Why Gears Must Always Slide Against Each Other, and How To Design A Gear For Any Shape
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- čas přidán 29. 04. 2024
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How do you design the perfect gear to partner with a given shape? It's tempting to think the way to do it is to treat both gears as if they're rolling on each other without slipping, but it turns out most gears by their very nature must slip as they spin. Why is that?
Playlist of Weird Wheel videos: • The Wonderful World of...
=Chapters=
0:00 - Wheels are not gears!
2:03 - What's wrong with wheels?
5:32 - Ground News ad
7:21 - How to design actual gears
12:07 - Envelopes
18:50 - Parametrizing an orbiting gear
22:04 - Computing the envelope
25:22 - Example gear pairs
29:05 - Resolving road-wheel clipping
30:39 - Outro
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This video was generously supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor, Mfriend, Carlos Herrado, James Spear
If you want to support the channel, you can become a patron at
/ morphocular
Thanks for your support!
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CREDITS
The music tracks used in this video are (in order of first appearance): Rubix Cube, Checkmate, Ascending, Orient, Falling Snow
The track "Rubix Cube" comes courtesy of Audionautix.com
The animation of the moving point of contact between two gears comes from Claudio Rocchini. Original source: commons.wikimedia.org/wiki/Fi...
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The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:
github.com/morpho-matters/mor...
Stay informed and get the full picture on every story by subscribing through the link ground.news/morphocular to get 40% off unlimited access with the Vantage subscription which is only $5/month.
Thanks. Even without knowing or having all the math skills, I still learned much.
very good video sir, but can you plz try to make video related to calculus and infinities , also matrix and why determinant as area moreover why cross product can be calculated as determinant, just what is linear algebra
Interesting video! I was wondering if you can create a gear pair for a fractal shape such as a Koch snowflake or the coastline of a country?
Make more vids for this
❤❤❤
As a mechanical engineer, I feel qualified enough to say this an amazing way to look at gear design. Definitely a different perspective than Ive seen, but I enjoy seeing it from someone with more of a math than engineering background
Obviously there are major things that this video doesn't take into account, but would this algorithm work at all for real-life gears, not caring about inefficiencies or wear?
@@nikkiofthevalley I will be printing gears tomorrow to find out lol
Just replying to stay updated
Interesting. Replying to stay updated, too.
This video is amazing, no qualifications needed.
It now makes a lot of sense why gearboxes are almost always lubricated - they need to slide past each other in order to work, even though they don't look like they're sliding!
That's also where a significant amount of driveline losses come from then! Lot'sa heat!
This is mostly wrong.
Gear shouldn't slide past each other. They would never last if that was the case.
@@electromummyfied1538did you watch the video lol
@@electromummyfied1538 mathematically wrong?
25:20 Great examples, but I kinda wish we saw them animated as actual gears too, in addition to the rolling versions
Didn't expect the envelope can be solved for a closed shape. That's so cool.
The long awaited sequel, I loved the road one.
18:50 - 25:20 Hmmmm I’m sensing a hidden connection to Fourier series and their epicycles when it comes to the construction of smooth gears. Seeing the formulas for the gears and then the algebraic construction of the gamma function parameterization with t in terms of s had had those ideas flowing through my head, Stellar work really sir.
You might be onto something… Epicycloids and hypocycloids can perfectly roll inside each other.
Cycloid gear already exist. They’re instrumental to clock making and are some of the few gears with zero sliding motion/friction. They must be spaced with extreme accuracy though, otherwise they go wonky. They most importantly can work without any lubrication, which is why watches and clocks can last so long.
The two parameter locus of motion (I.e. what you see in the thumbnail) for the generating gear is a field of epitrochoids for external spur gears and a field of hypotrochoids for internal spur gears.
So yes, the traced motion of each point on the generating gear is represented by the addition of two rotating vectors with some angular velocity ratio. You could call it a finite Fourier series if you wish.
