PETER AVITABILE is Professor Emeritus at the University of Massachusetts Lowell, the co-director of the Structural Dynamics and Acoustic Systems Laboratory, and the former President for the Society for Experimental Mechanics. In addition, he is the Associate Editor of the Handbook for Experimental Structural Mechanics. He has written hundreds of papers and articles on analytical and experimental modal analysis techniques, including the Modal Space article series published in SEM's Experimental Techniques.

• Preface xvAbout the CompanionWebsite xixPart I Overview of Experimental Modal Analysis using the Frequency Response Method 11 Introduction to ExperimentalModal Analysis: A Simple Non-mathematical Presentation 31.1 Could you Explain Modal Analysis to Me? 61.2 Just what are these Measurements called FRFs? 101.2.1 Why is Only One Row or Column of the FRF Matrix Needed? 131.3 What’s the Difference between a Shaker Test and an Impact Test? 171.3.1 What Measurements do we Actually make to Compute the FRF? 181.4 What’s the Most ImportantThing toThink about when Impact Testing? 211.5 What’s the Most ImportantThing toThink about when Shaker Testing? 221.6 Tell me More AboutWindows; They Seem Pretty Important! 241.7 So how do we get Mode Shapes from the Plate FRFs? 251.8 Modal Data and Operating Data 291.8.1 What is Operating Data? 291.8.2 So what Good is Modal Data? 331.8.3 So Should I Collect Modal Data or Operating Data? 341.9 Closing Remarks 362 General Theory of Experimental Modal Analysis 372.1 Introduction 372.2 Basic Modal AnalysisTheory – SDOF 382.2.1 Single Degree of Freedom System Equation 382.2.2 Single Degree of Freedom System Response due to Harmonic Excitation 402.2.3 Damping Estimation for Single Degree of Freedom System 422.2.4 Response Assessment with Varying Damping 432.2.5 Laplace Domain Approach for Single Degree of Freedom System 462.2.6 System Transfer Function 472.2.7 Different Forms of the Transfer Function 482.2.8 Residue of the SDOF System 492.2.9 Frequency Response Function for a Single Degree of Freedom System 492.2.10 Transfer Function/Frequency Response Function/S-plane for a Single Degree of Freedom System 512.2.11 Frequency Response Function Regions for a Single Degree of Freedom System 512.2.12 Different Forms of the Frequency Response Function 532.2.13 Complex Frequency Response Function 532.3 Basic Modal AnalysisTheory – MDOF 562.3.1 Multiple Degree of Freedom System Equations 572.3.2 Laplace Domain for Multiple Degree of Freedom System 662.3.3 The Frequency Response Function 682.3.4 Mode Shapes from Frequency Response Equations 682.3.5 Point-to-Point Frequency Response Function 712.3.6 Response of Multiple Degree of Freedom System to Harmonic Excitations 722.3.7 Example: Cantilever Beam Model with Three Measured DOFs 752.3.8 Summary of Time, Frequency, and Modal Domains 832.3.9 Response due to Forced Excitation using Mode Superposition 872.4 Summary 893 General Signal Processing andMeasurements Related to Experimental Modal Analysis 933.1 Introduction 933.2 Time and Frequency Domain 933.3 Some General Information Regarding Data Acquisition 963.4 Digitization of Time Signals 973.5 Quantization 973.5.1 ADC Underload 983.5.2 ADC Overload 1003.6 AC Coupling 1003.7 SamplingTheory 1013.8 Aliasing 1033.9 What is the Fourier Transform? 1053.9.1 Fourier Transform and Discrete Fourier Transform 1073.9.2 FFT: Periodic Signal 1083.9.3 FFT: Non-periodic Signal 1083.10 Leakage and Minimization of Leakage 1093.10.1 Minimization of Leakage 1113.11 Windows and Leakage 1113.11.1 RectangularWindow 1123.11.2 HanningWindow 1163.11.3 Flat TopWindow 1163.11.4 Comparison ofWindows withWorst Leakage Distortion Possible 1163.11.5 Comparison of Rectangular, Hanning and Flat TopWindow 1193.11.6 ForceWindow 1193.11.7 ExponentialWindow 1193.11.8 Convolution of theWindow in the Frequency Domain 1193.12 Frequency Response Function Formulation 1193.13 TypicalMeasurements 1233.13.1 Time Signal and Auto-power Functions 1233.13.2 TypicalMeasurement: Cross Power Function 1243.13.3 TypicalMeasurement: Frequency Response Function 1243.13.4 TypicalMeasurement: Coherence Function 1243.14 Time and Frequency Relationship Definition 1263.15 Input–Output Model with Noise 1273.15.1 H1 Formulation: Output Noise Only 1273.15.2 H2 Formulation: Output Noise Only 1283.15.3 H1 Formulation: Input Noise Only 1283.15.4 H2 Formulation: Input Noise Only 1283.16 Summary 1294 Excitation Techniques 1314.1 Introduction 1314.2 Impact Excitation Technique 1324.2.1 Impact Hammer 1324.2.2 Hammer Impact Tip Selection 1364.2.3 Useful Frequency Range for Impact Excitation 1374.2.4 ForceWindow for Impact Excitation 1374.2.5 Pre-trigger Delay 1374.2.6 Double Impact 1404.2.7 Response due to Impact 1404.2.8 Roving Hammer vs Stationary Hammer and Reciprocity 1434.2.9 Impact Testing: an Example Set of Measurements 1474.3 Shaker Excitation 1594.3.1 Modal Shaker Setup 1614.3.2 Historical Development of Shaker Excitation Techniques 1624.3.3 Swept Sine Excitation 1634.3.4 Pure Random Excitation 1634.3.5 Pure Random Excitation withWindows Applied 1654.3.6 Pure Random Excitation with Overlap Processing 1654.3.7 Pseudo-random Excitation 1674.3.8 Periodic Random Excitation 1674.3.9 Burst Random Excitation 1684.3.10 Sine Chirp Excitation 1704.3.11 Digital Stepped Sine Excitation 1704.4 