Engineering Vibration Fourth Edition DaniEl J. inman University of Michigan Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editorial Director, Computer Science and Engineering: Marcia J. Horton Acquisitions Editor: Norrin Dias Editorial Assistant: Sandia Rodriguez Senior Managing Editor: Scott Disanno Art Director: Jayne Conte Cover Designer: Bruce Kenselaar Project Manager: Greg Dulles Manufacturing Buyer: Lisa McDowell Senior Marketing Manager: Tim Galligan © 2014, 2008, 2001 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. The author and publisher 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 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.