What Gear Shape Meshes With a Square?

[@morphocular]

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[@prbmax]

Thanks. Even without knowing or having all the math skills, I still learned much.

[@lovishnahar1807]

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

[@iteragami5078]

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?

[@haipingcao2212_.]

Make more vids for this

[@haipingcao2212_.]

❤❤❤

[@guymcproblems7972]

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

[@nikkiofthevalley]

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?

[@guymcproblems7972]

@@nikkiofthevalley I will be printing gears tomorrow to find out lol

[@dantebroggi3734]

Interesting. Replying to stay updated, too.

[@samueldeandrade8535]

This video is amazing, no qualifications needed.

[@purewaterruler]

I am also replying for the notification

[@johnchessant3012]

25:20 Great examples, but I kinda wish we saw them animated as actual gears too, in addition to the rolling versions

I was thinking the same ! Still a great video ! Thanks.

[@leftaroundabout]

Ah, but that's easy to fix on your side: since the rolling has a constant angular velocity, you just need to stand in a hamster wheel rotating at the same angular velocity while watching the video, so it'll cancel out and you're just seeing the meshing gears as if they were rotating about fixed axles. ...what, you say I've spent too much time in a maths departement? No way...

[@jaredwonnacott9732]

Came to the comments to say that exact thing!

[@aguyontheinternet8436]

​@@leftaroundabout It would need to be more complicated than that. Typically in a hamster wheel, you don't move or change your angle at all, the wheel does. You'd need to set up a way for the hamster wheel to rotate your phone.

[@leftaroundabout]

@@aguyontheinternet8436 from your inability to sense satire, may I surmise you've spent even more time amongst mathematicians than I have? If you _stand_ in a Hamster wheel (perhaps best with hands and feet tied to the bars), and the wheel rotates, then yes you rotate with it. I didn't say it would be comfortable! ...though, still better than watching KZbin videos _on a phone,_ that's just ridiculous...

[@Codexionyx101]

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!

[@damiansmith5294]

That's also where a significant amount of driveline losses come from then! Lot'sa heat!

[@electromummyfied1538]

This is mostly wrong.

[@electromummyfied1538]

Gear shouldn't slide past each other. They would never last if that was the case.

[@Empika]

​@@electromummyfied1538did you watch the video lol

[@cewla3348]

@@electromummyfied1538 mathematically wrong?

[@Rudmin]

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.

[@trill_box]

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.

[@xenontesla122]

You might be onto something… Epicycloids and hypocycloids can perfectly roll inside each other.

[@1471SirFrederickBanbury]

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.

[@random-stuff814]

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).

[@yqisq6966]

Didn't expect the envelope can be solved for a closed shape. That's so cool.

[@pat9353]

19:30 my favourite trope of math lectures is the “be not afraid” portion. It happens so often and it’s never not funny

[@Green24152]

I feel like the Ellipse-related goofs in the previous videos fit that theme more than just adding √-1 to the number system. But alright.

[@two_squared]

The long awaited sequel, I loved the road one.

[@sebas31415]

(sequel)^2, there's 4 entries in this series

[@eliyahzayin5469]

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)

[@DigitalJedi]

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.

[@dragonmasterlangeweg7625]

​@@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

[@rodschmidt8952]

How have computers helped this situation?

[@pa0lo0_]

@@rodschmidt8952matlab

[@DanteTorn]

Even when playing with physics based creative videogames, gears absolutely SUCK to design. Some of my most frustrating machines to get to behave properly in LittleBigPlanet were anything where two parts were interacting this way.

[@zuthalsoraniz6764]

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.

[@GyroCannon]

The fundamental forces of the real world, the four in physics and then economic viability lol

[@Chrischi3TutorialLPs]

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.

[@Nicoder6884]

I couldn’t find anything on Google for “Mathesian gears”, but what you’re describing sounds like an intermittent mechanism

[@Chrischi3TutorialLPs]

@@Nicoder6884 Appearantly it's called a geneva drive in english, in German we call it Maltesergetriebe because one of the gears looks like a maltesian cross

[@buubaku]

Would clock hands be an example of this?

