The Mechanical Integrator - a machine that does calculus
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- เผยแพร่เมื่อ 3 พ.ค. 2024
- This video explains the function of the mechanical integrator, a mechanism crucial to the development of mechanical analog computers throughout the twentieth century.
This video is part of a project I have been working on in collaboration with Professor Michael Littman of Princeton University. One of the goals for developing this specific machine was to use it as a supplementary tool when teaching calculus and differential equations, so I made this video to demonstrate the machine's function with specific emphasis on its connection to calculus. - แนวปฏิบัติและการใช้ชีวิต
Over 50 years my parents used to run a business making leather bags and coats. One of the aspects of this was how the tannery measured the areas of the hides. They used a really basic and old device called a planimeter which physically did the integration. There were equally spaced toothed wheels which rotated and thad counters. The hide was placed on a big table and the planimeter was run over it. Knowing this viewers should be able to work out an expression for the approximate area and think about the effect of holes in the hide etc. I was recently writing a paper on working our the areas of arbitrary closed curves on a sphere using Stokes' Theorem (yes, it works and is better than an old approximation method NASA uses for determining oil spill areas from satellite data) and it brought back memories of the planimeter.
Of course it works. Its almost as if discrete mathematics (*1) is much more useful for any engineering than analytical because its bottom-up instead of top-down and will work directly with real data instead of a carefully planed model.
Aka, simulation instead of prediction.
*1 - (integrating using mechanical components is more akin to discrete, because its finite in precision, unlike pure calculus which is algebraic)
Maybe this comment is why I got recommended this video that one about the polar planimeter 😆
@@monad_tcp What do you mean when you say that discrete mathematics is bottom-up and analysis is top-down?
I saw that on Dirty Jobs years ago!
Fascinating, closed curves on a sphere! Makes me wish for time to sit and do thought/real life experiments..
I'm a mechanical engineer and I had my education decades ago. We were taught to use an instrument called a "poolplanimeter" which operates or more or less the same principle.
Like this one? -> th-cam.com/video/aLSx1eM27P4/w-d-xo.html
I have several planimeters in my collection of computers. The ones I have were made to calculate area on a map and were either designed for one scale, or had a scale selector. And they're still being made and sold. Fascinating tools.
Polar planimeter. It integrates area in a polar coordinate system. I've use one to measure the areas on a map. It gives the area in square inches and you multiply that by the square of the scale factor to get the land area.
This reminds me of my Chem lab. We were tasked with integrating an area beneath a complex curve that was the output of chemical measurements of some sort. "How would you solve this?" The main responses from students was: "Make a mathematical representation, and integrate it through matlab." His proposal was: "Not even. Trace of the graph to a sheet of paper. Measure and then weigh the paper. Then cut out the portion that is under the curve. Weigh that portion. Calculate the area of the curve using those figures. The error of your measurements and the error of your curve estimation is greater than the error in the variation in the sheet of paper and the error of your weighing. Make and include these error estimates in your paper, and turn them in. This is the way to get the most accurate estimate, and in the shortest period of time."
@@takotime Well if you're going to use a computer... just do a numerical integration. You could also plot the data on graph paper and count up the little squares. (There is a technique for adding up the partial squares crossed by the curve.) The error will typically be less than +/-2 squares.
I've heard of the weighing method used with photos of galaxies and nebulae to estimate their area. And it's an interesting way to teach Kepler's 2nd law of planetary motion (the equal areas in equal times). Using thick card or (if you can) metal plate makes for more accurate weighing.
bake a sheet cake, draw the curve on it, eat the area under the curve, weigh yourself before and after.
@@takotime oh yeah- but this was 1996, before we all had cell phones and digital cameras everywhere…
@@dont-want-no-wrench An integrEATor then! I don't want to think about how negative areas are handled...
It blew my student's minds when I showed them how to build an electronic analog computer using operational amplifiers. Seeing the equivalent in a mechanical medium was very enlightening!
I would be interested in trying this. Do you happen to have a link to instructions on how to reproduce this project?
