to be an integer power of 2. In this case, zero padding can provide a means of making N be an integer power of 2 without losing any information, which is discussed further in the following section. There are many computer programs available for performing the FFT. Among these, MATLAB users can use the functions listed in Table 2.1. Table 2.1 MATLAB functions for the FFT [156] Function

Syntax

fft ifft fftshift ifftshift

X ¼ fft(x) x ¼ ifft(X) Y ¼ fftshift(X) X ¼ ifftshift(Y)

Definition Returns the DFT of a vector x Returns the inverse DFT of a vector X Shifts zero-frequency component of DFT to the center of spectrum Undo fftshift

2.3 Aliasing 2.3.1 Aliasing Error According to Equation 2.12f, the DFT coefficients are symmetrically placed about the Nyquist frequency fNq ¼ 1/2Dt Hz when a signal is sampled at a uniform time interval Dt. In other words, the highest frequency that can be detected by DFT from the discrete time series sampled at a uniform time interval Dt is the Nyquist frequency fNq. If fmax is the highest frequency (called band frequency) of an actual signal, the Nyquist frequency fNq must be equal to or larger than fmax to accurately represent the signal as the DFT coefficients. This leads to an upper limit on the sampling time interval of Dt

1 2fmax

ð2:25Þ

18


This is called the Nyquist condition. As 1/fmax is the period of the highest frequency component of a signal, the Nyquist condition states that we must sample the signal more than twice within the period of the highest frequency associated with the signal. When we happen to violate the Nyquist condition (i.e., when we do not take at least two samples per period of the band frequency fmax) or when there are higher frequency components above the Nyquist frequency fNq in an actual signal, the higher frequency components are merged into the DFT spectrum as shown in Figure 2.4. This introduces an error into the DFT coefficients calculated for the frequencies below fNq and it is called aliasing.

Xn DFT Spectrum

True Spectrum

XN/2

0

Figure 2.4

ωN/2

XN

ωN

ωn

Aliasing error when the band frequency fmax exceeds the Nyquist frequency fNq

Example 2.1 Consider a harmonic function x(t) ¼ sin10pt, which is shown in Figure 2.5a. The highest frequency of the harmonic function is fmax ¼ 5 Hz and the required maximum sampling time interval must be smaller than Dtreq ¼ 0.1 s to satisfy the Nyquist condition, Equation 2.25. Thus, when we choose the sampling time as T ¼ 2 s, the required minimum number of samples must be larger than Nreq ¼ T/Dtreq ¼ 20. We will consider the following three cases. 1. When N ¼ 23 ¼ 8 Sampling time interval Dt ¼ T/N ¼ 2/8 ¼ 0.25 s Dtreq Thus, the Nyquist frequency fNq ¼ 1/2Dt ¼ 2 Hz fmax 2. When N ¼ 24 ¼ 16 Sampling time interval Dt ¼ T/N ¼ 2/16 ¼ 0.125 s Dtreq Thus, the Nyquist frequency fNq ¼ 1/2Dt ¼ 4 Hz fmax 3. When N ¼ 25 ¼ 32 Sampling time interval Dt ¼ T/N ¼ 2/32 ¼ 0.0625 s Dtreq Thus, the Nyquist frequency fNq ¼ 1/2Dt ¼ 8 Hz fmax The first two cases do not satisfy the Nyquist condition, while the last case satisfies it well. Hence, Figure 2.5 shows that aliasing appears for the first two cases, while it does not for the last case. In Figure 2.5, the thin lines represent the original harmonic function, while the thick lines represent the functions reconstructed from N samples by using the IDFT. The dots denote the sampled data.

19

Spectral Analysis of Signals Original signal

N=8 1

x(t )

x(t )

1 0 –1

0 –1

0

1

2

0

1

Time (s)

N = 16

1

x(t )

x(t )

1

2

Time (s)

0 –1

N = 32

0

–1

0

1

2

0

Time (s)

Figure 2.5

1

2

Time (s)

Illustration of aliasing for a harmonic function

Example 2.2 As the second example, consider a chirp signal x(t) ¼ sin[2p(5t þ 13t2)] (0 t 1) as shown in Figure 2.6a. The instant frequency of the chirp signal varies linearly from 5 to 31 Hz as t increase from 0 to 1 s, that is f(t) ¼ 5 þ 26t Hz. Assume that the sampling time interval is given by Dt ¼ 1/36 s. The Nyquist frequency is then given by fNq ¼ 1/2Dt ¼ 18 Hz. Thus, the

x(t )

1

(a) Original signal

0 –1 0

0.5

1

Time (s)

x(t )

1

(b) Aliased signal

0 –1 0

0.5

Time (s)

Figure 2.6

Illustration of aliasing for a chirp signal

1

20


Nyquist condition is satisfied during the time 0 t 1/2 as fmax fNq while it is not satisfied during the time 1/2 < t 1 as fmax fNq. Accordingly, aliasing is occurring as shown in Figure 2.6b, where the thin line represents the original chirp signal and the thick line represents the function reconstructed from 36 sampled data, denoted by dots.

2.3.2 Remedy for Aliasing To avoid or minimize aliasing errors, we can use the following remedies. 1. Remedy 1: Reduce the sampling time interval Dt sufficiently so that the Nyquist condition, Equation 2.25, can be satisfied. 2. Remedy 2: If the first remedy does not work well, use a low-pass filter to remove significant frequency components above the Nyquist frequency before the DFT is processed.

2.4 Leakage 2.4.1 Leakage Error In DFT applications, we must always deal with a restricted number of samples, such as N, in the time domain. Thus, if we set a sampling time interval Dt to a value that satisfies the Nyquist condition, the sampling time window will be fixed by Ts ¼ NDt. We must remember that, as the DFT theory was essentially developed for periodic functions, it can be accurately applied to the periodic time functions with period T. Thus, if the sampling time window Ts does not meet the periodicity condition of a signal under consideration, the error known as leakage can occur. The leakage introduces sidelobes into the frequency domain, whereas the aliasing introduces distortion of the frequency spectrum in the frequencies below the Nyquist frequency. The leakage can be observed in the following cases: 1. When the sampling time window Ts is not an integer multiple of the period T of the signal. 2. When the sampling time window Ts is not sufficiently large to see the signal decay towards zero. Figure 2.7 shows two different sampling time windows used for a harmonic signal. When we choose a sampling time window for a signal, the DFT treats the signal as if the sampled signal repeats for ever. Thus, if we choose the sampling time window so that it exactly matches the period of the signal, the DFT will treat the original signal accurately alongwith presenting its exact DFT spectrum, as illustrated in Figure 2.7a. However, if we fail to choose the sampling time window to match with the period of the signal, the DFT misunderstands the original signal as if it is no longer a harmonic signal while presenting an inaccurate DFT spectrum, as shown in Figure 2.7b. The sidelobes appearing around the true peak spectrum represent leakage errors. The leakage errors cannot be eliminated completely because in general most signals are not periodic (or it takes an extremely long time to decay towards zero), so that there are no time windows that can cover the whole

21


Xn

x(t)

t n

Ts

(a) Proper window Xn

x(t)

t n

Ts

(b) Improper window

Figure 2.7

Illustration of the dependence of the leakage on the sampling time window

time signal at one time, or the maximum sampling time might be restricted by various environmental conditions. To illustrate the leakage errors, we will consider the following two example cases.

Example 2.3 Consider a harmonic function x(t) ¼ sin6pt for which the period and maximum frequency are T ¼ 1/3 s and fmax ¼ 3 Hz, respectively. Assume that the sampling time interval is chosen to be less than Dt ¼ 1/6 s so that the Nyquist condition, Equation (2.25), is satisfied and the number of samples is fixed as N ¼ 24 ¼ 16. Now we will consider the following two cases for sampling time interval Dt. a. When Dt ¼ 1/16 s Sampling time window Ts ¼ NDt ¼ 16(1/16) ¼ 1 s Thus, Ts ¼ 3 T ¼ integer multiple of T b. When Dt ¼ 1/20 s Sampling time window Ts ¼ NDt ¼ 16(1/20) ¼ 0.8 s Thus, Ts ¼ 2.4 T 6¼ integer multiple of T When Dt ¼ 1/16 s, the sampling time window Ts is three times the period T of signal. Thus, as shown in Figure 2.8a, the DFT provides an accurate frequency spectrum from sampled data, with a unique spectral peak at 3 Hz, and the signal reconstructed from the DFT spectrum is exactly the same as the original harmonic function x(t). However, when Dt ¼ 1/20 s, the sampling time window Ts is not the integer multiple of the period T of the signal. Thus, Figure 2.8b shows that a non-zero frequency spectrum (i.e., sidelobes) appears around the spectral peak at 3 Hz and that the signal reconstructed from the DFT spectrum is no longer the same as the original harmonic function. This occurs because the energy associated with the 3 Hz spectral peak leaks into neighboring frequencies due to the improper sampling time window.

22

Spectral Element Method in Structural Dynamics 1 0

x(t )

|Xn|

10

5

–1

Ts 0

0

5

10

0

0.5

Frequency (Hz)

1

1.5

2

1.5

2

Time (s)

(a) when Ts = 3T 1

x(t )

|Xn|

10

5

0 –1

Ts 0

0

5

10

0

Frequency (Hz)

0.5

1

Time (s)

(b) when Ts = 2.4T

Figure 2.8

Illustrative example of leakage error

Example 2.4 Consider a damped vibration response of a one degree-of-freedom system given by x(t) ¼ e2tsin6pt. In fact, the vibration response is not a periodic function. The DFT spectrum of the vibration response is shown in Figure 2.9. As the band frequency fmax is infinite, the sampling time interval required to satisfy the Nyquist condition, Equation (2.25), is Dt ¼ 0 s, however this is impossible in practice. Thus, aliasing cannot be completely eliminated in this case. To investigate the leakage error as a dependence on the size of the sampling time window, we first set Dt ¼ 1/16 s and then consider two values of N, the number of samples, as follows. a. When N ¼ 23 ¼ 8 Sampling time window Ts ¼ NDt ¼ 8(1/16) ¼ 0.5 s b. When N ¼ 24 ¼ 16 Sampling time window Ts ¼ NDt ¼ 16(1/16) ¼ 1 s c. When N ¼ 25 ¼ 32 Sampling time window Ts ¼ NDt ¼ 32(1/16) ¼ 2 s As the size of the sampling time window is increased, the DFT spectrum of the sampled signal approaches the exact continuous Fourier spectrum of the vibration response, alongwith a decrease in the leakage error.

23


Continuous Fourier Transform

2

|Xn|

x (t )

0.5 0

1

–0.5

Ts 0

1

2

0

3

0

Time (s)

2

4

6

8

Frequency (Hz)

(a) when Ts = 0.5 s


2

|Xn|

x (t )

0.5 0

1

–0.5

Ts 0

1

2

0

3

0

2

4

6

8

Frequency (Hz)

Time (s)

(b) when Ts = 1 s


2

|Xn|

x (t )

0.5 0

1

–0.5

Ts 0

1

2

3

0

Time (s)

0

2

4

6

8

Frequency (Hz)

(c) when Ts = 2 s

Figure 2.9

Sampling time window dependence on leakage error

2.4.2 Artificial Damping When a system is lightly damped, its dynamic response takes a very long time to decay to a negligible value. If the sampling time window cannot be set up to be larger than the long decay time, leakage error will occur in the DFT processing. A possible remedy to reduce the leakage error is to use an exponential window, which will smoothly attenuate a signal at the ends of the sampled time history so that the windowed signal looks more periodic. We will show that artificial damping assumes the role of the exponential window in order to remedy leakage. Introducing artificial damping to a system will result in a faster decay of the dynamic response, which is a good sign for the reduction of leakage error for a prefixed sampling time window. The introduction of artificial damping is only an intermediate step in the DFT processing and the final dynamic response for the original unaltered system is recovered by removing the artificial damping at the final step.

24


Assume that the dynamic response for a system is x(t) and define the modified function by ðtÞ ¼ xðtÞeat x

ð2:26Þ

where a is a real positive constant that assumes the role of the artificial damping. Applying the DFT to Equation 2.26, we get n ¼ X ¼

N 1 X r¼0 N 1 X

r eivn tr ¼ x

N 1 X

½xr eatr eivn tr

r¼0

xr e

iðvn iaÞtr

ð2:27Þ

¼ Xn ðvn iaÞ

r¼0

Equation 2.27 shows that the DFT of the dynamic response x(t) of a system altered by artificial ðtÞ of the damping is obtained by the frequency shifting of the DFT of the dynamic response x original unaltered system. For undamped or very lightly damped systems, first derive the analytical form of Xn ¼ X(vn) n ¼ Xðvn iaÞ by replacing vn by (vn ia). We then compute x ðtÞ from and then compute X n by using the IDFT technique. Finally, the true dynamic response is recovered by removing X the artificial damping as follows: ðtÞeat xðtÞ ¼ x

ð2:28Þ

The appropriate value for the arbitrary constant a is suggested to be as follows a¼

ln100 ð1=sÞ Ts

ð2:29Þ

The constant a, given by Equation 2.29, ensures that the dynamic response x(t) is reduced by the factor 1/100 at the end of sampling time Ts.

Example 2.5 Consider a damped one degree-of-freedom dynamic system. Assume that it is initially at rest and its motion is represented by the equation of motion as x€ þ x_ þ 36p2 x ¼ 20dðtÞ where d(t) is the delta function. Its natural frequency is 3 Hz and the dynamic response can be obtained analytically as xExact ðtÞ ¼ 1:061e0:5t sinð5:998ptÞ The dynamic response will take more than 9.2 s to decay down by the factor 1/100. Figure 2.10 shows the dynamic responses obtained by using the DFT for t Ts ¼ 2 s. The DFT spectra used for the DFT processing are given from the equation of motion by 20 1 Xn ¼ ðn ¼ 0; 1; 2; . . . ; N 1Þ Dt 36p2 þ ivn v2n

25


Exact DFT

1

x (t )

x (t )

1

Exact DFT

0 –1

0 –1

0

1

2

0

1

Time (s)

Time (s)

(a) Artificial damping not used

(b) Artificial damping used

Figure 2.10

2

Artificial damping for leakage reduction

where N ¼ 1024 and Dt ¼ 1/512 s are used. Figure 2.10a shows that the dynamic response obtained without using the artificial damping is not accurate enough when compared with the exact dynamic response xExact(t). However, it is obvious from Figure 2.10b that the use of artificial damping improves the accuracy of the DFT application.

2.5 Picket-Fence Effect The DFT spectrum is a discrete frequency spectrum consisting of estimates of what the spectral level is at specific discrete frequencies, which are limited to integer multiples of the fundamental frequency defined by the reciprocal of the sampling time window. Thus the DFT cannot observe the frequency spectrum as a continuous function. This means that there may be, and probably are, peaks and valleys in the true frequency spectrum of a signal that are between two of the discrete frequencies. As the observation of the frequency spectrum with the DFT is analogous to looking at a mountain range through a picket fence, the true peaks and valleys may not be accurately observed by the DFT, known as the picket-fence effect, or often, the resolution bias error. The possible remedy for the picket-fence effect is to increase the number of DFT points by using zero padding while holding the original sampling time window fixed. The zero padding will be discussed in the next section.

2.6 Zero Padding Zero padding is a popular technique used in FFT applications. As the name implies, it means appending a string of zeros at the end, at the beginning, or split between the beginning and end of the finite number of sampled time series data. The common purposes of this process are: a. to extend the sampled data so that the number of samples N becomes an integer power of 2 (i.e., N ¼ 2p where p ¼ integer); b. to improve interpolation in the transformed domain; c. to remedy the wraparound error.

26


As discussed in the preceding section, making the number of samples N an integer power of 2 will maximize the computation efficiency of FFT. In the following, further discussion is given for the purposes (b) and (c).

2.6.1 Improving Interpolation in the Transformed Domain The spacing of data in the frequency domain, Df, is inversely proportional to the number of samples N in the time domain, that is, Df ¼ 1/T ¼ 1/NDt where T is the period of the signal and Dt is the sampling time interval. Thus, when we increase N by using zero padding, the frequency spectrum will contain more data points in the same frequency range. In other words, zero padding will introduce more points into the same frequency range to provide interpolation between points associated with the unpadded points. Hence, by increasing the number of zeros, we can improve the frequency spectrum using FFT to get closer to those from the continuous Fourier transform, alongwith curing the picket-fence effect. The zero padding can also be applied in the frequency domain. The inverse FFT will increase the data points in the time domain to provide better interpolation between points associated with the unpadded points.

Example 2.6 Consider a signal given by x(t) ¼ e2tsin6pt. Assume that the signal is sampled at the sampling time interval Dt ¼ 1/16 s for the first one second (i.e., Ts ¼ 1 s) so that the number of original samples becomes N ¼ 16. Figure 2.11 shows the effect of tripling the number of samples in the time domain. The FFT for the zero-padded data (white dots) contains the same information as the unpadded data (black dots), alongwith providing interpolation with two additional data between every three points.

x(t )

1

Original signal Sampled data Padded zeros

0

Ts –1

0

Zero-padding

0.5

1

1.5

2

2.5

3

Time (s)

|Xn|

4

Unpadded Zero-padded

2

0

0

1

2

3

4

5

6

7

8

Frequency (Hz)

Figure 2.11

The effect of zero padding in the time domain on the frequency-domain data

27


Example 2.7 Consider the unpadded case of Example 2.6. The frequency spectrum for the unpadded case is zero padded to three times its original length. Figure 2.12 shows the effect of zero padding in the frequency domain. We see that the original unpadded sampled data (black dots) are interpolated by the inverse FFT of the zero-padded frequency spectrum denoted by white dots.

|Xn |

4

Unpadded Padded zeros

2

Zero-padding 0

0

5

10

15

20

25

Frequency (Hz)

x (t )

1


0

–1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Figure 2.12

The effect of zero padding in the frequency domain on the time-domain data

2.6.2 Remedy for Wraparound Error The dynamic response x(t) of a system can be obtained from the convolution integral (or Duhamels integral), as given by ðt xðtÞ ¼ hðt tÞf ðtÞdt ð2:30Þ 0

where h(t) is the unit impulse response function and f(t) is the input forcing function. They satisfy the causality conditions h(t t) ¼ 0 if t < t and f(t) ¼ 0 if t < 0. The DFT representation of Equation 2.30 is given by simple multiplication as Xn ¼ H n F n

ðn ¼ 0; 1; 2; . . . ; N 1Þ

ð2:31Þ

where Xn, Hn and Fn are the DFT coefficients of x(t), h(t) and f(t), respectively. They are defined by ðXn ; Hn ; Fn Þ ¼

N 1 X

ðxr ; hr ; fr Þeivn tr

ð2:32Þ

r¼0

where xr ¼ x(tr), hr ¼ h(tr) and fr ¼ f(tr). From Equation 2.31, we obtain xr ffi

1 1 1 NX 1 NX Xn eivn tr ¼ Hn Fn eivn tr N n¼0 N n¼0

ð2:33Þ

28


By applying Equation 2.32 to Equation 2.33, we are able to obtain the discrete convolution theorem as [50] xr

r X

hn f r n þ

n¼0

N 1 X

hn fr n þ N

ðr ¼ 0; 1; 2; . . . ; N 1Þ

ð2:34Þ

n¼r þ 1

or x0 ½h0 f0 þ ½h1 fN 1 þ þ hN=2 fN=2 þ þ hN 1 f1 x1 ½h0 f1 þ h1 f0 þ ½h2 fN 1 þ þ hN=2 fN=2 þ 1 þ hN=2 þ 1 fN=2 þ þ hN 1 f2 .. .

ð2:35Þ

xN 2 ½h0 fN 2 þ h1 fN 3 þ þ hN=2 1 fN=2 1 þ þ hN 2 f0 þ ½hN 1 fN 1 xN 1 ½h0 fN 1 þ h1 fN 2 þ þ hN=2 1 fN=2 þ hN=2 fN=2 1 þ þ hN 1 f0 Equations 2.34 and 2.35 show that the DFT processing requires the values of h(t) and f(t) at the later times rather than the instant at which the value of x(t) is to be computed, which reveals an anomaly when compared with Equation 2.30. Thus, if the DFT requires the data outside of the window 0 t T, it will shift the data periodically into the window to cause a so called wraparound error. The remedy for wraparound error is to use the zero-padding technique. A fail-safe process is to pad the original N sampled data with N zeros in the time domain so that the total number of samples becomes 2N, which corresponds to an effective time window of 2T. In this case, all terms in the second bracket, that is, the second sum in Equation 2.34 indeed vanish because they all contain the zero pads of hr and fr.

Example 2.8 Revisit the signal x(t) ¼ e2tsin 6pt considered in Example 2.6. If we set the sampling time window as Ts ¼ 1 s, the DFT will treat the signal as if it is not zero for t < 0, which obviously violates the causality condition. To remedy this anomaly, we pad the sampling data with zeros to extend the sampling time window to 2Ts ¼ 2 s. Figure 2.13 compares the signals for the unpadded and zero-padded cases. It is obvious from Figure 2.13 that the signal with zero padding has been treated by the DFT as being a signal that was quiescent before the action of input force.

x (t )

1


0

–1 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

Time (s)

Figure 2.13

Zero padding used to satisfy the causality condition

1

29


2.7 Gibbs Phenomenon The Gibbs phenomenon is a mathematical phenomenon that always occurs when we apply the Fourier series to a function with discontinuities. To numerically illustrate the Gibbs phenomenon, we will consider a periodic function given by 8 < 1 1 < x < 0 xðtÞ ¼ 0 x¼0 ð2:36Þ : 1 0<x 2=Dt ¼ 200 Step 4: Compute the artificial damping factor a from Equation 2.44 as a ¼ 4:6=Ts ¼ 2:3ð1=sÞ We then follow the remaining steps 5–14 to compute the dynamic response in the time domain. In Figure 2.16, the dynamic responses computed by using the DFT processing, with and without considering artificial damping (AD), are compared with the exact dynamic

34

Spectral Element Method in Structural Dynamics 2 Exact

kx/f0

DFT(with AD) DFT(without AD)

1

0 0

0.5

1

1.5

2

Time (s)

Figure 2.16 Dynamic responses obtained by the DFT processing with and without considering artificial damping (AD)

response provided in a closed form. It shows that artificial damping may enable us to obtain a satisfactory result even though the sampling time window Ts chosen for DFT processing is not sufficiently large.

2.9 DFTs of Typical Functions 2.9.1 Product of Two Functions Consider a function z(t) that is defined by a product of two functions x(t) and y(t) as zðtÞ ¼ xðtÞyðtÞ

ð2:56Þ

It is obvious from Equation 2.56 that the discrete time series {zr} for z(t) are given by zr ¼ xr yr

ðr ¼ 0; 1; 2; . . . ; N 1Þ

ð2:57Þ

The discrete time series xr, yr and zr are related to the corresponding DFT spectrum Xn, Yn and Zn as follows: 1 1 NX Xn eivn tr N n¼0 1 1 NX yr ¼ Yn eivn tr N n¼0 1 1 NX zr ¼ Zn eivn tr N n¼0

xr ¼

ðr ¼ 0; 1; 2; . . . ; N 1Þ

ð2:58Þ

Substituting Equation 2.58 into Equation 2.57 and using Equations 2.12 and 2.13, we can get a discrete convolution formula as Zn ¼

1 1 NX Xj Y n j N j¼0

ðn ¼ 0; 1; 2; . . . ; N 1Þ

ð2:59aÞ

35


or n X

1 Zn ¼ N

Xj Yn j þ

j¼0

!

N 1 X

ðn ¼ 0; 1; 2; . . . ; N 1Þ

Xj YN þ n j

or n X

1 Zn ¼ N

ð2:59bÞ

j¼n þ 1

Xj Y n j þ

N=2 X j¼n þ 1

j¼0

Xj Yj n þ

N=2 1 X j¼1

! Xj Yn þ j

ðn ¼ 0; 1; 2; . . . ; N=2Þ

ZN n ¼ Zn

ð2:59cÞ

where the symbol ( ) denotes the complex conjugate. Thus, the DFT coefficients for a function z(t) can be readily computed from the DFT coefficients for x(t) and y(t) by using the discrete convolution formula of Equation 2.59. Similarly to the time-domain discrete convolution theorem represented by Equation 2.34, the DFT coefficients computed by Equation 2.59 can be contaminated by the wraparound error in the frequency domain (i.e., aliasing). This product aliasing error might be cured by using the zero padding in the frequency domain.

Example 2.10 Consider a function z(t) ¼ x(t) y(t) that is determined by two functions given by xðtÞ ¼ sin 2pt þ 0:5 sin 6pt þ cos 12pt þ 2 cos 14pt yðtÞ ¼ 2 cos 4pt þ cos 8pt þ 0:5 sin10pt þ 1:5 sin 16pt Figure 2.17 shows that the DFT coefficients obtained by using the formula in Equation 2.59 are identical with those obtained by applying the FFT algorithm directly to z(t). 40

x(t) y(t)

20

0 lIm(Xn)l

x(t) and y(t)

y(t)

lRe(Xn)l

x(t)

5

–5 0

0.5

1

1.5

0 20

2

0

Time (s)

lRe(Zn)l

z(t) = x(t) y(t)

5 0

lIm(Zn)l

z(t)

10

–5 0

0.5

1

Time (s)

5

10

15

Frequency (Hz)

1.5

2

40

Formula used FFT used

20 0 –20 0

5

10

15

Frequency (Hz)

Figure 2.17 DFT coefficients computed by using the discrete convolution formula and the FFT

36


2.9.2 Derivative of a Function Consider a Fourier series for a function x(t) as xðtÞ ¼

1 X n¼ 1

Xn eivn t

ð2:60Þ

where Xn is the nth Fourier coefficient. The Fourier series for the derivative of x(t) and its inverse Fourier transform are given by Equation 2.60 as _ xðtÞ ¼

1 X n¼ 1

1 X_ n ¼ T

ðT

ivn Xn eivn t

ivn t _ dt xðtÞe

1 X n¼ 1

X_ n eivn t

ðn ¼ 0; 1; 2; . . . ; 1Þ

ð2:61Þ

ð2:62Þ

0

_ Following the DFT theory where X_ n ¼ ivn Xn represent the Fourier coefficients of xðtÞ. introduced in Section 2.2.1, we can write Equations 2.61 and 2.62 into the DFT pair as x_ r ffi

1 1 NX X_ n eivn tr N n¼0

X_ n ¼

N 1 X

x_ r eivn tr

ðr ¼ 0; 1; 2; . . . ; N 1Þ

ð2:63Þ

ðn ¼ 0; 1; 2; . . . ; N 1Þ

ð2:64Þ

r¼0

Thus, the DFT coefficients for the derivative of a function x(t) are given by X_ n ¼ ivn Xn X_ N n ¼ X_ n

ðn ¼ 0; 1; 2; . . . ; N=2Þ

ð2:65Þ

Repeating the above process, the DFT coefficients for the kth derivative of a function x(t) are given by ðkÞ

Xn

¼ ðivn Þk Xn

ðkÞ ðkÞ XN n ¼ X_ n

ðn ¼ 0; 1; 2; . . . ; N=2Þ

ð2:66Þ

2.9.3 Other Typical Functions Table 2.2 lists the DFTs for some typical functions such as the delta function (unit impulse function) d(t), the unit step function u(t), and the sine and cosine functions, which we often meet during the DFT/FFT-based spectral analysis. By using the formula given in Table 2.2, we can obtain the DFT coefficients Xn for a typical function as follows: Xn ¼ gvien by the formula of Table 2:2 XN n ¼ Xn

ðn ¼ 0; 1; . . . ; N=2Þ

ð2:67Þ

For the functions (5), (6) and (7) in Table 2.2, the unit step function u(t) is introduced to emphasize that they have no values before the time t ¼ 0 and the constant parameter a plays a

37

Spectral Analysis of Signals Table 2.2 The DFT coefficients of typical functions Function x(t)

Sampled data xr if tr ¼ 0 otherwise

(1)

delta function d(t)

(2)

unit step function u(t)

(3)

d(t tshift)

(4)

u(t) u(t tshift)

(5)

eat uðtÞ

eatr

(6)

eat sin v t uðtÞ

eatr sin v tr

(7)

1

eat cos v t uðtÞ

1=Dt 0

1=Dt 0

if tr ¼ tshift otherwise

1 if 0 < tr < tshift 0 otherwise

eatr cos v tr

DFT coefficients Xn 1 Dt N 0

if n ¼ 0 otherwise

1 ivn tshift e Dt v t vn tshift tshift n shift i 2 sinc e Dt 2p

i Dtðvn iaÞ v

Dt ½ v2 ðvn iaÞ2 iðvn iaÞ Dt½ v2

ðvn iaÞ2

Note: Dt ¼ T/N, where T is the sampling time (or period) and N is the number of samples.

Figure 2.18

Time histories reconstructed by using the DFT formula from Table 2.2

38


role as the damping. Even if the constant parameter a is initially zero (i.e., no damping), it is still recommended to compute the DFT coefficients by using the formula given in the last column of the Table 2.2 by now considering the constant parameter a as the artificial damping parameter. In this instance, the original functions with a ¼ 0 should be recovered by taking off the artificial damping effect through using Equation 2.28. Figure 2.18 shows the time histories of x(t) ¼ e4.6tu(t) and x(t) ¼ u(t) by using the DFT formula (5) from Table 2.2 when the sampling time window is set as Ts ¼ 1 s. Owing to the discontinuities at t ¼ 0 and t ¼ 1 s, we can observe the Gibbs phenomenon near the discontinuities. The Gibbs phenomenon might be alleviated by increasing the number of samples N, and by increasing the sampling time window, for instance, twice as 2Ts and then ignoring the reconstructed time history for t Ts at the end.

Part Two Theory of Spectral Element Method


3 Methods of Spectral Element Formulation The formulation of the spectral element model for a structure element is fairly similar to the formulation of the conventional finite element models. However, the major difference is that in general the spectral element formulation begins with transforming the governing partial differential equations of motion from the time domain into the frequency domain by using the DFT. For one-dimensional (1-D) problems, as a result of the DFT processing, the time variable disappears and the frequency becomes a parameter to transform the original timedomain partial differential equations into the frequency-domain ordinary differential equations. The frequency-domain ordinary differential equations are then solved exactly and the exact wave solutions are used to derive frequency-dependent dynamic shape functions. The exact dynamic stiffness matrix called the spectral element matrix is finally formulated by using the dynamic shape functions in an analogous way to that commonly used in conventional FEM. In this chapter, three methods of spectral element formulation are introduced: the forcedisplacement relation method, the variational method, and the state-vector equation method. The methods of spectral element formulation are then applied to some representative 1-D structures, such as the axial rod, Bernoulli–Euler beam, Timoshenko beam, and extended Timoshenko beam.

3.1 Force-Displacement Relation Method In this section, the twin concepts of equilibrium and compatibility with exact solutions to the governing equations of motion, that is the direct relationship between nodal forces and nodal displacements, are used to formulate the spectral element model for a structure. The governing equations of motion of a structure undergoing forced vibration can be symbolically represented as L uðx; tÞ þ M€ u ðx; tÞ ¼ pðx; tÞ


ð3:1Þ


42

where L is the linear differential structural operator in the time t and spatial coordinate x domains and M is the inertial operator. The dots () denote the derivatives with respective to time t. u(x, t) and p(x, t) are the displacement fields vectors and external forces vector, respectively. Assume that the external forces can be represented in the spectral form by pðx; tÞ ¼

1 1 NX Pn ðx; vn Þeivn t N n¼0

ð3:2Þ

where Pn(x;v) are the spectral components (or Fourier coefficients) of the external forces p(x,t). The solution to Equation 3.1 can be then assumed in a form similar to 1 1 NX uðx; tÞ ¼ Un ðx; vn Þeivn t ð3:3Þ N n¼0 where Un(x;v) are the spectral components of the displacement fields u(x,t). By substituting Equations 3.2 and 3.3 into Equation 3.1 and assuming that the spectral components Un and Pn satisfy Equation 3.1 at each discrete frequency vn, we can obtain L Un ðx; vn Þ v2n M Un ðx; vn Þ ¼ Pn ðx; vn Þ

ð3:4Þ

where L is the linear differential structural operator in the spatial coordinate x domain. The frequency-dependent shape functions, which are termed dynamic shape functions and used to formulate the spectral element matrix, are obtained from the homogeneous equation L Un ðxÞ v2n M Un ðxÞ ¼ 0

ð3:5Þ

Equation 3.5 holds for all discrete frequencies and each discrete frequency vn then becomes aparameter. For brevity, the subscripts n that appeared in Equation 3.5 will be omitted in the following: L UðxÞ v2 M UðxÞ ¼ 0

ð3:6Þ

The same form of equation as Equation 3.6 can be derived by simply assuming harmonic solutions for the homogeneous Equation 3.1, which has been popularl in the literature associated with the dynamic stiffness method. The general solution to Equation 3.6 can be assumed in the form UðxÞ ¼ ceikx

ð3:7Þ

where c is a constant vector and k is the wavenumber. Substituting Equation 3.7 into Equation 3.6 yields an eigenvalue problem as Aðk; vÞc ¼ 0

ð3:8Þ

The conditions for the existence of the nontrivial solution c of Equation 3.8 requires that the determinant of A (k;v) must be zero as det Aðk; vÞ ¼ 0

ð3:9Þ

which yields a characteristic equation for the wavenumber k in the form kp þ aðp 1Þ ðvÞkðp 1Þ þ aðp 2Þ ðvÞkðp 2Þ þ þ a1 ðvÞk þ a0 ðvÞ ¼ 0

ð3:10Þ

Methods of Spectral Element Formulation

43

Equation 3.10 is called a dispersion relation or spectrum relation. Assume that k1, k2, . . ., kp are the distinct roots of Equation 3.10 at a discrete frequency v. The associated eigenvector ci can be then computed from Equation 3.8 as 1 ð3:11Þ ci ¼ ai ¼ ai fi ði ¼ 1; 2; . . . ; pÞ bi where the eigenvector ci is normalized so that a component of the normalized vector fi becomes unity and the other components are collected as the vector bi. The constants ai will be determined to satisfy the associated boundary conditions. Once the eigensolutions ki and ci are obtained to satisfy the eigenvalue problem in Equation 3.8, the general solutions to Equation 3.6 can be written as p X UðxÞ ¼ fi eiki ðvÞx ai Eðx; vÞa ð3:12Þ i¼1

where Eðx; vÞ ¼ ½f1 f2 f3 . . . fp Lðx; vÞ Lðx; vÞ ¼ diag ½ei ki ðvÞx a ¼ fa1 a2 a3 . . . ap gT

ð3:13Þ

For a finite element of length L, Equation 3.12 must satisfy the geometric and natural boundary conditions at two end-nodes x ¼ 0 and x ¼ L: the spectral nodal displacements and slopes (simply, nodal degrees-of-freedom, DOFs) vector d and the spectral nodal forces (simply, nodal forces) vector f. The displacement and slope are generally called the primary variables and their specification on the boundaries constitutes the geometric boundary conditions. The primary variables can be related to the displacement fields in the form DðxÞ ¼ LGB UðxÞ

ð3:14Þ

where LGB is the linear differential operator for the geometric boundary conditions. Substituting Equation 3.12 into Equation 3.14 and then considering the nodal DOFs d specified at two end-nodes of the finite element gives LGB Eðx; vÞjx¼0 Dð0Þ d¼ ¼ a ¼ HðvÞ a ð3:15Þ LGB Eðx; vÞjx¼L DðLÞ By eliminating the constant vector a from Equation 3.12 by using Equation 3.15, the general solutions can be rewritten in terms of the nodal DOFs vector as UðxÞ ¼ Nðx; vÞd

ð3:16Þ

where N(x,v) is the dynamic shape function defined by Nðx; vÞ ¼ Eðx; vÞH1 ðvÞ

ð3:17Þ

The internal forces and moment are generally called the secondary variables and their specification on the boundary constitutes the natural boundary conditions. The strength of


44

materials provides the relationship between the secondary variables and the displacement fields in the form of FðxÞ ¼ LNB UðxÞ

ð3:18Þ

where LNB is the linear differential operator for the natural boundary conditions. Substituting Equation 3.16 into Equation 3.18 and then considering the nodal forces fc, which are the concentrated loads specified at two end-nodes of the finite element, yields the spectral element equation LNB Nðx; vÞjx¼0 Fð0Þ ¼ fc d SðvÞd ð3:19Þ þLNB Nðx; vÞjx¼L þFðLÞ where

LNB Nðx; vÞjx¼0 ¼ GðvÞ H1 ðvÞ SðvÞ ¼ þLNB Nðx; vÞjx¼L

ð3:20Þ

with GðvÞ ¼

LNB Eðx; vÞjx¼0 þLNB Eðx; vÞjx¼L

ð3:21Þ

The matrix S(v) is the frequency-dependent exact dynamic stiffness matrix, which is termed the spectral element matrix in the literature. In Equation 3.19, the plus ( þ ) and minus () signs reflect the difference between the sign conventions used in the theory of the strength of materials and those commonly adopted in the element methods, such as the conventional FEM and the SEM. The recent development of symbolic computing software makes it easier to obtain the explicit analytical expression of the spectral element matrix. However, when this is formidable, computer implementation of the spectral element matrix formulation procedure can be accomplished numerically. In the following, the spectral element matrices are derived for the axial rod, the Bernoulli–Euler beam (simply the B-beam), the Timoshenko beam (simply the T-beam), and the extended Timoshenko beam (simply the ET-beam) elements as illustrative examples.

Example 3.1

Rod

The free longitudinal vibration of a uniform rod is represented by EAu00 rA€ u¼0

ð3:22Þ

where u(x,t) is the longitudinal displacement, E is the Young’s modulus, A is the crosssectional area, and r is the mass density. The prime (0) denotes the derivatives with respective to the spatial coordinate x. The internal axial force is given by Nt ðx; tÞ ¼ EAu0 ðx; tÞ where the subscript t is used to denote the quantity in the time domain.

ð3:23Þ


45

The solution to Equation 3.22 is assumed in the spectral form as uðx; tÞ ¼

1 1 NX Un ðx; vn Þ eivn t N n¼0

ð3:24Þ

Substituting Equation 3.24 into Equation 3.22 yields an eigenvalue problem for a specific discrete frequency (for example, v ¼ vn) as EAU 00 þ v2 rAU ¼ 0

ð3:25Þ

The general solution to Equation 3.25 is assumed to be UðxÞ ¼ aeikðvÞx

ð3:26Þ

By substituting Equation 3.26 into Equation 3.25, a dispersion relation can be obtained as k2 kL2 ¼ 0

ð3:27Þ

where kL is the wavenumber for the pure longitudinal wavemode and it is defined by rffiffiffiffiffiffi rA kL ¼ v ð3:28Þ EA Equation 3.27 gives two real roots as k1 ¼ k2 ¼ kL

ð3:29Þ

For a finite rod element of length L, the general solution to Equation 3.25 can be then obtained in the form UðxÞ ¼ a1 eikL x þ a2 eþikL x ¼ eðx; vÞa

ð3:30Þ

eðx; vÞ ¼ ½eikL x eþikL x a ¼ fa1 a2 gT

ð3:31Þ

where

The spectral nodal displacements of the finite rod element (see Figure 3.1) can be related to the displacement field as Uð0Þ U1 ð3:32Þ ¼ d¼ U2 UðLÞ

Figure 3.1 Sign convention for the rod element


46

Substituting Equation 3.30 into the right-hand side of Equation 3.32 gives eð0; vÞ d¼ a ¼ HR ðvÞa eðL; vÞ where

HR ðvÞ ¼

1

eikL L

1

ð3:33Þ

eþikL L

ð3:34Þ

By eliminating the constant vector a from Equation 3.30 by using Equation 3.33, we can represent the displacement field in the finite rod element in terms of the nodal DOFs as UðxÞ ¼ NR ðx; vÞd

ð3:35Þ

NR ðx; vÞ ¼ eðx; vÞH1 R ðvÞ ¼ ½NR1 NR2 cscðkL LÞsin½kL ðL xÞ NR1 ðx; vÞ ¼ cscðkL LÞsinðkL xÞ NR2 ðx; vÞ ¼

ð3:36Þ

where

From Equation 3.23, the spectral components of the axial force are related to U(x) by NðxÞ ¼ EAU 0 ðxÞ

ð3:37Þ

The spectral nodal axial forces defined for the finite rod element (see Figure 3.1) can be related to the forces defined by the strength of materials as N1 Nð0Þ fc ðvÞ ¼ ¼ ð3:38Þ N2 þNðLÞ Substituting Equations 3.35 and 3.37 into the right-hand side of Equation 3.38 gives SR ðvÞd ¼ fc ðvÞ

ð3:39Þ

where SR(v) is the spectral element matrix for the finite rod element and it is given by EA sR11 sR12 ð3:40Þ SR ðvÞ ¼ ¼ SR ðvÞT L sR12 sR22 where sR11 ¼ sR22 ¼ ðkL LÞ cotðkL LÞ sR12 ¼ ðkL LÞ cscðkL LÞ

Example 3.2

ð3:41Þ

Bernoulli–Euler Beam

The free bending (flexural) vibration of a Bernoulli–Euler beam (simply, B-beam) is represented by EIw0000 þ rA€ w¼0

ð3:42Þ


47

where w(x,t) is the transverse displacement, E is the Young’s modulus, A is the cross-sectional area, I is the area moment of inertia about the neutral axis, and r is the mass density. The internal transverse shear force and bending moment are given by Mt ðx; tÞ ¼ EIw00 ðx; tÞ;

Qt ðx; tÞ ¼ EIw000 ðx; tÞ

ð3:43Þ

Assume the solution to Equation 3.42 in the spectral form to be wðx; tÞ ¼

1 1 NX Wn ðx; vn Þ eivn t N n¼0

ð3:44Þ

Substituting Equation 3.44 into Equation 3.42 gives an eigenvalue problem for a specific discrete frequency (such as v ¼ vn) as EIW 0000 v2 rAW ¼ 0

ð3:45Þ

Assume the general solution to Equation 3.45 as WðxÞ ¼ aeikðvÞx

ð3:46Þ

Substituting Equation 3.46 into Equation 3.45 yields a dispersion relation k4 kF4 ¼ 0

ð3:47Þ

where kF is the wavenumber for the pure bending (flexural) wavemode defined by kF ¼

pffiffiffiffi rA 1=4 v EI

ð3:48Þ

Equation 3.47 gives two pure real roots and two pure imaginary roots as k1 ¼ k2 ¼ kF ;

k3 ¼ k4 ¼ ikF

ð3:49Þ

For the finite B-beam element of length L, the general solution of Equation 3.45 can be then obtained in the form Wðx; vÞ ¼ a1 eikF x þ a2 ekF x þ a3 eþ ikF x þ a4 eþ kF x ¼ eðx; vÞ a

ð3:50Þ

where eðx; vÞ ¼ ½e ikF x e kF x a ¼ fa1 a2 a3 a4 gT

eþ ikF x

eþ kF x

ð3:51Þ

The spectral nodal displacements and slopes of the finite B-beam element (see Figure 3.2) can be related to the displacement field by 9 9 8 8 W1 > > Wð0Þ > > > > > = = > < 0 < W ð0Þ Q1 ¼ d¼ > > > > > > WðLÞ > W2 > ; ; : : W 0 ðLÞ Q2

ð3:52Þ


48

Figure 3.2 Sign convention for the Bernoulli–Euler beam element

Substituting Equation 3.50 into the right-hand side of Equation 3.52 gives 2

3 eð0; vÞ 6 e0 ð0; vÞ 7 7 d ¼6 4 eðL; vÞ 5a ¼ HB ðvÞa e0 ðL; vÞ

ð3:53Þ

where 2

1 6 ik F HB ðvÞ ¼ 6 4 e ikF L ikF e ikF L

1 kF e kF L kF e kF L

1 ikF

þ ikF L

e ikF eþ ikF L

1 kF

þ kF L

e kF eþ kF L

3 7 7 5

ð3:54Þ

The displacement field within the finite B-beam element can be represented in terms of the nodal DOFs vector d by eliminating the constant vector a from Equation 3.50 by using Equation 3.53, thus WðxÞ ¼ NB ðx; vÞd

ð3:55Þ

where NB ðx; vÞ ¼ eðx; vÞHB1 ðvÞ ¼ ½NB1 NB2 NB3 NB4 x cos L coshðL x cos ðL Þ coshL Þ NB1 ðxÞ ¼ h 1 kF ½cos x

x sin L sinh ðL x þ sin ðL Þ sinh L Þ þ cosh x x þ cosh L sin ðL x Þ sin L Þ þ sin x NB2 ðxÞ ¼ h ½ cosh ðL x þ cos L sinh ðL x Þ sinh L Þ þ sinh x cos ðL 1

x cos L cosh x Þ cos x cosh L NB3 ðxÞ ¼ h 1 kF ½cos ðL Þ þ sin x sinh L sin L sinh x þ cosh ðL x þ cosh L sin x x sin L þ sin ðL Þ NB4 ðxÞ ¼ h 1 ½ cosh x sinh L þ cos L sinh x þ sinh ðL x Þ cos x cosh LÞ h ¼ 2kF ð1 cos L ¼ kF L ¼ kF x; L x

ð3:56Þ


49

From Equation 3.43, the spectral components of the bending moment and transverse shear force can be related to W(x) as QðxÞ ¼ EIW 000 ðxÞ;

MðxÞ ¼ EIW 00 ðxÞ

ð3:57Þ

The spectral nodal transverse shear forces and bending moments defined for the finite B-beam element (see Figure 3.2) can be related to the corresponding forces and moments defined by the strength of materials by 9 9 8 8 Qð0Þ > Q1 > > > > > > > = = < < Mð0Þ M1 ð3:58Þ ¼ fc ¼ þ QðLÞ > Q > > > > > ; : ; > : 2> þ MðLÞ M2 Substituting Equation 3.55 into Equation 3.57 and the results into the right-hand side of Equation 3.58 gives SB ðvÞd ¼ fc ðvÞ where SB(v) is the spectral element matrix for 2 sB11 sB12 EI 6 sB12 sB22 SB ðvÞ ¼ 3 6 L 4 sB13 sB23 sB14 sB24

the B-beam element given by 3 sB13 sB14 sB23 sB24 7 7 ¼ SB ðvÞT sB33 sB34 5 sB34 sB44

ð3:59Þ

ð3:60Þ

where sinh L þ sin L cosh LÞ 3 ðcos L sB11 ¼ sB33 ¼ DB L sinh L þ sin L cosh LÞ kF 2 ð cos L sB22 ¼ sB44 ¼ DB L 3

sinh L kF1 sin L sB12 ¼ sB34 ¼ DB L 3

3 ðsin L þ sinh LÞ sB13 ¼ DB L kF 1 ð cos L þ cosh LÞ sB14 ¼ sB23 ¼ DB L 3

ð3:61Þ

kF 2 ðsin L þ sinh LÞ sB24 ¼ DB L 3

1 cosh L 1 cos L L ¼ kF L DB ¼

Example 3.3

Timoshenko Beam

The free vibration of a uniform Timoshenko beam (simply, T-beam) is represented by w¼0 kGAðw00 u0 Þ rA€ 00 0 EIu þ kGAðw uÞ rI €u ¼ 0

ð3:62Þ


50

where w(x,t) and u(x,t) are the transverse displacement and the slope due to bending, respectively. E is the Young’s modulus, G is the shear modulus, k is the shear correction factor, which depends on the shape of the cross-section, A is the cross-sectional area, and I is the area moment of inertia about the neutral axis. The internal bending moment and transverse shear force are given by Qt ðx; tÞ ¼ kGA ½w0 ðx; tÞ uðx; tÞ Mt ðx; tÞ ¼ EIu0 ðx; tÞ

ð3:63Þ

Assume the solution of Equation 3.62 in the spectral form is wðx; tÞ ¼

1 1 NX Wn ðx; vn Þeivn t ; N n¼0

uðx; tÞ ¼

1 1 NX Qn ðx; vn Þ eivn t N n¼0

ð3:64Þ

Substituting Equation 3.64 into Equation 3.62 yields an eigenvalue problem as kGAðW 00 Q0 Þ þ rAv2 W ¼ 0 EIQ00 þ kGAðW 0 QÞ þ rIv2 Q ¼ 0

ð3:65Þ

Assume the general solutions to Equation 3.65 are WðxÞ ¼ ae ikðvÞx ;

QðxÞ ¼ bae ikðvÞx

Substitute Equation 3.66 into Equation 3.65 to obtain an eigenvalue problem as 0 1 kGAk2 rAv2 ikkGA ¼ b 0 ikkGA EIk2 þ kGA rIv2

ð3:66Þ

ð3:67Þ

Equation 3.67 gives a dispersion relation as k4 hkF4 k2 kF4 ð1 h1 kG4 Þ ¼ 0 where

pffiffiffiffi rA 1=4 kF ¼ v ; EI

pffiffiffiffi rA 1=4 kG ¼ v kGA

ð3:68Þ

ð3:69Þ

and h ¼ h 1 þ h2 ;

h1 ¼

rI ; rA

h2 ¼

EI kGA

Solving Equation 3.68 gives four roots as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k1 ¼ k2 ¼ pffiffiffi kF hkF2 þ h2 kF4 þ 4ð1 h1 kG4 Þ kt 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k3 ¼ k4 ¼ pffiffiffi kF hkF2 h2 kF4 þ 4ð1 h1 kG4 Þ ke 2 From the first line of Equation 3.67, we can obtain the wavemode ratio as 1 2 bp ðvÞ ¼ ðk kG4 Þ ¼ irp ðvÞ ðp ¼ 1; 2; 3; 4Þ ikp p

ð3:70Þ

ð3:71Þ

ð3:72Þ


51

where rp ðvÞ ¼

1 2 ðk kG4 Þ kp p

ð3:73Þ

Note that when h ¼ 0 and kG ¼ 0, Equation 3.68 is reduced to the disperison equation for the B-beam (see Equation 3.47) and the wavenumbers of Equation 3.71 become identical with those for the B-beam (see Equation 3.49). By using the four wavenumbers given by Equation 3.71, the general solutions of Equation 3.65 can be written as WðxÞ ¼ a1 eikt x þ a2 eþ ikt x þ a3 e ike x þ a4 eþ ike x ¼ ew ðx; vÞa QðxÞ ¼ b1 a1 e ikt x þ b2 a2 eþ ikt x þ b3 a3 e ike x þ b4 a4 eþ ike x ¼ eu ðx; vÞa

ð3:74Þ

where a ¼ fa1

a2

a3

a4 gT

ð3:75Þ

and ew ðx; vÞ ¼ ½e ikt x

eþ ikt x

e i ke x

eþ i ke x

eu ðx; vÞ ¼ ew ðx; vÞBðvÞ

ð3:76Þ

BðvÞ ¼ diag½bp ðvÞ The spectral nodal displacements and slopes of the finite T-beam element of length L (see Figure 3.3) can be related to the displacement fields by 9 9 8 8 W1 > > Wð0Þ > > > > > = < = > < Qð0Þ Q1 ¼ d¼ > > > WðLÞ > > > > W2 > ; : ; : QðLÞ Q2

Figure 3.3

Sign convention for the Timoshenko-beam element

ð3:77Þ


52

Substituting Equation 3.74 into the right-hand side of Equation 3.77 gives 2 3 ew ð0; vÞ 6 eu ð0; vÞ 7 7 d ¼6 4 ew ðL; vÞ 5a ¼ HT ðvÞa eu ðL; vÞ

ð3:78Þ

where 2

1 6 irt HT ðvÞ ¼ 6 4 et irt et

1 irt et 1 irt et 1

1 ire ee ire ee

3 1 ire 7 7 ee 1 5 ire ee 1

ð3:79Þ

with the use of following definitions: et ¼ e ikt L ; ee ¼ e ike L 1 1 rt ¼ ðkt2 kG4 Þ; re ¼ ðke2 kG4 Þ kt ke

ð3:80Þ

By using Equation 3.78, the constant vector a can be eliminated from Equation 3.74 to express the general solutions as WðxÞ ¼ Nw ðx; vÞd;

QðxÞ ¼ Nu ðx; vÞd

ð3:81Þ

where Nw ðx; vÞ ¼ ew ðx; vÞHT 1 ðvÞ Nu ðx; vÞ ¼ eu ðx; vÞHT 1 ðvÞ ¼ ew ðx; vÞBðvÞHT 1 ðvÞ

ð3:82Þ

From Equation 3.63, the spectral components of the bending moment and transverse shear force are related to W(x) and Q(x) by Q ¼ kGAðW 0 QÞ;

M ¼ EIQ0

ð3:83Þ

The spectral nodal transverse shear forces and bending moments defined for the finite T-beam element (see Figure 3.3) can be related to the corresponding forces and moments defined by the strength of materials as 9 9 8 8 Qð0Þ > Q1 > > > > > > > = = < < Mð0Þ M1 ð3:84Þ ¼ fc ðvÞ ¼ þ QðLÞ > Q2 > > > > > > > ; ; : : þ MðLÞ M2 Applying Equation 3.81 to Equation 3.83 and the results to the right-hand side of Equation 3.84 gives ST ðvÞd ¼ fc ðvÞ

ð3:85Þ


where ST(v) is the spectral element matrix given by 2 sT11 sT12 sT13 6 sT12 sT22 sT23 ST ðvÞ ¼ EI 6 4 sT13 sT23 sT33 sT14 sT24 s34

53

3 sT14 sT24 7 7 ¼ ST ðvÞT sT34 5 sT44

ð3:86Þ

where sT11 ¼ sT33 ¼ iDT A½ðe2t 1Þðe2e þ 1Þrt ðe2t þ 1Þðe2e 1Þre ðkt rt ke re Þ sT22 ¼ sT44 ¼ DT B½ ðe2t þ 1Þðe2e 1Þrt þ ðe2t 1Þðe2e þ 1Þre sT12 ¼ sT34 ¼ DT Afðe2t 1Þðe2e 1Þðkt re þ ke rt Þ ½ðe2t þ 1Þðe2e þ 1Þ 4et ee ðkt rt þ ke re Þg

ð3:87Þ

sT13 ¼ 2iDT A½ðe2t 1Þee rt ðe2e 1Þet re ð kt rt þ ke re Þ sT14 ¼ sT23 ¼ 2DT Aðet ee Þð1 et ee Þð kt rt þ ke re Þ sT24 ¼ 2DT B½ðe2e 1Þet rt ðe2t 1Þee re

A ¼ ð1=8ÞeF ð1 h1 kG4 Þ 1=2

B ¼ ð1=8ÞeF ½h2 kF4 þ 4ð1 h1 kG4 Þ1=2 8kF2 DT ¼ ieF ½2rt re fðe2t þ 1Þðe2e þ 1Þ 4et ee g ðr2t þ r2e Þðe2t 1Þðe2e 1Þ

Example 3.4

Extended Timoshenko Beam

The plane motion of an extended Timoshenko beam (simply, an ET-beam) is represented by [79] 8 09 2 38 9 8 9 rA 0 rR < €u = < 0 = < Nt = € 0 Q0 ¼ 4 0 rA 0 5 w ð3:88Þ : t0 ; :€; : ; Mt rR 0 rI u Qt where Nt(x, t), Qt(x, t), and Mt(x, t) are the axial tensile force, transverse shear force, and bending moment, respectively, and u(x, t), w(x, t), and u(x, t) are the axial displacement, transverse displacement, and slope, respectively. rA, rR, and rI are the effective mass per length, the first-order moment of inertia, and the second-order moment of inertia defined by ð ð ð rA ¼ rdA; rR ¼ rzdA; rI ¼ rz2 dA ð3:89Þ where r is the mass density per unit volume. The force-displacement relationship for an ETbeam is given by 8 9 2 9 38 EA C1 C2 < u0 = < Nt = ¼ 4 C1 kGA C3 5 w0 u ð3:90Þ Q : t; : ; u0 Mt C2 C3 EI


54

where C1, C2, and C3 are the coupling rigidities, which represent the couplings between axial, transverse shear, and bending deformations. When all the rR, C1, C2, and C3 vanish, the decoupled equations of motion for the classical Timoshenko beam and axial rod are recovered. Represent the displacement fields in the spectral forms as uðx; tÞ ¼

1 1 NX Un ðx; vn Þeivn t N n¼0

wðx; tÞ ¼

1 1 NX Wn ðx; vn Þeivn t N n¼0

uðx; tÞ ¼

1 1 NX Qn ðx; vn Þeivn t N n¼0

ð3:91Þ

Similarly, we represent the axial tensile force Nt, the transverse shear force Qt, and the bending moment Mt in the spectral forms as Nt ðx; tÞ ¼

1 1 NX Nn ðx; vn Þeivn t N n¼0

Qt ðx; tÞ ¼

1 1 NX Qn ðx; vn Þeivn t N n¼0

Mt ðx; tÞ ¼

1 1 NX Mn ðx; vn Þeivn t N n¼0

Substitution of Equations 3.91 and 3.92 into Equation 3.88 and 3.90 gives 8 09 2 38 9 8 9 rA 0 rR < U = < 0 = EA C1 C2 > U 00 > = < < = 6 7 6 7 0 kGA 5 W 0 4 C1 kGA C3 5 W 00 þ 4 0 > > ; ; : 00 > : 0> C2 C3 EI Q 0 Q C1 kGA ð3:95Þ 2 38 9 2 38 9 8 9 U> 0 0 0 rA 0 rR > U > > 0 > > < = < = < = 6 7 6 7 40 0 0 5 W þ v2 4 0 rA 0 5 W ¼ 0 > > ; ; : > ; > : > : > 0 0 kGA Q 0 rR 0 rI Q


55

Assume the general solutions of Equation 3.95 to be UðxÞ ¼ aaeikx ;

WðxÞ ¼ aeikx ;

QðxÞ ¼ bae ikx

ð3:96Þ

where k is the wavenumber. Substitution of Equation 3.96 into Equation 3.95 gives an eigenvalue problem as 8 9 8 9 > > > = < Q ¼ Q > > > > ; : M

ð3:219Þ

ð3:220Þ


72

and

2

0 6 0 AB ðvÞ ¼ 6 4 EIk4 F 0

1 0 0 0

0 0 0 1

3 0 1=EI 7 7 0 5 0

where kF ¼ v(rA/EI)1/2 is the wavenumber defined by Equation 3.48. By using AB of Equation 3.221, the transfer matrix TB can be obtained as TDD TDF AB L ¼ TB e TFD TFF 2 s1 s2 c2 3 c1 3 6 kF EIkF2 7 EIkF 6 c s1 7 2 7 c1 16 7 6 kF s2 EIkF 7 ¼ 6 EIkF2 7 26 EIkF2 c2 c1 kF s2 7 6 EIkF3 s1 5 4 s1 EIkF2 c2 EIkF s2 c1 kF where c1 ¼ cosðkF LÞ þ coshðkF LÞ c2 ¼ cosðkF LÞ coshðkF LÞ s1 ¼ sinðkF LÞ þ sinhðkF LÞ

ð3:221Þ

ð3:222Þ

ð3:223Þ

s2 ¼ sinðkF LÞ sinhðkF LÞ By substituting the submatrices of TB into Equation 3.209, we obtain the spectral element matrix SB(v), which is identical with Equation 3.60.

Example 3.11

Timoshenko Beam

Revisit the uniform T-beam considered in Example 3.3. In the frequency domain, the governing equations and force-displacement relations for the T-beam are given by ðGEÞ kGAðW 00 Q0 Þ þ rAv2 W ¼ 0 EIQ00 þ kGAðW 0 QÞ þ rIv2 Q ¼ 0 ðFDÞ

Q ¼ kGAðW 0 QÞ

ð3:224Þ

M ¼ EIQ0 Equation 3.224 can be transformed into four first-order ODEs as follows: W 0 ¼ Q þ Q=kGA Q0 ¼ M=EI Q0 ¼ kGAðW 00 Q0 Þ ¼ rAv2 W ¼ EIkF4 W

ð3:225Þ

M 0 ¼ EIQ00 ¼ kGAðW 0 QÞ rIv2 Q ¼ Q EIh1 kF4 Q Thus, the above equations can be written into the form of a state-vector equation as y0T ¼ AT ðvÞyT

ð0 x LÞ

ð3:226Þ


73

where yT ¼

and

D F

2

0 6 0 AT ðvÞ ¼ 6 4 EIk4 F 0

8 9 W > > > > = < Q ¼ > Q > > > ; : M

1 0 0 EIh1 kF4

1=kGA 0 0 1

ð3:227Þ

3 0 1=EI 7 7 0 5 0

ð3:228Þ

where kF and h1 are defined by the first part of Equation 3.69 and the middle part of Equation 3.70, respectively. By using the matrix AT of Equation 3.228, the transfer matrix TT can be derived as 2 3 T12 T13 T14 T11 6 T21 T22 T14 T24 7 7 ¼ TDD TDF TT eAT L ¼ 6 ð3:229Þ 4 T31 T32 T11 T21 5 TFD TFF T32 T42 T12 T22 where T11 ¼ T12 ¼

ðkG4 ke2 Þcoshðkt LÞ ðkG4 kt2 Þcoshðke LÞ kt2 ke2 ke ðhkF4 ke2 Þsinhðkt LÞ kt ðhkF4 kt2 Þsinhðke LÞ kt ke ðkt2 ke2 Þ

ke ½1 þ h2 ðh2 kF4 ke2 Þsinhðkt LÞ kt ½1 þ h2 ðh2 kF4 kt2 Þsinhðke LÞ EIkt ke ðkt2 ke2 Þ coshðkt LÞ coshðke LÞ ¼ EIðkt2 ke2 Þ

T13 ¼ T14

T21 ¼

kF4 ½ke sinhðkt LÞ kt sinhðke LÞ kt ke ðkt2 ke2 Þ

T22 ¼

ðh1 kF4 ke2 Þcoshðkt LÞ ðh1 kF4 kt2 Þcoshðke LÞ kt2 ke2

T24 ¼

ke ðh1 kF4 ke2 Þsinhðkt LÞ kt ðh1 kF4 kt2 Þsinhðke LÞ EIkt ke ðkt2 ke2 Þ

T31 ¼ T32 ¼

EIkF4 ½ke ðkG4 ke2 Þsinhðkt LÞ kt ðkG4 kt2 Þsinhðke LÞ kt ke ðkt2 ke2 Þ

EIkF4 ½coshðkt LÞ coshðke LÞ kt2 ke2

T42 ¼

EIkF4 fke ½1 þ h1 ðh1 kF4 ke2 Þsinhðkt LÞ kt ½1 þ h1 ðh1 kF4 kt2 Þsinhðke LÞg kt ke ðkt2 ke2 Þ ð3:230Þ


74

In Equation 3.230, kG is defined by the second part of Equation 3.69. By substituting the submatrices of TT into Equation 3.209, we can obtain the spectral element matrix ST(v), which is identical with Equation 3.86.

Example 3.12

Extended Timoshenko Beam

The governing equation and force-displacement relation for an ET-beam are given in the frequency domain by Equations 3.93 and 3.94, respectively. They are 8 09 2 38 9 8 9 rA 0 rR < U = < 0 = > > > > u_ 1 > = < 6 7< = 6 7 6 7 60 1 07 € _ þ 2 4 1 u u 5> 2 > 4 5> 2 > 4 > > ; > ; : : > € 0 1 4 0 0 1 u3 u_ 3 9 2 38 9 8 u1 > 3 2 0 > 0 > > > > > > = < = < 6 7 7 u2 ¼ þ 1006 2 3 1 0 4 5> > > > > > ; : ; > : > 0 1 3 f ðtÞ u3 subjected to the initial conditions fu1 ; u2 ; u3 g ¼ f1; 0; 0g

and

fu_ 1 ; u_ 2 ; u_ 3 g ¼ f0; 1; 0g

where f(t) ¼ 1 u(t 1), where u(t) is the unit step function. Figure 4.2(a) compares the solution for u3 obtained by using the present spectral analysis method (SAM) with the exact solution analytically obtained by using the modal analysis method (MAM) and with the solution obtained by using the Runge–Kutta method. To obtain the solutions within a 0.5% time-averaged error, the sampling time window T ¼ 4 s and the number of samples N ¼ 1024 are used for the SAM, while the time increment Dt ¼ 0.0039 s is used for the Runge–Kutta method. The solutions obtained by using the non-proportional damping-based approach are also confirmed to be identical with those obtained by using the proportional damping based approach. It is obvious from Figure 4.2(a) that the present SAM provides very reliable solutions when compared with the solutions by both the MAM and the

Figure 4.2

Comparison of the dynamic responses obtained by using different methods

Spectral Element Analysis Method

99

Table 4.1 Comparison of the CPU times and time-averaged errors Present SAM Solution methods CPU time (s) Time-averaged error (%)

u1(t) u2(t) u3(t)

Runge–Kutta method

N ¼ 1024

N ¼ 2048

Dt ¼ 3.91 (ms)

Dt ¼ 1.95 (ms)

0.203 0.214 0.232 0.168

0.296 0.087 0.120 0.085

0.234 0.502 0.265 0.264

0.344 0.283 0.236 0.230

Runge–Kutta method. Of course, as displayed in Figure 4.2(b), the accuracy of the SAM results can be controlled by varying the number of samples N for a chosen sampling time window T ¼ 4 s. Table 4.1 compares the CPU times and the time-averaged errors depending on the conditions used for the present SAM and the Runge–Kutta method.

4.5.2 Continuous Systems with Arbitrary Initial Conditions Consider a linear continuous system whose dynamics are represented in the general form as L uðx; tÞ þ M€ u ðx; tÞ ¼ pðx; tÞ

ð4:122Þ

where L is the linear differential structural operator, M is the linear differential inertial operator, p(x, t) is the excitation force vector, and u(x, t) is the displacement vector. Assume that the linear continuous system is subjected to arbitrary, non-null initial conditions given by uðx; 0Þ ¼ u0 ðxÞ;

_ 0Þ ¼ u_ 0 ðxÞ uðx;

ð4:123Þ

As discussed in the early part of this section, the complete dynamic response of a linear continuous system can be obtained by summing the forced vibration response part up(x, t) and the free vibration response part uh(x, t) as follows: uðx; tÞ ¼ up ðx; t; pÞ þ uh ðx; t; u0 ; u_ 0 Þ

ð4:124Þ

The forced vibration response part up(x, t) is determined solely by the excitation force p(x,t), while the free vibration response part uh(x, t) is determined solely by the initial conditions u0 and u_ 0 . Equation 4.124 implies that the linear continuous dynamic system when subjected to excitation forces, but initially at rest, will have only the forced vibration response part up(x, t). It is known that the convolution integral (or Duhamel’s integral) provides the forced vibration responses for the linear systems initially at rest. It is also shown in Section 4.4.2 that the discrete Fourier transformed convolution integral for a linear continuous system is equivalent to the spectral element equation, S(v)d ¼ f. Thus, the dynamic responses computed from d ¼ S(v)1f are just the forced vibration responses of the linear continuous system subjected to completely null initial conditions. Thus, to predict the dynamic responses of a linear continuous system subjected to arbitrary non-zero initial conditions by using the SEM, we may need a strategy. The strategy introduced herein is to first transform the original governing equation, Equation 4.122, subjected to arbitrary non-null initial conditions, into the new governing equation subjected to completely


100

null initial conditions by using the pseudo-force method, and next to apply the SEM to obtain the desired dynamic responses. 4.5.2.1 Pseudo-Force Method Pseudo-Force for the Initial Displacement Consider the free vibration of a continuous system, represented by the governing equation L u1 ðx; tÞ þ M€ u 1 ðx; tÞ ¼ 0

ð4:125Þ

and the initial conditions u1 ðx; 0Þ ¼ u0 ðxÞ;

u_ 1 ðx; 0Þ ¼ 0

ð4:126Þ

Assume the solution to Equation 4.125 is u1 ðx; tÞ ¼ u0 ðxÞ þ Du1 ðx; tÞ

ð4:127Þ

By using Equation 4.127, Equations 4.125 and 4.126 can be transformed into L Du1 ðx; tÞ þ MD€ u 1 ðx; tÞ ¼ fu0 ðx; tÞ

ð4:128Þ

and Du1 ðx; 0Þ ¼ 0;

Du_ 1 ðx; 0Þ ¼ 0

ð4:129Þ

In Equation 4.128, fu0 is the pseudo-force defined by fu0 ðx; tÞ ¼ L Lu0 ðxÞuðtÞ

ð4:130Þ

where u(t) is the unit step function. Pseudo-Force for the Initial Velocity Now consider a linear continuous system subjected to an impulse force u 2 ðx; tÞ ¼ iðxÞdðtÞ L u2 ðx; tÞ þ M€

ð4:131Þ

and the initial conditions u2 ðx; 0 Þ ¼ 0;

u_ 2 ðx; 0 Þ ¼ 0

ð4:132Þ

where d(t) is the Dirac delta function. The initial conditions in Equation 4.132 indicate that the linear continuous system is at rest before the application of an impulse force. The time t ¼ 0 emphasizes the instant of time just before the application of the impulse force, that is, the negative side of the time t ¼ 0. Integrating both sides of Equation 4.131 from t ¼ 0 to t ¼ Dt yields L u2 avg ðxÞ þ M½u_ 2 ðx; DtÞ u_ 2 ðx; 0 Þ ¼ iðxÞ

ð4:133Þ

where u2avg(x) is the time average of u2(x, t) for the short time duration Dt, which will become zero as Dt ! 0. By applying the null initial conditions of Equation 4.132 to Equation 4.133 and then setting Dt ! 0, we can obtain Mu_ 2 ðx; 0Þ ¼ iðxÞ

ð4:134Þ


101

Equation 4.134 implies that the impulse force applied at t ¼ 0 results in a sudden change in its velocity from zero, which is equal to M1i(x), without resulting in an appreciable change in its displacement, because the time duration required to develop the displacement is zero from Dt ! 0. Accordingly, the original governing equation (Equation 4.131) can be expressed as L u2 ðx; tÞ þ M€ u 2 ðx; tÞ ¼ 0

ð4:135Þ

subjected to the new initial conditions u2 ðx; 0Þ ¼ 0;

u_ 2 ðx; 0Þ ¼ M 1 iðxÞ u_ 0 ðxÞ

ð4:136Þ

The original set of equations 4.131 and 4.132 is equivalent to the new set of equations 4.135 and 4.136. Thus we can treat the system that has been subjected to the non-null initial velocity u_ 0 ðxÞ as the system subjected to the pseudo-impulse force fu_ 0 defined by fu_ 0 ðx; tÞ ¼ Mu_ 0 ðxÞdðtÞ

ð4:137Þ

In other words, the set of equations 4.135 and 4.136 can be replaced with the new set of equations L u2 ðx; tÞ þ M€ u 2 ðx; tÞ ¼ fu_ 0 ðx; tÞ

ð4:138Þ

and u2 ðx; 0Þ ¼ 0;

u_ 2 ðx; 0Þ ¼ 0

ð4:139Þ

Pseudo-Force for the General Case Finally, consider the general case when a continuous system is subjected to the excitation force p(x, t) and to the arbitrary non-null initial conditions u0(x) and u_ 0 ðxÞ; as represented by the governing equation (4.122) and the initial conditions (4.123). The complete dynamic responses are assumed by uðx; tÞ ¼ u0 ðxÞ þ Duðx; tÞ

ð4:140Þ

By substituting Equation 4.140 into the original governing equation (4.122) and initial conditions (4.123) and then by using the pseudo-forces developed previously, we can obtain a new set of governing equation and initial conditions for Du(x, t). They are given by L Duðx; tÞ þ MD€ u ðx; tÞ ¼ pðx; tÞ þ fu0 ðx; tÞ þ fu_ 0 ðx; tÞ

ð4:141Þ

and Duðx; 0Þ ¼ 0;

_ 0Þ ¼ 0 Duðx;

ð4:142Þ

where fuo and fu_ 0 are the pseudo-forces defined by fu0 ðx; tÞ ¼ L Lu0 ðxÞuðtÞ

ð4:143Þ

fu_ 0 ðx; tÞ ¼ Mu_ 0 ðxÞdðtÞ

ð4:144Þ

As the new governing equation (4.141) is now subjected to completely null initial conditions (Equation 4.142), we can use either the convolution integral or the SEM to compute Du(x, t).


102

Once Du(x, t) is obtained from Equations 4.141 and 4.142, the complete dynamic responses can be computed from Equation 4.140 by adding Du(x, t) to the initial displacements u0(x). 4.5.2.2 Application to the Transverse Vibration of a Beam As an illustrative example, consider a uniform Bernoulli–Euler beam subjected to an excitation force f(x, t). The transverse vibration w(x, t) of the beam is governed by the equation of motion EIw0000 þ rA€ w ¼ f ðx; tÞ

ð4:145Þ

and the initial conditions wðx; 0Þ ¼ w0 ðxÞ;

_ wðx; 0Þ ¼ w_ 0 ðxÞ

ð4:146Þ

where EI is the bending rigidity and rA is the mass per unit length of the beam. Transform of the Equation of Motion By comparing Equations 4.145 and 4.146 with Equations 4.122 and 4.123, we find the following relationships: L( ) EIð Þ0000 ;

M( ) rA

uðx; tÞ ( ) wðx; tÞ; pðx; tÞ ( ) f ðx; tÞ u0 ðxÞ ( ) w0 ðxÞ;

ð4:147Þ

u_ 0 ðxÞ ( ) w_ 0 ðxÞ

Assume the complete solution of Equation 4.145 to be wðx; tÞ ¼ w0 ðxÞ þ Dwðx; tÞ

ð4:148Þ

By using the relationships of Equation 4.147, the original equation of motion and initial conditions can be transformed into the new equation of motion EIDw0000 ðx; tÞ þ rAD€ w ðx; tÞ ¼ pðx; tÞ

ð4:149Þ

and the new initial conditions _ Dwðx; 0Þ ¼ Dwðx; 0Þ ¼ 0

ð4:150Þ

_ 0 dðtÞ pðx; tÞ ¼ f ðx; tÞ EIw0000 0 uðtÞ þ rAw

ð4:151Þ

where

Spectral Element Model for Dw(x, t) Represent the solution Dw(x, t) to Equation 4.149 and the excitation force f (x, t) in the spectral forms as Dwðx; tÞ ¼

f ðx; tÞ ¼

1 1 NX DWn ðxÞeivn t N n¼0

ð4:152Þ

1 1 NX Fn ðxÞeivn t N n¼0

ð4:153Þ


103

The spectral components F (omitting the subscripts n for brevity) can be derived from Equation 4.151 as _ 0 ðxÞuðvÞ PðxÞ ¼ FðxÞ EIw0000 0 ðxÞdðvÞ þ rAw

ð4:154Þ

where d(v) is the Dirac delta function and u(v) is the unit step function. Substituting Equations 4.152 and 4.153 into Equation 4.149 yields EIDW 0000 ðxÞ v2 rADWðxÞ ¼ PðxÞ

ð4:155Þ

The weak form of Equation 4.155 is given by ðl

ðEIDW 00 dDW 00 v2 rADWdDWÞdx

ðl

0

PðxÞdDWdx ¼ 0

ð4:156Þ

0

Equation 4.155 is essentially same as Equation 3.147 considered in Example 3.6. Thus we can use the same dynamic shape function NB(x,v) as used in Example 3.6 to express the general solution of Equation 4.155 as DWðxÞ ¼ NB ðx; vÞDd

ð4:157Þ

where Dd is the spectral DOFs vector defined by Dd ¼ fDW1

DQ1

DW2

DQ2 gT ¼ fDWð0Þ DW 0 ð0Þ DWðlÞ

DW 0 ðlÞgT

ð4:158Þ

Substituting Equation 4.157 into Equation 4.156, we can derive the spectral element equation as SðvÞDd ¼ f where SðvÞ ¼

ðl 0

ðEIN B00 T N B00 v2 rANTB NB Þdx

f¼

ðl 0

PðxÞNTB dx

ð4:159Þ

ð4:160Þ

ð4:161Þ

The spectral element matrix S(v) given by Equation 4.160 is identical with that given by Equation 3.156 in Example 3.6. Solution Procedure The spectral element equations are assembled, followed by the imposition of the boundary conditions to derive the global system dynamic equation in the form of Sg ðvÞDdg ¼ fg

ð4:162Þ

where Sg(v) is the global dynamic stiffness matrix, Ddg is the global perturbed spectral nodal DOFs vector, and fg is the global spectral nodal forces vector. The remaining solution procedure


104

is as follows: a. Compute Ddg from Equation 4.162. b. Compute the spectral components DWn(x) (n ¼ 0, 1, . . . , N 1) from Equation 4.157. c. Compute the perturbed displacement Dw(x, t) from DWn(x) (n ¼ 0, 1, . . . , N 1) by using the inverse-FFT algorithm. d. Finally compute the complete dynamic response w(x, t) from Equation 4.148 by adding Dw(x, t) to the initial displacement w0(x).

