Tapping dynamics for a column of particles and beyond

Abstract

The dynamics of a vertical stack of particles subject to gravity and a sequence of small, periodically applied taps is considered. First, the motion of the particles, assumed to be identical, is modeled as a system of ordinary differential equations, which is analyzed with an eye to observing connections with finite-dimensional Hamiltonian systems. Then, two approaches to obtaining approximate continuum models for large numbers of particles are described: the long-wave approximation that yields partial differential equations and the BSR method that employs integro-partial differential models. These approximate continuum models, which comprise infinite-dimensional dynamical systems, are studied with a focus on nonlinear wave type behavior, which naturally leads to investigating links to infinite-dimensional Hamiltonian systems. Several examples are solved numerically to show similarities among the solution properties of the finite-dimensional (lattice-dynamics), and the approximate long-wave and BSR continuum models. Extensions to higher dimensions and more general dynamically driven particle configurations are also sketched.