Stability And Bifurcation Of Mechanical Vibration System With Asymmetric Excitation

In many mechanical systems,the movement of some parts is limited by constraints.Because of the non-smoothness of the vector field and the discontinuity of Jacobi matrix in the collision mechanical vibration system,it has rich dynamic behavior,which is worth further study.Moreover,a large number of practical engineering problems urgently need people to have a deeper and more comprehensive understanding of the dynamic behavior of the vibration system of collision machinery.Therefore,the dynamic study of the vibration system of collision machinery has both theoretical significance and important engineering application value.There are many examples of mechanical vibration systems with asymmetric constraints such as asymmetric excitation or asymmetric clearance in engineering practice,such as gear transmission system,wheel-rail system running in curved track and so on.Firstly,the mechanical vibration system with asymmetric excitation is considered.The state space of the system is divided into three smooth sub-regions by the non-smooth interface.The smooth flow map of each sub-region and the switching map on the non-smooth interface are constructed,and the numerical method for calculating its Jacobi matrix is given,and then the Poincaré map and its Jacobi matrix of the periodic n attractor are synthesized.The direct numerical simulation method is used to identify the periodic attractor mode and parameter domain of the system in the two-parameter plane.The evolution of periodic attractors and Floquet multipliers are calculated and tracked by combining direct numerical simulation with extended shooting method.The stability and bifurcation of periodic attractors are judged based on Floquet theory.The attraction domain of coexisting attractors and the position of unstable attractors in the attraction domain are calculated by combining cell mapping method and shooting method.The migration characteristics of adjacent periodic 1 attractors in the two-parameter plane are studied,revealing the discontinuous bifurcation characteristics such as grazing-induced bifurcation and crisis,and the importance of unstable periodic attractors generated by saddle node bifurcation in the evolution of system dynamics.Under the condition of elastic collision,the saddle-node bifurcation induced by grazing produces a hysteresis domain in the transfer process of the attractor of adjacent period 1,while the period doubling bifurcation induced by grazing produces a subharmonic inclusion domain in the transfer process of the attractor of adjacent period 1.Both subharmonic inclusion domain and hysteresis domain are closed bifurcation curves.The unstable periodic attractor generated by saddle node bifurcation plays an important role in the evolution of system dynamics,and the unstable attractor is always located on the boundary of the attraction domain of coexisting attractors.When the attractor meets the coexisting stable periodic attractor,saddle node bifurcation occurs in the system,resulting in the disappearance of a pair of periodic attractors.When the attractor meets the coexisting chaotic attractor,the chaotic attractor suddenly changes its shape due to internal crisis or disappears due to boundary crisis.Finally,consider the mechanical vibration system with asymmetric excitation and rigid impact.The smooth flow map,the switching map between the fixed phase plane and the collision plane and the collision map of the system are constructed,and then the Poincaré map and its Jacobi matrix of the periodic n attractor are synthesized.The direct numerical simulation method is used to identify the periodic attractor mode and parameter domain of the system in the two-parameter plane.The two-parameter dynamics of the system is studied based on Floquet theory by combining direct numerical simulation,extended shooting method and cell mapping method.Under the rigid collision condition,the grazing bifurcation and saddle node bifurcation produce hysteresis domain in the transfer process of the attractors with adjacent period 1,while the grazing bifurcation and period doubling bifurcation produce subharmonic inclusion domain in the transfer process of the attractors with adjacent period 1.The global dynamics of coexisting attractors in hysteresis domain is studied by combining shooting method and cell mapping method.