Here’s an interesting problem I thought of today:

Given a gearbox with 2 or more stages, in general the first stage of the gearbox tends to have the best bull gear teeth-to-reduction ratio, because the motor pinion is usually much smaller than the pinions for the other stages.

However, if you make the bull gear (downstream mating gear) for the first gear larger by 1 tooth, you vary the pitch diameter, not the area.

The weight of a gear is almost directly proportional to the area, so at what point does it become more weight-efficient to use a second stage in a gearbox, disregarding the weight of shafts, bearings, and plate? What is the optimal ratio between the number of teeth in the bull gears?

The variables in play are the first stage pinion teeth or pitch diameter, and the second stage pinion teeth or pitch diameter. I have a suspicion the optimal tooth relationship between first and second stage bull gears is equal to the ratio between the squares of the pinion pitch diameters, but I’m not sure where to start to work this out. Does anybody have any ideas?