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

The 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 dimensionless peak acceleration Gamma, and the bead-plate collisions are characterized by the velocity restitution coefficient epsilon. Above the threshold dimensionless peak acceleration Gamma(s) approximately 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 v(2)(p)/(1-epsilon), where v(p) is the peak velocity of the plate, only in the limit Gamma>>Gamma(s). The threshold Gamma(s) is shown to decrease when the bead is subjected, in simulations, to additional nondissipative collisions occurring with the typical frequency nu(c). As a consequence, the bead energy scales like v(2)(p)/(1-epsilon) for all vibration strengths in the limit nu(c)>>nu(*)(c). From the experimental and numerical findings, an analytical expression of the bead energy as a function of the experimental parameters is proposed.