Preface xv
About the CompanionWebsite xix
Part I Overview of Experimental Modal Analysis using the Frequency Response Method 1
1 Introduction to ExperimentalModal Analysis: A Simple Non-mathematical Presentation 3
1.1 Could you Explain Modal Analysis to Me? 6
1.2 Just what are these Measurements called FRFs? 10
1.2.1 Why is Only One Row or Column of the FRF Matrix Needed? 13
1.3 What’s the Difference between a Shaker Test and an Impact Test? 17
1.3.1 What Measurements do we Actually make to Compute the FRF? 18
1.4 What’s the Most ImportantThing toThink about when Impact Testing? 21
1.5 What’s the Most ImportantThing toThink about when Shaker Testing? 22
1.6 Tell me More AboutWindows; They Seem Pretty Important! 24
1.7 So how do we get Mode Shapes from the Plate FRFs? 25
1.8 Modal Data and Operating Data 29
1.8.1 What is Operating Data? 29
1.8.2 So what Good is Modal Data? 33
1.8.3 So Should I Collect Modal Data or Operating Data? 34
1.9 Closing Remarks 36
2 General Theory of Experimental Modal Analysis 37
2.1 Introduction 37
2.2 Basic Modal AnalysisTheory – SDOF 38
2.2.1 Single Degree of Freedom System Equation 38
2.2.2 Single Degree of Freedom System Response due to Harmonic Excitation 40
2.2.3 Damping Estimation for Single Degree of Freedom System 42
2.2.4 Response Assessment with Varying Damping 43
2.2.5 Laplace Domain Approach for Single Degree of Freedom System 46
2.2.6 System Transfer Function 47
2.2.7 Different Forms of the Transfer Function 48
2.2.8 Residue of the SDOF System 49
2.2.9 Frequency Response Function for a Single Degree of Freedom System 49
2.2.10 Transfer Function/Frequency Response Function/S-plane for a Single Degree of Freedom System 51
2.2.11 Frequency Response Function Regions for a Single Degree of Freedom System 51
2.2.12 Different Forms of the Frequency Response Function 53
2.2.13 Complex Frequency Response Function 53
2.3 Basic Modal AnalysisTheory – MDOF 56
2.3.1 Multiple Degree of Freedom System Equations 57
2.3.2 Laplace Domain for Multiple Degree of Freedom System 66
2.3.3 The Frequency Response Function 68
2.3.4 Mode Shapes from Frequency Response Equations 68
2.3.5 Point-to-Point Frequency Response Function 71
2.3.6 Response of Multiple Degree of Freedom System to Harmonic Excitations 72
2.3.7 Example: Cantilever Beam Model with Three Measured DOFs 75
2.3.8 Summary of Time, Frequency, and Modal Domains 83
2.3.9 Response due to Forced Excitation using Mode Superposition 87
2.4 Summary 89
3 General Signal Processing andMeasurements Related to Experimental Modal Analysis 93
3.1 Introduction 93
3.2 Time and Frequency Domain 93
3.3 Some General Information Regarding Data Acquisition 96
3.4 Digitization of Time Signals 97
3.5 Quantization 97
3.5.1 ADC Underload 98
3.5.2 ADC Overload 100
3.6 AC Coupling 100
3.7 SamplingTheory 101
3.8 Aliasing 103
3.9 What is the Fourier Transform? 105
3.9.1 Fourier Transform and Discrete Fourier Transform 107
3.9.2 FFT: Periodic Signal 108
3.9.3 FFT: Non-periodic Signal 108
3.10 Leakage and Minimization of Leakage 109
3.10.1 Minimization of Leakage 111
3.11 Windows and Leakage 111
3.11.1 RectangularWindow 112
3.11.2 HanningWindow 116
3.11.3 Flat TopWindow 116
3.11.4 Comparison ofWindows withWorst Leakage Distortion Possible 116
3.11.5 Comparison of Rectangular, Hanning and Flat TopWindow 119
3.11.6 ForceWindow 119
3.11.7 ExponentialWindow 119
3.11.8 Convolution of theWindow in the Frequency Domain 119
3.12 Frequency Response Function Formulation 119
3.13 TypicalMeasurements 123
3.13.1 Time Signal and Auto-power Functions 123
3.13.2 TypicalMeasurement: Cross Power Function 124
3.13.3 TypicalMeasurement: Frequency Response Function 124
3.13.4 TypicalMeasurement: Coherence Function 124
3.14 Time and Frequency Relationship Definition 126
3.15 Input–Output Model with Noise 127
3.15.1 H1 Formulation: Output Noise Only 127
