ANALYTICAL NATURAL FREQUENCIES OF TAPERED WINGS

摘要

The bending frequencies of a wing are calculated based on the model of a beam clamped at the root and free at the tip; since the mass and the moment of inertia (per unit span) vary along the span, a non-uniform beam is considered. For a swept-back wing with straight leading- and trailing-edges, the chord is a linear function of the span; the same linear function of the span applies to thickness, in the case of constant thickness-to-chord ratio. Thus, the bending modes of a non-uniform beam are considered, with mass and moment of inertia respectively quadratic and quartic functions of the span. There is no exact solution expressible in finite terms using elementary functions, and thus power series expansions are used. The boundary conditions, that the wing is clamped at the root and free at the tip, lead to the natural bending frequencies. The fundamental bending frequency is calculated for a delta wing, and compared with a rectangular wing, with the same span, mean chord and thickness, mass density and Young modulus. It is shown that the fundamental frequency is higher by a factor 11.32 for the delta wing., i.e., it is stiffer because it has a higher proportion of the mass near the root; it is also shown that the case of the tapered swept-back wing is intermediate between the delta and the rectangular wing.