In Section 4.1, wings are assumed to be rigid and two-dimensional. That is, the airfoil geometry including incidence angle is independent of spanwise location, and the span is sufficiently large that lift and pitching moment are not functions of a spanwise coordinate. In turning our attention to wings that can be modeled as isotropic beams, the incidence angle now may be a function of the spanwise coordinate because of the possibility of elastic twist. We need the distributed lift force and pitching moment per unit span exerted by aerodynamic forces along a slender beam-like wing. At this stage, however, we ignore the three-dimensional tip effects associated with wings of finite length; the aerodynamic loads at a given spanwise location do not depend on those at any other.

The total applied, distributed, twisting moment per unit span about the elastic axis is denoted as M'(y), which is positive leading-edge-up and given by

M = Ma c + eL — Nmgd (4.35)

where L and M’ac are the distributed spanwise lift and pitching moment (i. e., the lift and pitching moment per unit length), mg is the spanwise weight distribution (i. e., the weight per unit length), and N is the “normal load factor” for the case in which the wing is level (i. e., the z axis is directed vertically upward). Thus, N can be written as

N = W =1 + 7 <436>

where Az is the z component of the wing’s inertial acceleration, W is the total weight of the aircraft, and Lis the total lift.

The distributed aerodynamic loads can be written in coefficient form as

L = qcce
Mac = qc cmac

where the freestream dynamic pressure, q, is

1 9

q = 2 P^u 2 (4.38)

Note that the sectional lift ct and moment cmac coefficients are written here in lower case to distinguish them from lift and pitching moment coefficients for a two­dimensional wing, which are normally written in upper case. Finally, the primes are included with L, M, and M^c to reflect that these are distributed quantities (i. e., per unit span).

The sectional lift and pitching-moment coefficients can be related to the angle of attack a by an appropriate aerodynamic theory as some functions ct(a) and cmac(a), where the functional relationship generally involves integration over the planform. To simplify the calculation, the wing can be broken up into spanwise segments of infinitesimal length, where the local lift and pitching moment can be estimated from two-dimensional theory. This theory, commonly known as “strip theory,” frequently uses a table for efficient calculation. Here, however, for small values of a, we may use an even simpler form in which the lift-curve slope is assumed to be a constant along the span, so that

ce(y) = aa(y) (4.39)

where a denotes the constant sectional lift-curve slope, and the sectional-moment coefficient cmac(a) is assumed to be a constant along the span.

The angle of attack is represented by two components. The first is a rigid con­tribution, ar, from a rigid rotation of the surface (plus any built-in twist, although none is assumed to exist here). The second component is the elastic angle of twist в (y). Hence

a( y) = ar + в (y) (4.40)

where, as is appropriate for strip theory, the contribution from downwash associated with vortices at the wing tip is neglected. Therefore, associated with the angle of attack at each infinitesimal section is a component of sectional-lift coefficient given by strip theory as ct (y) = a[ar + в (y)]