Engineering Vibroacoustic Analysis: Methods and Applications

Content: Wiley Series in Acoustics, Noise and Vibration xiv    List of Contributors xv    1 Overview 1    1.1 Introduction 1    1.2 Traditional Vibroacoustic Methods 2    1.2.1 Finite Element Method 2    1.2.2 Boundary Element Method 3    1.2.3 Statistical Energy Analysis 3    1.3 New Vibroacoustic Methods 4    1.3.1 Hybrid FE/SEA Method 4    1.3.2 Hybrid FE/TPA Method 4    1.3.3 Energy FE Analysis 4    1.3.4 Wave   Based Structural Analysis 5    1.3.5 Future Developments 5    1.4 Choosing Numerical Methods 5    1.4.1 Geometrical Discretization 5    1.4.2 Solution Frequency Ranges 6    1.4.3 Type of Application 7    1.5 Chapter Organization 9    References 9    2 Structural Vibrations 10    2.1 Introduction 10    2.2 Waves in Structures 11    2.2.1 Compressional and Shear Waves in Isotropic, Homogeneous Structures 11    2.2.2 Bending (Flexural) Waves in Beams and Plates 13    2.2.3 Bending Waves in Anisotropic Plates 17    2.2.4 Bending Waves in Stiffened Panels 20    2.2.5 Structural Wavenumbers 21    2.3 Modes of Vibration 22    2.3.1 Modes of Beams 22    2.3.2 Modes of Plates 25    2.3.3 Global and Local Modes of Stiffened Panels 28    2.3.4 Modal Density 30    2.4 Mobility and Impedance 30    2.4.1 Damping 34    2.5 Bending Waves in Infinite Structures 39    2.6 Coupled Oscillators, Power Flow, and the Basics of Statistical Energy Analysis 42    2.6.1 Equations of Motion 42    2.6.2 Power Input, Flow, and Dissipation 44    2.6.3 Oscillator-based Statistical Energy Analysis (SEA) 45    2.7 Environmental and Installation Effects 48    2.8 Summary 50    References 50    3 Interior and Exterior Sound 52    3.1 Introduction 52    3.2 Interior Sound 52    3.2.1 Acoustic Wave Equation 52    3.2.2 Boundary Conditions 54    3.2.3 Natural Frequencies and Mode Shapes 55    3.2.4 Forced Sound   Pressure Response 59    3.2.5 Steady   State Sound   Pressure Response 60    3.2.6 Enclosure Driven at Resonance 64    3.2.7 Random Sound   Pressure Response 66    3.2.8 Transient Sound   Pressure Response 68    3.3 Exterior Sound 70    3.3.1 Sound Radiation Measures 72    3.3.2 One   Dimensional Sound Radiation 73    3.3.3 Sound Radiation from Basic Sources and Radiators 75    3.3.3.1 Pulsating Sphere and Monopole Source 75    3.3.3.2 Oscillating Sphere and Dipole Source 77    3.3.4 Helmholtz and Rayleigh Integral Equations 78    3.3.5 Example Applications 81    3.3.5.1 Planar Baffled Vibrating Plate 81    3.3.5.2 Vibrating Crown Surface 84    3.4 Summary 86    References 86    4 Sound   Structure Interaction Fundamentals 88    4.1 Introduction 88    4.2 Circular Piston Vibrating against an Acoustic Fluid 89    4.3 Fluid Loading of Structures 95    4.4 Structural Waves Vibrating against an Acoustic Fluid 99    4.5 Complementary Problem: Structural Vibrations Induced by Acoustic Pressure Waves 105    4.6 Summary 113    References 113    5 Structural   Acoustic Modal Analysis and Synthesis 114    5.1 Introduction 114    5.2 Coupled Structural   Acoustic System 114    5.2.1 Acoustic Cavity Modal Expansion 115    5.2.2 Absorption Wall Impedance 117    5.2.3 Structural Modal Expansion 118    5.2.4 Coupled Structural   Acoustic Modal Expansions 120    5.3 Simplified Models 121    5.3.1 Helmholtz Resonator Model 121    5.3.2 Flexible Wall Model 122    5.3.3 Coupled Structural and Acoustic Modes 123    5.3.4 Dominant Structural Mode 125    5.3.5 Dominant Cavity Mode 127    5.4 Component Mode Synthesis 132    5.4.1 Coupled Structural   Acoustic Model 132    5.4.2 Coupled Structures 134    5.4.3 Coupled Cavities 138    5.5 Summary 142    References 143    6 Structural   Acoustic Finite   Element Analysis for Interior Acoustics 144    6.1 Introduction 144    