Handbook of Experimental Structural Dynamics

Preface and Introduction
Contents
About the Editors
Contributors
Part I Sensors and Measurements
1 Recent History of Experimental Structural Dynamics
Contents
Nomenclature
1 Introduction
2 Timeline History
3 Technology Developments
3.1 Sensors
3.1.1 Resistance Technology
3.1.2 Bonded Strain Gage Technology
3.1.3 Piezoelectric Crystal Technology
3.1.4 Inductance and Capacitance Technology
3.1.5 Optical Technology
3.2 Data Acquisition
3.2.1 Analog Technology
3.2.2 Digital Technology
4 Experimental Structural Dynamics Methods 1965–1985
4.1 Classification Methods
4.2 Data Acquisition Classification
4.2.1 Sinusoidal Input-Output Method
4.2.2 Frequency Response Function Method
4.2.3 Damped Complex Exponential Methods
4.2.4 Mathematical Input-Output Model Method
4.3 Summary
5 Conferences
5.1 ISMA
5.2 IMAC
5.3 Other Conferences
6 Publications and Books
7 Pioneers/Contributors
7.1 Pioneers: Experimental Structural Dynamics
7.2 Contributors: Handbook
7.3 Contributors: SEM IMAC
7.3.1 IMAC Advisory Board
8 Summary/Conclusions
References
2 Sensors and Their Signal Conditioning for Dynamic Acceleration, Force, Pressure, and Sound Applications
Contents
1 Sensing Technologies
1.1 Piezoelectricity
1.2 Metal Strain Gages
1.3 Semiconductor, Piezoresistive, and MEMS Strain Gages
1.4 Piezoelectric Strain Gages
1.5 Capacitance
2 Sensor Dynamic Models
2.1 Strain Gage
2.2 Pressure, Force, and Acceleration Transducers
3 Sensor Selection and Use
3.1 Piezoelectric Accelerometers
3.2 Force Sensors
3.3 Impedance Heads
3.4 Pressure Measurements
3.5 Pressure Transducers
4 Sensor Systems
4.1 Sensor System Architecture
5 Signal Conditioning
5.1 Types of Amplifiers
5.2 Filters
5.3 Analog-to-Digital Conversion
6 Other Considerations
6.1 Grounding
6.2 Cabling
6.3 Ground Loops
6.4 Triboelectric Effects
6.4.1 Connectors
7 Data Validation
7.1 Examples
References
3 Laser Doppler Vibrometry Measurements in StructuralDynamics
Contents
Nomenclature
1 Theory of Vibrometry
1.1 Lasers Sources and Doppler Effect
1.2 From Interferometers to Vibrometers
1.2.1 Optical Homodyne and Heterodyne
1.2.2 Michelson Interferometer
1.2.3 Mach-Zehnder Interferometer
1.2.4 Self-Mixing
1.3 Demodulation of Doppler Signals
1.4 Noise and Resolution
1.5 Critical Aspects in Laser Vibrometry
1.5.1 Backscatter Issues
1.5.2 Speckle Noise
1.5.3 Measuring in Media and Through Windows
1.6 Signal Enhancement Approaches
1.6.1 Tracking Filter
1.6.2 Diversity Combining
1.7 Uncertainty and Calibration
1.8 Laser Safety and Standards
2 Instrumentation, Measurement Issues, and Applications
2.1 Single-Point Vibrometers
2.2 Optical Fiber Vibrometers
2.3 Differential Vibrometers
2.4 Rotational Vibrometers
2.5 In-Plane Vibrometers
2.6 Scanning Vibrometers
2.6.1 Step Scan Vibrometers (Single DoF)
2.6.2 Special Solutions
Continuous-Scan LDV
LDV Strategies on Rotating/Moving Structures
Scanning Laser Doppler Vibrometry with Optical Derotator
Tracking Laser Doppler Vibrometry
2.6.3 3D Scanning Vibrometer
2.7 Multi-beam
3 Conclusions
References
4 Applied Digital Signal Processing
Contents
1 Overview
2 Deterministic Signals and Traditional Fourier Analysis
2.1 Periodic Signals and Fourier Series
2.2 Transient Signals and the Fourier Transform
2.3 Discrete-Time Signals and Digital Implementation
3 Experimental Signals and Stochastic Signal Modelling
3.1 Time-Varying Distributions: Ensemble Versus Time Quantities
3.2 Stationary Signal Model
3.3 Cyclostationary Signal Model
3.3.1 CS1 Model
3.3.2 CS2 Model
3.4 Cyclo-non-stationarity
3.5 Transient Signal Model
4 Scaling and Dimensions for Various Versions of the Fourier Transform
4.1 Periodic and Quasi-periodic Signals
4.2 Stationary Random Signals, Power Spectral Density (PSD)
4.3 Deterministic Transient Signals, Energy Spectral Density (ESD)
5 Choosing the Right Model for an Experimental Signal
5.1 Modal Analysis
5.2 Condition Monitoring for Rotating/Reciprocating Machines
5.2.1 Typical Rotor Problems
5.2.2 Gear Faults
5.2.3 Bearing Faults
5.2.4 Internal Combustion Engines
6 Signal Extraction and Separation
6.1 General Introduction
6.2 Signal Extraction
6.3 Reference-Based Filtering
6.4 Filtration
6.5 Blind Extraction
6.5.1 Blind Extraction of Impulsive Signals
6.5.2 Blind Extraction of Cyclostationary Signals
6.6 Blind Deconvolution
6.7 Discrete-Random Separation
6.7.1 Extraction of Periodic Signals with Known Periods: Synchronous Averaging
6.7.2 Extraction of Periodic Signals with Unknown Periods
6.8 Blind Source Separation
6.8.1 The Notion of a Source
6.8.2 Problem Statement
6.8.3 Types of Mixture
6.8.4 General Principles
Separation Operator
Separation Criteria
Optimization Algorithm
6.9 Blind Separation of Structural Modes
6.9.1 Context
6.9.2 Principle
6.9.3 Example of Application
7 Time-Frequency Representations
7.1 STFT
7.2 Wigner-Ville Distribution (WVD)
7.3 Wigner-Ville Spectrum (WVS)
7.4 Wavelets
7.4.1 Constant Percentage Bandwidth
7.4.2 Wavelet Packets
8 Specialized Analysis
8.1 Order Analysis
8.1.1 Computed Order Tracking
8.2 Second-Order Cyclostationary Indicators
8.3 The Envelope Spectrum
8.4 The Cyclic Modulation Spectrum
8.5 The Spectral Correlation Density
8.6 The Relationship Between Kurtosis, Envelope, and CS2 Indicators
9 Typical Diagnostic Examples
9.1 Rolling Element Bearings
9.2 Gears
9.3 Reciprocating Machines and Engines
10 Cepstrum
10.1 Background and Definitions
10.2 Cepstrum Liftering
10.2.1 Comb Notch Lifter
10.2.2 Exponential Lifter
10.2.3 Modal Suppression
10.2.4 Longpass Lifter
10.3 Cepstrum for Machine Diagnostics
10.3.1 Changes due to Forcing Functions