More details are available in my larger comment on this video (a comment + larger one broken in two as replies to myself).
I love this approach. Not a lot of new work on gear shapes in the last century, but modern 3D printing makes it easier than ever to play around with fun and nonstandard gear shapes. If you’re researching this, “conjugate action” is the technical term for gears moving at constant angular velocities. Also if anyone wants to know why involute gears are the global standard, it’s because of one more requirement which is constant pressure angle which also reduces vibrations.
Also sliding action is often desirable for real world gears. The gears in your car transmission for usually kept in an oil bath and have hydrodynamic contact with each other so that the gear teeth never actually touch, they slide on a microscopic layer of oil. If you look closely, the spot on the teeth that typically sees the most wear is actually the one spot where the sliding velocity hits zero because that’s where they make metal on metal contact.
As a mechanical engineering student who has an interest in knowing how mechanics equations are derived from first principles, this is a satisfying and informative video. Very awesome.
Despite gears being the posterchild of mechanical engineering and one of the first machines most kids are introduced to, they are absolutely one of the worst things to actually deal with in terms of designing (at least in terms of undergrad classes) There are an insane number of parameters you have to take into account and it quickly goes into a rabbit-hole of tables and equations. (At least if you want to design a set of gears that will last)
Yeah it absolutely sucks lol. My 3lb battlebot uses 3D printed gears in the drive train and they took forever to get running right. Making them herringbone was even harder.
@@DigitalJediyeah, gears kinda suck to make. I tried making herringbone gears for a small kinetic sculpture with a sla 3d printer, and I tried so many times before giving up because I was unable to make the gears work and have the right spacing to fit inside the gearbox I was using. I eventually gave up, and got rid of the nonfunctional 3d printed gears and the rest of the 3d printed parts. I probably still have the motor I was trying to use somewhere, but the rest of the stuff is gone
How have computers helped this situation?
One engineering solution to maintaining the same radial speed for meshed gears is to see the gear as 3-dimensional, and change the teeth from having their peak parallel to the gear's axis to being skewed, so when the gear is meshed with a similar (actually, mirror image) gear, the point of contact slides up or down in the direction of the gears' axes, but at a constant radius for both gears.
You're talking about helical gears, right? I kind of assumed that they were just normal gears twisted about the axis of rotation, and that if you untwisted them they'd work just like straight cut gears. I'm not sure if what you're saying means that assumption isn't true or not. Also, I thought the helical twist was mainly for noise and wear considerations.
Yeah, I'm in the same boat. I don't think helical gears suddenly are a whole different beast, but instead just twisted regular gears.
@@quinnobi42 While watching the video I thought of helical gears. It seems to me that those allow the point of contact to remain at the same radius from each axis.
Sweeping out negative space is essentially the way that some types of gear cutting machines work. You have a tool which is shaped like a gear with cutting teeth, and you rotate it together with a gear blank in the same way that two meshing gears would move. All the parts of the blank that aren't part of the matching gear shape get cut away.
"Babe, wake up, new Morphocular video just dropped"
Said no-one ever ;) Still I did find it very funny comment.
FreeSCAD library in 3...2...
The dot/cross product "trick" is because the complex numbers are the 2D Clifford algebra.
a•b + a∧b is the geometric product of vectors, but complex numbers are rotors not vectors
so this doesn't really explain it well
@@2fifty533 If you were to translate the common usage of complex numbers into geometric algebra terms, effectively what's going on is that all vectors are arbitrarily left-multiplied by e_x, which makes them into rotors.
e_x * v = e_x * (v_x * e_x + v_y * e_y) = v_x + v_y * e_xy = v_x + v_y * i
Complex conjugation corresponds to right-multiplication by e_x instead,
v_x - v_y * i = v_x - v_y * e_xy = v_x + v_y * e_yx = (v_x * e_x + v_y * e_y) * e_x = v * e_x
So his formula,
z^* * w
Effectively results in a geometric product,
= v1 * e_x * e_x * v2 = v1 * v2
It's just that that in common usage complex numbers are used to represent both rotors and vectors, the rotors are naturally identified with complex numbers, but the representation of vectors is a little bit strange when you translate it back into geometric algebra.
