Engineering Vibration

Engineering Vibration
اسم المؤلف
DaniEl J. inman
التاريخ
21 أكتوبر 2021
المشاهدات
التقييم
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Engineering Vibration
Fourth Edition
DaniEl J. inman
University of Michigan
Contents
PREFACE viii
1 INTRODUCTION TO VIBRATION AND THE FREE RESPONSE 1
1.1 Introduction to Free Vibration 2
1.2 Harmonic Motion 13
1.3 Viscous Damping 21
1.4 Modeling and Energy Methods 31
1.5 Stiffness 46
1.6 Measurement 58
1.7 Design Considerations 63
1.8 Stability 68
1.9 Numerical Simulation of the Time Response 72
1.10 Coulomb Friction and the Pendulum 81
Problems 95
MATLAB Engineering Vibration Toolbox 115
Toolbox Problems 116
2 RESPONSE TO HARMONIC EXCITATION 117
2.1 Harmonic Excitation of Undamped Systems 118
2.2 Harmonic Excitation of Damped Systems 130
2.3 Alternative Representations 144
2.4 Base Excitation 151
2.5 Rotating Unbalance 160
2.6 Measurement Devices 166iv Contents
2.7 Other Forms of Damping 170
2.8 Numerical Simulation and Design 180
2.9 Nonlinear Response Properties 188
Problems 197
MATLAB Engineering Vibration Toolbox 214
Toolbox Problems 214
3 GENERAL FORCED RESPONSE 216
3.1 Impulse Response Function 217
3.2 Response to an Arbitrary Input 226
3.3 Response to an Arbitrary Periodic Input 235
3.4 Transform Methods 242
3.5 Response to Random Inputs 247
3.6 Shock Spectrum 255
3.7 Measurement via Transfer Functions 260
3.8 Stability 262
3.9 Numerical Simulation of the Response 267
3.10 Nonlinear Response Properties 279
Problems 287
MATLAB Engineering Vibration Toolbox 301
Toolbox Problems 301
4 MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS 303
4.1 Two-Degree-of-Freedom Model (Undamped) 304
4.2 Eigenvalues and Natural Frequencies 318
4.3 Modal Analysis 332
4.4 More Than Two Degrees of Freedom 340
4.5 Systems with Viscous Damping 356
4.6 Modal Analysis of the Forced Response 362Contents v
4.7 Lagrange’s Equations 369
4.8 Examples 377
4.9 Computational Eigenvalue Problems for Vibration 389
4.10 Numerical Simulation of the Time Response 407
Problems 415
MATLAB Engineering Vibration Toolbox 433
Toolbox Problems 433
5 DESIGN FOR VIBRATION SUPPRESSION 435
5.1 Acceptable Levels of Vibration 436
5.2 Vibration Isolation 442
5.3 Vibration Absorbers 455
5.4 Damping in Vibration Absorption 463
5.5 Optimization 471
5.6 Viscoelastic Damping Treatments 479
5.7 Critical Speeds of Rotating Disks 485
Problems 491
MATLAB Engineering Vibration Toolbox 501
Toolbox Problems 501
6 DISTRIBUTED-PARAMETER SYSTEMS 502
6.1 Vibration of a String or Cable 504
6.2 Modes and Natural Frequencies 508
6.3 Vibration of Rods and Bars 519
6.4 Torsional Vibration 525
6.5 Bending Vibration of a Beam 532
6.6 Vibration of Membranes and Plates 544
6.7 Models of Damping 550
6.8 Modal Analysis of the Forced Response 556vi Contents
Problems 566
MATLAB Engineering Vibration Toolbox 572
Toolbox Problems 572
7 VIBRATION TESTING AND EXPERIMENTAL MODAL ANALYSIS 573
7.1 Measurement Hardware 575
7.2 Digital Signal Processing 579
7.3 Random Signal Analysis in Testing 584
7.4 Modal Data Extraction 588
7.5 Modal Parameters by Circle Fitting 591
7.6 Mode Shape Measurement 596
7.7 Vibration Testing for Endurance and Diagnostics 606
7.8 Operational Deflection Shape Measurement 609
Problems 611
MATLAB Engineering Vibration Toolbox 615
Toolbox Problems 616
8 FINITE ELEMENT METHOD 617
8.1 Example: The Bar 619
8.2 Three-Element Bar 625
8.3 Beam Elements 630
8.4 Lumped-Mass Matrices 638
8.5 Trusses 641
8.6 Model Reduction 646
Problems 649
MATLAB Engineering Vibration Toolbox 656
Toolbox Problems 656
APPENDIX A COMPLEX NUMBERS AND FUNCTIONS 657
APPENDIX B LAPLACE TRANSFORMS 663Contents vii
APPENDIX C MATRIX BASICS 668
APPENDIX D THE VIBRATION LITERATURE 680
APPENDIX E LIST OF SYMBOLS 682
APPENDIX F CODES AND WEB SITES 687
APPENDIX G ENGINEERING VIBRATION TOOLBOX
AND WEB SUPPORT 688
REFERENCES 690
ANSWERS TO SELECTED PROBLEMS 692
INDEX 69
index
A
Accelerance transfer function,
260–261
Acceleration, simple harmonic
motion, 10, 13w
Accelerometers, 166f, 167f, 578
Aircraft
base excitation example, 157–159
control tab, 113f
foot pedal model, 105f
jet engine with transverse
vibration, 290f
landing system, 106f
