Nonlinear Structures and Systems, Volume 1

Preface	6
Contents	7
1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint	10
1.1 Introduction	10
1.2 Non-internally Resonant Elastic Bar	11
1.3 Periodic Solutions and Nonsmooth Modal Analysis	13
1.4 Numerical Scheme	14
1.5 Spectrum of Nonsmooth Vibration	16
1.6 Response to Periodic External Forcing	17
1.7 Conclusions	19
References	19
2 A New Iwan/Palmov Implementation for Fast Simulation and System Identification	20
2.1 Introduction	20
2.2 Understanding the Algorithm	21
2.2.1 When the External Force is Non-zero	21
2.2.2 When the External Force is Zero	22
2.3 Application to Single Degree of Freedom System	23
2.3.1 Results and Analysis	24
Accuracy	24
Speed	28
2.4 Conclusions	28
References	29
3 Analysis of Transient Vibrations for Estimating Bolted Joint Tightness	30
3.1 Introduction	30
3.2 Background	31
3.3 Analysis	32
3.4 Conclusion	33
References	33
4 Spider Configurations for Models with Discrete Iwan Elements	34
4.1 Introduction	34
4.2 Modeling Approach	35
4.2.1 Hurty/Craig Bampton Reduction	35
4.2.2 Spidering	36
4.2.3 Whole-Joint Model	36
4.2.4 Linear Model Updating	37
4.2.5 Nonlinear Model Updating	37
4.3 Application to S4 Beam: Linear Updating	38
4.3.1 Single Beam Calibration	39
4.3.2 Whole-Joint Spring Calibration	39
4.4 Nonlinear Model Updating	42
4.5 Conclusion	46
Appendix: Full Interface RBAR Springs for all Models	46
References	47
5 Predicting S4 Beam Joint Nonlinearity Using Quasi-Static Modal Analysis	48
5.1 Introduction	48
5.2 Background	50
5.3 Model Creation	51
5.3.1 Bolt Preload	52
5.3.2 Modeling Surface Curvature	52
5.4 Test Cases	52
5.4.1 Solver Tolerances	54
5.4.2 Bolt Preload	54
5.4.3 Surface Contours	56
5.5 Conclusion and Future Work	58
References	60
6 The Best Linear Approximation of MIMO Systems: First Results on Simplified Nonlinearity Assessment	61
6.1 Introduction	61
6.2 Basics	62
6.3 Design of Experiment	62
6.3.1 Multisine Excitation and Detection of Nonlinearities	62
6.3.2 Multisines for Multiple Input Measurements	63
6.4 Best Linear Approximation	64
6.4.1 Theoretical Structure and the Basic Assumptions	64
6.4.2 Two-Dimensional Averaging	64
6.5 Measurement	65
6.6 Analysis	66
6.7 Conclusion	71
References	72
7 Forced Response of Nonlinear Systems Under Combined Harmonic and Random Excitation	73
Nomenclature	73
7.1 Introduction	74
7.2 The Fokker–Planck Equation	74
7.3 Galerkin-Type Method for Combined Harmonic and Random Excitation	75
7.4 Orthogonal Time-Dependent Correction Functions	78
7.5 Application	79
7.5.1 Duffing-Oscillator	80
7.5.2 Duffing-van der Pol-Oscillator	81
7.5.3 3-DOF-System	83
7.5.4 20-DOF Model of a Bladed Disk Assembly	84
7.6 Problems and Limitations	85
7.7 Conclusion and Outlook	87
References	87
8 Gerrymandering for Interfaces: Modeling the Mechanics of Jointed Structures	89
8.1 Introduction	89
8.2 Model Description	90
8.2.1 ZTE Meshes	90
8.2.2 Quasi Static Modal Analysis	91
8.3 Results	91
8.4 Conclusion	92
References	93
9 An Analysis of the Gimballed Horizontal Pendulum System for Use as a Rotary Vibrational Energy Harvester	94
9.1 Introduction	94
9.2 Analysis	94
9.3 Conclusion	96
References	96
10 On the Dynamic Response of Flow-Induced Vibration of Nonlinear Structures	97
10.1 Introduction	97
10.2 Experimental Setup	98
10.3 Results	98
10.4 Conclusion	100
References	100
11 Potential and Limitation of a Nonlinear Modal Testing Method for Friction-Damped Systems	101
11.1 Introduction and Method	101
11.2 Beam Test Rig with Friction Nonlinearity	102
11.3 Experimental Nonlinear Modal Analysis	102
11.4 Conclusion	103
References	103
12 Dynamics of a Magnetically Excited Rotational System	104
12.1 Introduction	104
12.2 System Description	104
12.3 Equilibria and Stabilities	105
12.4 Dynamic Response	106
12.5 Conclusion	107
References	107
13 Experimental Nonlinear Dynamics of a Post-buckled Composite Laminate Plate	108
