Nonlinear free periodic oscillations of a tethered satellite system

This paper deals with the non-linear dynamics of a tethered satellite system, with particular reference to periodic transverse oscillations. The motion is governed by two integro-differential equations with quadratic non-linear terms associated to gyroscopic forces. Notwithstanding the weakness of the non-linearities, the closeness of the in-plane and out-of-plane frequencies suggest that it would be interesting to study the periodic finite oscillations and analyze their stability. Since the system operates virtually in conditions of internal resonance, a certain number of modes are involved in the phenomenon. An asymptotic analysis of the partial differential equations of the continuous system makes it possible to take into account the fact that both the frequencies and the oscillation shapes are amplitude-dependent. Primary and secondary instability phenomena are investigated by using the Floquet theory and approximate analytical expressions of the unstable regions are obtained.