  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
Issued 6 times a year
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V. S. Aslanov, "The motion of a rotating body in a resisting medium," Mech. Solids. 40 (2), 2132 (2005) 
Year 
2005 
Volume 
40 
Number 
2 
Pages 
2132 
Title 
The motion of a rotating body in a resisting medium 
Author(s) 
V. S. Aslanov (Samara) 
Abstract 
The motion of a rotating rigid body in a resisting medium under the action of a
sinusoidal or biharmonic timedepending restoring torque and small perturbation
torques is considered in a nonlinear formulation. A justification of the
representation of the perturbations by slowly varying parameters and parameters
of small asymmetry is given. The solutions of the equations of nonperturbed
motion are presented in terms of Jacobi's elliptic functions. For the case
where the nutational torque biharmonically depends on the nutation angle, the
equations of nonperturbed motion are represented in terms of the angleaction
variables, which can be expressed in terms of complete elliptic integrals. The
averaged equations of motion of an axially symmetric body under the action of
the biharmonic torque and small damping torques are constructed. The equations
of perturbed motion of an asymmetric body are reduced to a standard
twofrequency system, and a partially averaged system is constructed. Necessary
and sufficient conditions of the stability of nonlinear resonances and those of
the Lyapunov stability of the motion in the neighborhood of a stationary point
under the action of small perturbations are obtained. A numerical example is
given to show that the stability of the resonance does not imply the stability
in the neighborhood of the stationary point, and vice versa. 
References 
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Moscow, 1968. 
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Moscow, Leningrad, 1944. 
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Kosmicheskie Issledovaniya [Cosmic Research], Vol. 30, No. 5, pp. 608614, 1992. 
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spacecraft during its reentry,"
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with the bilinear characteristic of the restoring torque,"
Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 1925, 1995. 
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under the action of a nonlinear torque,"
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[Journal of Computational Mathematics and Mathematical Physics],
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11.  V. S. Aslanov and I. A. Timbai, Motion of a Rigid Body in the Generalized
Lagrange's Case [in Russian], Izdvo SGAU, Samara, 2001. 
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[in Russian], Nauka, Moscow, 1988. 
13.  V. S. Aslanov and S. V. Myasnikov, "Stability of nonlinear resonant modes
of motion of a spacecraft in the atmosphere,"
Kosmicheskie Issledovaniya [Cosmic Research], Vol. 34, No. 6, pp. 626632, 1996. 
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Kosmicheskie Issledovaniya [Cosmic Research], Vol. 35, No. 6, pp. 659665, 1997. 
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of an asymmetric spacecraft in the atmosphere,"
Kosmicheskie Issledovaniya [Cosmic Research], Vol. 26, No. 2, pp. 220226, 1988. 
17.  L. D. Akulenko, T. A. Kozachenko, and D. D. Leshchenko,
"Perturbed rotations of a rigid body under the action of an unsteady
restoring torque depending on the nutation angle,"
in Mechanics of a Rigid Body [in Russian], No. 31, pp. 5762,
Int Prikl. Mat. i Mekh. NAN Ukrainy, Donetsk, 2001. 

Received 
8 April 2003 
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