Método de Melnikov, bandas de estocasticidad y no integrabilidad en un giróstato con momentos de inercia variables

Resumen

In this work we study the attitude dynamics of a deformable gyrostat (or dual-spin spacecraft) in absence of external forces. Here, deformable means that one of the moments of inertia is a periodic function of time. This model is a more realistic approximation to the motion of a gyrostat than the perfectly rigid model, and reveals chaotic phenomena in the dynamics of the system. In Chapter 1, we develop the Hamiltonian formulation corresponding to a generic free gyrostat with 3 rotors aligned with any different directions. We treat the problem in noncanonical variables: the components of the total angular momentum in the body frame. Then we focus on the particular case of a free triaxial deformable gyrostat whose Hamiltonian is a sum of an integrable part plus a timeperiodic perturbation. In Chapter 2, we consider the gyrostat when the rotors are at relative rest. By means of the Melnikov's method, we show that the gyrostat exhibits chaotic motion. The Melnikov function gives us an analytical estimation of the width of the stochastic layer generated by the perturbation. We check the validity of this analytical estimation calculating another numerical estimation. We find some deviations between both estimations due to the effect of nonlinear resonances. Chapters 3 and 4 focus on the gyrostat with one rotor in relative motion with respect to the platform. This spinning rotor is parallel to one of the principal axes of the gyrostat. By means of the Melnikov method and Poincaré surfaces of section we prove that the system also exhibits chaotic motion and that it can be removed if the spinning rotor reaches a relative angular momentum high enough. In Chapter 5 we study the reorientation process of the gyrostat and the effect of the perturbation in it. In order to show the influence of the perturbation, a suitable numerical parameter is introduced and it is related with the Melnikov function calculated in Chapter 2. Also we take into account the effect of nonlinear resonances to explain the sudden increments of the chaotic degree of the system.