Non-linear parametric resonance driven oscillations of dumbell satellite in elliptical orbit under the combined effects of magnetic field of the earth and oblateness of the earth

Parametric resonance driven oscillations of a dumbbell satellite in elliptical orbit in central gravitational field of force under the combined effects of perturbing forces Earth Magnetic field and Oblateness of the Earth has been studied. The system comprises of two satellite connected by a light, flexible and inextensible cable, moves like a dumbbell satellite in elliptical orbit, in central gravitational field of force. The gravitational field of the Earth is the main force governing the motion and magnetic field of the Earth and Oblateness of the Earth are considered to be perturbing forces, disturbing in nature. Non-linear oscillations of dumbbell satellite about the equilibrium position in the neighborhood of parametric resonance 1 2   , under the influence of perturbing forces, which is suitable for exploiting the asymptotic methods of Bogoliubov, Krilov and Metropoloskey has been studied, considering ‘e’ to be a small parameter. The Hamiltonian has been constructed for the problem and phase analysis has been applied to investigate the stability of the system.


Introduction
This paper is devoted to the analysis of non-linear parametric resonance driven oscillations of cable connected satellites system in elliptical orbit connected by a light, flexible and inextensible cable moving in the central gravitational field of the Earth under the combined effects of the Earth magnetic field and Oblateness of the Earth. The satellites are considered to be charged material particle and the motion of the system is studied relative to their centre of mass, under the assumption that the later moves along elliptical orbit. The cable connecting the two satellites is taut and nonelastic in nature such that, the system moves like a dumbbell satellite. Many space configurations of cable connected satellite system have been proposed and analysed by different authors like two satellite are connected by a rod (Celletti et al 2008), two or more satellites are connected by a tether (M. Krupa et al 2000Krupa et al & 2006, (Beletsky & Levin 1993), (Mishra & Modi 1982). All these authors have mentioned numerous important applications of system and stability of relative equilibrium, if the system moves in a circular and elliptical orbit. (Beletsky & Novikova 1969), studied the motion of a system of two satellite connected by a light, flexible and inextensible string in the central gravitational field of force relative to their centre of mass, which is itself assumed to more along a Keplerian elliptical orbit under the assumption that the two satellite are moving in the plane of the centre of mass. The same problem in its general form, was further investigated (Singh 1971(Singh , 1973, these works conducted the analysis of relative motion of the system for the elliptical orbit of the centre of mass in the two dimensional as well as three dimensional cases. (Narayan & Singh 1987, 1990, 1992, studied non-linear oscillations due to solar radiation pressure of the centre of mass of the system moves along an elliptical orbit. The different aspects of the problem of stability of satellites in low and high altitude orbit with different perturbation forces are studied by many scientists, (Sharma & Narayan 2001,2002, (Singh et al 1971(Singh et al ,1973(Singh et al ,1997, (Das et al 1976), and (Narayan et al 1987(Narayan et al ,1990(Narayan et al ,1992. Special references are mentioned (Sarychev et al 2000(Sarychev et al , 2007 studied the problem determining all equilibria of a satellite subject to gravitational and aerodynamic torque in circular orbit. All bifurcation values of the parameter corresponding to qualitative changes of stability domain are determined. (Palacian 2007), studied the dynamics of a satellites orbiting are Earth like planet at low altitude orbit and perturbation is caused by inhomogeneous potential due to the Earth. (Langbort 2002), studied bifurcation of relative equilibria in the main problem of artificial satellite theory for a prolate body. (Markeev et al 2003), studied the planar oscillation of a satellite in a circular orbit. (Ayub Khan et al 2011), investigated chaotic motion in problem of dumbbell satellite. The present paper deals with the non-linear parametric driven oscillation of dumbbell satellite in elliptical orbit under the combined effects of magnetic field of the Earth and oblateness of the Earth. The perturbing forces due to Earth magnetic field results from the interaction between space craft's residual magnetic field and the geomagnetic field. The perturbing force is arising due to magnetic moments, eddy current and hysteresis, out of these the space craft magnetic moment is usually the dominant source of disturbing effects.

Equation of motion
The combined effects of the geomagnetic field and Oblateness of the Earth on the motion and stability of the satellite connected by a light, flexible and inextensible cable, under the influence of the central gravitational field of the Earth have been considered. The analysis of Evolutional and Non-evolutional motion of dumbell satellite in elliptical orbit has been restricted to two Dimensional case, we have assumed that the satellites are moving in the orbital plane of the centre of mass of the system. The motion and stability of cable connected satellite system under the effects of Earth's magnetic field, (Das et al 1976), (Narayan et al 2004), and combined effects of Earth magnetic field and oblateness of the Earth, (Narayan and Pandey 2010), in elliptical and in low altitude orbit has been studied. The equation of two dimensional motion of one of the satellite under the rotating frame of reference in (Nechville's 1926) co-ordinate system, relative to their centre of mass, which moves along equatorial orbit under the combined influence of the Earth magnetic field and Oblateness of the Earth can be represented in (2.1) :  In this case the condition for constrained is given by the in equality: 22 the non-linear oscillations described by   2.6 , take place as long as inequality given below is satisfied.
Where v and e are respectively true anomaly of the centre of mass of the system and the eccentricity of the orbit of the system. The prime denotes differentiation with respect to true anomaly v.