10.3.2 Changes due to Structural Response
10.3.3 Example of Cepstrum Pre-whitening for the Diagnostic of Rolling Element Bearings in Variable Speed Conditions
10.4 Cepstrum for Modal Analysis
10.4.1 Full OMA Procedure
10.4.2 Cepstral Pre-processing for Other OMA Procedures
References
5 Introduction to Spectral and Correlation Analysis: Basic Measurements and Methods
Contents
1 Introduction
1.1 Properties of Linear Systems
1.2 Common Applications in Experimental Structural Dynamics
2 Signal Classes and Their Spectra
2.1 Periodic Signals
2.2 Random Signals
2.2.1 Correlation Functions
2.2.2 Spectral Density Functions
2.3 Transient Signals
2.4 Double-Sided Versus Single-Sided Spectra
3 Frequency Analysis
3.1 Spectrum Analysis Principle
3.2 The Discrete Fourier Transform (DFT)
3.2.1 Leakage and Windowing
3.2.2 Cyclic Convolution and Zero Padding
3.2.3 Window Scaling Factors
4 Block-Based Spectrum and Correlation Estimation
4.1 The Linear (RMS) Spectrum
4.2 The Phase Spectrum
4.3 Welch's PSD and CSD Estimates
4.4 Energy Spectral Density Estimates
4.5 Welch's Correlation Function Estimates
5 Periodogram-Based Spectrum and Correlation Estimation
5.1 The Periodogram
5.2 The Smoothed Periodogram PSD Estimate
5.3 The Periodogram Correlation Estimate
5.4 Dealing with Harmonics in Correlation Functions
6 Summary
References
6 Frequency Response Function Estimation
Contents
Nomenclature
1 Introduction
2 Frequency Response Function Development
3 Frequency Response Function Estimation
3.1 Noise/Error Minimization
3.2 Single Input FRF Estimation
3.2.1 H1 Algorithm: Minimize Noise on Output (η)
3.2.2 H2 Algorithm: Minimize Noise on Input (υ)
3.2.3 Hv Algorithm: Minimize Noise on Input and Output (η and υ)
3.2.4 Ordinary Coherence
3.3 Multiple Input FRF Estimation
3.3.1 Multiple Input Versus Single Input
3.3.2 H1 Algorithm: Minimize Noise on Output ( η )
3.3.3 H2 Algorithm: Minimize Noise on Input (υ)
3.3.4 Hv Algorithm: Minimize Noise on Input and Output (υ and η)
3.4 Coherence: Ordinary, Multiple, and Conditioned
3.4.1 Ordinary Coherence
3.4.2 Multiple Coherence
3.4.3 Conditioned Coherence
Conditioned Coherences Which Retain Physical Source Reference
Partial Coherence
Cumulative Coherence
Conditioned Coherences Which Utilize Virtual Source Reference
Fractional Coherence
Virtual Coherence
Two DOF Illustration
Ordinary coherence (output p and Input 1):
Ordinary coherence (output p and input 2):
Ordinary Coherence (Input 1 and Input 2):
Multiple Coherence
Summary of Methods
3.5 Multiple Input Force Analysis/Evaluation
3.5.1 Ordinary and Partial Coherence Functions
3.5.2 Principal/Virtual Input Forces (Virtual Forces)
3.5.3 Optimum Number Of Inputs
4 Averaging
4.1 General Averaging Methods
4.1.1 Linear Averaging
4.1.2 Magnitude Averaging
4.1.3 Root Mean Square (RMS) Averaging
4.1.4 Exponential Averaging
4.1.5 Stable Averaging
4.1.6 Peak Hold
4.2 Estimation of Frequency Response Functions
4.2.1 Asynchronous Signal Averaging
4.2.2 Synchronous Signal Averaging
4.2.3 Cyclic Signal Averaging
Theory of Cyclic Averaging
Practical Example
4.3 Special Types of Signal Averaging
4.3.1 Overlap Processing
4.3.2 Random Decrement
5 Excitation
5.1 Excitation Assumptions
5.2 Excitation Terminology and Nomenclature
5.2.1 Signal Type
5.2.2 Frequency Shaping
5.2.3 Contiguous Blocks
5.2.4 Capture Blocks
5.2.5 Window Function
5.2.6 Ensemble or Average
5.2.7 Excitation Signal
5.2.8 Burst Length
5.2.9 Power Spectral Average
5.2.10 Excitation Terminology Illustration
5.3 Classification of Excitation
5.4 Random Excitation Methods
5.4.1 Pure Random Signal
5.4.2 Pseudorandom Signal
5.4.3 Periodic Random Signal
5.4.4 Burst Random Signal
5.4.5 Slow Random Signal
5.4.6 MOOZ Random Signal
5.4.7 Hybrid Random Signal
5.5 Deterministic Excitation Methods
5.5.1 Slow Swept Sine Signal
5.5.2 Periodic Chirp Signal
5.5.3 Impact Signal
Impact Testing
Force Window
Response (Exponential) Window
Response (Exponential) Window Correction
5.5.4 Step Relaxation Signal
5.5.5 Summary of Excitation Signal Characteristics
5.6 Excitation Example: H-Frame
6 Structural Testing Conditions
7 Practical Measurement Considerations
8 Summary
References
7 Random Vibration and Mechanical Shock
Contents
Nomenclature
1 Introduction
2 Mathematical Foundations of Structural Dynamics
2.1 Single-Degree-of-Freedom Structures
2.1.1 Structural Dynamics in the Time Domain
2.1.2 Structural Dynamics in the Frequency Domain
2.2 Multiple-Degree-of-Freedom Structures
2.2.1 Structural Dynamics in the Time and Frequency Domains
2.3 Random Processes
2.4 Random Vibration
2.5 Mechanical Shock
3 Random Vibration Testing
3.1 Random Processes and the Autospectral Density
3.2 Random Test Control Loop
3.3 Test Setup and Instrumentation
3.4 Conducting a Random Vibration Test
4 Shock Testing on Shakers
4.1 Time History Synthesis
4.1.1 Basic Shaker Limitations
4.1.2 Classification of Waveforms
4.1.3 Matching a Required Shock Response Spectrum
4.1.4 Time History Synthesis Using Oscillatory Waveforms
4.1.5 Problems with Synthesis
4.2 Time History Reproduction
4.2.1 Classical Theory
4.2.2 Duration of a Transient Waveform
4.2.3 Measurement of the System Frequency Response Function
4.2.4 Why Things Do Not Always Work
4.2.5 Why Things Do Not Always Work, an Extreme Example
4.2.6 Improving Your Chances for a Good Test
4.3 Conclusion
5 Closure
References
8 DIC and Photogrammetry for Structural Dynamic Analysis and High-Speed Testing
Contents
1 Introduction and Relation to Prior Related Work on DIC and Point Tracking
1.1 Photogrammetry Techniques
1.1.1 Point Tracking
