Mechanical vibrations

Mechanical

Vibrations

Fifth Edition

Singiresu S. Rao

University of Miami

Prentice Hall

Upper Saddle River Boston Columbus San Francisco New York

Indianapolis London Toronto Sydney Singapore Tokyo Montreal

Dubai Madrid Hong Kong Mexico City Munich Paris Amsterdam Cape Town

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page i

Vice President and Editorial Director, ECS:

Marcia J. Horton

Senior Acquisitions Editor: Tacy Quinn

Editorial Assistant: Coleen McDonald

Senior Managing Editor: Scott Disanno

Production Liason: Irwin Zucker

Production Editor: Sangeetha Parthasarathy,

Laserwords

Senior Operations Specialist: Alan Fischer

Operations Specialist: Lisa McDowell

Senior Marketing Manager: Tim Galligan

Marketing Assistant: Mack Patterson

Art Director: Jayne Conte

Cover Designer: Suzanne Behnke

Art Editor: Greg Dulles

Media Editor: Daniel Sandin

Full-Service Project Management/Composition:

Laserwords Private Limited, Chennai, India.

Prentice Hall

is an imprint of

www.pearsonhighered.com

10 9 8 7 6 5 4 3 2 1

ISBN 13: 978-0-13-212819-3

ISBN 10: 0-13-212819-5

Copyright © 2011, 2004 Pearson Education, Inc., publishing as Prentice Hall, 1 Lake Street, Upper Saddle

River, NJ 07458.

All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright,

and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a

retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,

recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request

to Pearson Education, Inc., Permissions Department, imprint permissions address.

MATLAB is a registered trademark of The Math Works, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098

Many of the designations by manufacturers and seller to distinguish their products are claimed as trademarks.

Where those designations appear in this book, and the publisher was aware of a trademark claim, the

designations have been printed in initial caps or all caps.

The author and publisher of this book 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 make no warranty of any kind, expressed or implied, with regard to these programs or the

documentation contained in this book. 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

programs.

Library of Congress Cataloging-in-Publication Data

Rao, S. S.

Mechanical vibrations / Singiresu S. Rao. 5th ed.

p. cm.

Includes index.

ISBN 978-0-13-212819-3 (978-0-13-212819-3 : alk. paper) 1. Vibration. I. Title.

