The Perfect Road for a Square Wheel and How to Design It

Рет қаралды 1,309,446

Күн бұрын

How do you design a road that a square wheel will roll smoothly over? And what about other wheel shapes? How do you even approach such a problem?
=Chapters=
0:00 - Intro
1:36 - The Dynamics of Rolling
4:05 - Vertical Alignment Property
7:16 - Stationary Rim Property
8:29 - Describing the Road and Wheel
13:04 - The Road-Wheel Equations
17:02 - The Perfect Road for a Square Wheel
22:40 - Building the Road Visually
25:54 - Wrap Up
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Many of the ideas in this video came from, or were inspired by, "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). If you want a deeper dive (or if you want spoilers for the next video), I encourage you to read it yourself. As far as math papers go, it's fairly easy to read:
web.mst.edu/~lmhall/Personal/...
===============================
CREDITS
► The song at the beginning of this video is "Rubix Cube" and comes from Audionautix.com
► Thumbtack icon comes from Mister Pixel of the Noun Project.
===============================
Want to support future videos? Become a patron at / morphocular
Thank you for your support!
===============================
The animations in this video were mostly made with a homemade Python library called "Morpho".
I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:
github.com/morpho-matters/mor...

Пікірлер: 826

@morphocular 9 ай бұрын

There seem to be a lot of comments questioning the practicality/usefulness of square wheels, particularly whether you can turn side-to-side with them. The short answer is there's likely not much practical use for them and you can't turn side-to-side. To be clear, this video was mainly meant to be an interesting application of math and geometry to a fun problem and was not meant to be practical in the slightest.

@nyxalexandra-io 9 ай бұрын

yea

@bopcity5785 8 ай бұрын

I hope you've seen the Cody dock rolling bridge which now applies this math

@lukcurious 7 ай бұрын

7ýyy ÿt5vg 22:25

@lukcurious 7 ай бұрын

Ÿtyfus y😅ÿyÿy y yt 24:55

@lukcurious 7 ай бұрын

24:58

@kindoflame Жыл бұрын

I was going to mention that a second requirement for a smooth ride is that when the rotational speed of the wheel is constant, then horizontal speed of the axle is also constant. Otherwise, you could have a 'smooth ride' where the car constantly speeds up and stops short even when the wheels are not accelerating. However, the equation dx = r*d(theta) very simply shows that the only shape that could satisfy this new condition is a circle.

@kfawell Жыл бұрын

I thought of the same thing as I watched. And I imagined what it would be like to ride in such a car that's constantly jerking you forward and backwards. It made me laugh out loud. I think we would be used to a bumpy road going up and down. It would be somewhat tolerable at least. We experienced that walking and jogging for example. On the other hand, having our head jerked back and forth would be hilariously unpleasant or at least irritating. For example, as though somebody has grabbed our collar and is shaking us back and forth. I don't want to detract from the video. It was very enjoyable and solves the smoothness problem as defined.

@fghsgh Жыл бұрын

@@kfawell I've tried out one of those square wheel cars in a museum before. It was exactly like that.

@kfawell Жыл бұрын

@@fghsghI am laughing again thinking about that. Were you able to watch others doing that before you rode? If yes, I suppose you had to find out first hand. I just realized you had that memory while you watched the video. I wonder how you reacted when you saw the word smooth. I really appreciate that the creator specifically defined smooth. Thank you for telling me.

@fghsgh Жыл бұрын

@@kfawell I mean, you had to pedal yourself forward, and it was pretty slow so not too bad. It also mostly felt like variable resistance, not so much speed (because that's how inertia works). But yeah it seemed like it would not be entirely smooth from seeing others too. This was also at least 8 years ago so although my memory is pretty good, I can't give an exact description of the scene ;). But anyway I thought the lack-of-smoothness was just from the physical thing being imperfect, until this comment said otherwise.

