Abstract:
This paper studies the conditions under which the tippe top inverts in the presence of vibrations of the base along the vertical. A vibrational potential is constructed by averaging and it is shown that, when this potential is added to the system, the Jellett integral is preserved. This makes it possible to apply the modified Routh method and to find the effective potential to whose critical points permanent rotations or regular precessions of the tippe top correspond. Tippe top inversion is possible for a sufficiently large initial angular velocity under the condition that spinning with the lowest position of the center of gravity is unstable, spinning with the highest position of the center of gravity is stable, and that there are no precessions. Cases are found in which there is no inversion in the absence of vibrations, but it can be brought about by a suitable choice of the mean value of the squared velocity of the base. In particular, this type includes a ball with a spherical cavity filled with a denser substance.
\Bibitem{BorIva20}
\by Alexey V. Borisov, Alexander P. Ivanov
\paper Dynamics of the Tippe Top on a Vibrating Base
\jour Regul. Chaotic Dyn.
\yr 2020
\vol 25
\issue 6
\pages 707--715
\mathnet{http://mi.mathnet.ru/rcd1092}
\crossref{https://doi.org/10.1134/S1560354720060131}
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https://www.mathnet.ru/eng/rcd/v25/i6/p707
This publication is cited in the following 13 articles:
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D. D. Kulminskiy, M. V. Malyshev, “Experimental Study of the Accuracy of a Pendulum Clock with a Vibrating Pivot Point”, Rus. J. Nonlin. Dyn., 20:4 (2024), 553–563
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Alexander A. Kilin, Elena N. Pivovarova, “Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane”, Mathematics, 11:18 (2023), 3948
Alexey V. Borisov, Alexander P. Ivanov, “A Top on a Vibrating Base:
New Integrable Problem of Nonholonomic Mechanics”, Regul. Chaotic Dyn., 27:1 (2022), 2–10
Ivan A. Bizyaev, Ivan S. Mamaev, “Permanent Rotations in Nonholonomic Mechanics.
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Alexander A. Kilin, Elena N. Pivovarova, “Motion control of the spherical robot rolling on a vibrating plane”, Applied Mathematical Modelling, 109 (2022), 492
Adrián Ruiz, Cláudio H. C. Costa Basquerotto, “Reduced motion equations of an axisymmetric body spinning on a horizontal surface via Lie symmetries”, Acta Mech, 233:9 (2022), 3853
Alexander A. Kilin, Elena N. Pivovarova, “A Particular Integrable Case in the Nonautonomous Problem
of a Chaplygin Sphere Rolling on a Vibrating Plane”, Regul. Chaotic Dyn., 26:6 (2021), 775–786
S. Sailer, R. I. Leine, “Singularly perturbed dynamics of the tippedisk”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 477:2256 (2021), 20210536
S. Sailer, R. I. Leine, “Model reduction of the tippedisk: a path to the full analysis”, Nonlinear Dyn., 105:3 (2021), 1955–1975
Alexander A. Kilin, Elena N. Pivovarova, “Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base”, Regul. Chaotic Dyn., 25:6 (2020), 729–752
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