Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes

Stress, Strain, and Structural Dynamics

The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs and/or the documentation. MATLAB is a registered trademark of The Mathworks, Inc.

Stress, Strain, and Structural Dynamics An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes

Bingen Yang University of Southern California

Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald's Road, London WC1X 8RR, UK This book is printed on acid-free paper. © Copyright © 2005, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting "Customer Support" and then "Obtaining Permissions." Library of Congress Cataloging-in-Publication Data Yang, Bingen. Stress, strain, and structural dynamics. p. cm. Includes bibliographical references. 1. Strains and stresses. 2. Structural dynamics. I. Title. TA648.3.Y36 2004 624.1'7-dc22 2004022861 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-12-787767-3 For all information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 05 06 07 08 09 10 9 8 7

6 5 4

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Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org

To my parents, My wife Haiyan, and My daughters Sonia and Tanya

This Page Intentionally Left Blank

Contents

Preface

xv

Acknowledgements Chapter

Part I

1

Introduction 1.1 The Making of This Book 1.2 How to Use This Book

Strength of Materials

xvii 1 1 4

7

Chapter 2

Static Analysis of Euler-Bernoulli Beams 2.1 Getting Started 2.2 Beam Theory 2.2.1 Static Problem 2.2.2 Solution Methods 2.2.3 Fundamentals of Euler-Bernoulli Beams 2.3 Static Analysis by the Toolbox 2.3.1 System Setup 2.3.2 Response to External Forces 2.3.3 Response to Boundary Disturbances 2.3.4 Total Response 2.3.5 Exact Analytical Solutions 2.3.6 Other Useful Functions 2.4 Moments of Inertia of Beam Cross-Section Area 2.5 Quick Solution Guide 2.6 References

9 9 11 11 16 23 25 26 28 33 34 36 40 42 46 51

Chapter 3

Static Analysis of Bars, Shafts, and Strings 3.1 Getting Started 3.2 System Description 3.3 Static Analysis by the Toolbox 3.3.1 System Setup

53 53 55 59 59

VII

3.4

3.5 3.6 3.7

60 63 65 67 67 68 74 78 83

Chapter

4

Buckling Analysis of Columns 4.1 Getting Started 4.2 Uniform Columns 4.2.1 Column Buckling Theory 4.2.2 Solution by the Toolbox 4.2.3 Eccentric Loading 4.2.4 Beam-Column Problem 4.2.5 Geometric Imperfection 4.3 Stepped and Nonuniform Columns 4.3.1 Constrained Stepped Columns 4.3.2 Nonuniform Columns 4.4 The Distributed Transfer Function Method 4.5 Quick Solution Guide 4.6 References

85 85 87 87 91 96 100 104 109 109 116 123 127 133

Chapter

5

Stress Analysis in Two-Dimensional Problems 5.1 Getting Started 5.2 Plane Stress and Strain 5.2.1 In-Plane Stresses 5.2.2 In-Plane Strains 5.2.3 Strain Rosette 5.2.4 Hooke's Law 5.2.5 Criteria of Failure 5.3 Quick Solution Guide 5.4 References

135 135 136 136 142 147 149 151 155 156

Part II

Structural Mechanics

Chapter 6

viii

3.3.2 Response to External Forces 3.3.3 Response to Boundary Disturbances 3.3.4 Total Response Stepped Bars and Shafts 3.4.1 System Description 3.4.2 Solution by the Toolbox The Distributed Transfer Function Method Quick Solution Guide References

Contents

Static Analysis of Constrained Multispan Beams 6.1 Getting Started 6.2 System Description 6.3 Solution by the Toolbox 6.3.1 System Setup 6.3.2 Response to External Forces 6.3.3 Response to Boundary Disturbances 6.3.4 Response to Support Settlement 6.3.5 Total Response 6.3.6 Influence Lines 6.4 The Distributed Transfer Function Method

