كتاب Mechanical Vibrations

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Mechanical Vibrations
Sixth Edition in SI Units
Singiresu S. Rao
University of Miami
SI Conversion by
Philip Griffin
University of Limerick, Ireland

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Contents
Preface 16
Acknowledgments 21
List of Symbols 23
ChAPtEr 1
Fundamentals of Vibration 29
1.1 Preliminary Remarks 30
1.2 Brief History of the Study of Vibration 31
1.2.1 Origins of the Study of Vibration 31
1.2.2 From Galileo to Rayleigh 33
1.2.3 Recent Contributions 36
1.3 Importance of the Study of Vibration 38
1.3.1 Conversion of Vibrations to
Sound by the Human Ear 40
1.4 Basic Concepts of Vibration 43
1.4.1 Vibration 43
1.4.2 Elementary Parts of
Vibrating Systems 43
1.4.3 Number of Degrees of Freedom 44
1.4.4 Discrete and Continuous Systems 46
1.5 Classification of Vibration 46
1.5.1 Free and Forced Vibration 46
1.5.2 Undamped and Damped Vibration 47
1.5.3 Linear and Nonlinear Vibration 47
1.5.4 Deterministic and
Random Vibration 47
1.6 Vibration Analysis Procedure 48
1.7 Spring Elements 52
1.7.1 Nonlinear Springs 53
1.7.2 Linearization of a Nonlinear Spring 55
1.7.3 Spring Constants of Elastic Elements 57
1.7.4 Combination of Springs 60
1.7.5 Spring Constant Associated with the
Restoring Force due to Gravity 68
1.8 Mass or Inertia Elements 69
1.8.1 Combination of Masses 70
1.9 Damping Elements 74
1.9.1 Construction of Viscous Dampers 75
1.9.2 Linearization of a Nonlinear Damper 81
1.9.3 Combination of Dampers 81
1.10 Harmonic Motion 83
1.10.1 Vectorial Representation of
Harmonic Motion 85
1.10.2 Complex-Number Representation
of Harmonic Motion 86
1.10.3 Complex Algebra 87
1.10.4 Operations on Harmonic Functions 87
1.10.5 Definitions and Terminology 90
1.11 Harmonic Analysis 93
1.11.1 Fourier Series Expansion 93
1.11.2 Complex Fourier Series 95
1.11.3 Frequency Spectrum 96
1.11.4 Time- and Frequency-Domain
Representations 97
1.11.5 Even and Odd Functions 98
1.11.6 Half-Range Expansions 100
1.11.7 Numerical Computation
of Coefficients 101
1.12 Examples Using MATLAB 105
1.13 Vibration Literature 109
Chapter Summary 110
References 110
Review Questions 112
Problems 116
Design Projects 14910 ContEntS
ChAPtEr 2
Free Vibration of Single-Degree-of-Freedom
Systems 153
2.1 Introduction 155
2.2 Free Vibration of an Undamped
Translational System 158
2.2.1 Equation of Motion Using Newton’s
Second Law of Motion 158
2.2.2 Equation of Motion Using Other
Methods 159
2.2.3 Equation of Motion of a Spring-Mass
System in Vertical Position 161
2.2.4 Solution 162
2.2.5 Harmonic Motion 163
2.3 Free Vibration of an Undamped
Torsional System 176
2.3.1 Equation of Motion 177
2.3.2 Solution 178
2.4 Response of First-Order Systems
and Time Constant 181
2.5 Rayleigh’s Energy Method 183
2.6 Free Vibration with Viscous Damping 188
2.6.1 Equation of Motion 188
2.6.2 Solution 189
2.6.3 Logarithmic Decrement 198
2.6.4 Energy Dissipated in Viscous
Damping 199
2.6.5 Torsional Systems with Viscous
Damping 201
2.7 Graphical Representation of Characteristic Roots
and Corresponding Solutions 207
2.7.1 Roots of the Characteristic Equation 207
2.7.2 Graphical Representation of Roots and
Corresponding Solutions 208
2.8 Parameter Variations and Root Locus
Representations 209
2.8.1 Interpretations of vn, vd, z, and t in the
s-plane 209
2.8.2 Root Locus and Parameter
Variations 212
2.9 Free Vibration with Coulomb Damping 218
2.9.1 Equation of Motion 219
2.9.2 Solution 220
2.9.3 Torsional Systems with Coulomb
Damping 223
2.10 Free Vibration with Hysteretic Damping 225
2.11 Stability of Systems 231
2.12 Examples Using MATLAB 235
Chapter Summary 241
References 242
Review Questions 242
Problems 247
Design Projects 294
ChAPtEr 3
harmonically Excited Vibration 297
3.1 Introduction 299
3.2 Equation of Motion 299
3.3 Response of an Undamped System Under
Harmonic Force 301
3.3.1 Total Response 305
3.3.2 Beating Phenomenon 305
3.4 Response of a Damped System Under Harmonic
Force 309
3.4.1 Total Response 312
3.4.2 Quality Factor and Bandwidth 316
3.5 Response of a Damped System Under
F1t2 = F0eiVt 317
3.6 Response of a Damped System Under
the Harmonic Motion of the Base 320
3.6.1 Force Transmitted 322
3.6.2 Relative Motion 323
3.7 Response of a Damped System Under Rotating
Unbalance 326
3.8 Forced Vibration with Coulomb Damping 332
3.9 Forced Vibration with Hysteresis Damping 337
3.10 Forced Motion with Other Types
of Damping 339
3.11 Self-Excitation and Stability Analysis 340
3.11.1 Dynamic Stability Analysis 340
3.11.2 Dynamic Instability Caused by Fluid
Flow 344
3.12 Transfer-Function Approach 352
3.13 Solutions Using Laplace Transforms 356
3.14 Frequency Transfer Functions 359
3.14.1 Relation between the General Transfer
Function T(s) and the Frequency Transfer
Function T1iv2 361
3.14.2 Representation of Frequency-Response
Characteristics 362ContEntS 11
3.15 Examples Using MATLAB 365
Chapter Summary 371
References 371
Review Questions 372
Problems 375
Design Projects 402
ChAPtEr 4
Vibration Under General Forcing
Conditions 403
4.1 Introduction 404
4.2 Response Under a General
Periodic Force 405
4.2.1 First-Order Systems 406
4.2.2 Second-Order Systems 412
4.3 Response Under a Periodic Force
of Irregular Form 418
4.4 Response Under a Nonperiodic Force 420
4.5 Convolution Integral 421
4.5.1 Response to an Impulse 422
4.5.2 Response to a General Forcing
Condition 425
4.5.3 Response to Base Excitation 426
4.6 Response Spectrum 434
4.6.1 Response Spectrum for Base
Excitation 436
4.6.2 Earthquake Response Spectra 439
4.6.3 Design Under a Shock Environment 443
4.7 Laplace Transforms 446
4.7.1 Transient and Steady-State
Responses 446
4.7.2 Response of First-Order Systems 447
4.7.3 Response of Second-Order
Systems 449
4.7.4 Response to Step Force 454
4.7.5 Analysis of the Step Response 460
4.7.6 Description of Transient Response 461
4.8 Numerical Methods 467
4.8.1 Runge-Kutta Methods 469
4.9 Response to Irregular Forcing Conditions Using
Numerical Methods 471
4.10 Examples Using MATLAB 476
Chapter Summary 480
References 480
Review Questions 481
Problems 484
Design Projects 506
ChAPtEr 5
two-Degree-of-Freedom Systems 509
5.1 Introduction 510
5.2 Equations of Motion for Forced
Vibration 514
5.3 Free-Vibration Analysis of an Undamped
System 516
5.4 Torsional System 525
5.5 Coordinate Coupling and Principal
Coordinates 530
5.6 Forced-Vibration Analysis 536
5.7 Semidefinite Systems 539
5.8 Self-Excitation and Stability Analysis 542
5.9 Transfer-Function Approach 544
5.10 Solutions Using Laplace Transform 546
5.11 Solutions Using Frequency Transfer
Functions 554
5.12 Examples Using MATLAB 557
Chapter Summary 564
References 565
Review Questions 565
Problems 568
Design Projects 594
ChAPtEr 6
Multidegree-of-Freedom Systems 596
6.1 Introduction 598
6.2 Modeling of Continuous Systems as Multidegreeof-Freedom Systems 598
6.3 Using Newton’s Second Law to Derive Equations
of Motion 600
6.4 Influence Coefficients 605
6.4.1 Stiffness Influence Coefficients 605
6.4.2 Flexibility Influence Coefficients 610
6.4.3 Inertia Influence Coefficients 615
6.5 Potential and Kinetic Energy Expressions in
Matrix Form 617
6.6 Generalized Coordinates and Generalized
Forces 619
6.7 Using Lagrange’s Equations to Derive Equations
of Motion 62012 ContEntS
6.8 Equations of Motion of Undamped Systems in
Matrix Form 624
6.9 Eigenvalue Problem 626
6.10 Solution of the Eigenvalue Problem 628
6.10.1 Solution of the Characteristic (Polynomial)
Equation 628
6.10.2 Orthogonality of Normal Modes 634
6.10.3 Repeated Eigenvalues 637
6.11 Expansion Theorem 639
6.12 Unrestrained Systems 639
6.13 Free Vibration of Undamped Systems 644
6.14 Forced Vibration of Undamped Systems Using
Modal Analysis 646
6.15 Forced Vibration of Viscously Damped
Systems 653
6.16 Self-Excitation and Stability Analysis 660
6.17 Examples Using MATLAB 662
Chapter Summary 670
References 670
Review Questions 671
Problems 675
Design Projects 696
ChAPtEr 7
Determination of natural Frequencies
and Mode Shapes 699
7.1 Introduction 700
7.2 Dunkerley’s Formula 701
7.3 Rayleigh’s Method 703
7.3.1 Properties of Rayleigh’s Quotient 704
7.3.2 Computation of the Fundamental Natural
Frequency 706
7.3.3 Fundamental Frequency of Beams and
Shafts 708
7.4 Holzer’s Method 711
7.4.1 Torsional Systems 711
7.4.2 Spring-Mass Systems 714
7.5 Matrix Iteration Method 715
7.5.1 Convergence to the Highest Natural
Frequency 717
7.5.2 Computation of Intermediate Natural
Frequencies 718
7.6 Jacobi’s Method 723
7.7 Standard Eigenvalue Problem 725
7.7.1 Choleski Decomposition 726
7.7.2 Other Solution Methods 728
7.8 Examples Using MATLAB 728
Chapter Summary 731
References 731
Review Questions 733
Problems 735
Design Projects 744
ChAPtEr 8
Continuous Systems 745
8.1 Introduction 746
8.2 Transverse Vibration of a String or Cable 747
8.2.1 Equation of Motion 747
8.2.2 Initial and Boundary Conditions 749
8.2.3 Free Vibration of a Uniform String 750
8.2.4 Free Vibration of a String with Both Ends
Fixed 751
8.2.5 Traveling-Wave Solution 755
8.3 Longitudinal Vibration of a Bar or Rod 756
8.3.1 Equation of Motion and Solution 756
8.3.2 Orthogonality of Normal Functions 759
8.4 Torsional Vibration of a Shaft or Rod 764
8.5 Lateral Vibration of Beams 767
8.5.1 Equation of Motion 767
8.5.2 Initial Conditions 769
8.5.3 Free Vibration 769
8.5.4 Boundary Conditions 770
8.5.5 Orthogonality of Normal
Functions 772
8.5.6 Forced Vibration 776
8.5.7 Effect of Axial Force 778
8.5.8 Effects of Rotary Inertia and Shear
Deformation 780
8.5.9 Beams on Elastic Foundation 785
8.5.10 Other Effects 788
8.6 Vibration of Membranes 788
8.6.1 Equation of Motion 788
8.6.2 Initial and Boundary Conditions 790
8.7 Rayleigh’s Method 791
8.8 The Rayleigh-Ritz Method 794
8.9 Examples Using MATLAB 797
Chapter Summary 800
References 800
Review Questions 802
Problems 805
Design Project 818ContEntS 13
ChAPtEr 9
Vibration Control 819
9.1 Introduction 820
9.2 Vibration Nomograph and Vibration
Criteria 821
9.3 Reduction of Vibration at the Source 825
9.4 Balancing of Rotating Machines 826
9.4.1 Single-Plane Balancing 826
9.4.2 Two-Plane Balancing 829
9.5 Whirling of Rotating Shafts 835
9.5.1 Equations of Motion 835
9.5.2 Critical Speeds 837
9.5.3 Response of the System 838
9.5.4 Stability Analysis 840
9.6 Balancing of Reciprocating Engines 842
9.6.1 Unbalanced Forces Due to Fluctuations in
Gas Pressure 842
9.6.2 Unbalanced Forces Due to Inertia of the
Moving Parts 843
9.6.3 Balancing of Reciprocating Engines 846
9.7 Control of Vibration 848
9.8 Control of Natural Frequencies 848
9.9 Introduction of Damping 849
9.10 Vibration Isolation 851
9.10.1 Vibration Isolation System with Rigid
Foundation 854
9.10.2 Vibration Isolation System with Base
Motion 864
9.10.3 Vibration Isolation System with Flexible
Foundation 872
9.10.4 Vibration Isolation System with Partially
Flexible Foundation 874
9.10.5 Shock Isolation 875
9.10.6 Active Vibration Control 878
9.11 Vibration Absorbers 883
9.11.1 Undamped Dynamic Vibration
Absorber 884
9.11.2 Damped Dynamic Vibration
Absorber 891
9.12 Examples Using MATLAB 895
Chapter Summary 903
References 903
Review Questions 905
Problems 907
Design Project 922
ChAPtEr 10
Vibration Measurement and
Applications 924
10.1 Introduction 925
10.2 Transducers 927
10.2.1 Variable-Resistance Transducers 927
10.2.2 Piezoelectric Transducers 930
10.2.3 Electrodynamic Transducers 931
10.2.4 Linear Variable Differential Transformer
Transducer 932
10.3 Vibration Pickups 933
10.3.1 Vibrometer 935
10.3.2 Accelerometer 936
10.3.3 Velometer 940
10.3.4 Phase Distortion 942
10.4 Frequency-Measuring Instruments 944
10.5 Vibration Exciters 946
10.5.1 Mechanical Exciters 946
10.5.2 Electrodynamic Shaker 947
10.6 Signal Analysis 949
10.6.1 Spectrum Analyzers 950
10.6.2 Bandpass Filter 951
10.6.3 Constant-Percent Bandwidth and
Constant-Bandwidth Analyzers 952
10.7 Dynamic Testing of Machines and
Structures 954
10.7.1 Using Operational Deflection-Shape
Measurements 954
10.7.2 Using Modal Testing 954
10.8 Experimental Modal Analysis 954
10.8.1 The Basic Idea 954
10.8.2 The Necessary Equipment 954
10.8.3 Digital Signal Processing 957
10.8.4 Analysis of Random Signals 959
10.8.5 Determination of Modal Data from
Observed Peaks 961
10.8.6 Determination of Modal Data from
Nyquist Plot 964
10.8.7 Measurement of Mode Shapes 966
10.9 Machine-Condition Monitoring and
Diagnosis 969
10.9.1 Vibration Severity Criteria 969
10.9.2 Machine Maintenance Techniques 969
10.9.3 Machine-Condition Monitoring
Techniques 97014 ContEntS
10.9.4 Vibration Monitoring Techniques 972
10.9.5 Instrumentation Systems 978
10.9.6 Choice of Monitoring Parameter 978
10.10 Examples Using MATLAB 979
Chapter Summary 982
References 982
Review Questions 984
Problems 986
Design Projects 992
ChAPtEr 11
numerical Integration Methods in
Vibration Analysis 993
11.1 Introduction 994
