The aerodynamic theory of the lifting line was first solved by Multhopp^{1}. This solution uses a spanwise distribution of the control points accordingly a sinusfunction and is applicable to unswept wings. For the bellshaped lift distribution a direct solution of the lifting line formula can be derived and is described here. The integral for total lift can be written as (Truckenbrodt^{2}):
For a symetrical lift distribution this formula can be transformed:
If the spanwise control points are aranged along a "centralangle" similar to the Multhopp procedure, the following terms can be written:
This leads to the final integration and its solution:
The total required geometrical twist is, accordingly to Truckenbrodt, the sum of effective and induced angle of attack . Angles are calculated in [rad.]:
The local effective angle of attack can be found with the term:
This term is now used in Equation (Equ. 0001 b):
The solution for the induced angle of attack is more complex. Truckenbrodt gives the induced angle of attack for unswept wings:
Using the following terms, we can transform this integrartion in the formulation as given in Multhopp:
This formula now has to be solved for the bell shaped lift distribution:
For this the term is split up in several parts, which can be integrated seperately. The calculation is described in the Appendix A1 . The final solution is:
This gives the required calculation method for the geometrical twist , which can be transformed, using equation (Equ. 0003):
