Aerodynamic center

Within the assumptions embodied in thin airfoil theory, the aerodynamic center is located at the quarter-chord (25% chord position) on a symmetric airfoil while it is close but not exactly equal to the quarter-chord point on a cambered airfoil.

From thin airfoil theory:[2]

${\displaystyle \ c_{l}=2\pi \alpha }$

where ${\displaystyle c_{l}\!}$ is the section lift coefficient,

${\displaystyle \alpha \!}$ is the angle of attack in radian, measured relative to the chord line.

${\displaystyle \ {dc_{m,c/4} \over d\alpha }=m_{0}}$

where ${\displaystyle \ c_{m,c/4}}$ is the moment taken at quarter-chord point and ${\displaystyle \ m_{0}}$ is a constant.

${\displaystyle \ M_{ac}=L(cx_{ac}-c/4)+M_{c/4}}$

${\displaystyle \ c_{m,ac}=c_{l}(x_{ac}-0.25)+c_{m,c/4}}$

Differentiating with respect to angle of attack

${\displaystyle \ x_{ac}={-m_{0} \over {2\pi }}+0.25}$

For symmetrical airfoils ${\displaystyle \ m_{0}=0}$, so the aerodynamic center is at 25% of chord. But for cambered airfoils the aerodynamic center can be slightly less than 25% of the chord from the leading edge, which depends on the slope of the moment coefficient, ${\displaystyle \ m_{0}}$. These results obtained are calculated using the thin airfoil theory so the use of the results are warranted only when the assumptions of thin airfoil theory are realistic. In precision experimentation with real airfoils and advanced analysis, the aerodynamic center is observed to change location slightly as angle of attack varies. In most literature however the aerodynamic center is assumed to be fixed at the 25% chord position.

For longitudinal static stability:

${\displaystyle {\frac {dC_{m}}{d\alpha }}<0\quad {\text{and}}\quad {\frac {dC_{z}}{d\alpha }}>0}$

For directional static stability:

${\displaystyle {\frac {dC_{n}}{d\beta }}>0\quad {\text{and}}\quad {\frac {dC_{y}}{d\beta }}<0}$

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 cos(α) = 1 and sin(α) = α, β = 0, ${\displaystyle C_{z}=C_{L}-C_{d}*\alpha ,}$ ${\displaystyle C_{z}=C_{L}}$ simplifies to:

${\displaystyle {\begin{aligned}&X_{AC}=X_{\mathrm {ref} }+c{dC_{m} \over dC_{L}}\\&Y_{AC}=Y_{\mathrm {ref} }\\&Z_{AC}=Z_{\mathrm {ref} }\end{aligned}}}$

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

${\displaystyle {\begin{aligned}&X_{AC}=X_{\mathrm {ref} }+c{dC_{m} \over dC_{z}}+c{dC_{n} \over dC_{y}}\\&Y_{AC}=Y_{\mathrm {ref} }+c{dC_{l} \over dC_{z}}+c{dC_{n} \over dC_{x}}\\&Z_{AC}=Z_{\mathrm {ref} }+c{dC_{l} \over dC_{y}}+c{dC_{m} \over dC_{x}}\end{aligned}}}$

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

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

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
• α = angle of attack
• β = sideslip angle
• SM = Static Margin

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