steering-gear mechanism, 104f
vibration-induced fatigue, 479
wing
distributed-parameter
system, 502
engine mount, 107f
harmonic excitation example,
169–170
impulse response function, 288f
stability example, 71
torsional vibration, 420f
vibration examples, 49–52,
373–375
vibration models, 200f, 423f
Air damping, 179
Airfoil, 204f
Algebraic eigenvalue problem,
400, 402
Aliasing, 581–582
Amplitude, 8
Angular motion, 526f
Angular natural frequency, 8
Arbitrary input, general forced
response, 226–235
examples, 228–235
problems, 290–292
Arbitrary periodic input, general
forced response, 235–242
examples, 237–242
problems, 293–294
Argand plane plots, 591
Assumed mode method, 566
Asymmetric eigenvalue
problem, 332
Asymptotic stability, 69, 265, 267
Autocorrelation function, 249
Automobiles
base excitation example, 157–159
brake pedal model, 202f
drive train vibration analysis, 423f
frequency response function, 441f
single-degree-of-freedom
model, 440f
tires and resonance, 117
vibration isolation example,
446–448, 447f
vibration response example,
440–441
See also Suspension systems;
Vehicles
Average value, 20
B
banded matrix, 674
bar
distributed-parameter systems,
519–525
examples, 520–524
problems, 567–569
finite element method, 619–625
example, 624–625
problems, 649–650
two materials, 568f
See also Cantilevered bar
baseball bat, 42
base excitation, 151–160
examples, 156–160
harmonic excitation, 151f
problems, 205–208
beam bending vibration, 532–544
examples, 537–544
problems, 570
beam elements, finite element
method for, 630–638
examples, 634–638
problems, 652–653
beam–mass model, 126
beams
Euler–bernoulli, 533–540, 533f
shear deformation, 541f
single-finite-element model, 631f
Timoshenko, 540–544, 541f
tip mass, 568f
transverse vibration, 533f, 539t
bearing housing displacement, 609f
beat, 124
beats, two-degree-of-freedom
system, 317
bell-shaped curve, 253
bernoulli–Euler beams. See
Euler–bernoulli beams
bIbO (bounded-input, boundedoutput stability), 263, 265, 267
biharmonic operator, 549
bilinear systems, 83
borel’s theorem, 246
boundary value problem, 506
bounded-input, bounded-output
(bIbO) stable, 263, 265, 267
bridge, 651f
broadband vibration
absorption, 469f
buildings
ground motion, 207f
horizontal vibration example,
348–351, 349f, 351f
machine with rotating unbalance,
560f, 571f
C
Cable vibration, 504–507
example, 507
Camera mount, 222f
Cantilevered bar
finite element grids, 620f
longitudinal vibration, 555f, 620w
one-element model, 649f
three-element four-node
model, 620f
Cantilevered beam, 533
applied axial force, 649f
driving points, 599f700 Index
Cantilevered beam (Continued)
measurement points, 602f
spring-mass system attached, 652f
two-element, three-node
mesh, 637f
Center of percussion, 39, 40
Characteristic equation, 23, 311, 509
Cholesky decomposition, 318,
389–392
Circle fitting, 591–595
problems, 613–614
Clamped beam, 653f
Clamped–clamped bar, 555f, 640f
Clamped–free bar. See
Cantilevered bar
Clamped–free beam. See
Cantilevered beam
Clamped two-element beam
system, 654f
Clamped two-step aluminum
beam, 653f
Coherence function, 587, 588f
Coiled spring, stiffness of, 52–53, 53f
Complex arithmetic, 466w
Complex modes, 404
Complex modulus, 178
Complex stiffness, 178, 479, 480f
Computational eigenvalue problems
for vibration in MDOF
systems, 389–407
Computer-controlled vibration
endurance test, 607f
Computer disk drive motor, 452f
Computer software
eigenvalues, 392
numerical simulation, 72, 180
Consistent-mass matrices, 638
Constrained-layer damping, 482
Continuous systems. See
lumped- parameter systems
Conversation of energy
equations, 33
Convolution integral, 227, 228w, 233
Cooling fan, 483f
Coulomb damping, 81–82
Coulomb friction
free response, 84, 85f, 86f
harmonic excitation, 170–173
vibration, 81–88
examples, 85–88
problems, 114–115
Coupling device, 417f
Critical damping coefficient, 23
Critically damped motion, 27–31
response, 28f
Critical points, 471
Critical speeds, for vibration
suppression on rotating
disks, 485–491
examples, 488–491
problems, 500–501
Cross-correlation function, 585
Cross-spectral density, 586
Cutoff frequency, 577
D
Damped natural frequency, 24
Damped systems
eigenvalue problems, 396–399,
