Special Topics in Structural Dynamics, Volume 6

Preface	6
Contents	8
1 Vibration Class at GIST, Korea	10
1.1 Introduction	10
1.2 Class Contents	11
1.2.1 Single Degree of Freedom	11
1.2.2 Multi Degree of Freedoms	12
1.2.3 Experimental Modal Analysis	13
1.2.4 Model Updating Process	15
1.3 Class Project	15
1.3.1 Midterm Project	15
1.3.1.1 Cantilever Beam	16
1.3.1.2 Plate and Brake-Disk	16
1.3.1.3 Conclusion of Midterm Project	17
1.3.2 Individual Project	17
1.4 Conclusion	18
A.1 Appendix	19
A.1.1 Syllabus	19
A.1.2 Introduction to the Development of Laser Scanning Vibrometer	20
A.1.2.1 Motivation of LSV Development	20
A.1.2.2 Specification of LSV	20
A.1.2.3 Features of LSV	21
A.1.2.4 Software Features of LSV	22
A.1.2.5 Applications of LSV	22
References	23
2 Lab Exercises for a Course on Mechanical Vibrations	24
2.1 Introduction	24
2.2 Exercises	25
2.2.1 Mass Calibration	25
2.2.2 Accelerometer Mounting	26
2.2.3 SDOF Measurement and Analysis	27
2.2.4 Full-Scale Modal Analysis Test	28
2.3 Conclusions	28
References	29
3 Variational Foundations of Modern Structural Dynamics	30
3.1 Introduction	31
3.2 Economy in Nature and Basic Variational Formulations	32
3.3 Mathematical Physics and Hamilton's Principle	33
3.4 The Contributions of Ritz, Galerkin, and Trefftz	33
3.4.1 The Ritz Method	33
3.4.2 Galerkin's Method	34
3.4.3 Trefftz's Method	34
3.5 Automated Formulations in Structural Dynamics	35
3.5.1 The Finite Element Method	35
3.5.1.1 Finite Elements: The Building Blocks	35
3.5.1.2 Assembly of Structural System Models	35
3.5.2 Matrix Structural Analysis	36
3.5.2.1 Free Vibration and Modal Analysis	36
3.5.2.2 Proportional Damping Foolishness	36
3.5.2.3 Normal Modes and Mach's Interpretation of d'Alembert's Principle	37
3.5.2.4 Response to Dynamic Loads: Modal Truncation and d'Alembert's Remedy	37
3.5.2.5 Large Order Systems: Component Mode Synthesis	38
3.6 Additional Variational Principles and Applications	38
3.7 Conclusions	39
References	40
4 Some Cornerstones of Signal Analysis History	41
4.1 Introduction	41
4.2 Early Signal Processing History	41
4.3 Late Developments	42
4.3.1 The Sampling Theorem	42
4.3.2 Random Data Theory	43
4.3.3 Spectrum Analysis	43
References	44
5 Structural Dynamic Test-Analysis Correlation	45
5.1 Introduction	46
5.2 Relevant Finite Element Models	46
5.3 Initial Model Verification	47
5.4 Structural Dynamic Response Analysis and Model Correlation Requirements	47
5.5 Target Modes	49
5.6 Modal Kinetic and Strain Energy Distributions	49
5.7 Modal Test Planning	50
5.8 Test Planning Challenges Associated with Large-Order Models	52
5.9 Correlation of Test and Analysis Modal Data (Cross-Orthogonality)	52
5.10 Correlation of Test and Analysis Modal Data (Localization of Differences)	53
5.11 Correlation of Test and Analysis Modal Data (Non-Weighted Metrics)	53
5.12 Illustrative Example	54
5.13 Conclusions	56
References	59
6 A Brief History of 30 Years of Model Updating in Structural Dynamics	60
6.1 Introduction	60
6.2 Genesis and Concepts of the Finite Element Method	62
6.2.1 Early Genesis of the Finite Element Method	62
6.2.2 The Equations-of-Motion Considered	62
6.2.3 Basic Concepts of the Finite Element Method	63
6.2.4 Discussion of Implications for FE Model Updating	65
6.3 A Classification of Finite Element Model Updating Methods	66
6.3.1 Early Developments of Finite Element Model Updating	66
6.3.2 Three Broad Categories of Model Updating Techniques	67
6.3.3 The First Category: Optimum Matrix Updating (OMU)	68
6.3.4 The Second Category: Small Perturbation Updating (SPU)	70
6.3.5 The Third Category: Iterative Sensitivity-Based Updating (ISBU)	71
6.4 A Brief Discussion of Challenges to Updating Methods	73
6.4.1 Numerical Ill-Conditioning of Inverse Problems	73
6.4.2 Mismatch Between Measurement Locations and FE Discretization	74