@@sleeplessdev7204 th-cam.com/video/LF35eXfCMIQ/w-d-xo.html
^TH-cam series about theory and construction of analog electronic computers.
@@sleeplessdev7204 You can search for differentiators and integrators using op amps, which is what he is referring to.
Electronic analog computer? That's an oxymoron. At best that's an electrical analogy to a mechanical system. Glad I never had a "teacher" like you.
@@toomanyhobbies2011 Actually analog electronics are pretty important. Any time you have to convert a real-world signal into a digital signal, you need an analog front-end before the digital electronics. The analog computer just skips the digital back end and goes analog all the way. Not so useful in today's world, but simpler versions with a little bit of analog pre-conditioning can make it much easier for the digital side to deal with the incoming signals.
This is a great project! I think it would be cool if you can make a series of analog mathematical machines since there seems to be no one doing that. Good work!
The ability to make mechanical calculating machines such as this one, should certainly be preserved and disseminated. One large X class solar flare and coronal mass ejection that hits the earth, could take out every electronic computer in the world at the same time.
I second this! Such a great way to visualize abstract concepts. I fell in love with machines like these as a kid when my grandfather, a former WW2 seaman, would take me on museum ships and tell me the same story over and over about the first tsunami they encountered in the Pacific. While the big guns are cool, seeing the fire control computers blew my mind. I was somewhat familiar with the 6502 instruction set and basic (mainly from spending hours trying to figure out how to get my c64 to display loops of text and flashing colors), but the idea of rotational motion "computing" blew my mind, and still does. I only wish I had remembered that excitement and sense of wonder when I took geometry and trig in highschool. If anyone is interested, a couple folks have put up a number of old US navy training videos on mechanical computing systems. Definitely give them a watch.
Here here!
@@mbaladon TH-cam recommended this video after I watched those two excellent videos you mentioned. They were both really interesting. The second video included a machine similar to the one shown in this video.
@@partinthompson6724 would you tell us your grandfather's story about his first tsunami? Sounds interesting
I'm 51 and I have never understood mathematics. The issue is relatability - I just don't understand abstract words and phrases but if I can see it working, it's simple. This is probably the only time I understood the process of what I consider more complex math from start to finish. You need to write a book on how to teach abstract using physical models - absolutely brilliant demo - thanks.
I think this is the problem with most people & math. It's so goddamn abstract it's hard to picture a representation to modify according to the calculations needed.
Never had this problem with physics, chemistry or biology. There's usually some force, some amount of molecules or something at least remotely tangible you do you calculations on.
Another thing your brain probably does is automatically recognize arbitrary concepts, to include nomenclature, and dismisses them with any more care than some strangers name. I'm certain that once YOU understand how to calculate something using any number of notation, that when visually looked upon on paper, you would know what to do and where to begin. But if the same thing was in word-format with the textbook nomenclature used in lieu of a mathematical formula, you would probably be at a disadvantage. You would be able to also be able to solve real world problems with your brain calculating hypotheticals in logic format. It is very likely that you are smarter than you are told by academics.
@@iguess2739 probably why its hard for people to do word problems, many classmates were complaining about that in my last classes.
I needed to do ground exams for commercial pilot licence. First book was maths revision. I wish i had that book at school, it made maths so easier to understand.
@@flybobbie1449what was the full title of the book, may i ask sir?
Anyone who has read about WW1 & WW2 Battleships has wondered about these things. Thanks for very clear explanation.
I've heard integrals explained in so many ways... this one is the one that speaks the most to me and makes me conceptualize it in my head
If you heard integrals explained in so many ways before then you would hear they're just area under the function. Because this is the first thing everybody says
Man, if only today's Internet and TH-cam existed back when I was in college. My first day of Calc I was basically, "Here's some equations, move this stuff around and cancel this stuff out. Now do your homework". Not once did my teacher tell us even what Calculus was for, let alone what it meant, let alone what the curriculum was for the semester.
Right? I never really understood what calculus was trying to show until TH-cam.
