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

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 (? > 1)
1.1.3 The Underdamped Oscillator (? < 1)
1.1.4 The Critically Damped Oscillator (? = 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 ?–? 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