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).
The thing is, a lot of that math by hand is probably not long division, it's Greek style compass and straightedge stuff. There's an absolute ton of things you can do just tracing things out with a stick, and at adequate precision at that.
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.
It's cool to learn the term for this, I have used the same principle before with bicycle gearing, so the chain would wear more evenly. Not sure if it makes much difference but it was worth doing anyway.
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.
There’s a hotel I stay in sometimes that has a large decorative spur gear in the lobby. With straight sided teeth. Bugs me beyond all reason. I may have to find different lodging next time I’m in that city.
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.
The fact that it's a rolling contact, if the gears axis are parallel, and the pitch circles meet at one point, means that well made gear sets can run for decades, if properly maintained.
Some of the thousands of gears I've made will still be in service in the year 2100.
Once while visting another shop to buy some used equipment and tooling, I got into a long and fun yarn with the owner,
it was a vast shop and we were there alone
rumaging, and in comes a man, wanting a gear made "for his jeep" ,with the damaged one in hand,"can you do it" Earl listens and says "shure shure, leave it there by the door", we get back to our yarn, and after a bit the guy inerupts with "can you do it",Earl again "says sure sure leave it by the door", back to yarning, and the third time the guy interupts with "can you do it" Earl turns and says "Ive fucked up better jobs than this" and we wander off to rummage somewhere else.
Knowing the little bit I do about gear making, I class the skill to make gears as one of the great underrated skills.
And what almost nobody knows, is just how
complicated and expensive thd entire process becomes, when making critical gear sets for things like aircraft turbine engines, a small gear that you hold in your hand, costs as much as a car, not that anyone is going to let you hold one.
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.
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).
The thing is, a lot of that math by hand is probably not long division, it's Greek style compass and straightedge stuff. There's an absolute ton of things you can do just tracing things out with a stick, and at adequate precision at that.
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.
It's cool to learn the term for this, I have used the same principle before with bicycle gearing, so the chain would wear more evenly. Not sure if it makes much difference but it was worth doing anyway.
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.
This reminds me of the Stoneway Hardware store in Seattle and their bolt logo which pokes fun at this
https://www.stonewayhardware.com/
There’s a hotel I stay in sometimes that has a large decorative spur gear in the lobby. With straight sided teeth. Bugs me beyond all reason. I may have to find different lodging next time I’m in that city.
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.
The fact that it's a rolling contact, if the gears axis are parallel, and the pitch circles meet at one point, means that well made gear sets can run for decades, if properly maintained.
Some of the thousands of gears I've made will still be in service in the year 2100.
Once while visting another shop to buy some used equipment and tooling, I got into a long and fun yarn with the owner, it was a vast shop and we were there alone rumaging, and in comes a man, wanting a gear made "for his jeep" ,with the damaged one in hand,"can you do it" Earl listens and says "shure shure, leave it there by the door", we get back to our yarn, and after a bit the guy inerupts with "can you do it",Earl again "says sure sure leave it by the door", back to yarning, and the third time the guy interupts with "can you do it" Earl turns and says "Ive fucked up better jobs than this" and we wander off to rummage somewhere else. Knowing the little bit I do about gear making, I class the skill to make gears as one of the great underrated skills. And what almost nobody knows, is just how complicated and expensive thd entire process becomes, when making critical gear sets for things like aircraft turbine engines, a small gear that you hold in your hand, costs as much as a car, not that anyone is going to let you hold one.
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.
[0] https://www.youtube.com/watch?v=Wlp8RxFt0ec
The idea of shining polarized light through gears made of Lucite to view the stress lines at the point of tooth contact is genius!
This old training video is great too, it also mentions the idea touched upon in the article that gear teeth are levers:
Spinning Levers - How A Transmission Works (1936) https://www.youtube.com/watch?v=JOLtS4VUcvQ
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.