Mechanical Vibrations 5th Edition by S.S. Rao

1105 Pages·2012·38.38 MB·English

A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page i
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 ii
Vice President and Editorial Director, ECS: Operations Specialist: Lisa McDowell
Marcia J. Horton Senior Marketing Manager: Tim Galligan
Senior Acquisitions Editor: Tacy Quinn Marketing Assistant: Mack Patterson
Editorial Assistant: Coleen McDonald Art Director: Jayne Conte
Senior Managing Editor: Scott Disanno Cover Designer: Suzanne Behnke
Production Liason: Irwin Zucker Art Editor: Greg Dulles
Production Editor: Sangeetha Parthasarathy, Media Editor: Daniel Sandin
Laserwords Full-Service Project Management/Composition:
Senior Operations Specialist: Alan Fischer Laserwords Private Limited, Chennai, India.
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
Prentice Hall
is an imprint of
10 9 8 7 6 5 4 3 2 1
ISBN 13: 978-0-13-212819-3
www.pearsonhighered.com ISBN 10: 0-13-212819-5
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page iii
To Lord Sri Venkateswara
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page iv
Contents
Preface xi 1.7.5 Spring Constant Associated with the
Restoring Force due to Gravity 39
Acknowledgments xv 1.8 Mass or Inertia Elements 40
1.8.1 Combination of Masses 40
List of Symbols xvi 1.9 Damping Elements 45
1.9.1 Construction of Viscous Dampers 46
1.9.2 Linearization of a
CHAPTER 1 Nonlinear Damper 52
Fundamentals of Vibration 1 1.9.3 Combination of Dampers 52
1.10 Harmonic Motion 54
1.1 Preliminary Remarks 2
1.10.1 Vectorial Representation of
1.2 Brief History of the Study of Vibration 3
Harmonic Motion 56
1.2.1 Origins of the Study of Vibration 3
1.10.2 Complex-Number Representation
1.2.2 From Galileo to Rayleigh 6
of Harmonic Motion 57
1.2.3 Recent Contributions 9
1.10.3 Complex Algebra 58
1.3 Importance of the Study of Vibration 10
1.10.4 Operations on Harmonic Functions 59
1.4 Basic Concepts of Vibration 13
1.10.5 Definitions and Terminology 62
1.4.1 Vibration 13
1.4.2 Elementary Parts of 1.11 Harmonic Analysis 64
Vibrating Systems 13 1.11.1 Fourier Series Expansion 64
1.4.3 Number of Degrees of Freedom 14 1.11.2 Complex Fourier Series 66
1.4.4 Discrete and Continuous Systems 16 1.11.3 Frequency Spectrum 67
1.5 Classification of Vibration 16 1.11.4 Time- and Frequency-Domain
1.5.1 Free and Forced Vibration 17 Representations 68
1.5.2 Undamped and Damped Vibration 17 1.11.5 Even and Odd Functions 69
1.5.3 Linear and Nonlinear Vibration 17 1.11.6 Half-Range Expansions 71
1.5.4 Deterministic and 1.11.7 Numerical Computation
Random Vibration 17 of Coefficients 72
1.6 Vibration Analysis Procedure 18 1.12 Examples Using MATLAB 76
1.7 Spring Elements 22 1.13 Vibration Literature 80
1.7.1 Nonlinear Springs 23 Chapter Summary 81
1.7.2 Linearization of a References 81
Nonlinear Spring 25 Review Questions 83
1.7.3 Spring Constants of Elastic Elements 27 Problems 87
1.7.4 Combination of Springs 30 Design Projects 120
iv
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page v
CONTENTS v
CHAPTER 2 2.10 Free Vibration with Hysteretic Damping 192
Free Vibration of Single-Degree-of-Freedom 2.11 Stability of Systems 198