Example 4.3 Consider a uniform Bernoulli–Euler beam that is simply supported at its two ends. Assume that the beam is subjected to the initial conditions px _ ; wðx; 0Þ ¼ 0 wðx; 0Þ ¼ w0 sin L When a point unit impulse is applied at the middle point (x ¼ L/2) of the beam at time t ¼ t0, the dynamic response can be analytically obtained in a closed form by using the modal analysis method as wðx; tÞ ¼ wp ðx; tÞ þ wh ðx; tÞ where

8 >
: v rAL 2 L n¼1 n wh ðx; tÞ ¼ w0 sin

px cosv1 t L

ð0 t t0 Þ ðt t0 Þ

ðt 0Þ

and vn are the natural frequencies defined by sffiffiffiffiffiffiffiffiffiffiffi EI 2 2 vn ¼ n p ðn ¼ 1; 2; . . . ; ¥Þ rAL4 Figure 4.3 compares the dynamic response obtained by using the present SEM with the exact analytical solution obtained by using the modal analysis method (MAM). For a numerical comparison, L ¼ 0.5 m, rA ¼ 0.5 kg/m, EI ¼ 20 N m, t0 ¼ 0.003 s and w0 ¼ 5 mm are used. For the analytical solution by MAM, 100 modes are superposed to obtain a sufficiently converged dynamic response. For the SEM solution, the sampling time window T ¼ 0.1024 s and the number of samples N ¼ 212 are used. Figure 4.3 shows that the dynamic response obtained by using the present SEM certainly satisfies all initial conditions and is almost identical with that obtained by using MAM.

4.6 Dynamic Responses of Nonlinear Systems This section considers the spectral analysis method for the computation of the dynamic responses of the nonlinear systems subjected to arbitrary, non-null initial conditions.


105

Figure 4.3 Comparison of the dynamic responses obtained by the present SEM and the modal analysis method (MAM)

4.6.1 Discrete Systems with Arbitrary Initial Conditions Assume that a viscously damped, nonlinear m-DOFs system is represented by _ þ KuðtÞ þ N ðuðtÞ; uðtÞÞ _ M€ u ðtÞ þ CuðtÞ ¼ fðtÞ

ð4:163Þ

with the initial conditions uð0Þ ¼ u0

and

_ uð0Þ ¼ u_ 0

ð4:164Þ

_ is the nonlinear function of the unknown variable u(t). The pseudo-force where N ðu; uÞ concept introduced in Section 4.5.2 can be applied to the present discrete system in an analogous way. Comparing Equations 4.163 and 4.164 with Equations 4.122 and 4.123, we can find the relationships between the operators and variables used for the discrete and continuous systems as follows: ðContinuous systemÞ L M uðx; tÞ pðx; tÞ u0 ðxÞ u_ 0 ðxÞ

( ) ( ) ( ) ( ) ( ) ( )

ðDiscrete systemÞ Cdð Þ=dt þ K M uðtÞ fðtÞ u0 u_ 0

ð4:165Þ

We assume the solution of Equation 4.163 is uðtÞ ¼ u0 þ DuðtÞ

ð4:166Þ

_ can be expanded in a power series in Du By using Equation 4.166, the nonlinear term Nðu; uÞ and Du_ as follows: _ _ N ðuðtÞ; uðtÞÞ ¼ N ðu0 þ Du; DuÞ _ þ fNL ðDu; Du; _ u0 ÞuðtÞ ¼ KNL ðu0 ÞDuðtÞ þ CNL ðu0 ÞDuðtÞ

ð4:167Þ


106

_ u0 Þ contains all constant terms and nonlinear terms in either Du(t) or DuðtÞ. _ where fNL ðDu; Du; _ by Equation 4.167, and Substituting Equation 4.166 into Equation 4.163, replacing N ðu; uÞ then using the pseudo-force concept explained by Equations 4.143 and 4.144 in an analogous way, we can transform the original governing equation (4.163) and the initial conditions (4.164) into the perturbed governing equation uðtÞ fNL ðDu; Du; _ þ KDuðtÞ _ u0 ÞuðtÞ MD€ u ðtÞ þ CD ¼ fðtÞ

ð4:168Þ

and the null initial conditions Duð0Þ ¼ 0;

_ Duð0Þ ¼0

ð4:169Þ

where C ¼ C þ CNL ðu0 Þ K ¼ K þ KNL ðu0 Þ

ð4:170Þ

¼ fðtÞ Ku0 uðtÞ þ Mu_ 0 dðtÞ fðtÞ The solution Du(t) of the nonlinear perturbed governing equation (4.168) can be obtained by using an iteration method. Let us assume that the solution of the k-th iteration (previous k _ u0 Þ in iteration) is known. Then we can compute the nonlinear pseudo-force term fNL ðDu; Du; kþ1 advance so that we can obtain the new solution Du (t) from u_ k þ 1 þ KDu k þ 1 ¼ fðtÞ f k ðDuk ; Du_ k ; u0 ÞuðtÞ MDu€k þ 1 þ CD NL

ð4:171Þ

For the spectral analysis, we represent Du(t), fðtÞ and fNL(t) in the spectral forms as DuðtÞ ¼

1 1 NX DUn eivn t N n¼0

N 1 X ¼ 1 Fn eivn t fðtÞ N n¼0

fNL ðtÞ ¼

ð4:172Þ

1 1 NX FNLn eivn t N n¼0

By using Equation 4.172, we can transform Equation 4.171 into k Dðvn ÞDUkn þ 1 ¼ Fn FNLn

ð4:173Þ

Dðvn Þ ¼ K þ ivn C v2n M

ð4:174Þ

where

From Equation 4.173, we obtain k Þ DUkn þ 1 ¼ Dðvn Þ 1 ðFn FNLn

ð4:175Þ


107

The new time-domain solution Duk þ 1 can be computed from DUkn þ 1 (n ¼ 0, 1, . . ., N 1) by using the inverse-FFT algorithm as follows: D uk þ 1 ðtÞ

(

IFFTfDUkn þ 1 j n ¼ 0; 1; 2; . . . ; N 1g

ð4:176Þ

The iteration process is repeated until the solution converges to a required accuracy. The complete solution u(t) of the original governing equation is finally obtained by simply adding Du(t) computed from Equation 4.176 to the initial displacements u0, as shown by Equation 4.166.

4.6.2 Continuous Systems with Arbitrary Initial Conditions Consider a nonlinear continuous system represented by the governing equation _ tÞÞ ¼ pðx; tÞ L uðx; tÞ þ M€ u ðx; tÞ þ N ðuðx; tÞ; uðx;

ð4:177Þ

and the initial conditions uðx; 0Þ ¼ u0 ðxÞ;

_ 0Þ ¼ u_ 0 ðxÞ uðx;

ð4:178Þ

where L and M are the linear differential structural and inertial operators, respectively, and _ is the nonlinear function of u(x, t) and uðx; _ N ðu; uÞ tÞ. Following the pseudo-force approach introduced in Section 4.5.2, we assume the complete solution to be uðx; tÞ ¼ u0 ðxÞ þ Duðx; tÞ

ð4:179Þ

_ can be expanded in the form By using Equation 4.179, the nonlinear term N ðu; uÞ _ _ tÞÞ ¼ N ðu0 þ Du; DuÞ N ðuðx; tÞ; uðx; _ u0 ÞuðtÞ ¼ LNL Duðx; tÞ þ pNL ðDu; Du;

ð4:180Þ

_ u0 Þ contains all constant terms and nonlinear where LNL is the linear operator and pNL ðDu; Du; _ terms in either Du(x, t) or Duðx; tÞ. Substituting Equation 4.179 into Equation 4.177, replacing _ by Equation 4.180, and then using the pseudo-forces given by the nonlinear term N ðu; uÞ Equations 4.143 and 4.144, we can obtain a perturbed governing equation _ u0 ÞuðtÞ LDuðx; tÞ þ MD€ u ðx; tÞ ¼ pðx; tÞ pNL ðDu; Du;

ð4:181Þ

subjected to the null initial conditions Duðx; 0Þ ¼ 0;

_ 0Þ ¼ 0 Duðx;

ð4:182Þ

where L ¼ L þ LNL pðx; tÞ ¼ pðx; tÞ L u0 ðxÞuðtÞ þ Mu_ 0 ðxÞdðtÞ

ð4:183Þ

Similar to the nonlinear discrete systems, the iteration method can be used to solve the nonlinear perturbed governing Equation 4.181 for Du(x, t). Assuming that the nonlinear


108

_ u0 Þ can be computed in advance by using the k-th iteration pseudo-force term pNL ðDu; Du; (previous iteration) solution, we can obtain the new solution Duk þ 1(x, t) from LDuk þ 1 ðx; tÞ þ MD€ u k þ 1 ðx; tÞ ¼ pðx; tÞ pkNL ðDuk ; Du_ k ; u0 ÞuðtÞ

ð4:184Þ

_ u0 Þ in the spectral forms as Represent Duðx; tÞ; pðx; tÞ and pNL ðDu; Du; Duðx; tÞ ¼ pðx; tÞ ¼ _ u0 Þ ¼ pNL ðDu; Du;

1 1 NX DUn ðxÞeivn t N n¼0 1 1 NX n ðxÞeivn t P N n¼0

ð4:185Þ

1 1 NX PNLn ðxÞeivn t N n¼0

Following the procedure of spectral element formulation, we transform Equation 4.184 into the frequency domain by using Equation 4.185 and then relate DUn(x;vn) to the spectral nodal DOFs vector Ddn by DUn ðx; vn Þ ¼ Nðx; vn ÞDdn

ð4:186Þ

where N(x; vn) is the dynamic shape function matrix. From the weak form of the discrete Fourier transformed perturbed governing equation, we can derive the spectral element equation in the form of Sðvn ÞDdn ¼ Fn FNLn

ð4:187Þ

where S(vn) is the spectral element matrix for the perturbed governing equation and ðL Fn ¼ NT ðx; vn ÞPn ðxÞdx 0 ð4:188Þ ðL FNLn ¼

NT ðx; vn ÞPNLn ðxÞdx

0

The nonlinear equation (4.187) can be iteratively solved for Ddn (n ¼ 0, 1, . . . N 1) as follows: k Sðvn ÞDdnk þ 1 ¼ Fn FNLn

ð4:189Þ

From Equation 4.189, we can obtain k Þ Ddnk þ 1 ¼ Sðvn Þ 1 ðFn FNLn

ð4:190Þ

By using Ddnk þ 1 (n ¼ 0, 1, . . ., N 1) computed from Equation 4.190, we compute the spectral components DUkn þ 1 ðxÞ (n ¼ 0, 1, . . ., N 1) from Equation 4.186. Lastly we compute the new time-domain solution Duk þ 1(x, t) from DUkn þ 1 ðxÞ (n ¼ 0, 1, . . ., N 1) by using the inverseFFT algorithm as Duk þ 1 ðx; tÞ

( IFFTfDUkn þ 1 ðxÞj n ¼ 0; 1; 2; . . . ; N 1g

ð4:191Þ


109

The above iteration process will be repeated until the solution Du(x, t) converges to the required accuracy. As explained by Equation 4.179, the complete solution u(x, t) of the original governing equation is finally obtained by adding Du(x, t) computed from Equation 4.191 to the initial displacements u0(x).

Figure 4.4 Direct iteration based on the AFT method

Figure 4.4 shows a direct iteration method where the alternating frequency-time (AFT) method [25] is used. In this approach, we evaluate the nonlinear pseudo-force term pNL in the time domain by using the time history Du(x, t) available at the previous iteration and then switch to the frequency domain to compute its spectral components PNL by using the FFT algorithm for the use in the next iteration.

Figure 4.5 Phase plots for x1(t) and x2(t)


110

Example 4.4 Consider a nonlinear 2-DOFs system represented by the equations of motion m€ x þ cx_ þ kx þ k3 x3 ¼ fðtÞ and the initial conditions 0:2gT ;

x0 ¼ f0:5

x_ 0 ¼ f0

0gT

where xðtÞ ¼ m¼

1 0

0 ; 1

c¼

x1 ðtÞ ; x2 ðtÞ

0:4 0

0 ; 0:2

fðtÞ ¼ k¼

0:5 cos 2t 0

1 1

1 ; 4

k3 ¼

0:5 0

0 0:8

By assuming x(t) ¼ x0 þ Dx(t), we can write the above nonlinear equations of motion in the form fNL ðtÞ mD€ x þ cDx_ þ ðk þ kNL ÞDx ¼ fðtÞ subjected to the null initial conditions Dx0 ¼ f0

0gT ;

Dx_ 0 ¼ f0

0gT

where kNL ¼ 3k3 ½diagðx20 Þ ¼ fðtÞ kx0 uðtÞ þ mx_ 0 dðtÞ fðtÞ fNL ¼ k3 ðDx3 þ 3½diagðx0 ÞDx2 þ x30 ÞuðtÞ where [diag(n)] denotes the diagonal matrix whose diagonal components are identical with the components of the vector n. Figure 4.5 compares the phase plots for x1(t) and x2(t) computed by using the present spectral analysis method (SAM) and the Runge–Kutta (RK) method. The figure shows that the present SAM provides almost identical dynamic responses with those obtained by the Runge–Kutta method.

Part Three Applications of Spectral Element Method


5 Dynamics of Beams and Plates In this chapter, the spectral element method (SEM) is applied to the beams and plates subjected to distributed dynamic forces. In SEM, the distributed dynamic forces can be represented by the equivalent spectral nodal forces, which are defined by the integration of the multiplication of the discrete Fourier transformed distributed dynamic forces and the dynamic shape function matrix. As an alternative approach, the two-element method (TEM) can be applied to deal with the distributed dynamic forces acting on the beams or plates. For the TEM-based spectral element analysis, the distributed dynamic force acting on a beam is first divided into a sequence of point forces. The complete dynamic response is then obtained by superposing all dynamic responses excited by each point force. To compute the dynamic response due to a point force, the beam is represented by a two-element model so that the interconnection node becomes the point at which the point force is located. Unlike the 1-D beam structures, the plates show very complicated twodimensional wave characteristics. Thus, developing the spectral element models for the plates with general boundary conditions is not so straightforward. Therefore, in this chapter, the spectral element modeling is introduced only for the Levy-type plates. The TEM developed for the beams is then extended to the Levy-type plates subjected to the distributed dynamic forces.

5.1 Beams 5.1.1 Spectral Element Equation As the theory discussed in this section can be readily applied to any 1-D beam-type structure, we will consider a Bernoulli–Euler beam (B-beam) subjected to the distributed dynamic force as an illustrative example of the problem. The spectral element equation for a B-beam element is given by Equation 3.156 Sðvn Þdn ¼ fn

ðn ¼ 0; 1; 2; . . . ; N 1Þ

ð5:1Þ

where S(vn) is the spectral element matrix, dn is the spectral nodal degrees-of-freedom (DOFs) vector, and fn is the equivalent spectral nodal forces vector defined by ðL ð5:2Þ fn ¼ NT ðx; vn ÞFn ðxÞdx 0



114

where Fn(x; vn) is the spectral components of the distributed dynamic force f(x, t). Once the distributed dynamic force f(x, t) is specified, the corresponding equivalent spectral nodal forces fn can readily be computed from Equation 5.2 to complete the spectral element equation. There are two approaches to dealing with distributed dynamic forces. One approach is to represent the distributed dynamic forces by the equivalent spectral nodal forces computed from Equation 5.2. The other approach is to use the so-called two-element method that has been developed based on the linear superposition principle. In the next section, the two-element method developed for the beams is applied to the Levy-type plates.

5.1.2 Two-Element Method As shown in Figure 5.1, a distributed dynamic force f(x, t) acting on a beam can be represented by a sequence of equivalent dynamic point forces (simply, point forces). By using the linear superposition principle, the total dynamic response of the beam can be obtained by summing all dynamic responses excited by each of the point forces.

Figure 5.1

Linear superposition representation of a distributed dynamic force

Consider a beam subjected to a point force at x ¼ j. The point force is given by Df ðj; tÞ ¼ f ðj; tÞDj

ð5:3Þ

where Dj ¼ (b a)/Nd, where Nd is the total number of increments. The dynamic response excited by Df(j, t) during the time 0 t t can be obtained by using the convolution integral (or Duhamels integral) as ðt Dwðx; j; tÞ ¼ hðx; j; t tÞDf ðj; tÞdt ð5:4Þ 0

where h(x, j, t) is the unit impulse response function. Equation 5.4 can be discrete Fourier transformed as DWn ðx; j; vn Þ ¼ Hn ðx; j; vn ÞDFn ðj; vn Þ

ðn ¼ 0; 1; 2; . . . ; N 1Þ

ð5:5Þ

where DFn ðj; vn Þ ¼ Fn ðj; vn ÞDj

ð5:6Þ

Hn(x, j; vn) are the spectral components of h(x, j, t) (frequency response functions) and Fn(j; vn) are the spectral components of f(j, t). Equation 5.5 represents the spectral components of the incremental dynamic response Dw(x, j, t) excited by a point force Df(j, t) at x ¼ j.

Dynamics of Beams and Plates

115

Thus, by using the linear superposition principle, the spectral components of the total dynamic response w(x, t) excited by the whole dynamic force f(x, t) distributed over the region a x b, as shown in Figure 5.1, can be obtained from ð ðb ð5:7Þ Wn ðx; vn Þ ¼ DWn ðx; j; vn Þ ¼ Hn ðx; j; vn ÞFn ðj; vn Þdj a

The integration of Equation 5.7 can be readily accomplished by using a proper numerical integration method. When we use the Trapezoidal algorithm, for instance, Equation 5.7 can be written in a numerical form as 1 Wn ðx; vn Þ ¼ Dj½DWn ðx; a; vn Þ þ 2DWn ðx; a þ Dj; vn Þ 2 þ 2DWn ðx; a þ 2D; vn Þ þ þ DWn ðx; b; vn Þ

ð5:8Þ

When we obtain spectral components Wn(x; vn) from either Equation 5.7 or Equation 5.8, the time-domain dynamic response w(x, t) can be obtained by using the inverse-fast Fourier transform (FFT) algorithm as follows: wðx; tÞ ( IFFTfWn ðx; vn Þj n ¼ 0; 1; 2; . . . ; N 1g

ð5:9Þ

Equation 5.7 shows that we need Hn(x, j; vn) to compute spectral components Wn(x; vn) from either Equation 5.7 or Equation 5.8. Thus, we introduce the two-element method in the following for the computation of Hn(x,j; vn). To this end, we consider once again the same beam subjected to a point force DFn(j; vn) at x ¼ j. As there exists a point force at x ¼ j, we divide the beam into two segments as shown in Figure 5.1b to represent it by two spectral elements. Two spectral elements are connected at x ¼ j (node 2) at which the point force DFn(j; vn) is located. Assembling two spectral elements and then imposing associated boundary conditions, we can obtain a matrix equation as Sg ðj; vn ÞDdgn ðjÞ ¼ Dfgn ðjÞ where Ddgn

9 8 > = < Dd1n > ¼ Dd2n ; > > ; : Dd3n

J ¼ f0

Dfgn 1

9 8 > = < Df1n > ¼ Df2n ¼ DFn ðj; vn ÞJ > > ; : Df3n

ð5:10Þ

ð5:11Þ

0gT

and where Ddin (i ¼ 1, 2, 3) are the incremental spectral nodal DOFs at node i (remaining after imposing all boundary conditions) and Dfin (i ¼ 1, 2, 3) are the incremental spectral nodal forces corresponding to Ddin. J is the vector for which all components are zero, except for just one component that corresponds to the nodal DOF with which the equivalent point force is associated. It is worth mentioning that Sg and Dfgn are the functions of j (the location of point force) and thus Ddgn also. From Equation 5.10, we can obtain the global incremental nodal DOFs as Ddgn ðjÞ ¼ Sg ðj; vn Þ1 Dfgn ðjÞ ¼ Sg ðj; vn Þ1 JDFn ðj; vn Þ

ð5:12Þ


116

In the frequency domain, the dynamic response measured at x is given by Equation 3.55 as Wn ðx; vn Þ ¼ Nðx; vn Þdn

ð5:13Þ

where N(x; vn) is the dynamic shape function matrix and dn is the local spectral nodal DOFs vector for the spectral element in which the measurement point x is located. The local spectral nodal DOFs vector dn (and Ddn) can be related to the global nodal DOFs vector dgn (and Ddgn) by introducing a locator matrix Lx/j as dn ¼ Lx=j dgn and Ddn ¼ Lx=j Ddgn where, if x j

dn ¼

and, if x j

dn ¼

d1n ; d2n d2n ; d3n

Lx=j

I 0 ¼ 0 I

Lx=j

0 I ¼ 0 0

0 0

0 I

ð5:14Þ ð5:15Þ

ð5:16Þ

Equations 5.15 and 5.16 show that the locator matrix Lx/j depends on the location of measurement point x relative to the location of point force j. Substituting Equation 5.12 into the second part of Equation 5.14 gives Ddn ¼ Lx=j Sg ðj; vn Þ1 JDFn ðj; vn Þ

ð5:17Þ

From Equations 5.13 and 5.17, we can obtain the incremental frequency response as DWn ðx; vn Þ ¼ Nðx; vn ÞDdn ¼ Nðx; vn ÞLx=j Sg ðj; vn Þ1 JDFn ðj; vn Þ

ð5:18Þ

Comparing Equation 5.18 with Equation 5.5, we can obtain Hn ðx; j; vn Þ ¼ Nðx; vn ÞLx=j Sg ðj; vn Þ1 J

ð5:19Þ

Equation 5.19 provides the frequency response functions that are required to compute spectral components Wn(x; vn) from Equation 5.7. For Equation 5.19, the B-beam subjected to an equivalent dynamic point force is represented by two spectral elements: from here onwards, this approach is called the two-element method (TEM).

Example 5.1 As an illustrative example, consider a simply-supported B-beam subjected to a distributed dynamic force. The beam has the length L ¼ 2 m, EI ¼ 10 k Nm2 and rA ¼ 10 kg/m. The dynamic force of magnitude F0 ¼ 1 kN/m is uniformly distributed over the region L/4 x 3L/4 and it is applied from t ¼ 0 to t ¼ t0 ¼ 0.1 s. The vibration response at the middle of the beam is obtained by using four different methods: a. TEM-based SEM: The analysis results obtained by representing the distributed dynamic force by N equivalent dynamic point forces are denoted by SEM-TEM(N).


117

b. One-element SEM model based on Equation 5.2: In this method, the whole beam is represented by one spectral element and the distributed dynamic force is represented by the spectral nodal forces computed from Equation 5.2. The analysis results are denoted by SEM(1)-Eq. (5.2). c. Three-element SEM model based on Equation 5.2: In this method, the beam is represented by three spectral elements: the first element (without loading) in 0 x 1/4L, the second (middle) element (with uniform loading) in 1/4L x 3/4L, and the third element (without loading) in 3/4L x L. The uniformly distributed dynamic force on the middle element is represented by the spectral nodal forces computed from Equation 5.2. The analysis results are denoted by SEM(3)-Eq. (5.2). d. FEM: In the finite element method (FEM) the analysis results obtained by using N conventional finite elements are denoted by FEM(N). The finite element analysis is conducted in the frequency domain by transforming the finite element equation given in Appendix 5.A into the frequency domain as follows: ðK v2 MÞd ¼ f where d and f represent the spectral nodal DOFs and forces, respectively. e. Modal analysis method: The results are denoted by MAM and they are considered as the reference (exact) solution to evaluate the accuracy of other solution methods. The dynamic response analytically obtained by MAM is given by wðx; tÞ ¼

1 X n¼1

fn ðxÞqn ðtÞ

where pffiffiffi npx 2 sin L sffiffiffiffiffiffi np2 EI vn ¼ L rA

fn ðxÞ ¼

qn ðtÞ ¼

pffiffiffi 2 2 L 4 F0 5

ðnpÞ EI

sin

np np sin fð1 cos vn tÞuðtÞ ½1 cos vn ðt t0 Þuðt t0 Þg 2 4

The function u(t) is the unit step function. A sufficient number of modes are superposed to obtain fully converged, accurate dynamic responses by MAM. Figure 5.2a compares the frequency-domain dynamic response obtained by using the TEMbased SEM with the exact reference solution obtained by MAM. Figure 5.2b shows that, as the number of equivalent point forces used to represent the distributed dynamic force is increased, the time-averaged dynamic response obtained by the TEM-based SEM indeed converges to the exact reference solution. Figure 5.3 compares the frequency-domain dynamic responses obtained by the oneand three-element SEM models subjected to the spectral nodal forces computed from Equation 5.2 with the exact reference solution obtained by MAM. It is shown that the

118


Figure 5.2 Comparison of the dynamic responses obtained by the TEM(N)-based SEM, where N is the number of equivalent point forces used in the analysis

three-element SEM model provides the better frequency-domain dynamic response when compared with the one-element SEM model. This implies that, when Equation 5.2 is used for the distributed forces acting on a structure member, we may need to mesh the structure member into more finite elements to improve the analysis results if there are discontinuities in the distributed forces. Figure 5.4a compares the dynamic responses obtained by FEM with the exact solution obtained by MAM or three-element SEM model. The FEM results are obtained by varying the number of finite elements used for the finite element analysis. It is obvious from Figure 5.4b that the accuracy of the time-averaged dynamic response obtained by FEM improves as the number of finite elements is increased. However, the accuracy improvement of FEM seems to be restricted to a certain limit due to certain errors, such as the computational round-off errors and numerical errors, which are inevitable for most large-scale finite element models.

Figure 5.3 Comparison of the dynamic responses obtained by SEM(N), where N is the number of spectral elements used in the analysis


119

Figure 5.4 Comparison of the dynamic responses obtained by FEM(N), where N is the number of finite elements used in the analysis

5.2 Levy-Type Plates Unlike the 1-D structures, two-dimensional (2-D) plate structures show much more complicated wave characteristics. Thus, the spectral element formulation for the plates with general boundary conditions will not be so straightforward and some approximations may be necessary. One of possible approaches is first to transform the 2-D problem (in x y domain) into an equivalent 1-D problem (in the x-domain) by determining the displacement functions of y in analytical forms so as to exactly, or approximately, satisfy the boundary conditions along two parallel sides at y ¼ 0 and y ¼ Ly, and next to apply one of spectral element formulation methods discussed in Chapter 3 to the equivalent 1-D problem, where Ly is the dimension of the rectangular plate in the y-direction. The rectangular plates with at least two parallel sides simply supported are called Levy-type plates and it is well known in plate theory that exact solutions are available for the class of Levy-type plates. Thus the exact spectral element model will be introduced in this section for the Levy-type plates [92].

5.2.1 Equation of Motion We will consider a thin rectangular Levy-type plate whose two opposite sides parallel to the x-axis are simply supported and the other two opposite sides parallel to the y-axis are subjected to arbitrary boundary conditions, as shown in Figure 5.5. The Levy-type plate has the dimensions

Figure 5.5

Levy-type rectangular plate


120

Lx and Ly in the x- and y-directions, respectively. The small transverse (out-of-plane) vibration w (x, y, t) of a thin plate is represented by the classical plate equation 4

@ w @4w @4w D þ 2 þ þ rhw €¼ f ðx; y; tÞ ð5:20Þ @x4 @x2 @y2 @y4 where f(x, y, t) is the distributed dynamic force acting in the direction normal to the plate, h is the plate thickness, r is the mass density, and D is the bending rigidity of plate defined by Eh3/[12 (1 n2)], where E is Youngs modulus and n is the Poisson ratio. The resultant bending moments and transverse shear forces are given by 2

@ w @2w Mxx ðx; y; tÞ ¼ D þn 2 @x2 @y 3

@ w @3w Qxz ðx; y; tÞ ¼ D þ ð2 nÞ @x3 @x@y2 2

2 @ w @ w Myy ðx; y; tÞ ¼ D þn 2 2 @y @x 2

@ w @3w Qyz ðx; y; tÞ ¼ D þ ð2 nÞ 2 @y3 @x @y

ð5:21Þ

The subscripts of the resultant bending moments and transverse shear forces indicate the directions of stresses that cause them. For the Levy-type plates, the simply supported boundary conditions at y ¼ 0 and y ¼ Ly are given by wðx; 0; tÞ ¼ 0;

Myy ðx; 0; tÞ ¼ 0

wðx; Ly ; tÞ ¼ 0;

Myy ðx; Ly ; tÞ ¼ 0

ð5:22Þ

5.2.2 Spectral Element Modeling Represent the dynamic response of the plate and dynamic force in the spectral forms as wðx; y; tÞ ¼

1 1 NX Wn ðx; y; vn Þeivn t N n¼0

1 1 NX f ðx; y; tÞ ¼ Fn ðx; y; vn Þeivn t N n¼0

ð5:23Þ

where Wn(x, y) and Fn(x, y) are the spectral components of w(x, y, t) and f(x, y, t), respectively, and N is the sampling number. Substitution of Equation 5.23 into Equation 5.20 gives r4 Wn W2n Wn ¼ Fn ðx; y; vn Þ where W2n ¼ vn

pffiffiffiffiffiffiffiffiffiffiffi rh=D

ð5:24Þ

ð5:25Þ


121

Similarly, the resultant bending moments and transverse shear forces are written in the spectral forms as 2

@ Wn @ 2 Wn Mxxn ðx; y; vn Þ ¼ D þn @x2 @y2 3

@ Wn @ 3 Wn Qxzn ðx; y; vn Þ ¼ D þ ð2 nÞ @x3 @x@y2 ð5:26Þ 2

2 @ Wn @ Wn Myyn ðx; y; vn Þ ¼ D þn @y2 @x2 2

@ Wn @ 3 Wn Qyzn ðx; y; vn Þ ¼ D þ ð2 nÞ 2 @y3 @x @y where Mxxn, Qxzn, Myyn, and Qyzn are the spectral components of Mxx(x, y, t), Qxz(x, y, t), Myy(x, y, t), and Qyz(x, y, t), respectively. The simply supported boundary conditions in Equation 5.22 at y ¼ 0 and y ¼ Ly are also written in the spectral forms as Wn ðx; 0; vn Þ ¼ 0;

Myyn ðx; 0; vn Þ ¼ 0

Wn ðx; Ly ; vn Þ ¼ 0; Myyn ðx; Ly ; vn Þ ¼ 0

ð5:27Þ

The homogeneous equation can be reduced from Equation 5.24 as r4 Wn W2n Wn ¼ 0

ð5:28Þ

Assume the general solution of Equation 5.28 in the form of Wn ðx; y; vn Þ ¼ Xn ðxÞYn ðyÞ

ð5:29Þ

By substituting Equation 5.29 into the boundary conditions of Equation 5.27, we can obtain the conditions for the displacement functions Yn(y) as Yn ð0Þ ¼ 0;

00

Yn ð0Þ ¼ 0

ð5:30Þ

00

Yn ðLy Þ ¼ 0; Yn ðLy Þ ¼ 0 Assume the functions Yn(y) in the form of Yn ðyÞ ¼ an1 eikyn y þ an2 ekyn y þ an3 eþikyn y þ an4 eþkyn y

ð5:31Þ

By applying the conditions (5.30) to Equation 5.31, we can obtain an1 ¼ an3 ; and kynm ¼

an2 ¼ 0;

an4 ¼ 0

ð5:32Þ

mp ðm ¼ 0; 1; 2; 3; . . .Þ Ly

ð5:33Þ

Thus, the functions Yn(y) that satisfy the simply supported boundary conditions at y ¼ 0 and y ¼ Ly exactly can be obtained as Yn ðy; kynm Þ ¼

1 X m¼0

anm3 ðeikynm y eikynm y Þ

1 X m¼0

bnm sinðkynm yÞ

ð5:34Þ


122

where anm3 and bnm are constants. Assume that Equation 5.34 can be expressed in the spectral form as Yn ðy; kynm Þ ¼

1 1 MX Ynm e ikynm y M m¼0

ð5:35Þ

where Ynm are the constant spectral components of Yn(y) in the wavenumber ky-domain and M is the sampling number. Substituting Equation 5.35 into Equation 5.29 gives Wn ðx; y; vn Þ ¼

1 1 MX Wnm ðx; kynm ; vn Þeikynm y M m¼0

ð5:36Þ

where Wnm ðx; kynm ; vn Þ ¼ Xn ðx; kynm ; vn ÞYnm

ð5:37Þ

Wnm(x; kynm, vn) are the spectral components of Wn(x, y; vn) in the wavenumber ky-domain. Substitution of Equation 5.36 into Equation 5.28 gives 0000

00

2 4 Wnm ðxÞ þ kynm W2n Wnm ðxÞ ¼ 0 Wnm ðxÞ 2kynm

ð5:38Þ

or, by using Equation 5.37, 0000

00

2 4 Xn ðxÞ þ kynm W2n Xn ðxÞ ¼ 0 Xn ðxÞ 2kynm

ð5:39Þ

Assume the solution of Equation 5.38 or Equation 5.39 to be Wnm ðxÞ ¼ cnm eikxnm x

ð5:40Þ

Xn ðxÞ ¼ cnm eikxnm x

ð5:41Þ

or

where cnm ¼ cnm Ynm . Substituting Equation 5.40 into Equation 5.38 or Equation 5.41 into Equation 5.39 yields the identical dispersion relation to be 4 2 2 4 kxnm þ 2kynm kxnm þ kynm W2n ¼ 0

Equation 5.42 can be solved for the wavenumber kx as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; 2 kxnm2 ¼ Wn kynm kxnm1 ¼ Wn kynm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; k 2 kxnm3 ¼ i Wn þ kynm xnm4 ¼ i Wn þ kynm

ð5:42Þ

ð5:43Þ

By using four wavenumbers of Equation 5.43, the general solution to Equation 5.38 can be written as Wnm ðx; kynm ; vn Þ ¼ cnm1 eikxnm1 x þ cnm2 eikxnm2 x þ cnm3 eikxnm3 x þ cnm4 eikxnm4 x

ð5:44Þ

or, simply Wnm ðx; kynm ; vn Þ ¼ Enm ðx; kynm ; vn Þcnm

ð5:45Þ


123

where Enm ðx; kynm ; vn Þ ¼ ½eikxnm1 x

eikxnm2 x

eikxnm3 x

eikxnm4 x

ð5:46Þ

and cnm ¼ fcnm1

cnm2

cnm3

cnm4 gT

ð5:47Þ

To formulate the spectral element equation for the Levy-type plate element as shown in Figure 5.5, the same procedure of spectral element formulation as used for the 1-D structures in Chapter 3 can be applied. Assume that the spectral nodal DOFs (in the frequency v-domain) are specified on the boundary sides at x ¼ 0 and x ¼ Lx as follows: 9 8 9 8 W1n ðy; vn Þ > > > Wn ð0; y; vn Þ > > > > = > = < < 0 Q1n ðy; vn Þ Wn ð0; y; vn Þ ¼ ð5:48Þ dn ðvn Þ W ðy; vn Þ > > > Wn ðLx ; y; vn Þ > > > > ; > ; : 2n : 0 Q2n ðy; vn Þ Wn ðLx ; y; vn Þ where the prime denotes the derivative with respect to x. Represent W1n, Q1n, W2n, and Q2n in the spectral forms in the wavenumber ky-domain as fW1n ; Q1n ; W2n ; Q2n g ¼

1 1 MX fW1nm ; Q1nm ; W2nm ; Q2nm geikynm y M m¼0

By substituting Equation 5.36 and 5.49 into Equation 5.48, we can get 9 8 9 8 W1nm ðkynm ; vn Þ > > > Wnm ð0; kynm ; vn Þ > > > > = > = < < 0 Q1nm ðkynm ; vn Þ Wnm ð0; kynm ; vn Þ ¼ dnm ðkynm ; vn Þ W ðk ; v Þ > > W ðLx ; kynm ; vn Þ > > > > ; > ; : 2nm ynm n > : nm 0 Q2nm ðkynm ; vn Þ Wnm ðLx ; kynm ; vn Þ

ð5:49Þ

ð5:50Þ

Applying Equation 5.45 into the right-hand side of Equation 5.50 gives dnm ðkynm ; vn Þ ¼ Fnm ðkynm ; vn Þcnm where

2

1 6 ikxnm1 Fnm ðkynm ; vn Þ ¼ 6 4 enm1 ikxnm1 enm1

1 ikxnm2 enm2 ikxnm2 enm2

1 ikxnm3 enm3 ikxnm3 enm3

ð5:51Þ 3 1 ikxnm4 7 7 5 enm4 ikxnm4 enm4

ð5:52Þ

with enmj ¼ eikxnmj Lx

ðj ¼ 1; 2; 3; 4Þ

ð5:53Þ

By substituting Equation 5.36 into the first two equations of Equation 5.26, we obtain 1 1 MX Mxxnm ðx; kynm ; vn Þeikynm y M m¼0 1 1 MX Qxzn ðx; y; vn Þ ¼ Qxznm ðx; kynm ; vn Þeikynm y M m¼0

Mxxn ðx; y; vn Þ ¼

ð5:54Þ


124

where Mxxnm ðx; kynm ; vn Þ ¼ DðW

00

nm

Qxznm ðx; kynm ; vn Þ ¼ D½W

000

2 nkynm Wnm Þ nm

2 ð2 nÞkynm W

0

nm

ð5:55Þ

Represent the spectral components (in the frequency v-domain) of the resultant transverse shear force and bending moment specified on the side at x ¼ 0 by Q1n and M1n and those on the side at x ¼ Lx by Q2n and M2n. Then, by using the sign convention for the resultant transverse shear force and bending moment, we can write 9 8 9 8 Q1n ðy; vn Þ > > Qxzn ð0; y; vn Þ > > > > > > > > > = < M ðy; v Þ = < þM ð0; y; v Þ > 1n n xxn n ¼ ð5:56Þ fn ðvn Þ > > Q2n ðy; vn Þ > Qxzn ðLx ; y; vn Þ > > > > > > > > > ; : ; : M2n ðy; vn Þ þMxxn ðLx ; y; vn Þ Represent Q1n, M1n, Q2n, and M2n in the spectral forms in the wavenumber ky-domain as fQ1n ; M1n ; Q2n ; M2n g ¼

1 1 MX fQ1nm ; M1nm ; Q2nm ; M2nm ge ikynm y M m¼0

By substituting Equations 5.54 and 5.57 into Equation 5.56, we can obtain 9 8 9 8 Q1nm ðkynm ; vn Þ > > Qxznm ð0; kynm ; vn Þ > > > > > > > > > > = < M ðk ; v Þ = < þM 1nm ynm n xxnm ð0; kynm ; vn Þ ¼ fnm ðkynm ; vn Þ > > Q2nm ðkynm ; vn Þ > > > > > Qxznm ðLx ; kynm ; vn Þ > > > > ; > ; : : M2nm ðkynm ; vn Þ þMxxnm ðLx ; kynm ; vn Þ

ð5:57Þ

ð5:58Þ

Substitution of Equation 5.45 into Equation 5.55 gives 00

2 Mxxnm ðx; kynm ; vn Þ ¼ D½Enm ðxÞ nkynm Enm ðxÞcnm 00

0

2 Qxznm ðx; kynm ; vn Þ ¼ D½Enm ðxÞ ð2 nÞkynm E nm ðxÞcnm

Applying Equation 5.59 into the right-hand side of Equation 5.58 gives 3 2 000 0 2 Enm ð0Þ E nm ð0Þ ð2 nÞkynm 7 6 00 2 7 6 Enm ð0Þ Enm ð0Þ nkynm 7 6 fnm ðkynm ; vn Þ ¼ D6 000 7cnm 0 7 6 E nm ðLx Þ ð2 nÞk2 Enm ðL Þ x 5 ynm 4 00 2 Enm ðLx Þ Enm ðLx Þ nkynm

ð5:59Þ

ð5:60Þ

By using Equation 5.46, Equation 5.60 can be written as fnm ðkynm ; vn Þ ¼ Gnm ðkynm ; vn Þcnm

ð5:61Þ

where Gnm ¼ D½gij

ði; j ¼ 1; 2; 3; 4Þ

ð5:62Þ


125

with 2 2 þ ð2 nÞkynm g1j ¼ ikxnmj ½kxnmj 2 2 g2j ¼ ðkxnmj þ nkynm Þ

g3j ¼ g1j enmj

ð j ¼ 1; 2; 3; 4Þ

ð5:63Þ

g4j ¼ g4j enmj The constant vector cnm, which is equal to cnm Ynm , can be eliminated from Equation 5.61 by using Equation 5.51 to obtain the spectral element equation for the Levy-type thin plate element as Sðkynm ; vn Þdnm ¼ fnm

ð5:64Þ

1 ðkynm ; vn Þ Sðkynm ; vn Þ ¼ Gnm ðkynm ; vn ÞFnm

ð5:65Þ

where

The matrix S(kynm, vn) is the 4-by-4 spectral element matrix that is the function of frequency vn and wavenumber kynm. The spectral element equations for the Levy-type thin plate elements connected in the x-direction can be assembled in an analogous way to that used for the 1-D structures to form a global system equation.

5.2.3 Equivalent 1-D Structure Representation Figure 5.6a shows a Levy-type plate subjected to a distributed dynamic force f(x, y, t). The distributed dynamic force can be represented by a series of equivalent line forces that are parallel to the y-axis. If the incremental dynamic responses excited by individual equivalent line forces are available, the complete dynamic response excited by the whole distributed

Figure 5.6 (a) A Levy-type plate subjected to a distributed dynamic force; (b) two plate-elements model with an equivalent line force, and (c) the equivalent 1-D structure representation


126

dynamic force can be obtained by summing all incremental dynamic responses through the use of the linear superposition principle. For the computation of the incremental dynamic response excited by an equivalent line force, as shown in Figure 5.6b, we divide the plate into two plate elements so that the interconnecting node line coincides with the line on which the equivalent line force is applied. The equivalent line force on x ¼ j has the magnitude of f(j, y, t)Dj. The equivalent line force can be discrete Fourier transformed into the frequency v-domain and wavenumber ky-domain to obtain DFn ¼ Fn ðj; y; vn ÞDj ( FFTv ff ðj; y; tÞDjg DFnm ¼ Fnm ðj; kynm ; vn ÞDj ( FFTky fFn ðj; y; vn ÞDjg

ð5:66Þ

This procedure removes the y-dependence of the equivalent line force to transform the 2-D problem into an equivalent beam-like 1-D problem, as illustrated in Figure 5.6c. We benefit from this approach in that the two-element method developed for the beams in the previous section can be equally applied to the Levy-type plate vibration.

5.2.4 Computation of Dynamic Responses Consider a plate subjected to a typical equivalent line force. The equivalent line force f(j,y,t)Dj is acting on the inter-connecting node line at x ¼ j as shown in Figure 5.6b and its spectral components DFnm are given by Equation 5.66. Assembling two spectral plate elements and imposing the boundary conditions at x ¼ 0 and Lx, we can obtain the global system equation in the form Sg Ddgnm ¼ Dfgnm

ð5:67Þ

Dfgnm ¼ DFnm J

ð5:68Þ

where

with DFnm ¼ Fnm ðj; kynm ; vn ÞDj J ¼ f0 0

1

0

ð5:69Þ

0 0gT

Once DFnm or Dfgnm are computed, Ddgnm can be computed from Equation 5.67 as Ddgnm ðj; kynm ; vn Þ ¼ Sg1 Dfgnm ¼ Sg1 JDFnm ðj; kynm ; vn Þ

ð5:70Þ

The local spectral nodal DOFs vector dnm (and Ddnm) can be related to the global nodal DOFs vector dgnm (and Ddgnm) by introducing a locator matrix Lx/j as dnm ðj; kynm ; vn Þ ¼ Lx=j dgnm ðj; kynm ; vn Þ Ddnm ðj; kynm ; vn Þ ¼ Lx=j Ddgnm ðj; kynm ; vn Þ where, if x j

dnm ¼

d1nm ; d2nm

Lx=j

I 0 ¼ 0 I

0 0

ð5:71Þ

ð5:72Þ


and, if x j

127

dnm ¼

d2nm ; d3nm

Lx=j ¼

0 0

I 0 0 I

ð5:73Þ

The locator matrix Lx/j depends on the location of measurement point (x) relative to the location of equivalent line force (j). Substituting Equation 5.70 into Equation 5.71b gives Ddnm ðj; kynm ; vn Þ ¼ Lx=j Sg1 JDFnm ðj; kynm ; vn Þ

ð5:74Þ

By using Equation 5.51, Equation 5.45 can be written as Wnm ðx; j; kynm ; vn Þ ¼ Nnm ðx; kynm ; vn Þdnm ðj; kynm ; vn Þ

ð5:75Þ

1 ðkynm ; vn Þ Nnm ðx; kynm ; vn Þ ¼ Enm ðx; kynm ; vn ÞFnm

ð5:76Þ

where

From Equation 5.75, we can get DWnm ðx; j; kynm ; vn Þ ¼ Nnm ðx; kynm ; vn ÞDdnm ðj; kynm ; vn Þ

ð5:77Þ

Substituting Equation 5.74 into Equation 5.77 and using Equation 5.66, we obtain DWnm ðx; j; kynm ; vn Þ ¼ Hnm ðx; kynm ; vn ÞFnm ðj; kynm ; vn ÞDj

ð5:78Þ

Hnm ðx; j; kynm ; vn Þ ¼ Nnm ðx; kynm ; vn ÞLx=j Sg1 J

ð5:79Þ

where

Equation 5.78 represents the spectral components of the incremental vibration response excited by an equivalent line force acting on x ¼ j. The spectral components of the complete vibration response excited by the whole distributed dynamic force can then be obtained by superposing all incremental responses as follows: ð Wnm ðx; kynm ; vn Þ ¼ DWnm ðx; j; kynm ; vn Þ ðb ð5:80Þ ¼ Hnm ðx; j; kynm ; vn ÞFnm ðj; kynm ; vn Þdj a

The integration of Equation 5.80 can be conducted by using a numerical integration method. When the trapezoidal algorithm is applied, for instance, Equation 5.80 can be written in the form of 1 Wnm ðx; kynm ; vn Þ ¼ Dj DWnm ðx; a; kynm ; vn Þ þ DWnm ðx; a þ Dj; kynm ; vn Þ 2

(5.81)

þ 2DWnm ðx; a þ 2Dj; kynm ; vn Þ þ þ DWnm ðx; b; kynm ; vn Þ Once the spectral components Wnm are computed from either Equation 5.80 or Equation 5.81, the time history of the complete dynamic response can be obtained by using the inverse-FFT


128

algorithm, first with respect to wavenumber kynm, and next with respect to frequency vn as follows: Wn ðx; y; vn Þ ( IFFTky Wnm ðx; kynm ; vn Þj m ¼ 0; 1; 2; . . . ; M 1g wðx; y; tÞ ( IFFTv fWn ðx; y; vn Þj n ¼ 0; 1; 2; . . . ; N 1g

ð5:82Þ

Example 5.2 Consider a simply supported square thin plate as shown in Figure 5.7. The square plate has the side L ¼ 2 m, the mass density rh ¼ 150 kg/m2, and the bending rigidity of plate D ¼ 150 kN. To take into account the structural damping, the complex bending rigidity of plate D* ¼ D(1 þ ih) is used for computation, where h ¼ 0.03 is the loss factor. Two types of dynamic forces are considered. Figure 5.7a shows the plate subjected to a point dynamic force of magnitude F0 ¼ 50 kN, which is applied at the middle of the plate. Figure 5.7b shows the plate subjected to a dynamic force uniformly distributed over a square area centered at the middle of the plate. The magnitude of the force is F0 ¼ 50 kN/m2 and the square area has the side l ¼ 1 m. Both the point and distributed dynamic forces are applied from t ¼ 0 to t0 ¼ 0.1 s.

Figure 5.7 A simply supported square plate subjected to (a) a point dynamic force and (b) a partially distributed dynamic force

The dynamic responses analytically obtained by using the modal analysis method (MAM) for both the point force and the distributed dynamic force are given by wðx; y; tÞ ¼

1 X 1 X m¼1 n¼1

fmn ðx; yÞqmn ðtÞ

where fmn

2 mpx npy sin ; ¼ pffiffiffiffiffi sin L L L rh

vmn

sffiffiffiffiffi D mp2 np2 ¼ þ rh L L

8 F0 2 mp np > > pffiffiffiffiffi sin sin > 2 > 2 2 vmn L rh > > > > < fð1 cos vmn tÞuðtÞ ½1 cos vmn ðt t0 Þuðt t0 Þg qmn ðtÞ ¼ > F 1 8L mp mp np np 0 > > pffiffiffiffiffi sin sin sin sin > 2 2 > 2 4 2 4 vmn rh mnp > > > : fð1 cos vmn tÞuðtÞ ½1 cos vmn ðt t0 Þuðt t0 Þg

ðpoint forceÞ

ðdistributed forceÞ


129

where u(t) is the unit step function. A sufficient number of modes are superposed to obtain sufficiently accurate dynamic responses by MAM. For the FEM results, the 4-node 12-DOF nonconforming rectangular plate bending element is used. The finite element stiffness and mass matrices are available in Ref. [139]. Figures 5.8 and 5.9 compare the frequency-domain dynamic responses obtained by SEM and FEM with the exact reference solution obtained by MAM. Figure 5.10a compares the timedomain undamped dynamic response with damped dynamic response obtained by the TEMbased SEM for the case of the point dynamic force, while Figure 5.10b shows the case of the distributed dynamic force. Finally, the solution accuracy and central processing unit (CPU) times are compared in Figure 5.11 for the TEM-based SEM as a dependence on the number of equivalent line forces and for the FEM as a dependence on the number of finite elements used in the analysis.

Figure 5.8 Comparison of the dynamic responses excited by a point dynamic force obtained by the TEM-based SEM and the FEM(N), where N is the number of finite elements used in the analysis

Figure 5.9 Comparison of the dynamic responses excited by a distributed dynamic force obtained by TEM(M)-based SEM and FEM(N), where M is the number of equivalent line forces and N is the number of finite elements used in the analyses


130

Figure 5.10 Comparison of the time-domain undamped and damped dynamic responses obtained by TEM-based SEM for the point and distributed dynamic forces

Figure 5.11

Comparison of the solution accuracy and CPU times of the TEM-based SEM and FEM

Appendix 5.A: Finite Element Model of Bernoulli–Euler Beam The finite element equation for the Bernoulli–Euler beam model is given by Md€þ Kd ¼ fðtÞ

ð5:A:1Þ

where dðtÞ ¼ fw1 ðtÞ

u1 ðtÞ

w2 ðtÞ

fðtÞ ¼ fQ1 ðtÞ M1 ðtÞ Q2 ðtÞ

u2 ðtÞgT M2 ðtÞgT

ð5:A:2Þ

The finite element mass matrix M and stiffness matrix K can be formulated by using the displacement fields wðx; tÞ ¼ NðxÞdðtÞ;

uðx; tÞ ¼ w0 ðx; tÞ ¼ N 0 ðxÞdðtÞ

ð5:A:3Þ


131

where NðxÞ ¼ ½1 3ðx=LÞ2 þ 2ðx=LÞ3

x 2Lðx=LÞ2 þ Lðx=LÞ3

3ðx=LÞ2 2ðx=LÞ3

Lðx=LÞ2 þ Lðx=LÞ3

ð5:A:4Þ

The finite element mass matrix M and stiffness matrix K are given by 2

156 rAL 6 22L 6 M¼ 420 4 54 13L 2

12 EI 6 6L K¼ 36 L 4 12 6L

22L 4L2 13L 3L2

6L 4L2 6L 2L2

54 13L 156 22L 12 6L 12 6L

3 13L 3L2 7 7 22L 5 4L2

3 6L 2L2 7 7 6L 5 4L2

ð5:A:5Þ

ð5:A:6Þ

6 Flow-Induced Vibrations of Pipelines The pipelines in chemical plants, oil pipelines, pump discharges, propellant fluid lines of liquid-filled rockets, and the human circulatory system are typical examples of the transfer of high velocity pressured fluids. The time-varying conditions imposed by pump or valve operations may change the internal fluid fields. As the internal fluid interacts with the pipe wall, the changes in the fluid fields will keep changing the dynamic behavior of a pipeline system and vice versa. Sometimes such a fluid–structure interaction phenomenon may result in serious vibrations, which can cause structural failures [8, 133]. Therefore, it is important to accurately predict the structural dynamic characteristics of a pipeline and the fluid transients inside the pipeline during the early design phase. To this end, we need to take into account more realistic fluid–structure interaction mechanism in the pipe-dynamics theory [102, 166]. In general, the structure–fluid interactions are represented by a set of coupled governing equations for the dynamics of the pipeline and for the fluid transients inside the pipeline. Thus, this chapter introduces the spectral element models [93, 95] formulated from the coupled pipedynamics equations of the uniform pipelines conveying internal fluids derived by using Hamilton’s principle and the theory of fluid mechanics.

6.1 Theory of Pipe Dynamics We will consider a straight uniform pipeline having a small amplitude vibration as shown in Figure 6.1. The pipeline is initially inclined at angle u with respect to the ground. In Figure 6.1, (X, Z) represent the reference coordinates system fixed on the ground and (x, z) represent the local coordinates system fixed on the inclined pipeline. The transverse and axial displacements of the pipeline with respect to the local coordinates system (x, z) are represented by w(x, t) and u(x, t), respectively. Figure 6.2 shows an infinitesimal pipe–fluid element that is separated into the pipeline element and the fluid element (or control volume). In Figure 6.2b, c(x, t) is the average velocity of the fluid flow and p(x, t) is the average fluid pressure. The flow-induced resultant normal and tangential forces acting on the pipe wall are represented by FN and FT,


134


Figure 6.1

Vibration of an inclined uniform pipeline

respectively. The Hamilton principle is used to derive the equations of motion of the pipeline, while Newton’s second law and the law of mass conservation are used to derive the fluiddynamic equations for the internal fluid.

6.1.1 Equations of Motion of the Pipeline Hamilton’s principle is given by ð t2 t1

ðdT dV þ dWÞdt ¼ 0

where T and V are the kinetic and strain energies of the pipeline element given by ðL 1 T ¼ mp ðu_ 2 þ w_ 2 Þdx 2 0 and ð ð 1 L 1 L 2 V¼ EAp ðeT0 þ ex Þ2 dx þ EIp w00 dx 2 0 2 0 where 1 2 1 2 N e T0 ¼ ; ex ¼ u0 þ u0 þ w0 2 2 EAp

Figure 6.2

Free-body diagrams for (a) pipeline element and (b) fluid element

ð6:1Þ

ð6:2Þ

ð6:3Þ

ð6:4Þ

135

Flow-Induced Vibrations of Pipelines

In the above equations, E, L, Ap, mp, and Ip are Young’s modulus, the length of pipeline element, the cross-sectional area of pipeline, the pipeline mass per unit length, and the second moment of the cross-sectional area of pipeline, respectively. The first integral of Equation 6.3 represents and the the stain energy due to the strains generated by a constant axial tensile force N longitudinal motion, and the second integral represents the strain energy due to the bending motion. By substituting Equation 6.4 into Equation 6.3, neglecting some high-order terms, and ignoring a constant term that will not eventually appear in Hamilton’s principle, we obtain the strain energy as ð ð 1 L 1 L 2 0 02 02 02 V¼ ½Nð2u þ u þ w Þ þ EAp u dx þ EIp w00 dx ð6:5Þ 2 0 2 0 The virtual work done by the flow-induced forces FN and FT and also by the externally applied forces and moments is given by ðL dW ¼ ½ðFT þ FN w0 Þdu þ ðFT w0 FN Þdwdx 0

þ M1 dw0 ð0; tÞ þ M2 dw0 ðL; tÞ þ Q1 dwð0; tÞ þ Q2 dwðL; tÞ

ð6:6Þ

þ N1 duð0; tÞ þ N2 duðL; tÞ where Mi, Qi, and Ni (i ¼ 1, 2) are the moments, transverse shear forces, and axial forces, respectively, externally applied at the boundaries of the pipeline element, as shown in Figure 6.3. Substituting Equation 6.2, Equations 6.5 and 6.6 into the Hamilton principle 6.1 and taking the integral by parts, we obtain ð t ð L 00 þ mp w € FT w0 þ FN Þdwdx ½ðEAeff u00 mp € u þ FN w0 þ FT Þdu ðEIp w0000 Nw 0

0

EIp w00 dw0 jL0 þ M1 dw0 ð0; tÞ þ M2 dw0 ðL; tÞ L 0 ð0; tÞdwð0; tÞ þ ½Q2 Nw 0 ðL; tÞdwðL; tÞ þ EIp w000 dw0 þ ½Q1 þ Nw L tÞ þ ðN2 NÞduðL; tÞ dt ¼ 0 EAeff u0 duj0 þ ðN1 þ NÞduð0;

ð6:7Þ

where EAeff ¼ EAp þ N

ð6:8Þ

From Equation 6.7, we obtain the equation of motion for the axial vibration of a pipeline EAeff u00 mp € u þ FN w0 þ FT ¼ 0

Figure 6.3

Boundary forces and moments acting on the pipeline element

ð6:9Þ

136


and the equation of motion for the transverse vibration 00 þ mp w € FT w0 þ FN ¼ 0 EIp w0000 Nw

ð6:10Þ

with the boundary conditions Mð0; tÞ¼ M1 ðtÞ 0 ð0; tÞ Qð0; tÞ¼ Q1 ðtÞ Nw Nð0; tÞ¼ N1 ðtÞ N MðL; tÞ¼ þ M2 ðtÞ

or or

w0 ð0; tÞ ¼ u1 ðtÞ wð0; tÞ ¼ w1 ðtÞ

or or

uð0; tÞ ¼ u1 ðtÞ w0 ðL; tÞ ¼ u2 ðtÞ

0 ðL; tÞ or QðL; tÞ¼ þ Q2 ðtÞ Nw NðL; tÞ¼ þ N2 ðtÞ N or

ð6:11Þ

wðL; tÞ ¼ w2 ðtÞ uðL; tÞ ¼ u2 ðtÞ

For Equation 6.11, the following force-displacement relationships are applied Mðx; tÞ ¼ EIp w00 Qðx; tÞ ¼ EIp w000

ð6:12Þ

Nðx; tÞ ¼ EAeff u0

6.1.2 Fluid-Dynamics Equations Now consider the fluid element shown in Figure 6.2b. Using Newton’s second law, we can derive two fluid momentum equations as ðpAÞ0 þ FN w0 þ FT þ mw ðgX þ € u þ c_ þ cu_ 0 þ cc0 Þ ¼ 0 € þ 2cw_ 0 þ c2 w00 þ c_ w0 þ cc0 w0 Þ ¼ 0 ðpAw0 Þ0 þ FT w0 FN þ mw ðgY þ w

ð6:13Þ ð6:14Þ

where gX ¼ gsinu, gY ¼ gcosu, A is the constant cross-sectional wetted area of the pipeline through which the internal fluid flows, and mw ¼ rwA where rw is the density of fluid. By using the Darcy–Weisbach equation, the flow-induced tangential force FT can be related to the velocity of the fluid flow by f jcjc mw fD jcjc F T ¼ mw ð6:15Þ 2D where fD ¼ f/(2D), where D is the internal diameter of the pipe and f(Re) is the Darcy friction factor given by the function of Reynold’s number Re ¼ rwDc/m, where m is the viscosity of the fluid [163]. We can combine the above two fluid momentum equations into one equation by eliminating FN, neglecting the higher order terms, and using Equation 6.15 for FT as p0 A þ mw gY w0 þ mw fD jcjc þ mw ðgX þ €u þ c_ þ cu_ 0 þ cc0 Þ ¼ 0

ð6:16Þ

In addition to the fluid momentum in Equation 6.16, the continuity equation can be derived from the law of mass conservation, by taking into account the effects of the fluid bulk modulus of elasticity and the time rate of change of the cross-sectional area of the control volume, as _ þ cp0 A þ a2 mw ½c0 2nðu_ 0 þ cu00 Þ ¼ 0 pA

ð6:17Þ

137


where n is the Poisson ratio and a is the wave speed of fluid defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ev a¼ rw ½1 þ ðEv DÞ=ðEhÞ

ð6:18Þ

In above, Ev is the fluid bulk modulus of elasticity, rw is the mass density of fluid, and h is the thickness of the pipeline. By neglecting convective terms, we can simplify Equation 6.17 as _ þ a2 mw ðc0 2nu_ 0 Þ ¼ 0 pA

ð6:19Þ

6.1.3 Governing Equations for Pipe Dynamics By using Equations 6.13 and 6.14, we can eliminate the flow-induced forces FN and FT from the equations of motion 6.9 and 6.10 as follows: u p0 A mw ð_c þ cu_ 0 þ cc0 Þ mw gX ¼ 0 EAeff u00 m€

ð6:20Þ

00 þ p0 Aw0 þ pAw00 EIp w0000 þ m€ w Nw þ mw ð2cw_ 0 þ c2 w00 þ c_ w0 þ cc0 w0 Þ þ mw gY ¼ 0

ð6:21Þ

where m ¼ mp þ mw

ð6:22Þ

By using Equation 6.16 and ignoring the static deformation due to the pipeline weight, we can write Equation 6.20 as EAeff u00 mp € u þ mw gY w0 þ mw fD jcjc ¼ 0

ð6:23Þ

Similarly, we apply Equations 6.16–6.21. By neglecting some higher order terms and ignoring the static deformations due to the pipeline and fluid weights, we obtain 00 þ pAw00 þ mw c2 w00 mw fD jcjcw0 ¼ 0 EIp w0000 þ m€ w þ 2mw cw_ 0 Nw

ð6:24Þ

Thus, the governing equations of pipe dynamics consist of two equations of motion of the pipeline, Equations 6.23 and 6.24, and two fluid-dynamic equations, Equations 6.16 and 6.19. For the linearization, we assume cðx; tÞ ¼ c0 þ cd ðx; tÞ ðcd < c0 Þ pðx; tÞ ¼ p0 þ pd ðx; tÞ ðpd < p0 Þ

ð6:25Þ

where c0 and p0 are constant velocity of the fluid flow and fluid pressure, respectively. By substituting Equation 6.25 into Equations 6.23 and 6.24 and neglecting nonlinear terms, we obtain the linearized equations of motion of the pipeline as EIp w0000 T0 w00 mw fD c20 w0 þ 2mw c0 w_ 0 þ m€ w¼0

ð6:26Þ

EAeff u00 mp € u þ mw gY w0 þ 2mw fD c0 cd þ mw fD c20 ¼ 0

ð6:27Þ

138


where p0 A mw c20 T0 ¼ N

ð6:28Þ

Similarly, by substituting Equation 6.25 into Equations 6.16 and 6.19 and neglecting nonlinear terms, we obtain the linearized fluid-dynamic equations as P0d þ mw gY w0 þ mw fD c20 þ 2mw fD c0 cd þ mw ðgX þ €u þ c_ d þ c0 c0d þ c0 u_ 0 Þ ¼ 0

ð6:29Þ

P_ d þ a2 mw ðc0d 2nu_ 0 Þ ¼ 0

ð6:30Þ

Pd ðx; tÞ ¼ pd ðx; tÞA

ð6:31Þ

where

From Equation 6.30, we obtain mw cd0 ¼ a 2 P_d þ 2mw nu_ 0 €d þ 2mw n€u 0 mw c_d0 ¼ a 2 P mw c 00 ¼ a 2 P_ 0 þ 2mw nu_ 00 d

ð6:32Þ

d

Applying Equation 6.32 to the derivative of Equation 6.29 with respect to x, we obtain a wave form of the equation in P(x, t) as follows: €d 2a 2 fD c0 P_d þ mw gY w00 Pd00 a 2 c0 P_d0 a 2 P þ mw ð1 þ 2nÞð€ u 0 þ c0 u_ 00 Þ þ 4mw fD c0 n u_ 0 ¼ 0

ð6:33Þ

In summary, the governing equations consist of two equations of motion of the pipeline, Equations 6.26 and 6.27, and two fluid-dynamic equations, Equations 6.33 and 6.30.

6.2 Pipelines Conveying Internal Steady Fluid 6.2.1 Governing Equations If the velocity of the fluid flow and the fluid pressure have very small fluctuations with respect to their mean values, their effects on the motion of the pipeline may be negligible. In this situation, the perturbed values Pd and cd are negligible and we can assume that the velocity of the fluid flow and the fluid pressure are constant. Accordingly, we can obtain the equations of motion of the pipeline conveying internal steady fluid from Equations 6.26 and 6.27 as [93] EIp w0000 T0 w00 mw fD c20 w0 þ 2mw c0 w_ 0 þ m€ w¼0

ð6:34Þ

EAeff u00 mp € u þ mw gY w0 þ mw fD c20 ¼ 0

ð6:35Þ

with the boundary conditions given by Equation 6.11. The force–displacement relationships are given by Equation 6.12.

139


6.2.2 Spectral Element Modeling 6.2.2.1 Governing Equations in the Frequency Domain Represent the dynamic responses in the spectral forms as wðx; tÞ ¼

N 1X Wn ðxÞeivn t ; N n¼1

uðx; tÞ ¼

N 1X Un ðxÞeivn t N n¼1

ð6:36Þ

We also represent all boundary forces and moments in the spectral forms by 1 1 NX fQn ðxÞ; Mn ðxÞ; Nn ðxÞgeivn t N n¼0 1 1 NX fQi ðtÞ; Mi ðtÞ; Ni ðtÞg ¼ fQin ðtÞ; Min ðtÞ; Nin ðtÞgeivn t N n¼0 1 1 NX fwi ðtÞ; ui ðtÞ; ui ðtÞg ¼ fWin ; Qin ; Uin geivn t N n¼0

fQðx; tÞ; Mðx; tÞ; Nðx; tÞg ¼

ð6:37Þ

(6.37)

where i ¼ 1, 2. By substituting Equation 6.36 into the governing Equations 6.34 and 6.35 and applying the DFT formula of the unit step function (see Table 2.2) to the constant term in Equation 6.35, we obtain EIp W 0000 T0 W 00 mw fD c20 W 0 þ 2ivmw c0 W 0 v2 mW ¼ 0 00

EAeff U þ v mp U þ mw gY W 2

0

þ mw fD c20 Nd0n

¼0

ð6:38Þ ð6:39Þ

where Kronecker delta d0n has unit value when n ¼ 0, otherwise it is zero: that is, a constant in the time domain has only one spectral component at zero frequency (i.e., v0 ¼ 0). Similarly, substitution of Equations 6.36 and 6.37 into the boundary conditions of Equation 6.11 gives 0 ð0Þ or Wð0Þ ¼ W1 Qð0Þ ¼ Q1 NW Mð0Þ ¼ M1 0n Nð0Þ ¼ N1 NNd 0 ðLÞ QðLÞ ¼ þ Q2 NW MðLÞ ¼ þ M2 0n NðLÞ ¼ þ N2 NNd

or or or or or

W 0 ð0Þ ¼ Q1 Uð0Þ ¼ U1 WðLÞ ¼ W2 W 0 ðLÞ ¼ Q2 UðLÞ ¼ U2

ð6:40Þ

and substitution of Equations 6.36 and 6.37 into the force-displacement relationships 6.12 gives QðxÞ¼ EIp W 000 ðxÞ;

MðxÞ¼ EIp W 00 ðxÞ;

NðxÞ¼ EAeff U 0 ðxÞ

ð6:41Þ

In the above Equations 6.38–6.41, the subscripts n are omitted for brevity. 6.2.2.2 Spectral Nodal DOFs, Forces and Moments The spectral nodal displacements, slopes, forces, and moments are specified in Figure 6.4. The spectral nodal degrees-of-freedom (DOFs) vector is defined by d ¼ fW1 Q1 U1 W2 Q2 U2 gT ¼ fWð0Þ W 0 ð0Þ Uð0Þ WðLÞ W 0 ðLÞ

UðLÞgT

ð6:42Þ

140


Figure 6.4

Sign convention for the pipeline element

and the spectral nodal forces and moments vector by fc ¼ fQ1

M1

N 2 gT

N1 Q2 ¼ f Qð0Þ NW 0 ð0Þ

M2

Mð0Þ

ðLÞ QðLÞ þ NW

MðLÞ

0

0n Nð0Þ NNd 0n gT NðLÞ þ NNd

ð6:43Þ

6.2.2.3 Dynamic Shape Functions To derive the frequency-dependent dynamic shape functions, consider the homogeneous equation reduced from Equations 6.38 and 6.39 as EIp W 0000 T0 W 00 mw fD c20 W 0 þ 2ivmw c0 W 0 v2 mW ¼ 0

ð6:44Þ

EAeff U 00 þ v2 mp U þ mw gY W 0 ¼ 0

ð6:45Þ

Assume the general solutions of the homogeneous equation to be WðxÞ ¼ bae ikx ;

UðxÞ ¼ ae ikx

ð6:46Þ

Substituting Equation 6.46 into Equations 6.44 and 6.45, we obtain an eigenvalue problem b 0 0 K11 ¼ ð6:47Þ 1 0 K21 K22 where K11 ¼ EIp k4 þ T0 k2 þ iðmw fD c20 2ivmw c0 Þk v2 m K21 ¼ imw gY k K22 ¼ EAeff k2 v2 mp

ð6:48Þ

Equation 6.47 provides the dispersion relation EIp k4 þ T0 k2 þ iðmw fD c20 2ivmw c0 Þk v2 m ¼ 0

ð6:49Þ

EAeff k2 v2 mp ¼ 0

ð6:50Þ

and

We can numerically compute four wavenumbers k1, k2, k3, and k4 from Equation 6.49 and two wavenumbers k5 and k6 from Equation 6.50 as rffiffiffiffiffiffiffiffiffiffiffi mp k5 ¼ k6 ¼ v ð6:51Þ EAeff

141


For each wavenumber kj (j ¼ 1, 2, . . . , 6), we can compute the corresponding bj from Equation 6.47 as 8 2 2 < EAeff kj v mp ðj ¼ 1; 2; 3; 4Þ bj ¼ ð6:52Þ ikj mw gY : 0 ðj ¼ 5; 6Þ Define a diagonal matrix as B ¼ diag½bj ðj ¼ 1; 2; . . . ; 6Þ

ð6:53Þ

By using six wavenumbers kj (j ¼ 1, 2, . . ., 6) and the diagonal matrix B, we can write the general solutions of Equations 6.44 and 6.45 as 6 X WðxÞ ¼ bj aj e ikj x ¼ eðxÞBa j¼1

ð6:54Þ

6 X UðxÞ ¼ aj e ikj x ¼ eðxÞa j¼1

where eðx; vÞ ¼ ½eik1 x

eik2 x

eik3 x

eik4 x

eik5 x

eik6 x

ð6:55Þ

and a ¼ fa1

a2

a3

a4

a5

a6 gT

ð6:56Þ

Substituting Equation 6.54 into the right-hand side of Equation 6.42, we obtain d ¼ HðvÞa

ð6:57Þ

where 2 6 6 6 6 HðvÞ ¼ 6 6 6 4

b1 ik1 b1 1 b1 e 1 ik1 b1 e1 e1

b2 ik2 b2 1 b2 e 2 ik2 b2 e2 e2

b3 ik3 b3 1 b3 e 3 ik3 b3 e3 e3

b4 ik4 b4 1 b4 e 4 ik4 b4 e4 e4

ej ¼ e ikj L

ðj ¼ 1; 2; . . . ; 6Þ

0 0 1 0 0 e5

3 0 07 7 17 7 7 07 7 05 e6

ð6:58Þ

with ð6:59Þ

By using Equation 6.57, Equation 6.54 can be written as WðxÞ ¼ Nw ðx; vÞd;

UðxÞ ¼ Nu ðx; vÞd

ð6:60Þ

where Nw and Nu are the dynamic shape functions defined by Nw ðx; vÞ ¼ eðxÞBH 1 ðvÞ Nu ðx; vÞ ¼ eðxÞH 1 ðvÞ

ð6:61Þ

142


6.2.2.4 Weak Form of Governing Equation The weak form of the non-homogeneous governing Equations 6.38 and 6.39 in the frequency domain can be obtained from ðL ½fEIp W 0000 T0 W 00 mw fD c20 W 0 þ 2ivmw c0 W 0 v2 mWgdW 0

fEAeff U 00 þ v2 mp U þ mw gY W 0 þ mw fD c20 Nd0n gdUdx ¼ 0

ð6:62Þ

By applying the integral by parts, we obtain the weak form as ðL ½EIp W 00 dW 00 þ T0 W 0 dW 0 mw fD c20 W 0 dW v2 mWdW 0

þ ivmw c0 ðW 0 dW WdW 0 Þ þ EAeff U 0 dU 0 v2 mp UdU (6.63) 0n dU 0 gdx mw gY W 0 dU mw fD c20 Nd0n dU þ NNd 0 dW jL EIp W 000 þ NW 0 ÞdW L þ ivmw c0 WdWjL0 ðT0 NÞW 0 0 L 00 0 L 0 EIp W dW 0 ðEAeff U þ NNd0n ÞdU j0 ¼ 0 ð6:63Þ where the following relationship is used. ðL 0n dU 0 dx NNd 0n dUjL0 ¼ 0 NNd

ð6:64Þ

0

By using Equation 6.41 for Q(x), M(x), and N(x), Equation 6.63 can be written as ðL ½EIp W 00 dW 00 þ T0 W 0 dW 0 mw fD c20 W 0 dW v2 mWdW 0

þ ivmw c0 ðW 0 dW WdW 0 Þ þ EAeff U 0 dU 0 v2 mp UdU 0n dU 0 gdx mw gY W 0 dU mw fD c20 Nd0n dU þ NNd

ð6:65Þ

0 dW jL0 þ ivmw c0 WdWjL0 ðT0 NÞW 0 dW jL0 MðxÞdW 0 jL0 ½NðxÞ þ NNd 0n dU jL0 ¼ 0 ½QðxÞ þ NW 6.2.2.5 Spectral Element Equation Substituting Equation 6.60 into Equation 6.65, we obtain ð L T dd ½EIp Nw00T Nw00 þ T0 Nw0T Nw0 mw fD c20 NwT Nw0 v2 mNwT Nw 0

þ ivmw c0 ðNwT Nw0 Nw0T Nw Þ þ EAeff Nu0T Nu0 v2 mp NuT Nu wT N0w L d mw gY NuT N0w dx þ ivmw c0 NwT Nw ðT0 NÞN 0

(6.66)

¼ fc þ fd

ð6:66Þ

where fd ¼ Nd0n

ðL 0

u0 T Þdx ðmw fD c20 NuT NN

ð6:67Þ

143


and fc is defined by Equation 6.43. As dd is arbitrary by definition, from Equation 6.66 we obtain the spectral element equation as SðvÞd ¼ fc þ fd where SðvÞ ¼

ðL 0

ð6:68Þ

½EIp Nw00 T Nw00 þ T0 Nw0T Nw0 mw fD c20 NwT Nw0 v2 mNwT Nw

þ ivmw c0 ðNwT Nw0 Nw0T Nw Þ þ EAeff Nu0 T Nu0 v2 mp NuT Nu

ð6:69Þ

wT Nw0 L0 mw gY NuT Nw0 dx þ ½ivmw c0 NwT Nw ðT0 NÞN From Equation 6.61, we can get Nw0 ðx; vÞ ¼ ieðxÞKBH 1 ðvÞ Nu0 ðx; vÞ ¼ ieðxÞKH 1 ðvÞ Nw00 ðx; vÞ¼ eðxÞK BH 2

1

ð6:70Þ

ðvÞ

where K ¼ diag½ki ;

K2 ¼ diag½ki2

ð6:71Þ

Substitution of Equations 6.61 and 6.70 into Equation 6.69 yields SðvÞ ¼ H T ðvÞDðvÞH 1 ðvÞ

ð6:72Þ

where DðvÞ¼ EIp BK2 EK2 B T0 BKEKB þ imw fD c20 BEKB v2 mBEB þ vmw c0 BðEK KEÞB EAeff KEK v2 mp E þ imw gY EKB þ iðT0 NÞB EKB þ ivmw c0 BEB and where EðvÞ ¼

ðL

eT ðx; vÞeðx; vÞdx ¼ ½Ers ðvÞ

0 T

L rs ðvÞ EðvÞ ¼ e ðx; vÞeðx; vÞj0 ¼ ½E

with Ers ¼

8
> 1 q q21 þ q2 1 > > ðfor bending wave modeÞ > > < np 2p2 p4 ð7:176Þ cDn ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 > 1 q1 þ q1 þ q2 > > ðfor shear wave modeÞ : np 2p2 p4 where q1 ¼ n2 p2 ðp1 p2 þ p4 þ p3 p4 Þ þ L2 p2 q2 ¼ 4n2 p2 p2 p4 ½n2 p2 ðp1 þ p1 p3 Þ þ L2 p3

ð7:177Þ

Dynamics of Axially Moving Structures

191

Example 7.3 Consider a uniform Timoshenko beam that is moving axially over two simple supports as an illustrative example. The geometric and structural properties of the Timoshenko beam are given by L ¼ 2 m, EI ¼ 10 kN m2, kGA ¼ 80 106 N, rA ¼ 10 kg/m, and rI ¼ 30 106 kg m. Table 7.3 displays the natural frequencies obtained by SEM and FEM for different moving speeds and axial tensions. For the stationary Timoshenko beam (when c ¼ 0 m/s), the natural frequencies can be also computed from a theoretical formula [24] pffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 fn2 Timo ¼ fn2 BE 2 2 a a2 b ðn ¼ 1; 2; 3; . . .Þ 2n p where fn|BE are the natural frequencies of a stationary Bernoulli–Euler beam sffiffiffiffiffiffi n2 p EI fn jBE ¼ 2 2L rA and a¼

rA L2 rA kGA þ 1þ 2 2 ; rI EI n p rI

b¼4

rA kGA rI EI

Table 7.3 Natural frequencies of an axially moving Timoshenko beam Natural frequency (Hz) Moving speed (m/s) Axial tension (kN) Method 0

0

0

20

33.092 (0.5cD1)

20

66.184 (cD1)

20

Theory [24] SEM (1) FEM(20) FEM(100) SEM (1) FEM(20) FEM(100) SEM (1) FEM(50) FEM(300) SEM (1) FEM(50) FEM(300)

f1

f2

f3

f5

f10

11.630 11.630 11.630 11.630 16.421 16.421 16.421 13.591 13.774 13.621 0.000 3.386 0.000

46.498 46.498 46.498 49.498 51.957 51.957 51.957 49.601 49.818 49.638 42.037 43.035 42.209

104.53 104.53 185.65 104.53 110.16 110.17 110.16 108.07 108.30 108.11 101.67 102.60 101.83

289.60 289.60 289.71 289.60 295.33 295.45 295.33 293.45 293.68 293.49 287.80 288.70 287.95

1144.3 1144.3 1151.4 1144.4 1150.1 1157.2 1150.2 1148.3 1149.1 1148.3 1142.9 1144.3 1143.1

Note: () ¼ total number of the equal-length finite elements used in the analysis.

