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 |  موضوع: كتاب Mechanical Vibrations - Modeling and Measurement السبت 27 فبراير 2021, 2:03 am | |
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أخوانى فى الله
أحضرت لكم كتاب
Mechanical Vibrations - Modeling and Measurement
Second Edition
Tony L. Schmitz , K. Scott Smith
و المحتوى كما يلي :
Contents
1 Introduction 1
1.1 Mechanical Vibrations . 1
1.2 Types of Vibrations . 2
1.2.1 Free Vibration 2
1.2.2 Forced Vibration 3
1.2.3 Self-Excited Vibration 4
1.3 Damping 6
1.4 Modeling 6
1.5 Periodic Motion . 10
Exercises . 25
References 28
2 Single Degree of Freedom Free Vibration 29
2.1 Equation of Motion . 29
2.2 Energy-Based Approach . 40
2.3 Additional Information 46
2.3.1 Equivalent Springs . 46
2.3.2 Torsional Systems . 48
2.3.3 Nonlinear Springs . 49
2.4 Damped Harmonic Oscillator 51
2.4.1 Viscous Damping 51
2.4.2 Coulomb Damping . 52
2.4.3 Solid Damping 52
2.4.4 Damped System Behavior 52
2.4.5 Underdamped System . 54
2.4.6 Damping Estimate from Free Vibration Response . 65
2.4.7 Damping Estimate Uncertainty 68
2.5 Unstable Behavior 70
2.5.1 Flutter Instability 71
2.5.2 Divergent Instability 75
ix2.6 Free Vibration Measurement 81
Exercises . 83
References 87
3 Single Degree of Freedom Forced Vibration 89
3.1 Equation of Motion . 89
3.2 Frequency Response Function . 90
3.3 Evaluating the Frequency Response Function 95
3.4 Defining a Model from a Frequency Response Function
Measurement . 111
3.5 Rotating Unbalance . 115
3.6 Base Motion 119
3.7 Impulse Response 124
Exercises . 128
References 132
4 Two Degree of Freedom Free Vibration . 133
4.1 Equations of Motion 133
4.2 Eigensolution for the Equations of Motion 135
4.3 Time-Domain Solution 145
4.4 Modal Analysis 152
Exercises . 166
References 172
5 Two Degree of Freedom Forced Vibration . 173
5.1 Equations of Motion 173
5.2 Complex Matrix Inversion 175
5.3 Modal Analysis 182
5.4 Dynamic Absorber . 191
Exercises . 199
6 Model Development by Modal Analysis 205
6.1 The Backward Problem 205
6.2 Peak Picking 205
6.2.1 Single Degree of Freedom . 205
6.2.2 Two Degrees of Freedom 208
6.3 Building the Model . 209
6.4 Peak Picking for Multiple Degrees of Freedom . 220
6.5 Mode Shape Measurement 223
6.6 Shortcut Method for Determining Mass, Stiffness,
and Damping Matrices 229
6.6.1 Linearized Pendulum . 233
6.6.2 Automobile Suspension Model 237
Exercises . 242
References 253
x Contents7 Measurement Techniques 255
7.1 Frequency Response Function Measurement . 255
7.2 Force Input . 256
7.3 Vibration Measurement 258
7.3.1 Capacitance Probe . 258
7.3.2 Laser Vibrometer 259
7.3.3 Accelerometer 260
7.4 Impact Testing 265
7.5 Modal Truncation 274
Exercises . 280
Reference . 282
8 Continuous Beam Modeling 283
8.1 Beam Bending 283
8.2 Transverse Vibration Equation of Motion . 288
8.3 Frequency Response Function for Transverse Vibration . 289
8.3.1 Fixed-Free Beam 290
8.3.2 Free-Free Beam . 295
8.4 Solid Damping in Beam Models . 299
