## Analysis of the rolling motion of loaded hoops

##### Date

2008-03

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Stellenbosch : University of Stellenbosch

##### Abstract

This dissertation contains a detailed report on the results of a research project on the behaviour
of a dynamical system consisting of a hoop to which a heavy particle is fixed at the rim. This
loaded hoop rolls on a rough surface while remaining in the vertical plane. The motion of
the hoop consists of various, possibly alternating, phases consisting of rolling without slipping,
spinning or skidding motion and in some cases ends by hopping off the surface.
A general mathematical model is developed, consisting of a system of second order ordinary
differential equations, one for each of the three degrees of freedom. Analytic solutions are
obtained in some cases; otherwise numerical solutions are used.
Three specific applications of the general model are dealt with.
In the first application the problem of massless hoops is investigated. The main emphasis is on
the somewhat controversial question of what happens after the normal reaction becomes zero
in a position where the particle is still moving downwards. A new result shows that the hoop
can continue to move horizontally in a motion defined as skimming.
The second application deals with rigid hoops and a large number of detailed results are presented.
Classification schemes for the different types of behaviour are introduced and summarised
in the form of phase diagrams. Some emphasis is placed on the rather amazing number
of different patterns of motion that can be obtained by varying the parameters.
In the third application two elastic models are analysed, with the primary purpose of explaining
one aspect of the reported behaviour of experimental hoops, namely hopping while the particle
is moving downwards.
A chapter on experimental models rounds off the project.

##### Description

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008.

##### Keywords

Classical mechanics, Rolling motion, Hopping hoops, Loaded wheels, Dynamics, Equations of motion, Lagrange equations, Dissertations -- Applied mathematics, Theses -- Applied mathematics