11.2 Finite Difference Method 995
11.3 Central Difference Method for Single-Degree-ofFreedom Systems 996
11.4 Runge-Kutta Method for Single-Degree-ofFreedom Systems 999
11.5 Central Difference Method for Multidegree-ofFreedom Systems 1001
11.6 Finite Difference Method for Continuous
Systems 1005
11.6.1 Longitudinal Vibration of Bars 1005
11.6.2 Transverse Vibration of Beams 1009
11.7 Runge-Kutta Method for Multidegree-ofFreedom Systems 1014
11.8 Houbolt Method 1016
11.9 Wilson Method 1019
11.10 Newmark Method 1022
11.11 Examples Using MATLAB 1026
Chapter Summary 1032
References 1032
Review Questions 1033
Problems 1035
ChAPtEr 12
Finite Element Method 1041
12.1 Introduction 1042
12.2 Equations of Motion of an Element 1043
12.3 Mass Matrix, Stiffness Matrix, and Force
Vector 1045
12.3.1 Bar Element 1045
12.3.2 Torsion Element 1048
12.3.3 Beam Element 1049
12.4 Transformation of Element Matrices
and Vectors 1052
12.5 Equations of Motion of the Complete System
of Finite Elements 1055
12.6 Incorporation of Boundary Conditions 1057
12.7 Consistent- and Lumped-Mass Matrices 1066
12.7.1 Lumped-Mass Matrix for a Bar
Element 1066
12.7.2 Lumped-Mass Matrix for a Beam
Element 1066
12.7.3 Lumped-Mass Versus Consistent-Mass
Matrices 1067
12.8 Examples Using MATLAB 1069
Chapter Summary 1073
References 1073
Review Questions 1074
Problems 1076
Design Projects 1088
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 13-23ContEntS 15
13.6 Systems with Time-Dependent Coefficients
(Mathieu Equation) 13-24
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 relations and Material Properties 1092
APPEnDIx B
Deflection of Beams and Plates 1095
APPEnDIx C
Matrices 1097
APPEnDIx D
Laplace transform 1104
APPEnDIx E
Units 1112
APPEnDIx F
Introduction to MAtLAB 1116
Answers to Selected Problems 1126
Index 1135
Index
A
Accelerographs, 439–440
Accelerometer, 936–940
“Acoustics,” 34
Active vibration control, 878–883
Addition of harmonic motions, 89–90
Adjoint matrix, 1101
Advance, 927
Airfoil, dynamic instability of, 348–349
Amplitude, 90, 627, 856
Analysis, vibration, 48–51
equations, 48
mathematical modeling, 48
motorcycle, mathematical model of, 50
results, interpretation, 50
Annoyance, 41
Aristotle, 32
Aristoxenus, 32
Arrays with special structure, 1118
Attractor, 13-43–13-44
Asymptotically stable system, 231
Autocorrelation function, 14-9, 14-11, 14-32
Axial compressive force, beam subjected to, 779–783
Axial force effect, 778–780
B
Band-limited white noise, 14-25
Bandpass filter, 951–952
Bandwidth, 316–317
Bar element, 1045–1048
Base excitation
response spectrum for, 436–439
system response under, 367–369
Basic concepts of vibration, 43–46
Bathtub curve, 969
Beam deflections, 614
Beam element, 1049–1052
Beams, deflection of, 1095–1096
cantilever beam, 1095
fixed-fixed beam with end displacement, 1095
fixed-fixed beam, 1095
simply supported beam, 1095–1096
Beams, on elastic foundation, 785–788
Beams, fundamental frequency of, 708–710
Beating phenomenon, 92, 305–309
Belt friction system, 13-5
Bernoulli, Daniel, 34
Bifurcations, 13-46
Bivariate distributions, 14-8
Blast load on building frame, 432–433
Bode diagrams, 362–364
Bonaparte, Napoléon, 35
Boundary conditions, incorporation of, 1057–1066
Boundary curves, 13-28
Building frame response to an earthquake, 442
C
Cam-follower mechanism, 73–74, 102
spring mass system for, 155
Cannon analysis, 206
Cantilever beam, 1095
spring constants of, 57–58
center, 13-30, 13-40
Center of percussion, 180–181
Central difference method for multidegree-offreedom systems, 1001–1005
Centrifugal pump with rotating unbalance, 862–866
rattle space, 862–866
Cepstrum, 976
Chaos, 13-43–13-52
attractor, 13-43–13-44
bifurcations, 13-46
of Duffing’s equation, 13-47–13-52
functions with stable orbits, 13-45
functions with unstable orbits,
13-45–13-47
Poincaré section, 13-43–13-44
strange attractors, 13-46
Characteristic (polynomial) equation
solution, 628–633
Please note that references to pages in Chapters 41 and 42 appear in the form 13-1, 13-2, etc., and these chapters are provided on the Companion Website,
www.pearsonglobaleditions.com/Rao.1136 Index
Characteristic roots, graphical
representation, 207–209
Chimney, flow-induced vibration of, 350–352
Choleski decomposition, 726–728
Classification of vibration, 46–47
Clebsch, R. F. A., 36
Coefficients, numerical computation of, 101–105
Coherence function, 961
Column matrix, 1098
Column vector, 1117
Compacting machine, 427–428, 456–458
Complex algebra, 87
Complex damping, 339
Complex Fourier series, 95–96, 14-17–14-18
Complex frequency response, 318, 42-58
Complex numbers, 1119
harmonic motion representation, 86–87
Complex stiffness, 227
Complex vector representation of harmonic
motion, 319–320
Compound pendulum, 178–180, 604
natural frequency of, 178–181
Consistent mass matrices, 1066–1069
Constant bandwidth analyzers, 952–953
Constant damping, 219
Constant percent bandwidth, 951–952
Continuous systems, 46, 745–818, See also
Lateral vibration of beams; Longitudinal
vibration of bar or rod; Torsional vibration
of a shaft or rod
dynamic response of plucked string, 753–758
modeling as multidegree-of-freedom
systems, 598–599
transverse vibration of a string or
cable, 747–756
Continuous systems, finite difference method
for, 850–1005
longitudinal vibration of bars, 1005–1009
pinned-fixed beam, 1012
transverse vibration of beams, 1009–1012
Control, vibration, 819–923
criteria, 821–825
natural frequencies, control of, 848–849
nomograph, 821–825
ranges of vibration, 823
whirling of rotating shafts, 835–841
Conversion of units, 1112–1115
Conversion of vibration to sound
(by human ear), 40–43
Convolution integral, 405, 421–433, 1108–1111
blast load on building frame, 432–433
compacting machine under linear force, 431–432
rectangular pulse load, 429–430
response of a structure under double impact, 425
response of a structure under impact, 424
response to a general forcing condition, 425–426
response to an impulse, 422–425
response to base excitation, 426–433
step force on a compacting machine, 427–428
time-delayed step force, 428–429
Coordinate coupling, 530–535
Correlation functions of random process, 14-9–14-10
Coulomb damping, 75
Coulomb, Charles, 35
forced response of, using MATLAB, 366–367
forced vibration with, 332–336
free-vibration response of a system with, 238–239
free vibration with, 218–225
pulley subjected to, 224
Coupled differential equations, 512
Crane, equivalent k of, 65
Critical damping constant, 189
Critical speeds, 837
Critically damped system, 192
Cycle, 90
D
D’Alembert, Jean, 34
D’Alembert’s principle, 159
Damped dynamic vibration absorber, 891–895
Damped equation, 13-14
Damped response using numerical methods, 474–477
Damped single-degree-of-freedom system
Bode diagrams of, 363
transfer function, 354
Damped system, 155
forced vibration response of, MATLAB, 666–667
free-vibration response of, Laplace
transform, 546–549
Damped system response under F(t) = F0eivt, 317–320
Damped system response under harmonic force, 309–317,
See also under Harmonically excited vibration
under F(t) = F0eivt, 317–320
graphical representation, 310
under harmonic motion of base, 320–326
under rotating unbalance, 326–332
total response, 312–314
vectorial representation, 310Index 1137
Damped system response using Laplace
transform, 356–357
Damped vibration, 47
Damping, 849–850
damping matrix, 654
damping ratio, 189, 212
viscoelastic materials use, 849
Damping elements, 74–83
clearance in a bearing, 76–77
combination of dampers, 81–82
Coulomb or dry friction damping, 75
damping constant of journal bearing, 77–79
damping constant of parallel plates, 76
linearization of nonlinear damper, 81
material or solid or hysteretic damping, 75
piston-cylinder dashpot, 79–81
viscous damping, 74–75
viscous dampers construction, 75–81
De Laval, C. G. P., 36
Decibel, 93
Degree of freedom, 44–46
Delay time (td), transient response, 465
Design chart of isolation, 859–860
Determinant, 1099
Deterministic vibration, 47, 14-2
Diagonal matrix, 1098
Diesel engine, vibration absorber for, 888–889
Differential equations, 352, 1123–1125
Digital signal processing, 957–958
Dirac delta function, 421
Discrete systems, 46
Displacement method, 1057
Displacement transmissibility, 321–322, 856–857,
868–869
Dry friction damping, 75
Duffing’s equation, 13-13, 13-47–13-50
Duhamel integral, See Convolution integral
Dunkerley’s formula, 701–736
Dynamic coupling, 532
Dynamic instability caused by fluid flow, 344–350
of an airfoil, 348–350
flow-induced vibration of a chimney, 350–351
flow-induced vibration reduction, 346
Helical spoilers, 347
Stockbridge damper, 347
Dynamic response of plucked string, 753–754
Dynamic stability analysis, 340–344
Dynamic system, equations of motion of, 656–658
Dynamic testing of machines and structures, 954
Dynamical matrix, 628
E
Ear. See Human ear
Earthquake response spectra, 439–441
Eccentricity of rotor, probabilistic
characteristics of, 14-6
Eigenvalues/Eigenvalue problem, 626–627, 637–639
Eigenvectors, orthonormalization of, 635–637
Equilibrium states, stability, 13-37–13-40
Elastic coupling, 532
Elastic foundation, 785–788
Elastic potential energy, 617–619
Electric motor deflection due to rotating
unbalance, 329–330
Electrodynamic shaker, 947–949
Electrodynamic transducers, 931–932
Element matrices and vectors,
transformation, 1052–1055
Elementary parts of vibrating systems, 43–44
Energy dissipated in viscous damping, 199–201
Equation of motion, 177–178, 188, 219–220,
299–300
derivation, 620–624
of dynamic system, 656–658
of an element, 1043–1045
of finite elements, 1055–1057
for forced vibration, 514–515
of three-degree-of-freedom system, 629
of undamped systems in matrix form, 624–625
whirling of rotating shafts, 835–837
Equivalent linearized spring constant, 56
Equivalent mass of a system, 72–73
Equivalent rotational mass, 71
Equivalent translational mass, 71
Ergodic process, 14-13
Euler, Leonard, 34
Euler-Bernoulli theory, 768, 1049
Even functions, 98–100
Exciters, vibration, 946–949, 955
due to unbalanced force, 947
electrodynamic shaker, 947–949
mechanical exciters, 946–947
Expansion theorem, 639
Experimental modal analysis, 954–969
basic idea, 954
coherence function, 9611138 Index
Experimental modal analysis (continued)
digital signal processing, 957–958
modal data determination from observed peaks, 961–966
mode shapes measurement, 966–969
necessary equipment, 954–957
random signals analysis, 959–961
Explicit integration method, 997
F
Fast Fourier transform (FFT) method, 950, 956, 978
Finishing process, vibratory, 43
Finite difference method, 995–996
for continuous systems, 1005–1013
Finite element idealization, 38
Finite element method, 1041–1089
bar element, 1045–1048
beam element, 1049–1052
boundary conditions, incorporation of, 1057–1066
element matrices and vectors,
transformation, 1052–1055
equations of motion of, 1043–1045, 1055–1057
Euler-Bernoulli theory, 1049
force vector, 1045–1052
mass matrix, 1045–1052
stiffness matrix, 1045–1052
torsion element, 1048–1049
First-order systems, 181–182, 406–410
response of, 447–448
response under periodic force, 407–410
Fixed-free bar, free vibrations of, 760
Fixed-pinned beam, natural frequencies of,
774–777
Flexibility influence coefficients, 610–614
determination, 612
Flexibility matrix of a beam, 614
Flow-induced vibration
of a chimney, 350
reduction, 346
Flutter, 344
Focus, 13-40
Force transmissibility, 323
Force vector, 1045–1052
Forced system, steady-state response of, 658–662
Forced vibration, 46, 536–539, 776–777
steady-state response of spring-mass system,
536–539
of viscously damped systems, 653–659
Forging hammer
forced vibration response of, 651–656
response of anvil of, 202
Fourier analysis, 14-16–14-23
complex Fourier series expansion,
14-17–14-18
Fourier integral, 14-19–14-23
of triangular pulse, 14-22
Fourier integral, 14-19–14-23
Fourier series expansion, 93–95, 102
Cam-follower system, 102
complex Fourier series, 95–96
Gibbs phenomenon, 95
graphical representation using
MATLAB, 105–107
numerical Fourier analysis, 103–105
periodic function, 93
Fourth-order Runge-Kutta method,
1028–1029
Frahm tachometer, 36, 944
Francis water turbine, 330
Free vibration, 46
response of two-degree-of-freedom
system, 523–524
response using modal analysis, 649–651
Frequency domain representations, 97–98
Frequency-measuring instruments, 944–945
frequency-measuring instruments, 925
multireed instrument, 944
single-reed instrument, 944
stroboscope, 945
Frequency of damped vibration, 192
Frequency of oscillation, 91
Frequency or characteristic equation, 517
Frequency-response approach, 14-30,
14-33–14-39
mean square response, 14-34
power spectral density, 14-33
Frequency spectrum, 96–97
Frequency transfer functions, 359–364
frequency-response characteristics
representation, 362–364
general transfer function and, 361–362
physical system, 360
solutions using, 554–557
