Theoryof Vibration with Applications Fourth Edition William T. Thomson, Professor Emeritus Department of Mechanical and Environmental Engineering University of California Santa Barbara, Ca/ifornia [a-n[ SJ'RINGER-SCIENCE+ BUSINTISS MEDIA, BV. First edition 1972 Second edition 1981 Third edition 1988 Fourth edition 1993 Reprinted in paperback 1996 for distribution outside the USA and Canada © 1972, 1981, 1988 and 1993 by Springer Science+ Business Media Dordrecht Originally published by Chapman & Hall in 1993 ISBN 978-0-412-78390-6 ISBN 978-1-4899-6872-2 (eBook) DOI 10.1007/978-1-4899-6872-2 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page.
The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. IBM is a registered trademark of International Business Machines Corporation Contents PREFACE Ix THE SI SYSTEM OF UNITS 1 1 OSCILLATORY MOTION 5 1.3 Vibration Terminology 12 2 FREE VIBRATION 17 2.2 Equations of Motion: Natural Frequency 18 2.4 Rayleigh Method: Effective Mass 24 2.5 Principle of Virtual Work 26 2.6 Viscously Damped Free Vibration 28 2.8 Coulomb Damping 35 111 Iv Contents 3 HARMONICALL Y EXCITED VIBRATION 51 3.1 Forced Harmonie Vibration 51 3.4 Whirling of Rotating Shafts 61 3.7 Energy Dissipated by Damping 70 3.8 Equivalent Viscous Damping 73 3.10 Sharpness of Resonanee 77 3.11 Vibration-Measuring Instruments 78 4 TRANSIENT VIBRATION 92 4.3 Laplaee Transform Formulation 97 4.4 Pulse Excitation and Rise Time 100 4.5 Shock Response Speetrum 103 4.7 Finite Differenee Numerieal Computation 108 4.8 Runge-Kutta Method (Method 2) 117 5 SYSTEMS WITH TWO OR MORE DEGREES OFFREEDOM 130 5.1 The Normal Mode Analysis 131 5.4 Forced Harmonie Vibration 143 5.5 Digital Computation 145 Contents v 5.7 Centrifugal Pendulum Vibration Absorber 152 5.8 Vibration Damper 154 6 PROPERTIES OF VIBRATINO SYSTEMS 171 6.1 Flexibility Influence Coefficients 172 6.3 Stiffness Influence Coefficients 176 6.4 Stiffness Matrix of Beam Elements 179 6.5 Static Condensation for Pinned Joints 183 6.6 Orthogonality of Eigenvectors 185 6.8 Decoupling Forced Vibration Equations 189 6.9 Modal Damping in Forced Vibration 190 6.10 Normal Mode Summation 192 6.12 Unrestrained (Degenerate) Systems 197 7 UORANOE'S EQUATION 207 7.4 Kinetic Energy, Potential Energy, and Generalized Force in Terms of Generalized Coordinates q 221 7.5 Assumed Mode Summation 223 B COMPUTATIONAL METHODS 234 8.4 Convergence of the Iteration Procedure 240 vi Contents 8.5 Convergence to Higher Modes 241 8.6 The Dynamic Matrix 246 8.7 Transformation of Coordinates (Standard Computer Form) 247 8.8 Systems with Discrete Mass Matrix 248 8.11 Computer Program Notes 260 8.12 Description of Computer Programs 261 9 VIBRATION OF CONTINUOUS SYSTEMS 268 9.2 Longitudinal Vibration of Rods 271 9.3 Torsional Vibration of Rods 273 9.4 Vibration of Suspension Bridges 276 9.5 Euler Equation for Beams 281 9.6 Effect of Rotary Inertia and Shear Deformation 286 9.7 System with Repeated Identical Sections 289 10 INTRODUCTION TO THE FINITE ELEMENT METHOD 301 10.1 Element Stiffness and Mass 301 10.2 Stiffness and Mass for the Beam Element 306 10.3 Transformation of Coordinates (Global Coordinates) 309 10.4 Element Stiffness and Element Mass in Global Coordinates 312 10.5 Vibrations Involving Beam Elements 317 10.6 Spring Constraints on Structure 324 10.7 Generalized Force for Distributed Load 327 10.8 Generalized Force Proportional to Displacement 328 11 MODE-SUMMATION PROCEDURES FOR CONTINUOUS SYSTEMS 345 11.1 Mode-Summation Method 345 11.2 Beam Orthogonality