A Realistic Solution of the Symmetric Top Problem

Kane, T. R.; Levinson, David A.

doi:10.1115/1.3424439

Cited by 31 publications

(21 citation statements)

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“…Since it was established that sliding friction was necessary to explain the Tippe Top inversion [5,9,14], many studies have been dedicated to the analysis of models for tippe tops, involving linear stability analysis of the relative equilibria, numerical simulations, etc. Some studies have addressed the occurrence of transitions between rolling and sliding during the motion, see [13,15,18]. In this paper the presented mathematical results mainly reproduce those in [8,6,7,10,20,16] but our approach is inspired by the hands-on numerical approach as first attempted by Cohen in [9].…”

supporting

confidence: 69%

“…A debatable issue is whether transitions between sliding and rolling are possible during the motion of the top. As it was pointed out in [15], such transitions must also be considered when setting up a realistic model to describe the dynamics of the tippe top. To the different regimes there correspond different sets of equations.…”

Section: Further Remarksmentioning

confidence: 99%

“…Friction forces can be complicated, and there are various models in circulation. We adopt the assumption of viscous friction [15,16,24], and assume the friction force to be given by…”

mentioning

confidence: 99%

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Towards a Prototype of a Spherical Tippe Top

Ciocci¹,

Malengier²,

Langerock³

et al. 2012

Nelin. Dinam.

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Abstract.Among spinning objects, the tippe top exhibits one of the most bizarre and counterintuitive behaviours. The commercially available tippe tops basically consist of a section of a sphere with a rod. After spinning on its rounded body, the top flips over and continues spinning on the stem. It is the friction with the bottom surface and the position of the center of mass below the centre of curvature that cause the tippe top to rise its centre of mass while continuing rotating around its symmetry axis (through the stem). The commonly used simplified mathematical model for the tippe top is a sphere whose mass distribution is axially but not spherically symmetric, spinning on a flat surface subject to a small friction force that is due to sliding. Adopting a bifurcation theory point of view we reach a global geometric understanding of the phase diagram of this dynamical system. According to the eccentricity of the sphere and the Jellet invariant (which includes information on the initial angular velocity) three main different dynamical behaviours are distinguished: tipping, non-tipping, hanging (i.e. the top rises but converges to an intermediate state instead of rising all the way to the vertical state). Subclasses according to the stability of relative equilibria can further be distinguished. Since our concern is the degree of confidence in the mathematical model predictions, we applied 3D-printing and rapid prototyping to manufacture a '3-in-1 toy' that could catch the three main characteristics defining the three main groups in the classification of spherical tippe tops as mentioned above. This 'toy' is suitable to validate the mathematical model qualitatively and quantitatively.

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supporting

confidence: 69%

Section: Further Remarksmentioning

confidence: 99%

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Towards a Prototype of a Spherical Tippe Top

Ciocci¹,

Malengier²,

Langerock³

et al. 2012

Nelin. Dinam.

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“…It is well understood that the sliding friction is the main source for the tippe top inversion [2,3]. So we will ignore other possible frictions, such as, rolling friction [14] and rotational friction which is due to pure rotation about a vertical axis . Concerning the sliding friction, often used is a Coulomb law, which states that…”

Section: Equations Of Motion For Tippe Topsmentioning

confidence: 99%

“…The slip velocity of the contact point P necessarily vanishes at the steady state of the tippe top. In order to study the motion of the tippe top as realistically as possible and also to facilitate a linear stability analysis of steady states, we modify the expression of Coulomb friction (2.13) as 14) so that F is continuous in v P and vanishes at v P = 0. Here we choose Λ as a sufficiently small number with dimensions of velocity.…”

Section: Equations Of Motion For Tippe Topsmentioning

confidence: 99%

Motion of the Tippe Top: Gyroscopic Balance Condition and Stability

Ueda¹,

Sasaki²,

Watanabe³

2005

SIAM J. Appl. Dyn. Syst.

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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 v P at the point of contact and vanishes at v P = 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, to the inversion phenomenon of the tippe top. We introduce a variable ξ so that ξ = 0 corresponds to the GBC and analyze the behavior of ξ. 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 the GBC 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 θ f such that θ f < π, 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.

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