The Mathematics of Mechanisms (

Entry for the 2023 Summer of Math Exposition
Sources:
R. L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines
D. Eberly, Intersection of Linear and Circular Components in 2D, www.geometrict...
The code used to make the animations can be found at:
github.com/mti...

[@Axman6]

I’m only a few minutes in, but I wanted to say this video is beautiful; the colour scheme, the sizes of everything, the animations, the fading in and out. These are little details that are hard to get right, well done- subscribed ❤

[@YTomS]

Criminally underrated channel, what a nicely done video.

[@godfreypigott]

Based on ONE video posted 3 weeks ago, where your comment was posted two weeks ago?

[@slepenb]

The accent makes it tough to follow

[@godfreypigott]

@@slepenb It is easy at 75% speed.

[@bigbluebuttonman1137]

The math of mechanisms is super fascinating to me. Going into a machine shop is like being a kid in a candy store for me. So much stuff, and every little detail has its reasons for being there in one way or another.

[@magnuswootton6181]

yeh levers and cranks fit into maths perfectly.

[@thecalculusofexplanations]

Brilliant, I taught some of this stuff to engineers once upon a time, I wish I'd had this video to show them. Well done

[@JaredBrewerAerospace]

Perfect! Deep and simple is more essential than shallow and complex. It doesn't matter how many times I have taught or been taught the same topics, everyone at any level has something to gain from the way you present these fundamentals.

[@sundown456brick]

Im loving this movement, SoME is the best thing ive ever seen great to have found you, looking forward for more content, keep the good quality🎉❤

[@gianlaager1662]

Very nice animations and great video. Please keep it going with videos like this.

[@yadav-r]

Game developer here. I am not technically proficient as the ones working at Rockstar games. In one of their games red dead redemption, we see lot of horse animation, very realistic and I always wondered how they did it, found a research paper which described exactly this. It was too much for me to understand fully, but now I have some idea and where to look further. It's fascinating to see different fields working together to create a final product. Thankyou so much for creating such a wonderful and insightful video. God bless you brother.

[@PhilipMurphyExtra]

Such a well produced video, Glad KZbin suggested it.

[@zacharytoth1065]

Im taking a Mechanical Design class right now, and am definitely sharing this video with my friends. Its a very clear and concise recap of some of the topics covered in class, and will be helpful in getting a better grasp of the topic.

[@sonu-jangir]

So helpful video... 🎉🎉🎉 Thanks for sharing... ❤❤❤

[@ZimmervisionCZ]

This is really well done! Well-explained, beautifully designed and animated. This immediately makes me want to go out and program a 2D mechanism-based video game

[@wellscampbell9858]

@mtirado Excellent video, flows well while covering the topic completely enough to serve as video reference material. It's definitely going in my tech reference links. Thanks!

[@adissentingopinion848]

That last five bar linkage just threw me through a loop and subsequently jammed me such that √4ac = 0. Immaculate lesson into such a complex topic.

[@Jaylooker]

The discrete Fourier series describes a mechanism which can draw any closed curve using epicycles. If every coupler mechanism can only draw closed curves as well, then there must be an equivalence between two coupled discrete Fourier series and a single discrete Fourier series. Describing what mathematically represents the coupling between the two discrete Fourier series is difficult.

[@Mark-gd5yz]

More! Please. You have a rare talent: Use it.

[@agrathnam]

Beautiful graphics and great explanation. Looking forward to more videos from you.

[@bdzack2226]

This video is having too much knowledge and awesome way of representation. Crazy, keep up the great work. THANKS

[@mani_mincraft]

This is so cool! That circle approach is such an amazing method!

[@Haagimus]

Great video, very well explained mechanics, looking forward to your future content 🤙🏻

[@tenix6698]

OMG, That's something I've been thinking about for a long time, but never got to it. Thank you for providing such a good video on this topic!!