@Zildawolf Жыл бұрын

Well now I’m wondering what’s the shape that’d make the most speed inconsistency possible lol

@AsSiccAsPossibleASAP 2 жыл бұрын

I feel like I gained brain cells despite not understanding a word

@aartvb9443 Жыл бұрын

You didn't gain brain cells, you gained connections between brain cells ;)

@ANormalLemon Жыл бұрын

*Brain.exe has stopped working.*

@wildcard_772 Жыл бұрын

Same

@EverythingLvl Жыл бұрын

It's an illusion, still super dum

@BloonMan137 Жыл бұрын

@@aartvb9443 🤓

@cambridgehathaway3367 4 ай бұрын

We live in an astoundingly amazing age. One person is able to singlehandedly write, animate, narrate and publish such a polished, professional, easy to understand, and intriguing video. not to mention doing all the math and even providing a formal proof they crafted themselves. Such incredible talent has existed in past ages (rare tho it may be), but never before has the common man been able to so easily and readily benefit from it. I am astounded and humbled and grateful.

@juanroldan529 2 жыл бұрын

Awesome video! It's been just a few days since I have fallen in the rabbit hole of differential equations. I must say that I love your videos and that they inspire me to keep on improving and learning. Thank you!

@morphocular 2 жыл бұрын

That's great! I'm so glad you found these videos so valuable. One of my hopes for this channel was to inspire others to learn and love math, so it pleases me deeply to be succeeding in that. I wish you the best on your continuing studies :)

@redtortoise Жыл бұрын

@@morphocular first

@maxwellhavoc6996 9 ай бұрын

@@redtortoise I am confused by what you are trying to say.

@thomasrosebrough9062 Жыл бұрын

22:12 super hype to see my favorite curve show up in this video!! A Catenary Curve is also very commonly used in architecture for its even distribution of weight/pressure. The most famous catenary curve is the St Louis Arch which is over 600ft tall! It differs from the identity curve by having 0.01 in each exponent of e, as well as multiplying the entire equation by -68.8, resulting in a curve almost exactly as wide as it is tall!

@ethansmith876 Жыл бұрын

Saarinen my beloved

@csar07. Жыл бұрын

You ascend to a new level when you get your own favourite mathematical curve

@pulli23 Жыл бұрын

I'm late: but there's a single also important point to make a "square wheel" work. The very point that needs to stay at the height also needs to be the center of mass. Otherwise a wheel would give a force rolling back/forward during part of it's movement.

@mujtabaalam5907 Жыл бұрын

We can assume a powerful motor is spinning the wheel on a fixed gear system so the wheel's mass doesn't effect the motion

@whoisgliese Жыл бұрын

@@mujtabaalam5907 epic lateral thinking thanks

@gcewing Жыл бұрын

You can always achieve that by weighting the wheel appropriately, so it's not a constraint on the wheel's shape.

@johnmount5487 Жыл бұрын

That “force” exists even if the axle is at the center of mass. If the wheel is rotating at a constant angular speed the horizontal speed is by definition not constant (changing by a factor or r).the effect is exaggerated as the axle is moved away from the axle as the extremes of the bounds of the radius get larger. The wheels horizontal speed, speeds up and slows down constantly throughout its travel for any shape other than a circle

@aaaab384 Жыл бұрын

its*

@meade6291 Жыл бұрын

The flaw in this is a vehicle with a continuous force applied through is engine to the axle wouldn't experience bumps in the x axis, but it would experience lurches and lags in it's movement on the y axis. Therefore it still would not be a comfortable drive unless the wheels rotational speed was constantly adjusted.

@eventhisidistaken Жыл бұрын

Sure, but 'continuous force' was not specified. Yes, I'm an engineer.

@ob_stacle Жыл бұрын

and if there's any wheelspin at all you'll be on the worst road in existance

@meade6291 Жыл бұрын

@@ob_stacle holy shite I hadn't thought about that

@afoxwithahat7846 Жыл бұрын

I think you switched the axis, the axles aren't moving vertically at all.

@meade6291 Жыл бұрын

@@afoxwithahat7846 yep, and I teach coordinate plane. Shame on me

@RFVisionary 2 жыл бұрын

Great tutorial. Good didactic structure. Instructive, helpful and optically "super nice" to look at.

