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The aerodynamic center is the reference point about which an airfoil's pitching torque is approximately independent of angle of attack over a stated flow regime. More generally the defining test uses the pitching-moment coefficient at fixed flow conditions. The aerodynamic center simplifies replacing a distributed pressure field by an aerodynamic force and a nearly constant residual torque. It differs from the center of pressure, where the resultant's pitching torque is zero and whose position can move as angle of attack changes.^[ref]

A thin-airfoil model makes the distinction quantitative. Let $c$ be the airfoil chord, $S$ a reference area and $q_{\infty }=\rho V^2/2$ the dynamic pressure. Define $C_L=L/(q_{\infty }S)$ and $C_m=M/(q_{\infty }Sc)$. Take positive $x$ aft from the leading edge and positive pitching torque nose-up. In a chord-line lift model, moving the reference point givesConsequently, when $\partial C_L/\partial \alpha \ne 0$,Here $\alpha$ is angle of attack; differentiating holds the other test conditions fixed. A vanishing lift slope or strongly changing slopes defeats this simple inference.

For an original model choose $C_m(0)=-0.05-C_L/4$. Its aerodynamic center is $x_{\mathrm{ac}}=c/4$, with $C_m(x_{\mathrm{ac}})=-0.05$. In contrast, setting the pitching torque to zero givesThe center of pressure moves from $0.35c$ to $0.30c$ as $C_L$ increases from $0.5$ to $1$. With $c=1\mathrm{m}$ and $q_{\infty }S=1000\mathrm{N}$, the aerodynamic center's residual pitching torque remains $-50\mathrm{N}\mathrm{m}$ while lift doubles. At zero lift the nonzero pure torque cannot be represented by a finite lift application point.

A wind tunnel test can measure lift and pitching torque about a fixed support while stepping angle of attack. The measured slopes determine the inferred aerodynamic center; approximately flat moment data about that point test the model. NASA describes the quarter-chord position as characteristic of most low-speed airfoils. Flow separation, compressibility and three-dimensional wing geometry require their own assessment rather than transferring this location unchanged.^[ref]

An airfoil chord is the straight segment joining an airfoil leading edge to its airfoil trailing edge; its length is conventionally $c$. The airfoil chord is a geometric reference, not the curved distance along either surface or the curved mean camber line. The chord direction supplies the reference for angle of attack, while its length supplies a scale for reporting an airfoil's thickness and positions.^[ref]

If the edge positions in one section plane are ${\mathbf{r}}_L$ and ${\mathbf{r}}_T$, thenThe chordwise coordinate of a point $\mathbf{r}$ is its projection $x_c=(\mathbf{r}-{\mathbf{r}}_L)\cdot {\mathbf{e}}_c$. Thus a point a quarter of the way along the airfoil chord has $x_c/c=1/4$, regardless of how the profile is rotated in a drawing. The aerodynamic center's approximate quarter-chord location refers to this straight geometric reference.

For an original tilted section, choose edge positions $(0.02,0.01)$ and $(0.42,-0.02)$ in metres. ThenThe horizontal projection is $0.40\mathrm{m}$, while the airfoil chord is slightly longer. Its quarter point is $(0.12,0.0025)\mathrm{m}$. Using a ruler along the page's horizontal axis rather than along the segment would measure a projection, not $c$.

A finite wing generally has a local airfoil chord $c(y)$ varying along its span. For a stipulated unswept trapezoidal planform of span $b=10\mathrm{m}$, root chord $2\mathrm{m}$ and tip chord $1\mathrm{m}$, integrating the linear chord distribution gives areaIts span-average chord is $S/b=1.5\mathrm{m}$; that average is not the chord at every section. The mean aerodynamic chord is a different wing reference and should be identified explicitly when used. NASA's definitions distinguish section geometry from this spanwise wing geometry.^[ref]

For a scaled wind tunnel model, a specified pressure-tap position $x_c/c=0.20$ therefore remains a comparable geometric location as $c$ changes. Equal normalized geometry alone does not establish equal flow: comparisons must also consider Reynolds number, Mach number and the relevant boundary conditions.