2.12 Examples Using MATLAB 202
Systems 124
Chapter Summary 208
2.1 Introduction 126 References 209
2.2 Free Vibration of an Undamped Review Questions 209
Translational System 129 Problems 214
2.2.1 Equation of Motion Using Newtons Design Projects 256
Second Law of Motion 129
2.2.2 Equation of Motion Using Other
CHAPTER 3
Methods 130
Harmonically Excited Vibration 259
2.2.3 Equation of Motion of a Spring-Mass
System in Vertical Position 132 3.1 Introduction 261
2.2.4 Solution 133 3.2 Equation of Motion 261
2.2.5 Harmonic Motion 134 3.3 Response of an Undamped System
2.3 Free Vibration of an Undamped Under Harmonic Force 263
Torsional System 146 3.3.1 Total Response 267
2.3.1 Equation of Motion 147 3.3.2 Beating Phenomenon 267
2.3.2 Solution 148 3.4 Response of a Damped System Under
2.4 Response of First Order Systems Harmonic Force 271
and Time Constant 151 3.4.1 Total Response 274
2.5 Rayleighs Energy Method 153 3.4.2 Quality Factor and Bandwidth 276
2.6 Free Vibration with Viscous Damping 158 3.5 Response of a Damped System
2.6.1 Equation of Motion 158 UnderF(t) = F eiVt 278
0
2.6.2 Solution 158 3.6 Response of a Damped System Under the
2.6.3 Logarithmic Decrement 164 Harmonic Motion of the Base 281
2.6.4 Energy Dissipated in Viscous 3.6.1 Force Transmitted 283
Damping 166 3.6.2 Relative Motion 284
2.6.5 Torsional Systems with Viscous 3.7 Response of a Damped System Under Rotating
Damping 168 Unbalance 287
2.7 Graphical Representation of Characteristic Roots 3.8 Forced Vibration with Coulomb Damping 293
and Corresponding Solutions 174 3.9 Forced Vibration with Hysteresis Damping 298
2.7.1 Roots of the Characteristic Equation 174 3.10 Forced Motion with Other Types of
2.7.2 Graphical Representation of Roots and Damping 300
Corresponding Solutions 175 3.11 Self-Excitation and Stability Analysis 301
2.8 Parameter Variations and Root Locus 3.11.1 Dynamic Stability Analysis 301
Representations 176 3.11.2 Dynamic Instability Caused by Fluid
2.8.1 Interpretations of v , v , z, and t Flow 305
n d
in s-plane 176 3.12 Transfer-Function Approach 313
2.8.2 Root Locus and Parameter 3.13 Solutions Using Laplace Transforms 317
Variations 179 3.14 Frequency Transfer Functions 320
2.9 Free Vibration with Coulomb Damping 185 3.14.1 Relation Between the General Transfer
2.9.1 Equation of Motion 186 function T(s) and the Frequency Transfer
2.9.2 Solution 187 Function T(iv) 322
2.9.3 Torsional Systems with Coulomb 3.14.2 Representation of Frequency-Response
Damping 190 Characteristics 323
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page vi
vi CONTENTS
3.15 Examples Using MATLAB 326 Problems 444
Chapter Summary 332 Design Projects 465
References 332
Review Questions 333
CHAPTER 5
Problems 336
Two-Degree-of-Freedom Systems 467
Design Projects 362
5.1 Introduction 468
CHAPTER 4 5.2 Equations of Motion for Forced
Vibration 472
Vibration Under General Forcing
5.3 Free Vibration Analysis of an Undamped
Conditions 363
System 474
4.1 Introduction 364 5.4 Torsional System 483
4.2 Response Under a General 5.5 Coordinate Coupling and Principal
Periodic Force 365 Coordinates 488
4.2.1 First-Order Systems 366 5.6 Forced-Vibration Analysis 494
