On the Rotational Motion of a Body Re-Entering the Atmosphere

Abstract

The exact equations of motion of a body acted upon by aerodynamic and gravitational forces are formulated, using inertial axes fixed in a spherical, nonrotating earth. With the assumption that the spin rate of the vehicle is constant and of such a magnitude tha t Magnus forces may be ignored, expressions are developed for the applied forces and moments of forces. After considering the nature of a typical re-entry path, the equations of motion are linearized, and the translational and rotational modes are uncoupled. With the assumption that the translational equations may be independent^ solved, the rotational equations of motion are reduced to two coupled linear equations with known variable coefficients. A solution of the planar case is obtained, utilizing a modified WKBJ approximation method. This technique is extended to the coupled case. The stability of the oscillations is examined, and the conclusion is drawn that the oscillations are bounded for a statically stable, spinning vehicle if CLO. is positive. A brief investigation is made of the forced solutions, and finally a comparison of the results of this paper with those of previous studies is made.