Air-effect On Subharmonic Bifurcations Of Granular System Under Vertical Vibrations

Granular materials can exhibite lots of collective behaviors under vertical vibrated excitations, such as vibrating segregation of granular mixture(Brazil Nut Effect), convection, suface standing waves,"sandwich"structure, etc. When vibrated peirodically, the granular system would not give linear response to excitations if vibrating strengths exceed a critical value, and subharmonic motion can be observed. Reserches indicated that period doubling motion is controlled by normalized vibration acceleration. Meanwhile, air-effect on dynamic behaviors of the granular system can not be ignored, but the physical mechanism of air-effect on subharmonic motion is still unknown due to the complexity of these effects.In this paper, waht is studied is air-effect on the bifurcation process of the system in containers with solid-, vacuum-, and porous- bottom. How air-effect generates and its relationship with sizes of granular materials are analyzed, and the physical mechanism of this effect is clarified.In the solid-bottom container, the granular system can show subharmonic motion in the sequence of period-doubling, period-quadrpling, chaos, period-tripling, period-sextupling, chaos, period-quadrpling, period-octupling, chaos, etc. By using smaller and smaller particles, air-effect is enhanced, bifurcation points become larger, and the higher-order bifurcations disappear gradually. There is a critical size. Sizes larger than this value, interstitial air has little influence on the system. If sizes smaller, air flow plays an important role, the bifurcation processes are effected to change. When sizes are small enough, the system only exhibites period-doubling and period-quadrpling bifurcations. In the vacuum container, air-effect can be eliminated, differences in bifurcation process due to the different sizes would vanish, and the granular system with all sizes can undergo the same bifurcation sequence with sytems of larger granular.Theoretical analysis is mainly about adding air-effect expression to the completely inelastic bouncing ball model. By simulating the subharmonic motion of the granular system, motion trajectories can be obtained just like what are observed in experiments, and bifurcation processes caculated coincide well with experiments.