DaniEl J. inman University of Michigan
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© 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
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Library of Congress Cataloging-in-Publication Data on File
ISBN-13: 978-0-13-287169-3 ISBN-10: 0-13-287169-6
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
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
2.7 Other Forms of Damping 170
2.8 Numerical Simulation and Design 180
2.9 Nonlinear Response Properties 188
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
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
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
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
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
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
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
MATLAB Engineering Vibration Toolbox 656
Toolbox Problems 656
aPPendIx a comPlex numbers and functIons 657
aPPendIx b laPlace transforms 663
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
ansWers to selected Problems 692
This book is intended for use in a first course in vibrations or structural dynamics 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.
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.
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.
Chapter 3: New examples and equations have been added, as well as new problems. In particular, the explanation of impulse has been expanded. In addition, previous problems have been rewritten for clarity and precision. All examples and problems that referred to prior information in the text have been modified to pres- ent a more self-contained statement. All of the codes have been updated to current syntax, and older, obsolete commands have been replaced.
Chapter 4: Along with the addition of an entirely new example, many of the examples have been changed and modified for clarity and to include improved information. A new window has been added to clarify matrix information. A fig- ure has been removed and a new figure added. New problems have been added and older problems have been modified with the goal of making all problems and examples more self-contained. All of the codes have been updated to current syntax, and older, obsolete commands have been replaced. Several new plots intermixed in the codes have been redone to reflect issues with Mathematica and MATLAB’s automated time step which proves to be inaccurate when using singu- larity functions. Several explanations have been modified according to students’ suggestions.
Chapter 5: Section 5.1 has been changed, the figure replaced, and the example changed for clarity. The problems are largely the same but many have been changed or modified with different details and to make the problems more self-contained. Section 5.8 (Active Vibration Suppression) and Section 5.9 (Practical Isolation Design) have been removed, along with the associated problems, to make room for added material in the earlier chapters without lengthening the book. According to user surveys, these sections are not usually covered.
Chapter 6: Section 6.8 has been rewritten for clarity and a window has been added to summarize modal analysis of the forced response. New problems have been added and many older problems restated for clarity. Further details have been added to several examples. A number of small additions have been made to the to the text for clarity.
Chapters 7 and 8: These chapters were not changed, except to make minor corrections and additions as suggested by users.
This book uses SI units. The 1st edition used a mixture of US Customary and SI, but at the insistence of the editor all units were changed to SI. I have stayed with SI in this edition because of the increasing international arena that our engineering graduates compete in. The engineering community is now completely global. For instance, GE Corporate Research has more engineers in its research center in India than it does in the US. Engineering in the US is in danger of becoming the ‘gar- ment’ workers of the next decade if we do not recognize the global work place. Our engineers need to work in SI to be competitive in this increasingly international work place.
This text comes with a bit of support. In particular, MS PowerPoint presentations are available for each chapter along with some instructive movies. The solutions manual is available in both MS Word and PDF format (sorry, instructors only). Sample tests are available. The MS Word solutions manual can be cut and pasted into presentation slides, tests, or other class enhancements. These resources can be found at www.pearsonhighered.com and will be updated often. Please also email me at email@example.com with corrections, typos, questions, and suggestions. The book is reprinted often, and at each reprint I have the option to fix typos, so please report any you find to me, as others as well as I will appreciate it.
The best place to get help in studying this material is from your instructor, as there is nothing more educational than a verbal exchange. However, the book was writ- ten as much as possible from a student’s perspective. Many students critiqued the original manuscript, and many of the changes in text have been the result of sug- gestions from students trying to learn from the material, so please feel free to email me (firstname.lastname@example.org) should you have questions about explanations. Also I would appreciate knowing about any corrections or typos and, in particular, if you find an explanation hard to follow. My goal in writing this was to provide a useful resource for students learning vibration for the first time.
