Vibroacoustic Simulation: An Introduction to Statistical Energy Analysis and Hybrid Methods

Description

VIBROACOUSTIC SIMULATION Learn to master the full range of vibroacoustic simulation using both SEA and hybrid FEM/SEA methods Vibroacoustic simulation is the discipline of modelling and predicting the acoustic waves and vibration of particular objects, systems, or structures. This is done through finite element methods (FEM) or statistical energy analysis (SEA) to cover the full frequency range. In the mid-frequency range, both methods must be combined into a hybrid FEM/SEA approach. By doing so, engineers can model full frequency vibroacoustic simulations in complex technical systems used in aircraft, trains, cars, ships, and satellites. Indeed, hybrid approaches are increasingly used in the automotive, aerospace, and rail industries. Previously covered primarily in scientific journals, Vibroacoustic Simulation provides a practical approach that helps readers master the full frequency range of vibroacoustic simulation. Through a systematic approach, the book illustrates why both FEM and SEA are necessary in acoustic engineering and how both can be used in combination through hybrid methodologies. Striking a crucial balance between complex theories and practical applications, the text provides real-world examples of vibroacoustic simulation, such as fuselage simulation, interior-noise prediction for electric and combustion vehicles, train profiles, and more, to help elucidate the concepts described within. Vibroacoustic Simulation also features: A balance of complex theories with the nuts and bolts of real-world applicationsDetailed worked examples of junction equationsCase studies from companies like Audi and Airbus that illustrate how the methods discussed have been applied in real-world projects A companion website that provides corresponding Python codes for all examples, allowing readers to work through the examples on their own Vibroacoustic Simulation is a useful reference for acoustic and mechanical engineers working in the automotive, aerospace, defense, or rail industries, as well as researchers and graduate students studying acoustics. Vibroacoustic Simulation Contents Preface Acknowledgments Acronyms 1 Linear Systems, Random Process and Signals 1.1 The Damped Harmonic Oscillator 1.1.1 Homogeneous Solutions 1.1.2 The Overdamped Oscillator (𝜁 > 1) 1.1.3 The Underdamped Oscillator (𝜁 < 1) 1.1.4 The Critically Damped Oscillator (𝜁 = 1) 1.2 Forced Harmonic Oscillator 1.2.1 Frequency Response 1.2.2 Energy, Power and Impedance 1.2.3 Impedance and Response Functions 1.2.4 Damping 1.2.5 Damping in Real Systems 1.3 Two Degrees of Freedom Systems (2DOF) 1.3.1 Natural Frequencies of the 2DOF System 1.4 Multiple Degrees of Freedom Systems MDOF 1.4.1 Assembling the Mass Matrix 1.4.2 Assembling the Stiffness Matrix 1.4.3 Power Input into MDOF Systems 1.4.4 Normal Modes 1.5 Random Process 1.5.1 Probability Function 1.5.2 Correlation Coefficient 1.5.3 Correlation Functions for Random Time Signals 1.5.4 Fourier Analysis of Random Signals 1.5.5 Estimation of Power and Cross Spectra 1.6 Systems 1.6.1 SISO-System Response in Frequency Domain 1.6.2 System Response in Time Domain 1.6.3 Systems Excited by Random Signals 1.7 Multiple-input–multiple-output Systems 1.7.1 Multiple Random Inputs 1.7.2 Response of MIMO Systems to Random Load Bibliography 2 Waves in Fluids 2.1 Introduction 2.2 Wave Equation for Fluids 2.2.1 Conservation of Mass 2.2.2 Newton’s law – Conservation of Momentum 2.2.3 Equation of State 2.2.4 Linearized Equations 2.2.5 Acoustic Wave Equation 2.3 Solutions of theWave Equation 2.3.1 Harmonic Waves 2.3.2 Helmholtz equation 2.3.3 Field Quantities: Sound Intensity, Energy Density and Sound Power 2.3.4 Damping in Waves 2.4 Fundamental Acoustic Sources 2.4.1 Monopoles – Spherical Sources 2.5 Reflection of Plane Waves 2.6 Reflection and Transmission of Plane Waves 2.7 Inhomogeneous Wave Equation 2.7.1 Acoustic Green's Functions 2.7.2 Rayleigh integral 2.7.3 Piston in a Wall 2.7.4 Power Radiation 2.8 Units, Measures, and levels Bibliography 3 Wave Propagation in Structures 3.1 Introduction 3.2 Basic Equations and Definitions 3.2.1 Mechanical Strain 3.2.2 Mechanical Stress 3.2.3 Material Laws 3.3 Wave Equation 3.3.1 The One-dimensional Wave Equation 3.3.2 The Three-dimensional Wave Equation 3.4 Waves in Infinite Solids 3.4.1 Longitudinal Waves 3.4.2 Shear waves 3.5 Beams 3.5.1 Longitudinal Waves 3.5.2 Power, Energy, and Impedance 3.5.3 Bending Waves 3.5.4 Power, Energy, and Impedance 3.6 Membranes 3.7 Plates 3.7.1 Strain–displacement Relations 3.7.2 In-plane Wave Equation 3.7.3 Longitudinal Waves 3.7.4 Shear Waves 3.7.5 Combination of Longitudinal and Shear Waves 3.7.6 Bending Wave Equation 3.8 Propagation of Energy in Dispersive Waves 3.9 Findings Bibliography 4 Fluid Systems 4.1 One-dimensional Systems 4.1.1 System Response 4.1.2 Power Input 4.1.3 Pressure Field 4.1.4 Modes 4.2 Three-dimensional Systems 4.2.1 Modes 4.2.2 Modal Frequency Response 4.2.3 System Responses 4.3 Numerical Solutions 4.3.1 Acoustic Finite Element Methods 4.3.2 Deterministic Acoustic Elements 4.4 Reciprocity Bibliography 5 Structure Systems 5.1 Introduction 5.2 One-dimensional Systems 5.2.1 Longitudinal Waves in Finite Beams 5.2.2 Bending Wave in Finite Beams 5.3 Two-dimensional Systems 5.3.1 Bending Waves in Flat Plates 5.4 Reciprocity 5.5 Numerical Solutions 5.5.1 Normal Modes in Discrete Form Bibliography 6 Random Description of Systems 6.1 Diffuse Wave Field 6.1.1 Wave-Energy Relationships 6.1.2 Diffuse Field Parameter of One-Dimensional Systems 6.1.3 Diffuse Field Parameter of Two-Dimensional Systems 6.1.4 Diffuse Field Parameter of Three-Dimensional Systems 6.1.5 Topology Conclusions 6.1.6 Auto Correlation and Boundary Effects 6.1.7 Sources in the Diffuse Acoustic Field – the Direct Field 6.1.8 Some Comments on the Diffuse Field Approach 6.2 Ensemble Averaging of Deterministic Systems 6.3 One-Dimensional Systems 6.3.1 Fluid Tubes 6.4 Two-Dimensional Systems 6.4.1 Plates 6.4.2 Monte Carlo Simulation 6.5 Three-Dimensional Systems – Cavities 6.5.1 Energy and Intensity 6.5.2 Power Input to the Reverberant Field 6.5.3 Dissipation 6.5.4 Power Balance 6.5.5 Monte Carlo Simulation 6.6 Surface Load of Diffuse Acoustic Fields 6.7 Mode Wave Duality 6.7.1 Diffuse Field Energy 6.7.2 Free Field Power Input 6.8 SEA System Description 6.8.1 Power Balance in Diffuse Fields 6.8.2 Reciprocity Relationships 6.8.3 Fluid Analogy 6.8.4 Power Input 6.8.5 Engineering Units 6.8.6 Multiple Wave Fields Bibliography 7 Coupled Systems 7.1 Deterministic Subsystems and their Degrees of Freedom 7.2 Coupling Deterministic Systems 7.2.1 Fluid Subsystems 7.2.2 Fluid Structure Coupling 7.2.3 Deterministic Systems Coupled to the Free Field 7.3 Coupling Random Systems 7.3.1 Power Input to System (m) from the nth Reverberant Field 7.3.2 Power Leaving the (m)th Subsystem 7.3.3 Some Remarks on SEA Modelling 7.4 Hybrid FEM/SEA Method 7.4.1 Combining SEA and FEM Subsystems 7.4.2 Work Flow of Hybrid Simulation 7.5 Hybrid Modelling in Modal Coordinates Bibliography 8 Coupling Loss Factors 8.1 Transmission Coefficients and Coupling Loss Factors 8.1.1 𝜏–𝜂 Relationship from Diffuse Field Assumptions 8.1.2 Angular Averaging 8.2 Radiation Stiffness and Coupling Loss Factors 8.2.1 Point Radiation Stiffness 8.2.2 Point Junctions 8.2.3 Area Radiation Stiffness 8.2.4 Area Junctions 8.2.5 Line Radiation Stiffness 8.2.6 Line Junctions 8.2.7 Summary Bibliography 9 Deterministic Applications 9.1 Acoustic One-Dimensional Elements 9.1.1 Transfer Matrix and Finite Element Convention 9.1.2 Acoustic One-Dimensional