TA355.R37 2010

620.3 dc22 2010028534

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page ii

To Lord Sri Venkateswara

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page iii

Preface xi

Acknowledgments xv

List of Symbols xvi

CHAPTER 1

Fundamentals of Vibration 1

1.1 Preliminary Remarks 2

1.2 Brief History of the Study of Vibration 3

1.2.1 Origins of the Study of Vibration 3

1.2.2 From Galileo to Rayleigh 6

1.2.3 Recent Contributions 9

1.3 Importance of the Study of Vibration 10

1.4 Basic Concepts of Vibration 13

1.4.1 Vibration 13

1.4.2 Elementary Parts of

Vibrating Systems 13

1.4.3 Number of Degrees of Freedom 14

1.4.4 Discrete and Continuous Systems 16

1.5 Classification of Vibration 16

1.5.1 Free and Forced Vibration 17

1.5.2 Undamped and Damped Vibration 17

1.5.3 Linear and Nonlinear Vibration 17

1.5.4 Deterministic and

Random Vibration 17

1.6 Vibration Analysis Procedure 18

1.7 Spring Elements 22

1.7.1 Nonlinear Springs 23

1.7.2 Linearization of a

Nonlinear Spring 25

1.7.3 Spring Constants of Elastic Elements 27

1.7.4 Combination of Springs 30

iv

1.7.5 Spring Constant Associated with the

Restoring Force due to Gravity 39

1.8 Mass or Inertia Elements 40

1.8.1 Combination of Masses 40

1.9 Damping Elements 45

1.9.1 Construction of Viscous Dampers 46

1.9.2 Linearization of a

Nonlinear Damper 52

1.9.3 Combination of Dampers 52

1.10 Harmonic Motion 54

1.10.1 Vectorial Representation of

Harmonic Motion 56

1.10.2 Complex-Number Representation

of Harmonic Motion 57

1.10.3 Complex Algebra 58

1.10.4 Operations on Harmonic Functions 59

1.10.5 Definitions and Terminology 62

1.11 Harmonic Analysis 64

1.11.1 Fourier Series Expansion 64

1.11.2 Complex Fourier Series 66

1.11.3 Frequency Spectrum 67

1.11.4 Time- and Frequency-Domain

Representations 68

1.11.5 Even and Odd Functions 69

1.11.6 Half-Range Expansions 71

1.11.7 Numerical Computation

of Coefficients 72

1.12 Examples Using MATLAB 76

1.13 Vibration Literature 80

Chapter Summary 81

References 81

Review Questions 83

Problems 87

Design Projects 120

Contents

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page iv

CONTENTS v

CHAPTER 2

Free Vibration of Single-Degree-of-Freedom

Systems 124

2.1 Introduction 126

2.2 Free Vibration of an Undamped

Translational System 129

2.2.1 Equation of Motion Using Newton s

Second Law of Motion 129

2.2.2 Equation of Motion Using Other

Methods 130

2.2.3 Equation of Motion of a Spring-Mass

System in Vertical Position 132

2.2.4 Solution 133

2.2.5 Harmonic Motion 134

2.3 Free Vibration of an Undamped

Torsional System 146

2.3.1 Equation of Motion 147

2.3.2 Solution 148

2.4 Response of First Order Systems

and Time Constant 151

2.5 Rayleigh s Energy Method 153

2.6 Free Vibration with Viscous Damping 158

2.6.1 Equation of Motion 158

2.6.2 Solution 158

2.6.3 Logarithmic Decrement 164

2.6.4 Energy Dissipated in Viscous

Damping 166

2.6.5 Torsional Systems with Viscous

Damping 168

2.7 Graphical Representation of Characteristic Roots

and Corresponding Solutions 174

2.7.1 Roots of the Characteristic Equation 174

2.7.2 Graphical Representation of Roots and

Corresponding Solutions 175

2.8 Parameter Variations and Root Locus

Representations 176

2.8.1 Interpretations of and

in s-plane 176

2.8.2 Root Locus and Parameter

Variations 179

2.9 Free Vibration with Coulomb Damping 185

2.9.1 Equation of Motion 186

2.9.2 Solution 187

2.9.3 Torsional Systems with Coulomb

Damping 190

vn, vd, z, t

2.10 Free Vibration with Hysteretic Damping 192

2.11 Stability of Systems 198

2.12 Examples Using MATLAB 202

Chapter Summary 208

References 209

Review Questions 209

Problems 214

Design Projects 256

CHAPTER 3

Harmonically Excited Vibration 259

3.1 Introduction 261

3.2 Equation of Motion 261

3.3 Response of an Undamped System

Under Harmonic Force 263

3.3.1 Total Response 267

3.3.2 Beating Phenomenon 267

3.4 Response of a Damped System Under

Harmonic Force 271

3.4.1 Total Response 274

3.4.2 Quality Factor and Bandwidth 276

3.5 Response of a Damped System

Under 278

3.6 Response of a Damped System Under the

Harmonic Motion of the Base 281

3.6.1 Force Transmitted 283

3.6.2 Relative Motion 284

3.7 Response of a Damped System Under Rotating

Unbalance 287

3.8 Forced Vibration with Coulomb Damping 293

3.9 Forced Vibration with Hysteresis Damping 298

3.10 Forced Motion with Other Types of

Damping 300

3.11 Self-Excitation and Stability Analysis 301

3.11.1 Dynamic Stability Analysis 301

3.11.2 Dynamic Instability Caused by Fluid

Flow 305

3.12 Transfer-Function Approach 313

3.13 Solutions Using Laplace Transforms 317

3.14 Frequency Transfer Functions 320

3.14.1 Relation Between the General Transfer

function T(s) and the Frequency Transfer

Function 322

3.14.2 Representation of Frequency-Response

Characteristics 323

T(iv)

