حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition - صفحة 2
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منتدى هندسة الإنتاج والتصميم الميكانيكى
بسم الله الرحمن الرحيم

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نتمنى أن تقضوا معنا أفضل الأوقات
وتسعدونا بالأراء والمساهمات
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أو وإذا كانت هذة زيارتك الأولى للمنتدى فنتشرف بإنضمامك لأسرتنا
وهذا شرح لطريقة التسجيل فى المنتدى بالفيديو :
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 حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition

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حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition - صفحة 2 Empty
مُساهمةموضوع: حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition   حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition - صفحة 2 Emptyالثلاثاء 01 يناير 2013, 12:08 am

تذكير بمساهمة فاتح الموضوع :

أخواني في الله
أحضرت لكم كتاب
حل كتاب الاهتزازات الميكانيكية
Mechanical Vibrations Solution Manual
Fifth Edition
Singiresu S. Rao
University of Miami  

حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition - صفحة 2 M_v_s_13
و المحتوى كما يلي :


Prefacexi
Acknowledgmentsxv
List of Symbolsxvi
CHAPTER 1
Fundamentals of Vibration1
1.1Preliminary Remarks2
1.2Brief History of the Study of Vibration3
1.2.1Origins of the Study of Vibration 3
1.2.2From Galileo to Rayleigh 6
1.2.3Recent Contributions 9
1.3Importance of the Study of Vibration10
1.4Basic Concepts of Vibration13
1.4.1Vibration 13
1.4.2Elementary Parts of
Vibrating Systems 13
1.4.3Number of Degrees of Freedom 14
1.4.4Discrete and Continuous Systems 16
1.5Classification of Vibration16
1.5.1Free and Forced Vibration 17
1.5.2Undamped and Damped Vibration 17
1.5.3Linear and Nonlinear Vibration 17
1.5.4Deterministic and
Random Vibration 17
1.6Vibration Analysis Procedure18
1.7Spring Elements22
1.7.1Nonlinear Springs 23
1.7.2Linearization of a
Nonlinear Spring 25
1.7.3Spring Constants of Elastic Elements 27
1.7.4Combination of Springs 30
iv
1.7.5Spring Constant Associated with the
Restoring Force due to Gravity 39
1.8Mass or Inertia Elements40
1.8.1Combination of Masses 40
1.9Damping Elements45
1.9.1Construction of Viscous Dampers 46
1.9.2Linearization of a
Nonlinear Damper 52
1.9.3Combination of Dampers 52
1.10Harmonic Motion54
1.10.1Vectorial Representation of
Harmonic Motion 56
1.10.2Complex-Number Representation
of Harmonic Motion 57
1.10.3Complex Algebra 58
1.10.4Operations on Harmonic Functions 59
1.10.5Definitions and Terminology 62
1.11Harmonic Analysis64
1.11.1Fourier Series Expansion 64
1.11.2Complex Fourier Series 66
1.11.3Frequency Spectrum 67
1.11.4Time- and Frequency-Domain
Representations 68
1.11.5Even and Odd Functions 69
1.11.6Half-Range Expansions 71
1.11.7Numerical Computation
of Coefficients 72
1.12Examples Using MATLAB76
1.13Vibration Literature80
Chapter Summary81
References81
Review Questions83
Problems87
Design Projects120
ContentsCONTENTS v
CHAPTER 2
Free Vibration of Single-Degree-of-Freedom
Systems124
2.1Introduction126
2.2Free Vibration of an Undamped
Translational System129
2.2.1Equation of Motion Using Newton s
Second Law of Motion 129
2.2.2Equation of Motion Using Other
Methods 130
2.2.3Equation of Motion of a Spring-Mass
System in Vertical Position 132
2.2.4Solution 133
2.2.5Harmonic Motion 134
2.3Free Vibration of an Undamped
Torsional System146
2.3.1Equation of Motion 147
2.3.2Solution 148
2.4Response of First Order Systems
and Time Constant151
2.5Rayleigh s Energy Method153
2.6Free Vibration with Viscous Damping158
2.6.1Equation of Motion 158
2.6.2Solution 158
2.6.3Logarithmic Decrement 164
2.6.4Energy Dissipated in Viscous
Damping 166
2.6.5Torsional Systems with Viscous
Damping 168
2.7Graphical Representation of Characteristic Roots
and Corresponding Solutions174
2.7.1Roots of the Characteristic Equation 174
2.7.2Graphical Representation of Roots and
Corresponding Solutions 175
2.8Parameter Variations and Root Locus
Representations176
2.8.1Interpretations of and
in s-plane 176
2.8.2Root Locus and Parameter
Variations 179
2.9Free Vibration with Coulomb Damping185
2.9.1Equation of Motion 186
2.9.2Solution 187
2.9.3Torsional Systems with Coulomb
Damping 190
v t
n
, vd, z,
2.10Free Vibration with Hysteretic Damping192
2.11Stability of Systems198
2.12Examples Using MATLAB202
Chapter Summary208
References209
Review Questions209
Problems214
Design Projects256
CHAPTER 3
Harmonically Excited Vibration259
3.1Introduction261
3.2Equation of Motion261
3.3Response of an Undamped System
Under Harmonic Force263
3.3.1Total Response 267
3.3.2Beating Phenomenon 267
3.4Response of a Damped System Under
Harmonic Force271
3.4.1Total Response 274
3.4.2Quality Factor and Bandwidth 276
3.5Response of a Damped System
Under 278
3.6Response of a Damped System Under the
Harmonic Motion of the Base281
3.6.1Force Transmitted 283
3.6.2Relative Motion 284
3.7Response of a Damped System Under Rotating
Unbalance287
3.8Forced Vibration with Coulomb Damping293
3.9Forced Vibration with Hysteresis Damping298
3.10Forced Motion with Other Types of
Damping300
3.11Self-Excitation and Stability Analysis301
3.11.1Dynamic Stability Analysis 301
3.11.2Dynamic Instability Caused by Fluid
Flow 305
3.12Transfer-Function Approach313
3.13Solutions Using Laplace Transforms317
3.14Frequency Transfer Functions320
3.14.1Relation Between the General Transfer
function T(s) and the Frequency Transfer
Function 322
3.14.2Representation of Frequency-Response
Characteristics 323
T(iv)
F(t) = F0eiVtvi CONTENTS
3.15Examples Using MATLAB326
Chapter Summary332
References332
Review Questions333
Problems336
Design Projects362
CHAPTER 4
Vibration Under General Forcing
Conditions363
4.1Introduction364
4.2Response Under a General
Periodic Force365
4.2.1First-Order Systems 366
4.2.2Second-Order Systems 372
4.3Response Under a Periodic Force
of Irregular Form378
4.4Response Under a Nonperiodic Force380
4.5Convolution Integral381
4.5.1Response to an Impulse 382
4.5.2Response to a General Forcing
Condition 385
4.5.3Response to Base Excitation 386
4.6Response Spectrum394
4.6.1Response Spectrum for Base
Excitation 396
4.6.2Earthquake Response Spectra 399
4.6.3Design Under a Shock
Environment 403
4.7Laplace Transform406
4.7.1Transient and Steady-State
Responses 406
4.7.2Response of First-Order Systems 407
4.7.3Response of Second-Order Systems 409
4.7.4Response to Step Force 414
4.7.5Analysis of the Step Response 420
4.7.6Description of Transient
Response 421
4.8Numerical Methods428
4.8.1Runge-Kutta Methods 429
4.9Response to Irregular Forcing Conditions Using
Numerical Methods431
4.10Examples Using MATLAB436
Chapter Summary440
References440
Review Questions441
Problems444
Design Projects465
CHAPTER 5
Two-Degree-of-Freedom Systems467
5.1Introduction468
5.2Equations of Motion for Forced
Vibration472
5.3Free Vibration Analysis of an Undamped
System474
5.4Torsional System483
5.5Coordinate Coupling and Principal
Coordinates488
5.6Forced-Vibration Analysis494
5.7Semidefinite Systems497
5.8Self-Excitation and Stability
Analysis500
5.9Transfer-Function Approach502
5.10Solutions Using Laplace Transform504
5.11Solutions Using Frequency Transfer
Functions512
5.12Examples Using MATLAB515
Chapter Summary522
References523
Review Questions523
Problems526
Design Projects552
CHAPTER 6
Multidegree-of-Freedom Systems553
6.1Introduction555
6.2Modeling of Continuous Systems as Multidegreeof-Freedom Systems555
6.3Using Newton s Second Law to Derive Equations
of Motion557
6.4Influence Coefficients562
6.4.1Stiffness Influence Coefficients 562
6.4.2Flexibility Influence Coefficients 567
6.4.3Inertia Influence Coefficients 572
6.5Potential and Kinetic Energy Expressions in
Matrix Form574
6.6Generalized Coordinates and Generalized
Forces576
6.7Using Lagrange s Equations to Derive Equations
of Motion577CONTENTS vii
6.8Equations of Motion of Undamped Systems in
Matrix Form581
6.9Eigenvalue Problem583
6.10Solution of the Eigenvalue Problem585
6.10.1Solution of the Characteristic
(Polynomial) Equation 585
6.10.2Orthogonality of Normal Modes 591
6.10.3Repeated Eigenvalues 594
6.11Expansion Theorem596
6.12Unrestrained Systems596
6.13Free Vibration of Undamped Systems601
6.14Forced Vibration of Undamped Systems Using
Modal Analysis603
6.15Forced Vibration of Viscously Damped
Systems610
6.16Self-Excitation and Stability Analysis617
6.17Examples Using MATLAB619
Chapter Summary627
References627
Review Questions628
Problems632
Design Project653
CHAPTER 7
Determination of Natural Frequencies and
Mode Shapes654
7.1Introduction655
7.2Dunkerley s Formula656
7.3Rayleigh s Method658
7.3.1Properties of Rayleigh s Quotient 659
7.3.2Computation of the Fundamental Natural
Frequency 661
7.3.3Fundamental Frequency of Beams and
Shafts 663
7.4Holzer s Method666
7.4.1Torsional Systems 666
7.4.2Spring-Mass Systems 669
7.5Matrix Iteration Method670
7.5.1Convergence to the Highest Natural
Frequency 672
7.5.2Computation of Intermediate Natural
Frequencies 673
7.6Jacobi s Method678
7.7Standard Eigenvalue Problem680
7.7.1Choleski Decomposition 681
7.7.2Other Solution Methods 683
7.8Examples Using MATLAB683
Chapter Summary686
References686
Review Questions688
Problems690
Design Projects698
CHAPTER 8
Continuous Systems699
8.1Introduction700
8.2Transverse Vibration of a String or
Cable701
8.2.1Equation of Motion 701
8.2.2Initial and Boundary Conditions 703
8.2.3Free Vibration of a Uniform
String 704
8.2.4Free Vibration of a String with Both Ends
Fixed 705
8.2.5Traveling-Wave Solution 709
8.3Longitudinal Vibration of a Bar or Rod710
8.3.1Equation of Motion
and Solution 710
8.3.2Orthogonality of Normal
Functions 713
8.4Torsional Vibration of a Shaft or Rod718
8.5Lateral Vibration of Beams721
8.5.1Equation of Motion 721
8.5.2Initial Conditions 723
8.5.3Free Vibration 723
8.5.4Boundary Conditions 724
8.5.5Orthogonality of Normal
Functions 726
8.5.6Forced Vibration 730
8.5.7Effect of Axial Force 732
8.5.8Effects of Rotary Inertia and Shear
Deformation 734
8.5.9Other Effects 739
8.6Vibration of Membranes739
8.6.1Equation of Motion 739
8.6.2Initial and Boundary Conditions 741
8.7Rayleigh s Method742
8.8The Rayleigh-Ritz Method745
8.9Examples Using MATLAB748
Chapter Summary751
References751viii CONTENTS
Review Questions753
Problems756
Design Project768
CHAPTER 9
Vibration Control769
9.1Introduction770
9.2Vibration Nomograph and Vibration
Criteria771
9.3Reduction of Vibration at the Source775
9.4Balancing of Rotating Machines776
9.4.1Single-Plane Balancing 776
9.4.2Two-Plane Balancing 779
9.5Whirling of Rotating Shafts785
9.5.1Equations of Motion 785
9.5.2Critical Speeds 787
9.5.3Response of the System 788
9.5.4Stability Analysis 790
9.6Balancing of Reciprocating Engines792
9.6.1Unbalanced Forces Due to Fluctuations in
Gas Pressure 792
9.6.2Unbalanced Forces Due to Inertia of the
Moving Parts 793
9.6.3Balancing of Reciprocating
Engines 796
9.7Control of Vibration798
9.8Control of Natural Frequencies798
9.9Introduction of Damping799
9.10Vibration Isolation801
9.10.1Vibration Isolation System with Rigid
Foundation 804
9.10.2Vibration Isolation System with Base
Motion 814
9.10.3Vibration Isolation System with Flexible
Foundation 821
9.10.4Vibration Isolation System with Partially
Flexible Foundation 822
9.10.5Shock Isolation 824
9.10.6Active Vibration Control 827
9.11Vibration Absorbers832
9.11.1Undamped Dynamic Vibration
Absorber 833
9.11.2Damped Dynamic Vibration
Absorber 840
9.12Examples Using MATLAB843
Chapter Summary851
References851
Review Questions853
Problems855
Design Project869
CHAPTER 10
Vibration Measurement and
Applications870
10.1Introduction871
10.2Transducers873
10.2.1Variable Resistance Transducers 873
10.2.2Piezoelectric Transducers 876
10.2.3Electrodynamic Transducers 877
10.2.4Linear Variable Differential Transformer
Transducer 878
10.3Vibration Pickups879
10.3.1Vibrometer 881
10.3.2Accelerometer 882
10.3.3Velometer 886
10.3.4Phase Distortion 888
10.4Frequency-Measuring Instruments890
10.5Vibration Exciters892
10.5.1Mechanical Exciters 892
10.5.2Electrodynamic Shaker 893
10.6Signal Analysis895
10.6.1Spectrum Analyzers 896
10.6.2Bandpass Filter 897
10.6.3Constant-Percent Bandwidth and
Constant-Bandwidth Analyzers 898
10.7Dynamic Testing of Machines
and Structures900
10.7.1Using Operational Deflection-Shape
Measurements 900
10.7.2Using Modal Testing 900
10.8Experimental Modal Analysis900
10.8.1The Basic Idea 900
10.8.2The Necessary Equipment 900
10.8.3Digital Signal Processing 903
10.8.4Analysis of Random Signals 905
10.8.5Determination of Modal Data
from Observed Peaks 907
10.8.6Determination of Modal Data
from Nyquist Plot 910
10.8.7Measurement of Mode Shapes 912
10.9Machine Condition Monitoring
and Diagnosis915CONTENTS ix
10.9.1Vibration Severity Criteria 915
10.9.2Machine Maintenance Techniques 915
10.9.3Machine Condition Monitoring
Techniques 916
10.9.4Vibration Monitoring Techniques 918
10.9.5Instrumentation Systems 924
10.9.6Choice of Monitoring Parameter 924
10.10Examples Using MATLAB925
Chapter Summary928
References928
Review Questions930
Problems932
Design Projects938
CHAPTER 11
Numerical Integration Methods in
Vibration Analysis939
11.1Introduction940
11.2Finite Difference Method941
11.3Central Difference Method for Single-Degree-ofFreedom Systems942
11.4Runge-Kutta Method for Single-Degree-ofFreedom Systems945
11.5Central Difference Method for Multidegree-ofFreedom Systems947
11.6Finite Difference Method for Continuous
Systems951
11.6.1Longitudinal Vibration of Bars 951
11.6.2Transverse Vibration of Beams 955
11.7Runge-Kutta Method for Multidegree-ofFreedom Systems960
11.8Houbolt Method962
11.9Wilson Method965
11.10Newmark Method968
11.11Examples Using MATLAB972
Chapter Summary978
References978
Review Questions979
Problems981
CHAPTER 12
Finite Element Method987
12.1Introduction988
12.2Equations of Motion of an Element989
12.3Mass Matrix, Stiffness Matrix, and Force
Vector991
12.3.1Bar Element 991
12.3.2Torsion Element 994
12.3.3Beam Element 995
12.4Transformation of Element Matrices and
Vectors998
12.5Equations of Motion of the Complete System of
Finite Elements1001
12.6Incorporation of Boundary
Conditions1003
12.7Consistent- and Lumped-Mass Matrices1012
12.7.1Lumped-Mass Matrix for a Bar
Element 1012
12.7.2Lumped-Mass Matrix for a Beam
Element 1012
12.7.3Lumped-Mass Versus Consistent-Mass
Matrices 1013
12.8Examples Using MATLAB1015
Chapter Summary1019
References1019
Review Questions1020
Problems1022
Design Projects1034
Chapters 13 and 14 are provided as downloadable
files on the Companion Website.