@enbyarchmage Жыл бұрын

This video made me love catenaries even more, and I already considered them one of my favorite curves of all time! 🤩 I like catenaries bc they appear everywhere, from the Brachistochrone problem to architecture. For instance, Catalan architect Antoni Gaudi took pictures of carefully arranged sets of hanging chains and turned them upside down to model the structure of the most famous church he designed, bc upside-down catenaries make EXTREMELY stable arches. Isn't that beautiful? 🥰

@alriktimo644 Жыл бұрын

When I watched this video, I just realised that my intuition is strong that without even a mathematical description I can jump to the right conclusion, but at the same time I realised I lacked the ability to articulate since I didn't understand it mathematically or completely realising the fact that how this is so or 'How come?' in simple terms. I need to strengthen my mathematical comprehension of data into equations and other methods. Thanks 👍

@drmathochist06 Жыл бұрын

Maybe you get to this later, but the "stationary rim property" also follows from the pivot principle. When the point in the wheel is the contact point itself, then any line through that point can do for the reference line in the orthogonal motion property. Only one possible velocity could be orthogonal to every line: 0.

@amaarquadri Жыл бұрын

Great video! You took an idea that seemed complicated at first and explained it so well that it seemed almost obvious in hindsight.

@sozo8537 Жыл бұрын

The dopamine hit i got when i successfully calculated the equation of the road was something else. I thank you for presenting this problem.

@saintgermain6694 2 жыл бұрын

I never expected it to be that intuitive! Thanks for the really really great video.

@AJMansfield1 Жыл бұрын

19:45 It seems like the road shape depends on how you parameterize the wheel's rotation then -- the function I always instinctively reach for when parameterizing straight lines in polar coordinates is the secant function, and I'd have written that line as { r(t) = sec(t), θ(t) = t }

@AJMansfield1 Жыл бұрын

(In fact, you can choose *any* θ(t) parameterization you want, and just use r(t) = sec(θ(t)) to get a straight line for whatever speed you rotate the wheel at.)

@ChariotduNord Жыл бұрын

This is interesting. I suppose you can get from your parameterization to his by the change of variables t → tan(t'). I wonder if this freedom of parameterization has any physical meaning.

@AJMansfield1 Жыл бұрын

@@ChariotduNord I went and simulated it, and the resulting road curves *are* actually different from each other.

@ChariotduNord Жыл бұрын

@@AJMansfield1 Oh, how did you simulate it? On my end, starting with your parameterization, I ended up doing the standard integral of sec(t) which is ln(|sec(t)+tan(t)|). I then plotted this parametrically on Desmos (typing in "(ln(|sec(t)+tan(t)|),-sec(t))" on the first line) with the domain [-π/2,π/2] for t. It already looked close to the catenary shape. But to make sure, on the 2nd line I put in his solution of y=-cosh(x), and the curves stack on top of each other rather exactly.

@dyld921 Жыл бұрын

The parametrization of the road would change, but the shape (x-y relationship) wouldn't.

@danelyn.1374 Жыл бұрын

I feel like I've stumbled onto a video about a question that I never had in mind, and, along with an amazing explanation of the entire problem, has given me a solution that I am really satisfied by and solves that problem? plus the explanation is amazing so like, mad props

@philosophymikebill Жыл бұрын

Do you mind if I ask what programs/language/code you used to make this video? I'm attempting to learn this sort of simulation, but I'm not sure where to start. Thank you for making these videos. I've been trying to figure out this topic in my head for several years and this is the first meaningful insight I've come across in a good long while.

@morphocular Жыл бұрын

I actually used my own homemade software to make the animations in this video. You can find the software here if you want to play with it: github.com/morpho-matters/morpholib However, it's still largely just a personal project and the documentation is rather sparse. A more well-established and popular tool for making similar animations is called Manim, which you can find here: www.manim.community/ Hope this helps :)

@philosophymikebill Жыл бұрын

@@morphocular I really appreciate the advice and even sharing your program! Thank you for getting back to me

@alexv1129 Жыл бұрын

@@morphocular Math is interesting and fun - but I am subbing because of this right here. Amazing of you to be so kind and helpful. Good luck, creator!

@ianhickey3423 Жыл бұрын

@@morphocular This is so unbelievably cool

@LoganCralle Жыл бұрын

Incredible video. I just took a dynamics course at university and I learned so much. This is an incredible application of maths. Bravo 👏

@H3xx1st Жыл бұрын

You explained that beautifully! I am definitely looking forward to your future videos.

@mullactalk 2 жыл бұрын

This channel is a hidden gem of maths KZfaq

@bloomp7999 Жыл бұрын

I deeply agree with your channel description and the Poincaré quote, i'm in for what you do, keep the good work !