Non-linear non-resonance oscillations of the system about the position of equilibrium for small eccentricity
The non-linear oscillations of the system of cable-connected satellites under the influence of above mentioned forces described by equation   2.9 will be investigated for non-resonance cases. 2 ( sin ) 2 sin ν 4sin cos ν 2 cos sin 5 sin 2

 
will be assumed to be the order of e.
From the above solution, we concluded that amplitude 'a' remain constant up to the order of the square of the eccentricity. The phase of the oscillations of the system in this case of non-linear, non-resonance oscillations varies with respect to true anomaly. However the variation of the phase is of the order of the square of the eccentricity, which is a small quantity. We arrived at the conclusion that the system has main resonance at

Non-linear parametric driven oscillations of dumbell satellite system about the position of equilibrium for small eccentricity
The non-linear oscillations of the dumbell satellite under the influence of the above mentioned Forces described by (2.3), will be investigated for the parametric resonance case on the assumption That magnetic field parameter is of the order 'e' then, equation ( , will be assumed to be the order of e. The system described by equation (4.1), moves under the forced vibration due to the presence of the magnetic field of the earth and oblateness of the earth. this periodic sine force of erturbative nature as long as the period of oscillations of the system is different from the period of sine force for which solution is obtains. as the period of sine force is always changing, it may become equal to the sine force, in that case the periodic sine force plays vital role in the oscillatory motion of the system. while examining the nonresonance case, we conclude that the system experience parametric resonance behavior at and near 1 , 2

 
and hence the nonresonance solution fails. we are benefitted of the smallness of the eccentricity 'e' in equation (3.1), and hence the solution of the differential equation may be obtained by exploiting the bogoliubov, krilov and metropoloskey method.we constructs the asymptotic solutions of the system representing (4.1), in the most Where 00 ( 1) C n C   (4.14) indicates about three zones,one is stationary,allowable and forbidden region and it also indicated that there exists only two stationary regime of the amplitude and it is stable as the integral curves are closed curves. For any other initial condition we shall obtain periodic change in the amplitude 'a', which would be bounded. But the maximum value of 'a' in this case will always be greater than its value at the stationary regime. Therefore, for the gravity gradient stablisation of such a space system in elliptical orbit, we are required to bring the amplitude of oscillations near the stationary regime, which gives the smallest deflection of the system from the relative equilibrium position in comparison to any other regime of oscillations. In the Figure (4), the integral curves for  = 0.48, e =0.1, A = 0.005 and B = 0.001 has been plotted and it is found that the stationary regime of the amplitude 'a' exists in addition to allowable and forbidden region. In this case also there exists two stationary regime with a slight change in its position. Minute observation of the signature (3) and (4) suggested concluding that during the evolution of oscillations of the system as it approaches the parametric resonance, the stationary amplitude declines steadily with continuously changing phase.

Conclusion
We have discussed that the combined effects of the Earth Oblateness and the magnetic field of the Earth on the evolutional and non-evolutional motion of cable connected satellites system, connected by a light, flexible and inextensible cable in the central gravitational field of the Earth for the elliptical orbit of centre of mass of the system. The satellites are considered as charge material particle. The motion of each of their relative to the centre of mass has been studied. It is assumed that centre of mass moves along Keplerian orbit around oblate Earth in elliptical orbit. It is further assumed that satellites are subjected to absolutely nonelastic impacts as the cable tightened. Throughout our analysis, we assumed that the system moves like a dumbbell satellite. We further discussed the non-linear non-resonant and parametric resonant oscillations of dumbbell satellite about the equilibrium point of the system in elliptical orbit. The equations of motion have been derived in the required form. The non-resonant oscillations of the problem has been studied with the help of Bogoliubov, Krilov and Metropoloskey, method when the eccentricity 'e' of the orbit of centre of mass has been taken as the small parameter for the solution of the system.We arrived at the conclusion that the amplitude of oscillations of the system remains constant up to the second order of the approximation. The phase of oscillations varies with respect to true anomaly, but the rate of change is the function of the square of eccentricity of the orbit of the centre of mass. We also come to the conclusion that the system experience main resonance at It has also been established that the system will always move like a dumbbell satellite in the phase Plane ( , ) a  under consideration.
Thus, the oblateness of the Earth and magnetic field of the Earth will play important role in disturbing the attitude of the system of a dumbell satellite in elliptical orbit