Torque-radius relation of EUCs

[arvi]

Hi all,

it's been a while since I left college, so I got a bit rusty with my math and physics, but it it seems there's plenty of you who understand the EUC electro stuff very well.

It is clear that there is a relation between torque and wheel size. Small wheel diameters have much better climbing and accelerating abilities than greater once, given they use the same motor. E.g. Gotway ACM2S and MSuper V3S+ - they both have 1500 W motors, but ACM delivers significantly more torque, because it's only 16" in diameter, as opposed to 18" of MSuper. I've been wondering, whether this can be described more precisely with physical formulas.

Torque is function of angular velocity and power (M = P / w). But w is not dependent on radius (altghou linear distance is) and I am having trouble deriving some radius dependency. Is it at all possible? 

Thanks for the insight. 

[meepmeepmayer]

(Absolute value of) torque is force F times lever. The lever is the wheel radius r. Smaller radius means greater force at constant torque (which according to your formula is constant at constant speed and constant power input). The force of the tire on the ground (pushing forward) is what actually makes a strong vs weak wheel (the [maximum] torque is constant).

So M = F r = P/w, so you have F = P/(wr) which means double the radius, half the force.

Via F = m a you see that force and acceleration (which maybe is what you really feel) are proportional (m being the mass of wheel+rider), so you have the same radius dependency for acceleration (and mass dependency as well if you care):

a = P/(wrm)

So just from this the ACM should have 18/16 times the acceleration of the msuper (ignoring the weight difference).

Hope this is right, not sure but the basic formulae are simple enough. Don't know about other influences coming from the actual electrical stuff.

(You can get the same result via P = F v (power is force times speed) and v=wr, so F = P/(wr). Point is, the torque seems to be constant despite how we say high and low torque wheels, it's the force that changes and we should say high and low force wheels.)

[Scatcat]

Well, the radius of the motor should also count for something, or am I wrong? A simple archimedes principle would say that a bigger radius engine creates more torque than a smaller radius engine - all else being equal. So if for example the MS3 and The ACM both had the same motor-power (watts), but the MS3 had an engine with bigger radius, the differences in torque should even out.

Feel free to correct me if I'm wrong.

[meepmeepmayer]

The radius in the formulae is always the distance between center of rotation and the point where the force acts.

Overall, I think the geometry of the engine should not matter at all from this purely mechanical viewpoint, in the end you throw a fixed power and speed into there, and always rotate a tire with a fixed radius. I guess there are electrical differences once we no longer treat the motor as a black box?

And torque sems constant, but force on the ground (or acceleration) is not.

All I'm really doing is using basic formulae so don't ask me too much

[Scatcat]

1 minute ago, meepmeepmayer said:

The radius in the formulae is always the distance between center of rotation and the point where the force acts.

Overall, I think the geometry of the engine should not matter at all from this purely mechanical viewpoint, in the end you throw a fixed power and speed into there, and always rotate a tire with a fixed radius. I guess there are electrical differences once we no longer treat the motor as a black box?

And torque sems constant, but force on the ground (or acceleration) is not.

All I'm really doing is using basic formulae so don't ask me too much

Well, as I said, feel free to correct me if I'm wrong.

That said, listen to my reasoning and judge for yourself:

If the force between the windings on the stator and the magnets on two motors are the same, but the radius of the stator and magnet rails differ. Then the proportions of the distance between the center point and tire where the driving power is applied changes. If the power are applied closer to the rim, the archimedic principle should mean the lever force needed to create movement is less. Try to move a bicycle wheel by pressing on the spokes close to the hub or close to the rim, and the power needed for movement is very different.

Now suppose you have two wheels, both 18" and one with a 14" stator, the other with a 10" stator. Windings and magnets will probably have to compensate for the difference, but let's not go into that yet. Given the same power are feed to both wheels, shouldn't the wheel with the 14" stator have a stronger driving force where the tire meets the ground?

That at least seem intuitive.

[arvi]

@meepmeepmayer - I've come to the same conclusion. So simply put, torque is linearly proportional to radius (not squared, just radius). Treating all other variables as being the same, the torque is inversely proportional on radius. As you wrote it would be (taking diameter is ok for proportinality) 18/16, which is 1.125. It is inverse, so torque of MSuper would be 1/1.125 = 0.88 of ACM. I.e. ACM has 12,5% greater torque. But it seems more in reality, that's why I was wondering, whether I am not missing something.