6.2 Acoustic Finite   Element Analysis 144    6.2.1 Acoustic Finite   Element Formulation 144    6.2.2 Flexible and Absorbent Walls 147    6.2.3 Cavity Modal Analysis 148    6.2.4 Flexible Wall Excitation 150    6.2.5 Acoustic Impedance Modeling 151    6.2.6 Porous Material Modeling 152    6.3 Structural   Acoustic Finite   Element Analysis 155    6.3.1 Structural Finite   Element Formulation 155    6.3.2 Structural System Synthesis 158    6.4 Coupled Structural   Acoustic Finite   Element Formulation 159    6.4.1 Coupled Modes and Resonance Frequencies 160    6.4.2 Direct and Modal Frequency Response 161    6.4.3 Random Response 164    6.4.4 Participation Factors 166    6.4.5 Transient Response 171    6.4.5.1 Inverse Fourier Transform 171    6.4.5.2 Direct Transient Response 172    6.4.5.3 Modal Transient Response 172    6.4.6 Structural    and Acoustic   Response Variation 173    6.5 Summary 177    References 177    7 Boundary   Element Analysis 179    7.1 Theory   Assumptions 179    7.2 Theory   Overview of Theoretical Basis 180    7.3 Boundary   Element Computations 183    7.4 The Rayleigh Integral 184    7.5 The Kirchhoff   Helmholtz Equation 186    7.6 Nonexistence/Nonuniqueness Difficulties 191    7.7 Impedance Boundary Conditions 199    7.8 Interpolation 202    7.9 Applicability over Frequency and Spatial Resolution 205    7.10 Implementation     Software Required 208    7.11 Computer Resources Required 210    7.12 Inputs and How to Determine them 213    7.13 Outputs 213    7.14 Applications 214    7.15 Verification and Validation 220    7.16 Error Analysis 225    7.17 Summary 225    References 226    8 Structural and Acoustic Noise Control Material Modeling 230    8.1 Introduction 230    8.2 Damping Materials 231    8.2.1 Damping Mechanisms 231    8.2.2 Viscoelastic Damping 232    8.2.3 Representation of the Frequency   Dependent Properties of Viscoelastic Materials 233    8.2.4 Identification of the Dynamic Properties of VEM 234    8.2.5 Damping Design 235    8.2.6 Modeling Structures with added Viscoelastic Damping 238    8.2.7 Poroelastic Materials 241    8.2.8 Open   Cell Porous Materials 241    8.2.9 Acoustic Impedance 242    8.2.10 Models of Sound Propagation in a Porous Material 244    8.2.11 Fluids Equivalent to Porous Materials 244    8.2.12 Models for the Effective Density and the Bulk Modulus 245    8.2.13 Perforated Plates 247    8.2.14 Porous Materials having an Elastic Frame 249    8.2.15 Measurement of the Parameters Governing Sound Propagation in Porous Materials 249    8.2.15.1 Porosity 249    8.2.15.2 Flow Resistivity 250    8.2.15.3 Tortuosity 250    8.2.15.4 Characteristics Lengths 253    8.2.15.5 Mechanical Properties 257    8.3 Modeling Multilayer Noise Control Materials 257    8.3.1 Use of the Transfer Matrix Method 258    8.3.2 Modeling a Sound Package within SEA 263    8.3.3 Modeling a Sound Package within FE 264    8.4 Conclusion 265    References 265    9 Structural   Acoustic Optimization 268    9.1 Introduction 268    9.2 Brief Survey of Structural   Acoustic Optimization 269    9.3 Structural   Acoustic Optimization Procedures and Literature 271    9.3.1 Applications 271    9.3.2 Choice of Parameters 272    9.3.3 Constraints 273    9.3.4 Objective Functions 274    9.4 Process of Structural   Acoustic Optimization 277    9.4.1 Structural   Acoustic Simulation 277    9.4.2 Strategy of Optimization 279    9.4.2.1 Formulation of Optimization Problem 279    9.4.2.2 Multiobjective Optimization 280    9.4.2.3 Approximation Concepts and Approximate Optimization 280    9.4.2.4 Optimization Methods 282    9.4.3 Sensitivity Analysis 284    9.4.3.1 Global Finite Differences 284    9.4.3.2 Semi   Analytic Sensitivity Analysis 285    9.4.3.3 Adjoint Operators 286    