Motion of a Single Bead on a Bead Row: Theoretical Investigations

Abstract
The motion of a single bead on an inclined "line" made up of juxtaposed identical beads is analytically investigated. Initially, an entirely analytical method is presented in some detail: the problem is tractable with the usual analytical tools as long as any slip motion is ignored. Under these conditions, we show i) the existence of a steady regime within a certain range of line slopes and ii) that the mean velocity of the mobile bead may be calculated by induction. When compared to all the series of experiments (after the adjustment of the collisional parameter), this approach reflects well all the features displayed by experiments. The influence of slip is then numerically investigated, but, unfortunately, the unicity of solution to motion equations cannot be proved. Despite this uncertainty, energy considerations show that the main dissipation process is due to collisions. The analytical model is then compared to three alternative types of approach, namely a phenomenological approach, an heuristical model, and one based on molecular-dynamics model. The comparison throws light on the role of (solid) friction and collisions in granular materials: here, it is found that the macroscopic coefficient of friction (bulk resistance) is mainly due to geometrical characteristics and collisional effects.