Mean aerodynamic chord reference position

A mean aerodynamic chord reference position locates the reference segment whose length is the mean aerodynamic chord. Its chordwise origin and spanwise placement are additional geometric data. The length alone does not specify a point on a wing or determine its aerodynamic center.

For a symmetric wing planform, let $c(y)$ be the local airfoil chord, $x_{\mathrm{LE}}(y)$ its leading-edge coordinate and $S=2{\int }_0^{b/2}c(y)dy$ the wing area. A standard area-weighted half-wing placement isThe definition averages leading-edge positions themselves; it does not generally substitute the average station into a nonlinear leading-edge curve.^[ref]

The quarter-chord reference of a mean aerodynamic chord then hasThe last identity follows directly from the definition of $\bar{c}$. It is a geometric area-weighted average of the local quarter-chord positions, without assuming a uniform aerodynamic loading.

For an original curved-leading-edge example, take constant chord $c=0.4\mathrm{m}$, span $b=4\mathrm{m}$ and $x_{\mathrm{LE}}(y)=k{y}^2$ on the positive half-wing, with $k=0.05{\mathrm{m}}^{-1}$. Then $S=1.6{\mathrm{m}}^2$, $\bar{c}=0.4\mathrm{m}$, $y_{\mathrm{MAC}}=1\mathrm{m}$ andEvaluating the leading edge at the mean station would instead give $x_{\mathrm{LE}}(1\mathrm{m})=0.05\mathrm{m}$, moving the quarter reference forward by $16.667\mathrm{mm}$. For a straight leading edge the two operations agree, as in mean aerodynamic chord of a trapezoidal wing.

The positive half-wing station is a placement convention; the whole symmetric planform's spanwise centroid remains zero. Reporting a chord-reference position should identify its coordinate origin, planform-area convention and the chosen fraction of the reference chord.

The quarter-chord reference of a mean aerodynamic chord is the geometric point one quarter of the reference-chord length aft of its specified leading-edge position:It supplies a reproducible location for reporting positions and pitching moments. The length of the mean aerodynamic chord alone does not locate this point; its reference origin is also required.

For a symmetric wing planform using the area-weighted placement in mean aerodynamic chord reference position,Adding one quarter of the first integral to the second givesIt is consequently an area-weighted average of local quarter-chord coordinates under this convention. No aerodynamic-load distribution was needed for that identity.^[ref]

For an original trapezoidal example, let span $b=8$ m, root chord 3 m and tip chord 1 m. Then $S=16{\mathrm{m}}^2$, $\bar{c}=13/6$ m and the positive half-wing reference station is $y_{\mathrm{MAC}}=5/3$ m. With straight leading edge $x_{\mathrm{LE}}(y)=0.2\mathrm{m}+0.3y$, its reference leading edge is at 0.7 m. The quarter-chord point is therefore $x_{1/4}=1.241667$ m aft of the stated origin. Changing sweep changes this coordinate even if all spanwise chord lengths remain fixed.

The aerodynamic center is defined through the response of pitching moment to incidence, rather than through a geometric averaging rule. In ideal two-dimensional thin airfoil theory, the quarter-chord moment is independent of angle of attack, which motivates that familiar reference. It does not prove that every finite wing's aerodynamic center lies at this geometric point.^[ref]

Moment comparisons must also use the same reference. Moving the reference by a vector $\Delta \mathbf{r}$ changes the moment by $-\Delta \mathbf{r}\times \mathbf{F}$ for the same resultant force. A location convention thus has a measurable mechanical consequence even though it is chosen geometrically.