@RFVisionary 2 жыл бұрын

great tutorial. good didactic structure. instructive, helpful and optically "super nice" to look at.

@TheCynicalOne Жыл бұрын

I want an entire video, or at least a short, dedicated to the orthogonal movement principle. It’s a mess and I want to dive in with full understanding! Great video about the wheels too. I feel like many of the wheels shown would slip a lot on their roads, so I guess the dream of bumpy square wheeled roads is a long shot lol.

@DonkoXI Жыл бұрын

The proof he gave is actually pretty clean all things considered. If you are interested in understanding it, I highly recommend looking through it and trying to understand his reasoning one step at a time. You can ignore the algebraic details at first, but try to understand the concepts in the argument. If you understand the way complex numbers work well enough, it should all be pretty intuitive with some time. If you don't feel very comfortable with how complex numbers work, then stopping and thinking about each detail of this proof will actually be a pretty good way to get a better understanding of how they work. What feels clean to me is of course subjective though.

@thirockerr Жыл бұрын

Nice video ! I would be interested to see how you would present the optimal road shape taking into account a specific mass for the wheel, the gravitationnal force.

@phlapjakz Жыл бұрын

it always amazes me how e manages shows up everywhere even when the problem looks like it has nothing to do with it

@DitieBun Жыл бұрын

4:15 This is the most insane wheel I've ever seen, and I'm here for it

@duncanhw 2 жыл бұрын

Great video! Love how you started by making the equations and then deriving the shape from them! Can't wait for the next video. also, wouldn't the wine glasses in the thumbnail be knocked forward/backward due the second law of road-wheel motion?

@ineedtogetoutmore1848 Жыл бұрын

that “Pivotal Role” pun at 11:14 was painful, well done

@brucea9871 Жыл бұрын

Very interesting video and analysis. I'll be watching more of your videos. This one reminds me of an old comic strip. It was called BC and based in prehistoric times. Their only form of transportation (other than walking) was what they called the wheel. It was a circular wheel with an axle through the centre and they stood on the axle to ride the wheel. (How they propelled it - especially uphill - is beyond me.) In one of the strip's comics (presumably before they thought of using circular wheels and hence only had square wheels) one character declares to another he has derived an improvement to the square wheel and produces a triangular wheel. "Improvement?", the second character says, confused. The first character replied "It eliminates one bump". But of course if they designed their roads as you specified they could actually have square or triangular wheels with no bumps. (Somehow I think it would be easier to come up with a circular wheel.)

@gergonagy846 Жыл бұрын

I'm safe to say, that this is the most engaging video that I've ever watched.

@deathpigeon2 Жыл бұрын

While a flat ride is certainly an important thing for a smooth ride, I'm not convinced it's sufficient. It seems reasonable to describe a jerky ride as also a non-smooth ride. That is to say, given a constant torque applied to the wheel, the third derivative (the jerk) of the forward motion produced by the wheel spinning should be precisely equal to zero. Put another way, a linear acceleration of the rotation of the wheel should produce linear forward acceleration for the whole system. Now, I think the stationary rim principle should be sufficient to ensure that this is the case because it ensures that the rim speed and the axle speed are equal, but I think it'd be insufficient to consider only the flatness of a ride to determine if it's properly a smooth ride.

@klikkolee Жыл бұрын

We are used to vehicles which are propelled by the wheels. However, if the vehicle is moved by means unrelated to its wheels, then the criterion in the video is sufficient. For vehicles which are wheel-propelled, unless a fanciful control system regulates the wheel speed, your additional criterion is required to make the vehicle feel subjectively smooth to a real human occupant. The no-slip condition (stationary rim principle in this video) does *not* guarantee your criterion. The r in the no-slip equation is a function of t. Your criterion is only consistent with the no-slip condition if the radius is constant -- meaning a circular wheel.

@deathpigeon2 Жыл бұрын

@@klikkolee ...Right. I was thinking it'd ensure 0 jerk because it ensures that the rotational velocity at the touching point and the forward velocity at the axel are the same, but, for constant torque, the velocity at the touching point would be in part a function of the distance from the axel so you *need* at least some slipping to ensure a smooth ride unless you have a constant distance from the axel (ie being a circle as you said).