3.15.2 H2 Formulation: Output Noise Only 128
3.15.3 H1 Formulation: Input Noise Only 128
3.15.4 H2 Formulation: Input Noise Only 128
3.16 Summary 129
4 Excitation Techniques 131
4.1 Introduction 131
4.2 Impact Excitation Technique 132
4.2.1 Impact Hammer 132
4.2.2 Hammer Impact Tip Selection 136
4.2.3 Useful Frequency Range for Impact Excitation 137
4.2.4 ForceWindow for Impact Excitation 137
4.2.5 Pre-trigger Delay 137
4.2.6 Double Impact 140
4.2.7 Response due to Impact 140
4.2.8 Roving Hammer vs Stationary Hammer and Reciprocity 143
4.2.9 Impact Testing: an Example Set of Measurements 147
4.3 Shaker Excitation 159
4.3.1 Modal Shaker Setup 161
4.3.2 Historical Development of Shaker Excitation Techniques 162
4.3.3 Swept Sine Excitation 163
4.3.4 Pure Random Excitation 163
4.3.5 Pure Random Excitation withWindows Applied 165
4.3.6 Pure Random Excitation with Overlap Processing 165
4.3.7 Pseudo-random Excitation 167
4.3.8 Periodic Random Excitation 167
4.3.9 Burst Random Excitation 168
4.3.10 Sine Chirp Excitation 170
4.3.11 Digital Stepped Sine Excitation 170
4.4 Comparison of Different Excitations for aWeldment Structure 172
4.4.1 Random Excitation with NoWindow 172
4.4.2 Random Excitation with HanningWindow 173
4.4.3 Burst Random Excitation with NoWindow 173
4.4.4 Sine Chirp Excitation with NoWindow 174
4.4.5 Comparison of Random, Burst Random and Sine Chirp 175
4.4.6 Comparison of Random and Burst Random at Resonant Peaks 175
4.4.7 Linearity Check Using Sine Chirp 175
4.5 Multiple-input,Multiple-outputMeasurement 175
4.5.1 Multiple Input vs Single Input Testing 177
4.5.2 Multiple Input vs Single Input for aWeldment Structure 181
4.5.3 Multiple Input vs Single Input Testing 181
4.5.4 Comparison of Multiple Input and Single Input forWeldment Structure 182
4.5.5 MIMO Measurements on a Multi-component Structure 182
4.6 Summary 187
5 Modal Parameter Estimation Techniques 189
5.1 Introduction 189
5.2 ExperimentalModal Analysis 190
5.2.1 Least Squares Approximation of Data 190
5.2.2 Classification of Modal Parameter Estimation Techniques 193
5.3 Extraction of Modal Parameters 198
5.3.1 Peak Picking Technique 198
5.3.2 Circle Fitting – Kennedy and Pancu 199
5.3.3 SDOF Polynomial 200
5.3.4 Residual Effects of Out of Band Modes 200
5.3.5 MDOF Polynomial 201
5.3.6 Least Squares Complex Exponential 201
5.3.7 Advanced Forms of Time and Frequency Domain Estimators 203
5.3.8 General Time Domain Techniques 203
5.3.9 General Frequency Domain Techniques 203
5.3.10 General Consideration for Time vs Frequency Representation 204
5.3.11 Additional Remarks on Modal Parameter Estimation 204
5.3.12 Two Step Process for Modal Parameter Estimation 205
5.4 Mode Identification Tools 206
5.4.1 Summation Function 206
5.4.2 Mode Indicator Function 206
5.4.3 Complex Mode Indicator Function 207
5.4.4 Stability Diagram 208
5.4.5 PolyMAX 210
5.5 Modal Model Validation Tools 212
5.5.1 Synthesis of Frequency Response Functions using Extracted Parameters 212
5.5.2 Modal Assurance Criterion 213
5.5.3 Mode Participation Factors 215
5.5.4 Mode Overcomplexity 215
5.5.5 Mean Phase Co-linearity and Mean Phase Deviation 216
5.6 Operating Modal Analysis 216
5.7 Summary 219
Part II Practical Considerations for ExperimentalModal Testing 221
6 Test Setup Considerations 223
6.1 Test Plan? 224
6.2 How Many Modes Required? 225
6.3 Frequency Range of Interest? 228
6.4 Transducer Possibilities? 232
6.5 Test Configurations? 232
6.6 How Many Measurement Points Needed? 235
6.7 Excitation Techniques 238
6.8 Miscellaneous Items to Consider 238
6.9 Summary 245
7 Impact Testing Considerations 247
7.1 Hammer Impact Location 247
7.2 Hammer Tip and Frequency Range 248
7.3 Hammers for Different Size Structures 249
7.4 How Does Impact Skew and Deviation of Input Point Affect theMeasurement? 256
