# DIVERGENCE OF A LIFTING SURFACE

The central problem in steady-state aeroelasticity is the effect of elastic deformation on the lift distribution over lifting surfaces such as airplane wings and tails. At lower speeds of flight, the effect of elastic deformation is small. But at higher speeds of flight, the effect of elastic deformation may become so serious as to cause a wing to be unstable, or to render a control surface ineffective, or even to reverse the sense of control.

In this chapter, we shall treat in detail the problem of wing divergence, which is probably the simplest of all aeroelastic problems. Many funda­mental concepts and methods of solution can be illustrated in this connection.

In § 3.1, the phenomenon of divergence is explained by a two-dimen­sional example. In § 3.2, the problem of the divergence of an idealized three-dimensional wing is formulated in a general form. A “strip” assumption regarding the aerodynamic force is introduced to simplify the governing equations. The mathematical nature of the problem is then pointed out and illustrated by an example. Several methods of solution are discussed. First, in § 3.3, a solution based on a semirigid assumption is given. Second, in § 3.4, the semirigid assumption is reconsidered in the light of generalized coordinates. Third, in § 3.5, a process of successive approximation is discussed. The last method is mathematically more satisfactory; its convergence can be proved, and the relation between the successive approximations can readily be seen. Finally, in order to provide a practical numerical method of solution, the method of matrices is suggested. The basic concepts and definitions of matrix calculus is outlined in § 3.6, and the reduction of differential and integral equations into matrix equations according to the method of finite differences is explained in § 3.7. Some remarks regarding further refinements are made in the last section, § 3.8.

The unfortunate wing failure of Professor S. P. Langley’s famous monoplane in 1903, shortly before the Wright brothers’ successful flight, could probably be attributed to wing divergence.3-5 Monoplane designs in early aeronautical history often had divergence trouble.36 For modern aircraft, the critical speeds of flight at which divergence sets in are usually higher than those of flutter or other aeroelastic instabilities.

Hence the divergence speed itself is often, of minor importance. However, it is a convenient reference quantity for other aeroelastic phenomena; it enters into expressions for the effect of elastic deformation on lift distribu­tion, static and dynamic stability of the airplane, etc. Moreover, since the calculation of the divergence speed is relatively simple, it is generally made in the course of airplane design.