How to work out the area of an elliptical wing?

[Superhornet]

I am planning to measure the wing area of a Spitfire aircraft. I am going to use double integral formula, but firstly I need to derive the continuous function for the region of interest. Also,
How to do that? I searched the entire internet and only found out the wing span of that aircraft, can't find the width or the fuselage (part of the wing inside the aircraft, which doesn't count)

Thank you

 

[alw34]

Wikipedia says Vickers-Armstrongs succeeded manufacturer Supermarine Aviation.
If you can 't find the dimensions you need online, and there are many many google results with possibilities,
try e-mailing Vickers. Or an enthusiasts club. Bet they have plans.

https://en.wikipedia.org/wiki/Supermarine

 

[Superhornet]

Wikipedia says Vickers-Armstrongs succeeded manufacturer Supermarine Aviation.
If you can 't find the dimensions you need online, and there are many many google results with possibilities,
try e-mailing Vickers. Or an enthusiasts club. Bet they have plans.

https://en.wikipedia.org/wiki/Supermarine

how about maths? i don't know how to do the maths...

 

[alw34]

i don't know how to do the maths...

If you have ever found the area under a plot line, that's the idea.

In your case you have no zero line, no flat horizontal axis of zero, but rather continuous functions for two curves, the front and back of the wing. You also have a complex curve [limit] for the shape of the wing tip, but I guess the wing along the fuselage is close to straight??...if so, I'd use that as the lower limit for the two edges. That's rather a complicated integration...I'm guessing there are computer programs to do that kind of calculation...aeronautical engineering I guess.

 

[Superhornet]

If you have ever found the area under a plot line, that's the idea.

In your case you have no zero line, no flat horizontal axis of zero, but rather continuous functions for two curves, the front and back of the wing. You also have a complex curve [limit] for the shape of the wing tip, but I guess the wing along the fuselage is close to straight??...if so, I'd use that as the lower limit for the two edges. That's rather a complicated integration...I'm guessing there are computer programs to do that kind of calculation...aeronautical engineering I guess.

thanks, here is the wing: http://forum.keypublishing.com/attachment.php?attachmentid=12700&stc=1&d=1074055111
I think I will just work out the "wing area" (projected area), not the total surface area, because that's even more complicated, which including some 3d shapes and "curvy" bits...

 

[alw34]

I think I will just work out the "wing area" (projected area), not the total surface area, because that's even more complicated, which including some 3d shapes and "curvy" bits...

exactly. Why wings provide lift has been the subject of discussions in theses forums, and if I recall, t6here is a FAQ you might find as well. Is shape of the underside of the wing simply flat...I don't know.

 

[Superhornet]

exactly. Why wings provide lift has been the subject of discussions in theses forums, and if I recall, t6here is a FAQ you might find as well. Is shape of the underside of the wing simply flat...I don't know.

What is t6? the shape is simply flat i guess, depends how precise you want.

 

[SteamKing]

I am planning to measure the wing area of a Spitfire aircraft. I am going to use double integral formula, but firstly I need to derive the continuous function for the region of interest.

That's generally a bad idea. The aircraft's original designers didn't calculate the wing area in this manner, and neither should you.

If all you are interested in is calculating the area of the wing, there are much simpler numerical integration techniques which can accomplish the same result with fewer headaches, especially if you are not familiar with higher math. In particular, Simpson's Rule can be applied to this problem and you can calculate an area in about an hour or so.

http://smalltridesign.com/Trimaran-Articles/design/simpsons-rule.html

The article above is developed for calculating the areas and volumes of boats, which are composed of lots of curves. No one tries to develop all of the formulas for these curves because it's just too tedious. By measuring the offsets at a few key locations, Simpson's Rule still allows one to do plenty of calculations, even though the explicit formula for the shape of the vessel's hull is never known.

 

[Nidum]

http://www.rafmuseum.org.uk/research/default/archive-collection/supermarine-drawings.aspx

I've been to Cosford . Fascinating place - visit well recommended .

 

[Superhornet]

That's generally a bad idea. The aircraft's original designers didn't calculate the wing area in this manner, and neither should you.

If all you are interested in is calculating the area of the wing, there are much simpler numerical integration techniques which can accomplish the same result with fewer headaches, especially if you are not familiar with higher math. In particular, Simpson's Rule can be applied to this problem and you can calculate an area in about an hour or so.

http://smalltridesign.com/Trimaran-Articles/design/simpsons-rule.html

The article above is developed for calculating the areas and volumes of boats, which are composed of lots of curves. No one tries to develop all of the formulas for these curves because it's just too tedious. By measuring the offsets at a few key locations, Simpson's Rule still allows one to do plenty of calculations, even though the explicit formula for the shape of the vessel's hull is never known.

I have to use maths, I am doing a maths project.

 

[SteamKing]

I have to use maths, I am doing a maths project.

Numerical integration uses math. It's all numbers. It just doesn't use double dome math, which would be counter-productive in this case. In case you haven't noticed, the wing of the Spitfire is not a true ellipse anyway.