Nonlinear Dynamics, Volume 1

Preface	6
Contents	7
1 Interface Reduction on Hurty/Craig-Bampton Substructures with Frictionless Contact	10
1.1 Introduction	10
1.2 Theory	11
1.2.1 Nonlinear Hurty/Craig-Bampton Method	11
1.2.2 Interface Modal Basis	13
1.2.2.1 Extended System-Level Characteristic Constraint Modes About Preloaded State	13
1.2.2.2 Static Preload Deformation	14
1.2.2.3 Total SCCe Transformation Matrix	15
1.2.3 Normal Contact Model	16
1.2.4 Time Integration Schemes	16
1.3 Numerical Studies: Two-Beam Assembly with Frictionless Contact	17
1.3.1 Model Description	17
1.3.2 Static Preloading of Bolt Elements	17
1.3.3 Results: Full-Interface Model	18
1.3.4 Results: Interface-Reduced Model	19
1.4 Conclusions	23
References	25
2 Experimental Path Following of Unstable Static Equilibria for Snap-Through Buckling	26
2.1 Introduction	26
2.2 Experimental Design	27
2.2.1 Experimental Setup	27
2.2.2 Control System	27
2.3 Results	28
2.4 Conclusion	30
References	31
3 Direct Detection of Nonlinear Modal Interactions and Model Updating Using Measured Time Series	32
3.1 Method	32
3.2 Example	33
References	34
4 Pareto Optimization of a Nonlinear Tuned Mass Damper to Control Vibrations in Hand Held Impact Machines	35
4.1 Introduction	35
4.2 Engineering and Mathematical Models	36
4.3 Objective Functions and Design Parameters	38
4.4 Global Sensitivity Analysis	39
4.5 Multidisciplinary Design Optimization	39
4.5.1 Optimization Algorithm	40
4.5.2 Operational Scenarios	40
4.6 Results	40
4.6.1 GSA Results	41
4.6.1.1 Linear System	41
4.6.1.2 Nonlinear System	43
4.6.2 Multiobjective Optimization Results	44
4.6.3 Closed-Form Representation	47
4.7 Conclusion	49
References	51
5 Inverse Methods for Characterization of Contact Areas in Mechanical Systems	53
5.1 Introduction	53
5.2 Single Parameter Inverse Method	54
5.2.1 Inputs and Outputs	54
5.2.2 Contact Definitions	55
5.2.3 Contact Area Optimization	55
5.3 System Description	56
5.3.1 Experimental Set-Up	56
5.3.2 Finite Element Model	57
5.4 Results	58
5.4.1 Static Preload Analysis	58
5.4.2 Model Updating Results	61
5.5 Conclusions	63
References	64
6 Experimental Characterization of a New Benchmark Structure for Prediction of Damping Nonlinearity	65
6.1 Introduction	65
6.2 Theoretical Background	66
6.3 Experimental Methodology	67
6.3.1 Impact Hammer Testing	68
6.3.2 Joint Profile Testing	69
6.4 Experimental Results	70
6.4.1 Interface Characterization	70
6.4.2 Nonlinearity Detection and Analysis	73
6.5 Conclusion	81
Appendix A	84
Test Setup	84
Optimal Input Location	84
Appendix B	85
Mode 1 Results	85
References	86
7 Nonlinear System Identification for Joints Including Modal Interactions	87
Nomenclature	87
7.1 Introduction	88
7.2 Systems of Interest	89
7.2.1 2DOF System	89
7.2.2 3DOF System	90
7.2.3 Equations of Motion	90
7.2.4 Modal Transformation	90
7.2.5 Non-linear Elements	90
7.2.6 Generating Test Data	91
7.3 Methods of Capturing the Modal Coupling	91
7.3.1 Multi-modal Maps (MMM)	91
7.3.1.1 Signal Processing	91
7.3.1.2 Fitting the Map to the Data	92
7.3.1.3 Averaging Method to Simulate the Time Response	93
7.3.1.4 Computation of the Restoring Force	94
7.3.2 Restoring Force Surface (RFS)	94
7.3.2.1 Forms of the Model Used	95
7.3.2.2 Data Conditioning and Model Fitting	95
7.3.2.3 Parameter Reduction	96
7.3.2.4 Combining Multiple Data Sets	96
7.3.2.5 Time Integration	97
7.3.2.6 Numerical Computation of Instantaneous Natural Frequency and Damping	97
7.3.3 Error Metrics	98
7.3.3.1 Natural Frequency and Damping	99
7.3.3.2 Time Response	99
7.3.3.3 Restoring Force	99
7.4 Results	100