1.1.2 Digital Image Correlation (DIC)
1.1.3 Target-Less Approaches
1.2 DIC Hardware
1.3 DIC Software
1.4 Patterning
1.5 Calibration
1.6 Measurement and Applications
2 Overview of Modal Testing and Requirements
2.1 Frequency Response Function Measurement Considerations
2.2 Fourier Transformation and Leakage Considerations
2.3 Curve Fitting Considerations
3 The Distinction Between Operating Shapes and Mode Shapes
4 DIC Measurement Resolution in Relation to Structural Dynamic Testing/Modal Analysis
5 DIC Measurement Range in the Context of Structural Motion and Frequency
6 Analysis in the Temporal Versus Frequency Domains
7 Identifying the Number of Images Needed
7.1 Sampling Theory Relationships
7.2 Selecting Proper Sampling Parameters
7.3 Dealing with Long Sampling Requirements
8 Sources of Measurement Error and Best Practices
8.1 Modes of the Stereo System Hardware and iIts Measurement Effect
8.2 Aliasing
8.2.1 Description of Temporal Aliasing for Image Processing
8.2.2 Mitigating Aliasing with a Single Point Measurement
8.3 Artificial Aliasing to Enhance Measurement
8.3.1 Stroboscope Lights and High-Speed Measurements
8.3.2 Phase Stepping
8.4 Lighting Requirements, Shutter Time, and Lens Adjustment
8.4.1 Lens Adjustment
8.4.2 Blurring
8.4.3 Lighting
8.4.4 Heating Effects Due to Lighting
9 Excitation Strategies for Modal Testing and Application to DIC Measurements
9.1 Impact Testing
9.2 Shaker Testing
9.2.1 Sine Excitation
9.2.2 Swept Sine/Chirp Excitation
9.2.3 Pure Random Excitation
9.2.4 Pseudo Random Excitation
9.2.5 Periodic Random Excitation
9.2.6 Burst Random
9.3 Recommended Inputs for DIC Testing
10 DIC and Photogrammetry Measurement Range and Noise Floor
10.1 Pre-measurement Parameters
10.1.1 Camera Setup and Calibration
10.1.2 Blurring
10.1.3 Speckle Pattern and Target Shape
10.1.4 Camera Angle
10.1.5 Air Turbulence
10.2 Image Correlation and Data Processing Parameters
10.2.1 Data Processing Parameters
11 Strain Mode Shapes
12 Projected Patterns Pros and Cons
12.1 Projected Speckle Patterns
12.2 Deflectometry
13 Rotating Optical Measurements
13.1 Frequency of Measurement, Duration, and Shutter Time
13.2 Camera Setup
13.3 Rigid Body Correction
13.4 Mode Extraction Challenges and Effects of Harmonics
14 Some Experimental Case Studies
14.1 Comparison bBetween 3D Scanning Laser Doppler Vibrometry and 3D Stereo-DIC
15 DIC Comparison to Traditional Modal Analysis Sensing
16 DIC for High Rate Testing
16.1 Definition of High-Rate Testing
16.2 High-Rate Camera Selection
16.3 2D Versus 3D Stereo-DIC
16.4 Environmental Concerns
16.5 Camera Motion
16.6 Camera Protection
16.7 Extended Noise Floor Measurements
16.8 Camera Calibration
16.9 Lighting Techniques
16.9.1 Polarization
16.9.2 Motion Blur
16.10 Camera Synchronization
16.10.1 IRIG (Inter-Range Instrumentation Group) Timing
16.11 Painting Techniques
16.12 Conclusion
References
Part II Modal Model Development
9 Design of Modal Tests
Contents
Acronyms
1 Introduction
2 Excitation Techniques
2.1 General Considerations
2.1.1 Frequency Range
2.1.2 Excitation Level
2.1.3 Linearity of Structure
2.1.4 Damping of Structure
2.1.5 Simulation of Operational Loads
2.2 Artificial Input
2.2.1 Impulsive Inputs
Impact Inputs
Step Inputs
Damping and Nonlinearities
2.2.2 Controlled Inputs
2.2.3 Multiple Inputs
2.3 Natural or Operational Inputs
3 Response Measurements
3.1 Transducers
3.2 Number of Degrees of Freedom to Measure
3.3 Acquisition Methods
3.4 Geometry Definition
3.4.1 Accuracy of Location Measurements
3.4.2 Layout and Documentation
3.4.3 Test Display Model
4 Support Conditions
4.1 Approximating Free Boundary Conditions, and the Resulting Compromises
4.2 Suspension System Design, Low Spring Rates
4.3 Constrained Support, Built-In, and Other Boundary Conditions
4.4 Operating Environments
5 Measurement Quality Criteria
6 Modal Tests for Model Validation
6.1 Selecting Response Locations
6.1.1 Modal Kinetic Energy
6.1.2 Effective Independence
6.1.3 Min-MAC
6.1.4 Aerospace Cross-Orthogonality; TAMs
6.2 Selecting Input Locations, Directions, and Number
6.2.1 General Guidelines
6.2.2 Selection of Locations
6.3 Planning the Criteria for “Test Exit”
7 Closure
References
10 Experimental Modal Analysis Methods
Contents
Nomenclature
1 Introduction
2 Modal Parameter Estimation: Background
2.1 Assumptions, Definitions, and Concepts
2.1.1 Assumptions
2.1.2 Definition: Modal Parameters
Modal Vector Normalization
2.1.3 Definition: Degrees of Freedom (DOFs)
2.1.4 Concept: Experimental Modal Parameter Estimation
2.1.5 Concept: Data Domain
2.1.6 Concept: Characteristic Space
2.1.7 Concept: Data Dimensionality
2.1.8 Concept: Generalized Frequency
2.1.9 Concept: Kernel Equations
2.1.10 Concept: Overdetermined Linear Models
2.1.11 Concept: General (Two-Stage) Solution Procedure
2.1.12 Concept: Equation Normalization
2.2 Analytical Models
2.2.1 [M] [C] [K] Models
2.2.2 [A] [B] Models
2.2.3 [A] [B] [C] [D] Models
2.2.4 Eigen-Solutions, Orthogonality and Modal Scaling
Orthogonality
Modal Scaling
2.3 Experimental Models
2.3.1 Polynomial Models
Frequency Domain
Time Domain
2.3.2 Companion Matrix
2.3.3 Partial Fraction Models: Residues and Residuals
Residues
Residues from Single Reference FRFs
Residues from Multiple Reference FRFs
Residues from IRFs
Residuals from FRFs
2.3.4 Modal Vectors and Modal Scaling from Residues
2.3.5 Other Experimental Model Methods
3 Single Degree of Freedom Methods
3.1 SDOF Algorithms: Overview
3.2 Operating Vector (Peak-Pick) Estimation
3.2.1 Half-Power Bandwidth Method
3.2.2 Logarithmic Decrement Method
3.3 Complex Plot (Circle Fit) Method
