Theory of Vibration with Applications

512 Pages·7.661 MB·English

by  William T. Thomson

THEORY
OF
VIBRATION
WITH
APPLICATIONS
SECOND
EDITION
WILLIAM T. THOMSON, Professor Emeritus
Department of Mechanical and Environmental Engineering
University of California
Santa Barbara, California
London
GEORGE ALLEN & UNWIN
Sy<lney
First published in the USA by Prentice-Hall, Inc., 1981
First puhlished hy George Allen & Unwin 1983
This book is copyright under the Berne Convention. All rights
are reserved. Apart from any fair dealing for the purpose of
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the Copyright Act, 1956, no part of this publication may be
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sent to the publishers at the undermentioned address:
GEORGE ALLEN & UNWIN LTD
40 Museum Street, London WClA 1LU
<!:> 1981 by Prentice-Hall, Inc., Englewood Cliffs, NJ 07632
British Library Cataloguing in Publication Data
Thomson, William Tyrell
Theory of vibration with applications. - 2nd ed.
I. Vibration
I. Title
531'.32 QCI36
ISBN !~04-620012-6
Printed in Great Britain by Mackays of Chatham

CONTENTS
PREFACE
1 OSCILLATORY MOTION 1
1.1 Harmonic Motion 2
1.2 Periodic Motion 5
1.3 Vibration Terminology 8
2 FREE VIBRATION 13
2.1 Equation of Motion-Natural
Frequency 13
2.2 Energy Method 18
2.3 Viscously Damped Free Vibration 25
2.4 Logarithmic Decrement 30
2.5 Coulomb Damping 34
3 HARMONICALLY EXCITED VIBRATION 48
3.1 Forced Harmonic Vibration 48
3.2 Rotating Unbalance 52
Yl Contents
3.3 Balancing of Rotors 55
3.4 Whirling of Rotating Shafts 58
3.5 Support Motion 62
3.6 Vibration Isolation 64
3.7 Energy Dissipated by Damping 68
3.8 Equivalent Viscous Damping 72
3.9 Structural Damping 74
3.10 Sharpness of Resonance 76 )
3.11 Response to Periodic Forces 77
3.12 Vibration Measuring Instruments 78 r·i
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'.'
4
TRANSIENT VIBRATION 92
4.1 Impulse Excitation 92
4.2 Arbitrary Excitation 94
4.3 Laplace Transform Formulation 100
4.4 Response Spectrum 103 )
4.5 Finite Difference Numerical Computation 110
4.6 Runge-Kutta Method 119 .r~ i
j\
5
TWO DEGREES OF FREEDOM 132
5.1 Normal Mode Vibration 132
5.2 Coordinate Coupling 139
5.3 Forced Harmonic Vibration 143
5.4 Digital Computation 145
5.5 Vibration Absorber I
c,
5.6 Centrifugal Pendulum Vibration Absorber 151
.~~!
5.7 Vibration Damper 153
5.8 Gyroscopic Effect on Rotating 1\
Shafts 158 ~:
I
6 PROPERTIES OF VIBRATING SYSTEMS 174
6.1 Flexibility and Stiffness Matrix 174 \'
6.2 Reciprocity Theorem 182
I
6.3 Eigenvalues and Eigenvectors 183 c,
6.4 Equations Based on flexibility
1~7
Contents .U
6.5 Orthogonal Properties of Eigenvectors 188
6.6 kepeated Roots 190
6.7 Modal Matrix P 192
6.8 Modal Damping in Forced Vibration 196
6.9 Normal Mode Summation 197
7 NORMAL MODE VIBRATION
OF CONTINUOUS SYSTEMS 209
7.1 Vibrating String 210
7.2 Longitudinal Vibration of Rods 212
7.3 Torsional Vibration of Rods 215
7.4 Euler Equation for the Beam 218
7.5 Effect of Rotary Inertia and Shear
Deformation 221
7.6 Vibration of Membranes 223
7.7 Digital Computation 225
8 LAGRANGE'S EQUATION 238
8.1 Generalized Coordinates 238
8.2 Virtual Work 244
8.3 Kinetic Energy, Potential Energy,
and Generalized Force 247
8.4 Lagrange's Equations 252
8.5 Vibration of Framed Structures 256
8.6 Consistent Mass 258
9 APPROXIMATE NUMERICAL METHODS 268
9.1 Rayleigh Method 268
9.2 Dunkerley's Equation 276
9.3 Rayleigh-Ritz Method 281
9.4 Method of Matrix Iteration 285
9.5 Calculation of H1gher Modes 287

('onlentl lx
12.6 Pcrturba twn Method 380
12.7 Method of Iteration 383
\2.8 SeH-Ex~.:ited Oscillations 388
I 2.9 Analog Computer Circuits for Non-
linear Systems 390
12.10 Runge-Kutta Method 391
13 RANDOM VIBRATION 400
13.1 Random Phenomena 400
13.2 Time Averaging and Expected Value 402
13.3 Probability Distribution 404
13.4 Correlation 410
13.5 Power Spectrum and Power Spectral
Density 414
13.6 Fourier Transforms 420
13.7 Frequency Response Function 427
APPENDIXES
A SPECIFICATIONS OF VIBRATION BOUNDS 444
B INTRODUCTION TO
LAPLACE TRANSFORMATION 447
c
DETERMINANT AND MATRICES 453
I Determinant 453
II Matrices 455
Ill Rules of Matrix Operations 457
IV Determination of Eigenvectors 461
V Cholesky's Method of Solution 463
D NORMAL MODES OF UNIFORM BEAMS 465
ANSWERS TO SELECTED PROBLEMS 475
INDEX 487
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