A 15-minute intro to involute gears
A fun simple gear related concept is "hunting tooth".
https://en.wikipedia.org/wiki/Gear_train#Hunting_and_non-hun...
Basically you don't want any common denominators in teeth count, otherwise the same sets of teeth will engage at some frequency. If there is a small imperfection on a tooth it'll wear out quicker. E.g. a set with 5:14 teeth will theoretically wear better than a set with 5:15 teeth.
Circle involute is one of my favorite curves. It has a particularly nice Cesàro equation: κ = c√s. It also has the unusual property of being its own parallel curve. Thus, if you were to draw a curve using piecewise circle involutes, you'd have an exact mathematical offset.
It'll show up in a blog post soon, once I get back to having enough time to write.
It should be a part of mandatory training for graphic designers that keep drawing these gears that can't work. Lookup for "gears icon", it's appalling.
The simple of it: Gear teeth are made in such a way so that when their faces touch as the gear rotates, they "roll" against echother faces rather than slide.
Strain-wave gears are becoming popular in many areas including astronomy for sky-tracking telescope mounts, they're kind of odd but clever.
https://howtomechatronics.com/how-it-works/what-is-strain-wa...
If you search on YouTube there are a couple of good videos where people have 3D printed their own to demonstrate the principles
The simplest explanations I have found for involute gears is from one of those old 50's training films[0]. Not sure if this is the same one, but the first few minutes give a very intuitive explanation of where the involute shape comes from.
Really lovely piece.
As an aside, for those not familiar with the author's other posts, do take a look. They are lovingly illustrated and explained, and cover a wide variety of topics.
Reading this kind of stuff always makes me admire Victorian engineers even more, who had to do all the math by hand, to say nothing of the machine designs to cut and create these types of gears and mechanical systems.
I suppose they learned a lot from prior tech, like clockmaking, but even there it would seem that fundamental problems would have needed solutions from scratch (eg shear failure probably isnt much of a worry in most clocks but would be catastrophic in a steam engine).