Vibrations and Waves in Continuous Mechanical Systems

Preface xi

1 Vibrations of strings and bars 1

1.1 Dynamics of strings and bars: the Newtonian formulation 1

1.1.1 Transverse dynamics of strings 1

1.1.2 Longitudinal dynamics of bars 6

1.1.3 Torsional dynamics of bars 7

1.2 Dynamics of strings and bars: the variational formulation 9

1.2.1 Transverse dynamics of strings 10

1.2.2 Longitudinal dynamics of bars 11

1.2.3 Torsional dynamics of bars 13

1.3 Free vibration problem: Bernoulli’s solution 14

1.4 Modal analysis 18

1.4.1 The eigenvalue problem 18

1.4.2 Orthogonality of eigenfunctions 24

1.4.3 The expansion theorem 25

1.4.4 Systems with discrete elements 27

1.5 The initial value problem: solution using Laplace transform 30

1.6 Forced vibration analysis 31

1.6.1 Harmonic forcing 32

1.6.2 General forcing 36

1.7 Approximate methods for continuous systems 40

1.7.1 Rayleigh method 41

1.7.2 Rayleigh–Ritz method 43

1.7.3 Ritz method 44

1.7.4 Galerkin method 47

1.8 Continuous systems with damping 50

1.8.1 Systems with distributed damping 50

1.8.2 Systems with discrete damping 53

1.9 Non-homogeneous boundary conditions 56

1.10 Dynamics of axially translating strings 57

1.10.1 Equation of motion 58

1.10.2 Modal analysis and discretization 58

1.10.3 Interaction with discrete elements 61

Exercises 62

References 67

2 One-dimensional wave equation: d’Alembert’s solution 69

2.1 D’Alembert’s solution of the wave equation 69

2.1.1 The initial value problem 72

2.1.2 The initial value problem: solution using Fourier transform 76

2.2 Harmonic waves and wave impedance 77

2.3 Energetics of wave motion 79

2.4 Scattering of waves 83

2.4.1 Reflection at a boundary 83

2.4.2 Scattering at a finite impedance 87

2.5 Applications of the wave solution 93

2.5.1 Impulsive start of a bar 93

2.5.2 Step-forcing of a bar with boundary damping 95

2.5.3 Axial collision of bars 99

2.5.4 String on a compliant foundation 102

2.5.5 Axially translating string 104

Exercises 107

References 112

3 Vibrations of beams 113

3.1 Equation of motion 113

3.1.1 The Newtonian formulation 113

3.1.2 The variational formulation 116

3.1.3 Various boundary conditions for a beam 118

3.1.4 Taut string and tensioned beam 120

3.2 Free vibration problem 121

3.2.1 Modal analysis 121

3.2.2 The initial value problem 132

3.3 Forced vibration analysis 133

3.3.1 Eigenfunction expansion method 134

3.3.2 Approximate methods 135

3.4 Non-homogeneous boundary conditions 137

3.5 Dispersion relation and flexural waves in a uniform beam 138

3.5.1 Energy transport 140

3.5.2 Scattering of flexural waves 142

3.6 The Timoshenko beam 144

3.6.1 Equations of motion 144

3.6.2 Harmonic waves and dispersion relation 147

3.7 Damped vibration of beams 149

3.8 Special problems in vibrations of beams 151

3.8.1 Influence of axial force on dynamic stability 151

3.8.2 Beam with eccentric mass distribution 155

3.8.3 Problems involving the motion of material points of a vibrating beam 159

3.8.4 Dynamics of rotating shafts 163

3.8.5 Dynamics of axially translating beams 165

3.8.6 Dynamics of fluid-conveying pipes 168

Exercises 171

References 178

4 Vibrations of membranes 179

4.1 Dynamics of a membrane 179

4.1.1 Newtonian formulation 179

4.1.2 Variational formulation 182

4.2 Modal analysis 185

4.2.1 The rectangular membrane 185

4.2.2 The circular membrane 190

4.3 Forced vibration analysis 197

4.4 Applications: kettledrum and condenser microphone 197

4.4.1 Modal analysis 197

4.4.2 Forced vibration analysis 201

4.5 Waves in membranes 202

4.5.1 Waves in Cartesian coordinates 202

4.5.2 Waves in polar coordinates 204

4.5.3 Energetics of membrane waves 207

4.5.4 Initial value problem for infinite membranes 208

4.5.5 Reflection of plane waves 209

Exercises 213

References 214

5 Vibrations of plates 217

5.1 Dynamics of plates 217

5.1.1 Newtonian formulation 217

5.2 Vibrations of rectangular plates 222

5.2.1 Free vibrations 222

5.2.2 Orthogonality of plate eigenfunctions 228

5.2.3 Forced vibrations 229

5.3 Vibrations of circular plates 231

5.3.1 Free vibrations 231

5.3.2 Forced vibrations 234

5.4 Waves in plates 236

5.5 Plates with varying thickness 238

Exercises 239

References 241

6 Boundary value and eigenvalue problems in vibrations 243

6.1 Self-adjoint operators and eigenvalue problems for undamped free vibrations 243

6.1.1 General properties and expansion theorem 243

6.1.2 Green’s functions and integral formulation of eigenvalue problems 252

6.1.3 Bounds for eigenvalues: Rayleigh’s quotient and other methods 255

6.2 Forced vibrations 259

6.2.1 Equations of motion 259

6.2.2 Green’s function for inhomogeneous vibration problems 260

6.3 Some discretization methods for free and forced vibrations 261

6.3.1 Expansion in function series 261

6.3.2 The collocation method 262

6.3.3 The method of subdomains 266

6.3.4 Galerkin’s method 267

6.3.5 The Rayleigh–Ritz method 269

6.3.6 The finite-element method 272

References 288

7 Waves in fluids 289

7.1 Acoustic waves in fluids 289

7.1.1 The acoustic wave equation 289

7.1.2 Planar acoustic waves 294

7.1.3 Energetics of planar acoustic waves 295

7.1.4 Reflection and refraction of planar acoustic waves 297

7.1.5 Spherical waves 300

7.1.6 Cylindrical waves 305

7.1.7 Acoustic radiation from membranes and plates 307

7.1.8 Waves in wave guides 314

7.1.9 Acoustic waves in a slightly viscous fluid 318

7.2 Surface waves in incompressible liquids 320

7.2.1 Dynamics of surface waves 320

7.2.2 Sloshing of liquids in tanks 323

7.2.3 Surface waves in a channel 330

Exercises 334

References 337

8 Waves in elastic continua 339

8.1 Equations of motion 339

8.2 Plane elastic waves in unbounded continua 344

8.3 Energetics of elastic waves 346

8.4 Reflection of elastic waves 348

8.4.1 Reflection from a free boundary 349

8.5 Rayleigh surface waves 353

8.6 Reflection and refraction of planar acoustic waves 357

Exercises 359

References 361

A The variational formulation of dynamics 363

References 365

B Harmonic waves and dispersion relation 367

B.1 Fourier representation and harmonic waves 367

B.2 Phase velocity and group velocity 369

References 372

C Variational formulation for dynamics of plates 373

References 378

Index 379