Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2

This paper considers the rotational motion of a satellite equipped with flexible viscoelastic rods in an elliptic orbit. The satellite is modeled as a symmetric rigid body with a pair of flexible viscoelastic rods rigidly attached to it along the axis of symmetry. A planar case is studied, i. e., it is assumed that the satellite’s center of mass moves in a Keplerian elliptic orbit lying in a stationary plane and the satellite’s axis of rotation is orthogonal to this plane. When the rods are not deformed, the satellite’s principal central moments of inertia are equal to each other. The linear bending theory for thin inextensible rods is used to describe the deformations. The functionals of elastic and dissipative forces are introduced according to this model. The asymptotic method of motions separation is used to derive the equations of rotational motion reflecting the influence of the fluctuations, caused by the deformations of the rods. The method of motion separation is based on the assumption that the period of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time of these oscillations’ decay, which, in its turn, is much smaller than the characteristic time of the system’s motion as a whole. That is why only the oscillations induced by the external and inertial forces are taken into account when deriving the equations of the rotational motion. The perturbed equations are described by a third-order system of ordinary differential equations in the dimensionless variable equal to the ratio of the satellite’s absolute value of angular velocity to the mean motion of the satellite’s center of mass, the angle between the satellite’s axis of symmetry and a fixed axis and the mean anomaly. The right-hand sides of the equation depend on the mean anomaly implicitly through the true anomaly. A new slow angular variable is introduced in order to perform the averaging for the perturbed system near the 3:2 resonance, and the averaging is performed over the mean anomaly of the satellite’s center of mass orbit. In doing so the well-known expansions of the true anomaly and its sine and cosine in powers of the mean anomaly are used. The steady-state solutions of the resulting system of equations are found and their stability is studied. It is shown that, if certain conditions are fulfilled, then asymptotically stable solutions exist. Therefore, the 3:2 spin-orbital resonance capture is explained.