Memory in random bouncing ball dynamics

Laboratoire de Tribologie et Dynamique des Systèmes, UMR5513, CNRS/Ecole Centrale de Lyon/Univ Lyon/ENISE/ENTPE - 36 Avenue Guy de Collongue, F-69134 Ecully, France

Received: 13 July 2016
Accepted: 20 September 2016

Abstract

The bouncing of an inelastic ball on a vibrating plate is a popular model used in various fields, from granular gases to nanometer-sized mechanical contacts. For random plate motion, so far, the model has been studied using Poincaré maps in which the excitation by the plate at successive bounces is assumed to be a discrete Markovian (memoryless) process. Here, we investigate numerically the behaviour of the model for continuous random excitations with tunable correlation time. We show that the system dynamics are controlled by the ratio of the Markovian mean flight time of the ball and the mean time between successive peaks in the motion of the exciting plate. When this ratio, which depends on the bandwidth of the excitation signal, exceeds a certain value, the Markovian approach is appropriate; below, memory of preceding excitations arises, leading to a significant decrease of the jump duration; at the smallest values of the ratio, chattering occurs. Overall, our results open the way for uses of the model in the low-excitation regime, which is still poorly understood.