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The airfoil force-axis transformation expresses a section-plane aerodynamic force in chord-fixed axes or flow-fixed axes. Choose $x$ aft along the airfoil chord, $z$ toward the upper surface, and a flow direction with body components $(\cos \alpha ,\sin \alpha )$. Taking positive lift along $(-\sin \alpha ,\cos \alpha )$ givesHere $A$ and $N$ are the chord-axial aerodynamic force and chord-normal aerodynamic force. The convention for angle and axis direction must be stated before applying the signs. NASA's definitions make drag parallel and lift perpendicular to the flow.^[ref]

The rotation matrix is orthogonal: $R^{T}R=I$. Its inverse is therefore its transpose, givingIt also preserves $A^2+N^2=D^2+L^2$, a useful numerical check on a conversion. The two component pairs need not individually agree.

An original conversion uses measured $D=10$ N and $L=100$ N at $\alpha =10{}^{\circ }$. It gives $A\simeq -7.517$ N and $N\simeq 100.217$ N. Rotating these values back recovers the original flow-axis loads. Setting $A=D$ because the angle appears small would miss the term $L\sin \alpha$, which is large compared with the small drag in this example.

Angle uncertainty can likewise mix the large lift component into inferred drag. Holding body-axis readings fixed,For the example, an angular error of $0.1{}^{\circ }\simeq 0.001745$ radians contributes approximately 0.175 N to inferred drag. This is a coordinate-sensitivity calculation, not a prediction of the aerodynamic change caused by physically rotating the model. Changing its actual incidence can change the underlying load as well as the axes.

The same rotation applies to the lift coefficient, drag coefficient, chord-normal force coefficient and chord-axial force coefficient only when they share the same pressure and area normalization.

The chord-axial aerodynamic force $A$ is the component of the net aerodynamic force parallel to an airfoil chord, with positive direction here chosen aft. The perpendicular body component is the chord-normal aerodynamic force $N$. These two components resolve one load vector; flow-relative lift and drag resolve it in a different basis when the angle of attack is nonzero.

Let ${\mathbf{e}}_x$ point aft along the chord, and let $\mathbf{n}$ point outward from the body. Pressure and viscous stress giveNASA's primary explanation motivates integrating locally normal pressure forces before resolving the resultant into flow axes.^[ref] The explicitly outward normal in this equation fixes the pressure-force minus sign.

For a two-dimensional section whose upper and lower surfaces are graphs $z_u(x)$ and $z_l(x)$, the pressure-only axial load per unit span isThe surface slopes determine how much locally normal pressure projects onto the chord. A pressure force is normal to its local surface, rather than necessarily normal to the chord. Skin friction supplies a further contribution and is not included in $A_p^{\prime }$.

An original vector example takes $A=-5$ N, $N=80$ N and $\alpha =20{}^{\circ }$. The airfoil force-axis transformation gives $D\simeq 22.66$ N and $L\simeq 76.89$ N. A forward-directed axial component is therefore compatible with positive drag. It is not, by itself, evidence that the body supplies net propulsive power.

A proposed measurement compares a body-axis balance with integrated surface pressures after matching the angle of attack, sign conventions and support tare. Disagreement can include omitted shear or instrument effects; changing from body axes to flow axes does not remove those physical or measurement contributions.

The chord-normal force coefficient normalizes the chord-normal aerodynamic force:The reference dynamic pressure, wing area, body axis and sign convention accompany the coefficient. For sectional force per unit span, use $c_n=N^{\prime }/(qc)$ with airfoil chord $c$. Mixing total force and sectional normalization gives different units.

With stipulated $q=500\mathrm{Pa}$, $S=0.40{\mathrm{m}}^2$ and $N=100\mathrm{N}$, $C_N=0.50$. Reporting the same force with reference area $0.20{\mathrm{m}}^2$ gives $C_N=1.0$. This reference-area scaling of aerodynamic coefficients changes a reported number without changing the measured load.

When chord-axial force coefficient $C_A=A/(qS)$ uses the same references, the airfoil force-axis transformation gives
For $C_A=0.10$, $C_N=0.50$ and $\alpha =30{}^{\circ }$, the drag coefficient is approximately $0.3366$ and the lift coefficient is $0.3830$. Neither equals the normal coefficient.

Normalization does not remove measurement error. At fixed mass density and reference area, a small fractional speed error contributes approximately twice that fractional error to $q$, and hence the opposite fractional error to $C_N$ if $N$ is otherwise fixed. The usefulness of a dimensionless coefficient depends on retaining the measured references and their measurement uncertainty.

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]