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Analysis of Dynamics of the Tippe Top
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
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.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1500
##### National Category
Mathematical Analysis
##### Identifiers
ISBN: 978-91-7519-692-3 (print)OAI: oai:DiVA.org:liu-88316DiVA, id: diva2:602220
##### Public defence
2013-02-26, BL32 Nobel, Hus B, Campus Valla, Linköping University, Linköping, 10:15 (English)
##### Supervisors
Available from: 2013-02-01 Created: 2013-01-31 Last updated: 2013-11-14Bibliographically approved
##### List of papers
1. Tippe Top Equations and Equations for the Related Mechanical Systems
Open this publication in new window or tab >>Tippe Top Equations and Equations for the Related Mechanical Systems
2012 (English)In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 8, no 19Article in journal (Refereed) Published
##### Abstract [en]

The equations of motion for the rolling and gliding Tippe Top (TT) are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis 3ˆ during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT.

##### Place, publisher, year, edition, pages
Kyiv: National Academy of Science of Ukraine, 2012
##### Keyword
tippe top; rigid body, nonholonomic mechanics, integrals of motion, stability, gliding friction
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-76924 (URN)10.3842/SIGMA.2012.019 (DOI)000303833100001 ()
Available from: 2012-04-26 Created: 2012-04-26 Last updated: 2017-12-07
2. High frequency behaviour of a rolling ball and simplification of the Main Equation for the Tippe Top
Open this publication in new window or tab >>High frequency behaviour of a rolling ball and simplification of the Main Equation for the Tippe Top
##### 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 $\frac{2\pi}T$. These results have bearing for understanding the inverting motion of the Tippe Top, modeled by a rolling and gliding axisymmetric sphere.

##### Keyword
Tippe top; rigid body; rolling sphere; integrals of motion; elliptic integrals.
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-88314 (URN)
Available from: 2013-01-31 Created: 2013-01-31 Last updated: 2013-02-01Bibliographically approved
3. Dynamics of an inverting Tippe Top
Open this publication in new window or tab >>Dynamics of an inverting Tippe Top
2014 (English)In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 10, no 017Article in journal (Refereed) Published
##### Abstract [en]

We study an equivalent integrated form of the Tippe Top (TT) equations that leads to the Main Equation for the Tippe Top (METT), an equation describing time evolution of the inclination angle θ(t) of inverting TT. We study how the effective potential V(cos θ, D, λ) in METT deforms as TT is inverting and show that its minimum moves from a neighborhood of θ = 0 to a neighborhood of θ = π. We analyse behaviour of θ(t) and show that it oscillates and moves toward θ = π when the physical parameters of the TT satisfy 1 − α2 < γ < 1 and the initial conditions are such that Jellett’s integral satisfy

$\lambda > \lambda _t_h_r_e_s = \frac{\sqrt{mgR^3I_3\alpha}(1+\alpha)^2}{\sqrt{1+\alpha-\gamma}}$. Estimates for maximal value of the oscillation period of θ(t) are given.

##### Keyword
Tippe top, rigid body, nonholonomic mechanics, integrals of motion, gliding friction
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-88315 (URN)10.3842/SIGMA.2014.017 (DOI)000334516500001 ()
Available from: 2013-02-01 Created: 2013-01-31 Last updated: 2017-10-13Bibliographically approved

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Cite
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