Table 7.3 shows that the present spectral element model provides very accurate natural frequencies by representing the whole beam as a single spectral element. It also shows that the first natural frequency vanishes at divergence speed cD1 ¼ 66.184 m/s. The divergence speed cD1 predicted by using the Timoshenko beam theory is slightly lower than the value predicted


192

by using the Bernoulli–Euler beam theory, and vice versa for the natural frequencies: see Table 7.2. Figure 7.9 shows the instability boundaries as a function of the moving speed and axial tensile force. Figure 7.10 compares the dispersion curves at two different statuses: when the beam is moving axially and when the beam is not moving axially (stationary). Firstly, when the beam is stationary, k1 is pure realp atffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi all frequency, while k2 is pure imaginary at 0 f fc and pure real at f fc, where fc ¼ 2p kGA=rI is the cut-off frequency. Thus, k1 represents the forward traveling wave at all frequencies, while k2 represents the forward evanescent (nonpropagating) wave at f fc and the forward traveling wave at f fc. Secondly, when the beam is axially moving, k1 keeps its value at pure real at all frequencies so that it still represents the forward traveling wave. However, k2 changes its value from pure imaginary to pure real at 0 f fLc and from pure imaginary to complex at fLc f fc, although it keeps its value real at f fc, where fLc ( > > > > = < W 0 ð0; yÞ > n ð7:208Þ un ðyÞ ¼ > Wn ðL; yÞ > > > > > ; : 0 W n ðL; yÞ


200

where the prime (0 ) denotes the partial derivative with respect to x. Substitution of Equation 7.207 into Equation 7.208 gives 2 3 enm ð0Þ M 6 0 X 7 6 e nm ð0Þ 7Ym ðyÞanm un ðyÞ ¼ ð7:209Þ 4 enm ðLÞ 5 m¼1 0 e nm ðLÞ Define the generalized spectral nodal DOFs vector as ð 1 b un ðxÞYm ðyÞdy dnm I0m 0

ð7:210Þ

By substituting Equation 7.209 into Equation 7.210 and using the properties of Equations 7.192 and 7.195, we obtain dnm ¼ Hnm ðvÞanm where

3 2 enm ð0Þ 1 6 e0 nm ð0Þ 7 6 iknm1 7 6 Hnm ðvÞ ¼ 6 4 enm ðLÞ 5 ¼ 4 e1 e0 nm ðLÞ iknm1 e1 2

1 iknm2 e2 iknm2 e2

ð7:211Þ

1 iknm3 e3 iknm3 e3

3 1 iknm4 7 7 5 e4 iknm4 e4

ð7:212Þ

with ei ¼ eiknmi L

ði ¼ 1; 2; 3; 4Þ

ð7:213Þ

By using Equation 7.211, Equation 7.207 can be written as Wn ðx; yÞ ¼

M X

Nnm ðx; y; vn Þdnm ¼ Nn ðx; y; vn Þdn

ð7:214Þ

m¼1

where dn ¼ fdTn1

dTn2

dTn3 . . . dTnM g

T

Nn ðx; y; vn Þ ¼ ½Nn1 ðx; y; vn ÞNn2 ðx; y; vn Þ . . . NnM ðx; y; vn Þ

ð7:215Þ ð7:216Þ

with 1 Nnm ðx; y; vn Þ ¼ enm ðxÞHnm ðvn ÞYm ðyÞ

ðm ¼ 1; 2; . . . ; MÞ

ð7:217Þ

Nn(x,y;vn) is the 1-by-(4 M) dynamic shape function matrix and dn is the (4 M)-by-1 generalized spectral nodal DOFs vector. 7.4.2.3 Weak Form of Governing Equation The weak form of the governing Equation 7.186 in the frequency domain is obtained from ðb ðL @ 2 Wn @Wn Dr4 Wn ðNx rhc2 þrhgxÞ þð2ivn rhcrhgÞ 2 @x @x 0 0 rhv2n Wn Fn gdWn dxdy¼0

ð7:218Þ


201

Integrating Equation 7.218 by parts and using the force-displacement relationships (7.189), we can obtain the weak form as ðb ðL 2 @ Wn @ 2 dWn @ 2 Wn @ 2 dWn @ 2 Wn @ 2 dWn @ 2 Wn @ 2 dWn D þn 2 þn þ 2 2 2 @x @x @y @x @x2 @y2 @y2 @y2 0 0 @ 2 Wn @ 2 dWn @Wn @dWn @Wn @dWn Þ þðNx rhc2 Þ þrhgx @x@y @x@y @x @x @x @x @Wn @dWn dWn Wn rhv2n Wn dWn þivn rhc Fn dWn dxdy @x @x L ðb ðb @Wn @dWn L Vxn ðx;yÞþNx dy dWn dyþ Mxn ðx;yÞ @x @x 0 0 0 0 L ðb 2 @Wn þivn rhcWn dWn dy¼0 þ rhc @x 0 0

þ2ð1nÞ

ð7:219Þ

7.4.2.4 Spectral Element Equation By substituting Equation 7.214 into Equation 7.219 and applying the boundary conditions (7.187), we obtain ddTn

ð b ð L 2 T 2 2 T 2 @ Nn @ Nn @ 2 NTn @ 2 Nn @ Nn @ Nn @ 2 NTn @ 2 Nn D þ þn þ @x2 @x2 @y2 @y2 @x2 @y2 @y2 @x2 0 0 @ 2 NTn @ 2 Nn @NTn @Nn @NTn @Nn þ rhgx þ 2ð1 nÞ þ ðNx rhc2 Þ @x@y @x@y @x @x @x @x @Nn @NTn rhv2n NTn Nn þ ivn rhc NTn Nn dx dy @x @x L # ðb T @Nn 2 @Nn rhc þ ivn rhcNn dy dn ¼ ddTn fn þ @x 0 @x 0

ð7:220Þ

where fn ðvn Þ ¼ fcn ðvn Þ þ fdn ðvn Þ with fcn ðvn Þ ¼

ðb 0

fdn ðvn Þ ¼

NTn ð0;yÞVx1n ðyÞþ

ðb ðL 0 0

ð7:221Þ

@NTn ð0;yÞ @NTn ðL;yÞ Mx1n ðyÞþNTn ðL;yÞVx2n ðyÞþ Mx2n ðyÞ dy @x @x

NTn ðx;yÞFn ðx;yÞdxdy

ð7:222Þ

Since dd is arbitrary by definition, the spectral element equation can be derived from Equation 7.220 as SP ðvn Þdn ¼ fn ðvn Þ

ð7:223Þ


202

where SP ðvn Þ ¼ SI ðvn ÞþSII ðvn Þ

ð7:224Þ

with 2 T 2 ðb ðL 2 T 2 @ Nn @ Nn @ 2 NTn @ 2 Nn @ Nn @ Nn @ 2 NTn @ 2 Nn D þ þ þn SI ðvn Þ ¼ @x2 @x2 @y2 @y2 @x2 @y2 @y2 @x2 0 0 @ 2 NTn @ 2 Nn @NT @Nn @NT @Nn þrhgx n þ2ð1nÞ þðNx rhc2 Þ n @x@y @x@y @x @x @x @x @Nn @NTn Nn rhv2n NTn Nn þivn rhc NTn dxdy @x @x ðb T @Nn ðL;yÞ @Nn ðL;yÞ @NTn ð0;yÞ @Nn ð0;yÞ SII ðvn Þ ¼ rhc2 dy @x @x @x @x 0 þivn rhc

ðb 0

@NTn ðL;yÞ @x

Nn ðL;yÞ

@NTn ð0;yÞ @x

ð7:225Þ

ð7:226Þ

Nn ð0;yÞ dy

Substitution of Equation 7.216 into Equation 7.224 will give a (4 M)-by-(4 M) spectral element matrix in the partitioned matrix form as 2

SP11 SP12 6 SP21 SP22 6 .. 6 .. 6 . . SP ðvn Þ ¼ 6 6 SPi1 SPi2 6 6 . .. 4 .. . SPM1 SPM2

SP1j SP2j . .. . .. SPij . .. . .. SPMj

3 SP1M SP2M 7 7 .. 7 .. . 7 . 7 SPiM 7 7 .. 7 .. . 5 . SPMM

ð7:227Þ

In the above, SPij(vn) (i, j ¼ 1, 2, ... M) are the 4-by-4 matrices defined by SPij ðvn Þ ¼ SIij ðvn ÞþSIIij ðvn Þ

ð7:228Þ

where

ðb ðL 2 T 2 @ Nni @ Nnj @ 2 NTni @ 2 Nnj D þ SIij ðvn Þ ¼ @x2 @x2 @y2 @y2 0 0 2 T 2 @ Nni @ Nnj @ 2 NTni @ 2 Nnj @ 2 NTni @ 2 Nnj þn þ þ2ð1nÞ @x2 @y2 @y2 @x2 @x@y @x@y @NTni @Nnj @NT @Nnj þrhgx ni @x @x @x @x @NTni 2 T T @Nnj Nnj rhvn Nni Nnj þivn rhc Nni dx dy @x @x þðNx rhc2 Þ

ð7:229Þ


203

ðb T @Nni ðL;yÞ @Nnj ðL;yÞ @NTni ð0;yÞ @Nnj ð0;yÞ SIIij ðvn Þ ¼ rhc dy @x @x @x @x 0 ðb T @Nni ðL;yÞ @NTni ð0;yÞ Nnj ðL;yÞ Nnj ð0;yÞ dy þivn rhc @x @x 0 2

ð7:230Þ

By substituting Equation 7.217 into Equations 7.229 and 7.230 and applying the orthogonality properties (7.192), we obtain the spectral element matrix in the form of 2 3 SP11 0 0 6 0 SP22 0 7 6 7 SP ðvn Þ ¼ 6 .. ð7:231Þ .. 7 .. . . 4 . . 5 . . 0

0

SPMM

where SPii ðvn Þ ¼ SIii ðvn ÞþSIIii ðvn Þ ði ¼ 1; 2; 3; ...; MÞ

ð7:232Þ

SIii and SIIii are four-by-four matrices defined by SIii ðvn Þ ¼ HniT DIi Hni1 SIIii ðvn Þ¼ ðrhc2 Aþivn rhcBÞI0i

ð7:233Þ

where DIi ðvn Þ ¼ D½K2ni Eni K2ni I0i þEni I2i nðK2ni Eni þEni K2ni ÞI3i ð7:234Þ

2ð1nÞKni Eni Knj I1i ðNx rhc2 ÞKni Eni Kni I0i rhgKni Eni Knj I0i rhv2n Eni I0i þvn rhcðEni Kni Kni Eni ÞI0i and

2

0 6 0 A¼6 4 0 0

0 1 0 0

3 2 0 0 0 6 1 0 07 7; B ¼ 6 4 0 0 05 0 1 0

0 0 0 0

0 0 0 1

3 0 07 7 05 0

ð7:235Þ

with the use of following definitions Kni ¼ diag½kni1 kni2 kni3 kni4 2 2 2 2 K2ni ¼ diag½kni1 kni2 kni3 kni4 ÐL T Eni ðvn Þ ¼ 0 eni ðxÞeni ðxÞdx

ð7:236Þ

ni ðvn Þ ¼ eT ðLÞeni ðLÞeT ð0Þeni ð0Þ E ni ni I0i, I1i, and I2i are defined by Equation 7.195 and ðb I3i ¼ Yi Y 00i dy 0

ð7:237Þ


204

Similarly, by substituting Equations 7.216 and 7.217 into Equation 7.221, we obtain the generalized spectral nodal forces vector as 9 8 9 8 9 8 fcn1 > fdn1 > fn1 > > > > > > > > > > > > > > > = < fn2 = < fcn2 = < fdn2 > ¼ þ fn ðvn Þ ¼ .. .. .. > > > . > > > > > > . > > . > > > > > ; > ; > ; : : : fnM fcnM fdnM

ð7:238Þ

where fcni ðvn Þ ¼ Hni T feTni ð0Þ þ eTni ðLÞ fdni ðvn Þ ¼ Hni T

ðb 0

0

Ym ðyÞVx1n ðyÞdy þ e0 ni ð0Þ T

T Ym ðyÞVx2n ðyÞdy þ e0 ni ðLÞ

ðb ðL 0

ðb

0

ðb

ðb

Ym ðyÞMx1n ðyÞdy

0

Ym ðyÞMx2n ðyÞdyg

ð7:239Þ

0

eTni ðxÞYm ðyÞFn ðx; yÞdx dy

Substitution of Equations 7.231 and 7.238 into Equation 7.223 yields M independent equations as follows: SPii ðvn Þdni ¼ fni ðvn Þ

ði ¼ 1; 2; . . . ; MÞ

ð7:240Þ

As Equation 7.240 represents the independent relationships between the generalized spectral nodal DOFs and the generalized spectral nodal forces for each mode (beam functions) Yi(y) (i ¼ 1, 2, . . ., M), it is termed the “modal” spectral element equation. The modal spectral element Equation 7.240 can be assembled independently for each mode to obtain the global system equations for each mode as SPiig ðvÞdnig ¼ fnig

ði ¼ 1; 2; 3; . . . ; MÞ

ð7:241Þ

where SPiig(v) is the global dynamic stiffness matrix, dnig is the global spectral nodal DOFs vector, and fnig is the global spectral nodal forces vector for the ith mode.

7.4.3 Finite Element Model The weak form of the time-domain equation of motion (7.180) can be obtained from ðb ðL 0

0

2 2 @ w @2w 2@ w þc Dr w þ rh þ 2c @t2 @x@t @x2 4

@2w @w f ðx; y; tÞ dwdxdy ¼ 0 ðNx þ rhgxÞ 2 rhg @x @x

ð7:242Þ


205

Integrating Equation 7.242 by parts and using the force-displacement relationships (7.183), we obtain the weak form ðb ðL 2 2 @ w @ dw @ 2 w @ 2 dw @ 2 w @ 2 dw @ 2 w @ 2 dw D þ n þ n þ 2 @x2 @x2 @y2 @x2 @x2 @y2 @y @y2 0 0 @ 2 w @ 2 dw @w @dw @w @dw þ rhgx þ 2ð1 nÞ þ ðNx rhc2 Þ @x@y @x@y @x @x @x @x @2w @2w @w @dw dw þ rh 2 dw þ rhc f ðx; y; tÞdw dx dy ð7:243Þ @t @x@t @t @x L L ðb ðb @w @dw Vx ðx; y; tÞ þ Nx dy dw dy þ Mx ðx; y; tÞ @x @x 0 0 0 0 L ðb @w 2 @w þ rhc rhc þ dw dy ¼ 0 @t @x 0 0 We will consider the four-node 12-DOF rectangular plate bending element, which is commonly referred to as the ACM element [139]. At each node, there are three DOFs: one transverse displacement DOF w, and two rotational DOFs ux ¼ @w/@y and uy ¼ @w/@y. The displacement field is assumed to be wðx; y; tÞ ¼ Nðx; yÞdðtÞ

ð7:244Þ

where dðtÞ ¼ fw1 ðtÞ ux1 ðtÞ uy3 ðtÞ

w4 ðtÞ

uy1 ðtÞ

w2 ðtÞ

ux4 ðtÞ uy4 ðtÞg

ux2 ðtÞ uy2 ðtÞ

w3 ðtÞ

ux3 ðtÞ

T

ð7:245Þ

and Nðx; yÞ ¼ Bðx; yÞH 1

ð7:246Þ

with Bðx; yÞ ¼ ½1

x

y

x2

xy y2

x3

x2 y

xy2

y3

x3 y xy3

2

Rðx; yÞð312Þ

3 Bðx; yÞ 6 7 ¼ 4 @Bðx; yÞ=@y 5 @Bðx; yÞ=@x

2

Rð0; 0Þ

ð7:247Þ

3

6 RðL; 0Þ 7 6 7 Hð1212Þ ¼ 6 7 4 RðL; bÞ 5 Rð0; bÞ Substitution of Equation 7.244 into Equation 7.243 gives the finite element equation in the form of € þ ðCG þ CMT ÞdðtÞ _ þ ðKS þ KT þ Kg þ KMT ÞdðtÞ ¼ f ðtÞ þ f ðtÞ M dðtÞ c d

ð7:248Þ


206

where M ¼ rh

ðb ðL 0

NT Ndx dy ¼ MT

0

ðb ðL @NT T @N N dx dy ¼ CTG CG ¼ rhc N @x @x 0 0 ðb L CMT ¼ rhc NT N0 dy ¼ CTMT 0

2 T 2 @ 2 NT @ 2 N @ N @ N @ 2 NT @ 2 N @ 2 NT @ 2 N þ n þ þ @x2 @x2 @x2 @y2 @y2 @x2 @y2 @y2 0 0 @ 2 NT @ 2 N þ 2ð1 nÞ dx dy ¼ KTS @x@y @x@y ðb ðL @NT @N 2 dx dy ¼ KTT KN ¼ ðNx rhc Þ @x @x 0 0 ðb ðL @NT @N dx dy ¼ KTg Kg ¼ rhg x @x @x 0 0 L ðb T @N 2 KMT ¼ rhc N dy @x 0 0

KS ¼ D

ðb ðL

ð7:249Þ

and fc ðtÞ¼

ðb 0

fd ðtÞ¼

@NT ð0;yÞ @NT ðL;yÞ T Mx1 ðy;tÞþN ðL;yÞVx2 ðy;tÞþ Mx2 ðy;tÞ dy N ð0;yÞVx1 ðy;tÞþ @x @x

ðb ðL

T

NT f ðx;y;tÞdxdy

ð7:250Þ

0 0

ThefiniteelementmatricesM,C,andKaregiveninAppendix7.D.AshintedatfromEquation7.243, the mass transport term (the last integral) is determined only by the inlet (x ¼ 0) and outlet (x ¼ L) boundary conditions. Thus, during the assembly procedure, the mass transport matrices CMT and KMT will be considered only for the finite elements associated with the inlet and outlet boundaries.

Example 7.4 Consider a uniform thin plate that is moving upward passing through two simple supports as shown in Figure 7.11. The geometric and material properties of the plate are given by L ¼ 15 m, b ¼ 1 m, h ¼ 0.003 m, E ¼ 200 GPa, n ¼ 0.333, and r ¼ 7800 kg/m3. Table 7.4 compares the natural frequencies of the stationary plate (i.e., c ¼ 0 m/s) obtained by the present SEM with those obtained by FEM (m, n), where m and n denote the number of finite elements in the x- and y-directions, respectively. For SEM, the whole plate is treated as a single spectral element. It is assumed that the plate is subjected to a constant in-plane tensile force Nx ¼ 60 kN/m. When the effects of the gravity (g) are not considered (or when the plate surface is parallel to the ground), the SEM results are compared with the results computed by using the theoretical formula from Ref. [24]. Table 7.4 certainly shows that the present SEM


207

Table 7.4 Natural frequencies (Hz) of the stationary plate subjected to Nx ¼ 60 kN/m Without gravity (g ¼ 0)

with gravity (g ¼ 9.8 m/s2) FEM (m n)

Mode (p, q) (1, (1, (2, (2, (3, (5,

SEM (1 1)

Theory [24]

SEM (1 1)

(15 1)

(60 4)

(120 8)

Method A

Method B

1.688 1.799 3.378 3.599 5.071 9.027

1.688 1.799 3.378 3.599 5.072 9.028

1.458 1.800 2.939 3.602 4.456 9.036

1.711 1.818 3.423 3.638 5.139 9.125

1.712 1.821 3.426 3.643 5.143 9.137

1.710 1.819 3.423 3.641 5.141 9.139

1.713 1.823 3.428 3.648 5.145 9.144

1) 2) 1) 2) 1) 2)

provides very accurate results when compared with the results obtained by both FEM and theoretical formula. When the effects of the gravity are considered, the “Method B” based SEM seems to slightly overestimate the natural frequencies. It also shows that the effects of the gravity tend to increase the natural frequencies. Figure 7.12 displays the natural modes of the stationary plate computed by the present SEM. Mode (2,1)

Mode (1,2)

Mode (1,1) 1

1

1

0

0

0

15

–1 1 0.5

15

–1 1 0.5

10 5

15

–1 1 0.5

10 5

10 5

0 0

0 0

0 0

Mode (2,2)

Mode (3,1)

Mode (5,2)

1

1

0

15

–1 1 0.5

10 5 0 0

1

0

15

–1 1 0.5

10 5 0 0

Figure 7.12

0

15

–1 1 0.5

10 5 0 0

Natural modes of the stationary plate subjected to Nx ¼ 60 kN/m

Table 7.5 compares the natural frequencies obtained by SEM for two different moving speeds and axial tensions, with and without considering the effects of gravity. When the effects of gravity are considered, the “Method A” based SEM is used. It is clear from Table 7.5 that the natural frequencies decrease as the moving speed increases, while they increase as the axial tension increases. Table 7.5 also shows that the natural frequencies certainly increase due to gravity. Figure 7.13 shows the eigenvalues l ¼ iv of the lowest four vibration modes (1, 1), (1, 2), (2, 1), and (2, 2) as functions of the moving speed c. The static and dynamic instability zones are determined by the signs of the real and imaginary parts of the eigenvalues and they are indicated in the figure. It is shown that the divergence zone starts from cD ¼ 249.38 m/s, followed by the second narrow stable zone from cS ¼ 262.03 m/s and the dynamic instability “flutter” zone from


208

Table 7.5 Natural frequencies (Hz) of the plate passing through two simple supports c ¼ 5 m/s Nx ¼ 30 kN/m Mode (p, q) (1, (1, (2, (2, (3, (5,

1) 2) 1) 2) 1) 2)

c ¼ 10 m/s Nx ¼ 60 kN/m

Nx ¼ 30 kN/m

Nx ¼ 60 kN/m

Without gravity

With gravity

Without gravity

With gravity

Without gravity

With gravity

Without gravity

With gravity

1.170 1.325 2.344 2.652 3.523 6.673

1.204 1.355 2.413 2.713 3.630 6.835

1.672 1.783 3.345 3.568 5.023 8.951

1.696 1.806 3.394 3.613 5.096 9.069

1.089 1.254 2.181 2.512 3.282 6.337

1.130 1.289 2.240 2.565 3.376 6.498

1.618 1.733 3.238 3.468 4.863 8.707

1.644 1.757 3.296 3.520 4.957 8.833

Figure 7.13

The lowest four eigenvalues l (Hz) versus the moving speed c (m/s)

cF ¼ 169.19 m/s. The dynamic instability occurs as the coupled or mixed-mode flutter for which the natural frequencies of the mode (1, 1) and mode (2, 1) merge together near the flutter speed cF.

Example 7.5 Consider the same uniform thin plate as considered in Example 7.4. The plate is now moving upward passing through four simple supports at an equal distance of L/3, where L is the distance between the bottom support and the top support as shown in Figure 7.14. The natural frequencies obtained using the “Method A” based SEM for two different moving speeds and axial tensions are displayed in Table 7.6. The whole plate is represented by three spectral elements connected in the lengthwise direction. Table 7.6 shows that the natural frequencies decrease as the moving speed increases, while they increase as the axial tension increases. It is also shown that the effects of the gravity also increase the natural frequencies.


209 Nx

g

c

b

L

x y

z

Nx

Figure 7.14

Axially moving thin plate passing through four simple supports

Table 7.6 Natural frequencies (Hz) of the plate passing through four simple supports c ¼ 5 m/s Nx ¼ 30 kN/m

c ¼ 10 m/s Nx ¼ 60 kN/m

Nx ¼ 30 kN/m

Nx ¼ 60 kN/m

Mode (p, q)

Without gravity

With gravity

Without gravity

With gravity

Without gravity

With gravity

Without gravity

With gravity

(1,1) (2,1) (3,1) (1,2) (2,2) (4,2)

3.520 3.571 3.668 3.981 4.032 8.031

3.555 3.605 3.703 4.013 4.062 8.093

5.021 5.070 5.166 5.355 5.404 10.759

5.045 5.094 5.191 5.377 5.426 10.805

3.211 3.364 3.459 3.726 3.848 7.544

3.259 3.400 3.495 3.765 3.879 7.627

4.835 4.923 5.018 5.183 5.265 10.425

4.863 4.948 5.042 5.209 5.288 10.478

Appendix 7.A: Finite Element Matrices for Axially Moving String The finite element matrices in Equation 7.53 are given by " # " 0 rAL 2 1 M¼ G ¼ rAc 6 1 2 1 K¼

Nx rAc2 L "

CMT ¼ rAc

"

1

1 # 1 0 0

1

1

1

#

0

#

1 KMT

rAc2 ¼ L

"

1

1

1

1

# ð7:A:1Þ


210

Appendix 7.B: Finite Element Matrices for Axially Moving Bernoulli–Euler Beam The finite element matrices in Equation 7.103 are given by 2

156

54

4L2

13L

13L

156

3L2

22L

6 rAL 6 6 22L M ¼ 6 420 6 54 4 13L 2

0

6 rAc 6 6 6L G ¼ 6 30 6 30 4 6L 2

12 6 EI 6 6 6L KEI ¼ 3 6 L 6 12 4 6L 2

13L

22L

6L

30

0

6L

6L

0

L2

6L

7 3L2 7 7 7 22L 7 5 4L2

6L

1

0 0

12

4L

6L

6L

12

2L2

6L

4L2

36

3L

36

4L2

3L

3L

36

L2

3L

6L

2

3

0 6 7 60 2L 7 6 7 7 KMT ¼ rAc2 6 60 7 6L 5 4 2

0 3L

0

3

0 1

7 07 7 7 07 5

0 0

0

0 0

0

0

2

3L

2

3

6 7 6 0 L2 7 6 7 7 CMT ¼ rAc6 6 0 6L 7 4 5

6L

6 Nx rAc2 6 6 3L KN ¼ 6 30L 6 36 4

3

1

0 0

3

0

7 0 07 7 7 0 17 5

0

0 0

0

ð7:B:1Þ

3

7 L2 7 7 7 3L 7 5 4L2

Appendix 7.C: Finite Element Matrices for Axially Moving Timoshenko Beam The finite element matrices in Equation 7.166 are given by M ¼ MrA þ MrI C ¼ GrA þ GrI þ CMTrA þ CMTrI K ¼ K1 þ K2 þ K3 þ KMTrA þ KMTrI

ð7:C:1Þ


211

where 2 MrA ¼

6 rAL 6 6 840 6 4

4ð70þ7RþR2 Þ Lð35þ7Rþ2R2 Þ 4ð35þ7RþR2 Þ Lð35þ7Rþ2R2 Þ L2 ð7þR2 Þ

Lð35þ7Rþ2R2 Þ 4ð70þ7RþR2 Þ

symmetric 2 MrI ¼

6 rI 6 6 30L 6 4

36R2

3LRð5þ6RÞ

36R2

L2 ð1015Rþ9R2 Þ

3LRð56RÞ

6 rAc 6 6 GrA ¼ 30 6 4

Lð5þRÞ

30

Lð5þRÞ

0

Lð5þRÞ 0

GrI ¼

0

rIc 6 6 6 L 4

0

0

2R 0

Lð12RÞ 7 7 7 5 2R

skew symmetric 2

1

6 0 6 CMTrA ¼ rAc6 4 0 0

0

0 0

3

0 07 7 7 1 05

0

0 0

6 kGA 26 K1 ¼ ð1þRÞ 6 6 20L 4

20

2

0 10L

20

5L2

10L 20

6 Nx rAc2 6 6 K2 ¼ 6 60L 6 4

12ð5þR2 Þ

0

0

6 0 1 6 CMTrI ¼ rIc6 4 0 0

symmetric 2

ð7:C:2Þ

0

0 0 2

3

3

2R

2R

3

7 L2 R 7 7 Lð5þRÞ 7 5 0

skew symmetric 2

L2 ð7þR2 Þ

7 L2 ð515Rþ9R2 Þ 7 7 3LRð56RÞ 7 5 L2 ð1015Rþ9R2 Þ

36R2

0

7 7 7 Lð35þ7Rþ2R2 Þ 7 5 L2 ð7þR2 Þ

3LRð5þ6RÞ

symmetric 2

3

10L

0

0

3

0 0

07 7 7 05

0

1

3

7 5L2 7 7 10L 7 5 2 5L

6LR2

12ð5þR2 Þ

L2 ð5þ3R2 Þ

6LR2 12ð5þR2 Þ

symmetric

0

6LR2

3

7 L2 ð5þ3R2 Þ 7 7 7 2 7 6LR 5 2 2 L ð5þ3R Þ


212

2 6 EI rIc2 6 6 K3 ¼ 6 L3 6 4

6LR2

12R2

L2 ð1þ3R2 Þ

6LR2

12R2

12R2 symmetric

2 KMTrA ¼

2ð1RÞ

0 rAc2 6 6 6 2L 4 2ð1RÞ 0

3

6LR2

7 L2 ð1þ3R2 Þ 7 7 7 6LR2 7 5 2 2 L ð1þ3R Þ

Lð1þRÞ 2ð1RÞ Lð1RÞ 0

0

Lð1RÞ 0

2ð1RÞ 0

2

0 0 2 6 6R Lð1þ3RÞ rIc 6 KMTrI ¼ 2 6 L 4 0 0 6R Lð13RÞ

3

7 7 7 Lð1þRÞ 5 0

0

3 0 0 6R Lð13RÞ 7 7 7 5 0 0 6R

Lð1þ3RÞ

Appendix 7.D: Finite Element Matrices for Axially Moving Plate The finite element matrices in Equation 7.248 are given by 2 M¼

Lbrh 6 6 6 25200 4

M11

M12

M13

M22

M23 M33

M14

3

M24 7 7 7 M34 5

ð7:D:1Þ

M44

symmetric where 2 M11 ¼ 4

3454

461b 80b2

symm 2 M33 ¼ 4

3454

3 461L 63Lb 5 80L2

461b 80b2

symm 2

M12

1266 ¼ 4 199b 274L

199b 40b2 42Lb

3 461L 63Lb 5 80L2

2 M22 ¼ 4

3454 symm

2 M44 ¼ 4

3454

3 461b 461L 80b2 63Lb 5 80L2

461b 80b2

symm

3 2 274L 394 42Lb 5 M13 ¼ 4 166b 60L2 116L

166b 30b2 28Lb

3 461L 63Lb 5 80L2 3 116L 28Lb 5 30L2


2

M14

2

M24

394 ¼ 4 166b 116L

3 2 199L 1226 42Lb 5 M23 ¼ 4 274b 40L2 199L

274b 60b2 42Lb

1226 ¼ 4 274b 199L

3 116L 28Lb 5 30L2

166b 30b2 28Lb 2

CG ¼

213

bcrh 6 6 6 2520 4

CG11

2

M34

CG12 CG22

1226 ¼ 4 199b 274L CG13 CG23 CG33

skew

274b 60b2 42Lb

3 199L 42Lb 5 40L2

199b 40b2 42Lb

3 274L 42Lb 5 60L2

3 CG14 CG24 7 7 7 CG34 5

ð7:D:2Þ

CG44

symmetric

where 2

CG11

0 ¼4 0 175L

0 0 21Lb

2

CG12

936 ¼ 4 132b 175L 2

CG14

0 ¼4 0 77L

132b 24b2 21Lb

324 ¼ 4 78b 77L

3 2 175L 324 21Lb 5 CG13 ¼ 4 78b 28L2 77L

rhbc 6 6 6 420 4

CMT11

78b 18b2 14Lb

2

CG34

936 ¼ 4 132b 175L

0

0

CMT14

CMT22

CMT23 CMT33

0 0

symmetric

132b 24b2 21Lb

CMT11

156 ¼ 4 22b 0

3 2 22b 0 156 4b2 0 5 CMT33 ¼ 4 22b 0 0 0

7 7 7 5

CMT44

22b 4b2 0

3 175L 21Lb 5 28L2

3

where 2

3 77L 14Lb 5 14L2

CG23 ¼ CG14

3 77L 14Lb 5 14L2

78b 18b2 14Lb 2

CMT ¼

CG22 ¼ CG33 ¼ CG44 ¼ CG11

3 77L 14Lb 5 0

0 0 14Lb

2

CG24

3 175L 21Lb 5 0

3 0 05 0

ð7:D:3Þ


214

2

CMT14

54 4 ¼ 13b 0

13b 3b2 0 2

KS ¼

D 6 6 6 15L3 b3 4

3 0 05 0 KS11

CMT22 ¼ CMT11 CMT44 ¼ CMT33 CMT23 ¼ CMT14 KS12

KS13

KS22

KS23 KS33

symmetric

KS14

3

KS24 7 7 7 KS34 5

ð7:D:4Þ

KS44

where 2

3 60L4 6L2 b2 ð2n 7Þþ60b4 30L4 bþ3L2 b3 ð4n þ1Þ 30Lb4 3L3 b2 ð4n þ1Þ 5 KS11 ¼ 4 20L4 b2 4L2 b4 ðn 1Þ 15L3 b3 n 2 4 4 2 symmetric 20L b 4L b ðn 1Þ 2

3 60L4 6L2 b2 ð2n 7Þþ60b4 30L2 bþ3L2 b3 ð4n þ1Þ 30Lb4 þ3L3 b2 ð4n þ1Þ 5 KS22 ¼ 4 20L4 b2 4L2 b4 ðn 1Þ 15L3 b3 n 2 4 4 2 symmetric 20L b 4L b ðn 1Þ 2

3 60L4 6L2 b2 ð2n 7Þþ60b4 30L4 b3L2 b3 ð4n þ1Þ 30Lb4 þ3L3 b2 ð4n þ1Þ 5 KS33 ¼ 4 20L4 b2 4L2 b4 ðn 1Þ 15L3 b3 n 2 4 4 2 symmetric 20L b 4L b ðn 1Þ 2

3 60L4 6L2 b2 ð2n 7Þþ60b4 30L4 b3L2 b3 ð4n þ1Þ 30Lb4 3L3 b2 ð4n þ1Þ 5 KS44 ¼ 4 20L4 b2 4L2 b4 ðn 1Þ 15L3 b3 n 2 4 4 2 symmetric 20L b 4L b ðn 1Þ 2

3 30L4 þ6L2 b2 ð2n 7Þ60b4 15L4 b3L2 b3 ð4n þ1Þ 30Lb4 þ3L3 b2 ðn 1Þ 5 KS12 ¼ 4 10L4 b2 þ4L2 b4 ðn 1Þ 0 2 4 4 2 symmetric 10L b þL b ðn 1Þ 2

3 30L4 6L2 b2 ð2n 7Þ30b4 15L4 bþ3L2 b3 ðn 1Þ 15Lb4 3L3 b2 ðn 1Þ 5 KS13 ¼ 4 15L4 b3L2 b3 ðn 1Þ 5L4 b2 4L2 b4 ðn 1Þ 0 4 3 2 2 4 4 2 15Lb þ3L b ðn 1Þ 0 5L b L b ðn 1Þ


215

2

3 60L4 þ6L2 b2 ð2n 7Þþ30b4 30L4 b3L2 b3 ðn 1Þ 15Lb4 þ3L3 b2 ð4n þ1Þ 5 KS14 ¼ 4 30L4 bþ3L2 b3 ðn 1Þ 10L4 b2 þL2 b4 ðn 1Þ 0 4 3 2 2 4 4 2 15Lb þ3L b ð4n þ1Þ 0 10L b þ4L b ðn 1Þ 2

3 60L4 þ6L2 b2 ð2n 7Þþ30b4 30L4 b3L2 b3 ðn 1Þ 15Lb4 3L3 b2 ð4n þ1Þ 5 KS23 ¼ 4 30L4 bþ3L2 b3 ðn 1Þ 10L4 b2 þL2 b4 ðn 1Þ 0 4 3 2 2 4 4 2 15Lb 3L b ð4n þ1Þ 0 10L b þ4L b ðn 1Þ 2

3 30L4 6L2 b2 ð2n 7Þ30b4 15L4 bþ3L2 b3 ðn 1Þ 15Lb4 þ3L3 b2 ðn 1Þ 5 KS24 ¼ 4 15L4 b3L2 b3 ðn 1Þ 5L4 b2 L2 b4 ðn 1Þ 0 4 3 2 2 4 4 2 15Lb 3L b ðn 1Þ 0 5L b L b ðn 1Þ 2

3 30L4 þ6L2 b2 ð2n 7Þ60b4 15L4 bþ3L2 b3 ð4n þ1Þ 30Lb4 3L3 b2 ðn 1Þ 5 KS34 ¼ 4 10L4 b2 þ4L2 b4 ðn 1Þ 0 2 4 4 2 symmetric 10L b þL b ðn 1Þ 2 KN ¼

ðNx rhc2 Þb 6 6 6 1260L 4

KN11

KN12 KN22 symmetric

where

2

3 KN13 KN14 KN23 KN24 7 7 7 KN33 KN34 5

ð7:D:5Þ

KN44

3 2 3 66b 42L 552 66b 42L KN11 ¼ 4 KN22 ¼ 4 12b2 0 5 12b2 0 5 symm 56L2 symm 56L2 2 3 2 3 552 66b 42L 552 66b 42L KN33 ¼ 4 KN44 ¼ 4 12b2 0 5 12b2 0 5 2 symm 56L symm 56L2 2 3 2 3 552 66b 42L 204 39b 21L KN12 ¼ 4 66b 12b2 0 5 KN13 ¼ 4 39b 9b2 0 5 2 42L 0 14L 21L 0 7L2 2 3 2 3 204 39b 21L 204 39b 21L KN23 ¼ 4 39b 9b2 0 5 KN14 ¼ 4 39b 9b2 0 5 2 21L 0 28L 21L 0 28L2 2 3 2 3 204 39b 21L 552 66b 42L KN24 ¼ 4 39b 9b2 KN34 ¼ 4 66b 12b2 0 5 0 5 2 21L 0 7L 42L 0 14L2 2 3 Kg11 Kg12 Kg13 Kg14 Kg22 Kg23 Kg24 7 rhbg 6 6 7 Kg ¼ 6 7 5040 4 Kg33 Kg34 5 552

symmetric

Kg44

ð7:D:6Þ


216

where 2

3 1104 132b 175L Kg11 ¼ 4 24b2 21Lb 5 symm 56L2

2

3 1104 132b 7L Kg22 ¼ 4 24b2 21Lb 5 symm 168L2

2

3 1104 132b 7L Kg33 ¼ 4 24b2 21Lb 5 symm 168L2

2

3 1104 132b 175L Kg44 ¼ 4 24b2 21Lb 5 symm 56L2

2

3 2 3 1104 132b 7L 408 78b 7L Kg12 ¼ 4 132b 24b2 21Lb 5 Kg13 ¼ 4 78b 18b2 14Lb 5 175L 21Lb 28L2 77L 14Lb 14L2 2

3 2 3 408 78b 77L 408 78b 7L Kg14 ¼ 4 78b 18b2 14Lb 5 Kg23 ¼ 4 78b 18b2 14Lb 5 77L 14Lb 28L2 7L 14Lb 84L2 2

3 408 78b 77L Kg24 ¼ 4 78b 18b2 14Lb 5 7L 14Lb 14L2

2

3 1104 132b 175L Kg34 ¼ 4 132b 24b2 21Lb 5 7L 21Lb 28L2

2

3 KMT11 KMT12 KMT13 KMT14 7 rhbc2 6 6 KMT21 KMT22 KMT23 KMT24 7 KMT ¼ 6 7 420L 4 KMT31 KMT32 KMT33 KMT34 5 KMT41 KMT42 KMT43 KMT44 where 2

3 9 22b 147 KMT11 ¼ 4 b 4b2 21b 5 0 0 0

2

3 9 22b 147 KMT22 ¼ 4 b 4b2 21b 5 0 0 0

2

3 2 3 9 22b 147 9 22b 147 21b 5 KMT44 ¼ 4 b 4b2 21b 5 KMT33 ¼ 4 b 4b2 0 0 0 0 0 0 2

3 9 22b 0 KMT12 ¼ 4 b 4b2 0 5 0 0 0 2

3 9 13b 63 KMT14 ¼ 4 b 3b2 14b 5 0 0 0

2

3 9 13b 0 KMT13 ¼ 4 b 3b2 0 5 0 0 0 2

3 9 13b 63 KMT23 ¼ 4 b 3b2 14b 5 0 0 0

ð7:D:7Þ


2

3 9 13b 0 KMT31 ¼ 4 b 3b2 0 5 0 0 0 2

217

2

3 9 13b 63 KMT32 ¼ 4 b 3b2 14b 5 0 0 0

3 9 22b 0 KMT34 ¼ 4 b 4b2 0 5 0 0 0

2

3 9 13b 63 KMT41 ¼ 4 b 3b2 14b 5 0 0 0

KMT21 ¼ KMT12

KMT24 ¼ KMT13

KMT42 ¼ KMT31

KMT43 ¼ KMT34

8 Dynamics of Rotor Systems Rotating machines have been used extensively in diverse engineering applications such as motors, engines, turbines, and machine tools. In general, the rotating machines consist of multiple shafts and disks (or blades) that are connected to each other to form the rotor systems supported by multiple bearings. It is very important to predict the dynamic characteristics of the rotor systems accurately early during the design phase. Thus there have been extensive studies on the modeling and analysis of such rotor systems over many decades and detailed reviews on the subject can be found in Refs. [126, 143]. This chapter introduces a spectral element model for a multi-bearing flexible rotor system with rigid disks. In the first part of the chapter, the equations of motion for the rotor systems are derived using Hamilton’s principle and Lagrange’s equations. The spinning circular shafts of a rotor are represented by the uniform Timoshenko shaft models and the rigid disks are considered to be wobbling about their centers. The support-bearings are regarded as the flexible bearings and they are represented by effective translational and rotational springs. The mass unbalances of disks are taken into account in the spectral element model. The finite element model is also provided in Section 8.3 for comparison with the spectral element model introduced in this chapter.

8.1 Governing Equations A multi-bearing rotor consists of multiple flexible spinning shafts and interim disks with mass unbalances, supported by multiple supports. In this study, the spinning shafts are represented by the spinning Timoshenko shafts subjected to axial (longitudinal), torsional, and transverse vibrations and the interim disks are represented by the rigid, thin disks subjected to wobbling motions about their centers. In the following, the dynamic equations of motion for the spinning Timoshenko shafts and the wobbling rigid disks will be derived by using Hamilton’s principle and Lagrange’s equations, respectively.


220


8.1.1 Equations of Motion of the Spinning Shaft The equations of motion and relevant boundary conditions for a uniform spinning shaft represented by the Timoshenko shaft model can be derived by using Hamilton’s principle ð t2 ðdT dV þ dWÞ dt ¼ 0 ð8:1Þ t1

where T is the kinetic energy, V is the potential energy, and dW is the virtual work perfomed by external forces and moments. We assume that the spinning Timoshenko shaft has a uniform circular cross-section and it is spinning about its central axis at a constant speed of W (radian/s) to take a small amplitude vibration. As shown in Figure 8.1, we define v(x, t) as being the transverse displacement in the y-direction, w(x, t) as the transverse displacement in the z-direction, u(x, t) as the longitudinal (axial) displacement, u(x, t) as the twist angle about the central axis x, f(x, t) as the rotation angle about the y-axis, and c(x, t) as the rotation angle about z-axis.

Figure 8.1 Timoshenko-beam model for a uniform spinning shaft: (a) the boundary forces and moments, (b) the six degrees of freedom for the shaft, (c) the support-bearing models

The kinetic and potential energies are then given by ð ð 1 L 1 L 2 2 2 2 2 _ _ _ T ¼ rAðu þ v þ w Þ dx þ rIðf_ þ c_ Þ dx 2 0 2 0 ð 1 L _ 2 dx þ rJðW þ u_ fcÞ 2 0

ð8:2Þ

221

Dynamics of Rotor Systems

1 V ¼ 2 1 þ 2

ðL 0

ðL

1 EAu dx þ 2 02

0

ðL 0

ðL

1 GJu dx þ 2 02

0

2

EIðf0 þ c0 Þ dx 2

0

2

kGA ½ðv cÞ þ ðw þ fÞ dx þ

2 X 1

0

i¼1

where

Ksupport ðiÞ

2

Kyyi ¼ Kzyi

2

ð8:3Þ fvi wi gKsupport ðiÞ fvi wi g

T

Kyzi Kzzi

ði ¼ 1; 2Þ

ð8:4Þ

In above equations, the dot () and the prime (0 ) denote the derivatives with respect to the time t and axis x, respectively; EA is the axial rigidity, EI the bending (flexural) rigidity, GJ the torsional rigidity, kGA the transverse shear rigidity, rA the mass per length, rI the mass moment of inertia about the y- or z-axis, rJ the polar mass moment of inertia, and L is the length of shaft; Ksupport(1) and Ksupport(2) represent the stiffnesses of the support-bearings at the left end “1” and at the right end “2”, respectively; vi and wi (i ¼ 1, 2) are the transverse displacements specified at the left and right ends “1” and “2”. In Equation 8.2, the first integral represents the translational kinetic energy and the remaining two integrals present the rotational kinetic energy. In Equation 8.3, the first integral represents the strain energy for axial vibration, the second integral for torsional vibration, the third integral for transverse bending vibrations, the fourth integral for transverse shear deformations, and the last term represents the strain energy stored in the bearing supports at both ends. The virtual work dW is given by ðL dW ¼ ðpx du þ py dv þ pz dw þ tx du þ ty df þ tz dcÞ dx 0

þ Qx1 du1 þ Qx2 du2 þ Qy1 dv1 þ Qy2 dv2 þ Qz1 dw1 þ Qz2 dw2

ð8:5Þ

þ Mx1 du1 þ Mx2 du2 þ My1 df1 þ My2 df2 þ Mz1 dc1 þ Mz2 dc2 fv_ 1 dv1 fw_ 1 dw1 fv_ 2 dv2 fw_ 2 dw2 where Qxi, Qyi, Qzi, Mxi, Myi, and Mzi (i ¼ 1, 2) are the transverse shear forces, bending moments, and torques applied at two ends of the beam element, as shown in Figure 8.1, and px, py, pz, tx, ty, and tz are the forces, bending moments, and torques distributed along axis x. fv_ i and fw_ i (i ¼ 1, 2) are the viscous damping forces due to the bearing supports and they can be computed from Rayleigh’s dissipation function R as follows: fv_ i ¼

qR ; q_vi

where R¼

fw_ i ¼

2 X 1 i¼1

2

and where

ði ¼ 1; 2Þ

fv_ i w_ i gCsupportðiÞ f_vi w_ i gT

CsupportðiÞ ¼

qR qw_ i

Cyyi Czyi

Cyzi Czzi

ð8:6Þ

ð8:7Þ

ði ¼ 1; 2Þ

ð8:8Þ

222


Substituting Equations 8.2, 8.3 and 8.5 into Equation 8.1 and integrating by parts, we get the following equations of motion rA€ u EAu00 ¼ px € GJu00 ¼ tx rJð€ u f_ c_ fcÞ rA€v kGAðv00 c0 Þ ¼ py € rJðW þ u_ fcÞ _ f_ rJð€ € f EIc00 kGAðv0 cÞ ¼ tz rI c u f_ c_ fcÞ

ð8:9Þ

rA€ w kGAðw00 þ f0 Þ ¼ pz € þ rJðW þ u_ fcÞ _ c_ EIf00 þ kGAðw0 þ fÞ ¼ ty rI f and the boundary conditions Qx ð0; tÞ ¼ Qx1 Qx ðL; tÞ ¼ Qx2

or uð0; tÞ ¼ u1 ðtÞ

or uðL; tÞ ¼ u2 ðtÞ

Qy ð0; tÞ ¼ Kyy1 v1 þ ð1=2ÞðKyz1 þ Kzy1 Þw1 þ Cyy1 v_ 1

or vð0; tÞ ¼ v1 ðtÞ

þ ð1=2ÞðCyz1 þ Czy1 Þw_ 1 Qy1 Qy ðL; tÞ ¼ Kyy2 v2 ð1=2ÞðKyz2 þ Kzy2 Þw2 Cyy2 v_ 2

or vðL; tÞ ¼ v2 ðtÞ

ð1=2ÞðCyz2 þ Czy2 Þw_ 2 þ Qy2 Qz ð0; tÞ ¼ Kzz1 w1 þ ð1=2ÞðKyz1 þ Kzy1 Þv1 þ Czz1 w_ 1

ð8:10Þ

or wð0; tÞ ¼ w1 ðtÞ

þ ð1=2ÞðCyz1 þ Czy1 Þ_v1 Qz1 Qz ðL; tÞ ¼ Kzz2 w2 ð1=2ÞðKyz2 þ Kzy2 Þv2 Czz2 w_ 2

or

wðL; tÞ ¼ w2 ðtÞ

ð1=2ÞðCyz2 þ Czy2 Þ_v2 þ Qz2 Mx ð0; tÞ ¼ Mx1 Mx ðL; tÞ ¼ Mx2 My ð0; tÞ ¼ My1 My ðL; tÞ ¼ My2 Mz ð0; tÞ ¼ Mz1 Mz ðL; tÞ ¼ Mz2

or uð0; tÞ ¼ u1 ðtÞ or uðL; tÞ ¼ u2 ðtÞ or fð0; tÞ ¼ f1 ðtÞ or fðL; tÞ ¼ f2 ðtÞ or cð0; tÞ ¼ c1 ðtÞ or cðL; tÞ ¼ c2 ðtÞ

where the force-displacement relationships are defined by Qx ðx; tÞ ¼ EAu0 Qy ðx; tÞ ¼ kGAðv0 cÞ Qz ðx; tÞ ¼ kGAðw0 þ fÞ Mx ðx; tÞ ¼ EIu0 My ðx; tÞ ¼ EIf0 Mz ðx; tÞ ¼ EIc0

ð8:11Þ

223


Neglecting small coupling terms, we can further simplify Equation 8.9 as follows: rA€ u EAu00 ¼ px rJ € u GJu00 ¼ tx

ð8:12Þ

rA€v kGAðv00 c0 Þ ¼ py € rJWf_ EIc00 kGAðv0 cÞ ¼ tz rI c 00

0

rA€ w kGAðw þ f Þ ¼ pz € rI f þ rJWc_ EIf00 þ kGAðw0 þ fÞ ¼ ty

ð8:13Þ ð8:14Þ ð8:15Þ ð8:16Þ ð8:17Þ

The above simplified equations of motion are identical with those given in Refs. [150, 176].

8.1.2 Equations of Motion of Disks with Mass Unbalance 8.1.2.1 Thin Disk Figure 8.2 shows a thin rigid disk with mass unbalance. The mass of the disk itself is mD and the mass of the unbalance mass is me. Thus the total mass of the disk system is M ¼ mD þ me. Assume that unbalance mass me is located at de measured from the geometric center “O” where the spinning shafts are connected. Accordingly, the mass center of the disk system is located at d ¼ mede measured from the geometric center “O”, where me ¼ me/M. In Figure 8.2, the disk is denoted by “D” and its mass center by “C”. The global reference coordinates (x, y, z) and the moving coordinates (1, 2, 3) fixed on the disk are denoted by “R” and “S”, respectively. The unit direction vectors nx, ny, and nz are defined in the x-, y-, and z-directions, while n1, n2, and n3 are in the 1-, 2-, and 3-directions, respectively. The other unit direction vectors are also defined in Figure 8.2; u, v, and w are the displacements of the geometric center “O” in the x-, y-, and z-directions, respectively, and u, f, and c are the rotation angles about the x-, y-, and z-axis, respectively.

Figure 8.2

Dynamic model of a rigid disk with unbalanced mass

224


The position of the mass center “C” with respect to the global coordinates “R” can be then expressed by R C

r ¼ ½u þ df sin ðu þ WtÞ dc cos ðu þ WtÞnx þ ½v þ d cos ðu þ WtÞny þ ½w þ d sin ðu þ WtÞnz

ð8:18Þ

From Equation 8.18, we can obtain the velocity of the mass center as v ¼ ½u_ þ d f_ sinðu þ WtÞ d c_ cosðu þ WtÞ

R C

þ dðu_ þ WÞfc sinðu þ WtÞ þ f cosðu þ WtÞg nx

ð8:19Þ

þ ½_v dðu_ þ WÞsinðu þ WtÞ ny þ ½w_ þ dðu_ þ WÞ cosðu þ WtÞ nz The angular velocity of the disk with respect to the global coordinates “R” can be obtained from Figure 8.2 as R vD

_ þ f_ j þ c_ nz ¼ ðW þ uÞi _ n1 þ ½f_ cos ðu þ WtÞ þ c_ sin ðu þ WtÞ n2 ¼ ðW þ u_ fcÞ

ð8:20Þ

½f_ sin ðu þ WtÞ c_ cos ðu þ WtÞ n3 The kinetic energy of the disk system including the unbalanced mass is given by 1 1 R D T ¼ M R vC R vC þ R v D I D C v 2 2

ð8:21Þ

where ID C ¼ diag½I11 I22 I33 is the inertia dyadic of the disk evaluated with respect to its mass center “C”. Substituting Equations 8.19 and 8.20 into Equation 8.21 yields 1 _ _ f_ sin ðuþWtÞd c_ cos ðuþWtÞ þdðuþWÞ½c sin ðuþWtÞ T ¼ M fuþd 2 2 _ _ þf cos ðuþWtÞg2 þ v_ 2 þ w_ 2 þd 2 ðuþWÞ 2d v_ ðuþWÞ sin ðuþWtÞ h 1 _ _ _ 2 þI22 ff_ cos ðuþWtÞ _ uþWÞcos þ2d wð ðuþWtÞ þ I11 ðWþ uf cÞ 2 þ c_ sin ðuþWtÞg2 þI33 ff_ sin ðuþWtÞ c_ cos ðuþWtÞg2

ð8:22Þ (8.22)

Figure 8.3 shows the forces and moments acting on both sides of the disk. The virtual work dW done by these forces and moments is given by D D D D D dW ¼ QD x duþMx duþQy dvþMz dcþQz dwþMy df

ð8:23Þ

where D D QD j ¼ Qjl þQjr ;

MjD ¼ MjlD þMjrD

ðj ¼ x; y; zÞ

ð8:24Þ

225

Dynamics of Rotor Systems D Q Dyl Q yr D Mzr

D Q Dzl Q zr D M yr

M ylD

M zlD y

Q xlD M xlD

D M xr

D Q xr

z

Q xlD M xlD

x

D M xr

D Q xr

x

Figure 8.3 Forces and moments acting on the both sides of a rigid disk

By substituting Equations 8.22 and 8.23 into Lagrange’s equations and neglecting small nonlinear terms, we can obtain the equations of motion of the disk as M€ u ¼ QD x € JD u ¼ M D þM me x

x

me M€v ¼ QD y þfy

€ JD Wf_ ¼ M D þM me ID c z z

ð8:25Þ

me € ¼ QD Mw z þfz

€ þJD Wc_ ¼ M D þM me ID f y y where fyme and fzme are the centrifugal unbalance forces and Mxme ; Myme and Mzme are the unbalance moments, all induced by the unbalance mass. They are given by fyme ¼ me de W2 cos Wt fzme ¼ me de W2 sin Wt u M me ¼ me d 2 € x

Myme

ð8:26Þ

e

¼

2 € m W2 fþ2WcÞsin _ _ € me W2 c2WfÞsin me de2 ½ðf Wtðc Wtcos Wt e

_ cos 2 Wtðf € m W2 fþ2WcÞsin € me W2 c2WfÞ _ Wtcos Wt Mzme ¼ me de2 ½ðc e In most books and references associated with the theory of rotordynamics, the unbalance moments have been ignored or have not been considered. Thus, ignoring all unbalance moments, we can further simplify Equation 8.25 as follows: M€ u ¼ QD x € ¼ MD JD u x

me M€v ¼ QD y þfy

€ JD Wf_ ¼ M D ID c z

ð8:27Þ

me € ¼ QD Mw z þfz

€ þJD Wc_ ¼ M D ID f y The last four equations of Equation 8.27 are identical with those derived in Ref. [42] by using the Newton’s law of motion.

226


8.1.2.2 Thick Disk Consider a thick rigid disk, which has the thickness 2h as shown in Figure 8.4. The superscripts r and l denote the translational and rotational motions at the right- and left-hand sides of the disk, respectively. The position vector R rC of the mass center “C” and the angular velocity R vD of the thick disk are the same as in Equations 8.18 and 8.20, respectively. Thus the velocity R vC of the mass center “C” is also the same as in Equation 8.19.

Figure 8.4

Thick rigid disk

For the rigid thick disk, the following relationships are satisfied: u ¼ ul ¼ ur ;

v ¼ vl þ hcl ;

w ¼ wl hfl

u ¼ ul ¼ ur ;

f ¼ fl ¼ fr ;

c ¼ cl ¼ cr

ð8:28Þ

and vr ¼ vl þ 2hcl ;

wr ¼ wl 2hfl

ð8:29Þ

Substituting Equation 8.28 into Equations 8.19 and 8.20 gives l l v ¼ ½fu_ l þ d f_ sinðul þ WtÞ d c_ cosðul þ WtÞ

R C

l

þ dðu_ þ WÞfcl sinðul þ WtÞ þ fl cosðul þ WtÞg nx l l þ ½ð_vl þ hc_ Þ dðu_ þ WÞsinðul þ WtÞ ny

ð8:30Þ

l l þ ½ðw_ l hf_ Þ þ dðu_ þ WÞcosðul þ WtÞ nz

and R

l l l l vD ¼ ðW þ u_ fl c_ Þ n1 þ ½f_ cos ðul þ WtÞ þ c_ sin ðul þ WtÞ n2 l

l

½f_ sin ðul þ WtÞ c_ cos ðul þ WtÞ n3

ð8:31Þ

227


By using Equations 8.30 and 8.31, the kinetic energy is obtained from Equation 8.21 as 1 h l l T ¼ M fu_ l þ d f_ sin ðul þ WtÞ d c_ cos ðul þ WtÞ 2 l þ dðu_ þ WÞ½cl sin ðul þ WtÞ þ fl cos ðul þ WtÞg2 l

l

l

þ ð_vl þ hc_ Þ2 þ ðw_ l hf_ Þ2 þ d 2 ðu_ þ WÞ2

(8.32)

l l l þ 2dðu_ þ WÞ½ðw_ l hf_ Þ cos ðul þ WtÞ ð_vl þ hc_ Þ sin ðul þ WtÞ 1h l l l l þ I11 ðW þ u_ fl c_ Þ2 þ I22 ff_ cos ðul þ WtÞ þ c_ sin ðul þ WtÞg2 2 l l þ I33 ff_ sin ðul þ WtÞ c_ cos ðul þ WtÞg2

ð8:32Þ

The forces and moments acting on the both sides of the rigid disk are shown in Figure 8.4. The virtual work done by these forces and moments is given by l l D l D l D dW ¼ QD x du þ Qy dv þ Qz dw þ Mx du l l D D þ ðMyD 2hQD zr Þdf þ ðMz þ 2hQyr Þdc

ð8:33Þ

D where QD j and Mj (j ¼ x, y, z) defined by Equation 8.24. By substituting Equations 8.32 and 8.33 into the Lagrange’s equations and neglecting small nonlinear terms, we obtain the equations of motion of the thick rigid disk as

M€ u l ¼ QD x l me u ¼ MxD þ Mxl JD €

€ l Þ ¼ QD þ f me Mð€v l þ hc y y € l JD Wf_ l þ Mhð€v l þ hc € l Þ ¼ M D þ 2hQD þ M me ID c z yr zl

ð8:34Þ

€ l Þ ¼ Q D þ f me Mð€ w l hf z z € l Þ ¼ M D 2hQD þ M me € l þ JD Wc_ l Mhð€ ID f w l hf y zr yl where fyme ¼ me de W2 cos Wt fzme ¼ me de W2 sin Wt me u Mxl ¼ me de2 €

l

l

l

€ m W2 fl þ 2Wc_ Þ sin 2 Wt Mylme ¼ me de hW2 sin Wt me de2 ½ðf e € l me W2 cl 2Wf_ l Þ sin Wt cos Wt ðc € l me W2 cl 2Wf_ l Þ cos 2 Wt Mzlme ¼ me de hW2 cos Wt me de2 ½ðc € l m W2 fl þ 2Wc_ l Þ sin Wt cos Wt ðf e

ð8:35Þ

228


l l l Neglecting the unbalance moments Mmx , Mmy , and Mmz , the above equations (8.34) can be further simplified as

M€ u l ¼ QD x l

u ¼ MxD JD € l

€ Þ ¼ QD þ f me Mð€v l þ hc y y € l JD Wf_ l þ Mhð€v l þ hc € l Þ ¼ M D þ 2hQD ID c z yr

ð8:36Þ

€ l Þ ¼ Q D þ f me Mð€ w l hf z z € l Þ ¼ M D 2hQD € l þ JD Wc_ l Mhð€ ID f w l hf y zr If we use the disk thickness 2h ¼ 0, the above equations (8.36) return to the equations of motion for the thin rigid disks.

8.2 Spectral Element Modeling 8.2.1 Spectral Element for the Spinning Shaft The spectral element model for the spinning shaft is derived from the equations of motion 8.12–8.17. The equations of motion consist of two decoupled equations for longitudinal and torsional vibrations and four coupled equations for the transverse bending vibration. Thus the spectral element equations will be formulated for the longitudinal vibration, torsional vibration, and transverse bending vibration, separately, and then they will be combined to derive the spectral element equation for a spinning shaft subjected to axial, torsional, and transverse bending vibrations simultaneously. 8.2.1.1 Spectral Element for the Axial Vibration By using the DFT theory, express the solution and external force of Equation 8.12 in the spectral forms as fuðx; tÞ px ðx; tÞg ¼

1 1 NX fUn ðx; vn Þ Pxn ðx; vn Þgeivn t N n¼0

ð8:37Þ

Substitution of Equation 8.37 into Equation 8.12 yields EAU 00 þ v2 rAU þ Px ðxÞ ¼ 0

ð8:38Þ

where the subscript n is omitted for the brevity. The homogeneous solution to Equation 8.38 can be obtained as UðxÞ ¼ aL1 e ikL x þ aL2 eikL x ¼ eL ðx; vÞaL

ð8:39Þ

229


where aL ¼ {aL1aL2}T is the constant vector and eL ðx; vÞ ¼ ½e ikL x eikL x ;

kL ¼ v

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rA=EA

ð8:40Þ

The spectral nodal degrees-of-freedom (DOFs) are defined as U2 gT ¼ fUð0Þ UðLÞgT

dL ¼ fU1

ð8:41Þ

Applying Equations 8.39–8.41, we obtain 1 1 a1 dL ¼ ¼ HL ðvÞaL e ikL L eikL L a2

ð8:42Þ

By using Equation 8.42, Equation 8.39 can be rewritten as UðxÞ ¼ ½eL HL 1 dL ¼ NL ðx; vÞdL

ð8:43Þ

where NL(x; v) is the dynamic shape function. The weak form of Equation 8.38 can be derived from the weighted integral as ðL ðEAU 00 þ v2 rAU þ Px ÞdUdx ¼ 0

ð8:44Þ

0

where dU is the arbitrary variation in U. Applying the integral by parts to the first term of Equation 8.44, we get the weak form as ðL ðL ðL ð8:45Þ EA U 0 dU 0 dx v2 rA UdUdx Px dUdx ddTL fLc ¼ 0 0

0

0

where fLc is the spectral nodal axial forces vector defined by fLc ¼ fQx1 Qx2 gT

ð8:46Þ

Substituting Equation 8.43 into Equation 8.45, we get ddTL ½ðEA

ðL 0

T N 0 L N0 L dx v2 rA

ðL 0

NTL NL dxÞdL

ðL 0

Px NTL dx fLc ¼ 0

ð8:47Þ

As ddL is arbitrary, by definition, the spectral element equation can be obtained from Equation 8.47 as SL ðvÞdL ¼ fL where

cotðkL LÞ SL ðvÞ ¼ kL EA cscðkL LÞ and fL ¼ fLc þ

ðL 0

cscðkL LÞ ¼ ½sLij cotðkL LÞ

ð8:48Þ

ði; j ¼ 1; 2Þ

Px ðxÞNTL ðxÞdx ¼ fFu1 Fu2 gT

ð8:49Þ

ð8:50Þ

230


8.2.1.2 Spectral Element for the Torsional Vibration Represent the solution and external force of Equation 8.13 in the spectral forms as fuðx; tÞtx ðx; tÞg ¼

1 1 NX fQn ðx; vn ÞTxn ðx; vn Þgeivn t N n¼0

ð8:51Þ

Substituting Equation 8.51 into Equation 8.13, we get GJQ00 þ v2 rJQ þ Tx ðxÞ ¼ 0

ð8:52Þ

where the subscript n is again omitted for brevity. The homogeneous solution of Equation 8.52 is obtained as QðxÞ ¼ aT1 e ikT x þ aT2 eikT x ¼ eT ðx; vÞaT where aT ¼ {aT1aT2}T is the constant vector and

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kT ¼ v rJ=GJ

eT ðx; vÞ ¼ ½e ikT x eikT x ;

ð8:53Þ

ð8:54Þ

Define the spectral nodal DOFs as dL ¼ fQ1

Q2 gT ¼ fQð0Þ

QðLÞgT

ð8:55Þ

Substitution of Equation 8.53 into Equation 8.55 gives dT ¼

1

1

e ikT L

eikT L

aT1 aT2

¼ HT ðvÞaT

ð8:56Þ

By using Equation 8.56, Equation 8.53 can be rewritten as QðxÞ ¼ ½eT HT 1 dT ¼ NT ðx; vÞ dT where NT is the dynamic shape function. The weak form of Equation 8.52 can be derived from the weighted integral as ðL ðGJQ00 þ v2 rJQ þ Tx ÞdQdx ¼ 0

ð8:57Þ

ð8:58Þ

0

where dQ is the arbitrary variation in Q. Applying the integral by parts to the first term of Equation 8.58, we get the weak form as GJ

ðL 0

0

0

Q dQ dx v rJ 2

ðL

QdQdx

0

ðL 0

Tx dQdx ddTT fTc ¼ 0

ð8:59Þ

where fTc is the spectral nodal forces vector defined by fTc ¼ fMx1

Mx2 gT

ð8:60Þ

231


Substituting Equation 8.57 into Equation 8.59 yields ddTT ½ðGJ

ðL 0

T N0 T N0 T dx v2 rJ

ðL 0

NTT NT dxÞ dT

ðL 0

Tx NTT dx fTc ¼ 0

ð8:61Þ

As ddT is arbitrary, the spectral element equation can be derived from Equation 8.61 as ST ðvÞdT ¼ fT

ð8:62Þ

where

cscðkT LÞ ¼ ½sTij cotðkT LÞ

cotðkT LÞ ST ðvÞ ¼ kT GJ cscðkT LÞ and fT ¼ fTc þ

ðL 0

Tx ðxÞNTT ðxÞdx ¼ fFu1

ði; j ¼ 1; 2Þ

Fu2 gT

ð8:63Þ

ð8:64Þ

8.2.1.3 Spectral Element for the Transverse Bending Vibration As previously, we represent the solutions and external forces of Equation 8.14–8.17 in the spectral forms as fvðx; tÞwðx; tÞcðx; tÞfðx; tÞg 1 1 NX ¼ fVn ðx; vn ÞWn ðx; vn ÞYn ðx; vn ÞFn ðx; vn Þgeivn t N n¼0

ð8:65Þ

fpy ðx; tÞ pz ðx; tÞ ty ðx; tÞ tz ðx; tÞg 1 1 NX ¼ fPyn ðx; vn Þ Pzn ðx; vn Þ Tyn ðx; vn Þ Tzn ðx; vn Þgeivn t N n¼0

ð8:66Þ

Substituting Equations 8.65 and 8.66 into Equation 8.14–8.17, we get kGAðV 00 Y0 Þ þ rAv2 V þ Py ¼ 0 EIY00 þ ivWrJF þ kGAðV 0 YÞ þ rIv2 Y þ Tz ¼ 0 kGAðW 00 þ F0 Þ þ rAv2 W þ Pz ¼ 0

ð8:67Þ

EIF00 ivWrJY kGAðW 0 þ FÞ þ rIv2 F þ Ty ¼ 0 where the subscripts n are omitted for brevity. Similarly, substituting Equation 8.65 and the spectral forms of Qi(x, t) and Mi(x, t) (i ¼ y, z) into the associated force-displacement relationships from Equation 8.11, we get Qy ðxÞ ¼ kGAðV 0 YÞ;

My ðxÞ ¼ EIF0

Qz ðxÞ ¼ kGAðW 0 þ FÞ; Mz ðxÞ ¼ EIY0

ð8:68Þ

232


Assume the homogeneous solutions of Equation 8.67 to be VðxÞ ¼ ae ikx ;

YðxÞ ¼ rae ikx

WðxÞ ¼ tae ikx ;

FðxÞ ¼ t^rae ikx

ð8:69Þ

Substitution of Equation 8.69 into the homogeneous equations reduced from Equation 8.67 yields an eigenvalue problem as 2

s1 6s 6 3 6 4 0 0

s3 s2

0 0

0 s4

s1 s3

38 9 8 9 1 > >0> 0 > > > > > > > > > < 7 s4 7 r = < 0 = ¼ 7 >t> s3 5> > >0> > > > ; : > ; > : > 0 t^r s2

ð8:70Þ

where s1 ¼ k2 kGA v2 rA; s2 ¼ k2 EI þ kGA v2 rI s3 ¼ ikkGA;

s4 ¼ ivWrJ

ð8:71Þ

From Equation 8.70, we can get a dispersion equation as k8 2ðh1 kF4 þ kG4 Þk6 þ ðh21 kF8 þ 4h1 kG4 kF4 þ kG8 2kF h2 2 h23 W2 v2 Þk4 þð2h21 kG4 kF8 2h1 kG8 kF4 þ 2h1 kF8 þ 2kF8 þ 2kF4 h2 1 h23 W2 v2 Þk2

ð8:72Þ

þ h21 kG8 kF8 2h1 kG4 kF8 þ ð1 h23 W2 v2 ÞkF8 ¼ 0 where

pffiffiffiffi rA 1=4 pffiffiffiffi rA 1=4 v ; kG ¼ v EI kGA rI EI rJ ; h2 ¼ ; h3 ¼ h1 ¼ rA kGA kGA kF ¼

ð8:73Þ

Equation 8.72 gives eight wavenumbers as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi a ð1=2Þh2 1 ð b þ g h3 WvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k3 ¼ k4 ¼ a þ ð1=2Þh2 1 ð b þ g þ h3 WvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k5 ¼ k6 ¼ a ð1=2Þh2 1 ð b g þ h3 WvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k7 ¼ k8 ¼ a þ ð1=2Þh2 1 ð b g h3 WvÞ

k1 ¼ k2 ¼

where

ð8:74Þ

a ¼ ð1=2Þðh1 þ h2 ÞkF4 b ¼ ðh1 h2 Þ2 kG8 þ 4h2 kG4 þ h23 W2 v2 g ¼ 2ðh1 h2 ÞkG4 h3 Wv

ð8:75Þ

233


By substituting kj (i ¼ 1, 2, . . ., 8) into Equation 8.70, we can obtain rj ¼ ikj 1 ðkG4 kj2 Þ ^rj ¼ rj

ðj ¼ 1; 2; . . . ; 8Þ

ð8:76Þ

tj ¼ ðih3 WvÞ 1 rj 1 ½ikj þ ðh2 kj2 h1 kG4 Þrj By using eight wavenumbers kj (i ¼ 1, 2, . . ., 8) computed from Equation 8.74, the homogeneous solutions of Equation 8.67 can be expressed as follows: VðxÞ ¼ ev ðxÞaB ;

WðxÞ ¼ ew ðxÞaB

ð8:77Þ

FðxÞ ¼ ew ðxÞaB ; YðxÞ ¼ ec ðxÞaB where aB ¼ {a1, a2, . . ., a8}T is the constant vector and ev ðxÞ ¼ ½e ik1 x

eik1 x

e ik3 x

eik3 x

e ik5 x

eik5 x

e ik7 x

eik7 x

ec ðxÞ ¼ ev ðxÞR

ð8:78Þ

ew ðxÞ ¼ ev ðxÞT ef ðxÞ ¼ ev ðxÞTR where R and T are diagonal matrices defined by R ¼ diag½rj ;

T ¼ diag½tj

ðj ¼ 1; 2; . . . ; 8Þ

ð8:79Þ

Define the spectral nodal DOFs vector for the transverse bending vibration as dB ¼ f V 1

Y1

W1

F1

fVð0Þ Yð0Þ Wð0Þ

V2

Y2

W2

F2 g T

Fð0Þ VðLÞ YðLÞ WðLÞ FðLÞg T

ð8:80Þ

Applying Equation 8.77 to Equation 8.80, we can obtain dB ¼ HB ðvÞaB

ð8:81Þ

where HB ðvÞ ¼ ½eTv ð0Þ

eTc ð0Þ

eTw ð0Þ

eTf ð0Þ

eTv ðLÞ eTc ðLÞ eTw ðLÞ

eTf ðLÞ T

ð8:82Þ

By using Equation 8.81, Equation 8.77 can be written as VðxÞ ¼ Nv ðx; vÞ dB ;

WðxÞ ¼ Nw ðx; vÞ dB

YðxÞ ¼ Nc ðx; vÞ dB ; FðxÞ ¼ Nf ðx; vÞ dB

ð8:83Þ

where Nv, Nc, Nw, and Nf are the dynamic shape functions defined by Nv ðx; vÞ ¼ ev ðxÞ HB1 ;

Nw ðx; vÞ ¼ ew ðxÞ HB1

Nc ðx; vÞ ¼ ec ðxÞ HB1 ;

Nf ðx; vÞ ¼ ef ðxÞ HB1

ð8:84Þ

234


To formulate the spectral element equation, the weak form of Equation 8.67 is derived from ðL

f½kGAðV 00 Y0 Þ þ v2 rAV þ Py dV

0

þ ½EIY00 þ ivWrJF þ kGAðV 0 YÞ þ v2 rIY þ Tz dY 00

0

ð8:85Þ

þ ½kGAðW þ F Þ þ v rAW þ Pz dW 2

þ ½EIF00 ivWrJY kGAðW 0 þ FÞ þ v2 rIF þ Ty dFg dx ¼ 0 On integrating by parts and then using Equation 8.68, we can obtain the weak forms as ðL

0

0

0

ðL

0

EIðF dF þ Y dY Þdx þ

0

ðL

0

0

kGAðYdV þ V dYÞdx þ

0

þ

ðL

kGAðFdF þ YdYÞdx

kGAðFdW 0 þ W 0 dFÞdx

ðL

v2 rAðVdV þ WdWÞdx

ð8:86Þ

0

v2 rIðFdF þ YdYÞdx þ

0

ðL

ðL 0

0

ðL

kGAðV 0 dV 0 þ W 0 dW 0 Þdx

0

ðL

ivWrJðYdF FdYÞdx

0

ðPy dV þ Tz dY þ Pz dW þ Ty dFÞdx

0

Qy dVjL0 þ Mz dYjL0 þ Qz dWjL0 þ My dFjL0 ¼ 0 Substituting Equation 8.83 into Equation 8.86 and applying the discrete Fourier transformed boundary conditions Equation 8.10 gives ðL ðL T T T T ddTB ð EIðN0 f N0 f þ N0 c N0 c Þ dx þ kGAðN0 v N0 v þ N0 w N0 w Þ dx 0

ðL 0

þ

ðL 0

ðL 0

0

kGAðN0 v Nc þ NTc N0 v Þ dx þ T

kGAðNTf Nf þ NTc Nc Þ dx v2 rIðNTf Nf þ NTc Nc Þ dx þ

þ Ksupport þ ivCsupport ÞdB

ðL 0

ðL

kGAðN0 w Nf þ NTf N0 w Þ dx T

0

ðL 0

ðL 0

ð8:87Þ

v2 rAðNTv Nv þ NTw Nw Þ dx ivWrJðNTf Nc NTc Nf Þ dx

ðPy NTv

þ Tz NTc

þ Pz NTw

þ Ty NTf Þ dx fBc

¼0

235


where fBc is the spectral nodal point forces and moments vector defined by fBc ¼ f Qy1

Mz1

Qz1

My1

Qy2

Mz2

and Ksupport and Csupport are the matrices defined by K1 0 C1 Ksupport ¼ ; Csupport ¼ 0 K2 0 where

2

Kyyi

6 6 0 Ki ¼ 6 6 1 ðK þK Þ 4 2 yzi zyi 0 2

Cyyi 6 6 0 Ci ¼ 6 6 1 ðC þC Þ 4 2 yzi zyi 0

My2 g T

Qz2

0

1 2 ðKyzi þKzyi Þ

0

0

0

Kzzi

0

0

0

1 2 ðCyzi þCzyi Þ

0

0

0

Czzi

0

0

0 C2

0

ð8:88Þ

ð8:89Þ

3

7 07 7 07 5 0

0

3

ði ¼ 1; 2Þ

ð8:90Þ

7 07 7 07 5 0

As ddB is arbitrary by definition, we can obtain the spectral element equation for the transverse bending vibration from Equation 8.87 as SB ðvÞdB ¼ fB where

ðL

SB ðvÞ ¼

0

0 EIðN0T f Nf

ðL 0

þ

ðL 0

ðL 0

0 þ N0T c Nc Þ dx þ

ðL 0

ð8:91Þ

0 0T 0 kGAðN0T v Nv þ Nw Nw Þ dx

T 0 kGAðN0T v Nc þ Nc N v Þ dx þ

kGAðNTf Nf þ NTc Nc Þ dx v2 rIðNTf Nf þ NTc Nc Þ dx þ

ðL 0

ðL 0

T 0 kGAðN0T w Nf þ Nf Nw Þ dx

v2 rAðNTv Nv þ NTw Nw Þ dx

ðL 0

ð8:92Þ

ivWrJðNTf Nc NTc Nf Þ dx

þ Ksupport þ ivCsupport and fB ¼ fBc þ ¼ fFv1

ðL 0

½Py ðxÞNTv ðxÞ þ Tz ðxÞNTc ðxÞ þ Pz ðxÞNTw ðxÞ þ Ty ðxÞNTf ðxÞ dx

Fc1

Fw1

Fv2

Ff1

Fc2

Fw2

Ff2 g

ð8:93Þ

T

The last integral of Equation 8.92 forms a skew symmetric matrix, which represents the gyroscopic effect generated by the spinning of the shaft.

