Analysis of stability for uniform rotations of a dumbbell system in an elliptic orbit

— The evolution of space missions and related systems has been promoting innovation and creation of ideas for new technologies for decades. The search for innovative solutions that combine the optimization of resources and materials guide the recent research in the space area. The objective of this study is to analyze the behavior of two bodies connected by tethers and can be a solution for reducing costs in space missions, one of the concepts that have the potential to fulfill the objective of efficient transport space. In this paper it is discussed the motion of two massive bodies connected by tethers in keplerian motion in a central force field, their viability, the rotational dynamics and the system behavior in the space environment. Models will be created that simulate and explain the dynamics of the object and that analyzes the main parameters for determining the stability and the uniform rotations conditions.


INTRODUCTION
Tethered systems have many areas of application and has been studied in several published articles ([1]- [9]). Beletsky and Levin [1], begins by setting the scene for tethers in space summarizing possible applications and also discussing fact and fiction, analyzing clearly the main parameters and applications for Tethers Systems, as the density of the material the effective forces, orbital dynamics, mechanics models, attitude and possible disturbances for a flexible tethers with end masses, massless and massive variations.
The motion of tethers considering dumbbell oscillations, bodies in the central field of vibrating forces (Burov et al. ([10]- [14]), showing stability solutions for angles on the chaotic dynamics in elliptical orbit, analyzing the problem in another aspect of Moon-tethered pendulum, to considering the uniform rotations of a twobody tethered system in planar motion and the control the length of the tether [15]. In studies ( [15]- [21]) were suggested methods of controlling the geometric configurationand Moon-tethered system with variable tether length in restricted three-body problem ( [16] and The body consists of two-point masses ( 1 and 2 ) connected by a light tether where p is the focal parameter, e the eccentricity and the true anomaly of the orbit. The system coordinates are { 0 = cos( ) 0 = sin( ) 1 = 0 + 1 cos( + ) 1 = 0 + 1 sin( + ) 2 = 0 − 2 cos( + ) 2 = 0 − 2 sin( + ) (2)

Potential and Kinetic Energy
The potential energy of the system can be obtained by the following expression, where represents the mass of the point; ⃗ ⃗ the position of the point mass with respect to center of the Earth; 0 = ; is the universal gravitational constant and the mass of the Earth. Introduced the parameters, and , given by Substitute on Eq. (3)leads to The Eq. (7)is cumbersome and can be simplified, introducing a new parameter = , and assuming ≪ 1 the tether length is much smaller than the focal parameter . The Taylor Series expansion up to the 2 order of for the potential energy is = − 0 (1 + cos( )) + ( − 1) 0 (1 + cos( )) 3 (1 + 3 cos(2 )) 2 4 The kinetic energy of the system can be written as ( 2 2 (1 + 2 + 2 cos( )) 2 (1 + cos( )) 4 − 2( − 1) ( 2 (̇+̇) 2 + 2 )) (9)

Lagrange Equations of Motion
For the subsequent analysis, the generalized coordinates and are used and the system is assumed to be subject to the gravity-gradient forces.
The equation is rewritten as a function of the true anomaly (Eq. (11)-(12)) =̇= 0 (1 + cos( )) 2 The equations of motion of the spacecraft can be written as Suppose the following relation for the tether performance (Eq. \ref{equa16}), the Eq. \ref{equa14} with respect to the true anomaly takes the form: The analytical integration function is difficult but substituting values for the variable and solving the equation with respect to the variable , it is possible to obtain the following solutions ( ) closed-form no solution. For = 0 was examined in previous studies [12] where the relative equilibrium was studied.   The control laws are periodic in true anomaly ( ). In  An approximate solution for small eccentricities can be obtained using Taylor series in Eq. (17)and obtain applying the series of order 3, in the variable e obtain: One obtains the expression that has analytical integration, however this procedure introduces impossibilities in the system and restricts the solutions to the variable $\omega$, which after integration generates singularities for values: ω = (− .
A good numerical method to refine the closure of the orbit is essential to obtain an accurate monodromy matrix (A), which is obtained to analyze to stability with respect to small perturbations (δφ) of the orientation angle φ. It is possible to analyze the stability using the Floquet theory, since this equation is a differential equation of the second order. The linearized equation constricts this analysis to small variations of φ. The stability monodromy matrix (A) has some important properties, which are ( [13] and [19]):   4,4]. When = −1 there is no solution and when = 1 there is a small stable range ( Table 1). The Table 1 shows the complete set of stable solutions for the monodromy matrix, as a function of the eccentricity and of the true anomaly.        The stability analysis suggests the system's viability. Some parameters for the system are shown in monodromy matrix.

III. FORCES IN TETHERS
The force on the tethers can be calculated using the relation of the forces involved in the problem, applied to the tether, The force on the tether for a system composed of: direction is tether length (l) and sense 1 to 2 , and Tis magnitude of the tether force, solving the Eq.(16) with respect to l, it is possible to show that the cable suffers a large variation in tension and that these values are periodic and, in some cases, increase with grows to the eccentricity (Figures 19 --32).

IV. CONCLUSION
The uniform rotations of a dumbbell and with several possibilities are considered in the present study, as well as the stability analysis and the viable control laws. In some cases are possible obtain solutions closes-form, in other cases only numerical solutions for the control of the tether systems are available. The necessary conditions of stability for uniform rotations were analyzed using the parameters ω and the eccentricity of the orbit, generating the control laws for the cable length.