Оглавление
Alexander Peiffer. Vibroacoustic Simulation
Vibroacoustic Simulation. An Introduction to Statistical Energy Analysis and Hybrid Methods
Contents
List of Illustrations
List of Tables
Guide
Pages
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
1.2.3.1 Power Balance
1.2.4 Damping
1.2.5 Damping in Real Systems
1.2.5.1 Hysteretic Damping
1.3 Two Degrees of Freedom Systems (2DOF)
1.3.1 Natural Frequencies of the 2DOF System
1.3.1.1 Forced Vibration of the 2DOF System
1.3.1.2 Dynamic Vibration Absorber
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
1.4.4.1 Equation of Motion in Modal Coordinates
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
1.7.1.1 Fully Uncorrelated Signals – Rain on the Roof Excitation
1.7.1.2 Fully Correlated Signals
1.7.2 Response of MIMO Systems to Random Load
Bibliography
Notes
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 the Wave 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
2.4.1.1 Field Properties of Spherical Waves
2.4.1.2 Field Intensity, Power and Source Strength
2.4.1.3 Power and Radiation Impedance at the Surface Sphere
2.4.1.4 Point 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
2.7.3.1 Impedance Concept
2.7.3.2 Inertia Effects
2.7.4 Power Radiation
2.7.4.1 Radiation Efficiency
2.8 Units, Measures, and levels
Bibliography
Notes
3 Wave Propagation in Structures. 3.1 Introduction
3.2 Basic Equations and Definitions
3.2.1 Mechanical Strain
3.2.1.1 Mechanical Strain - Voigt Notation
3.2.1.2 Dilatation dilation – Relative Change in Volume
3.2.2 Mechanical Stress
3.2.3 Material Laws
3.2.3.1 Isotropic Materials
3.2.3.2 Ortotropic Solids
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
3.7.6.1 Cylindrical Solution of Bending Wave Equation
3.7.6.2 Power, Impedance, and Energy
Damping
3.8 Propagation of Energy in Dispersive Waves
3.9 Findings
Bibliography
Notes
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
4.1.4.1 Rigid Boundaries
4.1.4.2 Modal Coordinates and Matrix Representation
4.1.4.3 Modal Density
4.1.4.4 Damping
4.1.4.5 Modal Frequency Response
4.2 Three-dimensional Systems
4.2.1 Modes
4.2.1.1 Modal Density
4.2.2 Modal Frequency Response
4.2.3 System Responses
4.2.3.1 Point Sources
2.3.2 Radiation Impedance and Power
4.2.3.3 Rectangular Piston in the Wall
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
5.2.1.1 Modes
5.2.2 Bending wave in Finite Beams
5.2.2.1 Modes
5.2.2.2 Modal Response
5.3 Two-dimensional Systems
5.3.1 Bending Waves in Flat Plates
5.3.1.1 Modal Density
5.3.1.2 Modal Response
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
6.1.5.1 Dissipation in the Reverberant Field
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
6.3.1.1 Energy
6.3.1.2 Power Input to the Reverberant Field
6.3.1.3 Dissipation
6.3.1.4 Power Balance
6.3.1.5 Energy Ray Tracing
6.3.1.6 Load at the Boundaries due to the Reverberant Field
6.3.1.7 Monte Carlo Experiment
6.3.1.8 Input Power and Impedance
6.3.1.9 Random Fields
6.4 Two-Dimensional Systems
6.4.1 Plates
6.4.1.1 Energy
6.4.1.2 Power Input to the Reverberant Field
6.4.1.3 Dissipation and Power Balance in the Reverberant Field
6.4.1.4 Direct Field Correction
6.4.2 Monte Carlo Simulation
6.4.2.1 Input Power and Impedance
6.4.2.2 Random Displacement Field
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
6.5.5.1 Room Absorption
6.5.5.2 Input Power and Impedance
6.5.5.3 Random Pressure Field – Room 1
6.5.5.4 Random Pressure Field – Room2
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
6.7.2.1 Point Conductance
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
7.3.1.1 Power Radiated from Random Displacement
7.3.2 Power Leaving the (m)th Subsystem
7.3.2.1 Assembling the Hybrid SEA Matrix
7.3.3 Some Remarks on SEA Modelling
7.4 Hybrid FEM/SEA Method
7.4.1 Combining SEA and FEM Subsystems
7.4.1.1 Ensemble Average of Linear State Variable
7.4.1.2 Ensemble Average of Cross Spectral Density
7.4.1.3 Integration of FEM into the SEA Power Flow
7.4.2 Work Flow of Hybrid Simulation