Comparison of Different Excitations for aWeldment Structure 1724.4.1 Random Excitation with NoWindow 1724.4.2 Random Excitation with HanningWindow 1734.4.3 Burst Random Excitation with NoWindow 1734.4.4 Sine Chirp Excitation with NoWindow 1744.4.5 Comparison of Random, Burst Random and Sine Chirp 1754.4.6 Comparison of Random and Burst Random at Resonant Peaks 1754.4.7 Linearity Check Using Sine Chirp 1754.5 Multiple-input,Multiple-outputMeasurement 1754.5.1 Multiple Input vs Single Input Testing 1774.5.2 Multiple Input vs Single Input for aWeldment Structure 1814.5.3 Multiple Input vs Single Input Testing 1814.5.4 Comparison of Multiple Input and Single Input forWeldment Structure 1824.5.5 MIMO Measurements on a Multi-component Structure 1824.6 Summary 1875 Modal Parameter Estimation Techniques 1895.1 Introduction 1895.2 ExperimentalModal Analysis 1905.2.1 Least Squares Approximation of Data 1905.2.2 Classification of Modal Parameter Estimation Techniques 1935.3 Extraction of Modal Parameters 1985.3.1 Peak Picking Technique 1985.3.2 Circle Fitting – Kennedy and Pancu 1995.3.3 SDOF Polynomial 2005.3.4 Residual Effects of Out of Band Modes 2005.3.5 MDOF Polynomial 2015.3.6 Least Squares Complex Exponential 2015.3.7 Advanced Forms of Time and Frequency Domain Estimators 2035.3.8 General Time Domain Techniques 2035.3.9 General Frequency Domain Techniques 2035.3.10 General Consideration for Time vs Frequency Representation 2045.3.11 Additional Remarks on Modal Parameter Estimation 2045.3.12 Two Step Process for Modal Parameter Estimation 2055.4 Mode Identification Tools 2065.4.1 Summation Function 2065.4.2 Mode Indicator Function 2065.4.3 Complex Mode Indicator Function 2075.4.4 Stability Diagram 2085.4.5 PolyMAX 2105.5 Modal Model Validation Tools 2125.5.1 Synthesis of Frequency Response Functions using Extracted Parameters 2125.5.2 Modal Assurance Criterion 2135.5.3 Mode Participation Factors 2155.5.4 Mode Overcomplexity 2155.5.5 Mean Phase Co-linearity and Mean Phase Deviation 2165.6 Operating Modal Analysis 2165.7 Summary 219Part II Practical Considerations for ExperimentalModal Testing 2216 Test Setup Considerations 2236.1 Test Plan? 2246.2 How Many Modes Required? 2256.3 Frequency Range of Interest? 2286.4 Transducer Possibilities? 2326.5 Test Configurations? 2326.6 How Many Measurement Points Needed? 2356.7 Excitation Techniques 2386.8 Miscellaneous Items to Consider 2386.9 Summary 2457 Impact Testing Considerations 2477.1 Hammer Impact Location 2477.2 Hammer Tip and Frequency Range 2487.3 Hammers for Different Size Structures 2497.4 How Does Impact Skew and Deviation of Input Point Affect theMeasurement? 2567.4.1 Skewed Impact Force 2567.4.2 Inconsistent Impact Force Location 2567.5 Impact Hammer Frequency Bandwidth 2567.6 Accelerometer ICP Considerations for Low Frequency Measurements 2647.7 Considerations for Reciprocity Measurements 2647.8 Roving Hammer vs Roving Accelerometer 2677.9 Picking a Good Reference Location 2687.10 Multiple Impact Difficulties and Considerations 2687.10.1 Academic Structure 2697.10.2 LargeWind Turbine Blade 2717.11 What is “Filter Ring” during an Impact Measurement? 2747.12 Test Bandwidth MuchWider than Desired Frequency Range 2757.13 Why Does the Structure Response Need to Come to Zero at the End of the Sample Time? 2797.14 Measurements with no Overload but Transducers are Saturated 2827.14.1 Case 1: Sensitive Accelerometer with ExponentialWindow 2827.14.2 Case 2: Sensitive Accelerometer with NoWindow 2837.14.3 Case 3: Less Sensitive Accelerometer with NoWindow 2837.15 How much Roll Off in the Input Hammer Force Spectrum is Acceptable? 2867.16 Can the Hammer be Switched in the Middle of a Test to Avoid Double Impacts? 2897.17 Closing Remarks 2928 Shaker Testing Considerations 2938.1 General Hardware Related Issues 2938.1.1 General Information about Shakers and Amplifiers 2938.1.2 What is the Difference between the Constant Current and Constant Voltage Settings on the Shaker Amplifier? 2948.1.3 Some Shakers have a Trunnion: Is it Really Needed andWhy Do You Have It? 2948.1.4 Where is the Best Location to Place a Shaker for a Modal Test? 2958.1.5 How Should the Shaker be Constrained when Testing? 2968.1.6 What’s the BestWay to Support a Shaker for Lateral Vibration When it is Hung? 2968.1.7 What are the Most Common Practical Failures with Shaker Setup? 2978.1.8 What is the Correct Level of Shaker Excitation for Modal Testing? 2978.1.9 How many Shakers should I use in my Modal Test? 2978.1.10 Shaker and Stinger Alignment Issues 2978.1.11 When should the Shaker be Attached to the Structure? 2988.1.12 Should I Disconnect the Stingers while not Testing? 2988.1.13 Force Gage or Impedance Head must be Mounted on Structure Side of Stinger? 3008.1.14 What’s an Impedance Head? Why use it?Where does it go? 3018.2 Stinger Related Issues 3028.2.1 Why should Stingers be used? 3028.2.2 Can a Poorly Designed Shaker/Stinger Setup Produce Incorrect Results? 3038.2.3 Stingers and their Effect on Measured Frequency Response Functions 3068.2.3.1 Stinger Location 3078.2.3.2 Stinger Alignment 3078.2.3.3 Stinger Length 3088.2.3.4 Stinger Type 3108.2.3.5 Sleeved Stingers 3108.2.3.6 How do PianoWire StingersWork? How are they Pretensioned?? 3148.3 Shaker Related Issues 3148.3.1 Is MIMO needed for Structures with DirectionalModes? 