402–407
harmonic excitation, 130–144
examples, 134–144
problems, 201–204
single-degree-of-freedom, 23f
two-degree-of-freedom, 428f
Damping
air, 179
Coulomb, 81–82, 170–173
distributed-parameter systems,
550–555
examples, 551–555
problems, 570–571
harmonic excitation, 170–180
examples, 173–180
problems, 210–211
hysteretic, 176
modal, 356–362
models, 180t
proportional, 362
vibration absorber with, 463f
vibration absorption, 463–470
viscous, 21–31
Damping coefficient, 22, 60–61
amplitude of vibration example,
64–65
Damping ratio, 24, 60–61
Dashpot, 22f
Decibel (db), 20
Decoupling equations of motion
using modal analysis, 335f
Degree of freedom, 4
See also Multiple-degree-offreedom (MDOF) systems;
Single-degree-of-freedom
systems
Design, definition of, 436
Design considerations
harmonic excitation example,
185–187
modal approach, 407
range of, 448
robustness, 68, 463
rotor system example, 488–489
vibration, 63–68
examples, 64–68
problems, 111–112
vibration, acceptable levels of, 442
vibration absorbers, 458
vibration suppression, 435–501
Diagnostics, vibration testing for,
606–609
example, 607–608
Digital Fourier analyzer, 579
Digital Fourier transform
(DFT), 579
Digital representations of
signals, 581f
Digital signal processing, 579–584
example, 582
problems, 611
Digital spectral coefficients, 582
Dirac delta function, 219
Discrete systems. See
lumped-parameter systems
Discretization, 618
Disk drive motor of personal
computer, 452f
Disk–shaft system, 8w
critical speeds for vibration
suppression, 485–491
example, 37
harmonic excitation example,
149–150
torsional vibration, 47f
Displacement
simple harmonic motion,
10, 11w
vibration, 437t
Displacement transmissibility, 153,
154f, 156, 157f, 442, 443w
Distributed-parameter systems,
502–572
bar vibration, 519–525
beam bending vibration,
532–544
cable vibration, 504–507
damping models, 550–555
explanation, 503Index 701
forced response modal analysis,
556–566
membrane vibration, 544–550
modes, 508–518
natural frequencies, 508–518
plate vibration, 544–550
rod vibration, 519–525
string vibration, 504–507
torsional vibration, 525–532
Divergent instability, 69
Divergent response, 69, 69f
Diving board, 112f
Dot product, 306
Double pendulum, with generalized coordinates, 369f
Driving frequency, 118, 124
Driving point, 599
Duhamel integral, 228
Dynamically coupled
systems, 375
eigenvalue problems, 389–407
E
Effective mass, 35
Eigenfunctions, 510
Eigenvalue problems
computational, 389–407,
430–431
damped systems, 402–407
example, 404–407
dynamically coupled systems,
389–392
example, 391–392
two-degree-of-freedom system
example, 324–326
using codes, 392–399
Eigenvalues
distributed-parameter
systems, 510
MDOF systems, 318–332
examples, 320–332
problems, 418–420
Eigenvectors, 320w, 321, 324–326
Elastic damper, 476f
Elastic modulus, 46
complex data, 481t
measurement, 58
stress–strain curve, 59f
temperatures, 481f
Electric motor mount, 493f
Electronic cabinet with cooling
fan, 483f
Endurance, vibration testing for,
606–609
Energy methods for modeling
vibration, 31–46
examples, 34–46
problems, 104–108
Engineering Vibration Toolbox
distributed-parameter systems, 572
eigenvalue problems, 389
finite element method, 656
general forced response, 301–302
harmonic excitation, 214–215
MDOF systems, 433
Runge–Kutta method, 77
vibration, 115–116
vibration testing, 615–616
Ensemble average, 253
Equilibrium points/positions, 81,
83, 87–89, 89f
Equivalent viscous damping, 285f
Euler–bernoulli beam model, 543
Euler–bernoulli beams,
533–540, 533f
Euler method
linear and nonlinear equations,
90–91
numerical solution, 74–78
single-degree-of-freedom
system, 72
Euler relations, 18, 24, 25w
Exciters, 575–576
Expansion method, 346–351
examples, 348–351
Expansion theorem, 347, 566
Expected value, 253
Experimental modal analysis. See
Vibration testing
F
Fast Fourier transform (FFT),
579, 583
FEM. See Finite element method
Finite dimensional systems. See
lumped-parameter systems
Finite element analysis
(FEA), 619
Finite element mesh/grid, 618
Finite element method (FEM),
617–656
bar, 619–625
beam elements, 630–638
lumped-mass matrices, 638–641
model reduction, 646–649
three-element bar, 625–630
trusses, 641–646
