Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
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IssuesArchive of Issues2006-4pp.40-45

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A. K. Alekhin, "On the stability of plane motions of a heavy axially symmetric rigid body," Mech. Solids. 41 (4), 40-45 (2006)
Year 2006 Volume 41 Number 4 Pages 40-45
Title On the stability of plane motions of a heavy axially symmetric rigid body
Author(s) A. K. Alekhin (Moscow)
Abstract We consider the motion of a rigid body with one fixed point in a homogeneous gravitational field. We assume that the ellipsoid of inertia of the body about the fixed point is an ellipsoid of revolution.

We study the orbital stability of plane periodic motions of the body. In the unperturbed motion, the body oscillates or rotates around the axis of symmetry, which is always horizontal. The oscillation amplitude or the rotation frequency of the body is arbitrary.

We show that the problem depends on the following two parameters: the ratio of the nonequal principal moments of inertia and the energy constant in the unperturbed motion, which determines the oscillation amplitude or the rotation frequency.

In the plane of these parameters, we construct orbital stability and instability domains in the first (linear) approximation. We also perform nonlinear analysis of orbital stability in the stability domains in the first approximation and on their boundaries.
References
1.  A. P. Markeev, "On area-preserving mappings and their application in dynamics of systems with impacts," Izv. RAN. MTT [Mechanics of Solids], No. 2, pp. 37-54, 1996.
2.  A. P. Markeev, "Stability of equilibrium states of hamiltonian systems: a method of investigation," Izv. RAN. MTT [Mechanics of Solids], No. 6, pp. 3-12, 2004.
3.  A. P. Markeev, "On stability of plane motions of a rigid body in the Kovalevskaya case," PMM [Applied Mathematics and Mechanics], Vol. 65, No. 1, pp. 51-58, 2001.
4.  A. P. Markeev, "On pendulum-like motions of a rigid body in the Goryachev-Chaplygin case," PMM [Applied Mathematics and Mechanics], Vol. 68, No. 2, pp. 282-293, 2004.
Received 10 August 2004
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