157 159 159 161 166 166 171 176 178 181 183 189

6.5 6.6

Quick Solution Guide References

200

Chapter 7

Static Analysis of Plane Trusses 7.1 Getting Started 7.2 System Description 7.3 Solution by the Toolbox 7.3.1 System Setup 7.3.2 Response to External Forces 7.3.3 Support Settlement 7.3.4 Fabrication Errors and Thermal Effects 7.3.5 Total Response 7.4 The Stiffness Method 7.5 Quick Solution Guide 7.6 References

201 201 203 205 205 210 213 216 220 224 230 235

Chapter 8

Stati c Analysis of Plane Frames 8.1 Getting Started 8.2 System Description 8.3 Solution by the Toolbox 8.3.1 System Setup 8.3.2 Response to External Forces 8.3.3 Response to Settlement of Supports 8.3.4 Total Response 8.3.5 Response of Frame Members 8.4 The Distributed Transfer Function Method 8.5 Quick Solution Guide 8.6 References

237 237 240 243 243 248 252 256 260 263 268 276

Part III

Dynamics and Vibrations

Chapter

9

277

Dynamics of Particles and Rigid Bodies 9.1 Getting Started 9.2 Dynamics of Particles 9.2.1 Preliminaries 9.2.2 Kinematics 9.2.3 Kinetics of Single Particle 9.2.4 Particle Motion via Numerical Integration 9.2.5 Systems of Particles 9.2.6 Central Force Motion 9.3 Dynamics of Rigid Bodies in Plane Motion 9.3.1 Plane Motion 9.3.2 Mass Moments of Inertia 9.3.3 Kinetics 9.3.4 Simulation of Dynamic Response 9.4 Rigid Body Dynamics in Three Dimensions 9.4.1 Kinematics 9.4.2 Inertia Properties 9.4.3 Energy and Momentum

279 279 280 280 284 286 289 297 303 307 307 310 314 317 323 323 327 335

Contents

9.4.4 Equations of Motion 9.4.5 Rotation of Axisymmetric Bodies 9.5 Quick Solution Guide 9.6 References

x

336 340 349 350

Chapter 10

Vibration Analysis of One-Degree-of-Freedom Systems 10.1 Getting Started 10.2 System Description 10.3 Time Response 10.3.1 Free Response 10.3.2 Forced Response 10.3.3 Specific Forcing Functions 10.3.4 Total Response 10.3.5 Delayed Forcing Functions 10.3.6 Solution by Superposition 10.3.7 Mechanical Energy 10.3.8 Plotting Computed Response 10.4 Analytical Vibration Solutions 10.4.1 Analytical Methods 10.4.2 Solution by the Toolbox 10.5 Frequency Response 10.5.1 Harmonic Excitation 10.5.2 Base Excitation 10.5.3 Response under Rotating Unbalance 10.5.4 Vibration Isolation 10.6 Response to Periodic Excitation 10.6.1 Steady-State Solution in Fourier Series 10.6.2 Solution by the Toolbox 10.6.3 Fourier Coefficients of Certain Loads 10.7 Nonlinear Vibration 10.8 Quick Solution Guide 10.9 References

351 351 353 357 357 361 365 374 376 377 379 382 384 384 389 394 394 400 403 406 410 410 412 419 421 431 435

Chapter 11

Vibration and Control of Multiple-Degree-of-Freedom Systems 11.1 Getting Started 11.2 System Description 11.2.1 Introduction 11.2.2 Modes of Vibration 11.2.3 Solution by the Toolbox 11.3 Dynamic Response 11.3.1 Modal Analysis 11.3.2 Laplace Transform Method 11.3.3 Plotting Analytical Solutions 11.3.4 Runge-Kutta Algorithm 11.3.5 Solution by Superposition 11.3.6 Harmonic Excitation 11.4 Dynamic Vibration Absorption 11.4.1 Undamped Vibration Absorbers 11.4.2 Damped Vibration Absorbers