When I was studying to be engineer I eventually dropped out as there was so many bad teachers. Worst case was teacher that had his own engineering business. I drove hour to school,waited for couple hours and this guy could just drop message to someone that he’s out for the day. Other teacher made examples as class was going on and we could try to figure out something and after two hours in near end it was likely that he had fudged something and he had just said he’ll come back to issue at next class which could mean week after. There’s incredible amount of shitty teachers. Luckily there are some good ones,but they’re quite rare.
Waccoon? Now there's a name/character I haven't seen in years!
I had the same exact experience with my college calculus class. I got an A- but never understood what I was doing or how to use it in the real world.
My algebra one high school teacher was very much the same way. Some teachers are poor instructors.
This is so important to me, as I often struggle with the abstract nature of math. It's hard to build intuition around something that essentially only exists in my head. Seeing this concept as a physical machine really helped to ground the idea, freeing my mind for further thinking. Thank you for that.
My first exposure to a disk integrator was in a Roche QC lab sometime between 1989 and 1994. Probably closer to 1989. I actually taught a bunch of chemists what it was. It was part of an old strip chart recorder used for chromatography. Every Stem student should watch this video or one like it. This gets a like and a subscription. Nice work.
4 years of Engineering and today I learned clearly what integrals are!
similar
bro what
@@sdfggc4995 maybe he meant it by its pure definition, not how to do it or what to use it for.
In a way, I guess I'm the same too. EE but only knew how to do it, and only knew its application in approximation of values from graphs.
@@sdfggc4995in the current school system one is asked to pass a test by memorize and execute abstract procedures instead of really understand them and value their engineering potential. One can get a graduation scroll without understanding anything but being a good student, that is, working hard on subject's mindless but laborious tasks, hoping with faith to absorb some of the actual real knowledge from that method, many blessed are compliant with this system since it works for them just like that, because of their natural attention span and absorption of concepts, but others just want to finish study and start working to get money, something many, if not all schools dont realize.
Get your education in India, where everything is memorized?
Thanks a lot for such a high quality content. No annoying background music, no clickbait, no aggressive&meaningless talking. You deserve your every like and subscribe my friend!
The only thing omitted in these discussions is that the "follower", in this case the lead screw, must "follow" the curve to be integrated at a constant rate *horizontally. That is, it can't be permitted to "loiter" on some parts of the curve longer than others, or your total will be too high. The rate control is, in effect, your "dt".
The rotation of the disc must be coupled to the following of the curve, you may loiter around, as long as your disc is spinning slower/stopped during that time. If you had the curve plotted to follow it, so you push the paper sideways while moving the plate/follower up and down on the curve, the rate you push the paper is coupled to the speed of the disc.
Making this part fully mechanically would be pretty interesting, along with a dial to show the result at the end.
I really admire individuals like you. You not only grasp the concept but can turn around and teach it in such a simple form. Thank you for doing projects like this.
For those who want to learn this branch of mathematics, this video is very appropriate for the introductory stage, as it shows in an experimental and illustrative way how this process takes place. Congratulations to the author of the video and thanks for sharing this knowledge!
Hi Jack! This video is so enlightening! I'm a physics and engineering high school teacher and have never seen a mechanical integrator before! You've inspired me to want to construct one in class. Seriously this is great work.
que trabajo impresionante que has hecho, reduciendo al minimo las complicaciones tecnologicas, espectacular, gracias
I LOVE machines and I always have. I've spent my life around them, inside them, building them, repairing them, taking them apart and putting them back together. But this is the first time I've seen anything like these machines. Fascinating! Thank you.
It took me a minute to recognize how this works, but my mind was absolutely blown! It's so simple and elegant!
Awesome. I love mechanical computer tech. You should do a video on torque amplifiers too. They're an essential component in mechanical computers. They're used, for example, to take the weak output of a disk integrator and use it to drive the lead screw of another. A variety of clever mechanisms have been used.
This is beyond brilliant while staying humble and simple. Kudos! . 4 years of Engineering and today I learned clearly what integrals are!.
