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Need Algebra Help Calculating Area-rule For Wing  
User currently offlineLehpron From United States of America, joined Jul 2001, 7028 posts, RR: 21
Posted (6 years 8 months 3 weeks 3 days 17 hours ago) and read 4917 times:

Just to clarify, I cannot seem to derive a formula for a swept wing, but I have made approximations for delta wings.

For delta wings, I made a curve for the leading edge, y1(x); then an airfoil curve for a wing CS, y2(x); then assumed the wing's forward view CS was triangular and said area rule was half the base times the height, where height was the airfoil curve and base was the leading edge curve. This also assumes the wing CS is zero at the wing tip for simplicity, I could always subtract out a tip chord.

This formula worked as long as the wing ended when the airfoil curve went to zero at the trailing edge, but a swept wing's curve generally goes beyond that point, so I'm stuck.

So far, I know there will be two curves for leading and trailing edges and the airfoil curve. I'd like to still use the triangular forward cross section, half base times hieght, but I cannot just subtract the difference of the leading and trailing edges here.

Help please, this has been bugging me for quite some time.


The meaning of life is curiosity; we were put on this planet to explore opportunities.
9 replies: All unread, jump to last
 
User currently offlineRwessel From United States of America, joined Jan 2007, 2238 posts, RR: 2
Reply 1, posted (6 years 8 months 3 weeks 1 day 20 hours ago) and read 4844 times:
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What shape are the wings and which dimensions do you know? If it's a trapezoid, it's trivial if you know the lengths of the two parallel sides and the distance between them ((a+b)*d/2), there's also a straightforward formula if you know the lengths of all four sides.

If your wing is a general quadrilateral, you need to know enough to tell the shape (for example, the lengths of all four sides, plus the distance between an opposite pair of corners). If it's a more complex shape, life gets, well, more complex.  Wink


User currently offlineLehpron From United States of America, joined Jul 2001, 7028 posts, RR: 21
Reply 2, posted (6 years 8 months 3 weeks 1 day 9 hours ago) and read 4802 times:

Right now I'm simplifying the wing's airfoil as a box of 10% thickness, and the leading and trailing edges are parallel. We would let the chord length be a unit length of 1 and the half span be 3 units. The slope of the wing would be 2, or about 26-degrees.

The unit area can be calculated pretty easily, just a rhombus or parallelogram. But I need the area rule as a function of length, so the airfoil height has to multiply by the plan view edges somehow.

First thoughts were that the airfoil, for the simplified example, will stretch in accordance with the wing sweep. Problem is, as x increases from the front of the wing, the original chord ends at unit 1, but the wing keeps going until unit 2.5 (1.5 slope + 1 unit length tip chord).

[Edited 2007-08-01 14:52:02]


The meaning of life is curiosity; we were put on this planet to explore opportunities.
User currently offlineRwessel From United States of America, joined Jan 2007, 2238 posts, RR: 2
Reply 3, posted (6 years 8 months 3 weeks 1 day 4 hours ago) and read 4770 times:
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OK, I'm still not quite understanding your problem, but if you can simplify the wing so that it's a trapezoid, with the fuselage and wingtip being the parallel sides, with the fuselage interface having length A, the leading edge being swept back alpha, the trailing edge swept back beta, and the (half) span being B, the area would be something along the lines of:

area = ((A + (B/cot(alpha)) - (B/cot(beta))) + A) * B / 2

Or something along those lines...


User currently offlineLehpron From United States of America, joined Jul 2001, 7028 posts, RR: 21
Reply 4, posted (6 years 8 months 2 weeks 5 days 1 hour ago) and read 4715 times:

But I know the area, I want the area-rule diagram as a function of length. You do know what area-rule is right?  Wink


The meaning of life is curiosity; we were put on this planet to explore opportunities.
User currently offlinePoitin From , joined Dec 1969, posts, RR:
Reply 5, posted (6 years 8 months 2 weeks 4 days 21 hours ago) and read 4705 times:

Quoting Lehpron (Reply 4):
But I know the area, I want the area-rule diagram as a function of length. You do know what area-rule is right?

You might want to start with Wikipedia and look at the external references at the bottom. You can usually find the sort of information you want in the references. It is non-trivial as it also requires knowing the thickness of the wing cord.

http://en.wikipedia.org/wiki/Whitcomb_area_rule


User currently offlineJetlagged From United Kingdom, joined Jan 2005, 2532 posts, RR: 24
Reply 6, posted (6 years 8 months 2 weeks 4 days 19 hours ago) and read 4700 times:

Basically, you need to work out the spanwise chord as a function of x position.

Imagine your swept wing as being part of a delta. The leading edge is the same. The swept wing is missing a triangular tip (base = wing chord) and a cutout to form the trailing edge. All you need to do is calculate the spanwise chords of these cutouts as a function of x and subtract these numbers from your delta wing equation.

A single equation won't work. You'll need to break the wing into longitudinal sections and each section will have a different equation. The break points will be at the LE tip, TE tip and root TE.

Since your wing has no taper, and is of square section, the thickness is also constant, so area at a point x is

A = spanwise chord * 0.1 * chord.



The glass isn't half empty, or half full, it's twice as big as it needs to be.
User currently offlineLehpron From United States of America, joined Jul 2001, 7028 posts, RR: 21
Reply 7, posted (6 years 8 months 2 weeks 3 days 23 hours ago) and read 4670 times:

Quoting Jetlagged (Reply 6):
Since your wing has no taper, and is of square section, the thickness is also constant, so area at a point x is

A = spanwise chord * 0.1 * chord.

