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 |  موضوع: كتاب Dynamic Response of Linear Mechanical Systems - Modeling, Analysis and Simulation الأحد 27 أكتوبر 2024, 1:15 am | |
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أحضرت لكم كتاب
Dynamic Response of Linear Mechanical Systems - Modeling, Analysis and Simulation
Jorge Angeles
و المحتوى كما يلي :
Contents
1 The Modeling of Single-dof Mechanical Systems 1
1.1 Introduction . 1
1.2 Basic Definitions 3
1.3 The Modeling Process 7
1.4 The Newton-Euler Equations . 8
1.5 Constitutive Equations of Mechanical Elements 11
1.5.1 Springs 11
1.5.2 Dashpots 17
1.5.3 Series and Parallel Arrays of Linear Springs . 18
1.5.4 Series and Parallel Arrays of Linear Dashpots . 20
1.6 Planar Motion Analysis . 21
1.6.1 Lagrange Equations 25
1.6.2 Energy Functions . 26
1.6.3 Kinetic Energy . 26
1.6.4 Potential Energy 28
1.6.5 Power Supplied to a System and Dissipation Function 29
1.6.6 The Seven Steps of the Modeling Process 34
1.7 Hysteretic Damping . 53
1.8 Coulomb Damping 54
1.9 Equilibrium States of Mechanical Systems 59
1.10 Linearization About Equilibrium States. Stability . 66
1.11 Exercises . 76
References 84
2 Time Response of First- and Second-order Dynamical Systems . 85
2.1 Preamble . 85
2.2 The Zero-input Response of First-order LTIS . 88
2.3 The Zero-input Response of Second-order LTIS 91
2.3.1 Undamped Systems 91
2.3.2 Damped Systems . 97
xvxvi Contents
2.4 The Zero-State Response of LTIS 111
2.4.1 The Unit Impulse . 112
2.4.2 The Unit Doublet . 112
2.4.3 The Unit Step . 113
2.4.4 The Unit Ramp . 114
2.4.5 The Impulse Response . 115
2.4.6 The Convolution (Duhamel) Integral 126
2.5 Response to Abrupt and Impulsive Inputs . 130
2.5.1 First-order Systems 131
2.5.2 Second-order Undamped Systems . 132
2.5.3 Second-order Damped Systems . 136
2.5.4 Superposition . 140
2.6 The Total Time Response . 141
2.6.1 First-order Systems 141
2.6.2 Second-order Systems . 142
2.7 The Harmonic Response 145
2.7.1 The Unilateral Harmonic Functions . 147
2.7.2 First-order Systems 149
2.7.3 Second-order Systems . 152
2.7.4 The Response to Constant and Linear Inputs . 159
2.7.5 The Power Dissipated By a Damped
Second-order System 161
2.7.6 The Bode Plots of First- and Second-order Systems . 161
2.7.7 Applications of the Harmonic Response 164
2.7.8 Further Applications of Superposition 174
2.7.9 Derivation of zb(t) 179
2.8 The Periodic Response 181
2.8.1 Background on Fourier Analysis 182
2.8.2 The Computation of the Fourier Coefficients . 189
2.8.3 The Periodic Response of First- and
Second-order LTIS . 202
2.9 The Time Response of Systems with Coulomb Friction 207
2.10 Exercises . 210
References 231
3 Simulation of Single-dof Systems 233
3.1 Preamble . 233
3.2 The Zero-Order Hold (ZOH) . 234
3.3 First-Order Systems . 235
3.4 Second-Order Systems 239
3.4.1 Undamped Systems 239
3.4.2 Damped Systems . 245
3.5 Exercises . 257
Reference 262Contents xvii
4 Modeling of Multi-dof Mechanical Systems 263
4.1 Introduction . 263
4.2 The Derivation of the Governing Equations . 264
4.3 Equilibrium States 278
4.4 Linearization of the Governing Equations
About Equilibrium States . 283
4.5 Lagrange Equations of Linear Mechanical Systems 288
4.6 Systems with Rigid Modes . 298
4.7 Exercises . 302
References 305
5 Vibration Analysis of Two-dof Systems . 307
5.1 Introduction . 307
5.2 The Natural Frequencies and the Natural Modes
of Two-dof Undamped Systems 308
5.2.1 Algebraic Properties of the Normal Modes . 322
5.3 The Zero-Input Response of Two-dof Systems 323
5.3.1 Semidefinite Systems 324
