Abstract:
This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the plane of sections are presented.
Keywords:
integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber body model, permanent rotations, dynamics in a fixed reference frame, resonance, quadrature, unbounded motion.
This work was supported by Grant No. 18-08-00999-a of the Russian Foundation for Basic Research. The work of A. A. Kilin was carried out at MIPT under Project 5–100 for State Support for Leading Universities of the Russian Federation. The work of E.N. Pivovarova was carried out within the framework of the State Assignment of the Ministry of Education and Science of Russia (1.2404.2017/4.6) and was supported in part by the Moebius Contest Foundation for Young Scientists.
Citation:
Alexander A. Kilin, Elena N. Pivovarova, “Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 24:2 (2019), 212–233
\Bibitem{KilPiv19}
\by Alexander A. Kilin, Elena N. Pivovarova
\paper Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 2
\pages 212--233
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\crossref{https://doi.org/10.1134/S1560354719020072}
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https://www.mathnet.ru/eng/rcd455
https://www.mathnet.ru/eng/rcd/v24/i2/p212
This publication is cited in the following 4 articles:
Alexander A. Kilin, Elena N. Pivovarova, “Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane”, Nonlinear Dyn, 2024
Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of Rubber Chaplygin Sphere under Periodic Control”, Regul. Chaotic Dyn., 25:2 (2020), 215–236
Elizaveta M. Artemova, Yury L. Karavaev, Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass”, Regul. Chaotic Dyn., 25:6 (2020), 689–706
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem”, Regul. Chaotic Dyn., 24:5 (2019), 560–582
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