So a rectangular airfoil section that doesn't vary in curvature throughout the span is an oversimplification on my part. If I use the forward half section of a swept wing and treat it like a delta, the airfoil ends before reaching the trailing edge. You're right, there isn't a simple equation or way to do this. One thing I forgot to mention, that regardless of the foil CS, I assumed the tip foil thickness was zero, it was an attempt to maintain a triangular forward CS simplicity. Maybe instead I should keep the tip chord and thickness equal to the root?

I could but I don't want to build this in 3D and go in and pick point to map the area-rule, that would be insane, but do-able. FGD!



The meaning of life is curiosity; we were put on this planet to explore opportunities.
User currently offlineLehpron From United States of America, joined Jul 2001, 7028 posts, RR: 21
Reply 8, posted (6 years 8 months 1 week 3 days 21 hours ago) and read 4586 times:

Okay, I think I figured it out, but so far I had to make it a bit more complex to do so.

Only assumption here was that I maintained the wing's thickness from root to tip equal to the root-tip chord ratio (taper). I used a parabolic airfoil for simplicity. Picture of wing section I wanted to find:



I came to the conclusion that the leading edge governed the area rule formula and the wing had to be broken up into sections. I set up the equations to "watch" the changing area as forward looking strips of the airfoil CS.

The first in red is an area where the wing cross section grew. This was done by integrating the airfoil function across the half-span from trailing to leading edge.

The second section in dark yellow scaled the total foil's area from what it was at the red border down to the green border (distance in heights of y). These strips varied by the differences in distance from leading and trailing edges. So the formula here was the integral of a continually smaller scaled foil function.

The third section in green did a reverse of the red, the total area of a foil subtract the piece of the foil that was getting smaller, resulting in the chord length being the half-span minus trailing edge.

Took me all week during my breaks at work to figure this out, I'm pumping it up on a spreadsheet to see how accurate it is as you read this.



The meaning of life is curiosity; we were put on this planet to explore opportunities.
User currently offlineLehpron From United States of America, joined Jul 2001, 7028 posts, RR: 21
Reply 9, posted (4 years 3 months 3 weeks 2 days 23 hours ago) and read 4081 times:

Yes, I'm pulling this thread back from the dead just to update; it does shows up 2nd in a google search for the subject of "calculating area rule", so this is for anyone wanting to answer the question as I've long since answered it.  

Using the formulas in the picture from the previous reply #8 above as reference:

First set up a coordinate system; x-axis is chordwise, y-axis is spanwise, and z-axis is [chord] thickness-wise.

Definitions:
Yle is the y1(x) formula for the leading edge of the wing
Yte is the y2(x) formula for the trailing edge of the wing
b/2 is of course the half span of the wing.
I've set the y foil as the wing cross section to instead be z(x) = 0.4x - 0.4x^2, where the max thickness zmax = z(x at dz/dx=0) = 0.1

I assumed the chord length to vary with the span-wise distance y from leading to trailing edges, so there are three segments:

1) Yleading - 0
... y1(x) - 0 = x from x=0 to x=1,
2) Yleading - Y trailing
... y1(x) - y2(x) = x - (1.5x-1.5) = 1.5 - x from x=1 to x=2,
3) b/2 - Y trailing.
... b/2 - y2(x) = 2 - (1.5x-1.5) = 0.5 - 1.5x from x=2 to x=2.33

** It is easier to do this on a spreadsheet program like MS Excel as the above segments aren't continuous, it makes it difficult on paper or on a calculator. **

Since throughout each segment, the area is following along the wing cross section, that is the integration of the foil shape, we must scale the shape of the wing with this variation, as y1(x) - y2(x). This scale not only affects z(y(x)) plane chord length but also the thickness, which is zmax/chord * new chord scale, z(y(x)).

First integrate z(x) as z'(x) = 0.2x^2 - 0.1333x^3.

Then change formula by scale formula (and change of variable since plane of reference is normal to flight direction, we're assuming Mach 1):

z'(y(x)) = [(zmax/c) * (y1(x) - y2(x)]*[0.2*(x/(y1(x) - y2(x))^2 - 0.13333*(x/(y1(x) - y2(x))^3]

in our case, which chord = 1,

z'1(x) = [(0.1/1) * (x)]*[0.2*(x/x)^2 - 0.13333*(x/x)^3] from x = (0,1)
z'2(x) = [(0.1/1) * (1.5-x)]*[0.2*(x/(1.5-x))^2 - 0.13333*(x/(1.5-x))^3] from x = (1,2)
z'3(x) = [(0.1/1) * (2-1.5x)]*[0.2*(x/(2-1.5x))^2 - 0.13333*(x/(2-1.5x))^3] from x = (2,2.33)

When finally ploted out, it should look like this:



If wanting the volume of this wing section, simply integrate it (each individual z' within their ranges). I'm sure you can guess, the more complex airfoil along with complex leading and trailing edges, the crazier this math can get. Use a computer.

Note, with increasing Mach numbers, the oblique wave is angled away from 90-deg so it is that new slanted plane that "sees" the airplane and wing cross section, it isn't as simple as the above math.

Hope this helps anyone, if there are any questions, send me an email at lehpron@yahoo.com

[Edited 2009-12-26 23:24:31]


The meaning of life is curiosity; we were put on this planet to explore opportunities.
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