CHAPTER 13
Nonlinear Vibration13-1
13.1Introduction13-2
13.2Examples of Nonlinear Vibration Problems13-3
13.2.1Simple Pendulum 13-3
13.2.2Mechanical Chatter, Belt Friction
System 13-5
13.2.3Variable Mass System 13-5
13.3Exact Methods13-6
13.4Approximate Analytical Methods13-7
13.4.1Basic Philosophy 13-8
13.4.2Lindstedt s Perturbation Method 13-10
13.4.3Iterative Method 13-13
13.4.4Ritz-Galerkin Method 13-17
13.5Subharmonic and Superharmonic
Oscillations13-19
13.5.1Subharmonic Oscillations 13-20
13.5.2Superharmonic Oscillations 13-23
13.6Systems with Time-Dependent Coefficients
(Mathieu Equation)13-2413.7Graphical Methods13-29
13.7.1Phase-Plane Representation 13-29
13.7.2Phase Velocity 13-34
13.7.3Method of Constructing
Trajectories 13-34
13.7.4Obtaining Time Solution from Phase
Plane Trajectories 13-36
13.8Stability of Equilibrium States13-37
13.8.1Stability Analysis 13-37
13.8.2Classification of Singular
Points 13-40
13.9Limit Cycles13-41
13.10Chaos13-43
13.10.1Functions with Stable Orbits 13-45
13.10.2Functions with Unstable Orbits 13-45
13.10.3Chaotic Behavior of Duffing s Equation
Without the Forcing Term 13-47
13.10.6Chaotic Behavior of Duffing s Equation
with the Forcing Term 13-50
13.11Numerical Methods13-52
13.12Examples Using MATLAB13-53
Chapter Summary13-62
References13-62
Review Questions13-64
Problems13-67
Design Projects13-75
CHAPTER 14
Random Vibration14-1
14.1Introduction14-2
14.2Random Variables and Random Processes14-3
14.3Probability Distribution14-4
14.4Mean Value and Standard Deviation14-6
14.5Joint Probability Distribution of Several
Random Variables14-7
14.6Correlation Functions of a Random Process14-9
14.7Stationary Random Process14-10
14.8Gaussian Random Process14-14
14.9Fourier Analysis14-16
14.9.1Fourier Series 14-16
14.9.2Fourier Integral 14-19
14.10Power Spectral Density14-23
14.11Wide-Band and Narrow-Band Processes14-25
14.12Response of a Single-Degree-ofFreedom System14-28
14.12.1Impulse-Response Approach 14-28
14.12.2Frequency-Response Approach 14-30
14.12.3Characteristics of the Response
Function 14-30
14.13Response Due to Stationary Random
Excitations14-31
14.13.1Impulse-Response Approach 14-32
14.13.2Frequency-Response Approach 14-33
14.14Response of a Multidegree-of-Freedom
System14-39
14.15Examples Using MATLAB14-46
Chapter Summary14-49
References14-49
Review Questions14-50
Problems14-53
Design Project14-61
APPENDIX A
Mathematical Relationships and Material
Properties 1036
APPENDIX B
Deflection of Beams and Plates 1039
APPENDIX C
Matrices 1041
APPENDIX D
Laplace Transform 1048
APPENDIX E
Units 1056
APPENDIX F
Introduction to MATLAB 1059
Answers to Selected Problems 1069
Index 1077
x CONTENTSIndex
1077
A
Accelerographs, 399 400
Accelerometer, 882 886
Acoustics , 6
Active vibration control, 827 832
Addition of harmonic motions, 60 61
Adjoint matrix, 1045
Advance, 873
Airfoil, dynamic instability of, 309 310
Amplitude, 62, 584, 806
Analysis, vibration, 18 22
equations, 20
mathematical modeling, 18
motorcycle, mathematical model of, 20
results, interpretation, 20
Aristotle, 4
Aristoxenus, 4
Arrays with special structure, 1061
Attractor, 13-43 13-44
Asymptotically stable system, 198
Autocorrelation function, 14-9, 14-11, 14-32
Axial compressive force, beam subjected to,
733 737
Axial force effect, 732 734
B
Band-limited white noise, 14-25
Bandpass filter, 897 898
Bandwidth, 276 278
Bar element, 991 994
Base excitation
response spectrum for, 396 399
system response under, 328 330
Basic concepts of vibration, 13 16
Bathtub curve, 915
Beam deflections, 571
Beam element, 995 998
Beams, deflection of, 1039 1040
cantilever beam, 1039
fixed-fixed beam with end displacement,
1039
fixed-fixed beam, 1039
simply supported beam, 1039 1040
Beams, fundamental frequency of, 663 665
Beating phenomenon, 63, 267 271
Belt friction system, 13-5
Bernoulli, Daniel, 7
Bifurcations, 13-46
Bivariate distributions, 14-8
Blast load on building frame, 392 393
Bode diagrams, 323 324
Bonaparte, Napoléon, 7
Boundary conditions, incorporation of,
1003 1012
Boundary curves, 13-28
Building frame response to an earthquake, 402
C
Cam-follower mechanism, 44 45, 73
spring mass system for, 126
Cannon analysis, 173
Cantilever beam, 1039
spring constants of, 28
center, 13-30, 13-40
Center of percussion, 150 151
Central difference method for multidegree of
freedom systems, 947 951
Centrifugal pump with rotating unbalance,
812 816
rattle space, 812 816
Cepstrum, 922
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, 585 590
Characteristic roots, graphical representation, 174 176
Chimney, flow-induced vibration of, 311
Choleski decomposition, 681 683
Classification of vibration, 16 18
Clebsch, R. F. A., 8
Coefficients, numerical computation of,
72 76
Coherence function, 907
Column matrix, 1042
Column vector, 1060
Compacting machine, 387 388, 416 418
Complex algebra, 58
Complex damping, 300
Complex Fourier series, 66 67, 14-17 14-18
Complex frequency response, 279, 14-30
Complex numbers, 1062
harmonic motion representation, 57 58
Complex stiffness, 194
Complex vector representation of harmonic
motion, 280
Compound pendulum, 148 150, 561
natural frequency of, 148 151
Consistent mass matrices, 1012 1015
Constant bandwidth analyzers, 898 899
Constant damping, 186
Constant percent bandwidth, 898 899
Continuous systems, 16, 699 768, See also
Lateral vibration of beams; Longitudinal vibration of bar or rod; Torsional vibration of a shaft or rod
dynamic response of plucked string,
707 712
modeling as multidegree of freedom systems, 555 556
transverse vibration of a string or cable,
701 710
Continuous systems, finite difference
method for, 951 959
longitudinal vibration of bars, 951 955
pinned-fixed beam, 959
transverse vibration of beams, 955 959
Control, vibration, 769 869
criteria, 771 775
natural frequencies, control of, 798 799
nomograph, 771 775
ranges of vibration, 773
whirling of rotating shafts, 785 791
Conversion of units, 1056 1058
Convolution integral, 365, 381 393,
1052 1055
blast load on building frame, 392 393
compacting machine under linear force,
391 392
rectangular pulse load, 389 390
response of a structure under double
impact, 385
response of a structure under impact, 384
response to a general forcing condition,
385 386
response to an impulse, 382 385
response to base excitation, 386 393
step force on a compacting machine,
387 388
time-delayed step force, 388 389
Coordinate coupling, 488 493
Correlation functions of random process,
14-9 14-10
Coulomb damping, 46
forced response of, using MATLAB,
327 328
forced vibration with, 293 297
free vibration response of a system with,
205
free vibration with, 185 192
pulley subjected to, 191
Coulomb, Charles, 7
Please note that references to pages in chapters 13 and 14 appear in the form 13-1, 13-2, etc., and these chapters are provided on the companion Website.1078 INDEX
Coupled differential equations, 470
Crane, equivalent k of, 35
Critical damping constant, 159
Critical speeds, 787
Critically damped system, 162
Cycle, 62
DD
Alembert, Jean, 7
D Alembert s principle, 130
Damped dynamic vibration absorber,
840 843
Damped equation, 13-14
Damped response using numerical methods,
434 436
Damped single degree of freedom system
Bode diagrams of, 324
transfer function, 315
Damped system, 127
forced vibration response of, MATLAB,
623 624
free vibration response of, Laplace transform, 507 511
Damped system response under F(t) = F0ei*t,
278 281
Damped system response under harmonic
force, 271 278, See also under Harmonically excited vibration
under F(t) = F0ei*t, 278 281
graphical representation, 272
under harmonic motion of base, 281 287
under rotating unbalance, 287 293
total response, 274 276
vectorial representation, 272
Damped system response using Laplace
transform, 317 318
Damped vibration, 17
Damping, 799 800
damping matrix, 611
damping ratio, 159, 179
viscoelastic materials use, 799
Damping elements, 45 54
clearance in a bearing, 47 48
combination of dampers, 52 54
Coulomb or dry friction damping, 46
damping constant of journal bearing,
48 50
damping constant of parallel plates, 47
linearization of nonlinear damper, 52
material or solid or hysteretic damping, 46
piston-cylinder dashpot, 50 52
viscous damping, 45
viscous dampers construction, 46 52
De Laval, C. G. P., 9
Decibel, 63
Degree of Freedom, 14 16
Delay time (td), transient response, 425
Design chart of isolation, 809 810
Determinant, 1043
Deterministic vibration, 17 18, 14-2
Diagonal matrix, 1042
Diesel engine, vibration absorber for,
837 838
Differential equations, 313, 1066 1068
Digital signal processing, 903 904
Dirac delta function, 381
Discrete systems, 16
Displacement method, 1003
Displacement transmissibility, 282 283, 806
Dry friction damping, 46
Duffing s equation, 13-13, 13-47 13-50
Duhamel integral, See Convolution integral
Dunkerley s formula, 654 658
Dynamic coupling, 490
Dynamic instability caused by fluid flow,
305 313
of an airfoil, 309 310
flow-induced vibration of a chimney,
311
flow-induced vibration reduction, 307
Helical spoilers, 308
Stockbridge damper, 308
Dynamic response of plucked string, 707 712
Dynamic stability analysis, 301 305
Dynamic system, equations of motion of,
613 615
Dynamic testing of machines and structures,
900
Dynamical matrix, 585
E
Earthquake response spectra, 399 403
Eccentricity of rotor, probabilistic characteristics of, 14-6
Eigenvalues/Eigenvalue problem, 583 584,
594 596
Eigenvectors, orthonormalization of,
592 594
Equilibrium states, stability, 13-37 13-40
Elastic coupling, 490
Elastic potential energy, 574 576
Electric motor deflection due to rotating
unbalance, 290
Electrodynamic Shaker, 893 895
Electrodynamic transducers, 877 878
Element matrices and vectors, transformation, 998 1001
Elementary parts of vibrating systems,
13 14
Energy dissipated in viscous damping,
166 168
Equation of motion, 147 148, 158, 186 187,
261 262
derivation, 577 581
of dynamic system, 613 615
of an element, 989 991
of finite elements, 1001 1003
for forced vibration, 472 473
of three degree of freedom system, 586
of undamped systems in matrix form,
581 582
whirling of rotating shafts, 785 787
Equivalent linearized spring constant, 26
Equivalent mass of a system, 42 44
Equivalent rotational mass, 42
Equivalent translational mass, 42
Ergodic process, 14-13
Euler, Leonard, 7
Euler-Bernoulli theory, 722, 995
Even functions, 69 71
Exciters, vibration, 892 895, 901
due to unbalanced force, 893
electrodynamic shaker, 893 895
mechanical exciters, 892 893
Expansion theorem, 596
Experimental modal analysis, 900 915
basic idea, 900
coherence function, 907
digital signal processing, 903 904
modal data determination from observed
peaks, 907 912
mode shapes measurement, 912 915
necessary equipment, 900 903
random signals analysis, 905 907
Explicit integration method, 943
F
Fast Fourier Transform (FFT) method, 896,
902, 924
Finishing process, vibratory, 12
Finite difference method, 941 942
for continuous systems, 951 959
Finite element idealization, 10
Finite element method, 987 1035
bar element, 991 994
beam element, 995 998
boundary conditions, incorporation of,
1003 1012
element matrices and vectors, transformation, 998 1001
equations of motion of, 989 991,
1001 1003
Euler-Bernoulli theory, 995
force vector, 991 998
mass matrix, 991 998
stiffness matrix, 991 998
torsion element, 994 995
First order systems, 151 153, 366 372
response of, 407 408
response under periodic force, 367 372
Fixed-free bar, free vibrations of, 714
Fixed-pinned beam, natural frequencies of,
728 731
Flexibility influence coefficients, 567 571
determination, 569
Flexibility matrix of a beam, 571
Flow-induced vibration
of a chimney, 311
reduction, 307
Flutter, 305
Focus, 13-40
Force transmissibility, 283 284
Force vector, 991 998
Forced system, steady-state response of,
615 619
Forced vibration, 17, 494 497, 730 731
steady-state response of spring-mass
system, 495 497
of viscously damped systems, 610 616INDEX 1079
Forging hammer
forced vibration response of, 608 613
response of anvil of, 169
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, 64 66, 73
Cam-follower system, 73
complex Fourier series, 66 67
Gibbs phenomenon, 66
graphical representation using
MATLAB, 76 78
numerical Fourier analysis, 74 76
periodic function, 65
Fourth-order Runge-Kutta method, 974 975
Frahm tachometer, 9, 890
Francis water turbine, 291
Free vibration, 17
response of two degree of freedom
system, 481 482
response using modal analysis, 606 608
Frequency domain representations, 68 69
Frequency-measuring instruments, 890 891
frequency-measuring instruments, 891
multireed-instrument, 890
single-reed instrument, 890
stroboscope, 891
