Research On Different Types Of Co-dimension-two Grazing Bifurcations Of Two Kinds Of N-degree-of-freedom Vibro-impact System

The impact vibration system widely exists in the actual production and life,and is a kind of non-smooth system worthy of study.At present,most of the research focuses on some low degree of freedom systems.In this thesis,a class of n-degree-of-freedom vibro-impact system with symmetric elastic constraints and a class of n-degree-of-freedom vibro-impact system with unilateral rigid constraint are considered.These two classes of systems have more general research results,and have certain theoretical guiding significance for improving the operation stability,work efficiency and service life of related mechanical impact vibration equipment.The main results of the thesis are as follows:Firstly,for a class of n-degree-of-freedom impact vibration system with symmetric elastic constraints,the discontinuous mapping under two-sided elastic constraints is discussed.The global Poincaré mapping of double grazing periodic motion is obtained by combining the discontinuous mapping and smooth Poincaré mapping.And the stability criterion of double grazing periodic motion is analyzed based on the global Poincaré mapping.It is found that the grazing bifurcation can lead to instability in three cases.Furthermore,the expressions for a class of degenerate grazing bifurcation are obtained.According to the theoretical results,a two-degree-of-freedom vibro-impact system with symmetric elastic constraints is studied in detail.The grazing curve and the co-dimension-two grazing bifurcation point are presented via numerical simulation.Meanwhile,we show the unfolding diagram and phase portraits to display various types of periodic motions near the co-dimension-two grazing bifurcation points.Secondly,the dynamic behavior near the grazing periodic orbit is discussed for the n-degree-of-freedom vibro-impact system with unilateral rigid stop.The existence conditions of saddle-node bifurcation and period-doubling bifurcation for periodic motion with a collision in m-period are deduced theoretically.Furthermore,the existence conditions of co-dimension-two bifurcation are obtained.To prove the generality and validity of the theoretical results,a three degree of freedom unilateral rigid constraint system is selected as an example for analysis,and the conditions for simultaneous occurrence of saddle-node bifurcation,period-doubling bifurcation and grazing bifurcation are obtained.The periodic motion and chaotic motion of the system near co-dimension-two bifurcation points are studied by combining Lyapunov exponent and the local bifurcation diagram.The numerical results show that the periodic motion and chaotic motion of the system appear alternately in a certain range of parameters.Therefore,the numerical simulation is in good agreement with the theoretical analysis.