7.4.1 Results for a 2DOF System	100
7.4.1.1 Cubic Spring Non-linearity	100
7.4.1.2 Jenkins Element	101
7.4.1.3 Iwan Element	103
7.4.2 Results for a Symmetric 3DOF System	104
7.4.2.1 Cubic Spring Non-linearity	105
7.4.2.2 Iwan Element	106
7.5 Conclusions	107
References	107
8 Performance of Nonlinear Modal Model in Predicting Complex Bilinear Stiffness	108
Nomenclature	108
8.1 Introduction and Motivation	109
8.2 Experiments	109
8.2.1 Test Hardware Description	109
8.2.2 Linear Modal Test	110
8.2.3 High Level Shaker Testing	110
8.3 Modal Filtering	112
8.4 Nonlinear Model Form and Extraction of Nonlinear Parameters	112
8.5 Comparison to Physical Domain Data	114
8.6 Discussion	119
8.7 Conclusions	119
References	119
9 Low Order Nonlinear Dynamic Modelling of Fuel Supply Pipes	120
9.1 Introduction	120
9.1.1 A Short Summary of the Experimental Evaluation	120
9.1.1.1 Test Set Up	121
9.1.2 Basic Implicit Nonlinear Dynamic Pipe Model	122
9.1.2.1 Pipe Modelling	122
9.1.2.2 Clip Modelling	122
9.1.2.3 Nonlinear Dynamic Analysis	123
9.1.3 Model Updating	123
9.1.3.1 Model Verification	123
9.1.3.2 Optimisation and Validation	124
9.1.3.3 Summary	125
9.1.4 Results	125
9.1.4.1 Linear Validation	126
9.1.4.2 Nonlinear Validation	126
9.1.5 Discussion	126
9.1.6 Conclusion	127
References	127
10 System Identification to Estimate the Nonlinear Modes of a Gong	128
10.1 Introduction	128
10.2 Theoretical Development	129
10.2.1 System Identification Methods	130
10.3 Experimental Application	131
10.3.1 Linear System Identification	131
10.3.2 Force Appropriation	132
10.3.3 Nonlinear System Identification	133
10.4 Conclusions	139
References	142
11 An Enhanced Static Reduction Algorithm for Predictive Modeling of Bolted Joints	144
11.1 Introduction	144
11.2 Sample Results	145
References	146
12 Time-varying Spectral Submanifolds: Analytic Calculation of Backbone Curves and Forced Response	147
References	148
13 Operational Modal Analysis Based Stress Estimation in Friction Systems	149
13.1 Introduction	149
13.2 Theory	150
13.2.1 Operational Modal Analysis and the Frequency Domain Decomposition	150
13.2.2 Strain Estimation	150
13.2.3 Quality Measurements	151
13.3 Case Study	151
13.3.1 Numerical Simulation Case	151
13.3.2 Results of Operational Modal Analysis and Strain Estimation	154
13.4 Conclusion	156
References	158
14 Damping Estimation of Friction Systems in Random Vibrations	160
14.1 Introduction	160
14.2 Methodology	160
14.3 Application	161
14.4 Conclusions	162
References	162
15 System Identification of Jointed Structures: Nonlinear Modal Testing Vs. State-Space ModelIdentification	163
15.1 Introduction	163
15.2 Theoretical Background of the Identification Methods	164
15.2.1 EPR Testing	164
15.2.2 PNLSS Identification	164
15.3 Experimental Study	164
15.4 Conclusion	165
References	165
16 Effect of Boundary Conditions on Finite Element Submodeling	166
16.1 Introduction	166
16.2 Theoretical Development	168
16.3 Case Study	169
16.4 Results	170
16.5 Conclusion	172
References	173
17 On Euler Buckling and Snap-Through	174
17.1 Introduction	174
17.2 Computational Exploration of the Relationship Between Euler Buckling and Snap-Through	174
17.3 A Simple Approximation to Predict Force-Deformation Characteristics	176
17.4 Conclusions	177
References	177
18 Solitons in Cyclic and Symmetric Structures	178
18.1 Introduction	178
18.2 Physical System	178
18.3 Solution Approach	179
18.3.1 Non-conservative NLSE	179
18.3.2 Quasi-periodic HBM	179
18.4 Numerical Results	180
References	181
19 Experimental and Numerical Nonlinear Modal Analysis of a Beam with Impact: Part I – Numerical Investigation	182