7.4.1 Skewed Impact Force 256
7.4.2 Inconsistent Impact Force Location 256
7.5 Impact Hammer Frequency Bandwidth 256
7.6 Accelerometer ICP Considerations for Low Frequency Measurements 264
7.7 Considerations for Reciprocity Measurements 264
7.8 Roving Hammer vs Roving Accelerometer 267
7.9 Picking a Good Reference Location 268
7.10 Multiple Impact Difficulties and Considerations 268
7.10.1 Academic Structure 269
7.10.2 LargeWind Turbine Blade 271
7.11 What is “Filter Ring” during an Impact Measurement? 274
7.12 Test Bandwidth MuchWider than Desired Frequency Range 275
7.13 Why Does the Structure Response Need to Come to Zero at the End of the Sample Time? 279
7.14 Measurements with no Overload but Transducers are Saturated 282
7.14.1 Case 1: Sensitive Accelerometer with ExponentialWindow 282
7.14.2 Case 2: Sensitive Accelerometer with NoWindow 283
7.14.3 Case 3: Less Sensitive Accelerometer with NoWindow 283
7.15 How much Roll Off in the Input Hammer Force Spectrum is Acceptable? 286
7.16 Can the Hammer be Switched in the Middle of a Test to Avoid Double Impacts? 289
7.17 Closing Remarks 292
8 Shaker Testing Considerations 293
8.1 General Hardware Related Issues 293
8.1.1 General Information about Shakers and Amplifiers 293
8.1.2 What is the Difference between the Constant Current and Constant Voltage Settings on the Shaker Amplifier? 294
8.1.3 Some Shakers have a Trunnion: Is it Really Needed andWhy Do You Have It? 294
8.1.4 Where is the Best Location to Place a Shaker for a Modal Test? 295
8.1.5 How Should the Shaker be Constrained when Testing? 296
8.1.6 What’s the BestWay to Support a Shaker for Lateral Vibration When it is Hung? 296
8.1.7 What are the Most Common Practical Failures with Shaker Setup? 297
8.1.8 What is the Correct Level of Shaker Excitation for Modal Testing? 297
8.1.9 How many Shakers should I use in my Modal Test? 297
8.1.10 Shaker and Stinger Alignment Issues 297
8.1.11 When should the Shaker be Attached to the Structure? 298
8.1.12 Should I Disconnect the Stingers while not Testing? 298
8.1.13 Force Gage or Impedance Head must be Mounted on Structure Side of Stinger? 300
8.1.14 What’s an Impedance Head? Why use it?Where does it go? 301
8.2 Stinger Related Issues 302
8.2.1 Why should Stingers be used? 302
8.2.2 Can a Poorly Designed Shaker/Stinger Setup Produce Incorrect Results? 303
8.2.3 Stingers and their Effect on Measured Frequency Response Functions 306
8.2.3.1 Stinger Location 307
8.2.3.2 Stinger Alignment 307
8.2.3.3 Stinger Length 308
8.2.3.4 Stinger Type 310
8.2.3.5 Sleeved Stingers 310
8.2.3.6 How do PianoWire StingersWork? How are they Pretensioned?? 314
8.3 Shaker Related Issues 314
8.3.1 Is MIMO needed for Structures with DirectionalModes? 314
8.3.2 Shaker Force Levels and SISO vs MIMO Considerations 316
8.3.2.1 High Shaker Force Levels 316
8.3.2.2 High Shaker Force Levels 318
8.3.2.3 Effects of FRF Measurements in the Modal Parameter Estimation Process 320
8.4 Concluding Remarks 325
9 Insight intoModal Parameter Estimation 327
9.1 Introductory Remarks 327
9.2 Mode Indicator Tools Help Identify Modes 328
9.3 SDOF vsMDOF for a Simple System 330
9.4 Local vs Global: MACL Frame 332
9.5 Repeated Root: Composite Spar 334
9.6 Wind Turbine Blade: Same Geometry but Very Different Modes 335
9.7 Stability Diagram Demystified 337
9.8 Curvefitting Demystified 340
9.9 Curvefitting Different Bands for the Poles and Residues 343
9.10 Synthesizing the FRF from Parameters from Several Bands Stitched Together 344
9.11 A Large Multiple Reference Modal Test Parameter Estimation 346
9.11.1 Case 1: Use of All Measured FRFs 346
9.11.2 Case 2: Use of Selected Sets of Measured FRFs 350
9.11.3 Case 3: Use of PolyMAX 352
9.12 Operating Modal Analysis 357
9.13 Concluding Remarks 363
10 General Considerations 365
10.1 An ExperimentalModal Test: a Thought Process Divulged 369
