Quote:
Originally Posted by Paul Copioli
I can't find it now but I made a post a long time ago on this forum (maybe 2004 or 2005) titled "Gear ratio doesn't matter." The point of that post was that gear ratio alone doesn't matter, you have to take wheel diameter into consideration, too. In addition, weight has nothing to do with the optimal gear ratio either. Higher mass means that the wheel diameter for a given gear ratio (in this case a gear ratio of 1) must be smaller to generate enough force to counteract the weight (mass * g) and other acceleration of the mass. To simply show the relationship let's use simple F = ma. F in this case is T/Rw (torque divided by wheel radius) and a is g + your desired acceleration at max power. As you decrease your wheel radius you will increase your force which is needed for higher mass.
The bottom line is that the ability to climb the pole at all really only has to do with the motors and mass (in its simplest form). Motors represent the max available power. You just have to find the right combination to lift the mass.
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A more specific relationship than F=ma in terms of rotary acceleration is
Tau = I * Alpha
or
Alpha = Tau / I
Where
Alpha = rotational acceleration
Tau = Torque applied to the rotation
I = Moment of Inertia. Generally that's [mass, in kg] * [radius, in meters]^2
The radius^2 matters in this case. This is really because in almost all cases our sources of force are applied in the forms of torque from a motor. The generic moment of inertia works out because most of the time the mass at the radius far exceeds the mass averaged over the radius (e.g. a wheel weighs much less than the robot it pushes).
Since electric motor torque output is a function of its speed, and speed is a function of acceleration, the overall equation quickly becomes non-linear. So I've only ever put it into Excel to figure out the final numbers. Yet most of the errors I experienced went away when I swapped from trying a direct F=ma calculation to T=Ia (non-linearity induces round off error sometimes, so it's still just an estimate).