5.3.2 Systems with a Positive-Definite Frequency Matrix 333
5.3.3 The Beat Phenomenon . 344
5.4 The Classical Modal Method . 347
5.5 The Zero-State Response of Two-dof Systems 353
5.5.1 Semidefinite Systems 354
5.5.2 Definite Systems . 358
5.6 The Total Response of Two-dof Systems 363
5.6.1 The Classical Modal Method Applied to the
Total Response . 364
5.7 Damped Two-dof Systems 366
5.7.1 Total Response of Damped Two-dof Systems 375
5.8 Exercises . 381
Reference 388
6 Vibration Analysis of n-dof Systems 389
6.1 Introduction . 389
6.2 The Natural Frequencies and the Natural Modes of
n-dof Undamped Systems 390
6.2.1 Algebraic Properties of the Normal Modes . 405
6.3 The Zero-input Response of Undamped n-dof Systems . 406
6.3.1 The Calculation of the Zero-input Response of
n-dof Systems Using the Classical Modal Method . 409
6.4 The Zero-state Response of n-dof Systems 412
6.4.1 The Calculation of the Zero-state Response of
n-dof Systems Using the Classical Modal Method . 414
6.5 The Total Response of n-dof Undamped Systems . 415
6.6 Analysis of n-dof Damped Systems 415
6.7 Exercises . 417xviii Contents
7 Simulation of n-dof Systems 419
7.1 Introduction . 419
7.2 Undamped Systems . 420
7.3 The Discrete-Time Response of Undamped Systems . 421
7.3.1 The Numerical Stability of the Simulation
Algorithm of Undamped Systems . 426
7.3.2 On the Choice of the Time Step . 428
7.4 The Discrete-Time Response of Damped Systems 431
7.4.1 A Straightforward Approach 432
7.4.2 An Approach Based on the Laplace Transform 435
7.5 Exercises . 452
References 454
8 Vibration Analysis of Continuous Systems . 455
8.1 Introduction . 455
8.2 Mathematical Modeling 456
8.2.1 Bars Under Axial Vibration 456
8.2.2 Bars Under Torsional Vibration . 458
8.2.3 Strings Under Transverse Vibration . 460
8.2.4 Beams Under Flexural Vibration 462
8.3 Natural Frequencies and Natural Modes . 465
8.3.1 Systems Governed by Second-Order PDE 465
8.3.2 Systems Governed by Fourth-Order PDEs:
Beams Under Flexural Vibration 480
8.4 The Properties of the Eigenfunctions 484
8.4.1 Systems Governed by Second-Order PDEs . 484
8.5 Exercises . 494
References 496
A Matrix Functions 497
A.1 Introduction . 497
A.2 Preliminary Concepts . 497
A.3 Calculation of Analytic Matrix Functions of a Matrix Argument 499
A.3.1 Special Case: 2 × 2 Matrices 502
A.3.2 Examples . 505
A.4 Use of Mohr’s Circle to Compute Analytic Matrix Functions 516
A.4.1 Examples . 521
A.5 Shortcuts for Special Matrices 527
A.5.1 Example A.5.1 528
A.5.2 Example A.5.2 529
A.5.3 Example A.5.3 529
A.5.4 Example A.5.4 530
References 530Contents xix
B The Laplace Transform . 531
B.1 Introduction . 531
B.1.1 Properties of the Laplace Transform 533
B.2 Time Response via the Laplace Transform 535
B.2.1 The Inverse Laplace Transform via
Partial-Fraction Expansion 538
B.2.2 The Final- and the Initial-Value Theorems 548
Reference 550
Index . 551
Index
Symbols
2 × 2 identity matrix, 22
cosΩt, 525
ln(A), 512
K, 295
π, 193
sinΩt, 525
eA, 506
ith modal vector, 311
n-dof mechanical systems, 268
n-dof system, 145, 419
n-dof undamped system, 419
q, 28
q˙, 290
q, 267, 290
q˙, 290
q˙, 289
Ω2, 316
Ω2, 322
q˙, 267
p, 289, 290
q, 289
sgn, 54, 110
‘small’ perturbations of the equilibrium states,
283
A
A/D, see analog to digital
absolute acceleration, 11
abstraction, 7
accelerometer design, 172
active force, 29
additivity, 86, 128
algorithm
DnDOF, 434
UDnDOF, 426
Algorithm Damped-1dof, 247
aliasing, 428
analog system, 234
analog-to-digital (A/D) converter, 234
analysis, 6
analytic function f (λ ) of λ , 499
analytic function F(A) of A, 497