3148.3.2 Shaker Force Levels and SISO vs MIMO Considerations 3168.3.2.1 High Shaker Force Levels 3168.3.2.2 High Shaker Force Levels 3188.3.2.3 Effects of FRF Measurements in the Modal Parameter Estimation Process 3208.4 Concluding Remarks 3259 Insight intoModal Parameter Estimation 3279.1 Introductory Remarks 3279.2 Mode Indicator Tools Help Identify Modes 3289.3 SDOF vsMDOF for a Simple System 3309.4 Local vs Global: MACL Frame 3329.5 Repeated Root: Composite Spar 3349.6 Wind Turbine Blade: Same Geometry but Very Different Modes 3359.7 Stability Diagram Demystified 3379.8 Curvefitting Demystified 3409.9 Curvefitting Different Bands for the Poles and Residues 3439.10 Synthesizing the FRF from Parameters from Several Bands Stitched Together 3449.11 A Large Multiple Reference Modal Test Parameter Estimation 3469.11.1 Case 1: Use of All Measured FRFs 3469.11.2 Case 2: Use of Selected Sets of Measured FRFs 3509.11.3 Case 3: Use of PolyMAX 3529.12 Operating Modal Analysis 3579.13 Concluding Remarks 36310 General Considerations 36510.1 An ExperimentalModal Test: a Thought Process Divulged 36910.2 FFT Analyzer Setup 37710.2.1 General FFT Analyzer Setup 37710.2.2 Setup for Impact Testing 37810.2.3 Setup for Shaker Testing 37910.3 Log Sheets 37910.4 Practical Considerations: Checklists 37910.4.1 Checklist for Analyzer Setup 38010.4.2 Checklist for Impact Testing 38210.4.3 Checklist for Shaker Testing 38410.4.4 Checklist for Measurement Adequacy 38610.4.5 Checklist for Miscellaneous 38810.5 Summary 391Appendix: Logbook Forms 39211 Tips, Tricks, and Other Stuff 39511.1 Modal Testing Primer 39611.1.1 Impact Setup 39611.1.2 Shaker Setup 39711.1.3 Drive Point Measurements 39811.1.4 Reciprocity 39811.1.5 Inappropriate Reference Location 39911.1.6 Multiple-input,Multiple-output Testing 39911.1.7 Multiple Reference Testing 40011.2 Impact Hammer and Impulsive Excitation 40011.2.1 The Right Hammer for the Test 40011.2.2 Hammer – Get the Swing of it 40111.2.3 Hammer Tripod 40111.2.4 Hammer tip selection 40111.2.5 No Hammer: Improvise 40211.2.6 Pete’s Hammer Test Impact Ritual 40211.3 Accelerometer Issues 40311.3.1 Mass Loading 40311.3.2 Mass Loading Effects from Tri-axial Accelerometers 40411.3.3 Accelerometer Sensitivity Selection 40711.3.4 Tri-axial Accelerometers 40811.4 Curvefitting Considerations 41111.4.1 Should all Measurements be used when Curvefitting 41211.5 Blue Frame with Three Plate Subsystem 41411.6 Miscellaneous Issues 42211.6.1 Modal Test Axis Labels 42211.6.2 Testing Does Not Need to Start at point 1 42311.6.3 Test to aWider Frequency Range 42311.6.4 Ui times Uj; the key to many questions 42311.7 Summary 425A Linear Algebra: Basic Operations Needed forModal Analysis Operations 427A.1 Define a Matrix 427A.2 Define a Column Vector 427A.3 Define a Row Vector 428A.4 Define a Diagonal Matrix 428A.5 Define Matrix Addition 428A.6 Define Matrix Scalar Multiply 428A.7 Define Matrix Multiply 429A.8 Matrix Multiplication Rules 429A.9 Transpose of a Matrix 430A.10 Transposition Rules 430A.11 Symmetric Matrix Rules 430A.12 Define a Matrix Inverse 431A.13 Matrix Inverse Properties 431A.14 Define an Eigenvalue Problem 431A.15 Generalized Inverse 431A.16 Singular Value Decomposition 432B Example Using Two Degree of Freedom System: Eigenproblem 433C Pole, Residue, and FRF Problem for 2-DOF System 437D Example using Three Degree of Freedom System 443E DYNSYSWebsite Materials 451E.1 Technical Materials Developed 451E.1.1 Theoretical Aspects of First and Second Order Systems 452E.1.2 First Order Systems: Modeling Step with ODE and Block Diagram 452E.1.3 Second Order Systems: Modeling Step, Impulse, IC with ODE and Block Diagram 452E.1.4 MathematicalModeling Considerations 452E.1.5 Simulink and MATLAB Primer Materials 453E.1.6 Miscellaneous Materials 453E.2 DYNSYS.UML.EDUWebsite 453F Basic Modal Analysis Information 463F.1 SDOF Definitions 463F.1.1 Damping Estimates 463F.1.2 System Transfer Function 464F.1.3 Different Forms of the System Transfer Function 464F.1.4 Frequency Response Function 465F.2 MDOF Definitions 466Part III Collection of Sets of Modal Data Collected for Processing 467G Repeated Root Frame: Boundary Condition Effects 469G.1 Corner Supports Set #1 470G.2 Midlength Supports Set #2 474G.3 Modal Correlation between Set #1 and Set #2 474H Radarsat Satellite Testing 479H.1 Data Reduction Set 1: Reference BUS:109:Z, BUS:118:Z, PMS:217:X and PMS:1211:Y 479H.2 Data Reduction Set 2: Reference PMS:217:X and PMS:1211:Y 479I Demo Airplane Testing 487I.1 Impact Testing 487I.2 SIMO Testing with Skewed Shaker 487I.3 MIMO Testing with Two Vertical Modal Shakers 493J Whirlpool Dryer Cabinet Modal Testing 497K GM MTU Automobile Round Robin Modal Testing 501L UML Composite Spar Modal Testing 505M UML BUHModal Testing 509N Nomenclature 515Index 519

PETER AVITABILE is Professor Emeritus at the University of Massachusetts Lowell, the co-director of the Structural Dynamics and Acoustic Systems Laboratory, and the former President for the Society for Experimental Mechanics. In addition, he is the Associate Editor of the Handbook for Experimental Structural Mechanics. He has written hundreds of papers and articles on analytical and experimental modal analysis techniques, including the Modal Space article series published in SEM's Experimental Techniques.

