Nonlinear Dynamics, Volume 1

Preface	6
Contents	8
1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions	11
1.1 Introduction	11
1.2 Formulation of the Breathing Edge Crack Problem	12
1.3 Harmonic Balance Method	13
1.4 Case Study and Discussion	15
1.5 Conclusion	18
References	19
2 Enforcing Linear Dynamics Through the Addition of Nonlinearity	20
2.1 Introduction	20
2.2 Model	21
2.3 Calculation of Unforced, Undamped Response Using Nonlinear Normal Modes	22
2.4 Calculation of Forced, Damped Response Using Energy Balance	23
2.5 Enforcement of Linear Properties	24
2.5.1 Enforcing Force-Displacement Proportionality	24
2.5.2 Enforcing Straight Line Frequency Backbone	24
2.6 Beam Example	25
2.7 Conclusions	26
References	26
3 Experimental Analysis of a Softening-Hardening Nonlinear Oscillator UsingControl-Based Continuation	28
3.1 Introduction	28
3.2 Backbone Curve Identification Using CBC	29
3.2.1 Steady-State Periodic Solutions of the Forced System	29
3.2.2 Tracking the Backbone Curve	31
3.2.3 Multi-Harmonic Force Feedthrough	31
3.3 Description of the Experimental Set-Up	32
3.4 Experimental Results	33
3.4.1 Preliminary Tests	33
3.4.2 Forced Response and Backbone Curve	34
3.5 Conclusions	36
References	36
4 Experimental Nonlinear Dynamics of Laminated Quasi-Isotropic Thin Composite Plates	37
4.1 Introduction	37
4.2 Theoretical Model for Linear Free Vibrations	38
4.3 Experiment	40
4.3.1 Results	41
4.4 Conclusions and Future Work	42
References	44
5 Experimental Identification of a Structure with Internal Resonance	45
5.1 Introduction	45
5.2 Experimental Structure	46
5.3 Identification Method	48
5.3.1 Overview	48
5.3.2 Equivalent System	48
5.3.3 Parameter Identification	48
5.4 Results of Identification	49
5.4.1 ID1: Controlled Data, Sweep Near First Mode	51
5.4.2 ID2: Additional Low Level Data for Third Mode	51
5.4.3 ID3: Uncontrolled Harmonics, Only Fundamental Force Considered	51
5.4.4 ID4: Uncontrolled Harmonics, Harmonics in Force Considered	51
5.4.5 ID5: Isola Data	51
5.5 Conclusions	51
References	52
6 Shock Response of an Antenna Structure Considering Geometric Nonlinearity	54
6.1 Introduction	54
6.2 Mathematical Modeling	57
6.2.1 Equivalent Lumped Mass Model	57
6.2.2 Linear Continuous Model	58
6.2.3 Nonlinear Continuous Model	59
6.2.4 Finite Element Simulation by ANSYS	60
6.2.5 Approximate Methods	61
6.3 Case Studies	61
6.4 Conclusion	64
References	65
7 Investigation on Friction-Excited Vibration of Flexibly Supported Shafting System	67
7.1 Introduction	67
7.2 Experiment	68
7.2.1 Experimental Set Up	68
7.2.2 Experimental Results	68
7.3 Numerical Simulation	69
7.4 Conclusions	71
References	71
8 Resonant Analysis of Systems Equipped with Nonlinear Displacement-Dependent (NDD) Dampers	72
8.1 Introduction	72
8.2 Review on the NDD Damper Mechanism	74
8.3 Mathematical Formulation of the Forced-Resonant Mass-Spring-NDD Damper System	75
8.4 Forced-Resonant Vibration Analysis of the Mass-Spring-NDD Damper Using MSM	76
8.5 Numerical Examples	79
8.6 Results and Discussion	80
8.7 Conclusion	82
A.1 Appendix	83
References	85
9 Performance Comparison Between a Nonlinear Energy Sink and a Linear Tuned Vibration Absorber for Broadband Control	88
9.1 Introduction	88
9.2 Description of the Model	89
9.3 Linear Tuned Vibration Absorber	89
9.3.1 Deterministic Primary System	90
9.3.2 Uncertain Primary System	90
9.4 Theoretical Analysis of the Nonlinear Energy Sink	92
9.4.1 Order ε0	93
9.4.2 Order ε1	93
9.4.3 Detached Resonance Curve	94
9.5 Tuning of the NES	95
9.5.1 Deterministic Primary System	95
9.5.2 Uncertain Primary System	96
9.6 Performance Comparison of the NES and LTVA	97
9.6.1 Deterministic Primary System	97
9.6.2 Uncertain Primary System	99
9.7 Conclusion	100
References	100
10 Experimental and Numerical Investigation of the Nonlinear Bending-Torsion Coupling of a Clamped-Clamped Beam with Centre Masses	101
10.1 Introduction	101
10.2 Structure and Model Description	102
10.3 Linear Results	103
10.4 Nonlinear Results	105
10.5 Conclusion	107
