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Vibro-impact systems are extensively used in engineering and physics field, such as impact damper, particle accelerator, etc. These systems are most basic elements of many real world applications such as cars and aircrafts. Such vibro-impact systems possess both the continuous characteristics as continuous dynamical systems and discrete characteristics introduced by impacts at the same time. Thus, an appropriately developed discrete mapping system is required for such vibro-impact systems in order to simplify investigation on the complexity of motions. In this dissertation, a few vibro-impact oscillators will be investigated using discrete maps in order to understand the dynamics of vibro-impact systems. Before discussing the nonlinear dynamical phenomena and behaviors of these vibro-impact oscillators, the theory for nonlinear discrete systems will be applied to investigate a two-dimensional discrete system (Henon Map). And the complete dynamics of such a nonlinear discrete dynamical system will be presented using the inversed mapping method. Neimark bifurcations in such a discrete system have also drawn a lot of interest to the author. The Neimark bifurcations in such a system have actually formed a boundary dividing the stable solution of positive and negative maps (inversed mapping). For the first time, one is able to obtain a complete prediction of both stable and unstable solutions in such a discrete dynamical system. And a detailed parameter map will be presented to illustrate how changes of parameters could affect the different solutions in such a system. Then, the theory of discontinuous dynamical systems will be adopted to investigate the vibro-impact dynamics in several vibro-impact systems. First, the bouncing ball dynamics will be analytically discussed using a single discrete map. Different types of motions (periodic and chaotic) will be presented to understand the complex behavior of this simple model. Analytical condition will be expressed using switching phase of the system in order to easily predict stick and grazing motion. After that, a horizontal impact damper model will be studied to show how complex periodic motions could be developed analytically. Complete set of symmetric and asymmetric periodic motions can also be easily predicted using the analytical method. Finally, a Fermi-Accelerator being excited at both ends will be discussed in detail for application. Different types of motions will be thoroughly studied for such a vibro-impact system under both same and different excitations.
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.
Studies of vibro-impact dynamics falls into three main categories: modeling, mapping and applications. This text covers the latest in those studies plus selected deterministic and stochastic applications. It includes a bibliography exceeding 1,100 references.
The EUROMECH Colloquium "Dynamics of Vibro-Impact Systems" was held at th th Loughborough University on September 15 _18 , 1998. This was the flrst international meeting on this subject continuing the traditions of the series of Russian meetings held regularly since 1963. Mechanical systems with multiple impact interactions have wide applications in engineering as the most intensive sources of mechanical influence on materials, structures and processes. Vibro-impact systems are used widely in machine dynamics, vibration engineering, and structural mechanics. Analysis of vibro-impact systems involves the investigation of mathematical models with discontinuities and reveals their behaviour as strongly non-linear. Such systems exhibit complex resonances, synchronisation and pulling, bifurcations and chaos, exCitation of space coherent structures, shock waves, and solitons. The aim of the Colloquium was to facilitate the exchange of up-to-date information on the analysis and synthesis of vibro-impact systems as well as on the new developments in excitation, control and applications of vibro-impact processes.
A collection of especially written articles describing the theory and application of nonlinear dynamics to a wide variety of problems encountered in physics and engineering. Each chapter is self-contained and includes an elementary introduction, an exposition of the state of the art, as well as details of recent theoretical, computational and experimental results. Included among the practical systems analysed are: hysteretic circuits, Josephson circuits, magnetic systems, railway dynamics, rotor dynamics and nonlinear dynamics of speech. This book provides important information and ideas for all mathematicians, physicists and engineers whose work in R & D or academia involves the practical consequences of chaotic dynamics.
This book presents the theoretical frame for studying lumped nonsmoothdynamical systems: the mathematical methods are recalled, and adaptednumerical methods are introduced (differential inclusions, maximalmonotone operators, Filippov theory, Aizerman theory, etc.
The book presents the recent achievements on bifurcation studies of nonlinear dynamical systems. The contributing authors of the book are all distinguished researchers in this interesting subject area. The first two chapters deal with the fundamental theoretical issues of bifurcation analysis in smooth and non-smooth dynamical systems. The cell mapping methods are presented for global bifurcations in stochastic and deterministic, nonlinear dynamical systems in the third chapter. The fourth chapter studies bifurcations and chaos in time-varying, parametrically excited nonlinear dynamical systems. The fifth chapter presents bifurcation analyses of modal interactions in distributed, nonlinear, dynamical systems of circular thin von Karman plates. The theories, methods and results presented in this book are of great interest to scientists and engineers in a wide range of disciplines. This book can be adopted as references for mathematicians, scientists, engineers and graduate students conducting research in nonlinear dynamical systems. · New Views for Difficult Problems · Novel Ideas and Concepts · Hilbert's 16th Problem · Normal Forms in Polynomial Hamiltonian Systems · Grazing Flow in Non-smooth Dynamical Systems · Stochastic and Fuzzy Nonlinear Dynamical Systems · Fuzzy Bifurcation · Parametrical, Nonlinear Systems · Mode Interactions in nonlinear dynamical systems
- Models of vibro-impact systems are widely used in machine dynamics, vibration engineering, and structural mechanics. - Only monograph on this subject in English language. - Systematically presents the theory of vibro-impact systems by analysis of typical engineering applications. - Experimental data and computer simulations are presented. - Targeted to engineers and researchers in design and investigation of mechanical systems as well as to lecturers and advanced students.
"Bifurcation and Chaos in Discontinuous and Continuous Systems" provides rigorous mathematical functional-analytical tools for handling chaotic bifurcations along with precise and complete proofs together with concrete applications presented by many stimulating and illustrating examples. A broad variety of nonlinear problems are studied involving difference equations, ordinary and partial differential equations, differential equations with impulses, piecewise smooth differential equations, differential and difference inclusions, and differential equations on infinite lattices as well. This book is intended for mathematicians, physicists, theoretically inclined engineers and postgraduate students either studying oscillations of nonlinear mechanical systems or investigating vibrations of strings and beams, and electrical circuits by applying the modern theory of bifurcation methods in dynamical systems. Dr. Michal Fečkan is a Professor at the Department of Mathematical Analysis and Numerical Mathematics on the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, Slovakia. He is working on nonlinear functional analysis, bifurcation theory and dynamical systems with applications to mechanics and vibrations.
This book presents the theoretical frame for studying lumped nonsmooth dynamical systems: the mathematical methods are recalled, and adapted numerical methods are introduced (differential inclusions, maximal monotone operators, Filippov theory, Aizerman theory, etc.). Tools available for the analysis of classical smooth nonlinear dynamics (stability analysis, the Melnikov method, bifurcation scenarios, numerical integrators, solvers, etc.) are extended to the nonsmooth frame. Many models and applications arising from mechanical engineering, electrical circuits, material behavior and civil engineering are investigated to illustrate theoretical and computational developments.