Complex Bifurcations And Chaos In Vibro-impact Systems With A Rigid Stop

In the mechanical motion,impacting vibration is caused often by the clearance,and it generally introduces essential non-smoothness and non-linearity,which can exhibit complicated dynamical behavior such as bifurcations and chaos.The grazing is a kind of special phenomenon,where the oscillator tends to graze in impacting systems.The co-dimension-two grazing bifurcation and chaotic dynamics in vibro-impact systems with a rigid stop are considered in this thesis,and the contributions of main work are as follows:In the first part of the thesis,for a class of single degree of freedom vibro-impact system with a rigid stop and with a clearance,the global Poincare mapping of a grazing periodic motion is obtained by a combination of the discontinuity mapping and the local Poincare map derived in the neighborhood of the grazing point.It is used to discuss the stability of the grazing periodic motion,and the analytical criterion that makes co-dimension-two grazing bifurcation to happen in the vibro-impact system is derived.Finally the grazing curve and co-dimension-two grazing bifurcation points on this grazing curve are drawn by numerical simulation of the system,and the dynamic behaviors of bifurcation and chaos of the system in the vicinity of co-dimension-two grazing bifurcation points are analyzed.In the second part of this thesis,a general two degrees of freedom vibro-impact system with a rigid stop is investigated.At first,the local Poincare map near the grazing point is built,and by use of discontinuity mapping of grazing periodic motion,the global Poincare map of grazing periodic motion is obtained.The stability of grazing periodic trajectory is analyzed as well as the computational formula that the system must satisfy for co-dimension-two grazing bifurcation to occur is obtained.Then the co-dimension-two grazing bifurcation points of the system are drawn via the numerical simulation of the original system.Eventually,the unfolding diagram of the system is analyzed in the vicinity of co-dimension-two grazing bifurcation points,showing the phase portraits of different types of periodic motions and chaos produced by the system in different bifurcation sets.