Geodesic
Limit equations of motion for mechanical systems with vibrating elements
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2015), pp. 23-33 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper studies mechanical systems with rapidly oscillating elements. We suggested a method to obtain limiting equations of such systems when the oscillation frequency tends to infinity. We considered a mechanical system with an arbitrary number of degrees of freedom subject to rapid periodic vibrations with zero mean. Its motion can be described by second-order equations. Also we obtained an explicit form of the limiting equations of motion written in the form of a system of first-order differential equations. 
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A. A. Markeeva; M. A. Levin. Limit equations of motion for mechanical systems with vibrating elements. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2015), pp. 23-33. http://geodesic.mathdoc.fr/item/VMUMM_2015_5_a4/

[1] Bogolyubov H.H., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Fizmatgiz, M., 1963 | MR

[2] Stephenson A., “On a new type of dynamical stability”, Mem. and Proc. Manchester Literary and Phil. Soc. Part 2, 52:8 (1908), 1–10

[3] Kapitsa P.L., “Mayatnik s vibriruyuschim podvesom”, Uspekhi fiz. nauk, 44:1 (1951), 7–20 | DOI

[4] Kapitsa P.L., “Dinamicheskaya ustoichivost mayatnika pri koleblyuscheisya tochke podvesa”, Zhurn. eksperim. i teor. fiz., 21:5 (1951), 588–597 | MR

[5] Markeev A.P., “K teorii dvizheniya tverdogo tela s vibriruyuschim podvesom”, Dokl. RAN, 427:6 (2009), 771–775 | MR | Zbl

[6] Bornemann F. A., Schutte C., “Homogenization of Hamiltonian systems with a strong constraining potential”, Physica D, 102 (1997), 55–57 | DOI | MR

[7] Kugushev E.I., Sabitov D.I., “O ploskikh tonkikh uprugikh sterzhnyakh s bystromenyayuschimisya periodicheskimi kharakteristikami”, Vestn. Mosk. un-ta. Matem. Mekhan., 2009, no. 4, 42–48 | MR

[8] Zhuravlev V.F., Klimov D.M., Prikladnye metody v teorii kolebanii, Nauka, M., 1988 | MR

[9] Filatov A.I., Metody usredneniya v differentsialnykh i integrodifferentsialnykh uravneniyakh, FAN, Tashkent, 1971 | MR

[10] Sergeev I.N., Lektsii po differentsialnym uravneniyam, I semestr, Izd-vo TsPI pri mekh.-mat. f-te MGU, M., 2004

[11] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR