Nonlinear Dynamics, Volume 2

Preface	6
Contents	8
1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System	10
1.1 Introduction	10
1.2 The Experimental System	11
1.3 Techniques of Investigation	12
1.3.1 Time Lag Embedding	12
1.3.2 Poincaré Section	13
1.3.3 Stochastic Interrogation	13
1.4 Results	14
1.4.1 A First Look	14
1.4.2 Basins of Attraction	15
1.5 Conclusions	17
References	18
2 Complex Behavior of a Buckled Beam Under Combined Harmonic and Random Loading	19
2.1 Introduction	19
2.2 The Double-Well Duffing Equation	20
2.3 The Post-Buckled Beam	23
2.3.1 Results	24
2.4 Conclusions	25
References	26
3 The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems	27
3.1 Introduction	27
3.2 Harmonic Balance Method	28
3.2.1 Analytical Expression of the Nonlinear Terms and of the Jacobian Matrix of the System	29
3.2.2 Continuation Procedure	31
3.2.3 Stability Analysis	31
3.2.4 Detection and Tracking of Bifurcations	32
3.3 Validation of the Method on the Study of an Industrial, Complex Model with Strong Nonlinearities: The Smallsat	33
3.3.1 Case Study	33
3.3.2 Study of the Normal Modes	34
3.3.3 Study of the Forced Response	36
3.4 Bifurcation Tracking Coupled to the Harmonic Balance Method	37
3.4.1 Case Study	37
3.4.2 Tracking of LP and NS Bifurcations	38
3.5 Conclusions	41
References	41
4 Frequency Response Calculations of a Nonlinear Structure a Comparison of Numerical Methods	43
4.1 Introduction	43
4.2 Numerical Methods	44
4.2.1 The Fourth-Order Runge–Kutta Method	44
4.2.2 The Newmark Method for Nonlinear Problems	45
4.2.3 Pseudo Force in State Space	45
4.2.4 Multi-Harmonic Balance Method	46
4.3 Comparison of Numerical Results	48
4.3.1 Cantilever Beam with Cubic Spring at the End	48
4.4 Conclusion	50
References	52
5 A Framework for the Computational Dynamic Analysis of Coupled Structures Using Nonlinear Modes	53
5.1 Introduction	53
5.2 Equation of Motion and Dynamic Regime of Interest	54
5.3 Nonlinear Modal Analysis	54
5.4 Reduced Order Modeling Based on Nonlinear Modes	55
5.5 Selected Numerical Examples	56
5.5.1 Two-Degree-of-Freedom Oscillator with Friction Contact	56
5.5.2 Turbine Bladed Disk with Shroud Contact	58
5.6 Conclusions	60
References	60
6 Subspace and Nonlinear-Normal-Modes-Based Identification of a Beam with Softening-Hardening Behaviour	62
6.1 Introduction	62
6.2 Subspace Identification of Nonlinear Mechanical Systems in the Frequency Domain	63
6.2.1 Nonlinear Model Equations in the Physical Space	64
6.2.2 Feedback Interpretation of Nonlinear Structural Dynamics and State-Space Model	64
6.2.3 Formulation of an Output-State-Input System Equation	65
6.2.4 Determination of the System Order and of the Extended Observability Matrix	66
6.2.5 Estimation of the System Matrices	67
6.2.6 Conversion from Discrete-Time State Space to Continuous-Time Physical Space	67
6.3 Nonlinear Normal Modes	68
6.3.1 Framework and Definitions	68
6.3.2 Frequency-Energy Dependence	68
6.3.3 Numerical Computation	69
6.3.4 Experimental Evaluation	69
6.4 Experimental Tests Description	69
6.5 Frequency Nonlinear Subsystem Identification of the Beam	71
6.6 NNM-Based Identification of the Beam	73
6.7 Concluding Remarks	74
References	75
7 Model Updating of Nonlinear Structures	76
7.1 Introduction	76
7.2 Theory	77
7.3 Application of the Proposed Model Updating Approach to Simulated Test System	80
7.4 Discussion and Conclusions	87
References	87
8 Detection of Nonlinear Behaviour of Composite Components Before and After Endurance Trials	89
8.1 Introduction	89