8.5 Rotation Frequency Response Functions 305
8.6 Transverse Vibration FRF Measurement Comparisons 308
8.6.1 Fixed-Free Beam 308
8.6.2 Free-Free Beam . 310
8.6.3 Natural Frequency Uncertainty 312
8.7 Torsion Vibration 313
8.8 Axial Vibration 315
8.9 Timoshenko Beam Model 319
Exercises . 320
References 324
9 Finite Element Introduction 325
9.1 Introduction 325
9.2 Axial Element . 326
9.3 Transverse Element . 349
Exercises . 363
References 365
10 Receptance Coupling 367
10.1 Introduction 367
10.2 Two Component Rigid Coupling . 367
10.3 Two Component Flexible Coupling . 372
10.4 Two Component Flexible-Damped Coupling 380
10.5 Comparison of Assembly Modeling Techniques 381
10.5.1 Modal Analysis . 383
10.5.2 Complex Matrix Inversion . 385
10.5.3 Receptance Coupling . 386
Contents xi10.6 Advanced Receptance Coupling 390
10.7 Assembly Receptance Prediction . 396
10.7.1 Free-Free Beam Coupled to Rigid Support 397
10.7.2 Free-Free Beam Coupled to Fixed-Free Beam . 403
10.7.3 Comparison Between Model and BEP Measurement . 408
Exercises . 412
References 414
Appendix A: Beam Experimental Platform 415
Appendix B: Orthogonality of Eigenvectors 417
Index
Index
A
Acceleration, 18
Acceleration vector, 135
Accelerometer, 81, 281
coefficient, 264
equation, 264
equation of motion, 262
free body diagram, 262
piezoelectric material, 260, 261
seismic mass, 260, 261
spring-mass-damper system, 261, 262
vibration measurement, 260
Acousto-optic modulator (AOM), 260
Aircraft wing, 72, 78
Amplifiers, 255
Analog, 255, 259, 271
Analog-to-digital converter (ADC), 255
Anti-aliasing filter, 271
Anti-resonant frequencies, 302
Approximation, 28
Arbitrary argument, 35
Archimedes, 153
Argand “circle” quadrants, 105
Argand diagram
acceleration, 19, 21
damped free vibration response, 71
derivation, 17
position, 18, 19, 21
real/imaginary axis, 20, 22, 24, 25
rotating vector, 17
sine function, 12, 13
undamped free vibration response, 70
velocity, 18, 19, 21
vibration, 23
Assembly displacement-to-force tip receptance,
404
Assembly modeling techniques
complex matrix inversion, 385
flexible coupling, spring-mass-damper
systems, 383
Laplace variable, 382
modal analysis, 383, 384
receptance coupling, 386
Assembly receptance prediction
free-free beam coupled
fixed-free beam, 403, 404, 406, 408
rigid support, 397–400
model vs. BEP measurement, 408, 409
receptance coupling result vs. fixed-free
response, 399, 400
semi-logarithmic plot, 399
Asymptotic stability, 70, 80
Automobile response, 123
Automobile suspension model, 237, 238
Axial beam element
axial displacement, 330
cross-sectional area, 326
displacement and force, 327, 328
dynamic matrix, 337
eigenvalue problem, 335
equation of motion, 337
Euler-Lagrange equation, 331
fixed-free boundary conditions, 334, 342
free-free boundary conditions, 338, 346
frequency response functions, 336
generic case, 327
individual mass matrices, 333
individual stiffness matrices, 333Axial beam element (cont.)
