Calculating Hinge & Servo Torque - Simple (sorta) How To Maths

[Mid7night]

I recently did a video talking about my "Super Duper Scout 2XL" and how I did some math to estimate/calculate just how big of a servo I really need. I asked if anyone wanted to see the math, and most people said "YES!". So I did a follow-up video on the math, but wanted to also share the spreadsheet I made to do the calculations. I figure here is the best place to share that, so here we are!

Control Surface Torque Maths

EDIT UPDATE, 7-30-25:
I had an engineer-coworker of mine who specializes in Stability & Control take a look at my maths and compare to his AVL methods. He agreed that my "forces & moments" spreadsheet was a good approach if one lacks AVL or CFD access, but he DID find an error!

When I am transforming the deflected-control-surface based on its deflected-angle, I project the area using the Sin of the angle, and then later in the calculation I project the moment-arm of the area, also using Sin. This second projection is where I went wrong; Since I have already projected the area when calculating force using dynamic pressure, I don't need to project the moment-arm. The moment-arm of the dynamic pressure on the surface is simply "1/2 x Mean Chord". I was using "Sin 24° x (1/2 Mean Chord)", which ended up "halving" the arm again when it shouldn't have.

All that's needed to correct the calculation is to change the "Hinge Torque" cell-calculation to reference "Mean Chord" (C14) instead of the "Vertical Component..." (K24). This change results in a new Hinge Torque for the same surface & deflection of ~122 oz-in. This is indeed more than the ~50 oz-in I originally had, but with the servo I used in my example still results in a safety factor of 2 (instead of 5).

I've corrected this and updated the spreadsheet. I will also try to figure out how to post an update or "flag" on my Youtube video.

The spreadsheet is an Excel ".xls", and it is uploaded as a ZIP file for compatibility.

The blue fields are where you put in your own data. Everything else SHOULD be a working calculation. "Servo Arm Torque" is your main output, that's the number you're trying to size in order to pick a servo for the control surface in question.

It's a simple spreadsheet, but feel free to ask questions here and I'll do my best to answer.


Cheers!

 

[Bricks]

I do not see a place for control surface size such as a 3D plane versus trainer.

 

[Merv]

Similar question.
Will a couner balancing control surface change the calculation.

 

[Mid7night]

I do not see a place for control surface size such as a 3D plane versus trainer.

Type of plane doesn’t matter. The dimensions listed as “span”, “root”, “tip” are for the control surface itself, whatever control surface that is.

 

[Mid7night]

Similar question.
Will a couner balancing control surface change the calculation.

If you mean an “aero-balance”, like a Cessna has on its elevators, yes; that will change the calculation because that would offset some of the aero forces which would reduce the hinge moment. It would be the same math, just an additional step: projecting the area based on an angle, but since it’s on the “other side” of the hinge it would subtract from the forces pushing on the majority of the surface. If you look at it as a “sum of moments” equation, the balance torque is in the opposite direction.

A physical mass-balance technically would change the hinge moment slightly, but not in the same way and would not be an effective hinge-moment-reducer. A mass-balance (sized and positioned correctly) helps reduce control surface flutter.

 

[Bricks]

Type of plane doesn’t matter. The dimensions listed as “span”, “root”, “tip” are for the control surface itself, whatever control surface that is.

I was interpreting it as wing dimensions not control surface dimensions.

 

[Mid7night]

First post updated with video

 

[Merv]

Great video, thanks for the explanation.

 

[telnar1236]

Wasn't sure if I should say anything, but I don't want people potentially losing planes because of this. Unfortunately, this calculation doesn't really work. The vast majority of the moment on the control surface comes from the force along the control surface's lift vector. The end result of this is that the calculation is hugely anticonservative and the torque required is about 3x higher than predicted here.
For a section of wing with the control surface described with a free stream on all sides of it as shown below, I get a CFD prediction of 144 oz-in as the hinge torque. It's not a super realistic scenario, but it should more or less demonstrate the effect. And as a plausibility check, this online calculator gives a value of 133 oz-in with 25-degrees deflection which is about as well as I'd expect hand calcs to match for this sort of thing.

RC Airplane Calculator

RC Airplane Calculator. Determine flight characteristics of your plane including wing loading, stall speed. Calculate Propeller pitch and peak speeds.

www.radiocontrolinfo.com

You can pretty clearly see the reasons in the pressure contour below. As well as the high-pressure region above the control surface, you have a low-pressure region below it which drives a ton of force into the control surface (ignore the units, they default to m^2/s^2 - so Pa/air density - for pressure and I couldn't be bothered to convince Paraview to change to Pa).

