Reference-point transformation of a pitching moment coefficient

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.