Table Of ContentEngineering Vibration
Fourth Edition
DaniEl J. inman
University of Michigan
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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 theories and programs.
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data on File
ISBN-13: 978-0-13-287169-3
ISBN-10: 0-13-287169-6
Contents
PREFACE viii
1 INTRODUCTION TO VIBRATION AND THE FREE RESPONSE 1
1.1 Introduction to Free Vibration 2
1.2 Harmonic Motion 13
1.3 Viscous Damping 21
1.4 Modeling and Energy Methods 31
1.5 Stiffness 46
1.6 Measurement 58
1.7 Design Considerations 63
1.8 Stability 68
1.9 Numerical Simulation of the Time Response 72
1.10 Coulomb Friction and the Pendulum 81
Problems 95
MATLAB Engineering Vibration Toolbox 115
Toolbox Problems 116
2 RESPONSE TO HARMONIC EXCITATION 117
2.1 Harmonic Excitation of Undamped Systems 118
2.2 Harmonic Excitation of Damped Systems 130
2.3 Alternative Representations 144
2.4 Base Excitation 151
2.5 Rotating Unbalance 160
2.6 Measurement Devices 166
iii
iv Contents
2.7 Other Forms of Damping 170
2.8 Numerical Simulation and Design 180
2.9 Nonlinear Response Properties 188
Problems 197
MATLAB Engineering Vibration Toolbox 214
Toolbox Problems 214
3 GENERAL FORCED RESPONSE 216
3.1 Impulse Response Function 217
3.2 Response to an Arbitrary Input 226
3.3 Response to an Arbitrary Periodic Input 235
3.4 Transform Methods 242
3.5 Response to Random Inputs 247
3.6 Shock Spectrum 255
3.7 Measurement via Transfer Functions 260
3.8 Stability 262
3.9 Numerical Simulation of the Response 267
3.10 Nonlinear Response Properties 279
Problems 287
MATLAB Engineering Vibration Toolbox 301
Toolbox Problems 301
4 MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS 303
4.1 Two-Degree-of-Freedom Model (Undamped) 304
4.2 Eigenvalues and Natural Frequencies 318
4.3 Modal Analysis 332
4.4 More Than Two Degrees of Freedom 340
4.5 Systems with Viscous Damping 356
4.6 Modal Analysis of the Forced Response 362
Contents v
4.7 Lagrange’s Equations 369
4.8 Examples 377
4.9 Computational Eigenvalue Problems for Vibration 389
4.10 Numerical Simulation of the Time Response 407
Problems 415
MATLAB Engineering Vibration Toolbox 433
Toolbox Problems 433
5 DESIGN FOR VIBRATION SUPPRESSION 435
5.1 Acceptable Levels of Vibration 436
5.2 Vibration Isolation 442
5.3 Vibration Absorbers 455
5.4 Damping in Vibration Absorption 463
5.5 Optimization 471
5.6 Viscoelastic Damping Treatments 479
5.7 Critical Speeds of Rotating Disks 485
Problems 491
MATLAB Engineering Vibration Toolbox 501
Toolbox Problems 501
6 DISTRIBUTED-PARAMETER SYSTEMS 502
6.1 Vibration of a String or Cable 504
6.2 Modes and Natural Frequencies 508
6.3 Vibration of Rods and Bars 519
6.4 Torsional Vibration 525
6.5 Bending Vibration of a Beam 532
6.6 Vibration of Membranes and Plates 544
6.7 Models of Damping 550
6.8 Modal Analysis of the Forced Response 556
vi Contents
Problems 566
MATLAB Engineering Vibration Toolbox 572
Toolbox Problems 572
7 VIBRATION TESTING AND EXPERIMENTAL MODAL ANALYSIS 573
7.1 Measurement Hardware 575
7.2 Digital Signal Processing 579
7.3 Random Signal Analysis in Testing 584
7.4 Modal Data Extraction 588
7.5 Modal Parameters by Circle Fitting 591
7.6 Mode Shape Measurement 596
7.7 Vibration Testing for Endurance and Diagnostics 606
7.8 Operational Deflection Shape Measurement 609
Problems 611
MATLAB Engineering Vibration Toolbox 615
Toolbox Problems 616
8 FINITE ELEMENT METHOD 617
8.1 Example: The Bar 619
8.2 Three-Element Bar 625
8.3 Beam Elements 630
8.4 Lumped-Mass Matrices 638
8.5 Trusses 641
8.6 Model Reduction 646
Problems 649
MATLAB Engineering Vibration Toolbox 656
Toolbox Problems 656
APPENDIX A COMPLEX NUMBERS AND FUNCTIONS 657
APPENDIX B LAPLACE TRANSFORMS 663
Contents vii
APPENDIX C MATRIX BASICS 668
APPENDIX D THE VIBRATION LITERATURE 680
APPENDIX E LIST OF SYMBOLS 682
APPENDIX F CODES AND WEB SITES 687
APPENDIX G ENGINEERING VIBRATION TOOLBOX
AND WEB SUPPORT 688
REFERENCES 690
ANSWERS TO SELECTED PROBLEMS 692
INDEX 699
Preface
This book is intended for use in a first course in vibrations or structural d ynamics
for undergraduates in mechanical, civil, and aerospace engineering or engineer-
ing mechanics. The text contains the topics normally found in such courses
in accredited engineering departments as set out initially by Den Hartog and
refined by Thompson. In addition, topics on design, measurement, and computa-
tion are addressed.