Thank you for this visual demonstration of calculus. Please keep creating videos like this to educate people on complex math concepts.
You not only explain math you just show it...mind blowing..
Would be nice to see more of you digging into this thematic.
In 1958 I was a U.S. Navy Fire Control Technician. You showed a photo of the mechanical Mark 1A Fire Control Computer that I was trained on.
It was a unit about 3 foot x 4 foot and about 4 foot high. It had knobs and dials on the top.
Its output would aim the 5” guns.
It was a mechanical masterpiece.
What an amazing video, if a teacher would have explained to me how integrals work in this way, truly my uni days would have been easy
That was a super cool project. Had to use solid works to design the parts for the 3D printer, fully understand calculus far better than you think did when you just take the course, learn a bunch about mechanical components & encoders, learn how to program & control a stepper motor system, and finally how to data log & put it all together into some useful plots & data. Wicked job buddy! Hope you got 100% on your project. That was a ton of effort & stuff you had to learn 👍
This video demonstrates a very abstract process using an amazingly intuitive explanation. Well done!
I learned more in these few minutes than I did in my first 4 weeks of calculus in college and it makes a lot more sense. I wish we had stuff like this back in the 70s.
That was fantastic! I just wish my calculus teacher in high school in the 80’s would have found a way to demonstrate what it all meant, how it worked and what it was for rather than making it a frustrating and boring math class. This did more to spark my interest and understanding than a whole semester of class.
Great job , in construction my early days some estimators still used slide rules and profit wheels , many concrete companies and suppliers still give out concrete calculators that gives you quantity based on a couple parameters of the area , it’s quick cheap and it’s their calling card very clever and efficient
We used to use a "Manning Wheel", which was a circular slide rule for hydraulic calculations about open channel gravity flow in circular pipes.
I never would've thought of this, I love it! What's really neat is that the number of rotations really is a literal distance traveled!
Your explanation of integrals is so elegantly intuitive. Thank you for making it easier to understand it. These machines are so cool. I also like how intuitive you made the integral of a sine function. I probably memorized the integral when I went over it in college, but I doubt I understood why it was as such. This is a great demonstration of why and a phenomenal demonstration of these pretty cool machines. Thank you for showing all of this.
I actually had responsibility for a gunnery system which had a "Potters Wheel" device to provide integration. In spite of my surprise at the continued existence of such a device, it certainly worked and we hit lots of targets.
Potters wheel, in gunnery control. We might have trained in the same navy (RN?).
well, unlike electronic cumputing devices, a mechanical analog calculating machine wouldn't be vulnerable to an EMP burst
same for my underwater fire control system. Diesel boats forever...
Excellent work. I also have a planimeter story. My father was a landscape architect who designed homes, schools and country clubs. He used the planimeter on his scale layouts (pencil on vellum in those days) to get the area of very irregularly shaped lawns and golf courses, in order to specify the correct amount of grass seed. I still have it and amaze friends with it regularly.
I finally understood what integrals are about, what a clear and excellent explanation!
so this is more or less like taking the theoretical "car" for illustrating basic integration, making it real, putting it on a smoothly variable-speed treadmill, and then reading the odometer to get the result... that's so cool! it's such a simple concept it almost feels like taking a blunt hammer to the problem and just doing it for real, but the fact that you can do it with enough precision and control and strap enough of them together to get precise solutions for complicated problems is wild
I wish that you had been my maths teacher! Such an interesting topic, well done!
I think it's quite nice that you can just scale the output of the function you're integrating and directly use it to control the distance of the output wheel to the center of the time wheel since the function is the rate of change of its integral
You're an aweome young dude! I love your work and how you demystified calculus. I can hardly wait to see your continuation of these videos.
This video finally gave me an intuitive understanding of why the area under the curve is an integral.
Impressive! I was convinced the integrator was a 3d simulation... so well done
Now I am intrigued as to what you can use the intergra information for. Would be great if you could offer the mechanical integrator as a kit!