F(t) = F0e

iVt

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page v

vi CONTENTS

3.15 Examples Using MATLAB 326

Chapter Summary 332

References 332

Review Questions 333

Problems 336

Design Projects 362

CHAPTER 4

Vibration Under General Forcing

Conditions 363

4.1 Introduction 364

4.2 Response Under a General

Periodic Force 365

4.2.1 First-Order Systems 366

4.2.2 Second-Order Systems 372

4.3 Response Under a Periodic Force

of Irregular Form 378

4.4 Response Under a Nonperiodic Force 380

4.5 Convolution Integral 381

4.5.1 Response to an Impulse 382

4.5.2 Response to a General Forcing

Condition 385

4.5.3 Response to Base Excitation 386

4.6 Response Spectrum 394

4.6.1 Response Spectrum for Base

Excitation 396

4.6.2 Earthquake Response Spectra 399

4.6.3 Design Under a Shock

Environment 403

4.7 Laplace Transform 406

4.7.1 Transient and Steady-State

Responses 406

4.7.2 Response of First-Order Systems 407

4.7.3 Response of Second-Order Systems 409

4.7.4 Response to Step Force 414

4.7.5 Analysis of the Step Response 420

4.7.6 Description of Transient

Response 421

4.8 Numerical Methods 428

4.8.1 Runge-Kutta Methods 429

4.9 Response to Irregular Forcing Conditions Using

Numerical Methods 431

4.10 Examples Using MATLAB 436

Chapter Summary 440

References 440

Review Questions 441

Problems 444

Design Projects 465

CHAPTER 5

Two-Degree-of-Freedom Systems 467

5.1 Introduction 468

5.2 Equations of Motion for Forced

Vibration 472

5.3 Free Vibration Analysis of an Undamped

System 474

5.4 Torsional System 483

5.5 Coordinate Coupling and Principal

Coordinates 488

5.6 Forced-Vibration Analysis 494

5.7 Semidefinite Systems 497

5.8 Self-Excitation and Stability

Analysis 500

5.9 Transfer-Function Approach 502

5.10 Solutions Using Laplace Transform 504

5.11 Solutions Using Frequency Transfer

Functions 512

5.12 Examples Using MATLAB 515

Chapter Summary 522

References 523

Review Questions 523

Problems 526

Design Projects 552

CHAPTER 6

Multidegree-of-Freedom Systems 553

6.1 Introduction 555

6.2 Modeling of Continuous Systems as Multidegree￾of-Freedom Systems 555

6.3 Using Newton s Second Law to Derive Equations

of Motion 557

6.4 Influence Coefficients 562

6.4.1 Stiffness Influence Coefficients 562

6.4.2 Flexibility Influence Coefficients 567

6.4.3 Inertia Influence Coefficients 572

6.5 Potential and Kinetic Energy Expressions in

Matrix Form 574

6.6 Generalized Coordinates and Generalized

Forces 576

6.7 Using Lagrange s Equations to Derive Equations

of Motion 577

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page vi

CONTENTS vii

6.8 Equations of Motion of Undamped Systems in

Matrix Form 581

6.9 Eigenvalue Problem 583

6.10 Solution of the Eigenvalue Problem 585

6.10.1 Solution of the Characteristic

(Polynomial) Equation 585

6.10.2 Orthogonality of Normal Modes 591

6.10.3 Repeated Eigenvalues 594

6.11 Expansion Theorem 596

6.12 Unrestrained Systems 596

6.13 Free Vibration of Undamped Systems 601

6.14 Forced Vibration of Undamped Systems Using

Modal Analysis 603

6.15 Forced Vibration of Viscously Damped

Systems 610

6.16 Self-Excitation and Stability Analysis 617

6.17 Examples Using MATLAB 619

Chapter Summary 627

References 627

Review Questions 628

Problems 632

Design Project 653

