Other properties of the cambered airfoil may be determined as well. Recall that the aerodynamic center is the point on the airfoil about which the moment is independent of the angle of attack. Now, Eq. 5.31 may be written as:
1 2 2
and multiply through by — c, we find:
[ M ‘le 1 = – L’
The second equation implies that the integrated pressure force (i. e., the lift) effectively acts at the quarter-chord. Consider what happens at the quarter-chord point when the angle of attack changes and, consequently, Eq. 5.30 lift changes. Because the lift acts through the quarter-chord point, the lift-moment arm is zero. This means that as the lift changes, the moment due to lift at the quarter-chord point does not change. Thus, at the quarter-chord point the moment is independent of L’ and, therefore, of the angle of attack.
It follows that for an arbitrary airfoil, the aerodynamic center is located at:
Xac = 4. (5-33)
This is the same result as for the symmetrical airfoil. However, the moment about the quarter chord is not zero as before; there now is a camber contribution to moment. Consider the case in which aa = 0 so that L’ = Cg = 0. Equation 5.32 states that for this case, the moment about the leading edge is not zero. This means that although the downward-directed integrated pressure force in the lift direction on the upper surface is equal and opposite to the upward directed integrated pressure force, in the lift direction on the lower surface (i. e., the net lift is zero), these two forces are not collinear. Then a net moment must be present. Recall from statics that two equal parallel forces that are opposite in sense and are not collinear are called a couple. Recall also that the moment of a couple is the same about any point in the plane. Thus, evaluate the zero-lift moment at the leading edge and then transfer it unchanged to the quarter-chord point to yield the moment about the aerodynamic center. From Eq. 5.76 for zero-lift coefficient,
(C"LE L,- 4 c = I( ^ * )■ (5.34)
Equation 5.34 represents the moment coefficient about the aerodynamic center whether or not the lift is zero, because the resultant lift force acts at the aerodynamic center and, hence, creates no moment about that point.