Integrators are typically used in navigation systems based on relative information like odometry. You enter your number of steps and direction of each step, and out comes your position.
Beautiful clean description of maths by mechanical analogy. Please continue to teach the world Jack.
Awesome project and presentation. Thank you for taking the time to put this together and share it.
Nice video but I have a small comment:
in the introduction of "analog computers" a Curta calculater is shown. But a Curta is not analog! It is indeed mechanical but it is digital.
I came to say this as well! And it's not a small detail: a lot of people think electronic=digital and mechanical=analogue, but that's a misunderstanding that means you miss the philosophical beauty behind what digital and analogue mean.
The Hamann Manus R pictured just below the Curta is also digital.
I know of an integration method used by an analytical lab before computers were actually cool, they would use a highly precise scale, weigh a sheet of graph paper, and then when the results from the spectrograph printed out they would cut out the area under the curve and weigh it, then calculating the area from mass of the paper.
That is a extremely creative approach
That's wild! I love it.
This video literally blew up my mind. I've never thought of carrying out an integral with something purely mechanical.
Beautiful. Always wondered how mechanical calculators tackle calculus problems, and this is just the right video that answers that. Leaving a comment here to feed the algorithm, keep similar videos to be recommended.
When I was graduating in Software Engineering I attended to a class of basic Math where I was taught to calculate integral functions and I am sure that I was never told that an integral is the area underneath the curve (I JUST learned it after to watch this video). I think that attending to classes where you learn something you have no idea what is for is the reason because 99% of students hate to study.
i don't think there's any high school maths class that wouldn't teach you about the area underneath a curve part, you probably took a course that refreshes your knowledge on how to solve integrals, and not an introductory course like what you get in highschool, so you probably didn't listen in highschool
@@janzentwong8094 Sure.
Great video man!! I just get so mesmerized about earlier inventions.. like how do you think of something like that!!
As a young control system tech in the early 90s I had a fascination with mechanical PID controllers (proportional, derivative and integral) your video explains the working of the integrator controller very well.
Lovely video! I have always been intrigued by different forms of visualizations these mathematical concepts can take, and am always on the lookout for such avenues. This mechanical demonstration does a wonderful job in letting me think of integrals in more ways than one! Thank you for your work and good luck on your project!
Super cool! I'm not very good at math but how would DT be limited by the number of steps + gearing? I love the idea and learned a ton watching this video! Seriously so cool!
Hello Jack, this video is amazing! I'm doing my graduation project about make a mechanical integrator, but I'm stuck at how to realize the control of the machine by program, could you please give me some suggestions or recommendations? Thank you very much!
Interpret the step count of the stepper motor as you would with a 3d printer, working with steps per mm of linear movement. You just have to drive the stepper motor with the function you're evaluating.
linear interpolation function with the slope from 0 to 1 being interpolated to the stepper motor 0 to 1.
suppose that the analog values for the stepper motor go from 0 to 1024, and you want the slope to go from 0 to 100(in c++)
float slope;
float stepper_value;
//lerp function that i copied from another program
static float lerp(float a, float b, float f) {
return a + f * (b - a);
};
int main(){
stepper_value = lerp(0,1024,slope/100);
}
if you want the slope of the function to be variable, you just make it a variable and divide it by that variable instead of a constant;
this will get you only one timestep of the function, you can store all points in an aproximation of the function in an array or other data structure, and parse through each one of them over time, the array index would be T
in more abstracted terms:
lerp((minimum rotation of the stepper), (maximum rotation of the stepper), (x value of the function at this timestep) / (maximum value of the function))
@@RChero1010 Thank you very much!
@@eduardopupucon Thank you very much!
Currently a high school calculus student. This is an amazing video for students like me. I’m so happy to know that if all tech goes down, I can still build this device to solve for area!
Great video! This is the sort of informative and interesting clip that lifts my day. I'm now a new subscriber. Thank you.
