Vibrations and Acoustic Radiation of Thin Structures

Paul J. T. Filippi was a Directeur de Recherche at the Laboratoire de Mécanique et d'Acoustique (LMA-CNRS, Marseille) until the end of 1999. He was in charge of the Formation Doctorale d'Acoustique et Dynamique des Vibrations, which is, now, the Master de Mécanique at the University of Provence. He has published many papers on acoustics, vibrations of thin structures and vibro-acoustics.

• Preface 111 Equations Governing the Vibrations of Thin Structures 151.1 Introduction 151.1.1 General Considerations on Thin Structures 151.1.2 Overview of the Energy Method 161.2 Thin Plates 171.2.1 Plate with Constant Thickness 181.2.2 Plate with Variable Thickness 251.2.3 Boundary with an Angular Point 271.3 Beams 291.4 Circular Cylindrical Shells 311.5 Spherical Shells 381.5.1 Approximation of the Strain and Stress Tensors and Application of the Virtual Works Theorem 391.5.2 Regularity Conditions at the Apexes 461.6 Variational Form of the Equations Governing Harmonic Vibrations of Plates and Shells 491.6.1 Variational Form of the Plate Equation 501.6.2 Variational Form of the Shells Equations 511.7 Exercises 522 Vibratory Response of Thin Structures in vacuo: Resonance Modes, Forced Harmonic Regime, Transient Regime 532.1 Introduction 532.2 Vibrations of Constant Cross-Section Beams 552.2.1 Independent Solutions for the Homogenous Beam Equation 552.2.2 Response of an Infinite Beam to a Point Harmonic Force 572.2.3 Resonance Modes of Finite Length Beams 592.2.4 Response of a Finite Length Beam to a Harmonic Force 662.3 Vibrations of Plates 682.3.1 Free Vibrations of an Infinite Plate 682.3.2 Green’s Kernel and Green’s function for the Time Harmonic Plate Equation and Response of an Infinite Plate to a Harmonic Excitation 712.3.3 Harmonic Vibrations of a Plate of Finite Dimensions: General Definition and Theorems 732.3.4 Resonance Modes and Resonance Frequencies of Circular Plates with Uniform Boundary Conditions 762.3.5 Resonance Modes and Resonance Frequencies of Rectangular Plates with Uniform Boundary Conditions 842.3.6 Response of a Plate to a Harmonic Excitation: Resonance Modes Series Representation 972.3.7 Boundary Integral Equations and the Boundary Element Method 992.3.8 Resonance Frequencies of Plates with Variable Thickness 1172.3.9 Transient Response of an Infinite Plate with Constant Thickness 1192.4 Vibrations of Cylindrical Shells 1222.4.1 Free Oscillations of Cylindrical Shells of Infinite Length 1222.4.2 Green’s Tensor for the Cylindrical Shell Equation 1262.4.3 Harmonic Vibrations of a Cylindrical Shell of Finite Dimensions: General Definition and Theorems 1292.4.4 Resonance Modes of a Cylindrical Shell Closed by Shear Diaphragms at Both Ends 1302.4.5 Resonance Modes of a Cylindrical Shell Clamped at Both Ends 1332.4.6 Response of a Cylindrical Shell to a Harmonic Excitation: Resonance Modes Representation 1372.4.7 Boundary Integral Equations and Boundary Element Method 1382.5 Vibrations of Spherical Shells 1412.5.1 General Definition and Theorems 1412.5.2 Solution of the Time Harmonic Spherical Shell Equation 1432.6 Exercises 1453 Acoustic Radiation and Transmission by Thin Structures 1493.1 Introduction 1493.2 Sound Transmission Across a Piston in a One-Dimensional Waveguide 1513.2.1 Governing Equations 1513.2.2 Time Fourier Transform of the Equations – Response of the System to a Harmonic Excitation 1533.2.3 Response of the System to a Transient Excitation of the Piston 1593.3 A One-dimensional Example of a Cavity Closed by a Vibrating Boundary 1603.3.1 Equations Governing Free Harmonic Oscillations and their Reduced Form 1613.3.2 Transmission of Sound Across the Vibrating Boundary 1653.4 A Little Acoustics 1683.4.1 Variational Form of the Wave Equation and of the Helmholtz Equation 1683.4.2 Free-field Green’s Function of the Helmholtz Equation 1703.4.3 Series Expansions of the Free Field Green’s Function of the Helmholtz Equation 1703.4.4 Green’s Formula for the Helmholtz Operator and Green’s Representation of the Solution of the Helmholtz Equation 1723.4.5 Numerical Difficulties 1753.5 Infinite Structures 1763.5.1 Infinite Plate in Contact with a Single Fluid or Two Different Fluids 1763.5.2 Free Oscillations of an Infinite Circular Cylindrical Shell Filled with a vacuum and Immersed in a Fluid of Infinite Extent 1963.5.3 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 2023.6 Baffled Rectangular Plate 2033.6.1 General Theory: Eigenmodes, Resonance Modes, Series Expansion of the Response of the System 2033.6.2 Rectangular Plate Clamped along its Boundary: Numerical Approximation of the Resonance Modes 2093.6.3 Application: Transient Response of a Plate Struck by a Hammer 2223.7 General Method for the Harmonic Regime: Classical Variational Formulation and Green’s Representation of the Plate Displacement 2243.8 Baffled Plate Closing a Cavity 2283.8.1 Equations Governing the Harmonic Motion of the Plate-Cavity-External Fluid System 2293.8.2 Integro-differential Equation for the Plate Displacement and Matched Asymptotic Expansions 2323.8.3 Boundary Integral Representation of the Interior Acoustic Pressure 2373.8.4 Comparison between Numerical Predictions and Experiments 2383.9 Cylindrical Finite Length Baffled Shell Excited by a Turbulent Internal Flow 2433.9.1 Basic Equations and Green’s Representations of the Exterior and Interior Acoustic Pressures for a Normal Point Force 2453.9.2 Numerical Methods for Solving Equations (3.111) 2463.9.3 Comparison Between Numerical Results and Experimental Data 2483.10 Radiation by a Finite Length Cylindrical Shell Excited by an Internal Acoustic Source 2513.10.1 Statement of the Problem 2513.10.2 Boundary Integral Representations of the Radiated Pressure and of the Shell Displacement 2533.10.3 Green’s Representation of the Interior Acoustic Pressure and Matched Asymptotic Expansions 2563.10.4 Directivity Pattern of the Radiated Acoustic Pressure 2603.10.5 Numerical Method, Results and Concluding Remarks 2623.11 Diffraction of a Transient Acoustic Wave by a Line 2’ Shell 2643.11.1 Statement of the Problem 2663.11.2 Resonance Modes and Response of the System to an Incident Transient Acoustic Wave 2723.11.3 Numerical Method and Comparison between Numerical Prediction and Experimental Results 2743.12 Exercises 278Bibliography 279Notations 285Index 287