Nonlinear Dynamics, Volume 1

Preface	6
Contents	8
1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System	11
1.1 Introduction	11
1.2 Finite Element Model	12
1.3 Results	13
1.3.1 Onset of the Sliding	13
1.3.2 Phases Stability Analysis	15
1.4 Conclusions	19
References	19
2 Non-linear Dynamics of Jointed Systems Under Dry Friction Forces	21
2.1 Introduction	21
2.2 Mechanical Model	21
2.2.1 Dimensional Motion Equations	22
2.2.2 Dimensionless Motion Equations	23
2.2.3 Solution Method	24
2.3 Results	25
2.3.1 Bifurcation Diagrams	25
2.3.2 Time and Frequency Domain	27
2.3.3 Maximum Lyapunov Exponent	29
2.4 Concluding Remarks	30
References	31
3 Prediction of Nonlinear Forced Response on Ancillary Subsystem Components Attached to Reduced Linear Systems	32
Nomenclature	32
Symbols	32
Subscript	33
Superscript	33
Acronyms	33
3.1 Introduction	34
3.2 Theory	35
3.2.1 Equations of Motion for Multiple Degree of Freedom System	35
3.2.2 System Modeling and Mode Contribution	35
3.2.2.1 Physical Space System Modeling	36
3.2.2.2 Structural Dynamic Modification	36
3.2.3 General Reduction/Expansion Methodology and Model Updating	37
3.2.3.1 Expansion of System Modes from Uncoupled Component Modes	38
3.2.3.2 System Equivalent Reduction Expansion Process (SEREP)	38
3.2.4 System Forced Response Analysis	38
3.2.5 Expansion of Reduced Order Real Time Response	40
3.2.6 Time Response Correlation Tools	40
3.2.6.1 Modal Assurance Criterion (MAC)	40
3.2.6.2 Time Response Assurance Criterion (TRAC)	42
3.3 Model Description	42
3.4 Cases Studied	45
3.4.1 Case A: Two Nonlinear Contacts Between System 1 and System 2	46
3.4.1.1 Case A-1: Soft Contact Reference Solution	46
3.4.1.2 Component Mode Contribution – U12	46
3.4.1.3 Overview of Reduction Process	48
3.4.1.4 Case A-1.1: Soft Contact Reduced Model Solution with 12 Modes	48
3.4.1.5 Case A-1.2: Soft Contact Reduced Model Solution with 16 Modes	49
3.4.1.6 Case A-2: Hard Contact Reference Solution	49
3.4.1.7 Case A-2.1: Hard Contact Reduced Model Solution with 21 Modes	50
3.4.2 Case B: One Nonlinear Contact at System 2	53
3.4.2.1 Case B-1: Soft Contact Reference Solution	53
3.4.2.2 Case B-1.1: Soft Contact Reduced Model Solution with 22 Modes with ADOF at Ancillary	54
3.4.2.3 Case B-1.2: Soft Contact Reduced Model Solution with 18 Modes with No ADOF at Ancillary	56
3.5 Conclusions	57
References	59
4 Numerical Round Robin for Prediction of Dissipation in Lap Joints	61
4.1 Introduction	61
4.2 Approaches to Modeling Friction Joints	62
4.2.1 Iwan Model	62
4.2.2 Stuttgart Approach (Harmonic Balance Method)	63
4.2.3 Imperial Approach	64
4.3 The Lap Joint Test Case	65
4.3.1 The Finite Element Mesh	65
4.3.2 Nonlinear Static Analysis	65
4.3.3 Nonlinear Dynamic Model	66
4.4 Nonlinear Dynamic Results	67
4.5 Conclusion	71
References	71
5 The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems	73
5.1 Introduction	73
5.2 Harmonic Balance Method	74
5.2.1 Analytical Expression of the Nonlinear Terms and of the Jacobian Matrix of the System	75
5.2.2 Continuation Procedure	76
5.2.3 Stability Analysis	77
5.2.4 Detection of Bifurcations	78
5.2.5 Tracking of Bifurcations	80
5.3 Validation of the Method on the Study of an Industrial, Complex Model with Strong Nonlinearities: The SmallSat	81
5.3.1 Case Study: SmallSat Spacecraft	81
5.3.2 Nonlinear Dynamics and Bifurcations of the SmallSat	83
5.3.3 Influence of the Forcing Amplitude F and the Axial Damping caxon the Fold Bifurcations	85
5.3.4 Influence of the Axial Damping cax on the NS Bifurcations	87
5.4 Conclusions	87
References	89
