Motor Power Calculations

[DMetalKong]

First, a couple of disclaimers:

1. I know there is the JVN design calculator, but as far as I can tell it doesn't work well trying to mix and match motors in a single drivetrain.
2. I know there was a recent thread similar to this a little while ago. I was hoping this could go in a more general direction than slap x gearbox on y to make it match a CIM.
3. I know this is groupsourcing my math, but I think that it could be helpful for everyone to see the math behind the power calculations, if in fact the math is correct.
4. I am by no means confident in my math, I was hoping some more experienced heads could verify if what I am doing in correct.

With that being said
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Let's say that you want to have a drivetrain with 4 CIM motors and 4 RS-550 motors. The max power of the CIM is 340 W, and the max power of the RS-550 is 254 W. The total max power is

4 * 340 W + 4 * 254 W = 2,376 W

Now, there will be losses transmitting the power from the motors to the floor, let's approximate this at 80%. So the power getting to the floor is

2,376 W * 0.8 = 1900 W = 1406 ft-lb/s

(To get from W to ft-lb/s, multiply by 0.74)

Power = Torque * Angular Velocity, so Angular Velocity = Power / Torque.

Torque = Force * Distance. For 4in diameter (2in radius) wheels, the distance is 0.1667ft (2in). The force is equal to the Frictional force, which I'll say is

Ff = u * Fn = 1.0 * 165lb = 165lb

In the above, u is the coefficient of friction for the wheels on carpet (dependent on the wheels, I chose 1.0 as a typical number), and Fn is the normal force, which in most situations is the weight of the robot.

The torque is then

165lb * 0.1667ft = 27.5ft-lb

This means that the angular velocity of the wheels is

(1406 ft-lb/s) / (27.5 ft-lb) = 51.127 rad/s

Converting to rpm

51.127 rad/sec * rev/(2pi rad) * 60 sec/min = 488 rpm

This means that the CIM, which has a free rpm of 5310, should be geared down 5310/488 = 10.88:1.

The RS-550, which has a free rpm of 19300, should be geared down 19300/488 = 39.55:1.

Also, the RS-550 could be geared down to the free speed of the CIM - 19300/5310 = 3.63:1 - and then the two inputs merged, and the final 10.88:1 reduction applied.