Nonlinear dynamics and stability of hypersonic reentry vehicles.

Abstract

A theory of longitudinal dynamic stability of hypersonic aerospace vehicles in planar flight is introduced as a necessary step toward the construction of a general theory of three-dimensional flight for such vehicles. Regular perturbation methods are used as a fundamental tool for analysis. The influence of varying-thrust propulsion systems on the longitudinal stability is investigated for jet-, ramjet- and rocket-propelled vehicles in high-altitude, high-speed cruise flight. The spiral mode of ramjets and rockets is shown to be inherently unstable. A critical altitude for stability of airbreathing vehicles is identified and the condition for stability is expressed in terms of a reference cruise speed at sea level. The phugoid oscillations are analyzed for the critical case of shallow entry at near-circular speed, both in ballistic and gliding entry modes. The unsteady effects owing to the variations in speed and density are retained in the analysis. For a ballistic reference path, the phugoid is shown to be governed by a transformed confluent hypergeometric equation; a method due to George Boole is used to obtain analytic solutions to the problem. In gliding entry, the phugoid is determined by a forced hypergeometric equation for the flight path angle. A Liouville's transformation is used to help derive complete analytical solutions. The angle-of-attack mode is analyzed for the same critical entry cases. In ballistic entry, the pitching oscillations are governed by a Whittaker's equation with the density as independent variable. This leads to a convenient criterion for stability in pitch. For gliding entry, the angle of attack is governed by another hypergeometric equation with a forcing function. Another Liouville's transformation is used to assess stability. Analytical expressions are derived and elements of interest (frequencies, etc.) are obtained. A new set of high-order analytic solutions is introduced for unsteady reference trajectories. Three cases of special interest are treated: ballistic skip at supercircular speeds, and ballistic entries at either circular or subcircular speeds. The solutions are very accurate and fast enough computationally to be useful for guidance. They are also appropriate for mission planning and stability analysis.