A single bead bouncing on a vibrating plate

Collaboration with Claude Laroche

Introduction

We considered experimentally the case of a single bead bouncing on a vibrating plate. The plate oscillates sinusoidally along the vertical with the peak acceleration, G, and the bead-plate collisions are characterized by the velocity restitution coefficient, e. The situation has already been realized experimentally by many different groups since the early 50's. This system is probably one of the the most simple in which the period doubling route to chaos is observed. You can find a lot of informations on the subject in the nice website proposed by Piotr Pieranski. In order to discuss the thermalization of granular gases by vibrating boundaries, we determined experimentally the mean energy of the bead by tracking the bead-plate collision dates.

Results


Above a threshold acceleration
Gs = 0.85, which does not depend on the restitution coefficient, the bead energy is shown to initially increase linearly with the vibration amplitude, A, whereas it is found to scale like (Aw)2/(1-e) only in the limit G>>Gs (Fig.1).

Fig.1: Energy <E> vs. peak plate acceleration G.

The threshold acceleration Gs is shown to decrease when the bead is subjected, in simulations, to additional non-dissipative collisions occuring with the typical frequency nc. As a consequence, the bead energy scales like (Aw)2/(1-e) for all vibration strength in the limit nc >> 1.4 (1-e)1.5 n where n is the frequency of the plate oscillation. From the experimental and numerical findings, we propose an analytical expression of the bead energy as a function of the experimental parameters. The results are interesting for discussing the problem of the injection of energy by a vibrating boundary in the case of 2D granular gases.

Related publication

Energy of a single bead bouncing on a vibrating plate: experiments vs. numerical simulations,
Géminard J.-C., and Laroche C., Phys. Rev. E, 68 (2003) 031305.