solution of ODEs, 449
Numerical continuation, 298
O
ODE. See Ordinary differential equation
One-term EVP, 67
OpenGL, 452, 453, 455
Operator, 52
averaging, 390
derivative, 412
difference, 81
differential, 52, 121
effective stiffness, 434
Laplacian, 485
linear, 52
stiffness, 426, 428
unitary, 52
Orbital stability, 297
Orbits, phase plane, 105, 451
behavior near singular points, 110
for damped pendulum, 106
for undamped pendulum, 106, 124
heteroclinic, 107
homoclinic, 107
period-n, 328
regular, 107
singular, 107
topology, 111
Ordinary differential equation, 25
Orthogonality
general, 66
of eigenfunctions, 66
of modes, 7, 14
Schmidt-orthogonalization, 71
Orthonormality, 8, 69
Overshooting response, 254
P
Panel flutter, 372
Parametric
amplification, 142
anti-resonance, 209
attenuation, 142
damping, 142
excitation, 102, 103, 130, 142, 160, 212,
254, 481
pumping, 142
resonance, 132, 134, 140, 143, 209, 483
Partial differential equation, 25
Path following algorithms, 157, 299, 304
PDE. See Partial differential equation
Peak
response, 141, 159
amplitude, 141, 159
frequency, 141, 160
point, 141
Pendulum
analogy to Euler elastica, 379
Chelomei’s. See Chelomei’s pendulum
impacting, 368
in a magnetic field, 356
Kapitza’s, 239
parametrically excited, 102, 365
systems, chaotic, 364
vibration damping, 368
with a moving support, 102, 365, 393, 438
with external torque, 366
Period-doubling, 382
bifurcation, 293
route to chaos, 341, 348
Periodic
solutions, 127, 156, 233, 283, 284, 291,
292, 314
structure, 47
windows, 343, 345, 371
Period-n orbit, 328
Perturbation methods, 116
averaging, 124
multiple scales, 120
straightforward expansion, 118
Phase plane, 105, 327
chaotic, 327
orbits, 105
Phase quadrature, 307
Phase shift
spatial, 18
temporal, 2, 3, 7
Phase space, 327
contracting, 334
folding, 332
stretching, 332
Piecewise linear systems, 100
chaotic, 366
Pie dough, 337
Pin-jointed structures, 376
Pipe, Brumberg’s, 401
Pipe flow, 228, 371
and chaos, 371, 377
and HF effects, 426
vibration-induced, 258
Pitchfork bifurcation, 275
perturbation stability, 295
subcritical, 276
supercritical, 276
Pitchfork conditions, 282
Plasticity, 378
Plates, 493, 494
Play, 368, 376
Index
525
Numerical continuation, 298
O
ODE. See Ordinary differential equation
One-term EVP, 67
OpenGL, 452, 453, 455
Operator, 52
averaging, 390
derivative, 412
difference, 81
differential, 52, 121
effective stiffness, 434
Laplacian, 485
linear, 52
stiffness, 426, 428
unitary, 52
Orbital stability, 297
Orbits, phase plane, 105, 451
behavior near singular points, 110
for damped pendulum, 106
for undamped pendulum, 106, 124
heteroclinic, 107
homoclinic, 107
period-n, 328
regular, 107
singular, 107
topology, 111
Ordinary differential equation, 25
Orthogonality
general, 66
of eigenfunctions, 66
of modes, 7, 14
Schmidt-orthogonalization, 71
Orthonormality, 8, 69
Overshooting response, 254
P
Panel flutter, 372
Parametric
amplification, 142
anti-resonance, 209
attenuation, 142
damping, 142
excitation, 102, 103, 130, 142, 160, 212,
254, 481
pumping, 142
resonance, 132, 134, 140, 143, 209, 483
Partial differential equation, 25
Path following algorithms, 157, 299, 304
PDE. See Partial differential equation
Peak
response, 141, 159
amplitude, 141, 159
frequency, 141, 160
point, 141
Pendulum
analogy to Euler elastica, 379
Chelomei’s. See Chelomei’s pendulum
impacting, 368
in a magnetic field, 356
Kapitza’s, 239
parametrically excited, 102, 365
systems, chaotic, 364
vibration damping, 368
with a moving support, 102, 365, 393, 438
with external torque, 366
Period-doubling, 382
bifurcation, 293
route to chaos, 341, 348
Periodic
solutions, 127, 156, 233, 283, 284, 291,
292, 314
structure, 47
windows, 343, 345, 371
Period-n orbit, 328
Perturbation methods, 116
averaging, 124
multiple scales, 120
straightforward expansion, 118
Phase plane, 105, 327
chaotic, 327
orbits, 105
Phase quadrature, 307
Phase shift
spatial, 18
temporal, 2, 3, 7
Phase space, 327
contracting, 334
folding, 332
stretching, 332
Piecewise linear systems, 100
chaotic, 366
Pie dough, 337
Pin-jointed structures, 376
Pipe, Brumberg’s, 401
Pipe flow, 228, 371
and chaos, 371, 377
and HF effects, 426
vibration-induced, 258
Pitchfork bifurcation, 275
perturbation stability, 295
subcritical, 276
supercritical, 276
Pitchfork conditions, 282
Plasticity, 378
Plates, 493, 494
Play, 368, 376
Index
525