437 437 439 439 442 444 450 450 458 462 465 468 472 475 476 481

Contents

11.5 Transfer Function Formulation 11.5.1 Transfer Function and Green's Function 11.5.2 Open-Loop Transfer Function 11.6 Feedback Control 11.6.1 Vibration Control System 11.6.2 Position Control System 11.6.3 Solution by the Toolbox 11.7 Quick Solution Guide 11.8 References

Part IV

Structural Dynamics

487 491 497 497 501 506 515 518

519

Chapter 12

Dynamics and Control of Euler-Bernoulli Beams 12.1 Getting Started 12.2 System Description 12.2.1 Governing Equation 12.2.2 Eigenvalue Problem 12.2.3 System Setup by the Toolbox 12.2.4 Animation of Modes of Vibration 12.2.5 Distributed Transfer Function Method 12.3 Dynamic Response 12.3.1 Modal Expansion 12.3.2 Specification of Damping 12.3.3 Free Vibration 12.3.4 Forced Vibration 12.3.5 Total Response 12.3.6 Animation of Time Response 12.3.7 Frequency Response 12.4 Feedback Control 12.4.1 Control System Formulation 12.4.2 Solution by the Toolbox 12.5 Dynami cs and Control of Nonuniform Beams 12.5.1 Problem Statement 12.5.2 Rayleigh-Ritz Discretization 12.5.3 Modes of Vibration 12.5.4 Time Response 12.5.5 Control System Formulation 12.6 Quick Solution Guide 12.7 References

521 521 523 523 524 528 534 535 537 537 539 540 544 552 554 555 559 559 567 581 581 582 588 594 601 608 615

Chapter 13

Dynamic Analysis of Bars, Shafts, and Strings 13.1 Getting Started 13.2 System Description 13.2.1 Governing Equations 13.2.2 Eigenvalue Problem 13.2.3 System Setup by the Toolbox 13.2.4 Display of Modes of Vibration 13.3 Dynamic Response 13.3.1 Modal Expansion 13.3.2 Free Vibration

617 617 619 619 622 624 627 630 631 632

Contents

xi

Chapter 14

Part V

13.3.3 Forced Vibration 13.3.4 Total Response 13.3.5 Animation of Time Response 13.3.6 Frequency Response 13.4 Free Vibration of Stepped Systems 13.4.1 System Description 13.4.2 Free Vibration Analysis 13.4.3 Solution by the Toolbox 13.5 Quick Solution Guide 13.6 References

635 639 642 643 647 647 651 655 661 665

Dynamic Analysis of Constrained, Combined, and Stepped Beams 14.1 Getting Started 14.2 Constrained Beams 14.2.1 System Description 14.2.2 System Setup and Eigensolutions 14.2.3 Eigenvalue Locus 14.2.4 Frequency Response 14.2.5 Multispan Beam Structures 14.2.6 Transient Response 14.3 Combined Beams 14.3.1 System Description 14.3.2 System Setup and Eigensolutions 14.3.3 Transient Response 14.3.4 Frequency Response 14.3.5 Oscillators for Vibration Absorption 14.4 Stepped Beams 14.4.1 Free Vibration Analysis 14.4.2 Solution by the Toolbox 14.5 Quick Solution Guide 14.6 References

667 667 670 670 675 683 687 689 691 691 692 696 698 707 712 715 715 721 729 736

Two-Dimensional Elastic Continua

;er 1 5

Static Analysis of Linearly Elastic Bodies 15.1 Getting Started 15.2 Theory of Linear Elasticity 15.2.1 Stress and Strain 15.2.2 Basic Equations of Elasticity 15.2.3 Conversion of Elasticity Constants 15.2.4 Strain Energy 15.2.5 Principal of Minimum Potential Energy 15.3 Elasticity Problems in Two Dimensions 15.3.1 Plane Stress and Plane Strain 15.3.2 Governing Equations 15.3.3 Solution by Stress Function 15.3.4 Thermal Stresses 15.3.5 Elasticity Problems in Polar Coordinates 15.3.6 Stress Concentrations

