Page 49 - Aerodynamics for Engineering Students

32 Aerodynamics for Engineering Students

Now at moderate incidences, between say 3” and 7”:
CL = 0[2ocD] and cos a = 0[10 sin a]
where the symbol O[ ] means of the order of, i.e. CL is of the order of 20 times
CD. Then.
CL cos a = O[200 CD sin a]
and therefore CD sin a can be neglected compared with CL cos a. With this approx-
imation and the further approximation cos a = 1 ,

CM, = CMAC - CL (1.48)

Differentiating Eqn (1.48) with respect to CL gives

But the aerodynamic centre is, by definition, that point about which CM is independent
of CL, and therefore the first term on the right-hand side is identically zero, so that

(1.49)

(1 SO)
If, then, CM= is plotted against CL, and the slope of the resulting line is measured,
subtracting this value from a/c gives the aerodynamic centre position XAC/C.
In addition if, in Eqn (1.48), CL is made zero, that equation becomes
cMa CMAC (1.51)

i.e. the pitching moment coefficient about an axis at zero lift is equal to the constant
pitching moment coefficient about the aerodynamic centre. Because of this associa-
tion with zero lift, CM, is often denoted by CM,.

Example 1.4 For a particular aerofoil section the pitching moment coefficient about an axis
1/3 chord behind the leading edge varies with the lift coefficient in the following manner:

CL 0.2 0.4 0.6 0.8
CM -0.02 0.00 +0.02 +0.04
Find the aerodynamic centre and the value of CM, .
It is seen that CM varies linearly with CL, the value of dCM/dCL being
0.06
0.04 - (-0.02) = +-
0.80 - 0.20 0.60 = +0.10
Therefore, from Eqn (1.50), with u/c = 1/3

The aerodynamic centre is therefore at 23.3% chord behind the leading edge. Plotting CM
against CL gives the value of CM~, the value of CM when CL = 0, as -0.04.