Eigenvectors, 7, 50
left, 281
linearly dependent, 112, 115
linearly independent, 112
orthogonality, 51
Elastostatical chaos, 378
Embedding space method, 328
Energy pumping, 210
Engineering stability, 296
Equilibrium point, 106
Equivalent linear viscous damping coefficient,
33
Escape velocity, 359
Euler
column, 53
elastica, 378
loops, 379
Euler, Leonard, vii
Event function/handling, 183, 207, 450
EVP. See Eigenvalue problem
Experimental bifurcation analysis, 304
Explosive bifurcation, 294
External resonance. See Resonance
Extreme sensitivity to initial conditions, 350,
352
F
Fast
force, 388, 389, 392, 410, 412
motions, 388, 389, 391, 392, 394, 404, 406,
427
time, 121, 389, 392, 408, 428
Fast Fourier transform (FFT), 451
Feedback control, 364
Feigenbaum, 342, 348
constant, 342, 349
law, 349
number, 342, 348, 365
scenario, 348
Finite difference method, 74, 80
central differences, 80
difference operator, 81
higher order expressions, 83
Finite element method, 86, 87
First-order form, of ODEs, 449
Fixed point, 106, 342
Flexibility
coefficients, 38
matrix, 37
methods, 37, 38
Flexible joints, 377
Flip bifurcation, 293
Floquet theory, 481
Flow-induced vibrations, 171, 377
Fluid-conveying pipes. See pipe flow
Fluid forces, 100
Flutter, 57, 145, 233, 369
and divergence, 233
in panels, 372
soft/explosive, 228
Focus point, 115
Fold bifurcation, 277
Follower-loading
and circulatory forces, 43
double pendulum, 210, 228, 343, 361, 369
paradox of, 56
partial, 229
Force
circulatory, 43
classification, 42
completely dissipative, 43
dissipative, 43
gyroscopic, 43
instationary, 43
non-circulatory, 43
Force response, 217
for the autoparametric vibration absorber,
218, 219
for the follower-loaded double pendulum,
238
for the non-shallow arch, 226
Formal mode shape expansion, 43
Fourier
series, 10, 479
spectrum, 329
Fractal dimension, 339
Freeware, 454
Frequency
equation, 7, 60
locking, 217
Frequency-energy plot (FEP), 309
Frequency response, 139
equation, 156
for Duffing’s equation, 157–159
for slowly swept excitation, 254
for the autoparametric vibration absorber,
219
for the non-shallow arch, 226
for the pendulum with a moving support,
139, 142, 143
for the string with a sliding point mass, 252
plotting, 156
subharmonic, 166
superharmonic, 164
Frequency spectrum, 329, 451
broadband, chaotic, 329
dc-component, 329
Frequency-sweep, response to, 251
Index
521
left, 281
linearly dependent, 112, 115
linearly independent, 112
orthogonality, 51
Elastostatical chaos, 378
Embedding space method, 328
Energy pumping, 210
Engineering stability, 296
Equilibrium point, 106
Equivalent linear viscous damping coefficient,
33
Escape velocity, 359
Euler
column, 53
elastica, 378
loops, 379
Euler, Leonard, vii
Event function/handling, 183, 207, 450
EVP. See Eigenvalue problem
Experimental bifurcation analysis, 304
Explosive bifurcation, 294
External resonance. See Resonance
Extreme sensitivity to initial conditions, 350,
352
F
Fast
force, 388, 389, 392, 410, 412
motions, 388, 389, 391, 392, 394, 404, 406,
427
time, 121, 389, 392, 408, 428
Fast Fourier transform (FFT), 451
Feedback control, 364
Feigenbaum, 342, 348
constant, 342, 349
law, 349
number, 342, 348, 365
scenario, 348
Finite difference method, 74, 80
central differences, 80
difference operator, 81
higher order expressions, 83
Finite element method, 86, 87
First-order form, of ODEs, 449
Fixed point, 106, 342
Flexibility
coefficients, 38
matrix, 37
methods, 37, 38
Flexible joints, 377
Flip bifurcation, 293
Floquet theory, 481
Flow-induced vibrations, 171, 377
Fluid-conveying pipes. See pipe flow
Fluid forces, 100
Flutter, 57, 145, 233, 369
and divergence, 233
in panels, 372
soft/explosive, 228
Focus point, 115
Fold bifurcation, 277
Follower-loading
and circulatory forces, 43
double pendulum, 210, 228, 343, 361, 369
paradox of, 56
partial, 229
Force
circulatory, 43
classification, 42
completely dissipative, 43
dissipative, 43
gyroscopic, 43
instationary, 43
non-circulatory, 43
Force response, 217
for the autoparametric vibration absorber,
218, 219
for the follower-loaded double pendulum,
238
for the non-shallow arch, 226
Formal mode shape expansion, 43
Fourier
series, 10, 479
spectrum, 329
Fractal dimension, 339
Freeware, 454
Frequency
equation, 7, 60
locking, 217
Frequency-energy plot (FEP), 309
Frequency response, 139
equation, 156
for Duffing’s equation, 157–159
for slowly swept excitation, 254
for the autoparametric vibration absorber,
219
for the non-shallow arch, 226
for the pendulum with a moving support,
139, 142, 143
for the string with a sliding point mass, 252
plotting, 156
subharmonic, 166
superharmonic, 164
Frequency spectrum, 329, 451
broadband, chaotic, 329
dc-component, 329
Frequency-sweep, response to, 251
Index
521