Engineering Transactions, 46, 3-4, pp. 271–289, 1998

Perturbed Motions of a Rotating Symmetric Gyrostat

A.I. Ismail
Tanta University

T.S. Amer
Tanta University

M.O. Shaker
Tanta University

The aim of the present paper is to provide analytical solutions for the perturbed problem of the behaviour of a symmetric gyrostat about a fixed point. This gyrostat is acted upon by a gyrostatic momentum $l_s (s = 1, 2, 3)$, variable restoring moment $k$ and perturbing moments $M_i (i = 1, 2, 3)$. The moment $k$ is introduced in view of the rotation of gyrostat under the action of electro-magnetic field. The solutions are achieved when the third component of the gyrostatic momentum is different from zero $(l_3 \neq 0)$. The averaging method is applied to investigate the first order approximate solutions of resonant and non-resonant cases.
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I.G. MALKIN, Some problems in the theory of nonlinear oscillations, United States Atomic energy Commission, Technical Information Service, ABC 3766, 1959.

R. Cm and A. VIGUERAS, The analytical theory of the earth's rotation using a symmetrical gyrostat as a model, Acad. Ciencias Zaragoza, 45, pp. 83–93, 1990.

R. MOLINA and V. VIGUERAS, Analytical integration of a generalized Euler-Poisson problem: applications, IAV Symposium, No. 172, Paris 1995.

F.A. EI–BARKI and A.I. ISMAIL, Limiting case for the motion of a rigid body about a fixed point in the Newtonian force field, ZAMM, 75, 11, pp. 821–829, 1995.

D.D. LECHCHENKO, S.N. SALLAM, Perturbed rotational motions of a rigid body with mass distribution near to the case of Lagrange, Dep. Vokr NUUNTU, No 1656 ok 88, 28–06, 22, 1988.

D.D. LBCHCHBNKO, S.N. SALLAM, Perturbed rotational motions of a rigid body relative to fixed point IZV. ANUSS R., Mekh. Tvjord. Tela, 5, pp. 16–23, 1990.

D.D. LECHCHENKO, S.N. SALLAM, Perturbed rotational motions of a rigid body that are dose to regular precession PMM, 54, 2, pp. 224–232, 1990.

L.D. AKULENKO, D.D. LECHCHENKO and F.L. CHERNOUSHKO, Perturbed motions of a rigid body close to the Lagrange case, PMM, 43, 5, pp. 771–778, 1979.

A.H. NAYFEH, Introduction to perturbation techniques, Wiley-Interscience, New York 1981.

A.H. NAYFEH, Perturbation methods, Wiley-Interscience, New York 1973.

A.H. NAYFEH, Problems in perturbation, Wiley-Interscience, New York 1985

R. CID and A. VIGUERAS, Actas IV Asamblea Nacional de Astronomia y Astrofisica, Santiago de Compostela, pp. 912, 1983 (in Spanish).

R.F. GRANT, Analytical mechanics, Univ. of Utah, 1962.

V.S. SERGEEV, Periodic motions of a heavy rigid body with a fixed point, that is close to being dynamically symmetrical, PMM, 47, 1, pp. 163–166, 1983.

N.N. BOGOLIUBOV and U.A. MITROPOLSKI, Asymptotic method in the theory of nonlinear oscillations [in Russian], Nauka 1974.

V.I. ARNOL'D, Supplementary chapters in the theory of ordinary differential equations [in Russian], Nauka, Moscow 1970.

V.M. VOLOSOV and B.T. MORGUNOV, Method of averaging in the theory of nonlinear oscillatory systems [in Russian], Moscow, IZV–VOMGU, 1971.

V.N. KOSHLYAKOV, Some particular cases of integration of the dynamic Euler equations associated with motion of a gyroscope in a resistant medium, PMM 17, 2, pp. 137–148, 1953.

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