Inciuding Rotary Inertia and Shear Deformation 351 11.3 Normal Modes of Constrained Structures 353 11.4 Mode-Acceleration Method 358 11.5 Component-Mode Synthesis 360 12 CLASSICAL ItfETHODS 371 12.3 Rayleigh-Ritz Method 384 12.5 Digital Computer Program for the Torsional System 390 12.6 Myklestad's Method for Beams 393 12.7 Coupled Flexure-Torsion Vibration 397 12.9 Systems with Damping 400 12.12 Transfer Matrices for Beams 406 13 RANDOItf VIBRATIONS 419 13.2 Time Averaging and Expected Value 420 13.3 Frequency Response Function 422 13.6 Power Spectrum and Power Spectral Density 435 13.8 FTs and Response 448 vIII Contents 14 NONLINEAR VIBRATIONS 461 14.3 Stability of Equilibrium 466 14.4 Method of Isoclines 468 14.6 Method of Iteration 473 14.7 Self-Excited Oscillations 477 14.8 Runge-Kutta Method 479 APPENDICES A Specifications of Vibration Bounds 488 B Introduction to Laplace Transformation 490 C Determinants and Matrices 496 D Normal Modes of Uniform Beams 509 E Lagrange's Equation 518 F Computer Programs 521 ANSWERS TO SELECTED PROBLEMS 527 INDEX 541 Preface This book is arevision of the 3rd edition of Theory of Vibration with Appfications. The major addition is Chapter 8, "Computational Methods," wh ich presents the basic principles on wh ich most modern computer programs on vibration theory are developed. The new text is accompanied by a networked software for the PC to solve the vibration problems most frequently encountered.
The programs greatly expand the range of problems that can be solved for numerical solution. The author believes that problem solving is a vital part of the learning process and the reader should understand the computational process carried out by the computer. With this facility, the mass and stiffness matrices are inputed, and the lengthy calculations for the eigenvalues and eigenvectors are delegated to the computer. Besides the new chapter on computer methods, the material in other chap- ters is amplified and additional problems are introduced to take advantage of the computing programs offered by the computer disko The first four chapters, wh ich deal with single-degree-of-freedom systems, needed very few changes, and the simple physical approach of the previous edition is maintained.
An example on rotor balancing is introduced in Chapter 3, and the section on the shock spectrum and isolation is expanded in Chapter 4. In Chapter 5, "Systems with Two or More Degrees of Freedom," the importance of normal mode vibration is emphasized to demonstrate that a11 free vibrations are composed of normal mode vibrations and that the initial conditions play a determining influence in free vibrations. Forced vibrations are again presented in terms of the relationship of frequency ratio of forced to normal frequencies in the single degree of freedom response. The important application of vibration absorbers and dampers is retained unchanged.
Chapter 6, "Properties of Vibrating Systems," is completely rearranged for logical presentation. Stiffness of framed structures is again presented to bring out the introductory basics of the finite element method presented later in Chapter 10, Ix x Preface and an example of static condensation for pinned joints is added. Orthogonality of eigenvectors and the modal matrix and its orthonormal form enable concise presentation of basic equations for the diagonal eigenvalue matrix that forms the basis for the computation of the eigenvalue-eigenvector problem. They also pro- vide a background for the normal mode-summation method.