kinetic energy, 331
length, 326
mass matrix, 329, 332
MATLAB MOJO 9.1, 338
MATLAB MOJO 9.2, 341, 342
MATLAB MOJO 9.3, 345
natural frequencies, 336, 339, 342, 346
off-diagonal, 335
stiffness matrix, 327, 329
symmetric mass matrix, 332
time-dependent axial displacements, 332
two-element beam model, displacements,
333–335
Axial vibration, 315–317
B
Backward problem, 205, 213, 220
Bandwidth, 255
Base motion, 262
automobile suspension, 122
displacement transmissibility, 121
elastic supports, 119
equation and notation, 120
forcing frequency, 122
frequency ratio, 122
harmonic, 120
imaginary part, 121
MATLAB MOJO 3.4, 123
phase lag, 121
road excitation, 122
spring-mass-damper system, 120, 121
Beam bending
boundary conditions, 285–287
continuous beam transverse deflection, 284
deflection equation, 285
Euler-Bernoulli beam theory, 284, 286
force per unit length, 284, 285
integration, 284, 285
lumped parameter models, 283
second moment of area, 284
Beam experimental platform (BEP), 81, 82, 223
impact testing, 272–274
Beam model, 356, 358
Bending modes, 223, 224
Bending stiffness, 37
Boundary conditions, 139, 285–287, 291, 292,
295, 296, 298, 301, 302, 306, 309,
314–316, 320
C
Cantilever beam, 10, 137
Capacitance probe, 258, 259
Capacitive sensors, 258
Careless notation, 36
Centrifuge model, 128
Chain-type model, 133
Chain-type model format, 216
Chain-type spring-mass-damper model, 229
Characteristic equation, 33, 34, 136, 151, 161
Compatibility condition, 368, 370–373,
375, 377, 379, 380, 391,
393, 394
Complex coefficient, 35
Complex conjugates, 34, 57, 148
Complex matrix inversion, 175, 385
compact form, 175
cross FRF, 177
diagonals, 176
direct FRF, 176
external force, 176
freedom system, 175
FRF measurements, 178
magnitudes, 179
reciprocity, 177
Complex matrix inversion approach, 200
Complex modulus, 299, 301, 319
Complex plane representation, 35
Complex stiffness matrix, 408
Compliance, 256, 276
Concave cylindrical surface, 40
Consistent mass matrix, 356
Continuous beam transverse
deflection, 284
Continuous cross-section model, 325
Continuous models, 283
Convolution integral, 127
Coulomb damping, 52, 299
Counter-clockwise rotating vector, 35
Cramer’s rule, 292, 293, 307
Critical damping, 53
Cross FRF, 180, 245, 247–249
measurements, 211, 212
transformation, 211
Cross FRFs, 177, 178
Cutting tool-holder-spindle-machine
structure, 221
Cylinder height function, 42
Cylinder rolling, 84
Cylinder’s rotational velocity, 42
422 IndexD
Damped free vibration response, 55
Damped harmonic oscillator
Coulomb damping, 52
damped system behavior, 52–54
estimation, free vibration response, 65–68
solid damping, 52
uncertainty, free vibration response, 68–70
underdamped system, 54–58, 60
viscous damping, 51
Damped natural frequency, 55, 160
Damped period of vibration, 66
Damped system behavior, 52–54
Damping, 1, 6, 24
Damping matrix, 219, 229
Damping model, 30
Damping ratio
damped natural frequency, 56
definition, 54
equation of motion, 54
expression, 55
mechanical systems, 55
possibilities, 54
uncertainty, 68, 69
Deflection, 47
Degrees of freedom, 6
particle, 9
rigid body, 9
spring-mass-damper system, 10, 83–85
undamped, 27
vibratory motion, 25
Diagonal matrices, 156
Diagonal modal mass, 163
Diagonalization, 155
Diameter probe tip, 86
Differential equation, 32
Differential equation of harmonic motion, 20,
32, 43
Digital data acquisition, 280
Direct FRF, 177, 179, 247, 249, 250
measurement, 208, 212
Disk’s mass moment, 49
Displacement transmissibility, 121
