Nonlinear longitudinal dynamics of an orbital lifting vehicle

This paper presents an analytical study of the longitudinal dynamics of a thrusting, lifting, orbital vehicle in a nearly circular orbit. The translational motion is composed of a nonlinear oscillation, or phugoid, and a spiral mode which results in either decay or dilatation of the orbit depending on the perturbed initial conditions. The nonlinear effects on the phugoid period and damping are small in the altitude range considered. Elements of the orbit such as radial distance, velocity, and flight path angle were obtained explicitly as functions of time. The behavior of the variations of these elements is correctly predicted. Explicit expressions for period and damping of the angle-of-attack mode were derived. It is shown that a critical altitude may exist at which the phugoid mode and the angle-of-attack mode have nearly equal periods. Near this resonance altitude linearized solutions are no longer valid and a study of the nonlinear equations shows that there is a strong interactions between the translational and the rotational modes resulting in a switching of the two frequencies of oscillations.This work was supported by NASA Contract No. NASr 54(06).