Fullarton tachometer, 944
G
Galileo Galilei, 33–36
Galloping, 344–345
Gaussian random process, 14-14–14-16Index 1139
General forcing conditions, vibration under, 403–508,
See also General periodic force, response under;
Nonperiodic force, response under; Periodic force;
Response spectrum
General periodic force, response under, 405–418
first-order systems, 406–412
second-order systems, 407, 412–414
total response under harmonic base
excitation, 417–418
General transfer function and frequency transfer
function, 361–362
Generalized coordinates, 514, 530, 619–620
Generalized forces, 619–620
Generalized mass matrix, 618
Germain, Sophie, 36
Gibbs phenomenon, 95
Grid points, 995
H
Half power points, 316
Half-range expansions, 100–101
Harmonic analysis, 93–105, See also Fourier
series expansion
even functions, 98–100
frequency domain representations,
97–98
half-range expansions, 100–101
odd functions, 98–100
time domain representations, 97–98
Harmonic base excitation, total response
under, 417–418
Harmonic motion, 83–93, 163–175
addition of harmonic motions, 89
complex algebra, 87
complex number representation
of, 86–87
impact, free-vibration response due
to, 170–171
motion of, graphical representation, 165
natural frequency, 171–175
operations on harmonic functions, 88–90
Scotch yoke mechanism, 84
simple harmonic motion, 85
spring-mass system to initial condition,
response of, 168
undamped system, phase plane
representation, 167
vectorial representation of, 85–86
water tank, harmonic response of, 168–170
Harmonically excited vibration, 297–402
damped system response under F(t) = F0eivt,
317–320
damped system response under harmonic
force, 309–317, See also individual entries
equation of motion, 299–300
forced vibration with Coulomb damping, 332–336,
See also Coulomb damping
hysteresis damping, forced vibration with, 337–339
quadratic damping, 339
quality factor and bandwidth, 316–317
undamped system response under, 301–309
Helical spoilers, 347
Helicopter seat vibration reduction, 824–833
vibration at source, reduction, 825–826
Heterodyne analyzer, 952
History of vibration, 31–38
finite element idealization, 38
from Galileo to Rayleigh, 33–36
origin, 31–32
recent contributions, 36–37
theory of vibration of plates, 35
torsional vibration tests, 35
Hoisting drum, equivalent k of, 63–64
Holzer’s method, 711–715
resultant torque versus frequency, 712
spring-mass systems, 714–715
torsional systems, 712–714
Hooke, Robert, 33
Horizontal position, spring-mass system in, 155
Houbolt method, 1016–1019
for two-degree-of-freedom system, 1018
Human ear, 40–43
Hydraulic valve, periodic vibration of, 414–416
Hysteretic damping, 75
forced vibration with, 337–339
free vibration with, 225–231
I
Ideal white noise, 14-25
Identity matrix, 1098
Implicit integration methods, 1017
Impulse-response function, 422–423,
14-28–14-29
Inelastic collision, response to, 451–452
Inertia influence coefficients, 615–616
Influence coefficients, 605–616
flexibility influence coefficients, 610–614
flexibility matrix of a beam, 6141140 Index
Influence coefficients (continued)
inertia influence coefficients, 615–616
stiffness influence coefficient, 605–610
stiffness matrix of a frame, 609
Introduction to Harmonics, 32
Inverse Laplace transform, 1105
Inverse matrix, 1101
Inverse of the Matrix, 727
Irregular forcing conditions, response to, 471–475
Irregular forcing function, 418–420
Isolation, vibration, 851–883
with base motion, 864–872
damped spring mount, 851
pneumatic rubber mount, 851
system with flexible foundation, 872–873
system with partially flexible foundation, 873–875
types, 852–853
undamped spring mount, 851
with rigid foundation, 854–863, See also
Rigid foundation
Iteration method, 715–722, 13-13–13-16, See also
Matrices: matrix iteration method
J
Jacobi’s method, 723–725
eigenvalue solution using, 724–727, 729
standard eigenvalue problem, 725–728
joint probability distribution of random
variables, 14-7–14-9
bivariate distributions, 14-8
multivariate distribution, 14-8
univariate distributions, 14-8
Journal bearing, damping constant of, 77–79
Jump phenomenon, 13-16
K
Karman vortices, 344
Kinetic energy expressions in matrix form, 617–619
Kirchhoff, G. R., 36
Kronecker delta, 624
L
L’Hospital’s rule, 304
Lagrange, Joseph, 34
Lagrange’s equations, 620–624
Laplace transform, 352, 356–359, 404, 446–467,
546–554, 1104–1111
damped system response using, 356
definition, 1104–1105
first-order systems, response of, 447–448
inverse Laplace transform, 1105
partial fractions method, 1106–1108
second-order systems, response of, 449–454
shifting theorems, 1106
steady-state response using, 358–359
step force, response to, 454–460
transform of derivatives, 1105–1106
transient and steady-state responses, 446
transient response, 461–467, See also
individual entries
two-degree-of-freedom systems solutions
using, 546–554
Laplacian operator, 790
Lateral vibration of beams, 767–788
axial compressive force, beam subjected
to, 779–780
boundary conditions, 770–772
equation of motion, 767
fixed-pinned beam, natural frequencies
of, 774–777
forced vibration, 776–777
free vibration, 769–770
initial conditions, 769
orthogonality of normal functions, 772–774
simply supported beam, forced vibration, 777–780
Lathe, 510, 530–531
Left half-plane (LHP) yield, 231
Limit cycles, 13-41–13-43
Lindstedt’s perturbation method, 13-10–13-12
Linear algebraic equations, solution of, 1122
Linear coordinates, 598
Linear force, compacting machine under, 431–432
Linear springs, 53–54
Linear variable differential transformer (LVDT)
transducer, 932–933
Linear vibration, 47
Linearization of nonlinear spring, 55–56
Literature, vibration, 109–110
Local coordinate axis, 1052
Logarithmic decrement, 198–199
Longitudinal vibration of bar or rod, 756–764
bar carrying a mass, natural frequencies of,
761–762
bar subjected to initial force, vibrations of, 762–764
boundary conditions, 749–750
equation of motion and solution, 747–749
free vibrations of a fixed-free bar, 760
orthogonality of normal functions, 772–774Index 1141
Longitudinal vibration of bars, 1005–1013
Loops, 34
Lumped-mass matrices, 1066–1069
Lumped-mass model, 598
M
Machine condition monitoring techniques, 970–972
Machine maintenance techniques, 969–970
breakdown maintenance, 969
condition-based maintenance, 970
preventive maintenance, 969
Machine tool support, equivalent spring and
damping constants of, 81–83
Machine vibration monitoring techniques, 972–977
Magnification factor, 302, 311
Marine engine propeller system, 527–530
Mass matrix, 1045–1052
Mass or inertia elements, 69–74
Material damping, 75
Mathematical modeling, 48
Mathieu equation, 13-24–13-29
MATLAB, 365–370, 476–480, 557–564, 662–670,
728–731, 797–800, 895–902, 979–982, 1026–1031,
1069–1073, 1097–1103, 1116–1125
accelerometer equation plotting, 981–982
arrays and matrices, 1117
arrays with special structure, 1118
autocorrelation function plotting, 14-46–14-48
column vector, 1117
complex numbers, 1119
Coulomb damping, free-vibration
response of a system with, 238
damped system, forced vibration response
of, 666–667
differential equations solution,
1123–1125
eigenvalue problem solution, 557–558, 662, 728–729
finite element analysis of stepped bar, 1069
forced response of a system with
Coulomb damping, 366–367
forced vibration response of simply
supported beam, plotting, 797–800
Fourier series graphical representation using, 105–107
free-vibration response, plotting, 559
functions in, 1119
Gaussian probability distribution
function evaluation, 14-48–14-49
general eigenvalue problem, 730–731
impulse response of a structure, 477–478
matrix, 1117
matrix operations, 1118
M-files, 1119–1120
multidegree-of-freedom system, 662–670
nonlinear differential equation solution, 13-61
nonlinearly damped system solution, 13-57–13-59
nonlinear system under pulse loading solution, 13-59
numerical Fourier analysis using, 103
Nyquist circle plotting, 979–980
pendulum equation solution, 13-53–13-57
plotting of graphs, 1120–1121
program to generate characteristic
polynomial, 668
quartic equation roots, 558
railway cars, time response of, 560–561
response under a periodic force, 478–479
response under arbitrary forcing
function, 479
roots of a polynomial equation, 665
roots of a quartic equation, 558
roots of transcendental and nonlinear
equations, 799–800
row vector, 1119
solution of a single-degree-of-freedom system, 1026
solution of multidegree-of-freedom
system, 1027–1028
special matrices, 1118
spring-mass system, free-vibration
response of, 236–238
static deflection, variations of natural frequency
and period with, 235–236
steady-state response of viscously damped
system, 369–370
system response under base excitation, 367–369
total response of an undamped system using,
365–366
total response of system under base excitation,
476–477
transmissibility, plotting, 895
undamped system response, 237
variables, 1117
vibration amplitudes of vibration
absorber masses, 897–898
Matrices, 1097–1103, 1117 See also
individual entries
basic operations, 1102–1103, 1118
trace, 1099
transpose of, 1099
Maximum overshoot (Mp), 4631142 Index
Mean square response, 14-34
Mean value, 14-6–14-7, 14-32
Measurement and applications, vibration, 924–992
machine condition monitoring and
diagnosis, 969–979
measurement scheme, 926
Mechanical chatter, 13-5
Mechanical exciters, 946–947
Method of isoclines, 13-34–13-35
trajectories using, 13-36
Membranes, vibration of, 788–791
equation of motion, 788–790
free vibrations of rectangular
membrane, 791
initial and boundary conditions, 790–791
membrane under uniform tension, 789
Mersenne, Marin, 33–34
M-files, 1119–1120
Milling cutter, natural frequencies of, 766–767
Mindlin, R. D., 36
Modal analysis, 639
forced vibration of undamped systems
using, 646–653
free-vibration response using, 649–651
Modal damping ratio, 655
Modal matrix, 635
Modal testing, 954–969, See also
Experimental modal analysis
Modal vectors, 517
Mode shapes, 626
determination, 699–745
measurement, 966–969
of three-degree-of-freedom system, 633
Monochord, 32
Motor-generator set, absorber for, 889–895
Multidegree-of-freedom systems, 596–698,
14-39–14-46, See also Influence coefficients;
Three-degree-of-freedom system
central difference method for, 1001–1005
continuous systems modeling as,
598–599
equations of motion of undamped systems in
matrix form, 624–625
expansion theorem, 639
free vibration of undamped systems, 644–646
generalized coordinates, 619–620
generalized forces, 619–620
Lagrange’s equations to derive equations
of motion, 620–624
modal analysis, 646–653, See also individual entries
natural frequencies of free system, 641–643
Newton’s second law to derive equations
of motion, 600–605
potential and kinetic energy expressions in
matrix form, 617–619
repeated Eigenvalues, 637–639
self-excitation, 660–662
spring-mass-damper system, equations of
motion of, 600–603
stability analysis, 660–662
steady-state response of forced system, 658–662
trailer–compound pendulum system, equations
of motion of, 603
unrestrained systems, 639–641
Multivariate distribution, 14-8
N
Narrow-band process, 14-25–14-27
Natural frequencies, 92, 517
determination, 699–745, See also Dunkerley’s formula;
Holzer’s method; Jacobi’s method;
Rayleigh’s method
of free system, 641–643
of torsional system, 526, 713–718
Natural mode, two-degree-of-freedom systems, 513
Newmark method, 1022–1025
Newton, Isaac, 34
Newton’s second law, 158–159, 299, 600–603
Nodes, 34, 752, 13-40–13-41
Nomograph, vibration, 821–825
Nondeterministic vibration, 47
Nonlinear damper, linearization of, 81
Nonlinear differential equation solution, 13-61
Nonlinear equations, roots of, 1121
Nonlinear springs, 53–55
Nonlinear system under pulse loading
solution, 13-59
Nonlinear vibration, 47, 13-1–13-76
approximate analytical methods,
13-7–13-19
equilibrium states, stability, 13-37–13-40
exact methods for, 13-6–13-7
graphical methods, 13-29–13-37
iterative method, 13-13–13-16
Jump phenomenon, 13-16
limit cycles, 13-41–13-43
Lindstedt’s perturbation method,
13-10–13-12, 13-25Index 1143
nonlinear spring characteristics, 13-4
numerical methods, 13-52–13-53
Ritz-Galerkin method, 13-17–13-19
subharmonic oscillations, 13-20–13-22
superharmonic oscillations, 13-23–13-24
time-dependent coefficients, systems
with, 13-24–13-29
variable mass system, 13-5–13-6
Nonperiodic force, response under, 405, 420–421,
See also Convolution integral; Laplace transform;
Numerical methods
Normal modes, 513, 634–635
Number-decibel conversion line, 362
Numerical Fourier analysis, 103–105
Numerical integration methods, 993–1040
finite difference method, 995–996
single-degree-of-freedom systems, 996–999
Numerical methods, for response under
nonperiodic force, 405, 467–470
Nyquist circle plotting, 964–965
Nyquist plot, modal data determination from, 961–963
O
Octave, 93
Octave band analyzer, 950, 952
