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Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane

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Abstract

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slip** at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed.

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Data availability

This manuscript does not report data generation or analysis.

Notes

  1. Here and below, unless otherwise specified, all vectors will be referred to the moving coordinate system \(Cx_1x_2x_3\).

  2. We do not present here the phase portrait at the moment of bifurcation for \(\kappa =\kappa _c\) since visually it does not differ from the case \(\kappa >\kappa _c\).

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Acknowledgements

The authors thank Ivan Mamaev for useful comments and discussions.

Funding

The work of A. Kilin (Sects. 1–3) was carried out within the framework of the state assignment of the Ministry of Science and Higher Education of Russia (FEWS-2020-0009). The work of E. Pivovarova (Sects. 4 and 5) is supported by the Russian Science Foundation under Grant 23-71-01045.

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  1. Ural Mathematical Center, Udmurt State University, Izhevsk, Russia

    Alexander A. Kilin & Elena N. Pivovarova

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  1. Alexander A. Kilin
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A.K.: Conceptualization, Investigation, Writing—Review & Editing. E.P.: Investigation, Visualization, Writing—Original Draft. All authors reviewed the manuscript.

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Correspondence to Elena N. Pivovarova.

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Kilin, A.A., Pivovarova, E.N. Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09863-7

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