Finite element model, 619
Finite elements (FE), 618
First mode shape, 314, 348
Flexural vibrations, 532
Floor-mounted compressor, 494f
Fluid systems
example, 35
natural frequency example, 53–55
Flutter instability, 69, 70f, 267
Forced response. See General
forced response
Forced response modal analysis
distributed-parameter systems,
556–566
examples, 557–566
problems, 571–572
MDOF systems, 362–369
examples, 364–369
problems, 428–429
Force summation method, 32
Force transmissibility, 155, 157f,
442, 443w
Forcing frequency, 118
Formula error, 75–76
Fourier coefficients, 236
Fourier representations of
signals, 581f
Fourier series, 236, 579–580, 580w
Fourier transforms, 246, 579
Fragility, 436
Free response, 5, 10
Coulomb friction, 84, 85f, 86f
numerical simulation of time
response, 72–81
Free vibration, 1
Frequency
cutoff, 577
importance of concept, 15
two-degree-of-freedom
system, 317
vibration, 437
Frequency response approach
to harmonic excitation,
146–148
problems, 205–206
Frequency response curves for
mode shapes, 600–601f
Frequency response function, 147,
588, 588f
Friction coefficients, 82f, 82t702 Index
G
Gaussian distribution function, 253
General forced response, 216–302
arbitrary input response, 226–235
arbitrary periodic input response,
235–242
impulse response function,
217–226
nonlinear response properties,
279–287
numerical simulation, 267–279
random input response, 247–255
shock spectrum, 255–259
stability, 262–267
transfer functions, 259–262
transform methods, 242–247
Generalized eigenvalue
problem, 399
Generalized symmetric eigenvalue
problem, 331
Geometric approach to harmonic
excitation, 145–146
problems, 205–206
Gibbs phenomenon, 238
Global condition, 89
Global coordinate system, 641
Global mass matrix, 627
Global stiffness matrix, 627
Gravity, spring problems and,
16–17
H
Hammer
center of percussion, 42
impact, 576–578, 577f
impulse, 576
instrumented, 221, 222
Hanning window, 583, 584f
Harmonic excitation, 117–215
base excitation, 151–160
damped systems, 130–144
damping, forms of, 170–180
design considerations, 184–187
explanation, 118
frequency response approach,
146–148
geometric approach, 145–146
measurement devices, 166–170
nonlinear response properties,
188–197
numerical simulation, 180–187
rotating unbalance, 160–165
transform approach, 148–151
undamped systems, 118–130
Harmonic motion
examples, 16–21
problems, 99–101
representations, 19w
vibration, 13–15
Heaviside step functions, 224–225,
257, 244, 272–273
Helical spring
spring–mass system natural
frequency, 66
stiffness, 48
Helicopter. See Rotorcraft
Hertz (Hz), 15
Hooke’s law, 59
Houdaille damper, 469f
Humans
forearm vibration model, 199f
longitudinal vibration, 199f
Hysteresis loop, 175, 175f
Hysteretic damping, 176
Hysteretic damping constant, 176
I
Impact, 221, 223f, 226f, 577f
Impact hammer, 576–578
Impulse, definition of, 218
Impulse hammer, 576
Impulse response function
general forced response, 216–226
examples, 221–226
problems, 287–302
Inconsistent-mass matrices, 638
Inertia force, 32
Inertia matrix, 308
Infinite-dimensional systems, 503, 515
See also lumped-parameter
systems
Initial conditions, 10f
Inner product, 307
Input frequency, 118
International Organization of
Standards (ISO), 436, 437
Inverted pendulum, 112–113,
113f, 266
Isolation problems, 442
K
Kelvin–Voigt damping, 554
Kinetic energy, 2
Kronecker delta, 327w
L
lagrange’s equations, 32
energy method, 43–44
example, 43–45
MDOF systems, 369–377
examples, 371–377
problems, 429–431
lagrange stability, 265
laplace operator, 545
laplace transforms
common, 244t
convolution type evaluations, 233
Fourier transforms versus, 247
general forced response, 242–247
harmonic excitation, 148–151
laptop computers, 453
lathe
MDOF system example, 377–381
moving parts, 378f
leaf spring, transverse vibration of,
49, 49f
leakage, 583, 584f
legs, vibration example, 28–31
levers, vibration model of
coupled, 372f
linear systems, 7
local coordinate direction, 641
local stability, 90
logarithmic decrement, 60
longitudinal motion, 46
longitudinal vibration, 199f, 519f,
524t, 525t, 555f
loss coefficient, 174
loss factor, 174
lumped-mass matrices, 638–641
problems, 654–655
