An Old Mechanical Computer: The Harmonic Analyzer
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In this video you will learn more about Albert Michelson’s Harmonic Analyzer: a 100-year-old mechanical computer. Get some interesting facts, informations and explanations about this topic, easy to understand and compellingly presented.
Free PDF: www.engineerguy.com/fourier
"Albert Abraham Michelson was an American physicist known for his work on the measurement of the speed of light and especially for the Michelson-Morley experiment. In 1907 he received the Nobel Prize in Physics."
en.wikipedia.org/wiki/Albert_...
This video was made by another CZcams user and made available for the use under the Creative Commons licence "CC-BY". Source channel:
/ engineerguyvideo
Like all things genius this machine is amazingly simple and intuitive. Anyone can find an over complicated solution to a problem, but finding a simple one is what is truly impressive
This IS the Simple One!
It’s absolutely mind boggling to think that before this machine was built the inventor had a picture in his head of how it needs to be put together. UNBELIEVABLE!
Fourier Analysis is one of the most abstract subjects in Engineering. Using this machine during my university years would have opened up a new galaxy of knowledge. Thank you for this Prof Hammock.
The secret to this machine's success is the Fourier Inversion Theorem, where the same process of of synthesis does analysis. The synthesis is just using engineering tricks to generate cosines and add them together, but the analysis is mathematical beauty.
The knife edge was common for suspending pendulums on clocks and also on precision balances for chemists, and yes it has low friction and that's why it is used.
Do you mean like in The pit and the pendulum?
This device emphasizes why “hands-on” trades and technologies are so important today even though we have digital virtual computerized tech.
The gears and spacial dynamics are just a glimmer into how our hands and brain truly work.
This tactile need is strong in man.
We need to build things to understand the abstract at the cosmic scale and at the molecular scale.
Awesome....amazing...astounding.....and so humbling.
Thanks for all your efforts in making this video and the beautiful book, too!👏
Books are so simultaneously visual and tactile. They are necessary for us mere humans to learn from the geniuses.
The "inverse" operation is not as crazy as it seems. The Fourier transform is self-adjoint. This means that the inverse transform is essentially the transform. Essentially the machine only actually works one way. It's how you interpret the input and output which determines it's function. E.g., a sinc->square and square->sinc or squaresinc. If you interpret the square (wave) function as representing coefficients of harmonics then you'll get a sinc time function. If you interpret it as a time function then you will get a sinc frequency function.
That's cool. I haven't played with the math in a while, but I thought the Fourier transform gave a real and imaginary solution while the inverse required both as input. Are we talking about just the real magnitudes, then?
@@beenine5557 For real valued functions the imaginary part is irrelevant.
There is only one difference in the inverse(besides scale) and that is a sign. exp(-ipiwt) vs exp(ipiwt)
Yes, when you take the Fourier transform of a signal you will usually get a real and imaginary part but for real signals you do not need the imaginary part and it is thrown away as it is extraneous. Why? Because the real part has a degree of freedom since the imaginary part is not specified. This means the "phase information" is not defined for a real valued function. One can then, in effect, pretend an imaginary component actually existed in the input in a way that canceled the one of the components in the output. It is then similar to "taking the magnitude".
That is, there is no difference between the Fourier and inverse Fourier transform except one deals with frequencies in opposite direction(e.g., clockwise vs counterclockwise).
So you can literally use the Fourier transform to calculate an inverse transform.
This is what is meant by self adjoint(it's not exactly self adjoint due to a sign and sometimes a scale but that is simple to compensate).
If you take a square wave and take the Fourier transform of it you will get a sinc function, if you take an inverse Fourier transform of a square wave you will get a sinc function. The only difference will be sign and potential scaling factor.
The inverse Fourier transform *is* the Fourier transform(again, except for very minor differences). the point is that any Fourier transform device/algorithm will also work as an inverse Fourier transform device with almost no modification.
Check out the wiki articles on the Fourier and inverse Fourier transform.
While some functions will have imaginary components when transformed they can be effectively ignored as long as the function is absolutely integrate.
If you have int(f(t)*exp(-ipiwt)) and f(t) is real then the imaginary part only comes from the exponential and so, in some sense, if irrelevant to f(t).
In fact:
F(w) = int(f(t)*[cos(piwt) - i sin(piwt)]) =
int(f(t)*cos(piwt)) - i int(f(t)sin(piwt)) =
but F(-w) = int(f(t)*cos(piwt)) + i int(f(t)sin(piwt))
and so F(w) + F(-w) is a real valued function
So, in effect what we have are actually just cosine transformations and not Fourier transformations but the difference is only scales and signs.