3.4 Two-Point Finite Difference Formulation
3.5 Least-Squares (Local) SDOF Method
3.6 Least-Squares (Global) SDOF Method
3.7 Other SDOF Methods
4 Multiple Degree of Freedom Methods
4.1 General (Two-Stage) Solution Procedure
4.1.1 Consistency Diagrams
4.2 Current MPE Methods
4.3 Kernel Equations: Time Domain Algorithms
4.3.1 High-Order Methods
4.3.2 Low-Order Methods: First Order
4.3.3 Low-Order Methods: Second Order
4.4 Kernel Equations: Frequency Domain Algorithms
4.4.1 Generalized Frequency
4.4.2 High-Order Methods
4.4.3 Low-Order Methods: First Order
4.4.4 Low-Order Methods: Second Order
4.5 Residue (Modal Vector) Estimation
4.5.1 Time Domain Methods
4.5.2 Frequency Domain Methods
5 Differences in MPE Algorithms
5.1 Polynomial Coefficient Estimation
5.2 Generalized Frequency
5.2.1 Normalized Frequency
5.2.2 Orthogonal Polynomials
Discrete Orthogonal Polynomials
5.2.3 Complex Z Mapping
5.3 Data Sieving/Filtering/Decimation
5.4 Coefficient Condensation (Virtual DOFs)
5.4.1 Eigenvalue Decomposition
5.4.2 Singular Value Decomposition
5.4.3 Virtual FRFs
5.5 Equation Condensation
6 Summary
References
11 Experimental Modal Parameter Evaluation Methods
Contents
Nomenclature
1 Introduction
2 Background: Modal Parameter Estimation Methods
2.1 Assumptions, Definitions, and Concepts
2.1.1 Assumptions
2.1.2 Definition: Modal Parameters
Modal Vector Normalization
2.1.3 Definition: Degrees of Freedom (DOFs)
2.1.4 Concept: Experimental Modal Parameter Estimation
2.1.5 Concept: Experimental Modal Parameter Methods
2.1.6 Concept: Data Domain
2.1.7 Concept: Characteristic Space
2.1.8 Concept: Data Dimensionality
2.1.9 Concept: Generalized Frequency
2.1.10 Concept: Kernel Equations
Frequency Domain
Time Domain
2.1.11 Concept: Overdetermined Linear Models
2.1.12 Concept: General (Two-Stage) Solution Procedure
2.1.13 Concept: Equation Normalization
2.1.14 Concept: Modal Vectors, Modal Scaling, Residues
3 Modal Frequency Evaluation/Validation Tools
3.1 Model Order Relationships
3.2 Auto Moment Functions
3.3 Mode Indication Functions
3.3.1 Complex Mode Indication Function
3.3.2 Multivariate Mode Indication Function
3.4 Consistency Diagrams
3.4.1 Alternate Consistency Diagram
3.5 Pole Surface Consistency Plots
3.6 Modal Parameter Clustering
4 Modal Vector Evaluation/Validation Tools
4.1 Modal Vector Conditioning
4.1.1 Vector Normalization
4.1.2 Real Normalization
4.1.3 Central Axis Rotation
4.1.4 Vector Complexity
4.1.5 Modal Vector Complexity Plots
Case 6
Case 8
Case 10
4.2 Modal Vector Validation: eFRF
4.2.1 eFRF: Theoretical Definition
4.2.2 eFRF: Historical Development
4.2.3 eFRF: FRF SVD Development
4.3 Weighted Modal Vector Orthogonality
4.3.1 Weighted Orthogonality of Modal Vectors
4.4 Weighted Pseudo Orthogonality of Modal Vectors
4.5 Modal Vector Consistency
4.5.1 Modal Vector Linearity or Consistency
Consistency of Modal Vectors
Modal Assurance Criterion (MAC) Zero
Modal Assurance Criterion (MAC) Unity
4.5.2 Cross Modal Assurance Criterion (Cross MAC)
4.5.3 Pole-Weighted or State Vector MAC
4.5.4 Other Similar Assurance Criteria
Weighted Modal Analysis Criterion (WMAC)
Partial Modal Analysis Criterion (PMAC)
Modal Assurance Criterion Square Root (MACSR)
Scaled Modal Assurance Criterion (SMAC)
Modal Assurance Criterion Using Reciprocal Vectors (MACRV)
Modal Assurance Criterion with Frequency Scales (FMAC)
Coordinate Modal Assurance Criterion (COMAC)
Enhanced Coordinate Modal Assurance Criterion (ECOMAC)
Mutual Correspondence Criterion (MCC)
Modal Correlation Coefficient (MCC)
Inverse Modal Assurance Criterion (IMAC)
Frequency Response Assurance Criterion (FRAC)
Complex Correlation Coefficient (CCF)
Frequency Domain Assurance Criterion (FDAC)
Coordinate Orthogonality Check (CORTHOG)
4.5.5 Uses of the Modal Assurance Criterion
4.5.6 Misuse/Abuse of the Modal Assurance Criterion
5 Autonomous Modal Parameter Estimation
5.1 Current Approaches
5.2 Common Statistical Subspace Autonomous Mode Identification (CSSAMI)
6 Summary
References
12 Damping of Materials and Structures
Contents
Nomenclature
1 Classification and Survey
1.1 Introduction
1.2 The Notion of Damping
1.3 Classification of Damping Phenomena
1.4 Notes on Modern, Computer-Based Analytical and Measurement Programs
2 Damping of Solids
2.1 Physical Phenomena
2.2 Linear Models
2.2.1 Three-Parameter Models
2.2.2 Three-Parameter Models in the Standard Test
2.2.3 N-Parameter Model
2.2.4 Operator Notation
2.2.5 Creep and Relaxation
2.2.6 Harmonic Stress and Strain Function
2.2.7 Three-Dimensional Stress State
2.2.8 Temperature Dependence of Viscoelastic Material Properties
2.2.9 Thermo-Rhoelogical Simple Materials
2.3 Nonlinear Models
2.3.1 Models for Static Hysteresis
Point-Symmetrical Hysteresis Without Reversal Points
General Shape of Hysteresis Curves
2.3.2 Models for Nonlinear Viscoelasticity
2.3.3 Models for Static Hysteresis and Viscoelasticity
Rheological Models
Mathematical Model
3 Damping of Assemblies
3.1 From Material Description to Complete Homogeneous Component
3.2 Laminated Components
3.3 Damping in Joints
3.3.1 Description by a Functional Equation
3.3.2 Description in Terms of Springs and Coulomb Elements
3.3.3 Description in Terms of Equivalent Spring and Equivalent Damper
3.3.4 Description Using Finite Element Models
3.4 Damping Due to Fluids
3.4.1 Interaction Between a Structure and the Surrounding Medium
3.4.2 Radiation Efficiency, Logarithmic Radiation Efficiency, and Radiation Loss Factor
3.4.3 Elementary Radiators
Monopole or Zero-Order Spherical Radiator (Breathing Sphere)
Dipole or First-Order Radiator (Vibrating Rigid Sphere)