Displacement vector, 135
Displacement-to-couple receptance, 390
Displacement-to-force receptance, 396, 397,
403, 407, 413
Divergent instability, 71, 75, 77–80
Duffing spring, 49, 50, 83
Dynamic absorber, 194
absorber response, 196
damping, 194
freedom system, 193, 196
frequency, 196
FRF, 192
magnitude plots, 192
spring and mass, 194
Dynamic flexibility, 102
Dynamic matrix, 337, 358
Dynamic signal analyzer, 255
E
Earth’s gravitational acceleration value, 37
Eigensolution, 135, 154, 166
Eigenvalue problem, 135, 335
Eigenvalues, 135–137
Eigenvectors, 135, 219
coordinates, 155
equation, 143
expression, 141
Laplace-domain representation, 154
modes, 144
normalization, 141
ratios, 141, 143, 144
relationships, 146, 148
relative magnitude, 141, 155
type, 141
Elastic modulus, 48, 52
Energy-based approach
concave cylindrical surface, 40, 41
differential equation, 44
equation of motion, 43
free body diagram analysis, 45
gravitational potential energy, 45
kinetic energy equation, 42
natural frequency, 43
oscillating systems, 40
potential energy, 40, 42, 44
rotational velocity, 41
spring-mass system, 40, 44, 45
time derivative, 43
translational velocity, 41
Energy dissipation, 24
Equilibrium condition, 368–370, 372–376,
378, 394
Equivalent springs, 46, 47
Euler integration, 72, 73, 78, 265–268, 281,
282
Euler’s formula, 22, 23, 32, 58
Euler-Bernoulli beam equation, 319
Euler-Bernoulli beam model, 324
Euler-Bernoulli beam receptances, 397
Euler-Bernoulli beam theory, 284, 286, 319,
320, 325, 349, 352
Euler-Lagrange equation, 331, 356
Even function, 22
Excitation frequency, 114
Exponential notation, 22, 23
Index 423F
Finite element analysis
axial element (see Axial beam element)
continuous cross-section model, 325
Euler-Bernoulli beam theory, 325
transverse beam (see Transverse beam
element)
Finite stiffness, 9
Fixed-free beam, 246, 290–294, 301, 308, 310
Fixed-free beam mode shape calculation, 227
Fixed-free beam model, 321
Fixed-free boundary conditions, 341, 342
Fixed-free displacement-to-force tip receptance,
413
Flexible coupling
bolted connection, 372, 373
compatibility condition, 372, 373, 375, 377,
379
component displacements, 375, 377–379
component receptances, 374
components, 372–374, 376, 377, 379
cross receptances, 374, 376, 378, 380
direct receptances, 376, 378, 380
equilibrium condition, 373–376, 378, 379
viscously damping, 381
Flexible-damped coupling
compatibility condition, 380
compatibility equation, 381
component displacements, 380
equilibrium condition, 380
receptance coupling equations, 381, 382
velocity-dependent damping forces, 381
Flutter, 4, 5
Flutter instability, 71–74
Force balance, 230–232
Force input
impact hammer, 257
impulse, 256
random signal, 256
shaker, 256, 257
sine sweep test, 256
Force vectors, 282
Forced harmonic vibration, 128–130
Forced vibration
excitation, 4
frequency domain, 3
frequency-dependent nature, 4
periodic excitation, 3
rotating unbalance, 3
Forcing frequency, 3
Fourier coefficients, 12
Fourier series, 12, 14, 15, 25, 26
Free body diagram
force balance, 37, 45
force components, 77
inertial torque, 48
inverted pendulum, 76
lower mass, 134
rotating unbalance, 115, 116
spring constant, 46
spring-mass-damper system, 31, 37, 40, 44
three degree of freedom system, 150
torsional system, 49
Free vibration, 29, 85
damping, 4
equilibrium position, 2
initial conditions, 2
long-term external force, 5
motion, 2, 3
periodic response, 2
Free vibration magnitude, 39
Free vibration response
critically damped, 64, 65
