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IssuesArchive of Issues2012-4pp.373-379

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A.P. Markeev, "On the Motion of a Heavy Dynamically Symmetric Rigid Body with Vibrating Suspension Point," Mech. Solids. 47 (4), 373-379 (2012)
Year 2012 Volume 47 Number 4 Pages 373-379
DOI 10.3103/S0025654412040012
Title On the Motion of a Heavy Dynamically Symmetric Rigid Body with Vibrating Suspension Point
Author(s) A.P. Markeev (Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo†101, str.†1, Moscow, 119526 Russia, markeev@ipmnet.ru)
Abstract The motion of a dynamically symmetric rigid body in a homogeneous field of gravity is studied. One point lying on the symmetry axis of the body (the suspension point) performs high-frequency periodic or conditionally periodic vibrations of small amplitude. In the framework of approximate equations of motion obtained earlier, we find necessary and sufficient conditions for the stability of the body rotation about the vertical symmetry axis and study the existence and stability of regular precessions of the body in the coordinate system translationally moving together with the suspension point.
Keywords rigid body, vibration, precession, stability
References
1.  A. P. Markeev, "On the Theory of Motion of a Rigid Body with a Vibrating Suspension," Dokl. Ross. Akad. Nauk 427 (6), 771-775 (2009) [Dokl. Phys. (Engl. Transl.) 54 (8), 392-396 (2009)].
2.  I. G. Malkin, Theory of Motion Stability (Nauka, Moscow, 1966) [in Russian].
3.  G. K. Pozharitskii, "On the Construction of the Liapunov Functions from the Integrals of the Equations for Perturbed Motion," Prikl. Mat. Mekh. 22 (2), 145-154 (1958) [J. Appl. Math. Mech. (Engl. Transl.) 22 (2), 203-214 (1958)].
4.  N. G. Chetaev, "On Stability of Rotation of a Rigid Body with a Single Fixed Point in the Lagrangean Case," Prikl. Mat. Mekh. 18 (1), 123-124 (1954) [J. Appl. Math. Mech. (Engl. Transl.)].
5.  R. Grammel, The Gyroscope; Its Theory and Applications, Vol. 1 (Springer, Berlin, 1950; Izd-vo Inotstr. Liter., Moscow, 1952).
6.  L. V. Karapetyan and V. V. Rumyantsev, The Stability of Conservative and Dissipative Systems, in Advances in Science and Technology. General Mechanics, Vol. 6 (VINITI, Moscow, 1983) [in Russian].
Received 28 June 2010
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