A method of calculating the short-period longitudinal stability derivatives of a wing in linearised unsteady compressible flow

Abstract

A method is developed for the circulation of the pressure distribution and the aerodynamic forces and moments on a wing performing harmonic pitching and heaving oscillations. The calculation is based on the assumption of inviscid potential flow without shock waves and is restricted to small incidence, so that the linearized theory is valid. In contrast to other work in the field the theory applies to all Mach numbers. It is restricted to small values of the reduced frequency and should be valid for the usual range of short periods occurring at present in flight. The formal solution yields two integral equations for the parts of the load, which are in phase and go out of phase with the oscillation; these are of the same form as the corresponding equation in steady flow. The way is thus opened for solutions over the whole Mach number range at small frequencies, if the corresponding steady solutions can be found. The calculation is in fact easiest for M = 1 and has been done here for Delta wings to supplement a previous supersonic calculation made on different frequency assumptions, which broke down near M = 1. It appears from the two sets of results that the short-period oscillation will be unstable near M = 1, if the apex angle of the Delta wing is greater than about 60 deg. This confirms a now generally recognised trend. Such results near M = 1 must of course be invalidated to an unknown extent by thickness viscosity and shock waves at their maximum effect. Nevertheless it is unlikely that these factors will remove the critical nature of the transonic damping as calculated by this method. With all its obvious limitations this method, when extended to other planforms, should provide a useful tool in studying the effect of geometrical parameters on the stability of an aircraft at transonic speeds.