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 5
1.2.3 Recent Contributions 8
1.3 Importance of the Study of Vibration 10
1.3.1 Conversion of Vibrations to Sound by the Human Ear 12
1.4 Basic Concepts of Vibration 15
1.4.1 Vibration 15
1.4.2 Elementary Parts of Vibrating Systems 15
1.4.3 Number of Degrees of Freedom 16
1.4.4 Discrete and Continuous Systems 18
1.5 Classification of Vibration 18
1.5.1 Free and Forced Vibration 18
1.5.2 Undamped and Damped Vibration 19
1.5.3 Linear and Nonlinear Vibration 19
1.5.4 Deterministic and Random Vibration 19
1.6 Vibration Analysis Procedure 20
1.7 Spring Elements 24
1.7.1 Nonlinear Springs 25
1.7.2 Linearization of a Nonlinear Spring 27
1.7.3 Spring Constants of Elastic Elements 29
1.7.4 Combination of Springs 32
1.7.5 Spring Constant Associated with the Restoring Force due to Gravity 40
1.8 Mass or Inertia Elements 41
1.8.1 Combination of Masses 42
1.9 Damping Elements 46
1.9.1 Construction of Viscous Dampers 47
1.9.2 Linearization of a Nonlinear Damper 53
1.9.3 Combination of Dampers 53
1.10 Harmonic Motion 55
1.10.1 Vectorial Representation of Harmonic Motion 57
1.10.2 Complex-Number Representation of Harmonic Motion 58
1.10.3 Complex Algebra 59
1.10.4 Operations on Harmonic Functions 59
1.10.5 Definitions and Terminology 62
1.11 Harmonic Analysis 65
1.11.1 Fourier Series Expansion 65
1.11.2 Complex Fourier Series 67
1.11.3 Frequency Spectrum 68
1.11.4 Time- and Frequency-Domain Representations 69
1.11.5 Even and Odd Functions 70
1.11.6 Half-Range Expansions 72
1.11.7 Numerical Computation of Coefficients 73
1.12 Examples Using MATLAB 77
1.13 Vibration Literature 81
Chapter Summary 82
References 82
Review Questions 84
Problems 88
Design Projects 121
CHAPTER 2 Free Vibration of Single-Degree-of-Freedom Systems 125
2.1 Introduction 127
2.2 Free Vibration of an Undamped Translational System 130
2.2.1 Equation of Motion Using Newton’s Second Law of Motion 130
2.2.2 Equation of Motion Using Other Methods 131
2.2.3 Equation of Motion of a Spring-Mass System in Vertical Position 133
2.2.4 Solution 134
2.2.5 Harmonic Motion 135
2.3 Free Vibration of an Undamped Torsional System 148
2.3.1 Equation of Motion 149
2.3.2 Solution 150
2.4 Response of First-Order Systems and Time Constant 153
2.5 Rayleigh’s Energy Method 155
2.6 Free Vibration with Viscous Damping 160
2.6.1 Equation of Motion 160
2.6.2 Solution 161
2.6.3 Logarithmic Decrement 170
2.6.4 Energy Dissipated in Viscous Damping 171
2.6.5 Torsional Systems with Viscous Damping 173
2.7 Graphical Representation of Characteristic Roots and Corresponding Solutions 179
2.7.1 Roots of the Characteristic Equation 179
2.7.2 Graphical Representation of Roots and Corresponding Solutions 180
2.8 Parameter Variations and Root Locus Representations 181
2.8.1 Interpretations of ωn，ωd，ζ，and τ in the s-plane 181
2.8.2 Root Locus and Parameter Variations 184
2.9 Free Vibration with Coulomb Damping 190
2.9.1 Equation of Motion 191
2.9.2 Solution 192
2.9.3 Torsional Systems with Coulomb Damping 195
2.10 Free Vibration with Hysteretic Damping 197
2.11 Stability of Systems 203
2.12 Examples Using MATLAB 207
Chapter Summary 213
References 214
Review Questions 214
Problems 219
Design Projects 266
CHAPTER3 Harmonically Excited Vibration 269
3.1 Introduction 271
3.2 Equation of Motion 271
3.3 Response of an Undamped System Under Harmonic Force 273
3.3.1 Total Response 277
3.3.2 Beating Phenomenon 277
3.4 Response of a Damped System Under Harmonic Force 281