9.4.4 Special Techniques 287    9.4.4.1 General Aspects and Ideas 287    9.4.4.2 Efficient Reanalysis 288    9.4.4.3 Frequency Ranges 289    9.5 Minimization of Radiated Sound Power from a Finite Beam 289    9.5.1 General Remarks 289    9.5.2 Simulation Models 289    9.5.3 Noise Transfer Function of Original Configurations 291    9.5.4 Objective Function 293    9.5.5 Formulation of Optimization Problem 293    9.5.6 Optimization Strategy 293    9.5.7 Optimization Results 294    9.5.8 Discussion of Results 297    9.5.9 Optimization of Complex Models 298    9.6 Conclusions 298    References 299    10 Random and Stochastic Structural   Acoustic Analysis 305    10.1 Introduction 305    10.2 Uncertainty Quantification in Vibroacoustic Problems 308    10.2.1 Antioptimization Method 308    10.2.2 Possibilistic Method 309    10.2.3 Probabilistic Method 309    10.3 Random Variables and Random Fields 310    10.4 Discretization of Random Quantities 313    10.4.1 Karhunen   Loeve Expansion 313    10.4.2 Polynomial Chaos Expansion 314    10.5 Stochastic FEM Formulation of Structural Vibrations 317    10.5.1 General SFEM Formulation of Vibration Problems 319    10.5.2 Stochastic FEM Formulation of Vibroacoustic Problems 321    10.6 Numerical Simulation Procedures 322    10.6.1 Intrusive SFEM 322    10.6.2 Non   intrusive SFEM 323    10.7 Numerical Examples 324    10.7.1 Discrete 2   DOF Undamped System 324    10.7.2 Free Vibration of Orthotropic Plate with Uncertain Parameters 328    10.7.3 Random Equivalent Radiated Power 333    10.8 Summary and Concluding Remarks 335    References 335    11 Statistical Energy Analysis 339    11.1 Introduction 339    11.2 SEA Background 339    11.2.1 Characteristic Wavelengths 340    11.2.2 Modes and Complexity 341    11.2.3 Uncertainty 342    11.3 General Wave   Based SEA Formulation 343    11.3.1 Piston Coupled with a Single Room 344    11.3.2 Direct Field 344    11.3.3 Reverberant Field 345    11.3.4 Uncertainty 346    11.3.5 Piston Response 347    11.3.6 A Diffuse Reverberant Field 348    11.3.7 Reciprocity between Direct Field Impedance and Diffuse Reverberant Load 348    11.3.8 Coupling Power Proportionality 349    11.3.9 Reverberant Power Balance Equations 352    11.3.10 Recovering Local Responses 354    11.3.11 Numerical Example 354    11.3.12 An Arbitrary Number of Coupled Subsystems 355    11.3.13 Summary 356    11.4 Energy Storage 356    11.4.1 Energy Storage in 1D Waveguides 356    11.4.1.1 A Thin Beam 359    11.4.1.2 Higher   Order Wavetypes 360    11.4.2 Energy Storage in 2D Waveguides 361    11.4.2.1 A Thin Plate 363    11.4.2.2 A Singly Curved Shell 363    11.4.2.3 Higher Order Wavetypes 364    11.4.3 Energy Storage in 3D Waveguides 366    11.4.3.1 Numerical Example 368    11.4.4 Summary of Modal Density Formulas 369    11.5 Energy Transmission 370    11.5.1 Point Junctions 371    11.5.2 Line Junctions 373    11.5.3 Area Junctions 374    11.6 Power Input and Dissipation 377    11.7 Example Applications 378    11.7.1 Using SEA to Diagnose Transmission Paths 378    11.7.2 Industrial Applications 379    11.8 Summary 382    References 383    12 Hybrid FE   SEA 385    12.1 Introduction 385    12.2 Overview 385    12.2.1 Low   , Mid   , and High   Frequency Ranges 385    12.2.2 The Mid   Frequency Problem 386    12.3 The Hybrid FE   SEA Method 387    12.3.1 System 387    12.3.2 A Statistical Subsystem 387    12.3.3 Direct and Reverberant Fields 388    12.3.4 Ensemble Average Reverberant Loading 388    12.3.5 Coupling a Deterministic and Statistical Subsystem 389    12.4 Example 390    12.4.1 System 390    12.4.2 Deterministic Equations of Motion 390    12.4.3 Direct Field Dynamic Stiffness of SEA Subsystems 392    12.4.4 Ensemble Average Response 392    12.4.5 