CHAPTER 7

Determination of Natural Frequencies and

Mode Shapes 654

7.1 Introduction 655

7.2 Dunkerley s Formula 656

7.3 Rayleigh s Method 658

7.3.1 Properties of Rayleigh s Quotient 659

7.3.2 Computation of the Fundamental Natural

Frequency 661

7.3.3 Fundamental Frequency of Beams and

Shafts 663

7.4 Holzer s Method 666

7.4.1 Torsional Systems 666

7.4.2 Spring-Mass Systems 669

7.5 Matrix Iteration Method 670

7.5.1 Convergence to the Highest Natural

Frequency 672

7.5.2 Computation of Intermediate Natural

Frequencies 673

7.6 Jacobi s Method 678

7.7 Standard Eigenvalue Problem 680

7.7.1 Choleski Decomposition 681

7.7.2 Other Solution Methods 683

7.8 Examples Using MATLAB 683

Chapter Summary 686

References 686

Review Questions 688

Problems 690

Design Projects 698

CHAPTER 8

Continuous Systems 699

8.1 Introduction 700

8.2 Transverse Vibration of a String or

Cable 701

8.2.1 Equation of Motion 701

8.2.2 Initial and Boundary Conditions 703

8.2.3 Free Vibration of a Uniform

String 704

8.2.4 Free Vibration of a String with Both Ends

Fixed 705

8.2.5 Traveling-Wave Solution 709

8.3 Longitudinal Vibration of a Bar or Rod 710

8.3.1 Equation of Motion

and Solution 710

8.3.2 Orthogonality of Normal

Functions 713

8.4 Torsional Vibration of a Shaft or Rod 718

8.5 Lateral Vibration of Beams 721

8.5.1 Equation of Motion 721

8.5.2 Initial Conditions 723

8.5.3 Free Vibration 723

8.5.4 Boundary Conditions 724

8.5.5 Orthogonality of Normal

Functions 726

8.5.6 Forced Vibration 730

8.5.7 Effect of Axial Force 732

8.5.8 Effects of Rotary Inertia and Shear

Deformation 734

8.5.9 Other Effects 739

8.6 Vibration of Membranes 739

8.6.1 Equation of Motion 739

8.6.2 Initial and Boundary Conditions 741

8.7 Rayleigh s Method 742

8.8 The Rayleigh-Ritz Method 745

8.9 Examples Using MATLAB 748

Chapter Summary 751

References 751

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page vii

viii CONTENTS

Review Questions 753

Problems 756

Design Project 768

CHAPTER 9

Vibration Control 769

9.1 Introduction 770

9.2 Vibration Nomograph and Vibration

Criteria 771

9.3 Reduction of Vibration at the Source 775

9.4 Balancing of Rotating Machines 776

9.4.1 Single-Plane Balancing 776

9.4.2 Two-Plane Balancing 779

9.5 Whirling of Rotating Shafts 785

9.5.1 Equations of Motion 785

9.5.2 Critical Speeds 787

9.5.3 Response of the System 788

9.5.4 Stability Analysis 790

9.6 Balancing of Reciprocating Engines 792

9.6.1 Unbalanced Forces Due to Fluctuations in

Gas Pressure 792

9.6.2 Unbalanced Forces Due to Inertia of the

Moving Parts 793

9.6.3 Balancing of Reciprocating

Engines 796

9.7 Control of Vibration 798

9.8 Control of Natural Frequencies 798

9.9 Introduction of Damping 799

9.10 Vibration Isolation 801

9.10.1 Vibration Isolation System with Rigid

Foundation 804

9.10.2 Vibration Isolation System with Base

Motion 814

9.10.3 Vibration Isolation System with Flexible

Foundation 821

9.10.4 Vibration Isolation System with Partially

Flexible Foundation 822

9.10.5 Shock Isolation 824

9.10.6 Active Vibration Control 827

9.11 Vibration Absorbers 832

9.11.1 Undamped Dynamic Vibration

Absorber 833

9.11.2 Damped Dynamic Vibration

Absorber 840

9.12 Examples Using MATLAB 843

Chapter Summary 851

References 851

Review Questions 853