6 Nonlinear Vibrations of a Beam with a Breathing Edge Crack	91
6.1 Introduction	91
6.2 Mathematical Modeling	92
6.3 Application of Harmonic Balance Method	94
6.4 Results and Discussion	95
6.5 Conclusion	97
References	100
7 Stability Limitations in Simulation of Dynamical Systems with Multiple Time-Scales	101
7.1 Introduction	101
7.1.1 Motivation and Previous Work	101
7.2 Numerical Integration of Highly-Oscillatory Systems	102
7.2.1 Exponential Integrators	102
7.2.2 Trigonometric Integrators	103
7.2.2.1 Mollified Impulse Methods	103
7.3 A Proposed Exponential Integration Scheme	104
7.3.1 On the Choice of Nominal Deterministic Model	104
7.3.2 System Descritization	104
7.3.3 Implicit Implementation	105
7.3.4 Explicit Implementation	106
7.4 The Fermi-Pasta-Ulam Benchmark Problem	107
7.4.1 Linear Stability Limitations	109
7.4.2 Nonlinear Stability and Numerical Resonances	110
7.5 Concluding Results	111
References	112
8 Coupled Parametrically Driven Modes in Synchrotron Dynamics	114
8.1 Introduction	114
8.2 Model 1	115
8.3 Model 2	116
8.4 Results	117
8.4.1 Model 1	117
8.4.2 Model 2	117
8.5 Conclusion	118
References	119
9 Relating Backbone Curves to the Forced Responses of Nonlinear Systems	120
9.1 Introduction	120
9.2 Second-Order Normal form Technique	120
9.2.1 The Example System	120
9.2.2 Applying the Second-Order Normal form Technique	121
9.2.3 Finding the Backbone Curves of the Example System	124
9.3 Internal Resonance	126
9.3.1 Internal Resonance in the Example System	126
9.4 Conclusions	129
References	129
10 Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities	130
10.1 Introduction	130
10.2 In-Line Nonlinear 3-DOF Oscillators	131
10.3 Application of the Second-Order Normal Form Method	132
10.4 Backbone Curve and FRF Results	134
10.5 Conclusions	137
References	138
11 Passive Flutter Suppression Using a Nonlinear Tuned Vibration Absorber	139
11.1 Introduction	139
11.2 Problem Formulation	140
11.3 Elimination of Limit Cycles Through Stability Analysis	140
11.4 Enforcement of Supercritical Hopf Bifurcations Through Normal Form Analysis	141
11.4.1 Single Hopf Bifurcation	142
11.4.2 Two Intersecting Single Hopf Bifurcations	144
11.4.3 Proposed Tuning Rule for the Nonlinear Coefficient of the NLTVA	145
11.5 Reduction of the Amplitude of Limit Cycle Oscillations	147
11.5.1 Local Analysis	147
11.5.2 Global Analysis	147
11.6 Conclusions	149
References	149
12 Nonlinear Vibrations of a Flexible L-shaped Beam Using Differential Quadrature Method	151
12.1 Introduction	151
12.2 Modeling	152
12.3 Application of Differential Quadrature Method	155
12.4 Results	156
12.4.1 Linear Vibrations	156
12.4.2 Nonlinear Free Vibrations	157
12.5 Conclusion	159
References	159
13 Theoretical and Experimental Analysis of Bifurcation Induced Passive Bandgap Reconfiguration	161
13.1 Introduction	161
13.2 Math Model	162
13.2.1 Equations of Motion	162
13.2.2 Bandgap Calculations	163
13.3 Experimental System	164
13.3.1 Bandgap Predictions	164
13.3.2 Frequency Sweeps	165
13.3.3 Amplitude Sweep	165
13.4 Conclusion	166
References	168
14 A Model of Evolutionary Dynamics with Quasiperiodic Forcing	169
14.1 Introduction	169
14.2 The Model	170
14.2.1 Rock-Paper-Scissors Games with Quasiperiodic Forcing	170
14.2.2 Linearization	171
14.3 Floquet Theory	172
14.4 Harmonic Balance	173
14.5 Numerical Integration	174
14.6 Lyapunov Exponents	176
14.7 Conclusion	177
References	177
15 Experimental Demonstration of a 3D-Printed Nonlinear Tuned Vibration Absorber	178
15.1 Introduction	178
15.2 Primary System Description	179
