Research On Co-Dimension-Two Grazing Bifurcations Of Two Kinds Of Two-Degree-of-Freedom Vibro-Impact System

When the grazing bifurcation and the smooth bifurcations take place simultaneously,the co-dimension-two grazing bifurcations of the non-smooth system will occur.In this paper,the co-dimension-two grazing bifurcations of two kinds of two-degree-of-freedom vibro-impact systems are studied.The first is a two-degree-of-freedom vibro-impact system with unilateral constraints.Firstly,the existence conditions of grazing periodic motion are discussed.The global Poincaré mapping for the 1/n impact period motions is constructed by using the discontinuous mapping method,and the bifurcation conditions for the 1/n impact period motions are obtained.Secondly,an analytical expression of co-dimension-two grazing bifurcations is derived by combining the conditions of grazing bifurcation and the bifurcation conditions of the 1/n impact period motions.Based on the expression,the distribution of co-dimensional-two bifurcation points of the system under different periods is analyzed by numerical simulation.Finally,the effectiveness of theoretical analysis is verified by comparing the bifurcation diagrams obtained by the global Poincaré map and the differential system.The second is a two-degree-of-freedom vibro-impact system with symmetric constraints.Firstly,considering the periodic motion of the double-sided grazing,the existence condition of the periodic motion of thedouble-sided grazing was deduced theoretically.Using the discontinuous mapping method,the analytical expressions of saddle-node bifurcation and period doubling bifurcation for 1/1/n impact periodic motion are derived.Secondly,combining the existence condition of the grazing periodic motion and the bifurcation condition of the 1/1/n impact periodic motion,the analytical expression of co-dimensional-two grazing bifurcation was obtained,and The distribution of co-dimensional-two bifurcation points is analyzed when the period is 1.Finally,the dynamic behavior near the co-dimensional-two bifurcation points is numerically simulated.By comparing the bifurcation diagrams near the co-dimensional-two bifurcation points obtained from the global Poincaré map and the original system,the validity of the theoretical analysis is verified.