Subharmonic Motion of Particles in a Vibrating Tube
  We study the motion of strongly inelastic particles in a narrow vibrating tube using molecular dynamics simulation. At low frequency of the vibration, we observe qualitative changes of the motion, as the depth of the pile increases. The center of mass of the particle cloud can be described by a superposition of modes of different frequencies. For certain values of the depth, a single mode dominates. The frequency of the dominant mode is 1/2, 1/3, or 1/1 of the vibration. We suggest that the behavior can be understood in terms of a new time-scale $\tau$, reflecting the recompaction time for a finite-depth pile. 
Jysoo Lee