3.4.1 Total Response 284
3.4.2 Quality Factor and Bandwidth 288
3.5 Response of a Damped System Under F（t） = F0eiωt 289
3.6 Response of a Damped System Under the Harmonic Motion of the Base 292
3.6.1 Force Transmitted 294
3.6.2 Relative Motion 295
3.7 Response of a Damped System Under Rotating Unbalance 298
3.8 Forced Vibration with Coulomb Damping 304
3.9 Forced Vibration with Hysteresis Damping 309
3.10 Forced Motion with Other Types of Damping 311
3.11 Self-Excitation and Stability Analysis 312
3.11.1 Dynamic Stability Analysis 312
3.11.2 Dynamic Instability Caused by Fluid Flow 316
3.12 Transfer-Function Approach 324
3.13 Solutions Using Laplace Transforms 328
3.14 Frequency Transfer Functions 331
3.14.1 Relation between the General Transfer Function T（s） and the Frequency Transfer Function T（iω） 333
3.14.2 Representation of Frequency-Response Characteristics 334
3.15 Examples Using MATLAB 337
Chapter Summary 343
References 343
Review Questions 344
Problems 347
Design Projects 374
CHAPTER 4 Vibration Under General Forcing Conditions 375
4.1 Introduction 376
4.2 Response Under a General Periodic Force 377
4.2.1 First-Order Systems 378
4.2.2 Second-Order Systems 384
4.3 Response Under a Periodic Force of Irregular Form 390
4.4 Response Under a Nonperiodic Force 392
4.5 Convolution Integral 393
4.5.1 Response to an Impulse 394
4.5.2 Response to a General Forcing Condition 397
4.5.3 Response to Base Excitation 398
4.6 Response Spectrum 406
4.6.1 Response Spectrum for Base Excitation 408
4.6.2 Earthquake Response Spectra 411
4.6.3 Design Under a Shock Environment 415
4.7 Laplace Transforms 418
4.7.1 Transient and Steady-State Responses 418
4.7.2 Response of First-Order Systems 419
4.7.3 Response of Second-Order Systems 421
4.7.4 Response to Step Force 426
4.7.5 Analysis of the Step Response 432
4.7.6 Description of Transient Response 433
4.8 Numerical Methods 439
4.8.1 Runge-Kutta Methods 441
4.9 Response to Irregular Forcing Conditions Using Numerical Methods 443
4.10 Examples Using MATLAB 448
Chapter Summary 452
References 452
Review Questions 453
Problems 456
Design Projects 478
CHAPTER5 Two-Degree-of-Freedom Systems 481
5.1 Introduction 482
5.2 Equations of Motion for ForcedV ibration 486
5.3 Free-Vibration Analysis of an Undamped System 488
5.4 Torsional System 497
5.5 Coordinate Coupling and Principal Coordinates 502
5.6 Forced-Vibration Analysis 508
5.7 Semidetinite Systems 511
5.8 Self-Excitation and Stability Analysis 514
5.9 Transfer-Function Approach 516
5.10 Solutions Using Laplace Transform 518
5.11 Solutions Using Frequency Transfer Functions 526
5.12 Examples Using MATLAB 529
Chapter Summary 536
References 537
Review Questions 537
Problems 540
Design Projects 566
CHAPTER 6 Multidegree-of-Freedom Systems 568
6.1 Introduction 570
6.2 Modeling of Continuous Systems as Multidegree-of-Freedom Systems 570
6.3 Using Newton’s Second Law to Derive Equations of Motion 572
6.4 Influence Coefficients 577
6.4.1 Stiffness Influence Coefficients 577
6.4.2 Flexibility Influence Coefficients 582
6.4.3 Inertia Influence Coefficients 587
6.5 Potential and Kinetic Energy Expressions in Matrix Form 589
6.6 Generalized Coordinates and Generalized Forces 591
6.7 Using Lagrange’s Equations to Derive Equations of Motion 592
6.8 Equations of Motion of Undamped Systems in Matrix Form 596
6.9 Eigenvalue Problem 598
6.10 Solution of the Eigenvalue Problem 600
6.10.1 Solution of the Characteristic（Polynomial）Equation 600
6.10.2 Orthogonality of Normal Modes 606