damped, 58, 59
damping ratio, 86
initial displacement, 38, 86
overdamped, 60–63
spring-mass-damper system, 87
undamped, 69
underdamped, 63, 64
Freedom chain-type, 174, 213
Freedom damped system, 195
Freedom spring-mass-damper system, 199,
201, 244
Freedom spring-mass system, 200
Free-free beam, 295, 298, 310
Free-free boundary conditions, 332, 338, 339,
345, 346, 359
Free-free cylindrical beam, 323
Free-free receptance, 398, 407
Free-sliding beam model, 320–323
Frequency-dependent magnitude, 115
Frequency-domain displacement, 110
Frequency response functions (FRFs), 133, 175
ADC, 255
amplifiers, 255
bandwidth, 255
boundary conditions, 290
complex conjugates, 93
dynamic signal analyzer, 255
fixed-free beam, 290–294
force input, 256, 258
free-free beam, 295, 298
frequency domain, 90
frequency ratio, 92
Hooke’s law, 92
Laplace domain, 91
magnitude and phase, 93, 94, 129
measurement setup, 255, 256
physical systems, 90, 91
424 Indexposition-dependent vibration behavior, 289
real/imaginary parts, 93
resonance, 93
steady-state, 90, 91
time-dependence, 290
transfer function, 91
transient, 90
velocity, 90
vibration measurement
accelerometers, 260–262, 264, 265
capacitance probe, 258, 259
laser vibrometer, 259, 260
zero frequency, 92
FRF evaluation
approximation, 100, 107
Argand diagram, 105–107
associated natural frequency, 109
displacement lags, 103, 104
displacement magnitude, 111, 112
dynamic/static flexibility, 102
frequency ratio, 104
imaginary part vs. r plot, 98, 99
imaginary parts, 101, 103, 106, 110, 111
magnitude peak sharpness, 95
magnitude plot, 102
magnitude vs. r plot, 95
MATLAB MOJO 3.1, 96
MATLAB MOJO 3.2, 98
MATLAB MOJO 3.3, 108
peak-to-peak value, 100
phase plot, 102, 103
phase vs. r plot, 96
quadratic equation, 107, 108
real part vs. r plot, 98, 99
real parts, 101, 103, 105, 106, 110
single degree of freedom system, 109
spring-mass-damper system, 100, 101
time-domain representations, 97
viscous damping ratio, 100
zero frequency response, 100
FRF measurement, 206, 208
imaginary parts, 113
real parts, 112
spring-mass-damper model, 112–114
test structure, 112, 113
vibration magnitude, 114
G
Gaussian distribution, 312
Generalized displacement/rotation, 392
Generalized force/couple, 393
Generalized receptance matrix, 392
Gibbs’ phenomenon, 14
Gravitational potential energy, 43
Gravity force, 75, 78
H
Hammer motion, 38
Harmonic forced vibration, 129
Harmonic forcing function, 200
Harmonic free vibration, 170
Harmonic input force, 89, 90
Harmonic motion, 32
amplitude, 27
exponential function, 26
exponential notation, 23
sine function, 11
Harmonic torsion vibration, 313
Harmonic vibration
derivatives, 39
displacement and acceleration, 151
equation of motion, 38
identical approach, 39
Laplace-domain, 136
Hertz (Hz), 12, 18
Hooke’s law, 29, 30, 92
Hydraulic shakers, 256
Hyperbolic cosine, 290
Hyperbolic sine, 290
I
Identical geometry, 9
Imaginary axis, 20
Impact hammers, 257, 273, 280
Impact testing
aliasing
frequency-domain, 272
time-domain, 271
anti-aliasing filter, 271
BEP, 273, 274
current acceleration value, 266
displacement, 266
Euler integration, 267
force impacts, 272, 273
force profile, 267
frequency-domain force magnitudes, 272,
273
FRF, 265, 280
impact hammer, 257
instrumented hammer, 265
MATLAB MOJO 7.1, 267, 270
modal truncation, 274, 276
Nyquist frequency, 271
Nyquist-Shannon sampling theorem, 271
sampling frequency, 271
single degree, freedom spring-mass-damper
system, 265, 266
Index 425Impact testing (cont.)
