Dynamics of one-dimensional inelastic particle systems

Abstract

In this thesis, we investigate the dynamics of one-dimensional inelastic particle systems composed of rigid, frictionless, inelastic particles. Collisions between particles are assumed to be inelastic with constant coefficient of restitution, and between collisions the particles move with constant velocity. We consider two different models and in each case consider the dynamics of an arbitrary number of particles of arbitrary mass. First, we consider a system bounded by two walls, with external forcing from one of the walls. Second, we consider a system with periodic boundary conditions, that can also be thought of as a set of particles on a ring. We show that both systems exhibit surprising behavior that is completely absent in equivalent elastic systems. In the first case, we investigate continuous transitions between different periodic orbits. We show that continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. We give a full mechanical description of the system and explain why this is the case. Surprisingly, we also show that there are an infinite set of degenerate transitions of co-dimension 1. We provide a theoretical analysis that gives a simple criteria to classify which transitions are degenerate purely using the discrete set of collisions that occur in the orbits. Our analysis allows us to understand the nature of the degeneracy. We also show that higher degrees of degeneracy can occur, and provide an explanation. In the second case, we consider the dynamics of sequences of collisions that are self-similar in the sense that the relative positions return to their original relative positions after the collision sequence, while the relative velocities are reduced by a constant factor. For a given collision sequence, we develop the analytic machinery to determine the particle velocities and the locations of collisions, and show that the problem of determining self-similar orbits reduces to solving an eigenvalue problem to obtain the particle velocities and solving a linear system to obtain the locations of inter-particle collisions. For inelastic systems, we show that the collision locations can always be uniquely determined. We also show that this is in sharp contrast to the case of elastic systems in which infinite families of self-similar orbits can co-exist.

Date of Award	15 Jul 2011
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Jonathan James WYLIE (Supervisor)