1. 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 non-elastic 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 2000 & 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,
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
1971,1973,1997), (Das et al 1976), and (Narayan et al 1987,1990,1992). Special
references are mentioned (Sarychev et al 2000, 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.
2. 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) :
(2.1)
Fig. 1: Rotating
Frame of Reference
Here 
axis is in the direction
of position vector joining the centre of mass of the system and the attracting
centre and the 
axis is along the normal to the
position vector in the orbital plane of the centre of mass in the direction of the
motion of the satellite
where A is the Oblateness due to
the Earth and B is the magnetic field of the Earth.
Moreover:
(2.2)
The dipole of the Earth has its axis inclined from the
polar axis of the Earth by a value of 110
. The
angle f and W completely define the position of
, unit vector along the axis of magnetic
dipole of the Earth.
In this case the condition for constrained is given by
the in equality:
(2.3)
Fig. 2: Orientation
of
Where l denotes Lagrange’s multiplier
and m denotes product of the gravitational constant and the
mass of the Earth, where:
On the masses and
, where
v is the true anomaly of the centre of mass of the system in elliptical orbit.
And 
Where 
and 
are the focal parameter and the
eccentricity of the orbit of the centre of mass. In equation
, the prime denotes differentiation with respect
to v. When the motion of the satellite of the
system is determined with the help of equation (2.1), the motion of the
satellite is easily determined with the help of
identity.
(2.4)
Where 
and 
are the radius vectors of the satellites of
masses and respectively
with respect to the centre of mass of the system.
Obviously, the actual motion of the system will be
combination of three types of motion.
i) Free motion, i.e., 
ii) Constrained motion, i.e., 
iii) Evolutional motion (the combination of free and
constrained motion).
We only interested in the constrained motion because
free motion are bound to be converted into constrained motion with the lapse of
time. In case of constrained motion the equality sign holds in the equation
(2.3), i.e.; the particle is moving along the circle of variable radius given
by:
In order to discuss the non-linear planar oscillations
of the system, we transform the equation (2.1), into polar form by
substituting:
(2.5)
Where 
is
the angular deviation of the line joining the satellite with the stable
position of equilibrium? Solving with respect to y and
; we obtain:
(2.6)
The equation 
is the equation of
motion of a dumbbell satellite in the central gravitational field of the Earth
under the influence of the Earth magnetic field and oblateness of the Earth.
The equation determining the Lagrange’s multiplier is given by:
(2.7) the non-linear
oscillations described by, take place as long as inequality given below is
satisfied.
(2.8)
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. The system of
equation, oscillates
about the stable position of equilibrium in which it lies wholly along the
radius vector joining the centre of mass and the centre of force Narayan et al
[18], [19], [20]. Substituting
, the equation can be expressed as follows:
(2.9)
Equation 
describes non-linear oscillations of the
system in elliptical orbit in the central gravitational field of the oblateness
of the Earth together with the Earth magnetic field.
3. 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 will be investigated for non-resonance
cases.
. (3.1)
In the equation (3.1),
, and 
Moreover
the non-linearity 
will be assumed to be the order
of e.
The system described by equation (3.1), moves under the
forced vibration due to the presence of the magnetic field of the Earth and
oblaeness of the Earth. We are benefited of the smallness of the eccentricity
‘e’ in equation (3.1), and hence solution may be obtained by exploiting the
Bogoliubov, Krilov and Mitropolskey method (1961). For e = 0, the generating solutions
of zeroth order are:
.
Where the amplitude ‘a’ and phase 
are constant, which can be determined by
the initial conditions. The solution of equations (3.1) is obtained in the
form:
(3.2)
Where the amplitude ‘a’ and phase 
are determined by the differential equations.
(3.3)
(3.4)
From (3.2), we find 
and 
then substituting the value of
and 
in
equation (3.1), and equating the coefficients of like powers of ‘e’ we get:
(3.5)
(3.6)
Using Fourier expansion given by
(3.7)
Where 
stands for Bessel’s
function. Substituting these values in equation (3.5) and determining 
and 
in such
a way as 
should not contain resonance terms and
hence, equating the coefficients of 
and 
to zero, separately, we obtain:
(3.8)
With the help of the equation (3.8) it is not difficult
to obtain
in the form:
(3.9)
In order to obtain the second approximation of the
solution, we need to determine 
and 
and 
as obtained
in (3.8) and (3.9), in equation (3.6), and equating the coefficients of and to zero
with a view to eliminate resonance terms from
, we
obtain:
(3.10)
Thus, in the second approximation, the solution is
given by:
(3.11)
Where the amplitude ‘a’ and phase ‘
’ are
given by:
(3.12)
And in the second approximation the solution is
obtained as:
(3.13)
Where the amplitude ‘a’ and phase are given by:
Constant
+
(3.14)
Where 
is constant.
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
, and
parametric resonances at
,
for
these values of
, the solution fails as we get
singularity. The parametric resonance at arises due to non-linearity
condition.
4. 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 (2.3), can be put in the form:
(4.1)
Where, and 
More over the
non-linearity term 
,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 non-resonance case, we conclude that the system experience parametric
resonance behavior at and near 
and hence the
non-resonance 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 general case, which is valid at and near the main resonance
, exploiting the well known bogoliubov,
krilov and metropoloskey, method. the solution of equation (4.1), in the first
approximation will be sought in the form:
(4.2)
(4.3)
(4.4)
Where 
and 
are particular solution periodic with
respect to ‘’ of the system.
; (4.5)
Where 
and
,
is the coefficient of ‘e’ on the right hand side of equation (3.1).
Where 
Simple integration gives us:
(4.6)
Where 
is the Bessel function
of the first order?
(4.7)
Where 
stands for Bessel’s function.
The periodic solution of the system given by equations
(4.6) can obtain as:
(4.8)
Where the amplitude ‘a’ and phase ‘’ are the given by the system of
differential equations:
(4.9)
The system of equation can be written as:
(4.10)
Where 
(4.11)
Obviously, the system of the
equation (4.10) has first integral of the form:
; (4.12)
This reduced the problem to quadrature. Here 
, is the constant of integration. However,
it is preferable to analyse the integral urves in the phase plane
. In order to plot the integral curves reducing
the equation (4.11), in the form:
(4.13)
Where
(4.14)
Fig. 3: Oscillation
of the Dumbell Satellite in Elliptical Orbit
The integral curves (4.13), have been plotted in Figure
(3), for =0.52, e =.1, A=0.005 and B=0.001. The
integral curves drawn in the phase plane. Using MATLAB software 6.1 versions.
It 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.
Fig. 4: Oscillations
of the Dumbell Satellite in Elliptical Orbit
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.
5. 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 non-elastic
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 and parametric
resonances at ,, up to
the second order of approximation. We also obtained the general solution of the
non-linear oscillatory system based upon Bogoliubov, Krilov and Metropoloskey
method which is valid at near of the parametric resonance 
the
method of phase plane has been applied in order to obtain characteristic of the
amplitude and the phase of the oscillatory system. We come to the conclusion
that there exists two stationary stable regime of the amplitude for both 
and
.
However the stationary
amplitude declines steadily, when it passes through two values of given by and.
It has also been established
that the system will always move like a dumbbell satellite in the phase
Plane 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
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, 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.