Plane Radiator (Piston) 19:19-1:bib31
3.4.4 Damping of Bending Vibrations of Plates
Radiation loss Factor of Homogeneous, Constant Thickness Plates
Infinite Homogeneous Plate of Constant Thickness
Homogeneous Rectangular Plate of Constant Thickness
Rectangular Plates Supported on All Sides
Other Boundary Conditions
Ribbed Plates
3.4.5 Damping of Vibrating Pipes
Infinite Regular Cylindrical Pipe
Bending Vibrations of Long Regular Cylindrical Pipes
Bending Vibrations of Long Pipes with Elliptical or Rectangular Cross Section
3.4.6 References to Nonlinearities
3.5 Damping by Displacement
3.5.1 Damping by Air Displacement
3.5.2 Journal Bearings, Squeeze Film Dampers
Journal Bearings
3.5.3 Squeeze Film Dampers
3.6 Assemblies
4 Models for Damped Structures
4.1 Basic Model
4.1.1 Free Vibrations with F(t) = 0
4.1.2 Forced Vibrations Where F(t)≠0
4.1.3 Dynamic Compliance (Receptance)
4.1.4 Dynamic Stiffness
4.1.5 Mobility (Admittance)
4.1.6 Mechanical Impedance
4.1.7 Accelerance
4.1.8 Dynamic Mass or Inertance
4.2 Structures with a Finite Number of Degrees of Freedom
4.2.1 N-Parameter Model for Viscoelastic Material Behavior
Differential Operator Formulation
4.2.2 Memory Integral Formulation
4.2.3 2-Parameter Model According to Kelvin-Voigt, Viscous Damping
4.2.4 Damping with Given Frequency Dependence
4.2.5 Calculation of Viscoelastic Components by the Boundary Element Method
5 Experimental Techniques for the Determination of Damping Characteristics
5.1 Experimental Techniques
5.1.1 Basic Procedures
5.1.2 External Damping
5.1.3 Applicability of Results
5.2 Experimental Techniques and Types of Apparatus
5.2.1 Survey of Experimental Techniques
5.2.2 Quasi-Static Methods for the Determination of Material Properties
5.2.3 Experimental Determination of Damping in Solid Bodies with a Low Shear Modulus
5.2.4 Experimental Determination of Damping in Solid Bodies with a High Shear Modulus
5.2.5 Experimental Determination of Damping in Viscous Liquids
5.2.6 Determination of Damping in Uniformly Rotating Specimens
5.2.7 Determination of Damping in the Case of Free Vibrations with One Degree of Freedom
5.2.8 Determination of Damping via Specification of Harmonic Deformations
5.2.9 Measurement of the Oscillation Amplitude in Vicinity of Resonance (Determination of Halfwidth Value)
5.2.10 Measurement of Amplitudes and Phase Angles
5.2.11 Determination of Damping via Thermal Energy Balances
5.2.12 Energy Balances at the Subsystem Boundaries of Multicomponent Systems
5.2.13 Force and Displacement Measurements at Subsystem Boundaries
5.3 Special Experimental Techniques for Determining Damping Under Difficult Conditions
5.3.1 Systems with High Damping
5.3.2 Flexural Vibrations of Lamellar Specimens
Homogeneous Strips
Laminated Strips
5.3.3 Longitudinal Waves in Bars
5.4 Experimental Modal Analysis
5.4.1 Discrete Equivalent Model
5.4.2 Basic Principles in the Measurement of Complex Frequency Responses
5.4.3 Evaluation of Measured Frequency Responses at an Isolated Resonance Point
Idealization as Vibrator with One Degree of Freedom
Approximative Inclusion of the Other Degrees of Freedom
5.4.4 Approximation of Measured Frequency Responses in an Interval with Several Resonance Points
Incomplete Equivalent Model
Generalization of the Method for Isolated Resonance Points
General Approximation of the Frequency Response
5.5 Experimental Techniques for Measuring Soil Damping
6 Application of Fractional Calculus to Viscoelastically Damped Structures in the Finite Element Method
6.1 Grünwald Definition of Fractional Derivatives
6.2 Numerical Calculation of Fractional Derivatives
6.3 Fractional-Order Constitutive Equations
6.4 Finite Element Formulation and Implementation
6.5 Parameter Identification: A Case Study with DelrinTM
6.6 Finite Element Calculations and Comparison of the Different Concepts
7 Conclusion
Technical Standards
References
13 Modal Analysis of Nonlinear Mechanical Systems
Contents
1 Nonlinear Normal Modes: A Brief Historical Perspective
2 Nonlinear Normal Modes: What Are They?
2.1 Definition of a Nonlinear Normal Mode
2.1.1 Rosenberg's Definition
2.1.2 The Invariant Manifold Approach
2.2 Fundamental Properties
2.2.1 Frequency-Energy Dependence
2.2.2 Modal Interactions: Internally Resonant Nonlinear Normal Modes
2.2.3 Mode Bifurcations and Stability
3 Nonlinear Normal Modes: How to Compute Them?
3.1 Analytical Techniques
3.1.1 An Energy-Based Formulation
3.1.2 The Invariant Manifold Approach
3.2 Numerical Techniques
3.3 Assessment of the Different Methodologies
4 Nonlinear Normal Modes: Why Are They Useful?
4.1 ``Linear'' Modal Analysis
4.2 Nonlinear Modal Analysis
4.3 Reduced-Order Modeling
5 Closure
References
Part III Analytical/Experimental Modeling Applications
14 Substructuring Concepts and Component Mode Synthesis
Contents
1 Model Reduction: General Concepts
1.1 Reduction by Projection
1.2 The Guyan–Irons Method
1.3 Model Reduction Through Substructuring
2 Numerical Techniques for Model Reduction of Substructures
2.1 The Hurty/Craig–Bampton Method
