Stability solutions of a dumbbell-like system in an elliptical orbit

Abstract

A dumbbell-like system is analyzed, considering two mass points connected with a massless and rigid tether with length variations, and the center of mass described by Keplerian orbits. This kind of system, in a certain type of configuration, is a simple conceptualization of the space elevator. The system motion is obtained with the Lagrangian Formulation in a Central Gravitational Field, and the perturbations of motion are neglected. Laws of control are considered for the angle of systems rotation around the center of mass. Those include uniform rotations or no rotation at all. Stability conditions were obtained for the first case, analyzing its neighborhood and using Floquet Theory. The results shown there are regions of eccentricities were stability is found. Lastly, a dynamic numerical simulator was created, where the implementation of the results could be tested. The dynamic behavior of the system showed regular and chaotic properties.