In some sense the machine doesn't know time or frequency. It's just computing an algorithm on real values. So if those values you put in are suppose to represent time then the output will represent frequency. If what you are putting in is suppose to represent frequency then the output will represent time. The machine doesn't know or care what the units ofo your variables are(although they will always have to cancel out w*t has to be unitless: 1/s*s = 1).
The Fourier transform is really just a transform that is able to take a set of inputs and transform them to a set of outputs that can be reversed(it has an inverse). Such transforms are useful because they give us a different perspective on data.
In fact, basically all the transforms are essentially the same. the Mellin transform is a sort of more general Laplace transform and the Laplace transform is a more general form of the Fourier transform. All this stuff is really more general than "finding the area under the curve" as we are taught. They are actually symmetrical operators on abstract group theoretic structures and, in fact, known as functors(things that transform structure from one form to another but doesn't add or remove anything, at least when certain conditions are satisfied).
@@jsmdnq Thanks for the detailed info. I can't honestly say I understood everything you wrote, but it helped.
@@beenine5557 Well, again, just look at the formula: int(f(t)*exp(-i*pi*w*t)) vs int(f(t)*exp(i*pi*w*t)) (excluding scaling factors).
They are, effectively, the exact same(the minus sign is important for the mathematical invertability but it won't do anything for the mathematics).
If you literally reverse your function in time f(t) -> f(-t) you'll convert a Fourier transform in to an inverse transform. Same works for the frequency.
The idea is that they effectively are mirror images of each other sorta like how you are in a mirror. You aren't different, just "flipped". You might think it can't be that simple but it turns out it is. It's just a glorified addition and subtraction. Same how -3 is the additive inverse of 3.
It's only complicated looking but under the hood it's very simple. The idea is that we can take our function's values and distribute them on sinusoidal so that each sinusoid holds a "value"(the amplitude) and the frequencies separate those values. Then we do math on the sinusoids. Because of the property of sinusoids it helps us keep track of all the values we are transforming so they never get lost(sort of like a bijection) so we can invert them by undoing/unwinding the math.
If we just added all the numbers together, clearly it is not invertible. 2 + 3 = 5 = 4 + 1. So we have to sort of complify(add more dimensionality/degree's of freedom) to hold our data.
if you know anything about generating functions then it is the same idea. E.g., how 1/(1 - x) = sum(x^k) and 1/(1-x) "holds" the sequence of data 1,1,1,1,1..... and 1/(1 - x^2) holds the sequence 1,0,1,0,1,0.... and 1/(1-x)^2 holds 1,2,3,4,5,6...
By attaching our data(the values) to x^k we can keep track of the values so they are not lost in the computation like in simple addition.
In this case we compute the sum to find the "transform" and we can invert it by finding the sum(e.g., essentially compute the Taylor series).
The Fourier transform is very similar to this idea but instead of polynomials we are using sinusoids(but it's actually the same, it turns out. Not the same results but the same abstract machine underlying it all).
It's a huge subject but the thing to remember is that things sometimes are much simpler than they seem. The complexity stems from technicalities and/or "size". At the end of the day everything is simply addition and, at the end of the day all we have is 0 and 1(all mathematics comes from the complexification and analysis of how to combine and separate out 0's and 1's).
Synthesis. Very nice. 32:29 "magnalium" I translated "lightweight wheel" to mean: wheel with low moment of inertia. I have been working on a quantum computer and they involve many of these concepts, so I appreciated it from the perspective of history and that they went through such detail. Sincerely appreciate you for working with one and describing your practical experience with it.
This machine reminds me of the tide predicting machines of the late 1800's and early 1900's. In these machines up to 10 different components of various non-integer frequencies are added together to produce a tide chart for a particular location. Each location has different amplitudes for each component but once known they should stay the same for that particular location. It took into account rotations of the Earth, Moon, Sun and shorelines. With the tide analogue computer, they could generate a year's worth of tide data in about a half-hour of cranking the master wheel. See the wikipedia on the tide predicting machines.
A long time ago they had one of those at the Seattle Science Center. I don't know if they still have it on exhibit, but it is well worth checking out.