6.4.3 Localization of the Modeling Error	74
6.4.4 Information-Theoretic Limitations of Model Updating	74
6.4.5 Accounting for Experimental Variability and Uncertainty	75
6.4.6 Accounting for Numerical Uncertainty Caused by Truncation	75
6.4.7 Calibration of FE Representations for Nonlinear Dynamics	76
6.4.8 Closure	76
References	77
7 Techniques for Synthesizing FRFs from Analytical Models	79
7.1 Introduction	80
7.2 Synthesizing FRFs from Analytical Models	80
7.2.1 Full Space Method	81
7.2.2 Synthesizing FRFs from Reduced Order Models	81
7.2.3 Modal Superposition Method	81
7.2.4 Implementation Details	81
7.2.4.1 Retrieving System Matrices with ANSYS Workbench	82
7.2.4.2 Computing the FRF	82
7.3 Damping	82
7.3.1 Modeling Damping with ANSYS Workbench	82
7.4 Frequency Response Assurance Criteria (FRAC)	83
7.5 Experimental Example	83
7.5.1 FRF Comparison	83
7.5.2 Effect of Structural Damping	83
7.6 Conclusions	83
Reference	85
8 An Analytical Method and Its Extension for Linear Modal Analysis of Beam-Type Systems Carrying Various Substructures	86
8.1 Introduction	86
8.2 The Mathematical Model and Formulation	87
8.2.1 Evaluation of the Intermediate Discontinuities	88
8.2.2 General Formulations of Boundary Conditions	88
8.2.3 Consideration of Flexible Attachments	89
8.2.4 Condensation of the Eigenvalue Problem	89
8.3 Extension to More Complex Systems	90
8.4 Numerical Examples	90
8.4.1 Example 1: Validation of the Present Method	90
8.4.2 Example 2: Flexural Vibration of a Double-Beam Structure	90
8.4.2.1 Approach 1: The Analytical Solution	91
8.4.2.2 Approach 2: The Hybrid Analytical/FRF Solution	91
8.5 Conclusions	92
References	93
9 Computationally Efficient Nonlinear Dynamic Analysisfor Stress/Strain Applications	94
9.1 Introduction	95
9.2 Theory	96
9.2.1 Basic Equations of Motion	96
9.2.2 Model Reduction and Expansion	97
9.2.3 System Modeling and Mode Contribution	99
9.2.4 Uncoupled Expansion of Coupled Systems	100
9.2.5 Efficient Time Response Calculation	102
9.2.6 Correlation Tools	103
9.3 Cases Studied	103
9.3.1 Structure Description and General Modeling/Testing Performed	104
9.3.2 Case 1: Nonlinear Solution with Smaller Reduced Model	107
9.3.3 Case 2: Nonlinear Solution with Reduced Model Including Additional Modes	109
9.4 Conclusions	110
References	112
10 An Improved Expansion Process for Guyan Reduced Models: Technique for Improved Guyan Expansion Reconstruction (TIGER)	114
10.1 Introduction	115
10.2 Theory	115
10.2.1 General Reduction Techniques	115
10.2.2 Guyan Reduction Process	116
10.2.3 System Equivalent Reduction Expansion Process (SEREP)	116
10.2.4 Development of Matrices to Improve Guyan Results	117
10.3 Model Description and Cases Studied	118
10.3.1 Structure Description and General Modeling Performed	118
10.3.2 Case 1: Pinned–Pinned Beam with TIGER for Expansion	119
10.3.3 Case 2: Cantilever Beam with TIGER for Expansion	120
10.3.4 Case 3: Cantilever Beam with TIGER for Reduction and Expansion	121
10.4 Conclusion	122
References	123
11 Towards a Technique for Nonlinear Modal Reduction	126
11.1 Introduction	126
11.2 Second-Order Normal Form Technique	127
11.2.1 The Linear Modal Transform: (x →q)	127
11.2.2 The Nonlinear Near-Identity Transform: (q →u)	127
11.3 Example: A 3 Degree-of-Freedom Oscillator	128
11.4 Discussion	133
11.5 Conclusion	133
References	133
12 Identification of Independent Inputs and Their Spatial Positions	134
12.1 Introduction	134
12.2 On the Number of Inputs	134
12.2.1 Illustration	135
12.2.2 The Practical Case	135
12.2.3 Illustration	135
12.3 On the Location	136
12.4 Time Histories	137
12.5 Conclusions	137
References	138
13 Shock Response Fixture Developed from Analytical and Experimental Data and Customized Using Structural Dynamics Modification Techniques	139
13.1 Introduction	140
13.1.1 Background	140
13.1.2 Motivation	141
13.2 Methodology	141
