# How to calculate quadrotor propeller torque due to aerodynamic drag

1. Jul 12, 2012

### marcin123

I'd like to know the right formula to calculate torque of a quadrotor propeller (propeller pitch is constant) due to aerodynamic drag in terms of angular velocity ω. For the sake of my simulation I'm assuming APC 10x4,7 propellers. I'd like to refer to UIUC Propeller Data Site, so please assume only the set of data given there is available while answering. I know really little of aerodynamics, I'm planning to design a control system in Simulink later on, but would like to derive a non-linear mathematical model first.

Please note I need to use data from UIUC Propeller Data Site for APC 10x4,7, which means the propeller below. I'm guessing all is there to derive a formula τ(ω), I just don't understand how to do it.

http://img19.imageshack.us/img19/7838/apcsf10x47side.jpg [Broken]
http://img853.imageshack.us/img853/7319/apcsf10x47front.jpg [Broken]
http://img850.imageshack.us/img850/8931/apcsf10x47geom.png [Broken]

Static data from UIUC:
RPM CT CP
2377 0.1039 0.0431
2676 0.1058 0.0437
2947 0.1059 0.0437
3234 0.1083 0.0444
3494 0.1096 0.0450
3762 0.1121 0.0460
4029 0.1136 0.0466
4319 0.1155 0.0474
4590 0.1177 0.0484
4880 0.1199 0.0494
5147 0.1213 0.0500
5417 0.1228 0.0508
5715 0.1239 0.0513
5960 0.1253 0.0520
6226 0.1261 0.0524
6528 0.1274 0.0531

Last edited by a moderator: May 6, 2017
2. Jul 14, 2012

### viscousflow

Let me clarify a few things.

• You are simulating a quadrotor.
You want to develop a non-linear Thrust model. Torque isn't particularly useful in your simulation unless you need to know if your motor can spin the prop without overheating.
The three variables you should be concerned with are Thrust-RPM-Throttle (or current)

If you want to take a stab at this model you find a motor that will operate at some range of RPMs given on the table and use the formula

$C_T = \frac{T}{\rho n^2 D^4}$

Where T is thrust, D is the blade diameter, n is RPS (convert to RPM)

Then you can multiply by some efficiency factor $\eta$ ~0.95 or so to account for heat losses etc...

You will probably end up with a lookup table and not a formula

3. Jul 15, 2012

### marcin123

Thank you visciousflow. I had some problems with Imperial units, but were able to overcome those and am able to model both thrust and torque in terms of angular velocity now (inserted below, thrust in [g] on the left, torque in [Nm] on the right axis, angular vel. in RPM) with the UIUC data.

http://img594.imageshack.us/img594/1466/apcsf10x47characteristi.png [Broken]

My goal is to simulate all:
• dynamic response of a motor together with prop
• thrust
• torque

I'll ask a follow-up question when I get there, if I encounter problems. For now the case is solved - thanks all who read and gave a short thought.

Last edited by a moderator: May 6, 2017
4. Oct 8, 2012

### razorhash

Quad rotors are usually designed so that the props are counter-rotating and thus, the reaction torques are self compensated (for most maneuvers). For maneuvers that they are not self-compensated, they are negligible. I just completed my thesis and worked with the APC 12x9 prop. Trust me, it would not be worth adding to the simulation. Although, the model would look more complete if you do add them.

5. Oct 31, 2013

### poeli

Can you link me to your thesis razorhash? I would love to read it. It's an interesting topic, i'm also researching something similar.

6. Nov 1, 2013

### 3msinclair

I hope this isn't dragging up an old topic, but I'm currently looking for the governing equations for propellers and the relevant thrust/torque generated. I'm having a nightmare of a time trying to work out the relationship between RPM, pitch, diameter and generated torque/thrust. Could someone help point me in the right direction, please? I've tried asking on various RC plane forums, but their general answer is "50-75W per pound of craft is the minimum". While that's a helpful answer for a hobbyist, I'm working on a MEng project for uni so would really like something more substantial to justify my decisions.