Edited by Stephen A. Hambric Center for Acoustics & Vibration, Pennsylvania State University, State College, Pennsylvania, USA Shung H. Sung Consultant, Troy, Michigan, USA Donald J. Nefske Consultant, Troy, Michigan, USA

• Wiley Series in Acoustics, Noise and Vibration xivList of Contributors xv1 Overview 11.1 Introduction 11.2 Traditional Vibroacoustic Methods 21.2.1 Finite Element Method 21.2.2 Boundary Element Method 31.2.3 Statistical Energy Analysis 31.3 New Vibroacoustic Methods 41.3.1 Hybrid FE/SEA Method 41.3.2 Hybrid FE/TPA Method 41.3.3 Energy FE Analysis 41.3.4 Wave‐Based Structural Analysis 51.3.5 Future Developments 51.4 Choosing Numerical Methods 51.4.1 Geometrical Discretization 51.4.2 Solution Frequency Ranges 61.4.3 Type of Application 71.5 Chapter Organization 9References 92 Structural Vibrations 102.1 Introduction 102.2 Waves in Structures 112.2.1 Compressional and Shear Waves in Isotropic, Homogeneous Structures 112.2.2 Bending (Flexural) Waves in Beams and Plates 132.2.3 Bending Waves in Anisotropic Plates 172.2.4 Bending Waves in Stiffened Panels 202.2.5 Structural Wavenumbers 212.3 Modes of Vibration 222.3.1 Modes of Beams 222.3.2 Modes of Plates 252.3.3 Global and Local Modes of Stiffened Panels 282.3.4 Modal Density 302.4 Mobility and Impedance 302.4.1 Damping 342.5 Bending Waves in Infinite Structures 392.6 Coupled Oscillators, Power Flow, and the Basics of Statistical Energy Analysis 422.6.1 Equations of Motion 422.6.2 Power Input, Flow, and Dissipation 442.6.3 Oscillator-based Statistical Energy Analysis (SEA) 452.7 Environmental and Installation Effects 482.8 Summary 50References 503 Interior and Exterior Sound 523.1 Introduction 523.2 Interior Sound 523.2.1 Acoustic Wave Equation 523.2.2 Boundary Conditions 543.2.3 Natural Frequencies and Mode Shapes 553.2.4 Forced Sound‐Pressure Response 593.2.5 Steady‐State Sound‐Pressure Response 603.2.6 Enclosure Driven at Resonance 643.2.7 Random Sound‐Pressure Response 663.2.8 Transient Sound‐Pressure Response 683.3 Exterior Sound 703.3.1 Sound Radiation Measures 723.3.2 One‐Dimensional Sound Radiation 733.3.3 Sound Radiation from Basic Sources and Radiators 753.3.3.1 Pulsating Sphere and Monopole Source 753.3.3.2 Oscillating Sphere and Dipole Source 773.3.4 Helmholtz and Rayleigh Integral Equations 783.3.5 Example Applications 813.3.5.1 Planar Baffled Vibrating Plate 813.3.5.2 Vibrating Crown Surface 843.4 Summary 86References 864 Sound‐Structure Interaction Fundamentals 884.1 Introduction 884.2 Circular Piston Vibrating against an Acoustic Fluid 894.3 Fluid Loading of Structures 954.4 Structural Waves Vibrating against an Acoustic Fluid 994.5 Complementary Problem: Structural Vibrations Induced by Acoustic Pressure Waves 1054.6 Summary 113References 1135 Structural‐Acoustic Modal Analysis and Synthesis 1145.1 Introduction 1145.2 Coupled Structural‐Acoustic System 1145.2.1 Acoustic Cavity Modal Expansion 1155.2.2 Absorption Wall Impedance 1175.2.3 Structural Modal Expansion 1185.2.4 Coupled Structural‐Acoustic Modal Expansions 1205.3 Simplified Models 1215.3.1 Helmholtz Resonator Model 1215.3.2 Flexible Wall Model 1225.3.3 Coupled Structural and Acoustic Modes 1235.3.4 Dominant Structural Mode 1255.3.5 Dominant Cavity Mode 1275.4 Component Mode Synthesis 1325.4.1 Coupled Structural‐Acoustic Model 1325.4.2 Coupled Structures 1345.4.3 Coupled Cavities 1385.5 Summary 