737 739 739 742 742 749 753 755 755 759 759 760 761 764 765 769

15.4 Finite Element Method for 2-D Elasticity Problems 15.4.1 Finite Element Formulation 15.4.2 MATLAB Solutions in Rectangular Regions 15.4.3 MATLAB Solutions in Arbitrary-Shaped Regions 15.5 Quick Solution Guide 15.6 References

792 802 805

Chapter 16

Free Vibration of Membranes and Plates 16.1 Getting Started 16.2 Free Vibration of Membranes 16.2.1 Rectangular Membranes 16.2.2 Circular Membranes 16.3 Free Vibration of Rectangular Plates 16.3.1 Plate Theory 16.3.2 Eigenvalue Problem 16.3.3 Solution by the Toolbox 16.4 Free Vibration of Circular Plates 16.4.1 Equations in Polar Coordinates 16.4.2 Modes of Vibration 16.5 Quick Solution Guide 16.6 References

807 807 809 810 816 823 823 827 832 839 839 842 848 851

Appendix A

Commonly Used Mathematical Formulas A.l Algebraic Formulas A.2 Areas and Volumes of Common Shapes A.3 Trigonometry A.4 Hyperbolic Functions A.5 Derivatives and Integration A.6 Series Expansion A.7 Analytical Geometry A.8 Vector Analysis A.9 Matrix Theory A.10 Complex Numbers and Complex Functions A.ll Laplace Transforms A.12 Inverse Laplace Transform via Partial Fraction Expansion

853 853 855 857 860 860 862 864 868 872 878 879

Appendix B

MATLAB Basics B.l Getting Started B.2 Matrix and Vector Manipulations B.3 Graphics B.4 M-Files B.5 Control Flow B.6 Solution of Algebraic and Differential Equations B.7 Control System Toolbox

889 889 893 898 901 903 907 910

Appendix

The Distributed Transfer Function Method C.l DTFM for One-Dimensional Continua C.2 Transfer Function Synthesis of Multibody Structures

913 913 917

C

770 770 777

882

Contents

xiii

C.3 C.4

919 920

Appendix

D

Conversion of Units

923

Appendix

E

Mechanical Properties of Engineering Materials

925

Index

xiv

DTFM for Two- and Three-Dimensional Problems References

Contents

929

This book is intended to supply engineering professionals and students with a comprehensive and definitive reference to statics and dynamics of solids and structures. The book is for use as a resource and design tool in research and development, and for use as a study guide and learning aid in engineering education. The book is written to meet the needs for interactive computing in technical referencing and engineering education. These needs result from the design requirements for high-accuracy and high-performance structures, machines and devices in a variety of engineering applications, and from the trend of engineering curriculum development in response to today's environment of fast-changing technologies. Chapter 1 further explains the purpose and philosophy of this writing. A unique feature of this book is the integration of the development of principles, formulas, and solutions with user-friendly interactive computer programs, written in the powerful and popular software MATLAB. These programs produce instant engineering solutions, which cover pages of contents contained in a conventional handbook, and beyond. Different from and complementary to general-purpose numerical codes, these programs deliver analytical results pertaining to many topics covered in the book. Furthermore, with the rich resources of MATLAB, these programs allow in-depth exploration of the physics of deformation, stress and motion by analysis, simulation, graphics, and animation. This book contains five main parts: strength of materials, structural mechanics, dynamics and vibrations, structural dynamics, and two-dimensional elastic continua. Besides, the book presents feedback control of mechanical systems and flexible structures to a great extent. These subjects are covered in 15 chapters (Chapters 2 to 16), each being a self-contained package of subject review, fundamental theories, formulas, and a toolbox of MATLAB functions (computer programs) for numerical and analytical solutions. The MATLAB functions are stored in a CD-ROM that is attached to the book. In addition, Appendix A collects commonly used mathematical formulas for engineering analysis and Appendix B gives a tutorial on MATLAB-based computing and programming. There are two important points regarding this book. First, the book keeps a good balance between fundamental theory and technical computation. Such a balance, in the author's opinion, will best serve design engineers who often need a quick subject review, and instant engineering solutions at the same time. Second, the organization of the text closely follows a curriculum on solid mechanics and structural dynamics adopted by a typical engineering school. This naturally renders the book a useful study guide and learning aid for many courses.

