Hostname: page-component-5c6d5d7d68-txr5j Total loading time: 0 Render date: 2024-08-22T08:20:55.072Z Has data issue: false hasContentIssue false
  • English
  • Français

Planar Oscillation of the Satellite Near Parametric Resonance in the Elliptic Orbit

Published online by Cambridge University Press:  12 April 2016

Ram Binoy Singh
Ram Binoy Singh*
Affiliation:
Department of Mathematics, University of Bihar, Muzaffarpur

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is devoted to the study of motion of an artificial satellite relative to its centre of mass near parametric resonance in elliptic orbit. It is well known fact that the satellite, of the form of an ellipsoid with three unequal axes, while moving about the central planet, oscillates about the stable position of equilibrium (the longest axis of the satellite coinciding with the radius vector of its centre of mass). The oscillation of the satellite about this position of equilibrium in the orbital plane of its centre of mass is described by a well known second order nonlinear differential equation with a periodic sine force. Naturally there will be resonance cases (main as well as parametric) for such a systems. In the previous author’s work, it was discovered a series of parametric resonances for the system, which corresponds to n = ½k where k is a non-zero integer and n is a parameter depending on the shape of the satellite. The parametric resonance, for k = 1, has been considered here. The first approximate solution of the equation of motion has been obtained by Eogoliubov-Krilov method with e (the eccentricity of the orbit of the centre of mass of the satellite) as the small parameter. This method enables us to visualise the oscillation of the satellite for the resonance case as well as near the resonance. Three stationary values of the amplitudes and phase of oscillation have been obtained, out of which only one is stable near this particular parametric resonance. At the resonance there appear only one stationary regime of oscillation with a very small amplitude.

Type
Part V General Celestial Mechanics and Stellar Dynamics
Information
International Astronomical Union Colloquium , Volume 132: Instability, Chaos and Predictability in Celestial Mechanics and Stellar Dynamics , 1993 , pp. 391 - 397
Copyright
Copyright © Nova Science Publishers 1993

References

[1] Belatsky, V.V. (1965), Motion of Artificial Satellite Relative to its Centre of Mass (Russian), Nauka, Moscow. Google Scholar
[2] Bogoliubov, N.N. (1971), Asymptotic Methods in the Theory of Non-linear Oscillations (Gordon and Breach, New York).Google Scholar
[3] Chernousko, F.L. (1963), Resonance Phenomena in the Motion of Satellite Relative to its Centre of Mass, Zhurnal vichislitelnoi Mathematiki and Mat hematicheskoi physiki, Vol. 3, No.3, pp. 528–538.Google Scholar
[4] Jlataustov, B.A., Okhotsimsky, D.E., Sareechev, B.A., and Torzhevsky, A.P. (1964), Periodic Solution of the Problem of Planar Oscillation of the Satellite in Elliptic Orbit, Kosraicheskie Issledovania, Vol. 2, No.5, pp. 658–666.Google Scholar
[5] Singh, R.B. and Demin, V.G. (1973), Nonlinear Planar Oscillation of a Solid Body in Elliptic Orbit, Kosmicheskie Issledovania, Vol. 11, No.2.Google Scholar