The center of pressure locates the resultant of a distributed fluid pressure load relative to its moment. On a planar surface carrying parallel pressure forces, the first-moment ratio in center of pressure applies. On an airfoil, pressure variation and its resultant can change with the flow state, so the point is not a permanently fixed geometric centre. NASA's primary account explicitly distinguishes a moving center of pressure from retaining a force and pitching moment about an aerodynamic center.^[ref]

For an original hydrostatic plate example, take a vertical rectangle of width $w$, extending from the free surface to depth $H$ in a constant-density fluid. If the same atmospheric pressure acts on its other face, the net gauge pressure at depth $y$ is $p(y)=\rho gy$. This hydrostatic relation follows from vertical force balance.^[ref] The resultant and its depth areFor $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, $w=1\mathrm{m}$ and $H=2\mathrm{m}$, the net force is $19.62\mathrm{kN}$ at depth $1.333\mathrm{m}$. The area centroid is only one metre below the surface; the increasing pressure shifts the load's centroid downward.

A tank demonstration can infer the resultant line from the force and hinge moment on a small plate while changing the water depth. The plate's own gravitational moment and the pressure on both faces must be included. This tests a specified pressure-load model, not a general claim that the point always lies below every geometric centre.

For aerodynamic net loads, differences of pressures on opposite faces can have both signs. The convex-hull property proved for nonnegative contact pressure then need not apply, and a small resultant with a finite moment makes its location ill-conditioned. A curved surface or a general wrench may also require a residual couple rather than a single force that removes every component of torque. The resultant convention and moment reference are therefore part of the definition.

The mean aerodynamic chord is a reference length that weights each local airfoil chord by the area of its spanwise strip. For a symmetric aircraft wing with wingspan $b$, local chord $c(y)>0$ and wing area $S$, its geometric definition isThe extra factor of $c(y)$ distinguishes this from the mean geometric chord $S/b$.^[ref]

The weighting has a simple interpretation: a strip has area $dS=c(y)dy$, so $\bar{c}=(1/S)\int cdS$ is an area-weighted mean. If a simplified section model has the same local pitching moment coefficient $c_m$ and dynamic pressure $q$ everywhere, its strip moment is $dM=q{c}_{m}c(y{)}^{2}dy$. Integration then gives $M=q{c}_{m}S\bar{c}$. This explains its use as a moment reference; an actual three-dimensional flow need not have a uniform section coefficient.

For the mean aerodynamic chord of a trapezoidal wing, take root chord $c_r$, tip chord $c_t=\lambda c_r$ and a linear distribution $c(y)=c_r[1-2(1-\lambda )y/b]$. Integrating givesThe stipulated wing planform $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$ has $S=15{\mathrm{m}}^2$, $S/b=1.5\mathrm{m}$ and $\bar{c}=14/9\mathrm{m}\simeq 1.5556\mathrm{m}$. The wider strips receive more weight.

The Cauchy-Schwarz inequality gives a useful general check:Equality holds for a constant chord almost everywhere. A reported mean aerodynamic chord shorter than $S/b$ therefore contradicts this definition for the same span and area convention. Its length alone does not specify a location: the reference position of that chord and the aerodynamic center are additional geometric or aerodynamic data.

For a symmetric trapezoidal wing planform, let root chord be $c_r$, tip chord $c_t=\lambda c_r$, and wingspan $b$. The wing taper ratio $\lambda$ describes the linear local chord $c(y)=c_r[1-2(1-\lambda )y/b]$ on one half-wing. Direct integration givesA rectangular planform has $\lambda =1$ and $\overline{c}=c_r$; the triangular limit $\lambda =0$ gives $\overline{c}=2c_r/3$. These are lengths, not aerodynamic-center locations.