References	109
11 Tracking of Backbone Curves of Nonlinear Systems Using Phase-Locked-Loops	111
11.1 Introduction	111
11.2 Nonlinear Modal Analysis Using the Phase Resonance Method	112
11.3 Phase Resonance Testing Using the Phase-Locked-Loop	113
11.4 Parametric Study of a Numerical Example System	116
11.4.1 Effect of Internal Resonance on PLL Measurements	119
11.5 Experimental Demonstration for a Beam with Cubic Nonlinearities	120
11.6 Conclusion and Future Work	123
References	123
12 The Importance of Phase-Locking in Nonlinear Modal Interactions	125
12.1 Introduction	125
12.2 The Second-Order Normal Form Technique	126
12.2.1 The Example System	126
12.2.2 Applying the Second-Order Normal Form Technique to the Example System	127
12.3 The Backbone Curves of the Example System	130
12.3.1 The Backbone Curves of the Asymmetric Case	131
12.3.2 The Backbone Curves of the Symmetric Case	132
12.4 Conclusions	134
References	134
13 A Study of the Modal Interaction Amongst Three Nonlinear Normal Modes Using a Backbone Curve Approach	135
13.1 Introduction	135
13.2 The Nonlinear 3-DoF Oscillators Considered and Normal Form Method Application	136
13.3 Backbone Curve Calculating	138
13.4 Backbone Results	140
13.5 Conclusions	142
References	142
14 Investigating Nonlinear Modal Energy Transfer in a Random Load Environment	144
14.1 Introduction	144
14.2 Example Structure and Linear Analysis	146
14.3 Nonlinear Model Development	148
14.3.1 Nonlinear Reduced Order Models	148
14.3.2 Full-Order Computations	149
14.3.3 NLROM Creation and Validation	151
14.4 Parameter Sweep	152
14.5 Conclusion	154
References	156
15 Nonlinear Modal Testing Performed by Pulsed-Air Jet Excitation System	157
15.1 Introduction	157
15.2 Pulsed Air Jet Excitation System	158
15.3 Design and Manufacturing of the Composite Blades	159
15.3.1 Design Considerations	159
15.4 Experimental Work	161
15.4.1 Experimental Results	162
15.4.1.1 Unidirectional: 1st Flexural Mode	162
15.4.1.2 Unidirectional: 1st Torsional Mode	162
15.4.1.3 Unidirectional: 2nd Flexural Mode	163
15.4.2 Free Decay Procedure and Signal Processing	163
15.4.3 Free Decay Experimental Results	165
15.4.3.1 Unidirectional: 1st Flexural Mode	165
15.4.3.2 Unidirectional: 1st Torsional Mode	167
15.4.3.3 Unidirectional: 2nd Flexural Mode	169
15.5 Conclusions	169
References	171
16 EMA-FEA Correlation and Updating for Nonlinear Behaviour of an Automotive Heat-Shield	173
16.1 Introduction	173
16.2 Experimental Setup	174
16.3 Numerical Model	176
16.4 OFAT on FE Model and Validation	179
16.4.1 Model 2	179
16.4.2 Model 3	180
16.4.3 Model 4	180
16.4.4 Model 5	182
16.5 Final Model Updating and FRF Validation	182
16.6 Conclusions	183
References	185
17 Tutorial on Nonlinear System Identification	186
17.1 Introduction	186
Scope of the Presentation	188
References	189
18 Higher-Order Frequency Response Functions for Hysteretic Systems	191
18.1 Introduction	191
18.2 Bouc-Wen Model of Hysteresis	192
18.3 Removing the Hidden State from the Bouc-Wen Hysteretic Model	192
18.4 Higher-Order Frequency Response Functions for the Bouc-Wen Hysteretic Model	193
18.4.1 Volterra Series Representation of Nonlinear Systems	193
18.4.2 HFRFs for Bouc-Wen Models	193
18.4.3 Example Bouc-Wen HFRFs	197
18.5 Switching Between Higher-Order Frequency Response Functions for Non-Smooth Systems or Systems with Memory	199
18.6 Discussion and Conclusions	200
References	201
19 Model Upgrading T0 Augment Linear Model Capabilities into Nonlinear Regions	202
19.1 Introduction	202
19.1.1 Review on Nonlinear Identification Methods	203
19.2 Identification Process of Nonlinear System	204
19.3 Modular Framework and Upgrading Approach	204
19.4 The Multiple Beam Structure	205
19.4.1 Experimental Testing	205
19.4.2 Linear FE Model Validation	206
19.5 FE and EMA Correlation	209
19.6 Nonlinear Identification	209
19.6.1 Detection	210
19.6.2 Location and Characterisation	210
19.6.3 Quantification	211
19.7 Model Upgrading	212
19.8 Model Updating	213
19.9 Nonlinear Model Validation	215