xv

All the numerical results in the book are obtained through use of the attached MATLAB functions. All the artwork and tables in the book are created by the author; all the MATLAB functions of the book are generated by the author. For theories and formulas that are generally available, a list of references is provided at the end of each chapter. Specific formulas or results from certain resources have been duly acknowledged. Although much effort has been made to avoid errors, some of them are bound to escape detection. The author welcomes and appreciates any comments and suggestions from readers for necessary corrections and for further improvement of the quality of this work. Bingen Yang Los Angeles, California December 2004

Acknowledgements

I would like to express my appreciation to the staff at Elsevier, especially Joel Stein, Shoshanna Grossman, and Carl Soares, for their assistance, guidance, and high professional competence. Certain results and formulas presented in this book are the outcome of my previous research partially sponsored by the National Science Foundation, the US Army Research Office, and NASA's Jet Propulsion Laboratory. Finally, I wish to thank my wife Haiyan and my daughters Sonia and Tanya, for their love, dedication and encouragement. Without their endless support, this work could never have been completed.

xvii

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1

Introduction

Inside • The Making of This Book • How to Use This Book

1.1

The Making of This Book Computing is essentially important in both engineering education and engineering practice. The use of mathematical software packages can greatly enhance the learning of various topics in science and technology, and surely help increase the efficiency and accuracy of engineering designs. The recent technological advancements in computers, software, and telecommunications make it possible for individuals to enjoy an interactive and mobile computing environment in their study and work. Taking advantage of such a fast-changing environment, this book is intended to provide a new type of reference to statics and dynamics of solids and structures, with unique interactive computing capabilities. The purpose, approach, scope, and features of this writing are described under the following headlines. Who Will Find the Book Useful This book is ideal for both professionals and students dealing with aerospace, mechanical, and civil engineering, as well as naval architecture, biomechanics, robotics, and mechtronics. For engineers and specialists, the book is a valuable resource and handy design tool in research and development. For engineering students at both undergraduate and graduate levels, the book serves as a useful study guide and powerful learning aid in many courses. And for instructors, the book offers an easy and efficient approach to curriculum development and teaching innovation. Uniqueness This book differs from standard handbooks in that it integrates the development of formulas, fundamental theories, mathematical models, and solution methods with user-friendly interactive computer programs. Unlike the commonly adopted approach of "finish-the-bookfirst-and-add-software-later," the text-software integration is harmonically fabricated in the 1

book writing from scratch. This unique merger of technical referencing and interactive computing allows instant solution of a variety of engineering problems and in-depth exploration of the physics of deformation, stress, and motion by analysis, simulation, graphics, and animation. Interactive Computing with MATLAB

The computer programs for the book are written in the powerful and popular MATLAB, which is a premier software package that provides an interactive environment for technical computation. Different from many books that teach people how to use MATLAB in engineering analysis, this book shows how to obtain instant engineering solutions by hundreds of preprogrammed MATLAB functions from a CD-ROM that is attached to the book. These functions permit easy generation of data, figures, animation, and even analytical expressions, and produce results that are equivalent to the contents covered in pages of a conventional handbook and beyond. Motive for Writing