Problems 855

Design Project 869

CHAPTER 10

Vibration Measurement and

Applications 870

10.1 Introduction 871

10.2 Transducers 873

10.2.1 Variable Resistance Transducers 873

10.2.2 Piezoelectric Transducers 876

10.2.3 Electrodynamic Transducers 877

10.2.4 Linear Variable Differential Transformer

Transducer 878

10.3 Vibration Pickups 879

10.3.1 Vibrometer 881

10.3.2 Accelerometer 882

10.3.3 Velometer 886

10.3.4 Phase Distortion 888

10.4 Frequency-Measuring Instruments 890

10.5 Vibration Exciters 892

10.5.1 Mechanical Exciters 892

10.5.2 Electrodynamic Shaker 893

10.6 Signal Analysis 895

10.6.1 Spectrum Analyzers 896

10.6.2 Bandpass Filter 897

10.6.3 Constant-Percent Bandwidth and

Constant-Bandwidth Analyzers 898

10.7 Dynamic Testing of Machines

and Structures 900

10.7.1 Using Operational Deflection-Shape

Measurements 900

10.7.2 Using Modal Testing 900

10.8 Experimental Modal Analysis 900

10.8.1 The Basic Idea 900

10.8.2 The Necessary Equipment 900

10.8.3 Digital Signal Processing 903

10.8.4 Analysis of Random Signals 905

10.8.5 Determination of Modal Data

from Observed Peaks 907

10.8.6 Determination of Modal Data

from Nyquist Plot 910

10.8.7 Measurement of Mode Shapes 912

10.9 Machine Condition Monitoring

and Diagnosis 915

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page viii

CONTENTS ix

10.9.1 Vibration Severity Criteria 915

10.9.2 Machine Maintenance Techniques 915

10.9.3 Machine Condition Monitoring

Techniques 916

10.9.4 Vibration Monitoring Techniques 918

10.9.5 Instrumentation Systems 924

10.9.6 Choice of Monitoring Parameter 924

10.10 Examples Using MATLAB 925

Chapter Summary 928

References 928

Review Questions 930

Problems 932

Design Projects 938

CHAPTER 11

Numerical Integration Methods in

Vibration Analysis 939

11.1 Introduction 940

11.2 Finite Difference Method 941

11.3 Central Difference Method for Single-Degree-of￾Freedom Systems 942

11.4 Runge-Kutta Method for Single-Degree-of￾Freedom Systems 945

11.5 Central Difference Method for Multidegree-of￾Freedom Systems 947

11.6 Finite Difference Method for Continuous

Systems 951

11.6.1 Longitudinal Vibration of Bars 951

11.6.2 Transverse Vibration of Beams 955

11.7 Runge-Kutta Method for Multidegree-of￾Freedom Systems 960

11.8 Houbolt Method 962

11.9 Wilson Method 965

11.10 Newmark Method 968

11.11 Examples Using MATLAB 972

Chapter Summary 978

References 978

Review Questions 979

Problems 981

CHAPTER 12

Finite Element Method 987

12.1 Introduction 988

12.2 Equations of Motion of an Element 989

12.3 Mass Matrix, Stiffness Matrix, and Force

Vector 991

12.3.1 Bar Element 991

12.3.2 Torsion Element 994

12.3.3 Beam Element 995

12.4 Transformation of Element Matrices and

Vectors 998

12.5 Equations of Motion of the Complete System of

Finite Elements 1001

12.6 Incorporation of Boundary

Conditions 1003

12.7 Consistent- and Lumped-Mass Matrices 1012

12.7.1 Lumped-Mass Matrix for a Bar

Element 1012

12.7.2 Lumped-Mass Matrix for a Beam

Element 1012

12.7.3 Lumped-Mass Versus Consistent-Mass

Matrices 1013

12.8 Examples Using MATLAB 1015

Chapter Summary 1019

References 1019

Review Questions 1020

Problems 1022

Design Projects 1034

Chapters 13 and 14 are provided as downloadable

files on the Companion Website.