@Nuclear868 Жыл бұрын

What if, in case of a car, we make the distance between the front and the rear wheels such that front and rear wheels are offset - when the front wheels have the highest angular speed, the rear wheels have the lowest angular speed? Yes, they will not cancel out completely, but will reduce the 'jerk' feeling.

@eventhisidistaken Жыл бұрын

Who said the torque had to be constant? Stop trying to impose your roundism on the rest of us.

@klikkolee Жыл бұрын

@@eventhisidistaken It would be a substantial engineering challenge to create a vehicle where the torque applied by the wheels varies in perfect concert with the road shape. Without that perfection, a wheel-propelled vehicle can't have a smooth ride on an extreme road without slipping.

@adrianmisak07 2 жыл бұрын

fantastic video… cant even express how impressive this is to me, I try to do math recreationally after getting my masters in applied math…

@kdicus Жыл бұрын

Watching the shape of that ellipse move around makes me wonder if that visual perspective doesn’t unlock a thought on how to attack the unknown equation of the perimeter. For those of us who are visual, this was absolutely gorgeous to watch.

@LunaAlphaKretin Жыл бұрын

I'm curious what would happen if you impose the additional restriction of making the axle's horizontal speed (and, hence, velocity) constant (given constant rotation speed). I noticed the speed seemed to vary a lot with that particularly arbitrary-shaped wheel example at 4:18, which would probably be a disconcerting experience as a driver. Still I imagine the answer is that you can't have a road that does both - to prevent a change in horizontal speed you'd probably need a different road that causes vertical changes. What if we just say "constant velocity", allowing the vertical position of the axle to change as long as it feels like a smooth slope would for a circle-wheeled driver. I don't know how that would go, but it feels more likely to be possible.

@WaluigiisthekingASmith Жыл бұрын

The second equation says dx/dx =rdtheta/dt. Differentiating a second time d^2x/dt^2= dr/dt dtheta/dt +r dtheta^2/dt^2. Given your restriction dr/dt dtheta/dt = -r dtheta^2/dt^2. Thus r'/r =u'/u. Doing what any good physicist would do and pretending we can just cancel our differentials like fractions, we get ln(r *dtheta)= c and thus dtheta/dt =c/r

@joaogiorgini1326 Жыл бұрын

Make velocity constant with constant rotational speed? In other words, dx/dt=cte and d0/dt=cte. Meaning, in the second equation, r must also be a constant. In other words, the only shape that satisfies a truly smooth ride is a circle.

@bears7777777 Жыл бұрын

I think the only way this would be possible would be to allow wheel slip. The amount of slip would be the fastest angular speed - slowest angular speed. The slip would have to occur when the point of contact is farther than the minimum. For the square, this would be when the point of contact tends towards the corners as they are farther from the center then the center of a side. I’m not sure that’s even solvable though

@scifiordie Жыл бұрын

Nobody cares bro get a life

@dj_laundry_list Жыл бұрын

What the hell is this? It's awesome. I think it would be more complete/satisfying to state that the vertical alignment property relies on shapes being convex, but honestly this is one of the best math(s) videos i've seen for a while

@vikn331 Жыл бұрын

This is the perfect example of "I have no idea what this man is talking about, but I like it"

@arlyu606 9 ай бұрын

I reeeally love your content. Thank you for all your videos :-)

@NoOffenseAnimation Жыл бұрын

Great video, I like to wonder what this would look like in practise, if someone were to try this in the real world, but of course there would be a great deal of other things to consider

@TerrifyingBird Жыл бұрын

This problem (or rather a simpler version of the problem) ended up in an italian high school final exam, in 2017. It is to this day one of the most iconic problems to ever appear on the test.

@sriramn1809 Жыл бұрын

First video ive seen on this channel. Wondering why youtube took so long to recommend me stuff from here. This channel is amazing!

@tsar_asterov17 Жыл бұрын

This video is amazing, and all of his videos, ngl are basically 3b1b on light mode

@Happy_Abe 2 жыл бұрын

Why isn’t x equal to the hyperbolic sin plus a constant because we integrated Where’s the +c

@morphocular 2 жыл бұрын

Man! Nothing gets past you guys! :) Yes, technically a +C belongs there, but all it will do is shift the road forward or backward; it won't affect the actual shape of the road, which is what we were after. So I picked C = 0 in order to get the simplest and cleanest possible final answer.

