Heteroclinic bifurcation analysis of the tippedisk through the use of Melnikov theory

Abstract

The tippedisk is a mechanical–mathematical archetype for friction-induced instabilities caused by geometry and interaction with a frictional support. The instability leads to a counterintuitive rise of the centre of gravity when an unbalanced disc is spun rapidly about an in-plane axis. To understand the qualitative behaviour of the tippedisk, a nonlinear analysis is performed, revealing the singularly perturbed structure of the system equations. Application of singular perturbation theory shows that the long-term behaviour is dominated by a two-dimensional slow manifold, on which the asymptotic dynamics takes place. Moreover, Melnikov theory is used to derive a closed-form approximation of a heteroclinic bifurcation, which allows general statements to be made about the dynamic behaviour of the tippedisk.