Wing pitching moment decomposition into two terms

Question

Summary: Assuming constant angle of attack $\alpha$, speed $V$, and lift $L$ applied at the center of pressure CP (whose position is $x_{CP}$), the distance $(x_{CP} - x_O)$ between the moment reference point $O$ (whose position is $x_O$) and $CP$ determines the sign (positive, negative or zero) of the pitching moment as long as $L\neq 0$:

$$M_O = (x_{CP} - x_O)L$$

However, the moment $M_O$ depends on $\alpha$, i.e. $M_O=M_O(\alpha)$, since changing $\alpha$ while keeping the reference point $O$ position constant, varies the magnitude $M_O$ since both lift $L$ and the position $x_{CP}$ vary.

Interestingly, when $O=AC$ and $x_O=x_{AC}$, the moment does not change with changing $\alpha$:

$$ M_{O}(\alpha) = M_{AC} = (x_{CP} - x_{AC})L = constant$$

We can move lift $L$ from its application point $x_{CP}$ to point $x_{AC}$ while adding a constant free pitching $M_{AC}$ whose magnitude is $(x_{CP}-x_{AC})L$. The moment $M_{AC}=0$ for symmetric wings and $M_{AC}\neq0$ for cambered wings.

In the case of a cambered wing, the equation $$ M_{AC} = (x_{CP} - x_{AC})L$$ predicts that $$M_{AC}=0$$ when lift $L=0$. However, we know that a cambered wing has a a constant nonzero moment $M_{AC}$ for any $\alpha$ even when $L=0$ at the zero-lift $\alpha_0$. How do we transform the equation for $M_{AC}$ so it becomes equal to the sum of two terms, one solely due to camber and one solely due to lift: $$ M_{O}(\alpha) = M_{AC} = (x_{CP} - x_{AC})L= M_{camber}+M_{lift}$$

where the term $M_{camber}\neq 0$ for any $\alpha$?

Answer 1

Interestingly, when $O=AC$ and $x_O=x_{AC}$, the moment does not change with changing $α$

Yes, but only if your lift is also applied at the AC! That's the whole point of the aerodynamic center. You replace the pressure distribution with a lift force (actually a resultant force, but let's neglect drag here) and a moment. If you place the lift force at the center of pressure, then the resultant moment about that point is zero. Then, the moment about any other point $O$ is only "created" by the resultant force. So in that case indeed: $$M_O = (x_{CP} - x_O)L$$

If you now place the lift at your AC and also O=AC, then:

$$M_O = M_{AC}$$

This $M_{AC}$ is independent of $\alpha$. If you now move to any different O, you get: $$M_O = M_{AC} + L (x_O - x_{AC})$$

The first term $M_{AC}$ is what you are calling "$M_{camber}$" and the second term is your "$M_{lift}$".

Answer 2

In the case of a cambered wing, the equation $M_{AC} = (x_{CP} - x_{AC})\cdot L$ predicts that $M_{AC}=0$ when lift $L=0$.

No, it doesn't.

It only predicts that either $(x_{CP} - x_{AC})$ approaches infinity or that lift decreases to zero. Both conditions will satisfy $M_{AC} = (x_{CP} - x_{AC})\cdot L$, but only the first one will satisfy $M_{AC} = const.$ as well.

Plot from XFLR5 (own work). The overlaid line shows the center of pressure on a wing with washout at small angle of attack where the inner wing creates positive lift while the wingtips create negative lift. At approx. 75% of span the sign of the local lift force changes (= local lift is zero) and the center of pressure switches from negative infinity to positive infinity (apologies for the line not really going to infinity because it has been calculated only at discrete points along the span, but I hope the plot gets the point across).

Why would this happen? At $L = 0$ there is both positive and negative lift along the wing chord. An airfoil with positive camber will show negative lift in the forward part of the airfoil and positive lift in the rear part. Even if their sum is zero, there is enough local lift to create a sizeable pitching moment. Below you see the XFOIL results for a NACA 4409 airfoil. Note that lift is effectively zero while the pitching moment remains unchanged.

Blue is upper side pressure while red is lower side pressure relative to static pressure. The suction peak near the nose has traveled to the bottom side due to the low angle of attack while the rear part of the airfoil, which is less affected by AoA changes, still shows positive lift from camber. For illustration, the same thing with arrows indicating local pressure:

Answer 3

Thank you, Daniel and Peter. Just to wrap things up on this topic, the two-term moment decomposition for $M_O$ is mathematically possible when:

a) $L$ is moved to act at $AC$ instead of $CP$.

b) the moment $M_0$ generated by $L$ acting at $AC$ is calculated about a moment reference point $O\neq AC$

If a) and b) are satisfied, the pitching moment $M_O$ becomes:$$M_O= M_AC+L(x_O-x_{AC})$$ which is the sum of the two moment terms $M_AC$ and $L(x_O-x_{AC})$

However, if $L$ is not moved to $AC$, the moment expression calculated for $L$ acting at $CP$ about an arbitrary point $O$ is given by $$M_O= (x_{CP} - x_{O}) L$$

and this last expression cannot be converted into a sum of two moment terms (one solely due to camber and one solely due to lift).