On the stability of two-dimensional membrane wings

The stability of two-dimensional membrane wings is studied by an analytical solution, obtained for inviscid incompressible flow. The membrane is assumed to be extensible and of small camber, with constant tension along the membrane. The aerodynamic load along the airfoil is obtained by the unsteady thin airfoil theory, with special consideration of the wake vortices. It is shown that the stability of the membrane is controlled by the membrane mass ratio and the tension coefficient. Stability map is presented for various combinations of these two parameters, with comparison between the complete unsteady and the quasi-steady models. For light-weight membranes the unsteady analysis agrees with the membrane static solution, predicting stability loss by divergence, independent of the membrane mass. For heavy membranes the unsteady analysis predicts that membrane instability appears with neutrally-stable flutter, which occurs for a tension coefficient that increases with the membrane mass. In this case the membrane shape on the verge of instability is close to that of the second structural mode of the membrane, and as the membrane mass is increased the two shapes coincide. For very large values of mass ratio the aerodynamic load acts merely as a small perturbation to the structural system, that excites the first unstable structural mode of the membrane without affecting the frequency. This instability mechanism obtained for heavy membranes is not predicted by the quasi-steady model.