Monoidal natural transformation

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 .

$\frac{d{C}_m}{d{C}_L}=0$ where $C_L$ is the aircraft lift coefficient.

The concept of the aerodynamic center (AC) is important in aerodynamics. It is fundamental in the science of stability of aircraft in flight.

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: $\frac{d{C}_m}{d\alpha }<0$     and    $\frac{d{C}_z}{d\alpha }>0$

For directional static stability:   $\frac{d{C}_n}{d\beta }>0$     and    $\frac{d{C}_y}{d\beta }>0$

Where:

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

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

For A Force Acting Away at the Aerodynamic Center, which is away from the reference point:

$X_{AC}=X_{ref}+c\frac{d{C}_m}{d{C}_z}$

Which for Small Angles $cos(\alpha )=1$ and $sin(\alpha )$=$\alpha$, $\beta =0$, $C_z=C_L-C_d*\alpha$, $C_z=C_L$ simplifies to:

$X_{AC}=X_{ref}+c\frac{d{C}_m}{d{C}_L}$

$Y_{AC}=Y_{ref}$

$Z_{AC}=Z_{ref}$

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

$X_{AC}=X_{ref}+c\frac{d{C}_m}{d{C}_z}+c\frac{d{C}_n}{d{C}_y}$

.

$Y_{AC}=Y_{ref}+c\frac{d{C}_l}{d{C}_z}+c\frac{d{C}_n}{d{C}_x}$

.

$Z_{AC}=Z_{ref}+c\frac{d{C}_l}{d{C}_y}+c\frac{d{C}_m}{d{C}_x}$

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

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

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

See also

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