Abstract
The tippedisk is a mechanical-mathematical archetype for friction-induced instability phenomena that exhibits an interesting inversion phenomenon when spun rapidly. The inversion phenomenon of the tippedisk can be modeled by a rigid eccentric disk in permanent contact with a flat support, and the dynamics of the system can therefore be formulated as a set of ordinary differential equations. The qualitative behavior of the nonlinear system can be analyzed, leading to slow–fast dynamics. Since even a freely rotating rigid body with six degrees of freedom already leads to highly nonlinear system equations, a general analysis for the full system equations is not feasible. In a first step the full system equations are linearized around the inverted spinning solution with the aim to obtain a local stability analysis. However, it turns out that the linear dynamics of the full system cannot properly describe the qualitative behavior of the tippedisk. Therefore, we simplify the equations of motion of the tippedisk in such a way that the qualitative dynamics are preserved in order to obtain a reduced model that will serve as the basis for a following nonlinear stability analysis. The reduced equations are presented here in full detail and are compared numerically with the full model. Furthermore, using the reduced equations we give approximate closed form results for the critical spinning speed of the tippedisk.
Highlights
Various gyroscopic systems which are interacting with a horizontal frictional support, such as Euler’s disk [19,22,26], the rattleback [5,13] and the tippetop [6,10,25,29], form a scientific playground for frictioninduced instability phenomena
For ‘noninverted spinning’, the center of gravity (COG) is located below the geometric center and the disk is spinning with a constant velocity about the in-plane axis through the COG and the geometric center
In [30] we provide a suitable mechanical model of the tippedisk, which is able to describe the inversion phenomenon
Summary
Various gyroscopic systems which are interacting with a horizontal frictional support, such as Euler’s disk [19,22,26], the rattleback [5,13] and the tippetop [6,10,25,29], form a scientific playground for frictioninduced instability phenomena. The tippedisk is essentially an eccentric disk, for which the center of gravity (COG) does not coincide with the geometric center of the disk. For ‘noninverted spinning’, the COG is located below the geometric center and the disk is spinning with a constant velocity about the in-plane axis through the COG and the geometric center. The second stationary motion is referred to as ‘inverted spinning’, being similar to ‘noninverted spinning’, but with the COG located above the geometric center of the disk
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