19.1 Introduction	182
19.2 Numerical Procedure and Benchmark Structure	182
19.3 Numerical Results	183
19.4 Conclusion	184
References	185
20 Experimental and Numerical Nonlinear Modal Analysis of a Beam with Impact:Part II – Experimental Investigation	186
20.1 Introduction	186
20.2 Experimental Procedure and Benchmark Structure	186
20.3 Experimental Results	187
20.4 Conclusion	188
References	188
21 The Effect of Non-Flat Interfaces On System Dynamics	189
21.1 Introduction	189
21.2 Exprimental Setup	190
21.2.1 Test Specimens	190
21.2.2 Physical Setup and Data Acquisition	191
21.2.3 Parametric Study	191
21.3 Numerical Setup	192
21.3.1 Model Development	192
21.3.2 Quasi-Static Analysis	194
21.4 Results	195
21.4.1 Experimental Results	195
21.4.2 Numerical Results and Validation	196
21.5 Conclusions	197
References	199
22 Investigating Modal Contributions Using a Galerkin Model	200
22.1 Introduction	200
22.2 Galerkin Model Development	201
22.3 Model Reduction	203
22.3.1 Frequency Investigation	204
22.3.2 Amplitude Investigation	207
22.3.3 Backbone Curve	209
22.4 Conclusion	211
References	211
23 Acoustic Excitation of a Flanged Joint	212
23.1 Introduction	212
23.2 Fastened Joints Discussion	213
23.2.1 Lap Joints and Flange Joints	213
23.2.2 Mode Types and Damping	213
23.2.3 Methods of Excitation	216
23.2.4 Boundary Conditions	216
23.3 Experimental Setup	217
23.3.1 Hardware	217
23.3.2 Torquing	217
23.3.3 Boundary Conditions	217
23.3.4 Setup Procedures	219
23.4 Results	221
23.4.1 Damping	221
23.4.2 Steady-State	222
23.5 Conclusion	224
References	225
24 In Situ Measurements of Interfacial Contact Pressure During Impact Hammer Tests	226
24.1 Introduction	226
24.2 Experimental Setup	227
24.2.1 Impact Test Setup	227
24.2.2 Interface Pressure Measurements Setup	228
24.3 Influence of the Interface Measurement System on the Global Response	229
24.3.1 Comparison Between BRB and BRBS	230
24.3.2 Variation of Bolt Torque and Impact Force for the BRBS	230
24.4 Validation of the Interface Measurement System	232
24.5 Dynamic Interface Measurements	234
24.6 Conclusions and Perspectives	236
References	236
25 An Improved Shape Reconstruction Methodology for Long Rod Like Structures Using Cosserat Kinematics- Including the Poisson's Effect	238
25.1 Introduction	238
25.2 Geometry of the Deformed Configuration of the Beam	239
25.3 Kinematics of the Beam-Obtaining Uniaxial Scalar Strain	240
25.3.1 Strain Vector and Deformation Gradient Tensor	240
25.3.2 The Orientation of an Arbitrary Strain Gauge in the Deformed State (Measurement Model)	241
25.4 Solution Approach	243
25.4.1 An Approximate Solution for the Midcurve Position Vector and Director Triad	243
25.4.2 Approximate Solution to the Axial Strain	244
25.5 Simulation	244
25.6 Conclusion	245
References	246
26 Computing Nonlinear Normal Modes of Aerospace Structures Using the Multi-harmonic BalanceMethod	248
26.1 Introduction	248
26.2 Theory	250
26.2.1 Harmonic Balance Method Applied to Computing NNMs	250
26.3 Numerical Case Studies	251
26.3.1 Flat Beam	251
26.3.2 Discovery Experimental Panel	252
26.3.3 Curved Beam	253
26.4 Conclusion	255
References	259
27 Nonlinear Identification of an Aero-Engine Component Using Polynomial Nonlinear State Space Model	261
27.1 Introduction	261
27.2 Description of the Aero-Engine Casing	262
27.3 Experimental Setup and Linear Modal Analysis	262
27.4 Linear Analysis of the Casing and Plate Assembly	263
27.5 Non-linear Detection Based on Distortions in Time-Series and FRF Measurements	263
27.6 Identification Procedure	265