Odd functions, 98–100
Operational deflection shape measurements, 954
Optimally tuned vibration absorber, 894
Orthogonality of normal functions, 634–637, 759–764,
772–775
Orthonormalization of eigenvectors, 635–637
Overdamped system, 194, 458–460
P
Parameter variations, 209–218, See also under
Root locus representations
Parseval’s formula, 14-17, 14-21
Partial fractions method, 1106–1108
Peak time (tp), 461
Perfectly elastic collision, response to, 452–454
Periodic solutions using Lindstedt’s
perturbation method, 13-25
Period of beating , 307
Period of oscillation, 91
Periodic force, 405–418, See also General periodic force,
response under
Periodic vibration of a hydraulic valve, 414–416
Phase angle, 91, 627
Phase distortion, 942–944
Phase plane representation, nonlinear vibration,
13-29–13-34
phase velocity, 13-34
undamped nonlinear system, 13-32
undamped pendulum, 13-31
Phase plane trajectories, time solution from,
13-36–13-37
Phase velocity, 13-34
Philosophiae Naturalis Principia Mathematica, 34
Piezoelectric transducers, 930–931
Pinned-fixed beam, 1012
Piston-cylinder dashpot, 79–81
Plane milling cutter, 767
Plano-milling machine structure, 1043
Plates, deflection of, 1095–1096
Poincaré section, 13-43–13-44
Poisson, Simeon, 36
Positive definite matrix, 619
Positive definite quadratic forms, 619
Potential energy expressions in
matrix form, 617–619
Power spectral density, 14-23–14-25, 14-33
Precision electronic system, vibration control of, 880–881
Precision machine with base motion, design of
isolation for, 866–868
Principal coordinates, 514, 530–535
Principal mode, two-degree-of-freedom systems, 512
Principle of conservation of energy, 160
Principle of virtual displacements, 159
Probability density curve, 973–974
Probability distribution, 14-4–14-5
Propeller shaft, 62–63
Proportional damping, 598
Pseudo spectrum, 437
Pseudo velocity, 437
Pulley subjected to Coulomb damping, 224–225
Pulley system, 174–175
Pulse load, 429–430
response due to, 430
Pythagoras, 31–32
Q
Q factor/quality factor, 316–317
Quadratic damping, 339–340
Quefrency-domain analysis, 9761144 Index
R
Ramp function, first-order system response due to, 448
Random signals analysis, 959–961
Random vibration, 47, 14-1–14-61, See also Stationary
random process
band-limited white noise, 14-25
correlation functions of, 14-9–14-10
eccentricity of rotor, probabilistic
characteristics of, 14-6
Gaussian random process, 14-14–14-16
ideal white noise, 14-25
joint probability distribution, 14-7–14-9
mean value, 14-6–14-7
multidegree-of-freedom system response,
14-39–14-46
narrow-band process, 14-25–14-27
power spectral density, 14-23–14-25
probability distribution, 14-4–14-5
random processes, 14-3–14-4
random variables, 14-3–14-4
single-degree-of-freedom system
response, 14-28–14-31
standard deviation, 14-6–14-7
stationary process, 14-26
stationary random excitations, response due to,
14-31–14-39
wide-band process, 14-25–14-27
Rayleigh, Baron, 36
Rayleigh’s method, 183–188, 703–710, 746, 791–794
beams, fundamental frequency of, 708–710
effect of mass, 185–188
manometer for diesel engine, 183
Rayleigh’s quotient, properties of, 704–706
shafts, fundamental frequency of, 708–710
U-tube manometer, 184
Rayleigh-Ritz method, 746, 794–797
Reciprocating engines, balancing, 842–848
reciprocating engines, balancing,
846–848
unbalanced forces due to fluctuations in gas
pressure, 842–843
unbalanced forces due to inertia of the moving parts,
843–846
Recoil mechanism, 206
Rectangular pulse load, 429–430
response due to, 430
Recurrence formula, 997
Reference marks, 827–828
Relative motion, 323–326
Repeated Eigenvalues, 637–639
Resonance, 47
Resonant frequencies of vibration absorber, 899–900
Response spectrum, 434–446
for base excitation, 436–439
building frame response to an earthquake,
439–441
design under shock environment, 443–446
earthquake response spectra, 439–443
of sinusoidal pulse, 434–436
water tank subjected to base
acceleration, 438–439
Rigid bar
connected by springs, equivalent k of, 66
stability of, 234
Rigid foundation, vibration isolation system
with, 854–863
design chart of isolation, 859–860
isolator for stereo turntable, 860–862
machine member on, 854
resilient member on, 854
spring support for exhaust fan, 857–858
undamped isolator design, 858–860
vibratory motion of mass, reduction, 856–857
Rise time (tr), 461–464
Ritz-Galerkin method, 13-17–13-19
Rod, spring constants of, 57
Root locus representations, 209–218
and parameter variations, 212–218
roots study with variation of c, 214
z in s-plane, 209–212
t in s-plane, 209–212
vd in s-plane, 209–212
v
n in s-plane, 209–212
variation of mass, 218
variation of spring constant, 216
Rotary inertia effects, 780–785
Rotating machines, balancing, 826–835
single-plane balancing, 826–829
two-plane balancing, 829–835
Rotating unbalance, 326–332, 881–888, See also
under Damped system response under
harmonic force
Routh-Hurwitz criteria, 544, 840
Row matrix, 1098
Row vector, 1117
Runge-Kutta methods, 469–471Index 1145
S z
in s-plane, 209–212
t in s-plane, 209–212
vd in s-plane, 209–212
v
n in s-plane, 209–212
Saddle point, 13-40–13-41
Sample point, 14-3
Sample space, 14-3
Sauveur, Joseph, 34
Scotch yoke mechanism, 84
Second-order systems, 404, 412–414, 449–454
Seismograph, 32
Self-excitation, 340–352, 542–544,
660–662
Semidefinite systems, 539–542, 641
Settling time, transient response, 464
Shafts, fundamental frequency of, 708–710
Shear deformation effects, 780–785
Shearing stress (t), 78
Shock absorber for a motorcycle, 204–206
Shock environment, design under, 443–446
Shock isolation, 875–878
Shock loads, 444–447
Signal analysis, 949–953
Signum function, 220
Simple harmonic motion, 85
Simple pendulum, 44, 68, 13-3
Simply supported beam, 134
forced vibration, 776–777
natural frequencies of, 783–785, 787–788
Singing of transmission lines, 344
Single-degree-of-freedom systems, 44,
14-28–14-31
central difference method for, 996–999
characteristics of, 14-30–14-31
free vibration of, 153–296, See also Undamped
translational system, free vibration of
frequency-response approach, 14-30
impulse-response approach, 14-28–14-29
Single-plane balancing, 826–829
Singular point, 13-34
Sinusoidal pulse, response spectrum of, 434–436
Solid damping, 75
Sound (conversion from vibration by
human ear), 40–43
Space shuttle, vibration testing, 40
Special matrices, 1118
Spectrum analyzers, 950–951
Spring constants of elastic elements, 57–59
Spring elements, 52–69
deformation of spring, 52
equivalent linearized spring constant, 56
linear springs, 53–55
nonlinear springs, 53–55
spring constant associated with restoring force due to
gravity, 68
spring constants of elastic elements, 57–59
Spring-mass-damper system, 300, 600–603
Spring-mass systems, 155–157, 714–715
to initial condition, response of, 168
Springs, combination of, 60–68
equivalent k, 62–68
in parallel, 60
in series, 60
torsional spring constant of a propeller
shaft, 62–63
Spring-supported mass instability on
moving belt, 341
Square matrix, 1098
Stability analysis, 340–352, See also Dynamic
instability caused by fluid flow
two-degree-of-freedom systems, 542–544
dynamic, 340–344
multidegree-of-freedom systems,
660–662
whirling of rotating shafts, 835–836
Stability of systems, 231–235
asymptotically stable, 231–233
rigid bar, 234
stable, 231–233
unstable, 231–233
Stable focus, 13-40
Stable orbits, functions with, 13-45
Standard deviation, 14-6–14-7
Standard eigenvalue problem, 628
Static deflection, 301
Static equilibrium position, 161
Static unbalance, 826
Stationary random excitations, response due to,
14-31–14-39
Stationary random process, 14-10–14-14
Strange attractors, 13-46
Steady-state response, 446
of forced system, 658–659
using Laplace transform, 358–359
Step force, response to, 454–460, See also under
Laplace transform
Stepped bar, 1069, 10721146 Index
Stiffness influence coefficient, 605–610
Stiffness matrix, 609, 1045–1052
Stockbridge damper, 347
Stodola, Aurel, 36
Stroboscope, 945
Study of vibration, importance, 38–43
Subharmonic oscillations, 13-20–13-22
Superharmonic oscillations, 13-23–13-24
Suspension system, equivalent k of, 62
Symmetric matrix, 727–728, 1099
System response under base excitation, 367–369
T
Tapered beam, fundamental frequency of, 793–794
Taylor, Brook, 34
Taylor’s series expansion, 55, 349
Temporal averages, 14-14
Thick beam theory, 746
Thin beam theory, 768
Three-degree-of-freedom system, 45
equations of motion of, 629
fundamental frequency of, 706–708
mode shapes of, 633
natural frequencies of, 629–633, 718–722
Time constant, 181–182
Time-delayed step force, 428–429
Time-dependent coefficients, systems with,
13-24–13-29
Time domain analysis, 972
Time domain representations, 97–98
Timoshenko beam theory, 746, 780–781
Timoshenko, Stephen, 36
Torsion element, 1048–1049
Torsional pendulum, 177
Torsional spring constant of a propeller shaft, 62–63
Torsional system, 525–530, 711–714
with Coulomb damping, 223–225
with discs mounted on a shaft, 525
equations of motion of, 621–622
natural frequencies of, 526–527,
711–714
with viscous damping, 201–207
Torsional vibration of a shaft or rod, 764–767
Torsional vibration, 35, 176
Trace, 1099
Trajectories of simple harmonic oscillator,
13-29–13-30
Trailer–compound pendulum system,
equations of motion of, 603
Transducers, 927–933, 955
electric resistance strain gage, 927
electrodynamic transducers, 931–932
linear variable differential transformer (LVDT)
transducer, 932–933
piezoelectric transducers, 930–931
variable resistance transducers,
927–930
Transfer function approach, 352–355, 465–466,
544–545
Transient response, 299, 446, 461–465
Transition curves, 13-28
Transverse vibration of beams, 1009–1012
Transverse vibration of string or cable, 747–756,
See also under Continuous systems
Traveling-wave solution, 755–756
Triangular pulse, Fourier transform of, 14-22
Triple pendulum, 619
Tuned vibration absorber, 894
Two-degree-of-freedom systems, 45, 509–595, See also
Forced vibration; Laplace transform; Semidefinite
systems; Torsional system
automobile, frequencies and modes of, 534–535
coordinate coupling and principal
coordinates, 530–535
coupled differential equations, 512
equations of motion for forced vibration, 514–515
forced response of, 562–564
free-vibration response of, 523–524
Lathe, 511, 531–532
natural mode, 513
normal mode, 512
packaging of an instrument, 513
principal mode, 512
spring-mass-damper system, 514
transfer function approach, 544–545
Two-plane balancing, 829–835, 900–902
U
Undamped dynamic vibration absorber, 884–890
effect on the response of machine, 886
for diesel engine, 888–889
for motor-generator set, 889–890
Undamped equation, 13-13, 13-31
Undamped isolator design, 858–860
Undamped system, 155
free-vibration analysis, 516–524
free vibration of, 644–646
free-vibration response of, 546–554Index 1147
in matrix form, 624–625
response under harmonic force,
301–309, See also under
Harmonically excited vibration
total response of, using MATLAB, 365–366
Undamped torsional system, free vibration of,
176–181
Undamped translational system, free
vibration of, 158–175
auxiliary or characteristic equation, 163
D’Alembert’s principle, 159
eigenvalues or characteristic values, 163
mass under virtual displacement, 160
principle of conservation of energy, 159
principle of virtual displacements, 159
using Newton’s second law of motion,
158–159
Undamped vibration, 47
Underdamped system, 190, 454–456
response of due to initial conditions, 195–197
Uniform string, free vibration of, 750–751
Unit impulse response of second-order system, 447
Units, 1112–1115
Univariate distributions, 14-8
Unrestrained systems, 541–542, 639–643
Unstable focus, 13-40
Unstable orbits, functions with, 13-45–13-47
Unstable system, 231
V
Variable mass system, 13-5–13-6
Variable resistance transducers, 927–930
Vectorial representation of harmonic motion, 85
Velometer, 940–941
Vertical position, spring-mass system in, 161–162
Vibrating string, 748
Vibration absorbers, 883–895, 899–900, 891, See also
Damped dynamic vibration absorber; Undamped
dynamic vibration absorber
Vibration pickups, 933–935
Vibration severity of machinery, 823
Vibrometer, 935–936
Viscoelastic materials use, 849–850
Viscous damping, 74–75
Cannon analysis, 206
energy dissipated in, 199–201
forced transmission to the base, 314–315
forced vibration of, 653–659
free vibration with, 188–207
steady-state response of, 369–370
torsional systems with, 201–207
W
Wallis, John, 34
Whirling of rotating shafts, 835–841
critical speeds, 837
equations of motion, 835–837
shaft carrying an unbalanced rotor, 841
stability analysis, 840–841
system response, 838–840
Wide-band process, 14-25–14-27
Wiener-Khintchine formula, 14-23
Wilson method, 1019–1022
Wind-induced vibration, 39
Y
Young’s modulus, 172
Z
Zero matrix, 1098
Zhang Heng, 32

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كتاب Mechanical Vibrations - صفحة 2 761752