Pedagogy
Originally, a major difference between the pedagogy of this text and competing
texts is the use of high level computing codes. Since then, the other authors of
vibrations texts have started to embrace use of these codes. While the book is
written so that the codes do not have to be used, I strongly encourage their use.
These codes (Mathcad®, MATLAB®, and Mathematica®) are very easy to use,
at the level of a programmable calculator, and hence do not require any prereq-
uisite courses or training. Of course, it is easier if the students have used one or
the other of the codes before, but it is not necessary. In fact, the MATLAB®
codes can be copied directly and will run as listed. The use of these codes greatly
enhances the student’s understanding of the fundamentals of vibration. Just as
a picture is worth a thousand words, a numerical simulation or plot can enable a
completely dynamic understanding of vibration phenomena. Computer calcula-
tions and simulations are presented at the end of each of the first four chapters.
After that, many of the problems assume that codes are second nature in solving
vibration problems.
Another unique feature of this text is the use of “windows,” which are
distributed throughout the book and provide reminders of essential informa-
tion pertinent to the text material at hand. The windows are placed in the text at
points where such prior information is required. The windows are also used to
summarize essential information. The book attempts to make strong connections
to previous course work in a typical engineering curriculum. In particular, refer-
ence is made to calculus, differential equations, statics, dynamics, and strength of
materials course work.
viii
Preface ix
WHAT’S NEW IN THIS EDITION
Most of the changes made in this edition are the result of comments sent to me by
students and faculty who have used the 3rd edition. These changes consist of improved
clarity in explanations, the addition of some new examples that clarify concepts, and
enhanced problem statements. In addition, some text material deemed outdated and
not useful has been removed. The computer codes have also been updated. However,
software companies update their codes much faster than the publishers can update
their texts, so users should consult the web for updates in syntax, commands, etc. One
consistent request from students has been not to reference data appearing previously in
other examples or problems. This has been addressed by providing all of the relevant
data in the problem statements. Three undergraduate engineering students (one in
Engineering Mechanics, one in Biological Systems Engineering, and one in Mechanical
Engineering) who had the prerequisite courses, but had not yet had courses in vibra-
tions, read the manuscript for clarity. Their suggestions prompted us to make the fol-
lowing changes in order to improve readability from the student’s perspective:
Improved clarity in explanations added in 47 different passages in the text. In
addition, two new windows have been added.
Twelve new examples that clarify concepts and enhanced problem statements
have been added, and ten examples have been modified to improve clarity.
Text material deemed outdated and not useful has been removed. Two sections
have been dropped and two sections have been completely rewritten.
All computer codes have been updated to agree with the latest syntax changes
made in MATLAB, Mathematica, and Mathcad.
Fifty-four new problems have been added and 94 problems have been modi-
fied for clarity and numerical changes.
Eight new figures have been added and three previous figures have been modified.
Four new equations have been added.
Chapter 1: Changes include new examples, equations, and problems. New textual
explanations have been added and/or modified to improve clarity based on student sug-
gestions. Modifications have been made to problems to make the problem statement
clear by not referring to data from previous problems or examples. All of the codes have
been updated to current syntax, and older, obsolete commands have been replaced.
Chapter 2: New examples and figures have been added, while previous exam-
ples and figures have been modified for clarity. New textual explanations have also
been added and/or modified. New problems have been added and older problems
modified to make the problem statement clear by not referring to data from previ-
ous problems or examples. All of the codes have been updated to current syntax,
and older, obsolete commands have been replaced.
