Analyses Of The Dynamic Behaviors Of Several Physical Models In Discontinuous Dynamical Systems

In today’s mechanical engineering, there are many problems in discontinuous dynamical systems to be seen everywhere, and these problems have been widely concerned by the relevant experts and scholars. Because of the discontinuous of this system, we should obtain adequate theoretical analysis for engineering systems by considering discontinuous models. The theory study of the systems is useful to the normal operation of the machinery production. In this paper, we introduce two physical models in discontinuous dynamical systems. According to the discontinuity of the systems we define different domains and boundaries. And we have made the theoretical analysis and analytical predictions for them. The numerical simulations are given on the basis of selecting appropriate parameters. This paper is divided into three chapters.In Chapter 1, we introduce the research background and research significance of discontinuous dynamical systems, and give the corresponding flow switchability theories, concepts and several related lemmas. Consider a dynamical system of a sub-domain Ωα in a universal domain, X(α)=F(α)(X(α),t,Pα)∈Rn and the continuous flow in it X(α)(t)＝Φ(α)(X(α)(t0),t,Pα). The next research is flow switchability theories between two subdomains with com-mon boundaries.In Chapter 2, we study the dynamic behaviors of impact oscillators in an non-smooth inclined dynamical systems with a periodic excitation. It is a model with spring and damping, we give the sufficient and necessary conditions of the appearance and keeping of periodic motions, passable motions, stick motions and grazing motions. This physical model consists of the oscillator m and the base M. And the base is driven under periodically excitation F sin ωt. Finally we give numerical simulations of sliding motions by selecting the initial data.In Chapter 3, we use the flow switchability theories to study the dynamical behaviors of small object on incline translation belt with frictions. And this belt is with the periodically force. The frictional force changes with the change of the relative velocity between the small object and the belt, so the motion of the object is discontinuous. By defining the corresponding domains, the discontinuous boundaries are obtained. And we study several kinds of dynamic behaviors on the discontinuous boundary.