Chapter
1.2 Just what are these Measurements called FRFs?
1.2.1 Why is Only One Row or Column of the FRF Matrix Needed?
1.3 What's the Difference between a Shaker Test and an Impact Test?
1.3.1 What Measurements do we Actually make to Compute the FRF?
1.4 What's the Most Important Thing to Think about when Impact Testing?
1.5 What's the Most Important Thing to Think about when Shaker Testing?
1.6 Tell me More About Windows; They Seem Pretty Important!
1.7 So how do we get Mode Shapes from the Plate FRFs?
1.8 Modal Data and Operating Data
1.8.1 What is Operating Data?
1.8.2 So what Good is Modal Data?
1.8.3 So Should I Collect Modal Data or Operating Data?
Chapter 2 General Theory of Experimental Modal Analysis
2.2 Basic Modal Analysis Theory - SDOF
2.2.1 Single Degree of Freedom System Equation
2.2.2 Single Degree of Freedom System Response due to Harmonic Excitation
2.2.3 Damping Estimation for Single Degree of Freedom System
2.2.4 Response Assessment with Varying Damping
2.2.5 Laplace Domain Approach for Single Degree of Freedom System
2.2.6 System Transfer Function
2.2.7 Different Forms of the Transfer Function
2.2.8 Residue of the SDOF System
2.2.9 Frequency Response Function for a Single Degree of Freedom System
2.2.10 Transfer Function/Frequency Response Function/S-plane for a Single Degree of Freedom System
2.2.11 Frequency Response Function Regions for a Single Degree of Freedom System
2.2.12 Different Forms of the Frequency Response Function
2.2.13 Complex Frequency Response Function
2.3 Basic Modal Analysis Theory - MDOF
2.3.1 Multiple Degree of Freedom System Equations
2.3.2 Laplace Domain for Multiple Degree of Freedom System
2.3.3 The Frequency Response Function
2.3.4 Mode Shapes from Frequency Response Equations
2.3.5 Point-to-Point Frequency Response Function
2.3.6 Response of Multiple Degree of Freedom System to Harmonic Excitations
2.3.7 Example: Cantilever Beam Model with Three Measured DOFs
2.3.8 Summary of Time, Frequency, and Modal Domains
2.3.9 Response due to Forced Excitation using Mode Superposition
Chapter 3 General Signal Processing and Measurements Related to Experimental Modal Analysis