142References 1436 Structural‐Acoustic Finite‐Element Analysis for Interior Acoustics 1446.1 Introduction 1446.2 Acoustic Finite‐Element Analysis 1446.2.1 Acoustic Finite‐Element Formulation 1446.2.2 Flexible and Absorbent Walls 1476.2.3 Cavity Modal Analysis 1486.2.4 Flexible Wall Excitation 1506.2.5 Acoustic Impedance Modeling 1516.2.6 Porous Material Modeling 1526.3 Structural‐Acoustic Finite‐Element Analysis 1556.3.1 Structural Finite‐Element Formulation 1556.3.2 Structural System Synthesis 1586.4 Coupled Structural‐Acoustic Finite‐Element Formulation 1596.4.1 Coupled Modes and Resonance Frequencies 1606.4.2 Direct and Modal Frequency Response 1616.4.3 Random Response 1646.4.4 Participation Factors 1666.4.5 Transient Response 1716.4.5.1 Inverse Fourier Transform 1716.4.5.2 Direct Transient Response 1726.4.5.3 Modal Transient Response 1726.4.6 Structural‐ and Acoustic‐Response Variation 1736.5 Summary 177References 1777 Boundary‐Element Analysis 1797.1 Theory—Assumptions 1797.2 Theory—Overview of Theoretical Basis 1807.3 Boundary‐Element Computations 1837.4 The Rayleigh Integral 1847.5 The Kirchhoff–Helmholtz Equation 1867.6 Nonexistence/Nonuniqueness Difficulties 1917.7 Impedance Boundary Conditions 1997.8 Interpolation 2027.9 Applicability over Frequency and Spatial Resolution 2057.10 Implementation – Software Required 2087.11 Computer Resources Required 2107.12 Inputs and How to Determine them 2137.13 Outputs 2137.14 Applications 2147.15 Verification and Validation 2207.16 Error Analysis 2257.17 Summary 225References 2268 Structural and Acoustic Noise Control Material Modeling 2308.1 Introduction 2308.2 Damping Materials 2318.2.1 Damping Mechanisms 2318.2.2 Viscoelastic Damping 2328.2.3 Representation of the Frequency‐Dependent Properties of Viscoelastic Materials 2338.2.4 Identification of the Dynamic Properties of VEM 2348.2.5 Damping Design 2358.2.6 Modeling Structures with added Viscoelastic Damping 2388.2.7 Poroelastic Materials 2418.2.8 Open‐Cell Porous Materials 2418.2.9 Acoustic Impedance 2428.2.10 Models of Sound Propagation in a Porous Material 2448.2.11 Fluids Equivalent to Porous Materials 2448.2.12 Models for the Effective Density and the Bulk Modulus 2458.2.13 Perforated Plates 2478.2.14 Porous Materials having an Elastic Frame 2498.2.15 Measurement of the Parameters Governing Sound Propagation in Porous Materials 2498.2.15.1 Porosity 2498.2.15.2 Flow Resistivity 2508.2.15.3 Tortuosity 2508.2.15.4 Characteristics Lengths 2538.2.15.5 Mechanical Properties 2578.3 Modeling Multilayer Noise Control Materials 2578.3.1 Use of the Transfer Matrix Method 2588.3.2 Modeling a Sound Package within SEA 2638.3.3 Modeling a Sound Package within FE 2648.4 Conclusion 265References 2659 Structural–Acoustic Optimization 2689.1 Introduction 2689.2 Brief Survey of Structural–Acoustic Optimization 2699.3 Structural–Acoustic Optimization Procedures and Literature 2719.3.1 Applications 2719.3.2 Choice of Parameters 2729.3.3 Constraints 2739.3.4 Objective Functions 2749.4 Process of Structural–Acoustic Optimization 2779.4.1 Structural–Acoustic Simulation 2779.4.2 Strategy of Optimization 2799.4.2.1 Formulation of Optimization Problem 2799.4.2.2 Multiobjective Optimization 2809.4.2.3 Approximation Concepts and Approximate Optimization 2809.4.2.4 Optimization Methods 2829.4.3 Sensitivity Analysis 2849.4.3.1 Global Finite Differences 2849.4.3.2 Semi‐Analytic