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## Archive of Issues

Russia Nizhni Novgorod
Year
2012
Issue
4
Pages
140-145
 Section Mechanics Title On stability of the Chaplygin ball motion on a plane with an arbitrary friction law Author(-s) Ovsyannikov I.I.a Affiliations Research Institute for Applied Mathematics and Cybernetics, Nizhni Novgorod State Universitya Abstract The Chaplygin ball on a plane is considered under the action of the friction force which satisfies the following condition: $({\bf F}, {\bf u}) < 0$ as ${\bf u} \neq 0$ and ${\bf F} = 0$ as ${\bf u} = 0$, where ${\bf u}$ is the gliding velocity. The ball is supposed to have a point contact with the supporting plane (this means that the contact spot is absent and also there is no rotation friction torque). The main task of the paper is to determine a set of possible stationary (or final) motions and their stability. In the current paper it is shown that exactly three stationary motions are possible; these motions represent straightline uniform rolling motions of the ball without sliding, at that the ball is rotating around one of the primary axes of the inertia tensor. Rotation around the axis of the greatest moment of inertia is stable, around the middle one and the lowest one it is unstable. Keywords Chaplygin ball, stationary motions, stability UDC 531.011 MSC 37N15, 70E18, 70K20 DOI 10.20537/vm120411 Received 24 September 2012 Language Russian Citation Ovsyannikov I.I. On stability of the Chaplygin ball motion on a plane with an arbitrary friction law, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 140-145. References Chaplygin S.A. On rolling of a ball on a horizontal plane, Mat. Sb., 1903, vol. XXIV, pp. 76–101. Gallop E.G. On the rise of a spinning top, Proc. Cambr. Phylos. Soc., 1904, vol. 19, part 3, pp. 356–373. Kilin A.A. The dynamics of Chaplygin ball: the qualitative and computer analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291–306. Karapetyan A.V. Ustoichivost’ statsionarnykh dvizhenii (Stability of stationary motions), Moscow: Editorial URSS, 1998, 166 p. Moshchuk N.K. On the Chaplygin ball motion on a horizontal plane, Prikl. Mat. Mekh., 1983, vol. 47, no. 6, pp. 916–921. Moshchuk N.K. On a motion of a heavy rigid body on a horizontal plane with a viscous friction, Prikl. Mat. Mekh., 1985, vol. 49, no. 1, pp. 66–71. Karapetyan A.V. Global qualitative analysis of dynamics of the tippe-top, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 2008, no. 3, pp. 33–41. Markeev A.P. Dinamika tela, soprikasayushchegosya s tverdoi poverkhnost’yu (The dynamics of the body in contact with a solid surface), Moscow: Nauka, 1992, 336 p.; Second edition: Izhevsk: Regular and Chaotic Dynamics, 2011. Full text