The cover photo of the unmanned air vehicle is provided courtesy of General Atomics Aeronautical Systems, Inc., all rights reserved. Each chapter starts with two photos of different systems that vibrate to remind the reader that the material in this text has broad application across numerous sectors of human activity. These photographs were taken by friends, students, colleagues, relatives, and some by me. I am greatly appreciative of Robert Hargreaves (guitar), P. Timothy Wade (wind mill, Presidential helicopter), General Atomics (Predator), Roy Trifilio (bridge), Catherine Little (damper), Alex Pankonien (FEM graphic), and Jochen Faber of Liebherr Aerospace (landing gear). Alan Giles of General Atomics gave me an informative tour of their facilities which resulted in the photos of their products.
Many colleagues and students have contributed to the revision of this text through suggestions and questions. In particular, Daniel J. Inman, II; Kaitlyn DeLisi; Kevin Crowely; and Emily Armentrout provided many useful comments from the perspective of students reading the material for the first time. Kaitlyn and Kevin checked all the computer codes by copying them out of the book to
make sure they ran. My former PhD students Ya Wang, Mana Afshari, and Amin Karami checked many of the new problems and examples. Dr. Scott Larwood and the students in his vibrations class at the University of the Pacific sent many sug- gestions and corrections that helped give the book the perspective of a nonresearch insitution. I have implemented many of their suggestions, and I believe the book’s explanations are much clearer due to their input. Other professors using the book, Cetin Cetinkaya of Clarkson University, Mike Anderson of the University of Idaho, Joe Slater of Wright State University, Ronnie Pendersen of Aalborg University Esbjerg, Sondi Adhikari of the Universty of Wales, David Che of Geneva College, Tim Crippen of the University of Texas at Tyler, and Nejat Olgac of the University of Conneticut, have provided discussions via email that have led to improvements in the text, all of which are greatly appreciated. I would like to thank the review- ers: Cetin Cetinkaya, Clarkson University; Dr. Nesrin Sarigul-Klijn, University of California–Davis; and David Che, Geneva College.
Many of my former PhD students who are now academics cotaught this course with me and also offered many suggestions. Alper Erturk (Georgia Tech), Henry Sodano (University of Florida), Pablo Tarazaga (Virginia Tech), Onur Bilgen (Old Dominian University), Mike Seigler (University of Kentucky), and Armaghan Salehian (University of Waterloo) all contributed to clarity in this text for which I am grateful. I have been lucky to have wonderful PhD students to work with. I learned much from them.
I would also like to thank Prof. Joseph Slater of Wright State for reviewing some of the new materials, for writing and managing the associated toolbox, and constantly sending suggestions. Several colleagues from government labs and com- panies have also written with suggestions which have been very helpful from that perspective of practice.
I have also had the good fortune of being sponsored by numerous companies and federal agencies over the last 32 years to study, design, test, and analyze a large variety of vibrating structures and machines. Without these projects, I would not have been able to write this book nor revise it with the appreciation for the practice of vibration, which I hope permeates the text.
Last, I wish to thank my family for moral support, a sense of purpose, and for putting up with my absence while writing.
Daniel J. inman Ann Arbor, Michigan
Introduction to Vibration and the Free Response
1 Vibration is the subdiscipline of dynamics that deals with repetitive motion. Most of the examples in this text are mechanical or structural elements. However, vibration is prevalent in biological systems and is in fact at the source of communication (the ear vibrates to hear and the tongue and vocal cords vibrate to speak). In the case of music, vibrations, say of a stringed instrument such as a guitar, are desired. On the other hand, in most mechanical systems and structures, vibration is unwanted and even destructive. For example, vibration in an aircraft frame causes fatigue and can eventually lead to failure. An example of fatigue crack is illustrated in the circle in the photo on the bottom left. Everyday experiences are full of vibration and usually ways of mitigating vibration. Automobiles, trains, and even some bicycles have devices to reduce the vibration induced by motion and transmitted to the driver.
The task of this text is to teach the reader how to analyze vibration using principles of dynamics. This requires the use of mathematics. In fact, the sine function provides the fundamental means of analyzing vibration phenomena.
The basic concepts of understanding vibration, analyzing vibration, and predicting the behavior of vibrating systems form the topics of this text. The concepts and formulations presented in the following chapters are intended to provide the skills needed for designing vibrating systems with desired properties that enhance vibration when it is wanted and reduce vibration when it is not.
