lifting line theory

for a while i have been trying to understand the lifting line theory. this has many benefits over the basic aspect ratio corrections for the calculation of CL and CDi including:

• Non-elliptical planforms
• Twisted sections (AoA varies along span)
• Section changes along span
• Wing tips (probably - see below)
  

it took me a while to work it all out because i didn't want to pay for a textbook and most of the internet references i found would either:

• have no nomenclature
• have errors
• after stating the method equations declare the solution too hard
• give the results for a single (not very useful) case only
• not have a worked example to compare results to.
  
• obviously every website has minor differences in nomenclature and assumptions
  

Yesterday I had an epiphany and remembered what a Fourier series was and the pieces fell into place and allowed me finally understand what was going on. I won't repeat the procedure here but I will give 3 useful links for people who do want to attempt it themselves.

Best description of method
Example to compare your results to
Simple Matlab code

A few notes on solving Fourier terms:

• In the matlab code the line 'a=b\rhs' is matrix maths and means [An] = [b]^-1 . [RHS]
• Look up MMULT and MINVERSE in Excel help, makes solving linear equations sooo easy and makes anyone who has coded row reduction methods in excel feel stupid.
  

I can supply my spreadsheet by email if anyone is interested.

Note on wingtips analysis:

I am not sure whether this is valid or not yet. Probably valid enough for anyone satisfied with a Lifting-Line theory result.

1. Design a wing with wingtips
2. Fold these wingtips down so they are in plane with the wing, adding to its span
3. Twist the wingtips so that they produce zero lift, independent of the wings AoA