Geodesic
On the Stability of Pendulum-type Motions in the Approximate Problem of Dynamics of a Lagrange Top with a Vibrating Suspension Point
Russian journal of nonlinear dynamics, Tome 14 (2018) no. 2, pp. 243-263.

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This paper addresses the motion of a Lagrange top in a homogeneous gravitational field under the assumption that the suspension point of the top undergoes high-frequency vibrations with small amplitude in three-dimensional space. The laws of motion of the suspension point are supposed to allow vertical relative equilibria of the top’s symmetry axis. Within the framework of an approximate autonomous system of differential equations of motion written in canonical Hamiltonian form, pendulum-type motions of the top are considered. For these motions, its symmetry axis performs oscillations of pendulum type near the lower, upper or inclined relative equilibrium positions, rotations or asymptotic motions. Integration of the equation of pendulum motion of the top is carried out in the whole range of the problem parameters. The question of their orbital linear stability with respect to spatial perturbations is considered on the isoenergetic level corresponding to the unperturbed motions. The stability evolution of oscillations and rotations of the Lagrange top depending on the ratios between the intensities of the vertical, horizontal longitudinal and horizontal transverse components of vibration is described.
Keywords: high-frequency vibrations, pendulum-type motions, stability.
Mots-clés : Lagrange’s top 
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M. V. Belichenko. On the Stability of Pendulum-type Motions in the Approximate Problem of Dynamics of a Lagrange Top with a Vibrating Suspension Point. Russian journal of nonlinear dynamics, Tome 14 (2018) no. 2, pp. 243-263. http://geodesic.mathdoc.fr/item/ND_2018_14_2_a7/

[1] Mlodzieiowski, B., “Über permanente Axen bei der Bewegung eines schweren festen Körpers um einen Punkt”, Tr. Otdel. Fiz. Nauk Ob-va Lubit. Estestvozn., 7:1 (1894), 46–48 (Russian)

[2] Prikl. Mat. Mekh., 24:2 (1960), 294–302 (Russian) | DOI | MR

[3] Markeev, A. P., “Plane and Quasi-Plane Rotations of a Heavy Rigid Body about a Fixed Point”, Izv. AN SSSR. Mekh. Tverd. Tela, 1988, no. 4, 29–36 (Russian)

[4] Irtegov, V. D., “The Stability of the Pendulum-Like Oscillations of a Kovalevskaya Gyroscope”, Tr. Kazan. Aviats. Inst., 97 (1968), 38–40 (Russian)

[5] Prikl. Mat. Mekh., 53:6 (1989), 873–879 (Russian) | DOI | MR | Zbl

[6] Prikl. Mat. Mekh., 65:1 (2001), 51–58 (Russian) | DOI | MR | Zbl

[7] Izv. Akad. Nauk. Mekh. Tverd. Tela, 38:1 (2003), 3–9 (Russian) | MR

[8] Prikl. Mat. Mekh., 68:2 (2004), 282–293 (Russian) | DOI | MR

[9] Izv. Akad. Nauk. Mekh. Tverd. Tela, 2007, no. 2, 14–21 (Russian) | DOI

[10] Bardin, B. S., “On the Orbital Stability of Pendulum-Like Motions of a Rigid Body in the Bobylev – Steklov Case”, Regul. Chaotic Dyn., 15:6 (2010), 704–716 | DOI | MR | Zbl

[11] Bardin, B. S., Rudenko, T. V., and Savin, A. A., “On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev – Steklov Case”, Regul. Cahotic Dyn., 17:6 (2012), 533–546 | DOI | MR | Zbl

[12] Izv. Akad. Nauk. Mekh. Tverd. Tela, 41:4 (2006), 56–62 (Russian)

[13] Bardin, B. S. and Savin, A. A., “On the Orbital Stability of Pendulum-Like Oscillations and Rotations of a Symmetric Rigid Body with a Fixed Point”, Regul. Chaotic Dyn., 17:3–4 (2012), 243–257 | DOI | MR | Zbl

[14] Yudovich, V. I., “Vibrodynamics and Vibrogeometry in Mechanical Systems with Constraints”, Uspekhi Mekh., 4:3 (2006), 26–158 (Russian) | MR

[15] Dokl. Akad. Nauk, 427:6 (2009), 771–775 | DOI | MR | Zbl

[16] Prikl. Mat. Mekh., 75:2 (2011), 193–203 (Russian) | DOI | MR | Zbl

[17] Izv. Akad. Nauk. Mekh. Tverd. Tela, 2002, no. 1, 34–48 (Russian)

[18] Prikl. Mat. Mekh., 63:5 (1999), 785–796 (Russian) | DOI | MR | Zbl

[19] 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

[20] Dokl. Ross. Akad. Nauk, 375:5 (2000), 627–630 (Russian) | DOI | MR

[21] Prikl. Mat. Mekh., 64:5 (2000), 858–868 (Russian) | DOI | MR | Zbl

[22] Izv. Akad. Nauk. Mekh. Tverd. Tela, 2012, no. 4, 3–10 (Russian) | DOI

[23] Belichenko, M. V. and Kholostova, O. V., “On the Stability of Stationary Rotations in the Approximate Problem of Motion of Lagrange’s Top with a Vibrating Suspension Point”, Nelin. Dinam., 13:1 (2017), 81–104 (Russian) | DOI | MR | Zbl

[24] Markeev, A. P., Libration Points in Celestial Mechanics and Space Dynamics, Nauka, Moscow, 1978, 312 pp. (Russian)

[25] Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, Grundlehren Math. Wiss., 67, 2nd ed., rev., Springer, Berlin, 1971, XVI, 360 pp. | MR | Zbl

[26] Markeev, A. P., Theoretical Mechanics, R Dynamics, Institute of Computer Science, Izhevsk, 2007, 592 pp. (Russian)

[27] Gradshtein, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 7th ed., Acad. Press, Amsterdam, 2007, 1200 pp. | MR | Zbl