lumped-parameter systems, 503, 524
example, 639–641
M
Machinery
acceptable vibration levels, 438f
health monitoring, 607–608
rotating unbalance examples,
160f, 209f, 492f
rubber mount, 206f
vibration absorbers, 455
vibration isolation, 476f
vibration model, 371f
Marginal stability, 68
Mass, frequency of oscillation for
measuring, 62–63Index 703
Mass condensation, 648
Mass loading, 576
Mass matrix, 308
Mass moment of inertia, 58
Mass normalized stiffness, 319
Mass ratio versus natural
frequency, 461f
Mathcad
eigenvalues, 393–394, 405–407
general forced response,
269–271, 273, 274, 276, 282,
285–287
harmonic excitation, 181–182, 185
linear and nonlinear equations,
90–93, 195–197
MDOF systems, 408–409,
410–415
Runge–Kutta method, 79
Mathematica
eigenvalues, 394–396, 398–399,
406–407
general forced response, 271,
273–274, 276, 278, 283–284,
286–287
harmonic excitation, 187
linear and nonlinear equations,
92–93, 193, 196–197
MDOF systems, 410, 412, 414–415
Runge–Kutta method, 79
MATlAb
eigenvalues, 393, 397, 405–407
Engineering Vibration Toolbox,
77, 115–116, 214, 301, 389,
433, 501, 572, 615, 656
general forced response, 270,
272–274, 275, 277, 282, 286
harmonic excitation, 183–184, 186
linear and nonlinear equations,
91–92, 192, 196
MDOF systems, 409, 411, 414
Runge–Kutta method, 77
Matrix inverse, 310
Matrix of mode shapes, 335, 335f
Matrix square root, 318
MDOF systems. See Multipledegree-of-freedom (MDOF)
systems
Measurement
hardware, 574f, 575–579, 600f
problems, 611
harmonic excitation, 166–170
example, 169–170
problems, 210
transfer functions, 260–262
vibration
examples, 59–63
problems, 110–111
Membrane vibration, 544–550, 545f
example, 546–550
problems, 570
Method of undetermined
coefficients, 120
Mindlin–Timoshenko theory, 550
Min-max problem, 474
Mobility frequency response
function, 592f, 592w
Modal analysis
forced response distributed
parameter systems, 556–566
forced response MDOF systems,
362–369
MDOF systems, 332–340
problems, 420–422
See also Vibration testing
Modal coordinate system, 334, 336
Modal damping, 356–359, 361w,
404, 550–555
Modal data extraction, 588–591
example, 590–591
problems, 612–613
Modal equations, 334, 336, 551
Modal participation factors, 348
Modal testing. See Vibration testing
Modeling, definition of, 31
Modeling methods, vibration, 31–46
examples, 34–46
problems, 104–108
Model reduction, 646–649
example, 648–649
problems, 655
Modes, 355
distributed-parameter systems,
508–518
examples, 512–518
problems, 566–567
Mode shapes
clamped–pinned beam, 538f
definition, 304
eigenvectors, 326
explanation, 355
first, 314, 348
longitudinal vibration, 525f
measurement for vibration
testing, 596–606
examples, 599–606
problems, 614–615
nodes, 351
normalizing, 330–331
resonance, 366–367
second, 314
torsional vibration, 532t
vibrating string, 515f
Mode summation method
distributed-parameter systems,
522–524
forced response, 367–369
modal analysis, 346–351
examples, 348–351
modal damping, 358–361, 361w
Modulus data, 47t
Mounting bracket, 563f
Mounts
aircraft wing engine, 107f
base excitation and, 151–160
electric motor, 493f
Multiple-degree-of-freedom
(MDOF) systems, 303–434
computational eigenvalue
problems, 389–407
eigenvalues, 318–332
examples, 377–389
forced response modal analysis,
362–369
lagrange’s equations, 369–377
modal analysis, 332–340, 346–351
more than two degrees of
freedom, 340–346, 341f
examples, 343–346
problems, 422–426
natural frequencies, 318–332
numerical simulation, 407–415
two-degree-of-freedom model
(undamped), 304–318
viscous damping, 356–362
N
Natural frequency
aircraft wing, 49–50
angular, 8
damped, 24
distributed-parameter systems,
508–518
examples, 512–518
problems, 567–569
energy method, 42
fluid system, 38
human leg, 29–30
longitudinal vibration, 525t704 Index
Natural frequency (Continued)
mass ratio versus, 460f
MDOF systems, 304, 318–332
examples, 320–332
problems, 418–420
pendulum, 17, 40–42
spring–mass system, 9, 16, 37,
55–56
torsional system, 48
torsional vibration, 532t
wheel, 34–35
n-degree-of-freedom system, 341f
Neutral plane/surface, 549
Newton’s laws, 9–10, 32
Nodes, in finite element analysis,
618, 619
Nodes of a mode, 351, 515
Nonlinear response properties