8.2 Endurance Trials	90
8.2.1 Modal Testing of Pristine Components	91
8.2.2 Endurance Results	91
8.3 Quantification of Non-linear Behavior of Composites	92
8.3.1 Testing and Data Analysis Using Reverse Path Method	93
8.3.2 Test Results	93
8.3.3 Testing and Data Analysis Using Linearization Methods	95
8.3.4 Test Results	97
8.4 Discussion	98
8.5 Conclusions	100
References	101
9 Model Calibration of a Locally Non-linear Structure Utilizing Multi Harmonic Response Data	102
9.1 Introduction	102
9.2 Experimental Set-up	103
9.3 Computational Model	104
9.3.1 Modeling and Calibration of Supporting Frame	104
9.4 Nonlinear Calibration of Test Object	106
9.4.1 Nonlinear Model	106
9.4.2 Identifiability and Re-parameterization	107
9.4.3 Nonlinear Calibration Results	107
9.4.3.1 Nonlinear Calibration Result for the Case 1	109
9.4.3.2 Nonlinear Calibration Result for Case 2	110
9.5 Conclusion	112
References	114
10 Nonlinear Time Series Analysis Using Bayesian Mixture of Experts	115
10.1 Introduction	115
10.2 Variational Bayes Expectation Maximisation	116
10.2.1 The VBE and VBM Steps	116
10.3 Mixture of ARX Experts Model	117
10.3.1 Priors	118
10.3.2 Variational Inference	118
10.3.3 Variational Lower Bound	120
10.3.4 Predictive Distribution	120
10.4 Results	121
10.4.1 Case 1: Known nu and ny	121
10.4.2 Case 2: Unknown nu and ny	124
10.5 Conclusions	124
References	125
11 Identifying Robust Subspaces for Dynamically Consistent Reduced-Order Models	126
11.1 Introduction	126
11.2 Background	126
11.3 Model Reduction Framework and Theory	127
11.3.1 Model Reduction Subspace Selection Criteria	127
11.3.2 Smooth Subspace	128
11.3.3 Smooth Orthogonal Decomposition	129
11.3.4 Proper Orthogonal Decomposition	129
11.3.5 Lower-Dimensional Model	129
11.4 Forced Cantilever Beam in a Double-Well Potential Field	130
11.5 Discussion	130
References	132
12 Identification of Sub- and Higher Harmonic Vibrations in Vibro-Impact Systems	134
12.1 Introduction	134
12.2 Vibro-Impact System Representation	135
12.3 Harmonic Balance Method for Sub- and Higher Harmonic Response	135
12.3.1 Solution Method	137
12.4 Chaotic Motion	138
12.5 Results	139
12.6 Conclusion and Future Work	141
References	143
13 An Efficient Simulation Method for Structures with Local Nonlinearity	144
13.1 Introduction	144
13.2 Theory	145
13.3 Case Studies	146
13.3.1 A 3DOF System	147
13.3.1.1 Nonlinearity Between Two Masses	147
13.3.1.2 Nonlinearity Affecting Only One Mass	148
13.3.2 A clamped Beam with Gap Nonlinearity	148
13.4 Discussion on Numerical Results	150
13.5 Conclusions	152
References	152
14 Parametric Nonlinearity Identification of a Gearbox from Measured Frequency Response Data	153
14.1 Introduction	153
14.2 Theory	154
14.3 Experimental Study	155
14.3.1 Static Experiment	155
14.3.2 Dynamic Experiment	156
14.4 Conclusion	157
References	158
15 Nonlinear Gear Transmission System Numerical Dynamic Analysis and Experimental Validation	160
15.1 Introduction	160
15.2 Class of Mechanical Systems Examined: Equations of Motion	161
15.3 Review of Bayesian Formulation for Parameter Estimation and Model Class Selection	163
15.4 Application to Experimental Gearbox Simulator	164
15.5 Results	166
15.6 Conclusions	167
References	168
16 A Stochastic Framework for Subspace Identification of a Strongly Nonlinear Aerospace Structure	169
16.1 Introduction	169
16.2 Nonlinear Model Equations and Problem Statement	170
16.3 Generating Initial Parameter Estimates Using a Nonlinear Subspace Identification Method in the Frequency Domain	170