This writing is motivated by the following two needs in engineering education and technical referencing. Need for Interactive Computing Capabilities in Engineering Education The solution of a problem in an undergraduate engineering course usually requires knowledge in the following four areas: • The background material of the problem in consideration, such as strength of materials, vibrations, and structural dynamics; • Mathematical physics, including differential equations, linear algebra, and matrix theory; • Solution algorithms, which can be either analytical or numerical; and • Computer coding in programming languages like C, Fortran, and MATLAB. While computer coding is normally introduced in the freshman year, adequate knowledge in mathematical physics and solution algorithms is not available until the senior year or later. This lack of mathematical skills often limits undergraduate teaching to a few "classroom problems." Of course, commercial computer programs may be used for solution of complicated problems. The usage of those codes, however, still requires a background in mathematical physics and numerical algorithms. The current book fills this gap by offering adequate computing capabilities to many engineering courses. With this book, an undergraduate student in earlier years can solve various engineering problems without worrying about numerical algorithms. This allows the student to focus on important aspects of fundamental principles in engineering science and to explore the physical insight of practical problems. Moreover, with the interactive computing capabilities provided by this book, more advanced topics can be introduced to adapt an undergraduate or graduate curriculum to today's environment of emerging technologies. Need for Interactive Computing Capabilities in Technical Referencing A standard reference collects formulas and tables that normally cover a number of simple cases. Although general-purpose computer codes are available, they usually only deliver numerical results and are not integrated with many analytical formulas given in a standard reference. Quite often, an engineer or specialist would like to get a quick solution for verifying a design concept or a research idea. In this case, a reference with attached computer programs, which yield numerical or analytical solutions according to user-selected parameters, boundary conditions, and loads, would definitely be desirable.

2

STRESS, STRAIN, AND STRUCTURAL DYNAMICS

This book provides such needed interactive computing capabilities to technical referencing. With hundreds of preprogrammed MATLAB functions, numerical and analytical solutions of various engineering problems can be easily obtained. Besides facilitating quick concept proof in design and research, these solutions can serve as a benchmark for verification of numerical algorithms and computer codes developed by the user. Scope This book covers basic topics regarding solids and structures, including strength of materials, structural mechanics, elasticity, particle and rigid-body dynamics, vibrations, structural dynamics, and structural controls. As indicated by the table of contents, these topics are presented in five parts with a total of 15 chapters. Each chapter deals with a type of problem or a class of systems encountered in engineering. Each chapter is a self-contained package of subject review, fundamental theories, formulas, and a set (toolbox) of MATLAB functions for numerical and analytical solutions. For efficient utility of this book, no attempt has been made to include every topic in such a wide range of subjects. Instead, the following three criteria have been applied in selecting the book materials: (a) The problem in consideration is fundamentally important to engineering education and engineering practice. Examples include static analysis of Euler-Bernoulli beams (Chapter 2), stress and deformation of elastic bodies (Chapters 5 and 15), and rigid-body dynamics (Chapter 9). (b) The problem in consideration is representative of a wide class of engineering applications. Examples include columns (Chapter 4), trusses (Chapter 7), frames (Chapter 8), and multispan beam structures (Chapters 6 and 14). (c) The problem in consideration requires substantial analytical and numerical efforts for better understanding of its physics. Examples include vibration of multiple-degree-offreedom systems (Chapter 11), dynamics and control of Euler-Bernoulli beams (Chapter 12), and vibration of plates (Chapter 16). Special Features Besides its unique interactive computing capabilities, this book has several special features, some of which are not available in the existing references. New Formulas and Solutions solutions. Examples include

This book contains many new formulas and analytical

• Analytical expressions of static response of beams subject to general external loads and arbitrary boundary disturbances; • Exact static deflections and stresses of flexible frames under arbitrary external loads and support settlement; • Influence lines of statically indeterminate multispan beam structures; • Exact vibration solutions of one-degree-of-freedom systems subject to general forcing functions; • Exact expressions of normalized mode shapes (eigenfunctions) of beams, bars, shafts, and strings, under arbitrary boundary conditions; • Eigenfrequency loci of constrained beam structures; • Control system formulation and design for beams with feedback controllers; and • Exact free vibration solutions of plates with various boundary conditions. What makes these new results more useful is that they can be obtained easily through use of the attached MATLAB toolboxes.

