The subject of vibrations is of fundamental importance in engineering and technology. Discrete modelling is sufficient to understand the dynamics of many vibrating systems; however a large number of vibration phenomena are far more easily understood when modelled as continuous systems. The theory of vibrations in continuous systems is crucial to the understanding of engineering problems in areas as diverse as automotive brakes, overhead transmission lines, liquid filled tanks, ultrasonic testing or room acoustics.Starting from an elementary level, Vibrations and Waves in Continuous Mechanical Systems helps develop a comprehensive understanding of the theory of these systems and the tools with which to analyse them, before progressing to more advanced topics.
Author(s): Hagedorn P., DasGupta A.
Year: 2007
Language: English
Pages: 382
Vibrations and Waves in Continuous Mechanical Systems......Page 3
Contents......Page 7
Preface......Page 13
1.1.1 Transverse dynamics of strings......Page 17
1.1.2 Longitudinal dynamics of bars......Page 22
1.1.3 Torsional dynamics of bars......Page 23
1.2 Dynamics of strings and bars: the variational formulation......Page 25
1.2.1 Transverse dynamics of strings......Page 26
1.2.2 Longitudinal dynamics of bars......Page 27
1.2.3 Torsional dynamics of bars......Page 29
1.3 Free vibration problem: Bernoulli’s solution......Page 30
1.4.1 The eigenvalue problem......Page 34
1.4.2 Orthogonality of eigenfunctions......Page 40
1.4.3 The expansion theorem......Page 41
1.4.4 Systems with discrete elements......Page 43
1.5 The initial value problem: solution using Laplace transform......Page 46
1.6 Forced vibration analysis......Page 47
1.6.1 Harmonic forcing......Page 48
1.6.2 General forcing......Page 52
1.7 Approximate methods for continuous systems......Page 56
1.7.1 Rayleigh method......Page 57
1.7.2 Rayleigh–Ritz method......Page 59
1.7.3 Ritz method......Page 60
1.7.4 Galerkin method......Page 63
1.8.1 Systems with distributed damping......Page 66
1.8.2 Systems with discrete damping......Page 69
1.9 Non-homogeneous boundary conditions......Page 72
1.10 Dynamics of axially translating strings......Page 73
1.10.2 Modal analysis and discretization......Page 74
1.10.3 Interaction with discrete elements......Page 77
Exercises......Page 78
References......Page 83
2.1 D’Alembert’s solution of the wave equation......Page 85
2.1.1 The initial value problem......Page 88
2.1.2 The initial value problem: solution using Fourier transform......Page 92
2.2 Harmonic waves and wave impedance......Page 93
2.3 Energetics of wave motion......Page 95
2.4.1 Reflection at a boundary......Page 99
2.4.2 Scattering at a finite impedance......Page 103
2.5.1 Impulsive start of a bar......Page 109
2.5.2 Step-forcing of a bar with boundary damping......Page 111
2.5.3 Axial collision of bars......Page 115
2.5.4 String on a compliant foundation......Page 118
2.5.5 Axially translating string......Page 120
Exercises......Page 123
References......Page 128
3.1.1 The Newtonian formulation......Page 129
3.1.2 The variational formulation......Page 132
3.1.3 Various boundary conditions for a beam......Page 134
3.1.4 Taut string and tensioned beam......Page 136
3.2.1 Modal analysis......Page 137
3.2.2 The initial value problem......Page 148
3.3 Forced vibration analysis......Page 149
3.3.1 Eigenfunction expansion method......Page 150
3.3.2 Approximate methods......Page 151
3.4 Non-homogeneous boundary conditions......Page 153
3.5 Dispersion relation and flexural waves in a uniform beam......Page 154
3.5.1 Energy transport......Page 156
3.5.2 Scattering of flexural waves......Page 158
3.6.1 Equations of motion......Page 160
3.6.2 Harmonic waves and dispersion relation......Page 163
3.7 Damped vibration of beams......Page 165
3.8.1 Influence of axial force on dynamic stability......Page 167
3.8.2 Beam with eccentric mass distribution......Page 171
3.8.3 Problems involving the motion of material points of a vibrating beam......Page 175
3.8.4 Dynamics of rotating shafts......Page 179
3.8.5 Dynamics of axially translating beams......Page 181
3.8.6 Dynamics of fluid-conveying pipes......Page 184
Exercises......Page 187
References......Page 194
4.1.1 Newtonian formulation......Page 195
4.1.2 Variational formulation......Page 198
4.2.1 The rectangular membrane......Page 201
4.2.2 The circular membrane......Page 206
4.4.1 Modal analysis......Page 213
4.4.2 Forced vibration analysis......Page 217
4.5.1 Waves in Cartesian coordinates......Page 218
4.5.2 Waves in polar coordinates......Page 220
4.5.3 Energetics of membrane waves......Page 223
4.5.4 Initial value problem for infinite membranes......Page 224
4.5.5 Reflection of plane waves......Page 225
Exercises......Page 229
References......Page 230
5.1.1 Newtonian formulation......Page 233
5.2.1 Free vibrations......Page 238
5.2.2 Orthogonality of plate eigenfunctions......Page 244
5.2.3 Forced vibrations......Page 245
5.3.1 Free vibrations......Page 247
5.3.2 Forced vibrations......Page 250
5.4 Waves in plates......Page 252
5.5 Plates with varying thickness......Page 254
Exercises......Page 255
References......Page 257
6.1.1 General properties and expansion theorem......Page 259
6.1.2 Green’s functions and integral formulation of eigenvalue problems......Page 268
6.1.3 Bounds for eigenvalues: Rayleigh’s quotient and other methods......Page 271
6.2.1 Equations of motion......Page 275
6.2.2 Green’s function for inhomogeneous vibration problems......Page 276
6.3.1 Expansion in function series......Page 277
6.3.2 The collocation method......Page 278
6.3.3 The method of subdomains......Page 282
6.3.4 Galerkin’s method......Page 283
6.3.5 The Rayleigh–Ritz method......Page 285
6.3.6 The finite-element method......Page 288
References......Page 304
7.1.1 The acoustic wave equation......Page 305
7.1.2 Planar acoustic waves......Page 310
7.1.3 Energetics of planar acoustic waves......Page 311
7.1.4 Reflection and refraction of planar acoustic waves......Page 313
7.1.5 Spherical waves......Page 316
7.1.6 Cylindrical waves......Page 321
7.1.7 Acoustic radiation from membranes and plates......Page 323
7.1.8 Waves in wave guides......Page 330
7.1.9 Acoustic waves in a slightly viscous fluid......Page 334
7.2.1 Dynamics of surface waves......Page 336
7.2.2 Sloshing of liquids in tanks......Page 339
7.2.3 Surface waves in a channel......Page 346
Exercises......Page 350
References......Page 353
8.1 Equations of motion......Page 355
8.2 Plane elastic waves in unbounded continua......Page 360
8.3 Energetics of elastic waves......Page 362
8.4 Reflection of elastic waves......Page 364
8.4.1 Reflection from a free boundary......Page 365
8.5 Rayleigh surface waves......Page 369
8.6 Reflection and refraction of planar acoustic waves......Page 373
Exercises......Page 375
References......Page 377
A The variational formulation of dynamics......Page 379
References......Page 381
B.1 Fourier representation and harmonic waves......Page 383
B.2 Phase velocity and group velocity......Page 385
References......Page 388
C Variational formulation for dynamics of plates......Page 389
References......Page 394
Index......Page 395