16.3.1 Feedback Interpretation of Nonlinear Structural Dynamics and State-Space Model	170
16.3.2 Conversion from Discrete-Time State Space to Continuous-Time Physical Space	172
16.4 Optimising Initial Parameter Estimates in the Maximum Likelihood Framework	172
16.5 Demonstration on a Single-Degree-of-Freedom Duffing Oscillator	173
16.6 Application to a Full-Scale Strongly Nonlinear Satellite Structure	175
16.6.1 Description and Modelling of the SmallSat Spacecraft	175
16.6.2 Activation of a Single Nonlinearity of the WEMS Device	178
16.7 Conclusions	180
References	182
17 Composite Non-Linearity in High Cycle Fatigue Experimentation	183
17.1 Introduction	183
17.2 Methodology	184
17.3 Results	185
17.4 Conclusions	188
References	188
18 A Procedure to Identify the Handling Characteristics of Agricultural Tyre Through Full-Scale Experimental Tests	190
18.1 Introduction	191
18.2 Experimental Tests	191
18.3 Tyre Model	192
18.4 Vehicle Model	193
18.5 Identification Procedure	195
18.6 Results	197
18.7 Concluding Remarks	198
References	199
19 Nonparametric Analysis and Nonlinear State-Space Identification: A Benchmark Example	201
19.1 Introduction	201
19.1.1 Paper Outline	202
19.2 Nonparametric Identification	202
19.2.1 Periods: Noise Level Estimation and Other Benefits	202
19.2.2 Multisines and Their Advantages	202
19.2.3 The Fast Way: Measuring Several Periods of One Multisine Schoukens2005,schoukens2012exercices	203
19.2.4 The Robust Way: Measuring Several Realisations and Periods Schoukens2005,schoukens2012exercices	203
19.2.5 The Fast and Robust Methods Combined	204
19.3 Parametric Identification	204
19.3.1 Parametric BLA	204
19.3.2 Nonlinear State-Space Models	205
19.3.2.1 Advantages and Drawbacks	205
19.3.2.2 The General PNLSS Model	205
19.3.2.3 The Rank-Limited PNLSS Model	205
19.3.2.4 The Nonlinear LFR Model	206
19.4 Benchmark Application	206
19.4.1 Data Generation	206
19.4.2 Nonparametric Identification	207
19.4.2.1 Discussion	207
19.4.3 Parametric Identification	208
19.4.3.1 Numerical Estimation Results	208
19.4.3.2 Estimation Results: Discussion	209
19.4.3.3 Validation	209
19.4.3.4 Possible Pitfalls/ Possible Improvements	210
19.5 Conclusions	211
References	212
20 Nonlinear Black-Box Identification of a Mechanical Benchmark System	213
20.1 Introduction	213
20.2 The Considered Nonlinear Model	214
20.2.1 Description	214
20.2.2 Identification	215
20.3 The Simulations: Data and Results	215
20.3.1 Properties	215
20.3.2 Estimation and Validation Results	216
20.4 Conclusions	217
References	220
21 Suppression of Multiple Order Friction Torque Fluctuations with Modulated Actuation Pressure	221
21.1 Introduction	221
21.2 Problem Formulation	222
21.3 Pressure Modulation Experiment and Experimental Observations	222
21.4 Nonlinear Mathematical Model of the Pressure Modulation Experiment	225
21.5 Conclusion	227
References	228
22 Two-Dimensional Nonlinear Dynamics of Axially Accelerating Beam Based on DQM	229
22.1 Introduction	229
22.2 Governing Equation	230
22.3 Differential Quadrature Analogs	231
22.4 Numerical Results	233
22.5 Conclusion	235
References	237
23 Nonlinear Structural Coupling: Experimental Application	238
23.1 Introduction	238
23.2 Theory	239
23.3 Experimental Study and Verification	240
23.3.1 Experimental Setup	240
23.3.2 Computation of Nonlinear FRFs for the Test Structure	241
23.3.3 Comparison of Calculated and Measured Nonlinear FRFs	243
23.4 Discussion and Conclusions	243
References	244
