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High frequency behaviour of a rolling ball and simplification of the Main Equation for the Tippe Top
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The Chaplygin separation equation for a rolling axisymmetric ball has an algebraic expression for the effective potential V(z = cos θ, D, λ) which is difficult to analyse. We simplify this expression for the potential and find a 2-parameter family for when the potential becomes a rational function of z = cos θ. Then this separation equation becomes similar to the separation equation for the heavy symmetric top. For nutational solutions of a rolling sphere, we study a high frequency ω3-dependence of the width of the nutational band, the depth of motion above V(zmin, D, λ) and the ω3-dependence of nutational frequency . These results have bearing for understanding the inverting motion of the Tippe Top, modeled by a rolling and gliding axisymmetric sphere.

Keyword [en]
Tippe top; rigid body; rolling sphere; integrals of motion; elliptic integrals.
National Category
URN: urn:nbn:se:liu:diva-88314OAI: diva2:602201
Available from: 2013-01-31 Created: 2013-01-31 Last updated: 2013-02-01Bibliographically approved
In thesis
1. Analysis of Dynamics of the Tippe Top
Open this publication in new window or tab >>Analysis of Dynamics of the Tippe Top
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The Tippe Top is a toy that has the form of a truncated sphere with a small peg. When spun on its spherical part on a flat supporting surface it will start to turn upside down to spin on its peg. This counterintuitive phenomenon, called inversion, has been studied for some time, but obtaining a complete description of the dynamics of inversion has proven to be a difficult problem. This is because even the most simplified model for the rolling and gliding Tippe Top is a non-integrable, nonlinear dynamical system with at least 6 degrees of freedom. The existing results are based on numerical simulations of the equations of motion or an asymptotic analysis showing that the inverted position is the only asymptotically attractive and stable position for the Tippe Top under certain conditions. The question of describing dynamics of inverting solutions remained rather intact.

In this thesis we develop methods for analysing equations of motion of the Tippe Top and present conditions for oscillatory behaviour of inverting solutions.

Our approach is based on an integrated form of Tippe Top equations that leads to the Main Equation for the Tippe Top (METT) describing the time evolution of the inclination angle $\theta(t)$ for the symmetry axis of the Tippe Top.

In particular we show that we can take values for physical parameters such that the potential function $V(\cos\theta,D,\lambda)$ in METT becomes a rational function of $\cos\theta$, which is easier to analyse. We estimate quantities characterizing an inverting Tippe Top, such as the period of oscillation for $\theta(t)$ as it moves from a neighborhood of $\theta=0$ to a neighborhood of $\theta=\pi$ during inversion. Results of numerical simulations for realistic values of physical parameters confirm the conclusions of the mathematical analysis performed in this thesis.  

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2013. 24 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1500
National Category
Mathematical Analysis
urn:nbn:se:liu:diva-88316 (URN)978-91-7519-692-3 (ISBN)
Public defence
2013-02-26, BL32 Nobel, Hus B, Campus Valla, Linköping University, Linköping, 10:15 (English)
Available from: 2013-02-01 Created: 2013-01-31 Last updated: 2013-11-14Bibliographically approved

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