Brilliant machine, and video. Richard Feynman had mastered this machine at age 17 already - "A seventeen-year-old freshman, Theodore Welton, helped some of the older students operate the wind-tunnel display at the Massachusetts Institute of Technology’s Spring Open House in 1936. Like so many of his classmates he had arrived at the Tech knowing all about airplanes, electricity, and chemicals and revering Albert Einstein. ... With most of his first year behind him, he had lost none of his confidence. When his duties ended, he walked around and looked at the other exhibits. A miniature science fair of current projects made the open house a showcase for parents and visitors from Boston. He wandered over to the mathematics exhibit, and there, amid a crowd, his ears sticking out noticeably from a very fresh face, was what looked like another first-year boy, inappropriately taking charge of a complex, suitcase-size mechanical-mathematical device called a harmonic analyzer. This boy was pouring out explanations in a charged-up voice and fielding questions like a congressman at a press conference. The machine could take any arbitrary wave and break it down into a sum of simple sine and cosine waves. Welton, his own ears burning, listened while Dick Feynman rapidly explained the workings of the Fourier transform, the advanced mathematical technique for analyzing complicated wave forms, a piece of privileged knowledge that Welton until that moment had felt sure no other freshman possessed."
I just found this video at random but I watched it till the end because it was intriguing. I don't really understand these things, but I wanted to comment and give my respects because of how well made this video is. Very thorough and detailed.
Springs used as a summing device is the most interesting aspect.
Analogue computers are good because as said in the video they show physically what is the process whereas digital is much more abstract.
The x-y pen recorder I used at work 40 Years ago had a chopper stablised valve amplifier. Significantly the pen was a little cylinder container of ink with a filler hole and a inverted u shape capillary tube in which the tip barely touched the paper under the negligible weight of the capillary tube. The assembly was moved x-y motor and wire driven mechanism. Despite the age of the instrument it worked reliably but with a little jitter from the chopper stabiliser evident on the paper.
There was a pen lift leaver to lower the pen whilst you carried out a recording it also recorded Y-t (time) if you chose that.
It's hard to believe you completed all this work on the analyser in only two years. Thank you so much for the video. Very enjoyable. I'm sure Michelson would have been mightily impressed with the Sharpie! As a hydrologist of a certain age I remember river level pen chart recorders before the advent of the fibre tip pen . The pens consisted of a tubular nib connected to an open cylindrical ink reservoir which could be topped up with an ink dropper. They were horrible beasts to deal with, with many dribbled charts as well as dry scatchy tracks and very inky fingers! This was made worse by the hygroscopic nature of the ink designed to suck in atmospheric moisture to stop the ink drying up. Fine in a dry climate, not so good in Scotland where humidity can hover around 90%+ for days on end! Good old days? Ha ha.. give me a few megabytes of solid state memory to record data any time!
Incidentally 3blue1brown does a great explanation of Fourier analysis.
When I was in graduate school I used to attend the group meetings of a professor who specialized in quantum mechanics. His group used to look into things like this to solve equations since they were the physical embodiment of the principles involved and were far less complicated to do than program a computer.
Beautiful work! Thank you so much for taking the time to preserve this amazing example of mechanical analysis--a brilliant exposition of mathematics and mathematical insight.
This is quite an amazing machine! I was lucky enough to see one of these in summer school as a kid. i tried to explain this thing to people through the years, yet didn't know enough about it, nor could recall it's name, to find any information about it, to share it with anyone. Thank you for bringing this to video! Quite an amazing feat to build it!
@engineerguyvideo The flat peaks and troughs seen at 22:09 are intriguing. Even more interesting is the angle of the pen required to fix them.
I believe it is actually a sign that something they've done in the machine restoration is incorrect. A felt tip pen has more friction than a pencil (fibers versus graphite) and the video said a pencil has too much friction. Turning the Sharpie so that it points _against_ the motion of the paper (rightward) causes even more friction than other orientations. Imagine the fibers of the pen tip colliding into fibers of paper but unable to fold out of the way as they could if the pen was pointing to the left.
You'd think pointing to the right would be worse, but not all friction is bad. First notice that this friction is in the X-axis, not the Y-axis, so it does not directly affect the ability to go up and down. I hypothesize that in this case the restorers accidentally found a way to use the excessive friction in the X-axis as a way to reduce friction in the Y-axis. I believe that the pen is vibrating minutely away from the paper hundreds of times a second. When the pen jumps off the paper, it is able to freely move up or down, thus allowing a proper graph to be drawn.
This hypothesis could be easily tested. Simply try something with much less friction, like a paint brush, to see if it has the same orientation requirement as the Sharpie. Alternately, measure the vibration of the felt tip pen at different orientations.
I'm pretty sure the Harmonic Analyzer did not use a Sharpie originally. Perhaps it was a fountain pen as the video guesses, but I would suspect a well of ink and a glass tube, which would have almost no friction. If so the orientation of the pen would not have mattered. [Glass also fits with the evidence that this extraordinarily complex machine was missing only a single piece. It likely shattered at some point.]
