Damping (cont.)
over-damped, 3
power dissipation function, 21
quadratic, 99
ratio, 3
stiffness-proportional, 9, 18, 36
subcritical, 110
under-damped, 3
Dangerous bifurcation, 294
Decoupling, 9, 17
Definite
EVP, 66
matrix, 50, 51
Degenerate singular point, 273
Degrees of freedom, 1
Delayed-coordinate technique, 328
Densities of engineering materials, 496
Describing function, method of
Harmonic balance, method of, 127
Deterministic models, 324
Detuning parameter, 134, 234
Difference equation, 367
Differential algebraic equations, 21
Dimension
capacity, 339
correlation, 339
Hausdorff, 339
information, 339
Lyapunov, 339
pointwise, 339
topological, 330, 336, 338
Dirac’s delta, 479
Direct separation of motions, method of, 176,
388
generalized for continuous systems, 426
generalized for discrete systems, 403, 405
outline, 389
worked out examples, 393
Discontinuous bifurcation, 294
Discontinuous transformation, 175
Disk, rotating, 426
Disks, 494
Disordered patterns, 381
Dissipation function
power-, 21
Rayleigh-, 21
Dissipative
force, 43
system, 44
Distortion, harmonic, 173
Distributed systems, 22
Dither control, 435, 462
Divergence, 233, 369
and decaying oscillations, 233
bifurcation, 237
DOFs. See Degrees of freedom
Double pendulum, follower-loaded, 210, 228,
343, 361, 369
Dripping tap, 383
Dry friction. See Friction
Duffing
equation, 130, 146, 325, 356, 360, 364, 449
systems, 146
systems, chaotic, 363
Duhamel’s integral, 4
Dunkerley’s formula, 85
Dynamic bifurcation, 294
E
Earthquake excited structures, 368
Eccentric loading, 371
Effective
damping, 425, 446
friction force, 399, 400, 436, 437
natural frequency, 393, 396
nonlinearity, 397
potential energy, 396
restoring force, 395, 396, 446
stiffness, 396, 404, 412, 414, 429, 432, 434,
441
structural properties, 416
Eigenfunctions, 13, 52, 53, 64
expansion, 27
iteration, 74, 75
normalization of, 69
orthogonality, 69
properties of, 67
Eigenpair, 50, 52
Eigenspace, 285
Eigenvalue problems (EVPs), 7, 49
algebraic, 50
classes of, 67
closed form solutions, 75
definiteness of, 65
differential, 52
stability related, 53
vibration related, 58
Eigenvalues, 7, 50, 52
bounds on, 74, 76
complex conjugated, 111, 113
complex-valued, 110, 113
distinct, 63
minimum properties of, 70
properties of, 50, 67
real-valueness, 67
sign of, 68
simple, 63
spectrum of, 285
520
Index
over-damped, 3
power dissipation function, 21
quadratic, 99
ratio, 3
stiffness-proportional, 9, 18, 36
subcritical, 110
under-damped, 3
Dangerous bifurcation, 294
Decoupling, 9, 17
Definite
EVP, 66
matrix, 50, 51
Degenerate singular point, 273
Degrees of freedom, 1
Delayed-coordinate technique, 328
Densities of engineering materials, 496
Describing function, method of
Harmonic balance, method of, 127
Deterministic models, 324
Detuning parameter, 134, 234
Difference equation, 367
Differential algebraic equations, 21
Dimension
capacity, 339
correlation, 339
Hausdorff, 339
information, 339
Lyapunov, 339
pointwise, 339
topological, 330, 336, 338
Dirac’s delta, 479
Direct separation of motions, method of, 176,
388
generalized for continuous systems, 426
generalized for discrete systems, 403, 405
outline, 389
worked out examples, 393
Discontinuous bifurcation, 294
Discontinuous transformation, 175
Disk, rotating, 426
Disks, 494
Disordered patterns, 381
Dissipation function
power-, 21
Rayleigh-, 21
Dissipative
force, 43
system, 44
Distortion, harmonic, 173
Distributed systems, 22
Dither control, 435, 462
Divergence, 233, 369
and decaying oscillations, 233
bifurcation, 237
DOFs. See Degrees of freedom
Double pendulum, follower-loaded, 210, 228,
343, 361, 369
Dripping tap, 383
Dry friction. See Friction
Duffing
equation, 130, 146, 325, 356, 360, 364, 449
systems, 146
systems, chaotic, 363
Duhamel’s integral, 4
Dunkerley’s formula, 85
Dynamic bifurcation, 294
E
Earthquake excited structures, 368
Eccentric loading, 371
Effective
damping, 425, 446
friction force, 399, 400, 436, 437
natural frequency, 393, 396
nonlinearity, 397
potential energy, 396
restoring force, 395, 396, 446
stiffness, 396, 404, 412, 414, 429, 432, 434,
441
structural properties, 416
Eigenfunctions, 13, 52, 53, 64
expansion, 27
iteration, 74, 75
normalization of, 69
orthogonality, 69
properties of, 67
Eigenpair, 50, 52
Eigenspace, 285
Eigenvalue problems (EVPs), 7, 49
algebraic, 50
classes of, 67
closed form solutions, 75
definiteness of, 65
differential, 52
stability related, 53
vibration related, 58
Eigenvalues, 7, 50, 52
bounds on, 74, 76
complex conjugated, 111, 113
complex-valued, 110, 113
distinct, 63
minimum properties of, 70
properties of, 50, 67
real-valueness, 67
sign of, 68
simple, 63
spectrum of, 285
520
Index