Energy input and scaling laws for a single particle vibrating in one dimension

The one-dimensional motion of a single particle on a vibrating base is considered in the limit of high excitation (vibration frequency {} collision rate). An exact expression for the time averaged rate of energy input from the vibrating base to the particle is derived. By assuming a Gaussian form for the particle velocity distribution function, the expression can be numerically evaluated to obtain the one particle granular temperature as a function of the base velocity V and particle-base restitution coefficient {ε}. The granular temperature is shown to scale as V² and to scale approximately as (1-{ε})^-1. The velocity scaling is also shown to hold over a generic class of velocity distribution functions. Assuming sinusoidal excitation yields scaling behavior identical to the sawtooth excitations used in the analysis, two different stable states can exist [(i) particle bouncing and (ii) particle not bouncing] when the peak base acceleration is less than g.