The unsteady aerodynamic response to arbitrary modes of blade motion

Abstract

Analytical expressions are derived for predicting the unsteady aerodynamic response associated with small-amplitude, harmonic vibrations of the blades of a two-dimensional cascade. These expressions can be applied to predict the aeroelastic behavior of blades undergoing arbitrary, i.e. rigid and/or flexible, motions. In addition, a new local response function, termed the pressure-displacement function, has been identified. This function reveals the positions on a moving blade surface at which the local unsteady loads tend to support or suppress a prescribed blade motion and, hence, contribute to the stability or instability of that motion. The pressure-displacement function is continuous along a blade surface except at mean shock locations, where it exhibits a delta function behavior. The integral of this function over the entire blade surface, i.e. the aerodynamic work per cycle, determines the rate at which energy is transferred from an airstream to a moving blade. Example solutions are presented to illustrate the behavior of the global aerodynamic work per cycle and the local pressure-displacement function for a vibrating compressor-type cascade operating at subsonic and transonic Mach numbers.