Resonant Capture and Separatrix Crossing in Dual-Spin Spacecraft

We consider the rotational motion of a spacecraft composed of two bodies which are free to rotate relative to one another about a common shaft S. A motor on one of the bodies provides a small constant internal torque which influences the relative motion of the two bodies, and which may influence the orientation of their common shaft S. Resonant capture refers to the phenomenon that the spacecraft may end up in one of several possible orientations, including a nearly flat spin (transverse to S), in addition to the expected simple rotation about S.The method of averaging is used to treat the original equations of motion, and it is shown that the essential mathematical problem involves separatrix crossing in a problem with slowly moving separatrices. Energy changes represented by Melnikov integrals are used to supplement the averaged equations in the neighborhood of the heteroclinic motions. The method is used to predict which initial conditions lead to capture into each of three distinct capture regions. The asymptotic results are compared to those obtained by direct numerical integration of the equations of motion.