Analysis of the rolling motion of loaded hoops

[摘要] This dissertation contains a detailed report on the results of a research project on the behaviourof a dynamical system consisting of a hoop to which a heavy particle is fixed at the rim. Thisloaded hoop rolls on a rough surface while remaining in the vertical plane. The motion ofthe 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 ordinarydifferential equations, one for each of the three degrees of freedom. Analytic solutions areobtained 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 onthe somewhat controversial question of what happens after the normal reaction becomes zeroin a position where the particle is still moving downwards. A new result shows that the hoopcan 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 summarisedin the form of phase diagrams. Some emphasis is placed on the rather amazing numberof 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 explainingone aspect of the reported behaviour of experimental hoops, namely hopping while the particleis moving downwards.A chapter on experimental models rounds off the project.

[发布机构] Stellenbosch University