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Chapter 1 MECHANICAL SYSTEMS WITH ONE DEGREE OF FREEDOM 1

1.1 A simple mass-spring system 1

1.2 Free vibrations 6

1.3 Transient vibrations 12

1.4 Forced harmonic vibrations 21

1.5 Fourier series 25

1.6 Complex notation 27

Problems 31

Chapter 2 FREQUENCY DOMAIN 34

2.1 Introduction 34

2.2 Frequency response 36

2.3 Correlation functions 39

2.4 Spectral density 42

2.5 Examples of spectral density 44

2.6 Coherence 48

2.7 Time averages of power and energy 49

2.8 Frequency response and point mobility functions 55

2.9 Loss factor 61

2.10 Response of a 1-DOF system,a summary 66

Problems 69

Chapter 3 WAVES IN SOLIDS 72

3.1 Stresses and strains 72

3.2 Losses in solids 81

3.3 Transverse waves 86

3.4 Longitudinal waves 89

3.5 Torsional waves 93

3.6 Waves on a string 95

3.7 Bending or flexural waves-beams 96

3.8 Waves on strings and beams-a comparison 104

3.9 Flexural waves-plates 107

3.10 Orthotropic plates 114

3.11 Energy flow 116

Problems 118

Chapter 4 INTERACTION BETWEEN LONGITUDINAL AND TRANSVERSE WAVES 120

4.1 Generalised wave equation 120

4.2 Intensity 125

4.3 Coupling between longitudinal and transverse waves 126

4.4 Bending of thick beams/plates 131

4.5 Quasi longitudinal waves in thick plates 146

4.6 Rayleigh waves 149

4.7 Sandwich plates-general 150

4.8 Bending of sandwich plates 152

4.9 Equations governing bending of sandwich plates 153

4.10 Wavenumbers of sandwich plates 156

4.11 Bending stiffness of sandwich plates 158

4.12 Bending of Ⅰ-beams 160

Problems 163

Chapter 5 WAVE ATTENUATION DUE TO LOSSES AND TRANSMISSION ACROSS JUNCTIONS 165

5.1 Excitation and propagation of L-waves 166

5.2 Excitation and propagation of F-waves 170

5.3 Point excited infinite plate 175

5.4 Spatial Fourier transforms 180

5.5 Added damping 187

5.6 Losses in sandwich plates 193

5.7 Coupling between flexural and inplane waves 197

5.8 Transmission of F-waves across junctions,diffuse incidence 202

5.9 Transmission of F-waves across junctions,normal incidence 210

5.10 Atenuation due to change of cross section 211

5.11 Some other methods to increase attenuation 215

5.12 Velocity level differences and transmission losses 216

5.13 Measurements on junctions between beams 220

Problems 227

Chapter 6 LONGITUDINAL VIBRATIONS OF FINITE BEAMS 230

6.1 Free longitudinal vibrations in finite beams 230

6.2 Forced longitudinal vibrations in finite beams 240

6.3 The mode summation technique 247

6.4 Kinetic energy of vibrating beam 250

6.5 Mobilities 256

6.6 Mass mounted on a rod 259

6.7 Transfer matrices 263

Problems 269

Chapter 7 FLEXURAL VIBRATIONS OF FINITE BEAMS 271

7.1 Free flexural vibrations of beams 271

7.2 Orthogonality and norm of eigenfunctions 281

7.3 Forced excitation of F-waves 285

7.4 Mode summation and modal parameters 289

7.5 Point mobility and power 296

7.6 Transfer matrices for bending of beams 301

7.7 Infinite periodic structures 305

7.8 Forced vibration of periodic structures 311

7.9 Finite composite beam 316

Problems 322

Chapter 8 FLEXURAL VIBRATIONS OF FINITE PLATES 325

8.1 Free vibrations of simply supported plates 326

8.2 Forced response of a simply supported plate 332

8.3 Forced excitation of a rectangular plate with two opposite sides simply supported 338

8.4 Power and energy 343

8.5 Mobility of plates 348

8.6 The Rayleigh-Ritz method 351

8.7 Application of the Rayleigh-Ritz method 358

8.8 Non flat plates 365

8.9 The effect of an added mass or mass-spring system on plate vibrations 367

8.10 Small disturbances 371

8.11 Plates mounted on resilient layers 375

8.12 Vibration of orthotropic plates 380

8.13 Circular and homogeneous plates 381

8.14 Bending of plates in tension 387

Problems 390

References 393

Index 398