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أخوانى فى الله
أحضرت لكم كتاب
Mechanical Vibrations
Sixth Edition in SI Units
Singiresu S. Rao
University of Miami
SI Conversion by
Philip Griffin
University of Limerick, Ireland

و المحتوى كما يلي :

Contents
Preface 16
Acknowledgments 21
List of Symbols 23
ChAPtEr 1
Fundamentals of Vibration 29
1.1 Preliminary Remarks 30
1.2 Brief History of the Study of Vibration 31
1.2.1 Origins of the Study of Vibration 31
1.2.2 From Galileo to Rayleigh 33
1.2.3 Recent Contributions 36
1.3 Importance of the Study of Vibration 38
1.3.1 Conversion of Vibrations to
Sound by the Human Ear 40
1.4 Basic Concepts of Vibration 43
1.4.1 Vibration 43
1.4.2 Elementary Parts of
Vibrating Systems 43
1.4.3 Number of Degrees of Freedom 44
1.4.4 Discrete and Continuous Systems 46
1.5 Classification of Vibration 46
1.5.1 Free and Forced Vibration 46
1.5.2 Undamped and Damped Vibration 47
1.5.3 Linear and Nonlinear Vibration 47
1.5.4 Deterministic and
Random Vibration 47
1.6 Vibration Analysis Procedure 48
1.7 Spring Elements 52
1.7.1 Nonlinear Springs 53
1.7.2 Linearization of a Nonlinear Spring 55
1.7.3 Spring Constants of Elastic Elements 57
1.7.4 Combination of Springs 60
1.7.5 Spring Constant Associated with the
Restoring Force due to Gravity 68
1.8 Mass or Inertia Elements 69
1.8.1 Combination of Masses 70
1.9 Damping Elements 74
1.9.1 Construction of Viscous Dampers 75
1.9.2 Linearization of a Nonlinear Damper 81
1.9.3 Combination of Dampers 81
1.10 Harmonic Motion 83
1.10.1 Vectorial Representation of
Harmonic Motion 85
1.10.2 Complex-Number Representation
of Harmonic Motion 86
1.10.3 Complex Algebra 87
1.10.4 Operations on Harmonic Functions 87
1.10.5 Definitions and Terminology 90
1.11 Harmonic Analysis 93
1.11.1 Fourier Series Expansion 93
1.11.2 Complex Fourier Series 95
1.11.3 Frequency Spectrum 96
1.11.4 Time- and Frequency-Domain
Representations 97
1.11.5 Even and Odd Functions 98
1.11.6 Half-Range Expansions 100
1.11.7 Numerical Computation
of Coefficients 101
1.12 Examples Using MATLAB 105
1.13 Vibration Literature 109
Chapter Summary 110
References 110
Review Questions 112
Problems 116
Design Projects 14910 ContEntS
ChAPtEr 2
Free Vibration of Single-Degree-of-Freedom
Systems 153
2.1 Introduction 155
2.2 Free Vibration of an Undamped
Translational System 158
2.2.1 Equation of Motion Using Newton’s
Second Law of Motion 158
2.2.2 Equation of Motion Using Other
Methods 159
2.2.3 Equation of Motion of a Spring-Mass
System in Vertical Position 161
2.2.4 Solution 162
2.2.5 Harmonic Motion 163
2.3 Free Vibration of an Undamped
Torsional System 176
2.3.1 Equation of Motion 177
2.3.2 Solution 178
2.4 Response of First-Order Systems
and Time Constant 181
2.5 Rayleigh’s Energy Method 183
2.6 Free Vibration with Viscous Damping 188
2.6.1 Equation of Motion 188
2.6.2 Solution 189
2.6.3 Logarithmic Decrement 198
2.6.4 Energy Dissipated in Viscous
Damping 199
2.6.5 Torsional Systems with Viscous
Damping 201
2.7 Graphical Representation of Characteristic Roots
and Corresponding Solutions 207
2.7.1 Roots of the Characteristic Equation 207
2.7.2 Graphical Representation of Roots and
Corresponding Solutions 208
2.8 Parameter Variations and Root Locus
Representations 209
2.8.1 Interpretations of vn, vd, z, and t in the
s-plane 209
2.8.2 Root Locus and Parameter
Variations 212
2.9 Free Vibration with Coulomb Damping 218
2.9.1 Equation of Motion 219
2.9.2 Solution 220
2.9.3 Torsional Systems with Coulomb
Damping 223
2.10 Free Vibration with Hysteretic Damping 225
2.11 Stability of Systems 231
2.12 Examples Using MATLAB 235
Chapter Summary 241
References 242
Review Questions 242
Problems 247
Design Projects 294
ChAPtEr 3
harmonically Excited Vibration 297
3.1 Introduction 299
3.2 Equation of Motion 299
3.3 Response of an Undamped System Under
Harmonic Force 301
3.3.1 Total Response 305
3.3.2 Beating Phenomenon 305
3.4 Response of a Damped System Under Harmonic
Force 309
3.4.1 Total Response 312
3.4.2 Quality Factor and Bandwidth 316
3.5 Response of a Damped System Under
F1t2 = F0eiVt 317
3.6 Response of a Damped System Under
the Harmonic Motion of the Base 320
3.6.1 Force Transmitted 322
3.6.2 Relative Motion 323
3.7 Response of a Damped System Under Rotating
Unbalance 326
3.8 Forced Vibration with Coulomb Damping 332
3.9 Forced Vibration with Hysteresis Damping 337
3.10 Forced Motion with Other Types
of Damping 339
3.11 Self-Excitation and Stability Analysis 340
3.11.1 Dynamic Stability Analysis 340
3.11.2 Dynamic Instability Caused by Fluid
Flow 344
3.12 Transfer-Function Approach 352
3.13 Solutions Using Laplace Transforms 356
3.14 Frequency Transfer Functions 359
3.14.1 Relation between the General Transfer
Function T(s) and the Frequency Transfer
Function T1iv2 361
3.14.2 Representation of Frequency-Response
Characteristics 362ContEntS 11
3.15 Examples Using MATLAB 365
Chapter Summary 371
References 371
Review Questions 372
Problems 375
Design Projects 402
ChAPtEr 4
Vibration Under General Forcing
Conditions 403
4.1 Introduction 404
4.2 Response Under a General
Periodic Force 405
4.2.1 First-Order Systems 406
4.2.2 Second-Order Systems 412
4.3 Response Under a Periodic Force
of Irregular Form 418
4.4 Response Under a Nonperiodic Force 420
4.5 Convolution Integral 421
4.5.1 Response to an Impulse 422
4.5.2 Response to a General Forcing
Condition 425
4.5.3 Response to Base Excitation 426
4.6 Response Spectrum 434
4.6.1 Response Spectrum for Base
Excitation 436
4.6.2 Earthquake Response Spectra 439
4.6.3 Design Under a Shock Environment 443
4.7 Laplace Transforms 446
4.7.1 Transient and Steady-State
Responses 446
4.7.2 Response of First-Order Systems 447
4.7.3 Response of Second-Order
Systems 449
4.7.4 Response to Step Force 454
4.7.5 Analysis of the Step Response 460
4.7.6 Description of Transient Response 461
4.8 Numerical Methods 467
4.8.1 Runge-Kutta Methods 469
4.9 Response to Irregular Forcing Conditions Using
Numerical Methods 471
4.10 Examples Using MATLAB 476
Chapter Summary 480
References 480
Review Questions 481
Problems 484
Design Projects 506
ChAPtEr 5
two-Degree-of-Freedom Systems 509
5.1 Introduction 510
5.2 Equations of Motion for Forced
Vibration 514
5.3 Free-Vibration Analysis of an Undamped
System 516
5.4 Torsional System 525
5.5 Coordinate Coupling and Principal
Coordinates 530
5.6 Forced-Vibration Analysis 536
5.7 Semidefinite Systems 539
5.8 Self-Excitation and Stability Analysis 542
5.9 Transfer-Function Approach 544
5.10 Solutions Using Laplace Transform 546
5.11 Solutions Using Frequency Transfer
Functions 554
5.12 Examples Using MATLAB 557
Chapter Summary 564
References 565
Review Questions 565
Problems 568
Design Projects 594
ChAPtEr 6
Multidegree-of-Freedom Systems 596
6.1 Introduction 598
6.2 Modeling of Continuous Systems as Multidegreeof-Freedom Systems 598
6.3 Using Newton’s Second Law to Derive Equations
of Motion 600
6.4 Influence Coefficients 605
6.4.1 Stiffness Influence Coefficients 605
6.4.2 Flexibility Influence Coefficients 610
6.4.3 Inertia Influence Coefficients 615
6.5 Potential and Kinetic Energy Expressions in
Matrix Form 617
6.6 Generalized Coordinates and Generalized
Forces 619
6.7 Using Lagrange’s Equations to Derive Equations
of Motion 62012 ContEntS
6.8 Equations of Motion of Undamped Systems in
Matrix Form 624
6.9 Eigenvalue Problem 626
6.10 Solution of the Eigenvalue Problem 628
6.10.1 Solution of the Characteristic (Polynomial)
Equation 628
6.10.2 Orthogonality of Normal Modes 634
6.10.3 Repeated Eigenvalues 637
6.11 Expansion Theorem 639
6.12 Unrestrained Systems 639
6.13 Free Vibration of Undamped Systems 644
6.14 Forced Vibration of Undamped Systems Using
Modal Analysis 646
6.15 Forced Vibration of Viscously Damped
Systems 653
6.16 Self-Excitation and Stability Analysis 660
6.17 Examples Using MATLAB 662
Chapter Summary 670
References 670
Review Questions 671
Problems 675
Design Projects 696
ChAPtEr 7
Determination of natural Frequencies
and Mode Shapes 699
7.1 Introduction 700
7.2 Dunkerley’s Formula 701
7.3 Rayleigh’s Method 703
7.3.1 Properties of Rayleigh’s Quotient 704
7.3.2 Computation of the Fundamental Natural
Frequency 706
7.3.3 Fundamental Frequency of Beams and
Shafts 708
7.4 Holzer’s Method 711
7.4.1 Torsional Systems 711
7.4.2 Spring-Mass Systems 714
7.5 Matrix Iteration Method 715
7.5.1 Convergence to the Highest Natural
Frequency 717
7.5.2 Computation of Intermediate Natural
Frequencies 718
7.6 Jacobi’s Method 723
7.7 Standard Eigenvalue Problem 725
7.7.1 Choleski Decomposition 726
7.7.2 Other Solution Methods 728
7.8 Examples Using MATLAB 728
Chapter Summary 731
References 731
Review Questions 733
Problems 735
Design Projects 744
ChAPtEr 8
Continuous Systems 745
8.1 Introduction 746
8.2 Transverse Vibration of a String or Cable 747
8.2.1 Equation of Motion 747
8.2.2 Initial and Boundary Conditions 749
8.2.3 Free Vibration of a Uniform String 750
8.2.4 Free Vibration of a String with Both Ends
Fixed 751
8.2.5 Traveling-Wave Solution 755
8.3 Longitudinal Vibration of a Bar or Rod 756
8.3.1 Equation of Motion and Solution 756
8.3.2 Orthogonality of Normal Functions 759
8.4 Torsional Vibration of a Shaft or Rod 764
8.5 Lateral Vibration of Beams 767
8.5.1 Equation of Motion 767
8.5.2 Initial Conditions 769
8.5.3 Free Vibration 769
8.5.4 Boundary Conditions 770
8.5.5 Orthogonality of Normal
Functions 772
8.5.6 Forced Vibration 776
8.5.7 Effect of Axial Force 778
8.5.8 Effects of Rotary Inertia and Shear
Deformation 780
8.5.9 Beams on Elastic Foundation 785
8.5.10 Other Effects 788
8.6 Vibration of Membranes 788
8.6.1 Equation of Motion 788
8.6.2 Initial and Boundary Conditions 790
8.7 Rayleigh’s Method 791
8.8 The Rayleigh-Ritz Method 794
8.9 Examples Using MATLAB 797
Chapter Summary 800
References 800
Review Questions 802
Problems 805
Design Project 818ContEntS 13
ChAPtEr 9
Vibration Control 819
9.1 Introduction 820
9.2 Vibration Nomograph and Vibration
Criteria 821
9.3 Reduction of Vibration at the Source 825
9.4 Balancing of Rotating Machines 826
9.4.1 Single-Plane Balancing 826
9.4.2 Two-Plane Balancing 829
9.5 Whirling of Rotating Shafts 835
9.5.1 Equations of Motion 835
9.5.2 Critical Speeds 837
9.5.3 Response of the System 838
9.5.4 Stability Analysis 840
9.6 Balancing of Reciprocating Engines 842
9.6.1 Unbalanced Forces Due to Fluctuations in
Gas Pressure 842
9.6.2 Unbalanced Forces Due to Inertia of the
Moving Parts 843
9.6.3 Balancing of Reciprocating Engines 846
9.7 Control of Vibration 848
9.8 Control of Natural Frequencies 848
9.9 Introduction of Damping 849
9.10 Vibration Isolation 851
9.10.1 Vibration Isolation System with Rigid
Foundation 854
9.10.2 Vibration Isolation System with Base
Motion 864
9.10.3 Vibration Isolation System with Flexible
Foundation 872
9.10.4 Vibration Isolation System with Partially
Flexible Foundation 874
9.10.5 Shock Isolation 875
9.10.6 Active Vibration Control 878
9.11 Vibration Absorbers 883
9.11.1 Undamped Dynamic Vibration
Absorber 884
9.11.2 Damped Dynamic Vibration
Absorber 891
9.12 Examples Using MATLAB 895
Chapter Summary 903
References 903
Review Questions 905
Problems 907
Design Project 922
ChAPtEr 10
Vibration Measurement and
Applications 924
10.1 Introduction 925
10.2 Transducers 927
10.2.1 Variable-Resistance Transducers 927
10.2.2 Piezoelectric Transducers 930
10.2.3 Electrodynamic Transducers 931
10.2.4 Linear Variable Differential Transformer
Transducer 932
10.3 Vibration Pickups 933
10.3.1 Vibrometer 935
10.3.2 Accelerometer 936
10.3.3 Velometer 940
10.3.4 Phase Distortion 942
10.4 Frequency-Measuring Instruments 944
10.5 Vibration Exciters 946
10.5.1 Mechanical Exciters 946
10.5.2 Electrodynamic Shaker 947
10.6 Signal Analysis 949
10.6.1 Spectrum Analyzers 950
10.6.2 Bandpass Filter 951
10.6.3 Constant-Percent Bandwidth and
Constant-Bandwidth Analyzers 952
10.7 Dynamic Testing of Machines and
Structures 954
10.7.1 Using Operational Deflection-Shape
Measurements 954
10.7.2 Using Modal Testing 954
10.8 Experimental Modal Analysis 954
10.8.1 The Basic Idea 954
10.8.2 The Necessary Equipment 954
10.8.3 Digital Signal Processing 957
10.8.4 Analysis of Random Signals 959
10.8.5 Determination of Modal Data from
Observed Peaks 961
10.8.6 Determination of Modal Data from
Nyquist Plot 964
10.8.7 Measurement of Mode Shapes 966
10.9 Machine-Condition Monitoring and
Diagnosis 969
10.9.1 Vibration Severity Criteria 969
10.9.2 Machine Maintenance Techniques 969
10.9.3 Machine-Condition Monitoring
Techniques 97014 ContEntS
10.9.4 Vibration Monitoring Techniques 972
10.9.5 Instrumentation Systems 978
10.9.6 Choice of Monitoring Parameter 978
10.10 Examples Using MATLAB 979
Chapter Summary 982
References 982
Review Questions 984
Problems 986
Design Projects 992
ChAPtEr 11
numerical Integration Methods in
Vibration Analysis 993
11.1 Introduction 994