3.2 Time and Frequency Domain
3.3 Some General Information Regarding Data Acquisition
3.4 Digitization of Time Signals
3.9 What is the Fourier Transform?
3.9.1 Fourier Transform and Discrete Fourier Transform
3.9.2 FFT: Periodic Signal
3.9.3 FFT: Non-periodic Signal
3.10 Leakage and Minimization of Leakage
3.10.1 Minimization of Leakage
3.11.1 Rectangular Window
3.11.4 Comparison of Windows with Worst Leakage Distortion Possible
3.11.5 Comparison of Rectangular, Hanning and Flat Top Window
3.11.7 Exponential Window
3.11.8 Convolution of the Window in the Frequency Domain
3.12 Frequency Response Function Formulation
3.13 Typical Measurements
3.13.1 Time Signal and Auto-power Functions
3.13.2 Typical Measurement: Cross Power Function
3.13.3 Typical Measurement: Frequency Response Function
3.13.4 Typical Measurement: Coherence Function
3.14 Time and Frequency Relationship Definition
3.15 Input-Output Model with Noise
3.15.1 H1 Formulation: Output Noise Only
3.15.2 H2 Formulation: Output Noise Only
3.15.3 H1 Formulation: Input Noise Only
3.15.4 H2 Formulation: Input Noise Only
Chapter 4 Excitation Techniques
4.2 Impact Excitation Technique
4.2.2 Hammer Impact Tip Selection
4.2.3 Useful Frequency Range for Impact Excitation
4.2.4 Force Window for Impact Excitation
4.2.7 Response due to Impact
4.2.8 Roving Hammer vs Stationary Hammer and Reciprocity
4.2.9 Impact Testing: an Example Set of Measurements
4.3.2 Historical Development of Shaker Excitation Techniques
4.3.3 Swept Sine Excitation
4.3.4 Pure Random Excitation
4.3.5 Pure Random Excitation with Windows Applied
4.3.6 Pure Random Excitation with Overlap Processing
4.3.7 Pseudo-random Excitation
4.3.8 Periodic Random Excitation
4.3.9 Burst Random Excitation
4.3.10 Sine Chirp Excitation
4.3.11 Digital Stepped Sine Excitation
4.4 Comparison of Different Excitations for a Weldment Structure
4.4.1 Random Excitation with No Window
4.4.2 Random Excitation with Hanning Window
4.4.3 Burst Random Excitation with No Window
4.4.4 Sine Chirp Excitation with No Window
4.4.5 Comparison of Random, Burst Random and Sine Chirp
4.4.6 Comparison of Random and Burst Random at Resonant Peaks
4.4.7 Linearity Check Using Sine Chirp
4.5 Multiple-input, Multiple-output Measurement
4.5.1 Multiple Input vs Single Input Testing
4.5.2 Multiple Input vs Single Input for a Weldment Structure
4.5.3 Multiple Input vs Single Input Testing
4.5.4 Comparison of Multiple Input and Single Input for Weldment Structure
4.5.5 MIMO Measurements on a Multi-component Structure
Chapter 5 Modal Parameter Estimation Techniques
5.2 Experimental Modal Analysis
5.2.1 Least Squares Approximation of Data
5.2.2 Classification of Modal Parameter Estimation Techniques
5.3 Extraction of Modal Parameters
5.3.1 Peak Picking Technique
5.3.2 Circle Fitting - Kennedy and Pancu
5.3.4 Residual Effects of Out of Band Modes
5.3.6 Least Squares Complex Exponential
5.3.7 Advanced Forms of Time and Frequency Domain Estimators
5.3.8 General Time Domain Techniques
5.3.9 General Frequency Domain Techniques
5.3.10 General Consideration for Time vs Frequency Representation
5.3.11 Additional Remarks on Modal Parameter Estimation
5.3.12 Two Step Process for Modal Parameter Estimation
5.4 Mode Identification Tools
5.4.2 Mode Indicator Function
5.4.3 Complex Mode Indicator Function
5.5 Modal Model Validation Tools
5.5.1 Synthesis of Frequency Response Functions using Extracted Parameters
5.5.2 Modal Assurance Criterion
5.5.3 Mode Participation Factors
5.5.4 Mode Overcomplexity
5.5.5 Mean Phase Co-linearity and Mean Phase Deviation
5.6 Operating Modal Analysis
Part II Practical Considerations for Experimental Modal Testing
Chapter 6 Test Setup Considerations
6.2 How Many Modes Required?
6.3 Frequency Range of Interest?
6.4 Transducer Possibilities?
6.6 How Many Measurement Points Needed?
6.7 Excitation Techniques
6.8 Miscellaneous Items to Consider
Chapter 7 Impact Testing Considerations
7.1 Hammer Impact Location
7.2 Hammer Tip and Frequency Range
7.3 Hammers for Different Size Structures
7.4 How Does Impact Skew and Deviation of Input Point Affect the Measurement?
7.4.1 Skewed Impact Force
7.4.2 Inconsistent Impact Force Location
7.5 Impact Hammer Frequency Bandwidth
7.6 Accelerometer ICP Considerations for Low Frequency Measurements
7.7 Considerations for Reciprocity Measurements
7.8 Roving Hammer vs Roving Accelerometer
7.9 Picking a Good Reference Location
7.10 Multiple Impact Difficulties and Considerations
7.10.1 Academic Structure
7.10.2 Large Wind Turbine Blade
7.11 What is "Filter Ring" during an Impact Measurement?
7.12 Test Bandwidth Much Wider than Desired Frequency Range
7.13 Why Does the Structure Response Need to Come to Zero at the End of the Sample Time?
7.14 Measurements with no Overload but Transducers are Saturated
7.14.1 Case 1: Sensitive Accelerometer with Exponential Window
7.14.2 Case 2: Sensitive Accelerometer with No Window
7.14.3 Case 3: Less Sensitive Accelerometer with No Window
7.15 How much Roll Off in the Input Hammer Force Spectrum is Acceptable?
7.16 Can the Hammer be Switched in the Middle of a Test to Avoid Double Impacts?
Chapter 8 Shaker Testing Considerations
8.1 General Hardware Related Issues
8.1.1 General Information about Shakers and Amplifiers
8.1.2 What is the Difference between the Constant Current and Constant Voltage Settings on the Shaker Amplifier?
8.1.3 Some Shakers have a Trunnion: Is it Really Needed and Why Do You Have It?
