Dynamics of the Tippe Top on a Vibrating Base
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Dynamics of the Tippe Top on a Vibrating Base Alexey V. Borisov1, 2* and Alexander P. Ivanov1, 2** 1
Moscow Institute of Physics and Technology, Inststitutskii per. 9, 141700 Dolgoprudnyi, Russia 2 National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, 115409 Moscow, Russia Received September 14, 2020; revised October 22, 2020; accepted October 30, 2020
Abstract—This paper studies the conditions under which the tippe top inverts in the presence of vibrations of the base along the vertical. A vibrational potential is constructed by averaging and it is shown that, when this potential is added to the system, the Jellett integral is preserved. This makes it possible to apply the modified Routh method and to find the effective potential to whose critical points permanent rotations or regular precessions of the tippe top correspond. Tippe top inversion is possible for a sufficiently large initial angular velocity under the condition that spinning with the lowest position of the center of gravity is unstable, spinning with the highest position of the center of gravity is stable, and that there are no precessions. Cases are found in which there is no inversion in the absence of vibrations, but it can be brought about by a suitable choice of the mean value of the squared velocity of the base. In particular, this type includes a ball with a spherical cavity filled with a denser substance. MSC2010 numbers: 70E50, 70F40 DOI: 10.1134/S1560354720060131 Keywords: tippe top, dry friction, Jellett integral
1. INTRODUCTION As is well known, incorporation of friction into a conservative system changes its qualitative properties considerably. In the case of Coulomb friction the Cauchy problem for dynamics equations may have no solution at all [1, 2]. On the other hand, solutions can appear which are impossible in the absence of friction and therefore seem to be paradoxical [3–9]. We note that in the experiments carried out by Coulomb himself the body either executed translational motion or rolled in a fixed direction. It turned out that dissipation during rolling was by an order of magnitude smaller than dissipation during sliding. Further research showed that in the case of “pure” spinning the resistance is smaller by another order of magnitude. However, in the case of complex relative motion these types of friction cannot be regarded as independent. The role of spinning in tippe top dynamics was explained by Contensou [10]: at a high angular velocity of spinning the integral force of sliding friction is proportional to the velocity of the point of contact. Later this model was developed further in Refs. [11–15]. The mystery of the tippe top is that, spun fast enough in a stable equilibrium position, it inverts soon and spins for some time with the center of gravity at the highest point. Thomson was one of the first to draw the attention of colleagues to this unusual dynamics. This is why it is also called the Thomson top. A qualitative explanation of this phenomenon was given by Jellett [16], who showed that the top’s axis rises
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