Periodic Vibration And Bifurcation Of The Piecewise Smooth Vibration System

A two-degree-of-freeedom periodically-forced piecewise smooth vibration system with aclearance and friction is firstly considered in the paper. The correlative relationship andmatching law between dynamic performance and system parameters are analyzed. Two keyparameters of the system, the exciting frequencyω and clearanceδ, are first considered toanalyze the influence of the main factors on dynamic performance of the system. There aretwo state of being of the system’s periodic impact movements: pure slide and sliding-stictionstate, the transform between these two states of the system is called sliding bifurcation. Thefriction coefficient between the mass and the belt pulley and the constant running speed of thebelt pulley have an important influence on the existence and distribution of the sliding-stictionregion of the system’s periodic impact movements. A series of grazing bifurcations occurwith decreasing the exciting frequency so that the number p of impacts of the fundamentalgroup of motions increases one by one. With the increase of the impact number p, the impactvelocities become weaker and weaker by degrees in an excitation period and the region of theperiodic p/1motion becomes narrower. With the increase of the exciting frequency a seriesof subharmonic periotic motions1/n (1,2,3,...,)appear sequentially and the long-periodicor chaos impact motions exist between the adjacent subharmonic periotic movements of thewindow intermittently. A series of singular points between any two adjacent ones of thefundamental groups of motions are found, i.e., real-grazing ang bare-grazing bifurcationboundaries of the one that owns the smaller impact number, saddle-node andperiodic-doubling bifurcation boundaries of the other mutually cross themselves at the pointof intersection and creat the transition region: tongues-shaped regions as the paper called. Aseries of zones of complex and organized subharmonic impact motions are found to appear inthe tongues-shaped regions, which derive from the correlative fundamental p/1motions andsubhaomonic1/nmotions, associated with the number p of impacts of the fundamentalmotion and the number n of the exciting force periods of1/nmotion. The influence ofsystem parameters on impact velocities, existence regions and correlative distribution ofdifferent types of periodic-impact motions of the system is emphatically studied bymulti-parameter and multi-performance co-simulation analysis.The dynamic model of Railway passenger car bogie is built through the standpoint ofthe cartrack coupling dynamics theory and by applying the analysis method of the mordennonlinear dynamics. The process takes the elastic collision between wheel set and railway intoconsideration, namely considering the railway as another suspension component of thedynamic model of Railway passenger car bogie and using the piecewise function of the elastic collision force to express the influence of the railway elastic collision force to the wheelset.The reaserch is divided into two parts: the wheelset is newused and after subjected to fairwear. The paper firstly studies the dynamic performance of the system under selected basicparameters of the two stages. Then the influence of some important structure parametersincluding the primary suspension stiffness of the axle box, the coefficient of coulombfrication between the wheelset and railway, the wheelset tread gradient and the running speedof the system to the dynamic behavior is studied by the method of analyzing the impactvelocities, existence regions and correlative distribution of different types of periodic-impactmotions of the system in the parameter plane constituted by some variational parameter andthe exciting frequency in a certain range which has been stated clearly. The correlativerelationship and matching law between dynamic performance and some important structuresystem parameters and the running speed are analyzed. As a result, the theoretical basis ofimproving the dynamic performance of the system is provided, then a reasonable choice rangeof parameters is confirmed so that to improve the maximum running speed that the system’ssimplest periodic-impact motion can reach, namely the paramenter optimization can be done.