Frequency of damped vibration, 161
Frequency of oscillation, 62
Frequency or characteristic equation, 475
Frequency response approach, 14-30,
14-33 14-39
mean square response, 14-34
power spectral density, 14-33
Frequency spectrum, 67 68
Frequency transfer functions, 320 325
frequency response characteristics representation, 323 325
general transfer function and, 322 323
physical system, 321
solutions using, 512 515
Fullarton tachometer, 890
G
Galileo Galilei, 6 9
Galloping, 305 306
Gaussian random process, 14-14 14-16
General forcing conditions, vibration under,
363 466, See also General periodic
force, response under; Nonperiodic
force, response under; Periodic force;
Response spectrum
General periodic force, response under,
365 378
first-order systems, 366 372
second-order systems, 367, 372 374
total response under harmonic base excitation, 377 378
General transfer function and frequency
transfer function, 322 323
Generalized coordinates, 472, 488, 576 577
Generalized forces, 576 577
Generalized mass matrix, 575
Germain, Sophie, 8
Gibbs phenomenon, 66
Grid points, 941
H
Half power points, 276
Half-range expansions, 71 72
Harmonic analysis, 64 76, See also Fourier
series expansion
even functions, 69 71
frequency domain representations, 68 69
half-range expansions, 71 72
odd functions, 69 71
time domain representations, 68 69
Harmonic base excitation, total response
under, 377 378
Harmonic motion, 54 64, 134 146
addition of harmonic motions, 60
complex algebra, 58
complex number representation of,
57 58
impact, free vibration response due to,
141 142
motion of, graphical representation, 136
natural frequency, 142 146
operations on harmonic functions, 59 61
Scotch yoke mechanism, 55
simple harmonic motion, 56
undamped system, phase plane representation, 138
vectorial representation of, 56 57
water tank, harmonic response of,
139 141
Harmonically excited vibration, 259 362
damped system response under F(t) =
F0ei*t, 278 281
damped system response under harmonic
force, 271 278, See also individual entry
equation of motion, 261 262
forced vibration with Coulomb damping,
293 297, See also Coulomb
damping
hysteresis damping, forced vibration
with, 298 310
quadratic damping, 300
quality factor and bandwidth, 276 278
undamped system response under,
263 271
Helical spoilers, 308
Helicopter seat vibration reduction, 774 783
vibration at source, reduction, 775 776
Heterodyne analyzer, 899
History of vibration, 3 10
finite element idealization, 10
from Galileo to Rayleigh, 6 9
origin, 3 5
recent contributions, 9 10
theory of vibration of plates, 7
torsional vibration tests, 8
Hoisting drum, equivalent k of, 34 35
Holzer s method, 666 670
resultant torque versus frequency, 667
spring-mass systems, 669 670
torsional systems, 666 669
Hooke, Robert, 6
Horizontal position, spring-mass system in,
126
Houbolt method, 962 965
for a two degree of freedom system, 964
Hydraulic valve, periodic vibration of,
374 376
Hysteretic damping, 46
forced vibration with, 298 310
free vibration with, 192 198
I
Ideal white noise, 14-25
Identity matrix, 1042
Implicit integration methods, 963
Impulse response function, 382 385,
14-28 14-29
Inelastic collision, response to, 411 412
Inertia influence coefficients, 572 573
Influence coefficients, 562 573
flexibility influence coefficients,
567 571
flexibility matrix of a beam, 571
inertia influence coefficients, 572 573
stiffness influence coefficient, 562 567
stiffness matrix of a frame, 566
Introduction to Harmonics, 5
Inverse Laplace transform, 1049
Inverse matrix, 1045
Inverse of the Matrix, 682
Irregular forcing conditions, response to,
431 436
Irregular forcing function, 378 380
Isolation, vibration, 801 802
with base motion, 814 820
damped spring mount, 801
pneumatic rubber mount, 801
system with flexible foundation,
821 822
system with partially flexible foundation,
822 824
types, 802 803
undamped spring mount, 801
with rigid foundation, 804 813, See also
Rigid foundation
Iteration method, 670 677, 13-13 13-16,
See also Matrices: matrix iteration
method
J
Jacobi s method, 678 680
eigenvalue solution using, 679 682, 684
standard eigenvalue problem, 680 683
joint probability distribution of random variables, 14-7 14-9
bivariate distributions, 14-8
multivariate distribution, 14-8
univariate distributions, 14-81080 INDEX
Journal bearing, damping constant of, 48 50
Jump phenomenon, 13-16
K
Karman vortices, 305
Kinetic energy expressions in matrix form,
574 576
Kirchhoff, G. R., 8
Kronecker delta, 581
LL
Hospital s rule, 266
Lagrange, Joseph, 7
Lagrange s equations, 577 581
Laplace transform, 313, 317 320, 365,
406 427, 504 512, 1048 1055
damped system response using, 317
definition, 1048 1049
first-order systems, response of, 407 409
inverse Laplace transform, 1049
partial fractions method, 1050 1052
second order systems, response of,
409 414
shifting theorems, 1050
steady state response using, 319 320
step force, response to, 414 420
transform of derivatives, 1049 1050
transient and steady-state responses, 406
transient response, 421 427, See also
individual entry
two degree of freedom systems solutions
using, 504 512
Laplacian operator, 741
Lateral vibration of beams, 721 736
axial compressive force, beam subjected
to, 733 737
boundary conditions, 724 726
equation of motion, 721
fixed-pinned beam, natural frequencies
of, 728 731
forced vibration, 730 731
free vibration, 723 724
initial conditions, 723
orthogonality of normal functions,
726 729
simply supported beam, forced vibration,
731 734
Lathe, 469, 488 489
Left half-plane (LHP) yield, 198
Limit cycles, 13-41 13-43
Lindstedt s perturbation method, 13-10 13-12
Linear algebraic equations, solution of, 1065
Linear coordinates, 555
Linear force, compacting machine under,
391 392
Linear springs, 23 25
Linear variable differential transformer
(LVDT) transducer, 878 879
Linear vibration, 17
Linearization of nonlinear spring, 25 27
Literature, vibration, 80 81
Local coordinate axis, 998
Logarithmic decrement, 164 166
Longitudinal vibration of bar or rod,
710 718
bar carrying a mass, natural frequencies
of, 715 716
bar subjected to initial force, vibrations
of, 716 720
boundary conditions, 712 713
equation of motion and solution,
710 712
free vibrations of a fixed-free bar, 714
orthogonality of normal functions,
713 718
Longitudinal vibration of bars, 951 955
Loops, 6
Lumped mass matrices, 1012 1015
Lumped-mass model, 555
M
Machine condition monitoring techniques,
916 918
Machine maintenance techniques, 915 916
breakdown maintenance, 915
condition-based maintenance, 916
preventive maintenance, 915
Machine tool support, equivalent spring and
damping constants of, 52 53
Machine vibration monitoring techniques,
918 923
Magnification factor, 264, 273
Marine engine propeller system, 485 488
Mass matrix, 991 998
Mass or inertia elements, 40 45
Material damping, 46
Mathematical modeling, 18
Mathieu equation, 13-24 13-29
MATLAB, 326 332, 436 440, 515 522,
619 627, 683 686, 748 751,
843 850, 925 928, 972 978,
1015 1019, 1041 1047, 1059 1068
accelerometer equation plotting,
927 929
arrays and matrices, 1060
arrays with special structure, 1061
autocorrelation function plotting,
14-46 14-48
column vector, 1060
complex numbers, 1062
Coulomb damping, free vibration
response of a system with, 205
damped system, forced vibration
response of, 623 624
differential equations solution,
1066 1068
eigenvalue problem solution, 515 516,
619, 683 684
finite element analysis of stepped bar,
1015
forced response of a system with
Coulomb damping, 327 328
forced vibration response of simply supported beam, plotting, 748 751
Fourier series graphical representation
using, 76 78
free vibration response
frequency response, plotting, 519 520
functions in, 1062
Gaussian probability distribution function evaluation, 14-48 14-49
general eigenvalue proble, 685 686
impulse response of a structure, 437 438
matrix, 1060
matrix operations, 1061
M-files, 1062 1063
multidegree of freedom system, 619 626
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, 79
Nyquist circle plotting, 925 926
pendulum equation solution, 13-53 13-57
plotting of graphs, 1063 1064
program to generate characteristic polynomial, 625
quartic equation roots, 522
railway cars, time response of, 518 519
response under a periodic force, 438 439
response under arbitrary forcing
function, 439
roots of a polynomial equation, 622
roots of a quartic equation, 516
roots of transcendental and nonlinear
equations, 750 751
row vector, 1062
solution of a single degree of freedom
system, 972
solution of multidegree of freedom
system, 973 974
special matrices, 1061
spring-mass system, free vibration
response of, 203
static deflection, variations of natural
frequency and period with, 202
steady-state response of viscously
damped system, 330 331
system response under base excitation,
328 330
total response of an undamped system
using, 326 327
total response of system under base
excitation, 436 437
transmissibility, plotting, 843
undamped system response, 204
variables, 1060
vibration amplitudes of vibration
absorber masses, 845 846
Matrices, 1041 1047, 1060, See also individual entries
basic operations, 1046 1047, 1061
trace, 1043
transpose of, 1043INDEX 1081
Maximum overshoot (M
p
), 423
Mean square response, 14-34
Mean value, 14-6 14-7, 14-32
Measurement and applications, vibration,
870 938
machine condition monitoring and
diagnosis, 915 925
measurement scheme, 872
Mechanical chatter, 13-5
Mechanical exciters, 892 893
Method of isoclines, 13-34 13-35
trajectories using, 13-36
Membranes, vibration of, 739 742
equation of motion, 739 741
free vibrations of rectangular membrane,
742 744
initial and boundary conditions, 741 742
membrane under uniform tension, 740
Mersenne, Marin, 6 7
M-files, 1062 1063
Milling cutter, natural frequencies of, 720
Mindlin, R. D., 9
Modal analysis, 596
forced vibration of undamped systems
using, 603 610
free vibration response using, 606 608
Modal damping ratio, 612
Modal matrix, 592
Modal testing, 900 915, See also Experimental modal analysis
Modal vectors, 475
Mode shapes, 583
determination, 654 698
measurement, 912 915
of three degree of freedom system, 590
Monochord, 4 5
Motor-generator set, absorber for, 838 843
Multidegree of freedom systems, 553 653,
14-39 14-46, See also Influence
coefficients; Three degree of freedom
system
central difference method for, 947 951
continuous systems modeling as,
555 556
equations of motion of undamped systems in matrix form, 581 582
expansion theorem, 596
free vibration of undamped systems,
601 603
generalized coordinates, 576 577
generalized forces, 576 577
Lagrange s equations to derive equations
of motion, 577 581
modal analysis, 603 610, See also individual entry
natural frequencies of free system,
598 601
Newton s second law to derive equations
of motion, 557 562
potential and kinetic energy expressions
in matrix form, 574 576
repeated Eigenvalues, 594 596
self-excitation, 617 619
spring-mass-damper system, equations
of motion of, 557 560
stability analysis, 617 619
steady-state response of forced system,
615 619
trailer compound pendulum system,
equations of motion of, 560
unrestrained systems, 596 599
Multivariate distribution, 14-8
N
Narrow-band process, 14-25 14-27
Natural frequencies, 62, 475
determination, 654 698, See also
Dunkerley s formula; Holzer s
method; Jacobi s method;
Rayleigh s method
of free system, 598 601
of torsional system, 484, 668 673
Natural mode, two degree of freedom systems, 471
Newmark method, 968 971
Newton, Isaac, 6
Newton s second law, 129 130, 261,
557 562
Nodes, 6, 706, 13-40 13-41
Nomograph, vibration, 771 775
Nondeterministic vibration, 17 18
Nonlinear damper, linearization of, 52
Nonlinear differential equation solution, 13-61
Nonlinear equations, roots of, 1064
Nonlinear springs, 23 25
Nonlinear system under pulse loading solution, 13-59
Nonlinear vibration, 17, 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, 13-25
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, 365,
380 381, See also Convolution integral; Laplace transform; Numerical
methods
Normal modes, 470, 591 593
Number-decibel conversion line, 323
Numerical Fourier analysis, 74 76
Numerical integration methods, 939 986
finite difference method, 941 942
single degree of freedom systems,
942 946
Numerical methods, for response under nonperiodic force, 365, 428 431
Nyquist circle plotting, 925 926
Nyquist plot, modal data determination
from, 910 912
O
Octave band analyzer, 896, 899
Octave, 63
Odd functions, 69 71
Operational deflection shape
measurements, 900
Optimally tuned vibration absorber, 842
Orthogonality of normal functions, 591 593,
713 718, 726 729