analytic matrix function
method to compute, 500
angular deflection, 458
angular velocity, 9, 10
area moment of inertia, 459
asymptotic stability, 68
asymptotically stable, 284
asymptotically stable system, 68
asymptotically unstable system, 66
autonomous system, 31, 284
B
bandwidth, 171
bars
under axial vibration, 456
under torsional vibration, 456
BC, see boundary condition
beams
under flexural vibration, 456
beat phenomenon, 344
Belleville spring, 14
belt-pulley transmission, 328, 350
black-box, 127
black-box representation, 87
Bode plots, 147
of first- and second-order systems, 161
bogie-half-car, 8
boundary condition, 4, 491
boundary-value problem, 466
bump, 302
function, 243
BVP, see boundary-value problem
C
c.o.m, see center of mass
cable, 12
Cannon’s book, see Cannon, R.H.
Cannon, R.H., 304
cantilever beam, 13
Carl Sagan, 193
causal, 236
causality, 236
Cayley-Hamilton Theorem, 98, 241, 497
center of mass, 28
characteristic equation, 468
characteristic equation of A, 498
characteristic polynomial
of the dynamic matrix, 313
of A, 498
characteristic solutions, 468
characteristic values, 468
Cholesky factoring, 308
classical modal method, 347
applied to the total response, 364
commutativity, 129
compressible and incompressible fluid, 4
computation of the Fourier coefficients, 189
computer algebra, 156, 160
conditions
boundary, 458
initial, 458
configuration
-dependent damping coefficient, 32
of a system, 25
conservative forces, 31
constant-coefficient mechanical systems, 6
constitutive equation, 7, 464
constitutive equations
of mechanical elements, 11
contact, 193
continua, 4, 455
continuous
-time system, 234, 421
train of impulses, 128
continuous model, 455
continuous systems, 455, 456
continuous-time convolution, 237
continuum, 455
continuum mechanics, 4
controlled
forces or torques, 26
motion, 28
rates, 266
variables, 266
convolution, 127, 129, 442
Duhamel integral, 115, 355
for critically damped systems, 130
for overdamped systems, 130
of underdamped systems, 130
Coriolis and centrifugal forces, 35
cosine law, 37
Coulomb
damping, 54
dry-friction damping, 30
friction, 6
friction cum geometric nonlinearity, 56
critically damped system, 91, 509
cross product, 22
CT system, see continuous-time system
cycloidal slope, 215
D
damped n-dof system
discrete-time response, 431
eigenvalue problem, 435
simulation via an extension of single-dof
systems, 432
simulation via the Laplace transform, 435
damped natural frequency, 99
damped suspension
discrete time-response, 435
damped systems, 97, 245, 307
damped two-dof systems, 366
damping
constant, 18
matrix, 285
damping ratio, 69
dashpots, 5
db, see decibel
dec, see decade
decade, 164
decibel, 164
definite system, 358
degree of freedom, 7, 25, 263
delayed
impulse response, 128
input, 128
delta function, see Dirac function
derivative of the impulse response, 137
design of foundations, 167
design of pneumatic hammers, 167
deterministic, 26
difference equation, 238
differential-algebraic systems, 5
digital system, 234Index 553
Dirac function, 112
discrete Fourier transform, 192
discrete time, 234, 432
discrete-time
linear dynamical system, 234
system, 236, 248, 421
discrete-time response, 237
of n-dof undamped systems, 421, 423
of a damped suspension, 250
of first-order systems, 237
of undamped n-dof systems, 422
of undamped second-order systems, 241
discretization methods, 5
dissipation function, 29, 30
dissipative force, 29, 30
distributed normal stress, 463
distributed shear stress, 463
distributed-parameter models, 4, 5
distributed-parameter systems, 5
distributivity, 129