CHAPTER 13

Nonlinear Vibration 13-1

13.1 Introduction 13-2

13.2 Examples of Nonlinear Vibration Problems 13-3

13.2.1 Simple Pendulum 13-3

13.2.2 Mechanical Chatter, Belt Friction

System 13-5

13.2.3 Variable Mass System 13-5

13.3 Exact Methods 13-6

13.4 Approximate Analytical Methods 13-7

13.4.1 Basic Philosophy 13-8

13.4.2 Lindstedt s Perturbation Method 13-10

13.4.3 Iterative Method 13-13

13.4.4 Ritz-Galerkin Method 13-17

13.5 Subharmonic and Superharmonic

Oscillations 13-19

13.5.1 Subharmonic Oscillations 13-20

13.5.2 Superharmonic Oscillations 13-23

13.6 Systems with Time-Dependent Coefficients

(Mathieu Equation) 13-24

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page ix

13.7 Graphical Methods 13-29

13.7.1 Phase-Plane Representation 13-29

13.7.2 Phase Velocity 13-34

13.7.3 Method of Constructing

Trajectories 13-34

13.7.4 Obtaining Time Solution from Phase

Plane Trajectories 13-36

13.8 Stability of Equilibrium States 13-37

13.8.1 Stability Analysis 13-37

13.8.2 Classification of Singular

Points 13-40

13.9 Limit Cycles 13-41

13.10 Chaos 13-43

13.10.1 Functions with Stable Orbits 13-45

13.10.2 Functions with Unstable Orbits 13-45

13.10.3 Chaotic Behavior of Duffing s Equation

Without the Forcing Term 13-47

13.10.6 Chaotic Behavior of Duffing s Equation

with the Forcing Term 13-50

13.11 Numerical Methods 13-52

13.12 Examples Using MATLAB 13-53

Chapter Summary 13-62

References 13-62

Review Questions 13-64

Problems 13-67

Design Projects 13-75

CHAPTER 14

Random Vibration 14-1

14.1 Introduction 14-2

14.2 Random Variables and Random Processes 14-3

14.3 Probability Distribution 14-4

14.4 Mean Value and Standard Deviation 14-6

14.5 Joint Probability Distribution of Several

Random Variables 14-7

14.6 Correlation Functions of a Random Process 14-9

14.7 Stationary Random Process 14-10

14.8 Gaussian Random Process 14-14

14.9 Fourier Analysis 14-16

14.9.1 Fourier Series 14-16

14.9.2 Fourier Integral 14-19

14.10 Power Spectral Density 14-23

14.11 Wide-Band and Narrow-Band Processes 14-25

14.12 Response of a Single-Degree-of￾Freedom System 14-28

14.12.1 Impulse-Response Approach 14-28

14.12.2 Frequency-Response Approach 14-30

14.12.3 Characteristics of the Response

Function 14-30

14.13 Response Due to Stationary Random

Excitations 14-31

14.13.1 Impulse-Response Approach 14-32

14.13.2 Frequency-Response Approach 14-33

14.14 Response of a Multidegree-of-Freedom

System 14-39

14.15 Examples Using MATLAB 14-46

Chapter Summary 14-49

References 14-49

Review Questions 14-50

Problems 14-53

Design Project 14-61

APPENDIX A

Mathematical Relationships and Material

Properties 1036

APPENDIX B

Deflection of Beams and Plates 1039

APPENDIX C

Matrices 1041

APPENDIX D

Laplace Transform 1048

APPENDIX E

Units 1056

APPENDIX F

Introduction to MATLAB 1059

Answers to Selected Problems 1069

Index 1077

x CONTENTS

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page x

Preface

Changes in this Edition

This book serves as an introduction to the subject of vibration engineering at the undergraduate level. Favorable

reactions by professors and students to the fourth edition have encouraged me to prepare this fifth edition of the

book. I have retained the style of the prior editions, presenting the theory, computational aspects, and applications

of vibration in as simple a manner as possible, and emphasizing computer techniques of analysis. Expanded expla￾nations of the fundamentals are given, emphasizing physical significance and interpretation that build upon previ￾ous experiences in undergraduate mechanics. Numerous examples and problems are used to illustrate principles

and concepts.

In this edition some topics are modified and rewritten, many new topics are added and several new features

have been introduced. Most of the additions and modifications were suggested by users of the text and by reviewers.

Important changes include the following:

1. Chapter outline and learning objectives are stated at the beginning of each chapter.

2. A chapter summary is given at the end of each chapter.

3. The presentation of some of the topics is modified for expanded coverage and better clarity. These topics

include the basic components of vibration spring elements, damping elements and mass or inertia elements,

vibration isolation, and active vibration control.

4. Many new topics are presented in detail with illustrative examples. These include the response of first-order

systems and time constant, graphical representation of characteristic roots and solutions, parameter variations

and root locus representation, stability of systems, transfer-function approach for forced-vibration problems,

Laplace transform approach for the solution of free- and forced-vibration problems, frequency transfer-function

approach, Bode diagram for damped single-degree-of-freedom systems, step response and description of

transient response, and inelastic and elastic impacts.

5. I have added 128 new examples, 160 new problems, 70 new review questions, and 107 new illustrations.

6. The C++ and Fortran program-based examples and problems given at the end of every chapter in the pre￾vious edition have been deleted.

Features of the Book

Each topic in Mechanical Vibrations is self-contained, with all concepts fully explained and the derivations

presented in complete detail.

Computational aspects are emphasized throughout the book. MATLAB-based examples as well as sev￾eral general-purpose MATLAB programs with illustrative examples are given in the last section of every

xi

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page xi

chapter. Numerous problems requiring the use of MATLAB or MATLAB programs (given in the text) are

included at the end of every chapter.

