Geodesic
On stability of gyroscopic systems
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2013), pp. 3-13 Cet article a éte moissonné depuis la source Math-Net.Ru

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Mechanical systems described by the Lagrange equations of the second kind influenced by potential, dissipative, gyroscopic and nonconservative forces are considered. It is assumed that gyroscopic forces are dominant. This domination is accounted by in the presence of a large parameter as a coefficient at the vector of these forces. By the use of the Lyapunov function method the lower bounds for the large parameter values are obtained which permit to receive justified conclusions of asymptotic stability of the full system on the base of its decomposition on two subsystems. To provide this decomposition, two approaches are proposed. In the first approach, vector Lyapunov functions are applied, while the second one is based on using scalar Lyapunov functions. It is worth mentioning that compared with the known results the suggested approaches give opportunity to receive stability conditions in the case of nonstationary matrices of dissipative and positional forces. Moreover, systems with nonstationary unboundedly increasing with time parameter at gyroscopic forces are studied. The restrictions on the velocity of parameter increasing are found for which the evolution of the parameter does not destroy the asymptotic stability of the equilibrium position. The cases of linear dissipative forces and essentially nonlinear dissipative forces given by homogeneous Rayleigh functions are considered. Bibliogr. 18.
Keywords: mechanical systems, gyroscopic forces, stability, Lyapunov functions.
Mots-clés : decomposition 
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A. Yu. Aleksandrov; A. A. Kosov. On stability of gyroscopic systems. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2013), pp. 3-13. http://geodesic.mathdoc.fr/item/VSPUI_2013_2_a0/

[1] Zubov V. I., Analiticheskaya dinamika giroskopicheskikh sistem, Sudostroenie, L., 1970, 320 pp. | Zbl

[2] Merkin D. R., Giroskopicheskie sistemy, Nauka, M., 1974, 344 pp. | Zbl

[3] Kosov A. A., “Issledovanie ustoichivosti singulyarnykh sistem metodom vektor-funktsii Lyapunova”, Vestn. S.-Peterb. un-ta. Ser. 10: Prikladnaya matematika, informatika, protsessy upravleniya, 2005, no. 4, 123–129

[4] Kozlov V. V., “Ob ustoichivosti polozhenii ravnovesiya v nestatsionarnom silovom pole”, Prikl. matematika i mekhanika, 55:1 (1991), 12–19 | MR | Zbl

[5] Khatvani L., “O deistvii dempfirovaniya na svoistva ustoichivosti ravnovesii neavtonomnykh sistem”, Prikl. matematika i mekhanika, 65:4 (2001), 725–732 | MR

[6] Vorotnikov V. I., Rumyantsev V. V., Ustoichivost i upravlenie po chasti koordinat fazovogo vektora dinamicheskikh sistem: teoriya, metody i prilozheniya, Nauchnyi mir, M., 2001, 320 pp. | MR | Zbl

[7] Aleksandrov A. Yu., Kosov A. A., “Ob asimptoticheskoi ustoichivosti polozhenii ravnovesiya mekhanicheskikh sistem s nestatsionarnym veduschim parametrom”, Izv. RAN. Teoriya i sistemy upravleniya, 2008, no. 3, 8–22 | MR | Zbl

[8] Oziraner A. S., Rumyantsev V. V., “Metod funktsii Lyapunova v zadache ob ustoichivosti dvizheniya otnositelno chasti peremennykh”, Prikl. matematika i mekhanika, 36:2 (1972), 364–384 | MR | Zbl

[9] Krasovskii N. N., Nekotorye zadachi teorii ustoichivosti dvizheniya, Fizmatgiz, M., 1959, 212 pp. | MR

[10] Voronov A. A., Vvedenie v dinamiku slozhnykh upravlyaemykh sistem, Nauka, M., 1985, 352 pp. | MR

[11] Matrosov V. M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem, Fizmatlit, M., 2001, 384 pp.

[12] Chernousko F. L., Ananevskii I. M., Reshmin S. A., Metody upravleniya nelineinymi mekhanicheskimi sistemami, Fizmatlit, M., 2006, 328 pp.

[13] Agafonov S. A., “Ob ustoichivosti dvizheniya nekonservativnykh mekhanicheskikh sistem”, Prikl. matematika i mekhanika, 56:2 (1992), 212–217 | MR | Zbl

[14] Agafonov S. A., “Ustoichivost nekonservativnykh sistem i otsenka oblasti prityazheniya”, Prikl. matematika i mekhanika, 67:2 (2003), 239–243 | MR | Zbl

[15] Agafonov S. A., “Ob ustoichivosti i stabilizatsii dvizheniya nekonservativnykh mekhanicheskikh sistem”, Prikl. matematika i mekhanika, 74:4 (2010), 560–566

[16] Zorich V. A., Matematicheskii analiz, v 2 ch., v. 1, Nauka, M., 1981, 544 pp. | MR | Zbl

[17] Rouche N., Habets P., Laloy M., Stability theory by Liapunov's direct method | MR

[18] Agafonov S. A., “Ob asimptoticheskoi ustoichivosti nekonservativnykh sistem”, Izv. AN SSSR. Mekhanika tverdogo tela, 1988, no. 3, 3–8