This kind of content really scratches that curiosity based itch in my brain and I'm all for it
Another important consideration for real-world gears is mass-manufacturability and interchangeability. This is the reason that the involute gear shape is so dominant: Unlike other shapes for the gear teeth, there the precise gear shape depends only on the pressure angle (the angle that the line of contact makes with a line perpendicular to the line connecting the centers of the two gears), the number of teeth, and the pitch/module (respectively, the number of teeth per unit diameter, and the diameter divided by the number of teeth), but **not** on the details of the meshing gear (though obviously pitch/module and pressure angle have to be equal between two meshing gears). This, and the fact that almost all gears use the same pressure angle (20 degrees) and manufacturing tolerances means that only a small set of standardised gear cutters are required to cut all gears of a given module, no matter how many teeth they have or which tooth number gears they will mesh with.
I'm so glad this video came! The variable angular velocity was something that I had noticed in the previous videos and was bothering me, so seeing more of an in-depth exploration of that and the difference between the wheel pairs and the gear pairs is very satisfying! I've loved this whole series!
Jerkiness isn't always something you want to avoid. Look at Mathesian gears, for example, they convert a constant rotational speed into individual steps. It's useful in some cases.
Wow. I never thought about that but in retrospect it seems so obvious bc gears are either lubricated with lubricants, or made of inherently slick material like Teflon or nylon etc.
Wheels are generally maximally grippy
Nice to see another video on the series, i loved the series and am glad to see it return
I'm confused, how do you have a comment that is "2 hours ago" on this video that uploaded less than "2 hours ago?"
Weird, i uploaded it 40 minutes after the video went online
Exactly. Pressure angle is one of the two measures to know how much a gear should slip or "backlash" backwards.
It's a beautiful day when both Sebastian Lague and Morphocular release videos relating to Beziers ❤
That's what I was thinking lol - I spoiled myself by watching the font rendering video first and being reminded of beziers being a lerp'd point on a lerp'd line segment
Turns out I solved the envelope problem to draw very accurate involute gears for my own need recently. Being the caveman I am, I did it much less elegantly, brute-forcing it with algebra and questionable calculus. Your approach is so much more elegant
It was a doubt I had since long time ago and you solved it very nicely. Great video!
Oh wow, thank you! I've noticed the jerky motion in the last couple videos, and wondered what it would take to deal with it. And now you made a response to exactly that question, awesome!
I saw that I wasn't subscribed, despite thoroughly enjoying your videos. I made sure to remedy that mistake as soon as I discovered it.
I'm a computer scientist/software engineer. These videos are like candy to me. Thank you so much for covering these fascinating topics in an accessible manner!
Wow, such a great balance of show and science. Good graphics, just deep enough math, very good approach, humble person.
I've been wishing for this video since Pt3, and never expected my wish to be granted!
you've nailed teaching
The only part of the math I understood was the comparison to the check for extrema in calculus, but it was still a nice video and I do now know what envelopes are and that complex numbers are good for calculating something with rotation. And it was very interesting to see the various partner gears that different gear shapes produced.
Submit this to Summer of Math Exposition!
Fantastic video
so brilliant!! I'm astonished!
27:43 Incidentally, this internally meshed gear seems to be how the Wankel rotary engine is designed with a circular triangle inside forming an epitrochoid that the inside gear not only spin around, but also run the internal combustion cycle to run the engine.
Envelopes are like, my favorite thing, I particularly like the envelope I discovered independently of a line segment of constant length, with the endpoints bound to the x and y axes: the astroid, with equation x^(2/3) + y^(2/3) = 1, and somehow a length of exactly 6.