• Preface xvAbout the CompanionWebsite xixPart I Overview of Experimental Modal Analysis using the Frequency Response Method 11 Introduction to ExperimentalModal Analysis: A Simple Non-mathematical Presentation 31.1 Could you Explain Modal Analysis to Me? 61.2 Just what are these Measurements called FRFs? 101.2.1 Why is Only One Row or Column of the FRF Matrix Needed? 131.3 What’s the Difference between a Shaker Test and an Impact Test? 171.3.1 What Measurements do we Actually make to Compute the FRF? 181.4 What’s the Most ImportantThing toThink about when Impact Testing? 211.5 What’s the Most ImportantThing toThink about when Shaker Testing? 221.6 Tell me More AboutWindows; They Seem Pretty Important! 241.7 So how do we get Mode Shapes from the Plate FRFs? 251.8 Modal Data and Operating Data 291.8.1 What is Operating Data? 291.8.2 So what Good is Modal Data? 331.8.3 So Should I Collect Modal Data or Operating Data? 341.9 Closing Remarks 362 General Theory of Experimental Modal Analysis 372.1 Introduction 372.2 Basic Modal AnalysisTheory – SDOF 382.2.1 Single Degree of Freedom System Equation 382.2.2 Single Degree of Freedom System Response due to Harmonic Excitation 402.2.3 Damping Estimation for Single Degree of Freedom System 422.2.4 Response Assessment with Varying Damping 432.2.5 Laplace Domain Approach for Single Degree of Freedom System 462.2.6 System Transfer Function 472.2.7 Different Forms of the Transfer Function 482.2.8 Residue of the SDOF System 492.2.9 Frequency Response Function for a Single Degree of Freedom System 492.2.10 Transfer Function/Frequency Response Function/S-plane for a Single Degree of Freedom System 512.2.11 Frequency Response Function Regions for a Single Degree of Freedom System 512.2.12 Different Forms of the Frequency Response Function 532.2.13 Complex Frequency Response Function 532.3 Basic Modal AnalysisTheory – MDOF 562.3.1 Multiple Degree of Freedom System Equations 572.3.2 Laplace Domain for Multiple Degree of Freedom System 662.3.3 The Frequency Response Function 682.3.4 Mode Shapes from Frequency Response Equations 682.3.5 Point-to-Point Frequency Response Function 712.3.6 Response of Multiple Degree of Freedom System to Harmonic Excitations 722.3.7 Example: Cantilever Beam Model with Three Measured DOFs 752.3.8 Summary of Time, Frequency, and Modal Domains 832.3.9 Response due to Forced Excitation using Mode Superposition 872.4 Summary 893 General Signal Processing andMeasurements Related to Experimental Modal Analysis 933.1 Introduction 933.2 Time and Frequency Domain 933.3 Some General Information Regarding Data Acquisition 963.4 Digitization of Time Signals 973.5 Quantization 973.5.1 ADC Underload 983.5.2 ADC Overload 1003.6 AC Coupling 1003.7 SamplingTheory 1013.8 Aliasing 1033.9 What is the Fourier Transform? 1053.9.1 Fourier Transform and Discrete Fourier Transform 1073.9.2 FFT: Periodic Signal 1083.9.3 FFT: Non-periodic Signal 1083.10 Leakage and Minimization of Leakage 1093.10.1 Minimization of Leakage 1113.11 Windows and Leakage 1113.11.1 RectangularWindow 1123.11.2 HanningWindow 1163.11.3 Flat TopWindow 1163.11.4 Comparison ofWindows withWorst Leakage Distortion Possible 1163.11.5 Comparison of Rectangular, Hanning and Flat TopWindow 1193.11.6 ForceWindow 1193.11.7 ExponentialWindow 1193.11.8 Convolution of theWindow in the Frequency Domain 1193.12 Frequency Response Function Formulation 1193.13 TypicalMeasurements 1233.13.1 Time Signal and Auto-power Functions 1233.13.2 TypicalMeasurement: Cross Power Function 1243.13.3 TypicalMeasurement: Frequency Response Function 1243.13.4 TypicalMeasurement: Coherence Function 1243.14 Time and Frequency Relationship Definition 1263.15 Input–Output Model with Noise 1273.15.1 H1 Formulation: Output Noise Only 1273.15.2 H2 Formulation: Output Noise Only 1283.15.3 H1 Formulation: Input Noise Only 1283.15.4 H2 Formulation: Input Noise Only 1283.16 Summary 1294 Excitation Techniques 1314.1 Introduction 1314.2 Impact Excitation Technique 1324.2.1 Impact Hammer 1324.2.2 Hammer Impact Tip Selection 1364.2.3 Useful Frequency Range for Impact Excitation 1374.2.4 ForceWindow for Impact Excitation 1374.2.5 Pre-trigger Delay 1374.2.6 Double Impact 1404.2.7 Response due to Impact 1404.2.8 Roving Hammer vs Stationary Hammer and Reciprocity 1434.2.9 Impact Testing: an Example Set of Measurements 1474.3 Shaker Excitation 1594.3.1 Modal Shaker Setup 1614.3.2 Historical Development of Shaker Excitation Techniques 1624.3.3 Swept Sine Excitation 1634.3.4 Pure Random Excitation 1634.3.5 Pure Random Excitation withWindows Applied 1654.3.6 Pure Random Excitation with Overlap Processing 1654.3.7 Pseudo-random Excitation 1674.3.8 Periodic Random Excitation 1674.3.9 Burst Random Excitation 1684.3.10 Sine Chirp Excitation 1704.3.11 Digital Stepped Sine Excitation 1704.4 Comparison of Different Excitations for aWeldment Structure 1724.4.1 Random Excitation with NoWindow 1724.4.2 Random Excitation with HanningWindow 1734.4.3 Burst Random Excitation with NoWindow 1734.4.4 Sine Chirp Excitation with NoWindow 1744.4.5 Comparison of Random, Burst Random and Sine Chirp 1754.4.6 Comparison of Random and Burst Random at Resonant Peaks 1754.4.7 Linearity Check Using Sine Chirp 1754.5 Multiple-input,Multiple-outputMeasurement 1754.5.1 Multiple Input vs Single Input Testing 1774.5.2 Multiple Input vs Single Input for aWeldment Structure 1814.5.3 Multiple Input vs Single Input Testing 1814.5.4 Comparison of Multiple Input and Single Input forWeldment Structure 1824.5.5 MIMO Measurements on a Multi-component Structure 1824.6 Summary 1875 Modal Parameter Estimation Techniques 1895.1 Introduction 1895.2 ExperimentalModal Analysis 1905.2.1 Least Squares Approximation of Data 1905.2.2 Classification of Modal Parameter Estimation Techniques 1935.3 Extraction of Modal Parameters 1985.3.1 Peak Picking Technique 1985.3.2 Circle Fitting – Kennedy and Pancu 1995.3.3 SDOF Polynomial 2005.3.4 Residual Effects of Out of Band Modes 2005.3.5 MDOF Polynomial 2015.3.6 Least Squares Complex Exponential 2015.3.7 Advanced Forms of Time and Frequency