Appendix A3: Some Comments on the Bell-Shaped Lift Distribution

In many cases the discussion about the bell shaped lift distribution concentrates on the aerodynamic point of view. For the swept flying wing, this distribution has some advantages, but brings some drawbacks with respect to performance.

Unfortunately, one very important point is missed in all this discussion, the mathematics. Accordingly to Reimar HORTEN, the bell shaped lift distribution was already used on the H II, long before Multhopp presented his calvculation method. An exact solution of the (unswept) lifting line was only available for the elliptical lift distribution (, n=1) at this time. For arbitrary wing layout only approximate methods as Schrenk or Lippisch were available. A solution for other functions seem possible, but were not presented at this time. A solution of higher orders show that the lift distribution accordingly the power n=2 is physically not possible. It requires an indefinite twist at the wing tip. Next possibility is the power n=3, and trhis is the bell shaped lift distribution. The required wing twist includes the induced angle of attack which follows a simple function . Twist at the tip is the (negative) same on as at the root.

The reconstruction of this solution was tricky but for a mathematician like R. Horten possible. It enables a fast method to calculate the required basic twist, making the layout independent from a method like Multhiopp´s. Unfortunately we miss the final verification that Reimar Horten has used this simple method. The surviving calculations give no description how the twist was derived. Nevertheless, the twist fits well with the bell shape, at least for some of the airplanes.

Stress calculations then were performed using the well established Lippisch method.

 (C) R. STADLER, 96-01-15

 rev. 01: 98-01-05

rev. 03: 99-01-02