Stability And Unconventional Bifurcation Of A Class Of Non-smooth Systems

Non-smooth dynamical system is an indispensable part of nonlinear dynamical system,which is not only common in our daily life,but also widely exists in the engineering and industrial fields.The existence of non-smooth factors makes the non-smooth system have more complex dynamic behaviors.For example,some non-smooth factors such as the rigid constraint or gap of the collision and vibration system make the system experience impact,chattering and grazing phenomenon,which makes the mechanical system structure wear and output unstable.Therefore,in order to improve the service life and safety performance of mechanical equipment,it is of great practical value to study the complex motion behaviors such as impact,chattering and grazing of system,etc,and non-smooth systems have attracted the attention of many scholars in the academic world.Based on the bifurcation diagram,phase diagram and time response diagram obtained by numerical simulation,the stability,chattering phenomenon and grazing phenomenon of the collision vibration system with symmetric gap are studied in this paper,so as to provide a theoretical basis for the collision vibration system with gap.This paper mainly contains three parts: Firstly,the expression of Poincaré mapping was obtained by theoretical derivation of the system,then,the global bifurcation diagram and local partial bifurcation diagram of the system were drawn by numerical simulation.Hopf bifurcation parameters were found by means of Poincaré mapping diagram.Secondly,the chattering phenomenon of the system is theoretically derived.The stable manifold theory is used to obtain the expression of the time experienced by the complete chattering of the system.Then the system is numerically simulated to obtain the bifurcation diagram and the time response diagram of the system.According to the time response diagram,the time spent in the chattering can be known,and then the boundary point between the complete and the incomplete chattering can be obtained at a certain parameter value.Finally,the Poincarésection discontinuity mapping and zero-time discontinuity mapping are derived by using the theory of non-smooth system,and then the two discontinuities are combined with smooth mapping,so as to obtain the piecewise expression of the grazing mapping.By analyzing the jacobian matrix of the grazing mapping,we find that its trace is singular.By analyzing the jacobian matrix of the grazing mapping,we find that its trace is singular.It is found that there are discontinuity points on the bifurcation diagram obtained by the numerical simulation,and it is observed from the phase diagram and the time response diagram that the system has an grazing collision movement at this discontinuity point.