Table Of ContentTheory of
Vibration with
Appiications
Fourth Edition
William T. Thomson, Professor Emeritus
Department of Mechanical and Environmental Engineering
University of California
Santa Barbara, California
r.^ ^ Taylor S*.Francis
Taylor S*. Francis Group
LONDON AND NEW YORK
© 1972, 1981, 1988 and 1993 Prentice Hall, a Simon Schuster Company
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been asserted by him in accordance with the Copyright, Designs and Patents
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First edition published 1972
Second edition published 1981
Third edition published 1988
Fourth edition published 1993
Reprinted In 1998 by Stanley Thornes (Publishers) Ltd
Reprinted in 2003 by:
Published by Taylor & Francis
2 Park Square, Milton Park, Abingdon, Oxon, 0X14 4RN
270 Madison Ave, New York NY 10016
Transferred to Digital Printing 2010
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A catalogue record for this book is available from the British Library
ISBN 0 7487 4380 4
Publisher’s Note
The publisher has gone to great lengths to ensure the quality of this reprint
but points out that some imperfections in the original may be apparent.
PREFACE lx
THE SI SYSTEM OF UNITS
1 OSCILLATORY MOTION
1.1 Harmonic Motion 6
1.2 Periodic Motion 9
1.3 Vibration Terminology 12
2 FREE VIBRATION 17
2.1 Vibration Model 17
2.2 Equations of Motion: Natural Frequency 18
2.3 Energy Method 22
2.4 Rayleigh Method: Effective Mass 24
2.5 Principle of Virtual Work 26
2.6 Viscously Damped Free Vibration 28
2.7 Logarithmic Decrement 33
2.8 Coulomb Damping 35
Contents
3 HARMONICALLY EXCITED VIBRATION 51
3.1 Forced Harmonic Vibration 51
3.2 Rotating Unbalance 56
3.3 Rotor Unbalance 58
3.4 Whirling of Rotating Shafts 61
3.5 Support Motion 66
3.6 Vibration Isolation 68
3.7 Energy Dissipated by Damping 70
3.8 Equivalent Viscous Damping 73
3.9 Structural Damping 75
3.10 Sharpness of Resonance 77
3.11 Vibration-Measuring Instruments 78
TRANSIENT VIBRATION 92
4
4.1 Impulse Excitation 92
4.2 Arbitrary Excitation 94
4.3 Laplace Transform Formulation 97
4.4 Pulse Excitation and Rise Time 100
4.5 Shock Response Spectrum 103
4.6 Shock Isolation 108
4.7 Finite Difference Numerical Computation 108
4.8 Runge-Kutta Method {Method 2) 117
SYSTEMS WITH TWO OR MORE DEGREES
5
OF FREEDOM
130
5.1 The Normal Mode Analysis 131
5.2 Initial Conditions 135
5.3 Coordinate Coupling 138
5.4 Forced Harmonic Vibration 143
5.5 Digital Computation 145
Contents
5.6 Vibration Absorber 150
5.7 Centrifugal Pendulum Vibration Absorber 152
5.8 Vibration Damper 154
6 PROPERTIES OF VIBRATING SYSTEMS
171
6.1 Flexibility Influence Coefficients 172
6.2 Reciprocity Theorem 175
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
6.7 Modal Matrix P 187
6.8 Decoupling Forced Vibration Equations 189
6.9 Modal Damping in Forced Vibration 190
6.10 Normal Mode Summation 192
6.11 Equal Roots 195
6.12 Unrestrained (Degenerate) Systems 197
LAGRANGeS EQUATION 207
7
7.1 Generalized Coordinates 207
7.2 Virtual Work 212
7.3 Lagrange’s Equation 215
7.4 Kinetic Energy, Potential Energy, and Generalized
Force in Terms of Generalized Coordinates q 221
7.5 Assumed Mode Summation 223
a COMPUTATIONAL METHODS
234
8.1 Root Solving 235
8.2 Gauss Elimination 236
8.3 Matrix Iteration 238
8.4 Convergence of the Iteration Procedure 240
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
8.9 Cholesky Decomposition 249
8.10 Jacobi Diagonalization 253
8.11 Computer Program Notes 260
8.12 Description of Computer Programs 261
9 VIBRATION OF CONTINUOUS SYSTEMS 268
9.1 Vibrating String 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
Contents
vii
11 MODE^SUMMATION PROCEDURES FOR CONTINUOUS
SYSTEMS 345
11.1 Mode-Summation Method 345
11.2 Beam Orthogonality Including 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 METHODS 371
12.1 Rayleigh Method 371
12.2 Dunkerley’s Equation 379
12.3 Rayleigh-Ritz Method 384
12.4 Holzer Method 387
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
12.8 Transfer Matrices 399
12.9 Systems with Damping 400
12.10 Geared System 403
12.11 Branched Systems 404
12.12 Transfer Matrices for Beams 406
13 RANDOM VIBRATIONS 419
13.1 Random Phenomena 419
13.2 Time Averaging and Expected Value 420
13.3 Frequency Response Function 422
13.4 Probability Distribution 425
13.5 Correlation 431
13.6 Power Spectrum and Power Spectral
Density 435
13.7 Fourier Transforms 441
13.8 FTs and Response 448
Contents
14 NONLINEAR VIBRATIONS 461
14.1 Phase Plane 462
14.2 Conservative Systems 463
14.3 Stability of Equilibrium 466
14.4 Method of Isoelines 468
14.5 Perturbation Method 470
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
This book is a revision of the 3rd edition of Theory of Vibration with Applications.
The major addition is Chapter 8, “Computational Methods,” which presents the
basic principles on which 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 disk.
The first four chapters, which 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 all 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,