@Happy_Abe 2 жыл бұрын

@@morphocular Thanks! Yeah I didn’t think it was a problem for the solution, I just thought it should have been addressed. Amazing video, can’t wait for your channel to explode in popularity!😊

@mateuszbaginski5075 2 жыл бұрын

I can't really point to what it is in your videos that makes them one of the best I discovered through 3B1B's SoME. Whatever it is, you are grokking it, man.

@miguelcabaero5843 Жыл бұрын

I love the production quality

@blackboxlearning 5 ай бұрын

I was hoping to make a video on this exact topic, but I guess it has already been beautifully covered by this channel. While checking for that, I came across this channel and I love the animations and their interactivity. Already subbed. Expect a video soon covering more stuff, cus I'm not leaving the idea :)

@lenskihe Жыл бұрын

Awesome 👍 I tried to solve this problem on my own once. I'm glad I watched this video, because now I know that I would never have been able to do it 😂

@the25thdoctor Жыл бұрын

What I love about this is, it has a simple answer. Think gears, and a gear rack. But is far more complicated to preform

@agy3256 Жыл бұрын

This video is pure gold

@bitroix_ Жыл бұрын

This is an amazing video! Thank you.

@firiasu 8 ай бұрын

So good explanation!

@user-wv1in4pz2w Жыл бұрын

I am pretty sure you can easily derive the pivot principle from the fact the contact point is stationary: observation 1: the wheel is a 2D rigid body, so its motion is fully described by horizontal speed, vertical speed, and rotational speed, so it has 3 degrees of freedom. observation 2: the constraint that the contact point is stationary restricts 2 degrees of freedom, thus leaving 1 degree of freedom. observation 3: pivoting motion satisfies the stationary contact point constraints and has 1 degree of freedom. therefore pivoting motion is the only possible way to satisfy the stationary contact point constraint.

@kindlin Жыл бұрын

When he said it was _really hard to prove_ I was confused, as this is the only motion available due to the no-slip-condition and the rigid body motion. But honestly, the statement of the question itself is almost the proof of the question. You want to figure out how to prove that all points on the wheel move periductular around the contact point, well, proof by exhaustion, there are no other ways it could move around the contact point but to pivot, and the definition of pivoting, as noted in this video, is perpendicular motion about a point.

@user-hd9oh9bk8b Жыл бұрын

it's honestly fascinating how many titles and thumbnails this exact video's had. i've heard about this but never gotten to see it firsthand

@Adam-pj2qh 21 күн бұрын

thats so sick, finally some applied mathemathics!!!

@stuartl7761 Жыл бұрын

6:10 I love that the first and last terms cancelled happily :D Loved the proof too, I must remember to check through if complex numbers might help when I come across a problem.

@morphocular Жыл бұрын

A good hint that complex numbers might help is if your problem involves 2D rotation or 2D rotational symmetry. That's where complex numbers often come in handy!

@k7iq Жыл бұрын

This is fantastic ! 2 + 2 = 5 for large values of 2 But would a square wheel do good in snow or maybe even ice ?

@g10royalle Жыл бұрын

The animations are so satisfying

@iskallman5706 Жыл бұрын

This is as good as mathematics vidéos get. The pinacle.

@officiallyaninja 2 жыл бұрын

this video is so good. its criminal that you don't have hundreds of thousands of subs

@Happy_Abe 2 жыл бұрын

In time we’ll get this channel there

@Error-xl3ty Жыл бұрын

Videos like this are why I love math

@ANZEMusic Жыл бұрын

This is a really good video. The math is fascinating, and you present it clearly with exceptional visuals, and I greatly appreciate it

@Josephi_Krakowski Жыл бұрын

These are the type of videos I watch at 3 AM

@MF-dz7cp Жыл бұрын

I'm a sophomore in high school so I have no clue what this video is talking about but it's still interesting

@kinkinawesome 2 жыл бұрын

Exited for the next videos!

@vitorguilhermecoutinhodeba3253 Жыл бұрын

It is a nice video, even though I think some properties have different names in here. Instant center of rotation is the center (no pun intended) of all this procedure, and wasn’t mentioned. The animations were very good!

