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On the dynamic stability of flexible supersonic vehicles Drummond, Alastair Milne

Abstract

The dynamic stability of a long, slender-bodied vehicle with a flexible fuselage is examined analytically in the supersonic speed regime. The small aspect ratio lifting surfaces are considered to be rigid but dependence of their angles of attack on fuselage flexibility is accounted for. The amplitude of pitching oscillation is restricted to ±10° about the zero-lift line by the nature of the unsteady, supersonic aerodynamic theory used. The stability problem is formulated by a set of non-linear differential equations with the non-linear contributions arising from both the inertia and the aerodynamic forces. The present analysis accounts for non-linear contributions up to third degree in the rigid body angle of attack. The stability of the short period mode is investigated using Routh-Hurwitz criteria and an expression representing a stiffness criterion for dynamic stability is obtained. The analytical development is so presented as to make it easily applicable to a supersonic, flexible vehicle with or without wings, e.g. a supersonic transport or a missile. Moreover to facilitate the evaluation of the effect of flexibility and non-linearities on dynamic stability, four cases are considered separately: a. Rigid body equations of motion, without non-linear terms b. Rigid body equations of motion, with non-linear terms c. Flexible body equations of motion, without non-linear terms d. Flexible body equations of motion, with non-linear terms. A numerical example is presented towards the end which investigates the dynamic stability of a flexible, supersonic transport configuration. The conclusions from the example are: 1. The non-linearities can be safely neglected for rigid aircraft, but not for wingless vehicles. 2. Flexibility affects the stability through the lift and pitching moment and also by introducing two more possible equilibrium points. 3. The amount of work involved in finding a solution is markedly increased by the necessity of solution of more characteristic equations of higher degree. 4. The stiffness criterion can be used to adjust the stiffness distribution to one that can make an unstable configuration stable. The usefulness of the method is two-fold. For a flexible vehicle with known geometric, mass and elastic properties, the method can predict its dynamic stability. This feature is of considerable importance particularly in the design stage. On the other hand, if an aircraft with known geometry, total mass and centre-of-gravity location proves to be dynamically unstable, then the analysis provides a stiffness criterion by which it can be made stable. The analysis involves a considerable amount of computation and hence seems to be particularly suited for solution by a digital computer.

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