6.10.3 Repeated Eigenvalues 609
6.11 Expansion Theorem 611
6.12 Unrestrained Systems 611
6.13 Free Vibration of Undamped Systems 616
6.14 Forced Vibration of Undamped Systems Using Modal Analysis 618
6.15 Forced Vibration of Viscously Damped Systems 625
6.16 Self-Excitation and Stability Analysis 632
6.17 Examples Using MATLAB 634
Chapter Summary 642
References 642
Review Questions 643
Problems 647
Design Projects 668
CHAPTER7 Determination of Natural Frequencies and Mode Shapes 671
7.1 Introduction 672
7.2 Dunkerley’s Formula 673
7.3 Rayleigh’s Method 675
7.3.1 Properties of Rayleigh’s Quotient 676
7.3.2 Computation of the Fundamental Natural Frequency 678
7.3.3 Fundamental Frequency of Beams and Shafts 680
7.4 Holzer’s Method 683
7.4.1 Torsional Systems 683
7.4.2 Spring-Mass Systems 686
7.5 Matrix Iteration Method 687
7.5.1 Convergence to the Highest Natural Frequency 689
7.5.2 Computation of Intermediate Natural Frequencies 690
7.6 Jacobi’s Method 695
7.7 Standard Eigenvalue Problem 697
7.7.1 Choleski Decomposition 698
7.7.2 Other Solution Methods 700
7.8 Examples Using MATLAB 700
Chapter Summary 703
References 703
Review Questions 705
Problems 707
Design Projects 716
CHAPTER 8 Continuous Systems 717
8.1 Introduction 718
8.2 Transverse Vibration of a String or Cable 719
8.2.1 Equation of Motion 719
8.2.2 Initial and Boundary Conditions 721
8.2.3 Free Vibration of a Uniform String 722
8.2.4 Free Vibration of a String with Both Ends Fixed 723
8.2.5 Traveling-Wave Solution 727
8.3 Longitudinal Vibration of a Bar or Rod 728
8.3.1 Equation of Motion and Solution 728
8.3.2 Orthogonality of Normal Functions 731
8.4 Torsional Vibration of a Shaft or Rod 736
8.5 Lateral Vibration of Beams 739
8.5.1 Equation of Motion 739
8.5.2 Initial Conditions 741
8.5.3 Free Vibration 741
8.5.4 Boundary Conditions 742
8.5.5 Orthogonality of Normal Functions 744
8.5.6 Forced Vibration 748
8.5.7 Effect of Axial Force 750
8.5.8 Effects of Rotary Inertia and Shear Deformation 752
8.5.9 Beams on Elastic Foundation 757
8.5.10 Other Effects 760
8.6 Vibration of Membranes 760
8.6.1 Equation of Motion 760
8.6.2 Initial and Boundary Conditions 762
8.7 Rayleigh’s Method 763
8.8 The Rayleigh-Ritz Method 766
8.9 Examples Using MATLAB 769
Chapter Summary 772
References 772
Review Questions 774
Problems 777
Design Project 790
CHAPTER 9 Vibration Control 791
9.1 Introduction 792
9.2 Vibration Nomograph and Vibration Criteria 793
9.3 Reduction of Vibration at the Source 797
9.4 Balancing of Rotating Machines 798
9.4.1 Single-Plane Balancing 798
9.4.2 Two-Plane Balancing 801
9.5 Whirling of Rotating Shafts 807
9.5.1 Equations of Motion 807
9.5.2 Critical Speeds 809
9.5.3 Response of the System 810
9.5.4 Stability Analysis 812
9.6 Balancing of Reciprocating Engines 814
9.6.1 Unbalanced Forces Due to Fluctuations in Gas Pressure 814
9.6.2 Unbalanced Forces Due to Inertia of the Moving Parts 815
9.6.3 Balancing of Reciprocating Engines 818
9.7 Control of Vibration 820
9.8 Control of Natural Frequencies 820
9.9 Introduction of Damping 821
9.10 Vibration Isolation 823
9.10.1 Vibration Isolation System with Rigid Foundation 826
9.10.2 Vibration Isolation System with Base Motion 836
9.10.3 Vibration Isolation System with Flexible Foundation 844
9.10.4 Vibration Isolation System with Partially Flexible Foundation 846
9.10.5 Shock Isolation 847
9.10.6 Active Vibration Control 850
9.11 Vibration Absorbers 855
9.11.1 Undamped Dynamic Vibration Absorber 856