11.2 Finite Difference Method 995
11.3 Central Difference Method for Single-Degree-ofFreedom Systems 996
11.4 Runge-Kutta Method for Single-Degree-ofFreedom Systems 999
11.5 Central Difference Method for Multidegree-ofFreedom Systems 1001
11.6 Finite Difference Method for Continuous
Systems 1005
11.6.1 Longitudinal Vibration of Bars 1005
11.6.2 Transverse Vibration of Beams 1009
11.7 Runge-Kutta Method for Multidegree-ofFreedom Systems 1014
11.8 Houbolt Method 1016
11.9 Wilson Method 1019
11.10 Newmark Method 1022
11.11 Examples Using MATLAB 1026
Chapter Summary 1032
References 1032
Review Questions 1033
Problems 1035
ChAPtEr 12
Finite Element Method 1041
12.1 Introduction 1042
12.2 Equations of Motion of an Element 1043
12.3 Mass Matrix, Stiffness Matrix, and Force
Vector 1045
12.3.1 Bar Element 1045
12.3.2 Torsion Element 1048
12.3.3 Beam Element 1049
12.4 Transformation of Element Matrices
and Vectors 1052
12.5 Equations of Motion of the Complete System
of Finite Elements 1055
12.6 Incorporation of Boundary Conditions 1057
12.7 Consistent- and Lumped-Mass Matrices 1066
12.7.1 Lumped-Mass Matrix for a Bar
Element 1066
12.7.2 Lumped-Mass Matrix for a Beam
Element 1066
12.7.3 Lumped-Mass Versus Consistent-Mass
Matrices 1067
12.8 Examples Using MATLAB 1069
Chapter Summary 1073
References 1073
Review Questions 1074
Problems 1076
Design Projects 1088
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 13-23ContEntS 15
13.6 Systems with Time-Dependent Coefficients
(Mathieu Equation) 13-24
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 relations and Material Properties 1092
APPEnDIx B
Deflection of Beams and Plates 1095
APPEnDIx C
Matrices 1097
APPEnDIx D
Laplace transform 1104
APPEnDIx E
Units 1112
APPEnDIx F
Introduction to MAtLAB 1116
Answers to Selected Problems 1126
Index 1135
Index
A
Accelerographs, 439–440
Accelerometer, 936–940
“Acoustics,” 34
Active vibration control, 878–883
Addition of harmonic motions, 89–90
Adjoint matrix, 1101
Advance, 927
Airfoil, dynamic instability of, 348–349
Amplitude, 90, 627, 856
Analysis, vibration, 48–51
equations, 48
mathematical modeling, 48
motorcycle, mathematical model of, 50
results, interpretation, 50
Annoyance, 41
Aristotle, 32
Aristoxenus, 32
Arrays with special structure, 1118
Attractor, 13-43–13-44
Asymptotically stable system, 231
Autocorrelation function, 14-9, 14-11, 14-32
Axial compressive force, beam subjected to, 779–783
Axial force effect, 778–780
B
Band-limited white noise, 14-25
Bandpass filter, 951–952
Bandwidth, 316–317
Bar element, 1045–1048
Base excitation
response spectrum for, 436–439
system response under, 367–369
Basic concepts of vibration, 43–46
Bathtub curve, 969
Beam deflections, 614
Beam element, 1049–1052
Beams, deflection of, 1095–1096
cantilever beam, 1095
fixed-fixed beam with end displacement, 1095
fixed-fixed beam, 1095
simply supported beam, 1095–1096
Beams, on elastic foundation, 785–788
Beams, fundamental frequency of, 708–710
Beating phenomenon, 92, 305–309
Belt friction system, 13-5
Bernoulli, Daniel, 34
Bifurcations, 13-46
Bivariate distributions, 14-8
Blast load on building frame, 432–433
Bode diagrams, 362–364
Bonaparte, Napoléon, 35
Boundary conditions, incorporation of, 1057–1066
Boundary curves, 13-28
Building frame response to an earthquake, 442
C
Cam-follower mechanism, 73–74, 102
spring mass system for, 155
Cannon analysis, 206
Cantilever beam, 1095
spring constants of, 57–58
center, 13-30, 13-40
Center of percussion, 180–181
Central difference method for multidegree-offreedom systems, 1001–1005
Centrifugal pump with rotating unbalance, 862–866
rattle space, 862–866
Cepstrum, 976
Chaos, 13-43–13-52
attractor, 13-43–13-44
bifurcations, 13-46
of Duffing’s equation, 13-47–13-52
functions with stable orbits, 13-45
functions with unstable orbits,
13-45–13-47
Poincaré section, 13-43–13-44
strange attractors, 13-46
Characteristic (polynomial) equation
solution, 628–633
Please note that references to pages in Chapters 41 and 42 appear in the form 13-1, 13-2, etc., and these chapters are provided on the Companion Website,
www.pearsonglobaleditions.com/Rao.1136 Index
Characteristic roots, graphical
representation, 207–209
Chimney, flow-induced vibration of, 350–352
Choleski decomposition, 726–728
Classification of vibration, 46–47
Clebsch, R. F. A., 36
Coefficients, numerical computation of, 101–105
Coherence function, 961
Column matrix, 1098
Column vector, 1117
Compacting machine, 427–428, 456–458
Complex algebra, 87
Complex damping, 339
Complex Fourier series, 95–96, 14-17–14-18
Complex frequency response, 318, 42-58
Complex numbers, 1119
harmonic motion representation, 86–87
Complex stiffness, 227
Complex vector representation of harmonic
motion, 319–320
Compound pendulum, 178–180, 604
natural frequency of, 178–181
Consistent mass matrices, 1066–1069
Constant bandwidth analyzers, 952–953
Constant damping, 219
Constant percent bandwidth, 951–952
Continuous systems, 46, 745–818, See also
Lateral vibration of beams; Longitudinal
vibration of bar or rod; Torsional vibration
of a shaft or rod
dynamic response of plucked string, 753–758
modeling as multidegree-of-freedom
systems, 598–599
transverse vibration of a string or
cable, 747–756
Continuous systems, finite difference method
for, 850–1005
longitudinal vibration of bars, 1005–1009
pinned-fixed beam, 1012
transverse vibration of beams, 1009–1012
Control, vibration, 819–923
criteria, 821–825
natural frequencies, control of, 848–849
nomograph, 821–825
ranges of vibration, 823
whirling of rotating shafts, 835–841
Conversion of units, 1112–1115
Conversion of vibration to sound
(by human ear), 40–43
Convolution integral, 405, 421–433, 1108–1111
blast load on building frame, 432–433
compacting machine under linear force, 431–432
rectangular pulse load, 429–430
response of a structure under double impact, 425
response of a structure under impact, 424
response to a general forcing condition, 425–426
response to an impulse, 422–425
response to base excitation, 426–433
step force on a compacting machine, 427–428
time-delayed step force, 428–429
Coordinate coupling, 530–535
Correlation functions of random process, 14-9–14-10
Coulomb damping, 75
Coulomb, Charles, 35
forced response of, using MATLAB, 366–367
forced vibration with, 332–336
free-vibration response of a system with, 238–239
free vibration with, 218–225
pulley subjected to, 224
Coupled differential equations, 512
Crane, equivalent k of, 65
Critical damping constant, 189
Critical speeds, 837
Critically damped system, 192
Cycle, 90
D
D’Alembert, Jean, 34
D’Alembert’s principle, 159
Damped dynamic vibration absorber, 891–895
Damped equation, 13-14
Damped response using numerical methods, 474–477
Damped single-degree-of-freedom system
Bode diagrams of, 363
transfer function, 354
Damped system, 155
forced vibration response of, MATLAB, 666–667
free-vibration response of, Laplace
transform, 546–549
Damped system response under F(t) = F0eivt, 317–320
Damped system response under harmonic force, 309–317,
See also under Harmonically excited vibration
under F(t) = F0eivt, 317–320
graphical representation, 310
under harmonic motion of base, 320–326
under rotating unbalance, 326–332
total response, 312–314
vectorial representation, 310Index 1137
Damped system response using Laplace
transform, 356–357
Damped vibration, 47
Damping, 849–850
damping matrix, 654
damping ratio, 189, 212
viscoelastic materials use, 849
Damping elements, 74–83
clearance in a bearing, 76–77
combination of dampers, 81–82
Coulomb or dry friction damping, 75
damping constant of journal bearing, 77–79
damping constant of parallel plates, 76
linearization of nonlinear damper, 81
material or solid or hysteretic damping, 75
piston-cylinder dashpot, 79–81
viscous damping, 74–75
viscous dampers construction, 75–81
De Laval, C. G. P., 36
Decibel, 93
Degree of freedom, 44–46
Delay time (td), transient response, 465
Design chart of isolation, 859–860
Determinant, 1099
Deterministic vibration, 47, 14-2
Diagonal matrix, 1098
Diesel engine, vibration absorber for, 888–889
Differential equations, 352, 1123–1125
Digital signal processing, 957–958
Dirac delta function, 421
Discrete systems, 46
Displacement method, 1057
Displacement transmissibility, 321–322, 856–857,
868–869
Dry friction damping, 75
Duffing’s equation, 13-13, 13-47–13-50
Duhamel integral, See Convolution integral
Dunkerley’s formula, 701–736
Dynamic coupling, 532
Dynamic instability caused by fluid flow, 344–350
of an airfoil, 348–350
flow-induced vibration of a chimney, 350–351
flow-induced vibration reduction, 346
Helical spoilers, 347
Stockbridge damper, 347
Dynamic response of plucked string, 753–754
Dynamic stability analysis, 340–344
Dynamic system, equations of motion of, 656–658
Dynamic testing of machines and structures, 954
Dynamical matrix, 628
E
Ear. See Human ear
Earthquake response spectra, 439–441
Eccentricity of rotor, probabilistic
characteristics of, 14-6
Eigenvalues/Eigenvalue problem, 626–627, 637–639
Eigenvectors, orthonormalization of, 635–637
Equilibrium states, stability, 13-37–13-40
Elastic coupling, 532
Elastic foundation, 785–788
Elastic potential energy, 617–619
Electric motor deflection due to rotating
unbalance, 329–330
Electrodynamic shaker, 947–949
Electrodynamic transducers, 931–932
Element matrices and vectors,
transformation, 1052–1055
Elementary parts of vibrating systems, 43–44
Energy dissipated in viscous damping, 199–201
Equation of motion, 177–178, 188, 219–220,
299–300
derivation, 620–624
of dynamic system, 656–658
of an element, 1043–1045
of finite elements, 1055–1057
for forced vibration, 514–515
of three-degree-of-freedom system, 629
of undamped systems in matrix form, 624–625
whirling of rotating shafts, 835–837
Equivalent linearized spring constant, 56
Equivalent mass of a system, 72–73
Equivalent rotational mass, 71
Equivalent translational mass, 71
Ergodic process, 14-13
Euler, Leonard, 34
Euler-Bernoulli theory, 768, 1049
Even functions, 98–100
Exciters, vibration, 946–949, 955
due to unbalanced force, 947
electrodynamic shaker, 947–949
mechanical exciters, 946–947
Expansion theorem, 639
Experimental modal analysis, 954–969
basic idea, 954
coherence function, 9611138 Index
Experimental modal analysis (continued)
digital signal processing, 957–958
modal data determination from observed peaks, 961–966
mode shapes measurement, 966–969
necessary equipment, 954–957
random signals analysis, 959–961
Explicit integration method, 997
F
Fast Fourier transform (FFT) method, 950, 956, 978
Finishing process, vibratory, 43
Finite difference method, 995–996
for continuous systems, 1005–1013
Finite element idealization, 38
Finite element method, 1041–1089
bar element, 1045–1048
beam element, 1049–1052
boundary conditions, incorporation of, 1057–1066
element matrices and vectors,
transformation, 1052–1055
equations of motion of, 1043–1045, 1055–1057
Euler-Bernoulli theory, 1049
force vector, 1045–1052
mass matrix, 1045–1052
stiffness matrix, 1045–1052
torsion element, 1048–1049
First-order systems, 181–182, 406–410
response of, 447–448
response under periodic force, 407–410
Fixed-free bar, free vibrations of, 760
Fixed-pinned beam, natural frequencies of,
774–777
Flexibility influence coefficients, 610–614
determination, 612
Flexibility matrix of a beam, 614
Flow-induced vibration
of a chimney, 350
reduction, 346
Flutter, 344
Focus, 13-40
Force transmissibility, 323
Force vector, 1045–1052
Forced system, steady-state response of, 658–662
Forced vibration, 46, 536–539, 776–777
steady-state response of spring-mass system,
536–539
of viscously damped systems, 653–659
Forging hammer
forced vibration response of, 651–656
response of anvil of, 202
Fourier analysis, 14-16–14-23
complex Fourier series expansion,
14-17–14-18
Fourier integral, 14-19–14-23
of triangular pulse, 14-22
Fourier integral, 14-19–14-23
Fourier series expansion, 93–95, 102
Cam-follower system, 102
complex Fourier series, 95–96
Gibbs phenomenon, 95
graphical representation using
MATLAB, 105–107
numerical Fourier analysis, 103–105
periodic function, 93
Fourth-order Runge-Kutta method,
1028–1029
Frahm tachometer, 36, 944
Francis water turbine, 330
Free vibration, 46
response of two-degree-of-freedom
system, 523–524
response using modal analysis, 649–651
Frequency domain representations, 97–98
Frequency-measuring instruments, 944–945
frequency-measuring instruments, 925
multireed instrument, 944
single-reed instrument, 944
stroboscope, 945
Frequency of damped vibration, 192
Frequency of oscillation, 91
Frequency or characteristic equation, 517
Frequency-response approach, 14-30,
14-33–14-39
mean square response, 14-34
power spectral density, 14-33
Frequency spectrum, 96–97
Frequency transfer functions, 359–364
frequency-response characteristics
representation, 362–364
general transfer function and, 361–362
physical system, 360