Back in the day when I was studying calculus (over 50 years ago) my teacher suggested I was more suited to a trade when I asked for the practical applications of what we were learning. Times have changed for the better.
literally the instant the apparatus appeared on screen i could see how it worked and on the basis of what principles (points near edge of record moving faster than near center etc.) it worked.
what a brilliant idea.
I appreciate how he used an I-pad to write instead of paper. He literally tried to save the earth
Who here after vertasium's video :)
I think veritasium got the video idea from this
In 10 minutes you Taught me calculus in a way my professor from college never could and I finally understand
This is beyond brilliant while staying humble and simple. Kudos! 🙌🏻
Excellent and brilliant demonstration. Bravo!
One of the best videos I have ever watched. The explanation of the mechanical integrator is very clear. This video should be used to teach calculus.
Just WOW.
Best visual explanation of calculus I've ever seen
Amazing explanation! Thanks for putting this together. Especially after seeing Vertisium's video on computing tides with analogue computers.
Thanks for the great video and explanation of your machine! I never thought I would be able to understand even a tiny bit of calculus but you made it easy, BRAVO!
I wish I’d been shown this in A-level maths as this would have absolutely inspired me as a teenager. Thank you for making it all make sense 20 years later!
Didn't understand it before and now i 'm just amazed at how simple and beautiful it's mechanism is
thanks for the video, explain the integrals with simplicity and beauty
One of the mind-blowing videos I've seen in a while
I also tested analog computers that had a similar intergrator. This was many years ag. What a reminder.
Jack, This is an excellent introduction to mechanical computing devices. I hope you keep making videos.
This was absolutely amazing, would work wonderfully as a teaching aid, but I just love the history of mechanical computers
great work man !
Absolutely fantastic what a credit to you for explaining this subject so well with a great mechanical analog
What an excellent and yet concise explanation and demonstration of calculus.
Where were you 20 years ago when Calculus 1 and 2 were turning my nights into days? You explain Calculus better than any Prof I ever had!
The coding and design of the disk was genius. Great video!
A clear presentation of an elegant concept. Well done!
This is such a neat way to do it: taking advantage of the fact that the rotational velocity on the larger disk scales linearly with the radius to account for the height of your function in the interval.
Really well done video. Love your presentation style. Just the right pace to let the contents sink in for me.
Echt fantastische Umsetzung von Geisteswissenschaft zu Technik. Hat mich begeistert!
Never saw any content of your channel but this first one was remarkable for me. Simple and yet very demonstrative and objective with a perfect narration tone. Keep up! Congrats from a former helicopter pilot from Brazil.
it boggles my mind how creative and smart humans can be, like understanding calculus is one thing and to implement it in such a practical and precise way which is so obvious is such a satisfying thing...
Dude, this is really cool. It’s awesome to see how mathematics are directly related to physical phenomena.
I hope you do really well in your studies son. Thank you for your genuine from the heart video.
it's incredible. I have not learned calculus in any form yet but i feel like i learned the goal of calculus and some of it did make sense based on what i know. Math is awesome
That is so beautifully simple of a teaching tool! Well done!!
Very interesting and very entertaining. Also stated in a way which was clear and easily understandable. Hope to see more from your channel in the future
Thank you for this video. I am designing a mechanical integrator as part of a PID controller for controlling an air conditioning system that works entirely free of electricity. One of the problems I was having trouble with was the interface between the two wheels. Of course, this problem was solved long ago by using a ball on a cylinder. However, manufacturing a precision ball is not as straight forward as making the interface with two wheels. Your solution is a helpful innovation.
Being knowledgeable with a smile is even better! 👌 Thanks for sharing ! ❤
Excellent video. Explained integral calculus and the mechanical integrator very comprehensively. explained
This is a beautiful explanation with a beautiful mechanism. Thank you
Wonderful and crystal clear explanation!
A really beautifully-done machine!
This was a very cool demo and project.! Thanks for sharing.
That is so intuitive and tangible, thanks for a fantastic video
This is fantastic! Beautiful design and construction, and so well explained and easy to follow. I can't wait to see what other amazing things you can build!! :)
I like how you described this. Thank you for making this easy to understand!