236


By substituting the dynamic shape function matrices Equation 8.84 into Equation 8.92, the spectral element matrix SB(v) can be written in the form of sB11 sB12 T 1 SB ðvÞ ¼ HB DB HB þ Ksupport þ ivCsupport ð8:94Þ sB21 sB22 where DB ðvÞ ¼ EI

ðL 0

0 0T 0 ðe0T f ef þ ec ec Þ dx þ kGA

kGA

ðL 0

þ kGA

ðL 0

v2 rI

0

0 0T 0 ðe0T v ev þ ew ew Þ dx

T 0 ðe0T v ec þ ec ev Þ dx þ kGA

ðeTf ef

ðL 0

ðL

ðL

þ eTc ec Þ dx v2 rA

0

T 0 ðe0T w ef þ ef ew Þ dx

ðL 0

ðeTf ef þ eTc ec Þ dx þ ivWrJ

ð8:95Þ ðeTv ev

ðL 0

þ eTw ew Þ dx

ðeTf ec eTc ef Þ dx

and sB11, sB12, sB21, and sB22 are 4-by-4 submatrices of SB(v). Substitution of Equation 8.78 into Equation 8.95 yields DB ðvÞ ¼ EIðRTKEKTR þ RKEKRÞ kGAðKEK þ TKEKTÞ þ ikGAðKER þ REKÞ þ ikGAðTKETR þ RTEKTÞ þ kGAðRTETR þ RERÞ v2 rAðE þ TE TÞ

ð8:96Þ

v2 rIðRTETR þ RERÞ ivWrJðRTER RETRÞ where EðvÞ ¼

Elm ¼

8 < :

ðL 0

eTv ev dx ½Elm

ðl; m ¼ 1; 2; . . . ; 8Þ

h i i e iðkl þ km ÞL 1 kl þ km

if kl þ km 6¼ 0

L

if kl þ km ¼ 0

ð8:97Þ

ð8:98Þ

and K ¼ diag½kj ðj ¼ 1; 2; . . . ; 8Þ

ð8:99Þ

Similarly, the second term in the right-hand side of Equation 8.93 can be rewritten as fBd ¼ ¼

ðL 0

½Py ðxÞNTv ðxÞ þ Tz ðxÞNTc ðxÞ þ Pz ðxÞNTw ðxÞ þ Ty ðxÞNTf ðxÞ dx

ÐL HB T 0 ðPy I þ Tz RT

þ P z R þ Ty R T

T

ð8:100Þ

Þ eTv dx

8.2.1.4 Spectral Element Equation for the Spinning Shaft The spectral element equations (8.48), (8.62) and (8.91) derived for individual axial vibration, torsional vibration, and transverse bending vibration can be combined to derive the spectral

237


element equation for the spinning shafts subjected to the longitudinal, torsional, and transverse bending vibrations simultaneously as follows: S S ðvÞd S ¼ f S or

"

S S11 S S21

where

2

S S12 S S22

sLij

#(

)

d1S d2S

6 SSij ¼ 4 0

sTij

0

0

dj S ¼ fUj

Qj

Vj

fj S ¼ fFuj

Fuj

( ¼

f1S

) ð8:102Þ

f2S

3

0

0

ð8:101Þ

7 0 5

ði; j ¼ 1; 2Þ

ð8:103Þ

sBij

and Yj

Fvj

Fj g T

Wj

Fcj

Ffj gT

Fwj

ðj ¼ 1; 2Þ

ð8:104Þ

8.2.2 Spectral Element for the Disk 8.2.2.1 Thin Disk Represent the solutions to Equation 8.27 and the associated excitation forces in the spectral forms as fuðtÞ uðtÞ vðtÞ cðtÞ wðtÞ fðtÞg ¼

1 1 NX fUnD N n¼0

QD n

D D fQ D x ðtÞ Qy ðtÞ Qz ðtÞ

¼

1 N

f fyme ðtÞ

N 1 X

fQD xn

n¼0 fzme ðtÞg

QD yn ¼

VnD

YD n

MxD ðtÞ QD zn

WnD MyD ðtÞ

D Mxn

1 1 NX f F me N n¼0 yn

ivn t FD nge

D Myn

me Fzn geivn t

MzD ðtÞg

(8.105)

D Mzn g eivn t

ð8:105Þ

Substituting Equation 8.105 into Equation 8.27 yields v2 MU D ¼ QD x v2 JD QD ¼ MxD me v2 MV D ¼ QD y þ Fy

v2 ID YD ivJD WFD ¼ MzD

ð8:106Þ

me v2 MW D ¼ QD z þ Fz

v2 ID FD þ ivJD WYD ¼ MyD where the subscripts n are omitted for brevity. From Equation 8.106, we can obtain the spectral element equation for a thin rigid disk as SD ðvÞd D ¼ f

D

ð8:107Þ

238


where d D ¼ fU D

QD

f D ¼ fQ D x

MxD

YD

VD

WD

me QD y þ Fy

MzD

and

" D

S ðvÞ ¼ with

2

SD 11

M 6 ¼ v2 4 0 2

0

FDg

T

me QD z þ Fz

D S11

0

0

D S22

MyD gT

# ð8:109Þ

3 0 7 0 5 M

0 JD 0

v2 ID

0

ivWJD

0

v2 M

0

ivWJD

0

v2 ID

6 SD 22 ¼ ð 1Þ4

ð8:108Þ

3

ð8:110Þ

7 5

8.2.2.2 Thick Disk Represent the solutions to Equation 8.36 and the associated excitation forces in the spectral forms as fui ðtÞ ui ðtÞ vi ðtÞ ¼

1 N

fQ D xi ðtÞ ¼

N 1 X n¼0

fUinD

QD in

VinD

wi ðtÞ fi ðtÞg YD in

D D QD yi ðtÞ Qzi ðtÞ Mxi ðtÞ

1 1 NX fQD xin N n¼0

f fyme ðtÞ

ci ðtÞ

fzme ðtÞg ¼

QD yin

QD zin

1 1 NX fF me N n¼0 yn

WinD

ivn t FD in g e

MyiD ðtÞ MziD ðtÞg D Mxin

D Myin

ð8:111Þ

D Mzin g eivn t

me Fzn g eivn t

ð8:111Þ

where i ¼ r and l. Substituting Equation 8.111 into the equations of motion (8.36) and the kinematic relationships (8.28) and (8.29), we get v2 MUlD ¼ QD x D v2 JD QD l ¼ Mx D me v2 MðVlD þ hYD l Þ ¼ Q y þ Fy D D 2 D D D v2 ID YD l ivJD WFl v MhðVl þ hYl Þ ¼ Mz þ 2hQyr D me v2 MðWlD hFD l Þ ¼ Q z þ Fz D D 2 D D D v2 ID FD l þ ivJD WYl þ v MhðWl hFl Þ ¼ My 2hQzr

ð8:112Þ

239


and UrD UlD ¼ 0;

D QD r Ql ¼ 0

VrD VlD 2hYD l ¼ 0;

D YD r Yl ¼ 0

ð8:113Þ

D D WrD WlD þ 2h FD l ¼ 0; Fr Fl ¼ 0

From Equations 8.112 and 8.113, we can obtain the spectral element equation for a thick rigid disk as " #( ) ( ) dD fD SD 0 l 11 ð8:114Þ ¼ dD 0 SD SD r 21 22 where D dD l ¼ fUl

QD l

VlD

YD l

WlD

FD l g

T

D dD r ¼ fUr

QD r

VrD

YD r

WrD

FD r g

T

¼ fQD x

MxD

me QD y þ Fy

f

D

ð8:115Þ

MzD þ 2hQD yr

me QD z þ Fz

MyD 2hQD zr g

T

and 2

SD 11 SD 22

v2 m1 4 ¼ ð 1Þ 0 0 ¼ I6

3 0 0 v 2 m2 ivg 5; ivg v2 m3

2

SD 21

I2 ¼ ð 1Þ4 0 0

0 h1 0

3 0 05 h2

ð8:116Þ

where In is the n-by-n identity matrix and m1 ¼ m3 ¼ h1 ¼

M 0

0 ; JD

M Mh 1 0

Mh ; ID þ Mh2

2h ; 1

m2 ¼ g¼ h2 ¼

M Mh 0 0 1 0

Mh ID þ Mh2 0 WJD 2h 1

ð8:117Þ

8.2.3 Assembly of Spectral Elements The rotor systems may consist of multiple spinning shafts and interim disks, supported by multiple supports. The spectral element models for such rotor systems can be derived by assembling spectral elements for each of the spinning shaft elements and disks. The assembly can be readily conducted by adopting the same assembling procedure as used in the conventional FEM. As an illustrative example, we will consider the assembly for the shaft with an interim disk as shown in Figure 8.5.

240


Figure 8.5

Spinning shaft with an interim rigid disk

8.2.3.1 Shaft with a Thin Disk For a shaft with an interim thin disk, the spectral element equations for the shaft (1), the disk (D), and the shaft (2) are given by as follows: 2 1 38 9 8 9 S11 S112 < dl1 = < fl1 = 4 5 ¼ ðshaft 1Þ : 1; : 1; 1 1 fr S21 S22 dr S D ðvÞd D ¼ f D 2 2 38 9 8 9 S11 S212 < dl2 = < fl2 = 4 5 ¼ : 2; : 2; 2 2 fr S21 S22 dr

ð8:118Þ

ðdisk DÞ ðshaft 2Þ

Define the global spectral nodal DOFs as d1 d1l ¼ fU1

Q1

V1

d2 d1r ¼ dD ¼ d2l ¼ fU2 d3 d2r ¼ fU3

Q3

V3

Y1

W1

F1gT

V2

Y2

W3

F3gT

Q2 Y3

W2

F2gT ð8:119Þ

By using the above global spectral nodal DOFs, we can assemble the spectral element equations given in Equation 8.118 to obtain the global dynamic stiffness equation in the form 2 1 38 9 8 9 S11 S112 0 > = > = < d1 > < F1 > 6 1 1 D 2 2 7 ð8:120Þ 4 S21 S22 þ S þ S11 S12 5 d2 ¼ F2 > ; > ; : > : > 2 2 d3 F3 0 S21 S22 where F1 fl1 ¼ fFul1

1 Ful

F2 f 1r þ f 2l þ f

D

Fvl1

1 2 þ Fcl þ MzD Fcr 2 r

2 ¼ fFur

2 Fur

1 Fwl

1 T Ffl g

1 2 Fur þ Ful þ MxD

1 ¼ fFur þ Ful2 þ QD x

F3 f

1 Fcl

1 me Fvr þ Fvl2 þ QD y þ Fy

1 2 me Fwr þ Fwl þ QD z þ Fz

2 Fvr

2 Fcr

2 Fwr

2 T Ffr g

ð8:121Þ

1 2 Ffr þ Ffl þ MyD gT

ð8:121Þ

241


8.2.3.2 Shaft with a Thick Disk When the interim disk is a thick disk, the spectral element equations for the shaft (1), the disk (D), and the shaft (2) are given by as follows: " 1 #( 1 ) ( 1 ) fl S11 S112 dl ¼ ðshaft 1Þ 1 1 1 fr1 S21 S22 dr " D #( D ) ( D ) dl S11 0 f ðDisk DÞ ¼ ð8:122Þ D D D S21 S22 dr 0 " 2 #( 2 ) ( 2 ) fl S11 S212 dl ¼ ðshaft 2Þ 2 2 2 fr2 S21 S22 dr Define the global spectral nodal DOFs as Q1

V1

Y1

W1

F1 g T

d2 d1r ¼ dD l ¼ fU2

Q2

V2

Y2

W2

F2gT T

d1 d1l ¼ fU1 d3

dD r

¼

d2l

¼ fU3

d4 d2r ¼ fU4

Q4

Q3 V4

V3

Y3

W3

Y4

W4

F4gT

F3g

(8.123) ð8:123Þ

By using the global spectral nodal DOFs defined above, we can obtain assemble the spectral element equations given in Equation 8.122 in the form 9 38 9 8 2 1 d1 > > fl1 > S11 S112 0 0 > > > > > > > > > > > > 7> 6 1 1 D = > = < d2 > < f1 þ f D > 7> 6S S þ S 0 0 r 22 11 7 6 21 ¼ ð8:124Þ 7 6 2 7> 2 > > > 6 0 SD S222 þ SD > > > > d3 > > fl > 4 21 11 S12 5> > > > > > > ; > ; : > : d4 0 0 S221 S222 f 2r where f 1l ¼ fFul1

1 Ful

Fvl1

1 Fcl

1 Fwl

1 T Ffl g

1 f 1r ¼ fFur

1 Fur

1 Fvr

1 Fcr

1 Fwr

1 T Ffr g

f D ¼ fQD x

MxD

me QD x þ Fy

MzD þ 2hQD yr

f 2l ¼ fFul2

Ful2

Fvl2

2 Fcl

2 Fwl

2 f 2r ¼ fFur

2 Fur

2 Fvr

2 Fcr

2 Fwr

me QD z þ Fz

T MyD 2hQD zr g

(8.125)

2 T Ffl g

ð8:125Þ

2 T Ffr g

From Equation 8.113, we obtain the relationship D dD r ¼ TD d l

or

d3 ¼ TD d2

ð8:126Þ

242


where

2

I2 4 TD ¼ 0 0

0 h1 0

3 0 05 h2

ð8:127Þ

where I2 is the 2-by-2 identity matrix, h1 and h2 are the 2-by-2 matrices defined by Equation 8.117. By using Equation 8.126, Equation 8.124 can be condensed into the following form 2 1 38 9 8 9 S11 S112 0 > d1 > > F1 > 6 1 7< = < = 1 6S D þ S 2 S 2 7 d2 ¼ F2 ð8:128Þ S þ S 22 11 12 5> 4 21 ; > ; : > : > 2 d4 F4 0 S S222 21 where 2 ¼ TT S2 TD ; S D 11 11

2 ¼ TT S2 ; S D 12 12

2 ¼ S2 TD S 21 21

D ¼ SD þ TT SD þ TT SD TD S 11 D 21 D 22

ð8:129Þ

and F1 f 1l F2 f 1r þ f D þ TTD fl2 ¼ f F2U

F2Q

F2V

F2Y

F2W

F4 f 2r with

F2F gT ð8:130Þ

1 2 F2U ¼ Fur þ QD x þ Ful 1 2 F2Q ¼ Fur þ MxD þ Ful 1 me 2 F2V ¼ Fvr þ QD y þ Fy þ Fvl 1 2 2 F2Y ¼ Fcr þ MzD þ Fcl þ 2hQD yr þ 2hFvl

ð8:131Þ

1 me 2 F2W ¼ Fwr þ QD z þ Fz þ Fwl 1 2 2 F2F ¼ Ffr þ MyD þ Ffl 2hQD zr 2hFwl

If there are no external forces, except for the centrifugal unbalance forces, Equation 8.130 becomes F1 ¼ F4 ¼ 0 ð8:132Þ T F2 ¼ f D ¼ f 0 0 Fyme 0 Fzme 0 g If we set the disk thickness to 2h ¼ 0, Equation 8.128 for the thick rigid disk returns to Equation 8.120 for the thin rigid disk.

8.3 Finite Element Model Similar to the spectral element modeling, the finite element modeling of a rotor system with rigid disks also consists of the finite element modeling for the axial, torsion and transverse bending vibrations of the spinning shafts and the finite element modeling for the dynamics of the disks.

243


8.3.1 Finite Element for the Spinning Shaft 8.3.1.1 Finite Element for the Axial Vibration The equation of motion for the axial vibration of a shaft is given by Equation 8.12 and its weak form is given by ðL

EAu0 du0 dx þ

0

ðL

rA€ u du dx ¼ Qx1 du1 þ Qx2 du2 þ

0

ðL

px du dx

ð8:133Þ

0

where Qx1 and Qx2 are the nodal axial forces shown in Figure 8.1(a). The displacement field within the element is represented by uðx; tÞ ¼ NL ðxÞdL ðtÞ

ð8:134Þ

where NL ðxÞ ¼ ½1 ðx=LÞ x=L dL ðtÞ ¼ fu1 ðtÞ

u2 ðtÞgT ¼ fuð0; tÞ

uðL; tÞgT

ð8:135Þ

By substituting Equation 8.134 into Equation 8.133, we obtain ML € d L ðtÞ þ KL dL ðtÞ ¼ fL ðtÞ where

ð8:136Þ

" # rAL 2 1 ¼ ML ¼ ¼ mLij ði; j ¼ 1; 2Þ 6 1 2 0 " # ðL 1 1 EA T ¼ kLij ði; j ¼ 1; 2Þ KL ¼ EAN0 L N0 L dx ¼ L 1 1 0 ðL fL ðtÞ ¼ fQx1 ðtÞ Qx2 ðtÞg T þ NTL ðxÞpx ðx; tÞdx ðL

rANTL NL dx

ð8:137Þ

0

¼ f fu1 ðtÞ fu2 ðtÞg

T

8.3.1.2 Finite Element for the Torsional Vibration The equation of motion for the torsional vibration is given by Equation 8.13 and its weak form is given by ðL ðL ðL GJu0 du0 dx þ rJ € udu dx ¼ Mx1 du1 þ Mx2 du2 þ tx du dx ð8:138Þ 0

0

0

where Mx1 and Mx2 are the nodal torques shown in Figure 8.1(a). The displacement field u(x, t) is represented by uðx; tÞ ¼ NT ðxÞdT ðtÞ

ð8:139Þ

244


where NT ðxÞ ¼ ½ 1 ðx=LÞ x=L dT ðtÞ ¼ f u1 ðtÞ u2 ðtÞ gT ¼ f uð0; tÞ

uðL; tÞ gT

ð8:140Þ

By substituting Equation 8.139 into Equation 8.138, we obtain MT € d T ðtÞ þ KT dT ðtÞ ¼ fT ðtÞ

ð8:141Þ

where " # rJL 2 1 MT ¼ ði; j ¼ 1; 2Þ ¼ ¼ mTij 6 1 2 0 " # ðL 1 1 GJ T KT ¼ ði; j ¼ 1; 2Þ GJN0 T N0 T dx ¼ ¼ kTij L 1 1 0 ðL T fT ðtÞ ¼ fMx1 ðtÞ Mx2 ðtÞg þ NTT ðxÞtx ðx; tÞ dx ðL

rJNTT NT dx

ð8:142Þ

0

¼ f fu1 ðtÞ fu2 ðtÞg

T

8.3.1.3 Finite Element for the Transverse Bending Vibration The equations of motion for the transverse bending vibration are given by Equations 8.14–8.17 and their weak form is derived as ðL ðL EIðf0 df0 þ c0 dc0 Þdx þ kGAðv0 dv0 þ w0 dw0 Þ dx 0

ðL

0

kGAðcdv0 þ v0 dcÞdx þ

0

þ þ

ðL 0 ðL

kGAðfdw0 þ w0 dfÞ dx

0

kGAðfdf þ cdcÞdx þ € þ cdcÞdx € rIðfdf þ

0

ðL

ðL ðL

ðL

€ dwÞ dx rAð€v dv þ w

0

_ _ WrJðcdf fdcÞ dx

ð8:143Þ

0

ðpy dv þ tz dc þ pz dw þ ty dfÞ dx

0

ðQy1 dv1 þ Mz1 dc1 þ Qz1 dw1 þ My1 df1 þ Qy2 dv2 þ Mz2 dc2 þ Qz2 dw2 þ My2 df2 Þ ¼ 0 where Qyj and Myj (j ¼ 1, 2, 3) are the forces and moments shown in Figure 8.1(a). The displacement fields v(x, t), w(x, t), f(x, t), and c(x, t) are represented by v ¼ Nv ðxÞdB ðtÞ;

w ¼ Nw ðxÞdB ðtÞ

f ¼ Nf ðxÞdB ðtÞ; c ¼ Nc ðxÞdB ðtÞ

ð8:144Þ

245


where

dB ðtÞ ¼ fdB1 ðtÞ

dB2 ðtÞg T

dBj ðtÞ ¼ f vj ðtÞ cj ðtÞ and

wj ðtÞ

fj ðtÞ gT

Nv ðxÞ ¼ ð1 jÞð2 j j2 þ 6rÞðR=4Þ Lð1 j2 Þð1 j þ 3rÞðR=8Þ ð1 þ jÞð2 þ j j2 þ 6rÞðR=4Þ

Nw ðxÞ ¼ 0 Nf ðxÞ ¼ 0

0 0

Lð1 j2 Þð1 þ j þ 3rÞðR=8Þ

0 ð1 jÞð2 j j2 þ 6rÞðR=4Þ

0

ð8:145Þ

ðj ¼ 1; 2Þ

0

0

Lð1 j2 Þð1 j þ 3rÞðR=8Þ

0

ð1 þ jÞð2 þ j j2 þ 6rÞðR=4Þ Lð1 j2 Þð1 þ j þ 3rÞðR=8Þ

0

3ð1 j2 ÞðR=2LÞ

ð1 jÞð1 þ 3j 6rÞðR=4Þ

ð8:146Þ

0 0 3ð1 j ÞðR=2LÞ ð1 þ jÞð1 3j 6rÞðR=4Þ Nc ðxÞ ¼ 3ð1 j2 ÞðR=2LÞ ð1 jÞð1 þ 3j 6rÞðR=4Þ 0 0 2

0

ð8:146Þ

4EI 1 ; R¼ 1 ð0 x LÞ; r¼ L kGAL2 1 þ 3r Substitution of Equation 8.144 into Equation 8.143 gives € B ðtÞ þ GB d_ B ðtÞ þ KB dB ðtÞ ¼ f ðtÞ MB d B where ð ð

ð8:147Þ

3ð1 j2 ÞðR=2LÞ with j¼2

ð1 þ jÞð1 3j 6rÞðR=4Þ 0

x

L

MB ¼ ½mBij ¼

0

GB ¼ ½gBij ¼

ðL 0

KB ¼ ½kBij ¼

ðL 0

L

rAðNTv Nv þ NTw Nw Þ dx þ

WrJðNTf Nc

0

rIðNTf Nf þ NTc Nc Þ dx

NTc Nf Þ dx ðL

0 0T 0 EIðN0T f Nf þ Nc Nc Þ dx þ

0

ð8:148Þ

ð8:149Þ kGAðN0 v N0 v þ N0 w N0w T

T

þ NTf Nf þ NTc Nc þ N0 w Nf þ NTf N0w N0 v Nc NTc N v0 Þ dx T

and

T

fB ðtÞ ¼ fBc ðtÞ þ fBd ðtÞ f fB1 ðtÞ fB2 ðtÞ gT fBc ðtÞ ¼ f Qy1 ðtÞ Mz1 ðtÞ Qz1 ðtÞ My1 ðtÞ Qy2 ðtÞ ðL fBd ðtÞ ¼ ðNTv py þ NTw pz þ NTf ty þ NTc tz Þ dx

Mz2 ðtÞ Qz2 ðtÞ My2 ðtÞ gT ð8:150Þ

0

fBi ðtÞ ¼ f fvi ðtÞ

fci ðtÞ

fwi ðtÞ ffi ðtÞ gT

ði ¼ 1; 2Þ

MB and KB are the 8-by-8 symmetric matrices and GB is the 8-by-8 skew symmetric matrix. The matrices MB, KB, and GB are given in Appendix 8.A. For the next use, the matrices MB, KB, and GB and the vector fB are written in the partitioned forms as " # " # " # MB11 MB12 GB11 GB12 KB11 KB12 MB ¼ ; GB ¼ ; KB ¼ ð8:151Þ MB21 MB22 GB21 GB22 KB21 KB22

246


8.3.1.4 Finite Element Equation for the Spinning Shaft The finite element for a spinning shaft can be obtained by assembling the finite element equations for the axial, torsion, and transverse bending vibrations as follows: d_ S ðtÞ þ ðK S þ KÞd S ðtÞ ¼ f S ðtÞ MS € d S ðtÞ þ ðGS þ CÞ

ð8:152Þ

where

and

d S ðtÞ ¼ f u1 ðtÞ u1 ðtÞ

dB1 ðtÞ

f S ðtÞ ¼ f fu1 ðtÞ

fB1 ðtÞ fu2 ðtÞ

" S

M ¼

S M11

S M12

S M21

S M22

fu1 ðtÞ

#

" S

G ¼

;

dB2 ðtÞgT

u2 ðtÞ u2 ðtÞ

S G11

S G12

S G21

S G22

#

" S

; K ¼

ð8:153Þ

fB2 ðtÞgT

fu2 ðtÞ

S K11

S K12

S K21

S K22

# ð8:154Þ

with 2

mLij

6 MSij ¼ 4 0

0

3

0

0

2

0

7 6 0 5; GSij ¼ 4 0

mTij 0

0

MBij

0 0 0

0

3

2

kLij

7 6 0 5; KSij ¼ 4 0 0

GBij

0 kTij 0

3

0

7 0 5 KBij ð8:155Þ

are the stiffness and damping matrices for the bearing supports and C In Equation 8.152, K shown in Figure 8.1(c). They are given by 2

0 60 ¼6 K 40 0

0 K1 0 0

0 0 0 0

3 2 0 0 60 0 7 ¼6 7; C 40 0 5 K2 0

0 C1 0 0

0 0 0 0

3 0 0 7 7 0 5 C2

ð8:156Þ

where Ki and Ci (i ¼ 1, 2) are the 4-by-4 matrices defined by Equation 8.90.

8.3.2 Finite Element for the Disk 8.3.2.1 Thin Disk The equations of motion for a thin rigid disk are given by Equation 8.27 and they can be written in the matrix equation form as M D€ d D ðtÞ þ GD d_ D ðtÞ ¼ f D ðtÞ where

2

m1 MD ¼ 4 0 0

0 2 m 0

3 2 0 0 0 5; GD ¼ 4 0 3 m 0

ð8:157Þ 3 0 0 0 g5 g 0

ð8:158Þ

247


and d D ðtÞ ¼ fuD ðtÞ uD ðtÞ f D ðtÞ ¼ fQD x ðtÞ

MxD ðtÞ

me QD z ðtÞ þ fz ðtÞ

vD ðtÞ cD ðtÞ wD ðtÞ fD ðtÞg me QD y ðtÞ þ fy ðtÞ

MzD ðtÞ

T

ð8:159Þ

MyD ðtÞgT

The matrices m1 and g are defined by Equation 8.117 and M 0 3 ¼ 2 ¼ m m 0 ID

ð8:160Þ

8.3.2.2 Thick Disk The equations of motion for a thick rigid disk given by Equation 8.36 and the kinematic relationships given by Equations 8.28 and 8.29 can be written in the matrix equation form as D ( D ) D D D ( D ) € 0 0 dl dl d_ l f G 0 M 0 ¼ þ þ ð8:161Þ D D D I T d 0 0 0 0 0 € _ D 6 r dr dr where D D dD l ðtÞ ¼ f ul ðtÞ ul ðtÞ

D D vD l ðtÞ cl ðtÞ wl ðtÞ

fD l ðtÞ g

T

D D dD r ðtÞ ¼ f ur ðtÞ ur ðtÞ

D D vD r ðtÞ cr ðtÞ wr ðtÞ

fD r ðtÞ g

T

fD ðtÞ ¼ f QD x ðtÞ

MxD ðtÞ

me D D QD y ðtÞ þ fy ðtÞ Mz ðtÞ þ 2hQyr ðtÞ

ð8:162Þ

me D D T QD z ðtÞ þ fz ðtÞ My ðtÞ 2hQzr ðtÞ g

and

2

m1 MD ¼ 4 0 0

0 m2 0

3 2 0 0 0 5; GD ¼ 4 0 m3 0

3 0 0 0 g5 g 0

ð8:163Þ

The matrices m1, m2, m3, and g are defined by Equation 8.117 and TD by Equation 8.127. In Equation 8.161, the following kinematic relationship obtained from Equations 8.28 and 8.29 is used D dD ð8:164Þ r ðtÞ ¼ TD dl ðtÞ

8.3.3 Assembly of Finite Elements A thick rigid disk element “D” and its neighboring two shaft finite elements “1” and “2” can be assembled by using the same procedure as discussed in Section 8.2.3.2. The assembled finite element model can be represented by _ þK dðtÞ ¼ f ðtÞ € M dðtÞ þ G dðtÞ where

dðtÞ ¼ fd1 ðtÞ

d2 ðtÞ

fðtÞ ¼ ff 1 ðtÞ l

f 1r ðtÞ þ f D ðtÞ f 2l ðtÞ

ð8:165Þ

d3 ðtÞ d4 ðtÞgT f 2r ðtÞgT

ð8:166Þ

248


with d1 ðtÞ d 1l ðtÞ ¼ fu1 ðtÞ

u1 ðtÞ v1 ðtÞ

c1 ðtÞ

d2 ðtÞ d 1r ðtÞ ¼ d D l ðtÞ ¼ fu2 ðtÞ u2 ðtÞ

v2 ðtÞ

2 d3 ðtÞ d D r ðtÞ ¼ dl ðtÞ ¼ fu3 ðtÞ

v3 ðtÞ c3 ðtÞ

d4 ðtÞ dr2 ðtÞ ¼ fu4 ðtÞ u4 ðtÞ

u3 ðtÞ

f1 ðtÞgT

w1 ðtÞ

c2 ðtÞ w2 ðtÞ f2 ðtÞgT

v4 ðtÞ c4 ðtÞ

w3 ðtÞ

f3 ðtÞgT

ð8:167Þ

f4 ðtÞgT

w4 ðtÞ

and f 1l ðtÞ ¼ f ful1 ðtÞ ful1 ðtÞ

1 ðtÞ fvl1 ðtÞ fcl1 ðtÞ fwl

f 1r ðtÞ ¼ f fur1 ðtÞ

1 1 1 ðtÞ fwr ðtÞ ffr ðtÞ gT fvr1 ðtÞ fcr

f D ðtÞ ¼ fQD x ðtÞ

fur1 ðtÞ

1 ffl ðtÞ gT

(8.168)

me MxD ðtÞ QD y ðtÞ þ fy ðtÞ

MzD ðtÞ þ 2hQD yr ðtÞ

me QD z ðtÞ þ fz ðtÞ

fcl2 ðtÞ

T MyD ðtÞ 2hQD zr ðtÞ g

f 2l ðtÞ ¼ f ful2 ðtÞ

ful2 ðtÞ

fvl2 ðtÞ

2 2 fwl ðtÞ ffl ðtÞgT

f 2r ðtÞ ¼ f fur2 ðtÞ

fur2 ðtÞ

2 2 2 ðtÞ fwr ðtÞ ffr ðtÞgT fvr2 ðtÞ fcr

ð8:168Þ

and where 2

6 6 M1 6 21 M¼6 6 6 0 4 2

0

M122 þ MD

0

0

M211

0

M221

0 G111

6 6 G1 6 21 G¼6 6 6 0 4 2

M112

M111

G122 þ GD

0

0

G211

7 0 7 7 7 2 7 G12 7 5

0

G221

G222

6 6 K1 6 21 K¼6 6 6 0 4 0

0

K122

0

TD

I6 þ K211

0

K221

0

M222 3

0

K112

3

7 0 7 7 7 2 7 M12 7 5

G112

0 K111

0

0

ð8:169Þ

3

7 0 7 7 7 2 7 K12 7 5 K222

By using the kinematic relationship Equation 8.164, we can obtain ¼ RdðtÞ dðtÞ

ð8:170Þ

249


where 2

I6

6 60 6 R¼6 60 4 0

0 I6 TD 0

0

3

7 07 7 7 07 5 I6

ð8:171Þ

where I6 is the 6-by-6 identity matrix. Equation 8.165 can be condensed by using Equation 8.170 as follows: _ þ KdðtÞ ¼ fðtÞ M€ dðtÞ þ GdðtÞ

ð8:172Þ

where dðtÞ ¼ f d1 ðtÞ d2 ðtÞ M ¼ RT MR;

d4 ðtÞ gT ;

fðtÞ ¼ RTf ðtÞ

G ¼ RT GR; K ¼ RT KR

ð8:173Þ

8.4 Numerical Examples Example 8.1 Spinning Shaft without Disks Figure 8.6 shows a simply supported uniform Timoshenko shaft (T-shaft) without disks attached. The geometric and material properties used for numerical computation are: L ¼ 0.5 m, rA ¼ 4.0 kg/m, rI ¼ 1.5 104 kg m, rJ ¼ 3.0 104 kg m, EI ¼ 4.0 103 N m2, and kGA ¼ 3.5 107 N. In Tables 8.1–8.3, the natural frequencies and critical speeds obtained by both SEM and FEM are compared with the results obtained by using the formulae given by

Figure 8.6

Rotating shaft without disks

Table 8.1 Natural frequencies (Hz) of the non-spinning shaft FEM(N) Mode

N¼2

N ¼ 10

N ¼ 50

SEM(1)

Ref. [24]

1st mode 2nd mode

198.96 879.49

198.10 785.76

198.10 785.46

198.10 785.45

198.10 785.45

Note: (N) ¼ total number of finite elements used in the analysis.

250 Table 8.2

Spectral Element Method in Structural Dynamics Natural Frequencies (Hz) of the shaft spinning at W ¼ 3600 rpm FEM(N)

Modes 1st mode 2nd mode

Backward Forward Backward Forward

N¼2

N ¼ 10

N ¼ 50

SEM(1)

Ref. [176]

198.88 199.05 879.13 879.85

198.02 198.19 785.42 786.10

198.01 198.19 785.12 785.80

198.01 198.19 785.11 785.79

198.20 785.80

Note: (N) ¼ total number of finite elements used in the analysis. Table 8.3 Critical speeds (Hz) of the spinning shaft FEM(N) Modes 1st mode 2nd mode


N¼2

N ¼ 10

N ¼ 50

SEM(1)

Ref. [45]

198.96 199.26 874.29 884.78

197.82 198.40 781.33 790.26

197.81 198.39 781.03 789.96

197.81 198.39 781.02 789.95

197.81 198.39 780.91 789.99

Note: (N) ¼ total number of finite elements used in the analysis.

a. Natural frequencies of the non-spinning T-shaft [24] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fn ¼ fn jBE an

bn

b2n h11 h21

ðHzÞ

b. Natural frequencies of the spinning T-shaft (at speed W) [176] ffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h5 sin L s1 þ s21 4s2 h51 ¼0 2

ðHzÞ

c. Critical speeds of the spinning T-shaft [45] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 > a2n > > fn j ðHzÞ ðforward whirlingÞ > B > 2 < an þ h2 h1 Wn ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > a2n > > ðHzÞ ðbackward whirlingÞ : fn jB a2n þ h2 þ 3h1 where h1, h2, and h3 are defined by Equation 8.73 and fn jB are the natural frequencies of the simply supported, non-spinning uniform Bernoulli–Euler beam given by fn jB ¼

1 pffiffiffiffiffi 2pa2n h5

ðHzÞ

251


The other symbols used in above formulas are defined by h4 ¼

rI ; kGA

an ¼

L ; np

rA rJ ; h6 ¼ EI rA 1 bn ¼ h11 þ ð1 þ a2n h11 Þh21 2 h5 ¼

s1 ¼ ðh1 þ h2 Þv2 h6 Wv;

s2 ¼ h4 v4 h3 Wv3 v

Tables 8.1–8.3 show that the natural frequencies and critical speeds obtained by SEM are very close to the results obtained by using the theoretical formulae, and the FEM results certainly converge towards the SEM results as the total number of finite elements used for FEM is increased.

Example 8.2

Spinning Shaft with a Disk

Figure 8.7 shows a simply supported uniform T-shaft with a thick disk. The geometric and material properties for the T-shaft are the same as used in Example 8.1. The stiffness and damping coefficients of the bearing supports at boundaries are Kyy ¼ Kzz ¼ 1.0 109 N/m, Kyz ¼ Kzy ¼ 0 N/m, Cyy ¼ Czz ¼ 400 N s/m, and Cyz ¼ Czy ¼ 0 N s/m. The disk has the thickness 2h ¼ 20 mm, the uniform mass mD ¼ 2 kg, the moment of inertia ID ¼ 2.0 103 kg m2 and the polar moment of inertia JD ¼ 4.0 103 kg m2. The unbalance mass on the disk has the mass me ¼ 0.2 kg located at de ¼ 0.001 m.

Figure 8.7 Spinning shaft with a disk

Figure 8.8 shows the spinning speed (W) dependence of the natural frequencies obtained by SEM. The critical speeds (Wc) are the spinning speeds that coincide with the natural frequencies of the backward and forward whirling modes and they are indicated in Figure 8.8. The disk can be assembled with its neighboring two-shaft elements either by ignoring disk thickness (thin disk model, Equation 8.107) or by taking into account disk thickness (thick disk model, Equation 8.114). The critical speeds obtained by using the thick disk model with varying disk thickness, but holding all inertial properties constant, are displayed in Table 8.4. If the disk thickness 2h is reduced from its true value of 20 mm (thick disk) to 0 mm (thin disk), the critical speeds obtained by using the thick disk model certainly converge towards the results for the thin

252


Natural frequency (kHz)

0.8

Forward 0.6

2nd mode

Backward

0.4

0.2

0

1st

Forward Backward 0

0.2

mode

0.4

0.6

0.8

Spinning speed (kHz)

Figure 8.8 Natural frequency versus spinning speed of the shaft with a disk

Table 8.4

Disk model dependence of critical speeds (Hz) Thick disk ! Thin disk


Model 2h (mm)

20

10

5

0


113.92 114.03 608.41 664.78

113.92 114.03 622.67 685.44

113.92 114.03 631.32 698.33

113.92 114.03 637.09 707.12

disk. In general, the thin disk model seems to overestimate the critical speeds when compared with the thick disk model. Table 8.5 shows that the critical speeds obtained by FEM converge towards the SEM results as the number of finite elements used in FEM is increased. Figure 8.9 also shows that this is true for the vibration response calculated at spinning speed W ¼ 6000 rpm. These results prove the high accuracy of the present spectral element model.

Table 8.5

Critical speeds (Hz) of the shaft with a disk (2h ¼ 20 mm) FEM(N)



N¼2

N ¼ 10

N ¼ 50

SEM(2)

113.97 114.08 633.47 705.27

113.93 114.04 608.71 665.28

113.92 114.03 608.42 664.79

113.92 114.03 608.41 664.78

Note: (N) ¼ total number of the finite elements used in the analysis.

253

Dynamics of Rotor Systems 0.5

Amplitude (mm)

0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15

0.2

Time (s)

Amplitude (mm)

0.5 SEM (2 elements) FEM (6 elements) FEM (10 elements) FEM (40 elements)

0.4

0.3 0.08

0.09

0.1

0.11

0.12

0.13

Time (s)

Figure 8.9 Comparison of the vibration responses obtained by SEM and FEM (at spinning speed W ¼ 6000 rpm)

Appendix 8.A: Finite Element Matrices for the Transverse Bending Vibration a. Components of the matrix MB ¼ MTB ¼ ½mBij mB11 ¼ mB33 ¼ mB55 ¼ mB77 ¼ 12a1 ð26 þ 147r þ 210r2 Þ þ 36a2 mB12 ¼ mB34 ¼ mB56 ¼ mB78 ¼ a1 Lð44 þ 231r þ 315r2 Þ þ 3a2 Lð1 15rÞ mB15 ¼ mB37 ¼ 36a1 ð3 þ 21r þ 35r2 Þ 36a2 mB16 ¼ mB25 ¼ mB38 ¼ mB47 ¼ a1 Lð26 þ 189r þ 315r2 Þ þ 3a2 Lð1 15rÞ mB22 ¼ mB44 ¼ mB66 ¼ mB88 ¼ a1 L2 ð8 þ 42r þ 63r2 Þ þ a2 L2 ð4 þ 15r þ 90r2 Þ mB26 ¼ mB48 ¼ 3a1 L2 ð2 þ 14r þ 21r2 Þ a2 L2 ð1 þ 15r 45r2 Þ Others ¼ 0

(8.A.1) ð8:A1Þ

254


where a1 ¼

R2 rAL; 840

a2 ¼

R2 rI 30L

ð8:A2Þ (8.A.2)

b. Components of the matrix GB ¼ GTB ¼ ½gBij gB13 ¼ gB17 ¼ gB35 ¼ gB57 ¼ 36h gB14 ¼ gB18 ¼ gB23 ¼ gB27 ¼ gB36 ¼ gB45 ¼ gB58 ¼ gB67 ¼ 3hLð1 15rÞ gB24 ¼ gB68 ¼ hL2 ð4 þ 15r þ 90r2 Þ

ð8:A3Þ (8.A.3)

gB28 ¼ gB46 ¼ hL2 ð1 þ 15r 45r2 Þ Others ¼ 0 where h¼

R2 rJW 30L

ð8:A4Þ (8.A.4)

c. Components of the matrix KB ¼ KTB ¼ ½kBij kB11

¼ kB15 ¼ kB33 ¼ kB37 ¼ kB55 ¼ kB77 ¼ 12b1 þ 540b2 r2

kB12

¼ kB25 ¼ kB34 ¼ kB47 ¼ 6b1 L þ 270b2 Lr2

kB16

¼ kB38 ¼ kB56 ¼ kB78 ¼ 6b1 L þ 90b2 Lrð1 þ 3rÞ

kB22

¼ kB44 ¼ b1 L2 ð4 þ 6r þ 9r2 Þ þ 135b2 L2 r2

kB26

¼ kB48 ¼ b1 L2 ð2 6r 9r2 Þ þ 45b2 L2 rð1 þ 3rÞ

kB66

¼ kB88 ¼ b1 L2 ð4 þ 6r þ 9r2 Þ þ b2 L2 ð47 þ 210r þ 315r2 Þ

(8.A.5) ð8:A5Þ

Others ¼ 0 where b1 ¼

R2 EI; L3

b2 ¼

R2 kGA 60L

ð8:A6Þ (8.A.6)

9 Dynamics of Multi-Layered Structures Over the last half century, many types of multi-layered structures have been developed for various engineering applications. They include the elastic–elastic two-layer beams [53] and the passive constrained layer damping (PCLD) treated beams [9, 65, 66, 112, 122, 123, 172]. The bimetallic strip is a typical example of the elastic–elastic two-layer beam. The PCLD beam basically consists of an elastic base layer, a viscoelastic material (VEM) layer, and an elastic constraining layer and it is applied for the passive control of structural vibration and noise. For such complex multi-layer structures, more accurate analytical models and solution methods are mandatory to acquire satisfactory dynamic characteristics. In this chapter, the governing equations are derived for both an elastic–elastic two-layer beam and a PCLD beam by using Hamilton’s principle. The spectral element models are then formulated by using the variational approach. The conventional finite element models are also provided to evaluate the performance of the spectral element models. The governing partial differential equations of multilayer structures are usually fully coupled. The application of the classical modal analysis method to such coupled problems is rare in the literature. Thus, the spectral modal analysis method (SMAM) is introduced for the PCLD beam by deriving the orthogonality properties of exact damped normal modes.

9.1 Elastic–Elastic Two-Layer Beams 9.1.1 Equations of Motion The geometry and deformations of an elastic–elastic two-layer beam of length L are shown in Figure 9.1. The following assumptions are made for the derivation of the equations of motion. a. Geometric and material properties are uniform along the beam. b. Perfect continuity, but no slip, at the interfaces. c. Both layers take the same transverse displacement.



256

Figure 9.1 An elastic–elastic two-layer beam element

d. Small amplitude vibration. e. Linear elasticity is valid for both layers. f. Shear deformations and rotary inertias in both layers are negligible. In Figure 9.1, w(x, t) is the transverse displacement of the base beam, ub(x, t) is the axial displacement of the base beam, and uc(x, t) is the axial displacement of the constraining layer. hb and hc are the thicknesses of the base beam and constraining layer, respectively. In this chapter, the subscripts b and c are used to denote the quantities of the base beam and the constraining layer, respectively. From the perfect bonding condition, we can obtain a kinematic relationship uc ¼ ub ðh=2Þw0

ð9:1Þ

h ¼ hb þ h c

ð9:2Þ

where The prime (0 ) denotes the partial derivative with respect to spatial coordinate x. The equations of motion are derived by using Hamilton’s principle ð t2 ðdT dV þ dWÞdt ¼ 0 t1

ð9:3Þ

where T and V are the kinetic energy and potential energy, respectively, and dW is the virtual work. The strain energy V and the kinetic energy T are given by V T

1 ¼ 2 1 ¼ 2

ðL

½Eb Ab u0 b2 þ Eb Ib w00 þ Ec Ic w00 þ Ec Ac u0 c2 þ Fw0 dx 2

2

2

0

ðL 0

ð9:4Þ ½rb Ab ðu_ 2b

þ w_

2

Þ þ rc Ac ðu_ 2c

þ w_ Þdx 2

where E, A, I and r (for each layer) are the Young’s modulus, the cross-sectional area, the area moment of inertia about the neutral axis of each layer, and the mass density, respectively, and F is the constant axial tensile force applied to the beam. The dot () denotes the partial derivative with respect to time t. By eliminating uc from Equation 9.4 by using Equation 9.1, we can write the strain and kinetic energies as

Dynamics of Multi-Layered Structures

1 V ¼ 2 1 T ¼ 2 where

ðL 0

ðL 0

257

ðEAu0 b þ EIw00 2bw00 u0b þ Fw0 Þdx 2

ðrAu_ 2b

EA ¼ Eb Ab þ Ec Ac ; rA ¼ rb Ab þ rc Ac ; b ¼ ð1=2ÞEc Ac h;

2

2

0

ð9:5Þ

02

þ rAw_ 2aw_ u_ b þ g w_ Þdx 2

EI ¼ Eb Ib þ Ec Ic þ ð1=4ÞEc Ac h2 a ¼ ð1=2Þrc Ac h g ¼ ð1=4Þrc Ac h

ð9:6Þ

2

The virtual work dW done by the external forces, viscous damping forces, and boundary forces is given by ðL ðL ðL ð dW ¼ pðx; tÞdwdx þ tðx; tÞdub dx cb r_ b drb dAb dx 0

ðL ð 0

Ac

0

0

Ab

cc r_ c drc dAc dx þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ

ð9:7Þ

þ M1 ðtÞdw0 ð0; tÞ þ M2 ðtÞdw0 ðL; tÞ þ Q1 ðtÞdwð0; tÞ þ Q2 ðtÞdwðL; tÞ where rb ¼ ub nx þ wnz ;

rc ¼ uc nx þ wnz

ð9:8Þ

and where nx and nz are the unit direction vectors in the x- and z-directions, respectively. Ni, Qi, and Mi (i ¼ 1, 2) are the boundary forces and moments associated with ub, w, and u, respectively. cb and cc are viscous damping coefficients and p(x, t) and t(x, t) are the external forces applied along the beam. Substitute Equation 9.8 into Equation 9.7 and use Equation 9.1 to express uc in terms of ub. Next, apply the integral by parts to present the virtual work dW as ðL _ dW ¼ ½pðx; tÞdw þ tðx; tÞdub cAu_ b dub cAwdw þ c4 w_ 0 dub 0

þ c4 u_ b dw0 c1 w_ 0 dw0 dx þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ

ð9:9Þ

þ M1 ðtÞdw0 ð0; tÞ þ M2 ðtÞdw0 ðL; tÞ þ Q1 ðtÞdwð0; tÞ þ Q2 ðtÞdwðL; tÞ where cA ¼ cb Ab þ cc Ac ;

c1 ¼ ð1=4Þcc Ac h2 ;

c4 ¼ ð1=2Þcc Ac h

ð9:10Þ

Substitution of Equations 9.5 and 9.9 into Hamilton’s principle gives coupled two equations of motion EIw000 þ rA€ w þ cAw_

€ 00 þ c1 w_ 00 c4 u_ 0 b þ Fw00 þ pðx; tÞ ¼ a€ u 0 b þ bu000 þ g w

EAu00 b rA€ u b cAu_ b ¼ a€ w 0 þ bw000 c4 w_ 0 tðx; tÞ

ð9:11Þ


258

and the boundary conditions Qð0; tÞ ¼ Q1 ðtÞ Fw0 ð0; tÞ

or

wð0; tÞ ¼ w1 ðtÞ

Mð0; tÞ ¼ M1 ðtÞ Nð0; tÞ ¼ N1 ðtÞ

or or

w0 ð0; tÞ ¼ u1 ðtÞ ub ð0; tÞ ¼ u1 ðtÞ

QðL; tÞ ¼ þQ2 ðtÞ Fw0 ðL; tÞ MðL; tÞ ¼ þM2 ðtÞ

or or

wðL; tÞ ¼ w2 ðtÞ w0 ðL; tÞ ¼ u2 ðtÞ

NðL; tÞ ¼ þ N2 ðtÞ

or

ub ðL; tÞ ¼ u2 ðtÞ

ð9:12Þ

The resultant axial force N, bending moment M, and the transverse shear force Q are related to the displacement fields as € 0 þ c1 w_ 0 c4 u_ b Qðx; tÞ ¼ EIw w000 a€ u b þ bu00 b þ g w Mðx; tÞ ¼ EIw w00 bu0 b 0

ð9:13Þ

00

Nðx; tÞ ¼ EAu b bw

The structural damping of the base beam layer and constraining layer can be taken into account by using the complex modulus defined by Eb* ¼ Eb ð1 þ ihb Þ;

Ec* ¼ Ec ð1 þ ihc Þ

ð9:14Þ

where hb and hc are the loss factors of the base beam layer and the constraining layer, respectively.

9.1.2 Spectral Element Modeling 9.1.2.1 Governing Equations in the Frequency Domain As the first step of the spectral element formulation, the governing equations are transformed into the frequency domain by using the discrete Fourier transform (DFT) theory. To this end, we represent the dynamic responses in the spectral forms as fwðx; tÞ ub ðx; tÞg ¼

1 1 NX fWn ðx; vn Þ Un ðx; vn Þgeivn t N n¼0

ð9:15Þ

We also represent all forces and moments in the spectral forms as f pðx; tÞ tðx; tÞg ¼

1 1 NX fPn ðx; vn Þ Tn ðx; vn Þgeivn t N n¼0

fQðx; tÞ Mðx; tÞ Nðx; tÞg ¼

1 1 NX fQn ðx; vn Þ Mn ðx; vn Þ Nn ðx; vn Þgeivn t N n¼0

1 1 NX fQi ðx; tÞ Mi ðx; tÞ Ni ðx; tÞg ¼ fQin ðx; vn Þ Min ðx; vn Þ Nin ðx; vn Þgeivn t N n¼0

(9.16) ð9:16Þ


259

where i ¼ 1, 2. By substituting Equations 9.15 and 9.16 into Equations 9.11 and 9.12, we obtain the frequency-domain expressions of the equations of motion EIW 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ ivc1 W 00 ivc4 U 0 þ FW 00 þ PðxÞ

ð9:17Þ

EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 ivc4 W 0 TðxÞ and boundary conditions Qð0Þ ¼ Q1 FW 0 ð0Þ Mð0Þ ¼ M1

or or

Wð0Þ ¼ W1 W 0 ð0Þ ¼ Q1

Nð0Þ ¼ N1

or

Uð0Þ ¼ U1

QðLÞ ¼ þQ2 FW ðLÞ MðLÞ ¼ þM2

or or

WðLÞ ¼ W2 W 0 ðLÞ ¼ Q2

NðLÞ ¼ þN2

or

UðLÞ ¼ U2

0

ð9:18Þ

Similarly, substitution of Equations 9.15 and 9.16 into the force-displacement relationships (9.13) gives QðxÞ ¼ EIW 000 þ v2 aU þ bU 00 v2 gW 0 þ ivc1 W 0 ivc4 U MðxÞ ¼ EIW 000 bU 0 0

NðxÞ ¼ EAU bW

ð9:19Þ

00

In above, Equations 9.17–9.19, the subscripts n are omitted for brevity. 9.1.2.2 Spectral Nodal DOFs, Forces and Moments Figure 9.2 shows the spectral nodal degrees-of-freedom (DOFs), forces and moments. We define the spectral nodal DOFs vector as d ¼ fU1 W1 Q1 U2 W2 Q2 gT ¼ fUð0Þ Wð0Þ W 0 ð0Þ UðLÞ WðLÞ W 0 ðLÞgT

ð9:20Þ

and the spectral nodal forces and moments vector as fc ¼ fN1 Q1 M1 N2 Q2 M2 gT ¼ fNð0Þ Qð0Þ FW 0 ð0Þ Mð0Þ NðLÞ QðLÞ þ FW 0 ðLÞ MðLÞgT

Figure 9.2

Spectral element of the elastic–elastic two-layer beam

ð9:21Þ


260

9.1.2.3 Dynamic Shape Functions The frequency-dependent dynamic shape functions, which are required to formulate the spectral element equation, is derived from the homogeneous governing equations reduced from Equation 9.17 by setting P(x) ¼ 0 and T(x) ¼ 0. The homogeneous governing equations are given by EIW 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ ivc1 W 00 ivc4 U 0 þ FW 00 EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 ivc4 W 0

ð9:22Þ

The general solutions of Equation 9.22 are assumed to be Wðx; vÞ ¼ Aeikx ;

Uðx; vÞ ¼ rAeikx

Substitution of Equation 9.23 into Equation 9.22 yields an eigenvalue problem 0 1 X11 X12 ¼ X12 X22 r 0

ð9:23Þ

ð9:24Þ

where X11 ¼ k4 EI v2 rA þ ivcA v2 k2 g þ ivk2 c1 þ k2 F ð9:25Þ

X22 ¼ k2 EA þ v2 rA ivcA X12 ¼ vkc4 þ iv2 ka ik3 b The dispersion relation is obtained from Equation 9.24 as a1 k 6 þ a2 k 4 þ a3 k 2 þ a4 ¼ 0

ð9:26Þ

where a1 ¼ b2 EAEI a2 ¼ v2 ðEAg þ EIrA 2abÞ ivðEIcA þ EAc1 2bc4 Þ EAF a3 ¼ v4 ða2 grAÞ þ iv3 ðrAc1 þ gcA 2ac4 Þ

ð9:27Þ

þ v2 ðEArA þ cAc1 þ FrA c24 Þ ivcAðEA þ FÞ a4 ¼ rA2 v4 þ 2iv3 rAcA þ v2 cA2 Six wavenumbers kj (j ¼ 1, 2, . . . , 6) can be computed from Equation 9.26. For the jth wavenumber, the corresponding rj can be computed from Equation 9.24 as rj ¼

kj2 EA v2 rA þ ivcA vkj c4 þ iv2 kj a ikj3 b

ðj ¼ 1; 2; . . . ; 6Þ

ð9:28Þ

Define a diagonal matrix as R ¼ diag ½rj

ðj ¼ 1; 2; . . . ; 6Þ

ð9:29Þ


261

By using six wavenumbers kj (j ¼ 1, 2, . . . , 6), we can write the general solutions of Equation 9.22 in the forms 6 X Wðx; vÞ ¼ Aj e ikj x ¼ eðx; vÞa j¼1

Uðx; vÞ ¼

6 X

ð9:30Þ ikj x

r j Aj e

¼ eðx; vÞRa

j¼1

where

eðx; vÞ ¼ ½e ik1 x e ik2 x e ik3 x e ik4 x e ik5 x e ik6 x

ð9:31Þ

a ¼ fA1 A2 A3 A4 A5 A6 gT

ð9:32Þ

and

Applying Equation 9.30 to the spectral nodal DOFs defined by Equation 9.20, we obtain d ¼ HðvÞa where

2 6 6 6 HðvÞ ¼ 6 6 6 4

r1 1 ik1 r1 e1 e1 ik1 e1

r2 1 ik2 r2 e2 e2 ik2 e2

r3 1 ik3 r3 e 3 e3 ik3 e3

r4 1 ik4 r4 e4 e4 ik4 e4

ð9:33Þ r5 1 ik5 r5 e 5 e5 ik5 e5

3 r6 1 7 7 ik6 7 7 r6 e 6 7 7 e6 5 ik6 e6

ð9:34Þ

with ej ¼ e ikj L

ð9:35Þ

The constant vector a can be eliminated from Equation 9.30 by using Equation 9.33 to write the general solutions in terms of d as follows: Wðx; vÞ ¼ Nw ðx; vÞd;

Uðx; vÞ ¼ Nu ðx; vÞd

ð9:36Þ

where Nw and Nu are the dynamic shape functions defined by Nw ðx; vÞ ¼ eðx; vÞH 1 ðvÞ Nu ðx; vÞ ¼ eðx; vÞRH 1 ðvÞ

ð9:37Þ

9.1.2.4 Weak Form of Governing Equations The weak form of the governing Equation 9.17 in the frequency domain can be obtained from ðL ½ðEIW 0000 v2 rAW þ ivcAW ivc1 W 00 þ ivc4 U 0 v2 aU 0 0 ð9:38Þ bU 000 þ v2 gW 00 FW 0 0 PÞdW ðEAU 00 þ v2 rAU ivcAU v2 aW 0 bW 000 þ ivc4 W 0 þ TÞdUdx ¼ 0


262

By applying the integral by parts and using Equation 9.19 for the boundary-related terms generated by the integral by parts, we obtain the weak form as ðL ½EIW 00 dW 00 v2 rAWdW þ ivcAWdW þ ivc1 W 0 dW 0 0

ivc4 UdW 0 þ v2 aUdW 0 bU 0 dW 00 v2 gW 0 dW 0 þ FW 0 dW 0 þ EAU 0 dU 0 v2 rAUdU þ ivcAUdU þ v2 aW 0 dU bW 00 dU 0 ðL 0 ivc4 W dUdx ½PðxÞdW þ TðxÞdUdx

ð9:39Þ

0

f½QðxÞ þ FW 0 dWjL0 þ MðxÞdW 0 jL0 þ NðxÞdUjL0 g ¼ 0 9.1.2.5 Spectral Element Equation By substituting Equation 9.36 into Equation 9.39, we obtain ðL T ½EIN 00 w N w00 v2 rANTw Nw þ ivcANTw Nw þ ivc1 N 0Tw N w0 ddT 0

ivc4 N 0 w Nu þ v2 aN 0 Tw Nu bN 00 Tw N u0 v2 gN 0 Tw N w0 T

ð9:40Þ

þ FN 0 Tw N w0 þ EAN 0 Tu N u0 v2 rANTu Nu þ ivcANTu Nu þv

2

aNTu Nw0

where fd ðvÞ ¼

bN 0 Tu N 00w

ðL 0

ivc4 NTu N w0 dxd ðfc

þ fd Þ ¼ 0

½Pðx; vÞNTw ðx; vÞ þ Tðx; vÞNTu ðx; vÞdx

ð9:41Þ

As dd is arbitrary by definition, we can obtain the spectral element equation from Equation 9.40 as SðvÞd ¼ fðvÞ ð9:42Þ where ð SðvÞ ¼

L

0

fEAN 0 u N u0 þ EIN 00 w N 00w þ FN 0 w N w0 bðN 00 w N u0 þ N 0 u N 00w Þ T

T

T

T

T

v2 ½rAðNTw Nw þ NTu Nu Þ aðN0 w Nu þ NTu N w0 Þ þ gN0 w Nw0 T

þ iv½cAðNTw Nw and

T þ NTu Nu Þ þ c1 N0 w N w0

T

T c4 ðN0 w Nu

þ NTu N w0 Þgdx

fðvÞ ¼ fc ðvÞ þ fd ðvÞ

(9.43) ð9:43Þ ð9:44Þ

Substitution of the dynamic shape functions (9.37) into Equation 9.43 gives SðvÞ ¼ HT ðvÞDðvÞH1 ðvÞ

ð9:45Þ

where DðvÞ ¼ EARKEKR þ EIK2 EK2 ibðK2 EKR þ RKEK2 Þ v2 ½rAðE þ RERÞ þ iaðKER þ REKÞ gKEK þ iv½cAðE þ RERÞ c1 KEK þ ic4 ðKER þ REKÞ FRER

ð9:46Þ


263

and where K ¼ diag½ki ¼ KT ; K2 ¼ diag ½ki2 ¼ K2T ðj ¼ 1; 2; . . . ; 6Þ ðL EðvÞ ¼ eT ðx; vÞeðx; vÞdx ¼ ½Epq ðvÞ ðp; q ¼ 1; 2; . . . ; 6Þ

ð9:47Þ ð9:48Þ

0

with Epq ¼

8 > > < > > :

h i i eiðkp þ kq ÞL 1 kp þ kq

if kp þ kq 6¼ 0

L

if kp þ kq ¼ 0

ð9:49Þ

The spectral elements represented by Equation 9.42 can be assembled in an analogous way to that used for the conventional finite element method. After imposing the boundary conditions, the assembled global system equation can be obtained in the form Sg ðvÞdg ¼ fg ðvÞ

ð9:50Þ

where Sg(v) is the global dynamic stiffness matrix, dg is the global spectral DOFs vector, and fg is the global spectral nodal forces vector.

9.1.3 Spectral Modal Analysis For the modal analysis in the frequency domain (spectral modal analysis), the eigenfrequencies and eigenfuncions must be obtained in advance. The eigensolutions can be obtained from the eigenvalue problem given by Sg ðvÞdg ¼ 0

ð9:51Þ

a. Eigenfrequencies: (damped natural frequencies) vp (p ¼ 1, 2, . . . , ¥) are obtained by numerically computing the roots of the determinant of Sg(v). b. Wavenumbers: ki (i ¼ 1, 2, . . . , 6) for each eigenfrequency are computed from the dispersion relation (9.26). c. Eigenvectors: (spectral nodal DOFs vector) dgp (p ¼ 1, 2, . . . , ¥) corresponding to the eigenfrequencies vp are obtained so as to satisfy Equation 9.51. d. Eigenfunctions: (damped normal modes) Wp and Up are finally computed from Equation 9.36 by using the six wave numbers ki and eigenvectors dgp. 9.1.3.1 Orthogonality Properties of Normal Modes For the modal analysis, the orthogonality properties of normal modes are required. Let vp be the pth (damped) natural frequency and Wp and Up the corresponding (damped) normal modes of the bending and axial vibrations, respectively. The pth eigensolutions must satisfy EIWp000 v2p rAWp þ ivp cAWp ¼ v2p ðaUp0 gW 00p Þ þ ivp ðc1 W 00p c4 Up0 Þ þ bUp000 þ FW 00p EAU 00p

þ v2p rAUp

ivp cAUp ¼

v2p aW 0p

ivp c4 W 0p

þ bWp000

ð9:52Þ


264

By using Equation 9.19, Equation 9.52 can be written in the simple forms of v2p ðrAWp þ aUp0 gW 00p Þ þ ivp ðc1 W 00p c4 Up0 cAWp Þ þ FW 00p ¼ M 00p v2p ðrAUp aWp0 Þ þ ivp ðc4 W 0p cAUp Þ ¼ N 0p

ð9:53Þ

We multiply the first equation from Equation 9.53 by Wq and the second equation by Uq and then integrate from x ¼ 0 to x ¼ L, followed by integrating by parts. Finally, summing the final forms of the two equations gives ðL 2 vp ½rAðWp Wq þ Up Uq Þ aðUp Wq0 þ Uq Wp0 Þ þ gWp0 Wq0 dx 0 ðL ivp ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 c4 ðUp Wq0 þ Uq Wp0 Þ dx 0 ðL ð9:54Þ L L L ¼ Np Uq 0 ðQp þ FW 0 p ÞWq 0 Mp Wq0 0 þ Np Uq0 dx 0 ðL ðL 00 0 0 þ Mp W q dx þ FWp W q dx 0

0

Similarly, for the qth eigensolutions, we obtain ðL v2q ½rAðWp Wq þ Up Uq Þ aðUp Wq0 þ Uq Wp0 Þ þ gWp0 Wq0 dx 0 ðL ivq ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 W 0q c4 ðUp Wq0 þ Uq Wp0 Þdx 0 ðL L L L ¼ Nq Up 0 ðQq þ FW 0q ÞWp 0 Mq Wp0 0 þ Nq Up0 dx 0 ðL ðL 00 0 0 þ Mq W p dx þ FWq Wp dx 0

ð9:55Þ

0

Subtracting Equation 9.55 from Equation 9.54, applying homogenous boundary conditions and accomplishing a further manipulation, we obtain ðL ðv2p v2q Þ ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Uq Wp0 Þ 0 ðL ð9:56Þ þ gWp0 Wq0 dx iðvp vq Þ ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 0

c4 ðUp Wq0 þ Uq Wp0 Þdx ¼ 0

From Equation 9.56, we obtain the orthogonality properties as follows: ðL ½rAðWp Wq þ Up Uq Þ aðUp Wq0 þ Uq Wp0 Þ þ gWp0 Wq0 dx ¼ mp dpq 0 ðL ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 c4 ðUp Wq0 þ Uq Wp0 Þdx ¼ cp dpq

ð9:57Þ

0

where mp and cp are the pth modal mass and modal damping coefficient defined by ðL 2 mp ¼ ½rAðUp2 þ Wp2 Þ 2aUp Wp0 þ gW 0 q dx 0 ð9:58Þ ðL 2 2 02 0 cp ¼ ½cAðUp þ Wp Þ þ c1 W p 2c4 Up Wp dx 0


265

By applying the above orthogonality properties to Equation 9.54, we can also obtain ðL ðL L L L Nq Uq 0 ðQq þ FWq0 ÞWq 0 Mq Wq0 0 þ Nq Uq0 dx þ Mq W 00q dx 0 0 ðL 0 0 2 þ FWq Wq dx ¼ vq mq ivq cq 0 ðL ð9:59Þ L L 0 0 0 L Np Uq 0 ðQp þ FWp Þ Wq 0 Mp Wq 0 þ Np Uq dx 0 ðL ðL 00 0 0 þ Mp W q dx þ FWp Wq dx ¼ 0 0

0

9.1.3.2 Spectral Modal Equations Once the orthogonality properties have been derived, the forced vibration responses can be obtained by using the modal analysis method. The forced vibration responses are represented by superposing the normal modes as ¥ ¥ X X wðx; tÞ ¼ Wp ðxÞqp ðtÞ; uðx; tÞ ¼ Up ðxÞqp ðtÞ ð9:60Þ p¼1

p¼1

where qp(t) are the modal coordinates. The modal coordinates are written in the spectral form of qp ðtÞ ¼

1 1 NX q eivn t N n¼0 pn

ð9:61Þ

Substitution of Equation 9.61 into Equation 9.60 gives wðx; tÞ ¼

¥ N 1 X 1X Wp ðxÞqpn eivn t ; N p¼1 n¼0

uðx; tÞ ¼

¥ N 1 X 1X Up ðxÞqpn eivn t N p¼1 n¼0

ð9:62Þ

The external forces are also written in the spectral forms as pðx; tÞ ¼

1 1 NX Pn ðxÞeivn t ; N n¼0

tðx; tÞ ¼

1 1 NX Tn ðxÞeivn t N n¼0

ð9:63Þ

Substituting Equations 9.62 and 9.63 into the time-domain governing equations (9.11), we obtain ¥ ¥ X X ðM 00p FW 00p Þqpn v2n ðrAWp þ aUp0 gW 00p Þqpn p¼1

þ ivn

¥ X

p¼1

ðcAWp c1 W 00p þ c4 Up0 Þqpn ¼ Pn ðxÞ

p¼1 ¥ X Np0 qpn þ v2n ðrAUp p¼1 p¼1 ¥ X

¥ X

ð9:64Þ aWp0 Þqpn

ðcAUp c4 Wp0 Þqpn ¼ Tn ðxÞ

ivn

p¼1

where Equation 9.19 is used for Mp and Np. Multiply the first equation of Equation 9.64 by Wq and the second equation by Uq and then integrate from x ¼ 0 to x ¼ L, followed by integrating


266

by parts. Finally, summing the two equations gives ðL ¥ X 2 vn qpn ½rAðWp Wq þ Up Uq Þ aUp Wq0 þ gWp0 Wq0 aWp0 Uq dx 0

p¼1

ðL ¥ X ivn qpn ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 c4 Up Wq0 c4 Wp0 Uq dx

¼

¥ X

p¼1

qpn ½

p¼1

þ

ðL

0

L Qp þ FWp0 ÞWq 0

ðMp W 00q

þ FWp0 Wq0

L Mp Wq0 0

þ Np Uq0 Þdx

ð9:65Þ

L Np Uq 0 ðL

0

Pn ðxÞWq dx

ðL

0

Tn ðxÞUq dx

0

where Equation 9.19 is used for Qp. Applying the orthogonality properties (9.57) through (9.59) to Equation 9.65, we obtain the modal equations in the frequency domain as ðv2q v2n Þmq qqn iðvq vn Þcq qqn ¼ fqn

ðq ¼ 1; 2; 3; . . .Þ

ð9:66Þ

where fqn is the spectral modal force defined by fqn ¼

ðL

½Pn ðxÞWq þ Tn ðxÞUq dx

ðq ¼ 1; 2; 3; . . .Þ

ð9:67Þ

0

Equation 9.66 can be readily transformed into the time domain as mq € q q ðtÞ þ cq q_ q ðtÞ þ ðv2q mq ivq cq Þqq ðtÞ ¼ fq ðtÞ

ð9:68Þ

where fq is the modal force given by fq ðtÞ ¼

ðL

½ pðx; tÞ Wq þ tðx; tÞ Uq dx

ð9:69Þ

0

9.1.4 Finite Element Model The weak form of the equations of motion (9.11) is obtained as ðL _ € 0 dw0 ½EIw00 dw00 þ rA€ w dw þ cAwdw a€ u b dw0 bu0 b dw00 þ g w 0

þ c1 w_ 0 dw0 c4 u_ b dw0 þ Fw0 dw0 þ EAu0 bðdu0 b þ rA€u b dub þ cAu_ b dub L

a€ w 0 dub bw00 du0 b c4 w_ 0 du0 b dx ½pðx; tÞdw þ tðx; tÞdub dx 0 Nðx; tÞdub jL0 þ Mðx; tÞdw0 jL0 þ ½Qðx; tÞ þ Fw0 dwjL0 g ¼ 0

ð9:70Þ

The sign conventions of the nodal DOFs, forces, and moments defined in Figure 9.2 are also valid in the time domain and we define the nodal DOFs vector as dðtÞ ¼ fU1 ðtÞ W1 ðtÞ Q1 ðtÞ U2 ðtÞ W2 ðtÞ

Q2 ðtÞgT

ð9:71Þ


267

and the nodal forces and moments vector by fc ðtÞ ¼ fN1 ðtÞ Q1 ðtÞ M1 ðtÞ N2 ðtÞ Q2 ðtÞ M2 ðtÞgT ¼ f Nð0; tÞ Qð0; tÞ Fw0 ð0; tÞ Mð0; tÞ NðL; tÞ

QðL; tÞ þ Fw0 ðL; tÞ

The displacement fields are assumed to be ub ðx; tÞ ¼ Nu ðxÞdðtÞ;

MðL; tÞgT

wðx; tÞ ¼ Nw ðxÞdðtÞ

ð9:72Þ ð9:73Þ

where Nu ðxÞ ¼ ½Nu1 ðxÞ 0 0 Nu2 ðxÞ 0 0

ð9:74Þ

Nw ðxÞ ¼ ½0 Nw1 ðxÞ Nu1 ðxÞ 0 Nw2 ðxÞ Nu2 ðxÞ with Nu1 ðxÞ ¼ 1 ðx=LÞ; Nw1 ðxÞ ¼ 1 3ðx=LÞ2 þ 2ðx=LÞ3 ; Nu1 ðxÞ ¼ x½1 2ðx=LÞ þ ðx=LÞ2 ;

Nu2 ðxÞ ¼ x=L Nw2 ðxÞ ¼ 3ðx=LÞ2 2ðx=LÞ3 Nu2 ðxÞ ¼ x½ðx=LÞ ðx=LÞ2

ð9:75Þ

Substitution of Equation (9.73) into the weak form (9.70) gives the finite element equation as _ þ KdðtÞ ¼ f ðtÞ þ f ðtÞ M€ dðtÞ þ CdðtÞ c d

ð9:76Þ

where M¼

ðL 0

C ¼

ðL 0

K ¼

ðL

½rAðNTw Nw þ NTu Nu Þ aðNw0 Nu þ NTu Nw0 Þ þ gNw0 Nw0 dx T

T

½cAðNTw Nw þ NTu Nu Þ þ c1 Nw0 Nw0 c4 ðNw0 Nu þ Nu0 Nw0 Þdx T

T

T

T

ð9:77Þ

½EANu0 Nu0 þ EIN00w N00w bðN00w N0 u þ N0 u N00w Þ þ FNw0 Nw0 dx T

T

T

T

0

and fd ðtÞ ¼

ðL 0

½pðx; tÞNTw ðxÞ þ tðx; tÞNTu ðxÞdx

ð9:78Þ

The finite element matrices M, C, and K are given in Appendix 9.A.