solutions using, 554–557
Fullarton tachometer, 944
G
Galileo Galilei, 33–36
Galloping, 344–345
Gaussian random process, 14-14–14-16Index 1139
General forcing conditions, vibration under, 403–508,
See also General periodic force, response under;
Nonperiodic force, response under; Periodic force;
Response spectrum
General periodic force, response under, 405–418
first-order systems, 406–412
second-order systems, 407, 412–414
total response under harmonic base
excitation, 417–418
General transfer function and frequency transfer
function, 361–362
Generalized coordinates, 514, 530, 619–620
Generalized forces, 619–620
Generalized mass matrix, 618
Germain, Sophie, 36
Gibbs phenomenon, 95
Grid points, 995
H
Half power points, 316
Half-range expansions, 100–101
Harmonic analysis, 93–105, See also Fourier
series expansion
even functions, 98–100
frequency domain representations,
97–98
half-range expansions, 100–101
odd functions, 98–100
time domain representations, 97–98
Harmonic base excitation, total response
under, 417–418
Harmonic motion, 83–93, 163–175
addition of harmonic motions, 89
complex algebra, 87
complex number representation
of, 86–87
impact, free-vibration response due
to, 170–171
motion of, graphical representation, 165
natural frequency, 171–175
operations on harmonic functions, 88–90
Scotch yoke mechanism, 84
simple harmonic motion, 85
spring-mass system to initial condition,
response of, 168
undamped system, phase plane
representation, 167
vectorial representation of, 85–86
water tank, harmonic response of, 168–170
Harmonically excited vibration, 297–402
damped system response under F(t) = F0eivt,
317–320
damped system response under harmonic
force, 309–317, See also individual entries
equation of motion, 299–300
forced vibration with Coulomb damping, 332–336,
See also Coulomb damping
hysteresis damping, forced vibration with, 337–339
quadratic damping, 339
quality factor and bandwidth, 316–317
undamped system response under, 301–309
Helical spoilers, 347
Helicopter seat vibration reduction, 824–833
vibration at source, reduction, 825–826
Heterodyne analyzer, 952
History of vibration, 31–38
finite element idealization, 38
from Galileo to Rayleigh, 33–36
origin, 31–32
recent contributions, 36–37
theory of vibration of plates, 35
torsional vibration tests, 35
Hoisting drum, equivalent k of, 63–64
Holzer’s method, 711–715
resultant torque versus frequency, 712
spring-mass systems, 714–715
torsional systems, 712–714
Hooke, Robert, 33
Horizontal position, spring-mass system in, 155
Houbolt method, 1016–1019
for two-degree-of-freedom system, 1018
Human ear, 40–43
Hydraulic valve, periodic vibration of, 414–416
Hysteretic damping, 75
forced vibration with, 337–339
free vibration with, 225–231
I
Ideal white noise, 14-25
Identity matrix, 1098
Implicit integration methods, 1017
Impulse-response function, 422–423,
14-28–14-29
Inelastic collision, response to, 451–452
Inertia influence coefficients, 615–616
Influence coefficients, 605–616
flexibility influence coefficients, 610–614
flexibility matrix of a beam, 6141140 Index
Influence coefficients (continued)
inertia influence coefficients, 615–616
stiffness influence coefficient, 605–610
stiffness matrix of a frame, 609
Introduction to Harmonics, 32
Inverse Laplace transform, 1105
Inverse matrix, 1101
Inverse of the Matrix, 727
Irregular forcing conditions, response to, 471–475
Irregular forcing function, 418–420
Isolation, vibration, 851–883
with base motion, 864–872
damped spring mount, 851
pneumatic rubber mount, 851
system with flexible foundation, 872–873
system with partially flexible foundation, 873–875
types, 852–853
undamped spring mount, 851
with rigid foundation, 854–863, See also
Rigid foundation
Iteration method, 715–722, 13-13–13-16, See also
Matrices: matrix iteration method
J
Jacobi’s method, 723–725
eigenvalue solution using, 724–727, 729
standard eigenvalue problem, 725–728
joint probability distribution of random
variables, 14-7–14-9
bivariate distributions, 14-8
multivariate distribution, 14-8
univariate distributions, 14-8
Journal bearing, damping constant of, 77–79
Jump phenomenon, 13-16
K
Karman vortices, 344
Kinetic energy expressions in matrix form, 617–619
Kirchhoff, G. R., 36
Kronecker delta, 624
L
L’Hospital’s rule, 304
Lagrange, Joseph, 34
Lagrange’s equations, 620–624
Laplace transform, 352, 356–359, 404, 446–467,
546–554, 1104–1111
damped system response using, 356
definition, 1104–1105
first-order systems, response of, 447–448
inverse Laplace transform, 1105
partial fractions method, 1106–1108
second-order systems, response of, 449–454
shifting theorems, 1106
steady-state response using, 358–359
step force, response to, 454–460
transform of derivatives, 1105–1106
transient and steady-state responses, 446
transient response, 461–467, See also
individual entries
two-degree-of-freedom systems solutions
using, 546–554
Laplacian operator, 790
Lateral vibration of beams, 767–788
axial compressive force, beam subjected
to, 779–780
boundary conditions, 770–772
equation of motion, 767
fixed-pinned beam, natural frequencies
of, 774–777
forced vibration, 776–777
free vibration, 769–770
initial conditions, 769
orthogonality of normal functions, 772–774
simply supported beam, forced vibration, 777–780
Lathe, 510, 530–531
Left half-plane (LHP) yield, 231
Limit cycles, 13-41–13-43
Lindstedt’s perturbation method, 13-10–13-12
Linear algebraic equations, solution of, 1122
Linear coordinates, 598
Linear force, compacting machine under, 431–432
Linear springs, 53–54
Linear variable differential transformer (LVDT)
transducer, 932–933
Linear vibration, 47
Linearization of nonlinear spring, 55–56
Literature, vibration, 109–110
Local coordinate axis, 1052
Logarithmic decrement, 198–199
Longitudinal vibration of bar or rod, 756–764
bar carrying a mass, natural frequencies of,
761–762
bar subjected to initial force, vibrations of, 762–764
boundary conditions, 749–750
equation of motion and solution, 747–749
free vibrations of a fixed-free bar, 760
orthogonality of normal functions, 772–774Index 1141
Longitudinal vibration of bars, 1005–1013
Loops, 34
Lumped-mass matrices, 1066–1069
Lumped-mass model, 598
M
Machine condition monitoring techniques, 970–972
Machine maintenance techniques, 969–970
breakdown maintenance, 969
condition-based maintenance, 970
preventive maintenance, 969
Machine tool support, equivalent spring and
damping constants of, 81–83
Machine vibration monitoring techniques, 972–977
Magnification factor, 302, 311
Marine engine propeller system, 527–530
Mass matrix, 1045–1052
Mass or inertia elements, 69–74
Material damping, 75
Mathematical modeling, 48
Mathieu equation, 13-24–13-29
MATLAB, 365–370, 476–480, 557–564, 662–670,
728–731, 797–800, 895–902, 979–982, 1026–1031,
1069–1073, 1097–1103, 1116–1125
accelerometer equation plotting, 981–982
arrays and matrices, 1117
arrays with special structure, 1118
autocorrelation function plotting, 14-46–14-48
column vector, 1117
complex numbers, 1119
Coulomb damping, free-vibration
response of a system with, 238
damped system, forced vibration response
of, 666–667
differential equations solution,
1123–1125
eigenvalue problem solution, 557–558, 662, 728–729
finite element analysis of stepped bar, 1069
forced response of a system with
Coulomb damping, 366–367
forced vibration response of simply
supported beam, plotting, 797–800
Fourier series graphical representation using, 105–107
free-vibration response, plotting, 559
functions in, 1119
Gaussian probability distribution
function evaluation, 14-48–14-49
general eigenvalue problem, 730–731
impulse response of a structure, 477–478
matrix, 1117
matrix operations, 1118
M-files, 1119–1120
multidegree-of-freedom system, 662–670
nonlinear differential equation solution, 13-61
nonlinearly damped system solution, 13-57–13-59
nonlinear system under pulse loading solution, 13-59
numerical Fourier analysis using, 103
Nyquist circle plotting, 979–980
pendulum equation solution, 13-53–13-57
plotting of graphs, 1120–1121
program to generate characteristic
polynomial, 668
quartic equation roots, 558
railway cars, time response of, 560–561
response under a periodic force, 478–479
response under arbitrary forcing
function, 479
roots of a polynomial equation, 665
roots of a quartic equation, 558
roots of transcendental and nonlinear
equations, 799–800
row vector, 1119
solution of a single-degree-of-freedom system, 1026
solution of multidegree-of-freedom
system, 1027–1028
special matrices, 1118
spring-mass system, free-vibration
response of, 236–238
static deflection, variations of natural frequency
and period with, 235–236
steady-state response of viscously damped
system, 369–370
system response under base excitation, 367–369
total response of an undamped system using,
365–366
total response of system under base excitation,
476–477
transmissibility, plotting, 895
undamped system response, 237
variables, 1117
vibration amplitudes of vibration
absorber masses, 897–898
Matrices, 1097–1103, 1117 See also
individual entries
basic operations, 1102–1103, 1118
trace, 1099
transpose of, 1099
Maximum overshoot (Mp), 4631142 Index
Mean square response, 14-34
Mean value, 14-6–14-7, 14-32
Measurement and applications, vibration, 924–992
machine condition monitoring and
diagnosis, 969–979
measurement scheme, 926
Mechanical chatter, 13-5
Mechanical exciters, 946–947
Method of isoclines, 13-34–13-35
trajectories using, 13-36
Membranes, vibration of, 788–791
equation of motion, 788–790
free vibrations of rectangular
membrane, 791
initial and boundary conditions, 790–791
membrane under uniform tension, 789
Mersenne, Marin, 33–34
M-files, 1119–1120
Milling cutter, natural frequencies of, 766–767
Mindlin, R. D., 36
Modal analysis, 639
forced vibration of undamped systems
using, 646–653
free-vibration response using, 649–651
Modal damping ratio, 655
Modal matrix, 635
Modal testing, 954–969, See also
Experimental modal analysis
Modal vectors, 517
Mode shapes, 626
determination, 699–745
measurement, 966–969
of three-degree-of-freedom system, 633
Monochord, 32
Motor-generator set, absorber for, 889–895
Multidegree-of-freedom systems, 596–698,
14-39–14-46, See also Influence coefficients;
Three-degree-of-freedom system
central difference method for, 1001–1005
continuous systems modeling as,
598–599
equations of motion of undamped systems in
matrix form, 624–625
expansion theorem, 639
free vibration of undamped systems, 644–646
generalized coordinates, 619–620
generalized forces, 619–620
Lagrange’s equations to derive equations
of motion, 620–624
modal analysis, 646–653, See also individual entries
natural frequencies of free system, 641–643
Newton’s second law to derive equations
of motion, 600–605
potential and kinetic energy expressions in
matrix form, 617–619
repeated Eigenvalues, 637–639
self-excitation, 660–662
spring-mass-damper system, equations of
motion of, 600–603
stability analysis, 660–662
steady-state response of forced system, 658–662
trailer–compound pendulum system, equations
of motion of, 603
unrestrained systems, 639–641
Multivariate distribution, 14-8
N
Narrow-band process, 14-25–14-27
Natural frequencies, 92, 517
determination, 699–745, See also Dunkerley’s formula;
Holzer’s method; Jacobi’s method;
Rayleigh’s method
of free system, 641–643
of torsional system, 526, 713–718
Natural mode, two-degree-of-freedom systems, 513
Newmark method, 1022–1025
Newton, Isaac, 34
Newton’s second law, 158–159, 299, 600–603
Nodes, 34, 752, 13-40–13-41
Nomograph, vibration, 821–825
Nondeterministic vibration, 47
Nonlinear damper, linearization of, 81
Nonlinear differential equation solution, 13-61
Nonlinear equations, roots of, 1121
Nonlinear springs, 53–55
Nonlinear system under pulse loading
solution, 13-59
Nonlinear vibration, 47, 13-1–13-76
approximate analytical methods,
13-7–13-19
equilibrium states, stability, 13-37–13-40
exact methods for, 13-6–13-7
graphical methods, 13-29–13-37
iterative method, 13-13–13-16
Jump phenomenon, 13-16
limit cycles, 13-41–13-43
Lindstedt’s perturbation method,
13-10–13-12, 13-25Index 1143
nonlinear spring characteristics, 13-4
numerical methods, 13-52–13-53
Ritz-Galerkin method, 13-17–13-19
subharmonic oscillations, 13-20–13-22
superharmonic oscillations, 13-23–13-24
time-dependent coefficients, systems
with, 13-24–13-29
variable mass system, 13-5–13-6
Nonperiodic force, response under, 405, 420–421,
See also Convolution integral; Laplace transform;
Numerical methods
Normal modes, 513, 634–635
Number-decibel conversion line, 362
Numerical Fourier analysis, 103–105
Numerical integration methods, 993–1040
finite difference method, 995–996
single-degree-of-freedom systems, 996–999
Numerical methods, for response under
nonperiodic force, 405, 467–470
Nyquist circle plotting, 964–965
Nyquist plot, modal data determination from, 961–963
O
Octave, 93
Octave band analyzer, 950, 952
