We study the directional stability of rigid and deformable free-spinning satellites in terms of two attitude angles. The Euler equations of attitude motion are linearized relative to a uniform-spin reference solution which leads to a generic MGK linear system. This type of conservative system is at best oscillatory stable but the stability result cannot be extrapolated from the linear to the non-linear system. A practical method to establish sufficiency conditions for directional stability is provided by the Frobenius-Schur reduction formula. We show that this approach reproduces the well known McIntyre-Myiagi stability matrix which is the most useful tool when liquids are the deformable parts. Furthermore, we clarify the correspondence of these results with the Equivalent Rigid Body method. A few practical applications are worked out in fair detail, i.e. a spinning satellite augmented with a spring-mass system and a rigid body appended with two cables and tip masses. Finally, we study a spinning satellite under a constant axial thrust. First, the thrust is pointing along the spin axis and we add a particle mass that can move freely in a plane normal to this axis. This model represents a Solid Rocket Motor firing with slag motion. Next, the thrust is parallel but offset from the spin axis. Because the reference solution of the full non-linear system is unknown, we perform a linearization about the initial state and obtain bounded solutions. The non-linear system, however, can be shown to be unstable in general. We illustrate this situation by an instability that actually happened during a station-keeping maneuver of ESA's GEOS-I satellite in 1979.
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