Presents a systematic view of vibro-impact dynamics based on
the nonlinear dynamics analysis
Comprehensive understanding of any vibro-impact system is
critically impeded by the lack of analytical tools viable for
properly characterizing grazing bifurcation. The authors establish
vibro-impact dynamics as a subset of the theory of discontinuous
systems, thus enabling all vibro-impact systems to be explored and
characterized for applications.
Vibro-impact Dynamics presents an original theoretical
way of analyzing the behavior of vibro-impact dynamics that can be
extended to discontinuous dynamics. All topics are logically
integrated to allow for vibro-impact dynamics, the central theme,
to be presented. It provides a unified treatment on the topic
with a sound theoretical base that is applicable to both continuous
and discrete systems
Vibro-impact Dynamics:
* Presents mapping dynamics to determine bifurcation and chaos in
vibro-impact systems
* Offers two simple vibro-impact systems with comprehensive
physical interpretation of complex motions
* Uses the theory for discontinuous dynamical systems on
time-varying domains, to investigate the Fermi-oscillator
Essential reading for graduate students, university professors,
researchers and scientists in mechanical engineering.
Innehållsförteckning
Preface
Chapter 1 Introduction 1
1.1. Discrete and discontinuous systems 1
1.1.1 Discrete dynamical systems 2
1.1.2 Discontinuous dynamical systems 4
1.2 Fermi oscillator and impact problems 8
1.3 book layout 10
References 12
Chapter 2 Nonlinear Discrete Systems 19
2.1 Defintions 19
2.2 Fixed points and stability 21
2.3 Stability switching theory 34
2.4. Bifurcation theory 50
References 59
Chapter 3 Complete Dynamics and Fractality 61
3.1 Complete dynamics of discrete systems 61
3.2 Routes to chaos 69
3.2.1 One-dimensional maps 69
3.2.2 Two-dimensional maps 73
3.3 Complete Dynamics of Henon map 75
3.4 Simliarity and Multifractals 81
3.4.1 Similar Structures in period doubling 81
3.4.2 Fractality of chaos via PD bifurcation 86
3.4.3 An example 86
3.5 Complete dynamics of Logistic map 93
References 107
Chapter 4 Discontinuous Dynamical Systems 109
4.1 Basic concepts 109
4.2 G-functions 112
4.3 Passable flows 116
4.4 Non-passable flows 121
4.5 Grazing flows 135
4.6 Flow switching bifucations 149
References 162
Chapter 5 Nonlinear Dynamics of Bouncing Balls 163
5.1 Analytical dynamics of bouncing balls 163
5.1.1 Periodic motions 165
5.1.1 Stability and bifurcations 168
5.1.3 Numerical illustrations 175
5.2 Period-m motions 180
5.3 Complex dynamics 187
5.4 Complex periodic motions 192
References 200
Chapter 6 Complex Dynamics of Impact Pairs 201
6.1 Impact pairs 201
6.2 Analytical, simplest periodic motions 205
6.3 Possible impact notion sequences 216
6.4 Grazing dynamics and stick motions 220
6.5 Mapping structures and periodic motions 228
6.6 Stabilityand bifurcation 232
References 242
Chapter 7 Nonlinear Dynamics of Fermi Oscillators 243
7.1 Mapping dynamics 243
7.2 A Fermi oscillator 249
7.2.1 Absolute description 251
7.2.2 Relative description 257
7.3 Analytical conditions 258
7.4 Mapping structures and motions 260
7.4.1 Switching sets and generic mappings 260
7.4.2 Motions with mapping structures 263
7.4.3 Periodic motion and local stability 265
7.5 Predictions and similations 268
7.5.1 Bifurcation scenarios 268
7.5.2 Analytical predictions 271
7.5.3 Numberical illustractions 278
7.6 Appendix 291
References 295
Subject index 297
Om författaren
Professor Luo is currently a Distinguished Research
Professor at Southern Illinois University Edwardsville. He is an
international renowned figure in the area of nonlinear dynamics and
mechanics. For about 30 years, Dr. Luo’s contributions on
nonlinear dynamical systems and mechanics lie in (i) the local
singularity theory for discontinuous dynamical systems, (ii)
Dynamical systems synchronization, (iii) Analytical solutions of
periodic and chaotic motions in nonlinear dynamical systems, (iv)
The theory for stochastic and resonant layer in nonlinear
Hamiltonian systems, (v) The full nonlinear theory for a deformable
body. Such contributions have been scattered into 13 monographs and
over 200 peer-reviewed journal and conference papers. His new
research results are changing the traditional thinking in nonlinear
physics and mathematics. Dr. Luo has served as an editor for the
Journal ’Communications in Nonlinear Science and Numerical
simulation’, book series on Nonlinear Physical Science (HEP)
and Nonlinear Systems and Complexity (Springer). Dr. Luo is the
editorial member for two journals (i.e., IMe Ch E Part K Journal of
Multibody Dynamics and Journal of Vibration and Control). He also
organized over 30 international symposiums and conferences on
Dynamics and Control.
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