This first chapter examines vibration in its simplest form in which no external force is present (free vibration). This chapter introduces both the important concept of natural frequency and how to model vibration mathematically.
The Internet is a great source for examples of vibration, and the reader is encouraged to search for movies of vibrating systems and other examples that can be found there.
2 Introduction to Vibration and the Free Response Chap. 1
1.1 IntroductIon to Free VIbratIon
Vibration is the study of the repetitive motion of objects relative to a stationary frame of reference or nominal position (usually equilibrium). Vibration is evident everywhere and in many cases greatly affects the nature of engineering designs. The vibrational properties of engineering devices are often limiting factors in their per- formance. When harmful, vibration should be avoided, but it can also be extremely useful. In either case, knowledge about vibration—how to analyze, measure, and control it—is beneficial and forms the topic of this book.
Typical examples of vibration familiar to most include the motion of a guitar string, the ride quality of an automobile or motorcycle, the motion of an airplane’s wings, and the swaying of a large building due to wind or an earth- quake. In the chapters that follow, vibration is modeled mathematically based on fundamental principles, such as Newton’s laws, and analyzed using results from calculus and differential equations. Techniques used to measure the vibra- tion of a system are then developed. In addition, information and methods are given that are useful for designing particular systems to have specific vibrational responses.
The physical explanation of the phenomena of vibration concerns the inter- play between potential energy and kinetic energy. A vibrating system must have a component that stores potential energy and releases it as kinetic energy in the form of motion (vibration) of a mass. The motion of the mass then gives up kinetic en- ergy to the potential-energy storing device.
Engineering is built on a foundation of previous knowledge and the subject of vibration is no exception. In particular, the topic of vibration builds on pre- vious courses in dynamics, system dynamics, strength of materials, differential equations, and some matrix analysis. In most accredited engineering programs, these courses are prerequisites for a course in vibration. Thus, the material that follows draws information and methods from these courses. Vibration analysis is based on a coalescence of mathematics and physical observation. For example, consider a simple pendulum. You may have seen one in a science museum, in a grandfather clock, or you might make a simple one with a string and a marble. As the pendulum swings back and forth, observe that its motion as a function of time can be described very nicely by the sine function from trigonometry. Even more interesting, if you make a free-body diagram of the pendulum and ap- ply Newtonian mechanics to get the equation of motion (summing moments in this case), the resulting equation of motion has the sine function as its solution. Further, the equation of motion predicts the time it takes for the pendulum to repeat its motion. In this example, dynamics, observation, and mathematics all come into agreement to produce a predictive model of the motion of a pendulum, which is easily verified by experiment (physical observation).
Sec. 1.1 Introduction to Free Vibration 3
This pendulum example tells the story of this text. We propose a series of steps to build on the modeling skills developed in your first courses in statics, dy- namics, and strength of materials combined with system dynamics to find equations of motion of successively more complicated systems. Then we will use the tech- niques of differential equations and numerical integration to solve these equations of motion to predict how various mechanical systems and structures vibrate. The following example illustrates the importance of recalling the methods learned in the first course in dynamics.
Derive the equation of motion of the pendulum in Figure 1.1.
Figure 1.1 (a) A schematic of a pendulum. (b) The free-body diagram of (a).
Solution Consider the schematic of a pendulum in Figure 1.1(a). In this case, the mass of the rod will be ignored as well as any friction in the hinge. Typically, one starts with a photograph or sketch of the part or structure of interest and is immediately faced with having to make assumptions. This is the “art” or experience side of vibration analysis and modeling. The general philosophy is to start with the simplest model possible (hence, here we ignore friction and the mass of the rod and assume the motion remains in a plane) and try to answer the relevant engineering questions. If the simple model doesn’t agree with the experiment, then make it more complex by relaxing the assump- tions until the model successfully predicts physical observation. With the assumptions in mind, the next step is to create a free-body diagram of the system, as indicated in Figure 1.1(b), in order to identify all of the relevant forces. With all the modeled forces identified, Newton’s second law and Euler’s second law are used to derive the equa- tions of motion.