Example 9.1 Consider a cantilevered elastic–elastic two-layer beam as shown in Figure 9.3. The base beam is fully covered with a uniform elastic constraining layer. The geometric and material properties are given in Table 9.1. The high accuracy of the present spectral element model is shown in Table 9.2 by comparing the natural frequencies obtained by SEM with those obtained by FEM through varying the total number of finite elements used in the analysis. Figure 9.4 shows the natural frequencies as a function of the ratio of the constraining layer thickness hc to a fixed base beam thickness hb ¼ 5 mm.


268

Figure 9.3

An elastic–elastic two-layer beam

Table 9.1 Geometric and material properties of the elastic–elastic two-layer beam L Eb

b Ec

300 mm 70 GPa

20 mm 200 GPa

hb rb

hc rc

5 mm 2700 kg/m3

0.75 mm 7850 kg/m3

Table 9.2 Natural frequencies (Hz) of an elastic–elastic two layer beam SEM(N)

FEM(N)

Mode

N¼1

N¼5

N ¼ 10

N ¼ 20

N ¼ 40

N ¼ 100

1st 2nd 3rd 4th 5th 10th

56.5538 354.321 991.686 1942.06 3207.71 11385.70

56.6368 358.060 1018.43 2048.77 3527.25 13097.98

56.5745 355.230 997.855 1965.02 3270.37 12292.17

56.5590 354.547 993.191 1947.57 3222.38 11570.39

56.5551 354.378 992.054 1943.42 3211.32 11429.25

56.5540 354.330 991.738 1942.28 3208.29 11392.55

Note: (N) ¼ total number of the equal-length finite elements used in the analysis.

2nd axial mode

10

10th bending mode

4

1st axial mode

Natural frequency (Hz)

5th bending mode 4th bending mode

10

3rd bending mode

3

2nd bending mode

10

10

2

1st bending mode

1

0

0.05

0.1 0.15 0.2 Thickness ratio (hc /hb)

0.25

0.3

Figure 9.4 Thickness ratio dependence of the natural frequencies


Figure 9.5

269

An elastic–VEM–elastic three-layer (PCLD) beam element

9.2 Elastic–Viscoelastic–elastic Three-Layer (PCLD) Beams 9.2.1 Equations of Motion The passive constrained layered damping (PCLD) treated beams consist of three layers: an elastic base beam, a viscoelastic material (VEM) layer, and an elastic constraining layer. The geometry and deformations of a PCLD beam are shown in Figure 9.5. w(x, t) is the transverse displacement of the base beam, ub(x, t) is the axial displacement of the base beam, uv(x, t) is the axial displacement of the VEM layer, and uc(x, t) is the axial displacement of the constraining layer. hb, hv, and hc are the thicknesses of the base beam, VEM layer, and constraining layer, respectively, and c is the shear angle of the VEM layer. The subscripts b, v, and c are used to denote the quantities of the base beam, VEM layer, and constraining layer, respectively. The following assumptions are made for the derivation of the equations of motion [66]. a. Geometric and material properties are uniform along the beam. b. Perfect continuity, but no slip, at the interfaces. c. All layers take the same transverse displacement. d. Small amplitude vibration. e. Linear elasticity is valid for the base beam and constraining layer. f. Linear viscoelasticity is valid for the VEM layer. g. Shear deformations and rotary inertias in the base beam and constraining layer are negligible. Based on the above assumption, we obtain the kinematics relationships from Figure 9.5 as uv ¼ ub ðh1 =2Þw0 þ ðhv =2Þc uc ¼ ub ðh2 =2Þw0 þ hv c where

h1 ¼ hb þ hv ;

h2 ¼ hb þ 2hv þ hc

ð9:79Þ ð9:80Þ

The constitutive relationship for the one-dimensional viscoelastic materials can be represented in the following Stieltjes integral form [34] ðt _ tðx; tÞ ¼ g c ¼ gðt tÞcðx; tÞdt ð9:81Þ ¥

where t is the shear stress and g(t) is the relaxation function of VEM (the stress response to a unit-step strain input). This stress relaxation represents the energy loss from viscoelastic


270

materials, hence damping. Equation 9.81 can be transformed in the frequency domain by using the DFT as follows: DFT½g c ¼ GðvÞðivÞYðvÞ G* ðvÞYðvÞ

ð9:82Þ

where G(v) and Y (v) are the spectral components of g(t) and c(t), respectively, and G (v) ¼ ivG(v). The strain energy V and the kinetic energy T of the PCLD beam are given by ð 1 L 2 2 2 V ¼ ½Eb Ab u0b þ Eb Ib w00 þ Ev Av u0 v þ Ev Iv ðw00 c0 Þ2 þ Av ðg cÞc 2 0 ð9:83Þ þ Ec Ic w00 2 þ Ec Ac u0c 2 þ Fw0 2 dx ðL 1 _ 2 þ r Ac ðu_ 2 þ w_ 2 Þdx ½r Ab ðu_ 2b þ w_ 2 Þ þ rv Av ðu_ 2v þ w_ 2 Þ þ rv Iv ðw_ 0 cÞ T ¼ c c 2 0 b where E, I, A, and r (for each layer) are the Young’s modulus, area moment of inertia about the neutral axis, cross-sectional area, and the mass density, respectively. By eliminating uv and uc from Equation 9.83 by using Equation 9.79, the strain and kinetic energies can be written as ð 1 L 2 2 2 ½EAu0 b þ EIw w00 þ EIc c0 2bw00 u0b þ 2e4 u0 b c0 2e2 w00 c0 V ¼ 2 0 ð9:84Þ þ Av ðg cÞc þ Fw0 2 dx ðL 1 2 2 frAu_ 2b þ rAw_ 2 þ rI c_ 2aw_ 0 u_ b þ 2e3 u_ b c_ 2e1 w_ 0 c_ þ g w_ 0 gdx T ¼ 2 0 where EA EIw EIc rA rI a b g e1 e2 e3 e4

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

E b Ab þ Ev Av þ Ec Ac Eb Ib þ Ev Iv þ Ec Ic þ ð1=4ÞEv Av h21 þ ð1=4ÞEc Ac h22 Ev Iv þ ð1=4ÞEv Av h2v þ Ec Ac h2v r b Ab þ r v A v þ r c Ac rv Iv þ ð1=4Þrv Av h2v þ rc Ac h2v ð1=2Þðrv Av h1 þ rc Ac h2 Þ ð1=2ÞðEv Av h1 þ Ec Ac h2 Þ rv Iv þ ð1=4Þrv Av h21 þ ð1=4Þrc Ac h22 rv Iv þ ð1=4Þrv Av h1 hv þ ð1=2Þrc Ac h2 hv Ev Iv þ ð1=4ÞEv Av h1 hv þ ð1=2ÞEc Ac h2 hv ð1=2Þrv Av hv þ rc Ac hv ð1=2ÞEv Av hv þ Ec Ac hv

ð9:85Þ

The virtual work dW done by the external forces, viscous damping forces, and boundary forces is given by ðL ðL ðL dW ¼ pðx; tÞdwdx þ tðx; tÞdub dx þ gðx; tÞdcdx 0 0 0 ðL ð ðL ð ðL ð cb r_ b drb dAb dx cv r_ v drv dAv dx cc r_ c drc dAc dx ð9:86Þ 0

Ab

0

Av

0

Ac

þ N1 ðtÞ dub ð0; tÞ þ N2 ðtÞ dub ðL; tÞ þ M1 ðtÞ dw0 ð0; tÞ þ M2 ðtÞ dw0 ðL; tÞ þ Q1 ðtÞ dwð0; tÞ þ Q2 ðtÞ dwðL; tÞ þ R1 ðtÞ dcð0; tÞ þ R2 ðtÞ dcðL; tÞ


with

271

rb ¼ ub nx þ wnz rv ¼ ½uv zðw0 cÞnx þ wnz

ð9:87Þ

rc ¼ uc nx þ wnz where nx and nz are the unit direction vectors in the x- and z-directions, respectively. In Equation 9.86, Ni, Qi, Mi, and Ri (i ¼ 1, 2) are the boundary forces and moments associated with ub, w, u, and c, respectively. cb, cv, and cc are viscous damping coefficients and p(x, t), t(x, t) and r(x, t) are the external forces applied along the beam. Substitute Equation 9.87 into Equation 9.86 and then use Equation 9.79 to express uv and uc in terms of ub. Next, apply the integral by parts to write Equation 9.86 in the form ðL dW ¼ ½pðx; tÞdw þ tðx; tÞdub þ gðx; tÞdc 0

_ b c5 u_ b dc _ þ c4 w_ 0 dub c5 cdu cAu_ b dub cAwdw 0 0 0 _ _ 0 dx þ c4 u_ b dw c1 w_ dw0 þ c3 cdw þ c3 w_ dc c2 cdc þ M1 ðtÞdw0 ð0; tÞ þ M2 ðtÞdw0 ðL; tÞ þ Q1 ðtÞdwð0; tÞ

ð9:88Þ

þ Q2 ðtÞdwðL; tÞ þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ þ R1 ðtÞdcð0; tÞ þ R2 ðtÞdcðL; tÞ where cA ¼ cb Ab þ cv Av þ cc Ac c1 ¼ cv Iv þ ð1=4Þðcv Av h21 þ cc Ac h22 Þ c2 ¼ cv Iv þ ð1=4Þcv Av h2v þ cc Ac h2v

ð9:89Þ

c3 ¼ cv Iv þ ð1=4Þcv Av h1 hv þ ð1=2Þcc Ac h2 hv c4 ¼ ð1=2Þðcv Av h1 þ cc Ac h2 Þ c5 ¼ ð1=2Þcv Av hv þ cc Ac hv

Finally, by substituting Equations 9.84 and 9.88 into Hamilton’s principle (9.3), we obtain a set of coupled equations of motion as follows € 00 EIw w0000 þ rA€ w þ cAw_ ¼ a€ ub0 þ bu000 þ g w 0 € b þ e2 c000 þ c1 w_ 00 c4 u_ 0b c3 c_ 0 þ Fw00 þ pðx; tÞ e1 c

€ e4 c00 c4 w_ 0 þ c5 c_ tðx; tÞ EAu00 b rA€ u b cAu_ b ¼ a€ w 0 þ bw000 þ e3 c 00 € c2 c_ ¼ e1 w € 0 þ e2 w000 EIc c rI c

ð9:90Þ

þ e3 €u b e4 u00b c3 w_ 0 þ c5 u_ b þ Av ðg cÞ gðx; tÞ and the boundary conditions Qð0; tÞ ¼ Mð0; tÞ ¼ Nð0; tÞ ¼ Rð0; tÞ ¼ QðL; tÞ ¼ MðL; tÞ ¼ NðL; tÞ ¼ RðL; tÞ ¼

Q1 ðtÞ Fw0 ð0; tÞ M1 ðtÞ N1 ðtÞ R1 ðtÞ þ Q2 ðtÞ Fw0 ðL; tÞ þ M2 ðtÞ þ N2 ðtÞ þ R2 ðtÞ

or or or or or or or or

wð0; tÞ w0 ð0; tÞ ub ð0; tÞ cð0; tÞ wðL; tÞ w0 ðL; tÞ ub ðL; tÞ cðL; tÞ

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

w1 ðtÞ u1 ðtÞ u1 ðtÞ c1 ðtÞ w2 ðtÞ u2 ðtÞ u2 ðtÞ c2 ðtÞ

ð9:91Þ


272

The resultant forces and moments N, M, Q, and R are related to the displacement fields as follows: € þ e2 c00þ c1 w_ 0 c4 u_ b c3 c_ € 0 e1 c Qðx; tÞ ¼ EIw w000 a€ u b þ bu00b þ g w Mðx; tÞ ¼ EIw w00 bu0b e2 c0 Nðx; tÞ ¼ EAu0 b bw0 0 þ e4 c0

ð9:92Þ

Rðx; tÞ ¼ EIc c0 e2 w00 þ e4 u0b The structural damping of the base beam and elastic constraining layer can also be readily taken into account by using the complex modulus given by Equation 9.14. If all VEM related terms (i.e., e1, e2, e3, e4, and the terms with the subscript v) are removed, the equations of motion (9.90) and the force-displacement relationships (9.92) return to the equations of motion (9.11) and the force-displacement relationships (9.13) for the elastic– elastic two-layer beams considered in the previous section.

9.2.2 Spectral Element Modeling 9.2.2.1 Governing Equations in the Frequency Domain Following the same procedure of spectral element formulation as used for the two-layer beams in the previous section, we represent the dynamic responses in the spectral forms as 1 1 NX ð9:93Þ fwðx; tÞ ub ðx; tÞcðx; tÞg ¼ f Wn ðx; vn Þ Un ðx; vn Þ Yn ðx; vn Þ g eivn t N n¼0 We also represent all forces and moments and the voltage in the spectral forms fpðx; tÞ tðx; tÞ gðx; tÞg 1 1 NX fPn ðx; vn ÞTn ðx; vn ÞGn ðx; vn Þgeivn t ¼ N n¼0 fQðx; tÞ Mðx; tÞ Nðx; tÞ Rðx; tÞg 1 1 NX fQn ðx; vn Þ Mn ðx; vn Þ Nn ðx; vn Þ Rn ðx; vn Þgeivn t ¼ N n¼0

ð9:94Þ

fQi ðx; tÞ Mi ðx; tÞ Ni ðx; tÞ Ri ðx; tÞg 1 1 NX fQin ðx; vn Þ Min ðx; vn Þ Nin ðx; vn Þ Rin ðx; vn Þgeivn t ¼ N n¼0 where i, j ¼ 1, 2. Substitution of Equations 9.93 and 9.94 into Equations 9.90 and 9.91 yields the discrete Fourier transformed equations of motion EIw W 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU000 v2 gW 00 þ v2 e1 Y0 þ e2 Y000 þ ivc1 W 00 ivc4 U 0 ivc3 Y0 þ FW 00 þ PðxÞ EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 v2 e3 Y e4 Y00 ivc4 W 0 þ ivc5 Y TðxÞ 00 EIc Y þ v2 rIY ivc2 Y ¼ v2 e1 W 0 þ e2 W 000 v2 e3 U e4 U 00 ivc3 W 0 þ ivc5 U þ Av G Y GðxÞ

ð9:95Þ


273

and the associated boundary conditions Qð0Þ ¼ Q1 FW 0 ð0Þ

or

Wð0Þ ¼ W1

Mð0Þ ¼ M1 Nð0Þ ¼ N1

or or

W 0 ð0Þ ¼ Q1 Uð0Þ ¼ U1

Rð0Þ ¼ R1 QðLÞ ¼ þ Q2 FW 0 ðLÞ

or or

Yð0Þ ¼ Y1 WðLÞ ¼ W2

MðLÞ ¼ þ M2

or

W 0 ðLÞ ¼ Q2

NðLÞ ¼ þ N2 RðLÞ ¼ þ R2

or or

UðLÞ ¼ U2 YðLÞ ¼ Y2

ð9:96Þ

Equation 9.82 is applied to the convolution term in Equation 9.95. Similarly, by using Equations 9.93 and 9.94, the resultant force-displacement relationships (9.92) can be discrete Fourier transformed as QðxÞ ¼ EIw W 000 þ v2 aU þ bU 00 v2 gW 0 þ v2 e1 Y þ e2 Y00 þ ivc1 W 0 ivc4 U ivc3 Y MðxÞ ¼ EIw W 00 bU 0 e2 Y0 0

ð9:97Þ

0

00

NðxÞ ¼ EAU bW þ e4 Y RðxÞ ¼ EIc Y0 e2 W 00 þ e4 U 0 In Equations 9.95–9.97, the subscripts n are omitted for brevity. 9.2.2.2 Spectral Nodal DOFs, Forces, and Moments The spectral nodal DOFs, forces, and moments are specified in Figure 9.6. The spectral nodal DOFs vector is defined by d ¼ fU1 W1 Q1 Y1 U2 W2 Q2 Y2 gT ¼ fUð0Þ Wð0Þ W 0 ð0Þ Yð0Þ UðLÞ WðLÞ W 0 ðLÞ YðLÞgT

ð9:98Þ

and the spectral nodal forces and moments vector by fc ¼ fN1 Q1 M1 R1 N2 Q2 M2 R2 gT ¼ f Nð0Þ Qð0Þ FW 0 ð0Þ Mð0Þ Rð0Þ NðLÞ QðLÞ þ FW 0 ðLÞ MðLÞ RðLÞgT ð9:99Þ

Figure 9.6

Spectral element of PCLD beam


274

9.2.2.3 Dynamic Shape Functions The frequency-dependent dynamic shape functions can be derived from the homogeneous governing equations, which are reduced from Equation 9.95 as EIw W 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ v2 e1 Y0 þ e2 Y000 þ ivc1 W 00 ivc4 U 0 ivc3 Y0 þ FW 00 EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 v2 e3 Y e4 Y00 ivc4 W 0 þ ivc5 Y

ð9:100Þ

EIc Y00 þ v2 rIY ivc2 Y ¼ v2 e1 W 0 þ e2 W 000 v2 e3 U e4 U 00 ivc3 W 0 þ ivc5 U þ Av G Y The general solutions to Equation 9.100 are assumed to be Wðx; vÞ ¼ Ae ikx ;

Uðx; vÞ ¼ r1 Ae ikx ;

cðx; vÞ ¼ r2 Ae ikx

ð9:101Þ

By substituting Equation 9.101 into Equation 9.100, we obtain an eigenvalue problem 2 38 9 8 9 X11 X12 X13 < 1 = < 0 = 4 X12 X22 X23 5 r1 ¼ 0 ð9:102Þ : ; : ; 0 X13 X23 X33 r2 where X11 ¼ k4 EIw v2 rA þ ivcA v2 k2 g þ ivk2 c1 þ k2 F X12 ¼ vkc4 þ iv2 ka ik3 b; X13 ¼ vkc3 þ iv2 ke1 ik3 e2 X22 ¼ k2 EA þ v2 rA ivcA;

X23 ¼ v2 e3 k2 e4 ivc5

ð9:103Þ

X33 ¼ k2 EIc þ v2 rI ivc2 Av G The eigenvalue problem, Equation 9.102, yields a dispersion equation as a1 k 8 þ a 2 k 6 þ a3 k 4 þ a4 k 2 þ a5 ¼ 0 where a1 ¼ EAEIw EIc e24 EIw þ 2be4 e2 EAe22 EIc b2 a2 ¼ v2 ½ rIb2 þ 2be1 e4 g e24 2a EIc b þ 2e2 e3 b rAe22 þ gEAEIc 2EAe1 e2 þ 2ae2 e4 þ EIw ðrIEA þ rAEIc 2e3 e4 Þ iv ½ c2 b2 2c4 bEIc þ 2c3 e4 b cAe22 e24 c1 þ c1 EA EIc þ 2e2 ðe4 c4 c3 EA þ c5 bÞ þ EIw ðcA EIc 2e4 c5 þ EA c2 Þ þ e24 F EAEIw Av G EAEIc F þ Av G b2 a3 ¼ v4 ½ EAe21 2rAe2 e1 þ 2be3 e1 þ 2ae4 e1 2ab rI þ EArI g þ EIc ðg rA a2 Þ þ 2ae2 e3 þ EIw ðrArI e23 Þ 2g e3 e4 iv3 ½ 2cA e1 e2 þ 2e2 e3 c4 2rAe2 c3 þ 2ae2 c5 þ EArIc1 2e3 e4 c1 2c4 b rI þ 2e1 e4 c4 þ EIc ðg cA þ rA c1 2ac4 Þ 2EAe1 c3

ð9:104Þ


275

þ 2b e3 c3 þ 2ae4 c3 þ 2be1 c5 2ge4 c5 2abc2 þ EAg c2 þ EIw ðcArI 2e3 c4 þ rA c2 Þ þ v2 frA e24 þ 2F e3 e4 2c4 c3 e4 þ 2c1 c5 e4 þ EA c23 þ 2Av G ab gEA Av G EArI F EIc ½ c24 þ rA ðEA þ FÞ þ cA c1 þ 2cA e2 c3 2e2 c4 c5 2bc3 c5 EA c1 c2 þ 2b c5 c2 þ EIw ðc25 rA Av G cA c2 Þg þ iv ½ cA e24 2Fc5 e4 þ cA EIw Av G þ cA EIc ðEA þ FÞ þ c1 EA Av G 2Av G bc4 þ c2 EA F þ F EA Av G

ð9:105Þ

a4 ¼ v6 ½rAe21 2ae1 e3 þ g e23 þ rIða2 g rAÞ iv5 ½ c2 a2 2ac4 rI þ 2ae3 c3 cAe21 þ g cA rI þ c1 ðrArI e23 Þ 2g e3 c5 þ 2e1 ðe3 c4 c3 rA þ ac5 Þ þ g c2 rA þ v4 ½Av G a2 2ac3 c5 þ 2ac4 c2 þ F e23 þ rI c24 þ rA c23 þ g c25 g rA Av G EArArI FrArI rA2 EIc cArIc1 þ 2cAe1 c3 2e1 c4 c5 þ 2e3 ðrA e4 c4 c3 þ c1 c5 Þ g cAc2 rA c1 c2 þ iv3 ½ c2 c24 2ac4 Av G þ 2c3 c4 c5 cA c23 c1 c25 þ cAgAv G þ rIEA cA þ FcArI þ 2cArAEIc þ rAc1 Av G 2rA e4 c5 2e3 ðcAe4 þ Fc5 Þ þ c2 EArA þ F rAc2 þ cA c1 c2 þ v2 ðEIc cA2 þ cAc1 Av G 2e4 c5 cA þ EAc2 cA þ Fc2 cA Av G c24 Fc25 þ EArAAv G þ Av G F rAÞ ivðAv G EA cA þ Av G cAÞ a5 ¼ v6 rAðe23 rArIÞ þ iv5 ½ cAe23 2rAc5 e3 þ rAð2cA rI þ rAc2 Þ þ v4 ðrI cA2 2e3 c5 cA þ 2rAc2 cA þ Av G rA2 rA c25 Þ iv3 cAð c25 þ 2Av G rA þ cAc2 Þ v2 Av G cA2 We can compute eight wavenumbers ki (i ¼ 1, 2, . . . , 8) from the dispersion relation, Equation 9.104. For each wavenumber ki (i ¼ 1, 2, . . . , 8), we can obtain the corresponding r1i and r2i from Equation 9.102 as r1i ¼

bi d i c i e i ; bi c i ai e i

r2i ¼

ai di c2i bi c i ai e i

ði ¼ 1; 2; . . . ; 8Þ

ð9:106Þ

where ai ¼ ki2 EA þ v2 rA ivcA bi ¼ v2 e3 ki2 e4 ivc5 ci ¼ vki c4 iv2 ki a iki3 b di ¼ ei ¼

ki4 EIw þ v2 rA ivcA ivki2 c1 iv2 ki e1 þ iki3 e2 vki c3

ð9:107Þ þ v2 ki2 g

ki2 F


276

Define two diagonal matrices as R1 ¼ diag ½r1i ;

R2 ¼ diag ½r2i

ð9:108Þ

By using eight wavenumbers ki (i ¼ 1, 2, . . . , 8) and the matrices R1 and R2, we can write the general solutions to Equation 9.100 as Wðx; vÞ ¼

8 X Ai eiki x ¼ eðx; vÞa i¼1

8 X Uðx; vÞ ¼ r1i Ai eiki x ¼ eðx; vÞR1 a

Yðx; vÞ ¼

ð9:109Þ

i¼1 8 X

r2i Ai eiki x ¼ eðx; vÞR2 a

i¼1

where a ¼ fA1 A2 A3 A4 A5 A6 A7 A8 gT

ð9:110Þ

and eðx; vÞ ¼ ½eik1 x eik2 x eik3 x eik4 x eik5 x eik6 x eik7 x eik8 x

ð9:111Þ

Applying Equation 9.109 to the spectral nodal DOFs vector defined by Equation 9.98 gives d ¼ HðvÞa

ð9:112Þ

where 2 6 6 6 6 6 HðvÞ ¼ 6 6 6 6 6 4

r11 1 ik1 r21 r11 e1 e1 ik1 e1 r21 e1

r12 1 ik2 r22 r12 e2 e2 ik2 e2 r22 e2

r13 1 ik3 r23 r13 e3 e3 ik3 e3 r23 e3

r14 1 ik4 r24 r14 e4 e4 ik4 e4 r24 e4

r15 1 ik5 r25 r15 e5 e5 ik5 e5 r25 e5

r16 1 ik6 r26 r16 e6 e6 ik6 e6 r26 e6

r17 1 ik7 r27 r17 e7 e7 ik7 e7 r27 e7

3 r18 1 7 7 ik8 7 7 r28 7 7 r18 e8 7 7 e8 7 7 ik8 e8 5 r28 e8 ð9:113Þ

with ej ¼ eikj L

ð9:114Þ

The constant vector a can be eliminated from Equation 9.109 by using Equation 9.112 to write the general solutions as Wðx; vÞ ¼ Nw ðx; vÞd Uðx; vÞ ¼ Nu ðx; vÞd Yðx; vÞ ¼ Nc ðx; vÞd

ð9:115Þ


277

where Nw, Nu, and Nc are the dynamic shape functions defined by Nw ðx; vÞ ¼ eðx; vÞH 1 ðvÞ Nu ðx; vÞ ¼ eðx; vÞR1 H 1 ðvÞ Nc ðx; vÞ ¼ eðx; vÞR2 H

1

ð9:116Þ

ðvÞ

9.2.2.4 Weak Form of Governing Equations The weak form of the governing equations (9.95) can be obtained from ðL ½ðEIw W 0000 v2 rAW þ ivcAW ivc1 W 00 þ ivc4 U 0 v2 aU 0 0

bU 000 þ v2 gW 00 v2 e1 Y0 e2 Y000 þ ivc3 Y0 FW 00 PÞdW ðEAU 00 þ v2 rAU iv cAU v2 aW 0 bW 000 þ v2 e3 Y þ e4 Y00 þ ivc4 W 0 ivc5 Y þ TÞdU

ð9:117Þ

ðEIc Y00 þ v2 rIY ivj5 Y v2 e1 W 0 e2 W 000 þ v2 e3 U þ e4 U 00 þ ivc3 W 0 ivc5 U Av G Y þ GÞdYdx ¼ 0 By integrating Equation 9.117 by parts and using Equation 9.97, we can obtain the weak form as ðL ½EIw W 00 dW 00 v2 rAWdW þ ivcAWdW þ ivc1 W 0 dW 0 0

ivc4 UdW 0 þ v2 aUdW 0 bU 0 dW 00 v2 gW 0 dW 0 þ v2 e1 YdW 0 e2 Y0 dW 00 ivc3 YdW 0 þ FW 0 dW 0 þ EAU 0 dU 0 v2 rAUdU þ ivcAUdU þ v2 aW 0 dU bW 00 dU 0 v2 e3 YdU þ e4 Y0 dU 0 ivc4 W 0 dU þ ivc5 YdU þ EIc Y0 dY0 v2 rIYdY þ ivc2 YdY þ v2 e1 W 0 dY e2 W 00 dY0 v2 e3 UdY þ e4 U 0 dY0 ivc3 W 0 dY þ ivc5 UdY þ Av G YdYdx ðL ½PðxÞdW þ TðxÞdU þ GðxÞdYdx

ð9:118Þ

0

f½QðxÞ þ FW 0 dWjL0 þ MðxÞdW 0 jL0 þ NðxÞdUjL0 þ RðxÞdYjL0 g ¼ 0 9.2.2.5 Spectral Element Equation By substituting Equation 9.115 into Equation 9.118 and using Equation 9.99, we obtain ðL T 0 T ddT ½EIw N00 w N0w v2 rANTw Nw þ ivcANTw Nw þ ivc1 N0 w Nw0 0

ivc4 N0 w Nu þ v2 aN0 w Nu bN00 w Nu0 v2 gN0 w Nw0 T

T

T

T

T T T T þ v2 e1 N0 w Nc e2 N00 w Nc0 ivc3 N0 w Nc þ FN0 w Nw0 T þ EAN0 u Nu0 v2 rANTu Nu þ ivcANTu Nu þ v2 aNTu Nw0 T T bN0 u N00w v2 e3 NTu Nc þ e4 N0 u Nc0 ivc4 NTu Nw0 T þ ivc5 NTu Nc þ EIc N0 c N0 c v2 rINTc Nc þ ivc2 NTc Nc T T þ v2 e1 NTc N0 w e2 N0 c N00 w v2 e3 NTc Nu þ e4 N0 c N0 u

ivc3 NTc N0 w þ ivc5 NTc Nu þ Av G NTc Nc dxd ðfc þ fd Þ ¼ 0

ð9:119Þ


278

where fd ðvÞ ¼

ðL 0

½Pðx; vÞNTw ðx; vÞ þ Tðx; vÞNTu ðx; vÞ þ Gðx; vÞNTc ðx; vÞdx

ð9:120Þ

As dd is arbitrary, we can obtain the spectral element equation from Equation 9.119 as SðvÞd ¼ fðvÞ where SðvÞ ¼

ðL 0

ð9:121Þ

½EAN0 u N0 u þ EIw N00 w N00 w þ EIc N0 c Nc0 þ FN0 w Nw0 T

T

T

T

bðN00 w Nu0 þ N0 u N00 w Þ e2 ðN00 w Nc0 þ N0 c N00w Þ T

T

T

T

v2 frAðNTw Nw þ NTu Nu Þ þ rINTc Nc aðN0 w Nu þ NTu Nw0 Þ T

þ gN0 w Nw0 e1 ðN0 w Nc þ NTc Nw0 Þ þ e3 ðNTu Nc þ NTc Nu Þg T

T

þ e4 ðN0 u Nc0 þ N0 c N0u Þ þ ivfcAðNTw Nw þ NTu Nu Þ þ c1 N0 w Nw0 T

T

T

þ c2 NTc Nc c3 ðN0 w Nc þ NTc Nw0 Þ c4 ðN0 w Nu þ NTu Nw0 Þ T

T

þ c5 ðNTu Nc þ NTc Nu Þg þ Av G NTc Nc dx ¼ SðvÞT and

fðvÞ ¼ fc ðvÞ þ fd ðvÞ

ð9:122Þ ð9:123Þ

Substitution of the dynamic shape functions (9.116) into Equation 9.122 gives SðvÞ ¼ H T ðvÞDðvÞH 1 ðvÞ

ð9:124Þ

where DðvÞ ¼ EART1 KEKR1 þ EIw K2 EK2 EIc RT2 KEKR2 FKEK ibðK2 EKR1 þ RT1 KEK2 Þ ie2 ðK2 EKR2 þ RT2 KEK2 Þ v2 frAðE þ RT1 ER1 Þ þ rIRT2 ER2 þ iaðKER1 þ RT1 EKÞ gKEK þ ie1 ðKER2 þ RT2 EKÞ þ e3 ðRT1 ER2 þ RT2 ER1 Þg e4 ðRT1 KEKR2 þ RT2 KEKR1 Þ þ ivfcAðE þ RT1 ER1 Þ c1 KEK þ c2 RT2 ER2 þ ic3 ðKER2 þ RT2 EKÞ þ ic4 ðKER1 þ RT1 EKÞ þ c5 ðRT1 ER2 þ RT2 ER1 Þg þ Av G RT2 ER2

ð9:125Þ

and where K ¼ diag½ki ¼ KT ;

K2 ¼ diag½ki2 ¼ K2T

ð9:126Þ


EðvÞ ¼

ðL

279

eT ðx; vÞeðx; vÞdx ¼ ½Elm ðvÞ

ðl; m ¼ 1; 2; . . . ; 8Þ

ð9:127Þ

0

with Elm ¼

8 < :

h i i eiðkl þ km ÞL 1 kl þ km L

if kl þ km 6¼ 0

ð9:128Þ

if kl þ km ¼ 0

The spectral elements represented by Equation 9.121 can be assembled, with imposing the boundary conditions, in the form Sg ðvÞdg ¼ fg ðvÞ

ð9:129Þ


9.2.3 Spectral Modal Analysis The eigenvalue problem for the spectral PCLD element model can be reduced from Equation 9.129, by enforcing the forcing term to be zero, in the form of Sg ðvÞdg ¼ 0

ð9:130Þ

The (damped) natural frequencies vp (p ¼ 1, 2, . . . , ¥) can be computed from detSg(v) ¼ 0. Eight wavenumbers ki (i ¼ 1, 2, . . . , 8) can be obtained for each natural frequency from the dispersion relation Equation 9.104. The eigenvector dg corresponding to each natural frequency vp can be then computed from Equation 9.130. By using eight wavenumbers and eigenvector dg for each natural frequency, we can obtain the corresponding (damped) normal modes from Equation 9.115. It is worth remembering that the normal modes used in the present spectral modal analysis are the damped normal modes into which the effect of the VEM layer damping is smeared.

9.2.3.1 Orthogonality Properties of Normal Modes Assume that vp is the pth natural frequency and Wp, Up, and Yp are the corresponding normal modes, respectively. Then, the pth eigensolutions should satisfy EIw Wp0000 v2p rAWp þ ivp cAWp ¼ v2p ðaU 0p gW 00p þ e1 Yp0 Þ 00 þ ivp ðc1 W 00p c4 Up0 c3 Yp0 Þ þ bUp000 þ e2 Y000 p þ FW p

EAU 00p þ v2p rAUp ivp cAUp ¼ v2p ðaW 0 p e3 Yp Þ ivp ðc4 Wp0 c5 Yp Þ þ bW 00p e4 Y00p EIc Y00p þ v2p rIYp ivp c2 Yp ¼ v2p ðe1 Wp0 e3 Up Þ ivp ðc3 Wp0 c5 Up Þ þ e2 W 000p e4 U 00p þ Av G* Yp

ð9:131Þ


280

By using Equation 9.97, Equation 9.131 can be written in the simple forms as v2p ðrAWp þ aUp0 þ e1 Y0 p gW 00p Þ þ ivp ðc1 W 00p c3 Yp0 c4 Up0 cAWp Þ þ FW 00p ¼ M 00p v2p ðrAUp aWp0 þ e3 Yp Þ þ ivp ðc4 Wp0 c5 Yp cAUp Þ ¼ Np0

ð9:132Þ

v2p ðrIYp e1 Wp0 þ e3 Up Þ ivp ðc2 Yp c3 Wp0 þ c5 Up Þ Av GYp ¼ Rp0 Following the same procedure as used in Section 9.1.3, we first multiply the first equation in Equation 9.132 by Wq, the second equation by Uq, and the last equation by Yq and then integrate from x ¼ 0 to x ¼ L, followed by the application of the integral by parts. Finally, summing three equations gives ðL 2 vp ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Uq Wp0 Þ 0

e1 ðYp Wq0 þ Yq Wp0 Þ þ e3 ðYp Uq þ Yq Up Þ þ gWp0 Wq0 dx ðL ivp ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 þ c2 Yp Yq c3 ðYp Wq0 þ Yq Wp0 Þ 0

c4 ðUp Wq0 þ Uq Wp0 Þ þ c5 ðYp Uq þ Yq Up Þdx ðL L L L L ¼ Np Uq 0 ðQp þ FWp0 ÞWq 0 Mp Wq0 0 Rp Yq 0 þ Np Uq0 dx ðL ðL ðL ðL 0 þ Mp W 00q dx þ Rp Yq0 dx þ Av GYp Yq dx þ FWp0 Wq0 dx 0

0

0

ð9:133Þ

0

Similarly, for the qth eigensolutions, we obtain ðL v2q ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Uq Wp0 Þ 0

e1 ðYp Wq0 þ Yq Wp0 Þ þ e3 ðYp Uq þ Yq Up Þ þ gWp0 Wq0 dx ðL ivq ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 þ c2 Yp Yq 0

c3 ðYp Wq0 þ Yq Wp0 Þ c4 ðUp Wq0 þ Uq Wp0 Þ þ c5 ðYp Uq þ Yq Up Þdx ðL L L L L ¼ Nq Up 0 ðQq þ FWq0 ÞWp 0 Mq Wp0 0 Rq Yp 0 þ Nq Up0 dx 0 ðL ðL ðL ðL þ Mq W 00p dx þ Rq Yp0 dx þ Av GYq Yp dx þ FWq0 Wp0 dx 0

0

0

ð9:134Þ

0

By subtracting Equation 9.134 from Equation 9.133, applying the homogenous boundary conditions and accomplishing a further manipulation, we can obtain ðL ðv2p v2q Þ ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Uq Wp0 Þ 0

e1 ðYp Wq0 þ Yq Wp0 Þ þ e3 ðYp Uq þ Yq Up Þ þ gWp0 Wq0 dx ðL iðvp vq Þ ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 þ c2 Yp Yq c3 ðYp Wq0 þ Yq Wp0 Þ 0

c4 ðUp Wq0 þ Uq Wp0 Þ þ c5 ðYp Uq þ Yq Up Þdx ¼ 0

ð9:135Þ


281

From Equation 9.135, we can obtain the orthogonality properties as follows: ðL ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Uq Wp0 Þ 0

ðL 0

e1 ðYp Wq0 þ Yq Wp0 Þ þ e3 ðYp Uq þ Yq Up Þ þ gWp0 Wq0 dx ¼ mp dpq ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0

þ c2 Yp Yq c3 ðYp Wq0

ð9:136Þ

þ Yq Wp0 Þ

c4 ðUp Wq0 þ Uq Wp0 Þ þ c5 ðYp Uq þ Yq Up Þdx ¼ cp dpq

where ðL 2 mq ¼ ½rAðUq2 þ Wq2 Þ þ rIY2q 2aUq Wq0 2e1 Yq Wq0 þ 2e3 Yq Uq þ gW 0 q dx 0 ðL 2 cq ¼ ½cAðUq2 þ Wq2 Þ þ c1 Wq0 þ c2 Y2q 2c3 Yq Wq0 2c4 Uq Wq0 þ 2c5 Yq Uq dx

ð9:137Þ

0

The constants mp and cp are the pth modal mass and modal damping coefficient, respectively. By applying the above orthogonality properties to Equation 9.133, we can obtain L L L L ð L ðL 0 0 Nq Uq ðQq þ FWq ÞWq Mq Wq Rq Yq þ Nq Uq0 dx þ Mq W 00q dx 0 0 0 ð 0 ðL ð0 L 0 L 0 0 0 2 þ Rq Yq dx þ Av GYq Yq dx þ FWq Wq dx ¼ vq mq ivq cq 0

0

0

L L L L ð L ðL Np Uq ðQp þ FWp0 ÞWq Mp Wq0 Rp Yq þ Np Uq0 dx þ Mp W 00q dx 0 0 0 ð 0 ðL ð0 L 0 L 0 0 0 þ Rp Yq dx þ Av GYp Yq dx þ FWp Wq dx ¼ 0 0

0

ð9:138Þ

0

9.2.3.2 Spectral Modal Equations For the modal analysis, the forced vibration responses are represented by ¥ X fWp ðxÞ; Up ðxÞ; Yp ðxÞg qp ðtÞ fwðx; tÞ; uðx; tÞ; cðx; tÞg ¼

ð9:139Þ

p¼1

where qp(t) are the modal coordinates. Represent the modal coordinates in the spectral form as 1 1 NX q eivn t ð9:140Þ qp ðtÞ ¼ N n¼0 pn By substitution of Equation 9.140 into Equation 9.139, we obtain ¥ N 1 X 1X fWp ðxÞ; Up ðxÞ; Yp ðxÞg qpn eivn t fwðx; tÞ; uðx; tÞ; cðx; tÞg ¼ N p¼1 n¼0

ð9:141Þ

The external forces are also written in the spectral forms as f pðx; tÞ; tðx; tÞ; gðx; tÞg ¼

1 1 NX fPn ðxÞ; Tn ðxÞ; Gn ðxÞg eivn t N n¼0

ð9:142Þ


282

Substitution of Equations 9.141 and 9.142 into the equations of motion (9.90) yields three equations, for the nth spectral component, as follows: ¥ ¥ X X ðM 00p FW 00p Þ qpn v2n ðrAWp þ aUp0 þ e1 Yp0 gW 00p Þqpn p¼1

p¼1

¥ X

ðcAWp c1 W 00p þ c3 Yp0 þ c4 Up0 Þqpn ¼ Pn ðxÞ p¼1 ¥ ¥ X X Np0 qpn þ v2n ðrAUp aWp0 þ e3 Yp Þqpn p¼1 p¼1 ¥ X ivn ðcAUp c4 Wp0 þ c5 Yp Þqpn ¼ Tn ðxÞ p¼1 ¥ ¥ X X ðRp0 Av GYp Þ qpn þ v2n ðrIYp e1 Wp0 þ e3 Up Þqpn p¼1 p¼1 ¥ X ivn ðc2 Yp c3 Wp0 þ c5 Up Þ qpn ¼ Gn ðxÞ p¼1 þ ivn

ð9:143Þ

where Equation 9.97 is used for Mp, Np, and Rp. Multiply the first equation of Equation 9.143 by Wq, the second equation by Uq, and the last equation by Yq to integrate from x ¼ 0 to x ¼ L. By integrating by parts and summing the final forms of three equations, we can obtain ðL ¥ X 2 vn qpn ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Wp0 Uq Þ p¼1

0

þ gWp0 Wq0 ¥ X þ ivn

p¼1

e1 ðYp Wq0 þ Wp0 Yq Þ þ e3 ðYp Uq þ Up Yq Þdx ðL qpn ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0 þ c2 Yp Yq 0

ðc3 Yp Wq0

þ Wp0 Yq Þ c4 ðUp Wq0 þ Wp0 Uq Þ þ c5 ðYp Uq þ Up Yq Þdx L L L ¥ X L 0 0 ¼ qpn Np Uq þ ðQp þ FWp ÞWq þ Mp Wq þ Rp Uq 0 0 0 p¼1 0 ðL ðNp Uq0 þ Mp W 00q þ Rp Yq0 þ Av GYp Yq þ FWp0 Wq0 Þdx 0

þ

ðL 0

Pn ðxÞWq dx þ

ðL 0

Tn ðxÞUq dx þ

ðL

ð9:144Þ

Gn ðxÞYq dx

0

where Equation 9.97 is used for Qp. Finally, by applying the orthogonality properties from Equations 9.136–9.138 to Equation 9.144, we can obtain the spectral modal equations as ðv2q v2n Þmq qqn iðvq vn Þcq qqn ¼ fqn where fqn is the spectral modal force defined by ðL fqn ¼ ½Pn ðxÞWq þ Tn ðxÞUq þ Gn ðxÞYq dx 0

ð9:145Þ

ð9:146Þ


283

Equations 9.145 and 9.146 can be transformed into the time domain as mq € q q ðtÞ þ cq q_ q ðtÞ þ ðv2q mq ivq cq Þqq ðtÞ ¼ fq ðtÞ where fq is the modal force given by ðL fq ðtÞ ¼ ½pðx; tÞWq þ tðx; tÞUq þ rðx; tÞYq dx

ð9:147Þ

ð9:148Þ

0

9.2.4 Finite Element Model The weak form of the equations of motion (9.90) is given by ðL _ € 0 dw0 ½EIw w00 dw00 þ rA€ w dw þ cAwdw a€ u b dw0 bu0 b dw00 þ g w 0

€ 0 e2 c0 dw00 þ c1 w_ 0 dw0 c4 u_ b dw0 c3 cdw _ 0 þ Fw0 dw0 e1 cdw € b þ EAu0 b du0 b þ rA€ u b dub þ cAu_ b dub a€ w 0 dub bw00 du0 b þ e3 cdu _ b þ e4 c0 du0 b c4 w_ 0 du0 b þ c5 cdu _ € þ c2 cdc € 0 dc e2 w00 dc0 þ e3 €u b dc e1 w þ EIc c0 dc0 þ rI cdc

ð9:149Þ

þ e4 u0 b dc0 c3 w_ 0 dc þ c5 u_ b dc þ Av ðg cÞdcdx ðL ½pðx; tÞdw þ tðx; tÞdub þ gðx; tÞdcdx 0

fNðx; tÞdub jL0 þ Mðx; tÞdw0 jL0 þ ½Qðx; tÞ þ Fw0 dwjL0 þ Rðx; tÞdcjL0 g ¼ 0 The convolution term Av(gTc)dc in Equation 9.149 is approximately represented by using the GHM model [121], which is detailed in the following. The material behavior of a viscoelastic material can be represented by the mass–spring model as € mcðx; tÞ þ ðg cÞ ¼ f ðx; tÞ

ð9:150Þ

where the viscoelastic spring is described by a relaxation function g(x, t) and the convolution (gTc) is defined by Equation 9.81. Equation 9.150 can be transformed into the Laplace domain as þ sGðsÞcðsÞ ¼ f ðsÞ s2 mcðsÞ

ð9:151Þ

where s is the Laplace variable. The material modulus function sG(s), which is the complex modulus in the frequency domain, can be represented by either an analytical model sGðsÞ ¼ Ge þ

G1 b2

1 þ c1 ðifR =f1 Þ

þ ðifR =f1 Þ b1

or the GHM model sGðsÞ ¼ k 1 þ

R X r¼1

s2 þ 2^jr v ^ rs ar 2 ^ ^ rs þ v ^2 s þ 2j r v

ð9:152Þ

! ð9:153Þ

r

The parameters in Equations 9.152 and 9.153 are determined so as to fit experimentally measured frequency-dependent damping data [112]. The accuracy of the GHM model may depend on R, the


284

number of dissipation coordinates. For instance, the one-term GHM model (i.e., R ¼ 1) has one dissipation coordinate as follows ! s2 þ 2^jv ^s sGðsÞ ¼ k 1 þ a ð9:154Þ ^s þ v ^2 s2 þ 2^jv Substituting the one-term GHM model represented by Equation 9.154 into Equation 9.151, we obtain ! 2 ^jv s þ 2 ^ s ¼ f ðsÞ þk 1þa cðsÞ ð9:155Þ s2 mcðsÞ s2 þ 2^jv ^ sþv ^2 For the one-term GHM model, an auxiliary coordinate z (known as the dissipation coordinate) is introduced as v ^2 zðsÞ ¼ cðsÞ ð9:156Þ s2 þ 2^jv ^ sþv ^2 Thus, by using Equation 9.156, we can write Equation 9.155 as þ ð1 þ aÞkc ak zðsÞ ¼ f ðsÞ s2 mcðsÞ

ð9:157Þ

Equation 9.156 can be rewritten in the form þv ^ 2 zðsÞ ¼ 0 ^ s zðsÞ v ^ 2 cðsÞ s2zðsÞ þ 2^jv

ð9:158Þ

Equations 9.157 and 9.158 can be inverse Laplace transformed to obtain the time-domain expressions as follows: € mcðx; tÞ þ ð1 þ aÞkcðx; tÞ ak zðx; tÞ ¼ f ðx; tÞ

ð9:159Þ

€z ðx; tÞ þ 2^jv ^ z_ ðx; tÞ v ^ 2 cðx; tÞ þ v ^ 2 zðx; tÞ ¼ 0

ð9:160Þ

Comparing Equation 9.159 with Equation 9.150, we obtain ðg cÞ ffi ð1 þ aÞkcðx; tÞ ak zðx; tÞ

ð9:161Þ

Thus we can replace the last term in the integral of Equation 9.149 by Av ðg cÞdc ffi Av ½ð1 þ aÞkcðx; tÞ ak zðx; tÞ dc The weak form of Equation 9.160 is written as # !

ð L " Av ak 2Av ak^j €z þ z_ Av akc þ Av akz dzdx ¼ 0 v ^ v ^2 0

ð9:162Þ

ð9:163Þ

The sign convention for the nodal DOFs, forces and moments defined in Figure 9.6 are also valid in the time domain. Thus, we define the nodal DOFs vector as dðtÞ ¼ fU1 ðtÞ W1 ðtÞ Q1 ðtÞ Y1 ðtÞ U2 ðtÞ W2 ðtÞ Q2 ðtÞ Y2 ðtÞgT zðtÞ ¼ fz1 ðtÞ z2 ðtÞgT

ð9:164Þ


285

and the nodal forces and moments vector as fc ðtÞ ¼ fN1 ðtÞ Q1 ðtÞ M1 ðtÞ R1 ðtÞ N2 ðtÞ Q2 ðtÞ M2 ðtÞ R2 ðtÞgT ¼ f Nð0; tÞ Qð0; tÞ Fw0 ð0; tÞ Mð0; tÞ Rð0; tÞg QðL; tÞ þ Fw0 ðL; tÞ

NðL; tÞ

ð9:165Þ

MðL; tÞ RðL; tÞgT

The displacement fields are assumed as ub ðx; tÞ ¼ Nu ðxÞ dðtÞ; cðx; tÞ ¼ Nc ðxÞ dðtÞ;

wðx; tÞ ¼ Nw ðxÞ dðtÞ zðx; tÞ ¼ Nz ðxÞ zðtÞ

ð9:166Þ

where Nu ðxÞ ¼ ½Nu1 ðxÞ 0 0 0 Nu2 ðxÞ 0 0 0 Nw ðxÞ ¼ ½0 Nw1 ðxÞ Nu1 ðxÞ 0 0 Nw2 ðxÞ Nu2 ðxÞ 0 Nc ðxÞ ¼ ½0 0 0 Nu1 ðxÞ 0 0 0 Nu2 ðxÞ Nz ðxÞ ¼ ½Nz1 ðxÞ Nz2 ðxÞ

ð9:167Þ

with Nu1 ðxÞ ¼ Nz1 ðxÞ ¼ 1 ðx=LÞ Nu2 ðxÞ ¼ Nz2 ðxÞ ¼ x=L Nw1 ðxÞ ¼ 1 3ðx=LÞ2 þ 2ðx=LÞ3

ð9:168Þ

Nw2 ðxÞ ¼ 3ðx=LÞ2 2ðx=LÞ3 Nu1 ðxÞ ¼ x½1 2ðx=LÞ þ ðx=LÞ2 Nu2 ðxÞ ¼ x½ðx=LÞ ðx=LÞ2

Substituting Equation 9.166 into the weak forms of Equations 9.149 and 9.163 gives the finite element equation to be

Md 0

0 Mz

€ Cd d þ 0 €z

where Md ¼

ðL 0

0 Cz

Kd d_ þ KTdz z_

Kdz Kz

d fc þ fd ¼ z 0

½rAðNTw Nw þ NTu Nu Þ þ rINTc Nc aðN0 w Nu þ NTu Nw0 Þ T

þ gN0 w Nw0 e1 ðNTc Nw0 þ Nw0 Nc Þ þ e3 ðNTu Nc þ NTc Nu Þdx ðL T T Cd ¼ ½cAðNTw Nw þ NTu Nu Þ þ c1 N0 w Nw0 þ c2 NTc Nc c3 ðN0 w Nc T

T

0

þ NTc Nw0 Þ c4 ðN0 w Nu þ N0 u Nw0 Þ þ c5 ðNTu Nc þ NTc Nu Þdx ðL T T T Kd ¼ ½EAN0 u N0 u þ EIw N00w N00w þ EIc N0 c N0 c T

T

0

bðN00w N0 u þ N0 u N00w Þ e2 ðN00w N0 c þ N0 c N00w Þ T

T

T

T

þ e4 ðN0 u N0 c þ N0 c N0 u Þ þ FNw0 Nw0 þ Av ð1 þ aÞkNTc Nc dx T

T

T

ð9:169Þ


286

Kdz ¼ Av ak

ðL 0

Av ak Mz ¼ v ^2 Cz Kz

ðL 0

NTc Nz dx NTz Nz dx ð9:170Þ

ð 2Av ak^j L T ¼ Nz Nz dx v ^ 0 ðL ¼ Av ak NTz Nz dx 0

and fd ðtÞ ¼

ðL 0

½pðx; tÞNTw ðxÞ þ tðx; tÞNTu ðxÞ þ gðx; tÞNTc ðxÞdx

ð9:171Þ

The finite element matrices defined by Equation 9.170 are given in Appendix 9.B.

Example 9.2 Consider a cantilevered PCLD beam as shown in Figure 9.7. The base beam is fully covered with an elastic constraining layer. The geometric and material properties of the base beam and elastic constraining layer are the same as those given in Table 9.1. The VEM layer is represented by a one-term GHM model and its properties are given in Table 9.3.

Figure 9.7

An example PCLD beam

Figure 9.8 compares the frequency response function “receptance” obtained by using the present SEM with those obtained by using FEM and SMAM (spectral modal analysis method). As the number of finite elements used in FEM is increased, Figure 9.8 shows that the FEM result gets closer to the SEM result. Similarly, as more normal modes are superposed in the spectral modal analysis, the SMAM result also converges towards the SEM result. The lowest five

Table 9.3 Geometric and material properties of the VEM layer L Ev a

300 mm 0 GPa 6.0

b rv v ^

20 mm 1200 kg/m3 1 105 rad/s

hv k ^j

0.3 mm 5 105 Pa 4.0


287

Receptance (dB)

20

SEM ( 1 element) FEM ( 5 elements) FEM (10 elements) FEM (20 elements)

0 –20 –40 –60 –80 0

1 Frequency (kHz)

Receptance (dB)

20

2

SEM (1 element) SMAM (2 normal modes) SMAM (4 normal modes)

0 –20 –40 –60 –80 0

Figure 9.8

1 Frequency (kHz)

2

Frequency response functions obtained by SEM, FEM, and SMAM

Table 9.4 VEM-layer thickness dependence of the natural frequencies and modal damping ratios PCLD beam Mode number (n) 1 2 3 4 5

hv ¼ 0.3 mm fn (Hz) 49.465 280.11 734.16 1380.7 2237.1

jn 0.0881 0.0485 0.0164 0.0058 0.0024

hv ¼ 0.03 mm fn (Hz) 55.398 332.89 882.14 1634.6 2580.2

jn 0.0216 0.0249 0.0150 0.0045 0.0023

Two-layer beam (Table 9.2) hv ¼ 0.003 mm fn (Hz) 56.428 351.53 973.93 1880.7 3055.2

jn 0.0028 0.0037 0.0030 0.0026 0.0022

fn (Hz) 56.554 354.32 991.69 1942.1 3207.7

natural frequencies and corresponding modal damping ratios are shown in Table 9.4 whilst varying the VEM layer thickness. The modal damping ratios are computed from vnI jn ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2nR þ v2nI where vnR is the real part and vnI is the imaginary part of the nth eigenfrequency vn. Table 9.4 shows that as the VEM layer thickness is decreased, all natural frequencies tend to increase, converging towards the natural frequencies of the elastic–elastic two-layer beam considered in Example 9.1, while the modal damping ratios decrease.


288

Appendix 9.A: Finite Element Matrices for the Elastic–Elastic Two-Layer Beam The finite element matrices defined by Equation 9.76 are given by M ¼ MrA Ma þ Mg C ¼ CcA þ Cc1 Cc4 K ¼ KEA þ KEI Kb þ KF

ð9:A:1Þ

where " # rAL A1 A2 ¼ rA ¼ 420 AT A 0 3 2 " # ðL a A4 A5 T ¼ a ðN0 w Nu þ NTu Nw0 Þdx ¼ 12 AT A 0 ðL

MrA

Ma

ðNTw Nw

þ NTu Nu Þdx

6

5

Mg ¼ g

"

ðL

N0 w Nw0 dx ¼ T

0

C c1

A8 A9

" cAL A1 ¼ cA ¼ 420 AT 0 2 " # ðL c1 A7 A8 T ¼ c1 N0 w Nw0 dx ¼ 30L AT A 0 ðL

CCA

A7 g 30L AT 8

#

ðNTw Nw

Cc4 ¼ c4

0

KEA ¼ EA

"

ðN0 w Nu þ NTu Nw0 Þdx ¼ T

ðL

T Nu0 Nu0 dx

0

KEI ¼ EI

ðL

EA ¼ L

"

N00w N00w dx ¼ T

A4

c4 12 AT 5

C1

C1

CT1

C1 #

"

EI A10 L3 AT 11

A3

9

8

ðL

A5

A11

2

KF ¼ F

ðL 0

T Nw0 Nw0 dx

" F A13 ¼ 30L AT 14

ð9:A:2Þ

#

A6 #

A12 " ðL C2 b 00 T 0 0 T 00 Kb ¼ b ðN w Nu þ Nu N w Þdx ¼ L CT 0 0

#

A2

þ NTu Nu Þdx

A14 A15

#

C2 C2

#


with

2

3 140 0 0 A1 ¼ 4 0 156 22L 5 0 22L 4L2 2 3 140 0 0 A3 ¼ 4 0 156 22L 5 0 22L 4L2 2 3 0 6 L A5 ¼ 4 6 0 0 5 L 0 0 2 3 0 0 0 A7 ¼ 4 0 36 3L 5 0 3L 4L2 2 3 0 0 0 A9 ¼ 4 0 36 3L 5 0 3L 4L2 2 3 0 0 0 A11 ¼ 4 0 12 6L 5 0 6L 2L2 2 3 0 0 0 A13 ¼ 4 0 36 3L 5 0 3L 4L2 2 3 0 0 0 A15 ¼ 4 0 36 3L 5 0 3L 3 0 0 1 C2 ¼ 4 0 0 0 5 1 0 0 2

4L2

289

2

3 70 0 0 A2 ¼ 4 0 54 13L 5 0 13L 3L2 2 3 0 6 L A4 ¼ 4 6 0 05 L 0 0 2 3 0 6 L A6 ¼ 4 6 0 0 5 L 0 0 2 3 0 0 0 A8 ¼ 4 0 36 3L 5 0 2 0 4 A10 ¼ 0 0 2 0 4 A12 ¼ 0 0 2 0 A14 ¼ 4 0 0 2 1 C1 ¼ 4 0 0

3L L2 3 0 0 12 6L 5 6L 4L2 3 0 0 12 6L 5 6L 4L2 3 0 0 36 3L 5 3L L2 3 0 0 0 05

ð9:A:3Þ

0 0

Appendix 9.B: Finite Element Matrices for the Elastic–VEM–Elastic Three-Layer Beam The finite element matrices defined by Equation 9.169 are given by Md ¼ MrA þ MrI Ma þ Mg Me1 þ Me3 Cd ¼ CcA þ Cc1 þ Cc2 Cc3 Cc4 þ Cc5 Kd ¼ KEA þ KEIw þ KEIc Kb Ke2 þ Ke4 þ Kv þ KF Av akL T Av akL C 7 ; Mz ¼ Kdz ¼ C6 6 6^ v2 Av akzL Av akL C6 ; Kz ¼ C6 Cz ¼ 3^ v 6

ð9:B:1Þ


290

where

ðL

MrA ¼ rA

0

ðL

MrI ¼ rI

0

Ma ¼ a

ðL

Mg ¼ g

ðL 0

Me3 ¼ e3

ðNTc Nw0

ðL 0

CCA ¼ cA

0

Cc1 ¼ c1

ðL Nc 0

C c3 ¼ c3

ðL

Cc4 ¼ c4

0

C c5 ¼ c5

ðNTc Nu

KEIw

A17 A18

cAL A 1 ¼ T 420 A 2

2 A 3 A

C3 2C3

c3 A16 12 AT17

c4 A 4 ¼ T 12 A 5

þ NTu Nc Þdx

c5 L 2C4 ¼ 6 CT4

EA ¼ L

8 A 9 A

þ NTu Nw0 Þdx

T Nu0 Nu0 dx

C4 2C4

A17 A18

5 A 6 A C4 2C4

1 C C1 ¼ EA T 1 C C 0 1 ðL 11 EIw A10 A 00 T 00 ¼ EIw N w N w dx ¼ 3 T L A11 A12 0 ðL

KEA

5 A 6 A

e3 L 2C4 T 6 C4

c2 L 2C3 Nc dx ¼ CT3 6

T ðN0 w Nu

e1 A16 ¼ T 12 A17

c1 A 7 ¼ T 30L A 8

T

ðL 0

T

C3 2C3

2 A 3 A

8 A 9 A

ðNTc Nw0 þ N0 w Nc Þdx ¼

0

ðL

þ NTu Nu Þdx

T N0 w Nw0 dx

0

Cc2 ¼ c2

g A7 ¼ T 30L A 8

T þ N0 w Nc Þdx

ðNTw Nw

rAL A 1 T 420 A 2

a A 4 ¼ T 12 A 5

ðNTc Nu þ NTu Nc Þdx ¼

ðL

ðL

rIL 2C3 CT3 6

þ NTu Nw0 Þdx

T N0 w Nw0 dx

0

Me1 ¼ e1

NTc Nc dx ¼

T ðN0 w Nu

0

ðL

ðNTw Nw þ NTu Nu Þdx ¼

ð9:B:2Þ


291

C3 C3 KEIc ¼ EIc CT3 C3 0 ðL 2 b C2 C 00 T 0 0 T 00 Kb ¼ b ðN w Nu þ Nu N w Þdx ¼ T 2 L C 2 C 0 ðL e2 C 5 C5 T T Ke2 ¼ e2 ðN0 c N00w þ N00w N0 c Þdx ¼ T C4 L C4 0 ðL Av kð1 þ aÞL 2C3 C3 Kv ¼ Av kð1 þ aÞ NTc Nc dx ¼ T C3 2C3 6 0 ðL 14 F A13 A T KF ¼ F Nw0 Nw0 dx ¼ T 15 30L A14 A 0 ðL

and where

T N0 c N0 c dx

" k ¼ A " k ¼ C

and

2

0

60 6 A16 ¼ 6 40 2

0 0

60 6 A18 ¼ 6 40

C4

0 2 0 60 6 ¼ 6 40

C6 ¼

0

0

0 0

0 0

0

0

L 0 3 0 1 0 07 7 7 0 05

1 0 0 2 1 1 2

0

0T

0

Ck

0

0T

0

# ðk ¼ 1; 2; . . . ; 15Þ ð9:B:3Þ

#

3

67 7 7 L 5

6 L 0 3 0 0 0 0 0 67 7 7 0 0 L5 6 0 0

Ak

EIc ¼ L

ðk ¼ 1; 2Þ 2

0

60 6 A17 ¼ 6 40 2

0 0

60 6 C3 ¼ 6 40 0 2 0 60 6 C5 ¼ 6 40

0

C7 ¼

0

0

0 0

0 0

0

67 7 7 L5

0 0

L 0 3 0 0 0 07 7 7 0 05

0 0 0

0 0 0

0

0

6 0

3

ð9:B:4Þ

1 3 0 07 7 7 15

0 0 1 0 0 0 0 2 0 0 0 0 0 1 0 0

0 0

1 2

The matrices Ak (k ¼ 1, 2, . . . , 15) and Ck (k ¼ 1, 2) are given in Appendix 9.A.

10 Dynamics of Smart Structures Piezoelectric materials have received considerable attention over recent decades due to their potential applications to the active controls of structural vibration and noise. The converse piezoelectric effects are used for actuator designs while the direct piezoelectric effect are used for sensor designs. In order to use the piezoelectric effects, the piezoelectric materials are usually bonded onto the surfaces of the structures so that the structures become the active multi-layer laminate structures. In this chapter, we consider two types of active multi-layer laminate beams: the elastic–piezoelectric two-layer beams [36, 87] and the elastic–viscoelastic–piezoelectric three-layer (active constrained layered damping (ACLD)) beams [73, 88, 105, 112, 141, 147]. By using Hamilton’s principle, the coupled equations of motion are derived for both active multi-layer laminate beams. The effects of the structural damping and viscous damping of each layer are taken into account in the spectral element models formulated by using the variational method. The spectral element modeling is also introduced for an ACLD beam subjected to the feedback control. The orthogonality properties of the normal modes of a coupled structural dynamics problem are formulated to derive spectral modal equations for both active multi-layer laminate beams. The finite element models are also provided for use in the evaluation of the spectral element models.

10.1 Elastic–Piezoelectric Two-Layer Beams 10.1.1 Equations of Motion Figure 10.1 shows an elastic–piezoelectric two-layer beam element. w(x, t) is the transverse displacement of the base beam, ub(x, t) is the axial displacement of the base beam, and up(x, t) is the axial displacement of the piezoelectric material (PZT) layer. hb and hp are the thicknesses of the base beam and PZT layer, respectively. In this chapter, the subscripts b and p are used to denote the quantities of the base beam and PZT layer, respectively. For the derivation of the equations of motion, the following assumptions are made [87]. a. Geometric and material properties are uniform along the beam. b. Perfect continuity, but no slip, at the interfaces.


294


Figure 10.1 An elastic–PZT two-layer beam element

c. Both layers take the same transverse displacement. d. Small amplitude vibration. e. Linear elasticity and piezoelectricity are valid. f. Shear deformations and rotary inertias in both layers are negligible. g. Uniform voltage along the beam. Based on the above assumptions, we can obtain a kinematic relationship as up ¼ ub ðh=2Þw0

ð10:1Þ

h ¼ hb þ hp

ð10:2Þ

where

When the PZT layer is very thin, throughout its thickness the electrical displacement D3 can be considered to be constant. In this situation, the in-plane electrical fields will be negligible, especially when the voltage applied to the PZT layer is uniform along the axis of the beam. The constitutive relationship for a piezoelectric material under uni-axial loading is given by D s1 e1 h31 C11 ¼ ð10:3Þ E3 D3 h31 bs33 where s1 and e1 are the mechanical stress and strain in the x-direction, respectively; D3 is the electrical displacement (charge/area in the beam vertical direction) and E3 is the electrical field D (voltage/length along the vertical direction), all in the z-direction; C11 is the elastic stiffness; S b33 is the dielectric constant; and h31 is the piezoelectric constant. The material constants h31 and bS33 are related to the piezoelectric constant d31 through h231 bS33 1 ¼ d31 Ep

ð10:4Þ

D Ep ¼ C11 h231 bS33 1

ð10:5Þ

where

For the direct effect, applying a constant strain under the conditions of zero electric field (E3 ¼ 0) yields D3 ¼ h231 bS33 1 e1 ¼ d31 Ep u0p ¼ d31 Ep ½u0b ðh2 =2Þw00 þ hv c0

ð10:6Þ

Dynamics of Smart Structures

295

The equations of motion are derived by using Hamilton’s principle ð t2 ðdT dV þ dWÞdt ¼ 0 t1

ð10:7Þ

where T and V are the kinetic energy and potential energy, respectively, and dW is the virtual work. By using the constitutive relationship (10.3) and assuming that D3 is constant throughout the thickness of the PZT layer, the strain energy V and the kinetic energy T are obtained as ð 1 L D D V¼ ½Eb Ab u0b2 þ Eb Ib w002 þ C11 Ip w002 þ C11 Ap u0p2 2Ap h31 D3 u0p þ Ap bS33 D23 þ Fw02 dx 2 0 ð 1 L (10.8) T¼ ½r Ab ðu_ 2b þ w_ 2 Þ þ rp Ap ðu_ 2p þ w_ 2 Þdx 2 0 b ð10:8Þ where E, A, I, and r (for each layer) are the Young’s modulus, the cross-sectional area, the area moment of inertia about the neutral axis of each layer, and the mass density, respectively, and F is the constant axial tensile force applied to the beam. By eliminating up and D3 from Equation 10.8 by using Equations 10.1 and 10.6, respectively, we can write the strain and kinetic energies as ð 1 L 2 V¼ ðEAub0 þ EIw002 2bw00 u0b þ Fw02 Þdx 2 0 ð10:9Þ ð 1 L 2 2 0 02 T¼ ðrAu_ b þ rAw_ 2aw_ u_ b þ g w_ Þdx 2 0 where EA ¼ Eb Ab þ Ep Ap ; rA ¼ rb Ab þ rp Ap ; b ¼ ð1=2ÞEp Ap h;

D EI ¼ Eb Ib þ C11 Ip þ ð1=4ÞEp Ap h2 a ¼ ð1=2Þrp Ap h

g ¼ ð1=4Þrp Ap h

ð10:10Þ

2

The virtual work dW done by the external forces, viscous damping forces, boundary forces, and applied voltage is given by ðL ðL ðL ð dW ¼ pðx; tÞdwdx þ tðx; tÞdub dx cb r_ b drb dAb dx 0 0 0 Ab ðL ð cp r_ p drp dAp dx þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ ð10:11Þ 0 Ap þ M1 ðtÞdw0 ð0; tÞ þ M2 ðtÞdw0 ðL; tÞ þ Q1 ðtÞdwð0; tÞ þ Q2 ðtÞdwðL; tÞ ðL þ bVðtÞdD3 dx 0

where rb ¼ ub nx þ wnz ;

rp ¼ up nx þ wnz

ð10:12Þ

where nx and nz are the unit direction vectors in the x- and z-directions, respectively. In Equation 10.11, V(t) is the uniform voltage applied to the PZT layer and cb and cp are viscous damping coefficients. The definitions of the other symbols used in Equation 10.11 are the same as used in Equation 9.7. By applying Equations 10.1, 10.6 and 10.12 into Equation 10.11 and


296

taking the integral by parts, we can write the virtual work as ðL _ dW ¼ ½pðx; tÞdw þ tðx; tÞdub cAu_ b dub cAwdw þ c4 w_ 0 dub 0

þ c4 u_ b dw0 c1 w_ 0 dw0 dx þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ 0

ð10:13Þ

0

þ M1 ðtÞdw ð0; tÞ þ M2 ðtÞdw ðL; tÞ þ Q1 ðtÞdwð0; tÞ þ Q2 ðtÞdwðL; tÞ L L þ bd31 Ep VðtÞdub ð1=2Þbhd31 Ep VðtÞdw0 0

0

where cA ¼ cb Ab þ cp Ap ;

c1 ¼ ð1=4Þcp Ap h2 ;

c4 ¼ ð1=2Þcp Ap h

ð10:14Þ

By substituting Equations 10.8 and 10.13 into Hamilton’s principle (10.7), we obtain coupled two equations of motion EIw0000 þ rAw€ þ cAw_ ¼ a u€0b þ bu000 þ gw € 00 þ c1 w_ 00 c4 u_ 0b þ Fw00 þ pðx; tÞ u b cAu_ b ¼ aw€ 0 þ bw000 c4 w_ 0 tðx; tÞ EAu00b rA€

ð10:15Þ

and the boundary conditions Qð0; tÞ ¼ Q1 ðtÞ Fw0 ð0; tÞ Mð0; tÞ ¼ M1 ðtÞ M p ðtÞ

or wð0; tÞ ¼ w1 ðtÞ or w0 ð0; tÞ ¼ u1 ðtÞ

Nð0; tÞ ¼ N1 ðtÞ þ N p ðtÞ

or ub ð0; tÞ ¼ u1 ðtÞ

0

QðL; tÞ ¼ þQ2 ðtÞ Fw ðL; tÞ MðL; tÞ ¼ þM2 ðtÞ M p ðtÞ

or wðL; tÞ ¼ w2 ðtÞ or w0 ðL; tÞ ¼ u2 ðtÞ

NðL; tÞ ¼ þN2 ðtÞ þ N p ðtÞ

or ub ðL; tÞ ¼ u2 ðtÞ

M p ðtÞ ¼ ð1=2Þbhd31 Ep VðtÞ;

N p ðtÞ ¼ bd31 Ep VðtÞ

ð10:16Þ

where ð10:17Þ

where b ¼ bb ¼ bp is the width of the two-layer beam. The piezoelectric force Np(t) and moment Mp(t) are generated by the voltage V(t) applied to the PZT layer. The resultant axial force N(x, t), bending moment M(x, t) and the transverse shear force Q(x, t) are related to the displacement fields as follows: Qðx; tÞ ¼ EIw w000 a€ u b þ bu00b þ g w€ 0 þ c1 w_ 0 c4 u_ b ¼ Mðx; tÞ a€ u b þ g w€ 0 þ c1 w_ 0 c4 u_ b Mðx; tÞ ¼ EIw w00 bu0b

ð10:18Þ

Nðx; tÞ ¼ EAu0b bw00

The structural damping of each layer can be readily taken into account by using the complex modulus as Eb* ¼ Eb ð1 þ ihb Þ;

D* Ep* ¼ C11 h231 bS33 1 ;

D* D C11 ¼ C11 ð1 þ ihp Þ

ð10:19Þ

where hb and hp are the loss factors of the base beam and PZT layer, respectively. The governing Equations 10.15 for an elastic–PZT two-layer beam have exactly the same forms as the governing Equations 9.11 for an elastic–elastic two-layer beam, except for the use of


297

the structural and damping properties (Equations 10.10 and 10.14) defined for the elastic–PZT two-layer beam. Thus, a large part of the spectral element formulation carried out in Chapter 9 for the elastic–elastic two-layer beam can be shared by the elastic–PZT two-layer beam.