As for the motor dimensions/radius - @Scatcat - I think you are right, but the assumption was, that motors are completely the same (in fact I think they really are).

[Scatcat]

19 hours ago, arvi said:

@meepmeepmayer - I've come to the same conclusion. So simply put, torque is linearly proportional to radius (not squared, just radius). Treating all other variables as being the same, the torque is inversely proportional on radius. As you wrote it would be (taking diameter is ok for proportinality) 18/16, which is 1.125. It is inverse, so torque of MSuper would be 1/1.125 = 0.88 of ACM. I.e. ACM has 12,5% greater torque. But it seems more in reality, that's why I was wondering, whether I am not missing something.

As for the motor dimensions/radius - @Scatcat - I think you are right, but the assumption was, that motors are completely the same (in fact I think they really are).

Of course they are, the 100W difference on paper in the Monster, may just be slightly different windings or nothing at all.

My condition was the opposite, to have the same wheel-size but different motor radius, then archimedes principle should rule, shouldn't it?

[meepmeepmayer]

Sorry, wanted to answer yesterday, but was too distracted for the hard topics

@arvi I thought the same, the ACM probably has more than 12.5% more acceleration compared to msuper V3.

@Scatcat

I think it really comes down to what is constant. If you actually assume constant torque (same power input and same angular velocity), then the inner stator dimensions can be ignored. No matter the size of the stator, only the torque input and final outer radius of the tire will matter. Your accelerations only depend on the different outer radiuses.

So the question is, do you have constant torque with different radiuses (for tires or stators, both cases) or does a bigger radius somehow reduce torque. Aka do you (assuming a constant power input, which is kind of the prerequisite for comparisons) get the same angular velocity or does it get less with bigger radius in addition to the force loss from the bigger radius.
So there's a constitutive equation missing between w and r, and they may not be automatically independent of each other in real world use. No idea how this equation would look or where it comes from (the details of how the motor is built maybe?).

Anyways, there's a difference between constant force (your idea, press with the same strength at different radiuses) and constant torque (same power and rpm), and the reality will probably be somewhere in between.

TLDR, I have no idea really

[Gimlet]

As the magnets and windings on a euc are set just inside the rim that is the point where the force is applied.

The effect that the radius of the wheel would have on torque to be directly proportional to the distance between the axis and the contact point of the tyre and how close to the outer radius the magnet/coil interface is set.

Some of the bigger wheels have spokes or extensions between the motor and rim which will obviously reduce the available torque through the leverage and gearing this creates.

Still the biggest problem with bigger wheels is physically applying enough forward lean via the relatively small pedals to make use of the torque. It's easier to lean 5°forward on a small wheel with the axle close to your feet and the ground than it is when the axle is half way up your leg.

This is what I found with my Msuper V2 high torque anyway. I ended up attaching two lumps of rubber to the case just in front of my shins so that i could grip on them more easily to push forward.

[arvi]

On 6/12/2017 at 9:58 PM, meepmeepmayer said:

So the question is, do you have constant torque with different radiuses (for tires or stators, both cases) or does a bigger radius somehow reduce torque. Aka do you (assuming a constant power input, which is kind of the prerequisite for comparisons) get the same angular velocity or does it get less with bigger radius in addition to the force loss from the bigger radius.
So there's a constitutive equation missing between w and r, and they may not be automatically independent of each other in real world use. No idea how this equation would look or where it comes from (the details of how the motor is built maybe?).

Anyways, there's a difference between constant force (your idea, press with the same strength at different radiuses) and constant torque (same power and rpm), and the reality will probably be somewhere in between.

Since M= P/w and P is the same, than w will have to be less for bigger wheels, so yes, the bigger radius reduces torque. You get lower angular velocity, but (perhaps) same linear velocity (because linear velocity is dependent on angular velocity and radius). I've never thought that bigger wheel with the same motor power will have same rpms. So it is the first case you're mentioning - constant force.

The idea with possible constant might explain the absolute difference, but I am still missing its value, because all parameters of the equation are (assumed) the same but the radius.

Anyway, it's a pity that EUC manufacturers don't publish torque (maybe even graph with curve for different rpms), like car manufacturers do. Or perhaps it's ambiguos and depends on multiple factors... Or they just don't know (could measure it on brake though, just like cars :-).