


Simplex airfoil formula?
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 standlone 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... 






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/showairfoile.phtml?id=3 A18 airfoil design made with three logarithmic spiral arcs. 





Also:
http://www.mathsoft.com/mathresource...,,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." 





Quote:
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 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://wwwgroups.dcs.stand.ac.uk/~...uiangular.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) 


Last edited by JMP_blackfoot; May 11, 2005 at 11:00 AM.





"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 





Rdeis wrote :
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Quote:
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? 






Quote:
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. 






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 % 

Last edited by JMP_blackfoot; May 12, 2005 at 03:03 AM.




Rdeis,
Any luck with the spreadsheet ? 





Got it..
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 airfoilshaped 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 010. For odd camber numbers, it plots from 510 and only gives the back half of the airfoil? 





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. 





rdeis,
I reworked your spreadsheet a little, so as to spread the values better along the Xaxis :  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. 

Last edited by JMP_blackfoot; May 17, 2005 at 12:03 PM.