10.1.2 Spectral Element Modeling 10.1.2.1 Governing Equations in the Frequency Domain To transform the governing equations into the frequency domain, we represent all dynamic responses, forces, and moments in the spectral forms as given by Equations 9.15 and 9.16. In addition, we also represent piezoelectric forces, moments, and voltage in the spectral forms as fM p ðx; tÞN p ðx; tÞVðx; tÞg ¼

1 1 NX fMnp ðx; vn ÞNnp ðx; vn ÞVn ðx; vn Þgeivn t N n¼0

ð10:20Þ

By substituting Equations 9.15, 9.16 and 10.20 into Equations 10.15 and 10.16, we obtain the frequency-domain governing equations EIW 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ ivc1 W 00 ivc4 U 0 þ FW 00 þ PðxÞ

ð10:21Þ

EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 ivc4 W 0 TðxÞ and the boundary conditions Qð0Þ ¼ Q1 FW 0 ð0Þ Mð0Þ ¼ M1 M p

or Wð0Þ ¼ W1 or W 0 ð0Þ ¼ Q1

Nð0Þ ¼ N1 þ N p

or Uð0Þ ¼ U1 0

QðLÞ ¼ þQ2 FW ðLÞ MðLÞ ¼ þM2 M p

or WðLÞ ¼ W2 or W 0 ðLÞ ¼ Q2

NðLÞ ¼ þN2 þ N p

or UðLÞ ¼ U2

ð10:22Þ

where M p ¼ ð1=2Þbd31 Ep hVðvÞ;

N p ¼ bd31 Ep VðvÞ

ð10:23Þ

Similarly, we can transform the force-displacement relationships (10.18) into the frequency domain as QðxÞ ¼ EIW 000 þ v2 aU þ bU 00 v2 gW 0 þ ivc1 W 0 ivc4 U MðxÞ ¼ EIW 00 bU 0 0

NðxÞ ¼ EAU bW

ð10:24Þ

00

In the above equations, the subscripts n are omitted for brevity. 10.1.2.2 Spectral Nodal DOFs, Forces, and Moments The spectral nodal degrees-of-freedom (DOFs) and the mechanical spectral nodal forces and moments are specified in Figure 10.2. We define the spectral nodal DOFs vector as d ¼ fU1 W1 Q1 U2 W2 Q2 gT ¼ fUð0Þ Wð0Þ W 0 ð0Þ UðLÞ WðLÞ W 0 ðLÞgT

ð10:25Þ


298 W1

W2

U1

U2

N1

N2 M1 Q1

Figure 10.2

Q2 M2

L

Spectral element of the elastic–PZT two-layer beam

and the mechanical spectral nodal forces and moments vector as fcmech ¼ fN1 Q1 M1 N2 Q2 M2 gT

ð10:26Þ

10.1.2.3 Dynamic Shape Functions To formulate the spectral element equation, we consider the homogeneous governing equations reduced from Equation 10.21 to be EIW 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ ivc1 W 00 ivc4 U 0 þ FW 00 00

0

000

EAU þ v rAU ivcAU ¼ v aW þ bW ivc4 W 2

2

ð10:27Þ

0

As Equation 10.27 is identical with Equation 9.22, except for using the structural and damping properties defined by Equations 10.10 and 10.14, we can follow the same procedure as used in Section 9.1.2.3 to derive a dispersion relation a1 k 6 þ a2 k 4 þ a3 k 2 þ a4 ¼ 0

ð10:28Þ

where ai (i ¼ 1, 2, 3, 4) are given by Equation 9.27 with the structural and damping properties replaced by Equations 10.10 and 10.14. By using six wavenumbers ki (i ¼ 1, 2, . . ., 6) computed from Equation 10.28, we can obtain general solutions to Equation 10.27 in the form of Wðx; vÞ ¼ Nw ðx; vÞd;

Uðx; vÞ ¼ Nu ðx; vÞd

ð10:29Þ

where Nw and Nu are the dynamic shape functions given by Nw ðx; vÞ ¼ eðx; vÞH 1 ðvÞ;

Nu ðx; vÞ ¼ eðx; vÞRH 1 ðvÞ

ð10:30Þ

The matrices e(x;v), R(v) and H(v) are defined by Equations 9.31, 9.29 and 9.34, respectively. 10.1.2.4 Weak Form of Governing Equations As shown in Section 9.1.2.4, the weak form of the governing Equation 10.27 can be derived as ðL ½EIW 00 dW 00 v2 rAWdW þ ivcAWdW þ ivc1 W 0 dW 0 0

ivc4 UdW 0 þ v2 aUdW 0 bU 0 dW 00 v2 gW 0 dW 0 þ FW 0 dW 0 þ EAU 0 dU 0 v2 rAUdU þ ivcAUdU þ v2 aW 0 dU bW 00 dU 0 0

0

ivc4 W dUdx f½QðxÞ þ FW ðL ½PðxÞdW þ TðxÞdUdx ¼ 0 0

ðxÞdWjL0

þ MðxÞdW 0 jL0

þ NðxÞdUjL0 g

ð10:31Þ


299

10.1.2.5 Spectral Element Equation As shown in Section 9.1.2.5, substitution of Equation 10.29 into Equation 10.31 gives the spectral element equation as SðvÞd ¼ f ðvÞ

ð10:32Þ

The matrix S(v) is the spectral element matrix defined by SðvÞ ¼ H T ðvÞDðvÞH 1 ðvÞ

ð10:33Þ

where DðvÞ ¼ EARKEKR þ EIK2 EK2 ibðK2 EKR þ RKEK2 Þ v2 ½rAðE þ RERÞ þ iaðKER þ REKÞ gKEK

ð10:34Þ

þ iv½cAðE þ RERÞ c1 KEK þ ic4 ðKER þ REKÞ FRER In Equation 10.34, the following definitions are used. K ¼ diag ½ki ¼ KT ; K2 ¼ diag ½ki2 ¼ K2T ð j ¼ 1; 2; . . . ; 6Þ ðL EðvÞ ¼ eT ðx; vÞeðx; vÞdx ¼ ½Epq ðvÞ ðp; q ¼ 1; 2; . . . ; 6Þ

ð10:35Þ ð10:36Þ

0

where Epq ¼

8 > < > :

h i i e iðkp þ kq ÞL 1 kp þ kq L

if

kp þ kq 6¼ 0

if

k p þ kq ¼ 0

ð10:37Þ

The vector f(v) is the spectral nodal forces vector defined by fðvÞ ¼ fcmech ðvÞ þ fcpiezo ðvÞ þ fd ðvÞ

ð10:38Þ

fcpiezo ¼ f N p 0 M p N p 0 M p gT

ð10:39Þ

where

fd ðvÞ ¼

ðL 0

½Pðx; vÞNTw ðx; vÞ þ Tðx; vÞNTu ðx; vÞdx

ð10:40Þ

In the above, fcpiezo and fcmech represent the piezoelectric spectral nodal forces and moments generated by the applied voltage and the mechanical spectral nodal forces and moments vector defined by Equation 10.26, respectively. The spectral elements represented by Equation 10.32 can be assembled to form a global system equation as Sg ðvÞdg ¼ fg ðvÞ

ð10:41Þ



300

10.1.3 Spectral Element with Active Control Consider an elastic–PZT two-layer beam implemented with a proportional-derivative (PD) controller, as shown in Figure 10.3. Assume that the driving voltage applied to the PZT layer is determined by the vibration data measured at x ¼ x0 as _ 0 ; tÞ VðtÞ ¼ Kp wðx0 ; tÞ Kd wðx

ð10:42Þ

Driving voltage V (t ) +

–Kp w

– Kd w

Vibration response

Figure 10.3

An elastic–PZT two-layer beam with a PD controller

where Kp and Kd are the proportional and derivative gains, respectively. Equation 10.42 can be written in the spectral form as VðvÞ ¼ ðKp þ ivKd ÞWðx0 ; vÞ

ð10:43Þ

By using the first part of Equation 10.29, the measured displacement W(x0;v) can be related to the spectral nodal DOFs vector d as follows: Wðx0 ; vÞ ¼ Nw ðx0 ; vÞd

ð10:44Þ

The spectral nodal DOFs vector d in Equation 10.44 corresponds to the spectral element in which the sensor (i.e., x0) is located. Substitution of Equation 10.44 into Equation 10.43 gives VðvÞ ¼ ðKp þ ivKd ÞNw ðx0 ; vÞd

ð10:45Þ

By substituting Equation 10.45 into Equation 10.23, we obtain the piezoelectric forces and moments as N p ¼ bd31 Ep VðvÞ ¼ xN Nw ðx0 ; vÞd M p ¼ ð1=2Þbd31 Ep hVðvÞ ¼ xM Nw ðx0 ; vÞd

ð10:46Þ

where x N ¼ bd31 Ep ðKp þ ivKd Þ x M ¼ ð1=2Þbd31 Ep hðKp þ ivKd Þ

ð10:47Þ

Substitution of Equation 10.46 into Equation 10.39 gives fcpiezo ¼ Spiezo ðvÞd

ð10:48Þ


301

where 2 6 6 6 Spiezo ðvÞ ¼ 6 6 6 4

3 x N Nw ðx0 ; vÞ 7 0 7 þx M Nw ðx0 ; vÞ 7 7 þx N Nw ðx0 ; vÞ 7 7 5 0 x M Nw ðx0 ; vÞ

ð10:49Þ

By substituting Equation 10.48 into Equation 10.38 and the result into Equation 10.32, we can obtain the spectral element equation for the elastic–PZT two-layer beam element with a PD controller as follows: SðvÞCONTROL d ¼ fcmech þ fd

ð10:50Þ

SðvÞCONTROL ¼ SðvÞ þ Spiezo ðvÞ

ð10:51Þ

where

10.1.4 Spectral Modal Analysis As the governing equations for the elastic–PZT two-layer beam have exactly the same forms as those for the elastic–elastic two-layer beam, the spectral modal analysis in Section 9.1.3 is also valid for the elastic–PZT two-layer beam. 10.1.4.1 Orthogonality Properties of Normal Modes Let vp be the pth damped natural frequency and Wp and Up the corresponding normal modes. The pth eigensolutions (vp, Wp, Up) must satisfy an eigenvalue problem as EIWp0000 v2p rAWp þ ivp cAWp ¼ v2p ðaUp0 gWp00 Þ þ ivp ðc1 Wp00 c4 Up0 Þ þ bUp000 þ FWp00 EAUp00 þ v2p rAUp ivp cAUp ¼ v2p aWp0 ivp c4 Wp0 þ bWp000 ð10:52Þ Similarly, the qth eigensolutions (vq, Wq, Uq) must satisfy an eigenvalue problem as EIWq0000 v2q rAWq þ ivq cAWq ¼ v2q ðaUq0 gWq00 Þ þ ivq ðc1 Wq00 c4 Uq0 Þ þ bUq000 þ FWq00 EAUq00 þ v2q rAUq ivq cAUq ¼ v2q aWq0 ivq c4 Wq0 þ bWq000 ð10:53Þ Following the procedure used in Section 9.1.3.1, we can derive the orthogonality properties of the damped normal modes from Equations 10.52 and 10.53 as follows: ðL ½rAðWp Wq þ Up Uq Þ aðUp Wq0 þ Uq Wp0 Þ þ gWp0 Wq0 dx ¼ mp dpq 0 ð10:54Þ ðL 0 0 0 0 ½cAðWp Wq þ Up Uq Þ þ c1 Wp Wq c4 ðUp Wq þ Uq Wp Þdx ¼ cp dpq 0


302

where mp and cp are the modal mass and modal damping coefficients defined by ðL 2 mp ¼ ½rAðUp2 þ Wp2 Þ 2aUp Wp0 þ gWp0 dx 0 ðL 2 cp ¼ ½cAðUp2 þ Wp2 Þ þ c1 W 0p 2c4 Up Wp0 dx

ð10:55Þ

0

10.1.4.2 Spectral Modal Equations Based on the modal analysis method, the forced vibration responses are represented by superposing the normal modes as wðx; tÞ ¼

¥ X

Wp ðxÞqp ðtÞ;

uðx; tÞ ¼

p¼1

¥ X

Up ðxÞqp ðtÞ

ð10:56Þ

p¼1

where qp(t) are the modal coordinates. The modal coordinates can be written in the spectral form as 1 1 NX qp ðtÞ ¼ ð10:57Þ q eivn t N n¼0 pn By using Equation 10.57, Equation 10.56 can be written as wðx; tÞ ¼

¥ N 1 X 1X Wp ðxÞqpn eivn t ; N p¼1 n¼0

uðx; tÞ ¼

¥ N 1 X 1X Up ðxÞqpn eivn t N p¼1 n¼0

ð10:58Þ

The external forces are also written in the spectral forms as pðx; tÞ ¼

1 1 NX Pn ðxÞeivn t ; N n¼0

tðx; tÞ ¼

1 1 NX Tn ðxÞeivn t N n¼0

ð10:59Þ

By substituting Equations 10.58 and 10.59 into Equation 10.15, taking the same manipulation as carried out in Section 9.1.3.2, and applying the orthogonality properties (10.54), we can derive the modal equations in the frequency domain to be ðv2q v2n Þmq qqn iðvq vn Þcq qqn ¼ fqn where fqn ¼

ðL

½Pn ðxÞWq þ Tn ðxÞUq dx

ðq ¼ 1; 2; 3; . . .Þ

ðq ¼ 1; 2; 3; . . .Þ

ð10:60Þ

ð10:61Þ

0

The spectral modal Equations 10.60 can be transformed into the time domain as mpq€p ðtÞ þ cp q_ p ðtÞ þ ðv2p mp ivp cp Þqp ðtÞ ¼ fp ðtÞ where fp ðtÞ ¼

ðL 0

½pðx; tÞWp þ tðx; tÞUp dx

ð10:62Þ

ð10:63Þ


303

10.1.5 Finite Element Model The finite element equation can be formulated from the weak form of the time-domain governing Equations 10.15. The weak form can be derived in the form as given by Equation 9.70. By using the same nodal DOFs vector and displacement fields as used in Section 9.1.4, we can obtain the finite element equation as € þ CdðtÞ _ þ KdðtÞ ¼ f mech ðtÞ þ f piezo ðtÞ þ f ðtÞ MdðtÞ d c c where M¼ C ¼ K ¼

ðL ð0L ð0L 0

ð10:64Þ

½rAðNTw Nw þ NTu Nu Þ aðNw0 T Nu þ NTu Nw0 Þ þ gNw0 T Nw0 dx ½cAðNTw Nw þ NTu Nu Þ þ c1 N 0 w Nw0 c4 ðN 0w Nu þ N u0 Nw0 Þdx T

T

T

ð10:65Þ

½EAN 0u Nu0 þ EIN w00 Nw00 bðN w00 Nu0 þ N u0 Nw00 Þ þ FN w0 Nw0 dx T

T

T

T

T

and fcmech ¼ fN1 ðtÞ Q1 ðtÞ M1 ðtÞ N2 ðtÞ Q2 ðtÞ M2 ðtÞgT fcpiezo ¼ fN p ðtÞ 0 M p ðtÞ N p ðtÞ 0 M p ðtÞgT fd ðtÞ ¼

ðL 0

½pðx; tÞNTw ðxÞ þ tðx; tÞNTu ðxÞdx

ð10:66Þ

ð10:67Þ

The finite element matrices M, C, and K are same as given in Appendix 9.A, but replacing the structural and damping properties by Equations 10.10 and 10.14 defined for the elastic–PZT two-layer beams.

Example 10.1 Consider a cantilevered elastic–PZT two-layer beam as shown in Figure 10.4. The base beam is partially covered with a uniform PZT layer. The geometric and material properties are given in Table 10.1. The high accuracy of the present spectral element model is shown in Table 10.2 by comparing the natural frequencies obtained by both SEM and FEM. Table 10.3 compares the natural frequencies and modal damping ratios of the beam with and without the PD control, which is shown in Figure 10.3. It is shown that the beam with a PD controller has larger modal damping ratios than the beam without the PD controller to reduce the vibration quicker, as displayed in Figure 10.5. PZT layer hp

Base beam L1

Figure 10.4

L2

hb L3

A cantilevered beam partially covered with a PZT layer


304 Table 10.1

Geometric and material properties of the base beam and PZT layer

L1 b Eb d31

100 mm 20 mm 70 GPa 1.75 1010 m/V

L2 hb Ep rb

L3 hp D C11 rp

100 mm 5 mm 65 GPa 2700 kg/m3

100 mm 0.75 mm 75 GPa 7500 kg/m3

Table 10.2 Natural frequencies (Hz) of an elastic–PZT two-layer beam SEM(N)

FEM(N)

Mode

N¼1

N¼6

N ¼ 15

N ¼ 30

N ¼ 60

N ¼ 150

1st 2nd 3rd 4th 5th 10th

45.3380 289.987 800.120 1590.45 2621.51 9291.54

45.3412 290.089 802.295 1603.74 2662.93 11438.37

45.3385 289.995 800.285 1591.26 2623.85 9362.63

45.3381 289.988 800.154 1590.60 2621.85 9299.39

45.3380 289.987 800.128 1590.49 2621.58 9292.80

45.3380 289.987 800.121 1590.46 2621.52 9291.71

a

Note: (N) ¼ total number of the equal-length finite elements used in the analysis.

Table 10.3 Natural frequencies and modal damping ratios of the elastic–PZT two-layer beam with and without the PD controller Uncontrolled

Controlled Kp ¼ 5 10 V, Kd ¼ 100 Vs

Kp ¼ 5 107 V, Kd ¼ 300 Vs

7

Mode number (n) 1 2 3 4 5

fn (Hz)

fn (Hz)

xn

fn (Hz)

xn

45.338 289.99 800.12 1590.5 2621.5

48.960 289.35 800.46 1590.6 2618.5

0.0084 0.0117 0.0021 0.0012 0.0004

49.158 289.37 800.49 1590.6 2616.9

0.0240 0.0263 0.0059 0.0019 0.0009

Displacement (mm)

4

Uncontrolled Controlled: Kp = 50 MV, Kd = 100 Vs Controlled: Kp = 50 MV, Kd = 300 Vs

3 2 1 0 –1 –2 –3 0

Figure 10.5

0.1

0.2 Time (s)

0.3

0.4

Dynamic responses of an elastic–PZT two-layer beam with a PD controller


305

10.2 Elastic–Viscoelastic–Piezoelectric Three-Layer (ACLD) Beams 10.2.1 Equations of Motion The active constrained layered damping (ACLD) treated beams consist of three layers: an elastic base beam, a viscoelastic material (VEM) layer, and a piezoelectric material (PZT) layer. The geometry and deformations of an ACLD beam are shown in Figure 10.6. ub(x, t), uv(x, t), and up(x, t) are the axial displacements of the base-beam layer, VEM layer, and PZT layer, respectively. hb, hv, and hp are the thicknesses of the base beam, VEM layer, and PZT layer, respectively, and c is the shear angle of the VEM layer. The subscripts b, v, and p are used to denote the quantities of the base beam, VEM layer, and PZT layer, respectively. For the derivation of the equations of motion, the following assumptions are made [88]. a. Geometric and material properties are uniform along the beam. b. Perfect continuity, but no slip, at the interfaces. c. All layers take the same transverse displacement. d. Small amplitude vibration. e. Linear elasticity, viscoelasticity, and piezoelectricity are valid for the base beam, VEM layer, and PZT layer, respectively. f. Shear deformations and rotary inertias in the base beam and PZT layer are negligible. g. Uniform voltage along the beam. z Piezoelectric layer Viscoelastic

A-A section

A

layer up uv ub

hp

b hv A

w′

Base beam

hb w

x

Figure 10.6 An elastic–VEM-PZT three-layer (ACLD) beam element

From the geometry of Figure 10.6, we obtain the following kinematics relationships uv ¼ ub ðh1 =2Þw0 þ ðhv =2Þc up ¼ ub ðh2 =2Þw0 þ hv c

ð10:68Þ

where h1 ¼ hb þ hv ;

h2 ¼ hb þ 2hv þ hp

ð10:69Þ

The equations of motion of the ACLD beam are derived by using Hamilton’s principle ð t2 ðdT dV þ dWÞdt ¼ 0 ð10:70Þ t1


306

where T is the kinetic energy, V is the potential energy, and dW is the virtual work. The strain and kinetic energies are given by ð 1 L V¼ ½Eb Ab u0b2 þ Eb Ib w002 þ Ev Av u0v2 þ Ev Iv ðw00 c0 Þ2 þ Av ðg cÞc 2 0 D D Ip w002 þ C11 Ap u0p2 2Ap h31 D3 u0p þ Ap bS33 D23 þ Fw02 dx þ C11 ð10:71Þ ðL 1 _ 2 þ rp Ap ðu_ 2 þ w_ 2 Þ dx T¼ ½r Ab ðu_ 2b þ w_ 2 Þ þ rv Av ðu_ 2v þ w_ 2 Þ þ rv Iv ðw_ 0 cÞ p 2 0 b where the constitutive relationship in Equation 9.81 is applied for the VEM layer. By eliminating uv and up from Equation 10.71 by using Equation 10.68 and also D3 by using Equation 10.6, we can write the strain and kinetic energies as ð 1 L 2 ½EAu0b þ EIw w002 þ EIc c02 2bw00 u0b þ 2e4 u0b c0 2e2 w00 c0 V¼ 2 0 ð10:72Þ þ Av ðg cÞc þ Fw02 dx ðL 1 2 T ¼ frAu_ 2b þ rAw_ 2 þ rI c_ 2aw_ 0 u_ b þ 2e3 u_ b c_ 2e1 w_ 0 c_ þ g w_ 02 gdx 2 0 where EA ¼ Eb Ab þ Ev Av þ Ep Ap D EIw ¼ Eb Ib þ Ev Iv þ C11 Ip þ ð1=4ÞEv Av h21 þ ð1=4ÞEp Ap h22

EIc ¼ Ev Iv þ ð1=4ÞEv Av h2v þ Ep Ap h2v rA ¼ rb Ab þ rv Av þ rp Ap rI ¼ rv Iv þ ð1=4Þrv Av h2v þ rp Ap h2v a ¼ ð1=2Þðrv Av h1 þ rp Ap h2 Þ

ð10:73Þ

b ¼ ð1=2ÞðEv Av h1 þ Ep Ap h2 Þ g ¼ rv Iv þ ð1=4Þrv Av h21 þ ð1=4Þrp Ap h22 e1 ¼ rv Iv þ ð1=4Þrv Av h1 hv þ ð1=2Þrp Ap h2 hv e2 ¼ Ev Iv þ ð1=4ÞEv Av h1 hv þ ð1=2ÞEp Ap h2 hv e3 ¼ ð1=2Þrv Av hv þ rp Ap hv e4 ¼ ð1=2ÞEv Av hv þ Ep Ap hv

The virtual work dW done by external forces, viscous damping forces, boundary forces, and the voltage applied to the PZT layer is given by ðL ðL ðL dW ¼ pðx; tÞdwdx þ tðx; tÞdub dx þ gðx; tÞdcdx 0 0 0 ðL ð ðL ð ðL ð cb r_ b drb dAb dx cv r_ v drv dAv dx cp r_ p drp dAp dx 0

Ab

0

Av

0

Ap

0

þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ þ M1 ðtÞdw ð0; tÞ þ M2 ðtÞdw0 ðL; tÞ þ Q1 ðtÞdwð0; tÞ þ Q2 ðtÞdwðL; tÞ þ R1 ðtÞdcð0; tÞ þ R2 ðtÞdcðL; tÞ ðL bVðtÞdD3 dx þ 0

ð10:74Þ


where

307

rb ¼ ub nx þ wnz rv ¼ ½uv zðw0 cÞnx þ wnz

ð10:75Þ

rp ¼ up nx þ wnz and where nx and nz are the unit direction vectors in the x- and z-directions, respectively. In Equation 10.74, V(t) is the uniform voltage applied to the piezoelectric layer and cb, cv, and cp are viscous damping coefficients. The definitions of the other symbols used in Equation 10.74 are the same as those used in Equation 9.86. By applying Equations 10.6, 10.68 and 10.75 into Equation 10.74 and taking the integral by parts, we can write the virtual work as ðL _ dW ¼ ½pðx; tÞdw þ tðx; tÞdub þ gðx; tÞdc cAu_ b dub cAwdw þ c4 w_ 0 dub 0

_ b c5 u_ b dc þ c3 w_ 0 dc c2 cdc _ þ c4 u_ b dw0 c1 w_ 0 dw0 þ c3 cdw _ 0 dx c5 cdu þ M1 ðtÞdw0 ð0; tÞ þ M2 ðtÞdw0 ðL; tÞ þ Q1 ðtÞdwð0; tÞ þ Q2 ðtÞdwðL; tÞ

ð10:76Þ

þ N1 ðtÞdub ð0; tÞ þ N2 ðtÞdub ðL; tÞ þ R1 ðtÞdcð0; tÞ þ R2 ðtÞdcðL; tÞ þ bd31 Ep VðtÞdub jL0 ð1=2Þbh2 d31 Ep VðtÞdw0 jL0 þ ð1=2Þbhv d31 Ep VðtÞdcjL0 where cA ¼ cb Ab þ cv Av þ cp Ap c1 ¼ cv Iv þ ð1=4Þðcv Av h21 þ cp Ap h22 Þ c2 ¼ cv Iv þ ð1=4Þcv Av h2v þ cp Ap h2v c3 ¼ cv Iv þ ð1=4Þcv Av h1 hv þ ð1=2Þcp Ap h2 hv

ð10:77Þ

c4 ¼ ð1=2Þðcv Av h1 þ cp Ap h2 Þ c5 ¼ ð1=2Þcv Av hv þ cp Ap hv Finally, substitution of Equations 10.72 and 10.76 into Hamilton’s principle (10.70) gives three coupled equations of motion 0 0 EIw w0000 þrA€ w þcAw_ ¼ a€ u 0b þbu000 þg w€ 00 e1c€b þe2 c000 þc1 w_ 00 c4 u_ 0b c3 c_ þFw00 þpðx;tÞ _ EAu00 rA u€b cAu_ b ¼ a€ w 0 þbw000 þe3c€ e4 c00 c4 w_ 0 þc5 ctðx;tÞ b

EIc c00 rIc€ c2 c_ ¼ e1w€ 0 þe2 w000 þe3 u€b e4 u00b c3 w_ 0 þc5 u_ b þAv ðgcÞgðx;tÞ

(10.78)

and the boundary conditions Qð0;tÞ ¼ Q1 ðtÞFw0 ð0;tÞ p

or wð0;tÞ ¼ w1 ðtÞ

Mð0;tÞ ¼ M1 ðtÞM ðtÞ

or w0 ð0;tÞ ¼ u1 ðtÞ

Nð0;tÞ ¼ N1 ðtÞþN p ðtÞ

or ub ð0;tÞ ¼ u1 ðtÞ

p

Rð0;tÞ ¼ R1 ðtÞþR ðtÞ

or cð0;tÞ ¼ c1 ðtÞ

0

QðL;tÞ ¼ þQ2 ðtÞFw ðL;tÞ or wðL;tÞ ¼ w2 ðtÞ MðL;tÞ ¼ þM2 ðtÞM p ðtÞ

or w0 ðL;tÞ ¼ u2 ðtÞ

NðL;tÞ ¼ þN2 ðtÞþN p ðtÞ

or ub ðL;tÞ ¼ u2 ðtÞ

p

RðL;tÞ ¼ þR2 ðtÞþR ðtÞ

or cðL;tÞ ¼ c2 ðtÞ

ð10:79Þ


308

where

M p ðtÞ ¼ ð1=2Þbh2 d31 Ep VðtÞ N p ðtÞ ¼ bd31 Ep VðtÞ

ð10:80Þ

Rp ðtÞ ¼ bhv d31 Ep VðtÞ Np(t), Mp(t), and Rp(t) are the piezoelectric forces and moments generated by the voltage V(t) applied to the PZT layer, and b ¼ bb ¼ bv ¼ bp is the width of the ACLD beam. The resultant force-displacement relationships are given by Qðx;tÞ ¼ EIw w000 a€ u b þbu00b þg w€ 0 e1c€ þe2 c00 þc1 w_ 0 c4 u_ b c3 c_ Mðx;tÞ ¼ EIw w00 bu0b e2 c0 Nðx;tÞ ¼ EAu0b bw00 þe4 c0 Rðx;tÞ ¼ EIc c0 e2 w00 þe4 u0b

ð10:81Þ

If necessary, the structural damping of the base beam and PZT layer can be taken into account by using the complex modulus given in Equation 10.19. If all VEM related terms are removed, the equations of motion (10.78) and the forcedisplacement relationships (10.81) return to the equations of motion (10.15) and the forcedisplacement relationships (10.18) for the elastic–PZT two-layer beam, respectively. The equations of motion (10.78) are identical with the equations of motion (9.90) for the PCLD beam, except for using the structural and damping properties defined by (10.73) and (10.77) and the boundary conditions (10.79), which are different from those for the PCLD beams due to piezoelectric effects. Thus, a large part of the spectral element formulation conducted in Section 9.2 for the PCLD beams can be shared by the ACLD beams.

10.2.2 Spectral Element Modeling 10.2.2.1 Governing Equations in the Frequency Domain As with the PCLD beams in Section 9.2.2.1, we represent all dynamic responses, forces, and moments in the spectral forms given by Equations 9.93 and 9.94. In addition, we represent piezoelectric forces, moments, and voltage in the spectral forms as fM p ðx; tÞ N p ðx; tÞ Rp ðx; tÞ VðtÞg ¼

1 1 NX fMnp ðx; vn Þ Nnp ðx; vn Þ Rpn ðx; vn Þ Vn ðvn Þgeivn t N n¼0

ð10:82Þ

Substitution of Equations 9.93, 9.94 and 10.82 into Equations 10.78 and 10.79 gives the equations of motion in the frequency domain EIw W 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ v2 e1 Y0 þ e2 Y000 þ ivc1 W 00 ivc4 U 0 ivc3 Y0 þ FW 00 þ PðxÞ EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 v2 e3 Y e4 Y00 ivc4 W 0 þ ivc5 Y TðxÞ EIc Y00 þ v2 rIY ivc2 Y ¼ v2 e1 W 0 þ e2 W 000 v2 e3 U e4 U 00 ivc3 W 0 þ ivc5 U þ Av G Y GðxÞ

ð10:83Þ


309

and the associated boundary conditions Qð0Þ ¼ Mð0Þ ¼ Nð0Þ ¼ Rð0Þ ¼ QðLÞ ¼ MðLÞ ¼ NðLÞ ¼ RðLÞ ¼

Q1 FW 0 ð0Þ M1 M p N1 þ N p R1 þ Rp þQ2 FW 0 ðLÞ þM2 M p þN2 þ N p þR2 þ Rp

or or or or or or or or

Wð0Þ ¼ W1 W 0 ð0Þ ¼ Q1 Uð0Þ ¼ U1 Yð0Þ ¼ Y1 WðLÞ ¼ W2 W 0 ðLÞ ¼ Q2 UðLÞ ¼ U2 YðLÞ ¼ Y2

ð10:84Þ

where M p ¼ ð1=2Þbh2 d31 Ep VðvÞ N p ¼ bd31 Ep VðvÞ

ð10:85Þ

p

R ¼ bhv d31 Ep VðvÞ Similarly, the resultant force-displacement relationships (10.81) can be also transformed into the frequency domain as QðxÞ ¼ EIw W 000 þ v2 aU þ bU 00 v2 gW 0 þ v2 e1 Y þ e2 Y00 þ ivc1 W 0 ivc4 U ivc3 Y MðxÞ¼ EIw W 00 bU 0 e2 Y0 NðxÞ ¼ EAU 0 bW 00 þ e4 Y0 (10.86) RðxÞ ¼ EIc Y0 e2 W 00 þ e4 U 0 ð10:86Þ

In Equations 10.83–10.86, the subscripts n are omitted for brevity. 10.2.2.2 Spectral Nodal DOFs, Forces, and Moments

By using the spectral nodal DOFs, forces, and moments specified in Figure 10.7, we define the spectral nodal DOFs vector d ¼ fU1 W1 Q1 Y1 U2 W2 Q2 Y2 gT ¼ fUð0Þ Wð0Þ W 0 ð0Þ Yð0Þ UðLÞ WðLÞ W 0 ðLÞ YðLÞgT

ð10:87Þ

and the mechanical spectral nodal forces and moments vector fcmech ¼ fN1 Q1 M1 R1 N2 Q2 M2 R2 gT 1W1

2

1

W2

ð10:88Þ

2

U2

U1 N1

N2

M1 Q1 R1

Figure 10.7

L

R2 Q2 M2

Spectral element of the ACLD beam


310

10.2.2.3 Dynamic Shape Functions For the derivation of the dynamic shape function, we consider the homogeneous governing equations reduced from Equation 10.83 as EIw W 0000 v2 rAW þ ivcAW ¼ v2 aU 0 þ bU 000 v2 gW 00 þ v2 e1 Y0 þ e2 Y000 þ ivc1 W 00 ivc4 U 0 ivc3 Y0 þ FW 00 EAU 00 þ v2 rAU ivcAU ¼ v2 aW 0 þ bW 000 v2 e3 Y e4 Y00 ivc4 W 0 þ ivc5 Y EIc Y00 þ v2 rIY ivc2 Y ¼ v2 e1 W 0 þ e2 W 000 v2 e3 U e4 U 00 ivc3 W 0 þ ivc5 U þ Av G Y

(10.89)

The general solutions to Equation 10.89 are assumed to be Wðx; vÞ ¼ Ae ikx ;

Uðx; vÞ ¼ r1 Ae ikx ;

cðx; vÞ ¼ r2 Ae ikx

ð10:90Þ

By substituting Equation 10.90 into Equation 10.89 and following the same process as used in Section 9.2.2.3, we can derive a dispersion equation as a1 k 8 þ a 2 k 6 þ a3 k 4 þ a4 k 2 þ a5 ¼ 0

ð10:91Þ

where ai (i ¼ 1, 2, . . ., 5) are given by Equation 9.105 when the structural and damping properties are replaced by the values given by Equations 10.73 and 10.77. Eight wavenumbers ki (i ¼ 1, 2, . . ., 6) computed from Equation 10.91 can be used to obtain the general solutions to Equation 10.89 in the forms of Wðx; vÞ ¼ Nw ðx; vÞd;

Uðx; vÞ ¼ Nu ðx; vÞd;

Yðx; vÞ ¼ Nc ðx; vÞd

ð10:92Þ

where Nw, Nu, and Nc are the dynamic shape functions defined by Nw ðx; vÞ ¼ eðx; vÞH 1 ðvÞ Nu ðx; vÞ ¼ eðx; vÞR1 H 1 ðvÞ Nc ðx; vÞ ¼ eðx; vÞR2 H 1 ðvÞ

ð10:93Þ

The matrices e(x;v), R1(v), R2(v), and H(v) are defined in the same forms as given by Equations 9.111, 9.108 and 9.113, respectively. 10.2.2.4 Weak Form of Governing Equations Following the procedure used in Section 9.2.2.4, we can obtain the weak form of the governing Equation 10.83 as ðL ½EIw W 00 dW 00 v2 rAWdW þ ivcAWdW þ ivc1 W 0 dW 0 ivc4 UdW 0 0

þ v2 aUdW 0 bU 0 dW 00 v2 gW 0 dW 0 þ v2 e1 YdW 0 e2 Y0 dW 00 ivc3 YdW 0 þ FW 0 dW 0 þ EAU 0 dU 0 v2 rAUdU þ ivcAUdU þ v2 aW 0 dU bW 00 dU 0 v2 e3 YdU þ e4 Y0 dU 0 ivc4 W 0 dU þ ivc5 YdU þ EIc Y0 dY0 v2 rIYdY þ ivc2 YdY þ v2 e1 W 0 dY e2 W 00 dY0 v2 e3 UdY þ e4 U 0 dY0 ivc3 W 0 dY þ ivc5 UdY ðL þ Av G YdYdx ½PðxÞdW þ TðxÞdU þ GðxÞdYdx 0

f½QðxÞ þ FW ðxÞdWjL0 þ MðxÞdW 0 jL0 þ NðxÞdUjL0 þ RðxÞdYjL0 g ¼ 0 0

ð10:94Þ


311

10.2.2.5 Spectral Element Equation By substituting Equation 10.92 into Equation 10.94 and using the spectral nodal forces vector fc defined by Equation 10.88, we obtain the spectral element equation as SðvÞd ¼ fðvÞ

ð10:95Þ

The matrix S(v) is the spectral element matrix defined by SðvÞ ¼ H T ðvÞDðvÞH 1 ðvÞ

ð10:96Þ

where DðvÞ ¼ EART1 KEKR1 þ EIw K2 EK2 EIc RT2 KEKR2 FKEK

ibðK2 EKR1 þ RT1 KEK2 Þ ie2 ðK2 EKR2 þ RT2 KEK2 Þ v2 frAðE þ RT1 ER1 Þ þ rIRT2 ER2 þ iaðKER1 þ RT1 EKÞ gKEK þ ie1 ðKER2 þ RT2 EKÞ þ e3 ðRT1 ER2 þ RT2 ER1 Þg e4 ðRT1 KEKR2 þ RT2 KEKR1 Þ þ ivfcAðE þ RT1 ER1 Þ c1 KEK þc2 RT2 ER2 þ ic3 ðKER2 þ RT2 EKÞ þ ic4 ðKER1 þ RT1 EKÞ ð10:97Þ þc5 ðRT1 ER2 þ RT2 ER1 Þg þ Av G RT2 ER2

(10.97)

In the above equation, the following definitions are used. K ¼ diag ½ki ¼ KT ; EðvÞ ¼

ðL

K2 ¼ diag ½ki2 ¼ K2T

eT ðx; vÞeðx; vÞdx ¼ ½Elm ðvÞ

ð10:98Þ

ðl; m ¼ 1; 2; . . . ; 8Þ

ð10:99Þ

0

where Elm ¼

8 > < > :

iðk þ k ÞL i e l m 1 if kl þ km L

if

kl þ km 6¼ 0

ð10:100Þ

kl þ km ¼ 0

The vector f(v) is the spectral nodal forces vector defined by fðvÞ ¼ fcmech ðvÞ þ fcpiezo ðvÞ þ fd ðvÞ

ð10:101Þ

where fcpiezo ¼ f N p 0 M p fd ðvÞ ¼

ðL 0

R p N p 0 M p Rp g T

½Pðx; vÞNTw ðx; vÞ þ Tðx; vÞNTu ðx; vÞ þ Gðx; vÞNTc ðx; vÞdx

ð10:102Þ ð10:103Þ

The spectral elements represented by Equation 10.95 can be assembled, alongwith imposing the boundary conditions, in the form Sg ðvÞdg ¼ fg ðvÞ

ð10:104Þ



312

10.2.3 Spectral Element with Active Control Consider an ACLD beam element implemented with a proportional-derivative (PD) controller, as shown in Figure 10.8. Assume that the driving voltage applied to the PZT layer is determined by VðtÞ ¼ Kp w Kd w_

ð10:105Þ

Driving voltage V (t ) +

–Kp w

–Kd w

Vibration response

Figure 10.8

An ACLD beam with a PD controller

where Kp and Kd are the proportional and derivative gains, respectively. We can represent Equation 10.105 in the spectral form as VðvÞ ¼ ðKp þ ivKd ÞWðx; vÞ

ð10:106Þ

From Equation 10.92, the spectral components W(x; v) of the displacement measured at a point x ¼ x0 within an ACLD beam element can be related to the corresponding spectral nodal DOFs vector d as follows: Wðx0 ; vÞ ¼ Nw ðx0 ; vÞd

ð10:107Þ

Substitution of Equation 10.107 into Equation 10.106 gives VðvÞ ¼ ðKp þ ivKd ÞNw ðx0 ; vÞd

ð10:108Þ

By applying Equation 10.108 to Equation 10.85, the piezoelectric forces and moments are obtained as N p ¼ bd31 Ep VðvÞ ¼ xN Nw ðx0 ; vÞd M p ¼ ð1=2Þbd31 Ep hVðvÞ ¼ xM Nw ðx0 ; vÞd Rp ¼ bhv d31 Ep VðvÞ ¼ xR Nw ðx0 ; vÞd

ð10:109Þ

x N ¼ bd31 Ep ðKp þ ivKd Þ x M ¼ ð1=2Þbd31 Ep hðKp þ ivKd Þ x R ¼ bhv d31 Ep ðKp þ ivKd Þ

ð10:110Þ

where

Now, substitute Equation 10.109 into Equation 10.102 to obtain fcpiezo ¼ Spiezo ðvÞd

ð10:111Þ


313

where

2 6 6 6 6 6 6 Spiezo ðvÞ ¼ 6 6 6 6 6 6 4

3 x N Nw ðx0 ; vÞ 7 0 7 7 þx M Nw ðx0 ; vÞ 7 7 x R Nw ðx0 ; vÞ 7 7 þx N Nw ðx0 ; vÞ 7 7 7 0 7 7 x M Nw ðx0 ; vÞ 5

ð10:112Þ

þx R Nw ðx0 ; vÞ

Finally, by substituting Equation 10.111 into Equation 10.101 and then the result into Equation 10.95, we can obtain the spectral element equation for the ACLD beam element with a PD controller as being SðvÞCONTROL d ¼ fcmech þ fd ð10:113Þ where SðvÞCONTROL ¼ SðvÞ þ Spiezo ðvÞ ð10:114Þ

10.2.4 Spectral Modal Analysis The governing equations of the ACLD beam have exactly the same forms as those of the PCDL beam. Thus, we can follow the spectral modal analysis procedure introduced in Section 9.2.3. 10.2.4.1 Orthogonality Properties of Normal Modes Let vp be the pth natural frequency and Wp, Up, and Yp the corresponding normal modes, respectively. Then, the pth eigensolutions must satisfy the eigenvalue problem given by EIw Wp0000 v2p rAWp þivp cAWp ¼ v2p ðaUp0 gWp00 þe1 Y0p Þ 00 þivp ðc1 Wp00 c4 Up0 c3 Y0p ÞþbUp000 þe2 Y000 p þFWp

EAUp00 þv2p rAUp ivp cAUp ¼ v2p ðaWp0 e3 Yp Þivp ðc4 Wp0 c5 Yp ÞþbWp000 e4 Y00p EIc Y00p þv2p rIYp ivp c2 Yp ¼ v2p ðe1 Wp0 e3 Up Þivp ðc3 Wp0 c5 Up Þþe2 Wp000 e4 Up00 þAv G* Yp ð10:115Þ Similarly, the qth eigensolutions vq, Wq, Uq, and Yq must satisfy the eigenvalue problem given by EIw Wq0000 v2q rAWq þivq cAWq ¼ v2q ðaUq0 gWq00 þe1 Y0q Þ 00 þivq ðc1 Wq00 c4 Uq0 c3 Y0q ÞþbUq000 þe2 Y000 q þFWq

EAUq00 þv2q rAUq ivp cAUq ¼ v2q ðaWq0 e3 Yq Þivq ðc4 Wq0 c5 Yq ÞþbWq000 e4 Y00q

(10.116)

EIc Y00q þv2q rIYq ivq c2 Yq ¼ v2q ðe1 Wq0 e3 Uq Þivq ðc3 Wq0 c5 Uq Þ þe2 Wq000 e4 Uq00 þAv G* Yq

ð10:116Þ


314

By following the procedure used in Section 9.2.3.1, we can derive the orthogonality properties of the normal modes from Equations 10.115 and 10.116 as ðL ½rAðWp Wq þ Up Uq Þ þ rIYp Yq aðUp Wq0 þ Uq Wp0 Þ e1 ðYp Wq0 þ Yq Wp0 Þ 0

ðL 0

þ e3 ðYp Uq þ Yq Up Þ þ gWp0 Wq0 dx ¼ mp dpq ½cAðWp Wq þ Up Uq Þ þ c1 Wp0 Wq0

þ c2 Yp Yq c3 ðYp Wq0

(10.117) þ Yq Wp0 Þ

c4 ðUp Wq0 þ Uq Wp0 Þ þ c5 ðYp Uq þ Yq Up Þdx ¼ cp dpq

ð10:117Þ

where mp and cp are the modal mass and modal damping coefficients defined by ðL 2 mp ¼ ½rAðUp2 þ Wp2 Þ þ rIY2p 2aUp Wp0 2e1 Yp Wp0 þ 2e3 Yp Up þ gW 0 p dx ð10:118Þ ð0L 2 cp ¼ ½cAðUp2 þ Wp2 Þ þ c1 Wp0 þ c2 Y2p 2c3 Yp Wp0 2c4 Up Wp0 þ 2c5 Yp Up dx 0

10.2.4.2 Spectral Modal Equations Based on the modal analysis method, the forced vibration responses are assumed in the forms ¥ X fwðx; tÞ; uðx; tÞ; cðx; tÞg ¼ fWp ðxÞ; Up ðxÞ; Yp ðxÞgqp ðtÞ ð10:119Þ p¼1

where qp(t) are the modal coordinates. Represent the modal coordinates in the spectral form as qp ðtÞ ¼

1 1 NX q eivn t N n¼0 pn

ð10:120Þ

By substitution of Equation 10.120 into Equation 10.119, we obtain fwðx; tÞ; uðx; tÞ; cðx; tÞg ¼

¥ N 1 X 1X fWp ðxÞ; Up ðxÞ; Yp ðxÞgqpn eivn t N p¼1 n¼0

ð10:121Þ

The external forces are also written in the spectral forms as f pðx; tÞ; tðx; tÞ; gðx; tÞg ¼

1 1 NX fPn ðxÞ; Tn ðxÞ; Gn ðxÞgeivn t N n¼0

ð10:122Þ

Substitute Equations 10.121 and 10.122 into the time-domain governing Equations 10.78 and take the same manipulation as conducted in Section 9.2.3.2. Then, we can obtain the spectral modal equations as ðv2q v2n Þmq qqn iðvq vn Þcq qqn ¼ fqn where fqn are the spectral modal forces defined by ðL fqn ¼ ½Pn ðxÞWq þ Tn ðxÞUq þ Gn ðxÞYq dx 0

ð10:123Þ

ð10:124Þ


315

The spectral modal Equation 10.123 can be readily transformed into the time domain as mqq€q ðtÞ þ cq q_ q ðtÞ þ ðv2q mq ivq cq Þqq ðtÞ ¼ fq ðtÞ

ð10:125Þ

where fq ðtÞ ¼

ðL

½pðx; tÞWq þ tðx; tÞUq þ rðx; tÞYq dx

ð10:126Þ

0

10.2.5 Finite Element Model The weak form of the equations of motion (10.78) is given by ðL 0

€ 0 _ ½EIw w00 dw00 þ rA€ wdw þ cAwdw a€ ub dw0 bu0b dw00 þ g w€0 dw0 e1cdw _ 0 þ Fw0 dw0 þ EAu0 du0 þ rA€ e2 c0 dw00 þ c1 w_ 0 dw0 c4 u_ b dw0 c3 cdw u b dub b b 0 0 0 00 0 0 0 _ € þ cAu_ b dub aw€ dub bw du þ e3cdub þ e4 c du c4 w_ du þ c5 cdub b

b

_ e1w€ 0 dc e2 w00 dc0 þ e3u€b dc € þ c2 cdc þ EIc c0 dc0 þ rI cdc

b

þ e4 u0b dc0 c3 w_ 0 dc þ c5 u_ b dc þ Av ðg cÞdcdx ðL ½pðx; tÞdw þ tðx; tÞdub þ gðx; tÞdcdx fNðx; tÞdub jL0

(10.127)

0

þ Mðx; tÞdw0 jL0 þ ½Qðx; tÞ þ Fw0 ðx; tÞdwjL0 þ Rðx; tÞdcjL0 g ¼ 0

ð10:127Þ

The dynamic behavior of a viscoelastic material can be represented by using the GHM model [121]. As discussed in Section 9.2.4, if we adopt a one-term GHM model for the VEM layer, the convolution term Av(g c)dc in Equation 10.127 can be represented by Av ðg cÞdc ffi Av ½ð1 þ aÞkcðx; tÞ akzðx; tÞdc

ð10:128Þ

where z(t) is the dissipation coordinate governed by z€ðx; tÞ þ 2^jv ^ z_ ðx; tÞ v ^ 2 cðx; tÞ þ v ^ 2 zðx; tÞ ¼ 0

ð10:129Þ

The weak form of Equation 10.129 can be written in the form of ð L " 0

# !

Av ak 2Av ak^j z_ Av akc þ Av akz dzdx ¼ 0 z€þ v ^ v ^2

ð10:130Þ

The finite element equation can be formulated from the weak forms of Equations 10.127 and 10.130. By using the same nodal DOFs vector and displacement fields as used in Section 9.2.4, we can derive the finite element equation as

Md 0

0 Mz

Cd d€ þ 0 z€

0 Cz

Kd d_ þ KTdz z_

Kdz Kz

d fðtÞ ¼ z 0

ð10:131Þ


316

where ðL Md ¼

0

½rAðNTw Nw þ NTu Nu Þ þ rINTc Nc aðN0 w Nu þ NTu N0w Þ T

þ gN0 w N0w e1 ðNTc N0w þ N0 w Nc Þ þ e3 ðNTu Nc þ NTc Nu Þdx ð Av ak L T Mz ¼ N Nz dx v ^2 0 z ðL T T Cd ¼ ½cAðNTw Nw þ NTu Nu Þ þ c1 N0 w N0w þ c2 NTc Nc c3 ðN0 w Nc T

T

0

þ NTc N0w Þ c4 ðN0 w Nu þ N0 u N0w Þ þ c5 ðNTu Nc þ NTc Nu Þdx ð 2Av ak^j L T Cz ¼ Nz Nz dx v ^ 0 ðL T T T Kd ¼ ½EAN0 u N0u þ EIw N00 w N00w þ EIc N0 c N0c T

T

ð10:132Þ

0

bðNw00T Nu0 þ N 0 u Nw00 Þ e2 ðNw00T N0c þ N 0 c Nw00 Þ T

þ e4 ðN 0 u Nc0 þ N0 c Nu0 Þ þ FN 0 w Nw0 þ Av ð1 þ aÞkNTc Nc dx ðL ¼ Av ak NTc Nz dx 0 ðL ¼ Av ak NTz Nz dx T

Kdz Kz

T

T

T

0

and fðtÞ ¼ fcmech ðtÞ þ fcpiezo ðtÞ þ fd ðtÞ

ð10:133Þ

with the use of the definitions: fcmech ðtÞ ¼ fN1 ðtÞ Q1 ðtÞ M1 ðtÞ R1 ðtÞ N2 ðtÞ Q2 ðtÞ M2 ðtÞ R2 ðtÞgT fcpiezo ðtÞ ¼ fN p ðtÞ 0 M p ðtÞ Rp ðtÞ N p ðtÞ 0 M p ðtÞ Rp ðtÞgT ðL fd ðtÞ ¼ ½pðx; tÞNTw ðxÞ þ tðx; tÞNTu ðxÞ þ gðx; tÞNTc ðxÞdx

ð10:134Þ

0

The finite element matrices defined by Equation 10.132 are the same as given in Appendix 9.B, except for using the structural and damping properties defined by Equations 10.73 and 10.77.

Example 10.2 Consider a cantilevered ACLD beam as shown in Figure 10.9. The middle one-third part of the base beam is covered with a uniform VEM–PZT patch. The geometric and material properties of the base beam and PZT layer are same as those given in Table 10.1. The VEM layer is represented by a one-term GHM model and its geometric and material properties are the same as given in Table 9.3, except for the length. The vibration of the ACLD beam is controlled by


317

VEM layer hv

Base beam L2

L1

Figure 10.9

hp

PZT layer

hb L3

A cantilevered beam partially covered with a VEM–PZT patch

Table 10.4 Natural frequencies and modal damping ratios of an ACLD beam Uncontrolled

Controlled Kp ¼ 5 107 V, Kd ¼ 100 Vs

Mode number (n) 1 2 3 4 5

fn (Hz)

xn

fn (Hz)

xn

fn (Hz)

xn

44.098 264.62 760.62 1520.2 2428.5

0.0055 0.0112 0.0028 0.0015 0.0005

46.784 263.95 761.29 1520.4 2429.2

0.0101 0.0152 0.0028 0.0017 0.0006

46.766 264.04 761.36 1520.3 2429.2

0.0266 0.0286 0.0066 0.0021 0.0010

4 Displacement (mm)

Kp ¼ 5 107 V, Kd ¼ 300 Vs

Uncontrolled Controlled: Kp = 50 MV, Kd = 100 Vs Controlled: Kp = 50 MV, Kd = 300 Vs

3 2 1 0 –1 –2 –3 0

0.1

0.2 Time (s)

0.3

0.4

Figure 10.10 Dynamic responses of an ACLD beam with a PD controller

using a PD controller, which is shown schematically in Figure 10.8. The natural frequencies and modal damping ratios for the ACLD beam with and without the PD controller are compared in Table 10.4. It is shown that the PD controller certainly increases the modal damping ratios to suppress the vibration quicker, as shown in Figure 10.10.

11 Dynamics of Composite Laminated Structures It is well recognized that fiber reinforced composite materials have many advantages over isotropic materials due to their high strength-to-density ratios. Thus they have been widely used in many industrial applications [12, 29, 111, 144, 161]. Typical examples of are composite aircraft wings, helicopter blades, propeller blades, turbine blades, axles of vehicles and so on. In general, the composite laminated structures are fabricated by bonding two or more laminae together. The dynamic characteristics of composite laminated structures can be enhanced by appropriately tailoring the material and structural couplings determined by the specific lay-up and fiber orientations. Of particular interest, especially for the high aspect ratio composite laminated beams, are the coupling between the extensional (axial), bending, and torsional modes of deformation. As fibrous composite materials usually have very low shear modulus, the shear deformation should also be included in the analysis and design of such composite laminated beams. Thus, this chapter covers the dynamics of the axial–bending–shear coupled composite beams and the bending–torsion–shear coupled composite beams, all based on the first-order shear deformation theory (FSDT), which is the equivalent to the Timoshenko-beam theory for composite laminated beams.

11.1 Theory of Composite Mechanics 11.1.1 Three-Dimensional Stress–Strain Relationships For orthotropic materials, the stress–strain relationships in the coordinates aligned with its principal material coordinates (1, 2, 3) are given by 8 9 2 38 9 C11 C12 C13 «1 > s1 > 0 0 0 > > > > > > > > > > > > > 7 6 C s C C 0 0 0 «2 > > > > 2> 12 22 23 > > > > 7 6 = < = 6 < C13 C23 C33 s3 0 0 0 7 «3 7 6 ¼6 ð11:1Þ 0 0 C44 s4 > 0 0 7 «4 > > > > > > 7> 6 0 > > > > > 0 0 0 C55 s > 0 5> « > > 4 0 > > > > > ; ; : 5> : 5> 0 0 0 0 0 C66 s6 «6


320


where si (i ¼ 1, 2, . . ., 6) are the stress components, «i (i ¼ 1, 2, . . ., 6) are the strain components, Cij (i, j ¼ 1, 2, . . ., 6) are the stiffness coefficients. The strain–stress relationship can be also written as 8 9 2 38 9 S11 S12 S13 0 s1 > «1 > 0 0 > > > > > > 6 > > > > > 7> > > > > > 6 S12 S22 S23 0 > > «2 > 0 0 7> s2 > > > > > > > > > 7 6 > 6 > > > > > 7> = 6 S13 S23 S33 0 < «3 > < 0 0 7 s3 = 7 ¼6 ð11:2Þ 6 > 0 7 0 0 S44 0 > 6 0 > > «4 > > s4 > 7> > > > > > > > 7> 6 > > > > > > > 7> 6 0 > > > > « 0 s 0 0 0 S > > > > 5 55 5 5 4 > > > > > > > ; : ; : > «6 s6 0 0 0 0 0 S66 where Cij are the compliance coefficients determined from nine independent engineering constants as follows: 1 1 1 ; S22 ¼ ; S33 ¼ E1 E2 E3 1 1 1 ¼ ; S55 ¼ ; S66 ¼ G23 G31 G12 n21 n31 n32 ¼ ; S13 ¼ ; S23 ¼ E2 E3 E3

S11 ¼ S44 S12

ð11:3Þ

In the above, the subscripts are used to denote the quantities measured with respect to the principal material coordinates (1, 2, 3) and the following contracted notation for stresses and strains are used: s1 ¼ s11 ; s2 ¼ s22 ;

s3 ¼ s33 ;

s4 ¼ s23 ;

s5 ¼ s31 ;

s6 ¼ s12

«1 ¼ «11 ;

«3 ¼ «33 ;

«4 ¼ 2«23 ; «5 ¼ 2«31 ;

«6 ¼ 2«12

«2 ¼ «22 ;

ð11:4Þ

11.1.2 Stress–Strain Relationships for an Orthotropic Lamina For an orthotropic lamina, we assume that s3 ¼ 0. Thus, from Equation 11.2 we can obtain «3 ¼ S13 s1 þ S23 s2

ð11:5Þ

and Equation 11.2 can be reduced into the forms 8 9 2 38 9 S11 S12 0 > s1 > « > = < 1> < = 6 7 «2 ¼ 4 S12 S22 0 5 s2 > > ; ; : > : > «6 s6 0 0 S66 and

(

«4 «5

)

" ¼

S44

0

0

S55

#(

s4 s5

ð11:6Þ

) ð11:7Þ

321

Dynamics of Composite Laminated Structures

The stress–strain relationships for an orthotropic lamina can be readily obtained from Equations 11.6 and 11.7 as 8 9 2 38 9 Q11 Q12 s 0 > «1 > > = < 1> < = 6 7 ð11:8Þ s2 ¼ 4 Q12 Q22 0 5 «2 > > ; ; : > : > s6 «6 0 0 Q66 and (

s4

)

" ¼

s5

Q44

0

0

Q55

#(

«4

) ð11:9Þ

«5

where Qij are the reduced stiffnesses given by Q11 ¼

S22 E1 ¼ S11 S22 S212 1 n12 n21

Q22 ¼

S11 E2 ¼ 2 1 n12 n21 S11 S22 S12

Q12

ð11:10Þ

S12 n21 E1 n12 E2 ¼ ¼ ¼ 2 S11 S22 S12 1 n12 n21 1 n12 n21

Q44 ¼

1 ¼ G23 ; S44

Q55 ¼

1 ¼ G31 ; S55

Q66 ¼

1 ¼ G12 S66

Now consider global coordinates (x, y, z), which are used in the solution to a problem and, as shown in Figure 11.1, assume that the principal material coordinates (1, 2, 3) are obtained by rotating the xy-plane counterclockwise about the z-axis by an angle f. Then, we can derive the coordinate transformation rules as follows: 9 2 8 cos2 f sin2 f > = < sxx > 6 syy ¼ 4 sin2 f cos2 f > > ; : sxy sinfcosf sinfcosf syz cosf sinf s4 ¼ sinf cosf szx s5

z

38 9 > = < s1 > 7 sin2f 5 s2 > ; : > s6 cos2 f sin2 f sin2f

y

y

1

2

h

(k)

x

x b L

Figure 11.1 beam

ð11:11Þ

k th Layer

The global and principal material coordinate systems for a straight composite laminated

322


and 9 2 8 38 9 cos2 f sin2 f sinfcosf > > = = < «xx > < «1 > 6 7 2 2 «yy ¼ 4 sin f cos f sinfcosf 5 «2 > > > ; ; : : > gxy «6 sin2f sin2f cos2 f sin2 f gyz cosf sinf «4 ¼ gzx «5 sinf cosf

ð11:12Þ

where gxy ¼ 2«xy ;

gyz ¼ 2«yz ;

gzx ¼ 2«zx

ð11:13Þ

By applying the above coordinate transformation rules to Equations 11.8 and 11.9, we can get the stress–strain relationships with respect to the global coordinates (x, y, z) as 9 2 9 8 38 11 Q 12 Q 16 > «xx > Q > = = < sxx > < 6 22 Q 26 7 syy ¼ 4 Q ð11:14Þ Q 5 «yy 12 > > > > ; ; : : 16 Q 26 Q 66 gxy sxy Q and

syz szx

Q44 ¼ 45 Q

45 gyz Q 55 gzx Q

ð11:15Þ

ij are the transformed reduced stiffnesses defined by [144] where Q 11 ¼ Q11 cos4 f þ Q22 sin4 f þ 2ðQ12 þ 2Q66 Þsin2 fcos2 f Q 22 ¼ Q11 sin4 f þ Q22 cos4 f þ 2ðQ12 þ 2Q66 Þsin2 fcos2 f Q 66 ¼ Q66 ðsin4 f þ cos4 fÞ þ ðQ11 þ Q22 2Q12 2Q66 Þsin2 fcos2 f Q 12 ¼ Q12 ðsin4 f þ cos4 fÞ þ ðQ11 þ Q22 4Q66 Þsin2 fcos2 f Q 16 ¼ ðQ11 Q12 2Q66 Þsinfcos3 f þ ðQ12 Q22 þ 2Q66 Þsin3 fcosf Q 26 ¼ ðQ11 Q12 2Q66 Þsin3 fcosf þ ðQ12 Q22 þ 2Q66 Þsinfcos3 f Q 44 ¼ Q44 cos2 f þ Q55 sin2 f; Q 45 ¼ ðQ55 Q44 Þcosfsinf Q

ð11:16Þ

55 ¼ Q55 cos2 f þ Q44 sin2 f Q

11.1.3 Strain–Displacement Relationships Let u(x, y, z, t), v(x, y, z, t), and w(x, y, z, t) represent the displacements in the x-, y-, and z-directions, respectively. The strains are given by ›v ; ›y ›w ›v gyz ¼ 2«yz ¼ þ ; ›y ›z «xx ¼

›u ; ›x

«yy ¼

›v ›u þ ›x ›y ›w ›u þ ¼ ›x ›z

gxy ¼ 2«xy ¼ gzx ¼ 2«zx

ð11:17Þ

323


Assume that the displacements are given in the general forms as uðx; y; z; tÞ ¼ u0 ðx; y; tÞ zux ðx; y; tÞ ð11:18Þ

vðx; y; z; tÞ ¼ zuy ðx; y; tÞ wðx; y; z; tÞ ¼ w0 ðx; y; tÞ

where u0 and w0 are the displacements of a point on the reference plane z ¼ 0 in the x- and zdirections, and ux and uy are the negative of the slope along the x- and y-axes, respectively. Substitution of Equation 11.18 into Equation 11.17 gives «xx ¼ «0xx þ zx x ; ›w0 uy ; gyz ¼ ›y

«yy ¼ zx y ; gxy ¼ g0xy þ zx xy ›w0 ux gzx ¼ ›x

ð11:19Þ

where «0xx , «0yy , and «0xy are the in-plane strains defined by «0xx ¼

›u0 ; ›x

g0xy ¼

›u0 ›y

ð11:20Þ

and xx, xy, and xxy are the curvatures defined by ›ux ›uy ; xy ¼ ›x ›y ›uy ›ux þ ¼ ›x ›y

xx ¼ xxy

ð11:21Þ

11.1.4 Resultant Forces and Moments The resultant forces and moments can be obtained by integrating the stresses through the laminate thickness (h/2 z h/2) as ( ) ðh N ð zk ðkÞ X 2 Nx sxx sxx ¼ dz ¼ ð11:22Þ ðkÞ dz Qx kszx kszx h2 k¼1 zk 1 8 ðkÞ 9 8 9 8 9 = < sxx > < Mxx = ð h2 < sxx = N ð zk > X ðkÞ Myy ¼ syy zdz ¼ syy zdz > > : ; ; h2 : k¼1 zk 1 : ðkÞ ; Mxy sxy sxy

ð11:23Þ

where b is the width of the beam, h is the thickness of the beam, k is the shear correction factor, and zk and zk1 are defined in Figure 11.2. The stress–strain relationships for the kth lamina are taken from Equations 11.14 and 11.15 as 8 ðkÞ 9 2 ðkÞ ðkÞ ðkÞ 38 ðkÞ 9 > = 6 Q11 Q12 Q16 7> = < sxx > < «xx > ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ 7 ¼6 ð11:24Þ syy Q12 Q22 Q26 5 «yy 4 > > > ; ; : ðkÞ > : ðkÞ ðkÞ Q ðkÞ Q ðkÞ sxy gxy Q 16 26 66

324


Figure 11.2 Geometry of an N-layered laminate

sðkÞ zx

¼

h

ðkÞ Q 45

ðkÞ Q 55

( ) i gðkÞ yz ðkÞ

gzx

ð11:25Þ

By substituting Equation 11.19 into Equations 11.24 and 11.25 and the results into Equations 11.22 and 11.23, we can obtain the expressions Nx ¼ A11 «0xx þ A16 g0xy þ B11 x x þ B12 x y þ B16 x xy Qx ¼ kA45 9 2 8 M B11 > = < xx > 6 Myy ¼ 4 B12 > > ; : Mxy B16

›w0 ›w0 uy þ kA55 ux ›y ›x

3 2 B16 ( 0 ) D11 « 7 xx 6 B26 5 0 þ 4 D12 gxy B66 D16

D12 D22 D26

9 38 D16 > x x > = < 7 D26 5 xy > > ; : xxy D66

ð11:26Þ

ð11:27Þ

ð11:28Þ

where Aij ¼

N X k¼1

ðkÞ ðzk zk 1 Þ Q ij

Bij ¼

N 1X ðkÞ ðz2 z2 Þ Q k1 2 k¼1 ij k

Dij ¼

N 1X ðkÞ ðz3 z3 Þ Q k1 3 k¼1 ij k

ð11:29Þ

11.2 Equations of Motion for Composite Laminated Beams The equations of motion and associated boundary conditions for composite beams are derived from Hamiltons principle based on the first-order shear deformation theory (FSDT), which

325


is the equivalent of the Timoshenko-beam theory for composite laminated beams. The Hamiltons principle is given by ðt ð11:30Þ ðdT dU þ dWÞdt ¼ 0 0

where T and U are the kinetic and strain energies, respectively, and dW is the virtual work done by external forces.

11.2.1 Axial–Bending–Shear Coupled Vibration Consider the small amplitude axial–bending–shear coupled vibration of a straight composite laminated beam subjected to a constant axial tensile force P. The displacements of the beam are assumed as wðx; y; z; tÞ ffi w0 ðx; tÞ uðx; y; z; tÞ ffi uðx; z; tÞ ¼ u0 ðx; tÞ zuðx; tÞ

ð11:31Þ

Thus from Equations 11.19–11.21 we obtain ux ¼ u;

uy ¼ 0;

x x ¼ u0 ;

«0xx ¼ u00 ;

xy ¼ 0;

g0xy ¼ 0

xxy ¼ 0

ð11:32Þ

where the prime denotes the derivative with respective to the axial coordinate x. By using Equation 11.32, the resultant axial force, bending moment, and transverse shear force can be obtained from Equations 11.26–11.28 as follows: Nðx; tÞ bNx ¼ EAeq u00 KAeq u0 Qðx; tÞ bQx ¼ kGAeq ðw00 uÞ Mðx; tÞ bMxx ¼ EIeq u

0

ð11:33Þ

KAeq u00

where EAeq bA11 ; EIeq bD11 kGAeq bkA55 ; KAeq bB11

ð11:34Þ

The subscript “eq” is used to denote the apparent structural rigidities and mass inertia properties for a composite laminated beam, and the other terms are: the extensional rigidity (EA), the bending (flexural) rigidity (EI), the shear rigidity (kGA), the axial–bending material coupling rigidity (KA), the mass per unit length (rA), the first-order mass moment of inertia (rR), and the second-order mass moment of inertia (rI). Assume that the composite beam is subjected to the external forces and moments applied at two end boundaries as shown in Figure 11.3, the distributed loads are p(x, t), f(x, t), and t(x, t), and there is a constant axial tensile force P, which is assumed to act in the x-direction through the mass center of the cross-section. The strain energy U, the kinetic energy T, and the virtual

326

Spectral Element Method in Structural Dynamics w01

w02

Q1 N1

Q2

M(x,t )

θ1 M1

θ2

N(x,t ) u01

Figure 11.3

Q(x,t )

M2

N2

u02

L

Sign convention for the composite laminated beam element

work dW done by external forces and moments are given by ð 1 L U¼ fNu00 Mxx þ Qðw00 uÞ þ Pw02 0 g dx 2 0 ð 1 L 2 fEAeq u00 2 2KAeq u00 u0 þ EIeq u02 þ kGAeq ðw00 uÞ2 þ Pw0 0 g dx ¼ 2 0 ð ðh ð 1 L 2 1 L 2 _ 2 þ u_ 2 Þ dzdx ¼ rbð w ðrAeq w_ 20 þ rAeq u_ 20 2rReq u_ 0 u_ þ rIeq u_ Þ dx T¼ 2 0 h 2 0 2

dW ¼ N1 du0 ð0Þ þ N2 du0 ðLÞ þ M1 duð0Þ þ M2 duðLÞ þ Q1 dw0 ð0Þ þ Q2 dw0 ðLÞ ðL ð11:35Þ þ ½fu ðx; tÞdu0 þ fw ðx; tÞdw0 þ fu ðx; tÞdu dx 0

where

ðh rAeq b

2

ðh rReq b

rðkÞ ðzk zk 1 Þ

k¼1

rzdz ¼ h

N bX rðkÞ ðz2k z2k 1 Þ 2 k¼1

2

N X

2

2

ðh rIeq b

rdz ¼ b h

2

rz2 dz ¼ h 2

ð11:36Þ

N bX rðkÞ ðz3k z3k 1 Þ 3 k¼1

where rAeq is the apparent mass of the composite laminated beam, rReq is the apparent firstorder mass moment of inertia about the y-axis and rIeq is the apparent secondorder mass moment of inertia about the y-axis, all per unit length. By substituting Equation 11.35 into Hamiltons principle and integrating by parts,we can obtain ðt ðL ½ f½kGAeq ðw000 u0 Þ þ Pw000 rAeq w € 0 þ fw ðx; tÞdw0 0

0

þ ½EIeq u00 þ kGAeq ðw00 uÞ KAeq u000 þ rReq u€0 rIeq u€ þ fu ðx; tÞdu þ ½EAeq u000 KAeq u00 rAeq u€0 þ rReq u€ þ fu ðx; tÞdu0 gdx ð11:37Þ Nðx; tÞdu0 jL0 Mðx; tÞdujL0 ½Qðx; tÞ þ Pw00 dw0 jL0 þ N1 du0 ð0Þ þ N2 du0 ðLÞ þ M1 duð0Þ þ M2 duðLÞ þ Q1 dw0 ð0Þ þ Q2 dw0 ðLÞdt ¼ 0

327


where Equation 11.33 is used. From Equation 11.37, we get the equations of motion € 0 þ fw ðx; tÞ ¼ 0 kGAeq ðw000 u0 Þ þ Pw000 rAeq w EIeq u00 þ kGAeq ðw00 uÞ KAeq u000 þ rReq u€0 rIeq u€ þ fu ðx; tÞ ¼ 0 EAeq u000 KAeq u00 rAeq u€0 þ rReq u€ þ fu ðx; tÞ ¼ 0

ð11:38Þ

and the boundary conditions Nð0; tÞ ¼ N1 ðtÞ

or u0 ð0; tÞ ¼ 0


or u0 ðL; tÞ ¼ 0

Mð0; tÞ ¼ M1 ðtÞ

or w0 ð0; tÞ ¼ 0

MðL; tÞ ¼ þ M2 ðtÞ

or w0 ðL; tÞ ¼ 0

Qð0; tÞ ¼ Q1 ðtÞ Pw00 ð0; tÞ

or uð0; tÞ ¼ 0

þ Q2 ðtÞ Pw00 ðL; tÞ

or uðL; tÞ ¼ 0

QðL; tÞ ¼

ð11:39Þ

For a laminated beam with mid-plane symmetry, KAeq and rReq are equal to zero and the axial vibration (in-plane displacement u0) is decoupled with the bending–shear vibration. In this case, the bending–shear vibration and the axial vibration of the composite beam can be considered separately. The spectral element models for the pure bending–shear vibration (Timoshenko-beam model) and the pure axial vibration (rod model) were introduced in Chapter 3.

