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Title:  Dynamics of onedimensional inelastic particle systems 
Other Titles:  Yi wei fei tan xing li zi xi tong zhong de dong li xue yan jiu 一維非彈性粒子系統中的動力學研究 
Authors:  Yang, Rong ( 楊榮) 
Department:  Department of Mathematics 
Degree:  Doctor of Philosophy 
Issue Date:  2011 
Publisher:  City University of Hong Kong 
Subjects:  Dynamics of a particle. Deep inelastic collisions. 
Notes:  CityU Call Number: QA851 .Y36 2011 v, 97 leaves 30 cm. Thesis (Ph.D.)City University of Hong Kong, 2011. Includes bibliographical references (leaves [92]97) 
Type:  thesis 
Abstract:  In this thesis, we investigate the dynamics of onedimensional 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 codimension 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 codimension 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
selfsimilar 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 selfsimilar orbits reduces to solving an eigenvalue problem to obtain the
particle velocities and solving a linear system to obtain the locations of interparticle
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 selfsimilar orbits can coexist. 
Online Catalog Link:  http://lib.cityu.edu.hk/record=b4086911 
Appears in Collections:  MA  Doctor of Philosophy

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