Orthonormalization of Eigenvectors,
592 594
Overdamped system, 163, 418 420
P
Parameter variations, 176 185, See also
under Root locus representations
Parseval s formula, 14-17, 14-21
Partial fractions method, 1050 1052
Peak time (t
p
), 421
Perfectly elastic collision, response to,
412 414
Periodic solutions using Lindstedt s perturbation method, 13-25
Period of beating , 269
Period of oscillation, 62
Periodic force, 365 378, See also General
periodic force, response under
Periodic vibration of a hydraulic valve,
374 376
Phase angle, 62, 584
Phase distortion, 888 890
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, 6
Piezoelectric transducers, 876 877
Pinned-fixed beam, 959
Piston-cylinder dashpot, 50 52
Plane milling cutter, 721
Plano-milling machine structure, 989
Plates, deflection of, 1039 1040
Poincaré section, 13-43 13-44
Poisson, Simeon, 8
Positive definite matrix, 576
Positive definite quadratic forms, 576
Potential energy expressions in matrix form,
574 5761082 INDEX
Power spectral density, 14-23 14-25, 14-33
Precision electronic system, vibration control of, 829 830
Precision machine with base motion, design
of isolation for, 816 818
Principal coordinates, 472, 488 493
Principal mode, two degree of freedom systems, 470
Principle of conservation of energy, 131 132
Principle of virtual displacements, 130
Probability density curve, 919
Probability distribution, 14-4 14-5
Propeller shaft, 33
Proportional damping, 611
Pseudo spectrum, 397
Pseudo velocity, 397
Pulley subjected to Coulomb damping, 191
Pulley system, 145 146
Pulse load, 389 390
response due to, 390
Pythagoras, 3 4
Q
Q factor/Quality factor, 276 278
Quadratic damping, 300
Quefrency domain analysis, 922
R
Ramp function, first-order system response
due to, 408 409
Random signals analysis, 905 907
Random vibration, 17, 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, 8
Rayleigh s method, 153 158, 658 665, 700,
742 745
beams, fundamental frequency of,
663 665
effect of mass, 155, 157
manometer for diesel engine, 153
Rayleigh s quotient, properties of,
659 661
shafts, fundamental frequency of,
663 668
U-tube manometer, 154
Rayleigh-Ritz method, 700, 745 748
Reciprocating engines, balancing, 792 798
reciprocating engines, balancing,
796 798
unbalanced forces due to fluctuations in
gas pressure, 792 793
unbalanced forces due to inertia of the
moving parts, 793 796
Recoil mechanism, 173
Rectangular pulse load, 389 390
response due to, 390
Recurrence formula, 943
Reference marks, 777 778
Relative motion, 284 287
Repeated Eigenvalues, 594 596
Resonance, 11
Resonant frequencies of vibration absorber,
847 848
Response spectrum, 394 406
for base excitation, 396 399
building frame response to an earthquake, 402 403
design under shock environment,
403 406
earthquake response spectra, 399 403
of sinusoidal pulse, 394 397
water tank subjected to base acceleration, 398 399
Rigid bar
connected by springs, equivalent k of, 37
stability of, 201
Rigid foundation, vibration isolation system
with, 804 813
design chart of isolation, 809 810
isolator for stereo turntable, 810 813
machine member on, 804
resilient member on, 804
spring support for exhaust fan,
807 808
undamped isolator design, 808 810
vibratory motion of mass, reduction, 806
Rise time (t
r
), 422 423
Ritz-Galerkin method, 13-17 13-19
Rod, spring constants of, 27
Root locus representations, 176 185
and parameter variations, 179 185
roots study with variation of c, 181
+ in s-plane, 176 179
, in s-plane, 176 179
*d in s-plane, 176 179
*
n
in s-plane, 176 179
variation of mass, 183
variation of spring constant, 183
Rotary inertia effects, 734 739
Rotating machines, balancing, 776 785
single-plane balancing, 776 779
two-plane balancing, 779 785
Rotating unbalance, 287 293, 830 837, See
also under Damped system response
under harmonic force
Routh-Hurwitz criteria, 502, 790
Row matrix, 1042
Row vector, 1060
Runge-Kutta methods, 429 431
S+
in s-plane, interpretation, 176 179
, in s-plane, interpretation, 176 179
*d in s-plane, interpretation, 176 179
*
n
in s-plane, interpretation, 176 179
Saddle point, 13-40 13-41
Sample point, 14-3
Sample space, 14-3
Sauveur, Joseph, 6
Scotch yoke mechanism, 55
Second-order systems, 367, 372 374,
409 414
Seismograph, 5
Self-excitation, 301 313, 500 502, 617 619
Semidefinite systems, 497 500, 598
Settling time, transient response, 424
Shafts, fundamental frequency of, 663 668
Shear deformation effects, 734 739
Shearing stress (,), 49
Shock absorber for a motorcycle, 171 172
Shock environment, design under, 403 406
Shock isolation, 824 827
Shock loads, 403 406
Signal analysis, 895 899
Signum function, 187
Simple harmonic motion, 56
Simple pendulum, 14, 39, 13-3
Simply supported beam, 1039
forced vibration, 731 733
natural frequencies of, 737 742
Singing of transmission lines, 305
Single degree of freedom systems, 14,
14-28 14-31
central difference method for, 942 945
characteristics of, 14-30 14-31
free vibration of, 124 258, See also
Undamped translational system,
free vibration of
frequency response approach, 14-30
impulse response approach, 14-28 14-29
Single-plane balancing, 776 779
Singular point, 13-34
Sinusoidal pulse, response spectrum of,
394 397
Solid damping, 46
Space shuttle, vibration testing, 12
Special matrices, 1061
Spectrum analyzers, 896 897
Spring constants of elastic elements, 27 30
Spring elements, 22 39
deformation of spring, 22
equivalent linearized spring constant, 26
linear springs, 23 25
nonlinear springs, 23 25INDEX 1083
spring constant associated with restoring
force due to gravity, 39
spring constants of elastic elements,
27 30
Spring-mass-damper system, 262, 557 560
Spring-mass systems, 126 128, 669 670
Springs, combination of, 30 38
equivalent k, 32, 34 35, 37
in parallel, 30
in series, 31
torsional spring constant of a propeller
shaft, 33
Spring-supported mass instability on moving
belt, 302
Square matrix, 1042
Stability analysis, 301 313, See also
Dynamic instability caused by fluid
flow
two degree of freedom systems, 500 502
dynamic, 301 305
multidegree of freedom systems,
617 619
whirling of rotating shafts, 790 791
Stability of systems, 198 202
asymptotically stable, 198 200
rigid bar, 201
stable, 198 200
unstable, 198 200
Stable focus, 13-40
Stable orbits, functions with, 13-45
Standard deviation, 14-6 14-7
Standard eigenvalue problem, 585
Static deflection, 263
Static equilibrium position, 132
Static unbalance, 776
Stationary random excitations, response due
to, 14-31 14-39
Stationary random process, 14-10 14-14
Strange attractors, 13-46
Steady state response, 406
of forced system, 615 619
using Laplace transform, 319 320
Step force, response to, 414 420, See also
under Laplace transform
Stepped bar, 1015, 1018
Stiffness influence coefficient, 562 567
Stiffness matrix, 566, 991 998
Stockbridge damper, 308
Stodola, Aurel, 9
Stroboscope, 891
Study of vibration, importance, 10 13
Subharmonic oscillations, 13-20 13-22
Superharmonic oscillations, 13-23 13-24
Suspension system, equivalent k of, 32
Symmetric matrix, 682 683, 1043
System response under base excitation,
328 330
T
Tapered beam, fundamental frequency of,
744 750
Taylor, Brook, 7
Taylor s series expansion, 25, 310
Temporal averages, 14-14
Thick beam theory, 734
Thin beam theory, 722
Three degree of freedom system, 15
equations of motion of, 586
fundamental frequency of, 661 663
mode shapes of, 590
natural frequencies of, 587 589
natural frequencies of, 673 679
Time constant, 151 153
Time-delayed step force, 388 389
Time-dependent coefficients, systems with,
13-24 13-29
Time domain analysis, 918
Time domain representations, 68 69
Timoshenko beam theory, 734 735
Timoshenko, Stephen, 9
Torsion element, 994 995
Torsional pendulum, 148
Torsional spring constant of a propeller
shaft, 33
Torsional system, 483 488, 666 669
with Coulomb damping, 190 192
with discs mounted on a shaft, 483
equations of motion of, 578 579
natural frequencies of, 484 488,
668 673
with viscous damping, 168 174
Torsional vibration of a shaft or rod,
718 721
Torsional vibration, 8, 146
Trace, 1043
Trajectories of simple harmonic oscillator,
13-29 13-30
Trailer compound pendulum system, equations of motion of, 560
Transducers, 873 879, 901
electric resistance strain gage, 873
electrodynamic transducers, 877 878
linear variable differential transformer
(LVDT) transducer, 878 879
piezoelectric transducers, 876 877
variable resistance transducers, 873 876
Transfer function approach, 313 316,
425 426, 502 504
Transient response, 261 262, 406, 421 427
Transition curves, 13-28
Transverse vibration of beams, 955 959
Transverse vibration of string or cable,
701 710, See also under Continuous
systems
Traveling-wave solution, 709 710
Triangular pulse, Fourier transform of, 14-22
Triple pendulum, 576
Tuned vibration absorber, 842
Two degree of freedom systems, 15,
467 552, See also Forced vibration;
Laplace transform; Semidefinite systems; Torsional system
automobile, frequencies and modes of,
492 495
coordinate coupling and principal coordinates, 488 493
coupled differential equations, 470
equations of motion for forced vibration,
472 473
forced response of, 520 522
free vibration response of, 481 482
Lathe, 469, 488 489
natural mode, 471
normal mode, 470
packaging of an instrument, 471
principal mode, 470
spring-mass-damper system, 472
transfer function approach, 502 504
Two-plane balancing, 779 785, 848 850
U
Undamped dynamic vibration absorber,
833 839
effect on the response of machine, 835
for diesel engine, 837 838
for motor-generator set, 838 843
Undamped equation, 13-13, 13-31
Undamped isolator design, 808 810
Undamped system, 127
free vibration analysis, 474 482
free vibration of, 601 606
free vibration response of, 504 507
in matrix form, 581 582
response under harmonic force,
263 271, See also under Harmonically excited vibration
total response of, using MATLAB,
326 327
Undamped torsional system, free vibration
of, 146 151
Undamped translational system, free vibration of, 129 146
auxiliary or characteristic equation, 134
D Alembert s principle, 130
eigenvalues or characteristic values, 134
mass under virtual displacement, 131
principle of conservation of energy, 130
principle of virtual displacements, 130
using Newton s second law of motion,
129 130
Undamped vibration, 17
Underdamped system, 160, 414 416
Uniform string, free vibration of, 704 705
Unit impulse response of second-order
system, 409
Units, 1056 1058
Univariate distributions, 14-8
Unrestrained systems, 499 502, 596 599
Unstable focus, 13-40
Unstable orbits, functions with, 13-45 13-47
Unstable system, 198
V
Variable mass system, 13-5 13-6
Variable resistance transducers, 873 8761084 INDEX
Vectorial representation of harmonic motion,
56 57
Velometer, 886 887
Vertical position, spring-mass system in,
132 133
Vibrating string, 702
Vibration absorbers, 832 843, 847 848, See
also Damped dynamic vibration
absorber; Undamped dynamic vibration absorber
Vibration pickups, 879 890
Vibration severity of machinery, 773
Vibrometer, 881 882
Viscoelastic materials use, 799
Viscous damping, 45
Cannon analysis, 173
energy dissipated in, 166 168
forced vibration of, 610 616
free vibration with, 158 174
steady-state response of, 330 331
torsional systems with viscous damping,
168 174
torsional systems with, 168 174
W
Wallis, John, 6
Whirling of rotating shafts, 785 791
critical speeds, 787
equations of motion, 785 787
shaft carrying an unbalanced rotor, 791
stability analysis, 790 791
system response, 788 790
Wide-band process, 14-25 14-27
Wiener-Khintchine formula,
14-23
Wilson method, 965 968
Wind-induced vibration, 11
Y
Young s modulus, 142 143
Z
Zero matrix, 1042
Zhang Heng, 5


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حل كتاب الاهتزازات الميكانيكية
Mechanical Vibrations Solution Manual
Fifth Edition
Singiresu S. Rao
University of Miami  

حل كتاب الاهتزازات الميكانيكية - Mechanical Vibrations Solution Manual - Fifth Edition - صفحة 2 M_v_s_13
و المحتوى كما يلي :


Prefacexi
Acknowledgmentsxv
List of Symbolsxvi
CHAPTER 1
Fundamentals of Vibration1
1.1Preliminary Remarks2
1.2Brief History of the Study of Vibration3
1.2.1Origins of the Study of Vibration 3
1.2.2From Galileo to Rayleigh 6
1.2.3Recent Contributions 9
1.3Importance of the Study of Vibration10
1.4Basic Concepts of Vibration13
1.4.1Vibration 13
1.4.2Elementary Parts of
Vibrating Systems 13
1.4.3Number of Degrees of Freedom 14
1.4.4Discrete and Continuous Systems 16
1.5Classification of Vibration16
1.5.1Free and Forced Vibration 17
1.5.2Undamped and Damped Vibration 17
1.5.3Linear and Nonlinear Vibration 17
1.5.4Deterministic and
Random Vibration 17
1.6Vibration Analysis Procedure18
1.7Spring Elements22
