Aerodynamic Center1
Suppose we have the forces and moments specified about some reference location for the aircraft, and we want to restate them about some new origin.
N ~ L N ~ L
M AAM z ref new
x x ref xnew
M ref = Pitching moment about xref
M new = Pitching moment about xnew
xref = Original reference location
xnew = New origin N = Normal force ≈ L for small ∝ A = Axial force ≈ D for small ∝
Assuming there is no change in the z location of the two points:
MxxLMref=−() new − ref + new
Or, in coefficient form:
xx− new ref CCCmLM=− + ref cN new mean ac..
The Aerodynamic Center is defined as that location xac about which the pitching moment doesn’t change with angle of attack.
How do we find it?
1 Anderson 1.6 & 4.3 Aerodynamic Center
Let xnew= x ac
Using above: ()xx− CCC=−ac ref + M refc LM ac
Differential with respect to α :
∂CM xx− ∂C ref ac ref ∂CL M ac =− + ∂∂∂αααc By definition:
∂CM ac = 0 ∂α Solving for the above
∂C xxac− ref M ref ∂C =−L , or c ∂∂αα
∂C xxac− ref x ref M ref =− ccC∂ L
Example:
Consider our AVL calculations for the F-16C
• xref = 0 - Moment given about LE ∂C • Compute M ref for small range of angle of attack by numerical ∂CL differences. I picked αα= −=300 to 3. x • This gave ac ≈ 2.89 . c x ∂C • Plotting Cvsα about ac shows M ≈ 0 . M c ∂α
∂C Note that according to the AVL predictions, not only is M ≈=0 @ x 2.89 , but ∂α ac also that CM = 0 . The location about which CM = 0 is called the center of pressure.
16.100 2002 2 Aerodynamic Center
Center of pressure is that location where the resultant forces act and about which the aerodynamic moment is zero.
Changing “new” to “cp” and “ref” to “NOSE” to correspond to AVL:
xxcp − NOSE CCC=−+ MLMNOSE cp c
x C ⇒=−cp M NOSE cCL So if: CC∂ MMNOSE= NOSE ,
CCL ∂ L this will be true. This means that
CCML≈=0 at 0 .
16.100 2002 3 Cm vs Alpha for F16C from AVL (M=0)
1.5
1
0.5
0 -10 -5 0 5 10 15 20
-0.5
Xref = 0 -1 Cm Xref = Xac = 2.89
-1.5
-2
-2.5
-3
-3.5 Alpha Wind Tunnel Test Aerodynamic Center Characteristics for Washout and Rigid Wings (Altitude = 10,000 ft.)
50
45
40
35 Rigid Wing Washout Wing 30 A.C. (% MAC)
25
20
15 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Mach Number