time-domain response, 268, 270
time-domain simulation, 265
velocity, 266
Impulse response function, 125, 126
Impulsive force profile, 281
Impulsive forces, 127, 130
Inertance, 256
Initial conditions, 34
Initial displacements, 164, 165, 170
Initial velocities, 165
Instrumented hammer, 280
Interpolation/shape function, 351
Inverted pendulum, 75, 76
K
Kinetic energy, 40, 331, 354, 355
Kinetic energy equation, 42
Kinetic/potential energy derivatives, 43
L
Laplace domain, 91
Laplace notion, 143
Laplace variable, 32
Laser vibrometer, 259, 260
Level of confidence, 69
Linear algebra, 61
Linear differential equations, 32, 50
Linear variable differential transformer
(LVDT), 69
Linearized pendulum, 233, 234
Linearly dependent, 140
Logarithmic decrement, 66, 86
Lumped parameter model, 283
damping, 31
degrees of freedom, 51
free vibration, 29
kinetic energy, 44
linear spring, 29
physical masses, 29, 83
spring extension, 45
spring-mass-damper model, 30
torsional systems, 48
M
Machinist’s scale, 301
Maclaurin series, 22
Marginally stable, 70, 79
Mass matrix, 135, 246, 331, 354, 355, 363, 365
Mass moment of inertia, 42, 48, 76, 233
Material-dependent damping, 299
Matrix inversion, 60–62
Matrix multiplication, 157, 160, 163
Mechanical vibrations
categories, 2, 7, 24
external force/perturbation, 1
forced vibration, 3, 4
free vibration, 2, 3
self-excited vibration, 4–6
Mobility, 256, 260, 265, 280
Modal analysis, 151, 175, 383, 384
chain-type system, 153
coordinates, 152, 155
damping, 188
damping matrix, 158
damping ratio, 156
determinant, 154
eigenvalues, 154
eigenvalues and eigenvectors, 188
force vector, 189
freedom system, 183
frequency-domain representations, 184
mass matrix, 157
matrix, 153
modal matrix, 184
non-zero forcing frequency, 191
off-diagonal terms, 152
procedure, 153
proportional damping, 153
spring-mass-damper system, 182
steps, 160, 161, 163, 187
stiffness values, 191
vibration, 156
zero frequency, 190
Modal coordinates, 152, 159, 163
Modal damping matrix, 158
Modal damping ratio, 156, 158
Modal displacements, 160
Modal fitting parameters, 221
Modal matrix, 153, 159, 162, 163, 210, 215
Modal parameters, 209
Modal stiffness matrix, 157
Modal truncation
FRF, 274–277
modal fitting, 276, 277
Mode shape measurement
BEP, 226
damping matrix, 233
direct and cross FRFs, 227
eigenvector identification approach, 226
force balance, 233
off-diagonal term, 232
on-diagonal term, 230
rigid body modes, 224
steel rod, 225
stiffness matrix, 231
vibration frequency, 224
426 IndexMode shapes, 135, 139
boundary conditions, 139
cantilever beam, 137–139
components, 139
eigenvalues, 144
motion, 137, 138
natural frequency, 135, 142
normalize, 138, 139
system vibration, 142
Modeling
degrees of freedom, 6, 10
efforts, 9
elastic body, 9
ruler, 9
three-dimensional space, 6
vibratory system, 25
Motion equations
forcing frequencies, 173
freedom forced vibration analysis, 174
harmonic force, 173
N
Natural frequency, 3
Argand diagram, 33
concave surface, 44
damped, 55, 56, 69
eigenvalues, 163
equation of motion, 49
free vibration, 29
period of vibration, 39
stiff springs, 37
stiffness units, 33
system’s dynamic model, 73
undamped, 33, 55, 56
Natural frequency uncertainty, 312, 313
Nodes, 224, 242
Noncontact capacitive sensors, 258
Non-dimensionalized magnitude, 117
Nonlinear springs, 49, 50
Nyquist frequency, 271
Nyquist-Shannon sampling theorem, 271
O
Odd function, 14
Off-diagonal elements, 329
Off-diagonal term, 230, 239
On-diagonal elements, 329
P
Parallel springs, 46
Peak picking, 205, 209, 242
freedom model, 206, 207
freedom system, 207
minimum peak, 209
natural frequency, 208
parameters, 210
Peak picking approach, 111, 130
Period of vibration, 58
Periodic forcing function, 4
Periodic motion
acceleration, 18
Argand diagram, 12
circular frequency, 12
discontinuous function, 14
Euler’s formula, 23
exponential notation, 23
Fourier series, 12
frequency variable, 18
harmonic, 10, 11, 23
MATLAB MOJO 1.1, 12
odd function, 14
signals, 12
sine function, 10
sine functions, 14
square wave, 14
velocity, 18
vibrating frequency, 12, 25
Perpendicularity, 161
Phase calculation, 58
Phase values, 18
Piezoelectric, 260–262
Pink noise, 256
Poisson effects, 315
Poisson’s ratio, 48
Polar moment of inertia, 48
Potential energy, 40
Potential model, 229
Pounds-force, 37
Probe displacement, 86
Probe free vibration, 87
Proportional damping, 153, 161, 166, 170, 171,
200, 201, 214
Proportional damping relationship, 158
Q
Quadratic equation, 53, 136, 143
Quality factor Q, 95
R
Real axis, 20
Receptance coupling
assembly modeling techniques
(see Assembly modeling techniques)
assembly receptances, 390, 391, 393, 395
bending moments, 390
Index 427Receptance coupling (cont.)