[@akiamegami]

Isn't it this one that is used in film projectors?

[@Foivos_Apollon]

​​@@buubaku Not really. Clocks rely on either a timing wheel or a pendulum to create the stepping action. In a timing wheel system, there's a specially weighted wheel that swings back and forth to keep the time, that's powered by the watch's main spring. There's a piece that looks kind of like a fork if the middle tines were missing, and that ticks between 2 positions every time either the pendulum reaches the apex (highest point) of each swing, or when the timing wheel changes direction. Both of these forces are enough to make that little fork change what side it's leaning towards.

[@razdahooman]

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!

[@davidlindstrom4383]

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.

[@quinnobi42]

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.

[@TaserFish-qn2xy]

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.

[@richardmellish2371]

@@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.

[@kinexkid]

This kind of content really scratches that curiosity based itch in my brain and I'm all for it

[@thenimbo2]

The dot/cross product "trick" is because the complex numbers are the 2D Clifford algebra.

[@2fifty533]

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

[@georgechiporikov2297]

@@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.

[@SomeMrMindism]

@@2fifty533 Yeah but (xe1+ye2)e1 = x + ye1e2, so they are very naturally isomorphic

[@tulpjeeen]

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.

[@recursiveslacker7730]

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.

[@tobiaspeelen4395]

Nice to see another video on the series, i loved the series and am glad to see it return

[@vanouper]

I'm confused, how do you have a comment that is "2 hours ago" on this video that uploaded less than "2 hours ago?"

[@tobiaspeelen4395]

Weird, i uploaded it 40 minutes after the video went online

[@epremier20050]

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.

[@גיאדרי]

Exactly. Pressure angle is one of the two measures to know how much a gear should slip or "backlash" backwards.

[@ciCCapROSTi]

Wow, such a great balance of show and science. Good graphics, just deep enough math, very good approach, humble person.

[@penguinlordalan]

I work at a factory that produces plastic gears and it's part of my job to ensure that the gears are consistent and up to customer spec. I'm not an engineer though so it's neat to hear some of the math behind what I'm finding. Very good video!

[@featherofajay4667]

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.

[@lerarosalene]

"Babe, wake up, new Morphocular video just dropped"

[@jannegrey]

Said no-one ever ;) Still I did find it very funny comment.

[@MattHudsonAtx]

FreeSCAD library in 3...2...

[@corsaro0071]

Great work and great content

[@morphocular]

Thank you!

[@rodschmidt8952]

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

[@pedroalonso7606]

It was a doubt I had since long time ago and you solved it very nicely. Great video!

[@soranuareane]

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!

[@serkanmuhcu1270]

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.

[@Zakaarus]

Can’t you make a chain of 3+ wheels where the two at the end have a constant angular velocity while the ones in the middle don’t?

[@terdragontra8900]

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.

[@bluerendar2194]

Your intuition about some kind of "self-intersection" of the envelope is on the point for the artifacts. Just like how zero derivative is necessary but not sufficient for a maxima or minima, the envelope condition used is necessary but not sufficient for the type of envelope wanted here. If the curve traces out some kind of interior envelope, that will be caught too, and mess up the result. Additionally, the full failure is probably since not all positions of the gear necessarily have to correspond to being part of the envelope. That is, the gear at the positions for which the formula fails is entirely inside of the envelope, not touching it. I'm also not sure it would handle correctly the cases where multiple points or sections of the gear shape at a position are part of the envelope. In any of those cases however, the real-world implications is that the parameters set up are impossible to construct a normal gear for. Either the force transfer is not in the correct direction to couple the motions, and/or the gears would physically separate and not transfer motion. It may still be useful for things like cam systems, where the motion wanted is to pause (while the gears are not in contact), like in watch escapements or film projector reels, or if the intent for the gearing is to synchronize motion rather than transfer forces.