I love math but something about the music in these videos and your voice is soothing and makes me so sleepy sometimes. I’ll doze off until halfway through the video and then I have to go back several chapters 😅
We've all been waiting for the next episode, very fun to learn that way :)
26:17 this reminded me of the mathologer video about modulo times tables.
I bet that a gear that is just a line would pair with a cardioid gear.
Finally, my favorite wheel math content creator uploaded!
Excellent video. Thank you very much.
I loved watching the series prior to this video. Cool to see a new vid on it!
Yessssssss
Finally a new morphocular vid
Brilliant as usual
The one issue is that there are a whole category of gears with minimal to 0 sliding motion that do exist all around us. Cycloid all gears have for a long time been part of clock and watchmaking. Their contact allows them to have zero sliding friction as the gears themselves must have minimal friction and be never lubricated in order to prevent dirt build up. Other forms of cycloidal gears can be found in roots blowers and such. Having played which watch parts as a child and assembling a couple watches from parts, almost any sliding friction in watch wheels (gears) causes the rapid wearing out of gears that should never wear. This causes friction to increase rather exponentially until the watch spring can’t power the watch anymore, and in that case, every gear would need to be recut and at best, the plate that holds the jewel bearings be drilled again or tossed out.
But I believe those gears don't maintain a constant angular velocity.
I love how you played the algorithm and annoyingly me while im studying for my topology and fluid mechanics exams this week.
In an advanced calculus book, I saw a derivation of an integral equation which will give you the curve for the tooth of a partner gear, given any (reasonable) curve for the tooth of the first gear, under the explicit assumption that they roll on each other with no slipping
Thanks for making it clear that gears have to slide.
Especially around cycloidal gear teeth, there is a widespread misconception that the gear teeth are rolling against each other.
Yeah, learning there’s not just incidental/thermodynamically demanded energy loss from friction, but that sliding is literally necessary for smooth motion was an eye-opener.
Do you know what's most impressive to me? When someone shows me how to use basic tools and put them to real life use, in a very out of the box. I try to do this often, but it's very hard to come accross things like these!! How do you come across such things, and then also put it so beautifully in a video??
This is amazing.
YEEES A NEW EPISODE OF WEIRD WHEELS SERIES
20:00 - after seeing triangles and hexagons I believe it's the constant rate of change of R along the edges.
Since they're straight it helps; also in the limit with infinite sides it becomes a circle so more sides should make them more alike;
Awesome ⚙️s video. Thanks. Advertized it on X-platform.
I remember suggesting the clipping thing! not sure if you came up with it on your own before me, or my email was what gave u the idea, but either way im happy to see it
I saw the painted gear part and had to thumbs up and give a comment. That is so cool!
I really enjoyed this very mathematical take on the concept of gear engineering, very interesting, informative and fun. Also damn, that offset axle oval gear looks so interesting! I wonder if making it that much larger would produce more of the indents that it produced on a smaller scale, as currently it only has two.
As we are taught in undergrad mechanical engineering: theoretically, most gears have involute profile which perfectly roll over each other. But the speed ratio varies since the point of contact moves radially. I don't know why you did not mention this basic stuff. Clocks use cycloidal gears which often have constant speed ratios but have sliding and more strength which make more sound due to sliding.
i'll admit it, i wasn't expecting the parametric equations, the partial derivatives and specially the complex numbers
I was able to get Desmos's graphing calculator to make the envelopes and I think your analysis of what goes wrong with the ellipse is correct. As the distance between the axles decreases eventually the inner envelope starts to self intersect, which in this case indicates that there are moments where the source gear is no longer in contact with the partner gear assuming you shave off the areas created by the self-intersection. Interestingly, as you continue decreasing the distance the inner and outer envelopes meet and then each become discontinuous, forming two new curves - I think this is when the output is an error for you.
I plan on doing the same for a rack and pinion using a given pinion (and maybe vice versa, though that might be harder).
