The Effect of Tidal Forces on the Motion of Spacecraft. A Model of a Toroidal Satellite

Abstract

The model of a toroidal satellite is considered in the limiting case when the major radius of the torus (the distance from the center of the tube to the center of the torus) is much larger than its minor radius (the radius of the tube). Based on the Lagrange method, the equations of motion are derived. It is shown that the obtained mathematical equations describing the rotational motion do not contain small parameters, which implies that tidal forces have a significant effect on the rotational motion of the satellite. Numerical solutions of the obtained equations can be studied by students based on the use of differential equation solving procedures available in MATLAB and Octave environments. Numerical experiment in combination with analytical methods for studying these equations shows that with a sufficiently fast rotation of the satellite around the axis of symmetry, its orientation motion stabilizes and becomes quasi-periodic.