Sensitivity Analysis 2859.4.3.3 Adjoint Operators 2869.4.4 Special Techniques 2879.4.4.1 General Aspects and Ideas 2879.4.4.2 Efficient Reanalysis 2889.4.4.3 Frequency Ranges 2899.5 Minimization of Radiated Sound Power from a Finite Beam 2899.5.1 General Remarks 2899.5.2 Simulation Models 2899.5.3 Noise Transfer Function of Original Configurations 2919.5.4 Objective Function 2939.5.5 Formulation of Optimization Problem 2939.5.6 Optimization Strategy 2939.5.7 Optimization Results 2949.5.8 Discussion of Results 2979.5.9 Optimization of Complex Models 2989.6 Conclusions 298References 29910 Random and Stochastic Structural–Acoustic Analysis 30510.1 Introduction 30510.2 Uncertainty Quantification in Vibroacoustic Problems 30810.2.1 Antioptimization Method 30810.2.2 Possibilistic Method 30910.2.3 Probabilistic Method 30910.3 Random Variables and Random Fields 31010.4 Discretization of Random Quantities 31310.4.1 Karhunen–Loève Expansion 31310.4.2 Polynomial Chaos Expansion 31410.5 Stochastic FEM Formulation of Structural Vibrations 31710.5.1 General SFEM Formulation of Vibration Problems 31910.5.2 Stochastic FEM Formulation of Vibroacoustic Problems 32110.6 Numerical Simulation Procedures 32210.6.1 Intrusive SFEM 32210.6.2 Non‐intrusive SFEM 32310.7 Numerical Examples 32410.7.1 Discrete 2‐DOF Undamped System 32410.7.2 Free Vibration of Orthotropic Plate with Uncertain Parameters 32810.7.3 Random Equivalent Radiated Power 33310.8 Summary and Concluding Remarks 335References 33511 Statistical Energy Analysis 33911.1 Introduction 33911.2 SEA Background 33911.2.1 Characteristic Wavelengths 34011.2.2 Modes and Complexity 34111.2.3 Uncertainty 34211.3 General Wave‐Based SEA Formulation 34311.3.1 Piston Coupled with a Single Room 34411.3.2 Direct Field 34411.3.3 Reverberant Field 34511.3.4 Uncertainty 34611.3.5 Piston Response 34711.3.6 A Diffuse Reverberant Field 34811.3.7 Reciprocity between Direct Field Impedance and Diffuse Reverberant Load 34811.3.8 Coupling Power Proportionality 34911.3.9 Reverberant Power Balance Equations 35211.3.10 Recovering Local Responses 35411.3.11 Numerical Example 35411.3.12 An Arbitrary Number of Coupled Subsystems 35511.3.13 Summary 35611.4 Energy Storage 35611.4.1 Energy Storage in 1D Waveguides 35611.4.1.1 A Thin Beam 35911.4.1.2 Higher‐Order Wavetypes 36011.4.2 Energy Storage in 2D Waveguides 36111.4.2.1 A Thin Plate 36311.4.2.2 A Singly Curved Shell 36311.4.2.3 Higher Order Wavetypes 36411.4.3 Energy Storage in 3D Waveguides 36611.4.3.1 Numerical Example 36811.4.4 Summary of Modal Density Formulas 36911.5 Energy Transmission 37011.5.1 Point Junctions 37111.5.2 Line Junctions 37311.5.3 Area Junctions 37411.6 Power Input and Dissipation 37711.7 Example Applications 37811.7.1 Using SEA to Diagnose Transmission Paths 37811.7.2 Industrial Applications 37911.8 Summary 382References 38312 Hybrid FE‐SEA 38512.1 Introduction 38512.2 Overview 38512.2.1 Low‐, Mid‐, and High‐Frequency Ranges 38512.2.2 The Mid‐Frequency Problem 38612.3 The Hybrid FE‐SEA Method 38712.3.1 System 38712.3.2 A Statistical Subsystem 38712.3.3 Direct and Reverberant Fields 38812.3.4 Ensemble Average Reverberant Loading 38812.3.5 Coupling a Deterministic and Statistical Subsystem 38912.4 Example 39012.4.1 System 39012.4.2 Deterministic Equations of Motion 39012.4.3 Direct Field Dynamic Stiffness of SEA Subsystems 39212.4.4 Ensemble Average Response 