Certain topics are presented in a somewhat unconventional manner in particular, the topics of Chapters

9, 10 and 11. Most textbooks discuss isolators, absorbers, and balancing in different chapters. Since one of

the main purposes of the study of vibrations is to control vibration response, all topics related to vibration

control are given in Chapter 9. The vibration-measuring instruments, along with vibration exciters, exper￾imental modal analysis procedure, and machine-condition monitoring, are presented together in Chapter 10.

Similarly, all the numerical integration methods applicable to single- and multidegree-of-freedom systems,

as well as continuous systems, are unified in Chapter 11.

Specific features include the following:

More than 240 illustrative examples are given to accompany most topics.

More than 980 review questions are included to help students in reviewing and testing their understand￾ing of the text material. The review questions are in the form of multiple-choice questions, questions with

brief answers, true-false questions, questions involving matching of related descriptions, and fill-in-the￾blank type questions.

An extensive set of problems in each chapter emphasizes a variety of applications of the material cov￾ered in that chapter. In total there are more than 1150 problems. Solutions are provided in the instruc￾tor s manual.

More than 30 design project-type problems, many with no unique solution, are given at the end of vari￾ous chapters.

More than 25 MATLAB programs are included to aid students in the numerical implementation of the

methods discussed in the text.

Biographical information about 20 scientists and engineers who contributed to the development of the

theory of vibrations is presented on the opening pages of chapters and appendixes.

MATLAB programs given in the book, answers to problems, and answers to review questions can be

found at the Companion Website, www.pearsonhighered.com/rao. The Solutions Manual with solutions

to all problems and hints to design projects is available to instructors who adopt the text for their courses.

Units and Notation

Both the SI and the English system of units are used in the examples and problems. A list of symbols, along with

the associated units in SI and English systems, appears after the Acknowledgments. A brief discussion of SI units

as they apply to the field of vibrations is given in Appendix E. Arrows are used over symbols to denote column

vectors, and square brackets are used to indicate matrices.

Organization of Material

Mechanical Vibrations is organized into 14 chapters and 6 appendixes. Chapters 13 and 14 are provided as down￾loadable files on the Companion Website. The reader is assumed to have a basic knowledge of statics, dynamics,

strength of materials, and differential equations. Although some background in matrix theory and Laplace trans￾form is desirable, an overview of these topics is given in Appendixes C and D, respectively.

Chapter 1 starts with a brief discussion of the history and importance of vibrations. The modeling of practical

systems for vibration analysis along with the various steps involved in vibration analysis are discussed. A description

is given of the elementary parts of a vibrating system stiffness, damping, and mass (inertia). The basic concepts and

xii PREFACE

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page xii

PREFACE xiii

terminology used in vibration analysis are introduced. The free-vibration analysis of single-degree-of-freedom

undamped and viscously damped translational and torsional systems is given in Chapter 2. The graphical repre￾sentation of characteristic roots and corresponding solutions, the parameter variations, and root locus representa￾tions are discussed. Although the root locus method is commonly used in control systems, its use in vibration is

illustrated in this chapter. The response under Coulomb and hysteretic damping is also considered. The undamped

and damped responses of single-degree-of-freedom systems to harmonic excitations are considered in Chapter 3.

The concepts of force and displacement transmissibilities and their application in practical systems are outlined.

The transfer-function approach, the Laplace transform solution of forced-vibration problems, the frequency￾response and the Bode diagram are presented.

Chapter 4 is concerned with the response of a single-degree-of-freedom system under general forcing

function. The roles of Fourier series expansion of a periodic function, convolution integral, Laplace trans￾form, and numerical methods are outlined with illustrative examples. The specification of the response of an

underdamped system in terms of peak time, rise time, and settling time is also discussed. The free and forced

vibration of two-degree-of-freedom systems is considered in Chapter 5. The self-excited vibration and sta￾bility of the system are discussed. The transfer-function approach and the Laplace transform solution of

undamped and dampled systems are also presented with illustrative examples. Chapter 6 presents the vibra￾tion analysis of multidegree-of-freedom systems. Matrix methods of analysis are used for presentation of the

theory. The modal analysis procedure is described for the solution of forced-vibration problems in this chap￾ter. Several methods of determining the natural frequencies and mode shapes of discrete systems are outlined

in Chapter 7. The methods of Dunkerley, Rayleigh, Holzer, Jacobi, and matrix iteration are discussed with

numerical examples.

