Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 017, 18 pages      arXiv:1306.2470      https://doi.org/10.3842/SIGMA.2014.017

Dynamics of an Inverting Tippe Top

Stefan Rauch-Wojciechowski and Nils Rutstam
Department of Mathematics, Linköping University, Linköping, Sweden

Received September 05, 2013, in final form February 18, 2014; Published online February 27, 2014

Abstract
The existing results about inversion of a tippe top (TT) establish stability of asymptotic solutions and prove inversion by using the LaSalle theorem. Dynamical behaviour of inverting solutions has only been explored numerically and with the use of certain perturbation techniques. The aim of this paper is to provide analytical arguments showing oscillatory behaviour of TT through the use of the main equation for the TT. The main equation describes time evolution of the inclination angle $\theta(t)$ within an effective potential $V(\cos\theta,D(t),\lambda)$ that is deforming during the inversion. We prove here that $V(\cos\theta,D(t),\lambda)$ has only one minimum which (if Jellett's integral is above a threshold value $\lambda>\lambda_{\text{thres}}=\frac{\sqrt{mgR^3I_3\alpha}(1+\alpha)^2}{\sqrt{1+\alpha-\gamma}}$ and $1-\alpha^2$ < $\gamma=\frac{I_1}{I_3}$ < $1$ holds) moves during the inversion from a neighbourhood of $\theta=0$ to a neighbourhood of $\theta=\pi$. This allows us to conclude that $\theta(t)$ is an oscillatory function. Estimates for a maximal value of the oscillation period of $\theta(t)$ are given.

Key words: tippe top; rigid body; nonholonomic mechanics; integrals of motion; gliding friction.

pdf (585 kb)   tex (129 kb)

References

  1. Bou-Rabee N.M., Marsden J.E., Romero L.A., Tippe top inversion as a dissipation-induced instability, SIAM J. Appl. Dyn. Syst. 3 (2004), 352-377.
  2. Chaplygin S.A., On a motion of a heavy body of revolution on a horizontal plane, Regul. Chaotic Dyn. 7 (2002), 119-130.
  3. Cohen R.J., The tippe top revisited, Amer. J. Phys. 45 (1977), 12-17.
  4. Del Campo A.R., Tippe top (topsy-turnee top) continued, Amer. J. Phys. 23 (1955), 544-545.
  5. Ebenfeld S., Scheck F., A new analysis of the tippe top: asymptotic states and Liapunov stability, Ann. Physics 243 (1995), 195-217, chao-dyn/9501008.
  6. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, McGraw-Hill, New York - Toronto - London, 1953.
  7. Glad S.T., Petersson D., Rauch-Wojciechowski S., Phase space of rolling solutions of the tippe top, SIGMA 3 (2007), 041, 14 pages, nlin.SI/0703016.
  8. Jones E., Oliphant T., Peterson P., Open source scientific tools for Python, available at http://www.scipy.org.
  9. Karapetyan A.V., Qualitative investigation of the dynamics of a top on a plane with friction, J. Appl. Math. Mech. 55 (1991), 563-565.
  10. Karapetyan A.V., Rubanovskii V.N., On the stability of stationary motions of non-conservative mechanical systems, J. Appl. Math. Mech. 50 (1986), 30-35.
  11. Or A.C., The dynamics of a tippe top, SIAM J. Appl. Math. 54 (1994), 597-609.
  12. Rauch-Wojciechowski S., What does it mean to explain the rising of the tippe top?, Regul. Chaotic Dyn. 13 (2008), 316-331.
  13. Rauch-Wojciechowski S., Sköldstam M., Glad T., Mathematical analysis of the tippe top, Regul. Chaotic Dyn. 10 (2005), 333-362.
  14. Rutstam N., Study of equations for tippe top and related rigid bodies, Linköping Studies in Science and Technology, Theses No. 1106, Matematiska Institutionen, Linköpings Universitet, 2010, available at http://swepub.kb.se/bib/swepub:oai:DiVA.org:liu-60835.
  15. Rutstam N., Tippe top equations and equations for the related mechanical systems, SIGMA 8 (2012), 019, 22 pages, arXiv:1204.1123.
  16. Rutstam N., High frequency behavior of a rolling ball and simplification of the separation equation, Regul. Chaotic Dyn. 18 (2013), 226-236.
  17. Sturm C., Mémoire sur la résolution des équations numériques, Bull. de Ferussac 11 (1829), 419-425.
  18. Ueda T., Sasaki K., Watanabe S., Motion of the tippe top: gyroscopic balance condition and stability, SIAM J. Appl. Dyn. Syst. 4 (2005), 1159-1194, physics/0507198.

Previous article  Next article   Contents of Volume 10 (2014)