An investigation of forces on intruder in a granular material under vertical vibration

A rod in a vertically vibrated bed has an equilibrium height regardless of its initial placed position. The function relationship between the mass of the rod and the equilibrium height (mrod ~ hs) has an inflection point, that the absolute value of the slope of mrod ~ hs (|Δmrod/Δhs|) increases firstly, and then decreases along the bed depth. The point divides the bed into two zones - zone (I) where Frod increases along the bed depth and zone (II) where Frod decreases along the bed depth. At the zone I, the force on a completely immersed sphere increases along the bed depth; by contraries, at the zone II, it decrease. The immersed sphere has a max force at the boundary of two zones (inflection point). Therefore, a sphere can show different motion when initially placing it at different zones.