The low-pressure region is also most concentrated nearer the hinge point since that is the point of highest curvature and also due to flow separation after the sharp turn, so the aerodynamic force is also centered closer to the hinge point.
The good news here is that I can't imagine most planes fly around with the control surfaces fully deflected most of the time, so this works out to be most significant for flaps and probably a lot less important for a lot of other things. For ailerons, you probably don't need more than about 8-degrees of deflection on a scout anyways for most things

 

[TEAJR66]

So.....?????

Tower pro, plastic gear, 9g servos for everything! Got it.

 

[Mid7night]

Thank you all for your feedback, and especially thanks to telnar for challenging my math. I agree something seemed off, but I couldn't reconcile it. I know CFD and AVL will give more precise results, but even flat-plate approximations are just that. However, being off by a factor of 2.5 is not just "a few percent" and warrants investigation.

I had an engineer-coworker of mine who specializes in Stability & Control take a look at my maths and compare to his AVL methods. He agreed that my "forces & moments" spreadsheet was a good approach if one lacks AVL or CFD access, but he DID find an error!

When I am transforming the deflected-control-surface based on its deflected-angle, I project the area using the Sin of the angle, and then later in the calculation I project the moment-arm of the area, also using Sin. This second projection is where I went wrong; Since I have already projected the area when calculating force using dynamic pressure, I don't need to project the moment-arm. The moment-arm of the dynamic pressure on the surface is simply "1/2 x Mean Chord". I was using "Sin 24° x (1/2 Mean Chord)", which ended up "halving" the arm again when it shouldn't have.

All that's needed to correct the calculation is to change the "Hinge Torque" cell-calculation to reference "Mean Chord" (C14) instead of the "Vertical Component..." (K24). This change results in a new Hinge Torque for the same surface & deflection of ~122 oz-in. This is indeed more than the ~50 oz-in I originally had, but with the servo I used in my example still results in a safety factor of 2 (instead of 5).

I've corrected this and updated the spreadsheet in the first post. I will also try to figure out how to post an update or "flag" on my Youtube video.

Learn somethin' new every day! Count it a bad day when you don't!

 

[Mid7night]

Here is also my corrected whiteboard for reference, note the underlined corrections:

 

[Piotrsko]

Waaay too much math for me, but thanks. I would think that huge deflection is wasted on anything but a slow light goofball airplane for aerobatics or demos. Would be twitchy as all sin, too

 

[telnar1236]

Thank you all for your feedback, and especially thanks to telnar for challenging my math. I agree something seemed off, but I couldn't reconcile it. I know CFD and AVL will give more precise results, but even flat-plate approximations are just that. However, being off by a factor of 2.5 is not just "a few percent" and warrants investigation.

I had an engineer-coworker of mine who specializes in Stability & Control take a look at my maths and compare to his AVL methods. He agreed that my "forces & moments" spreadsheet was a good approach if one lacks AVL or CFD access, but he DID find an error!

When I am transforming the deflected-control-surface based on its deflected-angle, I project the area using the Sin of the angle, and then later in the calculation I project the moment-arm of the area, also using Sin. This second projection is where I went wrong; Since I have already projected the area when calculating force using dynamic pressure, I don't need to project the moment-arm. The moment-arm of the dynamic pressure on the surface is simply "1/2 x Mean Chord". I was using "Sin 24° x (1/2 Mean Chord)", which ended up "halving" the arm again when it shouldn't have.

All that's needed to correct the calculation is to change the "Hinge Torque" cell-calculation to reference "Mean Chord" (C14) instead of the "Vertical Component..." (K24). This change results in a new Hinge Torque for the same surface & deflection of ~122 oz-in. This is indeed more than the ~50 oz-in I originally had, but with the servo I used in my example still results in a safety factor of 2 (instead of 5).

I've corrected this and updated the spreadsheet in the first post. I will also try to figure out how to post an update or "flag" on my Youtube video.

Learn somethin' new every day! Count it a bad day when you don't!

View attachment 252508

That seems like a much more reasonable number. I'll probably keep a copy of this around since it's way quicker than CFD. This should be a really good tool.

As best I can tell there are a bunch of assumptions that all roughly cancel each other out here now, but since the number is a good match and everything should be pretty much linear I don't see any reason this shouldn't work for any reasonable control surface geometry and deflection.