11.2.2 Bending–Torsion–Shear Coupled Vibration Now consider the small amplitude bending–torsion–shear coupled vibration of a straight composite laminated beam subjected to a constant axial tensile force P. Assume that the x-axis is chosen to coincide with the elastic axis that passes through the shear center of the beam crosssection, and the beam takes the bending deflection h(x, t) along the x-axis and the twist a(x, t) about the x-axis. The displacements of the composite laminated beam are then assumed in the forms of wðx; y; z; tÞ ffi w0 ðx; y; tÞ ffi hðx; tÞ þ yaðx; tÞ uðx; y; z; tÞ ffi zux ðx; 0; tÞ zuðx; tÞ ›w0 vðx; y; z; tÞ ffi z zaðx; tÞ ›y

ð11:40Þ

y¼0

where u(x, t) represents the slope along the x-axis. As the in-plane displacement u0 is excluded from here onwards, from Equation 11.20 we get «0xx ¼ 0;

g0xy ¼ 0

ð11:41Þ

Assuming the moment Myy is zero and using Equation 11.41, we can eliminate the curvature xy from Equation 11.28 to obtain #( ) " 1 1 xx Mxx D16 D12 D26 D22 D11 D212 D22 ð11:42Þ ¼ 1 1 x xy Mxy D16 D12 D26 D22 D66 D226 D22

328


The curvatures can be obtained approximately as being 0

0

xx ¼ u x ¼ u ;

xxy

›2 w0 ffi 2 ¼ 2a0 ›x›yy¼0

ð11:43Þ

The resultant bending moment (M) and torque (T) are given by [161] Mðx; tÞ bMxx ;

Tðx; tÞ ¼ 2bMxy

ð11:44Þ

Thus, by applying Equations 11.42 and 11.43 into Equation 11.44, we can obtain #( 0 ) ( ) " EIeq KTeq M u ¼ KTeq GJeq T a0

ð11:45Þ

where 1 EIeq ¼ bðD11 D212 D22 Þ 1 Þ GJeq ¼ 4bðD66 D226 D22

KTeq ¼

ð11:46Þ

1 2bðD16 D12 D26 D22 Þ

and where KTeq represents the bending–torsion material coupling rigidity. By defining the beam deflection along the x-axis by h(x, t) ¼ w0(x, 0, t), from Equation 11.27 we can represent the transverse shear force approximately as Qðx; tÞ ffi kGAeq ðh0 uÞ

ð11:47Þ

kGAeq ¼ bkA55

ð11:48Þ

where

The strain energy U, the kinetic energy T, and the virtual work dW can be obtained by 1 U ¼ 2 ¼ T ¼ ¼

1 2 1 2 1 2

ðL 0

ðL 0

fMu0 þ Ta0 þ Qðh0 uÞ þ Pðh0 þ 2yCA h0 a0 þ r2g a0 Þg dx 2

2

fEIeq u0 þ 2KT eq a0 u0 þ GJeq a0 þ kGAeq ðh0 uÞ2 þ Pðh0 þ 2yCA h0 a0 þ r2g a0 Þg dx 2

ðL ð A

0

ðL

2

2

rðw_ 2 þ u_ 2 þ v_ 2 Þ dAdx

ð11:49Þ

ðrAeq h_ þ 2rAeq yCM h_ a_ þ rIeq u_ þ rJeq a_ 2 Þ dx 2

2

0

dW ¼ Q1 dhð0Þ þ Q2 dhðLÞ þ M1 duð0Þ þ M2 duðLÞ þ T1 dað0Þ þ T2 daðLÞ ðL _ c2 udu _ c3 adag _ dx þ ffh ðx; tÞdh þ fu ðx; tÞdu þ fa ðx; tÞda c1 hdh 0

2

329


where (c1, c2, c3) are the viscous damping coefficients and the following definitions are used: ð ð Aeq ¼ dA; Jeq ¼ ðy2 þ z2 ÞdA A ðA ðrAeq ; rIeq ; rJeq Þ ¼ rð1; z2 ; y2 þ z2 ÞdA ð11:50Þ A sffiffiffiffiffiffiffi ð ð Jeq 1 1 yCA ¼ ydA; yCM ¼ rydA; rg ¼ A A rAeq A Aeq where Aeq is the area of the cross-section of the composite laminate beam, Jeq is the polar area moment of inertia of the cross-section of the beam (about the x-axis), yCA is the distance between the elastic axis (x-axis) and the center of the area of the cross-section (CA), rAeq is the apparent mass per unit length, rIeq is the mass moment of inertia per unit length about the y-axis, rJeq is the apparent polar mass moment of inertia per unit length (about the x-axis), and yCM is the distance between the elastic axis and the center of mass (CM). If the mass density is constant over the whole cross-section, one may show from Equation 11.50 that rAeq ¼ rA, rJeq ¼ rIp, and yCA ¼ yCM. Substituting Equation 11.49 into Hamiltons principle and integrating by parts, we obtain ðt ðL ð ½fkGAeq ðh00 u0 Þ þ Pðh00 þ yCA a00 Þ rAeq h€ rAeq yCM a € c1 h_ þ fh ðx; tÞgdh 0

0

€ rAeq yCM h€ c3 a_ þ fa ðx; tÞgda þ fGJeq a00 þ KTeq u00 þ PðyCA h00 þ r2g a00 Þ rJeq a þ fEIeq u00 þ KTeq a00 þ kGAeq ðh0 uÞ rIeq u€ c2 u_ þ fu ðx; tÞgdudx fQðL; tÞ þ P½h0 ðLÞ þ yCA a0 ðLÞ Q2 gdhðLÞ þ fQð0; tÞ þ P½h0 ð0Þ þ yCA a0 ð0Þ þ Q1 gdhð0Þ fMðL; tÞ M2 gduðLÞ þ fMð0; tÞ þ M1 gduð0Þ fTðL; tÞ þ P½yCA h ðLÞ þ r2g a0 ðLÞ T2 gaðLÞ þ fTð0; tÞ þ P½yCA h0 ð0Þ þ r2g a0 ð0Þ þ T1 gdað0ÞÞdt

ð15:51Þ

0

¼0

when Equations 11.45 and 11.47 are used. From Equation 11.51, we obtain the equations of motion € c1 h_ þ fh ðx; tÞ ¼ 0 kGAeq ðh00 u0 Þ þ Pðh00 þ yCA a00 Þ rAeq h€ rAeq yCM a GJeq a00 þ KTeq u00 þ PðyCA h00 þ r2g a00 Þ rJeq a € rAeq yCM h€ c3 a_ þ fa ðx; tÞ ¼ 0 EIeq u00 þ KTeq a00 þ kGAeq ðh0 uÞ rIeq u€ c2 u_ þ fu ðx; tÞ ¼ 0

ð11:52Þ

and the boundary conditions Qð0; tÞ ¼ Q1 ðtÞ P½h0 ð0; tÞ þ yCA a0 ð0; tÞ

or hð0; tÞ ¼ 0

QðL; tÞ ¼ þ Q2 ðtÞ P½h0 ðL; tÞ þ yCA a0 ðL; tÞ Mð0; tÞ ¼ M1 ðtÞ

or hðL; tÞ ¼ 0 or uð0; tÞ ¼ 0

MðL; tÞ ¼ þ M2 ðtÞ Tð0; tÞ ¼ T1 ðtÞ P½yCA h0 ð0; tÞ þ r2g a0 ð0; tÞ

or uðL; tÞ ¼ 0 or að0; tÞ ¼ 0

TðL; tÞ ¼ þ T2 ðtÞ P½yCA h0 ðL; tÞ þ r2g a0 ðL; tÞ

or aðL; tÞ ¼ 0

ð11:53Þ

330


Equation 11.52 shows that the couplings between the bending, torsion, and shear displacements occurs as a result of the bending–torsion material coupling (i.e., KTeq) and the non-coincidence of the shear center (x-axis) and the mass center (CM) of the beam cross-section.

11.3 Dynamics of Axial–Bending–Shear Coupled Composite Beams 11.3.1 Equations of Motion The equations of motion and associated boundary conditions were derived in the previous section. They are given by kGAeq ðw00 u0 Þ þ Pw00 rAeq w € þ fw ðx; tÞ ¼ 0 EIeq u00 þ kGAeq ðw0 uÞ KA eq u00 þ rReq u€ rIeq u€ þ fu ðx; tÞ ¼ 0 EAeq u00 KA eq u00 rAeq u€ þ rReq u€ þ fu ðx; tÞ ¼ 0

ð11:54Þ

and Qð0; tÞ ¼ Q1 ðtÞ Pw00 ð0; tÞ or uð0; tÞ ¼ 0 QðL; tÞ ¼ þ Q2 ðtÞ Pw00 ðL; tÞ or uðL; tÞ ¼ 0 Mð0; tÞ ¼ M1 ðtÞ MðL; tÞ ¼ þ M2 ðtÞ

or wð0; tÞ ¼ 0 or wðL; tÞ ¼ 0

Nð0; tÞ ¼ N1 ðtÞ

or uð0; tÞ ¼ 0


or uðL; tÞ ¼ 0

ð11:55Þ

where the resultant axial tensile force N, the resultant moment M, and the resultant transverse shear force Q are defined by Qðx; tÞ ¼ kGAeq ðw0 uÞ Mðx; tÞ ¼ EIeq u0 KA eq u0 ð11:56Þ Nðx; tÞ ¼ EAeq u0 KA eq u0 In above equations, the subscripts “0” used for the in-plane displacements are omitted for brevity.

11.3.2 Spectral Element Modeling 11.3.2.1 Governing Equations in the Frequency Domain The displacement fields are assumed in the spectral forms to be 1 1 NX fWn ðx; vn Þ; Qn ðx; vn Þ; Un ðx; vn Þg eivn t fwðx; tÞ; uðx; tÞ; uðx; tÞg ¼ N n¼0

ð11:57Þ

Similarly, represent all forces and moments in the spectral forms as 1 1 NX fFwn ðx; vn Þ; Fun ðx; vn Þ; Fun ðx; vn Þgeivn t ffw ðx; tÞ; fu ðx; tÞ; fu ðx; tÞg ¼ N n¼0 1 1 NX fQn ðx; vn Þ; Mn ðx; vn Þ; Nn ðx; vn Þgeivn t ð11:58Þ N n¼0 1 1 NX fQkn ðvn Þ; Mkn ðvn Þ; Nkn ðvn Þgeivn t ðk ¼ 1; 2Þ fQk ðtÞ; Mk ðtÞ; Nk ðtÞg ¼ N n¼0

fQðx; tÞ; Mðx; tÞ; Nðx; tÞg ¼

331


Substituting Equations 11.57 and 11.58 into Equations 11.54 and 11.55 gives the frequencydomain equations of motion and boundary conditions as kGAeq ðW 00 Q0 Þ þ PW 00 þ v2 rAeq W þ Fw ¼ 0 EIeq Q00 þ kGAeq ðW 0 QÞ KAeq U 00 v2 rReq U þ v2 rIeq Q þ Fu ¼ 0

ð11:59Þ

EAeq U 00 KAeq Q00 þ v2 rAeq U v2 rReq Q þ Fu ¼ 0 and Qð0Þ ¼ Q1 PW 0 ð0Þ

or Qð0Þ ¼ 0

0

QðLÞ ¼ þ Q2 PW ðLÞ

or QðLÞ ¼ 0

Mð0Þ ¼ M1

or Wð0Þ ¼ 0

MðLÞ ¼ þ M2

or WðLÞ ¼ 0

Nð0Þ ¼ N1

or Uð0Þ ¼ 0

NðLÞ ¼ þ N2

or UðLÞ ¼ 0

ð11:60Þ

Similarly, substitution of Equations 11.57 and 11.58 into Equation 11.56 gives the frequencydomain force-displacement relationships as QðxÞ ¼ kGAeq ðW 0 QÞ MðxÞ ¼ EIeq Q0 KAeq U 0 0

NðxÞ ¼ EAeq U KAeq Q

ð11:61Þ 0

In above Equations 11.59–11.61, the subscripts n are omitted for brevity. 11.3.2.2 Spectral Nodal DOFs, Forces, and Moments By using the spectral nodal degrees-of-freedom (DOFs), forces, and moments specified in Figure 11.4, we define the spectral nodal DOFs vector as d ¼ ffU1 W1 Q1 U2 W2 Q2 gT ¼ fUð0Þ Wð0Þ Qð0Þ UðLÞ WðLÞ QðLÞgT

W1

ð11:62Þ

W2

U1

U2

Q1

Q(x )

Q2 N2

N(x )

N1 M1

M(x )

M2 L

Figure 11.4

Sign convention for the axial–bending–shear coupled composite beam element

332


and the spectral nodal forces and moments vector as ð11:63Þ

fc ¼ fN1 Q1 M1 N2 Q2 M2 gT 11.3.2.3 Dynamic Shape Functions

The homogeneous governing equations can be reduced from Equation 11.59 to give kGAeq ðW 00 Q0 Þ þ PW 00 þ v2 rAeq W ¼ 0 EIeq Q00 þ kGAeq ðW 0 QÞ KA eq U 00 v2 rReq U þ v2 rIeq Q ¼ 0 00

00

ð11:64Þ

EAeq U KA eq Q þ v rAeq U v rReq Q ¼ 0 2

2

Assume the general solution of Equation 11.64 to be WðxÞ ¼ ae ikx ;

QðxÞ ¼ aae ikx ;

UðxÞ ¼ bae ikx

ð11:65Þ

Substitution of Equation 11.65 into Equation 11.64 gives an eigenvalue problem as follows: 2 38 9 8 9 X11 X12 0 > 1 > > 0 > < = < = 6 7 ð11:66Þ 4 X12 X22 X23 5 a ¼ 0 > ; ; > : > : > 0 0 X23 X33 b where X11 ¼ ðkGAeq þ PÞk2 þ v2 rAeq X12 ¼ ikGAeq k ð11:67Þ

X22 ¼ EIeq k2 þ v2 rIeq kGAeq X23 ¼ X32 ¼ KA eq k v rReq 2

2

X33 ¼ EAeq k2 þ v2 rAeq Equation 11.66 yields a dispersion relation as ð11:68Þ

c1 kn6 þ c2 kn4 þ c3 kn2 þ c4 ¼ 0 where 2 c1 ¼ ðEAeq EIeq KAeq ÞðkGAeq þ PÞ 2 c2 ¼ ½ rAeq KAeq þ ðP þ kGAeq ÞðEIeq rAeq 2rReq KAeq Þ

þ EAeq ðEIeq rAeq þ kGAeq rIeq þ PrIeq Þv2 EAeq kGAeq P c3 ¼ ½ EIeq rA2eq kGAeq rAeq rIeq þ 2KAeq rAeq rReq þ kGAeq rR2eq EAeq rAeq rIeq þ PðrR2eq

rAeq rIeq Þv þ PkGAeq rAeq v 4

þ EAeq kGAeq rAeq c4 ¼ ðrA2eq rIeq rAeq rR2eq Þv6 kGAeq rA2eq v4

2

ð11:69Þ

333


From Equation 11.68, we can obtain six wavenumbers as follows rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k1;2 ¼ B3 þ B4 ðc2 =c1 Þ 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 1 k3;4 ¼ ðB3 þ B4 Þ ðc2 =c1 Þ þ i 23ðB3 B4 Þ 2 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi 1 3 1 k5;6 ¼ ðB3 þ B4 Þ 3 ðc2 =c1 Þ i 2 ðB3 B4 Þ 2 where

ð11:70Þ

i 1h 3ðc3 =c1 Þ ðc2 =c1 Þ2 9 i 1 h 9ðc2 c3 =c21 Þ 27ðc4 =c1 Þ 2ðc2 =c1 Þ3 B2 ¼ 54 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 B3 ¼ B2 þ B31 þ B22 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 B4 ¼ B2 B31 þ B22 B1 ¼

ð11:71Þ

By substituting wavenumber kp (p ¼ 1, 2, . . . 6) into Equation 11.66, we can obtain the corresponding ap and bp (p ¼ 1, 2, . . . , 6) as v2 rAeq kGAeq kp2 Pkp2 kGAeq kp ! ðp ¼ 1; 2; . . . ; 6Þ v2 rReq KAeq kp2 bp ¼ ap v2 rAeq EAeq kp2

ap ¼ i

ð11:72Þ

By using six wavenumbers given by Equation 11.70, we can write the general solution to Equation 11.64 as WðxÞ ¼

6 X

ap e ikp x ;

QðxÞ ¼

p¼1

6 X

ap ap e ikp x ;

p¼1

UðxÞ ¼

6 X i¼p

bp bp e ikp x

ð11:73Þ

or WðxÞ ¼ eðx; vÞa; where and

QðxÞ ¼ eðx; vÞAðvÞa;

UðxÞ ¼ eðx; vÞBðvÞa

eðx; vÞ ¼ e ik1 x e ik2 x e ik3 x e ik4 x e ik5 x e ik6 x AðvÞ ¼ diag ½ap ðvÞ;

BðvÞ ¼ diag ½bp ðvÞ

a ¼ f a1 a2 a3 a 4 a5 a6 g

T

ðp ¼ 1; 2; . . . ; 6Þ

ð11:74Þ ð11:75Þ

ð11:76Þ

Applying Equation 11.74 to the definition of the spectral nodal DOFs given by Equation 11.62 gives the relationship d ¼ HðvÞa ð11:77Þ

334


where,

2

b1

6 6 1 6 6 6 a1 HðvÞ ¼ 6 6b e 6 1 1 6 6 e 4 1 a1 e 1

b2

b3

b4

b5

1

1

1

1

a2

a3

a4

a5

b2 e 2

b 3 e3

b4 e 4

b5 e 5

e2

e3

e4

e5

a2 e 2

a 3 e3

a4 e 4

a5 e 5

b6

3

7 1 7 7 7 a6 7 7 b6 e 6 7 7 7 e6 7 5

ð11:78Þ

a6 e 6

with ep ¼ e ikp L

ðp ¼ 1; 2; . . . ; 6Þ

ð11:79Þ

By using Equation 11.77, the constant vector a can be removed from Equation 11.74 to write the general solutions in the form WðxÞ ¼ Nw ðx; vÞd;

QðxÞ ¼ Nu ðx; vÞd;

UðxÞ ¼ Nu ðx; vÞd

ð11:80Þ

where Nw, Nu, and Nu are the dynamic shape functions defined by Nw ðx; vn Þ ¼ eðx; vn ÞH 1 ðvÞ Nu ðx; vn Þ ¼ eðx; vn ÞAðvÞH 1 ðvÞ Nu ðx; vn Þ ¼ eðx; vn ÞBðvÞH

1

ð11:81Þ

ðvÞ

11.3.2.4 Weak Form of Governing Equations The weak form of the frequency-domain governing Equations 11.59 can be obtained from ðL

f½kGAeq ðW 00 Q0 ÞþPW 00 þv2 rAeq W þFw ðxÞdW

0

þ½EIeq Q00 þkGAeq ðW 0 QÞKAeq U 00 v2 rReq U þv2 rIeq QþFu ðxÞdQ

ð11:82Þ

þ½EAeq U 00 KA eq Q00 þv2 rAeq U v2 rReq QþFu ðxÞdUgdx ¼ 0 By applying the integral by parts and using Equation 11.61, we can obtain the weak form as ðL

fkGAeq W 0 dW 0 þkGAeq ðQdW 0 þW 0 dQÞkGAeq QdQ

0

EIeq Q0 dQ0 EAeq U 0 dU 0 þKAeq ðU 0 dQ0 þQ0 dU 0 ÞPW 0 dW 0 þv2 rAeq UdU þv2 rAeq WdW þv2 rIeq QdQv2 rReq ðUdQþQdUÞ þFu dU þFw dW þFu dQgdx i L h L L þNðxÞdU 0 þ QðxÞþPW 0 dW 0 þMðxÞdQ0 ¼ 0

ð11:83Þ

335


or, by applying the boundary conditions (11.60), Equations 11.62 and 11.63, ðL

fkGAeq W 0 dW 0 kGAeq ðQdW 0 þW 0 dQÞþkGAeq QdQ

0

þEIeq Q0 dQ0 þEAeq U 0 dU 0 KA eq ðU 0 dQ0 þQ0 dU 0 ÞþPW 0 dW 0

ð11:84Þ

v rAeq UdU v rAeq WdW v rIeq QdQþv rReq ðUdQþQdUÞgdx ÐL ¼ ddT fc þ 0 ðFu dU þFw dW þFu dQÞdx 2

2

2

2

11.3.2.5 Spectral Element Equation Substituting Equation 11.80 into Equation 11.84, we obtain dd

T

ðL 0

fkGAeq Nw0T Nw0 kGAeq ðNw0T Nu þ NTu Nw0 Þ þ kGAeq NTu Nu

0 0T 0 0T 0 þ EIeq Nu0T Nu0 þ EAeq N0T u Nu KA eq ðNu Nu þ Nu Nu Þ

þ PNw0T Nw0

v

2

rAeq ðNTu Nu

ð11:85Þ

þ NTw Nw Þ v2 rIeq NTu Nu

þ v2 rReq ðNTu Nu þ NTu Nu Þgdxd ¼ ddT ðfc þ fd Þ where fd ¼

ðL 0

ðFu NTu þ Fw NTw þ Fu NTu Þdx

ð11:86Þ

As dd is arbitrary by definition, we can obtain the spectral element equation from Equation 11.85 as SðvÞd ¼ fc þ fd

ð11:87Þ

where SðvÞ ¼

ÐL 0

½kGAeq Nw0T Nw0 kGAeq ðNw0T Nu þ NTu Nw0 Þ þ kGAeq NTu Nu

0 0T 0 0T 0 þ EIeq Nu0T Nu0 þ EAeq N0T u Nu KAeq ðNu Nu þ Nu Nu Þ

þ PNw0T Nw0 v2 rAeq ðNTu Nu þ NTw Nw Þ v2 rIeq NTu Nu

ð11:88Þ

þ v2 rReq ðNTu Nu þ NTu Nu Þdx By substituting Equation 11.81 into Equation 11.88, the spectral element matrix can be expressed in the form SðvÞ ¼ H T ðvÞDðvÞH 1 ðvÞ

ð11:89Þ

336


where DðvÞ ¼

ðL

½kGAeq e0 e0 kGAeq ðe0 eA þ AT eT e0 Þ þ kGAeq AT eT eA T

T

0

þ EIeq AT e0 T e0 A þ EAeq BT e0 T e0 B KA eq ðAT e0 T e0 B þ BT e0 T e0 AÞ 0T 0

ð11:90Þ

þ Pe e v rAeq ðB e eB þ e eÞ v rIeq A e eA T T

2

T

T T

2

þ v2 rReq ðAT eT eB þ BT eT eAÞdx From Equation 11.75, we can obtain the derivative of e(x;v) with respect to x as e0 ðx; vÞ ¼ ieðx; vÞKðvÞ

ð11:91Þ

where K ¼ diag ½kp

ðp ¼ 1; 2; . . . ; 6Þ

ð11:92Þ

Substitution of Equations 11.75 and 11.91 into Equation 11.90 gives DðvÞ ¼ kGAeq ½KEK þ iðKEA þ AT EKÞ þ AT EA EIeq AT KEKA EAeq BT KEKB þ KA eq ðAT KEKB þ BT KEKAÞ PKEK

ð11:93Þ

v2 ½rAeq ðBT EB þ EÞ þ rIeq AT EA rReq ðAT EB þ BT EAÞ where EðvÞ ¼

ðL

eT ðx; vÞeðx; vÞdx ¼ ½Ers ðvÞ

ðr; s ¼ 1; 2; . . . ; 6Þ

ð11:94Þ

0

with

8
= = > 6 7 ð11:118Þ 4 T21 T22 T23 5 a ¼ 0 > ; ; > : > : > 0 b T31 T32 T33

342


where T11 ¼ ðkGAeq þ PÞk2 þ v2 rAeq ivc1 T12 ¼ T21 ¼ ikGAeq k T13 ¼ T31 ¼ PyCA k2 þ v2 rAeq yCM T22 ¼ EIeq k2 þ ikGAeq k þ v2 rIeq ivc2 T23 ¼ T32 ¼ KTeq k2

ð11:119Þ

T33 ¼ ðGJeq þ Pr2g Þk2 þ v2 rJeq ivc3 Equation 11.118 yields a dispersion relation as follows: b1 kn6 þ b2 kn5 þ b3 kn4 þ b4 kn3 þ b5 kn2 þ b6 kn þ b7 ¼ 0

ð11:120Þ

where 2 2 b1 ¼ EIeq PðPyCA Pr2g GJeq Þ þ kGAðKTeq EIeq GJeq EIeq Pr2g Þ þ KTeq P

b2 ¼ ikGA½GJeq kGA þ Pr2g ðkGA þ PÞ þ PðGJeq Py2CA Þ 2 þ EIeq GJeq EIeq Pr2g Þ b3 ¼ kGA2 ðGJeq þ Pr2g Þ þ ifc1 ðKTeq

c2 ½PðPr2g þ Py2CA GJeq ÞP þ kGAðGJeq Pr2g Þ c3 EIeq ðP kGAÞgv þ fEIeq ðGJeq rAeq þ rAeq Pr2g PyCM yCA rAeq þ rJeq P þ rJeq kGAÞ 2 þ rIeq PðPr2g Py2CA þ kGAr2g Þ þ GJeq rIeq ðP þ kGAÞ KTeq rAeq gv2

b4 ¼ fc1 kGAðGJeq þ Pr2g Þ þ c3 kGAðkGA þ PÞgv þ ikGAfrAeq ð2PyCA yCM GJeq Pr2g Þ rJeq ðkGA þ PÞgv2 b5 ¼ ic3 kGA2 v

ð11:121Þ

þ fc1 c3 EIeq þ c1 c2 GJeq þ c2 c3 P þ c1 c2 Pr2g þ kGAðc2 c3 kGArJeq Þgv2 þ ifc1 ðGJeq rIeq þ rIeq Pr2g þ EIeq rJeq Þ þ c2 ½rAeq ðGJeq þ Pr2g 2yCA yCM PÞ þ rJeq ðP þ kGAÞ þ c3 ½EIeq rAeq þ rIeq ðkGA þ PÞgv3 þ fEIeq rAeq ðy2CM rIeq Þ rAeq rIeq ðGJeq Pr2g þ 2yCA yCM PÞ rIeq rJeq ðP þ kGAÞgv4 b6 ¼ ic1 c3 kGAv2 þ kGAðc1 rJeq þ c3 rAeq Þv3 þ ikGArAeq ðrJeq iy2CM rAeq Þv4 b7 ¼ ic1 c2 c3 v3 ðc2 c3 rAeq c1 c3 rIeq c1 c2 rJeq Þv4 þ ifrAeq ðc2 y2CM rAeq c3 rIeq c2 rJeq Þ c1 rIeq rJeq gv5 þ rAeq rIeq ðrJeq y2CM rAeq Þv6 From Equation 11.120, we can numerically compute six wavenumbers kp (p ¼ 1, 2, . . ., 6). The coefficients ap and bp for each wavenumbers kp (p ¼ 1, 2, . . ., 6) can be obtained from

343


Equation 11.118 as 1 ðKTeq kp2 bp þ ikGAeq kp Þ m1p m2p bp ¼ m3p

ap ¼

ðp ¼ 1; 2; . . . 6Þ

ð11:122Þ

where m1p ¼ EIeq kp2 þ ikGAeq kp þ v2 rIeq ivc2 m2p ¼ ðkGAeq þ PÞKTeq kp3 þ ikGAeq PyCA kp2 KTeq ðv2 rAeq ivc1 Þkp ikGAeq v2 rAeq yCM m3p ¼ PyCA KTeq kp3 ikGAeq ðGJeq þ Pr2g Þkp2 þ v2 rAeq yCM KTeq kp

ð11:123Þ

þ kGAeq vðivrJeq þ c3 Þ The general solutions of Equation 11.116 can be then be written in the forms HðxÞ ¼

6 X

ap e ikp x ;

QðxÞ ¼

p¼1

6 X

ap ap e ikp x ;

AðxÞ ¼

p¼1

6 X

ap bp e ikp x

p¼1

ð11:124Þ

or HðxÞ ¼ eðx; vÞa;

QðxÞ ¼ eðx; vÞAa;

AðxÞ ¼ eðx; vÞBa

ð11:125Þ

where a ¼ fa1

a2

a3

a4

a5

ð11:126Þ

a6 gT

and eðx; vÞ ¼ ½e ik1 x

e ik2 x

e ik3 x

e ik4 x

AðvÞ ¼ diag½ap ðvÞ

e ik5 x

e ik6 x

ðp ¼ 1; 2; . . . ; 6Þ

BðvÞ ¼ diag½bp ðvÞ

ð11:127Þ ð11:128Þ

By applying Equation 11.125 to the definition of the spectral nodal DOFs (11.114), we can obtain the relationship d ¼ GðvÞa where

2

1

6 a 6 1 6 6 b1 GðvÞ ¼ 6 6 e 6 1 6 4 a1 e 1 b1 e 1

ð11:129Þ

1

1

1

1

a2 b2

a3 b3

a4 b4

a5 b5

e2

e3

e4

e5

a2 e 2 b2 e 2

a 3 e3 b 3 e3

a4 e4 b4 e4

a5 e 5 b5 e 5

1

3

7 7 7 7 7 e6 7 7 7 a6 e 6 5 a6 b6

b6 e 6

ð11:130Þ

344


with

ep ¼ e ikp L

ðp ¼ 1; 2; . . . ; 6Þ

ð11:131Þ

By using Equation 11.129, the constant vector a can be eliminated from Equation 11.125 to obtain the expressions HðxÞ ¼ Nh ðx; vÞd;

QðxÞ ¼ Nu ðx; vÞd;

AðxÞ ¼ Na ðx; vÞd

ð11:132Þ

where Nh(x; vn), Nu(x; vn), Na(x; vn) are the dynamic shape functions defined by Nh ðx; vÞ ¼ eðx; vÞG 1 ðvÞ Nu ðx; vÞ ¼ eðx; vÞAG 1 ðvÞ Na ðx; vÞ ¼ eðx; vÞBG

1

ð11:133Þ

ðvn Þ

11.4.2.4 Weak Form of Governing Equations To formulate the spectral element equation in the next subsection, we first derive the weak form of Equation 11.111 from ðL f½kGAeq ðH 00 Q0 Þ þ PðH 00 þ yCA A00 Þ þ v2 rAeq H þ v2 rAeq yCM A 0

ivc1 H þ Fh dH þ ½EIeq Q00 þ KTeq A00 þ kGAeq ðH 0 Q0 Þ þ v2 rIeq Q

ð11:134Þ

ivc2 Q þ Fu dQ þ ½GJeq A00 þ KTeq Q00 þ PðyCA H 00 þ r2g A00 Þ þ v2 rJeq A þ v2 rAeq yCM H ivc3 A þ Fa ÞdAgdx ¼ 0 Applying the integral by parts to some terms in Equation 11.134 and then using the forcedisplacement relationships (11.113), we can obtain the weak form as ðL ½kGAeq H 0 dH 0 þ kGAeq QdQ kGAeq ðQdH 0 þ H 0 dQÞ þ EIeq Q0 dQ0 0

þ ðGJeq þ Pr2g ÞA0 dA0 þ KTeq ðA0 dQ0 þ Q0 dA0 Þ þ PyCA ðA0 dH 0 þ H 0 dA0 Þ þ PH 0 dH 0 þ ðivc1 v2 rAeq ÞHdH þ ðivc2 v2 rIeq ÞQdQ þ ðivc3 v2 rJeq ÞAdA v2 rAeq yCM ðAdH þ HdAÞdx L L L ¼ MðxÞdQ þ ½QðxÞ þ PðH 0 þ yCA A0 ÞdH þ ½TðxÞ þ PðyCA H 0 þ r2g A0 ÞdA 0 0 0 ðL þ ½Fh ðxÞdH þ Fu ðxÞdQ þ Fa ðxÞdAdx

ð11:135Þ

0

or, by applying the boundary conditions (11.112) and then using the definitions of Equations 11.114 and 11.115, ðL ½kGAeq H 0 dH 0 þ kGAeq QdQ kGAeq ðQdH 0 þ H 0 dQÞ þ EIeq Q0 dQ0 0

þ ðGJeq þ Pr2g ÞA0 dA0 þ KTeq ðA0 dQ0 þ Q0 dA0 Þ þ PyCA ðA0 dH 0 þ H 0 dA0 Þ þ PH 0 dH 0 þ ðivc1 v2 rAeq ÞHdH þ ðivc2 v2 rIeq ÞQdQ þ ðivc3 v rJeq ÞAdA v rAeq yCM ðAdH þ HdAÞdx ðL h i Fh ðxÞdH þ Fu ðxÞdQ þ Fa ðxÞdA dx ¼ ddT fc þ 2

0

2

ð11:136Þ

345


11.4.2.5 Spectral Element Equation Substitution of Equation 11.132 into Equation 11.136 gives ðL T 0 T 0T T 0 dd fkGAeq N0T h Nh þ kGAeq Nu Nu kGAeq ðNh Nu þ Nu Nh Þ 0

0 0T 0 0T 0 þ EIeq Nu0T Nu0 þ ðGJeq þ Pr2g ÞN0T a Na þ KTeq ðNu Na þ Na Nu Þ 0 0T 0 0T 0 T 2 þ PyCA ðN0T h Na þ Na Nh Þ þ PNh Nh þ ðivn c1 vn rAeq ÞNh Nh þ ðivn c2 v2n rIeq ÞNTu Nu þ ðivn c3 v2n rJeq ÞNTa Na

ð11:137Þ

v2n rAeq yCM ðNTh Na þ NTa Nh Þgdxd i h ÐL ¼ ddT fc þ 0 ðFh ðxÞNTh þ Fu ðxÞNTu þ Fa ðxÞNTa Þdx dd is arbitrary by definition. Thus, we can obtain the spectral element equation from Equation 11.137 as being ð11:138Þ SðvÞd ¼ fc þ fd where ðL 0 T 0T T 0 SðvÞ ¼ fkGAeq N0T h Nh þ kGAeq Nu Nu kGAeq ðNh Nu þ Nu Nh Þ 0

0 0T 0 0T 0 þ EIeq Nu0T Nu0 þ ðGJeq þ Pr2g ÞN0T a Na þ KTeq ðNu Na þ Na Nu Þ 0 0T 0 0T 0 T 2 þ PyCA ðN0T h Na þ Na Nh Þ þ PNh Nh þ ðivn c1 vn rAeq ÞNh Nh

ð11:139Þ

þ ðivn c2 v2n rIeq ÞNTu Nu þ ðivn c3 v2n rJeq ÞNTa Na v2n rAeq yCM ðNTh Na þ NTa Nh Þgdx and fd ¼

ðL 0

½Fh ðxÞNTh þ Fu ðxÞNTu þ Fa ðxÞNTa dx

ð11:140Þ

By substituting Equation 11.133 into Equation 11.139, the spectral element matrix S(v) can be expressed in the form ð11:141Þ SðvÞ ¼ G T ðvÞDðvÞG 1 ðvÞ where DðvÞ ¼

ðL

½kGAeq e0 e0 þ kGAeq AeT eA kGAeq ðe0 eA þ AeT e0 Þ T

T

0

þ EIeq Ae0 T e0 A þ ðGJeq þ Pr2g ÞBe0 T e0 B þ KTeq ðAe0 T e0 B þ Be0 T e0 AÞ þ PyCA ðe0 T e0 B þ Be0 T e0 Þ þ Pe0 T e0

ð11:142Þ

þ ðivn c1 v2n rAeq ÞeT e þ ðivn c2 v2n rIeq ÞAeT eA þ ðivn c3 v2n rJeq ÞBeT eB v2n rAeq yCM ðeT eB þ BeT eÞdx The derivative of e(x;v) with respect to x can be obtained from Equation 11.127 as e0 ðx; vÞ ¼ ieðx; vÞKðvÞ

ð11:143Þ

where K ¼ diag½kp

ðp ¼ 1; 2; . . . ; 6Þ

ð11:144Þ

346


Substitution of Equations 11.127 and 11.143 into Equation 11.142 gives DðvÞ ¼ kGAeq ½ KEK þ AEA þ iðKEA þ AEKÞ EIeq AKEKA ðGJeq þ Pr2g ÞBKEKB KTeq ðAKEKB þ BKEKAÞ PyCA ðKEKB þ BKEKÞ PKEK þ ðivn c1 v2n rAeq ÞE

ð11:145Þ

þ ðivn c2 v2n rIeq ÞAEA þ ðivn c3 v2n rJeq ÞBEB v2n rAeq yCM ðEB þ BEÞ where the same definition for E(v) as that given by Equation 11.94 is used.

11.4.3 Finite Element Model We can obtain the weak form of the equations of motion (11.106) in the form of ðL ½kGAeq h0 dh0 þ kGAeq udu kGAeq ðudh0 þ h0 duÞ þ EIeq u0 du0 0

þ ðGJeq þ Pr2g Þa0 da0 þ KTeq ða0 du0 þ u0 da0 Þ þ PyCA ða0 dh0 þ h0 da0 Þ þ Ph0 dh0 þ ðc1 h_ þ rAeq h€Þdh þ ðc2 u_ þ rIeq u€Þdu þ ðc3 a_ þ rJeq a € Þda € dh þ rAeq yCM h€dadx þ zdrAeq yCM a L L L ¼ ½Qðx; tÞ þ Pðh0 þ yCA a0 Þdh þ Mt du þ ½Tðx; tÞ þ PðyCA h0 þ r2g a0 Þda 0 0 0 ðL ðL þ fh dhdx þ fu dudx þ fa dadx 0

ð11:146Þ

0

We use the same convention for the nodal DOFs, forces, and moments in the time domain as specified in Figure 11.5 and we define the nodal DOFs vector by dðtÞ ¼ fh1 ðtÞ

u1 ðtÞ a1 ðtÞ

h2 ðtÞ

u2 ðtÞ a2 ðtÞgT

ð11:147Þ

and the nodal forces and moments vector by fc ðtÞ ¼ fQ1 ðtÞ

M1 ðtÞ T1 ðtÞ

Q2 ðtÞ

M2 ðtÞ T2 ðtÞgT

ð11:148Þ

We apply the boundary conditions (11.107) to the right-hand side of Equation 11.146 and then use the definitions (11.147) and (11.148) to write Equation 11.146 as ðL ½kGAeq h0 dh0 þ kGAeq udu kGAeq ðudh0 þ h0 duÞ þ EIeq u0 du0 0

þ ðGJeq þ Pr2g Þa0 da0 þ KTeq ða0 du0 þ u0 da0 Þ þ PyCA ða0 dh0 þ h0 da0 Þ € € þ Ph0 dh0 þ ðc1 h_ þ rAeq hÞdh € Þda þ ðc2 u_ þ rIeq uÞdu þ ðc3 a_ þ rJeq a € dh þ rAeq yCM h€dadx þ rAeq yCM a ðL ðL T ¼ dd fc ðtÞ þ fh dhdx þ fu dudx þ fa dadx 0

0

ð11:149Þ

347


Assume the displacement fields within the composite beam element to be hðx; tÞ ¼ Nh ðxÞdðtÞ;

uðx; tÞ ¼ Nu ðxÞdðtÞ;

aðx; tÞ ¼ Na ðxÞdðtÞ

ð11:150Þ

where Nh ðxÞ ¼ ½N1 ðxÞ

N2 ðxÞ 0

N3 ðxÞ N4 ðxÞ

0

Nu ðxÞ ¼ ½N5 ðxÞ

N6 ðxÞ 0

N7 ðxÞ N8 ðxÞ

0

Na ðxÞ ¼ ½0

0

N9 ðxÞ 0

0

ð11:151Þ

N10 ðxÞ

with N1 ðxÞ ¼ Rð1 jÞð1 þ j 2j2 þ dÞ N2 ðxÞ ¼ ð1=2ÞRLjð1 jÞð2 2j þ dÞ N3 ðxÞ ¼ Rjð3j 2j2 þ dÞ;

N4 ðxÞ ¼ ð1=2ÞRLjð1 jÞð2j þ dÞ

N5 ðxÞ ¼ 6ðR=LÞjð1 jÞ;

N6 ðxÞ ¼ Rð1 jÞð1 3j þ dÞ

N7 ðxÞ ¼ 6ðR=LÞjð1 jÞ;

N8 ðxÞ ¼ Rjð 2 þ 3j þ dÞ

N9 ðxÞ ¼ 1 j;

N10 ðxÞ ¼ j

ð11:152Þ

where x j¼ ; L

R¼

1 ; 1þd

d¼

12r ; L2

r¼

EI kGA

ð11:153Þ

By substituting Equation 11.150 into Equation 11.149, we can obtain the finite element equation in the form of _ þ KdðtÞ ¼ f ðtÞ þ f ðtÞ M d€ðtÞ þ CdðtÞ c d

ð11:154Þ

where ðL ðL NTh Nh dx þ rIeq NTu Nu dx þ rJeq NTa Na dx 0 0 0 ðL T T þ rAeq yCM ðNh Na þ Na Nh Þdx 0 ðL ðL ðL T T C ¼ c1 Nh Nh dx þ c2 Nu Nu dx þ c3 NTa Na dx

M ¼ rAeq

0

ðL

ðL

0

0

ðL

ðL

ð11:155Þ

0 T 0 NTu Nu dx kGAeq ðN0T K ¼ kGAeq N0T h Nh dx þ kGAeq h Nu þ Nu Nh Þdx 0 0 0 ðL ðL ðL 0 0 N dx þ K ðNu0T N0a þ N0T þ EIeq Nu0T Nu0 dx þ ðGJeq þ Pr2g Þ N0T Teq a a a Nu Þdx 0 0 0 ðL ðL 0 0T 0 0 þ PyCA ðN0T N þ N N Þdx þ P N0T h a a h h Nh dx 0

0

348


and fd ðtÞ ¼

ðL 0

½fh ðx; tÞNTh ðxÞ þ fu ðx; tÞNTu ðxÞ þ fa ðx; tÞNTa ðxÞdx

ð11:156Þ

The finite element mass matrix M, stiffness matrix K, and damping matrix C are given in Appendix 11.B.

Example 11.3 Consider a cantilever glass–epoxy composite beam from Refs. [12, 111] as an illustrative example. The beam is of rectangular cross-section and is made up of unidirectional plies with fiber angles in each ply set to a single angle þ15 . Thus the composite beam can be considered as a single thick ply. The beam has the length L ¼ 0.1905 m and its structural and inertia properties are taken from Refs. [12, 111] and they are given by EIeq ¼ 0:2865 N m2

kGAeq ¼ 6343:3 N

GJeq ¼ 0:1891 N m2

KTeq

¼ 0:1143 N m2

rAeq ¼ 0:0544 kg=m

rIeq

¼ 4:584 10 8 kg m

rJeq ¼ 0:777 10 6 kg m with yCA ¼ yCM ¼ 0 m and the damping coefficients set to c1 ¼ c2 ¼ c3 ¼ 0. The first five natural frequencies of the composite beam subjected to an axial tensile force (P ¼ 7.5 N) are computed by using the present spectral element model and the finite element model, and in Table 11.3 the results are compared with the results from Refs. [12, 111]. The one-element model is used for the SEM results, while the number of finite elements used for FEM is varied in order to investigate the solution convergence to the SEM results. Similarly, Table 11.4 is for the same composite beam subjected to a compressive axial force (P ¼ 7.5 N). The present SEM results are shown to be almost equal to the results obtained by an analytical method in Ref. [111] and the dynamic stiffness method in Ref. [12]. The FEM results are shown to converge towards the SEM results as the number of finite elements used in the FEM is increased. As expected, Tables 11.3 and 11.4 also show that the natural frequencies of the composite beam subjected to a compressive axial force are lower than those of the composite beam subjected to a tensile axial force. Table 11.3

Natural frequencies of the cantilever composite beam subjected to an axial tensile force FEM(N)

Mode

SEM (1)

N¼2

N ¼ 10

N ¼ 30

N ¼ 50

N ¼ 200

Ref. [111]

Ref. [12]

1 2 3 4 5

37.10 197.7 525.7 648.5 992.9

37.19 205.6 590.3 654.6 1259

37.10 198.2 530.0 649.1 1011

37.10 197.7 526.2 648.6 994.9

37.10 197.7 525.7 648.5 993.6

37.10 197.7 525.7 648.5 992.9

37.10 197.7 525.7 648.5 992.9

37.10 197.7 525.6 648.6 —

Note: N ¼ total number of the equal-length finite elements used in the analysis.

349

Dynamics of Composite Laminated Structures Table 11.4 Natural frequencies of the cantilever composite beam subjected to an axial compressive force FEM(N) Mode

SEM(1)

N¼2

N ¼ 10

N ¼ 50

N ¼ 150

N ¼ 200

Ref. [111]

Ref. [12]

1 2 3 4 5

21.99 181.5 511.8 648.0 979.5

22.07 189.4 576.9 653.9 1247

21.99 182.0 516.2 648.6 997.9

21.99 181.5 512.0 648.1 980.2

21.99 181.5 511.8 648.1 979.6

21.99 181.5 511.8 648.0 979.5

21.98 181.5 511.8 648.0 979.5

21.99 181.5 511.9 648.0 —

Note: N ¼ total number of the equal-length finite elements used in the analysis.

Appendix 11.A: Finite Element Matrices for Axial–Bending–Shear Coupled Composite Beams The finite element mass and stiffness matrices defined in Equation 11.104 are given by M ¼ MrA þ MrI MrR

ð11:A:1Þ

K ¼ KkGA þ KEI þ KEA KK Aeq þ KP

where 2

MrA

6 6 6 6 rAeq L 6 6 ¼ 6 840 6 6 6 6 4

280

0

0

4ð70 þ 7R þ R Þ 2

Lð35 þ 7R þ 2R2 Þ L2 ð7 þ R2 Þ

Symmetric

140 0 0 280

0

0

3

7 4 35 7R R2 L 35 7R 2R2 7 7 7 2 2 2 7 L 35 7R 2R L 7R Þ 7 7 7 0 0 7 7 L 35 þ 7R þ 2R2 7 4 70 þ 7R þ R2 5 2 2 L 7þR

350


2

MrI

6 6 6 rIeq 6 6 ¼ 30L 6 6 6 4

0

0 3LRð5 6RÞ

0 0

0 36R2

L2 ð10 15R þ 9R2 Þ 0

0 36R2

Symmetric

6 6 6 rReq 6 6 ¼ 12 6 6 6 4

0

6R 0

0 3LRð5 6RÞ

3LRð5 6RÞ

0

2

M3

0 36R2

Lð4 3RÞ 0

0 6R

0

Lð2 3RÞ

0

0

6R 0

7 7 2 2 7 L ð5 15R þ 9R Þ 7 7 7 0 7 7 5 3LRð5 6RÞ

L2 ð10 15R þ 9R2 Þ 3 Lð2 3RÞ 7 0 7 7 7 0 7 ð11:A:2Þ Lð4 3RÞ 7 7 7 5 0

6R 0

Symmetric

3

0 and 2 6 6 6 kGAeq 26 KkGA ¼ ðI RÞ 6 6 4L 6 6 4 2

KEI

6 6 6 EIeq 6 ¼ 3 6 L 6 6 6 4 2

KEA

6 6 EAeq 6 6 ¼ L 6 6 4

0

0

0

0

0

4

2L

0

4

L2

0 0

2L 0

Symmetric 0 2

6LR

2

L2 ð1 þ 3R2 Þ

0

0

0

12R

0 0

6LR2 0

0 0

1 0 0 1

0 0 0

0 0 0 0 0

Symmetric

6 6 KAeq 6 6 ¼ L 6 6 4

0

0 0

Symmetric

1 0 0

0 0 1 0

3

2L 7 7 7 L2 7 7 0 7 7 7 2L 5 L2 3

0

2

12R2

Symmetric

1

0

4

0

12R

2

K KA

0

7 7 2 2 7 L ð1 3R Þ 7 7 7 0 7 7 5 6LR2 6LR2

L2 ð1 þ 3R2 Þ 3 0 07 7 07 7 ð11:A:3Þ 07 7 05 0 0 0 0 0 0

3 1 0 7 7 0 7 7 1 7 7 0 5 0

351


2 6 6 P 6 6 KP ¼ 60L 6 6 4

0

0 12ð5 þ R2 Þ

0 0 6LR2 0 L2 ð5 þ 3R2 Þ 0 0

0 12ð5 þ R2 Þ 6LR2 0 12ð5 þ R2 Þ

Symmetric

3 0 2 7 6LR 7 L2 ð5 3R2 Þ 7 7 7 0 7 5 6LR2 L2 ð5 þ 3R2 Þ

Appendix 11.B: Finite Element Matrices for Bending–Torsion–Shear Coupled Composite Beams The finite element matrices defined in Equation 11.154 are given by M ¼ MrA þ MrI þ MrJ þ MrAyCM C ¼ Cc1 þ Cc2 þ Cc3

ð11:B:1Þ

K ¼ KkGA þ KEI þ KGJ þ Pr2g þ KKT þ KPyCA þ KP where 2

MrA

6 6 6 6 rAeq L 6 6 ¼ 840 6 6 6 6 4

4ð70 þ 7R þ R2 Þ

Lð35 þ 7R þ 2R2 Þ

0

L ð7 þ R Þ

0

2

2

0 Symmetric 4 35 7R R2 L 35 7R 2R2

L 35 7R 2R2 L2 7 R 2 Þ

0 4 70 þ 7R þ R2

2 6 6 6 rIeq 6 6 MrI ¼ 30L 6 6 6 4

36R2

3Lð5R 6R2 Þ L2 ð10 15R þ 9R2 Þ

L 35 þ 7R þ 2R2 L 2 7 þ R2

3

7 07 7 7 07 7 7 07 7 7 07 5 0

0 0

36R2 3Lð5R 6R2 Þ

3Lð5R 6R2 Þ L2 ð5 15R þ 9R2 Þ

0

0

0

36R Symmetric

0

0

2

3Lð5R 6R2 Þ L2 ð10 15R þ 9R2 Þ

3 0 07 7 7 07 7 07 7 7 05 0

352


2

MrJ

6 6 rJeq L 6 6 ¼ 6 6 6 4

0

0 0

0 0 0 0

6 6 rAeq yCM L 6 6 ¼ 120 6 6 4

0

0 0

2ð20 þ RÞ Lð5 þ RÞ 0

3 0 07 7 17 7 07 7 05 2

0 0 0 0 0

Symmetric 2

MrAyCM

0 0 2

0 0 2ð10 RÞ 0

Symmetric

ð11:B:2Þ

0 0 Lð5 RÞ 0 0

3 2ð10 RÞ Lð5 RÞ 7 7 7 0 7 2ð20 þ RÞ 7 7 Lð5 þ RÞ 5 0

and 2 6 6 6 c1 L 6 6 Cc1 ¼ 840 6 6 6 4

4ð70 þ 7R þ R2 Þ Lð35 þ 7R þ 2R2 Þ 0 L2 ð7 þ R2 Þ

0 0

Symmetric 4 35 7R R2 L 35 7R 2R2 L2 7 R 2 Þ L 35 7R 2R2 0 0 2 4 70 þ 7R þ R L 35 þ 7R þ 2R2 L2 7 þ R2

0

3

7 07 7 07 7 7 07 7 7 05 0

2 6 6 c2 6 6 C c2 ¼ 6 30L 6 6 4 2 6 6 c3 L 6 6 C c3 ¼ 6 6 6 6 4

36R2

3Lð5R 6R2 Þ L2 ð10 15R þ 9R2 Þ

0 36R2 0 3Lð5R 6R2 Þ 0 0 36R2

Symmetric 0

0 0

Symmetric

0 0 2

0 0 0 0

0 0 0 0 0

3Lð5R 6R2 Þ L2 ð5 15R þ 9R2 Þ 0 3Lð5R 6R2 Þ L2 ð10 15R þ 9R2 Þ 3 0 07 7 17 7 7 07 7 05 2

3 0 07 7 07 7 7 07 7 05 0

ð11:B:3Þ

353


and 2 6 6 6 6 kGAeq ð1 RÞ2 6 KkGA ¼ 6 4L 6 6 4

4

2L

0

4

2L

L2

0 0

2L 0

L2 0

4

2L L2

Symmetric

0

3

07 7 7 07 7 07 7 7 05 0

2

KEI

6 6 EIeq 6 ¼ 3 6 L 6 6 4

12R2

6 6 GJeq þ Pr2g 6 6 ¼ 6 L 6 4 2

K KT

6 6 KTeq 6 6 ¼ L 6 6 4

0

6 6 PyCA 6 6 ¼ L 6 6 4 2

6 6 P 6 6 KP ¼ 60L 6 6 4

0

0 0

0 0 1

0 0

0

12ð5 þ R2 Þ

0 0 0 0

Symmetric 0 1 0

0 0 0 0

Symmetric 2

KPyCA

6LR2 2 L ð1 3R2 Þ 0 6LR2 L2 ð1 þ 3R2 Þ

Symmetric 2

KGJ þ Pr2g

12R2 6LR2 0 12R2

6LR2 0 2 L ð1 þ 3R2 Þ 0 0

0 0

1 0 0

0 0 1 0 0

0 0 1 0

Symmetric

6LR2 2 L ð5 þ 3R2 Þ

Symmetric

0 0 0

12ð5 þ R2 Þ 6LR2 0 12ð5 þ R2 Þ

3 0 07 7 07 7 07 7 05 0 3 0 0 7 7 17 7 0 7 7 0 5 1

0 0 0 0 0 3 0 17 7 0 7 7 0 7 7 1 5 0 0 0 0 0 0

ð11:B:4Þ

3 1 0 7 7 0 7 7 1 7 7 0 5 0

6LR2 2 L ð5 3R2 Þ 0 6LR2 L2 ð5 þ 3R2 Þ

3 0 07 7 07 7 07 7 05 0

12 Dynamics of Periodic Lattice Structures Lattice structures are usually composed of many identically constructed cell units connected end-to-end to form spatially periodic arrays, ranging in size from nanometers to many kilometers. A carbon nanotube is an example of a periodic nanostructure, while the space station is an example of a large periodic space structure. In general, most of the periodic lattice structures are very flexible, containing a huge number of elements and nodes. Thus, a significant computation cost is inevitable in order to obtain sufficiently reliable dynamic characteristics, through the use of matrix methodologies such as the finite element method (FEM). In this chapter, two alternative solution methods are introduced: the continuum modeling method and the spectral transfer matrix method. In the continuum modeling method, a periodic lattice structure is transformed into an equivalent continuum model by determining the effective continuum structural properties in an appropriate way, which enables us to apply existing solution methods directly to the equivalent continuum model. The spectral transfer matrix method is a spectral element method (SEM) based transfer matrix method useful for beam-like one-dimensional (1-D) lattice structures. In this method, we first derive transfer matrices for each of the lattice cells in a lattice structure by using SEM and then use the classical transfer matrix method to conduct dynamic analysis for the whole lattice structure.

12.1 Continuum Modeling Method In the continuum modeling approach, a periodic lattice structure is transformed into an equivalent continuum model by determining the effective structural properties of the equivalent continuum model from the appropriate relationshipships between the geometric and material properties of a periodic lattice structure and its equivalent continuum model. Because there can be numerous concepts and approaches to determining equivalent structural properties in terms of the geometric and material properties of the original periodic lattice structure, the continuum modeling approach will not be unique in nature. In the literature, the equivalence of a periodic lattice structure and its equivalent continuum model has been established with the



356

requirement that both have the same kinetic and strain energies [79, 129], constitutive relationships [125], static deformation characteristics [152], or dispersion curves [1] when they take identical global dynamic behaviors. In this section, we will consider a continuum modeling method that was developed based on the transfer matrix equivalence [80]. Figure 12.1 shows some typical examples of beam-like (1-D) periodic lattice structures and their representative lattice cells. As the continuum modeling method introduced in this chapter can be readily extended to the 2-D and 3-D periodic lattice structures, we will consider a periodic plane truss, as shown in Figure 12.1(b), as an illustrative example.

(a) Carbon nanotube

(b) Plane lattice beam

(c) Single-bay double-laced lattice beam

Figure 12.1

Examples of beam-like periodic lattice structures

12.1.1 Transfer Matrix for the Representative Lattice Cell (RLC) 12.1.1.1 RLC to Continuum Beam Element Figure 12.2(a) shows a representative lattice cell (RLC) isolated from the example periodic plane truss. The RLC consists of a total of six structure members connecting four corner nodes and it has length L and height H. We will consider a continuum modeling method by which the RLC can be transformed into an equivalent continuum beam element as shown in Figure 12.2(b). The global shear deformation will be significant for discrete structures such as

M1

W1 Q1

W3 Q3

M3 N3 U3

N1 U1

Nc1 Uc1

H N2 U2 M2

N4 U4

L W2 Q2

W4 Q4

Mc1

Wc2 Qc2

1

Nc2 Uc2

2

L

Mc2

M4

(a) Representative lattice cell (RLC)

Figure 12.2

Wc1 Qc1

(b) Continuum ET-beam model

Continuum modeling of an RLC isolated from a plane truss

Dynamics of Periodic Lattice Structures

357

the periodic lattice structures, and the couplings between the axial, shear, and bending deformations may occur if a periodic lattice structure is not perfectly axisymmetric about its central axis. Thus, to take account of the global shear deformation and the couplings between the axial, shear, and bending deformations, the extended Timoshenko beam (ET-beam) model will be adopted as the equivalent continuum beam element. The ET-beam model is a generalization of the classical Timoshenko-beam model to include the couplings between the axial, shear, and bending deformations, which will be detailed in the next section.

12.1.1.2 Spectral Beam Element Representation of the Structure Members in RLC The structure members in RLC can be represented by the rod models (axial bar models) or beam models, depending on the type of joints at each corner of the RLC. Assuming that the structural members in the RLC shown in Figure 12.2(a) are all clamped at the joints, we will represent them by the beam models, which take the longitudinal (axial) and bending vibrations simultaneously. The spectral element model for the axial–bending beam model can be obtained by combining the spectral rod element given by Equation 3.39 and the spectral beam element given by Equation 3.59 in the form sðvÞd ¼ f ð12:1Þ where

and

d ¼ fU1

W 1

1 Q

U2

W 2

f ¼ fN1

Q1

M 1

N2

Q2

2

sR11

6 6 0 6 6 6 0 6 sðvÞ ¼ 6 6 sR 6 21 6 6 0 4 0

2 gT Q T M 2 g

0

0

sR12

0

sB11

sB12

0

sB13

sB21

sB22

0

sB23

0

0

sR22

0

sB31

sB32

0

sB33

sB41

sB42

0

sB43

0

ð12:2Þ

3

7 sB14 7 7 7 sB24 7 7 7 0 7 7 7 sB34 7 5 B s44

ð12:3Þ

where the superscripts R and B denote the components of the spectral element matrices for the rod element and Bernoulli–Euler beam element, respectively, the over-bars denote the quantities measured with respect to the local (or element) coordinates, d is the spectral nodal degrees-of-freedom (DOFs) vector, and f is the spectral nodal forces and moments vector: U i , i (i ¼ 1, 2, 3, 4) are the spectral components of the nodal axial displacements, i , and Q W transverse displacements, and rotations (slopes), respectively, and Ni , Qi , and M i (i ¼ 1, 2, 3, 4) are the spectral components of the nodal resultant axial forces, transverse shear forces, and bending moments, respectively. The spectral nodal DOFs d in the local coordinates can be related to the spectral nodal DOFs d in the global (or reference) coordinates as d ¼ Tl=g d

ð12:4Þ


358

where Tl/g is the “local-to-global” coordinate 2 cos f sin f 6 sin f cos f 6 6 0 0 Tl=g ¼ 6 6 0 0 6 4 0 0 0 0

transformation matrix given by 3 0 0 0 0 0 0 0 07 7 1 0 0 07 7 0 cos f sin f 0 7 7 0 sin f cos f 0 5 0 0 0 1

ð12:5Þ

where f is the rotation angle of the local coordinates measured counterclockwise from the global coordinates. The coordinate transformation matrix Tl/g is an orthogonal matrix 1 satisfying Tl=g ¼ TTl=g . Similarly, the spectral nodal forces and moments f in the local coordinates can be related to the spectral nodal forces and moments f in the global coordinates by using the same transformation matrix T as f ¼ Tl=g f ð12:6Þ By substituting Equations 12.4 and 12.6 into Equation 12.1, we can obtain the spectral element equation in the global coordinates as follows: sðvÞd ¼ f

ð12:7Þ

sðvÞ ¼ TTl=gsðvÞTl=g

ð12:8Þ

d ¼ fU1 W1 Q1 U2 W2 Q2 gT f ¼ fN1 Q1 M1 N2 Q2 M2 gT

ð12:9Þ

where

and

It is worth mentioning that, if a structure member is pin connected at its joints, the structure member may be represented by the rod element. In this instance, the rotational DOFs and the associated bending moments will all be removed from Equation 12.7. 12.1.1.3 Spectral Element Modeling of the RLC in Global Nodal DOFs The example RLC consists of six structure members: two parallel members, two vertical members, and two diagonal members. Each structure member can be modeled by using one spectral beam element represented by Equation 12.7. The spectral nodal DOFs, forces, and moments in the global coordinates are specified at each node of the RLC in Figure 12.2(a). By assembling six spectral beam elements for six structure members, we can readily obtain the global system matrix equation in the form as Sg ðvÞdg ¼ fg

ð12:10Þ

where S (v) is the assembled global dynamic stiffness matrix, dg is the global spectral nodal DOFs vector, and fg is the global spectral nodal forces and moments vector. By rearranging the order of the nodal DOFs and forces, Equation 12.10 can be written in a partitioned form as fgL SgLL ðvÞ SgLR ðvÞ dgL ¼ ð12:11Þ fgR SgRL ðvÞ SgRR ðvÞ dgR


359

where dgL ¼ fU1 W1 Q1 U2 W2 Q2 gT dgR ¼ fU3 W3 Q3 U4 W4 Q4 gT fgL ¼ fN1 Q1 M1 N2 Q2 M2 gT fgR ¼ fN3 Q3 M3 N4 Q4 M4 gT

ð12:12Þ

The subscripts L and R denote the spectral values at the nodes on the left-section and the rightsection of the RLC, respectively. 12.1.1.4 Spectral Element Modeling of the RLC in Continuum Nodal DOFs The RLC will be transformed in an equivalent continuum ET-beam element. We assume that the centerline of the continuum ET-beam element is aligned with the centerline of the RLC. The spectral nodal DOFs for the continuum ET-beam element are defined in Figure 12.2(b) and are denoted by dcL ¼ fUc1

Wc1

Qc1 gT

dcR ¼ fUc2

Wc2

Qc2 gT

ð12:13Þ

where dcL and dcR are the continuum nodal DOFs specified at the left and right nodes of the continuum ET-beam element, respectively. Similarly, the spectral nodal forces and moments are defined by fcL ¼ fNc1

Qc1

Mc1 gT

fcR ¼ fNc2

Qc2

Mc2 gT

ð12:14Þ

where fcL and fcR are the continuum nodal forces and moments specified at the right and left nodes of the continuum ET-beam element, respectively. The spectral nodal DOFs vector dgL can be related to the continuum nodal DOFs dcL as dgL ¼ Tg=c dcL

ð12:15Þ

where 2

Tg=c

1 60 6 60 ¼6 61 6 40 0

0 1 0 0 1 0

3 0:5H 0 7 7 1 7 7 þ 0:5H 7 7 0 5 1

ð12:16Þ

Similarly, dgR can be related to the continuum nodal DOFs dcR as dgR ¼ Tg=c dcR From Equations 12.15 and 12.17, we obtain Tg=c dgL ¼ 0 dgR

0 Tg=c

ð12:17Þ

dcL dcR

ð12:18Þ


360

The spectral nodal forces and moments of the continuum ET-beam element can be related to the spectral nodal forces and moments of the RLC as fcL ¼ TTg=c fgL

ð12:19Þ

fcR ¼ TTg=c fgR

ð12:20Þ

and

From Equations 12.19 and 12.20, we obtain " T Tg=c fcL ¼ fcR 0

0 TTg=c

#

fgL fgR

ð12:21Þ

By applying Equations 12.18 and 12.21 to Equation 12.11, we can obtain dcL ScLL ðvÞ ScLR ðvÞ fcL ¼ ScRL ðvÞ ScRR ðvÞ dcR fcR

ð12:22Þ

where

ScLL ðvÞ ScRL ðvÞ

" T Tg=c ScLR ðvÞ ¼ ScRR ðvÞ 0

0 TTg=c

#

SgLL ðvÞ SgRL ðvÞ

SgLR ðvÞ SgRR ðvÞ

Tg=c 0

0 Tg=c

ð12:23Þ

12.1.1.5 Transfer Matrix for the RLC in Continuum Nodal DOFs The sign convention defined for the transfer matrix is the same as that used in the strength of materials and it is shown in Figure 12.3. Comparing the sign convention used for the spectral ET-beam element with that for the transfer matrix, we find that only fcL is negative to each other. Thus, by changing the sign of fcL only, Equation 12.22 can be written in terms of the continuum nodal DOFs and forces defined for the transfer matrix as dcL ScLL ðvÞ ScLR ðvÞ fcL ¼ ð12:24Þ ScRL ðvÞ ScRR ðvÞ dcR þ fcR Wc1

Wc2

Uc1

Uc2

Qc1

Qc2

Nc1

Nc2 Mc1

Figure 12.3

Mc2

Sign convention for the transfer matrix


361

Equation 12.24 is the force-to-displacement relationship and it can be transformed into the input-to-output relationship as dcR dcL ¼ TRLC ðvÞ ð12:25Þ fcR fcL where

1 ScLL ScLR TRLC ðvÞ ¼ 1 ScRL ScRR ScLR ScLL

1 ScLR 1 ScRR ScLR

ð12:26Þ

TRLC(v) is the transfer matrix for the RLC in the continuum nodal DOFs.

12.1.2 Transfer Matrix for an ET-Beam Element The ET-beam model is a generalization of the classical Timoshenko-beam model to take account of the couplings between the axial, shear, and bending deformations. The dynamics of the ET-beam is represented by the equations of motion 8 9 2 9 8 9 38 N= rA 0 rR < q2 u=qt2 = < 0 = < q Q ¼ 4 0 rA 0 5 q2 w=qt2 0 ð12:27Þ : 2 ; : ; qx : ; M rR 0 rI Q q u=qt2 in conjunction with the force–displacement relationships 9 8 9 2 38 EA C1 C2 < qu=qx = 0 0 0 > > > > > > > 7 6 > = < Ku1 > 6 0 1=a1 0 0 7 7 6 ¼ 6 ð14:19Þ 7Fb > > 6 0 1=a6 0 7 > > > Kw2 > 5 4 0 > > > > ; : 0 0 0 1=a6 Ku2 From the second row of Equation 14.17, we obtain 1 ðFe Sem Dm Þ Du ¼ Seu

ð14:20Þ

By substituting Equation 14.20 into the first row of Equation 14.17, the unknown boundary force Fb can be obtained in terms of the known quantities Dm and Fe as follows: 1 1 Sem ÞDm þ Sbu Seu Fe Fb ¼ ðSbm Sbu Seu

ð14:21Þ

Finally, the unknown stiffnesses of the boundary springs can be computed by substituting Equation 14.21 into Equation 14.19.


400

Example 14.1 An experiment is conducted for a one-end supported aluminum beam as shown in Figure 14.2. The exciting force at node 2 and the accelerations measured at nodes 2 and 3 are transformed into the frequency domain by using the fast Fourier transform (FFT) algorithm and they are used to compute the transverse inertance at node 1 and the rotational inertances at nodes 1,2, and 3 by using Equation 14.10. Figure 14.4 shows the measured transverse inertances (FRFs) at nodes 2 and 3, while Figure 14.5 shows the predicted rotational inertances (FRFs) at nodes 2 and 3. The equivalent boundary springs are identified by using Equation 14.9 and they are displayed in Figure 14.6. The identified equivalent boundary springs have frequencydependent complex stiffnesses.

a (dB re 1 ms–2/N)

80

40

0 Node 2 Node 3 0

0.2

0.4

0.6

0.8

Frequency (kHz)

Figure 14.4

Measured transverse FRFs

Figure 14.5

Predicted rotational FRFs

1

1

0.5

Real Imaginary

0 0

0.2

0.4

0.6

0.8

401

K (dB re 1 Nm/rad)

KW (x10–2 dB re 1 N/m)

Identification of Structural Boundaries and Joints

1

Real Imaginary

80 40 0 –40 0

0

0.2

Figure 14.6

0.4

0.6

0.8

1

Frequency (kHz)

Frequency (kHz)

Predicted boundary spring stiffness

Figure 14.7 compares the transverse inertance measured at node 3 with the results predicted by using identified boundary conditions and ideal boundary conditions. When identified boundary conditions are used, the predicted transverse inertance is shown to be almost identical with the measured data. However, when ideal boundary conditions are used instead, the predicted transverse inertance deviates significantly from the measured data, especially at high frequency. Similarly, Figure 14.8 compares the measured time history of the acceleration with the results predicted by using identified boundary conditions and ideal boundary conditions. The time history predicted by using identified boundary conditions is shown to be very close to the measured data. A small discrepancy is observed with the lapse of time, which seems to be in part due to the inherent numerical errors or the lack of the higher frequency spectral components beyond the Nyquist frequency. In contrast to the prediction by using identified boundary conditions, Figure 14.8 certainly shows that a fairly large discrepancy exists between the measure data and the prediction when using ideal boundary conditions.


80 60 40 20

Measured Predicted : Identified BCs Predicted : Ideal BCs

0 0

Figure 14.7

0.2

0.4 0.6 Frequency (kHz)

0.8

1

Measured and predicted transverse FRFs at node 3


Acceleration (x10–2 m/s2)

402


4 2 0 –2 –4 0

0.025

0.05

Time (s)

Measured and predicted time responses at node 3

Figure 14.8

Example 14.2 An experiment is also conducted for a two-ends supported aluminum beam as shown in Figure 14.3. The transverse inertances are measured at nodes 2, 3, 4, and 5 and only the transverse inertances at nodes 2 and 4 are displayed in Figure 14.9 for brevity. By using the measured transverse inertances, the transverse inertances at nodes 1 and 6 and the rotational inertances at nodes 1 through 6 are predicted from Equation 14.20 and only two rotational inertances at nodes 2 and 4 are displayed in Figure 14.10. Similarly, the transverse and rotational boundary spring stiffness predicted at node 1 are displayed in Figure 14.11. The transverse inertance and time history measured at node 4 are compared with those predicted by using identified boundary conditions and ideal boundary conditions in Figures 14.12 and 14.13, respectively. The dynamic responses predicted by using identified boundary conditions are shown to be almost identical with the measured data. This may imply the importance of the use of accurately identified non-ideal boundary conditions.

a (dB re 1 ms–2 /N)

80

40

0 Node 2 Node 4 0

0.2

0.4

0.6

0.8

Frequency (kHz)

Figure 14.9


1


K (x10–2dB re 1 Nm/rad)


1.2

0.8

0.4 Real Imaginary

0 0

0.2

0.4

0.6

0.8

1

1.2

0.8

Real Imaginary

0.4 0

0.2

Figure 14.11

0.4

0.6

Frequency (kHz)

Frequency (kHz)

Predicted boundary spring stiffness at node 1

60



Figure 14.10

403

40 20 0 Measured Predicted : Identified BCs Predicted : Ideal BCs

–20 0

0.2

0.4

0.6

0.8

1

Frequency (kHz)

Figure 14.12


0.8

1


Acceleration (x10–2 m/s2)

404


1.5 1 0.5 0 –0.5 –1 –1.5 0

0.025

0.05

Time (s)

Figure 14.13


14.2 Identification of Joints A cantilevered beam structure shown in Figure 14.14 is considered as an example problem. The beam structure consists of two uniform beams that are connected to each other by using a bolt-joint at the midway point. To apply the spectral element analysis technique to the vibration analysis of the beam structure, two uniform beams are represented by the spectral Timoshenko beam (T-beam) element models that were introduced in Chapter 3. Accordingly, the joint part will be represented by the equivalent spectral element model characterized by four parameters to be determined by using experimentally measured FRFs [82].

Figure 14.14

A cantilevered beam structure with a bolt-joint

14.2.1 Spectral T-Beam Element Model for Uniform Beam Parts The uniform beam parts to the left and right of the joint part will be considered as the uniform T-beam elements and they will be represented by the spectral T-beam elements. For a uniform T-beam element, the spectral element equation is given by ST ðvÞ d ¼ f

ð14:22Þ

where ST(v) ¼ [sij(v)] is the 4-by-4 spectral element matrix given by Equation 3.86, d is the spectral nodal DOFs vector and f is the associated spectral nodal forces and moments vector. Once the joint part has been represented by equivalent spectral element model, which will be detailed in the next section, it can be assembled with two spectral T-beam elements to form a global dynamic stiffness equation as an example beam structure.


405

14.2.2 Equivalent Spectral Element Model of the Joint Part Consider an example joint part as shown in Figure 14.15(a). The joint part will be represented by an equivalent four-parameter model as shown in Figure 14.15(b). The four-parameter model consists of a lumped mass M, a lumped mass moment of inertia I, an effective extensional spring Kw, and an effective torsional spring Ku. The inertia parameters M and I are connected to the spring parameters Kw and Ku in series. The identification of the joint part will be completed by determining four parameters M, I, Kw, and Ku by using experimentally measured vibration responses, which is detailed in the following.

Figure 14.15

A joint part and its equivalent four-parameter model

14.2.2.1 Transfer Matrix for the Joint Model To derive the transfer matrix of the joint model, we use the sign convention defined in Figure 14.15(a), which is commonly adopted in the transfer matrix method [138]. From the kinematics continuity and dynamic equilibrium of the mass-system, the transfer matrix equation for the mass-system can be derived as 9 2 9 8 38 Q1 > QJ > 1 Mv2 0 0 > > > > > > = 6 = < < W1 WJ 0 1 0 0 7 6 7 ¼4 ð14:23Þ 2 5 M > 0 M > 0 1 Iv > > > > ; ; : 1> : J> Q1 QJ 0 0 0 1 where QJ, WJ, MJ, and QJ are the quantities at the junction “J” between the inertia-system and the spring-system, as shown in Figure 14.15(b). Similarly, the transfer matrix equation for the spring-system can be derived as 9 2 9 8 38 1 0 0 0 > QJ > Q2 > > > > > > = 6 = < < 1=Kw 1 WJ 0 07 7 W2 ¼6 ð14:24Þ 0 1 0 5> M > 4 0 M > > > > ; ; : J> : 2> 0 0 1=Ku 1 QJ Q2 By combining Equations 14.23 and 14.24, we can obtain a transfer matrix equation that relates the nodal values at the left of the joint part (node 1) to those at the right of the joint part (node 2) as follows: 9 2 9 8 38 Q2 > Q1 > 1 Mv2 0 0 > > > > > > = 6 < 7< W1 = W2 1=Kw 1 Mv2 =Kw 0 0 6 7 ¼ ð14:25Þ M > 4 0 M > 0 1 Iv2 5> > > > ; ; : 2> : 1> 2 Q2 Q1 0 0 1=Ku 1 Iv =Ku


406

or 9 2 8 W2 > 1 Mv2 =Kw > > > > = 6 0 2 6 ¼6 2 > > 4 Q Mv 2 > > > > ; : M2 0

0

1=Kw

1 Iv2 =Ku 0

0 1

Iv2

0

9 38 W1 > > > > > > = < 1=Ku 7 7 Q1 7 0 5> > > Q1 > > > ; : M1 1 0

ð14:26Þ

14.2.2.2 Equivalent Spectral Element Model for the Joint Model The input–output relationship of Equation 14.26 can be written in the force-displacement relationship as 9 8 9 2 38 W1 > > Q 1 > Kw þ Mv2 0 Kw 0 > > > > > > > = > = < <M > 6 0 Ku þ Iv2 0 Ku 7 1 6 7 Q1 ¼ ð14:27Þ 6 7 > 4 W2 > Q2 > Kw 0 Kw 0 5> > > > > > > > > ; : ; : Q2 M2 0 Ku 0 Ku In Equation 14.27, the sign convention defined for the transfer matrix is used for the nodal DOFs, forces, and moments. As shown in Figure 14.16, the sign conventions of Q1 and M1 defined for the transfer matrix are opposite to those defined for the spectral element matrix. Thus, by simply changing the signs of Q1 and M1 in Equation 14.27, we can obtain the spectral element equation for the joint part in the form of

Figure 14.16

Sign convention defined for: (a) transfer matrix; (b) spectral element matrix

SJ ðvÞ dJ ¼ fJ where

2 6 6 SJ ðvÞ ¼ 6 4

ð14:28Þ

Kw Mv2

0

Kw

0 Kw

Ku Iv2 0

0 Kw

0

Ku

0

0

3

Ku 7 7 7 0 5

ð14:29Þ

Ku

and dJ ¼fW1 Q1 W2 Q2 gT fJ ¼fQ1 M1 Q2 M2 gT

ð14:30Þ


407

The frequency-dependent symmetric matrix SJ(v) is the spectral element matrix for the joint part and it contains four unknown parameters M, I, Kw, and Ku. The four parameters will be determined from experimentally measured FRFs.