13.3 Theoretical Background	142
13.3.1 Equation of Motion and Modal Space Representation	142
13.3.2 Time Response Computation with Modal Superposition Technique in Time Domain	143
13.3.3 Time Response Computation with Frequency Response Function (FRF) in Frequency Domain	144
13.3.4 Shock Response Spectrum (SRS)	145
13.3.5 Contribution Analysis and Mass Sensitivity Analysis (MSA)	147
13.3.6 Structural Dynamics Modification (SDM)	148
13.4 Analytical Example for 5-DOF System	149
13.4.1 Trial of the Methodology for Customizing SRS	149
13.4.2 Proper Configurations for SRS Computation	156
13.5 Analytical/Experimental Example for Beam Structure	158
13.5.1 Experimental SRS Measurement/EMA for Acquiring Modal Parameter	158
13.5.2 FE Model Development Using Measured Modal Parameters	161
13.5.3 SRS Computation with Time Response from Measured Data/EMA/FEA	163
13.5.4 Trial of the Methodology for Customizing SRS	165
13.6 Conclusion	172
References	172
14 Parameter Identification for Nonlinear Dynamic Systems via Multilinear Least Square Estimation	173
14.1 Introduction	173
14.2 Nonlinear Resonant Decay Method (NLRDM)	174
14.2.1 Least Square Estimation	174
14.2.2 Coefficient of Determination Criteria	176
14.3 Problem Formulation	177
14.3.1 Curve Fitting of Numerical Problem	180
14.3.2 Cantilever Beam	181
14.3.3 Modal Space Equation	184
14.3.4 Implementation of Curve Fitting Algorithm	184
14.4 Concluding Remarks	186
References	186
15 Support Systems for Developing System Models	187
15.1 Introduction	187
15.2 Guidelines for Selecting Support Systems	188
15.2.1 Separation of Elastic Modes from Rigid Body Modes	188
15.2.2 Location of Supports	188
15.2.3 Isolation of Test Structure from the Support System and External Noise Sources	188
15.3 Types of Support Systems	188
15.3.1 Shock Cords	188
15.3.2 Soft Foam	189
15.3.3 Coil Springs	189
15.3.4 Air Springs	189
15.4 Experimental Examples	189
15.4.1 Formula SAE Racecar Space Frame	190
15.4.2 Rectangular Steel Plate	190
15.4.2.1 Unconstrained Boundary Condition Testing	191
15.4.2.2 Constrained Boundary Condition Testing	191
15.4.3 Circular Aluminum Plate	193
15.5 Conclusions	195
References	197
16 Nonlinear Modeling for Adaptive Suppression of Axial Drilling Vibration	198
16.1 Introduction	198
16.2 Model Development	199
16.3 Experimental Study of Model Parameters	201
16.3.1 Experimental Study of Independent Model Parameters	201
16.3.2 Parameter Estimation	201
16.3.3 Experimental Study of Drill Bit Model Parameters	203
16.4 Test Analysis Correlation to Experiments	205
16.4.1 Linear Modal Analysis	205
16.4.2 Nonlinear Acceleration Response to Drilling	205
16.5 Adaptive Positive Position Feedback Controller	207
16.6 Results	208
16.6.1 Results of Non-adaptive Controller Suppression	209
16.6.2 Future Sensor Placement and Actuator Type	211
16.7 Conclusion	211
References	212
17 A Regenerative Approach to Energy Efficient Hydraulic Vibration Control	214
17.1 Introduction	214
17.2 Active Control	215
17.3 Hydraulic Buck Converter	215
17.4 Hydraulic Component Sizing	219
17.5 Results	220
17.6 Conclusions	221
References	222
18 Virtual Sensing of Acoustic Potential Energy Through a Kalman Filter for Active Control of Interior Sound	224
18.1 Introduction	225
18.2 Development of a Numerical Model of the Piezo-Structural-Acoustic System	226
18.2.1 Finite Element Model	226
18.2.2 State Space Model of the Plant	227
18.3 Virtual Sensing of Acoustic Potential Energy	228
18.3.1 Development of a Steady State Kalman Filter	228
18.3.2 Development of Radiation Filter	230
18.4 Numerical Example	231
18.4.1 Selection of Optimal Locations of Sensor and Actuator	231
18.4.2 Eigen Value Analysis	233
18.4.3 Results and Discussion	233
18.4.3.1 Case 1. Virtual Sensing for Impulsive Disturbances	233
18.4.3.2 Case 2. Virtual Sensing for Random Disturbances	234
