Pitching Moment

Consider a thin wing with unswept quarter-chord as shown in Fig. 6.22.

A small section of the wing of thickness dy contributes to the lift force according to

1 2

dL = pU Г( y )dy = 2 pU 2Cf (y)c( y )dy (6.78)

and to the pitching moment about the y-axis as

dM, o = dM, a.c. Xa. c.dL (6.79)

The term dM, a.c. is the contribution to the pitching moment at the aerodynamic center of the profile and is given by

dM, a.c. = 2 pU2 c2( y)Cm, a.c.( y)dy (6.80)

In thin airfoil theory the aerodynamic center pitching moment coefficient depends on the relative camber, as Cm, a.c. = —nd(y)/c(y). The negative sign is for a nose down moment. Combining the two results and integrating along the span yields

a. c.( y)c2( y )dy — 2 pU 2 Xa. c

Cl(y)c(y)dy (6.81)

Cm

Assuming for simplicity that the profile relative camber is constant along the span, the moment coefficient at the aerodynamic center is constant and the integration can be performed as

12

M, o — 2pU ScCm, a.c

Here, we have introduced the average aerodynamic chord c. It is used, with the wing area S, to define the pitching moment coefficient

xax.

CM, o = CM, a.c. = CL

c

The wing aerodynamic center pitching moment coefficient is defined more gen­erally as