1.7.1Nonlinear Springs 23
1.7.2Linearization of a
Nonlinear Spring 25
1.7.3Spring Constants of Elastic Elements 27
1.7.4Combination of Springs 30
iv
1.7.5Spring Constant Associated with the
Restoring Force due to Gravity 39
1.8Mass or Inertia Elements40
1.8.1Combination of Masses 40
1.9Damping Elements45
1.9.1Construction of Viscous Dampers 46
1.9.2Linearization of a
Nonlinear Damper 52
1.9.3Combination of Dampers 52
1.10Harmonic Motion54
1.10.1Vectorial Representation of
Harmonic Motion 56
1.10.2Complex-Number Representation
of Harmonic Motion 57
1.10.3Complex Algebra 58
1.10.4Operations on Harmonic Functions 59
1.10.5Definitions and Terminology 62
1.11Harmonic Analysis64
1.11.1Fourier Series Expansion 64
1.11.2Complex Fourier Series 66
1.11.3Frequency Spectrum 67
1.11.4Time- and Frequency-Domain
Representations 68
1.11.5Even and Odd Functions 69
1.11.6Half-Range Expansions 71
1.11.7Numerical Computation
of Coefficients 72
1.12Examples Using MATLAB76
1.13Vibration Literature80
Chapter Summary81
References81
Review Questions83
Problems87
Design Projects120
ContentsCONTENTS v
CHAPTER 2
Free Vibration of Single-Degree-of-Freedom
Systems124
2.1Introduction126
2.2Free Vibration of an Undamped
Translational System129
2.2.1Equation of Motion Using Newton s
Second Law of Motion 129
2.2.2Equation of Motion Using Other
Methods 130
2.2.3Equation of Motion of a Spring-Mass
System in Vertical Position 132
2.2.4Solution 133
2.2.5Harmonic Motion 134
2.3Free Vibration of an Undamped
Torsional System146
2.3.1Equation of Motion 147
2.3.2Solution 148
2.4Response of First Order Systems
and Time Constant151
2.5Rayleigh s Energy Method153
2.6Free Vibration with Viscous Damping158
2.6.1Equation of Motion 158
2.6.2Solution 158
2.6.3Logarithmic Decrement 164
2.6.4Energy Dissipated in Viscous
Damping 166
2.6.5Torsional Systems with Viscous
Damping 168
2.7Graphical Representation of Characteristic Roots
and Corresponding Solutions174
2.7.1Roots of the Characteristic Equation 174
2.7.2Graphical Representation of Roots and
Corresponding Solutions 175
2.8Parameter Variations and Root Locus
Representations176
2.8.1Interpretations of and
in s-plane 176
2.8.2Root Locus and Parameter
Variations 179
2.9Free Vibration with Coulomb Damping185
2.9.1Equation of Motion 186
2.9.2Solution 187
2.9.3Torsional Systems with Coulomb
Damping 190
v t
n
, vd, z,
2.10Free Vibration with Hysteretic Damping192
2.11Stability of Systems198
2.12Examples Using MATLAB202
Chapter Summary208
References209
Review Questions209
Problems214
Design Projects256
CHAPTER 3
Harmonically Excited Vibration259
3.1Introduction261
3.2Equation of Motion261
3.3Response of an Undamped System
Under Harmonic Force263
3.3.1Total Response 267
3.3.2Beating Phenomenon 267
3.4Response of a Damped System Under
Harmonic Force271
3.4.1Total Response 274
3.4.2Quality Factor and Bandwidth 276
3.5Response of a Damped System
Under 278
3.6Response of a Damped System Under the
Harmonic Motion of the Base281
3.6.1Force Transmitted 283
3.6.2Relative Motion 284
3.7Response of a Damped System Under Rotating
Unbalance287
3.8Forced Vibration with Coulomb Damping293
3.9Forced Vibration with Hysteresis Damping298
3.10Forced Motion with Other Types of
Damping300
3.11Self-Excitation and Stability Analysis301
3.11.1Dynamic Stability Analysis 301
3.11.2Dynamic Instability Caused by Fluid
Flow 305
3.12Transfer-Function Approach313
3.13Solutions Using Laplace Transforms317
3.14Frequency Transfer Functions320
3.14.1Relation Between the General Transfer
function T(s) and the Frequency Transfer
Function 322
3.14.2Representation of Frequency-Response
Characteristics 323
T(iv)
F(t) = F0eiVtvi CONTENTS
3.15Examples Using MATLAB326
Chapter Summary332
References332
Review Questions333
Problems336
Design Projects362
CHAPTER 4
Vibration Under General Forcing
Conditions363
4.1Introduction364
4.2Response Under a General
Periodic Force365
4.2.1First-Order Systems 366
4.2.2Second-Order Systems 372
4.3Response Under a Periodic Force
of Irregular Form378
4.4Response Under a Nonperiodic Force380
4.5Convolution Integral381
4.5.1Response to an Impulse 382
4.5.2Response to a General Forcing
Condition 385
4.5.3Response to Base Excitation 386
4.6Response Spectrum394
4.6.1Response Spectrum for Base
Excitation 396
4.6.2Earthquake Response Spectra 399
4.6.3Design Under a Shock
Environment 403
4.7Laplace Transform406
4.7.1Transient and Steady-State
Responses 406
4.7.2Response of First-Order Systems 407
4.7.3Response of Second-Order Systems 409
4.7.4Response to Step Force 414
4.7.5Analysis of the Step Response 420
4.7.6Description of Transient
Response 421
4.8Numerical Methods428
4.8.1Runge-Kutta Methods 429
4.9Response to Irregular Forcing Conditions Using
Numerical Methods431
4.10Examples Using MATLAB436
Chapter Summary440
References440
Review Questions441
Problems444
Design Projects465
CHAPTER 5
Two-Degree-of-Freedom Systems467
5.1Introduction468
5.2Equations of Motion for Forced
Vibration472
5.3Free Vibration Analysis of an Undamped
System474
5.4Torsional System483
5.5Coordinate Coupling and Principal
Coordinates488
5.6Forced-Vibration Analysis494
5.7Semidefinite Systems497
5.8Self-Excitation and Stability
Analysis500
5.9Transfer-Function Approach502
5.10Solutions Using Laplace Transform504
5.11Solutions Using Frequency Transfer
Functions512
5.12Examples Using MATLAB515
Chapter Summary522
References523
Review Questions523
Problems526
Design Projects552
CHAPTER 6
Multidegree-of-Freedom Systems553
6.1Introduction555
6.2Modeling of Continuous Systems as Multidegreeof-Freedom Systems555
6.3Using Newton s Second Law to Derive Equations
of Motion557
6.4Influence Coefficients562
6.4.1Stiffness Influence Coefficients 562
6.4.2Flexibility Influence Coefficients 567
6.4.3Inertia Influence Coefficients 572
6.5Potential and Kinetic Energy Expressions in
Matrix Form574
6.6Generalized Coordinates and Generalized
Forces576
6.7Using Lagrange s Equations to Derive Equations
of Motion577CONTENTS vii
6.8Equations of Motion of Undamped Systems in
Matrix Form581
6.9Eigenvalue Problem583
6.10Solution of the Eigenvalue Problem585
6.10.1Solution of the Characteristic
(Polynomial) Equation 585
6.10.2Orthogonality of Normal Modes 591
6.10.3Repeated Eigenvalues 594
6.11Expansion Theorem596
6.12Unrestrained Systems596
6.13Free Vibration of Undamped Systems601
6.14Forced Vibration of Undamped Systems Using
Modal Analysis603
6.15Forced Vibration of Viscously Damped
Systems610
6.16Self-Excitation and Stability Analysis617
6.17Examples Using MATLAB619
Chapter Summary627
References627
Review Questions628
Problems632
Design Project653
CHAPTER 7
Determination of Natural Frequencies and
Mode Shapes654
7.1Introduction655
7.2Dunkerley s Formula656
7.3Rayleigh s Method658
7.3.1Properties of Rayleigh s Quotient 659
7.3.2Computation of the Fundamental Natural
Frequency 661
7.3.3Fundamental Frequency of Beams and
Shafts 663
7.4Holzer s Method666
7.4.1Torsional Systems 666
7.4.2Spring-Mass Systems 669
7.5Matrix Iteration Method670
7.5.1Convergence to the Highest Natural
Frequency 672
7.5.2Computation of Intermediate Natural
Frequencies 673
7.6Jacobi s Method678
7.7Standard Eigenvalue Problem680
7.7.1Choleski Decomposition 681
7.7.2Other Solution Methods 683
7.8Examples Using MATLAB683
Chapter Summary686
References686
Review Questions688
Problems690
Design Projects698
CHAPTER 8
Continuous Systems699
8.1Introduction700
8.2Transverse Vibration of a String or
Cable701
8.2.1Equation of Motion 701
8.2.2Initial and Boundary Conditions 703
8.2.3Free Vibration of a Uniform
String 704
8.2.4Free Vibration of a String with Both Ends
Fixed 705
8.2.5Traveling-Wave Solution 709
8.3Longitudinal Vibration of a Bar or Rod710
8.3.1Equation of Motion
and Solution 710
8.3.2Orthogonality of Normal
Functions 713
8.4Torsional Vibration of a Shaft or Rod718
8.5Lateral Vibration of Beams721
8.5.1Equation of Motion 721
8.5.2Initial Conditions 723
8.5.3Free Vibration 723
8.5.4Boundary Conditions 724
8.5.5Orthogonality of Normal
Functions 726
8.5.6Forced Vibration 730
8.5.7Effect of Axial Force 732
8.5.8Effects of Rotary Inertia and Shear
Deformation 734
8.5.9Other Effects 739
8.6Vibration of Membranes739
8.6.1Equation of Motion 739
8.6.2Initial and Boundary Conditions 741
8.7Rayleigh s Method742
8.8The Rayleigh-Ritz Method745
8.9Examples Using MATLAB748
Chapter Summary751
References751viii CONTENTS
Review Questions753
Problems756
Design Project768
CHAPTER 9
Vibration Control769
9.1Introduction770
9.2Vibration Nomograph and Vibration
Criteria771
9.3Reduction of Vibration at the Source775
9.4Balancing of Rotating Machines776
9.4.1Single-Plane Balancing 776
9.4.2Two-Plane Balancing 779
9.5Whirling of Rotating Shafts785
9.5.1Equations of Motion 785
9.5.2Critical Speeds 787
9.5.3Response of the System 788
9.5.4Stability Analysis 790
9.6Balancing of Reciprocating Engines792
9.6.1Unbalanced Forces Due to Fluctuations in
Gas Pressure 792
9.6.2Unbalanced Forces Due to Inertia of the
Moving Parts 793
9.6.3Balancing of Reciprocating
Engines 796
9.7Control of Vibration798
9.8Control of Natural Frequencies798
9.9Introduction of Damping799
9.10Vibration Isolation801
9.10.1Vibration Isolation System with Rigid
Foundation 804
9.10.2Vibration Isolation System with Base
Motion 814
9.10.3Vibration Isolation System with Flexible
Foundation 821
9.10.4Vibration Isolation System with Partially
Flexible Foundation 822
9.10.5Shock Isolation 824
9.10.6Active Vibration Control 827
9.11Vibration Absorbers832
9.11.1Undamped Dynamic Vibration
Absorber 833
9.11.2Damped Dynamic Vibration
Absorber 840
9.12Examples Using MATLAB843
Chapter Summary851
References851
Review Questions853
Problems855
Design Project869
CHAPTER 10
Vibration Measurement and
Applications870
10.1Introduction871
10.2Transducers873
10.2.1Variable Resistance Transducers 873
10.2.2Piezoelectric Transducers 876
10.2.3Electrodynamic Transducers 877
10.2.4Linear Variable Differential Transformer
Transducer 878
10.3Vibration Pickups879
10.3.1Vibrometer 881
10.3.2Accelerometer 882
10.3.3Velometer 886
10.3.4Phase Distortion 888
10.4Frequency-Measuring Instruments890
10.5Vibration Exciters892
10.5.1Mechanical Exciters 892
10.5.2Electrodynamic Shaker 893
10.6Signal Analysis895
10.6.1Spectrum Analyzers 896
10.6.2Bandpass Filter 897
10.6.3Constant-Percent Bandwidth and
Constant-Bandwidth Analyzers 898
10.7Dynamic Testing of Machines
and Structures900
10.7.1Using Operational Deflection-Shape
Measurements 900
10.7.2Using Modal Testing 900
10.8Experimental Modal Analysis900
10.8.1The Basic Idea 900
10.8.2The Necessary Equipment 900
10.8.3Digital Signal Processing 903
10.8.4Analysis of Random Signals 905
10.8.5Determination of Modal Data
from Observed Peaks 907
10.8.6Determination of Modal Data
from Nyquist Plot 910
10.8.7Measurement of Mode Shapes 912
10.9Machine Condition Monitoring
and Diagnosis915CONTENTS ix
10.9.1Vibration Severity Criteria 915
10.9.2Machine Maintenance Techniques 915
10.9.3Machine Condition Monitoring
Techniques 916
10.9.4Vibration Monitoring Techniques 918
10.9.5Instrumentation Systems 924
10.9.6Choice of Monitoring Parameter 924
10.10Examples Using MATLAB925
Chapter Summary928
References928
Review Questions930
Problems932
Design Projects938
CHAPTER 11
Numerical Integration Methods in
Vibration Analysis939
11.1Introduction940
11.2Finite Difference Method941
11.3Central Difference Method for Single-Degree-ofFreedom Systems942
11.4Runge-Kutta Method for Single-Degree-ofFreedom Systems945
11.5Central Difference Method for Multidegree-ofFreedom Systems947
11.6Finite Difference Method for Continuous
Systems951
11.6.1Longitudinal Vibration of Bars 951
11.6.2Transverse Vibration of Beams 955
11.7Runge-Kutta Method for Multidegree-ofFreedom Systems960
11.8Houbolt Method962
11.9Wilson Method965
11.10Newmark Method968
11.11Examples Using MATLAB972
Chapter Summary978
References978
Review Questions979
Problems981
CHAPTER 12
Finite Element Method987
12.1Introduction988
12.2Equations of Motion of an Element989
12.3Mass Matrix, Stiffness Matrix, and Force
Vector991
12.3.1Bar Element 991
12.3.2Torsion Element 994
12.3.3Beam Element 995
12.4Transformation of Element Matrices and
Vectors998
12.5Equations of Motion of the Complete System of
Finite Elements1001
12.6Incorporation of Boundary
Conditions1003
12.7Consistent- and Lumped-Mass Matrices1012
12.7.1Lumped-Mass Matrix for a Bar
Element 1012
12.7.2Lumped-Mass Matrix for a Beam
Element 1012
12.7.3Lumped-Mass Versus Consistent-Mass
Matrices 1013
12.8Examples Using MATLAB1015
Chapter Summary1019
References1019
Review Questions1020
Problems1022
Design Projects1034
Chapters 13 and 14 are provided as downloadable
files on the Companion Website.