18.4.3.3 Case 3. Influence of Unmodeled Plant Dynamics on Virtual Sensing	234
18.5 Conclusions	236
References	244
19 Wavenumber Decomposition Applied to a Negative Impedance Shunts for Vibration Suppression on a Plate	245
19.1 Introduction	245
19.2 Metacomposite Design	246
19.3 Experimental Setup	248
19.4 Analysis of 2D Waveguide	248
19.5 Conclusions	251
References	252
20 Modal Parameter Estimation of a Two-Disk- Shaft System by the Unified Matrix Polynomial Approach	253
20.1 Introduction	253
20.2 Theoretical Analysis	254
20.2.1 Frequency Response Function Matrix of the Multiple Disk Shaft System	254
20.2.2 Modal Parameters of the Multiple Disk Shaft System	257
20.3 Results and Discussion	259
20.4 Conclusions	262
References	264
21 System Identification of an Isolated Structure Using Earthquake Records	266
21.1 Introduction	266
21.2 Building and Instrumentation Description	267
21.3 Seismic Events Records	268
21.4 Models and Identification Methods	268
21.4.1 Kelvin Voight Model and Maxwell Model	268
21.4.2 Non-Linear Elastic Model	269
21.4.3 Bouc-Wen Models	269
21.4.4 Biaxial Hysteretic Restoring Force Model	269
21.4.5 Multiple Shear Spring Models	270
21.5 Identification Isolation Bearing with No-Linear Behavior	270
21.6 Modal Properties Identification Results	274
21.7 Conclusion	275
References	275
22 Design of an Inertial Measurement Unit for Enhanced Training	276
22.1 Introduction	276
22.2 The Designed IMU	277
22.3 The Experimental Results	278
22.3.1 Experimental Setup	278
22.3.2 Test 1: GPS and IMU Calibration	280
22.3.3 Test 2: Sprints and Changes of Direction	280
22.3.4 Test 3: Endurance Test	282
22.4 Conclusions	282
References	286
23 A Parameter Optimization for Mode Shapes Estimation Using Kriging Interpolation	287
23.1 Introduction	287
23.2 Framework for Optimal Sensor Configuration	288
23.2.1 Optimal Sensor Placement (OSP) Methods	288
23.2.2 Kriging	289
23.3 Case Studies	289
23.3.1 Shear Building Model	289
23.3.2 Northampton Steel Bridge	290
23.4 Conclusion	292
References	293
24 Determination of Principal Axes of a Wineglass Using Acoustic Testing	294
24.1 Introduction	294
24.2 Dynamic Characteristics of a Wineglass	295
24.3 Methodology to Determine the Principal Axes	296
24.4 Acoustic Testing to Verify the Methodology	297
24.5 Conclusions	297
References	298
25 Remote Placement of Magnetically Coupled Ultrasonic Sensors for Structural Health Monitoring	299
25.1 Introduction	299
25.2 Remote Sensor Placement Device	300
25.3 Internal Ballistics Models	301
25.3.1 External Ballistics Modeling	303
25.3.2 Combine Internal Modeling and Inverse Problem	303
25.3.3 Bench Test Setup	303
25.3.4 Sensor Package	305
25.3.5 Computer Vision	305
25.3.5.1 Concept	305
25.3.5.2 Object Tracking Methods	306
25.3.5.3 Feature Based Tracking	306
25.3.6 Color Based CAMShift Tracking	307
25.3.7 Algorithm Improvements	308
25.3.8 Implementation Improvements	308
25.4 Multicopter	308
25.4.1 Control	308
25.5 Conclusion	310
References	310
26 Modular System for High-Speed 24-Bit Data Acquisition of Triaxial MEMS Accelerometers for Structural Health Monitoring Research	311
26.1 Introduction	311
26.2 Requirements and Network Architecture	311
26.3 Aggregator Board Overview	313
26.4 Front End Section Design	314
26.5 ADC Section Design	315
26.6 Digital Section Design	315
26.7 Noise Measurements	315
26.7.1 Front End Breadboard Noise	315
26.7.2 MITEI Aggregator Board Noise	315
26.8 Power	316
26.9 LABVIEW GUI for Host PC	316
26.10 Conclusions	317
27 Mode Shape Comparison Using Continuous-Scan Laser Doppler Vibrometry and High Speed 3D Digital Image Correlation	319
27.1 Introduction	319
27.2 Measurements	320
27.2.1 CSLDV Theory/Mode Shape Extraction Description	320
27.2.2 DIC Theory/Mode Shape Extraction Description	321
27.3 Experimental Setup	321
27.3.1 Beam Description	321