19.10 Conclusion	215
References	216
20 Obtaining Nonlinear Frequency Responses from Broadband Testing	217
20.1 Introduction	217
20.2 Frequency-Domain Nonlinear Subspace Identification	218
20.2.1 Identification Problem Formulation	218
20.2.2 The Output-State-Input Equation	219
20.2.3 Estimation of the State Matrices and the Order of the System	219
20.3 Harmonic Balance-Based Continuation Method	220
20.3.1 Harmonic Balance Formulation	220
20.3.2 Continuation of Periodic Solutions	221
20.4 Numerical Example	222
20.5 Conclusion	224
References	225
21 Experimental Study of Isolated Response Curves in a Two-Degree-of-Freedom Nonlinear System	226
21.1 Introduction	226
21.2 An Experimental Two-Degree-of-Freedom System with Hardening Springs	226
21.3 Experimental Forced Responses	228
21.4 Conclusions	230
References	232
22 Nonlinear Response of a Thin Panel in a Multi-Discipline Environment: Part I—Experimental Results	233
22.1 Introduction & Background	233
22.2 Experimental History and Overview	236
22.3 Preliminary Results and Discussion	239
22.4 Conclusions and Future Work	243
References	244
23 Nonlinear Dynamic Response Prediction of a Thin Panel in a Multi-Discipline Environment: Part II—Numerical Predictions	245
23.1 Introduction	245
23.2 Computational Model	246
23.2.1 Finite Element Model	246
23.2.2 ROM Formulation	247
23.2.3 ROM of RC-19 Panel	247
23.3 PSP Data Post-Processing	248
23.4 Simulation of Structural Response	249
23.5 Summary	251
References	253
24 Stability Analysis of Curved Panels	254
24.1 Introduction	254
24.2 Static Stability Analysis	255
24.2.1 Numerical Procedure	255
24.2.2 Benchmark Panel	256
24.2.3 Orthogrid Panels	257
24.3 Dynamic Analysis	259
24.4 Concluding Remarks	259
References	260
25 Optimal Representation of a Varying Temperature Field for Coupling with a Structural Reduced Order Model	262
25.1 Introduction	262
25.2 Structural-Thermal Nonlinear Reduced Order Modeling	263
25.3 Optimum Thermal Modes	264
25.4 Validation Example	266
References	272
26 Basis Identification for Nonlinear Dynamical Systems Using Sparse Coding	274
26.1 Introduction and Problem Statement	274
26.2 Method of Solution	275
26.2.1 Full Order Models	275
26.2.2 Generation of the Reduced Order Modes	276
26.2.2.1 Proper Orthogonal Decomposition	276
26.2.2.2 Sparse Coding	276
26.2.2.3 Galerkin Projection	277
26.2.2.4 Energy Balance Equations	278
26.3 Results	279
26.3.1 Comparison Between POD and Sparse Modes	279
26.3.2 Predicted Instantaneous TKE	280
26.3.3 Energy Balance Analysis	281
26.4 Conclusions	283
References	287
27 Interaction Between Aerothermally Compliant Structures and Boundary Layer Transition	289
27.1 Introduction	289
27.2 Methodology	290
27.2.1 Aerothermoelastic Model	290
27.2.2 Transitional Aerothermodynamic Loads	291
27.2.2.1 Intermittency	291
27.2.2.2 Heat Flux	292
27.2.2.3 Fluctuating Pressure	293
27.2.3 Problem Description	294
27.3 Results and Analysis	295
27.4 Conclusions and Future Work	296
References	297
28 Simultaneous Vibration Isolation and Energy Harvesting: Simulation and Experiment	299
28.1 Introduction and Background	299
28.2 Theory	300
28.2.1 Static Equilibrium	300
28.2.2 Dynamic Equations of Motion	301
28.2.3 Electrical Modeling	302
28.3 Experiment	303
28.3.1 Description of Experimental Set-Up	303
28.3.2 Empirically-Derived Parameters	303
28.4 Results and Discussion	305
28.4.1 Vibration Isolation	305
28.4.2 Energy Harvesting	305
28.4.3 Design Considerations	306
28.5 Conclusions	309
References	309
29 Nonlinear Dynamic Interaction in a Coupled Electro-Magneto-Mechanical System: Experimental Study	310
29.1 Introduction	310
29.2 Experimental Setup	310
29.3 Multi-Physics Coupled Oscillators Model	311
29.4 Experimental Tests	312
29.5 Conclusions	315
References	315
30 Hysteresis Identification Using Nonlinear State-Space Models	316
30.1 Introduction	316
30.2 Synthetic Generation of Hysteretic Data	317
30.2.1 The Bouc-Wen Model of Hysteresis	317
30.2.2 Excitation Signal	318
30.2.3 Time Integration	319
30.2.4 Noise Assumptions	320