15.3 Linear and Nonlinear Den Hartog's Equal-Peak Methods	180
15.3.1 The Linear Tuned Vibration Absorber (LTVA)	180
15.3.2 The Nonlinear Tuned Vibration Absorber (NLTVA)	181
15.4 Practical Realization of LTVA and NLTVA Using Cantilever and Doubly-Clamped Beams	182
15.4.1 LTVA Cantilever Beam	183
15.4.2 NLTVA Doubly-Clamped Beam	183
15.4.3 Experimental Characterization of the Manufactured Absorbers	184
15.5 Experimental Demonstration of LTVA and NLTVA Performance	185
15.6 Concluding Remarks	186
References	188
16 The Effect of Gravity on a Slender Loop Structure	189
16.1 Introduction	189
16.2 The Experimental System	190
16.3 Results	191
16.3.1 Results in Dimensional Terms	191
16.3.2 Results in Non-dimensional Terms	192
16.3.3 An Interesting Feature	193
16.4 Conclusions	194
References	194
17 Wave Propagation in a Materially Nonlinear Rod: Numerical and Experimental Investigations	195
17.1 Introduction	195
17.2 Modeling	196
17.3 Parametric Study	197
17.4 Experiments	198
17.5 Conclusion and Future Work	199
References	200
18 Experimental Nonlinear Dynamics and Chaos of Post-buckled Plates	202
18.1 Introduction	202
18.2 Experimental Results	203
18.3 Conclusions	205
References	205
19 Control-Based Continuation of a Hybrid Numerical/Physical Substructured System	206
19.1 Introduction	206
19.2 Experimental Set-Up	207
19.2.1 A Test Problem	207
19.2.2 Real-Time Dynamic Substructuring and Control-Based Continuation	208
19.3 Results	208
References	209
20 Towards Finite Element Model Updating Based on Nonlinear Normal Modes	211
20.1 Introduction	211
20.2 Model Updating Methodology	212
20.3 Harmonic Balance Method for NNM Calculation	213
20.4 Parametric Study of an Example System	214
20.5 Model Updating Results for Beam with Cubic Nonlinearities	216
20.6 Conclusion and Future Work	219
References	219
21 Experimental Modal Analysis of Nonlinear Structures Using Broadband Data	220
21.1 Introduction	220
21.2 Brief Review of Nonlinear Normal Modes (NNMs) and Identification Using Phase Resonance	221
21.3 Identification Methodology of Nonlinear Normal Modes (NNMs) Under Broadband Forcing	222
21.3.1 Experimental Identification of an Undamped Nonlinear State-Space Model	225
21.3.1.1 Nonlinear Model Equations in the Physical Space	225
21.3.1.2 Feedback Interpretation and State-Space Model Equations	225
21.3.1.3 Conversion from State Space to Physical and Modal Space	226
21.3.1.4 Removal of Damping Terms in the Identified State-Space Model	227
21.3.2 Computation of the Individual NNMS in the State Space Using Numerical Continuation	228
21.4 Numerical Demonstration Using a Cantilever Beam Possessing a Cubic Nonlinearity	229
21.4.1 Identification Using the FNSI Method Under a Multisine Excitation	230
21.4.1.1 Selection of the Nonlinear Basis Functions	230
21.4.1.2 Selection of the Model Order	231
21.4.1.3 Estimation of the Underlying Linear Properties	232
21.4.1.4 Estimation of the Nonlinear Coefficient	232
21.4.1.5 Removal of Damping Terms	233
21.4.2 Computation of the First Two NNMS Using Continuation	234
21.5 Comparison with NNMs Identified Using Nonlinear Phase Resonance	235
21.6 Conclusion	237
References	240
22 Measurement of Nonlinear Normal Modes Using Mono-harmonic Force Appropriation: Experimental Investigation	242
22.1 Introduction	242
22.2 Nonlinear Normal Modes	243
22.2.1 Numerically Calculated NNMs	243
22.2.2 Measuring NNMs with Force Appropriation	245
22.3 Structure Description	246
22.3.1 Beam Description	246
22.3.2 Plate Description	247
22.3.3 Experimental Setup	247
22.4 Results	248
22.4.1 Clamped-Clamped Beam	248
