# What is the total lift equation of a helicopter with 3 Flettner rotors?

I'm doing research regarding the possiblity of using Flettner rotors on helicopters.

I would like to derive the total lift equation for a helicopter that consist of 3 Flettner rotors. Given that:

1. The general lift equation is

$$F = \frac{1}{2} \rho V^2 S_{ref} C_L$$

where F is the lift force.

1. The Kutta-Joukowski Lift equation is

$$\frac{F}{L} = \rho V G$$

where F is the lift force and L is the length of the tube.

Is the total lift equation equal to?

$$F = \frac{1}{2} \rho V^2 S_{ref} C_L + 3\left(L \rho V G\right)$$

Interesting technical idea, at least by a theoretical point of view.

To distinguish the spinning of the cilinder around its own axis from its rotation around the hub, let's call the former $$\omega_{spin}$$ and the latter $$\omega_{hub}$$.

The lift per unit length generated by a spinning cylinder is:

$$l = 2 \rho V \pi r^2 \omega_{spin}$$

This has to be summed up (integrated) along the cylinder's span taking into account that in this case $$V$$ changes from root to tip due to the rotation around the hub, so its value is $$V= \omega_{hub}\cdot x$$ with $$x$$ going from 0 to L.

The sum (integration) along the span gives therefore:

Lift per cylinder $$= \int_0^L {2 \rho\omega_{hub} x \pi r^2 \omega_{spin} }\, dx = \rho \pi r^2 \omega_{spin} \omega_{hub}^2 L^2$$

This multiplied by the number of cylinders per rotor gives the total thrust.

That's for the thrust. Regarding the drag, as a first approximation the drag coefficient of a non-spinning cylinder could be used, which is more or less 1. This is a lot: the same coefficient for an airfoil with a thickness equals to the diameter of the cylinder is some 10 times lower. The drag coefficient of 1 has to be multiplied as usual by the dynamic pressure and by the diameter of the cylinder and summed up (integrated) spanwise from 0 to L, like done for the lift.

These simplified aerodynamic results have to be corrected to take into account the several aerodynamic phenomena typical of the rotating wings. And afterward there are all the dynamic, flight dynamic and structural issues to be investigated.