Vibrations and Acoustic Radiation of Thin Structures: Physical Basis, Theoretical Analysis and Numerical Methods

Preface 11

1 Equations Governing the Vibrations of Thin Structures 15

1.1 Introduction 15

1.1.1 General Considerations on Thin Structures 15

1.1.2 Overview of the Energy Method 16

1.2 Thin Plates 17

1.2.1 Plate with Constant Thickness 18

1.2.2 Plate with Variable Thickness 25

1.2.3 Boundary with an Angular Point 27

1.3 Beams 29

1.4 Circular Cylindrical Shells 31

1.5 Spherical Shells 38

1.5.1 Approximation of the Strain and Stress Tensors and Application of the Virtual Works Theorem 39

1.5.2 Regularity Conditions at the Apexes 46

1.6 Variational Form of the Equations Governing Harmonic Vibrations of Plates and Shells 49

1.6.1 Variational Form of the Plate Equation 50

1.6.2 Variational Form of the Shells Equations 51

1.7 Exercises 52

2 Vibratory Response of Thin Structures in vacuo: Resonance Modes, Forced Harmonic Regime, Transient Regime 53

2.1 Introduction 53

2.2 Vibrations of Constant Cross-Section Beams 55

2.2.1 Independent Solutions for the Homogenous Beam Equation 55

2.2.2 Response of an Infinite Beam to a Point Harmonic Force 57

2.2.3 Resonance Modes of Finite Length Beams 59

2.2.4 Response of a Finite Length Beam to a Harmonic Force 66

2.3 Vibrations of Plates 68

2.3.1 Free Vibrations of an Infinite Plate 68

2.3.2 Green’s Kernel and Green’s function for the Time Harmonic Plate Equation and Response of an Infinite Plate to a Harmonic Excitation 71

2.3.3 Harmonic Vibrations of a Plate of Finite Dimensions: General Definition and Theorems 73

2.3.4 Resonance Modes and Resonance Frequencies of Circular Plates with Uniform Boundary Conditions 76

2.3.5 Resonance Modes and Resonance Frequencies of Rectangular Plates with Uniform Boundary Conditions 84

2.3.6 Response of a Plate to a Harmonic Excitation: Resonance Modes Series Representation 97

2.3.7 Boundary Integral Equations and the Boundary Element Method 99

2.3.8 Resonance Frequencies of Plates with Variable Thickness 117

2.3.9 Transient Response of an Infinite Plate with Constant Thickness 119

2.4 Vibrations of Cylindrical Shells 122

2.4.1 Free Oscillations of Cylindrical Shells of Infinite Length 122

2.4.2 Green’s Tensor for the Cylindrical Shell Equation 126

2.4.3 Harmonic Vibrations of a Cylindrical Shell of Finite Dimensions: General Definition and Theorems 129

2.4.4 Resonance Modes of a Cylindrical Shell Closed by Shear Diaphragms at Both Ends 130

2.4.5 Resonance Modes of a Cylindrical Shell Clamped at Both Ends 133

2.4.6 Response of a Cylindrical Shell to a Harmonic Excitation: Resonance Modes Representation 137

2.4.7 Boundary Integral Equations and Boundary Element Method 138

2.5 Vibrations of Spherical Shells 141

2.5.1 General Definition and Theorems 141

2.5.2 Solution of the Time Harmonic Spherical Shell Equation 143

2.6 Exercises 145

3 Acoustic Radiation and Transmission by Thin Structures 149

3.1 Introduction 149

3.2 Sound Transmission Across a Piston in a One-Dimensional Waveguide 151

3.2.1 Governing Equations 151

3.2.2 Time Fourier Transform of the Equations – Response of the System to a Harmonic Excitation 153

3.2.3 Response of the System to a Transient Excitation of the Piston 159

3.3 A One-dimensional Example of a Cavity Closed by a Vibrating Boundary 160

3.3.1 Equations Governing Free Harmonic Oscillations and their Reduced Form 161

3.3.2 Transmission of Sound Across the Vibrating Boundary 165

3.4 A Little Acoustics 168

3.4.1 Variational Form of the Wave Equation and of the Helmholtz Equation 168

3.4.2 Free-field Green’s Function of the Helmholtz Equation 170

3.4.3 Series Expansions of the Free Field Green’s Function of the Helmholtz Equation 170

3.4.4 Green’s Formula for the Helmholtz Operator and Green’s Representation of the Solution of the Helmholtz Equation 172

3.4.5 Numerical Difficulties 175

3.5 Infinite Structures 176

3.5.1 Infinite Plate in Contact with a Single Fluid or Two Different Fluids 176

3.5.2 Free Oscillations of an Infinite Circular Cylindrical Shell Filled with a vacuum and Immersed in a Fluid of Infinite Extent 196

3.5.3 A Few Remarks on the Free Oscillations of an Infinite Circular Cylindrical Shell containing a Fluid and Immersed in a Second Fluid of Infinite Extent 202

3.6 Baffled Rectangular Plate 203

3.6.1 General Theory: Eigenmodes, Resonance Modes, Series Expansion of the Response of the System 203

3.6.2 Rectangular Plate Clamped along its Boundary: Numerical Approximation of the Resonance Modes 209

3.6.3 Application: Transient Response of a Plate Struck by a Hammer 222

3.7 General Method for the Harmonic Regime: Classical Variational Formulation and Green’s Representation of the Plate Displacement 224

3.8 Baffled Plate Closing a Cavity 228

3.8.1 Equations Governing the Harmonic Motion of the Plate-Cavity-External Fluid System 229

3.8.2 Integro-differential Equation for the Plate Displacement and Matched Asymptotic Expansions 232

3.8.3 Boundary Integral Representation of the Interior Acoustic Pressure 237

3.8.4 Comparison between Numerical Predictions and Experiments 238

3.9 Cylindrical Finite Length Baffled Shell Excited by a Turbulent Internal Flow 243

3.9.1 Basic Equations and Green’s Representations of the Exterior and Interior Acoustic Pressures for a Normal Point Force 245

3.9.2 Numerical Methods for Solving Equations (3.111) 246

3.9.3 Comparison Between Numerical Results and Experimental Data 248

3.10 Radiation by a Finite Length Cylindrical Shell Excited by an Internal Acoustic Source 251

3.10.1 Statement of the Problem 251

3.10.2 Boundary Integral Representations of the Radiated Pressure and of the Shell Displacement 253

3.10.3 Green’s Representation of the Interior Acoustic Pressure and Matched Asymptotic Expansions 256

3.10.4 Directivity Pattern of the Radiated Acoustic Pressure 260

3.10.5 Numerical Method, Results and Concluding Remarks 262

3.11 Diffraction of a Transient Acoustic Wave by a Line 2’ Shell 264

3.11.1 Statement of the Problem 266

3.11.2 Resonance Modes and Response of the System to an Incident Transient Acoustic Wave 272

3.11.3 Numerical Method and Comparison between Numerical Prediction and Experimental Results 274

3.12 Exercises 278

Bibliography 279

Notations 285

Index 287