Axial Velocity Solution for Spinning-Up Rigid Bodies Subject to Constant Forces

Abstract

An approximate analytical solution is presented for the axial velocity of a spinning-up spacecraft subject to constant body-fixed forces and moments. We assume that the rigid body spins about the maximum or the minimum principal moment of inertia. Also, it is assumed that the deviation of the spin axis from an inertially fixed direction is small. Our closed-form solution for axial velocity applies to axisymmetric, nearly axisymmetric, and in some cases asymmetric rigid bodies. Furthermore, asymptotic limits are determined for large spin rates and for some geometric limits such as a sphere, a thin rod, and a flat disk. When the axial body-fixed force is nonzero, the axial velocity is dominated by the expected secular term of axial acceleration multiplied by time. However, when the axial force is zero, then the much more complicated effects of the transverse forces come into play: there are no longer any secular terms, but rather there is an asymptotic limit to the axial velocity accrued from the projection of the transverse accelerations onto the inertial direction of the initial spin axis. The axial velocity solution consists of Fresnel integrals, integrals of Fresnel integrals, and many new families of related integrals. Numerical simulation demonstrates the accuracy of the analytical solution in a practical example.