Reverberant Power Balance 393    12.4.6 Computing the Coupled Response 394    12.5 Implementation and Algorithms 395    12.5.1 Overview 395    12.5.2 Point Connection 395    12.5.3 Line Connection 396    12.5.4 Area Connection 396    12.6 Application Examples 397    12.6.1 Simple Numerical Example 397    12.6.2 Industrial Applications 398    12.7 Summary 403    References 403    13 Hybrid Transfer Path Analysis 406    13.1 Introduction 406    13.2 Transfer Path Analysis 406    13.3 Hybrid Transfer Path Analysis 408    13.4 Vibro   Acoustic Transfer Function 409    13.4.1 Measured VATF 409    13.4.2 Predicted VATF 411    13.5 Operating Powertrain Loads 412    13.5.1 Measured Stiffness Method 412    13.5.2 Matrix Inversion Method 415    13.5.3 Predicted Powertrain Loads 416    13.6 HTPA Applications 417    13.6.1 Predicted Operating Loads + Measured VATFs 417    13.6.1.1 Predicted Powertrain Loads 418    13.6.1.2 Measured VATFs 419    13.6.1.3 Predicted Interior SPL 421    13.6.2 Predicted VATFs + Measured Operating Loads 424    13.6.2.1 Predicted VATFs 424    13.6.2.2 Measured Operating Loads 426    13.6.2.3 Predicted Interior SPL 426    13.6.2.4 Structural Modification 427    13.7 Vibrational Power Flow 429    13.8 Summary 430    References 431    14 Energy Finite Element Analysis 433    14.1 Overview of Energy Finite Element Analysis 433    14.2 Developing the Governing Differential Equations in EFEA 435    14.2.1 Derivation of the Governing Differential Equation for an Acoustic Space 436    14.2.2 Derivation of the Governing Differential Equation for the Bending Response of a Plate 439    14.3 Power Transfer Coefficients 441    14.3.1 Power Transfer Coefficients between Two Plates 441    14.3.2 Power Transfer Coefficients between a Plate and an Acoustic Space 444    14.3.2.1 Power Transmission from Plate to Acoustic Space 445    14.3.2.2 Power Transmission from Acoustic Space to Plate 447    14.4 Formulation of Energy Finite Element System of Equations 447    14.4.1 Finite Element Formulation of EFEA System of Equations 447    14.4.2 EFEA Joint Matrix 448    14.4.3 Input Power 450    14.4.4 EFEA System of Equations for a Simple Plate   Acoustic System 451    14.5 Applications 455    14.5.1 Automotive Application 455    14.5.2 Aircraft Application 461    14.5.3 Naval Application 464    References 470    15 Wave   based Structural Modeling 472    15.1 General Approach 472    15.1.1 Background 473    15.1.2 Advantages/Limitations 474    15.2 Theoretical Formulation 475    15.2.1 Elementary Rod Theory 475    15.2.2 Straight Beams, Timoshenko Beam Theory 477    15.2.3 Reflections at Boundaries 479    15.2.4 Wave Propagation Solution 480    15.2.5 Spectral Element Method 481    15.3 Wave   based Spectral Finite Element Formulation 483    15.3.1 Dynamic Stiffness Matrix of a substructure 483    15.3.2 State Vector Formulation and the Eigenvalue Problem 484    15.3.3 Relations between Dynamic Stiffness and Transfer Matrices 485    15.3.4 Derivation of a Numerical Spectral Matrix for Beam Problems 487    15.3.5 Numerical Spectral Matrix for General Periodic Structures 489    15.4 Applications 491    15.4.1 Beam Analysis via Analytical Approaches 491    15.4.2 Beam Analysis via Numerical Approach (WSFEM) 491    15.4.3 General Periodic Structure Analysis via Numerical Approach (WSFEM) 495    15.4.4 Range of Applicability 499    15.4.5 Implementation   Software Required 500    15.4.6 Computer Resources Required 500    15.4.7 Inputs and How to Determine Them 501    15.4.8 Forces/Enforced Displacements 501    15.4.9 Boundary Conditions 501    15.4.10 Material Properties 502    15.4.11 Outputs 502    15.4.12 Verification and Validation 502    15.5 Conclusion/Summary 503    References 503    Index 506