On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform

TL;DR

This paper analytically investigates the inelastic collapse of a bouncing ball on a randomly vibrating platform, revealing how the distributions of collapse time and flights change from power-law to exponential tails with inelasticity.

Contribution

It provides an exact analysis of the transition from universal power-law to nonuniversal exponential tail distributions in inelastic bouncing dynamics.

Findings

Distributions of flights and collapse time have exponential tails in the inelastic case.

Decay exponents depend on the coefficient of restitution and vanish at the elastic limit.

Explicit expression for the decay exponent is derived for a specific noise distribution.

Abstract

We study analytically the dynamics of a ball bouncing inelastically on a randomly vibrating platform, as a simple toy model of inelastic collapse. Of principal interest are the distributions of the number of flights n_f till the collapse and the total time \tau_c elapsed before the collapse. In the strictly elastic case, both distributions have power law tails characterised by exponents which are universal, i.e., independent of the details of the platform noise distribution. In the inelastic case, both distributions have exponential tails: P(n_f) ~ exp[-\theta_1 n_f] and P(\tau_c) ~ exp[-\theta_2 \tau_c]. The decay exponents \theta_1 and \theta_2 depend continuously on the coefficient of restitution and are nonuniversal; however as one approches the elastic limit, they vanish in a universal manner that we compute exactly. An explicit expression for \theta_1 is provided for a particular case of the platform noise distribution.