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¢"Lo¨ve 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