Domain Estimators 2035.3.8 General Time Domain Techniques 2035.3.9 General Frequency Domain Techniques 2035.3.10 General Consideration for Time vs Frequency Representation 2045.3.11 Additional Remarks on Modal Parameter Estimation 2045.3.12 Two Step Process for Modal Parameter Estimation 2055.4 Mode Identification Tools 2065.4.1 Summation Function 2065.4.2 Mode Indicator Function 2065.4.3 Complex Mode Indicator Function 2075.4.4 Stability Diagram 2085.4.5 PolyMAX 2105.5 Modal Model Validation Tools 2125.5.1 Synthesis of Frequency Response Functions using Extracted Parameters 2125.5.2 Modal Assurance Criterion 2135.5.3 Mode Participation Factors 2155.5.4 Mode Overcomplexity 2155.5.5 Mean Phase Co-linearity and Mean Phase Deviation 2165.6 Operating Modal Analysis 2165.7 Summary 219Part II Practical Considerations for ExperimentalModal Testing 2216 Test Setup Considerations 2236.1 Test Plan? 2246.2 How Many Modes Required? 2256.3 Frequency Range of Interest? 2286.4 Transducer Possibilities? 2326.5 Test Configurations? 2326.6 How Many Measurement Points Needed? 2356.7 Excitation Techniques 2386.8 Miscellaneous Items to Consider 2386.9 Summary 2457 Impact Testing Considerations 2477.1 Hammer Impact Location 2477.2 Hammer Tip and Frequency Range 2487.3 Hammers for Different Size Structures 2497.4 How Does Impact Skew and Deviation of Input Point Affect theMeasurement? 2567.4.1 Skewed Impact Force 2567.4.2 Inconsistent Impact Force Location 2567.5 Impact Hammer Frequency Bandwidth 2567.6 Accelerometer ICP Considerations for Low Frequency Measurements 2647.7 Considerations for Reciprocity Measurements 2647.8 Roving Hammer vs Roving Accelerometer 2677.9 Picking a Good Reference Location 2687.10 Multiple Impact Difficulties and Considerations 2687.10.1 Academic Structure 2697.10.2 LargeWind Turbine Blade 2717.11 What is “Filter Ring” during an Impact Measurement? 2747.12 Test Bandwidth MuchWider than Desired Frequency Range 2757.13 Why Does the Structure Response Need to Come to Zero at the End of the Sample Time? 2797.14 Measurements with no Overload but Transducers are Saturated 2827.14.1 Case 1: Sensitive Accelerometer with ExponentialWindow 2827.14.2 Case 2: Sensitive Accelerometer with NoWindow 2837.14.3 Case 3: Less Sensitive Accelerometer with NoWindow 2837.15 How much Roll Off in the Input Hammer Force Spectrum is Acceptable? 2867.16 Can the Hammer be Switched in the Middle of a Test to Avoid Double Impacts? 2897.17 Closing Remarks 2928 Shaker Testing Considerations 2938.1 General Hardware Related Issues 2938.1.1 General Information about Shakers and Amplifiers 2938.1.2 What is the Difference between the Constant Current and Constant Voltage Settings on the Shaker Amplifier? 2948.1.3 Some Shakers have a Trunnion: Is it Really Needed andWhy Do You Have It? 2948.1.4 Where is the Best Location to Place a Shaker for a Modal Test? 2958.1.5 How Should the Shaker be Constrained when Testing? 2968.1.6 What’s the BestWay to Support a Shaker for Lateral Vibration When it is Hung? 2968.1.7 What are the Most Common Practical Failures with Shaker Setup? 2978.1.8 What is the Correct Level of Shaker Excitation for Modal Testing? 2978.1.9 How many Shakers should I use in my Modal Test? 2978.1.10 Shaker and Stinger Alignment Issues 2978.1.11 When should the Shaker be Attached to the Structure? 2988.1.12 Should I Disconnect the Stingers while not Testing? 2988.1.13 Force Gage or Impedance Head must be Mounted on Structure Side of Stinger? 3008.1.14 What’s an Impedance Head? Why use it?Where does it go? 3018.2 Stinger Related Issues 3028.2.1 Why should Stingers be used? 3028.2.2 Can a Poorly Designed Shaker/Stinger Setup Produce Incorrect Results? 3038.2.3 Stingers and their Effect on Measured Frequency Response Functions 3068.2.3.1 Stinger Location 3078.2.3.2 Stinger Alignment 3078.2.3.3 Stinger Length 3088.2.3.4 Stinger Type 3108.2.3.5 Sleeved Stingers 3108.2.3.6 How do PianoWire StingersWork? How are they Pretensioned?? 3148.3 Shaker Related Issues 3148.3.1 Is MIMO needed for Structures with DirectionalModes? 3148.3.2 Shaker Force Levels and SISO vs MIMO Considerations 3168.3.2.1 High Shaker Force Levels 3168.3.2.2 High Shaker Force Levels 3188.3.2.3 Effects of FRF Measurements in the Modal Parameter Estimation Process 3208.4 Concluding Remarks 3259 Insight intoModal Parameter Estimation 3279.1 Introductory Remarks 3279.2 Mode Indicator Tools Help Identify Modes 3289.3 SDOF vsMDOF for a Simple System 3309.4 Local vs Global: MACL Frame 3329.5 Repeated Root: Composite Spar 3349.6 Wind Turbine Blade: Same Geometry but Very Different Modes 3359.7 Stability Diagram Demystified 3379.8 Curvefitting Demystified 3409.9 Curvefitting Different Bands for the Poles and Residues 3439.10 Synthesizing the FRF from Parameters from Several Bands Stitched Together 3449.11 A Large Multiple Reference Modal Test Parameter Estimation 3469.11.1 Case 1: Use of All Measured FRFs 3469.11.2 Case 2: Use of Selected Sets of Measured FRFs 3509.11.3 Case 3: Use of PolyMAX 3529.12 Operating Modal Analysis 3579.13 Concluding Remarks 36310 General Considerations 36510.1 An ExperimentalModal Test: a Thought Process Divulged 36910.2 FFT Analyzer Setup 37710.2.1 General FFT Analyzer Setup 37710.2.2 Setup for Impact Testing 37810.2.3 Setup for Shaker Testing 37910.3 Log Sheets 37910.4 Practical Considerations: Checklists 37910.4.1 Checklist for Analyzer Setup 38010.4.2 Checklist for Impact Testing 38210.4.3 Checklist for Shaker Testing 38410.4.4 Checklist for Measurement Adequacy 38610.4.5 Checklist for Miscellaneous 38810.5 Summary 391Appendix: Logbook Forms 39211 Tips, Tricks, and Other Stuff 39511.1 Modal Testing Primer 39611.1.1 Impact Setup 39611.1.2 Shaker Setup 39711.1.3 Drive Point Measurements 39811.1.4 Reciprocity 39811.1.5 Inappropriate Reference Location 39911.1.6 Multiple-input,Multiple-output Testing 39911.1.7 Multiple Reference Testing 40011.2 Impact Hammer and Impulsive Excitation 40011.2.1 The Right Hammer for the Test 40011.2.2 Hammer – Get the Swing of it 40111.2.3 Hammer Tripod 40111.2.4 Hammer tip selection 