Simplex airfoil formula?

[rdeis]

Hello, all. I'm looking for a formula that I can use to plot a simplex (logarithmic spiral) airfoil for various cambers in a spreadsheet.

I found a stand-lone plot utility, but I'm a mac user- and I'd rather understand the math a little anyway.

A quick google found lots of references to simplex templates and to logarithmic spirals, but I didn't get anything that related the spiral to camber...

[Ollie]

http://www.modelresearchlabs.com/pricelist.htm
Expanding Logarithmic Spiral Patterns
Makes tapered wings easy, draws any airfoil from 1% to 10% thick and 1" to 20" chord
$5.00

Example:
http://www.nasg.com/afdb/show-airfoil-e.phtml?id=3
A18 airfoil design made with three logarithmic spiral arcs.

[Ollie]

Also:
http://www.mathsoft.com/mathresources/constants/wellknown/article/0,,1941,00.html

"Logarithmic spirals are found throughout nature, for example, the shell of a chambered nautilus, the tusks of an elephant and patterns in sunflowers and pine cones.

The Mathcad file logspira.mcd discusses how we obtained the diagram presented earlier and further topics."

[JMP_blackfoot]

A quick google found lots of references to simplex templates and to logarithmic spirals, but I didn't get anything that related the spiral to camber...
I did not have time to check it thoroughly, but I think camber can be calculated thus :
Camber (% chord) = 100 x Sin(A) x e^(-A/tan(A))
where A is the generator angle of the logarithmic spiral, in radians.
If the angle is in degrees, divide it by 57.3
For A = 6°, I find Camber = 3.83 %, which sounds like a good start ?
There remains to figure A from the Camber :cool:

I started from this formula :
Rho = Chord x e^(Theta/tan(A))
where Rho is the radius, and Theta is the angle of the current radius line.
(Look here for reference : http://www-groups.dcs.st-and.ac.uk/~history/Curves/Equiangular.html)
I would use Theta = -90° as a start point (radius line vertical at the leading edge)
and go to Theta = 0° (radius line horizontal through L.E. and T.E. points).
At each angle step between -90° and 0°, calculate X and Y from Rho and Theta :
X = Rho x Cos(Theta) and Y = Rho x Sin(-Theta)

(The high camber point is where the tangent to the airfoil is horizontal, which occurs when Theta = -A, hence my camber formula above)

[Ollie]

"This little program creates logarithmic spiral airfoils based on the thickness you choose. Output can be sent to printer or plotter."
http://www.smallflyingarts.com/Downloads/Downloads.htm

[JMP_blackfoot]

Rdeis wrote :
I found a stand-lone plot utility, but I'm a mac user- and I'd rather understand the math a little anyway.
Obviously, the download program is the easy way, but it does not help with understanding the math :rolleyes:

[rdeis]

"This little program

That's the one I found- works great on a windows box, but I don't have any windows boxes with print capability. Only my macs are allowed to talk to the printers. (-:

So if I understand you guys correctly, I should be able to use a logarithmic spiral formula to plot (r,theta), and for the correct camber I can solve the eqn "Camber (% chord) = 100 x Sin(A) x e^(-A/tan(A))" for A, then start the spiral plot with theta = A.

Or, perhaps easier, plot the given equation and pick an A off the curve that corresponds to the desired camber.

Right?

[JMP_blackfoot]

So if I understand you guys correctly, I should be able to use a logarithmic spiral formula to plot (r,theta), and for the correct camber I can solve the eqn "Camber (% chord) = 100 x Sin(A) x e^(-A/tan(A))" for A, then start the spiral plot with theta = A.

Or, perhaps easier, plot the given equation and pick an A off the curve that corresponds to the desired camber.

Right?
Yes, that's exactly the idea. Once you've solved the equation for A as a function of Camber, would you let us know ? Thank you in advance.
I think that, given the small angles A involved, it will be possible to use a simpler polynomial approximation for A = f(Camber) and not lose any significant precision. I'll have a look on my side.

[JMP_blackfoot]

Approximation function :
A (degrees) = 1.554 x Camber(%)
is accurate enough.

It is accurate within 1% from 0 to over 10% camber.
For example :
A = 1° -> Camber = 0.6421 %; Approximation = 0.644 %;
Relative error = 0.644/0.6421 - 1 = 0.3%
A = 7.8° -> Camber = 5.0233 %; Approximation = 5.0232 %;
Relative error = 5.0232/5.0233 - 1 = 0 %
A = 16° -> Camber = 10.4078 %; Approximation = 10.3040 %;
Relative error = 10.3040/10.4078 - 1 = - 1 %;

This is good enough for all practical purposes in my opinion.
Relative error is nearly zero (<0.2 %) for cambers between 5~10 %

[JMP_blackfoot]

Rdeis,

Any luck with the spreadsheet ?

[rdeis]

Not yet, got distracted pulling the motor out of my car.. back to you soon.. (-:

[rdeis]

Excell spreadsheet attached, I can't upload .xls, so I renamed the file with a legal extension. You'll have to rename it with a .xls extension to use it, attachements are itendical in all but name in case the upload protocol depends on name.

It produces airfoil-shaped plots, but I haven't carefully scaled one for printing yet. Chord, Camber, and resolution are set from cells in the upper left.

Interesting tidbit: I have it set for a 10 <insert units> chord. For even camber % numbers, it produces a smooth plot from 0-10. For odd camber numbers, it plots from 5-10 and only gives the back half of the airfoil?

[rdeis]

The .doc works, I just downloaded it to a WIntel box, renamed it with a .xls extension, and opened it in excell. Neat.

[rdeis]

Ahah! You can fix the print scaling easily by using hte "print selected plot" feature, measuring the length of the printed axis, then manually setting the X and Y axis scales to go from 0 to the length. After doing that and making two prints I could overlay them and show that a 4" 10% camber plot does indeed match the first half of an 8" 10% camber plot. Fantastic!

Thanks for your help, everyone.

[JMP_blackfoot]

rdeis,

I reworked your spreadsheet a little, so as to spread the values better along the X-axis :
- introduced an X0 column between Rho and X, used for calculating Theta intervals
- Theta is calculated so as to spread the intervals in a more practical manner
- Y is modified for calculation with -Theta
It works with any reasonable value of Camber, even fractional.
Change the .txt extension to .xls after downloading

Hope you like it.

[DICKEYBIRD]

Wow, you guys really know your stuff on math & spreadsheets! I've been looking for a method to do simplex airfoils and whaddyknow, it's right here. I want to do a very light backyard electric with an elliptical wing, sliced ribs and a simplex airfoil.

Just for the halibut, I printed a simplex 7% from the little program above, scanned it as a .bmp into CAD, traced it, scaled it to 1" and converted it into the .cor format for CompuFoil. I attached the files if anybody wants them.