Some Non-linear Parametric Resonance Oscillations Of Dumbell Satellite In Elliptical Orbit

Some non-linear parametric resonance oscillations of Dumbell satellite in elliptical orbit in the central gravitational field of the Earth under the combined influence of Earth magnetic field ,oblateness of the Earth and some external periodic forces of general in nature has been studied.The system comprises of two charged material particles connected by a light flexible and inextensible cable ,moves with taut cable like a dumbell satellite around the Earth in elliptical orbit.The central gravitational field of the Earth is the main force governing the motion of the system and various perturbing forces influencing the system are disturbing in nature.Non-linear oscillations of dumbbell satellite about the equilibrium position in the neighbourhood of the parametric resonance 1 2   , has been investigated, exploiting the well known asymptotic method due to Bogoliubov,Krilov and Metropoloskey , considering 'e' to be a small parameter .The analysis of stability of the system has been discussed due to Poincare method.