Aerodynamics: Wing Pitching Moment

[Keen]

Hi,

I've been designing a wing lately, and have been trying to figure out something. Hope someone could help.

Considerations of small trim deflection have led to a design requirement of Cmac for the wing. I've been trying to figure out ways to reduce this to a Cmac value for the airfoil. In general, they are not the same.

Would anyone know of any ways?

Thanks very much.

 

[enigma]

Hi Keen,

welcome to the forums!

Let me make sure I'm understanding your question correctly (I'm a space track aerohead, so your question is out of my area of focus...)

You're designing a wing which needs a low AoA when it's in equilibrium. Because of that, you're trying to minimize the moment about the aerodynamic center for the wing. You're trying to find out a way to get the CMac for the airfoil from the required CMac for the wing requirements?

Is that correct?

I'm not sure exactly how to do it (easily). Is the wing straight, or do you have a wing sweep? Is the airfoil constant throughout the length of the wing? Is the chordlength changing significantly over the length of the wing?

If you've got no's to most of those questions (or at least very small values), you should be able to approximate by simply dividing. Otherwise, I don't know how to do it short of designing it in a solid modeling package and extracting the values.

 

[Keen]

Hi Enigma,

Thanks for your welcome and reply.

The wing is swept, tapered, constant airfoil and untwisted. I recently came across an equation in Datcom (Section 4.1.4) and also used in Roskam's "Airplane Design" Part 6 which relates the Cm of the airfoil to the Cm of the wing, given aspect ratio and sweep.

Would anyone know about the equation and how it is derived? I don't think it's from lifting-line theory which doesn't give info about force distributions in the chordwise direction.

Regards

 

[ken]

Hi Keen

Is this a flying wing design?

Some cambered airfoils gain pos Cmo values by reflexing the trailing edge. Your swept planform would allow you to transition the section from no reflex at the root chord to a reflex section at the tip.
This will be stable with less total reflex than employing the same reflex over the total span. (more efficient)
Note beta for root and tip will then be different so you have to employ some geometric twist to keep zero lift aligned for all sections
along the span. That way the whole wing works at the same CL.
Effectively there would be no aerodynamic twist.

You will have to be careful with camber to prevent tip stalling
Reflex sections generally have limited CL if memory serves. That might limit the root camber. I don't know if you particularly want
a high camber though. What's the application?

 

[ken]

Hi Keen

I read you question again and I guess I didn't really help.

I assume, the formulae you have does not lend itself to rearanging
to make airfoil Cm the subject.

Can you work back the other way, from an airfoil Cm determine the wing Cm?

If you can do this, the problem can be solved itteratively.
Choose 3 airfoil Cm values and calculate the wing Cm.

Now you can establish a relationship between airfoil Cm and Wing Cm.
You could just graph this but if you are looking for a program solution, you can use:

Y = a * X^2 + b * X + c (for parabolic relationships)
Y = a * X^-2 + b * X^-1 +c (for exponential relationships)

Plug the Cm values into 3 equations for X and Y and solve a, b and c by simultaneous equations or by matrix algebra, there is another method, I can't recall it's name. If you like, I have a basic program I wrote many years ago for any number of points but it's best not to use too many.

Now you have an equation for estimating airfoil Cm from wing Cm.

Use the estimated airfoil cm to calculate wing Cm. If this is not close enough. Pick the 3 airfoil Cms that yield wing Cms closest to what you want and use them for another iteration.

Repeat until your estimate is close enough.



Obviously I don't know the formulae you have and can only guess at it's derivation. As it is aspect ratio based, I would guess it is to do with aerodynamic twist caused by tip vortex resulting in a spanwise variation in Cl and Cm. If that where the case, I'd expect plan form to be a factor as well.

I would have thought wind Cm could be estimated by dividing the span into strips, calculating the moments of each strip. Summing the strip moments about wing AC and reducing the overall moment back to a Cmac for the wing using Cm = 2 * M/( p * V^2 * S * C ).

Does a calculous solution applied to this approach sound likely?

I'm only guessing at the derivation though.