Vibrations and Waves in Continuous Mechanical Systems

Vibrations and Waves in Continuous Mechanical Systems......Page 4
Contents......Page 8
Preface......Page 14
1.1.1 Transverse dynamics of strings......Page 18
1.1.2 Longitudinal dynamics of bars......Page 23
1.1.3 Torsional dynamics of bars......Page 24
1.2 Dynamics of strings and bars: the variational formulation......Page 26
1.2.1 Transverse dynamics of strings......Page 27
1.2.2 Longitudinal dynamics of bars......Page 28
1.2.3 Torsional dynamics of bars......Page 30
1.3 Free vibration problem: Bernoulli’s solution......Page 31
1.4.1 The eigenvalue problem......Page 35
1.4.2 Orthogonality of eigenfunctions......Page 41
1.4.3 The expansion theorem......Page 42
1.4.4 Systems with discrete elements......Page 44
1.5 The initial value problem: solution using Laplace transform......Page 47
1.6 Forced vibration analysis......Page 48
1.6.1 Harmonic forcing......Page 49
1.6.2 General forcing......Page 53
1.7 Approximate methods for continuous systems......Page 57
1.7.1 Rayleigh method......Page 58
1.7.2 Rayleigh–Ritz method......Page 60
1.7.3 Ritz method......Page 61
1.7.4 Galerkin method......Page 64
1.8.1 Systems with distributed damping......Page 67
1.8.2 Systems with discrete damping......Page 70
1.9 Non-homogeneous boundary conditions......Page 73
1.10 Dynamics of axially translating strings......Page 74
1.10.2 Modal analysis and discretization......Page 75
1.10.3 Interaction with discrete elements......Page 78
Exercises......Page 79
References......Page 84
2.1 D’Alembert’s solution of the wave equation......Page 86
2.1.1 The initial value problem......Page 89
2.1.2 The initial value problem: solution using Fourier transform......Page 93
2.2 Harmonic waves and wave impedance......Page 94
2.3 Energetics of wave motion......Page 96
2.4.1 Reflection at a boundary......Page 100
2.4.2 Scattering at a finite impedance......Page 104
2.5.1 Impulsive start of a bar......Page 110
2.5.2 Step-forcing of a bar with boundary damping......Page 112
2.5.3 Axial collision of bars......Page 116
2.5.4 String on a compliant foundation......Page 119
2.5.5 Axially translating string......Page 121
Exercises......Page 124
References......Page 129
3.1.1 The Newtonian formulation......Page 130
3.1.2 The variational formulation......Page 133
3.1.3 Various boundary conditions for a beam......Page 135
3.1.4 Taut string and tensioned beam......Page 137
3.2.1 Modal analysis......Page 138
3.2.2 The initial value problem......Page 149
3.3 Forced vibration analysis......Page 150
3.3.1 Eigenfunction expansion method......Page 151
3.3.2 Approximate methods......Page 152
3.4 Non-homogeneous boundary conditions......Page 154
3.5 Dispersion relation and flexural waves in a uniform beam......Page 155
3.5.1 Energy transport......Page 157
3.5.2 Scattering of flexural waves......Page 159
3.6.1 Equations of motion......Page 161
3.6.2 Harmonic waves and dispersion relation......Page 164
3.7 Damped vibration of beams......Page 166
3.8.1 Influence of axial force on dynamic stability......Page 168
3.8.2 Beam with eccentric mass distribution......Page 172
3.8.3 Problems involving the motion of material points of a vibrating beam......Page 176
3.8.4 Dynamics of rotating shafts......Page 180
3.8.5 Dynamics of axially translating beams......Page 182
3.8.6 Dynamics of fluid-conveying pipes......Page 185
Exercises......Page 188
References......Page 195
4.1.1 Newtonian formulation......Page 196
4.1.2 Variational formulation......Page 199
4.2.1 The rectangular membrane......Page 202
4.2.2 The circular membrane......Page 207
4.4.1 Modal analysis......Page 214
4.4.2 Forced vibration analysis......Page 218
4.5.1 Waves in Cartesian coordinates......Page 219
4.5.2 Waves in polar coordinates......Page 221
4.5.3 Energetics of membrane waves......Page 224
4.5.4 Initial value problem for infinite membranes......Page 225
4.5.5 Reflection of plane waves......Page 226
Exercises......Page 230
References......Page 231
5.1.1 Newtonian formulation......Page 234
5.2.1 Free vibrations......Page 239
5.2.2 Orthogonality of plate eigenfunctions......Page 245
5.2.3 Forced vibrations......Page 246
5.3.1 Free vibrations......Page 248
5.3.2 Forced vibrations......Page 251
5.4 Waves in plates......Page 253
5.5 Plates with varying thickness......Page 255
Exercises......Page 256
References......Page 258
6.1.1 General properties and expansion theorem......Page 260
6.1.2 Green’s functions and integral formulation of eigenvalue problems......Page 269
6.1.3 Bounds for eigenvalues: Rayleigh’s quotient and other methods......Page 272
6.2.1 Equations of motion......Page 276
6.2.2 Green’s function for inhomogeneous vibration problems......Page 277
6.3.1 Expansion in function series......Page 278
6.3.2 The collocation method......Page 279
6.3.3 The method of subdomains......Page 283
6.3.4 Galerkin’s method......Page 284
6.3.5 The Rayleigh–Ritz method......Page 286
6.3.6 The finite-element method......Page 289
References......Page 305
7.1.1 The acoustic wave equation......Page 306
7.1.2 Planar acoustic waves......Page 311
7.1.3 Energetics of planar acoustic waves......Page 312
7.1.4 Reflection and refraction of planar acoustic waves......Page 314
7.1.5 Spherical waves......Page 317
7.1.6 Cylindrical waves......Page 322
7.1.7 Acoustic radiation from membranes and plates......Page 324
7.1.8 Waves in wave guides......Page 331
7.1.9 Acoustic waves in a slightly viscous fluid......Page 335
7.2.1 Dynamics of surface waves......Page 337
7.2.2 Sloshing of liquids in tanks......Page 340
7.2.3 Surface waves in a channel......Page 347
Exercises......Page 351
References......Page 354
8.1 Equations of motion......Page 356
8.2 Plane elastic waves in unbounded continua......Page 361
8.3 Energetics of elastic waves......Page 363
8.4 Reflection of elastic waves......Page 365
8.4.1 Reflection from a free boundary......Page 366
8.5 Rayleigh surface waves......Page 370
8.6 Reflection and refraction of planar acoustic waves......Page 374
Exercises......Page 376
References......Page 378
A The variational formulation of dynamics......Page 380
References......Page 382
B.1 Fourier representation and harmonic waves......Page 384
B.2 Phase velocity and group velocity......Page 386
References......Page 389
C Variational formulation for dynamics of plates......Page 390
References......Page 395
Index......Page 396