[@321oob]

I don't really fully understand these videos, but I still love the stuff it goes over. Very nice, thanks morphocular

[@hallucinogender]

Some very niche conspiracy theorist out there is watching just the first part of this video and shouting "Yes! I knew gears aren't real! They're mathematically impossible, even the engineers say so! All advanced machines work with ropes and pulleys!"

[@Will-et3ee]

25:29 Title question. You're welcome

[@tAbes314]

Hi. I am a gear theoretician who studies this topic and has applied it industrially for years. I just wanted to say this video is excellent. The theory of enveloping is one way of determining conjugate gear geometries and is the one I use every day. As far as I know, it was popularized in the US by Dr. Litvin. It can be read about extensively in the book that he and Dr. Fuentes wrote, called "Applied Gear Geometry and Theory". The profile of one gear is first described with two surface parameters. The meshing process is simulated via a coordinate system transformation (complex numbers are not commonly used to my knowledge because everything can be kept Euclidean) that is a function of a generalized parameter of motion. Simulating the mesh for different combinations of these parameters leads to the development of a family of curves/surfaces. The envelope is determined by the fact that the normal vector is orthogonal to the sliding velocity. In the world of gearing, this is known as an equation of mesh. This process can be further leveraged to simulate manufacturing. Gears had to be precisely manufactured long before CNC technology, the process for doing so was to develop analog computers to do this. If you would ever like to learn more about how this is done within the world of mechanical engineering,or get a specific example to expand this topic, feel free to reach out!

[@SumangalSinhaRoy]

There are gear theorists? Wow what kind of problems do you study?

[@Crystallize-s6i]

I have read litvit's book and still have no idea how to implement them in autocad lol. Can I ask what are the most notable equations to draw the gears?

[@SheepUndefined]

I think the screenshake just before, 2:40 or so is literally the first time screenshake has made me nauseated lmao

[@dotXyore]

Weak genes.

[@Lynx86]

It's a beautiful day when both Sebastian Lague and Morphocular release videos relating to Beziers ❤

[@coolreader18]

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

[@abhimanyukar]

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.

[@uploadforgraduation9]

This comment is hilariously incorrect. Like, the opposite of everything you said is true. Involute gears do not perfectly roll over each other, there is a sliding action as the teeth mesh and the point of contact changes. However the contact angle remains the same and the point of contact moves in a straight line, which keeps torque constant during rotation and eliminates torque ripple/variation in speed ratio. You can literally see this in the Wikipedia animation. In contrast cycloidal gears do not have sliding contact points (they roll smoothly over each other because the teeth are alternating segments of epicycloids and hypocycloids) but the contact angle varies throughout the rotation and does not move in a straight line, so you absolutely have torque ripple during rotation.

[@1471SirFrederickBanbury]

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.

[@queueeeee9000]

But I believe those gears don't maintain a constant angular velocity.

[@elliott614]

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

[@DylanPiep]

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.

[@logan5018]

This seems like it could be a really cool process for generating unique spirographs since they dont really have any of the other requirements relating to force transfer that you mentioned. too bad im not knowledgable enough to utilize it

[@BleachWizz]

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;

[@donwald3436]

27:01 Imagine trying to build that thing, immediate jam when the egg gets in the crevice. That's why it doesn't work lol.

[@dillpikkle7550]

Step one. Draw a circle big enough to contain your shape entirely. Step 2. Fill the empty space with something easily bendable. Step 3. Take away your shape. Step 4. Turn the ring inside out. Step five. Make fun of me in the replies when this method enevidably fails, because I'm 13. Step six. Profit.

[@naturecomics]

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).

[@alexandredevert4935]

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

[@aaravs524]

My sheer happiness to see bezier curves on a video about gears

[@asdfghyter]

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

[@YdenPL]

I really, truly suck at academic math, for some reason. But here, with everything presented graphically and with a clear objective, I actually got a decent grasp on the stuff I attempted to learn at uni. Thank you!