In this example Euler’s second law takes the form of summing moments about point O. This yields
ΣMO = J𝛂
4 Introduction to Vibration and the Free Response Chap. 1
where MO denotes moments about the point O, J = ml2 is the mass moment of inertia of the mass m about the point O, l is the length of the massless rod, and 𝛂 is the angu- lar acceleration vector. Since the problem is really in one dimension, the vector sum of moments equation becomes the single scalar equation
Jα(t) = -mgl sin θ(t) or ml2θ $ (t) + mgl sin θ(t) = 0
Here the moment arm for the force mg is the horizontal distance l sin θ, and the two overdots indicate two differentiations with respect to the time, t. This is a second-order ordinary differential equation, which governs the time response of the pendulum. This is exactly the procedure used in the first course in dynamics to obtain equations of motion.
The equation of motion is nonlinear because of the appearance of the sin(θ) and hence difficult to solve. The nonlinear term can be made linear by approximating the sine for small values of θ(t) as sin θ ≈ θ. Then the equation of motion becomes
θ $ (t) +
l θ(t) = 0
This is a linear, second-order ordinary differential equation with constant coefficients and is commonly solved in the first course of differential equations (usually the third course in the calculus sequence). As we will see later in this chapter, this linear equa- tion of motion and its solution predict the period of oscillation for a simple pendulum quite accurately. The last section of this chapter revisits the nonlinear version of the pendulum equation.
Since Newton’s second law for a constant mass system is stated in terms of force, which is equated to the mass multiplied by acceleration, an equation of motion with two time derivatives will always result. Such equations require two constants of integration to solve. Euler’s second law for constant mass systems also yields two time derivatives. Hence the initial position for θ(0) and velocity of θ
# (0) must be
specified in order to solve for θ(t) in Example 1.1.1. The term mgl sin θ is called the restoring force. In Example 1.1.1, the restoring force is gravity, which provides a potential-energy storing mechanism. However, in most structures and machine parts the restoring force is elastic. This establishes the need for background in strength of materials when studying vibrations of structures and machines.
As mentioned in the example, when modeling a structure or machine it is best to start with the simplest possible model. In this chapter, we model only sys- tems that can be described by a single degree of freedom, that is, systems for which Newtonian mechanics result in a single scalar equation with one displacement coor- dinate. The degree of freedom of a system is the minimum number of displacement coordinates needed to represent the position of the system’s mass at any instant of time. For instance, if the mass of the pendulum in Example 1.1.1 were a rigid body, free to rotate about the end of the pendulum as the pendulum swings, the angle of rotation of the mass would define an additional degree of freedom. The problem would then require two coordinates to determine the position of the mass in space, hence two degrees of freedom. On the other hand, if the rod in Figure 1.1 is flexible,
Sec. 1.1 Introduction to Free Vibration 5
its distributed mass must be considered, effectively resulting in an infinite number of degrees of freedom. Systems with more than one degree of freedom are dis- cussed in Chapter 4, and systems with distributed mass and flexibility are discussed in Chapter 6.
The next important classification of vibration problems after degree of freedom is the nature of the input or stimulus to the system. In this chapter, only the free response of the system is considered. Free response refers to analyzing the vibration of a system resulting from a nonzero initial displacement and/or velocity of the system with no external force or moment applied. In Chapter 2, the response of a single-degree-of-freedom system to a harmonic input (i.e., a sinusoidal applied force) is discussed. Chapter 3 examines the response of a sys- tem to a general forcing function (impulse or shock loads, step functions, random inputs, etc.), building on information learned in a course in system dynamics. In the remaining chapters, the models of vibration and methods of analysis become more complex.
The following sections analyze equations similar to the linear version of the pen- dulum equation given in Example 1.1.1. In addition, energy dissipation is introduced, and details of elastic restoring forces are presented. Introductions to design, measure- ment, and simulation are also presented. The chapter ends with the introduction of high-level computer codes (MATlAb®, Mathematica, and Mathcad) as a means to visualize the response of a vibrating system and for making the calculations required to solve vibration problems more efficiently. In addition, numerical simulation is intro- duced in order to solve nonlinear vibration problems.