  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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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), 2128 (2006) 
Year 
2006 
Volume 
41 
Number 
4 
Pages 
2128 
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 l_{1} and l_{2} is constant (l_{1} is
always attached to the gyrostat and l_{2} 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
GoryachevChaplygin 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, 810]). 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. 317, 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. 755762, 1986. 
3.  E. V. Verkhovod and G. V. Gorr,
"Precessionisoconic motions of a rigid body with a fixed point,"
PMM [Applied Mathematics and Mechanics], Vol. 57, No. 4, pp. 3139, 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. 2630, 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. 385411, 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. 34, pp. 271281, 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. 573587, 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. 451458, 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. 1620, 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. 6771, 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. 15, 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. 95100, 1968. 

Received 
30 June 2004 
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