[@cherylchui4510]

please make another video it has been 6 months

[@Konstamonsta]

27:20 i have not tried to prove it but from how the shape looks like i'd assume that if you took the negative space and tried to roll the shape around it with the given parameters you'd get points in time where you actually have no contact points for the shape because the backswing moved into those contact points erasing them from the final negative space

[@Mulakulu]

16:15 shouldn't that be "tangent" rather than "parallel"? Parallel requires two straight lines and tangent is at least 1 curved line. Am I wrong?

[@ПендальфСерый-б3ф]

Technically, "parallel to the curve at a point" can be defined as "parallel to the tangent line at a point", so no problem (the same way "perpendicular to the curve" is defined, because we don't have another word like "tangent" to describe precisely the concept of perpendicularity for curves only)

[@Mulakulu]

@@ПендальфСерый-б3ф i guess you could define it like that. Feels a bit unnecessary though, and when we talk about parallel it's usually between two straight lines like you said "parallel to the tangent line", but why say parallel when creating a hypothetical tangent line is the only way to make it make sense, when you could just say that tangent line? It's not communicating precisely. We already have a word for it. It's like saying "it's a rectangle with all equal sides". Just call it a square

[@morphocular]

I actually considered using the word "tangent" instead of "parallel" there, but I wanted to emphasize the fact that both vectors being tangent at the same point implies they are parallel to each other, and so I thought using the same word "parallel" for everything would help with that.

[@Mulakulu]

​@@morphocularthat would make more sense if you compared the two vectors with each other at that point, rather than with the envelope curve, but I see what you mean. It's nitpicking anyway 😂

[@CoasterCam-k3x]

I had no idea this was going to be this technical.

[@ettorem7829]

Amazing video! It could be mentioned that Euler, the great mathematician, first proved that no gears with constant ratio of angular velocities, but for perfect circles, could roll without slipping. He also invented involute gears.

[@clydehawkins7194]

I love how some of them ends up as geneva mechanisms. Also great video. Gonna experiment myself with a custom slicer based on the knowledge you provided me and attempt to 3D print them :)

[@neopalm2050]

26:50 what happens if you use a better cos^-1? cos(ix) = cosh(x), so you can have x such that cos(x) > 1 ( e.g. cos(i) = (e + 1/e)/2 = 1.543... )

[@erikhaag4250]

Is the code for the gear generation on github? I'm trying to recreate the expressions in desmos and I'm not getting the correct contours

[@telotawa]

you should fix it to allow inverse cos of >1 cause that'll give you complex solutions that might still be useful

[@piratepartyftw]

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.

[@TheSilentLightning]

I think what helped me understand this the most was finding that no-slip gears pay a cost for precision: durability. Great strain gets applied to the material of the gear from no-slipage. That is why it is not always desirable to exclusively use no-slip gears. Slip gears are useful in resisting wear on the materials due to the lenience in the motion. Idk. I had trouble understanding and for some reason that's all I was missing to get it. 👍

[@EdbertWeisly]

I watched the whole ad to support you

[@bigyeet5587]

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 😅

[@mandarinduck5485]

Would it still work with overhangs?

[@rickwoods5274]

I *think* the note onscreen at 27:22 is equivalent to what you are saying verbally (that is, that the longer segment moves backward when it's facing toward the center of the partner gear)

[@gcewing]

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.

[@TemPo_ACCOUNCO]

Camus' theorem would give good insight. 27:00 The error can be interpreted as being caused by the gear gets inside-out in some point. It is interesting problem that how much the gear's projection can gouge out its pair-gear without causing errors or slipping through.

[@nicerknifes9465]

30:00 would it also work to mirror the overlapping envelop at the road and cut that mirrored version out of the triangle wheel?

[@UltimateAzumanger]

I feel like there should be a video about designing a road in such a way a wheel of a given shape can roll with a constant velocity and vice versa. That would really complete the series.