dof
see degree of freedom, 145
dot product, see scalar product
doublet function, 112
doublet response, 137
of second-order damped systems, 136
of second-order systems, 134
of second-order undamped systems, 132
drill for deep-boring, 303
driving force, 29
Duhammel integral, 129
dynamic equilibrium, 460
dynamic matrix, 312
dynamical system, 4
E
eigenfrequencies, 525
eigenfunctions, 468
properties, 484
eigenmatrix, 322
eigenvalue problem, 312, 313
eigenvalues, 468
eigenvalues of Ω, 525
eigenvalues of A, 498
elastic potential energy, 29
elastica, 463
energy functions, 26
engineering approximation, 8
epicyclic gear train, 33
equilibrium configuration, 292
equilibrium analysis
of the eccentric plate, 62
of the overhead crane, 60
equilibrium configuration, 67
equilibrium configurations
of the gantry robot, 279
of the two-link robot, 278
equilibrium states, 278
of mechanical systems, 59
of the actuator mechanism, 61
equivalent dashpot coefficient, 20
equivalent stiffness of parallel array, 20
Euclidean norm, 242
Euler’s Law, 459
Euler-Bernoulli beam, 462
even function, 182
excitations, 3
existence and unicity of the solution, 358
exponential of At, 98, 528
F
fast Fourier transform, 193
FBD, see free-body diagram
FFT, see fast Fourier transform
FFT analysis, 193
final-value theorem, 548
first law of thermodynamics, 194
first-order
LTI dynamical systems, 203
ODEs, 6
systems, 116, 235
flexural rigidity, 464
floating-point arithmetic, 241
floor function, 193
flow-induced drag, 30
fluid-clutch system, 542
force
-controlled source, 29
-driven overhead crane, 42
-driven system, 42
sources, 29
forced response, 86
Fourier algorithm, 195
Fourier analysis, 182, 183
of a monotonic function, 187
of a square wave, 186
of a train of impulses, 184
of the pyr(x) function, 187
Fourier coefficients, 192
Fourier expansion, 183
Fourier series, 145
Fourier transform, 145
free response, 86
free-body diagram, 18
frequency matrix, 310, 500
frequency radius, 315, 525554 Index
frequency ratio, 153
frequency response, 145, 147
functional, 236
FVT, see final-value theorem
G
gear transmission, 304
generalized, see generalized velocity
active force, 30
coordinate, 6, 25, 264
Coriolis and centrifugal forces, 267
damping matrix, 263
dissipative force, 265
driving force, 265
forces, 30, 289
mass, 28
mass matrix, 263
momentum, 266
velocity, 30
generalized coordinate, 28
Geneva mechanism, 131
Geneva wheel, 134, 303
discrete-time response, 254
geometric BCs, 481
governing equation, 38, 271, 464
gyroscopic effects, 290
gyroscopic forces, 284
H
harmonic
excitation, 146
functions of a positive-definite matrix, 525
motions of single-dof systems, 313
oscillator, 92
harmonic response, 145
applications, 164
of first-order systems, 149
of second-order systems, 155
of undamped systems, 152, 154, 155
Heaviside function, 113
Hessian matrices, 289
high-pass filter, 174
Hooke’s Law, 457
hydraulic clutch, 51
hysteretic damping, 30, 53
I
iconic model, 8
idealization, 7
identification of damping from the time
response, 100
improper orthogonal matrix, 241
impulse, 128
impulse response, 115, 120, 127
of n-dof damped systems, 441
of first- and second-order LTI systems, 115
IMSL, 372
independent generalized coordinates, 25
independent generalized speeds, 25
indexing mechanism of a production machine,
302
inertia matrix, 9
inertial measurement units (IMU), 6
initial-value problem, 466
initial-value theorem, 548
input, 3
instability, 158
invariant quantity, 503
inverse Laplace transform
via partial-fraction expansion, 538
inviscid and viscous fluid, 4
IVT, see initial-value theorem