13.1 Introduction	108
13.2 Pre-buckled Analysis	109
13.2.1 Analytical Solution	109
13.2.2 Experimentation	110
13.2.3 Comparison of Analytical Solution and Experimental Results	111
13.3 Post-buckled Analysis	112
13.3.1 Experimentation	113
13.3.2 Asymmetric Buckle	114
13.3.3 Chaotic Snap Through	115
13.3.4 Low Amplitude Modal Sweeps	115
13.4 Summary and Future Work	119
References	119
14 Simulation of a Self-Resonant Beam-Slider-System Considering Geometric Nonlinearities	120
14.1 Introduction	120
14.2 Natural Frequency of Clamped-Clamped Beam with Bonded Slider	120
14.3 Self-Resonant Process	121
14.4 Conclusion	122
References	123
15 Reinforcement Learning for Active Damping of Harmonically Excited Pendulum with Highly Nonlinear Actuator	124
15.1 Introduction	124
15.2 Methods	125
15.3 Results	125
15.4 Conclusion	128
References	128
16 Investigation of Nonlinear Dynamic Phenomena Applying Real-Time Hybrid Simulation	129
16.1 Introduction	129
16.2 Nonlinear Vibrations in Hybrid Simulation	130
16.3 Control of Transfer System	131
16.4 Real-Time Hybrid Testing of a Duffing Oscillator	133
16.5 Conclusion	134
References	135
17 Experimental and Numerical Aeroelastic Analysis of Airfoil-Aileron System with NonlinearEnergy Sink	136
17.1 Present Work	136
References	138
18 On the Modal Surrogacy of Joint Parameter Estimates in Bolted Joints	139
18.1 Introduction	139
18.2 Methodology	140
18.3 Results and Conclusions	142
References	142
19 Vehicle Escape Dynamics on an Arbitrarily Curved Surface	143
19.1 Introduction	143
19.2 Background	143
19.3 Models	144
19.4 Methods	146
19.5 Results	147
19.6 Conclusion	149
19.7 Future Work	150
References	150
20 Nonlinear Dynamical Analysis for Coupled Fluid-Structure Systems	152
20.1 Introduction	152
20.2 Description of the Computational Model	152
20.3 Numerical Application	153
20.4 Conclusions	154
References	154
21 Experimental Nonlinear Vibration Analysis of a Shrouded Bladed Disk Model on a Rotating Test Rig	155
21.1 Introduction	155
21.2 Bladed Disk Model	156
21.3 Rotating Test Rig	157
21.4 Excitation Mechanism	157
21.5 Experimental Results	159
21.6 Conclusions	163
References	163
22 The Measurement of Tangential Contact Stiffness for Nonlinear Dynamic Analysis	164
22.1 Introduction	164
22.2 Oxford Rig	164
22.3 Imperial College Rig	165
22.4 Comparison of Tangential Contact Stiffness	165
22.5 Discussion and Conclusions	166
References	166
23 Investigating Nonlinearity in a Bolted Structure Using Force Appropriation Techniques	167
23.1 Approach	167
23.2 Results	168
References	169
24 Techniques for Nonlinear Identification and Maximizing Modal Response	170
24.1 Introduction	170
24.2 Theory	171
24.2.1 Nonlinear Pseudo-Modal Model	171
24.2.2 Modal Filter Theory	172
24.2.3 Experimental Applied Force	172
24.3 Experimental Set-Up	174
24.3.1 Hardware Description	174
24.3.2 Linear Model Extraction	174
24.4 Nonlinear Modal Model Extraction Overview	175
24.5 Experimental Results to Modified Windowed Sinusoidal Inputs	176
24.5.1 Experimental Results from Cases A–D	177
24.5.2 Experimental Results from Cases E–F	179
24.6 Experimental Results from Cases E and H	180
24.7 Model Comparison	181
24.8 Conclusions and Future Work	183
References	185
25 Influences of Modal Coupling on Experimentally Extracted Nonlinear Modal Models	186
25.1 Introduction	186
25.2 Experimental Theoretical Background	187
25.2.1 Pseudo – Modal Models	187
25.2.2 Hilbert Analysis	188
25.3 Experimental Methodology	188
25.3.1 Test Structure	188
25.3.2 Linear Testing	189
25.3.3 Nonlinear Shaker Testing	189
25.3.4 Shaker Experimental Results	191
25.3.5 Nonlinear Hammer Testing	194
25.4 Conclusions	196
*-30pt	198
Appendix B—Additional Hammer Data Plots	200
References	201
26 Dynamic Response of a Curved Plate Subjected to a Moving Local Heat Gradient	202