[@ciana42]

At 26:42 what's happening is that the inner gear fails to be a stard domain, so the boundary can't be expressed as a function of the angle

[@varunahlawat169]

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??

[@nicreven]

Leaving another comment partially because of the algorithm to commend you for having this video be fully subtitled. Hell yeah.

[@kaisalmon1646]

I've been wishing for this video since Pt3, and never expected my wish to be granted!

[@kyleblake7522]

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

[@jacksonrocks4259]

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?

[@oafkad]

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...

[@oafkad]

Oh I guess you are doing this with smart pants math stuff.

[@fs2000]

Great video! The last detail you shared about the envelope is referred to as undercut in gear design and also needed there.

[@NateDulcimer]

18:50 just an aside: would that graph be described as a45 degree rotated parabola? My mind went in the direction of a graph for an equation similar to y=1/x where x is greater than 0. I don’t remember the specifics or how to test in this instance, but would that actually satisfy the requirements for a parabola?

[@doctorkiwano]

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?

[@bartleyhoran9833]

I love how you played the algorithm and annoyingly me while im studying for my topology and fluid mechanics exams this week.

[@uploadforgraduation9]

Great video. This is definitely a different approach than I've seen when talking about gears; much more of a mathematician's approach than an engineer's approach (I don't mean that pejoratively! I mean it as a compliment.) The more common treatment I've heard usually starts with the properties of involutes and then describes why making the teeth involutes makes a 'good' gear.

[@woodenpotato7550]

i'll admit it, i wasn't expecting the parametric equations, the partial derivatives and specially the complex numbers

[@iveeuwu]

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

[@BleachWizz]

30:38 - the contact point jumps to the envelope then comes back. so it's kinda janky, BUT IT IS touching constantly

[@whisper3856]

Finally, my favorite wheel math content creator uploaded!

[@ABaumstumpf]

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).

[@julia_qwerty]

YEEES A NEW EPISODE OF WEIRD WHEELS SERIES

[@Raye938]

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?

[@zygzag]

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

[@JaredBrewerAerospace]

@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.

[@Navaxco]

2:15 I am now craving beef jerky

[@redyau_]

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!

[@nathanielscreativecollecti6392]

You should do a short video on why a circular wheel is most often used. I suspect because as the size of the wheel relative to the path traveled increases the less a circle moves up on down on any path.

[@MH_Binky]

I'd love to see some 1:1 gear partners

[@Gregregorovich]

30:31 - Clearly, instead of calculating the envelope to remove from the road, the other envelope where the corners of the triangle intersect must be calculated instead and removed from the triangle! ...Which would necessitate then re-calculating the road to account for those rounded edges. ...And then it wouldn't be a triangle wheel anymore, only most of one...

[@zoryion3867]

We've all been waiting for the next episode, very fun to learn that way :)

[@ykyjohn]

the gear itself does not slide, but the teeth of the gear uses slide properties to make it work smooth and maintain constant rotation. So to say it slide or not is more about the perspective in what are you looking at. If one gear is rotating clock wise the other has to rotate in the opposite direction, in this sense they don't ever slide. How the gear really works in a micro perspective they do use sliding properties on the teeth to make the rotations constant. It is all about perspective. It is weird as to make a gear not slide it must slide? I guess so, both things are right, depending on what are you referring to.

[@edwardblair4096]

One other condition you completely failed to discuss is that the relative angular moments of the two component gears need to be rationally related. In other words, when the "bigger gear" rotates through a whole circle, the smaller gear needs to rotate through an integer number of revolutions. You could break this requirement somewhat if you are able to take advantage of internal symetry of one (or more) of the gears such that, for example when the larger gear spins once, the smaller gear spins 5 1/2 times. As the smaller gear makes the second circuit, the contact point between the gears will be exactly halfway around the smaller gear than it was during the first revolution. The allowed fractional components are determined based on the symetry present in the given shape. Maybe you discussed this issue in one of your "rolling" videos?

[@electromummyfied1538]

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.