general forced response, 279–287
examples, 280–287
harmonic excitation, 188–197
examples, 189–197
problems, 213–214
Nonlinear systems, 7
Coulomb friction, 81–88
general forced response
problems, 299–301
nonlinear pendulum equations,
89–95
Nonperiodic forces, 217–218
Normalization of eigenvectors,
324–326
Nose cannon, 498f
Numerical simulation
general forced response, 267–279
examples, 269–279
problems, 298–299
harmonic excitation, 180–187
examples, 181–187
problems, 211–213
MDOF systems, 407–415
examples, 408–415
problems, 431–433
vibration and free response, 72–81
examples, 74–81
problems, 114–115
Numerical solutions
concept of, 72–73
Euler method examples, 73–77
sources of error, 75
Nyquist circles, 591
Nyquist frequency, 582
Nyquist plots, 591, 594f
O
Operational deflection shape (ODS)
measurement, 609–611
Optical table with vibration
absorber,456f
Optimization in vibration
suppression
examples, 474–479
problems, 498–500
vibration suppression, 471–479
Orthogonality, 236, 322, 325, 521
Orthogonal matrices, 326
Orthonormal vectors, 322, 326, 326w
Oscillation
decay in, 21
frequency of, for measuring mass
and stiffness, 61–63
natural frequency examples,
34–35, 38–39, 48
Oscillatory motion, 10
Overdamped motion, 26–27, 27f
Overshoot, 230
P
Package, vibration model of
dropped, 289f
Parts sorting machine, 496f
Peak amplitude method, 590f
Peak frequency, 143
Peak time, 230
Peak value, 19
Pendulum
compound, 39–42, 40f, 41f
damped, 202f
double, with generalized
coordinates, 369f
equilibrium positions, 89f
examples, 2–4, 35–36, 39–42
inverted, 70–71, 266
nonlinear systems, 89–94
problems, 115
swinging, 8w
Performance robustness, 463
Periodic forces, 217
Period T, 15, 124
Personal computer disk drive
motor, 452f
Phase, 8
Physical constants for common
materials, 47t
Piezoelectric accelerometers, 167f,
169, 578
Pinned beam, 535
Pitch, 341f
Pivot point, 42
Plate vibration, 544–550
Positive definite matrix, 390w
Positive semidefinite matrix, 390w
Potential energy, 2
Power-line pole with
transformer, 296f
Power spectral density (PSD),
249–251, 585
Printed circuit board, 495f
Probability density function, 253
Proportional damping, 362
Pulse input function, 281f
Punch press
base excitation example, 159
machine schematic, 386f
MDOF system example, 385–389
three-element-bar problem, 651f
vibration model, 386f
Q
Quadratic damping, 179
Quadrature peak picking
method, 589f
R
Radial saw, 461f
Radius of deflection, 488f
Radius of gyration, 40
Random input, general forced
response, 247–255
examples, 252–255
problems, 295
Random signal analysis, 584–587
Random vibration analysis, 585w
Rattle space, 448
Rayleigh dissipation function, 107
Receptance matrix, 596–597
Receptance transfer function, 592f
Rectilinear system, 46t
Reduced-order modeling. See
Model reduction
Resonance
damped systems, 138, 143
distributed systems, 503
explanation, 117–118
importance of concept, 121
MDOF systems, 366–367
modal testing, 575
undamped systems, 125, 125fIndex 705
Response
divergent, 69, 69f
free, 5, 10, 72–81, 84, 85f, 86f
steady-state, 133, 137–138
transient, 133, 137–138
See also Forced response modal
analysis; General forced
response
Response spectrum. See Shock
spectrum
Rigid-body modes, 352–355, 380
example, 352–355
Robustness, of designs, 68, 463
Rod vibration, 519–525
examples, 520–524
problems, 569–570
Roll, 341f
Rolling disk vibration model, 108f
Root-mean-square value, 20, 249, 437
Rotating disk critical speeds,
485–491, 485f
Rotating unbalance
equation, 486w
harmonic excitation, 160–165
examples, 163–165
problems, 208–210
model of disk–shaft system, 485f
model of machine, 160f, 209f, 492f
model of machine in building,
560f, 571f
model of motor, 209f
Rotational kinetic energy, 34
Rotational system, 46t
Rotorcraft
and resonance, 117
rotating unbalance example,
164–165
thrust directions, 164f
Round-off error, 75
Runge–Kutta method, 75–76
examples, 77–80
general forced response, 272–278
linear and nonlinear equations, 92
Mathematica, 79
MDOF systems, 410–412
single-degree-of-freedom
system, 72
S
Saddle point, 472, 473f
Sample function, 247–255
Sampling theorem, 582
Scalar product, 307
Scanning laser doppler vibrometer
(SlDV), 579
Second mode shape, 314
