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- Rutstam, Nils, 1982-
(författare)
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Analysis of Dynamics of the Tippe Top
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
- Rutstam, Nils, 1982-
(författare)
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Study of equations for Tippe Top and related rigid bodies
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The Tippe Top consist of a small truncated sphere with a peg as a handle. When it is spun fast enough on its spherical part it starts to turn upside down and ends up spinning on the peg. This counterintuitive behaviour, called inversion, is a curious feature of this dynamical system that has been studied for some time, but obtaining a complete description of the dynamics of inversion has proved to be a difficult problem.The existing results are either numerical simulations of the equations of motion or asymptotic analysis that shows that the inverted position is the only attractive and stable position under certain conditions.This thesis will present methods to analyze the equations of motion of the Tippe Top, which we study in three equivalent forms that each helps us to understand different aspects of the inversion phenomenon.Our study of the Tippe Top also focuses on the role of the underlying assumptions in the standard model for the external force, and what consequences these assumptions have, in particular for the asymptotic cases.We define two dynamical systems as an aid to understand the dynamics of the Tippe Top, the gliding heavy symmetric top and the gliding eccentric cylinder. The gliding heavy symmetric top is a natural non-integrable generalization of the well-known heavy symmetric top. Equations of motion and asymptotics for this system are derived, but we also show that equations for the gliding heavy symmetric top can be obtained as a limit of the equations for the Tippe Top.The equations for the gliding eccentric cylinder can be interpreted as a special case of the equations for the Tippe Top, and since it is a simpler system, properties of the Tippe Top equations are easier to study. In particular, asymptotic analysis of the gliding eccentric cylinder reveals that the standard model seems to have inconsistencies that need to be addressed.
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| 3. |
- Rutstam, Nils, 1982-
(författare)
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Tippe Top Equations and Equations for the Related Mechanical Systems
- 2012
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Ingår i: SIGMA. Symmetry, Integrability and Geometry. - Kyiv : National Academy of Science of Ukraine. - 1815-0659 .- 1815-0659. ; 8:19
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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