Effect of frictional force on subharmonic bifurcations of a completely inelastic ball bouncing on a vibrating table

The behavior of a completely inelastic ball bouncing on a vertically vibrating table in the presence of frictional force is investigated. The frictional force is assumed to be constant. It is found that the sequence of bifurcation, controlled solely by the normalized vibration acceleration Gamma, is the same as that in the absence of frictional force, but the value of each bifurcation point becomes larger. In the bifurcation diagram of ball flight time, the structure consisting of an infinity of bifurcation cascades in a narrow range of Gamma is observed. Compared with that of no frictional force, it is longitudinally compressed and transversely stretched, and has a different fractal property. A comparison with the bifurcations observed in vertically vibrated granular beds is also made. When the fractional force is set to be 20%-30% of the whole weight of the particles, the results from the bouncing ball model are in good agreement with experimental observations.