Effect of an unsteady three-dimensional wake on elastic blade-flapping eigenvalues in hover

The formulation of a finite-state inflow model based on an acceleration potential and a helical wake geometry is described. The states of the model are coefficients of an inflow expansion in terms of a Fourier series (azimuthally) and of special polynomials (radially). The integrals over the wake are done in closed form to obtain a set of ordinary differential equations for the inflow coefficients. The forcing functions for these equations are generalized forces which are integrals of the blade loading exactly as in structural dynamics. This model implicitly includes (for the hover case) Prandt-Goldstein tip losses, dynamic inflow, and Theodorsen/Loewy lift deficiency. Thus, it is a fully three-dimensional unsteady wake model. The model is coupled with elastic blade equations in hover, and eigenvalues are calculated. The results show that the three-dimensional wake has a large effect on the flap damping of all modes.