Introduction

3

Exact Solution via the Distributed Transfer Function Method This book presents exact static and dynamic responses of beams, bars, shafts, columns, and frames, which are determined by the Distributed Transfer Function Method (DTFM). The DTFM is a closed-form analytical solution technique for modeling, analysis, and control of flexible structures. The DTFM is flexible in dealing with different geometric configurations and boundary conditions, and convenient in computer coding. The DTFM is introduced in Appendix C and its application to specific problems is given in related chapters. Instant Animation of Motion and Vibration The MATLAB toolboxes of the book have functions for animating the modes of vibration and transient response of beams, bars, shafts, constrained and combined beam structures, and plates, the motion of rigid bodies in two and three dimensions, and the response of lumped parameter systems and flexible beams under feedback control. This animation functionality, which takes advantage of the rich resources of MATLAB, helps better understand the physics of motion and vibration and makes learning of difficult subjects a fun experience. Integrated System Modeling and Controller Design Feedback control has wide applications in machines and structures. This important topic is addressed in the current book, for lumped dynamic systems (Chapter 11) and flexible beams (Chapter 12). For these systems, the book presents major steps in control system design, including system modeling, dynamic analysis, control system formulation, controller design and numerical simulation, and provides MATLAB functions for each of the steps. This integration of modeling, analysis, design, and simulation for feedback control of machines and structures is not available in any other reference on structural dynamics.

1.2

How to Use This Book Chapters 2 to 16 cover topics in strength of materials, structural mechanics, elasticity, particle and rigid-body dynamics, vibrations, structural dynamics, and structural controls. Each of the chapters has the following parts: • • • • • •

Getting started Fundamental principles, formulas, and solutions MATLAB functions Examples Quick Solution Guide References

Each chapter has a toolbox of MATLAB functions contained in the attached CD-ROM. In addition, Appendices A to E will be useful for engineering design and analyses. Some key points in using this book are given below. Getting Started

To start, find the right chapter from the Contents for the problem or system in consideration, then go to the first section of the chapter, titled Getting Started. This section tells what the chapter is about, how to install the MATLAB Toolbox, how to use the Toolbox through a tutorial example (in most chapters), and what to do next. Fundamental Principles

The fundamental principles of each subject covered are briefly reviewed. Some derivations of theories and mathematical models are provided. For detailed information on these basic issues, a list of references is given at the end of each chapter.

4

STRESS, STRAIN, AND STRUCTURAL DYNAMICS

Formulas and Solutions Formulas and solutions in a few special cases can be directly found from the text of a chapter. Formulas and solutions in general cases of geometric configurations, boundary conditions, and loadings can be obtained through use of the MATLAB toolbox for the chapter. This requires computing and programming with MATLAB. MATLAB MATLAB is a software product of The MathWorks, Inc., headquartered in Natick, Massachusetts. MATLAB is a registered trademark of The MathWorks, Inc. To obtain MATLAB, visit the company's Web site http://www.mathworks.com/. A quick tutorial and brief summary of computing and programming with MATLAB is given in Appendix B of this book, which is useful for both beginners and advanced users. MATLAB Functions in CD-ROM Attached to this book is a CD-ROM with hundreds of preprogrammed MATLAB functions. These functions form 15 toolboxes, one for each of Chapters 2 to 16. The license agreement and limited warranty about the software package is given at the end of the book. For possible updated versions of the MATLAB toolboxes contained in the CD-ROM, check the publisher's Web site. Windows "Windows" are used to summarize the purpose and utility of the MATLAB functions from the attached CD-ROM. The windows are distributed in the text flow so that they are naturally related to the formulas and solutions presented. Examples The windows are normally followed by step-by-step examples demonstrating how the MATLAB functions can be used in analysis, simulation, graphics, and animation. Furthermore, each toolbox has a function Run Ex, which, when launched, displays all the numerical examples contained in the chapter. Quick Solution Guide Each chapter has a section titled Quick Solution Guide. This section briefly describes the problem or system in consideration, lists the MATLAB functions from the toolbox with window or section numbers for easy reference, and outlines the solution procedure in the MATLAB-based computation. This section is especially convenient to those who are familiar with the material covered in the chapter and would like to engage in technical computation directly. References At the end of each chapter is a list of references for further reading. These references are mostly textbooks and monographs. Unit Conversion In all the examples, quantities are given in either the standard international system (SI) of units or nondimensional units. For conversion between SI system and the U.S. customary system, refer to Appendix D.