Thanks, you gave me an idea of a breakthrough in one of my math-heavy projects, I will spend countless hours researching and it will all be your fault. Sincere thanks.
23:39 i think some of the expressions might become simpler or at least more intuitive if you go back to vector representation somewhere here. in particular, Re[f’(s)/|f’(s)|*f(s)] is just the projection of f(s) onto f’(s).
-in other words, it’s the radial component of the derivative-
you might also be able to eliminate the cos-1, since we immediately take the cosine of it afterwards, but maybe not, since we’re multiplying it with things in the meantime
Finding the perfect gear partner is like finding the perfect dance partner - they have to mesh well together!
Great video! I think it would be cool to see the last animations with both gears at fixed points to see what they would look like in real life.
The well anticipated sequel finally comes.
Best animations I've seen, if some4 will come out, you can easily win
That "let's shift gears" joke made my day lol
Thanks doc
Great work and great content
Thank you!
15:44 This part is really clever!
I think one of the reasons for the commen gear shape is its compatibility with multiple different sized partners.
The one thing missing from this video are animations of the weird gears turning in place.
Love that you are educational. Hate that your explanations are not conducive to dyslexic individuals. Too many definitive words (representing variables) not represented by any visuals, therefore nothing to hold onto within my mind before you prove your algorithm.
Hey morpho! I think it would be easier to say that, if velocity vectors of changing s and t are parallel [17:11], then del gamma/del s = (lambda) * del gamma/del t
I solved the example envelopes as well as the general equation using the lambda parameter and it doesn’t involve the “unusual” albeit beautiful step of pulling out f’(s) from the Re{.} part (which you did in the complete derivation).
Both the conditions are essentially the same but i thought i would share this. Great video btw!
A true popularization masterclass! Thank you
The serie remind me slightly of the news of a team that invented an algorithm to create a 3D shape that would follows any predetermined path (trajectoïd)! Maybe an idea for a futur video? :)
watching this while playing Epidemic Playstation (1995) BGM in another tab was the best thing ever
@24:00 I'm astonished that the solution stays closed form when I imagine all of the different types of gears in my head in particular, square and triangular teeth. To no surprise, as you developed your solution your mathematics are starting to look more like the equations used for cams and lobes. At the end of the day, all mechanisms are going to be an inclined plane, lever arm, wedge, pulley or some combination.
Hum actually, your own animation of the gear show that the surface of engagement roll one on another since the point of contact move for both surface at the same time. The only sliding there is in a gear set is in practical application since you need a tiny clearance gap and bad teeth number engagment can create a jumping phenomenon.
Interesting domain for that solution! In the parens we have dot(normalized tangent, radius vector), so all in all this means "gear radius projected on tangent to contact point is no greater part of R than w'/(w+w')". It's sort of a lever rule, but for angular speeds, and reflects the common design that the gears' average radii are in ratio with their number of teeth (in that case you can make all the teeth the same).
This incredible! I'm curious if there's a way to solve for f(s) such that, we could find a function whose gear partner envelope is the original function, probably with some angular offset. I know a circle is a trivial solution to this, but, I wonder if there's a whole family of functions.
Reminds me of the `Moving sofa problem'
With the internally meshing gears, is it possible to stack multiple gears to create a sort of rotary engine? My understanding is that you could give the shape of a single rotor and create the housing and then another internal gear inside the rotor for the crankshaft. Rotary engines commonly use a gear ratio of 2:3 between the spinning rotor and the crankshaft, but i wonder if there are any other ratios that would work
Is there some sort of gear equivalent to an ideal road? Like, a road which maintains constant angular velocity on a gear-wheel and constant forward velocity relative to the road.
