Nonlinear longitudinal dynamics of an orbital lifting vehicle

Abstract

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 Near this resonance altitude linearized solutions are no longer valid and a study of the nonlinear equations shows that there is a strong interaction between the translational and the rotational modes resulting in a switching of the two frequencies of oscillations.

Phugoid oscillations in optimal reentry trajectories

“…It is larger at high altitude and decreases as the altitude decreases. This is to be expected since when Z~0, the motion is Keplerian and the period of oscillation is the orbital period while at low altitude, an approximate expression for the phugoid period is T = ~rx/2(2)V/g (Vinh and Dobrzelecki, 1969). Hence, in terms of 0, the phugoid period decreases from 2~r and tends to ~r~v/(2v) at low speed.…”

Phugoid motion for grazing-entry trajectories at near-circular speeds

“…Extension to orbital cruise of hypervelocity vehicles was carried out first by Etkin 8 and then by Vinh and Dobrzelecki. 9 This analysis was done for the case of a steady reference trajectory (initial cruise at constant altitude with constant speed). The case for a re-entry trajectory is more difficult to analyze because an accurate analytic solution for the unsteady reference trajectory has not been available.…”