8.1.4 Where is the Best Location to Place a Shaker for a Modal Test?
8.1.5 How Should the Shaker be Constrained when Testing?
8.1.6 What's the Best Way to Support a Shaker for Lateral Vibration When it is Hung?
8.1.7 What are the Most Common Practical Failures with Shaker Setup?
8.1.8 What is the Correct Level of Shaker Excitation for Modal Testing?
8.1.9 How many Shakers should I use in my Modal Test?
8.1.10 Shaker and Stinger Alignment Issues
8.1.11 When should the Shaker be Attached to the Structure?
8.1.12 Should I Disconnect the Stingers while not Testing?
8.1.13 Force Gage or Impedance Head must be Mounted on Structure Side of Stinger?
8.1.14 What's an Impedance Head? Why use it? Where does it go?
8.2 Stinger Related Issues
8.2.1 Why should Stingers be used?
8.2.2 Can a Poorly Designed Shaker/Stinger Setup Produce Incorrect Results?
8.2.3 Stingers and their Effect on Measured Frequency Response Functions
8.2.3.2 Stinger Alignment
8.2.3.6 How do Piano Wire Stingers Work? How are they Pretensioned??
8.3 Shaker Related Issues
8.3.1 Is MIMO needed for Structures with Directional Modes?
8.3.2 Shaker Force Levels and SISO vs MIMO Considerations
8.3.2.1 High Shaker Force Levels
8.3.2.2 High Shaker Force Levels
8.3.2.3 Effects of FRF Measurements in the Modal Parameter Estimation Process
Chapter 9 Insight into Modal Parameter Estimation
9.2 Mode Indicator Tools Help Identify Modes
9.3 SDOF vs MDOF for a Simple System
9.4 Local vs Global: MACL Frame
9.5 Repeated Root: Composite Spar
9.6 Wind Turbine Blade: Same Geometry but Very Different Modes
9.7 Stability Diagram Demystified
9.8 Curvefitting Demystified
9.9 Curvefitting Different Bands for the Poles and Residues
9.10 Synthesizing the FRF from Parameters from Several Bands Stitched Together
9.11 A Large Multiple Reference Modal Test Parameter Estimation
9.11.1 Case 1: Use of All Measured FRFs
9.11.2 Case 2: Use of Selected Sets of Measured FRFs
9.11.3 Case 3: Use of PolyMAX
9.12 Operating Modal Analysis
Chapter 10 General Considerations
10.1 An Experimental Modal Test: a Thought Process Divulged
10.2.1 General FFT Analyzer Setup
10.2.2 Setup for Impact Testing
10.2.3 Setup for Shaker Testing
10.4 Practical Considerations: Checklists
10.4.1 Checklist for Analyzer Setup
10.4.2 Checklist for Impact Testing
10.4.3 Checklist for Shaker Testing
10.4.4 Checklist for Measurement Adequacy
10.4.5 Checklist for Miscellaneous
Chapter 11 Tips, Tricks, and Other Stuff
11.1 Modal Testing Primer
11.1.3 Drive Point Measurements
11.1.5 Inappropriate Reference Location
11.1.6 Multiple-input, Multiple-output Testing
11.1.7 Multiple Reference Testing
11.2 Impact Hammer and Impulsive Excitation
11.2.1 The Right Hammer for the Test
11.2.2 Hammer - Get the Swing of it
11.2.4 Hammer tip selection
11.2.5 No Hammer: Improvise
11.2.6 Pete's Hammer Test Impact Ritual
11.3 Accelerometer Issues
11.3.2 Mass Loading Effects from Tri-axial Accelerometers
11.3.3 Accelerometer Sensitivity Selection
11.3.4 Tri-axial Accelerometers
11.4 Curvefitting Considerations
11.4.1 Should all Measurements be used when Curvefitting
11.5 Blue Frame with Three Plate Subsystem
11.6 Miscellaneous Issues
11.6.1 Modal Test Axis Labels
11.6.2 Testing Does Not Need to Start at point 1
11.6.3 Test to a Wider Frequency Range
11.6.4 Ui times Uj; the key to many questions
Appendix A Linear Algebra: Basic Operations Needed for Modal Analysis Operations
A.2 Define a Column Vector
A.4 Define a Diagonal Matrix
A.5 Define Matrix Addition
A.6 Define Matrix Scalar Multiply
A.7 Define Matrix Multiply
A.8 Matrix Multiplication Rules
A.9 Transpose of a Matrix
A.11 Symmetric Matrix Rules
A.12 Define a Matrix Inverse
A.13 Matrix Inverse Properties
A.14 Define an Eigenvalue Problem
A.16 Singular Value Decomposition
Appendix B Example Using Two Degree of Freedom System: Eigenproblem
Appendix C Pole, Residue, and FRF Problem for 2-DOF System
Appendix D Example using Three Degree of Freedom System
Appendix E DYNSYS Website Materials
E.1 Technical Materials Developed
E.1.1 Theoretical Aspects of First and Second Order Systems
E.1.2 First Order Systems: Modeling Step with ODE and Block Diagram
E.1.3 Second Order Systems: Modeling Step, Impulse, IC with ODE and Block Diagram
E.1.4 Mathematical Modeling Considerations
E.1.5 Simulink and MATLAB Primer Materials
E.1.6 Miscellaneous Materials
E.2 DYNSYS.UML.EDU Website
Appendix F Basic Modal Analysis Information
F.1.2 System Transfer Function
F.1.3 Different Forms of the System Transfer Function
F.1.4 Frequency Response Function
F.1.4 F.1.5 System Transfer Function/Frequency Response Function/S-Plane
Part III Collection of Sets of Modal Data Collected for Processing
Appendix G Repeated Root Frame: Boundary Condition Effects
G.1 Corner Supports Set #1
G.2 Midlength Supports Set #2
G.3 Modal Correlation between Set #1 and Set #2
Appendix H Radarsat Satellite Testing
H.1 Data Reduction Set 1: Reference BUS:109:Z, BUS:118:Z, PMS:217:X and PMS:1211:Y
H.2 Data Reduction Set 2: Reference PMS:217:X and PMS:1211:Y
Appendix I Demo Airplane Testing
I.2 SIMO Testing with Skewed Shaker
I.3 MIMO Testing with Two Vertical Modal Shakers
Appendix J Whirlpool Dryer Cabinet Modal Testing
Appendix K GM MTU Automobile Round Robin Modal Testing
Appendix L UML Composite Spar Modal Testing
Appendix M UML BUH Modal Testing
Modal Testing
：A Practitioner's Guide
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