The mean aerodynamic chord reference position requires a placement convention. On the positive half-span, define the area-weighted stationFor a straight leading edge with $x_{\mathrm{LE}}(y)=x_0+y\tan {\Lambda }_{\mathrm{LE}}$, its corresponding chord reference starts at $x_{\mathrm{LE},\mathrm{MAC}}=x_0+y_{\mathrm{MAC}}\tan {\Lambda }_{\mathrm{LE}}$. A geometric quarter-chord reference is then $x_{1/4}=x_{\mathrm{LE},\mathrm{MAC}}+\overline{c}/4$. These are the standard planform integrals, not a guarantee about the aerodynamic center of every finite wing.^[ref]

For the stipulated $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$, the values are $\overline{c}=14/9\mathrm{m}$ and $y_{\mathrm{MAC}}=20/9\mathrm{m}$. With $x_0=0.10\mathrm{m}$ and ${\Lambda }_{\mathrm{LE}}=20{}^{\circ }$, the leading-edge reference is $0.908823\mathrm{m}$ aft of the chosen origin, and the quarter-chord reference is $1.297712\mathrm{m}$. Changing the wing sweep moves these reference positions without changing $\overline{c}$ when the same spanwise chords are retained.

The displayed $y_{\mathrm{MAC}}$ belongs to one half-wing; the spanwise centroid of the whole symmetric planform is instead zero. A half-wing placement convention and a whole-wing centroid are different quantities.

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.

A pitching moment is a torque about an aircraft pitch axis, associated with rotation of the nose upward or downward. It requires a reference point, an axis and a sign convention; the numerical value is not an intrinsic property of a force alone.^[ref]

For a planar chord-line example, take $x$ positive aft along the airfoil chord, $z$ positive toward the upper surface, and a nose-up moment as positive. An upward chord-normal aerodynamic force $N$ applied at $x_f$ has moment about a chord-line reference $x_0$,Thus a stipulated upward $N=100\mathrm{N}$ acting $0.050\mathrm{m}$ behind the leading edge gives $M(0)=-5.0\mathrm{N}\mathrm{m}$, a nose-down moment. Moving the reference aft to $x_0=0.020\mathrm{m}$ gives $-3.0\mathrm{N}\mathrm{m}$ without changing the force.

For a distributed aerodynamic load, the same translation law givesDividing by the same $qSc$ yields the reference-point transformation of a pitching moment coefficient. Replacing $N$ by lift is an approximation unless the body and flow axes coincide; with a stated angle of attack, the full vector transformation accounts for drag as well.

Net force can vanish while a pitching moment remains. Two stipulated chord-normal forces, $+100\mathrm{N}$ at $x=0.20\mathrm{m}$ and $-100\mathrm{N}$ at $x=0$, have $N=0$ and $M=-20\mathrm{N}\mathrm{m}$. They form a couple; no finite location of a single zero resultant represents that load.

When $N\ne 0$ and a chord-line resultant is meaningful, its center of pressure satisfies $x_{\mathrm{cp}}=x_0-M(x_0)/N$. This zero-moment location differs from an aerodynamic center, where moment is approximately insensitive to angle of attack in a specified regime. The latter can have a nonzero moment, especially for a cambered airfoil.^[ref]

The pitching moment coefficient expresses an aerodynamic pitching torque relative to a chosen flow-load and geometry scale:Here $M$ is the moment about a stated reference point and axis, $q_{\infty }$ the freestream dynamic pressure, $S$ a reference area, and $\bar{c}$ a reference chord length, commonly the mean aerodynamic chord for a finite wing. The denominator has units $\mathrm{Pa}{\mathrm{m}}^2\mathrm{m}=\mathrm{N}\mathrm{m}$, so $C_m$ is dimensionless. The reference point, axis, sign, area and length are all necessary metadata.

An original wind-tunnel example uses $\rho =1.2kg/{\mathrm{m}}^3$, $U_{\infty }=30m/s$, $S=0.80{\mathrm{m}}^2$ and $\bar{c}=0.25\mathrm{m}$. Then $q_{\infty }=540\mathrm{Pa}$ and $q_{\infty }S\bar{c}=108\mathrm{N}\mathrm{m}$. A measured nose-down moment $M=-5.4\mathrm{N}\mathrm{m}$ gives $C_m=-0.050$ when nose-up is positive. At twice the speed, the same coefficient would imply $M=-21.6\mathrm{N}\mathrm{m}$. That scaling is conditional on equal coefficients; a changed Reynolds number, Mach number or flow state can change the coefficient itself.