CHAPTER 13
Nonlinear Vibration13-1
13.1Introduction13-2
13.2Examples of Nonlinear Vibration Problems13-3
13.2.1Simple Pendulum 13-3
13.2.2Mechanical Chatter, Belt Friction
System 13-5
13.2.3Variable Mass System 13-5
13.3Exact Methods13-6
13.4Approximate Analytical Methods13-7
13.4.1Basic Philosophy 13-8
13.4.2Lindstedt s Perturbation Method 13-10
13.4.3Iterative Method 13-13
13.4.4Ritz-Galerkin Method 13-17
13.5Subharmonic and Superharmonic
Oscillations13-19
13.5.1Subharmonic Oscillations 13-20
13.5.2Superharmonic Oscillations 13-23
13.6Systems with Time-Dependent Coefficients
(Mathieu Equation)13-2413.7Graphical Methods13-29
13.7.1Phase-Plane Representation 13-29
13.7.2Phase Velocity 13-34
13.7.3Method of Constructing
Trajectories 13-34
13.7.4Obtaining Time Solution from Phase
Plane Trajectories 13-36
13.8Stability of Equilibrium States13-37
13.8.1Stability Analysis 13-37
13.8.2Classification of Singular
Points 13-40
13.9Limit Cycles13-41
13.10Chaos13-43
13.10.1Functions with Stable Orbits 13-45
13.10.2Functions with Unstable Orbits 13-45
13.10.3Chaotic Behavior of Duffing s Equation
Without the Forcing Term 13-47
13.10.6Chaotic Behavior of Duffing s Equation
with the Forcing Term 13-50
13.11Numerical Methods13-52
13.12Examples Using MATLAB13-53
Chapter Summary13-62
References13-62
Review Questions13-64
Problems13-67
Design Projects13-75
CHAPTER 14
Random Vibration14-1
14.1Introduction14-2
14.2Random Variables and Random Processes14-3
14.3Probability Distribution14-4
14.4Mean Value and Standard Deviation14-6
14.5Joint Probability Distribution of Several
Random Variables14-7
14.6Correlation Functions of a Random Process14-9
14.7Stationary Random Process14-10
14.8Gaussian Random Process14-14
14.9Fourier Analysis14-16
14.9.1Fourier Series 14-16
14.9.2Fourier Integral 14-19
14.10Power Spectral Density14-23
14.11Wide-Band and Narrow-Band Processes14-25
14.12Response of a Single-Degree-ofFreedom System14-28
14.12.1Impulse-Response Approach 14-28
14.12.2Frequency-Response Approach 14-30
14.12.3Characteristics of the Response
Function 14-30
14.13Response Due to Stationary Random
Excitations14-31
14.13.1Impulse-Response Approach 14-32
14.13.2Frequency-Response Approach 14-33
14.14Response of a Multidegree-of-Freedom
System14-39
14.15Examples Using MATLAB14-46
Chapter Summary14-49
References14-49
Review Questions14-50
Problems14-53
Design Project14-61
APPENDIX A
Mathematical Relationships and Material
Properties 1036
APPENDIX B
Deflection of Beams and Plates 1039
APPENDIX C
Matrices 1041
APPENDIX D
Laplace Transform 1048
APPENDIX E
Units 1056
APPENDIX F
Introduction to MATLAB 1059
Answers to Selected Problems 1069
Index 1077
x CONTENTSIndex
1077
A
Accelerographs, 399 400
Accelerometer, 882 886
Acoustics , 6
Active vibration control, 827 832
Addition of harmonic motions, 60 61
Adjoint matrix, 1045
Advance, 873
Airfoil, dynamic instability of, 309 310
Amplitude, 62, 584, 806
Analysis, vibration, 18 22
equations, 20
mathematical modeling, 18
motorcycle, mathematical model of, 20
results, interpretation, 20
Aristotle, 4
Aristoxenus, 4
Arrays with special structure, 1061
Attractor, 13-43 13-44
Asymptotically stable system, 198
Autocorrelation function, 14-9, 14-11, 14-32
Axial compressive force, beam subjected to,
733 737
Axial force effect, 732 734
B
Band-limited white noise, 14-25
Bandpass filter, 897 898
Bandwidth, 276 278
Bar element, 991 994
Base excitation
response spectrum for, 396 399
system response under, 328 330
Basic concepts of vibration, 13 16
Bathtub curve, 915
Beam deflections, 571
Beam element, 995 998
Beams, deflection of, 1039 1040
cantilever beam, 1039
fixed-fixed beam with end displacement,
1039
fixed-fixed beam, 1039
simply supported beam, 1039 1040
Beams, fundamental frequency of, 663 665
Beating phenomenon, 63, 267 271
Belt friction system, 13-5
Bernoulli, Daniel, 7
Bifurcations, 13-46
Bivariate distributions, 14-8
Blast load on building frame, 392 393
Bode diagrams, 323 324
Bonaparte, Napoléon, 7
Boundary conditions, incorporation of,
1003 1012
Boundary curves, 13-28
Building frame response to an earthquake, 402
C
Cam-follower mechanism, 44 45, 73
spring mass system for, 126
Cannon analysis, 173
Cantilever beam, 1039
spring constants of, 28
center, 13-30, 13-40
Center of percussion, 150 151
Central difference method for multidegree of
freedom systems, 947 951
Centrifugal pump with rotating unbalance,
812 816
rattle space, 812 816
Cepstrum, 922
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, 585 590
Characteristic roots, graphical representation, 174 176
Chimney, flow-induced vibration of, 311
Choleski decomposition, 681 683
Classification of vibration, 16 18
Clebsch, R. F. A., 8
Coefficients, numerical computation of,
72 76
Coherence function, 907
Column matrix, 1042
Column vector, 1060
Compacting machine, 387 388, 416 418
Complex algebra, 58
Complex damping, 300
Complex Fourier series, 66 67, 14-17 14-18
Complex frequency response, 279, 14-30
Complex numbers, 1062
harmonic motion representation, 57 58
Complex stiffness, 194
Complex vector representation of harmonic
motion, 280
Compound pendulum, 148 150, 561
natural frequency of, 148 151
Consistent mass matrices, 1012 1015
Constant bandwidth analyzers, 898 899
Constant damping, 186
Constant percent bandwidth, 898 899
Continuous systems, 16, 699 768, See also
Lateral vibration of beams; Longitudinal vibration of bar or rod; Torsional vibration of a shaft or rod
dynamic response of plucked string,
707 712
modeling as multidegree of freedom systems, 555 556
transverse vibration of a string or cable,
701 710
Continuous systems, finite difference
method for, 951 959
longitudinal vibration of bars, 951 955
pinned-fixed beam, 959
transverse vibration of beams, 955 959
Control, vibration, 769 869
criteria, 771 775
natural frequencies, control of, 798 799
nomograph, 771 775
ranges of vibration, 773
whirling of rotating shafts, 785 791
Conversion of units, 1056 1058
Convolution integral, 365, 381 393,
1052 1055
blast load on building frame, 392 393
compacting machine under linear force,
391 392
rectangular pulse load, 389 390
response of a structure under double
impact, 385
response of a structure under impact, 384
response to a general forcing condition,
385 386
response to an impulse, 382 385
response to base excitation, 386 393
step force on a compacting machine,
387 388
time-delayed step force, 388 389
Coordinate coupling, 488 493
Correlation functions of random process,
14-9 14-10
Coulomb damping, 46
forced response of, using MATLAB,
327 328
forced vibration with, 293 297
free vibration response of a system with,
205
free vibration with, 185 192
pulley subjected to, 191
Coulomb, Charles, 7
Please note that references to pages in chapters 13 and 14 appear in the form 13-1, 13-2, etc., and these chapters are provided on the companion Website.1078 INDEX
Coupled differential equations, 470
Crane, equivalent k of, 35
Critical damping constant, 159
Critical speeds, 787
Critically damped system, 162
Cycle, 62
DD
Alembert, Jean, 7
D Alembert s principle, 130
Damped dynamic vibration absorber,
840 843
Damped equation, 13-14
Damped response using numerical methods,
434 436
Damped single degree of freedom system
Bode diagrams of, 324
transfer function, 315
Damped system, 127
forced vibration response of, MATLAB,
623 624
free vibration response of, Laplace transform, 507 511
Damped system response under F(t) = F0ei*t,
278 281
Damped system response under harmonic
force, 271 278, See also under Harmonically excited vibration
under F(t) = F0ei*t, 278 281
graphical representation, 272
under harmonic motion of base, 281 287
under rotating unbalance, 287 293
total response, 274 276
vectorial representation, 272
Damped system response using Laplace
transform, 317 318
Damped vibration, 17
Damping, 799 800
damping matrix, 611
damping ratio, 159, 179
viscoelastic materials use, 799
Damping elements, 45 54
clearance in a bearing, 47 48
combination of dampers, 52 54
Coulomb or dry friction damping, 46
damping constant of journal bearing,
48 50
damping constant of parallel plates, 47
linearization of nonlinear damper, 52
material or solid or hysteretic damping, 46
piston-cylinder dashpot, 50 52
viscous damping, 45
viscous dampers construction, 46 52
De Laval, C. G. P., 9
Decibel, 63
Degree of Freedom, 14 16
Delay time (td), transient response, 425
Design chart of isolation, 809 810
Determinant, 1043
Deterministic vibration, 17 18, 14-2
Diagonal matrix, 1042
Diesel engine, vibration absorber for,
837 838
Differential equations, 313, 1066 1068
Digital signal processing, 903 904
Dirac delta function, 381
Discrete systems, 16
Displacement method, 1003
Displacement transmissibility, 282 283, 806
Dry friction damping, 46
Duffing s equation, 13-13, 13-47 13-50
Duhamel integral, See Convolution integral
Dunkerley s formula, 654 658
Dynamic coupling, 490
Dynamic instability caused by fluid flow,
305 313
of an airfoil, 309 310
flow-induced vibration of a chimney,
311
flow-induced vibration reduction, 307
Helical spoilers, 308
Stockbridge damper, 308
Dynamic response of plucked string, 707 712
Dynamic stability analysis, 301 305
Dynamic system, equations of motion of,
613 615
Dynamic testing of machines and structures,
900
Dynamical matrix, 585
E
Earthquake response spectra, 399 403
Eccentricity of rotor, probabilistic characteristics of, 14-6
Eigenvalues/Eigenvalue problem, 583 584,
594 596
Eigenvectors, orthonormalization of,
592 594
Equilibrium states, stability, 13-37 13-40
Elastic coupling, 490
Elastic potential energy, 574 576
Electric motor deflection due to rotating
unbalance, 290
Electrodynamic Shaker, 893 895
Electrodynamic transducers, 877 878
Element matrices and vectors, transformation, 998 1001
Elementary parts of vibrating systems,
13 14
Energy dissipated in viscous damping,
166 168
Equation of motion, 147 148, 158, 186 187,
261 262
derivation, 577 581
of dynamic system, 613 615
of an element, 989 991
of finite elements, 1001 1003
for forced vibration, 472 473
of three degree of freedom system, 586
of undamped systems in matrix form,
581 582
whirling of rotating shafts, 785 787
Equivalent linearized spring constant, 26
Equivalent mass of a system, 42 44
Equivalent rotational mass, 42
Equivalent translational mass, 42
Ergodic process, 14-13
Euler, Leonard, 7
Euler-Bernoulli theory, 722, 995
Even functions, 69 71
Exciters, vibration, 892 895, 901
due to unbalanced force, 893
electrodynamic shaker, 893 895
mechanical exciters, 892 893
Expansion theorem, 596
Experimental modal analysis, 900 915
basic idea, 900
coherence function, 907
digital signal processing, 903 904
modal data determination from observed
peaks, 907 912
mode shapes measurement, 912 915
necessary equipment, 900 903
random signals analysis, 905 907
Explicit integration method, 943
F
Fast Fourier Transform (FFT) method, 896,
902, 924
Finishing process, vibratory, 12
Finite difference method, 941 942
for continuous systems, 951 959
Finite element idealization, 10
Finite element method, 987 1035
bar element, 991 994
beam element, 995 998
boundary conditions, incorporation of,
1003 1012
element matrices and vectors, transformation, 998 1001
equations of motion of, 989 991,
1001 1003
Euler-Bernoulli theory, 995
force vector, 991 998
mass matrix, 991 998
stiffness matrix, 991 998
torsion element, 994 995
First order systems, 151 153, 366 372
response of, 407 408
response under periodic force, 367 372
Fixed-free bar, free vibrations of, 714
Fixed-pinned beam, natural frequencies of,
728 731
Flexibility influence coefficients, 567 571
determination, 569
Flexibility matrix of a beam, 571
Flow-induced vibration
of a chimney, 311
reduction, 307
Flutter, 305
Focus, 13-40
Force transmissibility, 283 284
Force vector, 991 998
Forced system, steady-state response of,
615 619
Forced vibration, 17, 494 497, 730 731
steady-state response of spring-mass
system, 495 497
of viscously damped systems, 610 616INDEX 1079
Forging hammer
forced vibration response of, 608 613
response of anvil of, 169
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, 64 66, 73
Cam-follower system, 73
complex Fourier series, 66 67
Gibbs phenomenon, 66
graphical representation using
MATLAB, 76 78
numerical Fourier analysis, 74 76
periodic function, 65
Fourth-order Runge-Kutta method, 974 975
Frahm tachometer, 9, 890
Francis water turbine, 291