10.2 FFT Analyzer Setup 377
10.2.1 General FFT Analyzer Setup 377
10.2.2 Setup for Impact Testing 378
10.2.3 Setup for Shaker Testing 379
10.3 Log Sheets 379
10.4 Practical Considerations: Checklists 379
10.4.1 Checklist for Analyzer Setup 380
10.4.2 Checklist for Impact Testing 382
10.4.3 Checklist for Shaker Testing 384
10.4.4 Checklist for Measurement Adequacy 386
10.4.5 Checklist for Miscellaneous 388
10.5 Summary 391
Appendix: Logbook Forms 392
11 Tips, Tricks, and Other Stuff 395
11.1 Modal Testing Primer 396
11.1.1 Impact Setup 396
11.1.2 Shaker Setup 397
11.1.3 Drive Point Measurements 398
11.1.4 Reciprocity 398
11.1.5 Inappropriate Reference Location 399
11.1.6 Multiple-input,Multiple-output Testing 399
11.1.7 Multiple Reference Testing 400
11.2 Impact Hammer and Impulsive Excitation 400
11.2.1 The Right Hammer for the Test 400
11.2.2 Hammer – Get the Swing of it 401
11.2.3 Hammer Tripod 401
11.2.4 Hammer tip selection 401
11.2.5 No Hammer: Improvise 402
11.2.6 Pete’s Hammer Test Impact Ritual 402
11.3 Accelerometer Issues 403
11.3.1 Mass Loading 403
11.3.2 Mass Loading Effects from Tri-axial Accelerometers 404
11.3.3 Accelerometer Sensitivity Selection 407
11.3.4 Tri-axial Accelerometers 408
11.4 Curvefitting Considerations 411
11.4.1 Should all Measurements be used when Curvefitting 412
11.5 Blue Frame with Three Plate Subsystem 414
11.6 Miscellaneous Issues 422
11.6.1 Modal Test Axis Labels 422
11.6.2 Testing Does Not Need to Start at point 1 423
11.6.3 Test to aWider Frequency Range 423
11.6.4 Ui times Uj; the key to many questions 423
11.7 Summary 425
A Linear Algebra: Basic Operations Needed forModal Analysis Operations 427
A.1 Define a Matrix 427
A.2 Define a Column Vector 427
A.3 Define a Row Vector 428
A.4 Define a Diagonal Matrix 428
A.5 Define Matrix Addition 428
A.6 Define Matrix Scalar Multiply 428
A.7 Define Matrix Multiply 429
A.8 Matrix Multiplication Rules 429
A.9 Transpose of a Matrix 430
A.10 Transposition Rules 430
A.11 Symmetric Matrix Rules 430
A.12 Define a Matrix Inverse 431
A.13 Matrix Inverse Properties 431
A.14 Define an Eigenvalue Problem 431
A.15 Generalized Inverse 431
A.16 Singular Value Decomposition 432
B Example Using Two Degree of Freedom System: Eigenproblem 433
C Pole, Residue, and FRF Problem for 2-DOF System 437
D Example using Three Degree of Freedom System 443
E DYNSYSWebsite Materials 451
E.1 Technical Materials Developed 451
E.1.1 Theoretical Aspects of First and Second Order Systems 452
E.1.2 First Order Systems: Modeling Step with ODE and Block Diagram 452
E.1.3 Second Order Systems: Modeling Step, Impulse, IC with ODE and Block Diagram 452
E.1.4 MathematicalModeling Considerations 452
E.1.5 Simulink and MATLAB Primer Materials 453
E.1.6 Miscellaneous Materials 453
E.2 DYNSYS.UML.EDUWebsite 453
F Basic Modal Analysis Information 463
F.1 SDOF Definitions 463
F.1.1 Damping Estimates 463
F.1.2 System Transfer Function 464
F.1.3 Different Forms of the System Transfer Function 464
F.1.4 Frequency Response Function 465
F.2 MDOF Definitions 466
Part III Collection of Sets of Modal Data Collected for Processing 467
G Repeated Root Frame: Boundary Condition Effects 469
G.1 Corner Supports Set #1 470
G.2 Midlength Supports Set #2 474
G.3 Modal Correlation between Set #1 and Set #2 474
H Radarsat Satellite Testing 479
H.1 Data Reduction Set 1: Reference BUS:109:Z, BUS:118:Z, PMS:217:X and PMS:1211:Y 479
H.2 Data Reduction Set 2: Reference PMS:217:X and PMS:1211:Y 479
I Demo Airplane Testing 487
I.1 Impact Testing 487
I.2 SIMO Testing with Skewed Shaker 487
I.3 MIMO Testing with Two Vertical Modal Shakers 493
J Whirlpool Dryer Cabinet Modal Testing 497
K GM MTU Automobile Round Robin Modal Testing 501
L UML Composite Spar Modal Testing 505
M UML BUHModal Testing 509
N Nomenclature 515
Index 519
