Analysis Of Non-smooth System Map With Grazing Bifurcation,Chattering And Viscous

The Vibro-Impact system is a typical non-smooth system.As a typical non-smooth factor,impact is common in mechanical engineering.In the process of operation,impact will affect the working accuracy and efficiency of the mechanical system,impact will cause greater abras on the mechanical system and further shorten the service life of the equipment.The study of the Vibro-Impact system can further optimize the mechanical structure and find the optimal parameters,so as to improve the safety and reliability.In this paper,a class of two-degree-of-freedom collision vibration system with one-sided rigid constraint is taken as an object to study the following aspects: Firstly,force analysis is carried out on the two mass respectively,according to Newton's second law,the differential equation of motion of the system is established,after the dimensionless transformation,the regular matrix method is used to decouple the dimensionless differential equation of motion and then the analytical solution is obtained,According to the analytical solution of the system in stable state,impact equation,the definition of collision recovery coefficient and boundary condition,the parameters in the analytic solution are obtained.Considering the system's disturbed motion,similarly,the stability analysis is carried out.By constructing the Poincarémap of this system,its Jacobi matrix can be obtained.Secondly,the dynamic behavior near the grazing point is analyzed,including the critical value of the gap when the grazing bifurcation occurs,the conditions to be satisfied when the periodic motion of grazing occurs,zero-time discontinuity map and Poincaré discontinuity map near the point of grazing.The singularity of Jacobi matrix is also discussed.It is found that there is no singularity in the determinant of the mapping.Then the chatting of the system is derived in detail,under the condition that the completion point of flutter is known,the time required to complete the chatting is obtained.Through the numerical simulation of this two-degree of freedom Vibro-Impact system,it can be seen that the number of force cycles and the number of impact of the system under this parameter will change with the change of gap and force frequency from the bifurcation diagram,phase diagram and time response diagram,and chattering and viscosity may occur at low frequencies.