Analysis Of Dynamical Behaviors In A 2-DOF System With Asymmetric Damping

In actual engineering,discontinuous dynamical systems can be seen everywhere.Strong nonlinear factors such as impact,clearance and friction will inevitably cause collision and vibration.However,using continuous systems to approximate discontinuous systems has bigger errors.In order to analyze and characterize the engineering dynamical system more accurately,scholars propose that discontinuous systems need to be described by discontinuous models.The analysis of the mechanical system with friction and collision is helpful to improve the accuracy and reduce the noise of the mechanical system.In recent years,the theory of flow switchability in discontinuous dynamical systems has been put forward,which has made new progress in the research of discontinuous dynamical systems.Among them,G-function is used as the main research tool to show the motion switching of flow on discontinuous boundary from a new perspective.Based on this new discontinuous dynamics theory,this paper studies the dynamic behavior of a twodegree-of-freedom system with asymmetric damping in the coexistence of friction and collision.The phase space of the system is divided,and then some research results of the two-degree-of-freedom system are given,such as the analytical conditions of motion switching and numerical simulations.This paper mainly includes the following contents:In Chapter 1,there search background and current situation of the system with friction and collision are described.The first chapter briefly gives the definition of G-function in the flow switchability theory of discontinuous dynamical system and the criterion of flow switching at the discontinuous boundary.In Chapter 2,first of all,the problem studied in this paper is given,that is,a two-degree-of-freedom oscillator with asymmetric damping.Through the analysis of the physical model,several possible motion situations of the oscillator in this system are presented:When the two masses M1 and M2 are not in contact,both masses M1 and M2 have free motion or stick motion,one of masses M1 and M2 has stick motion;when the two masses M1 and M2 come into contact,there will be collision or stuck motion between the masses M1 and M2.Then,the domains and boundaries of the system's phase space are defined in absolute and relative coordinates,respectively.Since the static friction coefficient is considered to be greater than the dynamic friction coefficient in this paper,the flow barriers vector field is introduced on the velocity boundary.Furthermore,based on the theory of flow switchability,the G-functions are defined on the discontinuous boundaries of the two-degree-of-freedom system,and the G-functions are used to obtain the analytical conditions of motion switching at discontinuous boundaries,such as passable motion,stick motion,grazing motion and stuck motion.In addition,using the mapping theory,the mapping structure and periodic motion of the system are presented,and the stability of the periodic motion is simply analyzed.Finally,we use MATLAB software to provide numerical simulations of several typical motions of the system,such as the time histories of displacement,velocity,G-functions;and stick bifurcation scenario for varying driving frequency and amplitude are also displayed.Chapter 3 summarizes the research contents of this article,and looks forward to future research directions and research issues.