# Thread: Calculate Motor and Wheel Size

1. ## Re: Calculate Motor and Wheel Size

As I suspected a combination of logic, transcription and arithmetic errors made my examples give bad results. The mathematics are completely sound.

The most basic error is in post #6. In fact a solution only exists if the ratio of r0 to r1 is greater than 4, not less than as I said there. This makes more logical sense, since a large ratio results from stronger and faster motors. I also said that the example bot wouldn't work because the ratio was 6.5. But I had used the wrong numbers there. The ratio was really 3.8, so the bot still won't meet the goals but because the ratio is less than 4, not because it's greater.

Finally, because of stupid arithmetic errors I thought that the equation for r could sometimes result in values smaller than r0, which should be impossible. I was wrong -- that can't happen. So it all appears consistent, as it should be.

As penance, let's compute how heavy koas116's bot can be given the motors he has. Again we set the ideal ratio between r0 and r1 to 4 and solve for mass.
r1 = 4 * r0

t / f = 4 * r0

g * m * sin(a) = t / (4 * r0)

m = t / (4 * r0 * g * sin(a))
This works out to 1.6 Kg (3.5 lbs), with wheel radius of 2 * r0, or 0.16m (about 12" diameter). The wheels have to be big to generate the desired speed given the low RPM, but then the bot can't be too heavy since large wheels mean low pushing force.

2. ## Re: Calculate Motor and Wheel Size

I thought I would take a crack at the question of acceleration -- how long does it take a wheeled bot to get to its maximum speed? For the idealized bot it turns out the answer is: never. On a level surface (and in the absence of friction) a bot's max speed s given by the unloaded RPM of the motor p0 and wheel radius r.
s = p0 * r / 10
At full speed there is zero torque on the wheels -- the full power of the motor is being converted to RPM. At a standing start, however, the motors are stalled and giving their full torque. As the bot speeds up the torque drops, and the rate of acceleration approaches zero as the bot reaches its full speed. No matter how fast the bot goes, there's never quite enough torque to get it to full speed.

Mathematically this is represented by a linear differential equation. Torque is related to velocity such that at 0 we have full torque and at speed we have zero. I'm using uppercase T for torque here since t is time. (I was using tau in my notes, but I don't have a tau key.)
T(v) = T0 * (1 - v / s)
Acceleration (the derivative of velocity) is given by force divided by mass. Force is torque divided by radius. Multiplying it out we get:
v' = T0 * (1 - v / s) / (r * m)
I'll spare you the tedious details, but by doing some old-fashioned calculus we can determine a closed form solution for velocity as a function of time:
v(t) = s - s * exp ( -t * T0 / (s * r * m))
So at time zero the speed is zero. It rises rapidly after that but then levels off and becomes asymptotic to the max speed. Of course for practical purposes there's a point where it's going about as fast as it can. The bot will be at 95&#37; when the exponent reaches -3:
-t * T0 / (s * r * m) = -3
Solving for t:
t = 3 * r * m * s / T0
Alternately if we have a desired time we can rearrange to solve for torque:
T0 = 3 * r * m * s / t

3. ## Re: Calculate Motor and Wheel Size

Let me put these last formulas to practical use. Suppose you're working on your heavyweight combat robot to end all combat robots. It weighs 220 lbs, has 10-inch tires, and you want it to have a max speed of 35 mph and get up to speed in 2 seconds from a standing start. What motors do you need?
m = 100 Kg
r = 0.15 m
s = 15 m/s
t = 2 s
From the max speed we can compute the RPM:
p0 = 10 * s / r
1000 RPM. From the desired time we can compute the stall torque:
T0 = 3 * r * m * s / t
Required torque is 250 ft-lbs (340 Nm). Better get out your checkbook. In combat of course you want to both speed and strength. If your bot is pushing another bot at half its max speed, then half of the motor power is going to torque. What would that look like? Half speed would be 18 mph. Half torque would be a force of:
f = 0.5 * T0 / r
250 pounds of force (1100 N). That ought to be enough to throw the competition around.

4. ## Re: Calculate Motor and Wheel Size

τ

http://forums.trossenrobotics.com/sh...7612#post27612

Greek Alphabet as cut-pastable 8bit characters...

Thanks for your continued efforts, Meta!

5. ## Re: Calculate Motor and Wheel Size

I've completed an introductory tutorial based on insights from this analysis. Still using T instead of tau though.

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