Geodesic
On the stability of stationary rotations in the approximate problem of motion of Lagrange’s top with a vibrating suspension point
Russian journal of nonlinear dynamics, Tome 13 (2017) no. 1, pp. 81-104.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the motion of Lagrange’s top with a suspension point performing the specified highfrequency periodic motion with small amplitude in three-dimensional space. The approximate autonomous system of equations of motion written in the form of canonical Hamiltonian equations is investigated. The problem of the existence and number of stationary rotations of the top about its dynamical symmetry axis is solved. The study of stability of the corresponding equilibrium positions of the reduced two-degree-of-freedom system for fixed values of the cyclic integral constant depending on the angular velocity of rotation is carried out. For suspension points’ motions allowing for stationary rotations about the vertical, a detailed linear and nonlinear stability analysis of these rotations and rotations about inclined axes is carried out. For a number of other cases of the suspension point motions a linear stability analysis is carried out.
Mots-clés : Lagrange’s top
Keywords: “sleeping” top, high-frequency vibrations, stability. 
@article{ND_2017_13_1_a5,
     author = {M. V. Belichenko and O. V. Kholostova},
     title = {On the stability of stationary rotations in the approximate problem of motion of {Lagrange{\textquoteright}s} top with a vibrating suspension point},
     journal = {Russian journal of nonlinear dynamics},
     pages = {81--104},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2017_13_1_a5/}
}
TY  - JOUR
AU  - M. V. Belichenko
AU  - O. V. Kholostova
TI  - On the stability of stationary rotations in the approximate problem of motion of Lagrange’s top with a vibrating suspension point
JO  - Russian journal of nonlinear dynamics
PY  - 2017
SP  - 81
EP  - 104
VL  - 13
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2017_13_1_a5/
LA  - ru
ID  - ND_2017_13_1_a5
ER  - 
%0 Journal Article
%A M. V. Belichenko
%A O. V. Kholostova
%T On the stability of stationary rotations in the approximate problem of motion of Lagrange’s top with a vibrating suspension point
%J Russian journal of nonlinear dynamics
%D 2017
%P 81-104
%V 13
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2017_13_1_a5/
%G ru
%F ND_2017_13_1_a5
M. V. Belichenko; O. V. Kholostova. On the stability of stationary rotations in the approximate problem of motion of Lagrange’s top with a vibrating suspension point. Russian journal of nonlinear dynamics, Tome 13 (2017) no. 1, pp. 81-104. http://geodesic.mathdoc.fr/item/ND_2017_13_1_a5/

[1] Lagrange J. L., Méchanique analitique, Desaint, Paris, 1788, 530 pp.

[2] Routh E. J., The advanced part of a treatise on the dynamics of a system of rigid bodies: Being part II of a treatise on the whole subject, 6th ed., Dover, New York, 1955, 484 pp. | MR

[3] Tournaire L.-M., “Mémoire sur la rotation des corps pesant”, C. R. Acad. Sci. Paris, 50:1 (1860), 476–481

[4] Skimel' V. N., “On the problems of stability of motion of a solid rigid body about a stationary point”, Prikl. Mat. Mekh., 20:1 (1956), 130–132 (Russian) | MR | Zbl

[5] Chetaev N. G., “On stability of rotation of a rigid body with a single fixed point in the Lagrangean case”, Prikl. Mat. Mekh., 18:1 (1954), 123–124 (Russian) | Zbl

[6] Rose N. V., Rigid body dynamics, Kubuch, Leningrad, 1932 (Russian)

[7] Grammel R., Der Kreisel: Seine Theorie und seine Anwendungen, 2nd ed., Springer, Berlin, 1950, XI+284 pp. | Zbl

[8] Magnus K., Kreisel: Theorie und Anwendungen, Springer, Berlin, 1971, XII + 493 pp. | MR | Zbl

[9] Gorr G. A., Kudryashova L. W., Stepanova L. A., Classical problems of rigid body dynamics: Development and present state, Naukova Dumka, Kiev, 1978 (Russian) | MR

[10] Borisov A. V., Mamaev I. S., Dynamics of a rigid body: Hamiltonian methods, integrability, chaos, 2nd ed., R Dynamics, Institute of Computer Science, Izhevsk, 2005 (Russian) | MR

[11] Zhuravlev V. Ph., Klimov D. M., Applied methods in oscillation theory, Nauka, Moscow, 1988 (Russian) | MR

[12] Chelnokov Yu. N., “Motion of a heavy symmetric rigid body with a mobile point of a support”, Mech. Solids, 25:4 (1990), 1–7

[13] Koshlyakov V. N., “Structural transformations of dynamical systems with gyroscopic forces”, J. Appl. Math. Mech., 61:5 (1997), 751–756 | DOI | MR | Zbl

[14] Savchenko Ya. A., Kanibolotskiy V. V., “On the stability of a unbalanced system of two Lagrange's gyroscopes with a vibrating suspension point”, Mekh. Tverd. Tela, 1991, no. 2, 101–104 (Russian) | Zbl

[15] Kholostova O. V., “On the periodic motion of Lagrange's top with vibrating suspension”, Mech. Solids, 2002, no. 1, 26–38

[16] Kholostova O. V., “The dynamics of a Lagrange top with a vibrating suspension point”, J. Appl. Math. Mech., 63:5 (1999), 741–750 | DOI | MR | Zbl

[17] Kholostova O. V., “A case of periodic motion for a Lagrange gyroscope with a vibrating suspension”, Dokl. Phys., 45:12 (2000), 690–693 | DOI | MR

[18] Kholostova O. V., “On a case of periodic motions of the Lagrangian top with vibrating fixed point”, Regul. Chaotic Dyn., 4:4 (1999), 81–93 | DOI | MR | Zbl

[19] Kholostova O. V., “The stability of a «sleeping» lagrange for with a vibrating suspension point”, J. Appl. Math. Mech., 64:5 (2000), 821–831 | DOI | MR | Zbl

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

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

[22] Markeyev A. P., “The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point”, J. Appl. Math. Mech., 75:2 (2011), 132–139 | DOI | MR | Zbl

[23] Markeev A. P., Theoretical mechanics, R Dynamics, Institute of Computer Science, Izhevsk, 2007 (Russian)

[24] Markeev A. P., Libration points in celestial mechanics and space dynamics, Nauka, Moscow, 1978 (Russian)

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

[26] Kholostova O. V., “On motions of a pendulum with a vibrating suspension point”, Teor. Mekh., 24, 2003, 157–167 (Russian)