27.3.2 Experimental Setup	322
27.4 Results	324
27.4.1 Modal Hammer Test	324
27.4.2 Mode Shape Comparison: Sine Dwell Experiments	324
27.4.3 Natural Frequencies and Damping/Mode Shapes from Random Excitation	325
27.5 Conclusion	328
References	329
28 Triaxial Accelerometer, High Frequency Measurement and Temperature Stability Considerations	330
28.1 Introduction	330
28.1.1 High Frequency Response Characterization in Each Orthogonal Axis	330
28.1.2 Low Sensitivity Temperature Coefficient and High Stiffness Properties of PiezoStar® Crystals	333
28.2 Summary	335
References	337
29 Laser Speckle in Dynamic Sensing Applications	338
29.1 Introduction	338
29.2 Background	339
29.3 Experiment	340
29.4 Results	341
29.5 Conclusions	342
References	342
30 Sensor Placements for Damage Localization with the SDLV Approach	343
30.1 Introduction	343
30.2 The SDLV Approach	343
30.3 Sensor Placement Strategies	345
30.3.1 Effective Independence Method	345
30.3.2 Linear Independence Stress Field	345
30.4 Criterion to Compare Performance	346
30.5 Numerical Examination	346
30.5.1 Sensor Placements Results	347
30.5.2 Damage Scenarios	347
30.5.3 Summary of Results	347
30.5.4 Discussion	348
30.6 Concluding Comments	349
References	349
31 Diaphragm Flexibility in Floor Spectra	350
31.1 Introduction	350
31.2 Experimental Examination	351
31.3 A Simplified Model	351
31.4 Results	351
31.5 Conclusions	353
References	354
32 Use of Zernike Polynomials for Modal Vector Correlation of Small Turbine Blades	355
32.1 Introduction	356
32.2 Theoretical Background	356
32.2.1 Zernike Polynomials	356
32.2.2 Zernike Moment Descriptors	357
32.2.3 Additional Processing	358
32.2.4 Modal Assurance Criterion	358
32.2.5 Mode Shape Decomposition	358
32.3 Test Case Studied: Analytical	359
32.3.1 Model Description	359
32.4 Nominal Model with Two Element Width and Height Difference	360
32.5 Test Case Studied: Experimental	361
32.5.1 Structure Description	361
32.5.2 Correlation of IGT Blade 1 to IGT Blade 2 Alignment 1	363
32.5.3 Correlation of IGT Blade 1 to IGT Blade 2 Alignment 2	364
32.5.4 Correlation of IGT Blade 1 to IGT Blade 2 Alignment 3	365
32.5.5 Comparison of Results	368
32.6 Conclusion	371
References	371
33 Modeling of Flexible Tactical Aerospace Vehicle for Hardware-in-Loop Simulations	372
33.1 Introduction	373
33.2 Mathematical Model	375
33.2.1 Lagrange's Equations	376
33.2.2 Kinetic Energy	377
33.2.3 Potential Energy	377
33.2.4 Damping Terms	379
33.2.5 Non-conservative Forces and Moments	379
33.2.5.1 Aerodynamic Forces and Moments	379
33.2.5.2 Propulsive Forces and Moments	382
33.2.6 Equations of Motion and Bending	383
33.2.7 Accelerations and Angular Rates at Sensor Locations	387
33.3 Simulations and Results	387
33.4 Conclusion	388
References	392
34 Modal Test of Six-Meter Hypersonic Inflatable Aerodynamic Decelerator	393
34.1 Introduction	393
34.2 Test Article Setup	394
34.2.1 Test Instrumentation	395
34.2.2 Excitation Techniques	395
34.3 Modal Test Results	396
34.3.1 Predicted vs. Experimental Results	397
34.3.2 Influence of the Straps	398
34.3.3 Influence of Tri-Torus	401
34.3.4 Comparison of Tri-Torus and Straps Influences	403
34.3.5 Influence of Varying Toroid Pressure	403
34.4 Conclusions	405
References	405
35 Modal Testing of Space Exploration Rover Prototypes	406
35.1 Introduction	406
35.2 Description of Rover and Mast Assembly	407
35.3 Test Sequence	408
35.4 Developmental Activities	409
35.5 Modal Testing of Rover with Portable Exciter	410
35.5.1 Test Configuration and Instrumentation	410
35.5.2 Estimation of Modal Parameters	413
35.6 Modal Testing of Rover and Mast Assembly with Portable Exciter	413
35.6.1 Test Configuration and Instrumentation	414
35.6.2 Estimation of Modal Parameters	416
35.7 Lessons Learned	416
35.8 Summary	417
Reference	418

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