Seismic accelerometer, 166f
Self-excited vibrations, 69
Semidefinite systems, 380
Separation of variables, 508
solutions method, 518w
Settling time, 230
Shaft and disk. See Disk–shaft
system
Shaker, 260
Shakers, 575–576
Shannon’s sampling theorem, 582
Shape functions, 621
Shear coefficient, 541
Shear modulus, 541f
Ship, fluid system example, 53–54
Shock, 255, 441–442, 442f
Shock loading, 217
Shock pulse, 446
Shock spectrum, general forced
response, 255–259
examples, 256–259
problems, 295–296
Signal conditioners, 578
Signal processing. See Digital signal
processing
Signals, representations of, 581f
Signum function, 83
Simple harmonic motion, 10, 13w
Simple harmonic oscillator, 10
Simple machine part, vibration
model of, 371f
Simple sine function, 248f
Simply supported beam, 535
Sine function, 1
Single-degree-of-freedom curve
fit, 588
Single-degree-of-freedom systems,
5, 219w
compliance frequency response
function, 589f
damped, 23f
example, 8w
external force, 119f
response, 219w
undamped, 10
Sinusoidal vibration, acceptable
limits, 438f
Sliding boundary, 535
Sloshing, 39
Software
eigenvalues, 392
numerical simulation, 72, 180
Solid damping, 176
Specific damping capacity, 174
Spectral matrix, 328
Spring–mass–damper system
deterministic and random
excitations, 254w, 255
excitation response, 586w
general applied force, 280f
magnitude plot, 261f
potentially nonlinear
elements, 189f
square input, 272f
total time response, 242f
truck suspension system
example, 232–233
Spring–mass system, 8w
examples, 11, 37–38, 43
gravitational field, 33f
harmonic excitation examples,
122–130, 141–142
helical spring natural frequency, 66
kinetic coefficient of friction, 82f
natural frequency example, 55–57
nonnegligible mass, 37f
problems, 95–98
response of, 9f
vibration, 5–13
vibration absorbers, 455
Springs
coiled, 52–53, 53f
constants, 53t, 56
helical
natural frequency, 66
stiffness, 51
leaf, 49–50, 49f
manufacture of, 56
static deflection, 6f, 57, 67
stiffness calculation rules, 55f, 56
Stability
asymptotic, 69, 265, 266
bIbO, 263, 265, 267
general forced response, 262–267
examples, 266–267
problems, 298
local, 90
marginal, 68
vibration, 68–71, 263w
examples, 70–71
problems, 112–113
response, 69f706 Index
State matrix, 77, 401
State variables, 77
State vector, 77, 401
Static coupling, 375
Static deflection of spring, 6f, 57, 67
Stationary signals, 248f
Steady-state response, 133, 137–138
Steam-pipe system with
absorber, 495f
Steel, elastic modulus, 59f
Step function, 228, 229f
Stereo turntable
frequency response function, 441f
single-degree-of-freedom
model, 440f
vibration response example,
440–441
Stiffness, 46–57
calculation rules for parallel and
series springs, 55f, 56
coiled spring, 49–50, 50f
definition, 6
examples, 48–57
frequency of oscillation for
measuring, 62–63
helical spring, 48
problems, 108–110
twist, 48
Stiffness matrix, 308
Stinger, 576
Strain gauges, 578
Strain rate damping, 554
Stress–strain curve for elastic
modulus, 59f
String equation, 506
String vibration, 504–507, 504f
Structural damage
acceptable vibration levels, 438f
vibration measurements, 609
Structural damping, 176
Structural Health Monitoring
(SHM), 574
Subway car coupling device, 417f
Superposition, 95, 118, 217, 636
Support motion. See base excitation
Suspension systems
arbitrary input response, 232
base excitation, 159
chassis dynamometer, 289f
damped, 108f
design of, 42
examples, 11, 48–49, 67–68, 158
harmonic excitation, 150
mass of occupants, 207f
model of, 289f
multiple-degree-of-freedom
systems, 340
speed bump, 291f
torsional, 149–150
torsion rod, 100f
trifilar, 58f
two-degree-of-freedom
model, 417f
vertical, 67–68
Symmetric eigenvalue problem,
321–322, 320w, 332, 401
Symmetric matrix, 308
Synchronous whirl, 487
T
Tangent line method, 73
Telephone lines, 456
Tennis racket, 42
Tensile test, 59
Thin plate theory, 550
3-db down point, 589
Three-element bar, finite element
method for, 625–630
examples, 627–630
problems, 652–653
Time history of impulse force, 218f
Time response
lumped- versus distributedparameter systems, 524
MDOF systems, 407–415
vibration and free response, 72–81
examples, 74–81
Time to peak, 230
Timoshenko beam model, 543