[@yaacheese8643]

You need to make more videos on Mechanisms! Awesome video, I subscribed hoping to see more from you in the near future!

[@danielpitts6913]

Very nice. Makes me want to write a simulator for this. One more project to the backlog lol. It doesn’t seem like it would be too difficult to calculate some physical properties for these after determining big positions based on the constraints. Like torque or linear force.

[@pierreabbat6157]

I've written a program to simulate the Chebyshëv linkage, which traces the Nilla curve. The bottom is nearly flat, while the top is nearly an arc. At four equally spaced times, it's at three points in a line on the bottom and at the middle of the top. It looks like the cross section through the middle of a Nilla cookie.

[@onadja]

Excellent animation and great explanation! What editing software did you use? THANKS !

[@jormando2002]

Wow... It is amazing, thank you so much for this video ❤

[@senthilkr1970]

Fantastic videos, amazingly done. 👏👏👏

[@moralboundaries1]

so interesting and enjoyable, thank you for the lesson!

[@guillegilcriado6879]

This video is so well produced. Great explanation, simple yet complete. The animations are so cool and well made. Overall, amazing video!!! New sub here! ^^

[@shivabalaji6668]

Extraordinary 🤩🤩🤩🤩😍 pls upload many more videos like this

[@zzznah]

Congratulations on making this very informative and beautiful video! As an aspiring KZbinr I know how much hard work it takes

[@_krzysio_6910]

In Robotics those are so simple mechanisms... We have really great methods there - check it out. We just use matrixes for everything.

[@geraldopontes37]

Excelente vídeo! Thanks you

[@derektauffner8828]

Fantastic video and loved the animations. Well done.

[@hjfreyer]

Very nice! I'd love to learn more about how you disambiguate between the cases with multiple solutions. Like, for each place with ambiguity do you just have to pick either the positive or negative root?

[@ico-theredstonesurgeon4380]

This video Is really well done! I would love It if you could also talk about the forces that act on the mechanism. I am a robotic enthusiast and that would be really helpful

[@alfredovillal5263]

Exelent Video,, very nice.

[@sahhaf1234]

exactly the video i was looking for.. pls continue..

[@databang]

Thanks for the video. Very nice illustrative presentation that’s easy on the eyes and labeled well. I’m curious what software you use to construct models and animate them, is it Adobe AE or something more specific?

[@sashiyendamuri1018]

Very nicely explained with simple graphics...

[@emil_richard]

This is so well produced! Can you recommend any program where anybody can test such configurations easily?

[@TheMagicFellow]

Beautiful; breath-taking

[@arkadiusz4133]

I would be very pleased if you will make a few videos how to solve some practical tasks about power, inertia, moments etc. In mechanisms

[@PeterNerlich]

Fascinating video! I'm most interested in the inverse problem, finding a mechanism that produces a certain path. In your example, you show a how to derive a solution of an easy instance of this problem, where a simple four bar linkage is sufficient, and using only three "samples" of position+rotation of a segment that should be reached by the mechanism. But how would one go about synthesising for a path like the one in 13:32?

[@farhatali3634]

Its a beautiful video. Thanks for all the effort and thanks for sharing with all of us. Simply amazing. Kind request to share which software or programming language you have used for creating those beautiful animations. Regards.

[@ToMMiTTo]

Big clap per your video! Awesome.. please do follow up videos. I would suggest to use a math editor for formulas (latex or similar), so they are more easily readable

[@bubbaloo8049]

Gran video, el mejor por lejos. Muy bueno !!!

[@mohammadkaheel973]

Amazing 👏

[@jerry-yu7yi]

i really really love this video

[@nttn3666]

This is so cool, please make more videos on this topic.

[@장종훈-u1t]

깔끔하고 멋지네요. 감사합니다~

[@AtharavD]

Please make more video's like this. ( like if any one wants video's like this )

[@chienbanane3168]

This is great for developing walker linkages!

