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 Γ, and the bead-plate collisions are characterized by the velocity restitution coefficient ε. Above the threshold dimensionless peak acceleration Γₛ0.85, 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 vₚ²/(1-ε), where vₚ is the peak velocity of the plate, only in the limit ΓΓₛ. The threshold Γₛ is shown to decrease when the bead is subjected, in simulations, to additional nondissipative collisions occurring with the typical frequency νc. As a consequence, the bead energy scales like vₚ²/(1-ε) for all vibration strengths in the limit νcνc*. From the experimental and numerical findings, an analytical expression of the bead energy as a function of the experimental parameters is proposed.