[patrickrd]

I haven't designed a gearbox for FIRST (at least in a few years), but if I were to do so: I'd first need to know the the transmitted force is on the gear teeth. I'd then decide on a material (or a couple candidate materials) that are available, reasonably priced, and appropriate for the torque and speeds expected.

Once I have that, I'd look at several different pitches. For each pitch, I'd calculate the minimum required gear thickness to handle the load without tooth deformation. This can be found using the formula:

b = F/(mYs) (mm)
where:
b = tooth width (mm)
m = 25.4/P (P=pitch diameter in mm)
Y = lewis form factor (which is a function of the pitch, which can be looked up in an emperical chart in a machine design book)
s = yield strength of the material i've chosen (Pa)

Note that the formula does not take into account fatigue or misalignment, or other factors that might damage gears.

Finally you add in a safety factor (usually > 2), so you multiply your thickness by at least two. Then you can look at all the gears you want (and know that they are strong enough) and find the one that is cheapest, easiest to integrate into the gearbox, or whatever other criteria you desire.

This is how I chose the gears used on Cornell's RoboCup team. We ended up using plastic gears with 32 english pitch, if my memory is correct.