2.2 Substructure Reduction Using Free Interface Modes
2.2.1 Rubin Method (RM)
2.2.2 MacNeal Method (MNM)
2.2.3 Dual Craig–Bampton Method (DCBM)
2.3 Numerical Examples of Different Substructure Reduction Techniques
2.4 Other Reduction Techniques for Substructures
3 Interface Reduction with the Hurty/Craig–Bampton Method: Characteristic Constraint Modes
3.1 Interface Reduction Approaches
3.2 System-Level Characteristic Constraint (S-CC) Modes
References
15 Finite Element Model Correlation
Contents
Acronyms
Nomenclature
1 Introduction/Background
2 Theory
2.1 Model Reduction
2.1.1 Guyan Reduction
2.1.2 Improved Reduced System (IRS)
2.1.3 Dynamic Condensation
2.1.4 System Equivalent Reduction Expansion Process (SEREP)
2.1.5 Modal TAM
2.1.6 Hybrid
2.2 Model Expansion (Vector Expansion)
2.2.1 Guyan Expansion
2.2.2 Improved Reduced System (IRS)
2.2.3 Dynamic Expansion
2.2.4 System Equivalent Reduction Expansion Process (SEREP)
2.2.5 Modal
2.2.6 Hybrid
2.2.7 Model Reduction Considerations for Sensor Locations
2.3 Test Data Considerations
2.4 Vector Correlation
2.4.1 Modal Assurance Criteria (MAC)
2.4.2 Orthogonality Checks
2.4.3 Coordinate Modal Assurance Criteria (CoMAC)
2.4.4 Frequency Response Assurance Criteria (FRAC)
2.4.5 Response Vector Assurance Criteria (RVAC)
2.4.6 Test Response Assurance Criteria (TRAC)
2.4.7 CORTHOG
3 Closing Remarks
References
16 Model Updating
Contents
Nomenclature
1 Introduction
2 Parameter Estimation
3 Modeling Errors and Measurement Inaccuracy
4 Sensitivity Analysis
4.1 Undamped Eigenvalue Residual
4.2 Undamped Mode-Shape Residual
4.3 Frequency-Domain Displacement Response Residual
5 Regularization
5.1 Example: Two Degree-of-Freedom Statically Loaded System
6 Parameterization
6.1 Mass, Damping, and Stiffness Matrix Multipliers
6.2 Material Properties, Thicknesses, and Sectional Properties
6.3 Offset Nodes
6.3.1 Example: Parameterization of a “T” Joint
6.4 Generic Elements
6.4.1 Example: Eigenvalue Decomposition of a Beam Element
6.4.2 Example: Generic Element Parameters for a Pinned-Pinned Beam
6.4.3 Example: Updating a System of Three Beams with Offset Central Span
7 Stochastic Model Updating
7.1 Example: Stochastic Model Updating of a Three Degree-of-Freedom System
7.2 Example: Parameter Selection for Stochastic Model Updating
8 Validation of Updated Models
8.1 Benchmark Data
8.2 Summary of Model Validation Results
9 Industrial Example Problem
9.1 Automotive Example Problem
9.1.1 Component Level
9.1.2 Subassembly Level
10 Conclusions
References
17 Nonlinear System Analysis Methods
Contents
Nomenclature
1 Introduction
2 Experimental Setup
3 Methods for Nonlinear Characterization
3.1 Coherence
3.1.1 Example: Undamaged Panel Versus Panel with Disbond
3.2 Frequency Response Function Distortion
3.2.1 Example: Undamaged Panel Versus Panel with Disbond
3.3 Higher-Order FRFs
3.3.1 Example: Panel with Disbond
3.4 Hilbert Transform in the Time Domain
3.5 Hilbert Transform in the Frequency Domain
3.6 Restoring Force Method
3.6.1 Example: Panel with Disbond
3.7 Vibro-Acoustic Modulation
3.7.1 Example: Undamaged Panel Versus Panel with Disbond
4 Methods for Parameter Estimation
4.1 Nonlinear Autoregressive Moving Average with Exogenous Inputs (NARMAX)
4.1.1 Example: Panel with Disbond
4.2 Direct Parameter Estimation
4.3 Reverse Path
4.3.1 Example: Panel with Disbond
4.4 Nonlinear Identification Through Feedback of the Outputs (NIFO)
4.4.1 Example: Panel with Disbond
5 Summary
References
18 Structural Health Monitoring and Damage Identification
Contents
1 Introduction
1.1 Motivation and Definition of SHM
1.2 Statistical Pattern Recognition Approach to SHM
1.2.1 Operational Evaluation
1.2.2 Data Acquisition, Normalization, and Cleansing
1.2.3 Feature Extraction and Information Condensation
1.2.4 Statistical Model Development
1.3 Fundamental Axioms of SHM
1.4 Historical Overview
2 SHM Data and Damage-Sensitive Features
2.1 Vibration
2.1.1 Ways to Measure Vibration
2.1.2 Damage-Sensitive Features from Vibration
2.1.3 Coherence
2.2 Acoustic Emissions (AE)
2.3 Guided Waves
2.4 Performance, Operational, and Environmental Parameters
3 Advanced Topics in Signal Processing and Feature Extraction
3.1 Inference via the Kalman Filter
3.1.1 Inference for Parameter Identification
3.2 Recursive Estimation in the Nonlinear Case
4 Statistical Pattern Recognition for Damage Identification
4.1 Pattern Recognition for Feature Discrimination
4.2 Data-Driven Models in SHM: Learning and Prediction
4.2.1 Acoustic Emission Dataset
4.3 Outlier Analysis for Damage Identification
4.3.1 Statistical Outlier Analysis
4.3.2 Outlier Analysis as One-Class Classifiers
4.4 The Problem of Inclusive Outliers: Robust Outlier Analysis
4.4.1 The Minimum Covariance Determinant
4.5 Probabilistic Classification through Supervised Learning
4.6 The Problem of Feature Dimensionality
4.7 Outstanding Challenges in Data-Driven SHM
5 SHM in Changing Environmental and Operational Conditions
5.1 Removing Confounding Influences
5.2 Linear and Nonlinear Cointegration
5.2.1 Using Cointegration for SHM
