Modal Analysis of Nonlinear Mechanical Systems (eBook)

Preface 6
Contents 7
Definition and Fundamental Properties of Nonlinear Normal Modes 8
1 A Brief Historical Perspective 8
2 Nonlinear Normal Modes: What Are They ? 9
2.1 Definition of a Nonlinear Normal Mode 10
2.2 Fundamental Properties 11
3 Nonlinear Normal Modes: How to Compute Them ? 17
3.1 Analytical Techniques 17
3.2 Numerical Techniques 21
3.3 Assessment of the Different Methodologies 22
4 Nonlinear Normal Modes: Why Are They Useful ? 24
4.1 ‘Linear’ Modal Analysis 24
4.2 Nonlinear Modal Analysis 24
5 Conclusion 27
Bibliography 27
Invariant Manifold Representations of Nonlinear Modes of Vibration 54
1 Introduction 54
2 Basic Concepts 55
2.1 Some Definitions 55
2.2 Example Illustrating Invariant Manifolds 57
2.3 Key Points About Invariant Manifolds 59
2.4 Nonlinear Vibration Equations of Motion 59
3 Normal Modes as Invariant Manifolds for Linear Systems 60
4 Normal Modes for Nonlinear Systems 67
4.1 State Space Formulation 68
4.2 Amplitude/Phase Coordinates 70
4.3 Systems with Harmonic Excitation 70
4.4 Multi-Nonlinear Mode Models 71
4.5 Continuous Systems 73
4.6 Example: A Nonlinear Gyroscopic System 74
5 Discussion 78
Acknowledgements 78
Bibliography 79
Normal form theory and nonlinear normal modes: Theoretical settings and applications 82
1 Introduction and notations 82
2 Normal form theory 84
2.1 Problematic 85
2.2 Example study 87
2.3 General case: Poincaré and Poincaré-Dulac theorems 92
2.4 Application to vibratory systems, undamped case 93
2.5 NNMs and Normal form 97
2.6 Single-mode motion 102
2.7 Classification of nonlinear terms, case of internal resonance 103
2.8 Damped systems 104
2.9 Closing remarks 110
3 Hardening/softening behaviour 110
3.1 Definition 111
3.2 A two dofs example 113
3.3 Application to shells 115
3.4 Influence of the damping 126
3.5 Closing remarks 129
4 Reduced-order models for resonantly forced response 129
4.1 Derivation of the reduced-order model 130
4.2 Application : the case of a doubly-curved panel with in-plane inertia 134
4.3 Application : the case of a closed circular cylindrical shell 142
4.4 Comparison with the Proper Orthogonal Decomposition method 148
Bibliography 155
A Expressions of the coefficients for the normal transform in the conservative case 161
A.1 Quadratic coefficients 161
A.2 Cubic coefficients 161
Nonlinear normal modes in damped forced systems 168
1 Introduction 168
2 Linear modal analysis of damped systems 169
3 NNMs and their bifurcations in damped nonlinear systems. 174
4 Time scale separation and a problem of targeted energy transfer. 177
5 Targeted energy transfer in forced systems. 186
6 Concluding remarks 218
Bibliography 219
Computation of Nonlinear Normal Modes through Shooting and Pseudo-Arclength Computation 221
1 Introduction 221
2 Numerical Computation of NNMs 222
2.1 Shooting Method 222
2.2 Continuation of Periodic Solutions 226
2.3 An Integrated Approach for the NNM Computation 229
3 Numerical Experiments 232
3.1 Weakly Nonlinear 2DOF System 232
3.2 Nonlinear Bladed Disk System 233
3.3 Nonlinear Cantilever Beam 235
4 Conclusion 235
Bibliography 236
Numerical computation of nonlinear normal modes using HBM and ANM 257
1 Introduction to HBM and ANM on a toy model 257
1.1 Harmonic Balance Method (HBM) 257
1.2 Continuation 258
2 Continuation with Asymptotic Numerical Method 259
2.1 How to compute the serie efficiently 262
2.2 Determining the range of utility of the series 264
2.3 Comparison with predictor-corrector algorithm 265
2.4 MANLAB and DIAMANLAB sofware 266
2.5 Simple bifurcation detection and branch switching using Taylor series 268
2.6 Conclusion 272
3 High order harmonic balance method 272
3.1 The harmonic balance principle 272
3.2 A key point: the quadratic recast 273
3.3 The harmonic balance method applied to a quadratic system 275
3.4 The case of a periodically forced system 277
3.5 The case of conservative nonlinear normal modes 279
3.6 Stability 281
3.7 Numerical results on selected examples 282
4 Application : NNMs of elastic structures with contactors 289
4.1 NNMS of a bar with an unilateral contactor 289
4.2 First nonlinear mode 291
4.3 Beam with a bilateral contactor 292
4.4 Conclusion 295
Bibliography 296
Elements of Nonlinear System Identification of Broad Applicability 299
1 Introduction 299
2 Slow flows and empirical mode decomposition 300
3 Frequency-energy plots, wavelet transforms and damped transitions 307
4 Global and local aspects of the NSI methodology 309
5 Applications 311
6 Conclusion 316
Bibliography 317
Acoustic mitigation 331
1 INTRODUCTION 331
2 EXPERIMENTAL SET-UP AND ASSOCIATED MODELS 332
2.1 The classical 2 d.o.f. mechanical system 332
2.2 The vibro-acoustic experimental set-up 333
2.3 Associated models 334
3 EXPERIMENTAL RESULTS 336
3.1 Different regimes under sinusoidal forcing 336
3.2 Free oscillations 338
3.3 Frequency responses 340
3.4 Discussion 343
Bibliography 344