4.2.2 Second-Order Systems 372 5.7 Semidefinite Systems 497
4.3 Response Under a Periodic Force 5.8 Self-Excitation and Stability
of Irregular Form 378 Analysis 500
4.4 Response Under a Nonperiodic Force 380 5.9 Transfer-Function Approach 502
4.5 Convolution Integral 381 5.10 Solutions Using Laplace Transform 504
4.5.1 Response to an Impulse 382 5.11 Solutions Using Frequency Transfer
4.5.2 Response to a General Forcing Functions 512
Condition 385 5.12 Examples Using MATLAB 515
4.5.3 Response to Base Excitation 386 Chapter Summary 522
4.6 Response Spectrum 394 References 523
4.6.1 Response Spectrum for Base Review Questions 523
Excitation 396 Problems 526
4.6.2 Earthquake Response Spectra 399 Design Projects 552
4.6.3 Design Under a Shock
Environment 403
CHAPTER 6
4.7 Laplace Transform 406
Multidegree-of-Freedom Systems 553
4.7.1 Transient and Steady-State
Responses 406 6.1 Introduction 555
4.7.2 Response of First-Order Systems 407 6.2 Modeling of Continuous Systems as Multidegree-
4.7.3 Response of Second-Order Systems 409 of-Freedom Systems 555
4.7.4 Response to Step Force 414 6.3 Using Newtons Second Law to Derive Equations
4.7.5 Analysis of the Step Response 420 of Motion 557
4.7.6 Description of Transient 6.4 Influence Coefficients 562
Response 421 6.4.1 Stiffness Influence Coefficients 562
4.8 Numerical Methods 428 6.4.2 Flexibility Influence Coefficients 567
4.8.1 Runge-Kutta Methods 429 6.4.3 Inertia Influence Coefficients 572
4.9 Response to Irregular Forcing Conditions Using 6.5 Potential and Kinetic Energy Expressions in
Numerical Methods 431 Matrix Form 574
4.10 Examples Using MATLAB 436 6.6 Generalized Coordinates and Generalized
Chapter Summary 440 Forces 576
References 440 6.7 Using Lagranges Equations to Derive Equations
Review Questions 441 of Motion 577
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page vii
CONTENTS vii
6.8 Equations of Motion of Undamped Systems in 7.8 Examples Using MATLAB 683
Matrix Form 581 Chapter Summary 686
6.9 Eigenvalue Problem 583 References 686
6.10 Solution of the Eigenvalue Problem 585 Review Questions 688
6.10.1 Solution of the Characteristic Problems 690
(Polynomial) Equation 585 Design Projects 698
6.10.2 Orthogonality of Normal Modes 591
6.10.3 Repeated Eigenvalues 594
6.11 Expansion Theorem 596
CHAPTER 8
6.12 Unrestrained Systems 596
Continuous Systems 699
6.13 Free Vibration of Undamped Systems 601
6.14 Forced Vibration of Undamped Systems Using 8.1 Introduction 700
Modal Analysis 603 8.2 Transverse Vibration of a String or
6.15 Forced Vibration of Viscously Damped Cable 701
Systems 610 8.2.1 Equation of Motion 701
6.16 Self-Excitation and Stability Analysis 617 8.2.2 Initial and Boundary Conditions 703
6.17 Examples Using MATLAB 619 8.2.3 Free Vibration of a Uniform
Chapter Summary 627 String 704
References 627 8.2.4 Free Vibration of a String with Both Ends
Review Questions 628 Fixed 705
Problems 632 8.2.5 Traveling-Wave Solution 709
Design Project 653 8.3 Longitudinal Vibration of a Bar or Rod 710
8.3.1 Equation of Motion
and Solution 710
CHAPTER 7
8.3.2 Orthogonality of Normal
Determination of Natural Frequencies and
Functions 713
Mode Shapes 654
8.4 Torsional Vibration of a Shaft or Rod 718
