Tippe-top on visco-elastic plane: steady-state motions, generalized smale diagrams and overturns

Abstract

We consider the dynamics of a tippe-top on a visco-elastic plane with dry friction. Tippetop is a rigid dynamically symmetric sphere with the center of mass that lies on the dynamical axis of symmetry but does not coincide with the center of the sphere. It was shown earlier [1] that the full mechanical energy conserves only on the steady-state motions of the tippe-top while the linear on the pseudovelocities Jellet integral remains constant for a quite general model of friction, for example, viscous friction. It allows studying the stability of the steady-state motions and constructing the generalized Smale diagrams [1]. Here we consider another model of friction—so-called distributed dry friction proposed in [2] that is more suitable for the convex bodies on a rough plane. In this case, both energy and Jellet function change their value on solutions. The earlier investigated case is considered as generating. Steady state motions transform in pseudo-steady motions with slow changing parameters. Calculations illustrate the analytical investigation.