On the motion of a bead on a rough hoop freely rotating around a vertical diameter

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Abstract

We consider the problem of the motion of a heavy bead strung on a rough heavy hoop freely rotating around a vertical diameter. Non-isolated sets of steady state motions of the system are identified, and their bifurcation diagrams are constructed. The dependence of these solutions on an essential parameter of the problem—the constant of the cyclic integral—is studied. The results obtained are compared with the results obtained previously for the case when a rough hoop rotates around a vertical diameter with a constant angular velocity. Characteristic phase portraits are constructed for various combinations of system parameters.

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About the authors

А. А. Burov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Author for correspondence.
Email: jtm@yandex.ru
Russian Federation, Moscow

V. I. Nikonov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Email: nikon_v@list.ru
Russian Federation, Moscow

Е. А. Nikonova

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences; Sirius University of Science and Technology Sirius Federal territory

Email: nikonova.ekaterina.a@gmail.com
Russian Federation, Moscow; Sochi

References

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Supplementary files

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2. Fig. 1. Bead on a hoop.

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3. Fig. 2. Bifurcation diagrams in the absence of friction for different combinations of parameters: on the plane on the left; on the plane on the right.

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4. Fig. 3. Regions of sign constancy of submodular expressions of inequality (3.4).

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5. Fig. 4. Subregions of regions corresponding to solutions of inequality (3.4).

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6. Fig. 5. Bifurcation diagram in the presence of friction for different combinations of parameters. Here , .

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7. Fig. 6. Dependence on the friction coefficient: left, center, right.

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8. Fig. 7. Phase portrait for pre-bifurcation combinations of parameters.

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9. Fig. 8. Phase portrait for post-bifurcation combinations of parameters.

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10. Fig. 9. Neighborhood of sets for post-bifurcation combinations of parameters.

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11. Fig. 10. Neighborhood of sets for post-bifurcation combinations of parameters.

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