9.11.2 Damped Dynamic Vibration Absorber 863
9.12 Examples Using MATLAB 867
Chapter Summary 875
References 875
Review Questions 877
Problems 879
Design Project 894
CHAPTER 10 Vibration Measurement and Applications 896
10.1 Introduction 897
10.2 Transducers 899
10.2.1 Variable-Resistance Transducers 899
10.2.2 Piezoelectric Transducers 902
10.2.3 Electrodynamic Transducers 903
10.2.4 Linear Variable Differential Transformer Transducer 904
10.3 Vibration Pickups 905
10.3.1 Vibrometer 907
10.3.2 Accelerometer 908
10.3.3 Velometer 912
10.3.4 Phase Distortion 914
10.4 Frequency-Measuring Instruments 916
10.5 Vibration Exciters 918
10.5.1 Mechanical Exciters 918
10.5.2 Electrodynamic Shaker 919
10.6 Signal Analysis 921
10.6.1 Spectrum Analyzers 922
10.6.2 Bandpass Filter 923
10.6.3 Constant-Percent Bandwidth and Constant-Bandwidth Analyzers 924
10.7 Dynamic Testing of Machines and Structures 926
10.7.1 Using Operational Deflection-Shape Measurements 926
10.7.2 Using Modal Testing 926
10.8 Experimental Modal Analysis 926
10.8.1 The Basic Idea 926
10.8.2 The Necessary Equipment 926
10.8.3 Digital Signal Processing 929
10.8.4 Analysis of Random Signals 931
10.8.5 Determination of Modal Data from Observed Peaks 933
10.8.6 Determination of Modal Data from Nyquist Plot 936
10.8.7 Measurement of Mode Shapes 938
10.9 Machine-Condition Monitoring and Diagnosis 941
10.9.1 Vibration Severity Criteria 941
10.9.2 Machine Maintenance Techniques 941
10.9.3 Machine-Condition Monitoring Techniques 942
10.9.4 Vibration Monitoring Techniques 944
10.9.5 Instrumentation Systems 950
10.9.6 Choice of Monitoring Parameter 950
10.10 Examples Using MATLAB 951
Chapter Summary 954
References 954
Review Questions 956
Problems 958
Design Projects 964
CHAPTER 11 Numerical Integration Methods in Vibration Analysis 965
11.1 Introduction 966
11.2 Finite Difference Method 967
11.3 Central Difference Method for Single-Degree-of-Freedom Systems 968
11.4 Runge-Kutta Method for Single-Degree-of-Freedom Systems 971
11.5 Central Difference Method for Multidegree-of-Freedom Systems 973
11.6 Finite Difference Method for Continuous Systems 977
11.6.1 Longitudinal Vibration of Bars 977
11.6.2 Transverse Vibration of Beams 981
11.7 Runge-Kutta Method for Multidegree-of-Freedom Systems 986
11.8 Houbolt Method 988
11.9 Wilson Method 991
11.10 Newmark Method 994
11.11 Examples Using MATLAB 998
Chapter Summary 1004
References 1004
Review Questions 1005
Problems 1007
CHAPTER 12 Finite Element Method 1013
12.1 Introduction 1014
12.2 Equations of Motion of an Element 1015
12.3 Mass Matrix，Stiffness Matrix，and Force Vector 1017
12.3.1 Bar Element 1017
12.3.2 Torsion Element 1020
12.3.3 Beam Element 1021
12.4 Transformation of Element Matrices and Vectors 1024
12.5 Equations of Motion of the Complete System of Finite Elements 1027
12.6 Incorporation of Boundary Conditions 1029
12.7 Consistent- and Lumped-Mass Matrices 1038
12.7.1 Lumped-Mass Matrix for a Bar Element 1038
12.7.2 Lumped-Mass Matrix for a Beam Element 1038
12.7.3 Lumped-Mass Versus Consistent-Mass Matrices 1039
12.8 Examples Using MATLAB 1041
Chapter Summary 1045
References 1045
Review Questions 1046
Problems 1048
Design Projects 1060
APPENDIX A 1064
Mathematical Relations and Material Properties 1064
APPENDIX B 1067
Deflection of Beams and Plates 1067
APPENDIX C 1069
Matrices 1069
APPENDIX D 1076
Laplace Transform 1076
APPENDIX E 1084
Units 1084
APPENDIX F 1087
Introduction to MATLAB 1087
Answers to Selected Problems 1097
Index 1106