14.2.3 Determination of Joint Parameters A uniform structure member can be represented by one spectral element, regardless of its length, without degrading the solution accuracy. However, if any discontinuities exist in the structural or material properties, the structure member needs to be discretized into multiple spectral elements. The present example beam structure contains a bolt-joint and it is excited by a force applied at node 1 as shown in Figure 14.17. Thus, we perfom a first trial to discretize the beam structure into four segments: three T-beam elements and a joint element. In Figure 14.17, three T-beam

Figure 14.17

Spectral element modeling for a cantilevered beam structure with a bolt-joint

elements are denoted by [1], [2], [3] and the joint element by [J]. As each node of the spectral T-beam element has two nodal DOFs, there are a total of eight nodal DOFs: four transverse nodal DOFs (W1, W2, W3, and W4) and four rotational nodal DOFs (Q1,Q2,Q3, and Q4). Assembling three spectral T-beam elements and the spectral joint element and then imposing the clamped boundary conditions at the root of the beam structure, we can obtain an 8-by-8 global dynamic stiffness matrix equation in the form of SðvÞ d ¼ f

ð14:31Þ

where d ¼fW1

Q1

f ¼fFðvÞ 0

W2

Q2

W3

0

0 0

0

Q3 0

W4 0gT

Q4 gT

ð14:32Þ

where F(v) represents the spectral components of the excitation force. As it is somewhat easier to measure the transverse displacements than to measure the rotations, we may prefer to measure some or all of transverse DOFs experimentally. For the present example problem, we assume that all transverse DOFs (W1, W2, W3, and W4) are measured experimentally and grouped into the measured DOFs. Accordingly, the remaining rotational DOFs (Q1,Q2,Q3, and Q4) will be grouped into the unmeasured DOFs. In this case, there are a total of eight unknowns including four joint parameters (M, I, Kw, and Ku) and four unmeasured DOFs (Q1,Q2,Q3, and Q4). As the number of unknowns is equal to the number of equations provided by Equation 14.31, the first trial of structural discretization is successful


408

for the formulation of a well-posed inverse problem. However, if for any reason there are more unknowns, we may need another trial to discretize the beam structure into more segments so that more equations can be acquired to obtain a well-posed inverse problem. For the present example problem, assuming that all transverse DOFs are measured, the first trial to discretize the beam structure into four segments as shown in Figure 14.16 suffices to obtain a well-posed inverse problem. For short, we represent the measured DOFs by dm and the unmeasured DOFs by du as dm ¼ fW1

W2

W3

W 4 gT

du ¼ fQ1

Q2

Q3

Q4 gT

ð14:33Þ

By rearranging the nodal DOFs and forces, Equation 14.31 can be written in a partitioned form as dm 0 S0m S0u ¼ ð14:34Þ Sem Seu du fe where 0 is the zero vector and fe ¼ fFðvÞ 0 and

2

s243

0

Kw

s311 þ Kw

0

0

s321

s232

s234

0

s244 þ ðKu Iv2 Þ

Ku

0

s312

Ku

s322 þ Ku 3

0 s133 þ s211

s213

0

s223

0

0

s331

7 0 7 7 7 s333 5

0

0

s341

s343

s134 þ s212

s214

0

0

s224

0

0

s332

0

s342

6 1 6 s43 þ s221 Sem ¼6 6 0 4 2

ð14:35Þ

Kw

6 2 6 s42 S0u ¼6 6 4 0 2

0gT

s233 þ ðKw Mv2 Þ

s231

6 2 6 s41 S0m ¼6 6 4 0 2

0

6 1 6 s þ s2 22 6 44 Seu ¼6 6 0 4 0

0

0

3

7 0 7 7 7 s313 5

0

s323 3

7 0 7 7 7 s314 5

ð14:36Þ

s324

3

7 0 7 7 7 3 7 s34 5 s344

The superscripts 1, 2 and 3 in Equation 14.36 denote the finite element number defined in Figure 14.17.


409

By using the transverse and rotational inertances defined by Equation 14.6, we can write Equation 14.34 as S0m S0u Dm 0 ¼ ð14:37Þ Sem Seu Du Fe where v2 dm ¼fa1 a2 a3 a4 gT F v2 Du ¼ du ¼fa1 a2 a3 a4 gT F v2 T Fe ¼ fe ¼fv2 0 0 0g F

Dm ¼

ð14:38Þ

From the second row of Equation 14.37, the unmeasured nodal inertances Du can be obtained as 1 ðFe Sem Dm Þ Du ¼ Seu

ð14:39Þ

Equation 14.39 shows that unmeasured nodal inertances Du can be predicted by using the excitation force Fe and measured nodal inertances Dm. From the first row of Equation 14.37, the joint parameters can be explicitly expressed in terms of the measured and unmeasured inertances as follows: 1 ðs3 a3 þ s312 a3 þ s313 a3 þ s314 a4 Þ a2 a3 11 1 ðs3 a3 þ s322 a3 þ s323 a4 þ s324 a4 Þ KQ ¼ a2 a3 21 1 Kw ða2 a3 Þ þ s231 a1 þ s232 a1 þ s233 a2 þ s234 a2 M¼ 2 a2 v 1 I¼ Ku ða2 a3 Þ þ s241 a1 þ s243 a2 þ s242 a1 þ s244 a2 2 a2 v

KW ¼

ð14:40Þ

The unmeasured inertances (a1, a2, a3, a4) are first predicted from Equation 14.39 and the results are then substituted into Equation 14.40 to obtain four unknown joint parameters. Once the joint part is identified in terms of four parameters, the vibration analysis can be readily accomplished for the beam structure by using the standard spectral element analysis procedure. In above discussion, the ideal “clamped” boundary condition is applied to the root of the example beam structure. However, as in most practical situations, if the boundaries of a structural system cannot be represented by using such ideal boundary conditions, the present method of joint identification can be applied in conjunction with the method of boundary conditions identification discussed in the previous section, in order to identify the non-ideal boundary conditions simultaneously.

Example 14.3 Experiments are conducted for an aluminum beam structure with a bolt-joint as shown in Figure 14.18. Two uniform beams of equal length are connected at the mid-point by a bolt-joint.


410 Force Transducer

Charge Amplifier

1

2

3

4

Signal Analyzer

Accelerometer Charge Amplifier

Figure 14.18

Schematic diagram of the experimental setup

The transverse inertances are measured at nodes 1–4 and, among them, the values at nodes 2 and 4 are shown in Figure 14.19. Figure 14.20 shows the rotational inertances at nodes 2 and 4 predicted by using measured inertances. Figures 14.21 and 14.22 show the predicted inertia properties and spring stiffness of the joint model, respectively. The transverse inertance at node

a (x10–2 dB re 1 ms–2/N)

1

0.8

0.6

0.4

0.2

Node 2 Node 4 0

0.2

0.4 0.6 Frequency (kHz)

0.8

Figure 14.19


Figure 14.20


1


411 1

M I

0.5

0.5

0

0

–0.5

–0.5

–1 0

0.2

0.4

0.6

0.8

1

I (x10–2 dB re 1 kgm2)

M (x10–2 dB re 1 kg)

1

–1

Frequency (kHz)

Predicted inertia properties of the joint

1.8

1.8

1.2

1.2

0.6

0.6

KW 0

0

0.2

0.4

0.6

0.8

1

(x10–2 dB re 1 Nm/rad)


Figure 14.21

0

Frequency (kHz)

Figure 14.22

Predicted spring stiffness of the joint

a (x10–2 dB re 1 ms–2/N)

1

0.6

Measured Predicted : Identified Joint 0.2 0

0.2

0.4

0.6

0.8

1

Frequency (kHz)

Figure 14.23



412

Acceleration (m/s2)

4

Measured Predicted : Identified Joint

–2 0 –2 –4 0

0.025

0.05

Time (s)

Figure 14.24


4 (free-end) is predicted by using identified joint parameters and it is compared with the measured data in Figure 14.23. It is shown that the predicted and measured inertances are in good agreement with each other. Similarly, the time history of the transverse acceleration at node 4 is also predicted by using identified joint parameters and the result is compared with the measured data in Figure 14.24. The predicted and measured time histories are shown to be very close to each other, but with a small discrepancy. As the predicted time history is computed from the predicted FRF by using the inverse-FFTalgorithm, the small discrepancy may, in part, be due to neglecting the high frequency spectral components beyond the Nyquist frequency and also in part due to the inherent computation errors.

15 Identification of Structural Damage The structural damage generated within a structure may lead to changes in the dynamic responses and characteristics of the structure, such as the natural frequency, mode shape, and the modal damping, which, in turn, can be used to detect, locate, and quantify the damage. In the literature, there are a number of structural damage identification methods (SDIMs), which can be classified into several categories depending on the types of experimental data used in the methods. The experimental data used in the literature include the modal data, strain energy, transfer function parameters, flexibility matrix, residual forces, wave characteristics, mechanical impedances, frequency response function (FRF) data, and so forth [2, 35, 37, 46, 134, 145, 154, 170]. As the FRF data are measured directly from the structures, they will be free from any modal extraction errors. Furthermore, the FRF data can provide much more damage information in a desired frequency range when compared with the modal data, because the modal data are extracted mainly from a very limited number of FRF data around the resonance [160]. Motivated by the advantages of the FRF data over the modal data, this chapter introduces a spectral elment method (SEM) based SDIM in which the FRF data are used for structural damage identification. For the inverse problem, how to reduce the size of the problem has been an important research issue over a long period of time. Thus, a domain-reduction method is also introduced as a potential candidate approach.

15.1 Spectral Element Modeling of a Damaged Structure 15.1.1 Assembly of Spectral Elements The spectral element equation for an intact structure is given in the form sk ðvÞdk ¼ fk ðvÞ

ð15:1Þ

where sk, dk, and fk are the spectral element matrix, the spectral nodal degrees-of-freedom (DOFs) vector and the spectral nodal forces vector, respectively, for the kth element of the intact structure. When the structure has been damaged, but is still subjected to the same external forces, the spectral element equation is given in the form sk ðvÞdk ¼ fk ðvÞ Spectral Element Method in Structural Dynamics Usik Lee © 2009 John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82374-3

ð15:2Þ


414

where the over-bar indicates that the structure is damaged. The spectral elements represented by Equations 15.1 or 15.2 can be assembled in an analogous way to those used in the conventional finite element method (FEM) to obtain a global system equation for the intact structure as Sg ðvÞdg ¼ fg ðvÞ ðintactÞ

ð15:3Þ

where Sg ðvÞ ¼

N X k¼1

LTk sk ðvÞLk

ð15:4Þ

and for the damaged structure as Sg ðvÞdg ¼ fg ðvÞ ðdamagedÞ

ð15:5Þ

where Sg ðvÞ ¼

N X k¼1

LTk sk ðvÞLk

ð15:6Þ

The matrix Lk (k ¼ 1, 2, . . ., N) is the locator matrix, which locates the kth spectral element matrices sk and sk in the global dynamic stiffness matrices Sg(v) and Sg ðvÞ, respectively, and N is the total number of finite elements used to represent a structure. The vector fg(v) is the global spectral nodal forces vector.

15.1.2 Imposition of Boundary Conditions By applying the boundary conditions, the spectral nodal DOFs can be reduced to dg ¼ TB dB ðdamagedÞ

dg ¼ TB dB ðintactÞ;

ð15:7Þ

where dB and dB are the reduced spectral nodal DOFs vectors for the intact and damaged structures, respectively, and TB is the coordinates transformation matrix to impose the boundary conditions. By using the first part of Equation 15.7, Equation 15.3 can be transformed into the form SB ðvÞdB ¼ fB ðvÞ

ðintactÞ

ð15:8Þ

where SB ðvÞ ¼ TTB Sg ðvÞTB ;

fB ðvÞ ¼ TTB fg ðvÞ

ð15:9Þ

Similarly, by using the second part of Equation 15.7, Equation 15.5 can be transformed into the form SB ðvÞdB ¼ f ðvÞ ðdamagedÞ ð15:10Þ B

where SB ðvÞ ¼ TTB Sg ðvÞTB ;

fB ðvÞ ¼ TTB fg ðvÞ

ð15:11Þ

Identification of Structural Damage

415

15.1.3 Reordering of Spectral Nodal DOFs The spectral nodal DOFs can be divided into two groups: the first group dm contains the spectral nodal DOFs (called “master” DOFs) to be measured or to be excited by an external force and the second group ds contains the remaining spectral nodal DOFs (called “slave” DOFs). The spectral nodal DOFs of dB and dB can be reordered to form a new spectral nodal DOFs vectors as dm d d¼ ðintactÞ; d ¼ m ðdamagedÞ ð15:12Þ ds ds The spectral nodal DOFs vectors dB and dB can be related to the reordered spectral nodal DOFs by introducing a transformation matrix TO, as follows: vectors d and d, dB ¼ T O d

dB ¼ TO d ðdamagedÞ

ðintactÞ;

ð15:13Þ

By using the first part of Equation 15.13, Equation 15.8 can be transformed into the form SðvÞd ¼ fðvÞ ðintactÞ

ð15:14Þ

SðvÞ ¼ TTO SB ðvÞTO ¼ TTBO Sg ðvÞTBO

ð15:15Þ

fðvÞ ¼ TTO fB ðvÞ ¼ TTBO fg ðvÞ

ð15:16Þ

TBO ¼ TB TO

ð15:17Þ

where

and

with the use of the definition

Similarly, by using the second part of Equation 15.13, Equation 15.10 can be transformed into the form d ¼ fðvÞ SðvÞ

ðdamagedÞ

ð15:18Þ

where SðvÞ ¼ TTO SB ðvÞTO ¼ TTBO Sg ðvÞTBO Equations 15.14 and 15.18 can be written in the partitioned forms as dm Smm ðvÞ Sms ðvÞ f ðvÞ ðintactÞ ¼ m Ssm ðvÞ Sss ðvÞ 0 ds

ð15:19Þ

ð15:20Þ

and

Smm ðvÞ Ssm ðvÞ

Sms ðvÞ Sss ðvÞ

dm ðvÞ ds ðvÞ

fm ðvÞ ¼ 0

ðdamagedÞ

ð15:21Þ

Assuming that the structure is excited by an external force of spectral magnitude F(v), the global spectral nodal forces vector fg can be written as fg ðvÞ ¼ f0

FðvÞ 0gT ¼ FðvÞd

ð15:22Þ


416

where d ¼ f0

1

0gT

By using Equation 15.22, Equation 15.16 can be written as fm ðvÞ fðvÞ ¼ ¼ TTBO fg ðvÞ ¼ FðvÞTTBO d 0

ð15:23Þ

ð15:24Þ

Introduce a transformation matrix Tf such that fm ðvÞ ¼ Tf fðvÞ

ð15:25Þ

Substitution of Equation 15.23 into Equation 15.24 gives fm ðvÞ ¼ FðvÞTf TTBO d ¼ FðvÞdm

ð15:26Þ

dm ¼ Tf TTBO d

ð15:27Þ

where

15.2 Theory of Damage Identification 15.2.1 Uniform Damage Representation As the location, geometry, and severity of the damage in a structure element are not known in advance, in most practical situations it is impossible to assign definitive representation for the stiffness at the damaged sites. Thus, as illustrated in Figure 15.1, a simple and rational approach might be to represent a damaged structure element by the elastic modulus degraded as follows: [175] ¼ Eð1 DÞ E

ð15:28Þ

is Young’s modulus at a damaged state. D where E is Young’s modulus for an intact state and E is the damage magnitude, which is assumed to be uniform over a structure element: D ¼ 0 implies the intact state, while D ¼ 1 implies complete material rupture due to damage.

E

E = E(1–D)

Damage

(a) True damage state

Figure 15.1

(b) Uniform damage representation

Uniform damage representation of the local damage in a structure element


417

Local damage may give rise to certain nonlinearities, for example, clearance or bilinear stiffness effects in systems with cracks. As early damage, long before structural failure, is mostly localized on very small spots rather than being spread wide over the structure, the nonlinearity due to such small local damage will be weak and confined to small areas surrounding the local damage. Thus, at the early stage of damage growth, the damage-induced nonlinearity may be negligible. Assume that the spectral element matrix for an intact structure element is given by s(v). The spectral element matrix for a damaged element can be obtained by simply replacing E by ¼ Eð1 DÞ as follows: E sðvÞ ¼ sðvÞjE¼E

ð15:29Þ

The perturbed spectral element matrix D s(v) can be computed from DsðvÞ ¼ sðvÞ sðvÞ

ð15:30Þ

Dsapprox ðvÞ ffi Dsðv ¼ 0Þ ¼ Dk

ð15:31Þ

We can approximate Ds(v) as

The accuracy of Dsapprox(v) may be improved by shortening the length of the spectral element and by lowering the frequencies chosen for damage identification [96].

15.2.2 Damage Identification Algorithms Equation 15.21 can be written as Smm dm þ Sms ds ¼ fm

ð15:32Þ

Ssm dm þ Sss ds ¼ 0

ð15:33Þ

ds ¼ tðvÞdm

ð15:34Þ

tðvÞ ¼ Sss 1 Ssm

ð15:35Þ

From Equation 15.33, we obtain

where

Substitution of Equations 15.26 and 15.34 into Equation 15.32 gives FðvÞdm ¼ ½Smm þ SmstðvÞdm

ð15:36Þ

We define the perturbed quantities as follows: SðvÞ DSðvÞ ¼ SðvÞ

ð15:37Þ

or

DSmm DSsm

DSms DSss

S ¼ mm Ssm

S Sms mm Ssm Sss

Sms Sss

ð15:38Þ


418

and DtðvÞ ¼ tðvÞ tðvÞ

ð15:39Þ

tðvÞ ¼ Sss1 Ssm

ð15:40Þ

where

Applying Equations 15.37 and 15.39 into Equation 15.36 yields YðD; vÞdm ¼ FðvÞdm XðvÞdm

ð15:41Þ

XðvÞ ¼ Smm þ Sms tðvÞ

ð15:42Þ

YðD; vÞ ¼ DSmm þ DSms t þ Sms Dt þ DSms Dt

ð15:43Þ

where

and D represents the damage magnitudes as D ¼ fDk ; k ¼ 1; 2; . . . ; Ng

ð15:44Þ

Define the inertance FRF as am ðvÞ ¼ v2

dm FðvÞ

ð15:45Þ

Equation 15.41 can be then be rewritten in terms of inertance FRF as YðD; vÞ am ðvÞ ¼ v2 dm þ XðvÞam ðvÞ

ð15:46Þ

The right-hand side of Equation 15.46 does not depend on the damage state because it is determined by the structural model (spectral element model) at the intact state, the inertance FRF measured at the damaged state, and the pre-specified excitation force. On the other hand, the left-hand side of Equation 15.46 does depend on the unknown damage magnitudes D. If D ¼ 0, Equation 15.46 is reduced to v2 dm þ XðvÞam ðvÞ ¼ 0

ð15:47Þ

Thus, applying Equation 15.47 to the right-hand side of Equation 15.46 gives ZðD; vÞ am ðvÞ ¼ Dam ðvÞ

ð15:48Þ

ZðD; vÞ ¼ XðvÞ 1 YðD; vÞ

ð15:49Þ

am ðvÞ am ðvÞ Dam ðvÞ ¼

ð15:50Þ

where

and


419

15.2.2.1 Linear Damage Identification Algorithm Applying Equations 15.35 and 15.40 into Equation 15.39 gives DtðvÞ ¼ tðvÞ tðvÞ ¼ Sss1 Ssm þ Sss1 Ssm

ð15:51Þ

By using Equation 15.38, we can show that Sss1 ¼ ¼

ðSss þ DSss Þ 1 ¼ ðI þ Sss1 DSss Þ 1 Sss1

¥ X p¼0

ð Sss1 DSss Þp Sss1 ffi Sss1 Sss1 DSss Sss1

ð15:52Þ

where I is the identity matrix. Substituting Equation 15.52 into Equation 15.51 gives DtðvÞ ffi ðSss1 Sss1 DSss Sss1 ÞðSsm þ DSsm Þ þ Sss1 Ssm

¼ Sss1 DSss Sss1 Ssm Sss1 DSsm þ Sss1 DSss Sss1 DSsm ffi

Sss1 DSss Sss1 Ssm

ð15:53Þ

Sss1 DSsm

Neglecting a small term DSmsDt from Equation 15.43 and then substituting Equation 15.53, Y(D, v) can be approximated as YðD; vÞ ffi DSmm þ DSms t þ Sms Dt ffi DSmm þ DSms t þ tT DSss t þ tT DSsm ¼ where

TTC ðvÞDSðvÞTC ðvÞ

ð15:54Þ

Yapprox ðD; vÞ

I I TC ðvÞ ¼ ¼ Sss1 Ssm tðvÞ

ð15:55Þ

Applying Equations 15.15 and 15.19 into Equation 15.37 gives SðvÞ ¼ TT DSg ðvÞTBO DSðvÞ ¼ SðvÞ BO

ð15:56Þ

Next, by substituting Equation 15.56 into Equation 15.54, we can obtain Yapprox ðD; vÞ ¼ TT ðvÞDSg ðvÞTðvÞ

ð15:57Þ

TðvÞ ¼ TBO TC ðvÞ

ð15:58Þ

where

By using Equations 15.4 and 15.6, we obtain DSg ðvÞ ¼ Sg ðvÞ Sg ðvÞ N N X X ¼ LTk ðsk sk ÞLk ¼ LTk Dsk ðvÞLk k¼1

ð15:59Þ

k¼1

Applying Equation 15.31 into Equation 15.59 gives DSg ðvÞ ffi

N X k¼1

LTk ðDk kk ÞLk ¼

N X k¼1

LTk kk Lk Dk

ð15:60Þ


420

By substituting Equation 15.60 into Equation 15.57, we can linearize Yapprox(D, v) with respect to the damage magnitudes D ¼ {Dk; k ¼ 1, 2, . . ., N} as ! N X T T Yapprox ðD; vÞ ffi T Lk kk Lk Dk T k¼1

N X ¼ ðTT LTk kk Lk TÞDk YLinear ðD; vÞ

ð15:61Þ

k¼1

We then replace Y(D, v) by YLinear(D, v) to write Equation 15.48 as am ðvÞ ffi Dam ðvÞ ZLinear ðvÞ

ð15:62Þ

ZLinear ðD; vÞ ¼ XðvÞ 1 YLinear ðD; vÞ N X ¼ XðvÞ 1 ðTT LTk kk Lk TÞDk

ð15:63Þ

XðvÞ ¼ Smm Sms Sss1 Ssm

ð15:64Þ

where

k¼1

and

The left-hand side of Equation 15.62 can be then be written as ZLinear ðD; vÞ am ðvÞ ¼ XðvÞ 1

N X ðTT LTk kk Lk T am ÞDk k¼1

¼ XðvÞ

1

AðvÞD

N X hk ðvÞDk

ð15:65Þ

k¼1

where D ¼ fD1 D2 D3 DN gT

ð15:66Þ

AðvÞ ¼ XðvÞ 1 HðvÞ

ð15:67Þ

and

with the use of the following definitions HðvÞ ¼ ½h1 h2 h3 hN

ð15:68Þ

am hk ðvÞ ¼ TT LTk kk Lk T

ð15:69Þ

Lastly, by using Equation 15.65, we can obtain the linear damage identification algorithm from Equation 15.62 as follows: AðvÞðMNÞ DðN1Þ ffi bðvÞðM1Þ

ð15:70Þ


421

where bðvÞðM1Þ ¼ Dam ðvÞðM1Þ

ð15:71Þ

The dimensions of the matrix A(v) are M by N and the dimensions of the vector b(v) are M by 1, where N is the number of unknown damage magnitudes, which is equal to the number of finite elements used to represent a complete structure under inspection, and M is the number of the spectral nodal DOFs to be measured for damage identification. Equations 15.70 and 15.71 show that A(v) and b(v) are determined by the spectral element model at the intact state and the damage-induced changes of inertance FRFs measured at M nodes. As explained by Equation 15.21, the spectral nodal DOFs at which neither vibration responses are measured nor excitation force is applied are considered as the “slave” nodal DOFs to be removed, while all other spectral nodal DOFs are considered as the “master” nodal DOFs to be retained. Thus, in order to excite a damaged structure, an excitation force should be applied to one of the M nodes at which the vibration responses will be measured. In general, the number of unknown damage magnitudes to be determined will be larger than the number of measured nodal DOFs (or the number of linear algebraic equations): that is, N > M. In such a situation, we may need to acquire more equations from Equation 15.70 to make the number of equations equal to or larger than the number of unknowns. Equation 15.70 is valid for any frequency. Thus, by varying the excitation frequency, we can obtain as many sets of linear algebraic equations as are needed to form a system of linear algebraic equations as follows: AðvÞ DðN1Þ ¼ bðvÞ ð15:72Þ ðQMNÞ

where

3 Aðv1 ÞðMNÞ 7 6 Aðv2 Þ 6 ðMNÞ 7 7; 6 ¼6 .. 7 5 4 . AðvQ ÞðMNÞ

ðQM1Þ

2

AðvÞ ðQMNÞ

bðvÞðQM1Þ

9 8 bðv1 ÞðM1Þ > > > > > > > = < bðv2 ÞðM1Þ > ¼ . .. > > > > > > > > ; : bðv Þ Q ðM1Þ

ð15:73Þ

where Q is the number of chosen excitation frequencies and it is so determined that the number of equations is equal to or larger than the number of unknowns: that is, (Q M) N. Solving Equation 15.72 for D is equal to estimating the damage distribution over N finite elements. This again implies identifying locations and magnitudes of multiple damages simultaneously. The matrix AðvÞ might be very sensitive to the noises in the measured FRF data and to the chosen multiple frequencies (when N > M). Thus it is recommended to check the condition number of AðvÞ during the matrix inversion process. One may use the algorithms such as the singular value decomposition (MATHLAB function “svd”) and the non-negative least squares (MATHLAB function “nnls”) to solve Equation 15.72 for D [7]. 15.2.2.2 Nonlinear Damage Identification Algorithm For the derivation of the linear damage identification algorithm, Equation 15.70 or Equation 15.72, Y(D, v) is linearized with respect to damage magnitudes D through the approximations of Equations 15.54 and 15.61. Owing to the approximations involved, the linear damage identification algorithm might be very sensitive to the chosen frequencies and unknown damage magnitudes and provide poor identification results. Thus, in this subsection,


422

we will consider two iteration methods to solve the nonlinear damage identification algorithms: one is the direct iteration method and the other is the Newton–Raphson method. Direct Iteration Method We represent Z(D, v) from Equation 15.49 as ZðD; vÞ ¼ ZLinear ðD; vÞ þ ZNonlinear ðD; vÞ

ð15:74Þ

where ZLinear(D, v) is defined by Equation 15.63. Thus, from Equation 15.74, we get ZNonlinear ðD; vÞ ¼ ZðD; vÞ ZLinear ðD; vÞ ¼ XðvÞ 1 ½YðD; vÞ YLinear ðD; vÞ

ð15:75Þ

Remember that exact definitions of Y(D, v) and YLinear(D, v) are given by Equations 15.43 and 15.61, respectively. Substituting Equation 15.74 into the exact equality, Equation 15.48, gives ZLinear ðD; vÞ am ðvÞ ¼ Dam ðvÞ ZNonlinear ðD; vÞam ðvÞ

ð15:76Þ

Applying Equations 15.65 and 15.75 to Equation 15.76 gives an iterative non-linear damage identification algorithm as AðvÞDr þ 1 ¼ bNL ðDr ; vÞ

ð15:77Þ

AðvÞ ¼ XðvÞ 1 HðvÞ

ð15:78Þ

where and

am ðvÞ bNL ðDr ; vÞ ¼ Dam ðvÞ ZNonlinear ðDr ; vÞ ¼ Dam ðvÞ þ XðvÞ 1 ½YðDr ; vÞ YLinear ðDr ; vÞam ðvÞ ¼ Dam ðvÞ þ XðvÞ

1

r

YðD ; vÞ am ðvÞ þ AðvÞD

ð15:79Þ

r

In above equations, Dr and Dr þ 1 represent the damage magnitudes computed at the rth and (r þ 1)th iterations, respectively. If Q excitation frequencies are chosen, Equation 15.77 can be written as ðQMNÞ Dr þ 1 ¼ bNL ðDr ÞðQM1Þ A ðN1Þ where

3 Aðv1 ÞðMNÞ 6 Aðv2 Þ 7 6 ðMNÞ 7 ðQMNÞ ¼ 6 7 A .. 6 7 4 5 . AðvQ ÞðMNÞ 9 8 bNL ðDr ; v1 ÞðM1Þ > > > > > > > = < bNL ðDr ; v2 ÞðM1Þ > bNL ðDr ÞðQM1Þ ¼ . .. > > > > > > > > ; : b ðDr ; v Þ NL Q ðM1Þ

ð15:80Þ

2

ð15:81Þ

The iteration of Equation 15.80 is continued until a pre-specified convergence criterion is satisfied.


423

Newton–Raphson Method Equation 15.48 can be rewritten in the form of rðD; vÞ ¼ ZðD; vÞ am ðvÞ Dam ðvÞ ¼ 0

ð15:82Þ

where r(D, v) denotes the residual. Let Dr and Dr þ 1 be the damage magnitudes computed at the rth and (r þ 1)th iterations, respectively. Applying the Taylor series expansion to r(Dr þ 1, v) about Dr yields rðDr þ 1 ; vÞ ¼ rðDr ; vÞ þ KðDr ; vÞDD þ OðDD2 Þ ¼ 0

ð15:83Þ

DD ¼ Dr þ 1 Dr

ð15:84Þ

where

and K(Dr, v) is the tangent stiffness matrix defined by ›rðD; vÞ 1 ›yðD; vÞ r KðD ; vÞ ¼ ¼ XðvÞ ›D D¼Dr ›D D¼Dr

ð15:85Þ

where yðD; vÞ ¼ YðD; vÞam ðvÞ

ð15:86Þ

Neglecting the higher order terms from Equation 15.83 gives rðDr ; vÞðM1Þ ffi KðDr ; vÞðMNÞ DDðN1Þ

ð15:87Þ

If Q excitation frequencies are chosen such that Q M ¼ N, Equation 15.87 provides rÞ rðDr ÞðQM1Þ ffi KðD ðQMNÞ DDðN1Þ

ð15:88Þ

where

rðDr ÞðQM1Þ

9 8 9 8 ZðDr ; v1 Þam ðv1 Þ Dam ðv1 Þ > rðDr ; v1 Þ > > > > > > > > > > = < rðDr ; v2 Þ = < ZðDr ; v2 Þam ðv2 Þ Dam ðv2 Þ > ¼ ¼ .. .. > > > > . . > > > > > > > ; > ; : : r r rðD ; vQ Þ ZðD ; vQ Þam ðvQ Þ Dam ðvQ Þ ðQM1Þ 2

3 KðDr ; v1 Þ 6 KðDr ; v2 Þ 7 6 7 rÞ ¼ KðD 6 7 .. ðQMNÞ 4 5 . KðDr ; vQ Þ

ð15:89Þ

ðQMNÞ

By using Equation 15.84, we can derive the nonlinear damage identification algorithm from Equation 15.88 as r Þ 1rðDr Þ Dr þ 1 ¼ Dr KðD ð15:90Þ The iteration of Equation 15.90 is continued until a pre-specified convergence criteria is satisfied or the residual r(D, v) becomes less than a certain preselected value.


424

Example 15.1 Figure 15.2 shows a cantilevered uniform beam with three local areas damaged. The beam has the length L ¼ 0.4 m, the bending stiffness EI ¼ 15 N m2, and the mass per length rA ¼ 0.3 kg/ m. Three local damaged areas of uniform magnitudes D1 ¼ 0.2, D2 ¼ 0.3 and D3 ¼ 0.4 are prespecified on the regions 7.0 cm x 7.5 cm, 19.5 cm x 20 cm, and 32.5 cm x 33 cm, respectively. An impulse load is applied at x ¼ 28 cm and the vibration responses are analytically calculated for damage identification. D3 = 0.4 D2 = 0.3 D 1 = 0.2

Impulse

ROOT

FREE END

0

10

Figure 15.2

20 X (cm)

30

40

Cantilevered uniform beam with three local damaged areas

In Figure 15.3, the vibration response (inertance FRF) at x ¼ 15 cm when the beam is at intact state is compared with that at a damaged state. It is shown that the FRF peaks are shifted to slightly lower frequencies due to the existence of the damage. That is, the natural frequencies are slightly reduced because the damage tends to degrade the Young’s modulus and the structural stiffness.

Inertance (dB re 1 ms–2/N)

120 Intact state Damaged state 80

40

0

–40

0

Figure 15.3

0.4

0.8 Frequency (kHz)

1.2

1.6

Inertance FRFs at the intact and damaged states

For a numerical test of the damage identification, the damaged beam shown in Figure 15.2 is represented by 80 finite elements of equal length. The inertance FRFs are then analytically computed at all nodes, in place of the experimental measurement. By using the analytically computed inertance FRF data at 150 Hz, the damage identification analyses are conducted


425

by using both the linear and nonlinear damage identification algorithms. As expected, Figure 15.4 shows that the nonlinear algorithm certainly provides improved results when compared with the results obtained by using the linear algorithm.

Damage magnitudes D

0.6

Linear algorithm Nonlinear algorithm 0.4

0.2

0

Figure 15.4

0

10

20 Damage location (cm)

30

40

Damage identification results from the linear and nonlinear algorithm

15.3 Domain-Reduction Method There have been continuous efforts to reduce the size of inverse problems such as the structural system identification and the structural damage identification. An approach commonly used in much of the existing literature is to reduce the number of DOFs used to formulate the inverse problems [62, 118]. As an alternative approach, we will introduce the domain-reduction method (DRM) of structural damage identification [83]. In the DRM, a three-step process is applied to reduce the initial spatial domain of damage inspection by iteratively searching out and removing damage-free zones during the damage identification analysis.

15.3.1 Domain-Reduction Method Assume that, for the damage identification analysis, a damaged structure is divided into a total of N elements and only n elements among them are damaged: that is n < N. Thus, Dn þ 1 ¼ Dn þ 2 ¼ . . . ¼ DN ¼ 0. Accordingly, the summation in Equation 15.60 can be made only for n damaged elements as follows: DSg ðvÞ ¼

N X

n X

ðn < NÞ

ð15:91Þ

Yapprox ðD; vÞ ffi YLinear ðD; vÞ ! n n X X ¼ TT LTk kk Lk Dk T ¼ ðTT LTk kk Lk TÞDk

ð15:92Þ

k¼1

LTk kk Lk Dk ¼

k¼1

LTk kk Lk Dk

Substitution of Equation 15.91 into Equation 15.57 gives

k¼1

k¼1


426

Applying Equation 15.92 to Equation 15.49, the left-hand side of Equation 15.48 can be written as LHS of Equation 15:48 ¼ ZðD; vÞam ðvÞ n X ffi XðvÞ 1 ðTT LTk kk Lk T am ÞDk k¼1

¼ XðvÞ

1

n X

ð15:93Þ

^ ^ hk ðvÞDk ¼ AðvÞ D

k¼1

where ^ ¼ fD1 D2 D3 Dn gT D

ð15:94Þ

^ ^ AðvÞ ¼ XðvÞ 1 HðvÞ

ð15:95Þ

^ HðvÞ ¼ ½h1 h2 h3 hn

ð15:96Þ

and

using the definitions

am hk ðvÞ ¼ TT LTk kk Lk T

ðk ¼ 1; 2; . . . ; nÞ

ð15:97Þ

Substituting Equation 15.93 into Equation 15.48 gives ^ ^ AðvÞ ðMnÞ Dðn1Þ ¼ bðvÞðM1Þ

ð15:98Þ

^ If Equation 15.98 represents M linear algebraic equations in n unknown damage magnitudes D. M < n, we may need to acquire more equations from Equation 15.98 by varying the excitation frequency v to obtain a square matrix equation as ^ ^ AðvÞ ðQMnÞ Dðn1Þ ¼ bðvÞðQM1Þ

ð15:99Þ

where 3 ^ 1Þ Aðv ðMnÞ 7 6^ 6 Aðv2 ÞðMnÞ 7 7 6 ^ ðvÞ ¼ A ðQMnÞ 6 7 .. 5 4 . ^ QÞ Aðv ðMnÞ 2

9 8 bðv1 ÞðM1Þ > > > > > > > = < bðv2 ÞðM1Þ > bðvÞðQM1Þ ¼ .. > > > > . > > > > ; : bðv Þ Q ðM1Þ

ð15:100Þ

and where Q is determined such that (Q M) n. Equation 15.99 shows that, by removing all damage-free elements from the spatial domain under inspection, the size of the inverse problem can be reduced significantly. In theory, such domain reduction does not degrade the accuracy of damage identification. Thus, instead of inspecting the whole initial spatial domain under investigation all of the time to search out local damage (i.e., full-domain method), it will be more efficient to keep reducing the spatial domain, iteration after iteration, by removing definite damage-free zones from the spatial domain inspected at the previous iteration (i.e., domain-reduction method, simply DRM).


427

15.3.2 Three-Step Process To begin the DRM, the locations and sizes of some damage-free zones should be known in advance. However, unfortunately this is impossible in most practical situations. Thus, we need a special technique to search out zones that are definitely damage-free during the iteration process of damage identification. Assume that we start by dividing a damaged structure (the initial spatial domain of the problem) into a finite number of elements. Conducting damage identification analysis by using Equation 15.72, we obtain the first estimation of damage in each element. Next, we increase the number of elements, followed by the damage identification analysis to get an improved second damage estimation. By repeating such damage identification analysis, we may observe that the damage magnitudes estimated in the truly damaged elements become larger and converge to true damage magnitudes as the total number of elements is increased, while the damage magnitudes estimated in the truly intact elements become smaller and converge towards zero. This observation will be natural provided the structure model and the FRF data used for damage identification are sufficiently reliable. From the above observation, the DRM can be realized by iteratively using a three-step process as illustrated in Figure 15.5. The details are as follows: a. The First Step: (Figure 15.5(a)) Divide the initial spatial domain of damage inspection into p elements and make the first damage estimation for p elements; the results are represented by Di (i ¼ 1, 2, . . ., p). b. The Second Step: (Figure 15.5(b)) Divide each element in the first step into q finer elements to produce a total of (p q) elements and then make the second damage estimation for (p q) elements; the results are represented by Dij (i ¼ 1, 2 . . ., p; j ¼ 1, 2, . . ., q). c. The Third Step: (Figure 15.5(c)) Judge whether the jth fine element in the ith element for the first step is damage-free if Dij < Di, while it could possibly be damaged if Dij > Di.

D2

D1

D3

D4

(a) First step (initial meshing): 4 elements D11

D12

D21

D22

D31

D32

D41

D42

(b) Second step (mesh refining): 8 elements

-

+

-

-

(c) Third step (judgment):

Figure 15.5

-

+

-

-

If Dij < Di, damage-free (–) If Dij ≥ Di, damaged (+)

Three-step process of the domain-reduction method: in (a), & indicates local damage

Following the judgment in the third step, we remove all elements judged to be damage-free from the initial spatial domain of inspection to consider only the remaining elements in the next iteration, which will start again from the first step of the three-step process. By applying

428


the three-step process repeatedly, all damage-free elements can be searched out and removed from the initial spatial domain to eventually leave only damaged elements: this means the identification of multiple local damage. The DRM will be valid when the above three-step (or equivalent) process can be successfully applied. The three-step process may work well when the noise in the FRF data are below a certain level, which can be met by well-prepared measurements.

16 Other Applications As discussed in Chapter 1, the spectral element method (SEM) has many advantages because it can share the meshing and assembly features with the conventional finite element method (FEM) and the solution exactness feature with the dynamic stiffness method. However, in addition to such advantages, the SEM also has some difficulties in dealing with local discontinuities of geometry or material properties and also in avoiding the FFT errors (e.g., aliasing and leakage errors) to improve the accuracy of time-domain solutions. Because of such difficulties, application of the SEM has been limited mostly to simple one-dimensional (1-D) or 1-D-like problems. Although there are continuing efforts to apply the SEM to plates and shells, most studies have been confined to very specific boundary conditions or geometries. As application of the SEM to extended areas is still in itsinfancy, efforts to overcome associated difficulties are worth undertaking. In the previous chapters, the SEM has been applied to some specific problems: dynamics of beams and plates, pipelines conveying internal fluids, axially moving structures, rotor systems, multi-layered structures, smart structures, lattice structures, blood flow analysis, and the identification of structural boundaries, joints and damage. Thus, this chapter is devoted to a brief summary of other possible SEM applications, which could not be fully covered in this book.

16.1 SEM–FEM Hybrid Method As the spectral elements and the conventional finite elements are all stiffness formulated, the same assembly technique can be applied in both SEM and FEM. The only the difference is that the conventional finite elements are formulated and assembled in the time domain, while the spectral elements are formulated in the frequency domain and assembled repeatedly at all discrete frequencies. The spectral element model represented by an exact dynamic stiffness matrix can be formulated when exact frequency-domain wave solutions to the governing differential equations are available in analytical forms. As the spectral element model is formulated by using exact wave solutions, it can represent the mass distribution exactly. Thus it can capture the wave characteristics within a spatial domain of the problem exactly, of course within the accuracy limit of the governing differential equations adopted to formulate the spectral element


430


model. Accordingly, a regular structure member (without discontinuities in geometry or material properties) of any length can be modeled by using a single spectral element. This will significantly reduce the total number of DOFs used in the spectral element analysis. Despite such advantages of the SEM, unfortunately it is not always possible to obtain exact wave solutions, especially for most complex multi-dimensional problems. Thus the development and application of spectral element models have been limited mostly to simple onedimensional (1-D) or one-dimension-type problems. In contrast to the spectral element formulation, the conventional finite element formulation does not require exact time-domain solutions to the governing differential equations. Instead, simple polynomials are used as the shape (interpolation) functions to formulate finite element models. Thus, at present, it will be easier and more convenient to apply the FEM to more complicated geometries and multi-dimensional problems. There are many practical problems that consist of multiple 1-D members and multiple complex 2-D or 3-D members. In such problems, it will be very effective to apply the spectral element modeling to 1-D members, and the finite element modeling to complex 2-D or 3-D members: this combination approach is thus called the “SEM–FEM hybrid method.” A typical example is the cardiovascular system, which consists of a large number of blood vessels connected at many bifurcations as shown in Figure 16.1. The blood flow characteristics through the bifurcation regions will be more complicated than those through the blood vessels. Thus, for efficient blood flow analysis, one may apply the 1-D spectral element modeling (see Chapter 13) to most of the blood vessels, while the 2-D or 3-D finite element modeling to the bifurcation regions. The SEM–FEM hybrid method can be also applied to the multiple waveguides connected by arbitrarily shaped joints (e.g., pipeline networks and large space

Figure 16.1

Application of the SEM–FEM hybrid method to the cardiovascular system

431

Other Applications

lattice structures) and to the structures containing local non-uniformities such as holes and cracks [40]. For the SEM–FEM hybrid method, the finite element models formulated in the time domain must be transformed into the frequency domain by using the discrete Fourier transform (DFT) theory. The finite element equation for a specific problem with viscous damping can be derived in a standard form as :: MdðtÞ þ Cd_ ðtÞ þ KdðtÞ ¼ fðtÞ ð16:1Þ where d ðtÞ is the nodal DOFs vector, f ðtÞ is the nodal forces vector, and M, C, and K are the finite element mass, damping, and stiffness matrices, respectively. Throughout this section, the over bars (–) are used to denote the quantities derived from the finite element formulation. Based on the DFT theory, d ðtÞ and f ðtÞ are expressed in the spectral forms as 1 1 NX dðtÞ ¼ dn eivn t ; N n¼0

1 1 NX f ðtÞ ¼ fn eivn t N n¼0

ð16:2Þ

By substituting Equation 16.2 into Equation 16.1 and omitting the subscripts n for brevity, we can obtain S ðvÞdðvÞ ¼ f ðvÞ

ð16:3Þ

S ðvÞ ¼ K þ ivC v2 M

ð16:4Þ

where

Thus, Equation 16.3 shows that the time-domain finite element Equation 16.1 can be transformed into the standard form of the frequency-domain spectral element equation SðvÞdðvÞ ¼ fðvÞ

ð16:5Þ

where d(v) are the spectral nodal DOFs, f(v) are the spectral nodal forces, and S(v) is the spectral element matrix (exact dynamic stiffness matrix). As an illustrative example, Figure 16.2(a) shows a 2-D structural joint connected to three 1-D beam members. The structural joint region is meshed with the 2-D finite elements and each 2-D finite element can be represented by the frequency-domain finite element in Equation 16.3. By assembling all 2-D finite elements in the structural joint region, we can obtain the global system equation in the form of G ðvÞdG ðvÞ ¼ fG ðvÞ S

ð16:6Þ

where the subscripts G denote the global quantities obtained by assembling all finite elements. Equation 16.6 can provide detailed local dynamic behaviors within the structural joint region. We can write Equation 16.6 in the partitioned form as fGm SGmm SGmi dGm ¼ ð16:7Þ 0 SGim SGii dGi where dGm are the spectral nodal DOFs defined at the nodes located on the interfaces between the structural joint region and the 1-D beam members, dGi are the spectral nodal DOFs defined

432


Figure 16.2 A structural joint region represented by a 2-D spectral super-element and three 1-D spectral beam elements

at the internal nodes, and fGm are the spectral nodal forces associated with dGm : they are all defined in the global coordinates. Applying the condensation procedure to Equation 16.7 without any approximation, dGm can be related to fGm as follows: S super dGm ¼ fGm

ð16:8Þ

1 S Gim super ¼ S Gmm S Gmi S S Gii

ð16:9Þ

where

Equation 16.8 implies that the 2-D structural joint region can be regarded as a single equivalent spectral element represented in terms of the spectral nodal DOFs dGm and the spectral nodal forces fGm . The equivalent spectral element is called the “spectral super-element” in the literature [40]. The remaining procedure involves assembling the spectral super-element along with its neighboring three-beam members, all represented by the 1-D spectral beam elements. As shown in Figure 16.2(b), the spectral super-element has multiple nodes across each interface, while the 1-D spectral beam element has a single node on the corresponding interface. Thus, for assembly, it is necessary to reduce the spectral nodal DOFs across each interface of the spectral super-element so as to make the spectral DOFs calculated at the center of each interface (termed the connection node) equal to the values at the same point on the interface of the

433

Other Applications

corresponding spectral beam element side. One of the easy-to-use approximate approaches is as follows. Assume that the kth interface of the 2-D spectral super-element has Nk nodes and each ^ k W ^ k gT (n ¼ 1,2, . . ., N ; k ¼ 1, 2, 3), k ¼ fU node has two (in-plane) nodal displacements d^ k n n n where the super hats (^) denote the quantities with respect to the local coordinates. The k in the local coordinates can be related to the values in the global nodal displacements d^ n coordinates by dnk ¼ T k d^nk

ð16:10Þ

where dkn ¼ are the nodal displacements in the global coordinates and Tk is the local-to-global coordinates transformation matrix. It is worth mentioning that the present approach can be readily extended to more complicated situations. Thus, to simplify the following discussion, we assume that the 1-D beam members of the illustrative example are the axially deformable Bernoulli–Euler beams. Accordingly, we define four spectral DOFs at the connection node on the kth interface of the spectral superelement side and assume that they are identical with those defined at the connection node of the kth spectral beam element side as T kn g kn W fU

^ k gT ^ Ck Q ^ Ck W d^kC ¼ fU C dCk

T ¼ fUCk WCk QkC g

ðin local coordinatesÞ ðin global coordinatesÞ

ð16:11Þ

The spectral DOFs vector d^Ck can be related to the spectral DOFs vector dCk by introducing a coordinates transformation matrix T k as follows: d^ kC ¼ T k dCk

ð16:12Þ

^ ^ ^kÞ kn Þ can be related to the spectral DOFs ðU kn ; W ^ Ck ; W ^ Ck ; Q The nodal displacements ðU C through the kinematic relationships as ^ ^k ; kn ffi U ^ Ck zkn Q U C

^ k ffi W ^ Ck W n

ð16:13Þ

or k ¼ Rk ðzk Þd^ k d^ n n n C

ð16:14Þ

where zkn

is the distance of the nth node from the kth connection node. By using Equation 16.12, Equation 16.14 can be written as k ¼ Rk T k d k d^ n n C

ð16:15Þ

Similarly, by using Equation 16.15, Equation 16.10 can be written as d kn ¼ Qkn dCk

ð16:16Þ

Qkn ¼ Tk Rkn ðzkn ÞT k

ð16:17Þ

where

434


The spectral nodal DOFs dGm of Equation 16.8 can be written as 8 9 < dGm;1 = dGm ¼ dGm;2 : ; d Gm;3

ð16:18Þ

where

dGm;k ¼

d1k dk

9 > > > > > =

.. > > . > > > : dk

> > > > > ;

8 > > > > >
n,

where G(v)

þ

is the pseudo-inverse of the non-square matrix G(v) and it is defined by GðvÞ þ ¼ ðGH GÞ 1 GH

ð16:33Þ

where the superscript H denotes the Hermitian transpose. Once f(v) is computed from the deconvolution formulation in Equations 16.31 or 16.32 by using the measured dynamic responses d(v) and the FRF matrix G(v), the force-history of the impact force can be reconstructed in the time domain by using an inverse-FFT (fast Fourier transform). The FRF matrix G(v) for an undamped or lightly damped structure tends to be illconditioned near the resonance frequencies, inducing a significant computation error during the matrix inversion process. In such instances, we can implement regulation filters, such as the truncated singular value decomposition filter and the Tikhonov filter in the above least-squares scheme, Equation 16.32, to overcome inversion instability. Further detailed discussion on this issue can be found in Ref. [114]. In addition, random noise can be added to G(v) in order to avoid the loss of information near the anti-resonance frequencies where G(v) is zero or near to zero [40]. The FRF matrix G(v) can be obtained either analytically or experimentally. The advantage of an analytically formulated FRF matrix is obvious: G(v) can be readily computed at any location on a structure. This may enhance the effectiveness of the frequency-domain forcehistory and force-location identification methods.

16.2.2 Force-Location Identification For the SEM-model based force-location identification, consider two illustrative examples as shown in Figure 16.3. One is a one-dimensional beam element that has the length L and the other one is a two-dimensional plate element that has the lengths Lx and Ly in the x-direction and y-direction, respectively. Nodes are placed at all sensor locations where the dynamic responses will be measured (i.e., the sensor nodes 1 and 2 for the beam and the sensor nodes 1–8 for the plate) and at the impact location (i.e., the internal node i for both beam and plate). By assembling two spectral beam elements (one has the length x and the other one has the length Lx x) for the complete beam of length L and similarly four spectral plate elements for the complete plate of dimensions (Lx Ly), we can obtain the global dynamic stiffness equation in a general form as _

_

_

S ðv; xÞd ðvÞ ¼ f ðvÞ _

_

ð16:34Þ

where S ðv;_xÞ is the global dynamic stiffness matrix, d ðvÞ is the global spectral nodal DOFs vector, and f ðvÞ is the global spectral nodal forces vector. The global dynamic stiffness matrix

437

Other Applications p (t) i

1

2

x L (a) 1-D structure

6

7

8

p (t ) (x,y)

Ly

4

5

i

y

1

3

2 Lx

x

(b) 2-D structure

Figure 16.3

Force identifications for 1-D and 2-D structures

_

S ðv; xÞ is the function of impact location: that is, x ¼ x for the beam and x ¼ (x, y) for the plate. The matrix in Equation 16.34 can be written in a partitioned form as "_ #( _ ) ( _ ) _ Sss ðv; xÞ Ssi ðv; xÞ f s ðvÞ ds ¼ _ ð16:35Þ _ _ _ di f i ðvÞ Sis ðv; xÞ Sii ðv; xÞ _

_

the where ds ðvÞ are the spectral nodal DOFs specified on the sensor nodes and_ d i ðvÞ are _ spectral nodal DOFs _ specified on_the internal node i. The spectral nodal forces f s ðvÞ and f i ðvÞ are associated with d s ðvÞ and d i ðvÞ, respectively. _ We set f s ðvÞ ¼ 0 because no external forces will be applied at the sensor nodes. The internal _ spectral nodal DOFs d i ðvÞ can be then condensed out from Equation 16.35 to obtain an expression as _

_

_

S * ðv; xÞds ðvÞ ¼ f * ðv; xÞ

ð16:36Þ

where _*

_

_

_1

_

S ðv; xÞ ¼ S ss ðv; xÞ S si ðv; xÞS ii ðv; xÞS is ðv; xÞ

ð16:37Þ

and _*

_

_1

_

f ðv; xÞ ¼ S si ðv; xÞS ii ðv; xÞ f i ðvÞ

ð16:38Þ

438

Spectral Element Method in Structural Dynamics _

The matrix S *_ðv; xÞ is the condensed dynamic stiffness matrix. Equation 16.38 shows that the impact force f i ðvÞ applied on the internal node i can be replaced by equivalent nodal forces _* f ðv; xÞ placed on the nodes surrounding the internal node i. To make the discussion simple, we will consider the beam shown in Figure 16.3(a) as an illustrative example. Assume that an impact _ force is applied on the internal node i in the transverse direction only. Then we can write f i ðvÞ as _ Qi 1 f i ðvÞ ¼ ¼ PðvÞ PðvÞd ð16:39Þ Mi 0 where Qi and Mi are the transverse shear forces and bending moments, respectively. P(v) are the spectral components of the impact force p(t) and d ¼ {1 0}T. Applying Equation 16.39 to Equation 16.38 and the result to Equation 16.36, we can obtain _ d s ðvÞ as _

ds ðvÞ ¼ Gðv; xÞ zðv; xÞPðvÞ

ð16:40Þ

where _

Gðv; xÞ ¼ S * ðv; xÞ 1 ½Glm ðv; xÞ _

zðv; xÞ ¼ S

_ 1 si ðv; xÞS ii ðv; xÞd

ðl; m ¼ 1; 2; 3; 4Þ

fz1 z2 z3 z4 gT

ð16:41Þ

Equation 16.40 can be written as _ d sl ðvÞ

¼

4 X

Glm ðv; xÞ zm ðv; xÞPðvÞ

ð16:42Þ

m¼1

Thus, from Equation 16.42, we can obtain _

PðvÞ ¼

4 X

d sl ðvÞ

ð16:43Þ

Glm ðv; xÞ zm ðv; xÞ

m¼1

Once the transfer functions Glm(v, x) (m ¼ 1, 2, 3, 4) are provided in analytical forms and _ the dynamic response dsl ðvÞ is measured from the lth sensor node, the impact force P(v) at a guessed impact location x can be computed from Equation 16.43. The force-history can be then reconstructed from P(v) by using the inverse-FFT. A good guess of the force-location will _ provide matching impact forces from each dynamic response dsl ðvÞ measured at the sensor nodes, while a poor guess will provide dissimilar impact forces. Thus the amount of agreement between the estimated impact forces in the time- or frequency-domain can be used as a measure of how accurate a guessed force-location is. Based on this concept, the correlation function between the estimated impact forces from each sensor pair can be used to iteratively search for an accurate force-location [116]. Although the theory from Equations 16.39–16.43 is described for the case of the example beam shown in Figure 16.3(a), we can readily generalize it for the application to other more complex structures, including the example plate as shown in Figure 16.3(b).

Other Applications

439

16.3 Other Applications Some other spectral element modeling and applications, which could not be discussed fully in the previous chapters, are listed in the following. Although not all of the related literature can be cited herein due to a lack of information, the following brief summary will be useful for readers to understand the areas where the SEM has been, or can be, applied. . . . . . . .

. . . . . . . .

Curved beams: Issa [61] Twisted beams: Banerjee [15] Helical springs: Lee and Thompson [78], Leung [107], Pearson and Wittrick [136] Beams of non-uniform cross-section: Banerjee and Williams [17], Eisenberger [43], Gopalakrishinan and Doyle [51], Huang and Liu [56] Layered solids: Al-Khoury et al. [5], Doyle [39] Sandwich structures: Benerjee [14], Howson and Zare [55], Kim and Lee [66], Lee and Kim [87, 88], Wang and Wereley [158] Plates (isotropic, orthotropic, composite, honeycomb): Berçin [22], Berçin and Langley [23], Casimir et al. [27], Kim et al. [68], Lee and Lee [92], Leung and Zhou [110], Massimo and Panos [119] Thin-walled structures: Friberg [48], Kim et al. [69], Leung and Zeng [109] Aircraft wings (flutter analysis): Banerjee et al. [21] Structures on elastic foundation: Issa [61], Williams and Kennedy [167] Functionally graded material (FGM) structures: Chakraborty and Gopalakrishinan [28] Acoustic radiation: Doyle [41] Power flow analysis: Ahmida and Arruda [3], Langley [75] Structural intensity analysis: Oya et al. [132] Continuum modeling of periodic lattice structures: Lee and Lee [90]

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Index Abel-Ruffini theorem, 79 ACLD beam, see smart structure ACM element, 205 acoustic radiation, 439 active control feedback, 293 gain, 300, 312 aircraft wing, 9, 439 aliasing, 8, 17, 20, 32, 35, 87 429 alternating frequency-time (AFT) method, 109 analogue-to-digital converter, 7, 11 analysis equation, 12 approximate dynamic shape function, 8 approximate spectral element model, 8 artificial damping, 23, 31, 34 assembly, 4, 6, 77, 82, 239, 247, 413 axial force, 9, 44, 135, 172, 179, 180, 181, 256, 295, 325, 327, 348, 361 axial vibration, 44, 60, 135, 146, 219, 221, 228, 236, 243, 246, 263, 327, 338 axially moving structures Bernoulli-Euler beam, 172 string, 9, 163 thin plate, 192 Timoshenko beam, 181 Band frequency, 17, 22 base beam, 256, 258, 267, 269, 272, 286, 293, 296, 303, 304, 305, 308, 316 beams axial-bending-shear coupled composite, 330 axially moving Bernoulli-Euler, 172 axially moving Timoshenko, 181 bending-torsion-shear coupled composite, 339

Bernoulli-Euler, 9, 46, 61, 71, 84, 87, 102, 104, 113, 116, 130, 172, 357, 433 composite laminated, 9, 319, 325, 327 curved, 9, 439 elastic-elastic two-layer, 255, 267, 288 elastic-piezoelectric two-layer, 303, 304 elastic-viscoelastic-elastic three-layer (PCLD), 273, 289 elastic-viscoelastic-piezoelectric three-layer (ACLD), 305 extended Timoshenko, 53, 57, 65, 74, 357, 361 sandwich, 9 Timoshenko, 9, 49, 63, 72, 181, 191, 319, 325, 327, 338 twisted, 439 bearing support, 219, 221, 246, 251 Bernoulli-Euler beam (B-beam), see beams bifurcation conditions, 386 bimetallic strip, 255 binary branch, 385 bisection method, 79 blood flow analysis, 373, 429 blood flow rate, 375, 379, 385, 387, 389, 390 blood pressure, 375, 389, 390 blood waves, 376, 384 diastole phase, 375 diastole pressure, 374 one-dimensional theory, 373 blood vessel bifurcated, 389, 390 finite spectral element, 376 semi-infinite spectral element, 376


450

boundary conditions equivalent boundary spring, 394 ideal, 393, 394, 401, 402, 409 identification, 393 non-ideal, 393, 394, 397, 402, 409 non-reflecting, 7 Cardiovascular disease, 373 cardiovascular system, 430 causality conditions, 27, 28 Cayley-Hamiton theorem, 69 centrifugal unbalance force, 225, 242 characteristic equation, 42, 78, 85, 95 chirp signal, 19 companion matrix method, 80 complex modulus, 258, 272, 283 complex stiffness, 400 composite aircraft wing, 319 composite laminated beams angle-ply symmetric, 338 axial-bending-shear coupled vibration, 325 bending-torsion-shear coupled vibration, 327 cross-ply antisymmetric, 338 composite laminated structure, 319 composite mechanics compliance coefficients, 320 first-order shear deformation theory, 319, 324 orthotropic lamina, 320, 321 orthotropic material, 319 principal material coordinates, 319–321 stress-strain relation, 319 theory, 319 transformed reduced stiffness, 322 condensation, 432 condensed dynamic stiffness matrix, 438 condition number, 421 connection node, 432, 433 conservative system, 59 constitutive relation, 269, 294, 356, 374 constraining layer, 255, 256, 258, 267, 269, 272, 286 continuity equation, 136, 147, 373, 385 continuous Fourier transform, see Fourier transform continuum beam element, 356 continuum element, 4 continuum method, 6 continuum modeling method, 355, 356, 364, 439 continuum nodal DOFs, 359, 360, 361 continuum nodal forces, 359

Index

convolution integral, 87, 88, 89, 90, 99, 101, 114, 435 Cooley-Tukey algorithm, 17 coordinate transformation matrix, 358, 433 correlation function, 438 coupling rigidities, 54, 325, 328 cracks, 417, 431 critical fluid velocity, 146 critical speed, 171, 178, 179, 180, 249, 250, 252 Damped normal modes, 255, 263, 279, 293, 301, 313 damping non-proportional, 91 proportional, 95 decay time, 23 deconvolution, 436 determinant search method, 79 DFT coefficients arrangement, 15 formula, 36 induced errors, 30 pair, 14, 15, 36, 92, 95 processing, 5, 14, 23, 28, 30, 61, 63, 66 theory, 6, 11, 77, 81, 82, 228, 258, 377, 431 digital signal processing, 9 direct iteration method, 109, 422 discrete convolution, 28, 34 discrete Fourier transform, see DFT discrete frequencies, 4, 5, 6 discrete sampled data, 11 discrete system, 77, 90, 105, 107 discrete time series, 13, 34 discrete times, 5 disk equations of motion, 223 thick, 226, 238, 241, 247, 251 thin, 223, 225, 237, 240, 246, 251 dispersion curve, 192, 193, 356, 380 dispersion equation, see dispersion relation dispersion relation, 8, 43, 45, 47, 50, 55, 78, 79, 122, 140, 149, 166, 175, 185, 199, 232, 260, 263, 274, 275, 279, 298, 310, 332, 342, 380, 384 dissipation coordinates, 284 distributed dynamic force, 58, 113, 114, 120, 125, 127, 128, 129 divergence fluid velocity, 145, 156 speed, 179, 180, 181, 189, 190, 191 zone, 181, 207

451

Index Do-Loop routine, 4, 83 Duhamel’s integral, see convolution integral dynamic instability, see flutter dynamic shape function, 4, 42, 43, 57, 60, 77, 78, 81, 88, 103, 108, 140, 141, 148, 150, 166, 167, 175, 176, 185, 186, 197, 200, 229, 230, 233, 260, 261, 262, 274, 277, 278, 298, 310, 334, 341, 344, 380 dynamic stiffness matrix, 6, 9, 75, 77, 83, 84, 86, 90, 103, 299, 311, 358 dynamic stiffness method (DSM), 3, 4 Eigenvalue problem linear, 75, 76, 83, 84 transcendental, 9, 76 elastic foundation, 9, 439 electrical displacement, 294 electrical field, 294 element method, 6, 44 equivalent 1-D structure representation, 125 equivalent boundary spring, 394 equivalent continuum structural properties, 362 equivalent line force, 125, 126, 127 equivalent point force, 113, 114, 115, 117 equivalent spectral element, 404, 405 equivalent spectral nodal force, see equivalent point spectral nodal force Euler identities, 12 exact dynamic stiffness matrix, 4, 6, 9, 44, 76 exponential window, 23 extended form of Hamilton’s principle, see Hamilton’s principle extended Timoshenko beam (ET-beam), see beams extension-bending-torsion coupled beam, 9 Fast Fourier transform (FFT), 5, 11, 16, 81, 87, 109, 384, 400 finite element method (FEM), 3 finite element models axial-bending-shear coupled composite beam, 336, 349 axially moving Bernoulli-Euler beam, 178, 210 axially moving string, 170, 209 axially moving thin plate, 204, 212 axially moving Timoshenko beam, 188, 210 bending-torsion-shear coupled composite beam, 346, 351

blood flow, 386, 391 disk, 246 elastic-elastic two-layer beam, 266, 288 elastic-piezoelectric two-layer beam, 303 elastic-viscoelastic-elastic three-layer (PCLD) beam, 283, 289 elastic-viscoelastic-piezoelectric three-layer (ACLD) beam, 315 pipeline conveying internal steady fluid, 144, 157 pipeline conveying internal unsteady fluid, 134, 159 rotor systems, 247 spinning Timosenko shaft, 243, 253 finite strip method, 195 flexibility matrix, 413 flow-induced vibration, 133 fluid-structure interaction, 7, 133, 373 fluid bulk modulus of elasticity, 136, 137, 145 fluids continuity equation, 136, 147 Darcy friction factor, 136 Darcy-Weisbach equation, 136 internal fluids, 429 momentum equation, 136 Reynold’s number, 136 velocity profile, 374, 380 flutter coupled-mode, 146, 208 speed, 181, 182, 208 zone, 181, 207 force-displacement relation method, 41 four-parameter model, 405 Fourier series, 8, 11, 29, 36 Fourier transform, 5, 12 frequency-dependent shape function, see dynamic shape function frequency-domain method, 5, 8, 393 frequency response function (FRF), 7, 394, 400, 404, 407, 412, 413, 418, 421, 424, 427, 428, 436 functionally graded material (FGM) structures, 439 Gaussian elimination method, 84 GHM model, 283, 284, 286, 315, 316 Gibbs phenomenon, 29 Green’s function, 435 gyroscopic effect, 235

452 Hamilton’s principle extended form of, 164, 172, 173, 183, 192, 194 heat conduction, 8 helical spring, 439 helicopter blade, 319 Identification impact force, 7, 434 joint, 404, 429 structural boundaries, 393, 429 structural damage, 413, 429 impact force identification force-history, 435, 436, 438 force-location, 435, 436, 438 impedance boundary condition, 7 in-plane displacement, 327, 330 in-plane strain, 323 in-plane tensile force, 194, 206 inertance, 396, 418, 421, 424 infinite-domain problem, 7 instant frequency, 19 interelement continuity condition, 82 interpolation function, see dynamic shape function inverse-FFT (IFFT), 6, 8, 26, 87, 91, 94, 98, 104, 107, 108, 115, 127, 429, 436, 438 inverse discrete Fourier transform (IDFT), 14, 16, 17, 18, 24 inverse Fourier transform, 5, 36 inverse problem, 7, 397, 408, 413, 425, 426 inverse quadratic interpolation method, 79 iteration method, 7, 8, 79, 106, 107, 376, 384, 422 Joint bolt, 404, 407, 409 four-parameter model, 405 structural, 7, 430, 431, 432 Kantorovich method, 195, 197, 199 Lagrange’s equations, 219, 225, 227 LAPACK, 81 lattice structure, 355, 429 law of mass conservation, 134, 136 layered solids, 439 leakage, 8, 20, 30, 32, 87, 429 least-squares scheme, 436 locator matrix, 86, 116, 126, 414 locking-free method, 7 longitudinal vibration, see axial vibration

Index

low-pass filter, 20 lumped mass method, 75 lumped parameters model, 373 Mass transport, 163, 170, 171, 172, 177, 178, 188, 189, 206 mass unbalance, 219, 223 material modulus function, 283 matrix inversion instability, 436 process, 421, 436 mechanical impedance, 413 mobility matrix, 436 modal analysis method (MAM), 4, 95, 98, 104, 117, 128, 265, 302 modal damping, 264, 281, 287, 302, 303, 314, 413 modal mass, 264, 281, 302, 314 modal spectral element equation, 204 momentum balance equation, 373 multi-layered structures elastic-elastic two-layer beam, 255 elastic-viscoelastic-elastic three-layer (PCLD) beam, 269, 308 Newton-Raphson method, 79, 422, 423 non-conservative force, 164 non-negative least squares, 421 non-uniform beam, 9 nonlinear pseudo-force, 106, 107, 108, 384, 386 nonlinear system, 104 normal mode, 255, 263, 265, 279, 286, 293, 301, 313 numerical integration method, 4, 115, 127 Nyquist condition, 18, 29, 30 Nyquist frequency, 16, 401, 412 Orthogonality property, 95, 197, 203, 255, 263, 264, 265, 266, 281, 282, 293, 301, 302, 314 PCLD beam, see multi-layered structures periodic lattice structure, 355, 439 phase speed, 180 picket-fence effect, 25, 26, 32 piezoelectric material, 293, 294 piezoelectricity, 274, 305 pipe-dynamics fluid element, 136 pipeline element, 134 steady fluid, 138 theory, 133 unsteady fluid, 146

Index plates axially moving thin, 192 Levy-type, 8, 113, 119, 120, 123, 125, 126, 195 polynomial eigenvalue problem (PEP), 80, 81 polynomial root-finding method, 79 power flow analysis, 439 principle of superposition, 8 pseudo-force method, 100 pseudo-force, 8, 100, 101, 105, 106, 107, 376, 377, 386 pseudo-impulse force, 101 pseudo-inverse, 436 pseudo-static problem, 58, 83 QR method, 81 Rayleigh-Ritz method, 195 Rayleigh-Timoshenko beam, 9 Rayleigh’s dissipation function, 221 regulation filter, 436 relaxation function, 269, 283 representative lattice cell (RLC), 356 residual forces, 413 rotor-dynamics, 225 rotor systems, 219, 239, 242, 429 Runge-Kutta method, 98, 99, 110 SAM, see spectral analysis method sandwich structure, 439 SDIM, see structural damage identification method secant method, 79 SEM-FEM hybrid method, 429 semi-infinite spectral element, 7, 376, 385 sensor node, 436, 437, 438 shear correction factor, 50, 323 shear locking, 7 shell, 429 sign count, 84, 85, 86 singular value decomposition (SVD), 81, 83, 421 smart structures elastic-piezoelectric two-layer beam, 293 elastic-viscoelastic-piezoelectric three-layer (ACLD) beam, 293, 305 spectral analysis method (SAM), 4, 6, 9, 77, 89, 98, 104, 110 spectral element equation, 44, 59, 77, 78, 87, 88, 89, 90, 103, 108, 113, 114, 123, 125, 142, 143, 151, 152, 167, 168, 176, 187, 201, 204, 228,

453 229, 231, 234, 235, 236, 237, 239, 240, 241, 260, 262, 277, 278, 298, 299, 301, 311, 313, 335, 344, 345, 381, 382, 386, 404, 406, 413 spectral element method (SEM), 5 spectral element models axial-bending-shear coupled composite beam, 330 axially moving Bernoulli-Euler beam, 174 axially moving string, 165 axially moving thin plate, 195 axially moving Timoshenko beam, 183 bending-torsion-shear coupled composite beam, 339 Bernoulli-Euler beam, 46, 61, 71, 102 blood flow, 374, 376 disk, 237 elastic-elastic two-layer beam, 258 elastic-piezoelectric two-layer beam, 297 elastic-viscoelastic-elastic three-layer (PCLD) beam, 272 elastic-viscoelastic-piezoelectric three-layer (ACLD) beam, 308 extended Timoshenko beam, 53, 65, 74 Levy-type plate, 120 nonlinear, 373 pipeline conveying internal steady fluid, 143 pipeline conveying internal unsteady fluid, 152 rod, 44, 60, 70 rotor system, 239 spinning Timosenko shaft, 228 Timoshenko beam, 49, 63, 72 spectral finite element method, see spectral element method spectral modal analysis, 255, 263, 279, 301, 313 spectral modal equation, 265, 281 282, 293, 302, 314, 315 spectral super-element, 432, 433, 434 spectral transfer matrix method (STMM), 355, 365, 366 spinning shaft, 219, 220, 223, 228, 236, 237, 239, 242, 243, 246, 249, 251 stability analysis, 179, 189, 190 state-vector equation method, 41, 68 state-vector equation, 68, 70, 71, 72, 74, 362 state-vector, 68, 362, 367, 368 static eigenvalue problem, 179, 189, 190 static instability, see divergence Stieltjes integral form, 269 string: 9, 163

454

structural damage identification method domain-reduction method, 413 full-domain method, 425, 426, 427 linear algorithm, 425 non-linear algorithm, 425 three-step process, 425, 427, 428 structural damage, 7, 413, 434 structural health monitoring, 435 structural intensity analysis, 439 structural system identification, 425 structural transfer function, 435 Sturm’s theorem, 74, 75 symbolic computation algorithm, 69 synthesis equation, 12, 14 system identification, 7 Tangent stiffness matrix, 423 thin-walled structures, 439 three-node finite element, 387 through-off element, see semi-infinite spectral element Tikhonov filter, 436 time-domain method, 4 Timoshenko beam (T-beam), see beams Timoshenko shaft (T-shaft), 220, 249 transfer function parameters, 413 transfer matrix method (TMM), 355, 365, 405, transfer matrix, 69, 71, 72, 73, 74, 366, 405, 406 Trapezoidal algorithm, 115, 127 truncated singular value decomposition filter, 436 two-element method (TEM), 82, 113, 114, 126 Unbalance mass, 223, 225 uniform damage representation, 416 unit impulse response, 27, 30, 114

Index Variational method, 41, 58, 68, 82 virtual momentum transport, 164, 173, 182 virtual work, 81, 135, 164, 173, 182, 194, 220, 224, 256, 257, 270, 295, 296, 306, 307, 325, 328 viscoelastic material (VEM), 269, 286, 305, 315, 255 viscoelasticity, 269, 305 viscous damping, 82, 91, 221 viscously damped m-DOFs system linear, 90, 98 nonlinear, 105 Wave propagation evanescent wave, 192 traveling wave, 192 wave matrix, 79, 81 waveguide, 430 wavemode, 78, 79, 81 wave propagation, 7, 10 weak form, 58, 59, 61, 62, 63, 66, 88, 103, 108, 142, 144, 151, 153, 154, 167, 170, 176, 178, 186, 187, 188, 200, 201, 204, 205, 229, 230, 234, 243, 244, 261, 262, 266, 267, 277, 283, 284, 285, 298, 303, 310, 315, 334, 336, 344, 346, 378, 381, 386 weighted-integral statement, 58, 62, 63, 66, 144, 153, 155 whirling, backward, 251 forward, 251 Wittrick-William algorithm, 9, 83, 84, 85 wraparound error, 28, 30, 32, 35 Zero-padding, 17, 25, 28, 35

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