The amazing variety of motions of a loaded hoop

• Published in: Mathematical and computer modelling Vol. 47; no. 9; pp. 1077 - 1088
• Main Authors: Theron, W.F.D., Maritz, M.F.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.05.2008; Elsevier Science
• Subjects: Classical mechanics; Eccentric hoops; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis. Scientific computation; Rolling hoops; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Classical mechanics; Eccentric hoops; Rolling hoops; Scientific computation; Applied mathematics; Friction coefficient; Numerical simulation; Mathematical model; Computer aided analysis
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/j.mcm.2007.06.031

The subject of loaded hoops, and their ability to hop under certain conditions, received some attention at the turn of the century, inter alia by Tokieda, Pritchett and Theron. In this paper the emphasis is rather on extending the analysis of the motion before the hoop hops, including those cases where the hoop does not hop. The results of numerical simulation based on a classical mechanical model of a loaded hoop is presented. Results show that the motion consists of an amazing variety of alternating phases of rolling and slipping, some of which may be absent. A list of thirty-six possible sequences is included. Phase diagrams are presented to illustrate the relationship between the various slipping and rolling phases, as well as how they depend on the eccentricity, the friction coefficient and the initial angular velocity.