Simple Mathematical Models of Elastic and Flexible Wings in the Extreme Ground Effect

The elasticity and flexibility of the lifting surface can play a particular role in ground-effect aerodynamics due to the expected increase of dimensions of wing-in-ground-effect vehicles, use of light materials and fabric, etc. Usually, to account for elastic properties and/or flexibility of the wing, we have to consider simultaneously the equations of aerodynamics and elasticity. In a more profound analysis, the formulations should also cover the equations of the dynamics of the vehicle. As a relationship, linking the deformations of a lifting surface with aerodynamic loading, we normally use equations of unsteady bending of an elastic plate accounting for forces, acting in its camber plane; see Bisplinghoff et al. [151]. In nondimensional form,[71] [72] [73] [74] this equation can be written as

2JS‘U + + + =*- -J>+- <12-»

The following notations are used in (12.1) for functions and parameters: / are vertical displacements of points of the elastic plate, pw is the density of the material of the plate, p is the density of air, 5t is the thickness of the plate, D = 2D/pUqCq, D is the bending (cylindrical) stiffness, Nxx and Nzz are factors that characterize the forces acting in the x and z directions, respectively, Nxx – 2Nxx/pU%C0, and =

Equation (12.1) should be solved with boundary conditions, correspond­ing to the method of fixing the edges of an elastic and/or flexible lifting surface, as well as with initial conditions that define the initial magnitudes of the deflection and the rate of deflection of the surface.

The formulations for an elastic/flexible wing can be used both for eval­uating the variation in aerodynamic properties due to the deformation of the lifting surface and for predicting the occurence of static (divergence) and dynamic (flutter) instability. In what follows, some simplified schemes are considered for analyzing conditions of the aeroelastic instability of a wing in the extreme ground effect. The last paragraph contains approximate predic­tions of the influence of the flexibility of the foil upon its lift coefficient and some effects of the porosity of a soft foil.