J
Joseph Fourier (1768–1830), 145
journal bearings, 24
KK
onig’s Theorem, ¨ 27
Konig’s theorem, ¨ 44
kinematic analysis, 28
kinetic energy, 26, 265
of the system, 266
L
L’Hospital’s rule, 158, 178, 515
Lagrange equations, 25, 266
of linear mechanical systems, 288
vector form, 263
Lagrangian, 26, 267
Laplace transform, 531
additivity, 533
basic properties, 534
denominator with repeated roots, 539
double-sided, 531
inverse, 532, 538
linear homogeneity, 533
numerator with distinct roots, 539
of cosωt, 535
of sinωt, 534
of second-order underdamped systems, 544
of the convolution, 534Index 555
of the decaying exponential, 532
of the delay, 534
of the unit step function, 533
one-sided, 531
pairs, 532
properties, 533
quadratic factors, 545
superposition, 533
the impulse function, 535
the ramp function, 535
LHS, 119
linear dashpots, 30
linear homogeneity, 129
linear mechanical systems, 5
linear springs, 29, 30
linear time-invariant system, 85
linear transformation, 531
linear, soft, and hard springs, 15
linearity, 115, 127
linearity and time-invariance, 176, 179
linearization
about equilibrium states, 66
of the governing equations, 283
linearized
equation of a one-dof system, 68
model of a two-link robot, 285
linearized model
of a gantry robot, 286
linearly elastic, 456
linearly homogeneous, 86
linearly viscous damping, 30
load, 29
locomotive wheel array, 35
logarithmic decrement, 101
low-pass filters, 164
LTIS, see linear time-invariant system
lumped-parameter models, 5
M
Macsyma, 372
magnification factor, 162
of the transmitted force, 166
of the transmitted motion, 169
magnitude, 146
Maple, 372
marginal stability, 68
marginally stable, 284
equilibrium state, 66
mass density, 459
mass matrix, 266, 285
mass subjected to a time-varying force, 253
mass-spring-dashpot system in a gravity field,
74
mass-transit system
undamped discrete-time response, 429
Mathematica, 372
mathematical model, 35, 233
mathematical modeling, 8
Matlab, 193, 372
matrix
exponential, 497
function F(A), 500
functions, 497
notation, 21
with a repeated eigenvalue, 515
mean frequency, 315, 525
mechanical modeling, 7
mechanical system, 4
mechanical system configuration, 264
mechanical transmissions, 4
memory, 4
memoryless systems, 4
modal analysis, 318
of a damped test pad, 372
of a two-dof gantry robot, 318
modal coordinates, 364
modal equation, 314
modal matrix, 322
modal vectors, 311, 313, 316, 430
mode, 314
mode-orthogonality, 489
model, 233
model for the vertical vibration of mass-transit
cars, 295
modeling, 6
modeling process, 7
Mohr circle, 307, 313, 497, 516, 518
of cosΩt, 525
of sinΩt, 525
of Ω, 525
momentum-preserving system, 355
revisited, 365
motion, 25
motion sources, 29
motion-controlled source, 29
motion-driven system, 42
motor-cam transmission, 58
multibody systems, 7, 307
N
n-tuplet function, 113
natural BCs, 482
natural frequencies, 310, 525
natural frequencies of two-dof undamped
systems, 308
natural frequency, 69556 Index
natural modes of two-dof undamped systems,
308
negative-definite matrix, 284
negative-semidefinite matrix, 284
neutral axis, 463
Newton equation, 464
Newton-Euler equations, 8, 10
nominal behavior, 6
non-zero initial conditions, 141
non-zero input, 141
nonlinear spring, 15
normal form
of the governing equations, 310
of the mathematical model, 420
normal stress, 462
numerical quadrature, 189
O
oct, see octave
octave, 164
odd function, 182
ODE, see ordinary differential equations
ordinary differential equations, 5, 6
orthogonal matrix, 22
orthogonality and normality, 484
orthogonality condition, 491