40111.2.5 No Hammer: Improvise 40211.2.6 Pete’s Hammer Test Impact Ritual 40211.3 Accelerometer Issues 40311.3.1 Mass Loading 40311.3.2 Mass Loading Effects from Tri-axial Accelerometers 40411.3.3 Accelerometer Sensitivity Selection 40711.3.4 Tri-axial Accelerometers 40811.4 Curvefitting Considerations 41111.4.1 Should all Measurements be used when Curvefitting 41211.5 Blue Frame with Three Plate Subsystem 41411.6 Miscellaneous Issues 42211.6.1 Modal Test Axis Labels 42211.6.2 Testing Does Not Need to Start at point 1 42311.6.3 Test to aWider Frequency Range 42311.6.4 Ui times Uj; the key to many questions 42311.7 Summary 425A Linear Algebra: Basic Operations Needed forModal Analysis Operations 427A.1 Define a Matrix 427A.2 Define a Column Vector 427A.3 Define a Row Vector 428A.4 Define a Diagonal Matrix 428A.5 Define Matrix Addition 428A.6 Define Matrix Scalar Multiply 428A.7 Define Matrix Multiply 429A.8 Matrix Multiplication Rules 429A.9 Transpose of a Matrix 430A.10 Transposition Rules 430A.11 Symmetric Matrix Rules 430A.12 Define a Matrix Inverse 431A.13 Matrix Inverse Properties 431A.14 Define an Eigenvalue Problem 431A.15 Generalized Inverse 431A.16 Singular Value Decomposition 432B Example Using Two Degree of Freedom System: Eigenproblem 433C Pole, Residue, and FRF Problem for 2-DOF System 437D Example using Three Degree of Freedom System 443E DYNSYSWebsite Materials 451E.1 Technical Materials Developed 451E.1.1 Theoretical Aspects of First and Second Order Systems 452E.1.2 First Order Systems: Modeling Step with ODE and Block Diagram 452E.1.3 Second Order Systems: Modeling Step, Impulse, IC with ODE and Block Diagram 452E.1.4 MathematicalModeling Considerations 452E.1.5 Simulink and MATLAB Primer Materials 453E.1.6 Miscellaneous Materials 453E.2 DYNSYS.UML.EDUWebsite 453F Basic Modal Analysis Information 463F.1 SDOF Definitions 463F.1.1 Damping Estimates 463F.1.2 System Transfer Function 464F.1.3 Different Forms of the System Transfer Function 464F.1.4 Frequency Response Function 465F.2 MDOF Definitions 466Part III Collection of Sets of Modal Data Collected for Processing 467G Repeated Root Frame: Boundary Condition Effects 469G.1 Corner Supports Set #1 470G.2 Midlength Supports Set #2 474G.3 Modal Correlation between Set #1 and Set #2 474H Radarsat Satellite Testing 479H.1 Data Reduction Set 1: Reference BUS:109:Z, BUS:118:Z, PMS:217:X and PMS:1211:Y 479H.2 Data Reduction Set 2: Reference PMS:217:X and PMS:1211:Y 479I Demo Airplane Testing 487I.1 Impact Testing 487I.2 SIMO Testing with Skewed Shaker 487I.3 MIMO Testing with Two Vertical Modal Shakers 493J Whirlpool Dryer Cabinet Modal Testing 497K GM MTU Automobile Round Robin Modal Testing 501L UML Composite Spar Modal Testing 505M UML BUHModal Testing 509N Nomenclature 515Index 519

PETER AVITABILE is Professor Emeritus at the University of Massachusetts Lowell, the co-director of the Structural Dynamics and Acoustic Systems Laboratory, and the former President for the Society for Experimental Mechanics. In addition, he is the Associate Editor of the Handbook for Experimental Structural Mechanics. He has written hundreds of papers and articles on analytical and experimental modal analysis techniques, including the Modal Space article series published in SEM's Experimental Techniques.

• Preface xvAbout the CompanionWebsite xixPart I Overview of Experimental Modal Analysis using the Frequency Response Method 11 Introduction to ExperimentalModal Analysis: A Simple Non-mathematical Presentation 31.1 Could you Explain Modal Analysis to Me? 61.2 Just what are these Measurements called FRFs? 101.2.1 Why is Only One Row or Column of the FRF Matrix Needed? 131.3 What’s the Difference between a Shaker Test and an Impact Test? 171.3.1 What Measurements do we Actually make to Compute the FRF? 181.4 What’s the Most ImportantThing toThink about when Impact Testing? 211.5 What’s the Most ImportantThing toThink about when Shaker Testing? 221.6 Tell me More AboutWindows; They Seem Pretty Important! 241.7 So how do we get Mode Shapes from the Plate FRFs? 251.8 Modal Data and Operating Data 291.8.1 What is Operating Data? 291.8.2 So what Good is Modal Data? 331.8.3 So Should I Collect Modal Data or Operating Data? 341.9 Closing Remarks 362 General Theory of Experimental Modal Analysis 372.1 Introduction 372.2 Basic Modal AnalysisTheory – SDOF 382.2.1 Single Degree of Freedom System Equation 382.2.2 Single Degree of Freedom System Response due to Harmonic Excitation 402.2.3 Damping Estimation for Single Degree of Freedom System 422.2.4 Response Assessment with Varying Damping 432.2.5 Laplace Domain Approach for Single Degree of Freedom System 462.2.6 System Transfer Function 472.2.7 Different Forms of the Transfer Function 482.2.8 Residue of the SDOF System 492.2.9 Frequency Response Function for a Single Degree of Freedom System 492.2.10 Transfer Function/Frequency Response Function/S-plane for a Single Degree of Freedom System 512.2.11 Frequency Response Function Regions for a Single Degree of Freedom System 512.2.12 Different Forms of the Frequency Response Function 532.2.13 Complex Frequency Response Function 532.3 Basic Modal AnalysisTheory – MDOF 562.3.1 Multiple Degree of Freedom System Equations 572.3.2 Laplace Domain for Multiple Degree of Freedom System 662.3.3 The Frequency Response Function 682.3.4 Mode Shapes from Frequency Response Equations 682.3.5 Point-to-Point Frequency Response Function 712.3.6 Response of Multiple Degree of Freedom System to Harmonic Excitations 722.3.7 Example: Cantilever Beam Model with Three Measured DOFs 752.3.8 Summary of Time, Frequency, and Modal Domains 832.3.9 Response due to Forced Excitation using Mode Superposition 872.4 Summary 893 General Signal Processing andMeasurements Related to Experimental Modal Analysis 933.1 Introduction 933.2 Time and Frequency Domain 933.3 Some General Information Regarding Data Acquisition 963.4 Digitization of Time Signals 973.5 Quantization 973.5.1 ADC Underload 983.5.2 ADC Overload 1003.6 AC Coupling 1003.7 SamplingTheory 1013.8 Aliasing 1033.9 What is the Fourier Transform? 