| Gaps in mechanical dynamical systems often lead to collisions between components or between components and constraints during motion.The collisions make mechanical systems with gaps a typical class of non-smooth dynamical systems and exhibit rich dynamical behavior.Many examples of collisional vibration systems with gaps exist in engineering practice.Since the study of single-degree-of-freedom collision vibration systems has both theoretical and practical significance,this paper considers the dynamics of single-degree-of-freedom linear vibration systems and Duffing vibration systems under plastic collision conditions,applies numerical simulation methods and cell mapping method to study the types of periodic motion and existence domains of the systems in the two-dimensional parameter space,and the behaviors of the shock vibration systems such as the codimension-1 and codimension-2 sliding bifurcations as well as the the chaotic boundary crisis are revealed.The critical speed of the serpentine motion of a high-speed rolling stock is studied,and its vibration characteristics are simulated when it runs on tracks with different curve radii at different initial speeds.First,the mechanical model of a single-degree-of-freedom linear vibration system under plastic collision conditions is considered.Theoretical analysis of the boundary conditions for the occurrence of grazing-sliding,switching-sliding,crossing-sliding and multi-sliding bifurcations of the system is carried out.The numerical simulation method is applied to calculate the type of periodic motion modes and the occurrence regions of the system in the two-parameter space.The bifurcation characteristics of single shock cycle motion are discussed in combination with single-parameter bifurcation diagrams,phase diagrams and time trajectories diagrams.The non-smooth bifurcations such as crossing-sliding,switching-sliding and codimension-2 sliding bifurcations as well as the discontinuous bifurcations such as grazing bifurcation and boundary crisis are revealed.There are two types of transition domains in the transition between the periodic motion(1,0,0)and periodic motion(1,1,0): the hysteresis domain and the subharmonic inclusion domain.The connection points of the hysteresis and subharmonic inclusion domains are the codimension-2 point of grazing bifurcation of the periodic motion(1,0,0)and the point of the codimension-2 point of double-saddle-node bifurcation of the periodic motion(1,1,0).The non-sticking periodic motion and sticking periodic motion transit into each other via a crossing-sliding bifurcation.In the two-dimensional parameter region of the low frequency and small clearance,the periodic motions(1,1,1)and(1,2,1)are distributed alternately.The boundary between the two types of periodic motions is a switching-sliding bifurcation curve.The boundary crisis of chaos occurs leading the chaotic attractor and its basin to vanish.Then,the mechanical model of the Duffing vibration system under plastic collision conditions is considered.The numerical simulation method is applied to calculate the type of periodic motion modes and occurrence regions of the system in the two-parameter space.The bifurcation characteristics of single shock cycle motion are discussed in combination with single-parameter bifurcation diagrams,phase diagrams and time trajectories diagrams.The non-smooth bifurcations such as crossing-sliding,switching-sliding and codimension-2sliding bifurcations as well as the discontinuous bifurcations such as grazing bifurcation and boundary crisis are revealed.In the low-frequency small gap region,the periodic motions(1,1,0)and(1,1,1)transit each other through the sliding bifurcation.There exists a singularity point on the boundary curve of its two-parameter domain.This singularity point is the codimension-2 point of grazing-crossing-sliding bifurcation point for periodic motion(1,1,0)and the codimension-2 point of crossing-switching-sliding bifurcation point for periodic motion(1,1,1).Finally,the integrated vehicle-line model was established by using the multi-body dynamics software UM Input,and then the vehicle operation was simulated by using the UM Simulation simulation software,and the critical velocity of the serpentine motion of the vehicle was solved by using the scanning method and the downward method successively.The transverse and vertical vibration displacement and vibration acceleration of the vehicle are simulated when it runs on the track with different curve radius at different speeds,and the transverse and vertical Sperling smoothness indices of the vehicle are calculated and analyzed to comprehensively analyze the vehicle dynamics performance.The results show that the smoothness of this type of rolling stock is better when running on a track with a radius of6000 m at a speed of 300km/h. |