The reference-point transformation of a pitching moment coefficient follows the wrench translation law. If ${\mathbf{r}}_{AB}$ points from old reference $A$ to new reference $B$ in common axes,Project onto the chosen pitch axis before normalizing. In the chord-line model used in aerodynamic center, take $x$ positive aft, nose-up moment positive, and resultant $N$ perpendicular to the chord. ThenFor $C_m(x_A)=-0.050$, $C_N=0.60$ and an aft reference shift of $0.10\bar{c}$, the new coefficient is $+0.010$. The loads did not change; their reported moment did. Replacing $C_N$ by a lift coefficient requires the appropriate axis transformation or an explicitly justified small-angle approximation.

The aerodynamic center is a reference location about which moment is approximately independent of angle of attack in the stated regime. The center of pressure instead places a resultant so that the relevant moment is zero. NASA distinguishes these locations and reports the near-quarter-chord aerodynamic center as a low-speed approximation for many airfoils.^[ref] An approximately constant nonzero $C_m$ at the aerodynamic center is therefore entirely possible.

For sectional data, moment per unit span $m^{\prime }$ is often normalized by $q_{\infty }{c}^2$. That is consistent with using unit span and area $c$ times that span, but should not be interchanged silently with a finite-wing coefficient using total area and mean aerodynamic chord. A comparison between datasets requires matching these conventions.

A proposed apparatus test mounts a model on a force-and-moment balance, measures freestream speed and density, subtracts support tare, and steps angle of attack while retaining a fixed moment reference. Translating the measured wrench to another reference tests the moment-shift relation without changing the flow. Repeat measurements should track balance uncertainty and the sensitivity of $q_{\infty }$ to speed: a fractional speed error contributes approximately twice that fractional error to dynamic pressure when density is fixed.

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.

A reference-point transformation of a pitching moment coefficient changes the location about which the aerodynamic torque is reported while leaving the physical loads unchanged. The pitching moment coefficient therefore requires a reference point as well as a sign, axis and normalization.

Let ${\mathbf{r}}_{AB}$ point from the old reference $A$ to the new reference $B$, with force $\mathbf{F}$ and moment vectors expressed in common axes. The wrench translation law isIt follows immediately by replacing each old force arm ${\mathbf{r}}_i$ with ${\mathbf{r}}_i-{\mathbf{r}}_{AB}$ before summing their cross products. The primary Drake spatial-force documentation gives the same old-to-new offset convention and unchanged force component.^[ref]

For a planar example, choose $x$ positive aft, $z$ positive upward and the right-handed pitch axis so positive moment is nose-up. Write $\mathbf{F}=(X,0,N)$ and ${\mathbf{r}}_{AB}=(\Delta x,0,\Delta z)$. Dividing the pitch component by $q_{\infty }S\bar{c}$ givesHere $q_{\infty }$ is dynamic pressure, $S$ the reference area and $\bar{c}$ the reference chord. The equation assumes those normalizations remain the same.

In a constructed case, $C_{m,A}=-0.04$, $C_N=0.70$ and an aft shift $\Delta x=0.15\bar{c}$ at the same height give $C_{m,B}=0.065$. Adding an upward shift $0.02\bar{c}$ with $C_X=0.05$ changes it to 0.064. The aerodynamic loading has not changed between these reports.

A chordwise zero-moment point, when $C_N\ne 0$, satisfiesThis is a center of pressure construction, not the definition of an aerodynamic center. If the normal force vanishes while the moment does not, no finite chordwise shift removes that moment. NASA distinguishes the force-plus-moment representation from placing a resultant at the center of pressure.^[ref] Replacing $C_N$ by a lift coefficient additionally requires transforming between chord-normal and freestream-normal axes; changing a reference point alone performs no such rotation.