1053.9.1 Fourier Transform and Discrete Fourier Transform 1073.9.2 FFT: Periodic Signal 1083.9.3 FFT: Non-periodic Signal 1083.10 Leakage and Minimization of Leakage 1093.10.1 Minimization of Leakage 1113.11 Windows and Leakage 1113.11.1 RectangularWindow 1123.11.2 HanningWindow 1163.11.3 Flat TopWindow 1163.11.4 Comparison ofWindows withWorst Leakage Distortion Possible 1163.11.5 Comparison of Rectangular, Hanning and Flat TopWindow 1193.11.6 ForceWindow 1193.11.7 ExponentialWindow 1193.11.8 Convolution of theWindow in the Frequency Domain 1193.12 Frequency Response Function Formulation 1193.13 TypicalMeasurements 1233.13.1 Time Signal and Auto-power Functions 1233.13.2 TypicalMeasurement: Cross Power Function 1243.13.3 TypicalMeasurement: Frequency Response Function 1243.13.4 TypicalMeasurement: Coherence Function 1243.14 Time and Frequency Relationship Definition 1263.15 Input–Output Model with Noise 1273.15.1 H1 Formulation: Output Noise Only 1273.15.2 H2 Formulation: Output Noise Only 1283.15.3 H1 Formulation: Input Noise Only 1283.15.4 H2 Formulation: Input Noise Only 1283.16 Summary 1294 Excitation Techniques 1314.1 Introduction 1314.2 Impact Excitation Technique 1324.2.1 Impact Hammer 1324.2.2 Hammer Impact Tip Selection 1364.2.3 Useful Frequency Range for Impact Excitation 1374.2.4 ForceWindow for Impact Excitation 1374.2.5 Pre-trigger Delay 1374.2.6 Double Impact 1404.2.7 Response due to Impact 1404.2.8 Roving Hammer vs Stationary Hammer and Reciprocity 1434.2.9 Impact Testing: an Example Set of Measurements 1474.3 Shaker Excitation 1594.3.1 Modal Shaker Setup 1614.3.2 Historical Development of Shaker Excitation Techniques 1624.3.3 Swept Sine Excitation 1634.3.4 Pure Random Excitation 1634.3.5 Pure Random Excitation withWindows Applied 1654.3.6 Pure Random Excitation with Overlap Processing 1654.3.7 Pseudo-random Excitation 1674.3.8 Periodic Random Excitation 1674.3.9 Burst Random Excitation 1684.3.10 Sine Chirp Excitation 1704.3.11 Digital Stepped Sine Excitation 1704.4 Comparison of Different Excitations for aWeldment Structure 1724.4.1 Random Excitation with NoWindow 1724.4.2 Random Excitation with HanningWindow 1734.4.3 Burst Random Excitation with NoWindow 1734.4.4 Sine Chirp Excitation with NoWindow 1744.4.5 Comparison of Random, Burst Random and Sine Chirp 1754.4.6 Comparison of Random and Burst Random at Resonant Peaks 1754.4.7 Linearity Check Using Sine Chirp 1754.5 Multiple-input,Multiple-outputMeasurement 1754.5.1 Multiple Input vs Single Input Testing 1774.5.2 Multiple Input vs Single Input for aWeldment Structure 1814.5.3 Multiple Input vs Single Input Testing 1814.5.4 Comparison of Multiple Input and Single Input forWeldment Structure 1824.5.5 MIMO Measurements on a Multi-component Structure 1824.6 Summary 1875 Modal Parameter Estimation Techniques 1895.1 Introduction 1895.2 ExperimentalModal Analysis 1905.2.1 Least Squares Approximation of Data 1905.2.2 Classification of Modal Parameter Estimation Techniques 1935.3 Extraction of Modal Parameters 1985.3.1 Peak Picking Technique 1985.3.2 Circle Fitting – Kennedy and Pancu 1995.3.3 SDOF Polynomial 2005.3.4 Residual Effects of Out of Band Modes 2005.3.5 MDOF Polynomial 2015.3.6 Least Squares Complex Exponential 2015.3.7 Advanced Forms of Time and Frequency Domain Estimators 2035.3.8 General Time Domain Techniques 2035.3.9 General Frequency Domain Techniques 2035.3.10 General Consideration for Time vs Frequency Representation 2045.3.11 Additional Remarks on Modal Parameter Estimation 2045.3.12 Two Step Process for Modal Parameter Estimation 2055.4 Mode Identification Tools 2065.4.1 Summation Function 2065.4.2 Mode Indicator Function 2065.4.3 Complex Mode Indicator Function 2075.4.4 Stability Diagram 2085.4.5 PolyMAX 2105.5 Modal Model Validation Tools 2125.5.1 Synthesis of Frequency Response Functions using Extracted Parameters 2125.5.2 Modal Assurance Criterion 2135.5.3 Mode Participation Factors 2155.5.4 Mode Overcomplexity 2155.5.5 Mean Phase Co-linearity and Mean Phase Deviation 2165.6 Operating Modal Analysis 2165.7 Summary 219Part II Practical Considerations for ExperimentalModal Testing 2216 Test Setup Considerations 2236.1 Test Plan? 2246.2 How Many Modes Required? 2256.3 Frequency Range of Interest? 2286.4 Transducer Possibilities? 2326.5 Test Configurations? 2326.6 How Many Measurement Points Needed? 2356.7 Excitation Techniques 2386.8 Miscellaneous Items to Consider 2386.9 Summary 2457 Impact Testing Considerations 2477.1 Hammer Impact Location 2477.2 Hammer Tip and Frequency Range 2487.3 Hammers for Different Size Structures 2497.4 How Does Impact Skew and Deviation of Input Point Affect theMeasurement? 2567.4.1 Skewed Impact Force 2567.4.2 Inconsistent Impact Force Location 2567.5 Impact Hammer Frequency Bandwidth 2567.6 Accelerometer ICP Considerations for Low Frequency Measurements 2647.7 Considerations for Reciprocity Measurements 2647.8 Roving Hammer vs Roving Accelerometer 2677.9 Picking a Good Reference Location 2687.10 Multiple Impact Difficulties and Considerations 2687.10.1 Academic Structure 2697.10.2 LargeWind Turbine Blade 2717.11 What is “Filter Ring” during an Impact Measurement? 2747.12 Test Bandwidth MuchWider than Desired Frequency Range 2757.13 Why Does the Structure Response Need to Come to Zero at the End of the Sample Time? 2797.14 Measurements with no Overload but Transducers are Saturated 2827.14.1 Case 1: Sensitive Accelerometer with ExponentialWindow 2827.14.2 Case 2: Sensitive Accelerometer with NoWindow 2837.14.3 Case 3: Less Sensitive Accelerometer with NoWindow 2837.15 How much Roll Off in the Input Hammer Force Spectrum is Acceptable? 2867.16 Can the Hammer be Switched in the Middle of a Test to Avoid Double Impacts? 2897.17 Closing Remarks 2928 Shaker Testing Considerations 2938.1 General Hardware Related Issues 2938.1.1 General Information about Shakers and Amplifiers 2938.1.2 What is the Difference between the Constant Current and Constant Voltage Settings on the Shaker Amplifier? 