@Danker1248 Жыл бұрын

this was a very enjoyable video

@karllenc 4 ай бұрын

amaizing! just amazing video! Thanks

@iamtraditi4075 2 жыл бұрын

I know I'm late, but this is really good!

@jursamaj Жыл бұрын

The horizontal motion of the axle is necessary for a smooth ride, but not sufficient. It needs to be *smooth* horizontal motion, not jerking forward and back. That, in turn, requires the wheels to rotate at highly variable speed. But that's not how driven axles tend to work. Additionally, when the wheel is moving up the slope, the wheel will be moving too fast at any given moment. Combined with the uphill configuration, you basically guarantee slippage. You face a similar problem on the downhill side, but inverse. Those novelty tourist attractions tend to reduce both these effects by having the front & back wheels be exactly a half wave out of phase. That way the slippage either way hopefully cancels out, and one axle can be speeding up while the other is slowing down.

@abcdaiy Жыл бұрын

KEEP IT UP BRO U ARE DOING GREAT WORK ❤❤❤❤❤

@WeeIrishLaddie1 Жыл бұрын

I'd be interested in a sister video where "smooth" was defined as "constant velocity" rather than "constant axel height", ie changing the axel height in the wheel as it rolls to keep it moving horizontally at constant speed

@jorgec98 Жыл бұрын

I'm kinda proud of myself I grasped the first analytical definition more easily than the second visual one

@user-le6hf4ys7y Жыл бұрын

When you try to stop the wheel, wouldnt it be unable to stop at some certain points though? For example, for the square wheel, if you trying to stop just a bit further forward from the x value of top of a "bump", wouldnt the wheel roll toward the next "ditch"?

@dergecko92 Жыл бұрын

True

@convincingmountain Жыл бұрын

very nice video, i really enjoyed the small steps taken each time to get to the answer. and even then, there's so much more to discover! well presented and paced, didn't feel like half an hour. your consistent use of both visual and verbal explanations for each new idea is great.

@johnnyvishnevskiy8090 Жыл бұрын

I'm more interested in how this transfers over to 3 dimensions and how turning affects how the shape of the road is made.

@josiphanak9130 11 ай бұрын

noone cares...youtube is in 2D

@nerdsgalore5223 Жыл бұрын

This is a beautiful video.

@redyau_ Жыл бұрын

The way you use - I assume - MAnim is absolutely outstanding. I bet you come to understand every concept you explain in an incredible depth as you code these. Really impressive!

@archie1490 Жыл бұрын

This was a nice brain teaser before I go off to do maths at uni. GL everyone off to uni in Septemember!

@pastadcasta Жыл бұрын

I have a way I like to think about it, if you take the path that the axle takes when the shape is rolled continuously over a flat surface, and use that for the road surface, the shape will roll smoothly. It's cool to see the algebraic representation of that though. Very cool video! ^^

@steffahn Жыл бұрын

A square wheel rolled over a flat surface will actually just pivot around each of the 4 corners. Thus, the axle would take a path composed of a series of arcs (i.e. sections of the perimeter of a circle), which is definitely *different* from the series of catenaries that are shown in this video to be the shape of road that you need.

@TomatoBulb Жыл бұрын

I have absolutely no idea what any of this means but I find it interesting

@ZotyLisu Жыл бұрын

this should have way more views

@benjaminstandfest6265 Жыл бұрын

This is awesome

@nalat1suket4nk0 Жыл бұрын

Awesome!!! You used math and physics so cool

@ParadoxProblems Жыл бұрын

If you want a non circular wheel that moves with constant speed, you can give the wheel a non uniform mass density such that when the wheel would slow down, the part on the bottom that is moving slower is made more massive. It's momentum is transfered to the entire wheel body, maintaining a constant velocity. Most likely the mass distribution would be such that every dTheta slice around the axle has the same mass regardless of radius. (Constant moment of inertia)

@KOZMOuvBORG Жыл бұрын

Assuming lateral motion in respect to the axle is constant, wouldn't that mean the rotation of the 'square wheel' varies depending on which part of the catenary is in contact with it? The radius of the corners being √2 (1.414) more than on the middle of its sides. If rotation was constant, wouldn't lateral motion undulate?

@tracy449 2 жыл бұрын

Thanks for the video. I learned a lot. Also, I have a question: If the axle moves at a constant velocity, does the wheel rotate with a constant angular velocity?

