The Tippedisk: a Tippetop Without Rotational Symmetry
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The Tippedisk: a Tippetop Without Rotational Symmetry Simon Sailer1* , Simon R. Eugster1** , and Remco I. Leine1*** 1
Institute for Nonlinear Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
Received July 24, 2020; revised September 28, 2020; accepted October 16, 2020
Abstract—The aim of this paper is to introduce the tippedisk to the theoretical mechanics community as a new mechanical-mathematical archetype for friction induced instability phenomena. We discuss the modeling and simulation of the tippedisk, which is an inhomogeneous disk showing an inversion phenomenon similar but more complicated than the tippetop. In particular, several models with different levels of abstraction, parameterizations and force laws are introduced. Moreover, the numerical simulations are compared qualitatively with recordings from a high-speed camera. Unlike the tippetop, the tippedisk has no rotational symmetry, which greatly complicates the three-dimensional nonlinear kinematics. The governing differential equations, which are presented here in full detail, describe all relevant physical effects and serve as a starting point for further research. MSC2010 numbers: 70E18, 70K20, 70E50 DOI: 10.1134/S1560354720060052 Keywords: gyroscopic system, friction, contact modelling, dynamics, inversion phenomenon
1. INTRODUCTION Various gyroscopic systems which are interacting with a horizontal frictional support, such as rolling and sliding disks [3, 5, 15, 25, 27], Euler’s disk [14, 16, 20], the rattleback [4, 8] and the tippetop [6, 7, 13, 19, 28], form a scientific playground for research in theoretical mechanics. The tippetop belongs to a subclass of gyroscopic systems which shows inversion phenomena. The tippetop is a rotationally symmetric top, consisting of a spherical body and a stem attached to it, see Fig. 1. The center of gravity (COG) does not coincide with the geometric center, such that the stem points upwards as the top rotates slowly with noninverted orientation. When the top is spun fast around its axis of symmetry, the gravitational, normal, and friction forces acting on the top cause the top to invert its orientation, such that it finally balances on its stem. This phenomenon of inversion also occurs for other axisymmetric bodies with rotational symmetry in inertia and geometry, for example, hard-boiled spinning eggs [21, 22] or the science toy called “PhiTOP”1) , which is basically the symmetric equivalent of a hard-boiled spinning egg. But what happens if rotational symmetry does not exist? This is the topic of the present paper. The “orbit spinning top” is a commercial toy consisting of a thin disk and an eccentrically attached sphere. If this top is spun around an in-plane axis, the center of gravity rises until the top is spinning in an inverted configuration2) . Similar to the orbit spinning top, an eccentric disk also shows this kind of inversion and faced a huge interest on social media3),4) . In Fig. 2 the inversion process of the tippedisk is depicted in a stroboscopic photo sequence, whi
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