27.7 Parametric and Non-parametric Analysis of the Nonlinear Distortions	266
27.8 BLA Estimation of the Casing Assembly	266
27.9 Non-linear Identification Based on Polynomial Nonlinear State Space Models	268
27.9.1 Identification Results Based on BLA Initialisation	269
27.10 PNLSS Model Validation	272
27.11 Conclusion	273
References	273
28 Curved Structures That Can Elastically Snap-Through	274
28.1 Introduction	274
28.2 The Geometry	274
28.3 Theoretical Results	275
28.3.1 Analysis Based on Shape	275
28.3.2 Thermal Loading	276
28.4 Some 3D Printed Shapes	276
28.5 Concluding Remarks	276
References	276
29 Experimental Analysis of Non-Linear Damping Performance in Composites Materials Thanks to Local Transduction-Dissipation Phenomenon	277
References	279
30 Subspace-Based Identification of a Distributed Nonlinearity in Time and Frequency Domains	280
30.1 Introduction	280
30.2 Beam Under Large Lateral Oscillations	281
30.3 Nonlinear System Identification	281
References	282
31 Reduced Order Modelling for Non-linear Rotating Systems in ALE Formulation with Contact	283
31.1 Introduction	283
31.2 Model Order Reduction	283
31.2.1 Linear Systems	284
31.2.1.1 Second Order Modal Truncation	284
31.2.1.2 Second Order Krylov Subspace Method	285
31.2.2 Craig-Bampton Method for Non-linear Systems	286
31.3 ALE Formulation with Contact Mechanics	287
31.3.1 Kinematic Framework	287
31.3.2 Contact Mechanics	288
31.4 Dynamics of Gyroscopic Systems	288
31.4.1 Eigenvectors as Travelling Waves	288
31.4.2 Frequency Split	289
31.5 Application	290
31.5.1 Rotating Disc Under Forcing Excitation	290
31.5.1.1 ROM by Second Order Modal Truncation	292
31.5.1.2 ROM by Krylov Subspace Technique	293
31.5.2 Rotating Disc with Normal Contact	294
31.5.2.1 Full System	294
31.5.2.2 ROM by Craig-Bampton Substructuring Technique	295
31.6 Conclusion	297
References	297
32 Damage Precursor Indicator for Aluminum 7075-T6 Based on Nonlinear Dynamics	299
32.1 Introduction	299
32.2 Experimental Approach	299
32.3 Nonlinear Dynamic Parameter Development	301
32.4 Results and Discussion	302
32.5 Conclusions and Future Work	309
References	309
33 Application of Control-Based Continuation to a Nonlinear System with Harmonically Coupled Modes	310
References	311
34 Numerically Assessing the Relative Significance of Nonlinear Normal Modes to Forced Responses	312
34.1 Introduction	312
34.2 The Nonlinear Dynamics of a Pinned-Pinned Beam	313
34.2.1 A Nonlinear Pinned-Pinned Beam	313
34.2.2 The Nonlinear Normal Modes of a Pinned-Pinned Beam	315
34.3 Energy Transfer Due to a Phase Shift in Forcing	316
34.3.1 The Relationship Between Energy and Phase	316
34.3.2 Energy-Phase Relationships for the NNMs of the Example System	319
34.4 Settling Time After a Delay	320
34.5 Conclusions	321
References	321
35 Direct Frequency Domain Identification of Time Varying Systems	322
35.1 Background	322
35.2 Mathematical Methods	323
35.3 Experimental Methods	325
35.4 Results	325
35.5 Discussion	328
35.6 Conclusions	328
References	329
36 Reduced-Order Modelling for Investigating Nonlinear FEM Systems	330
Nomenclature	330
36.1 Introduction	331
36.2 Methodology	331
36.2.1 Nonlinear Modal Force	332
36.2.2 Truncation	332
36.2.3 Coefficient Identification	334
36.3 Test Case	335
36.4 Results	338
36.4.1 Case a	339
36.4.2 Case b	340
36.5 Conclusions	343
References	344
37 Nonlinear Forced Response of a Composite Fan Blade Actuated by Piezoelectric Patches: Simulation and Testing	346
37.1 Introduction	346
37.2 Experimental Set-Up	347
37.2.1 General Description	347
37.2.2 Test-Rig Excitation System	347
37.2.3 Vibration Measurement	348