7.1 Introduction 655 8.5 Lateral Vibration of Beams 721
7.2 Dunkerleys Formula 656 8.5.1 Equation of Motion 721
7.3 Rayleighs Method 658 8.5.2 Initial Conditions 723
7.3.1 Properties of Rayleighs Quotient 659 8.5.3 Free Vibration 723
7.3.2 Computation of the Fundamental Natural 8.5.4 Boundary Conditions 724
Frequency 661 8.5.5 Orthogonality of Normal
7.3.3 Fundamental Frequency of Beams and Functions 726
Shafts 663 8.5.6 Forced Vibration 730
7.4 Holzers Method 666 8.5.7 Effect of Axial Force 732
7.4.1 Torsional Systems 666 8.5.8 Effects of Rotary Inertia and Shear
7.4.2 Spring-Mass Systems 669 Deformation 734
7.5 Matrix Iteration Method 670 8.5.9 Other Effects 739
7.5.1 Convergence to the Highest Natural 8.6 Vibration of Membranes 739
Frequency 672 8.6.1 Equation of Motion 739
7.5.2 Computation of Intermediate Natural 8.6.2 Initial and Boundary Conditions 741
Frequencies 673 8.7 Rayleighs Method 742
7.6 Jacobis Method 678 8.8 The Rayleigh-Ritz Method 745
7.7 Standard Eigenvalue Problem 680 8.9 Examples Using MATLAB 748
7.7.1 Choleski Decomposition 681 Chapter Summary 751
7.7.2 Other Solution Methods 683 References 751
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page viii
viii CONTENTS
Review Questions 753 References 851
Problems 756 Review Questions 853
Design Project 768 Problems 855
Design Project 869
CHAPTER 9
Vibration Control 769 CHAPTER 10
Vibration Measurement and
9.1 Introduction 770
Applications 870
9.2 Vibration Nomograph and Vibration
Criteria 771 10.1 Introduction 871
9.3 Reduction of Vibration at the Source 775 10.2 Transducers 873
9.4 Balancing of Rotating Machines 776 10.2.1 Variable Resistance Transducers 873
9.4.1 Single-Plane Balancing 776 10.2.2 Piezoelectric Transducers 876
9.4.2 Two-Plane Balancing 779 10.2.3 Electrodynamic Transducers 877
9.5 Whirling of Rotating Shafts 785 10.2.4 Linear Variable Differential Transformer
9.5.1 Equations of Motion 785 Transducer 878
9.5.2 Critical Speeds 787 10.3 Vibration Pickups 879
9.5.3 Response of the System 788 10.3.1 Vibrometer 881
9.5.4 Stability Analysis 790 10.3.2 Accelerometer 882
9.6 Balancing of Reciprocating Engines 792 10.3.3 Velometer 886
9.6.1 Unbalanced Forces Due to Fluctuations in 10.3.4 Phase Distortion 888
Gas Pressure 792 10.4 Frequency-Measuring Instruments 890
9.6.2 Unbalanced Forces Due to Inertia of the 10.5 Vibration Exciters 892
Moving Parts 793 10.5.1 Mechanical Exciters 892
9.6.3 Balancing of Reciprocating 10.5.2 Electrodynamic Shaker 893
Engines 796 10.6 Signal Analysis 895
9.7 Control of Vibration 798 10.6.1 Spectrum Analyzers 896
9.8 Control of Natural Frequencies 798 10.6.2 Bandpass Filter 897
9.9 Introduction of Damping 799 10.6.3 Constant-Percent Bandwidth and
9.10 Vibration Isolation 801 Constant-Bandwidth Analyzers 898
9.10.1 Vibration Isolation System with Rigid 10.7 Dynamic Testing of Machines
Foundation 804 and Structures 900
9.10.2 Vibration Isolation System with Base 10.7.1 Using Operational Deflection-Shape
Motion 814 Measurements 900
9.10.3 Vibration Isolation System with Flexible 10.7.2 Using Modal Testing 900
Foundation 821 10.8 Experimental Modal Analysis 900
9.10.4 Vibration Isolation System with Partially 10.8.1 The Basic Idea 900