In your example with triangle wheel at the end you mentioned it would not be smooth because it's not touching, but it is -- the touching point instantaneously swaps to the tip in all points at which it is unsupported. Doesn't this provide stability since the triangle can't leave the trough it's currently in?
yay another video
If this isn't already known in the literature, I feel like this might be publishable. Some engineers would find this useful. You might consider emailing some engineering professor who would know and offering to coauthor the paper with them.
i didn't quite get it why this solves the switch problem, you are using another type of switch? also kudos for the project, nicely done!
I've seen your 3D Euler's formula video too and both in that video and in this you use linear algebra and complex numbers to manipulate rotations and transformations. I thought that maybe you should consider looking into a topic called geometric algebra, I promise you wont regret it. For example when I saw the equation at 22:52 I laughed knowing how natural of an interpretation it has in geometric algebra. It generalises a lot of things. So may I urge you to maybe do your next video about lets say a rotational topic using geometric algebra perhaps?
I am yet to be a mechanical engineer, and ai find this very cool, I think this can be used in improving rotary engine design if they didn't already use such a technique for doing so.
You seemed to me to imply that you can arbitrarily define both rotational velocities as well as the axle distance separately while maintaining a fixed shape & scale for the arbitrary gear. To my understanding the shape & scale with a given rotation ratio should require a specific axle separation and the shape & scale with a given axle separation should require a specific rotation ratio. Something something conservation of torque & energy
To clarify I'm more curious than anything
i thought so too, it didn't seem like you could specify the angular velocity of the partner gear as a free parameter, it seemed dependent on your axle separation and source gear shape and velocity
Mathematician: "Mathematically Impossible"
Engineer: Oh Yeah? What if I decide to put wheels where they contact?
I feel so smart for catching that the condition was jacobian determinant 0 during that pause.
I'm imagining an iterative process where we start with a gear and assign it a number of "teeth", select some other number of teeth to construct the partner gear with an appropriate ratio of angular velocities (there seems to be some flexibility in selecting R, which might yield an interesting constraint to explore), construct the partner gear, and then repeat with another number of teeth (again there's flexibility here, making for another interesting tweakable attribute on the iterative process), etc.
It seems obvious that for suitably chosen R, the collection of circles would make something of a fixed point for a dynamical system constructed around such an iterative process; is it attractive? What's its basin of attraction? Are there other attractive fixed points? Do any of them closely resemble gear profiles currently in widespread use? What about repulsive fixed points?
I was thinking of a weird way to do this. Create a line, where the height of any point on the line matches the length of a line from the center of the gear to the edge of the gear, rotate until you return to the starting position.
Once you do this you have a line the length of the border of the original gear.
Then you start a new object. Start drawing a line that is as far from the center as the height of this first line. Rotate and keep drawing at that height value. Once you return to the start you will have drawn the second gear. Just keep restarting at the beginning of your height line any time you reach the end.
I wonder if i deacribed that well enough...
Oh I guess you are doing this with smart pants math stuff.
may i suggest looking into geometric algebra? it gives a very clean and simple intuition for the complex conjugate formula for 2d cross and dot products around 23:02
What happens if you place a spring between the axles of the rolling wheels, so that their distance need not be constant? Then can you get non-jerky rotation without slipping?
4:33 I’m thinking about those weird solids of constant width that aren’t spheres but still roll perfectly smoothly. Would a 2D analog of that work here in addition to circles?
Hey @Morphocular, can you see my long comment from yesterday? It is full of gearing theory literature references to answer some of the questions you posed! EDIT: I split it in two in reply to this comment and it shows up (CZcams hides the full response if left whole). Hope you can check it out!
It starts as follows because it originally didn't post due to having links to gearing theory literature:
"On further review - this comment was originally broken/silent-deleted because I had links to Litvin gearing theory references, which are now removed. Original comment (gearing theory-heavy for @Morphocular in particular):"
I even mentioned an obscure gearing theory paper that you've replicated here 😉. I hope you can see it, I accidentally did quite a deep dive lol.
One thing I intended to mention: you'd probably enjoy learning and visualizing the Euler-Savary equations as they relate to gearing and kinematics applications.