Introduction

5

Mathematical Formulas For convenience in engineering analyses, this book collects commonly used mathematical formulas in algebra, trigonometry, analytical geometry, calculus, vector analysis, matrix theory, complex analysis, differential equations, and Laplace transforms; see Appendix A. Mechanical Properties of Engineering Materials For convenience in engineering designs, the mechanical properties of selected engineering materials are given in Appendix E. Comments and Technical Questions For comments on this book and technical questions about the attached MATLAB toolboxes, please contact the author by the following mail and e-mail address: Professor Bingen Yang Department of Aerospace and Mechanical Engineering University of Southern California 3650 McClintock Avenue, Room 430 Los Angeles, CA 90089-1453 E-mail: [email protected]

6

STRESS, STRAIN, AND STRUCTURAL DYNAMICS

PARTI Strength of Materials

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2

Static Analysis of Euler-Bernoulli Beams

Inside • Getting Started • Beam Theory • Static Analysis by the Toolbox • Moment of Inertia of Beam Cross-Section Area • Quick Solution Guide • References

2.1

Getting Started What Is in This Chapter This chapter is a package of subject review, fundamental theories, formulas, solution methods, and a set (toolbox) of MATLAB functions for static analysis of Euler-Bernoulli beams. System Requirements for the MATLAB Toolbox • PC with Win 98SE/NT/2000 and XP or Mac with OS 9.x and up • The software MATLAB (version 5.x and up) installed on computer Software Installation and Test (i) Drag the Toolbox folder from the CD onto a hard disk of your computer; (ii) Launch MATLAB and set a path to the Toolbox folder on your hard disk;1 and (iii) Test the toolbox by typing TBdemo in the MATLAB command window, which will launch a demo program showing how the Toolbox works. The demo ends with a message: "The Toolbox works properly." At this stage, the Toolbox is properly installed, and it is ready for use. If the M-files of the toolbox are put in the MATLAB work folder, there is no need to set a path.

9

Quick Tutorial To show how to use the Toolbox, consider a simply-supported beam in Fig. 2.1.1, which has length L = 1 and bending stiffness EI = 25, and is subject to a pointwise forcefy = 1.2 at its midpoint. In the MATLAB command window, type: » EI = 2 5 ; L = 1; BCJpec = [1 1 ] ; » setbeam(EI, L, BC_Spec) » Load_Spec = [0 0.5 1.2]; » y = beamf(Load_Spec); » plotbeam(y) where » i s the prompt in the MATLAB command window. This yields the spatial distributions of the displacement (transverse deflection), rotation, bending moment, and shear force of the beam as plotted in Fig. 2.1.2, and the maximum response and the reactions as shown below.

t /o 3& EI

FIGURE 2.1.1

A simply-supported beam under a pointwise force.

Maximum Beam Response in Absolute Value Max Max Max Max

displacement = 0.001, location x = 0.5 rotation (degrees) = 0.17189, location x = 0 bending moment = -0.3, location x = 0.5 shear force = -0.6, location x = 0

x

0

i g ' 3 displacement, w(x)

0.5

x10" 3

1

0

bending moment, M(x)

'

-0.3 -0.4

FIGURE 2.1.2

10

0.5 shear force, Q(x)

71 --/--\

-0.1 -0.2

slope, theta(x)

---/-

0.5

1

A.

J

:

1

0.5

Beam response distribution.

STRESS, STRAIN, AND STRUCTURAL DYNAMICS

0 Qg

.1 0.5

1

Reactions at Two Ends of the Beam Sign convention for support reactions: * Positive moment Mc: counterclockwise * Positive force Re: upward At left boundary (x = 0 ) : Mc = 0, Re = -0.6 At right boundary (x = L ) : Mc = 0, Re = -0.6 Also, the command »

mathf(Lbad_Spec)

produces the analytical expressions of the beam response as follows Beam response f o r 0 H

L

-[H(x)][Nb]eW -S\

x