Vol. 6, No. 1-4, 2011

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Tapping dynamics for a column of particles and beyond

Denis Blackmore, Anthony Rosato, Xavier Tricoche, Kevin Urban and Vishagan Ratnaswamy

Vol. 6 (2011), No. 1-4, 71–86

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.

This paper is respectfully dedicated to Marie-Louise and Charles Steele.

Newtonian models, periodic taps, Hamiltonian system, long-wave limit, BSR approximation
Received: 18 September 2010
Revised: 22 November 2010
Accepted: 22 November 2010
Published: 28 June 2011
Denis Blackmore
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics
New Jersey Institute of Technology
University Heights
Newark, NJ 07102-1982
United States
Anthony Rosato
Department of Mechanical Engineering
New Jersey Institute of Technology
Newark, NJ 07102-1982
United States
Xavier Tricoche
Department of Computer Science
Purdue University
West Lafayette, IN 47907-2107
United States
Kevin Urban
Center for Solar-Terrestrial Research
New Jersey Institute of Technology
Newark, NJ 07102-1982
United States
Vishagan Ratnaswamy
Department of Mechanical Engineering
New Jersey Institute of Technology
Newark, NJ 07102-1982
United States