Dynamic Behaviors Analysis Of A Dry Friction System

Dry friction system has complex dynamic behaviors,in addition to the bifurcations of smooth systems such as fold bifurcations and Hopf bifurcations,dry friction systems may also exhibit irregular bifurcations such as grazing bifurcations and so on.Moreover,the multi-time scales effect may influence the behaviors of dynamical systems,resulting in multi-modal mixed oscillation.In this paper,the complex dynamic behavior of a kind of dry friction system is studied.Considering the multi-scale effect under periodic excitation,the bifurcation mechanism and transition mechanism of the bursting oscillation are analyzed.Firstly,a conveyor belt system with dry friction and periodic external excitation is established.The external excitation frequency is far less than the natural frequency of the system to construct a generalized autonomous system with two-time scales,and the external excitation term is regarded as a slowly varying parameter.Numerical simulation shows that the system will produce bursting oscillation under appropriate parameters.The dynamic behavior of the system under three groups of parameters and the evolution process with the change of external excitation amplitude are given.Due to the non-smooth characteristics of the system,when the system trajectory contacts the interface,the unconventional bifurcation may occur,which may result in special mode.Moreover the sliding region of the system,the stability of the equilibrium points and the conditions of some bifurcations are analyzed and given,and the bifurcation mechanism of the transition between the spiking states and the quiescent states in a periodic oscillation under three groups of parameters is revealed.In addition,the influence of the change of equilibrium points types and the inherent structure of the system on the existence of the local structure of the trajectory is studied.On the basis of the above mode,we introduced a periodic parameter excitation,and let the frequency ratio of the parameter excitation and external be 1:2.The frequency of both excitation has a significant difference in magnitude with the natural frequency of the system.A multi-scale dry friction system with combining the external and parametric excitation is constructed.To analyze it with the traditional fast and slow analysis method,a new parameter is introduced,which is used to represent the two excitation terms respectively,and then the critical conditions for Hopf and fold bifurcation of the fast subsystem are given.The influence of the external excitation amplitude on the bursting oscillation of the system is investigated.With the numerical simulation under different groups of parameters,it is found that the system will produce multiple modes of bifurcation with the change of the external excitation amplitude.In the process of equilibrium transition,there may be many structures in the spiking state,but it is certain that there are always a pair of subcritical Hopf bifurcation points,and the fold bifurcation points are located on the unstable equilibrium curve.In addition,the time for the system to slide times on the non-smooth interface is given.