Engineering Vibroacoustic Analysis

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 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–Loève 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