Memory in random bouncing ball dynamics

TL;DR

This study explores how the memory effects in a bouncing ball model are influenced by the correlation time of the vibrating plate's motion, revealing a transition from Markovian to memory-dependent dynamics.

Contribution

It introduces a numerical analysis of the bouncing ball model with continuous random excitation, highlighting the impact of excitation bandwidth on system memory and dynamics.

Findings

Memory effects depend on the ratio of flight time to excitation peaks.

Markovian approximation is valid above a certain ratio threshold.

Low excitation regimes exhibit complex behaviors like chattering.

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 Poincar{\'e} 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.