Dynamics of an elastic ball bouncing on an oscillating plane and the oscillon

Abstract

The oscillon is a highly localized dynamical phenomena occurring in a thin horizontal layer of granular material, which rests on a rigid metal plate and the plate oscillates in the vertical direction. It is axially symmetric and physically resembles a splash of liquid due to a falling drop, except that it continually perpetuates itself and does not generate a spreading wave, as is the case for a liquid splash. If the plate vibrates with amplitude A and period T=2π/ ω, then the oscillon moves from “peak” to “crater” in time T 1 and “crater” to “peak” time T 2, such that the time from “peak” to “peak” or “crater” to “crater” is twice the period of the oscillating plate namely T 1+ T 2=2 T. At present the physics of granular phenomena is not properly understood and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behavior. Here we present an elementary analysis of a single elastic ball bouncing on an oscillating plate, and we demonstrate that under certain circumstances the ball can perform a “big” bounce followed by a “little” bounce, and then simply repeat the sequence ad infinitum. For a perfectly elastic ball initially at rest on the oscillating plate, the theory with T 1= T 2 predicts oscillonic behavior with an acceleration amplitude Γ= Aω 2/ g ( g is the acceleration due to gravity) of about 4.6, while experimentally oscillons have been observed to occur for Γ around 2.5. However, for T 1≠ T 2 the theory predicts oscillonic behavior for values of Γ which are well in accord with those observed experimentally. The elementary analysis presented here at least provides specific alternative Γ values for future experimentation, as well providing some insight into what is otherwise a complex physical phenomena.