Free vibration, 17
response of two degree of freedom
system, 481 482
response using modal analysis, 606 608
Frequency domain representations, 68 69
Frequency-measuring instruments, 890 891
frequency-measuring instruments, 891
multireed-instrument, 890
single-reed instrument, 890
stroboscope, 891
Frequency of damped vibration, 161
Frequency of oscillation, 62
Frequency or characteristic equation, 475
Frequency response approach, 14-30,
14-33 14-39
mean square response, 14-34
power spectral density, 14-33
Frequency spectrum, 67 68
Frequency transfer functions, 320 325
frequency response characteristics representation, 323 325
general transfer function and, 322 323
physical system, 321
solutions using, 512 515
Fullarton tachometer, 890
G
Galileo Galilei, 6 9
Galloping, 305 306
Gaussian random process, 14-14 14-16
General forcing conditions, vibration under,
363 466, See also General periodic
force, response under; Nonperiodic
force, response under; Periodic force;
Response spectrum
General periodic force, response under,
365 378
first-order systems, 366 372
second-order systems, 367, 372 374
total response under harmonic base excitation, 377 378
General transfer function and frequency
transfer function, 322 323
Generalized coordinates, 472, 488, 576 577
Generalized forces, 576 577
Generalized mass matrix, 575
Germain, Sophie, 8
Gibbs phenomenon, 66
Grid points, 941
H
Half power points, 276
Half-range expansions, 71 72
Harmonic analysis, 64 76, See also Fourier
series expansion
even functions, 69 71
frequency domain representations, 68 69
half-range expansions, 71 72
odd functions, 69 71
time domain representations, 68 69
Harmonic base excitation, total response
under, 377 378
Harmonic motion, 54 64, 134 146
addition of harmonic motions, 60
complex algebra, 58
complex number representation of,
57 58
impact, free vibration response due to,
141 142
motion of, graphical representation, 136
natural frequency, 142 146
operations on harmonic functions, 59 61
Scotch yoke mechanism, 55
simple harmonic motion, 56
undamped system, phase plane representation, 138
vectorial representation of, 56 57
water tank, harmonic response of,
139 141
Harmonically excited vibration, 259 362
damped system response under F(t) =
F0ei*t, 278 281
damped system response under harmonic
force, 271 278, See also individual entry
equation of motion, 261 262
forced vibration with Coulomb damping,
293 297, See also Coulomb
damping
hysteresis damping, forced vibration
with, 298 310
quadratic damping, 300
quality factor and bandwidth, 276 278
undamped system response under,
263 271
Helical spoilers, 308
Helicopter seat vibration reduction, 774 783
vibration at source, reduction, 775 776
Heterodyne analyzer, 899
History of vibration, 3 10
finite element idealization, 10
from Galileo to Rayleigh, 6 9
origin, 3 5
recent contributions, 9 10
theory of vibration of plates, 7
torsional vibration tests, 8
Hoisting drum, equivalent k of, 34 35
Holzer s method, 666 670
resultant torque versus frequency, 667
spring-mass systems, 669 670
torsional systems, 666 669
Hooke, Robert, 6
Horizontal position, spring-mass system in,
126
Houbolt method, 962 965
for a two degree of freedom system, 964
Hydraulic valve, periodic vibration of,
374 376
Hysteretic damping, 46
forced vibration with, 298 310
free vibration with, 192 198
I
Ideal white noise, 14-25
Identity matrix, 1042
Implicit integration methods, 963
Impulse response function, 382 385,
14-28 14-29
Inelastic collision, response to, 411 412
Inertia influence coefficients, 572 573
Influence coefficients, 562 573
flexibility influence coefficients,
567 571
flexibility matrix of a beam, 571
inertia influence coefficients, 572 573
stiffness influence coefficient, 562 567
stiffness matrix of a frame, 566
Introduction to Harmonics, 5
Inverse Laplace transform, 1049
Inverse matrix, 1045
Inverse of the Matrix, 682
Irregular forcing conditions, response to,
431 436
Irregular forcing function, 378 380
Isolation, vibration, 801 802
with base motion, 814 820
damped spring mount, 801
pneumatic rubber mount, 801
system with flexible foundation,
821 822
system with partially flexible foundation,
822 824
types, 802 803
undamped spring mount, 801
with rigid foundation, 804 813, See also
Rigid foundation
Iteration method, 670 677, 13-13 13-16,
See also Matrices: matrix iteration
method
J
Jacobi s method, 678 680
eigenvalue solution using, 679 682, 684
standard eigenvalue problem, 680 683
joint probability distribution of random variables, 14-7 14-9
bivariate distributions, 14-8
multivariate distribution, 14-8
univariate distributions, 14-81080 INDEX
Journal bearing, damping constant of, 48 50
Jump phenomenon, 13-16
K
Karman vortices, 305
Kinetic energy expressions in matrix form,
574 576
Kirchhoff, G. R., 8
Kronecker delta, 581
LL
Hospital s rule, 266
Lagrange, Joseph, 7
Lagrange s equations, 577 581
Laplace transform, 313, 317 320, 365,
406 427, 504 512, 1048 1055
damped system response using, 317
definition, 1048 1049
first-order systems, response of, 407 409
inverse Laplace transform, 1049
partial fractions method, 1050 1052
second order systems, response of,
409 414
shifting theorems, 1050
steady state response using, 319 320
step force, response to, 414 420
transform of derivatives, 1049 1050
transient and steady-state responses, 406
transient response, 421 427, See also
individual entry
two degree of freedom systems solutions
using, 504 512
Laplacian operator, 741
Lateral vibration of beams, 721 736
axial compressive force, beam subjected
to, 733 737
boundary conditions, 724 726
equation of motion, 721
fixed-pinned beam, natural frequencies
of, 728 731
forced vibration, 730 731
free vibration, 723 724
initial conditions, 723
orthogonality of normal functions,
726 729
simply supported beam, forced vibration,
731 734
Lathe, 469, 488 489
Left half-plane (LHP) yield, 198
Limit cycles, 13-41 13-43
Lindstedt s perturbation method, 13-10 13-12
Linear algebraic equations, solution of, 1065
Linear coordinates, 555
Linear force, compacting machine under,
391 392
Linear springs, 23 25
Linear variable differential transformer
(LVDT) transducer, 878 879
Linear vibration, 17
Linearization of nonlinear spring, 25 27
Literature, vibration, 80 81
Local coordinate axis, 998
Logarithmic decrement, 164 166
Longitudinal vibration of bar or rod,
710 718
bar carrying a mass, natural frequencies
of, 715 716
bar subjected to initial force, vibrations
of, 716 720
boundary conditions, 712 713
equation of motion and solution,
710 712
free vibrations of a fixed-free bar, 714
orthogonality of normal functions,
713 718
Longitudinal vibration of bars, 951 955
Loops, 6
Lumped mass matrices, 1012 1015
Lumped-mass model, 555
M
Machine condition monitoring techniques,
916 918
Machine maintenance techniques, 915 916
breakdown maintenance, 915
condition-based maintenance, 916
preventive maintenance, 915
Machine tool support, equivalent spring and
damping constants of, 52 53
Machine vibration monitoring techniques,
918 923
Magnification factor, 264, 273
Marine engine propeller system, 485 488
Mass matrix, 991 998
Mass or inertia elements, 40 45
Material damping, 46
Mathematical modeling, 18
Mathieu equation, 13-24 13-29
MATLAB, 326 332, 436 440, 515 522,
619 627, 683 686, 748 751,
843 850, 925 928, 972 978,
1015 1019, 1041 1047, 1059 1068
accelerometer equation plotting,
927 929
arrays and matrices, 1060
arrays with special structure, 1061
autocorrelation function plotting,
14-46 14-48
column vector, 1060
complex numbers, 1062
Coulomb damping, free vibration
response of a system with, 205
damped system, forced vibration
response of, 623 624
differential equations solution,
1066 1068
eigenvalue problem solution, 515 516,
619, 683 684
finite element analysis of stepped bar,
1015
forced response of a system with
Coulomb damping, 327 328
forced vibration response of simply supported beam, plotting, 748 751
Fourier series graphical representation
using, 76 78
free vibration response
frequency response, plotting, 519 520
functions in, 1062
Gaussian probability distribution function evaluation, 14-48 14-49
general eigenvalue proble, 685 686
impulse response of a structure, 437 438
matrix, 1060
matrix operations, 1061
M-files, 1062 1063
multidegree of freedom system, 619 626
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, 79
Nyquist circle plotting, 925 926
pendulum equation solution, 13-53 13-57
plotting of graphs, 1063 1064
program to generate characteristic polynomial, 625
quartic equation roots, 522
railway cars, time response of, 518 519
response under a periodic force, 438 439
response under arbitrary forcing
function, 439
roots of a polynomial equation, 622
roots of a quartic equation, 516
roots of transcendental and nonlinear
equations, 750 751
row vector, 1062
solution of a single degree of freedom
system, 972
solution of multidegree of freedom
system, 973 974
special matrices, 1061
spring-mass system, free vibration
response of, 203
static deflection, variations of natural
frequency and period with, 202
steady-state response of viscously
damped system, 330 331
system response under base excitation,
328 330
total response of an undamped system
using, 326 327
total response of system under base
excitation, 436 437
transmissibility, plotting, 843
undamped system response, 204
variables, 1060
vibration amplitudes of vibration
absorber masses, 845 846
Matrices, 1041 1047, 1060, See also individual entries
basic operations, 1046 1047, 1061
trace, 1043
transpose of, 1043INDEX 1081
Maximum overshoot (M
p
), 423
Mean square response, 14-34
Mean value, 14-6 14-7, 14-32
Measurement and applications, vibration,
870 938
machine condition monitoring and
diagnosis, 915 925
measurement scheme, 872
Mechanical chatter, 13-5
Mechanical exciters, 892 893
Method of isoclines, 13-34 13-35
trajectories using, 13-36
Membranes, vibration of, 739 742
equation of motion, 739 741
free vibrations of rectangular membrane,
742 744
initial and boundary conditions, 741 742
membrane under uniform tension, 740
Mersenne, Marin, 6 7
M-files, 1062 1063
Milling cutter, natural frequencies of, 720
Mindlin, R. D., 9
Modal analysis, 596
forced vibration of undamped systems
using, 603 610
free vibration response using, 606 608
Modal damping ratio, 612
Modal matrix, 592
Modal testing, 900 915, See also Experimental modal analysis
Modal vectors, 475
Mode shapes, 583
determination, 654 698
measurement, 912 915
of three degree of freedom system, 590
Monochord, 4 5
Motor-generator set, absorber for, 838 843
Multidegree of freedom systems, 553 653,
14-39 14-46, See also Influence
coefficients; Three degree of freedom
system
central difference method for, 947 951
continuous systems modeling as,
555 556
equations of motion of undamped systems in matrix form, 581 582
expansion theorem, 596
free vibration of undamped systems,
601 603
generalized coordinates, 576 577
generalized forces, 576 577
Lagrange s equations to derive equations
of motion, 577 581
modal analysis, 603 610, See also individual entry
natural frequencies of free system,
598 601
Newton s second law to derive equations
of motion, 557 562
potential and kinetic energy expressions
in matrix form, 574 576
repeated Eigenvalues, 594 596
self-excitation, 617 619
spring-mass-damper system, equations
of motion of, 557 560
stability analysis, 617 619
steady-state response of forced system,
615 619
trailer compound pendulum system,
equations of motion of, 560
unrestrained systems, 596 599
Multivariate distribution, 14-8
N
Narrow-band process, 14-25 14-27
Natural frequencies, 62, 475
determination, 654 698, See also
Dunkerley s formula; Holzer s
method; Jacobi s method;
Rayleigh s method
of free system, 598 601
of torsional system, 484, 668 673
Natural mode, two degree of freedom systems, 471
Newmark method, 968 971
Newton, Isaac, 6
Newton s second law, 129 130, 261,
557 562
Nodes, 6, 706, 13-40 13-41
Nomograph, vibration, 771 775
Nondeterministic vibration, 17 18
Nonlinear damper, linearization of, 52
Nonlinear differential equation solution, 13-61
Nonlinear equations, roots of, 1064
Nonlinear springs, 23 25
Nonlinear system under pulse loading solution, 13-59
Nonlinear vibration, 17, 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, 13-25
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, 365,
