On a Model for Reentry Roll Resonance

Abstract

This paper concerns a mathematical model for the problem of roll buildup encountered in the high altitude flight of reentry vehicles with slight aerodynamic and mass asymmetries. The sixth order system of approximate equations governing this system is presented and the mechanism leading to roll buildup is pointed out. It is shown that this phenomenon occurs when two slowly varying natural frequencies in the coupled system become equal. In order to establish the techniques needed to solve this problem, a simplified mathematical model having all the essential features of the main problem is introduced and discussed in detail. It is shown that the solution can be constructed in approximate analytical form by matching two multiple-scale expansions. The first multiple-scale expansion represents the solution for times when the two slowly varying frequencies are not nearly equal. This so-called outer expansion involves two fast time scales and one slow time scale, and can be calculated using a generalization of a technique discussed by the author in an earlier work. Near resonance, i.e., when the two slowly varying fundamental frequencies are nearly equal, the solution is developed in terms of two other appropriate time scales and the result is then matched with the outer expansion. It is shown that the phenomenon of buildup is an exceptional case corresponding to a solution curve passing through a saddle point, and the conditions corresponding to this case are related to the initial conditions before resonance by the matching. In the case of buildup the motion evolves such that the two fundamental frequencies remain equal for a sufficiently long time. Otherwise, the solution exhibits no anomalous behavior after passage through the resonance region. In this case, which characterizes almost all sets of initial conditions, the two slowly varying fundamental frequencies remain nearly equal for only a short time.