# Relation between lift force and torque produced in propeller used for vertical take-off and landing aircrafts

There are many answers for this question but none of them are specific. I need to find the exact relation between the lift force of a propeller of specific pitch, attack angle , dimensions and material with torque to be produced in a propeller at a specific speed at a certain altitude of a vertical take-off and landing aircraft to hover and fly forward. Can some experts in the field derive this relation.

• There are too many variables to catch this in one equation, there is some background in this answer and in this one Dec 8, 2021 at 1:04
• Plus - some experts in the field have derived the many equations describing this, notably Prouty (Helicopter Performance, Stability & Control) and Leishman ( Principles of Helicopter Aerodynamics.) In the first couple of chapters, you will find that rotor diameter, blade profile, blade twist, blade AoA, forward speed, altitude and a few other parameters are involved. In other words: your question is very broad. Dec 24, 2021 at 5:40

What you are looking for is Blade Element Theory:

The expression for the torque of the whole propeller is therefore

$$Q = \frac{1}{2} \rho V^2 B \int_0^RQ_c \, dr$$

• Q - total torque required by propeller
• p (rho) - density of atmosphere the propeller is acting in
• V - free stream velocity (aka velocity of airplane w.r.t air)
• B - number of blades on propeller
• R - radius of propeller
• Qc - Torque required by a segment of a propeller blade at dr
• dr - small delta of radius

Qc is the force dF (see fig.2) of the section dr that we are calculating for, multiplied by the radius R at which the section is located.

fig.2

As we can see from the figure, dF is the total aerodynamic force (lift + drag) of the section of the propeller blade dr we are calculating for, projected onto the plane of the propeller. df can thus be calculated by the equation: where:

• ф = arcsin(V/(2*pi/n)) is the angle of the travel of blade with respect to air
• V - freestream velocity
• n - rpm
• - angle between the lift component and the resultant aerodynamic force dR (*note - should be dD/DL)
• dL and dD are calculated the same as for a wing profile. (*note - be sure to use correct lift and drag coefficients Cl and Cd at alpha = beta - ф)
• Can you explain what the terms in the equation mean and how to apply this formula? Dec 8, 2021 at 12:17
• The equation boils down to Q = Q. Dec 8, 2021 at 13:41
• @Koyovis that's like saying F=m*a boils down to F=F... well yeah, but that's an oversimplification. Dec 20, 2021 at 13:42
• What is the relation between lift force L and torque Q? Dec 21, 2021 at 6:19
• @Koyovis A propeller is essentially a rotating wing. When it rotates, each section of each blade, produces lift and drag. The vector formed by summing L + D of a section of a blade, and projecting onto the plane (disk) of the propeller, forms a force acting against the rotation of the propeller. The sum of these forces along the length of each blade, multiplied by the radius at which these forces are acting, is the total torque required to turn the propeller. Dec 21, 2021 at 9:33

The relationship of torque produced by the engine and lift force will be the drag of the rotor. Increase in lift force by increasing rpm or AOA will require a proportional increase in power output from the engine. Air density also plays a role.

With a common helicopter, torque is counter balanced by increasing or decreasing torque of the tail rotor.

• Thanks for replying @RobertDiGiovanni, Can you please put that into a formula considering all proportionality constants and what if an electric motor is used to drive the rotor? Mar 13, 2021 at 14:29
• @Mohammed Siddique using an electric motor is a good idea. Starting on a small scale, power input, lift and torque are easily measured. Mar 13, 2021 at 16:04
• Well, I am designing a Motor and, i have calculated the Lift force to be produced by the rotor, i just need to figure out how much torque does my motor should produce to achieve that lift. Can you help me @RobertDiGiovanni. Mar 13, 2021 at 16:22
• Not without knowing the lift/drag ratio of the airfoil you are using for your rotor. Depending on the drone, you may be able to "reverse engineer" from existing drones (at the hobby store). Mar 13, 2021 at 20:43
• How is lift/drag ratio depends on torque!? Can you explain me a little further. @RobertDiGiovanni Mar 18, 2021 at 8:23