Mathcad, 456
Mathematica, 456
Mathieu equation, 203, 212, 222, 241, 481
damped, 481, 482
nonlinear, 130
Mathieu-Hill equation, 481
MATLAB, 452, 456
Maxwell’s reciprocity theorem, 42
MDSM. See Direct separation of motions,
method of
Melnikov
criterion, 353
function, 353
method, 350
Membranes, 433, 492
Meta-stability, 296
Microbeam, 468
Midplane stretching, 101, 147, 149, 364
Modal
amplitudes, 9, 15
analysis, 17
coefficients, 15
factors, 15
interaction, 209
Mode
coupling, nonlinear, 209, 220
nonlinear normal, 7, 209, 308
normal, 308
participation factors, 7, 15, 26
saturation, 217, 225
shape, 7, 13
for beams, 61, 487
for membranes, 492
for plates, 493
for rings, 491
for rods, 59, 486, 487
for strings, 486
shape expansion, 16, 25, 243, 429
shape expansion, formal, 43
Modulation equations, 133, 168, 388
Monogenic forces, 22
Moon’s beam, 325, 356, 358
Multiple scales, method of, 120
compared to the MDSM, 388
Multiple stationary solutions, 140
Multiplicity of eigenvalues, 63
N
Natural coordinates, 20
Natural frequency
commensurate, 210
damped, 3, 17
for beams, 61, 487
for membranes, 492
for plates, 493
for rings, 491
for rods, 59, 486, 487
for strings, 486
undamped, 2, 7
Natural parameter continuation, 298
Near-commensurate frequencies, 210
Near-resonance, 132
Negative damping, 171, 172
Negative definite
EVP, 66
matrix, 51
Neimark bifurcation, 293
Neutral stability, 107, 115, 296
Newton, Sir Isaac, 324
Node, 18, 113
Non-circulatory force, 43
Non-conservative system, 21, 23, 44
Non-critical variables, 286
Non-degenerate singular point, 273
Nondimensional equations of motion, 29
Non-gyroscopic system, 44
Non-hyperbolic singular point, 273
Non-ideal power source, 100
Non-invertible map, 342
Nonlinear
damping, 99, 171, 364
inertia, 97
interaction, 209, 238
stretching, 101
Nonlinearity
body forces, 99
geometrical, 97
hardening, 98, 139, 140, 147, 152
material, 98
midplane stretching, 101
physical configuration, 100
piece-wise linear, 100
softening, 98, 139, 140
strong, 170
weak, 116
Nonlinear Normal Mode (NNM), 7, 209, 308
Non-resonant excitation, 160
Non-transverse intersection, 281
Nontrivial solution, 50
Normal
coordinates, 15
form reduction, 288, 291
forms, 292
mode method, 27
modes, 7, 61, 209, 308
Numerical
libraries, 454
simulation, 449
524
Index
Mathematica, 456
Mathieu equation, 203, 212, 222, 241, 481
damped, 481, 482
nonlinear, 130
Mathieu-Hill equation, 481
MATLAB, 452, 456
Maxwell’s reciprocity theorem, 42
MDSM. See Direct separation of motions,
method of
Melnikov
criterion, 353
function, 353
method, 350
Membranes, 433, 492
Meta-stability, 296
Microbeam, 468
Midplane stretching, 101, 147, 149, 364
Modal
amplitudes, 9, 15
analysis, 17
coefficients, 15
factors, 15
interaction, 209
Mode
coupling, nonlinear, 209, 220
nonlinear normal, 7, 209, 308
normal, 308
participation factors, 7, 15, 26
saturation, 217, 225
shape, 7, 13
for beams, 61, 487
for membranes, 492
for plates, 493
for rings, 491
for rods, 59, 486, 487
for strings, 486
shape expansion, 16, 25, 243, 429
shape expansion, formal, 43
Modulation equations, 133, 168, 388
Monogenic forces, 22
Moon’s beam, 325, 356, 358
Multiple scales, method of, 120
compared to the MDSM, 388
Multiple stationary solutions, 140
Multiplicity of eigenvalues, 63
N
Natural coordinates, 20
Natural frequency
commensurate, 210
damped, 3, 17
for beams, 61, 487
for membranes, 492
for plates, 493
for rings, 491
for rods, 59, 486, 487
for strings, 486
undamped, 2, 7
Natural parameter continuation, 298
Near-commensurate frequencies, 210
Near-resonance, 132
Negative damping, 171, 172
Negative definite
EVP, 66
matrix, 51
Neimark bifurcation, 293
Neutral stability, 107, 115, 296
Newton, Sir Isaac, 324
Node, 18, 113
Non-circulatory force, 43
Non-conservative system, 21, 23, 44
Non-critical variables, 286
Non-degenerate singular point, 273
Nondimensional equations of motion, 29
Non-gyroscopic system, 44
Non-hyperbolic singular point, 273
Non-ideal power source, 100
Non-invertible map, 342
Nonlinear
damping, 99, 171, 364
inertia, 97
interaction, 209, 238
stretching, 101
Nonlinearity
body forces, 99
geometrical, 97
hardening, 98, 139, 140, 147, 152
material, 98
midplane stretching, 101
physical configuration, 100
piece-wise linear, 100
softening, 98, 139, 140
strong, 170
weak, 116
Nonlinear Normal Mode (NNM), 7, 209, 308
Non-resonant excitation, 160
Non-transverse intersection, 281
Nontrivial solution, 50
Normal
coordinates, 15
form reduction, 288, 291
forms, 292
mode method, 27
modes, 7, 61, 209, 308
Numerical
libraries, 454
simulation, 449
524
Index