24 State Estimation in Nonlinear Structural Systems	246
24.1 Introduction	246
24.2 Theoretical Background	247
24.3 Damage Quantification Model	248
24.4 Numerical Simulations	248
24.4.1 System and Model Description	248
24.4.2 Implementation of Nonlinear Filters	250
24.4.3 CASE 1: Simulation with No Model Class Error	250
24.4.4 CASE 2: Simulation with Model Class Error	251
24.4.5 CASE 3: Simulation with Model Class Error and Suboptimal Model	252
24.5 Conclusion	252
References	253
25 An Efficient Exponential Integrator for Large Nonlinear Stiff Systems Part 1: Theoretical Investigation	255
25.1 Introduction	255
25.2 Theoretical Background	256
25.3 Proposed Integration Scheme	258
25.3.1 Discretization and Numerical Integration	258
25.3.2 Jacobian Approximation for Large-Scale Systems	259
25.3.2.1 Reduction Based on a Specific Frequency Bandwidth and Modal Correlation Matrix	260
25.3.2.2 Reduction Based on the Modal Contribution to the I/O Relation	260
25.4 Conclusion	262
References	263
26 An Efficient Exponential Integrator for Large Nonlinear Stiff Systems Part 2: Symplecticity and Global Error Analysis	265
26.1 Introduction	265
26.2 Structure-Preserving Integration and Hamiltonian Dynamical Systems	266
26.2.1 Hamiltonian Dynamical Systems	266
26.2.2 Symplectic Structure Preservation	267
26.2.3 Energy and Momentum Conservation	268
26.2.4 Splitting Integration Schemes	268
26.2.4.1 Leapfrog Method	269
26.2.4.2 Modified Leapfrog for Viscously Damped Systems	269
26.3 Symplecticity and Global Error Analysis of the Presented Exponential Integrator	270
26.3.1 A Conservative Example: Planar Pendulum	270
26.3.2 A Dissipative Example: Planar Pendulum	272
26.4 Efficiency Analysis of the Presented Exponential Integrator: A Large-Scale Problem	272
26.4.1 Implementation Issues	274
26.4.2 Computational Efficiency Analysis	275
26.5 Conclusion	275
References	276
27 Vibration Suppression of a Flexible Parallel Kinematic Manipulator	277
27.1 Introduction	277
27.2 Robot Description	278
27.2.1 Kinematics Analysis	278
27.3 Compliance Maps	279
27.4 Trajectory Generation Technique	281
27.4.1 Circular Paths	282
27.4.2 Trajectories Whose Circular Path Coincides with the Arrival Circumference	282
27.4.3 Trajectories Whose Circular Path Doesn't Coincide with the Arrival Circumference	283
27.4.4 Rectilinear Paths	284
27.4.5 Performance Evaluation	285
27.5 Numerical Analysis and Results	285
27.6 The Non-Ideal Robot Model	286
27.6.1 Effects of Different Positioning Path and Changing Mass ME on System Behaviour	289
27.7 Conclusions	290
References	290
28 Analysis of Nonlinear System Response to an Impulse Excitation	292
28.1 Introduction	292
28.2 Impulse Response of a Linear System	293
28.3 Volterra Series Representation of a Nonlinear System	294
28.4 Impulse Response of a Nonlinear System	295
28.5 Comparison Between Simulation and Volterra Series Expressions	299
28.6 Discussion and Conclusions	302
References	302
29 Experimental Evaluation of Veering Crossing and Lock-In Occurring in Parameter Varying Systems	304
29.1 Introduction	304
29.2 Analytical Developments	305
29.2.1 Veering	305
29.2.2 Lock-In	307
29.2.3 Crossing	308
29.3 Experimental Evaluation of Veering and Crossing	309
29.3.1 Experimental Set-Up and Methodology	309
29.3.2 Veering in Low-Damped Coupled Systems	310
29.3.3 Crossing in Asymmetrically-Damped Coupled Systems	311
29.4 Mode Lock-In in the Beam on Disc Set-Up	311
29.5 Conclusions	313
References	314

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