22.4.2 Circular Plate	250
22.5 Conclusion	253
References	255
23 Nonlinear System Identification Through Backbone Curves and Bayesian Inference	256
23.1 Introduction	256
23.2 Nonlinear Normal Forms and Backbone Curves	257
23.3 Identification with Bayesian Inference	258
23.4 Nonlinear Identification of a 2-DOF Nonlinear Oscillator	260
23.5 Conclusions	261
References	262
24 Experimental Nonlinear Identification of an Aircraft with Bolted Connections	264
24.1 Introduction	264
24.2 Description of the Aircraft and of the Experimental Setup	265
24.3 Nonlinear Subspace Identification in the Frequency Domain	266
24.4 Linear Analysis at Low Level and Nonlinearity Detection	268
24.5 Nonlinearity Identification at High Level	269
24.5.1 Selection of Appropriate Basis Functions	271
24.5.2 Choice of the Processed Bandwidth	272
24.5.3 Determination of the Model Order	272
24.5.4 Estimation of the Underlying Linear Properties	272
24.5.5 Estimation of the Nonlinear Coefficients	274
24.6 Proposed Strategy for Removing Spurious Poles and Improved Results	275
24.7 Conclusions	278
References	279
25 Non linear Finite Element Model Validation of a Lap-Joint	280
25.1 Introduction	280
25.2 Testing and Modelling	283
25.2.1 Broadband and Stepped Sine Tests	283
25.2.2 Models	283
25.3 Nonlinear Analysis	285
25.3.1 Detection	285
25.3.2 Location	286
25.3.3 Characterisation	286
25.3.4 Quantification	287
25.4 Correlation and Models Updating	288
25.4.1 SDOF Correlation and Updating	288
25.4.2 Nonlinear FE Model Validation	289
25.5 Conclusions	291
References	292
26 Experimental Validation of Pseudo Receptance Difference (PRD) Method for Nonlinear Model Updating	293
26.1 Introduction	293
26.2 Theory	294
26.3 Experimental Study	296
26.3.1 Experimental Setup	296
26.3.2 First Set of Experiments	297
26.3.2.1 Application of the PRD Method at Excitation Frequency 39 Hz	298
26.3.2.2 Application of the PRD Method at Excitation Frequency 40 Hz	299
26.3.3 Second Set of Experiments	300
26.4 Application of the PRD Method	301
26.5 Model Updating of the Test System and Verification of the Updated Model	302
26.6 Summary and Conclusions	304
References	305
27 Systems with Bilinear Stiffness: Extraction of Backbone Curves and Identification	307
27.1 Introduction	307
27.2 Backbone Curves from Experimental Data	308
27.3 Experimental Example	308
27.4 Conclusions	312
References	313
28 Simplifying Transformations for Nonlinear Systems: Part I, An Optimisation-Based Variant of Normal Form Analysis	314
28.1 Introduction	314
28.2 Simplifying Transformation Strategy and Differential Evolution	314
28.3 An Example	316
28.4 Conclusion	317
References	318
29 Simplifying Transformations for Nonlinear Systems: Part II, Statistical Analysisof Harmonic Cancellation	320
29.1 Introduction	320
29.2 Differential Evolution	320
29.3 An Example	321
29.3.1 Monte Carlo Simulation	321
29.3.2 Discussion	322
29.4 Conclusion	324
References	325
30 Considerations for Indirect Parameter Estimation in Nonlinear Reduced Order Models	326
30.1 Introduction	326
30.2 Theoretical Development	327
30.2.1 Selection of Modes	328
30.2.1.1 Axial Vibration Modes	329
30.2.1.2 Dual Modes	329
30.2.2 Scaling of Modes	330
30.2.3 Identification of Nonlinear Stiffness Coefficients	330
30.2.3.1 Least Squares Method	330
30.2.3.2 Least Squares, Constrained Method	331
30.2.3.3 RANSTEP Method	332
30.3 Numerical Results: Comparison of Parameter Estimation Method and Scaling Method	332
30.4 Numerical Results: Axial Modes Versus Dual Modes	336
30.5 Numerical Results: FEA Model of Exhaust Cover Plate	338