The chapter con- cIudes with modal damping and examples of equal roots and degenerate systems. Chapter 7 presents the cIassic method of Lagrange, wh ich is associated with virtual work and generalized coordinates. Added to this chapter is the method of assumed modes, which enables the determination of eigenvalues and eigenvectors of continuous systems in terms of smaller equations of discrete system equations. The Lagrangian method offers an aIJ-encompassing view of the entire field of dynamics, a knowledge of which should be acquired by aIJ readers interested in a serious study of dynamics.
Chapter 8, "Computational Methods," examines the basic methods of com- putation that are utilized by the digital computer. Most engineering and science students today acquire knowledge of computers and programming in their fresh- man year, and given the basic background for vibration calculation, they can easily foIJow computer programs for the calculation of eigenvalues and eigenvectors. Presented on the IBM computer disk are four basic Fortran programs that cover most of the calculations encountered in vibration problems. The source programs written as subroutines can be printed out by typing ".For" (for Fortran) after the file name; i.
The user needs only to input the mass and stiffness matrices and the printout will contain the eigenvalues and eigenvectors of the problem. Those wishing additional information can modify the command instruc- tions preceding the computation. In Chapter 9, "Vibration of Continuous Systems," a section on suspension bridges is added to iIlustrate the application of the continuous system theory to simplified models for the calculation of natural frequencies. By discretizing the continuous system by repeated identical sections, simple analytic expressions are available for the natural frequencies and mode shapes by the method of difference equations.
The method exercises the disciplines of matching boundary conditions. Chapter 10, "Introduction to the Finite Element Method," remains essen- tially unchanged. A few helpful hints have been injected in some pi aces and the section on generalized force proportional to displacement has been substantially expanded by detailed computation of rotating helicopter blades. Brought out by this example is the advantage of forming equal element sections of Iength I = 1 (all l's can be arbitrarily equated to unity inside of the mass and stiffness matrices when the elements are of equal lengths) for the compiling of the mass and stiffness matrices and converting the final results to those of the original system only after the computation is completed.
Chapters 9, 11 and 12 of the former edition are consolidated into new chapter 11, "Mode-Summation Procedures for Continuous Systems," and Chapter 12, "Classical Methods." This was done mainly to leave undisturbed Chapter 13, "Random Vibrations," and Chapter 14, "Nonlinear Vibrations," and in no way Preface xl implies that Chapters 11 and 12 are of lesser importance. As one finds in the finite element method, the equation of motion soon becomes large in order to obtain acceptable accuracies for higher modes, and the methods of new Chapters 11 and 12 yield these resuits with considerably simpler caIculations. This book can be used at the undergraduate or graduate level of instruction. Chapters 1 through 6 can be covered in a first course on vibration, although parts of other chapters might be appropriately introduced.
The subject of vibration and dynamics, fascinating to the author for over most of his academic career, offers a wide range of opportunities for applying various mathematical techniques to the solution of vibration problems, and is presented with the hope that the subject matter will be enjoyed by others. Finally, the author wishes to acknowledge his indebtedness to those who have contributed to the writing of the computer programs on disko Of these, Dr. Grant Johnson of the Mechanical Engineering Department has generously aided the author for the past few years, and Derek Zahl, also of the Mechanical Engineering Department, carefully compiled the disk that is encIosed with the text. Thanks also are due to David Bothman and Tony Peres for the photos of some of the equipment used in our Undergraduate Laboratory.
Thomson QUANTIlY ENGLISH SYSTEM S. SYSTEM force Ilb 4.448 Newtons (N) mass Ilb .59 kg (kilogram) length 1ft 0.3048 meters (m) mass density Ilb/ft 3 16.02 kg/m 3 torque or moment 11b .81 m/s 2 spring constant k 1Ib/in.1 N/m spring constant K 1 Ib. rn/rad damping constant c 11b. s/m mass moment of inertia 1 Ib.1129 kg m 2 modulus of elasticity 10 6 Ib/in.895 x 10 9 N/m 2 modulus of elasticity of steel 29 X 10 6 Ib/in.3 radian The SI System of Unlts THE SI SYSTEM OF UNITS The English system of units that has dominated the Uni ted States from historical times is now being replaced by the SI system of units.