On the stability of the specific motions of a heavy rigid body due to fast vertical vibrations of one of its points

Olga V. Kholostova

Russian journal of nonlinear dynamics, Tome 11 (2015) no. 1, pp. 99-116.

Résumé

We consider the motion of a heavy rigid body with one point performing the specified high-frequency harmonic oscillations along the vertical. In the framework of an approximate autonomous system of differential equations of motion two new types of permanent rotations of the body about the vertical are found. These motions are affected by presence of fast vibrations and do not exist in the case of a body with a fixed point. The problem of stability of the motions is investigated.

Détail

Keywords: rigid body, fast vibrations, stability, resonance.
Mots-clés : permanent rotations

Bibliographie

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

[2] Kapitsa P. L., “Mayatnik s vibriruyuschim podvesom”, UFN, 44:5 (1951), 7–20

[3] Kapitsa P. L., “Dinamicheskaya ustoichivost mayatnika pri koleblyuscheisya tochke podvesa”, ZhETF, 21:5 (1951), 588–597

[4] Bardin B. S., Markeev A. P., “Ob ustoichivosti ravnovesiya mayatnika pri vertikalnykh kolebaniyakh tochki podvesa”, PMM, 59:6 (1995), 922–929

[5] Kholostova O. V., “Ob ustoichivosti periodicheskikh dvizhenii mayatnika s gorizontalno vibriruyuschei tochkoi podvesa”, MTT, 1997, no. 4, 35–39

[6] Kholostova O. V., “O dvizheniyakh mayatnika s vibriruyuschei tochkoi podvesa”, Sb. nauch.-metod. st., Teoreticheskaya mekhanika, 24, MGU, M., 2003, 157–167

[7] Strizhak T. G., Metody issledovaniya dinamicheskikh sistem tipa «mayatnik», Nauka, Alma-Ata, 1981, 253 pp.

[8] Bogolyubov N. N., “Teoriya vozmuschenii v nelineinoi mekhanike”, Sb. tr. In-ta stroit. mekh. AN USSR, 14, 1950, 9–34

[9] Markeev A. P., “O dinamike sfericheskogo mayatnika s vibriruyuschim podvesom”, PMM, 63:2 (1999), 213–219

[10] Kholostova O. V., “O dvizheniyakh dvoinogo mayatnika s vibriruyuschei tochkoi podvesa”, MTT, 2009, no. 2, 25–40

[11] Vishenkova E. A., Kholostova O. V., “K dinamike dvoinogo mayatnika s gorizontalno vibriruyuschei tochkoi podvesa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 2, 114–129

[12] Kholostova O. V., Dinamika volchka Lagranzha s nepodvizhnoi i vibriruyuschei tochkoi podvesa, MAI, M., 2000, 84 pp.

[13] Yudovich V. I., “Vibrodinamika i vibrogeometriya mekhanicheskikh sistem so svyazyami”, Uspekhi mekhaniki, 4:3 (2006), 26–158

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

[15] Markeev A. P., “Ob uravneniyakh priblizhennoi teorii dvizheniya tverdogo tela s vibriruyuschei tochkoi podvesa”, PMM, 75:2 (2011), 193–203

[16] Kholostova O. V., “Ob ustoichivosti ravnovesii tverdogo tela s vibriruyuschei tochkoi podvesa”, Vestn. RUDN. Matem. Informatika. Fizika, 2011, no. 2, 111–122

[17] Markeev A. P., “O dvizhenii tyazhelogo dinamicheski simmetrichnogo tverdogo tela s vibriruyuschei tochkoi podvesa”, MTT, 2012, no. 4, 3–10

[18] Staude O., “Über permanente Rotationsaxen bei der Bewegung eines schweren Körpers um einen festen Punkt”, J. Reine Angew. Math., 113:4 (1894), 318–334

[19] Mlodzeevskii B. K., “O permanentnykh osyakh v dvizhenii tyazhelogo tverdogo tela vokrug nepodvizhnoi tochki”, Tr. otd. fiz. nauk Obschestva lyubitelei estestvoznaniya, 7:1 (1894), 46–48

[20] Grammel R., Giroskop, ego teoriya i primeneniya, V 2-kh tt., IL, M., 1952, 351 pp.; 318 pp.

[21] Rumyantsev V. V., “Ob ustoichivosti vrascheniya tyazhelogo tverdogo tela s odnoi nepodvizhnoi tochkoi v sluchae S. V. Kovalevskoi”, PMM, 18:4 (1954), 457–458

[22] Rumyantsev V. V., “Ustoichivost permanentnykh vraschenii tyazhelogo tverdogo tel”, PMM, 20:1 (1956), 51–66

[23] Magnus K., Giroskop. Teoriya i primenenie, Mir, M., 1974, 526 pp.

[24] Kovalev A. M., Savchenko A. Ya., “Ustoichivost ravnomernykh vraschenii tverdogo tela vokrug glavnoi osi”, PMM, 39:4 (1975), 650–660

[25] Sergeev V. S., “Ob ustoichivosti permanentnykh vraschenii tyazhelogo tverdogo tela vokrug nepodvizhnoi tochki”, PMM, 40:3 (1976), 408–416

[26] Kovalev A. M., Savchenko A. Ya., “Ustoichivost statsionarnykh dvizhenii gamiltonovykh sistem pri nalichii rezonansa chetvertogo poryadka”, Mekhanika tverdogo tela, 1977, no. 9, 40–44

[27] Kholostova O. V., Issledovanie ustoichivosti permanentnykh vraschenii Shtaude, Regulyarnaya i khaoticheskaya dinamika, M.–Izhevsk, 2008, 128 pp.

[28] Karapetyan A. V., Rumyantsev V. V., Ustoichivost konservativnykh i dissipativnykh sistem, Itogi nauki i tekhniki. Ser. Obschaya mekhanika, 6, VINITI, M., 1983, 132 pp.

[29] Arnold V. I., Kozlov V. V., Neishtadt A. I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, Editorial URSS, M., 2009, 416 pp.

[30] Markeev A. P., Tochki libratsii v nebesnoi mekhanike i kosmodinamike, Nauka, M., 1978, 312 pp.