[@disaffected_npc]

So, I've been trying to figure out how to visually represent some stuff - I'm quite hypermobile/have some pretty peculiar stuff going on with my nervous system and I want to find a way to create a hard map of the range of motion of my bones/joints, and then somehow overlay that with my internal/imagined map of my body. For most of my life I've had a bunch of involuntary tics, and since I was a child they were dismissed as a baked in problem of being a flappy autistic person and thus to be pretty much ignored - but upon realising that they were a manifestation of problems with connective tissue/rooted in weird stuff with my nervous system, I started engaging with/adjusting some of them - one in particular had been constantly subluxing my jaw and had (as wild as this sounds) resulted in me losing an enormous range of my sense of touch/pressure detection. Fixed the issue with my jaw and trained myself to pay attention to what my body was actually trying to do and over the past two years my sense of touch/proprioceptive map has exploded outwards from my neck/shoulders/spine. I feel like this kind of map of where my body actually can move, and being able to mark onto that which ranges - while possible, were destructive/overstretched joints would be incredibly useful. It also feels like something that someone must already have done to some degree. Do you have any suggestions on resources to look into? I'm not a mathematician/programmer of any kind - but this feels like the most promising tool with which to build the physical/mechanical part of what I need to create to make useful/discrete statements about what's been happening

[@polyhistorphilomath]

The discussion of jamming position was interesting. I have to wonder if there is a way to limit or constrain the configuration space during synthesis such that the number of degrees of freedom can only ever increment or decrement (by one). Similar to the K-map concept the intent would be to prevent simultaneous changes and thus minimize undesirable or indeterminate behavior.

[@AllenKnutson]

¡Que rico! And while manim has its place I'm especially pleased to see explorations of other visual options. (The rectangular boundary is an especially unusual choice and I wish I'd thought of it!)

[@1022darkar]

exelente video sigue con tu contenido

[@mrshodz]

Great explanation.

[@HannyDart]

Some time ago i was trying to analyze a rather complicated 3d mechanism using this "distance & circles" approach but for some reasons my equations were no longer symbolically solvable. Ive verified my numerically obtained solutions several times and they were correct so the equations had to be correct too. Since then I was interested in a proper way to do the math behind it...

[@simongross3122]

This is so clever and fascinating

[@sifatahmed1413]

Excellent

[@petrkisselev5085]

Great presentation !

[@squantaai]

Beautiful

[@tombouie]

Interresting, Giving an arbitrary output motion, ?might the entire linkage soultion set be solved for?

[@nahuelpiguillem2949]

Wowwwww mannnnnn, it's greatttt. Pretty clear

[@tonyfarah7685]

Nice video, but i just wanted to understand more about equations, so i hope you will explain it in detail please I liked your visualization ❤

[@Garglicious]

Cannot wait for more videos from you !

[@Spiegelradtransformation]

Well Done.

[@anandjoshi9716]

Really good

[@SimpleLangSolution]

God tier video and explanation.

[@christianprice4049]

This is GORGEOUS!!!

[@MissPiggyM976]

Very interesting, thanks!

[@MrAminmohamed]

Please make another video of more examples of building mechanics without anything just basic geometry. This way kids in middle schools will be able to use their compas and rulers to draw prototypes

[@juancarlossanchezveana1812]

Amazing

[@jakobr_]

Can complex numbers be applied to this math? I’m curious because there’s a lot of rotation, and complex numbers seem to fit in wherever you see rotation.

[@mtirado]

Complex numbers are just 2D vectors, so yes!

[@derektauffner8828]

This is close but not entirely true. There is an isomorphism between 2D vectors and complex numbers. And you need to be careful on how you treat the two if you want one to be the other! There is a fantastic answer on Math Stack Exchange if you google 2D vectors as complex numbers. @@mtirado

[@dsgowo]

You can also use conformal geometric algebra to describe not just rotation, but also translations as well as the circles defining the possible positions of P2 and P3 (or similar circularly constrained points in a linkage). Many of the calculations done in this video, such as finding the intersections of two circles or constructing a circle from three points on its perimeter, are expressed very elegantly in this language. To top it off, it generalizes very elegantly to 3D and higher dimensions, so you can get all the benefits of the complex numbers as well as quaternions and dual quaternions inside CGA.

[@crimsnblade8555]

they are used commonly, actually

[@francomaccaroni795]

very nice video, good job

[@pavelperina7629]

I once tried to simulate heusinger gear of steam engine and failed at combination lever. If I recall problem is that contrain is something like end and mid point are allowed to move on two circles and distance is defined by distance of mechanical joints. Third point is on some curve which I cannot properly describe. Other link has the same or similar contrains and intersection of these curves is a solution. Maybe it can be solved for tens of possible positions, drawing line segments between solutions, repeating for other links, finding intersections of line segments approximating these two curves and subdividing intervals to get more precise result. I just can't imagine how people designed that 150 years ago or so, because solving something like 4-5 equations with trigonometric functions is hard. Maybe tthey did not need to know precise position of joint, they just made sure that it satisfies number of degrees of freedom and that it combined movements of two levers with a proper ratio and made some smaller model from sheets of metal with holes and rivets.

[@jairoc.peralta]

Buen video, compa

[@herbertattema9890]

the algorithm did you bad, how am I only now finding this channel

[@bradhayes8294]

A crank slider is a four-bar linkage also.

[@mtirado]

Yes but it has a prismatic joint. I focused on revolute joints only.

[@bradhayes8294]

@@mtirado I had a mechanisms class as an undergraduate mechanical engineering student and an advanced analysis and synthesis of mechanisms class as a graduate student. We used primarily the vector loop-closure method for mechanism analysis. We used both analytical and graphical methods for mechanism synthesis. One of the final projects we had was to derive the position, velocity, and acceleration equations for a 10-bar John Deere level-lift mechanism. I also had an advanced dynamics of machinery class as a grad. student. One of my favorite analysis methods was the Chace vector analysis method for 3-dimensional mechanisms.

[@TechTerminater]

Keep it up . Very good video

[@alcyonecrucis]

Wow good job. Do these paths have equations ??

[@mtirado]

They have, but I didn't take the time to obtain them.

[@realcygnus]

Quite nifty !

[@labkome]

Please create more videos bro, respect from indonesia

[@SimplexonYt]

at some parts of the video u have to get the square of a vector or multiply two vectors with each other But how are you supposed do do that should i multiply/square the single components of the vectors, should i take the cross product of them, should i use the dot product or something different?

[@FrostyHandled]

anyone else feel bad for the universe for having to do so much computation

[@xenorzy9331]

Nice.

[@seeker4430]

Could you please make more such videos

[@hbenzd5301]

Gooood..

[@BradleySlavik]

At 9:43 the |U|^2 is represented by |U| in the bottom equation that threw me until I saw the mistake.

[@Maria-ig6yd]

But egual identicall component on a movements pedals not have a problem, on a movements, but if not is, have a problem

[@M0rshu1]

This is essentially the math you will be doing in the last 3 semesters of a mechanical engineering bachelor’s.

[@RajSingh-ln1mn]

Bolo zuban kesari , I really needed this video , really helpful and informative, keep sharing these . ❤

[@foxprojects247]

I love this!

[@raylopez99]

the old 4-bar linkage mechanism.

[@nad2040]

i think arglin kampling likes this stuff

[@coledavidson5630]

Machine dynamics!

[@PSS_1]

I need help with a certain mechanism

[@xhulioballa8606]

When can I find the book

[@migfed]

This is a superb video. Just one suggestion with all due respect, try to vocalise better, there were some parts that I could not get it. And another thing, try to use an equation editor for displaying your algebra. Even PowerPoint has got one. So once again, let me congratulate you for that great video.