A study of permanent rotations of a heavy dynamically symmetric rigid body with a vibrating suspension point

E. A. Vishenkova   ; O. V. Kholostova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 4, pp. 590-607

Résumé

The motion of a dynamically symmetric rigid body in a uniform gravity field is considered for the case of vertical high-frequency harmonic oscillations of small amplitude of one of its points (the suspension point). The investigation is carried out within the framework of an approximate autonomous system of differential equations of motion written in the canonical Hamiltonian form. A detailed description of admissible arcs of permanent rotations of the body about vertical axes is given. Special cases of motions of the body are found which are caused by fast vibrations of the suspension point. One of these cases is studied when the rotation axis lies in the principal plane of inertia which does not contain the center of mass of the body and does not coincide with the equatorial plane of inertia. A complete nonlinear stability analysis of the corresponding equilibrium position of the two-degree-of-freedom system is carried out. For all admissible values of the three-dimensional parameter space, regions of linear stability are found. Cases of resonances of the third and fourth orders, as well as degeneration cases, are considered.

Détail

Mots-clés : Staude's permanent rotations
Keywords: high-frequency oscillations, rigid body, dynamic symmetry, stability, resonance.

Bibliographie

[1] Staude O., “Ueber permanente Rotatiosaxen bei der Bewegung eines schweren Körpers um einen festen Punkt”, Journal für die Reine und Angewandte Mathematik, 1894, no. 113, 318–334 | DOI | MR | Zbl

[2] Mlodzeevskii B. K., “About a permanent axis in the motion of a heavy rigid body around a fixed point”, Proceedings of the Department of Physical Sciences of the Society of Naturalists, 7:1 (1894), 46–48 (in Russian)

[3] Kholostova O. V., Investigation of stability of permanent rotations of Staude, Regular and Chaotic Dynamics, M.–Izhevsk, 2008, 128 pp.

[4] Grammel R., Der kreisel. Seine theorie und seine anwendungen, v. 1, 2, Berlin, 1950

[5] Rumyantsev V. V., “Stability of permanent rotations of a heavy rigid bodies”, Prikl. Mat. Mekh., 20:1 (1956), 51–66 (in Russian) | Zbl

[6] Rumyantsev V. V., “On stability of the rotation of a heavy rigid body with one fixed point in the case of S. V. Kovalevskaya”, Prikl. Mat. Mekh., 18:4 (1954), 457–458 (in Russian) | Zbl

[7] Magnus K., Kreisel. Theorie und anwendungen, Springer, Berlin, 1971, 507 pp. | MR | Zbl

[8] Kovalev A. M., Savchenko A. Ia., “Stability of uniform rotations of a rigid body about a principal axis”, Journal of Applied Mathematics and Mechanics, 39:4 (1975), 623–633 | DOI | MR | Zbl

[9] Sergeev V. S., “On the stability of permanent rotation of a heavy solid body about a fixed point”, Journal of Applied Mathematics and Mechanics, 40:3 (1976), 370–378 | DOI | Zbl

[10] Kovalev A. M., Savchenko A.Ya., “Stability of stationary motions of Hamiltonian systems in the presence of fourth order resonance”, Solid Mechanics, 9, Naukova Dumka, Kiev, 1977, 40–44 (in Russian)

[11] Yudovich V. I., “Vibrodynamics and vibrogeometry of mechanical systems with constraints”, Uspekhi Mekhaniki, 4:3 (2006), 26–158 (in Russian) | Zbl

[12] Blekhman I. I., Vibration mechanics, Fizmatlit, M., 1994, 400 pp.

[13] Strizhak T. G., Methods of research of dynamic systems of the type “pendulum”, Nauka, Alma-Ata, 1981, 253 pp.

[14] Kholostova O. V., Problems of dynamics of solids with vibrating suspension, Regular and Chaotic Dynamics, M.–Izhevsk, 2016, 308 pp.

[15] Markeev A. P., “On the theory of motion of a rigid body with a vibrating suspension”, Doklady Physics, 54:8 (2009), 392–396 | DOI | MR | Zbl

[16] Markeyev A. P., “The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point”, Journal of Applied Mathematics and Mechanics, 75:2 (2011), 132–139 | DOI | MR | Zbl

[17] Markeev A. P., “On the motion of a heavy dynamically symmetric body with vibrating suspension point”, Mechanics of Solids, 47:4 (2012), 373–379 | DOI

[18] Kholostova O. V., “On stability of relative equilibria of a rigid body with a vibrating point of support”, Vestn. Ross. Univ. Dr. Nar. Ser. Mat. Inform. Fiz., 2011, no. 2, 111–122 (in Russian)

[19] Belichenko M. V., Kholostova O. V., “On the stability of stationary rotations in the approximate problem of motion of Lagrange's top with a vibrating suspension point”, Nelineinaya Dinamika, 13:1 (2017), 81–104 (in Russian) | DOI | Zbl

[20] Vishenkova E. A., “Stability of special motions (permanent rotations) of a heavy rigid body with a suspension point vibrating along the vertical”, Nelineinaya Dinamika, 11:3 (2015), 459–474 (in Russian) | DOI | Zbl

[21] Vishenkova E. A., Kholostova O. V., “On the influence of vertical vibrations on the stability of permanent rotations of a rigid body about axes lying in the main plane of inertia”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 27:1 (2017), 98–120 (in Russian) | DOI | Zbl

[22] Kholostova O. V., “On the stability of the specific motions of a heavy rigid body due to fast vertical vibrations of one of its points”, Nelineinaya Dinamika, 11:1 (2015), 99–116 (in Russian) | DOI | Zbl

[23] Markeev A. P., Libration points in celestial mechanics and cosmodynamics, Nauka, M., 1978, 312 pp.

[24] Arnold V. I., Kozlov V. V., Neishtadt A. I., Mathematical aspects of classical and celestial mechanics, Springer-Verlag, Berlin–Heidelberg, 2006 | DOI | MR | Zbl