Flexible Foundation 822 10.8.2 The Necessary Equipment 900
9.10.5 Shock Isolation 824 10.8.3 Digital Signal Processing 903
9.10.6 Active Vibration Control 827 10.8.4 Analysis of Random Signals 905
9.11 Vibration Absorbers 832 10.8.5 Determination of Modal Data
9.11.1 Undamped Dynamic Vibration from Observed Peaks 907
Absorber 833 10.8.6 Determination of Modal Data
9.11.2 Damped Dynamic Vibration from Nyquist Plot 910
Absorber 840 10.8.7 Measurement of Mode Shapes 912
9.12 Examples Using MATLAB 843 10.9 Machine Condition Monitoring
Chapter Summary 851 and Diagnosis 915
A01_RAO08193_05_SE_FM.QXD 8/21/10 12:25 PM Page ix
CONTENTS ix
10.9.1 Vibration Severity Criteria 915 12.3.1 Bar Element 991
10.9.2 Machine Maintenance Techniques 915 12.3.2 Torsion Element 994
10.9.3 Machine Condition Monitoring 12.3.3 Beam Element 995
Techniques 916 12.4 Transformation of Element Matrices and
10.9.4 Vibration Monitoring Techniques 918 Vectors 998
10.9.5 Instrumentation Systems 924 12.5 Equations of Motion of the Complete System of
10.9.6 Choice of Monitoring Parameter 924 Finite Elements 1001
10.10 Examples Using MATLAB 925 12.6 Incorporation of Boundary
Chapter Summary 928 Conditions 1003
References 928 12.7 Consistent- and Lumped-Mass Matrices 1012
Review Questions 930 12.7.1 Lumped-Mass Matrix for a Bar
Problems 932 Element 1012
Design Projects 938 12.7.2 Lumped-Mass Matrix for a Beam
Element 1012
CHAPTER 11
12.7.3 Lumped-Mass Versus Consistent-Mass
Numerical Integration Methods in Matrices 1013
Vibration Analysis 939 12.8 Examples Using MATLAB 1015
Chapter Summary 1019
11.1 Introduction 940
References 1019
11.2 Finite Difference Method 941
Review Questions 1020
11.3 Central Difference Method for Single-Degree-of-
Problems 1022
Freedom Systems 942
Design Projects 1034
11.4 Runge-Kutta Method for Single-Degree-of-
Freedom Systems 945
Chapters 13 and 14 are provided as downloadable
11.5 Central Difference Method for Multidegree-of-
files on the Companion Website.
Freedom Systems 947
11.6 Finite Difference Method for Continuous
CHAPTER 13
Systems 951
Nonlinear Vibration 13-1
11.6.1 Longitudinal Vibration of Bars 951
11.6.2 Transverse Vibration of Beams 955 13.1 Introduction 13-2
11.7 Runge-Kutta Method for Multidegree-of- 13.2 Examples of Nonlinear Vibration Problems 13-3
Freedom Systems 960 13.2.1 Simple Pendulum 13-3
11.8 Houbolt Method 962 13.2.2 Mechanical Chatter, Belt Friction
11.9 Wilson Method 965 System 13-5
11.10 Newmark Method 968 13.2.3 Variable Mass System 13-5
11.11 Examples Using MATLAB 972 13.3 Exact Methods 13-6
Chapter Summary 978 13.4 Approximate Analytical Methods 13-7
References 978 13.4.1 Basic Philosophy 13-8
Review Questions 979 13.4.2 Lindstedts Perturbation Method 13-10
Problems 981 13.4.3 Iterative Method 13-13
13.4.4 Ritz-Galerkin Method 13-17
CHAPTER 12
13.5 Subharmonic and Superharmonic
Finite Element Method 987
Oscillations 13-19
12.1 Introduction 988 13.5.1 Subharmonic Oscillations 13-20
12.2 Equations of Motion of an Element 989 13.5.2 Superharmonic Oscillations 13-23
12.3 Mass Matrix, Stiffness Matrix, and Force 13.6 Systems with Time-Dependent Coefficients
Vector 991 (Mathieu Equation) 13-24