30.6 Conclusion	340
References	340
31 Nonlinear Model Updating Methodology with Application to the IMAC XXXIII Round Robin Benchmark Problem	342
31.1 Introduction	342
31.2 Proposed Model Updating Methodology	343
31.3 Application to the Benchmark Problem	343
31.4 Concluding Remarks	346
References	346
32 Bridging the Gap Between Nonlinear Normal Modes and Modal Derivatives	347
32.1 Introduction	347
32.2 Governing Equations	348
32.3 Galerkin Projection	348
32.3.1 Vibration Modes	349
32.3.2 Modal Derivatives	349
32.4 Reduction with Quadratic Manifold	350
32.5 Nonlinear Normal Modes	351
32.6 Numerical Examples	352
32.6.1 2DOF Example	352
32.6.2 4DOF Example	354
32.7 Conclusions	356
References	359
33 Validation of Nonlinear Reduced Order Models with Time Integration Targeted at Nonlinear Normal Modes	360
33.1 Introduction	360
33.2 Theoretical Development	362
33.2.1 Review of Reduced Order Modeling	362
33.2.2 Review of Nonlinear Normal Modes	363
33.2.3 Proposed Procedure for Generating a Valid ROM	363
33.3 Numerical Results	365
33.3.1 Clamped-Clamped Beam from [18, 21]	365
33.3.2 Cantilevered Plate	367
33.4 Conclusion	370
References	372
34 Model Order Reduction of Nonlinear Euler-Bernoulli Beam	373
34.1 Introduction	373
34.2 Nonlinear Model Reduction	374
34.3 Model Reduction Subspace Selection	374
34.3.1 Dynamical Consistency	374
34.3.2 Subspace Robustness	375
34.4 Nonlinear Euler-Bernoulli Beam	375
34.4.1 Full-Dimensional System Model	376
34.5 Results and Discussion	377
34.5.1 Full-Dimensional Model	377
34.5.2 Reduced Order Model	377
34.6 Conclusion	381
References	381
35 Identification of Dynamic Nonlinearities of Bolted Structures Using Strain Analysis	382
35.1 Introduction	382
35.2 Structure Design (CAD) and Manufacturing	384
35.3 Experimental Model Validation- Linear Case	384
35.3.1 Modal Analysis	384
35.3.1.1 Numerical Linear Finite Element Model	384
35.3.1.2 Experimental Linear Modal Testing	386
35.3.1.3 Updating Process	387
35.4 Nonlinearity Criticality Modal Ranking Criteria	388
35.5 Nonlinear Modal Testing of the Flange Structure	390
35.5.1 Experimental Set Up	391
35.5.2 Modal Testing	392
35.5.2.1 Modal Testing Without Control	392
35.6 Nonlinear Dynamic Investigation Using Strain Analysis	395
35.6.1 Experimental Strain Measurements	395
35.6.1.1 Strain Gages Setup	395
35.6.1.2 Experimental Process	396
35.6.2 Signal Post-processing	398
35.6.2.1 Notations	398
35.6.2.2 Post-processing Objectives	398
35.6.3 Strain Results Analysis	399
35.6.3.1 Mode 4 (T2): 166 Hz	399
35.6.3.2 Mode 8 (T4): 359 Hz	400
35.7 Linear Correlation for Model Validation	401
35.7.1 Linear FRF Correlation	401
35.7.2 Linear Modal Strains Correlation	402
35.7.3 Numerical Time Domain Nonlinear Analysis	402
35.7.3.1 Nonlinear Model Parameters	403
35.7.3.2 Time-Domain Results for the Critical Modes	405
35.8 Conclusions	408
References	408
36 The Effects of Boundary Conditions, Measurement Techniques, and Excitation Type on Measurements of the Properties of Mechanical Joints	410
36.1 Introduction	410
36.2 Effects of Boundary Conditions	412
36.3 Effects of Excitation and Measurement Conditions	414
36.4 The Jointed Beam	417
36.5 Conclusions	421
References	426
37 Numerical Model for Elastic Contact Simulation	427
37.1 Introduction	427
37.2 Notation and Convention	428
37.3 Normal Contact Problem	428
37.3.1 Method of Solution	428
37.4 Tangential Contact Problem	431
37.4.1 Method of Solution	431
37.5 Results	432
37.6 Conclusion	432
References	434
38 Efficient and Accurate Consideration of Nonlinear Joint Contact Within Multibody Simulation	435
