Chattering And Grazing Bifurcation Characteristics Of Non-linear Vibration Systems Based On Discontinuous Mapping

The Vibro-Impact phenomenon is the most common phenomenon in life.As one of the representatives of nonlinear systems,the research on it has become an important topic for scholars at home and abroad.In this paper,two different models of collisional vibration systems are discussed.The dynamics knowledge is used to study the flutter behavior of the two systems.The singularity of the Jacobian matrix and the stability of the periodicity of the rubbing cycle are studied in depth.Firstly,this paper studies a mechanical model of a vibrating system with a gap collision,and theoretically deduces the flutter behavior of the system,and obtains the flutter completion point of the system and the time it takes for the chattering phenomenon to occur.And the conditions for the existence of the wiping track when the system is subjected to the wiping motion are derived.Then zero-time discontinuous mapping is derived near the frictional orbit,and the singularity of Jacobi matrix is analyzed and deduced in detail by using the zero-time discontinuous mapping method.It was found that the singularity of the Jacobi matrix is included in the trace of the Jacobi matrix under certain conditions.Secondly,this paper studies the mechanical model of the collision vibration system with two-sided constraints,and also derives the flutter theory of the system,and analyzes and deduces the existence of the system’s rubbing cycle motion.Then,the local dynamic behavior near the grazing orbit is analyzed by Nordmark discontinuous mapping method,and the expression of discontinuous mapping is deduced.The stability of the grazing periodic orbit is analyzed according to the expression of discontinuous mapping.Because of the singularity of the mapping at the grazing point,the Jacobi matrix cannot be directly calculated.By numerically iterating the obtained local Poincaré map,it is shown that the eigenvalue of the linear matrix is real and greater than zero,and that the grazing periodic trajectory is stable when the certain conditions are satisfied.Finally,the variation of the stability of the system motion as the excitation frequency changes is studied by numerical simulation analysis.