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Motion of the tippe top: gyroscopic balance condition and stability. (English) Zbl 1145.70304

Summary: We reexamine a very classical problem, the spinning behavior of the tippe top on a horizontal table. The analysis is made for an eccentric sphere version of the tippe top, assuming a modified Coulomb law for the sliding friction, which is a continuous function of the slip velocity \(\mathbf v_P\) at the point of contact and vanishes at \(\mathbf v_P=\mathbf 0\). We study the relevance of the gyroscopic balance condition (GBC), which was discovered to hold for a rapidly spinning hard-boiled egg by Moffatt and Shimomura [see H. K. Moffatt, Y. Shimomura and M. Branicki, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460, No. 2052, 3643–3672 (2004; Zbl 1105.70005); ibid. 461, No. 2058, 1753–1774 (2005; Zbl 1139.70303)], to the inversion phenomenon of the tippe top. We introduce a variable \(\xi\) so that \(\xi=0\) corresponds to the GBC and we analyze the behavior of \(\xi\). Contrary to the case of the spinning egg, the GBC for the tippe top is not fulfilled initially. But we find from simulation that for those tippe tops which will turn over, the GBC will soon be satisfied approximately. It is shown that theGBC and the geometry lead to the classification of tippe tops into three groups: The tippe tops of Group I never flip over however large a spin they are given. Those of Group II show a complete inversion, and the tippe tops of Group III tend to turn over up to a certain inclination angle \(\theta_f\) such that \(\theta_f < \pi\), when they are spun sufficiently rapidly. There exist three steady states for the spinning motion of the tippe top. Giving a new criterion for stability, we examine the stability of these states in terms of the initial spin velocity \(n_0\). And we obtain a critical value \(n_c\) of the initial spin which is required for the tippe top of Group II to flip over up to the completely inverted position.

MSC:

70E18 Motion of a rigid body in contact with a solid surface
34D20 Stability of solutions to ordinary differential equations
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37N05 Dynamical systems in classical and celestial mechanics
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