30.3 Nonparametric Analysis of Nonlinear Distortions	320
30.4 Nonlinear State-Space Identification	321
30.4.1 The Polynomial Nonlinear State-Space Model Structure	322
30.4.2 Identification Methodology	322
30.4.2.1 Initial Linear Model	322
30.4.2.2 Full Nonlinear Model	323
30.4.3 Identification Results	323
30.5 Model Validation Under Sine-Sweep Excitations	327
30.6 Conclusions	329
References	330
31 Nonholonomically Constrained Dynamics of Rolling Isolation Systems	332
31.1 Introduction	332
31.2 The Model	332
31.3 Gauss's Principle of Least Constraint	333
31.4 The Unconstrained Dynamics	333
31.5 The Constraints and the True Equations of Constrained Motion	336
31.6 Simulation	337
31.7 Conclusion	339
References	339
32 Parameter Estimation on Nonlinear Systems Using Orthogonal and Algebraic Techniques	340
32.1 Introduction	340
32.2 Nonlinear Vibrating Mechanical System	341
32.3 Nonlinearity Index	343
32.4 Some Illustrative Cases	344
32.5 Conclusions	346
References	347
33 Online State and Parameter Estimation of a Nonlinear Gear Transmission System	348
33.1 Introduction	348
33.2 Class of Mechanical Systems Examined: Equations of Motion	349
33.3 Review of Unscented Kalman Filter Formulation for State and Parameter Estimation	350
33.4 Application to Experimental Gearbox Simulator	352
33.5 Results	353
33.6 Conclusions	356
References	356
34 Model Updating of a Nonlinear System: Gun Barrel of a Battle Tank	358
34.1 Introduction	358
34.2 Theory	359
34.3 Experimental Study	360
34.3.1 Experimental Setup	361
34.3.2 First Set of Experiments	361
34.3.2.1 Application of PRD Method at Scaled Frequency 1 Hz for Identifying Nonlinearity	364
34.3.2.2 Application of PRD Method at Scaled Frequency 0.933 Hz for Identifying Nonlinearity	367
34.3.2.3 Application of PRD Method at Scaled Frequency 0.95 Hz for Identifying Nonlinearity	368
34.3.2.4 Application of PRD Method at Scaled Frequency 0.966 Hz for Identifying Nonlinearity	368
34.3.2.5 Application of PRD Method at Scaled Frequency 0.983 Hz for Identifying Nonlinearity	368
34.3.2.6 Comparison of Nonlinearities Identified by Using PRD Method at Different Frequencies	369
34.3.3 Second Set of Experiments	370
34.3.4 Application of PRD Method for Obtaining Linear FRFs	371
34.4 Model Updating of the Test System and Verification of the Updated Model	372
34.4.1 FE Modeling of the Gun Barrel	373
34.4.2 Mesh Sensitivity Analysis for the FE Model of Gun Barrel	374
34.4.3 Model Updating of the FE Model of Gun Barrel System	375
34.5 Summary and Conclusions	378
References	380
35 Experimental Passive Flutter Mitigation Using a Linear Tuned Vibrations Absorber	381
35.1 Introduction	381
35.2 Primary System	382
35.2.1 Experimental Setup	382
35.2.2 Reduced Order Model	383
35.3 Aeroelastic Analysis	385
35.4 LTVA Design	387
35.4.1 LTVA Equations of Motion	387
35.4.2 Experimental LTVA Design and Identification	388
35.5 Numerical and Experimental Aeroelastic Analysis of the Primary System Coupled with the LTVA	389
35.5.1 Effect of the LTVA on the Flutter Speed	389
35.5.2 Effect of the LTVA on the Subcritical Response of the System	391
35.5.3 Effect of the LTVA on the Supercritical Response of the System	391
35.6 Conclusions	394
References	394
36 Adaptive Harmonic Balance Analysis of Dry Friction Damped Systems	396
36.1 Introduction	396
36.2 Adaptive Harmonic Balance Analysis	398
36.2.1 Transformation of Harmonics	398
36.2.2 Criteria for Selecting Harmonics	399
36.2.2.1 Approach 1: Estimation of Response Displacement Harmonics	399
36.2.2.2 Approach 2: Estimation of Partial Derivatives	400
36.3 Numerical Results	401
36.4 Conclusions	403
References	404
37 Dynamics of an MDOF Rotor Stator Contact System	406
37.1 Introduction	406
37.2 Synchronisation as the Cause of Bouncing Orbits	407
37.3 Test System	408
37.4 Comparison with Simulation	408
37.4.1 Light Forcing	408
37.4.2 Moderate Forcing	410
37.4.3 Harder Forcing	410
37.5 Conclusions and Future Work	412
References	414

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