37.2.4 Experimental Protocol	348
37.3 Numerical Simulation	348
37.3.1 Numerical Modeling	348
37.3.2 Condensation of Piezoelectric Actuator Potential DOFs	350
37.3.3 Reduction and Resolution of the System	352
37.3.4 Influence Studies on Resolution Parameters	353
37.4 Correlation Between Test and Simulation Results	354
37.4.1 Correlation on the Linear Simulations	354
37.4.2 Influence Study on Friction Coefficient	355
37.4.3 Correlation for the Three Targeted Modes	355
37.5 Conclusion	357
References	357
38 Locating Nonlinearity in Mechanical Systems: A Dynamic Network Perspective	358
38.1 Introduction	358
38.2 Methodology	358
38.2.1 Step 1: BLA Analysis from References to Nodes	359
38.2.2 Step 2: BLA Analysis from Nodes to Nodes	359
38.2.3 Step 3: Residual Analysis and Nonlinearity Location	360
38.3 Application to a Three-Degree-of-Freedom Mechanical System	360
38.4 Conclusions and Perspectives	361
References	362
39 Tracing a Prescribed Force-Displacement Curve Using Topology Optimization	363
39.1 Introduction	363
39.2 Optimization Formulas	363
39.3 Simulation	364
39.4 Conclusion	365
References	365
40 Model Updating of a Wing-Engine Structure with Nonlinear Connections	366
References	367
41 Modal Analysis of Axially Deforming Rods with Isolated Lap Joints	368
References	370
42 Identification of Nonlinear Viscoelastic Parameters Based on an Enhanced Oberst Beam Method	371
42.1 Introduction	371
42.2 Enhanced Oberst Beam Method	372
42.3 Conclusion	373
References	373
43 A General Framework for Time Domain Finite Element Analysis of Viscoelastically Damped Structures	375
43.1 Introduction	375
43.2 Integration of the Viscoelastic Models and Finite Element Implementation	375
43.3 Numerical Example	376
43.4 Conclusions	377
References	377
44 Nonlinear Structural, Inertial and Damping Effects in an Oscillating Cantilever Beam	378
Nomenclature	378
44.1 Introduction	379
44.2 Theoretical Background	379
44.2.1 The Equation of Motion	379
44.2.2 Damping	380
44.3 Experimental System	380
44.4 Results and Discussion	381
44.4.1 Effect of Structural Damping and Drag Coefficient on Computational Results	382
44.4.2 Comparing Experimental and Computational Results	387
44.4.3 Nonlinear Effects and Hysteresis Near Second Natural Frequency	388
44.5 Summary and Conclusions	389
Nonlinear Terms of the Computational Model	390
References	391
45 Reduced Order Modeling of Structures with Preloaded Bolted Joints by the Use of Trial VectorDerivatives	392
45.1 Introduction	392
45.2 Theory	392
45.3 Numerical Example	393
45.3.1 Dynamic Convergence Analysis	393
45.4 Conclusion	394
References	395
46 Towards the Development of a Model for Nonlinear Elements in Machine Tools	396
46.1 Introduction	396
46.2 Methodology	397
46.3 Development of Test Fixture	399
46.4 Collection of Data	399
46.5 Results	400
46.5.1 Study of Acceleration Surface: Full Content Data	403
46.5.2 Investigation of Harmonics	405
46.6 Conclusion	407
References	408
47 Nonlinear Characterization of a Machine Tool Energy Absorber	409
47.1 Introduction	409
47.2 Collection of Data	410
47.3 Results	413
47.4 Conclusions	415
References	416
48 Tutorial: Bolted Joints and Tribomechadynamics	417
48.1 Introduction	417
48.2 Experimental Considerations	418
48.3 State of Practice	418
48.3.1 Linear Spring Models	418
48.3.2 Coulomb Friction	418
48.3.3 Jenkins Elements	418
48.4 State of Art	419
48.4.1 Iwan Models	419
48.4.2 Bouc-Wen Models	419
48.5 Concluding Remarks	419
48.5.1 The Surrogate System Hypothesis	419
48.5.2 Coupled Normal Contact	420
References	420

RkJQdWJsaXNoZXIy MTMzNzEzMQ==