Timoshenko beams, 540–544, 541f
Timoshenko shear coefficient, 541
Tires, and resonance, 117
Torsional constant, 527–528
Torsional motion, 46
Torsional system
natural frequency, 48–49
two degrees of freedom, 416f
Torsional vibration
boundary conditions, 528t
distributed-parameter systems,
525–532
examples, 528–532
problems, 569–570
shaft, 47f
transform approach, 149–150
Transducers, 166, 575, 578
Transfer functions, 149, 260t
general forced response, 260–262
problems, 296–297
Transformations, 332
Transform methods
general forced response, 242–247
examples, 243–247
problems, 294–295
harmonic excitation, 148–151
problems, 204–205
Transient response, 133, 137
Transmissibility
base excitation example, 157–158
displacement, 153, 154f, 155, 157f
force, 155, 157f
formulas, 443w
Transmissibility ratio, 442–455, 446f
Transmission lines, 456
Transpose of matrix, 308
Transverse motion, 46
Transverse vibrations, 532
Truck
hitting object, 571f
loading dirt, 232f
pipe stacking, 105f
spring–mass–damper system
example, 232–233
Trusses, 641–646
problems, 654–655
three-element, 655f
Turntable. See Stereo turntable
Twist, 48
Two-degree-of-freedom model
damped, 428f
MDOF systems, 304–318,
305f, 328f
examples, 307–318
problems, 415–418
rigid-body translation, 352f
vehicle example, 382–385
viscous damping, 361f
Two-member framed structure, 642f
U
Undamped motion, 68–69
Undamped systems
harmonic excitation, 118–130
examples, 122–130
problems, 197–201
lagrange stability, 265
single-degree-of-freedom, 10Index 707
two-degree-of-freedom, 304–318
examples, 307–318
Underdamped motion, 24–26
Underdamped solution, 25w
Underdamped systems
forced response, 140w, 364w
response, 26f, 60f
vibration example, 30–31
Unrestrained degree of
freedom, 352
U-tube manometer, 38f
V
Valve and rocker arm
system, 102f
Variance, 249
Vector equation, 307
Vehicles
side section, 382f
two-degree-of-freedom system
example, 382–385
See also Automobiles
Velocity, simple harmonic motion,
13, 14w
Velocity-squared damping, 179,
193–194, 284–285
Vibration, 1–116
acceptable levels, 436–442
examples, 438–442
problems, 491
consequences of, 436
Coulomb friction, 81–95
description, 1–2
design considerations, 63–68
displacement amplitude, 437t
energy methods, 31–46
explanation, 2
frequency ranges, 437t
harmonic motion, 13–21
measurement, 58–63
modeling methods, 31–46
nonlinear, 81
nonlinear pendulum equations,
89–94
numerical simulation of time
response, 72–81
performance standards, 436
shock versus, 442
spring–mass model, 5–13
stability, 68–71
stiffness, 46–57
viscous damping, 21–31
Vibration absorbers, 455–463
damping, 463f
examples, 461–462
problems, 495–496
viscous, 469f
Vibration absorption damping,
463–470
problems, 496–498
Vibration dampers, 463
Vibration isolation, 442–455
examples, 446–455
optimization, 476–478
problems, 491–494
transmissibility formulas, 443w
Vibration model
airplane wing, 50f, 373f
coupled levers, 372f
punch press, 386f
simple machine part, 371f
Vibration suppression, 435–501
optimization, 471–479
rotating disk critical speeds,
485–491
vibration absorbers, 455–463
vibration absorption damping,
463–470
vibration isolation, 442–455
Vibration testing, 573–616
circle fitting, 591–595
digital signal processing, 579–584
endurance and diagnostics,
606–609, 607f
measurement hardware, 575–579,
575f, 600f
modal data extraction, 588–591
mode shape measurement,
596–606
operational deflection shape
(ODS) measurement,
609–611
random signal analysis, 584–587
uses of, 574
Virtual displacements, 370
Virtual work, 370
Viscoelastic, definition of, 480
Viscous damping, 21–31
critically damped motion,
27–31
equivalent, 285f
examples, 27–31
MDOF systems, 356–362, 376
examples, 357–362, 376–377
problems, 426–428
overdamped motion, 26–27
problems, 101–103
two-degree-of-freedom
system, 361f
underdamped motion, 24–26
vibration, 21–31
See also Damped systems;
Undamped systems;
Underdamped systems
Viscous vibration absorbers,
469f, 470f
W
Washing machine, 492f
Wave hitting seawall, 292f
Wave speed, 506
Whirling, 485, 487
Window function, 583, 584f
Wing. See under Aircraft
Y
Yaw, 341f
Young’s modulus. See Elastic
modulus
Z
Zero mode, 380

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