Publisher Description

The book describes analytical methods (based primarily on classical modal synthesis), the Finite Element Method (FEM), Boundary Element Method (BEM), Statistical Energy Analysis (SEA), Energy Finite Element Analysis (EFEA), Hybrid Methods (FEM-SEA and Transfer Path Analysis), and Wave-Based Methods. The book also includes procedures for designing noise and vibration control treatments, optimizing structures for reduced vibration and noise, and estimating the uncertainties in analysis results. Written by several well-known authors, each chapter includes theoretical formulations, along with practical applications to actual structural-acoustic systems. Readers will learn how to use vibroacoustic analysis methods in product design and development; how to perform transient, frequency (deterministic and random), and statistical vibroacoustic analyses; and how to choose appropriate structural and acoustic computational methods for their applications. The book can be used as a general reference for practicing engineers, or as a text for a technical short course or graduate course.

Back Cover

ENGINEERING VIBROACOUSTIC ANALYSIS Methods and Applications Computational vibroacoustic methods are commonly applied in engineering design. Engineers must therefore understand the fundamentals of the available methods and how to apply them. Taking a practical approach, this book describes methods for simulating the structural vibration, interior acoustics, and sound radiation of mechanical and transportation vehicle systems for low, mid, and high frequencies. The book describes analytical methods (based primarily on classical modal synthesis), the finite element method (FEM), boundary element method (BEM), statistical energy analysis (SEA), energy finite element analysis (EFEA), hybrid methods (FEM-SEA and transfer path analysis), and wave-based methods. The book also includes procedures for designing noise and vibration control treatments, optimizing structures for reduced vibration and noise, and estimating the uncertainties in analysis results. Each chapter includes theoretical formulations, along with practical applications to actual structural-acoustic systems. Readers will learn how to use vibroacoustic analysis methods in product design and development; how to perform transient, frequency (deterministic and random), and statistical vibroacoustic analyses; and how to choose appropriate structural and acoustic computational methods for their applications. The book can be used as a general reference for practicing engineers or as a text for a technical short course or graduate course.

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ENGINEERING VIBROACOUSTIC ANALYSIS Methods and Applications Computational vibroacoustic methods are commonly applied in engineering design. Engineers must therefore understand the fundamentals of the available methods and how to apply them. Taking a practical approach, this book describes methods for simulating the structural vibration, interior acoustics, and sound radiation of mechanical and transportation vehicle systems for low, mid, and high frequencies. The book describes analytical methods (based primarily on classical modal synthesis), the finite element method (FEM), boundary element method (BEM), statistical energy analysis (SEA), energy finite element analysis (EFEA), hybrid methods (FEM-SEA and transfer path analysis), and wave-based methods. The book also includes procedures for designing noise and vibration control treatments, optimizing structures for reduced vibration and noise, and estimating the uncertainties in analysis results. Each chapter includes theoretical formulations, along with practical applications to actual structural-acoustic systems. Readers will learn how to use vibroacoustic analysis methods in product design and development; how to perform transient, frequency (deterministic and random), and statistical vibroacoustic analyses; and how to choose appropriate structural and acoustic computational methods for their applications. The book can be used as a general reference for practicing engineers or as a text for a technical short course or graduate course.

Author Biography

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

Table of Contents

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 WaveBased 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 SoundPressure Response 59 3.2.5 SteadyState SoundPressure Response 60 3.2.6 Enclosure Driven at Resonance 64 3.2.7 Random SoundPressure Response 66 3.2.8 Transient SoundPressure Response 68 3.3 Exterior Sound 70 3.3.1 Sound Radiation Measures 72 3.3.2 OneDimensional 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 SoundStructure 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 StructuralAcoustic Modal Analysis and Synthesis 114 5.1 Introduction 114 5.2 Coupled StructuralAcoustic 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 StructuralAcoustic 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 StructuralAcoustic Model 132 5.4.2 Coupled Structures 134 5.4.3 Coupled Cavities 138 5.5 Summary 142 References 143 6 StructuralAcoustic FiniteElement Analysis for Interior Acoustics 144 6.1 Introduction 144 6.2 Acoustic FiniteElement Analysis 144 6.2.1 Acoustic FiniteElement 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 StructuralAcoustic FiniteElement Analysis 155 6.3.1 Structural FiniteElement Formulation 155 6.3.2 Structural System Synthesis 158 6.4 Coupled StructuralAcoustic FiniteElement 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 AcousticResponse Variation 173 6.5 Summary 177 References 177 7 BoundaryElement Analysis 179 7.1 Theory—Assumptions 179 7.2 Theory—Overview of Theoretical Basis 180 7.3 BoundaryElement 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 FrequencyDependent 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 OpenCell 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 SemiAnalytic 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 Nonintrusive SFEM 323 10.7 Numerical Examples 324 10.7.1 Discrete 2DOF 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 WaveBased 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 HigherOrder 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 FESEA 385 12.1 Introduction 385 12.2 Overview 385 12.2.1 Low, Mid, and HighFrequency Ranges 385 12.2.2 The MidFrequency Problem 386 12.3 The Hybrid FESEA 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 VibroAcoustic 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 PlateAcoustic 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 Wavebased 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 Wavebased 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

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