orthogonality of the eigenvectors, 314
oscillating follower, 229
output, 3, 87
overdamped system, 91, 510
overhead crane, 39
P
Parseval’s identity, see Parseval’s Theorem
Parseval’s Theorem, 193
partial derivatives
fourth-order, 465
second-order, 465
partial differential equation, 4, 458
partial-fraction expansion, 538
of a rational function, 538
particular solution of the second-order damped
system, 155
PDE, see partial differential equations
PDE of the fourth order, 465
perfectly elastic shock, 338
periodic
function, 182
input, 202
response, 181
response of an air compressor, 205
response of first- and second-order LTIS,
202
persistent time-varying input, 145
PFE, see partial-fraction expansion
phase, 146
planar motion, 21, 265
pneumatic hammer, 164
Poisson ratio, 15
positive-definite
matrix, 284
square root of M, 309
positive-definite matrix
harmonic functions, 525
positive-semidefinite
frequency matrix, 354
matrix, 284
potential, 31
energy: elastic and gravitational, 28
potentiometers, 4
power
developed by a moment, 29
dissipated by a damped second-order
system, 161
supplied to a system, 26, 29
Principle of Conservation of Energy, 194
proper orthogonal matrix, 241
proper rational function, 538
proportional damping, 368
pulse, 140
pulse-like input, 174
purely flexible mode, 329
Q
quadratic expressions, 266
quadratic form, 288
quadruplet function, 113
quick-return cam mechanism, 229
R
ramp response, 139
of first-order systems, 132
of an overdamped second-order system,
139
of second-order undamped systems, 135
reflection, 241
resonance, 155
response, 3
to abrupt and impulsive inputs, 130
of a damped, second-order system to a unit
doublet, 137
of a second-order underdamped system to a
ramp, 139Index 557
of a second-order, critically damped system
to a ramp, 139
of first-order systems, 129
of second-order damped systems, 130
of second-order undamped systems, 130
to constant and linear inputs, 159
to the unilateral cosine function, 152
to the unilateral sine function, 156
Riemann integral, 129
rigid and deformable solid, 4
rigid mode, 263, 295, 323, 473
rigid ring suspended from a pin, 23
rms, see root-mean-square
rod, 12
root-mean-square, 194
rotation, 241, 420
rotational damping coefficient, 18
rotational dashpot, 18
roundoff errors, 241
S
sampled signal, 234
sampling interval, 234, 243
saturation function, 55
scalar moment of inertia, 10
scalar product, 23
Scotch yoke, 223
second-order
damped systems, 120
ODEs, 6
systems, 152, 239
undamped systems, 119, 204
second-order damped systems
discrete-time response, 249
seismograph design, 173
semidefinite stiffness matrix, 295
semidefinite systems, 354
semigraphical approach, 372
separation of variables, 465
series
equivalent of two springs, 19
expansion of cosωnt, 92
expansion of sinωnt, 92
series and parallel array, 20
of linear dashpots, 20
of linear springs, 18
seven steps, 34, 35, 268
seven-step procedure, 35
shaft, 13
Shannon’s Theorem, 428
shear deformation, 458
shear force, 463
shear modulus, 459
shortcuts for special matrices, 527
sign-indefinite matrix, 284
signed angles, 25
signed distances, 25
simulation, 233
of n-dof systems, 419
of single-dof systems, 233
time-step choice, 428
simulation scheme
for undamped second-order systems, 240
single-degree-of-freedom system, 6, 25, 307
skew-symmetric matrix, 22, 527
small-perturbation analysis, 6
small-slope assumption, 460, 463
spectral analysis, 145, 183
of the displacement of an air compressor,
195
spring stiffness, 12
springs, 5, 11
square matrix
analytic function, 497
square root of M, 309
square-root factoring, 308
stability, 66