I'll try it here split in two: "On further review - this comment was originally broken/silent-deleted because I had links to Litvin gearing theory references, which are now removed. Original comment (gearing theory-heavy for @Morphocular in particular):
Very impressive! Instant subscribe (instant "created this channel just to comment", actually - you got me that good).
However, I'd like to note - gearing theory is a very old discipline that very much exists and developed closed-form analytical formulas for envelopes ~100 years ago (there's a long history - some specific cases long before, more general theory a bit later in the early-mid 20th century). See a modern text: Gear Geometry and Applied Theory (Litvin). Below, I have a bunch of comments related to your video and all the problems you identified (which appear in gearing theory). I hope you enjoy the discussion!
A much older text by Litvin is available from the NASA NTRS site: Theory of Gearing - Faydor Litvin, 19900010277. See chapter 4, and section 4.4-4.5 in particular regarding content in this video. One can use these equations with any parametrically defined generating tooth profile to generate a conjugate envelope (mating gear). This one is free, so I refer to it more below.
Gearing theory goes far, far beyond 2D external spur gear shapes (though I see you snuck in a planetary setup - hypotrochoidal generation). Manufacturing gears with rack cutters / hobs / using gearing theory to make gears, crossed-axis, skew-axis, etc. - 3D profiles are a mess! What you did at the end with the "road" is basically generating a rack profile.
No numerical computation is needed because F(s) is some continuously differentiable function, i.e. at least C^1. Numerical enveloping is needed in many practical circumstances. The only thing needed for analytical enveloping is a parametric generating curve, center locations (assuming circular centrodes), the angular velocity ratio, and the ability to calculate a normal vector on the generating curve (i.e. C^1). See 4.5.8. in Litvin (2nd link, Theory of Gearing)
The self-intersection you observe is called undercutting. This occurs when the radius of curvature changes sign by crossing zero. It is also when the relative velocity of the point of meshing on the generating profile (wrt to the stationary profile) and the contact point velocity on the generated profile are equal. Essentially, it is a stationary point of a parametric curve, meaning that it can go in any direction next and makes an infinitely sharp point. See section 6.5 (Litvin, Theory of Gearing, i.e. page 120).
If you want to visualize it, maybe this works: if you are the envelope, the contact point of the generating curve is in motion, and the generating curve is always slipping along that contact point (if not, it is at the pitch point and can make a singularity, i.e. hypocycloids). If you're in the generating curve frame, the contact point moves along the existing curve, i.e. the "rolling" velocity or contact point velocity (or "transfer velocity"). The contact point velocity is always in the direction of the slipping (general theorem of planar gearing, 4.4.16, Litvin Theory of Gearing, page 82), so if the magnitudes are opposite, then the contact point velocity for the stationary envelope/generated profile will be zero for the given point in the motion of the generating curve. I'm not sure if that made it harder or easier to think about!
Gearing is a union of the geometry of space curves and surfaces with kinematics, so as you've mentioned it's all about relative motion and reference frames. Give yourself some credit; the vast majority of mechanical engineers know absolutely nothing about gearing theory and also lack skills of actual mathematical analysis."
Part 2 of original comment: "You can use equations of meshing and generating profiles to generate contact lines as well - the generated gear frame is not special, there is also the stationary frame. There's a great deal of theory regarding undercutting and how to avoid it. If you derive an explicit formula for curvature K (or the radius of curvature 1/K), you can see how generating parameters impact it. Literature is somewhat case-specific. For instance, cycloidal gearing and classical involute gearing have typical methods of analysis specific to their geometries, usage, etc.
Regarding curvature, if you perform a normal vector offset of your generating curve, it is the same as rolling a circle on it - this offsets the radius of curvature by the magnitude of the offset. You can predict precisely when this will cause undercutting.
