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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.

Mohammed Siddique
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    There are too many variables to catch this in one equation, there is some background in this answer and in this one – Koyovis Dec 08 '21 at 01:04
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    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. – Koyovis Dec 24 '21 at 05:40

2 Answers2

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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

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: dF 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
  • г (gamma) - angle between the lift component and the resultant aerodynamic force dR (*note - should be dD/DL)
  • 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)
MishaP
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    Can you explain what the terms in the equation mean and how to apply this formula? – Bianfable Dec 08 '21 at 12:17
  • The equation boils down to Q = Q. – Koyovis Dec 08 '21 at 13:41
  • @Koyovis that's like saying F=m*a boils down to F=F... well yeah, but that's an oversimplification. – MishaP Dec 20 '21 at 13:42
  • What is the relation between lift force L and torque Q? – Koyovis Dec 21 '21 at 06:19
  • @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. – MishaP Dec 21 '21 at 09:33
  • Sorry. I missunderstood your last question. I will update the answer a bit later today – MishaP Dec 21 '21 at 09:38
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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.

Robert DiGiovanni
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  • 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? – Mohammed Siddique Mar 13 '21 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. – Robert DiGiovanni Mar 13 '21 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. – Mohammed Siddique Mar 13 '21 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). – Robert DiGiovanni Mar 13 '21 at 20:43
  • How is lift/drag ratio depends on torque!? Can you explain me a little further. @RobertDiGiovanni – Mohammed Siddique Mar 18 '21 at 08:23
  • The lift/drag ratio of the rotor. It is a spinning wing. – Robert DiGiovanni Mar 18 '21 at 11:34
  • Yeah! I don't know, how to explain you but "how much torque and speed does my propeller need to generate to get the expected lift/drag... " Can you please state the answer for this question in mathematical form!?... Or do you know any website to learn this specific concept!... Please give me the formula for this or if there's nothing like it... explain me why or give me a website name which will explain me this question! @RobertDiGiovanni – Mohammed Siddique Mar 20 '21 at 07:20
  • @Mohammed Siddique Well, appreciate that we can keep working on it. The formula for lift and drag are similar: all have same rho (air density), same Area, same velocity (squared) Lift = area x rho x V$^2$ x coefficient of lift (or drag of prop). Coefficient is airfoil type at a given Angle of Attack (and so is Drag). Once you have those values, you know how much rotor torque there will be (as with a helicopter). Then you know how much counter torque is required by the tail rotor. Try Airfoil Tools for starters on the net. – Robert DiGiovanni Mar 20 '21 at 11:54
  • treat your rotor as a rotating airfoil. The average drag as expressed over its length (remember V$^2$!) will be its torque. Because of V$^2$, the rotor angular momentum "sweet spot" is not halfway between each end (but can be controlled by varying AOA along the blade). Look at a windmill. – Robert DiGiovanni Mar 20 '21 at 12:02
  • that is center of drag load can be controlled by varying AOA along blade – Robert DiGiovanni Mar 20 '21 at 12:09
  • Thank you so much! @RobertDiGiovanni. Now, i can proceed with my work full-fledged. – Mohammed Siddique Mar 21 '21 at 03:22