Flexible-rigid dynamics and vibration suppression of slender structures on partial space elevator

This paper studied the coupled flexible-rigid dynamics of a partial space elevator with slender and flexible structural appendages on the main satellite with the goal of suppressing flexural vibration of the appendages caused by climber movement. The partial space elevator is modeled as a two-piece dumbbell system, with the local dynamics of the main satellite simplified to attitude and flexural motions of appendages using modal decomposition. Flexible-rigid dynamic coupling occurs at the main satellite, where its attitude and flexural deflection of appendages interact with the orbital motion of the partial space elevator induced by the climber's movement. However, the main satellite’s influence on overall libration and orbital dynamics is negligible due to the large tether-to-satellite size ratio, allowing local decoupling of its dynamics. Accordingly, the dynamic model for the main satellite is locally decoupled from the libration and orbital dynamics of the partial space elevator to analyze the dynamic characteristics. This model reveals two distinct features: (i) steady state of flexural deformation of appendages (under which, appendages will be in a bending deformation state without vibration) occurs from gravity gradient difference across the partial space elevator, an effect absent in standalone satellites with similar slender and flexible structural appendages; and (ii) a steady state for the partial space elevator does not guarantee a steady state for the flexible appendages, even with zero initial deflection. By treating the flexible-rigid coupling effect and modeling approximation errors as disturbances, a sliding mode control law based on modal decomposition of a beam is developed to suppress flexural vibration of appendages during climber transfer by exclusively adjusting the tether length at the end body. The stability of the control law is proved by Lyapunov theory. Numerical simulations demonstrate that the proposed control law effectively suppresses the flexural motion of flexible structures on the main satellite throughout the climber transfer.