Also, the reason that they all have flower shapes is simple - every single profile being generated this way is by a circle rolling on another circle. By definition these are epitrochoidal curves of various forms, and the locus of motion (i.e. the two-parameter generation space) is a field of epitrochoidal curves traced by each point in the generating curve. An epitrochoid is just two vectors added together, rotating in the same direction with some angular velocity ratio that is constant. In the internal gearing case you briefly showed, it is the same, but one vector rotates the opposite direction, and it is a field of hypotrochoids.
Worth noting that trochoids generally have self-intersecting "looping" behavior when viewed as a circle rolling on another (with the point being on, inside, or outside the radius of the circle that is rolling), where the point on the rolling circle has a larger radius than the rolling circle itself. Further related, something you touched on briefly - spur gears all behave like circles rolling on each other. These are generally called "centrodes", as they represent a point of equal velocity for the gears when they mesh (an instantaneous center of rotation where relative velocity is 0). When you decrease the center distance between gears as you mentioned (getting odd results), you effectively increase the radius of the points on your field of trochoidal rolling circles. So they start to have looping motion. Looping motion does not precisely correspond to impossibility of meshing, but basically it can present a problem and cause self-intersection.
Another tidbit - you can easily calculate sliding velocities at all points using the pitch point (instantaneous center) and meshing equation in the stationary frame (contact line equation). The envelope is simply the equation of meshing in the 2nd gear reference frame, as I mentioned above.
Final bonus/addendum - upon re-watching, I had deja-vu looking at your final equation for "t". This is because you actually exactly replicated the primary result of a favorite paper of mine: Gert Bar - Explicit Calculation Methods for Conjugate Profiles (Semantic Scholar has a free text link). See page 4 (listed as page 204). Not coincidentally, this paper was written by a mathematician, outside of the typical gearing theory research world (gearing theory is a very mathematical branch of mechanical science and engineering); you seem to have that in common regarding this very nice complex representation of the problem. It also has some very unique discussion on contact lines, i.e. the stationary frame representation of the meshing equation (your equation for t).
Anyways, it was fun to see your take on the mathematics and to see you explore your way into some classic gearing problems! I hope some of the above is of interest or use to you. I DID NOT plan on writing this much lmao, but it was fun seeing you work through this and I couldn't resist sharing since you seem like the rare type of nerd who will actually look at my references and love it (I mean that as the highest compliment). I genuinely hope I said something that sparks your curiosity a bit further."
I'd add that the Gert Bar result that you've also come by is almost certainly an older result by decades - a lot of gearing literature is not in English, and the complex algebra approach is quite nice and straightforward.
The interlocking wheel are gears - a gear is a spinning device using mechanical interlocks to transmit power. There is no requirement on slipping or continuity and there are gears that are specifically designed to give non-uniform rotation even to the point of not rotating at all for large parts (geneva drive).
I think allowing the axle away from the centre of the gear allows non sliding gears -- such as in an external cycloidal gear set. This means at least one of the gears must be on a crank.
If we flatten gear shape (same way as from circular coordinate system) and calculate R-"shapefunction" will it give us flatten form of shape we need? Or there is a problems with neighboring collisions or revolutions speed??
Sorry to disappoint but you CAN achieve constant speed using circle involutes WITHOUT SLIPPING. The gears shown in min 5.08 are made from circle involute which actually roll without slipping this is because the force applied is normal to the surface at every point of contact (except at the end where "clipping" may occur, that where envelopes come in handy). A quick search on using the generation principle may help clarify any doubts.
Wow this so bizarre I was also working on trying to find the partner gear of gears of arbitrary shapes and you posted a video on this subject the same week. You solution using complex numbers is definitely more elegant than mine which involved using the epitrochoid curves in cartesian coordinates, but I think the results would be the same.
Now the question I was trying to answer, and very similar to the one you answered for partner wheels : what is the _family_ of gears that are their own partner gears ? I'm going to try to find it myself, but I think the challenge could please you.