7.4.2.1 Setting up the System Configuration
7.4.2.2 Setting up the System Matrices and Coupling Loss Factors
7.4.2.3 Apply External Loads
7.4.2.4 Solving the Linear System of Equations
7.4.2.5 Combining Both Results
7.5 Hybrid Modelling in Modal Coordinates
Bibliography
Notes
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
8.1.2.1 Two‐Dimensional Systems
8.1.2.2 Three‐Dimensional Systems
8.2 Radiation Stiffness and Coupling Loss Factors
8.2.1 Point Radiation Stiffness
8.2.1.1 Bars and Tubes
8.2.1.2 Beams
8.2.1.3 Plates
8.2.1.4 Cavities
8.2.2 Point Junctions
8.2.2.1 Impedance Tube Example
8.2.2.2 Connected Bars
8.2.2.3 Connected Beams
8.2.2.4 Plates Connected via Springs
8.2.3 Area Radiation Stiffness
8.2.3.1 Cavities – Discrete Space Domain
8.2.3.2 Cavities – Wavenumber Domain
8.2.3.3 Cavities – the Concept of Shape Stiffness
8.2.3.4 Cavities – Radiation Efficiency of Finite, Rectangular Shapes
8.2.3.5 Plates – Wavenumber Domain
8.2.3.6 Plates – Discrete Space Domain
8.2.4 Area Junctions
8.2.4.1 Fluid–Fluid Coupling
8.2.4.2 Diffuse Field Transmission from Wavenumber Space
8.2.4.3 Plate–Fluid Connection
8.2.4.4 Approximation from Radiation Efficiency
8.2.4.5 Fluid–Plate–Fluid Connection
8.2.5 Line Radiation Stiffness
8.2.5.1 Plates – Wavenumber Domain
8.2.6 Line Junctions
8.2.6.1 Coupling Loss Factor of in‐plane and out‐of‐plane Waves
8.2.6.2 Diffuse Field Coupling Loss Factor
8.2.7 Summary
Bibliography
Notes
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
9.1.2.1 Properties of the System Matrices
9.1.3 The Acoustic Pipe
9.1.4 Volumes and Closed Pipes
9.1.5 Limp Layer
9.1.5.1 Mass
9.1.5.2 Stiffness
9.1.5.3 Viscous Damping
9.1.6 Membranes
9.1.7 Perforated Sheets
9.1.7.1 Example for a Micro Perforated Grid
9.1.8 Branch Lumped Elements
9.1.9 Boundary Conditions
9.1.10 Performance Indicators
9.1.10.1 Transfer and Insertion Coefficients
9.1.10.2 Absorption
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
9.2.5.1 Expansion Chamber
9.2.5.2 Open end Conditions
9.2.5.3 Realistic End Conditions
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
9.5.1.1 Double Wall of Limp Mass
9.5.2 Limp Double Walls with Fiber
9.5.3 Two Plates with Fiber
9.5.4 Conclusion on Double Walls
Bibliography
Notes
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
10.5.1.1 Ideal Situation
10.5.1.2 Concrete Walls
10.5.2 Noise Control Treatments
10.5.2.1 Damping Loss due to Noise Control Treatment
10.5.2.2 Coupling Loss due to Noise Control Treatment
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
Notes
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. 11.2.4.1 Step 4.1 – Power Input due to FEM System Excitation
11.2.4.2 Setp 4.2 – Solve the SEA Equations with SEA and FEM Power Input from Step 4.1
11.2.4.3 Step 4.3 – Calculate FE Response due to Energies in SEA Subsystems from Step 4.2
11.2.5 Step 5 – Adding the Results
11.2.5.1 Conlusions
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
12.2.3.1 Double Wall Simulation Strategy
12.2.3.2 Fuselage Panels
12.2.3.3 Windows
12.2.3.4 Lining and Insulation
12.2.3.5 Transmission Loss
12.2.4 SEA Model of a Fuselage Section
12.2.4.1 Fuselage Structure and Lining
12.2.4.2 Cavities
12.2.4.3 Force excitation
12.2.4.4 Turbulent Boundary Layer Excitation
12.2.4.5 Conclusions
12.3 Automotive
12.3.1 Simulation Strategy
12.3.2 Excitation
12.3.3 Rear Carbody
12.3.3.1 Transmission Loss
12.3.3.2 Force Excitation
12.3.4 Full Scale SEA Models
12.3.4.1 Cavities
12.3.4.2 Carbody
12.3.4.3 Noise Control
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
12.4.5.1 Carbody Elements from Extruded Profiles
12.4.6 Carbody Sections – High Speed Applications
12.4.6.1 Roof Structures
12.4.6.2 Cab Structures
12.5 Summary
Bibliography
Notes
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
Appendix 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
Notes
Appendix 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
Notes
Appendix C Symbols
Index
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