Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem
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This paper addresses the problem of the rolling of a spherical shell with a frame rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire system is at the geometric center of the shell.
For the rubber rolling model and the classical rolling model it is shown that, if the angular velocities of rotation of the frame and the rotors are constant, then there exists a noninertial coordinate system (attached to the frame) in which the equations of motion do not depend explicitly on time. The resulting equations of motion preserve an analog of the angular momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the problem reduces to investigating a two-dimensional Poincaré map.
The case of the rubber rolling model is analyzed in detail. Numerical investigation of its Poincaré map shows the existence of chaotic trajectories, including those associated with a strange attractor. In addition, an analysis is made of the case of motion from rest, in which the problem reduces to investigating the vector field on the sphere S2.
Keywordsnonholonomic mechanics Chaplygin ball rolling without slipping and spinning strange attractor straight-line motion stability limit cycle balanced beaver-ball
MSC2010 numbers37J60 37C10
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The work of I. A. Bizyaev (Section 2 and Section 4) was supported by the Russian Science Foundation (project 18-71-00110). The work of A. V. Borisov and I. S. Mamaev was supported by the RFBR Grant No. 18-29-10051 mk and was carried out at MIPT under project 5–100 for state support for leading universities of the Russian Federation. The work of A. V. Borisov (Section 1 and Appendix A) was supported by the Russian Science Foundation (project 15-12-20035).
- 11.Borisov, A. V., Ivanova, T. B., Kilin, A. A., and Mamaev, I. S. Nonholonomic Rolling of a Ball on the Surface of a Rotating Cone, Nonlinear Dynam., 2019, vol. 97, no. 2, pp. 1635–1648.Google Scholar
- 17.Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics, De Gruyter Stud. Math. Phys., vol. 52, Berlin: De Gruyter, 2018.Google Scholar
- 24.Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., Dynamical Systems with Non-Integrable Constraints: Vaconomic Mechanics, Sub-Riemannian Geometry, and Non-Holonomic Mechanics, Russian Math. Surveys, 2017, vol. 72, no. 1, pp. 1–32; see also: Uspekhi Mat. Nauk, 2017, vol. 72, no. 5(437), pp. 3–62.MathSciNetzbMATHGoogle Scholar
- 34.Ehlers, K. M. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2 — 3 — 5 Distributions, in Proc. IUTAM Symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, Russia, 25–30 August 2006), pp. 469–480.Google Scholar
- 37.Fedorov, Yu. N., Motion of a Rigid Body in a Spherical Suspension, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1988, no. 5, pp. 91–93 (Russian).Google Scholar
- 39.Golubev, V. V., Chaplygin, Izhevsk: Institute of Computer Science, 2002 (Russian).Google Scholar
- 48.Markeev, A. P., Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1986, vol. 21, no. 1, pp. 64–65 (Russian).Google Scholar