Energy of a single bead bouncing on a vibrating plate: Experiments and numerical simulations

International audienceThe energy of a single bead bouncing on a vibrating plate is determined in simulations and experiments by tracking the bead-plate collision times. The plate oscillates sinusoidally along the vertical with the dimension-less peak acceleration ⌫, and the bead-plate collisions are characterized by the velocity restitution coefficient ⑀. Above the threshold dimensionless peak acceleration ⌫ s Ӎ0.85, 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 v p 2 /(1Ϫ⑀), where v p is the peak velocity of the plate, only in the limit ⌫ӷ⌫ s. The threshold ⌫ s is shown to decrease when the bead is subjected, in simulations, to additional nondissipative collisions occurring with the typical frequency ␯ c. As a consequence, the bead energy scales like v p 2 /(1Ϫ⑀) for all vibration strengths in the limit ␯ c ӷ␯ c *. From the experimental and numerical findings, an analytical expression of the bead energy as a function of the experimental parameters is proposed. As they exhibit a wide range of unusual behaviors, granu-lar materials are the subject of intensive investigations ͓1–3͔. Because of the inelastic nature of the contacts between the grains, these systems are intrinsically dissipative. In order to explore experimentally the effects of the dissipation on the properties of granular systems, it is convenient to produce stationary states; they are achieved by continuously providing the system with energy, compensating the energy intrin-sically lost when the grains are in motion. Among these studies , one can note the experimental realization of two-dimensional (2D) granular gases, consisting of inelastic beads constituting less than one-layer coverage on a vertically shaken, horizontal plate ͓4 –8͔ ͑the experimental situation has been the framework of molecular-dynamics simulations by Nie et al. ͓9͔͒. The velocity distributions, granular temperature, pressure, as well as phase transitions have been studied as functions of the vibration strength, usually characterized by the peak plate acceleration ⌫. Nevertheless, detailed analysis of the clustering transition and of the pressure indicates that the vibrating boundary becomes inefficient to thermalize the system when the acceleration or the density of the gas are decreased. The energy input by the vibrating boundary has been the subject of several theoretical studies ͓10–13͔; the scaling law for the energy as a function of the vibration strength has been shown to depend on the shape of the boundary vibration ͑sinusoidal, sawtooth, etc.͒ and on the nature of the dissipation within the gas ͑viscous, inelastic͒. In these studies, the assumption was made that the bead impinges randomly on the boundary; we note that this is not the case when the bead collides more than once with the boundary between two collisions with another bead. One can easily show that the velocity is exponentially correlated between two successive collisions in this case. It is hence relevant to study the dynamical behavior of a single bead bouncing on a vibrating plate, and to determine the mean energy ͗E͘ of th