Effects Of Wall Friction On Subharmonic Bifurcations In Vibrated Granular System

Scatter system is an assembly of large number of small solid particles (whose scale isμm), such as gains, sands, minerals, concretes etc, under agitation of external periodic vibrations, can show some non-linear and non-equilibrium behaviors. For example, when the vibrated tension exceed some critical value, the impact strength, controlled by the normalized vibration acceleration, will undergo a series of subharmonic bifurcations in the sequence of period-doubling, period-quadrupling, cluster region, period-tripling, period-sextupling, cluster region, and so on. The physical mechanism of the bifurcation is the frequent inelastic collisions between granular bed and the vibrated plate, in vibrated granular systems. So we consider the granular bed as a completely inelastic ball, and set up the bouncing ball model. Shown that the theory results agree well with the low order subharmonic bifurcations, but lower than the experimental high order subharmonic bifurcations. That's because the air and wall effects are not regarded. In the bouncing ball model, besides gravity, there also be air and wall friction during the ball flying.In this paper, as an initial approach, we assume that all external force of the bouncing ball (including the wall frictional force and air resistance force) is constant and no link with the velocity, in the earth frame of reference. Then we build the bouncing ball flying equation in vibrated system. According it we use the software of Matlab draw the diagrams of the bifurcation motion and the formula of the bifurcation points, and provide the bifurcation of the bouncing ball. All stimulations demonstrate that a constant resistance force acting on the ball during its flight will not change the sequence of bifurcation, but it can enlarge the value of bifurcation points,which make the theory value is agree with the experimental values. In the bifurcation diagrams of the ball flight time and relative velocity, structure consisting of densely accumulated cascades of period-doubling bifurcation also exists.Then we put the vibrated plate as the frame of reference,and build the flying equation, comprising the wall frictional force. We simulate bifurcation motion and the value of the bifurcation motion by using Matlab, which get a good agreement with the experiment. Shown that,for the system of larger granular (≥0.5mm),when the control parameter is larger, a good agreement is obtained when the frictional force from the container wall accounted by the bouncing ball model, is set as a constant one and assigned a value of 20-30% of the total granular plate?s gravity and is a leading key which effects the global motion of the granular system. A comparison with the period-doubling bifurcations observed in the vibrated plate is made. In the motion of the granular system, the direction of the wall frictional force will change, after entering into the cluster regions, this change will be more continuously, and it will cause the flying height lower and flying time shorter of the granular system. This is the fundamental cause of the change of subharmonic bifurcations when the wall friction is considered. After the wall frictional force is considered and view vibrated plate as frame of reference, we can give the period-doubling bifurcation process in vibrated granular system a better explanation.