One thing I'm wondering about: do opposing gears always need to have the same form for the teeth? Also, what are the constraints that lead to the form of the teeth? I can imagine that one constraint is that the teeth must "roll" onto each other. But are there more constraints?
The "rounded" design allows constant contact to be maintained between two gears. If they were of the simpler design with flat faces on the teeth, they would simply wear to this shape anyway, but in the process they would also end up with slack between them. A helical twist is also usually employed which improves the transition between adjacent teeth, and consequently reduces noise. Ever hear a transmission with a loud whine in reverse gear? IIRC, it's because they didn't use the helical twist trick on the reverse gear. I have no idea why they wouldn't though. If we're lucky, an ME will be along in a bit and explain all of this for us much better.
Yep. Straight cut gears are cheaper to make. Since people don't do as much reversing as going forward, it's probably a good compromise on cost vs comfort.
Typically most real world gears vary in teeth shape. I mean among the gear pairs. Some are more pointy and some more round. What we see here is probably optimum teeth form.
I have done some calculations about them. But the manual I used was so vague about everything that I can't honestly say why and how for certain. My current understanding is that first you select gear ratio, then you get somewhat good nominal distance for the axels, and then you adjust teeth shape to accommodate all that.
Opposing gears don't have the same form unless they have the same number of teeth. This isn't obvious when you look at the first gear pair, but look at the pair doing the 10:1 reduction. The teeth of the small gear are undercut quite a bit while the teeth on the big gear aren't undercut at all.
The only real constraint is that the teeth must roll into one another as you concluded.
Take a look at the "Ikona" non-involute gear form. The contact forces form a curve instead of a line, with zero backlash, and multiple tooth contacts before/after top dead center. Amazing:
There are other tooth forms that also satisfy the fundamental law of gearing (constant ratio of angular velocity). E.g. cycloid gearing used typically in mechanical clocks. The involute is popular because its easy to manufacture. It can be auto-generated on a gear hobbing machine- probably less important now with CNC machines, but very important in a pre-digital machine era. The involute is also less sensitive to gear center to center spacing.
