Bifurcation Analysis Of The Dry Friction Collision System Based On Differential Inclusion

Dry friction collision is a significant type of non-smooth dynamical system,which widely exists in the engineering design and the machine manufacturing.Because of the existence of dry friction collision increases the systemic uncertainty and the investigative complexity,the traditional research methods cannot be directly applied to dry friction collision system.Differential inclusion theory is an important tool in investigating non-smooth dynamical systems,which is also used into the control of dynamic system,polytopic system and impulsive system.Based on differential inclusion theory,this paper analyses the dynamic characteristics of the collision-dry friction system on the belt pulley and explores the bifurcation resulted from different parameters.Meanwhile,by combining the above theory,the existing conditions of different sports are found and the simulation is conducted by virtue of C programming language and Matlab.This paper is divided into six chapters,and its main contents are as follows:Chapter I: The chapter briefly introduces the research background,significance of the collision-dry friction system,main research method,existing problems and the application with the theory of differential inclusions in collision-dry friction system.And it also presents the primary coverage in the paper.Chapter II: Firstly,set-valued mapping,the theory of differential inclusions and some other basic definitions and theorems are introduced,in which the importance is attached to the stability and existences of the solution of differential inclusions in the non-smooth dynamical systems.What's more,the movement model of the non-smooth dynamical systems is established,and the necessary conditions for the existence of synovial membrane solutions in the system as well as the jump matrix of jumper motion are computed.Finally,the chapter introduces the Poincaré map,and the relationship between the stability of dynamical system and Poincaré section diagrams.Chapter III: In this chapter,the dynamic characteristics of two kinds of dry friction systems with single freedom degree are analyzed.The equations of system have been standardized by the convex hull and set-valued mapping.The stability of the equilibrium point is studied;the conditions of synovial jumping movement,the bounds of the trajectory synovial membrane,the jumper motion are computed.The conclusion that the trajectory without external excitation cannot jump over the non-smooth surface is obtained.Through the numerical simulation methods,the influence of the amplitude on the synovial jumping movement is considered.The bifurcation diagram of the jumping movement is shown,from which we can get the conclusion that the solution of curve tends to the jumping movement with the amplitude increasing.Finally,the effect of the external excitation amplitude and frequency on the stability of system is considered.Chapter IV: This chapter analyzes the two-freedom-degree collision model,and establishes the differential equation about collision.The transfer matrix is deduced in the mass collision process.According to the trajectory of the transversal index,the trajectory cannot jump over collision surface in the collision system.Through the numerical simulation of the bifurcation diagram under different parameters,the conclusion can be found that the system is more sensitive to amplitude and frequency parameters.Chapter V: The differential equation model of the two-degree-freedom collision-dry friction system is built,and the equation is standardized with differential inclusion by convex hull and set-valued mapping.The dynamic behavior of non-smooth phenomenon is also considered,and the transfer matrixes and the jump matrixes are deduced in the dry friction collision process.The single and two parameters bifurcation diagrams are presented by the method of the numerical simulation,and the influence of different parameters on dry friction collision system is discussed.Chapter VI: This part summarizes the main contents of the paper and gives the further research direction of parameters matching in the dry friction collision system.