Networks 9.1.3 The Acoustic Pipe 9.1.4 Volumes and Closed Pipes 9.1.5 Limp Layer 9.1.6 Membranes 9.1.7 Perforated Sheets 9.1.8 Branch Lumped Elements 9.1.9 Boundary Conditions 9.1.10 Performance Indicators 9.2 Coupled One-Dimensional Systems 9.2.1 Change in Cross Section 9.2.2 Impedance Tube 9.2.3 Helmholtz Resonator 9.2.4 Quarter Wave Resonator 9.2.5 Muffler System 9.2.6 T-Joint 9.2.7 Conclusions of 1D-Systems 9.3 Infinite Layers 9.3.1 Plate Layer 9.3.2 Lumped Elements Layers 9.3.3 Fluid Layer 9.3.4 Equivalent Fluid – Fiber Material 9.3.5 Performance Indicators 9.3.6 Conclusions on Layer Formulation 9.4 Acoustic Absorber 9.4.1 Single Fiber Layer 9.4.2 Multiple Layer Absorbers 9.4.3 Absorber with Perforate 9.4.4 Single Degree of Freedom Liner 9.5 Acoustic Wall Constructions 9.5.1 Double Walls 9.5.2 Limp Double Walls with Fiber 9.5.3 Two Plates with Fiber 9.5.4 Conclusion on Double Walls Bibliography 10 Application of Random systems 10.1 Frequency Bands for SEA Simulation 10.2 Fluid Systems 10.2.1 Twin Chamber 10.3 Algorithms of SEA 10.4 Coupled Plate Systems 10.4.1 Two Coupled Plates 10.5 Fluid-Structure Coupled Systems 10.5.1 Twin Chamber 10.5.2 Noise Control Treatments 10.5.3 Transmission Loss of Trimmed Plate 10.5.4 Free Field Radiation into Half Space 10.5.5 Isolating Box 10.5.6 Rules of Noise Control Bibliography 11 Hybrid Systems 11.1 Hybrid SEA Matrix 11.2 Twin Chamber 11.2.1 Step 1 – Setting up System Configurations 11.2.2 Step 2 – Setting up System Matrices and Coupling Loss Factors 11.2.3 Step 3 – External Loads 11.2.4 Step 4 – Solving System Matrices 11.2.5 Step 5 – Adding the Results 11.3 Trim in Hybrid Theory 11.3.1 The Trim Stiffness Matrix 11.3.2 Hybrid Modal Formulation of Trim and Plate 11.3.3 Modal Space 11.3.4 Plate Example with Trim Bibliography 12 Industrial Cases 12.1 Simulation Strategy 12.1.1 Motivation 12.1.2 Choice of Simulation Method 12.2 Aircraft 12.2.1 Excitation 12.2.2 Simulation Strategy 12.2.3 Fuselage Sidewall 12.2.4 SEA Model of a Fuselage Section 12.3 Automotive 12.3.1 Simulation Strategy 12.3.2 Excitation 12.3.3 Rear Carbody 12.3.4 Full Scale SEA Models 12.4 Trains 12.4.1 Structural Design 12.4.2 Interior Design 12.4.3 Excitation and Transmission Paths 12.4.4 Simulation Strategy 12.4.5 Applications to Rail Structures – Double Walls 12.4.6 Carbody Sections – High Speed Applications 12.5 Summary Bibliography 13 Conclusions and Outlook 13.1 Conclusions 13.2 What Comes Next? 13.3 Experimental Methods 13.3.1 Transfer Path Analysis 13.3.2 Experimental Modal Analysis 13.3.3 Correlation Between Test and Simulation 13.3.4 Experimental or Virtual SEA 13.4 Further Reading on Simulation 13.4.1 Advances in SEA and Hybrid FEM/SEA Methods 13.5 Energy Flow Method and Influence Coefficient 13.5.1 More Realistic Systems 13.5.2 Anisotropic Material 13.5.3 Porous Elastic Material 13.5.4 Composite Material 13.5.5 Sandwich 13.5.6 Shell Theory 13.5.7 Wave Finite Element Method (WFE) 13.5.8 The High Frequency Limit 13.6 Vibroacoustics Simulation Software Bibliography A Basic Mathematics A.1 Fourier Analysis A.1.1 Fourier Series A.1.2 Fourier Transformation A.1.3 Dirac Delta Function A.1.4 Signal Power A.1.5 Fourier Transform of Real Harmonic Signals A.1.6 Useful Properties of the Fourier Transform A.1.7 Fourier Transformation in Space A.2 Discrete Signal Analysis A.2.1 Fourier Transform of Discrete Signals A.2.2 The Discrete Fourier Transform A.2.3 Windowing A.3 Coordinate Transformation of Discrete Equation of Motion Bibliography B Specific Solutions B.1 Second Moments of Area B.2 Wave Transmission B.2.1 The Blocked Forces Interpretation B.2.2 Bending Waves B.2.3 Longitudinal Waves B.2.4 Shear Waves B.2.5 In-plane Waves B.3 Conversion Formulas of Transfer Matrix B.3.1 Derivation of Stiffness Matrix from Transfer Matrix Bibliography C Symbols Index EULA