@morphocular 2 жыл бұрын

Thanks for watching! To answer your question: Not necessarily! The second Road-Wheel equation says the axle's velocity is dx/dt = r dθ/dt, where dθ/dt is the angular velocity. So the only way both the axle velocity and the angular velocity can be constant is if the wheel has a constant radius, meaning this will only happen for the case of a circular wheel.

@TheHuesSciTech Жыл бұрын

@@morphocular Fascinating -- I *believe* an involute rack and pinion has the property of dx/dt = k dθ/dt, where k is a fixed property of a given gear (the radius of the gear's "pitch circle", or half the "pitch diameter", perhaps?). This would appear to contradict the statement you made above, but I believe that might be because you're assuming no slippage between the wheel and road in your video, whereas an involute rack and pinion does have slippage?

@cheshire1 Жыл бұрын

@@TheHuesSciTech The equation you gave is approximately true, since a gear is pretty close to a circle.

@TheHuesSciTech Жыл бұрын

@@cheshire1 It's approximately true for all gears, yes. But I believe it's *precisely* true for an involute gear. (Neglecting real-world clearances and manufacturing tolerances, of course.)

@cheshire1 Жыл бұрын

@@TheHuesSciTech You may be right, involute gears do have slippage (and the contact point jumps around instead of staying on a vertical line), so the argument from the video doesn't work in their case.

@oskarandreasolsen495 Жыл бұрын

When describing the small timesteps in the visual prrof at the end, maybe it would be more specific using a Δt->dt for smaller and smaller time steps :) But great video! really liked it!

@TheGiuse45 Жыл бұрын

This was a question on the high school finals in italy a few years back

@Cesar-ey7wu Жыл бұрын

You would actually feel "bumps" in a square wheeled car because for a constant speed, the rotational speed of square varies (you can see it in the video : it's accelerating when it gets to the side of the square and slowing down on the corner). So if your engine is putting a constant torque to the wheel, the car's acceleration would vary four times for each wheel rotation, which wouldn't be comfy at all.

@elitestryker5709 Жыл бұрын

Its so much more understandable than PHysics I and Technischemechanik at ETH together eith explaining all the concepts

@WAMTAT Жыл бұрын

Great video. You've earned a subscriber

@idiotsincarsalberta Жыл бұрын

man if my classes were taught like this, I'd actually show up to lectures

@RagingBadger68 4 ай бұрын

While I don’t understand 80% of what I’ve been told here, I did finally understand the purpose of imaginary numbers here. I’ve struggled through so many math classes which could never just explain it so effectively.

@imad_uddin Жыл бұрын

Nice to meet you Grant Anderson Junior!

@rebeccastevens2903 5 ай бұрын

Wheels are paradise. The wheel groups form paradise!

@darealmrog Жыл бұрын

Congrats! You just reinvented a train!

@manifaridi9200 Жыл бұрын

I hate mathematics but man... look how beautiful it is.

@WAMTAT Жыл бұрын

You don't hate math, you just hate how it was taught to you.

@blackbeast9268 Жыл бұрын

Trust me nobody hates math. I used to have E and was on my way to F but then i moved school and my teacher was amazing and i got A because he explained everything so well and got me motivated. Math is a language with rules and if you're teacher doesn't explain the rules in details it will be boring because you rely on common methods and formulas instead of understanding why they work. It's very fun and i would argue chemistry or physics are much harder then any math expect super high level .

@cut2000trees Жыл бұрын

To demonstrate the stationary rim property, you could imagine a wheel with a notch cit out on the edge. If, say, a squirrel was in the road, and it positioned itself to line up with the notch, it would be safe. Because the notch wouldn't move, it would be stationary

@nano_shinonome Жыл бұрын

so cool!!

@Smitology Жыл бұрын

Can the pivot principle also be explained as being necessary from the no-slip principle and the fact that the shape never deforms ie all distances stay the same? If all distances stay the same, then the rate of change of the distance between a particular point and the contact point must be 0. Therefore, the displacement can only change direction, not magnitude. Therefore the rate of change of displacement (aka relative velocity) must be perpendicular to the displacement vector as that's the condition for circular motion Therefore because of the no-slip principle, the absolute velocity of that point must be perpendicular to the displacement vector.