Problems of longitudinal stability below minimum drag speed, and theory of stability under constraint

Abstract

Practical difficulties in cruising below minimum drag speed have long been known but not fully explained. The reasoning proposed by Painleve in 1910 purported that flight below minimum drag speed should be fundamentally unstable, so that any speed error would lead to a divergence. This reasoning is shown to be invalid on the ground of the general theory of dynamic stability in uncontrolled flight, Painleve's criterion being a grossly inadequate approximation to the condition of phugoid stability. However, the criterion may be fully vindicated for the case of flight controlled by the elevator in such a way as to maintain constant height. In this form, the criterion seems not only to explain qualitatively the troubles encountered in slow cruising, but also to lead to a good quantitative estimate of speed variation following an initial disturbance. The criterion also applies to the problem of ultimate height response to an elevator deflection. The concept of stability of partially controlled flight is further developed, leading to a general theory of 'stability with constraint', i.e., when a control (elevator, throttle, etc.) is used to suppress one component of the disturbance. The theory may be useful as giving approximate solutions of problems in which the pilot moves his control so as to keep one component of the disturbance always as small as possible. The principles of the theory are set out in section 4.1, and several examples given in sections 4.2, 4.3 and 4.4. Flight tests are needed to explore further the validity of this method of approximation.