I don't think there's a single rule, there's too much variance - not just the material you're making the gears with (steel? Nylon?), but also the accuracy of your manufacturing process (CNC milling? FDM?), the flex of the axles the gears are mounted on, and so on.
There definitely are industry standards and rules of thumb, but you probably want to start from there to figure out what works for you. The most important parameters are: tooth pitch, which depends to a large extent on the scale of the item you're making and the accuracy of your tooling; tooth count, which is a function of your application; tooth height, where you can start with the standard formula; pressure angle, where you can default to some value and then see if you're getting shapes that look good and are easy to make; profile shift, which is worth bothering with mostly if you're making gears with tooth counts below 20 or so; and backlash / bottom clearance, which again depends mostly on process accuracy.
Thanks for the response! I am an economist, so don’t know anything about engineering. But I imagined that gears being such an old technology that there would have been attempts to add in as many of these sorts of variables into a single function, and then to optimize over all of them simultaneously, and this would have been done long ago. Of course it is as you say: there are a lot of such variables.
Nice writeup! A small correction about your comment on profile shift: it doesn't necessarily need to be complementary. A shifted spur gear can mate with a non-shifted one just fine if the axial distance is adjusted.
If you just add the amount of shift to the axial distance you usually get close-enough, but it adds a little bit of backlash. Getting the backlash-free axial distance with profile shift is a little more involved but possible...
Is there a single formula that optimizes over all of a material's variables: friction, shear, expansion, etcetera?
I don't think there's a single rule, there's too much variance - not just the material you're making the gears with (steel? Nylon?), but also the accuracy of your manufacturing process (CNC milling? FDM?), the flex of the axles the gears are mounted on, and so on.
There definitely are industry standards and rules of thumb, but you probably want to start from there to figure out what works for you. The most important parameters are: tooth pitch, which depends to a large extent on the scale of the item you're making and the accuracy of your tooling; tooth count, which is a function of your application; tooth height, where you can start with the standard formula; pressure angle, where you can default to some value and then see if you're getting shapes that look good and are easy to make; profile shift, which is worth bothering with mostly if you're making gears with tooth counts below 20 or so; and backlash / bottom clearance, which again depends mostly on process accuracy.
Thanks for the response! I am an economist, so don’t know anything about engineering. But I imagined that gears being such an old technology that there would have been attempts to add in as many of these sorts of variables into a single function, and then to optimize over all of them simultaneously, and this would have been done long ago. Of course it is as you say: there are a lot of such variables.
Nice writeup! A small correction about your comment on profile shift: it doesn't necessarily need to be complementary. A shifted spur gear can mate with a non-shifted one just fine if the axial distance is adjusted.
If you just add the amount of shift to the axial distance you usually get close-enough, but it adds a little bit of backlash. Getting the backlash-free axial distance with profile shift is a little more involved but possible...
I made a youtube series about gears some time ago : https://youtu.be/nrsCoQN6V4M?si=-VcLBF4xa73DpKsl
(I know the music is terrible, I take responsibility for my mistakes. The 2nd and 3d vids are not that bad.)
And also working on a gears project in python for those interested:
https://github.com/GarryBGoode/gggears
As for the "always ends in tears" part, look no further than image search: https://www.google.com/search?&q=gear+icon&udm=2