Aerodynamic center

The distribution of forces on a wing in flight are both complex and varying. This image shows the forces for two typical airfoils, a symmetrical design on the left, and an asymmetrical design more typical of low-speed designs on the right. This diagram shows only the lift components; the similar drag considerations are not illustrated. The aerodynamic center is shown, labeled "c.a."

The torques or moments acting on an airfoil moving through a fluid can be accounted for by the net lift applied at some point on the airfoil, and a separate net pitching moment about that point whose magnitude varies with the choice of where the lift is chosen to be applied. The aerodynamic center is the point at which the pitching moment coefficient for the airfoil does not vary with lift coefficient (i.e. angle of attack), so this choice makes analysis simpler .^[1]

${dC_{m} \over dC_{L}}=0$ where $C_{L}$ is the aircraft lift coefficient.

Forces (lift/drag) can be summed up to act through a single point, the center of pressure, about which sum of all moments equal zero. The center of pressure location however, changes significantly with a change in angle of attack and is thus impractical for analysis. Thus the 25% chord position, or assumed aerodynamic center, is taken about which the forces and moment are generated. At the 25% chord position the moment generated was found and proven to be nearly constant with varying angle of attack. The concept of the aerodynamic center (AC) is important in aerodynamics. It is fundamental in the science of stability of aircraft in flight.

Please note that in highly theoretical/analytical analysis the aerodynamic center does vary slightly and changes location. In most literature however the aerodynamic center is taken at the 25% chord position. This conversely means that if one keeps the assumed AC fixed at 25% chord, that the moment about 25% chord can be 'not constant' depending on the airfoil. A large portion of cambered airfoils have non constant moments about the 25% chord position because the AC does vary slightly. For most analysis the non constant moment about the 25% chord position is not significant enough to warrant consideration, but is important to keep in mind.

For symmetric airfoils in subsonic flight the aerodynamic center is located approximately 25% of the chord from the leading edge of the airfoil. This point is described as the quarter-chord point. This result also holds true for 'thin-airfoils'. For non-symmetric (cambered) airfoils the quarter-chord is only an approximation for the aerodynamic center.

A similar concept is that of center of pressure. The location of the center of pressure varies with changes of lift coefficient and angle of attack. This makes the center of pressure unsuitable for use in analysis of longitudinal static stability. Read about movement of centre of pressure.

Role of aerodynamic center in aircraft stability

For longitudinal static stability: ${dC_{m} \over d\alpha }<0$     and    ${dC_{z} \over d\alpha }>0$

For directional static stability:   ${dC_{n} \over d\beta }>0$     and    ${dC_{y} \over d\beta }>0$

Where:

${\displaystyle C_{z}=C_{L}\cos(\alpha )+C_{d}\sin(\alpha )}$

${\displaystyle C_{x}=C_{L}\sin(\alpha )-C_{d}\cos(\alpha )}$

For a force acting away from the aerodynamic center, which is away from the reference point:

${\displaystyle X_{AC}=X_{\mathrm {ref} }+c{dC_{m} \over dC_{z}}}$

Which for small angles ${\displaystyle \cos(\alpha )=1}$ and ${\displaystyle \sin(\alpha )=\alpha }$, $\beta =0$, $C_{z}=C_{L}-C_{d}*\alpha$, $C_{z}=C_{L}$ simplifies to:

${\displaystyle X_{AC}=X_{\mathrm {ref} }+c{dC_{m} \over dC_{L}}}$

${\displaystyle Y_{AC}=Y_{\mathrm {ref} }}$

${\displaystyle Z_{AC}=Z_{\mathrm {ref} }}$

General Case: From the definition of the AC it follows that

${\displaystyle X_{AC}=X_{\mathrm {ref} }+c{dC_{m} \over dC_{z}}+c{dC_{n} \over dC_{y}}}$

.

${\displaystyle Y_{AC}=Y_{\mathrm {ref} }+c{dC_{l} \over dC_{z}}+c{dC_{n} \over dC_{x}}}$

.

${\displaystyle Z_{AC}=Z_{\mathrm {ref} }+c{dC_{l} \over dC_{y}}+c{dC_{m} \over dC_{x}}}$

The Static Margin can then be used to quantify the AC:

$SM={X_{{AC}}-X_{{CG}} \over c}$

where:

$C_{n}$ = yawing moment coefficient

$C_m$ = pitching moment coefficient

$C_{l}$ = rolling moment coefficient

$C_{x}$ = X-force ~= Drag

$C_{y}$ = Y-force ~= Side Force

$C_{z}$ = Z-force ~= Lift

ref = reference point (about which moments were taken)

c = reference length

S = reference area

q = dynamic pressure

$\alpha$ = angle of attack

$\beta$ = sideslip angle

SM = Static Margin

References

1. Benson, Tom (2006). "Aerodynamic Center (ac)". The Beginner's Guide to Aeronautics. NASA Glenn Research Center. Retrieved 2006-04-01.

See also

• Aircraft flight mechanics
• Flight dynamics
• Longitudinal static stability
• Thin-airfoil theory
• Joukowsky transform