5.3 Nonlinear Cointegration
6 Physics-Based Models in SHM
6.1 Inverse Model-Driven SHM
6.2 Forward Model-Driven SHM
7 Summary
7.1 Applications
References
19 Experimental Dynamic Substructures
Contents
Nomenclature
1 Introduction
2 Experimental Substructure Technology
2.1 Connecting Substructures with Compatibility and/or Equilibrium Equations
2.2 Connecting Substructures in the Physical Domain
2.3 Connecting Substructures in the Modal Domain: Component Mode Synthesis
2.4 Connecting Substructures in the Frequency Domain: Frequency-Based Substructuring
3 Dealing with Experimental Difficulties
3.1 Common Experimental Difficulties
3.1.1 Measuring Rigid Body Modes
3.1.2 Modal Fitting of Nonlinear Response
3.1.3 Mass Errors Introduced by Sensors
3.1.4 Modal Truncation Errors
3.1.5 Measuring Rotational DOF Motion and Forces
3.1.6 Continuity of the Attachment Interface
3.1.7 Difficulty in Mounting Sensors at Connection Locations
3.1.8 Dynamic Effects in the Joints
3.1.9 Experimental Errors
3.2 The Transmission Simulator Approach to Mitigate Traditional Experimental Difficulties
3.3 An Example Using the Transmission Simulator Approach
3.4 Transmission Simulator Theory
3.4.1 Preparation to Implement Transmission Simulator Theory
3.4.2 Transmission Simulator Method Using CMS
3.4.3 Transmission Simulator Method Using FBS
3.5 Practical Guidance Using the Transmission Simulator Approach
References
20 Structural Dynamics Modification and Modal Modeling
Contents
1 Modal Models
2 Design Modifications
3 Eigenvalue Modification
4 Measurement Chain to Obtain an EMA Modal Model
4.1 Critical Issues in the Measurement Chain
4.2 Calculating FRFs from Experimental Vibration Data
4.3 Sensing Force and Motion
4.4 Sensitivity Flatness
4.5 Transverse Sensitivity
4.6 Sensor Linearity
4.7 Sensor Mounting
4.8 Leakage Error
4.9 Finite Length Sampling Window
4.10 Leakage-Free Spectrum
4.11 Leakage-Free Signals
4.12 Reduced Leakage
4.13 Linear Versus Nonlinear Dynamics
4.14 Random Excitation and Spectrum Averaging
4.15 Curve Fitting FRFs
4.16 Modal Models and SDM
5 Structural Dynamic Models
5.1 Structural Resonances
5.2 Truncated Modal Model
5.3 Sub-structuring
5.4 Rotational DOFs
6 Time Domain Dynamic Model
6.1 Finite Element Analysis (FEA)
6.2 FEA Modes
7 Frequency Domain Dynamic Model
8 Parametric Models Used for Curve Fitting
8.1 Rational Fraction Polynomial Model
8.2 Partial Fraction Expansion Model
8.3 Experimental FRFs
9 FRF-Based Curve Fitting
9.1 Modal Frequency and Damping
9.2 Modal Residue
10 Transformed Equations of Motion
11 Dynamic Model in Modal Coordinates
11.1 Damping Models
11.2 Lightly Damped Structures
12 Scaling Mode Shapes to Unit Modal Masses
12.1 Modal Mass Matrix
12.2 Modal Stiffness Matrix
12.3 Modal Damping Matrix
12.4 Unit Modal Masses
13 SDM Dynamic Model
14 SDM Equations Using UMM Mode Shapes
15 Scaling Residues to UMM Mode Shapes
15.1 Driving Point FRF Measurement
15.2 UMM Mode Shape
15.3 Triangular FRF Measurements
16 Integrating Residues to Displacement Units
17 Effective Mass
18 Diagonal Mass Matrix
18.1 Checking the Engineering Units
19 Effective Mass Example
20 SDM Example
20.1 Cap Screw Stiffnesses
21 EMA Mode Shapes of the Plate
22 FEA Mode Shapes of the Plate
22.1 Mode Shape Comparison
22.2 Modal Frequency Comparison
22.3 Hybrid Modal Model
22.4 RIB FEA Model
22.5 RIB Impact Test
22.6 Hybrid Modal Model of the RIB
23 Substructure Modal Model
23.1 Block Diagonal Format
24 Calculating New Modes with SDM
25 SDM Versus FEA Modes: Plate and RIB
25.1 SDM Mode Shapes
26 SDM Versus EMA Modes: Plate and RIB
27 Conclusions
28 Modeling a Tuned Vibration Absorber with SDM
29 Adding a Tuned Absorber to the Plate and RIB
30 Modal Sensitivity Analysis
30.1 EMA Modes of the Plate and RIB
30.2 Using SDM to Explore Joint Stiffnesses
30.3 Current Versus Target Frequency
30.4 Solution Space
31 FEA Modal Updating
32 Difference Between Modal Sensitivity and FEA Model Updating
References
21 Toward Robust Response Models: Theoretical and Experimental Issues
Contents
1 Introduction
1.1 Foreword
1.2 Classification of the Models
1.2.1 The Spatial Model
1.2.2 The Response Model
1.2.3 The Modal Model
1.2.4 Relation Among the Models
2 Coupling/Uncoupling Techniques
2.1 Coupling
2.2 Uncoupling
3 Measurement of Rotational Degrees of Freedom
3.1 Introduction
3.2 Experimental Methods for Measuring Rdofs
3.2.1 Indirect Techniques for Measuring Rdofs
Block Excitation
Mass Additive Technique
Finite Difference Technique
Finite Difference Transformation Matrices for First-Order Approximation
Finite Difference Transformation Matrices for Second-Order Approximation
Laser Doppler Vibrometer
PZTs and Strain Gauge Transducers
3.2.2 Direct Techniques
Angular Transducers
Micro-Electro-Mechanical-Sensor (MEMS)
Direct Piezoelectric Rotational Accelerometers
3.3 Experimental Methods for Exciting Rdofs
3.3.1 Indirect Techniques to Apply a Moment Excitation
3.3.2 Direct Techniques to Apply a Moment Excitation
4 Condensation (Reduction) Versus Expansion
4.1 Model Reduction
