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IssuesArchive of Issues2006-4pp.21-28

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E. K. Uzbek, "Semiregular second type precessions of a gyrostat acted upon by potential and gyroscopic forces," Mech. Solids. 41 (4), 21-28 (2006)
Year 2006 Volume 41 Number 4 Pages 21-28
Title Semiregular second type precessions of a gyrostat acted upon by potential and gyroscopic forces
Author(s) E. K. Uzbek (Donetsk)
Abstract We study conditions for the existence of precession motions of a gyrostat with a fixed point under the constant spin velocity condition in a problem described by differential equations of the Kirchhoff class [1, 2]. We obtain new cases of such motions which complete the results obtained in [3, 4] and correspond to new solutions of the equations of motion.

The precession motions of a gyrostat with a fixed point, whose property is that the angle between the axes l1 and l2 is constant (l1 is always attached to the gyrostat and l2 is immovable in space), are widely used in the theory of gyro systems, which is very important in engineering. Ishlinskii([5, p. 353] noted: "After the nutation dies, the further slow motion of the rotor axis, known as precession, agrees with the precession equations in gyro theory very accurately..." A classical example of precession motion is the regular precession of the Lagrange gyro around the vertical axis in the gravitational field. Precessions of nonsymmetric bodies were studied by Appelrot [6] and Grioli [7]. Appelrot proved that the gyros similar to the Kovalevskaya gyro and the Goryachev-Chaplygin gyro cannot have precession motions other than pendulum motions. Grioli discovered regular precession of a nonsymmetric body around an inclined axis. In the problem of motion of a heavy rigid body with a fixed point and in its various generalizations, many results were obtained by Gorr (e.g., see [3, 8-10]). Most of them pertain to the study of precessions in the problem of gyrostat motion under the action of electric, magnetic, and Newtonian fields on a magnetized and charged gyrostat [2]. The most interesting point in the study of precession is that it is described by the Kirchhoff type differential equations [1, 2] and, by the hydrodynamic analogy, all results can be transferred to the problem on the motion of a heavy rigid body in an ideal incompressible fluid.

At present, all regular precessions [11] and first type semiregular precessions [12, 13] in this problem have been studied completely. Second type semiregular precessions have been studied only in several special cases [3, 4]. In the present paper, these precessions are studied for an admissible special parametrization of one of the algebraic equations determining the existence conditions for this class of precessions.
References
1.  P. V. Kharlamov, G. V. Mozalevskaya, and M. E. Lesina, "On various representations of the Kirchhoff equations," Mekh. Tverd. Tela, Naukova Dumka, Kiev, No. 31, pp. 3-17, 2001.
2.  H. M. Yehia, "On the motion of a rigid body acted upon by potential and gyroscopic forces," J. Théor. Appl. Méch., Vol. 5, No. 5, pp. 755-762, 1986.
3.  E. V. Verkhovod and G. V. Gorr, "Precession-isoconic motions of a rigid body with a fixed point," PMM [Applied Mathematics and Mechanics], Vol. 57, No. 4, pp. 31-39, 1993.
4.  A. V. Maznev, "On semiregular precession of the second type in the generalized problem of dynamics of rigid bodies," Mekh. Tverd. Tela, Naukova Dumka, Kiev, No. 25, pp. 26-30, 1993.
5.  A. Yu. Ishlinskii, Orientation, Gyroscopes, and Inertial Navigation [in Russian], Nauka, Moscow, 1976.
6.  G. G. Appelrot, "Determining classes of kinetically symmetric heavy gyroscopes admitting simplified motions close to the inertial motion or to some simplified motion of the Lagrange gyroscope," Izv. AN SSSR. Ser. Fiz., No. 3, pp. 385-411, 1938.
7.  G. Grioli, "Esistenza e determinazione delle precessioni regolari dinamicamente possibili per un solido pesante asimmetrico," Ann. Mat. Pura Appl,. Ser. 4, Vol. 26, Nos. 3-4, pp. 271-281, 1947.
8.  G. V. Gorr, "Precession motions in the dynamics of a rigid body and in the dynamics of systems of connected rigid bodies," PMM [Applied Mathematics and Mechanics], Vol. 67, No. 4, pp. 573-587, 2003.
9.  G. V. Gorr, "Several properties of precession motions around the vertical axis of a heavy rigid body with a single fixed point," PMM [Applied Mathematics and Mechanics], Vol. 38, No. 3, pp. 451-458, 1974.
10.  G. V. Gorr, A. A. Ilyukhin, A. M. Kovalev, and A. Ya. Savchenko, Nonlinear Analysis of Behavior of Mechanical Systems [in Russian], Naukova Dumka, Kiev, 1984.
11.  G. V. Gorr and N. V. Kurganskii, "On regular precession around the vertical axis in a problem of dynamics of solids," in Mekh. Tverd. Tela, No. 19, pp. 16-20, Naukova Dumka, Kiev, 1987.
12.  N. V. Kurganskii, "On semiregular precession of the first type around the vertical axis in a problem of dynamics of solids," in Mekh. Tverd. Tela, No. 20, pp. 67-71, Naukova Dumka, Kiev, 1988.
13.  G. V. Mozalevskaya and L. N. Oreshkina, "Motions of a rigid body without nutations," in Mekh. Tverd. Tela, No. 23, pp. 1-5, Naukova Dumka, Kiev, 1991.
14.  A. I. Dokshevich, "Integrable cases of the problem about the motion of a heavy rigid body around a fixed point," PMM [Applied Mathematics and Mechanics], Vol. 4, No. 11, pp. 95-100, 1968.
Received 30 June 2004
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