stability analysis
of the actuator mechanism, 70
of the eccentric plates, 72
of the overhead crane, 69
stable system, 66
staircase approximation, 235, 246
state, 4, 25
-variable form, 375, 431
-variable vector, 26, 264
variable, 25, 264
static equilibrium, 460
steady-state
part, 146
response, 203
response of an undamped system to a
sinusoidal input, 158
Steiner’s Theorem, 9, 10
step response
of critically damped systems, 139
of first-order system, 131
of overdamped systems, 139
of second-order undamped systems, 135
of the second-order underdamped system,
138
of underdamped systems, 138
stiffness matrix, 285
strain, 457
stress, 457
strings
under transverse vibration, 456558 Index
superposition, 6, 86, 115, 140, 203, 326, 332
system, 3
impulse response in state-variable form,
144
matrix, 419
output, 433
with a time-varying equilibrium state, 65
with positive-definite frequency matrix,
333
system with a time-varying equilibrium state,
64
T
tangential stress, 463
test pad, 122
revisited, 447
two-dof, 338
theory of beams, 462
time constant, 90
time delay, 87
time invariance, 87
time response, 85
of critically damped systems, 99
of first- and second-order systems, 85
of overdamped systems, 100
of underdamped second-order system to a
constant input, 160
via the Laplace transform, 535
time-invariance, 115, 127, 246
time-invariant, 6
time-series, 192
time-varying equilibrium state, 280
time-varying inputs, 145
torque, 30
torque source, 29
torsional spring, 14
torsional stiffness, 14, 460
torsional vibrations of aircraft wings, 294
total generalized force, 31
total kinetic energy, 26, 28
total potential energy, 26
total response
of two-dof system, 363
of damped two-dof systems, 375
of the system, 144
total time response
of dynamical systems, 141
of first-order systems, 141
of second-order damped systems, 143
of second-order undamped systems, 142
trace of A, 503
transducers, 4
transient part, 146
translational
damping coefficient, 18
dashpot, 18
spring, 13
stiffness, 13
transmitted force, 165
transverse motion, 460
trapezoidal rule, 189
triplet function, 113
two-dimensional form, 22
two-dof gantry robot, 273
two-dof model of a terrestrial vehicle, 302
two-dof systems, 307
two-dof test pad, 348
two-dof undamped linear mechanical system,
308
two-link robotic arm, 269
U
undamped
linear mechanical systems, 307
second-order system, 120
suspension discrete-time response, 242
systems, 91, 239
terrestrial vehicle, 175
undamped two-dof systems, x
underdamped
impulse response second-order system, 121
system, 91, 508
unilateral harmonic functions, 147
unilateral sine input, 156
unit doublet, 112
unit eigenvectors, 310
unit impulse, 111, 112, 126
unit ramp function, 114
unit step function, 113
unstable system, 66
V
Vandermonde matrix, 501
vector
notation, 21
of generalized coordinates, 263
of generalized forces, 267
of generalized speeds, 263
of modal coordinates, 347
velocity meter design, 169
vibration absorber, 360
vibration analysis of two-dof systems, 307
viscous damping, 30Index 559
W
weighted magnitude, 312
whirling of shafts, 290, 376
Willis’ formula, 23
Y
Young modulus of elasticty, 456
Z
zero-input response, 86, 307
of damped systems in state-variable form,
143
of first-order LTIS, 88
of second-order damped systems, 507
of second-order LTIS, 91
of two-dof systems, 323
zero-order hold, 234, 419, 422, 432
zero-state response, 86, 127
in state-variable form, 143
of LTIS, 111
of two-dof systems, 353
to arbitrary input, 144
ZOH, see zero-order hold
ZSR, see zero-state response
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