380 381, See also Convolution integral; Laplace transform; Numerical
methods
Normal modes, 470, 591 593
Number-decibel conversion line, 323
Numerical Fourier analysis, 74 76
Numerical integration methods, 939 986
finite difference method, 941 942
single degree of freedom systems,
942 946
Numerical methods, for response under nonperiodic force, 365, 428 431
Nyquist circle plotting, 925 926
Nyquist plot, modal data determination
from, 910 912
O
Octave band analyzer, 896, 899
Octave, 63
Odd functions, 69 71
Operational deflection shape
measurements, 900
Optimally tuned vibration absorber, 842
Orthogonality of normal functions, 591 593,
713 718, 726 729
Orthonormalization of Eigenvectors,
592 594
Overdamped system, 163, 418 420
P
Parameter variations, 176 185, See also
under Root locus representations
Parseval s formula, 14-17, 14-21
Partial fractions method, 1050 1052
Peak time (t
p
), 421
Perfectly elastic collision, response to,
412 414
Periodic solutions using Lindstedt s perturbation method, 13-25
Period of beating , 269
Period of oscillation, 62
Periodic force, 365 378, See also General
periodic force, response under
Periodic vibration of a hydraulic valve,
374 376
Phase angle, 62, 584
Phase distortion, 888 890
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, 6
Piezoelectric transducers, 876 877
Pinned-fixed beam, 959
Piston-cylinder dashpot, 50 52
Plane milling cutter, 721
Plano-milling machine structure, 989
Plates, deflection of, 1039 1040
Poincaré section, 13-43 13-44
Poisson, Simeon, 8
Positive definite matrix, 576
Positive definite quadratic forms, 576
Potential energy expressions in matrix form,
574 5761082 INDEX
Power spectral density, 14-23 14-25, 14-33
Precision electronic system, vibration control of, 829 830
Precision machine with base motion, design
of isolation for, 816 818
Principal coordinates, 472, 488 493
Principal mode, two degree of freedom systems, 470
Principle of conservation of energy, 131 132
Principle of virtual displacements, 130
Probability density curve, 919
Probability distribution, 14-4 14-5
Propeller shaft, 33
Proportional damping, 611
Pseudo spectrum, 397
Pseudo velocity, 397
Pulley subjected to Coulomb damping, 191
Pulley system, 145 146
Pulse load, 389 390
response due to, 390
Pythagoras, 3 4
Q
Q factor/Quality factor, 276 278
Quadratic damping, 300
Quefrency domain analysis, 922
R
Ramp function, first-order system response
due to, 408 409
Random signals analysis, 905 907
Random vibration, 17, 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, 8
Rayleigh s method, 153 158, 658 665, 700,
742 745
beams, fundamental frequency of,
663 665
effect of mass, 155, 157
manometer for diesel engine, 153
Rayleigh s quotient, properties of,
659 661
shafts, fundamental frequency of,
663 668
U-tube manometer, 154
Rayleigh-Ritz method, 700, 745 748
Reciprocating engines, balancing, 792 798
reciprocating engines, balancing,
796 798
unbalanced forces due to fluctuations in
gas pressure, 792 793
unbalanced forces due to inertia of the
moving parts, 793 796
Recoil mechanism, 173
Rectangular pulse load, 389 390
response due to, 390
Recurrence formula, 943
Reference marks, 777 778
Relative motion, 284 287
Repeated Eigenvalues, 594 596
Resonance, 11
Resonant frequencies of vibration absorber,
847 848
Response spectrum, 394 406
for base excitation, 396 399
building frame response to an earthquake, 402 403
design under shock environment,
403 406
earthquake response spectra, 399 403
of sinusoidal pulse, 394 397
water tank subjected to base acceleration, 398 399
Rigid bar
connected by springs, equivalent k of, 37
stability of, 201
Rigid foundation, vibration isolation system
with, 804 813
design chart of isolation, 809 810
isolator for stereo turntable, 810 813
machine member on, 804
resilient member on, 804
spring support for exhaust fan,
807 808
undamped isolator design, 808 810
vibratory motion of mass, reduction, 806
Rise time (t
r
), 422 423
Ritz-Galerkin method, 13-17 13-19
Rod, spring constants of, 27
Root locus representations, 176 185
and parameter variations, 179 185
roots study with variation of c, 181
+ in s-plane, 176 179
, in s-plane, 176 179
*d in s-plane, 176 179
*
n
in s-plane, 176 179
variation of mass, 183
variation of spring constant, 183
Rotary inertia effects, 734 739
Rotating machines, balancing, 776 785
single-plane balancing, 776 779
two-plane balancing, 779 785
Rotating unbalance, 287 293, 830 837, See
also under Damped system response
under harmonic force
Routh-Hurwitz criteria, 502, 790
Row matrix, 1042
Row vector, 1060
Runge-Kutta methods, 429 431
S+
in s-plane, interpretation, 176 179
, in s-plane, interpretation, 176 179
*d in s-plane, interpretation, 176 179
*
n
in s-plane, interpretation, 176 179
Saddle point, 13-40 13-41
Sample point, 14-3
Sample space, 14-3
Sauveur, Joseph, 6
Scotch yoke mechanism, 55
Second-order systems, 367, 372 374,
409 414
Seismograph, 5
Self-excitation, 301 313, 500 502, 617 619
Semidefinite systems, 497 500, 598
Settling time, transient response, 424
Shafts, fundamental frequency of, 663 668
Shear deformation effects, 734 739
Shearing stress (,), 49
Shock absorber for a motorcycle, 171 172
Shock environment, design under, 403 406
Shock isolation, 824 827
Shock loads, 403 406
Signal analysis, 895 899
Signum function, 187
Simple harmonic motion, 56
Simple pendulum, 14, 39, 13-3
Simply supported beam, 1039
forced vibration, 731 733
natural frequencies of, 737 742
Singing of transmission lines, 305
Single degree of freedom systems, 14,
14-28 14-31
central difference method for, 942 945
characteristics of, 14-30 14-31
free vibration of, 124 258, See also
Undamped translational system,
free vibration of
frequency response approach, 14-30
impulse response approach, 14-28 14-29
Single-plane balancing, 776 779
Singular point, 13-34
Sinusoidal pulse, response spectrum of,
394 397
Solid damping, 46
Space shuttle, vibration testing, 12
Special matrices, 1061
Spectrum analyzers, 896 897
Spring constants of elastic elements, 27 30
Spring elements, 22 39
deformation of spring, 22
equivalent linearized spring constant, 26
linear springs, 23 25
nonlinear springs, 23 25INDEX 1083
spring constant associated with restoring
force due to gravity, 39
spring constants of elastic elements,
27 30
Spring-mass-damper system, 262, 557 560
Spring-mass systems, 126 128, 669 670
Springs, combination of, 30 38
equivalent k, 32, 34 35, 37
in parallel, 30
in series, 31
torsional spring constant of a propeller
shaft, 33
Spring-supported mass instability on moving
belt, 302
Square matrix, 1042
Stability analysis, 301 313, See also
Dynamic instability caused by fluid
flow
two degree of freedom systems, 500 502
dynamic, 301 305
multidegree of freedom systems,
617 619
whirling of rotating shafts, 790 791
Stability of systems, 198 202
asymptotically stable, 198 200
rigid bar, 201
stable, 198 200
unstable, 198 200
Stable focus, 13-40
Stable orbits, functions with, 13-45
Standard deviation, 14-6 14-7
Standard eigenvalue problem, 585
Static deflection, 263
Static equilibrium position, 132
Static unbalance, 776
Stationary random excitations, response due
to, 14-31 14-39
Stationary random process, 14-10 14-14
Strange attractors, 13-46
Steady state response, 406
of forced system, 615 619
using Laplace transform, 319 320
Step force, response to, 414 420, See also
under Laplace transform
Stepped bar, 1015, 1018
Stiffness influence coefficient, 562 567
Stiffness matrix, 566, 991 998
Stockbridge damper, 308
Stodola, Aurel, 9
Stroboscope, 891
Study of vibration, importance, 10 13
Subharmonic oscillations, 13-20 13-22
Superharmonic oscillations, 13-23 13-24
Suspension system, equivalent k of, 32
Symmetric matrix, 682 683, 1043
System response under base excitation,
328 330
T
Tapered beam, fundamental frequency of,
744 750
Taylor, Brook, 7
Taylor s series expansion, 25, 310
Temporal averages, 14-14
Thick beam theory, 734
Thin beam theory, 722
Three degree of freedom system, 15
equations of motion of, 586
fundamental frequency of, 661 663
mode shapes of, 590
natural frequencies of, 587 589
natural frequencies of, 673 679
Time constant, 151 153
Time-delayed step force, 388 389
Time-dependent coefficients, systems with,
13-24 13-29
Time domain analysis, 918
Time domain representations, 68 69
Timoshenko beam theory, 734 735
Timoshenko, Stephen, 9
Torsion element, 994 995
Torsional pendulum, 148
Torsional spring constant of a propeller
shaft, 33
Torsional system, 483 488, 666 669
with Coulomb damping, 190 192
with discs mounted on a shaft, 483
equations of motion of, 578 579
natural frequencies of, 484 488,
668 673
with viscous damping, 168 174
Torsional vibration of a shaft or rod,
718 721
Torsional vibration, 8, 146
Trace, 1043
Trajectories of simple harmonic oscillator,
13-29 13-30
Trailer compound pendulum system, equations of motion of, 560
Transducers, 873 879, 901
electric resistance strain gage, 873
electrodynamic transducers, 877 878
linear variable differential transformer
(LVDT) transducer, 878 879
piezoelectric transducers, 876 877
variable resistance transducers, 873 876
Transfer function approach, 313 316,
425 426, 502 504
Transient response, 261 262, 406, 421 427
Transition curves, 13-28
Transverse vibration of beams, 955 959
Transverse vibration of string or cable,
701 710, See also under Continuous
systems
Traveling-wave solution, 709 710
Triangular pulse, Fourier transform of, 14-22
Triple pendulum, 576
Tuned vibration absorber, 842
Two degree of freedom systems, 15,
467 552, See also Forced vibration;
Laplace transform; Semidefinite systems; Torsional system
automobile, frequencies and modes of,
492 495
coordinate coupling and principal coordinates, 488 493
coupled differential equations, 470
equations of motion for forced vibration,
472 473
forced response of, 520 522
free vibration response of, 481 482
Lathe, 469, 488 489
natural mode, 471
normal mode, 470
packaging of an instrument, 471
principal mode, 470
spring-mass-damper system, 472
transfer function approach, 502 504
Two-plane balancing, 779 785, 848 850
U
Undamped dynamic vibration absorber,
833 839
effect on the response of machine, 835
for diesel engine, 837 838
for motor-generator set, 838 843
Undamped equation, 13-13, 13-31
Undamped isolator design, 808 810
Undamped system, 127
free vibration analysis, 474 482
free vibration of, 601 606
free vibration response of, 504 507
in matrix form, 581 582
response under harmonic force,
263 271, See also under Harmonically excited vibration
total response of, using MATLAB,
326 327
Undamped torsional system, free vibration
of, 146 151
Undamped translational system, free vibration of, 129 146
auxiliary or characteristic equation, 134
D Alembert s principle, 130
eigenvalues or characteristic values, 134
mass under virtual displacement, 131
principle of conservation of energy, 130
principle of virtual displacements, 130
using Newton s second law of motion,
129 130
Undamped vibration, 17
Underdamped system, 160, 414 416
Uniform string, free vibration of, 704 705
Unit impulse response of second-order
system, 409
Units, 1056 1058
Univariate distributions, 14-8
Unrestrained systems, 499 502, 596 599
Unstable focus, 13-40
Unstable orbits, functions with, 13-45 13-47
Unstable system, 198
V
Variable mass system, 13-5 13-6
Variable resistance transducers, 873 8761084 INDEX
Vectorial representation of harmonic motion,
56 57
Velometer, 886 887
Vertical position, spring-mass system in,
132 133
Vibrating string, 702
Vibration absorbers, 832 843, 847 848, See
also Damped dynamic vibration
absorber; Undamped dynamic vibration absorber
Vibration pickups, 879 890
Vibration severity of machinery, 773
Vibrometer, 881 882
Viscoelastic materials use, 799
Viscous damping, 45
Cannon analysis, 173
energy dissipated in, 166 168
forced vibration of, 610 616
free vibration with, 158 174
steady-state response of, 330 331
torsional systems with viscous damping,
168 174
torsional systems with, 168 174
W
Wallis, John, 6
Whirling of rotating shafts, 785 791
critical speeds, 787
equations of motion, 785 787
shaft carrying an unbalanced rotor, 791
stability analysis, 790 791
system response, 788 790
Wide-band process, 14-25 14-27
Wiener-Khintchine formula,
14-23
Wilson method, 965 968
Wind-induced vibration, 11
Y
Young s modulus, 142 143
Z
Zero matrix, 1042
Zhang Heng, 5


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