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.
~L
N~
z
A
Mref
x
~L
N~
Mnew
xref
A
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:
M ref = −( xnew − xref ) L + M new
Or, in coefficient form:
Cm ref

x −x
ref
= −  new
cN

 mean
 a .c.


 CL + CM
new



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 = xac
Using above:
( x − xref )
CM ref = − ac
CL + CM ac
c
Differential with respect to α :
∂CM ref
 x − xref  ∂CL  ∂CM ac 
= −  ac
 ∂α +  ∂α 
∂α
c




By definition:
 ∂CM ac 

=0
 ∂α 
Solving for the above
xac − xref
c
 ∂CM ref   ∂CL 
= −
 
 , or
∂
∂
α
α




xac − xref
c
=
 ∂CM ref 
−

c
 ∂CL 
xref
Example:
Consider our AVL calculations for the F-16C
• xref = 0 - Moment given about LE
•
•
•
Compute
∂CM ref
∂CL
for small range of angle of attack by numerical
differences. I picked α = −30 to α = 30 .
x
This gave ac ≈ 2.89 .
c
x
∂CM
Plotting CM vs α about ac shows
≈0.
c
∂α
∂CM
≈ 0 @ xac = 2.89 , but
∂α
also that CM = 0 . The location about which CM = 0 is called the center of
pressure.
Note that according to the AVL predictions, not only is
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:
CM NOSE
⇒
xcp
c
So if:
CM NOSE


= −


=−
=
xcp − xNOSE 
 C L + C M cp
c 
CM NOSE
CL
∂CM NOSE
,
CL
∂CL
this will be true. This means that
CM ≈ 0 at CL = 0 .
16.100 2002
3
Cm
-10
-5
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0
Alpha
5
10
15
Xref = 0
Xref = Xac = 2.89
Cm vs Alpha for F16C from AVL (M=0)
20
A.C. (% MAC)
15
20
25
30
35
40
45
50
0.5
0.6
0.7
0.8
Mach Number
0.9
1
1.1
1.2
Rigid Wing
Washout Wing
1.3
Wind Tunnel Test Aerodynamic Center Characteristics for Washout and Rigid Wings (Altitude
= 10,000 ft.)