component receptances, 394
cross receptances, 394, 396
direct receptances, 396
displacement-to-force analyses, 396
equilibrium condition, 394
joining spring-mass-damper systems, 385
MATLAB MOJO 10.1, 388
modal analysis vs. complex matrix
inversion, 387
rotations, 390
solid cylinder-prismatic cantilever beam
assembly, 390, 391
stiffness matrix, 396
Regeneration of waviness, 6
Resonance, 3, 93
Rigid body mode
rotational, 224
translational, 223
Rigid body modes, 223, 224, 310, 316
Rigid body motion, 52
Rigid coupling
assembly response, 370
compatibility condition, 368, 370, 371
component receptances, 369, 371
components, 367, 368, 370, 371
cross receptances, 371, 372
direct receptances, 371
equilibrium condition, 368–370
free-free cylinder, 397
frequency-domain displacements, 370
substructures, 367
Rolling cylinder problem, 49
Rotating unbalance, 3
derivatives, 115
frequency-domain vibration response, 118
magnitude, 118
mass distribution, 115
Rotation frequency response functions
boundary conditions, 306
Cramer’s rule, 307
Euler-Bernoulli beam FRFs
fixed-free, 308, 309
free-free boundary conditions, 308, 309
fixed-free beam, 305, 308
fixed-free beam, harmonic bending couple,
307
rotation vibration, 305, 306
transverse deflection, 305, 306
Rotation-to-couple receptance, 390, 399, 414
Rotation-to-force receptance, 390, 414
Round-off error, 216
Rumble strips, 124
S
Sampling frequency, 271
Self-excited vibration, 72
aeroelastic applications, 5
behavior, 5
chatter, 5
feedback, 5
flutter, 4
magnitude, 5
Series spring, 47
Shaker, 256, 257
Shape factor, 319
Shear force, 284–287, 289
Shear modulus, 48
Shortcut method, 229–242
SI units, 49
Sine function, 10, 11, 27
Sine sweep test, 256
Sine wave, 17
Single degree of freedom forced vibration
base motion, 119
FRF (see Frequency response function
(FRF))
impulse response, 124, 125
rotating unbalance, 115–117, 119
spring-mass-damper model, 89, 90
Single degree of freedom free vibration
damped oscillator, 70
initial conditions, 38
lumped parameter model, 44
mass, 31
modeling techniques, 83
natural frequency, 33, 36
physical single, 67
spring-mass-damper system, 52
torsional systems, 48
underdamped, 66
velocity-dependent aerodynamic force, 72
Slinky, 94
Solid damping
coulomb, 299
differential equation of motion, 299
direct FRF
beam, 305
free end of the beam, 302
second moment of area, 304
semi-logarithmic format, 302, 303
fixed-free beam, 301, 302
fixed-free boundary conditions, 301
FRF, 299, 300
FRF value, 300
material-dependent, 299
MATLAB MOJO 8.2, 302
428 Indexnatural frequencies, 305
second moment of area, 301
steel alloys, 301
viscous, 299
Solid/structural damping, 52
Spring-mass-damper model, 2, 10, 31
Spring-mass-damper system, 174, 182, 261,
262, 281
Square matrix, 153
Square wave, 14, 15
Squared scaling, 151
Stable behavior, 72
Static flexibility, 102
Static force sum, 31
Static free body diagram, 46
Steady-state behavior, 3
Steady-state magnitude, 132
Steady-state solution, 90
Stiffness, 1
Stiffness denominator, 49