Nomenclature	435
38.1 Introduction	436
38.2 Theory	436
38.2.1 Brief Review of Joint Trial Vectors Based on Trial Vector Derivatives	436
38.2.2 Consideration of Contact Forces in the Framework of Multibody Simulation	438
38.3 Numerical Example	439
38.3.1 Evaluation of the Reduction Base	439
38.3.2 Dynamic Simulation	440
38.4 Conclusion	442
References	442
39 Model Reduction for Nonlinear Multibody Systems Based on Proper Orthogonal- and Smooth Orthogonal Decomposition	443
Nomenclature	443
39.1 Introduction	443
39.2 Multibody System Modelling	444
39.2.1 Full Order Modelling	444
39.2.2 Reduced Order Modeling	445
39.3 Reduction Methods	445
39.3.1 Proper Orthogonal Decomposition	445
39.3.2 Smooth Orthogonal Decomposition	446
39.4 Numerical Example	447
39.4.1 Model Data	447
39.4.2 Results	448
39.5 Conclusion	450
References	451
40 Cam Geometry Generation and Optimization for Torsion Bar Systems	452
40.1 Introduction	452
40.2 Modelling of the System	454
40.2.1 Torsion Bar System Components	454
40.2.2 Mechanical Behavior of the Torsion Bar	455
40.2.3 System Variables	456
40.2.4 Balance Equations of the System	456
40.2.5 Set of Equations Defining the Cam Profile Curve	458
40.2.6 Overall System Equations	459
40.3 Solution and Optimization of the System Variables	460
40.3.1 Constraints and Objective Function of the Optimization Problem	460
40.4 Optimal Solution and Results	461
40.4.1 System Parameters for a Specific Case	461
40.4.2 Optimal Solution for the Given System	461
40.5 Conclusion	462
References	464
41 Dynamics Modeling and Accuracy Evaluation of a 6-DoF Hexaslide Robot	465
41.1 Introduction	465
41.2 Hexaslide Kinematics	466
41.3 Multibody Model	467
41.3.1 Links Modeling	467
41.3.2 Belt Modeling	467
41.3.3 Screw Modeling	468
41.3.3.1 Screw-Nut Coupling Modeling	468
41.4 Numerical Results	469
41.5 Conclusions	469
References	470
42 A Belt-Driven 6-DoF Parallel Kinematic Machine	472
42.1 Introduction	472
42.2 Hexaglide Robot	473
42.2.1 Inverse Kinematics	473
42.2.2 Inverse Dynamics	473
42.3 Dynamic Analysis of BDS	473
42.4 Error Evaluation	474
42.5 Numerical Results	477
42.6 Conclusion	479
References	480
43 Bearing Cage Dynamics: Cage Failure and Bearing Life Estimation	481
43.1 Introduction	481
43.2 Cage Dynamics	481
43.2.1 Free Flight	483
43.2.2 Race Contact	484
43.2.3 Ball Contact	485
43.3 Ball Dynamics	487
43.3.1 Pure Stick Motion	487
43.3.2 Stick-Slip Motion	488
43.4 Results	489
43.4.1 Kinematic Results	489
43.4.2 Dynamic Results	490
43.4.3 Probabilistic Analysis	491
43.5 Concluding Remarks	492
References	494
44 Bias Errors of Different Simulation Methods for Linear and Nonlinear Systems	495
44.1 Introduction	495
44.2 Theoretical Background	496
44.2.1 Aliasing Error	497
44.2.2 Bias Error	498
44.2.2.1 Linear Systems	498
44.2.2.2 Application to Nonlinear Systems	500
44.3 Numerical Examples	505
44.3.1 A Linear System	505
44.3.2 A Nonlinear System	505
44.4 Conclusion	509
References	510
45 Internal Resonance and Stall Flutter Interactions in a Pitch-Flap Wing in the Wind-Tunnel	511
45.1 Introduction	511
45.2 Experimental Setup	512
45.3 Results	514
45.3.1 Wind-Off Modal Analysis of the System	514
45.3.2 Wind-on Study of the System	515
45.3.2.1 Variation of the Modal Parameters with Airspeed	515
45.3.2.2 Limit Cycle Oscillation Amplitude and Shape Variation with Airspeed	516
45.3.2.3 Limit Cycle Oscillation Frequency Variation with Airspeed	517
45.4 Discussion of the Results	519
45.5 Conclusion	520
References	520

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