# Analysis of the rolling motion of loaded hoops

 dc.contributor.advisor Maritz, M. F. en_ZA dc.contributor.author Theron, Willem F. D. en_ZA dc.contributor.other University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics. dc.date.accessioned 2008-06-23T12:47:27Z en_ZA dc.date.accessioned 2010-06-01T08:15:02Z dc.date.available 2008-06-23T12:47:27Z en_ZA dc.date.available 2010-06-01T08:15:02Z dc.date.issued 2008-03 dc.identifier.uri http://hdl.handle.net/10019.1/1206 dc.description Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008. dc.description.abstract This dissertation contains a detailed report on the results of a research project on the behaviour en_ZA 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. dc.language.iso en en_ZA dc.publisher Stellenbosch : University of Stellenbosch dc.subject Classical mechanics en_ZA dc.subject Rolling motion en_ZA dc.subject Hopping hoops en_ZA dc.subject Loaded wheels en_ZA dc.subject Dynamics dc.subject Equations of motion dc.subject Lagrange equations dc.subject Dissertations -- Applied mathematics dc.subject Theses -- Applied mathematics dc.title Analysis of the rolling motion of loaded hoops en_ZA dc.type Thesis en_ZA dc.rights.holder University of Stellenbosch
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