4.1.1 Guyan Reduction
4.1.2 Dynamic Reduction
4.1.3 Improved Reduction System (IRS)
4.1.4 System Equivalent Reduction Expansion Process (SEREP)
4.1.5 Modal Truncation
4.1.6 Component Mode Synthesis
4.1.7 Sum of Weighted Accelerations Technique (SWAT)
4.1.8 Reduction of Damped Models
4.2 Expansion of Measured Data
4.2.1 Kidder's Method
4.2.2 Expansion Using Analytical Modes
4.2.3 Expansion of Frequency Response Functions (FRF)
5 Transmissibility as a Means to Estimate the Dynamic Response
5.1 Introduction
5.2 Theoretical Description
5.2.1 Fundamental Formulation
5.2.2 Alternative Formulation
5.2.3 Transmissibility Properties
5.3 Other Possible Applications of Transmissibility
References
22 Linear Modal Substructuring with Nonlinear Connections
Contents
Nomenclature
1 Introduction
2 Theory
2.1 Equations of Motion and Modal Space Representation
2.2 Model Reduction and Model Expansion
2.3 Structural Dynamic Modification and System Modeling
2.4 Mode Contribution Matrix
2.5 Response of Linear Components Interconnected with Nonlinear Connection Elements
2.6 Expansion of Transient Time Response from Reduced Order System Models
3 Test Cases
3.1 Nonlinear Response Prediction (Thibault and Marinone)
3.2 Nonlinear Response Prediction with Expansion for Full Field Dynamic Strain Prediction (Harvie)
3.3 Nonlinear Response Prediction with Embedded Subcomponent Models (Obando)
4 Conclusions
References
Part IV Applications and Miscellaneous Topics
23 Civil Structural Testing
Contents
Nomenclature
1 Why Structural Health Monitoring (SHM) for Civil Structures
2 Basic Considerations on Loading and Response
3 Data Acquisition and Sensors
3.1 Sensors
3.2 Selection of Data Acquisition Systems
4 Basic Procedures for SHM in Civil Engineering
4.1 Initial Conditions Testing
4.2 Force Vibration with Shaker
4.3 Ambient Vibrations
5 Identification Methods
5.1 The Fourier Transform and Time-Frequency Analysis
5.2 Extended Logarithmic Decay
5.2.1 Linear Adjustment of the Decay
5.2.2 Nonlinear Adjustment of the Decay
5.3 Eigensystem Realization Algorithm
5.4 The Periodogram
5.5 Frequency Domain Decomposition
5.6 Natural Excitation Technique ERA
5.7 Stochastic Subspace Identification
5.8 Multivariable Output-Error State sPace
5.9 Iterative Modal Identification
6 Stability and Cluster Diagrams
7 Modal Estimation
7.1 Distance Matrices
7.2 Spurious Pole Elimination Using Hard Stability Criteria
7.3 Modal Characterization Based on Parametric Procedures
7.4 Modal Characterization Using Clustering
7.5 Assessing the Number of Modes Using Cluster Validity
7.6 Cluster Representatives as Mode Features' Estimates
8 Modal Tracking
8.1 Problem Statement and Motivation
8.2 Tracking Based on Boundaries
8.3 Cluster-Based Modal Tracking
9 An Important Note to Increase the Speed of Identification
10 Application Examples
10.1 Building Excited by Ambient Vibration
10.1.1 Temporary Testing
10.1.2 Permanent Ambient Vibration Monitoring
10.2 Buildings Earthquake Vibrations
10.3 Base-Isolated Buildings
10.4 Bridges Under Operational and Ambient Excitation
10.4.1 The Suspended 25 de Abril Bridge in Lisbon
10.4.2 Structural Monitoring System and Data
10.4.3 Modal Estimation and Tracking
11 Concluding Remarks
References
24 Aerospace Perspective for Modeling and Validation
Contents
1 Theoretical Foundations
1.1 Categories of Dynamic Systems
1.2 Variational Principles
1.3 The Finite Element Method
2 Structural Dynamic Models
2.1 Modal Analysis
2.2 Dynamic Bandwidth
2.3 Effective Modeling Guidelines
2.4 Further Thoughts on Structural Dynamic Modeling
3 Matrix Structural Dynamic Analysis
3.1 Guyan Reduction
3.2 The Hurty-Craig-Bampton Method
3.3 The Benfield-Hruda Method
3.4 The MacNeal-Rubin Method
3.5 Application of Hurty-Craig-Bampton and MacNeal-Rubin Methodology
3.6 Detailed Structural Dynamic Loads and the Mode Acceleration Method
4 Verifiction and Validation of Structural Dynamic Models
4.1 System Dynamic Model
4.2 Modal Test Planning and the Test Analysis Model (TAM)
4.3 Target Modes
4.4 Automated Response DOF Selection for Mapping of Experimental Modes
4.5 Measured Data Acquisition, Data Analysis, Experimental Modal Analysis
4.6 Modal Test-Analysis Correlation and U.S. Government Standards
4.7 Overview of Efficient Structural Dynamic Sensitivity Analysis
4.8 Residual Mode Augmentation (RMA)
5 Concluding Remarks
References
25 Applied Math for Experimental Structural Dynamics
Contents
Acronyms
1 Domains and Transforms
1.1 Frequency Domain
1.1.1 Integral Fourier Transform
2 Linear Algebra
2.1 Basic Concepts and Definitions
2.2 Transposition Rules
2.3 Special Matrix Forms
2.4 Symmetric Matrix Rules
2.5 Matrix Measures (Determinant and Trace)
2.6 Vector Space
2.7 Vector Space Applied to Matrices
2.8 Spectral Decomposition
2.9 Singular Value Decomposition
2.10 Eigen Solutions
2.11 Inverse Problems
2.11.1 Solution of Determined Equations
2.12 LU Decomposition
2.12.1 Solution of Underdetermined Equations
2.12.2 Solution of Overdetermined Equations
2.13 Moore-Penrose Generalized Inverse
3 Summary
References
Index