While the equations of motion of discrete systems are in the form of ordinary differential equations, those

of continuous or distributed systems are in the form of partial differential equations. The vibration analysis of

continuous systems, including strings, bars, shafts, beams, and membranes, is given in Chapter 8. The method

of separation of variables is presented for the solution of the partial differential equations associated with con￾tinuous systems. The Rayleigh and Rayleigh-Ritz methods of finding the approximate natural frequencies are

also described with examples. Chapter 9 discusses the various aspects of vibration control, including the prob￾lems of elimination, isolation, and absorption. The vibration nomograph and vibration criteria which indicate

the acceptable levels of vibration are also presented. The balancing of rotating and reciprocating machines and

the whirling of shafts are considered. The active control techniques are also outlined for controlling the response

of vibrating systems. The experimental methods used for vibration-response measurement are considered in

Chapter 10. Vibration-measurement hardware and signal analysis techniques are described. Machine-condition

monitoring and diagnosis techniques are also presented.

Chapter 11 presents several numerical integration techniques for finding the dynamic response of discrete and

continuous systems. The central difference, Runge-Kutta, Houbolt, Wilson, and Newmark methods are discussed

and illustrated. Finite element analysis, with applications involving one-dimensional elements, is discussed in

Chapter 12. Bar, rod, and beam elements are used for the static and dynamic analysis of trusses, rods under tor￾sion, and beams. The use of consistent- and lumped-mass matrices in the vibration analysis is also discussed in

this chapter. Nonlinear vibration problems are governed by nonlinear differential equations and exhibit phenom￾ena that are not predicted or even hinted at by the corresponding linearized problems. An introductory treatment

of nonlinear vibration, including a discussion of subharmonic and superharmonic oscillations, limit cycles, sys￾tems with time-dependent coefficients, and chaos, is given in Chapter 13. The random vibration of linear vibration

systems is considered in Chapter 14. The concepts of random process, stationary process, power spectral density,

autocorrelation, and wide- and narrow-band processes are explained. The random vibration response of single- and

multidegree-of-freedom systems is discussed in this chapter.

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page xiii

Appendixes A and B focus on mathematical relationships and deflection of beams and plates, respectively.

The basics of matrix theory, Laplace transform, and SI units are presented in Appendixes C, D, and E, respectively.

Finally, Appendix F provides an introduction to MATLAB programming.

Typical Syllabi

The material of the book provides flexible options for different types of vibration courses. Chapters 1 through 5,

Chapter 9, and portions of Chapter 6 constitute a basic course in mechanical vibration. Different emphases/orien￾tations can be given to the course by covering, additionally, different chapters as indicated below:

Chapter 8 for continuous or distributed systems.

Chapters 7 and 11 for numerical solutions.

Chapter 10 for experimental methods and signal analysis.

Chapter 12 for finite element analysis.

Chapter 13 for nonlinear analysis.

Chapter 14 for random vibration.

Alternatively, in Chapters 1 through 14, the text has sufficient material for a one-year sequence of two vibra￾tion courses at the senior or dual level.

Expected Course Outcomes

The material presented in the text helps achieve some of the program outcomes specified by ABET (Accreditation

Board for Engineering and Technology):

Ability to apply knowledge of mathematics, science, and engineering:

The subject of vibration, as presented in the book, applies knowledge of mathematics (differential equa￾tions, matrix algebra, vector methods, and complex numbers) and science (statics and dynamics) to solve

engineering vibration problems.

Ability to identify, formulate, and solve engineering problems:

Numerous illustrative examples, problems for practice, and design projects help the student identify various

types of practical vibration problems and develop mathematical models, analyze, solve to find the response,

and interpret the results.

Ability to use the techniques, skills, and modern engineering tools necessary for engineering practice:

The application of the modern software, MATLAB, for the solution of vibration problems is illustrated

in the last section of each chapter. The basics of MATLAB programming are summarized in Appendix F.

The use of the modern analysis technique, the finite element method, for the solution of vibration prob￾lems is covered in a separate chapter (Chapter 12). The finite element method is a popular technique

used in industry for the modeling, analysis, and solution of complex vibrating systems.

Ability to design and conduct experiments, as well as to analyze and interpret data:

The experimental methods and analysis of data related to vibration are presented in Chapter 10. Discussed

also are the equipment used in conducting vibration experiments, signal analysis and identification of sys￾tem parameters from the data.

xiv PREFACE

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page xiv