Stiffness matrix, 135, 143, 163, 171, 219, 239,
240, 246, 247, 327, 329, 353, 356,
363, 364, 396
Stinger, 257
String’s natural frequency, 4
Superposition, 173
advantages, 174
linear, 182
Symbolic expression, 200
Symmetric, 143, 151
Symmetric mass matrix, 332
System stiffness matrix, 232
System vibration, 53
System’s free vibration, 170
T
Term-by-term multiplication, 61
Time-domain displacement, 282
Time-domain representation, 35, 36
Time-domain responses, 145–151, 165
Time-domain simulation, 265
Time-domain solutions, 166
Time period, 10
Time step, 73, 74, 78, 266, 267
Time-varying signal, 17
Timoshenko beam model, 319, 324
Torsion vibration, 313–315
Torsional systems, 48
Transient solutions, 90
Transpose operation, 147
Transverse beam element
beam end FRFs, 358, 359
beam model, 356, 358
boundary conditions, 350, 351
cross-sectional second moment of area, 349
dynamic matrix, 358
Euler-Bernoulli beam theory, 349, 352
free-free boundary conditions, 359
interpolation/shape function, 351
kinetic energy, 354, 355
mass matrix, 354–356
MATLAB MOJO 9.4, 359, 360
rotation function, 349
stiffness matrix, 353, 354, 356
Transverse deflection, 284, 305, 316, 320
Transverse vibration
cross-section Euler-Bernoulli beam, 289
forces, 288
frequency response function
boundary conditions, 290
fixed-free beam, 290–294
free-free beam, 295, 298
position-dependent vibration behavior,
289
time-dependence, 290
moments, 288
shear force, 289
Transverse vibration FRF measurement
comparisons
experimental impact testing setup, 311
fixed-free beam, 308, 310
free-free beam, 310
measured vs. predicted fixed-free beam,
BEP, 310
measured vs. predicted free-free FRFs, 310,
311
natural frequency uncertainty, 312, 313
Trivial solution, 32, 52, 136
Two degree of freedom free vibration
characteristic equation, 136
eigenvectors, 140, 142
equations of motion, 142, 153
free oscillation, 148
FRF, 133, 134
linear combination, 148
local vs. modal coordinates, 152
Lumped parameter, 133, 134
modal analysis, 160
modal matrices, 166
mode shapes, 138, 144, 145
parameters and initial conditions, 142
spring-mass-damper system, 167–170
time-domain responses, 151
time-responses, 161
U
Unbalanced mass, 115, 116, 119
Uncoupled equations, 156
Index 429Undamped free vibration, 36
Undamped model, 52
Undamped natural frequency, 33, 56, 158
Underdamped single degree of freedom, 159
Underdamped system, 54–58, 60
Unstable behavior, 71
asymptotically stable, 70
divergent instability, 75, 77, 78, 80
flutter instability, 71–74
free vibration measurement, 81
marginally stable, 70
unstable behavior, 71
V
Variance, 312
Vector representation, 36
Velocity, 18
Vibration, 1, 137
Vibration measurement
capacitance probe, 258, 259
laser vibrometer, 259, 260
Viscous damping, 299, 300
coefficient, 30, 113, 130
equivalent, 68
forces, 51, 75
linear differential equations, 31
modeling choice, 52
vibration prevention, 53
W
Waveform, 25
White noise, 256
Wing’s natural frequency, 5
430 Index
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