2948.1.3 Some Shakers have a Trunnion: Is it Really Needed andWhy Do You Have It? 2948.1.4 Where is the Best Location to Place a Shaker for a Modal Test? 2958.1.5 How Should the Shaker be Constrained when Testing? 2968.1.6 What’s the BestWay to Support a Shaker for Lateral Vibration When it is Hung? 2968.1.7 What are the Most Common Practical Failures with Shaker Setup? 2978.1.8 What is the Correct Level of Shaker Excitation for Modal Testing? 2978.1.9 How many Shakers should I use in my Modal Test? 2978.1.10 Shaker and Stinger Alignment Issues 2978.1.11 When should the Shaker be Attached to the Structure? 2988.1.12 Should I Disconnect the Stingers while not Testing? 2988.1.13 Force Gage or Impedance Head must be Mounted on Structure Side of Stinger? 3008.1.14 What’s an Impedance Head? Why use it?Where does it go? 3018.2 Stinger Related Issues 3028.2.1 Why should Stingers be used? 3028.2.2 Can a Poorly Designed Shaker/Stinger Setup Produce Incorrect Results? 3038.2.3 Stingers and their Effect on Measured Frequency Response Functions 3068.2.3.1 Stinger Location 3078.2.3.2 Stinger Alignment 3078.2.3.3 Stinger Length 3088.2.3.4 Stinger Type 3108.2.3.5 Sleeved Stingers 3108.2.3.6 How do PianoWire StingersWork? How are they Pretensioned?? 3148.3 Shaker Related Issues 3148.3.1 Is MIMO needed for Structures with DirectionalModes? 3148.3.2 Shaker Force Levels and SISO vs MIMO Considerations 3168.3.2.1 High Shaker Force Levels 3168.3.2.2 High Shaker Force Levels 3188.3.2.3 Effects of FRF Measurements in the Modal Parameter Estimation Process 3208.4 Concluding Remarks 3259 Insight intoModal Parameter Estimation 3279.1 Introductory Remarks 3279.2 Mode Indicator Tools Help Identify Modes 3289.3 SDOF vsMDOF for a Simple System 3309.4 Local vs Global: MACL Frame 3329.5 Repeated Root: Composite Spar 3349.6 Wind Turbine Blade: Same Geometry but Very Different Modes 3359.7 Stability Diagram Demystified 3379.8 Curvefitting Demystified 3409.9 Curvefitting Different Bands for the Poles and Residues 3439.10 Synthesizing the FRF from Parameters from Several Bands Stitched Together 3449.11 A Large Multiple Reference Modal Test Parameter Estimation 3469.11.1 Case 1: Use of All Measured FRFs 3469.11.2 Case 2: Use of Selected Sets of Measured FRFs 3509.11.3 Case 3: Use of PolyMAX 3529.12 Operating Modal Analysis 3579.13 Concluding Remarks 36310 General Considerations 36510.1 An ExperimentalModal Test: a Thought Process Divulged 36910.2 FFT Analyzer Setup 37710.2.1 General FFT Analyzer Setup 37710.2.2 Setup for Impact Testing 37810.2.3 Setup for Shaker Testing 37910.3 Log Sheets 37910.4 Practical Considerations: Checklists 37910.4.1 Checklist for Analyzer Setup 38010.4.2 Checklist for Impact Testing 38210.4.3 Checklist for Shaker Testing 38410.4.4 Checklist for Measurement Adequacy 38610.4.5 Checklist for Miscellaneous 38810.5 Summary 391Appendix: Logbook Forms 39211 Tips, Tricks, and Other Stuff 39511.1 Modal Testing Primer 39611.1.1 Impact Setup 39611.1.2 Shaker Setup 39711.1.3 Drive Point Measurements 39811.1.4 Reciprocity 39811.1.5 Inappropriate Reference Location 39911.1.6 Multiple-input,Multiple-output Testing 39911.1.7 Multiple Reference Testing 40011.2 Impact Hammer and Impulsive Excitation 40011.2.1 The Right Hammer for the Test 40011.2.2 Hammer – Get the Swing of it 40111.2.3 Hammer Tripod 40111.2.4 Hammer tip selection 40111.2.5 No Hammer: Improvise 40211.2.6 Pete’s Hammer Test Impact Ritual 40211.3 Accelerometer Issues 40311.3.1 Mass Loading 40311.3.2 Mass Loading Effects from Tri-axial Accelerometers 40411.3.3 Accelerometer Sensitivity Selection 40711.3.4 Tri-axial Accelerometers 40811.4 Curvefitting Considerations 41111.4.1 Should all Measurements be used when Curvefitting 41211.5 Blue Frame with Three Plate Subsystem 41411.6 Miscellaneous Issues 42211.6.1 Modal Test Axis Labels 42211.6.2 Testing Does Not Need to Start at point 1 42311.6.3 Test to aWider Frequency Range 42311.6.4 Ui times Uj; the key to many questions 42311.7 Summary 425A Linear Algebra: Basic Operations Needed forModal Analysis Operations 427A.1 Define a Matrix 427A.2 Define a Column Vector 427A.3 Define a Row Vector 428A.4 Define a Diagonal Matrix 428A.5 Define Matrix Addition 428A.6 Define Matrix Scalar Multiply 428A.7 Define Matrix Multiply 429A.8 Matrix Multiplication Rules 429A.9 Transpose of a Matrix 430A.10 Transposition Rules 430A.11 Symmetric Matrix Rules 430A.12 Define a Matrix Inverse 431A.13 Matrix Inverse Properties 431A.14 Define an Eigenvalue Problem 431A.15 Generalized Inverse 431A.16 Singular Value Decomposition 432B Example Using Two Degree of Freedom System: Eigenproblem 433C Pole, Residue, and FRF Problem for 2-DOF System 437D Example using Three Degree of Freedom System 443E DYNSYSWebsite Materials 451E.1 Technical Materials Developed 451E.1.1 Theoretical Aspects of First and Second Order Systems 452E.1.2 First Order Systems: Modeling Step with ODE and Block Diagram 452E.1.3 Second Order Systems: Modeling Step, Impulse, IC with ODE and Block Diagram 452E.1.4 MathematicalModeling Considerations 452E.1.5 Simulink and MATLAB Primer Materials 453E.1.6 Miscellaneous Materials 453E.2 DYNSYS.UML.EDUWebsite 453F Basic Modal Analysis Information 463F.1 SDOF Definitions 463F.1.1 Damping Estimates 463F.1.2 System Transfer Function 464F.1.3 Different Forms of the System Transfer Function 464F.1.4 Frequency Response Function 465F.2 MDOF Definitions 466Part III Collection of Sets of Modal Data Collected for Processing 467G Repeated Root Frame: Boundary Condition Effects 469G.1 Corner Supports Set #1 470G.2 Midlength Supports Set #2 474G.3 Modal Correlation between Set #1 and Set #2 474H Radarsat Satellite Testing 479H.1 Data Reduction Set 1: Reference BUS:109:Z, BUS:118:Z, PMS:217:X and PMS:1211:Y 479H.2 Data Reduction Set 2: Reference PMS:217:X and PMS:1211:Y 479I Demo Airplane Testing 487I.1 Impact Testing 487I.2 SIMO Testing with Skewed Shaker 487I.3 MIMO Testing with Two Vertical Modal Shakers 493J Whirlpool Dryer Cabinet Modal Testing 497K GM MTU Automobile Round Robin Modal Testing 501L UML Composite Spar Modal Testing 505M UML BUHModal Testing 509N Nomenclature 515Index 519