Odd functions, 98–100
Operational deflection shape measurements, 954
Optimally tuned vibration absorber, 894
Orthogonality of normal functions, 634–637, 759–764,
772–775
Orthonormalization of eigenvectors, 635–637
Overdamped system, 194, 458–460
P
Parameter variations, 209–218, See also under
Root locus representations
Parseval’s formula, 14-17, 14-21
Partial fractions method, 1106–1108
Peak time (tp), 461
Perfectly elastic collision, response to, 452–454
Periodic solutions using Lindstedt’s
perturbation method, 13-25
Period of beating , 307
Period of oscillation, 91
Periodic force, 405–418, See also General periodic force,
response under
Periodic vibration of a hydraulic valve, 414–416
Phase angle, 91, 627
Phase distortion, 942–944
Phase plane representation, nonlinear vibration,
13-29–13-34
phase velocity, 13-34
undamped nonlinear system, 13-32
undamped pendulum, 13-31
Phase plane trajectories, time solution from,
13-36–13-37
Phase velocity, 13-34
Philosophiae Naturalis Principia Mathematica, 34
Piezoelectric transducers, 930–931
Pinned-fixed beam, 1012
Piston-cylinder dashpot, 79–81
Plane milling cutter, 767
Plano-milling machine structure, 1043
Plates, deflection of, 1095–1096
Poincaré section, 13-43–13-44
Poisson, Simeon, 36
Positive definite matrix, 619
Positive definite quadratic forms, 619
Potential energy expressions in
matrix form, 617–619
Power spectral density, 14-23–14-25, 14-33
Precision electronic system, vibration control of, 880–881
Precision machine with base motion, design of
isolation for, 866–868
Principal coordinates, 514, 530–535
Principal mode, two-degree-of-freedom systems, 512
Principle of conservation of energy, 160
Principle of virtual displacements, 159
Probability density curve, 973–974
Probability distribution, 14-4–14-5
Propeller shaft, 62–63
Proportional damping, 598
Pseudo spectrum, 437
Pseudo velocity, 437
Pulley subjected to Coulomb damping, 224–225
Pulley system, 174–175
Pulse load, 429–430
response due to, 430
Pythagoras, 31–32
Q
Q factor/quality factor, 316–317
Quadratic damping, 339–340
Quefrency-domain analysis, 9761144 Index
R
Ramp function, first-order system response due to, 448
Random signals analysis, 959–961
Random vibration, 47, 14-1–14-61, See also Stationary
random process
band-limited white noise, 14-25
correlation functions of, 14-9–14-10
eccentricity of rotor, probabilistic
characteristics of, 14-6
Gaussian random process, 14-14–14-16
ideal white noise, 14-25
joint probability distribution, 14-7–14-9
mean value, 14-6–14-7
multidegree-of-freedom system response,
14-39–14-46
narrow-band process, 14-25–14-27
power spectral density, 14-23–14-25
probability distribution, 14-4–14-5
random processes, 14-3–14-4
random variables, 14-3–14-4
single-degree-of-freedom system
response, 14-28–14-31
standard deviation, 14-6–14-7
stationary process, 14-26
stationary random excitations, response due to,
14-31–14-39
wide-band process, 14-25–14-27
Rayleigh, Baron, 36
Rayleigh’s method, 183–188, 703–710, 746, 791–794
beams, fundamental frequency of, 708–710
effect of mass, 185–188
manometer for diesel engine, 183
Rayleigh’s quotient, properties of, 704–706
shafts, fundamental frequency of, 708–710
U-tube manometer, 184
Rayleigh-Ritz method, 746, 794–797
Reciprocating engines, balancing, 842–848
reciprocating engines, balancing,
846–848
unbalanced forces due to fluctuations in gas
pressure, 842–843
unbalanced forces due to inertia of the moving parts,
843–846
Recoil mechanism, 206
Rectangular pulse load, 429–430
response due to, 430
Recurrence formula, 997
Reference marks, 827–828
Relative motion, 323–326
Repeated Eigenvalues, 637–639
Resonance, 47
Resonant frequencies of vibration absorber, 899–900
Response spectrum, 434–446
for base excitation, 436–439
building frame response to an earthquake,
439–441
design under shock environment, 443–446
earthquake response spectra, 439–443
of sinusoidal pulse, 434–436
water tank subjected to base
acceleration, 438–439
Rigid bar
connected by springs, equivalent k of, 66
stability of, 234
Rigid foundation, vibration isolation system
with, 854–863
design chart of isolation, 859–860
isolator for stereo turntable, 860–862
machine member on, 854
resilient member on, 854
spring support for exhaust fan, 857–858
undamped isolator design, 858–860
vibratory motion of mass, reduction, 856–857
Rise time (tr), 461–464
Ritz-Galerkin method, 13-17–13-19
Rod, spring constants of, 57
Root locus representations, 209–218
and parameter variations, 212–218
roots study with variation of c, 214
z in s-plane, 209–212
t in s-plane, 209–212
vd in s-plane, 209–212
v
n in s-plane, 209–212
variation of mass, 218
variation of spring constant, 216
Rotary inertia effects, 780–785
Rotating machines, balancing, 826–835
single-plane balancing, 826–829
two-plane balancing, 829–835
Rotating unbalance, 326–332, 881–888, See also
under Damped system response under
harmonic force
Routh-Hurwitz criteria, 544, 840
Row matrix, 1098
Row vector, 1117
Runge-Kutta methods, 469–471Index 1145
S z
in s-plane, 209–212
t in s-plane, 209–212
vd in s-plane, 209–212
v
n in s-plane, 209–212
Saddle point, 13-40–13-41
Sample point, 14-3
Sample space, 14-3
Sauveur, Joseph, 34
Scotch yoke mechanism, 84
Second-order systems, 404, 412–414, 449–454
Seismograph, 32
Self-excitation, 340–352, 542–544,
660–662
Semidefinite systems, 539–542, 641
Settling time, transient response, 464
Shafts, fundamental frequency of, 708–710
Shear deformation effects, 780–785
Shearing stress (t), 78
Shock absorber for a motorcycle, 204–206
Shock environment, design under, 443–446
Shock isolation, 875–878
Shock loads, 444–447
Signal analysis, 949–953
Signum function, 220
Simple harmonic motion, 85
Simple pendulum, 44, 68, 13-3
Simply supported beam, 134
forced vibration, 776–777
natural frequencies of, 783–785, 787–788
Singing of transmission lines, 344
Single-degree-of-freedom systems, 44,
14-28–14-31
central difference method for, 996–999
characteristics of, 14-30–14-31
free vibration of, 153–296, See also Undamped
translational system, free vibration of
frequency-response approach, 14-30
impulse-response approach, 14-28–14-29
Single-plane balancing, 826–829
Singular point, 13-34
Sinusoidal pulse, response spectrum of, 434–436
Solid damping, 75
Sound (conversion from vibration by
human ear), 40–43
Space shuttle, vibration testing, 40
Special matrices, 1118
Spectrum analyzers, 950–951
Spring constants of elastic elements, 57–59
Spring elements, 52–69
deformation of spring, 52
equivalent linearized spring constant, 56
linear springs, 53–55
nonlinear springs, 53–55
spring constant associated with restoring force due to
gravity, 68
spring constants of elastic elements, 57–59
Spring-mass-damper system, 300, 600–603
Spring-mass systems, 155–157, 714–715
to initial condition, response of, 168
Springs, combination of, 60–68
equivalent k, 62–68
in parallel, 60
in series, 60
torsional spring constant of a propeller
shaft, 62–63
Spring-supported mass instability on
moving belt, 341
Square matrix, 1098
Stability analysis, 340–352, See also Dynamic
instability caused by fluid flow
two-degree-of-freedom systems, 542–544
dynamic, 340–344
multidegree-of-freedom systems,
660–662
whirling of rotating shafts, 835–836
Stability of systems, 231–235
asymptotically stable, 231–233
rigid bar, 234
stable, 231–233
unstable, 231–233
Stable focus, 13-40
Stable orbits, functions with, 13-45
Standard deviation, 14-6–14-7
Standard eigenvalue problem, 628
Static deflection, 301
Static equilibrium position, 161
Static unbalance, 826
Stationary random excitations, response due to,
14-31–14-39
Stationary random process, 14-10–14-14
Strange attractors, 13-46
Steady-state response, 446
of forced system, 658–659
using Laplace transform, 358–359
Step force, response to, 454–460, See also under
Laplace transform
Stepped bar, 1069, 10721146 Index
Stiffness influence coefficient, 605–610
Stiffness matrix, 609, 1045–1052
Stockbridge damper, 347
Stodola, Aurel, 36
Stroboscope, 945
Study of vibration, importance, 38–43
Subharmonic oscillations, 13-20–13-22
Superharmonic oscillations, 13-23–13-24
Suspension system, equivalent k of, 62
Symmetric matrix, 727–728, 1099
System response under base excitation, 367–369
T
Tapered beam, fundamental frequency of, 793–794
Taylor, Brook, 34
Taylor’s series expansion, 55, 349
Temporal averages, 14-14
Thick beam theory, 746
Thin beam theory, 768
Three-degree-of-freedom system, 45
equations of motion of, 629
fundamental frequency of, 706–708
mode shapes of, 633
natural frequencies of, 629–633, 718–722
Time constant, 181–182
Time-delayed step force, 428–429
Time-dependent coefficients, systems with,
13-24–13-29
Time domain analysis, 972
Time domain representations, 97–98
Timoshenko beam theory, 746, 780–781
Timoshenko, Stephen, 36
Torsion element, 1048–1049
Torsional pendulum, 177
Torsional spring constant of a propeller shaft, 62–63
Torsional system, 525–530, 711–714
with Coulomb damping, 223–225
with discs mounted on a shaft, 525
equations of motion of, 621–622
natural frequencies of, 526–527,
711–714
with viscous damping, 201–207
Torsional vibration of a shaft or rod, 764–767
Torsional vibration, 35, 176
Trace, 1099
Trajectories of simple harmonic oscillator,
13-29–13-30
Trailer–compound pendulum system,
equations of motion of, 603
Transducers, 927–933, 955
electric resistance strain gage, 927
electrodynamic transducers, 931–932
linear variable differential transformer (LVDT)
transducer, 932–933
piezoelectric transducers, 930–931
variable resistance transducers,
927–930
Transfer function approach, 352–355, 465–466,
544–545
Transient response, 299, 446, 461–465
Transition curves, 13-28
Transverse vibration of beams, 1009–1012
Transverse vibration of string or cable, 747–756,
See also under Continuous systems
Traveling-wave solution, 755–756
Triangular pulse, Fourier transform of, 14-22
Triple pendulum, 619
Tuned vibration absorber, 894
Two-degree-of-freedom systems, 45, 509–595, See also
Forced vibration; Laplace transform; Semidefinite
systems; Torsional system
automobile, frequencies and modes of, 534–535
coordinate coupling and principal
coordinates, 530–535
coupled differential equations, 512
equations of motion for forced vibration, 514–515
forced response of, 562–564
free-vibration response of, 523–524
Lathe, 511, 531–532
natural mode, 513
normal mode, 512
packaging of an instrument, 513
principal mode, 512
spring-mass-damper system, 514
transfer function approach, 544–545
Two-plane balancing, 829–835, 900–902
U
Undamped dynamic vibration absorber, 884–890
effect on the response of machine, 886
for diesel engine, 888–889
for motor-generator set, 889–890
Undamped equation, 13-13, 13-31
Undamped isolator design, 858–860
Undamped system, 155
free-vibration analysis, 516–524
free vibration of, 644–646
free-vibration response of, 546–554Index 1147
in matrix form, 624–625
response under harmonic force,
301–309, See also under
Harmonically excited vibration
total response of, using MATLAB, 365–366
Undamped torsional system, free vibration of,
176–181
Undamped translational system, free
vibration of, 158–175
auxiliary or characteristic equation, 163
D’Alembert’s principle, 159
eigenvalues or characteristic values, 163
mass under virtual displacement, 160
principle of conservation of energy, 159
principle of virtual displacements, 159
using Newton’s second law of motion,
158–159
Undamped vibration, 47
Underdamped system, 190, 454–456
response of due to initial conditions, 195–197
Uniform string, free vibration of, 750–751
Unit impulse response of second-order system, 447
Units, 1112–1115
Univariate distributions, 14-8
Unrestrained systems, 541–542, 639–643
Unstable focus, 13-40
Unstable orbits, functions with, 13-45–13-47
Unstable system, 231
V
Variable mass system, 13-5–13-6
Variable resistance transducers, 927–930
Vectorial representation of harmonic motion, 85
Velometer, 940–941
Vertical position, spring-mass system in, 161–162
Vibrating string, 748
Vibration absorbers, 883–895, 899–900, 891, See also
Damped dynamic vibration absorber; Undamped
dynamic vibration absorber
Vibration pickups, 933–935
Vibration severity of machinery, 823
Vibrometer, 935–936
Viscoelastic materials use, 849–850
Viscous damping, 74–75
Cannon analysis, 206
energy dissipated in, 199–201
forced transmission to the base, 314–315
forced vibration of, 653–659
free vibration with, 188–207
steady-state response of, 369–370
torsional systems with, 201–207
W
Wallis, John, 34
Whirling of rotating shafts, 835–841
critical speeds, 837
equations of motion, 835–837
shaft carrying an unbalanced rotor, 841
stability analysis, 840–841
system response, 838–840
Wide-band process, 14-25–14-27
Wiener-Khintchine formula, 14-23
Wilson method, 1019–1022
Wind-induced vibration, 39
Y
Young’s modulus, 172
Z
Zero matrix, 1098
Zhang Heng, 32

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