Study On Existence, Coexistence And Stability Of Periodic Motions In An Impact Vibrating System Of Two Degrees Of Freedom

Vibro-impact system is one of the focuses in the research of nonlinear dynamics.The periodic impact motions in a vibro-impact system of two degrees of freedom are presented in this dissertation.By using the analytic and numerical methods,existence and coexistence of single impact period-n responses in the above system are obtained and a discontinuity mapping which can describe the motions of the system is deduced.Furthermore the stability of the periodic motions of the system is considered.Firstly,a two-degree-of-freedom vibro-impact system with more complicated impacting conditions and uncertain impacting position is studied in this paper by the theory of non-smooth dynamical systems.The existence and non-existence theorems of single impact periood-n responses in some parameter regions are presented through a great deal of computation and detailed theoretical deducing,with this provides an analytic method to find out the coexistence of periodic trajectories in the multi-degree-of-freedom vibro-impact systems. In this way,dynamics in particular parameter regions of the systems can be known well and so is the problem of how to guarantee that other more impacts could not happen so that single impact periodic responses can occur. Subsequently simulation illustrates the theoretical results.After that,by using the theory of manifold and the method of composition mapping , the discontinuity mapping of the system is deduced and it composites the process of motion manifold and the process of impact.On the basis of a particular known motion trajectory (such as a trajectory of single impact period-n responses),the future motion states of a given point on the trajectory and other points in the proximity of it can be determined subsequently.Finally, in this paper the Poincare mapping with definite phase surface as Poincare section is established ,then stability of periodic motions of the system is investigated by using the deduced discontinuity mapping and the method of perturbation analysis.