39212.4.5 Reverberant Power Balance 39312.4.6 Computing the Coupled Response 39412.5 Implementation and Algorithms 39512.5.1 Overview 39512.5.2 Point Connection 39512.5.3 Line Connection 39612.5.4 Area Connection 39612.6 Application Examples 39712.6.1 Simple Numerical Example 39712.6.2 Industrial Applications 39812.7 Summary 403References 40313 Hybrid Transfer Path Analysis 40613.1 Introduction 40613.2 Transfer Path Analysis 40613.3 Hybrid Transfer Path Analysis 40813.4 Vibro‐Acoustic Transfer Function 40913.4.1 Measured VATF 40913.4.2 Predicted VATF 41113.5 Operating Powertrain Loads 41213.5.1 Measured Stiffness Method 41213.5.2 Matrix Inversion Method 41513.5.3 Predicted Powertrain Loads 41613.6 HTPA Applications 41713.6.1 Predicted Operating Loads + Measured VATFs 41713.6.1.1 Predicted Powertrain Loads 41813.6.1.2 Measured VATFs 41913.6.1.3 Predicted Interior SPL 42113.6.2 Predicted VATFs + Measured Operating Loads 42413.6.2.1 Predicted VATFs 42413.6.2.2 Measured Operating Loads 42613.6.2.3 Predicted Interior SPL 42613.6.2.4 Structural Modification 42713.7 Vibrational Power Flow 42913.8 Summary 430References 43114 Energy Finite Element Analysis 43314.1 Overview of Energy Finite Element Analysis 43314.2 Developing the Governing Differential Equations in EFEA 43514.2.1 Derivation of the Governing Differential Equation for an Acoustic Space 43614.2.2 Derivation of the Governing Differential Equation for the Bending Response of a Plate 43914.3 Power Transfer Coefficients 44114.3.1 Power Transfer Coefficients between Two Plates 44114.3.2 Power Transfer Coefficients between a Plate and an Acoustic Space 44414.3.2.1 Power Transmission from Plate to Acoustic Space 44514.3.2.2 Power Transmission from Acoustic Space to Plate 44714.4 Formulation of Energy Finite Element System of Equations 44714.4.1 Finite Element Formulation of EFEA System of Equations 44714.4.2 EFEA Joint Matrix 44814.4.3 Input Power 45014.4.4 EFEA System of Equations for a Simple Plate‐Acoustic System 45114.5 Applications 45514.5.1 Automotive Application 45514.5.2 Aircraft Application 46114.5.3 Naval Application 464References 47015 Wave‐based Structural Modeling 47215.1 General Approach 47215.1.1 Background 47315.1.2 Advantages/Limitations 47415.2 Theoretical Formulation 47515.2.1 Elementary Rod Theory 47515.2.2 Straight Beams, Timoshenko Beam Theory 47715.2.3 Reflections at Boundaries 47915.2.4 Wave Propagation Solution 48015.2.5 Spectral Element Method 48115.3 Wave‐based Spectral Finite Element Formulation 48315.3.1 Dynamic Stiffness Matrix of a Substructure 48315.3.2 State Vector Formulation and the Eigenvalue Problem 48415.3.3 Relations between Dynamic Stiffness and Transfer Matrices 48515.3.4 Derivation of a Numerical Spectral Matrix for Beam Problems 48715.3.5 Numerical Spectral Matrix for General Periodic Structures 48915.4 Applications 49115.4.1 Beam Analysis via Analytical Approaches 49115.4.2 Beam Analysis via Numerical Approach (WSFEM) 49115.4.3 General Periodic Structure Analysis via Numerical Approach (WSFEM) 49515.4.4 Range of Applicability 49915.4.5 Implementation–Software Required 50015.4.6 Computer Resources Required 50015.4.7 Inputs and How to Determine Them 50115.4.8 Forces/Enforced Displacements 50115.4.9 Boundary Conditions 50115.4.10 Material Properties 50215.4.11 Outputs 50215.4.12 Verification and Validation 50215.5 Conclusion/Summary 503References 503Index 506