26.1 Introduction	202
26.2 Experimental Setup	203
26.2.1 Introduction	203
26.2.2 Thermal Loading	203
26.3 Results	203
26.3.1 Quasi-Static Response	203
26.3.2 Dynamic Response	204
26.4 Future Work	205
References	205
27 A Test-Case on Continuation Methods for Bladed-Disk Vibration with Contact and Friction	206
27.1 Introduction	206
27.2 Methodology and Analysis	207
27.2.1 Framework	208
27.2.2 Preliminary Results	208
27.3 Conclusion	208
References	208
28 Dynamics of Geometrically-Nonlinear Beam Structures, Part 1: Numerical Modeling	210
28.1 Introduction	210
28.2 Background	211
28.3 Results	211
28.4 Conclusion	212
References	213
29 Dynamics of Geometrically-Nonlinear Beam Structures, Part 2: Experimental Analysis	214
29.1 Introduction	214
29.2 Beam Under Moderately Large Lateral Oscillations	215
29.3 Nonlinear System Identification	215
29.4 Experimental Tests and Results	215
29.5 Conclusion	217
References	217
30 Constructing Backbone Curves from Free-Decay Vibrations Data in Multi-Degrees of Freedom Oscillatory Systems	218
30.1 Introduction	218
30.2 Method Overview and Numerical Example	219
30.3 Conclusion	220
References	220
31 Nonlinear 3D Modeling and Vibration Analysis of Horizontal Drum Type Washing Machines	221
31.1 Introduction	221
31.2 Analysis	222
31.3 Conclusion	223
References	224
32 Comparison of Linear and Nonlinear Modal Reduction Approaches	225
32.1 Harmonic Vibration Analysis of Nonlinear MDOF Systems	225
32.2 Case Study	226
32.3 Conclusions	228
References	228
33 Reduced Order Modeling of Bolted Joints in Frequency Domain	230
33.1 Introduction	230
33.2 Theory	231
33.2.1 Nonlinear Static Analysis of Brake-Reuß Beam	231
33.2.2 Representative High Fidelity Model	231
33.2.3 Reduced Order Joint Model	231
33.3 Results and Discussion	232
References	233
34 Comparison of ANM and Predictor-Corrector Method to Continue Solutions of Harmonic Balance Equations	234
34.1 Introduction	234
34.1.1 Alternating-Frequency-Time Scheme with Predictor-Corrector Continuation (AFT-PreCo)	235
34.1.2 Classical Harmonic Balance with Asymptotic Numerical Method (cHB-ANM)	235
34.2 Results	236
34.3 Conclusions	237
References	237
35 A Priori Methods to Assess the Strength of Nonlinearities for Design Applications	238
35.1 Introduction	238
35.2 Defining Strength of Nonlinearity	239
35.3 Parameter Extractions	239
35.3.1 Amplitude Dependent Frequency and Damping	239
35.3.2 Contact Pressure and Modal Strain	240
35.4 Correlation of Numerical and Experimental Parameters	240
35.5 Conclusion	241
References	241
36 Predictive Modeling of Bolted Assemblies with Surface Irregularities	242
36.1 Introduction	242
36.2 System Description	243
36.3 Single Beam Structure	244
36.3.1 Experimental Modal Analysis	244
36.3.2 Model Updating	245
36.3.2.1 Density	245
36.3.2.2 Young's Modulus	245
36.3.2.3 Surface Geometry	246
36.4 Bolted Beam Assembly	246
36.4.1 Experimental Modal Analysis	247
36.4.2 Computational Preload Analysis	247
36.4.3 Modal Analysis Validation	250
36.5 Conclusions	252
References	253
37 A Novel Computational Method to Calculate Nonlinear Normal Modes of Complex Structures	254
37.1 Introduction	254
37.2 Theory	255
37.2.1 Describing Function Method	255
37.2.2 Dynamic Stiffness Formulation to Obtain NNMs	255
37.3 Results	256
37.4 Conclusion	256
References	257
38 Experimental-Numerical Comparison of Contact Nonlinear Dynamics Through Multi-level Linear Mode Shapes	258
38.1 Introduction	258
38.2 Experimental Set-Up	259
38.3 Multi-Phi Method	261
38.4 Comparison Between Prediction and Experimental Data	262
38.5 Conclusion	265
References	265
39 Dynamic Behavior and Output Charge Analysis of a Bistable Clamped-Ends Energy Harvester	267
39.1 Introduction and Model Description	267
39.2 Results and Discussion	269
39.3 Conclusion	271
References	271

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