About those mysterious holes in the rocker arms (see 28:00)...try removing transfer bars and reattaching them to the "inner" hole for 10x magnification on the y-axis. Come to think of it, this gives you a range from -100 to +100 for each component wave
What an elegant and beautiful machine. If they're not doing it already, they should bring it out during the lectures on fourier analysis.
Every Engineering Student should be given the diagrams to build their own with a 3D Printer, and put it up with their Antikythera Mechanism above the Desk.
I would walk past this exact machine almost every week in college, it was always so facinating
What an amazing piece of work! Not only the machine but also the video and the book!
What a beautiful machine, it make mathematical functions visible and very understandable!
Elegant work on a beautiful machine... Thank you for your dedication in producing this work.
This channel did not make this piece. It was made by Engineer Guy, who can be found here: czcams.com/users/engineerguyvideo
He makes a lot of great and original content.
To add the components with springs certainly the idea that distinguishes this mechanical masterpiece from the tide predicting machines. There, you add up with a string passing multiple pulleys. The friction adds up in this process. Here, this is not the case. Despite all the disadvantages springs usually have compared to weights, here they are the winning move.
This is extremely well done, thanks for your work.
Can't you align the notches in the gears with a thin strip of wood. That fits through all the notches. Seems more accurate than doing it by eye.
I wanted to see what the summing springs were doing during the square wave. I agree about the eccentric hubs. They're commonly used in steam engines too.
I always found it interesting that while Fourier analysis is the inverse of synthesis, it's ultimately the same operation.
They are slightly different, you need to invert the x value, but yes, it's pretty remarkable.
Michaelson & Morley . I am fascinated about the „easiness“ to deskribier Natur.
What an incredibly clever machine! And excellent video(s). Thank you!
Amazing machine!
The holes in the rocker arms are probably there so that the arms could be machined to match.
Machinist's parallels have similar holes for matching up during grinding and machining.
Extremely fascinating
Thank you for the narrative story.
I have an underhand, an exfended underhand and an overhand pencil grip.
Thank you so much for that. I enjoyed every second of it.
To be fair the first harmonic analyser was made by Lord Kelvin for his Tide Predictor, and this one seems very similar in design. But still yes Michelson was an amazing physicist and apparently engineer..
22:29 Why are the peaks and troughs flat when the pen is oriented with the direction of travel of the paper? I would have thought that the pen offered the same amount of resistance in that case than it did when it was orientated against the direction of travel of the paper.
you have to marvel at the genius
I can't explain myself how it's possible that it doesn't have any more views. Amazing. Thanks
Obviously, the people back then did their best.
But, with so many linkages between input and output and the dimensional or alignment error adding from each, there must be significant error in the final output.
However it's good to see the old mechanical analog computers. Prior to these YT videos on them I'd thought that the Babbage machine was the only one of significance.
Oscillators in digital synthesizers have been setup for doing such additive synthesis. Back in the 80s this was very expensive so Yamaha used fm synthesis instead where so called operators could be used as argument for other operators. So you could get stuff like sin(sin(t))
Oh cool. Engineer guy
A practical use and application of this machine in civil engineering, aeronautics, statistical analysis, even the census, would be helpful to the layman viewing this video. The sines and cosine look like radio waves.
In the beginning you say "mechanical computer" but I feel you should have emphasized that it's an analogue computer. Many people already know about mechanical computers that calculate using discrete movements based around numbers and digits.
Very clear and detailed explanation.
Well done, and an amazing device.
I always wondered if this machine existed. I had heard that there was a machine that could do fourier analysis, but I never found it... Has anyone attempted to come up with CNC plans for a model of this calculator? Perhaps 3D printed or cut from acrylic? Except for the springs and a few bits to inset for the knife edge pivot it look like there are no difficult parts. The main big of precision seems to be with the balanced spring tension system. Is the friction of the amplitude bars with the rocker arms critical?
seen one of this kind in the science museum of London, in the mecanical calculator section
It's coming right up. I'm currently drawing one and will be publishing it soon. This video gave me so much inspiration to do it !
@@jackcarterdrelias Did you ever finish? Can you link us to your models?
@@beenine5557 I don't think my answer was posted due to link. I hosted it on Instructables as "Harmonic Analyzer : Mechanical Lasercut Signal Plotter". It was already almost 3 years ago ! Time goes quickly. It is really simpler than I had in mind back in time, but it does the job with not much hardware, just a few bearings and springs.
Thanks Bill. What a beautiful machine....👍
I think some kind of Machine like this (mecanical Fourrier analysis), is on display in the Science Museum in London, and (if my memory is accurate enough) i remember reading this could be used to "scan " recordings of history line graph of sea level" to calculate exact tides fourrier parameters to produce accurate table of tide amplitude for complete years in the future (related to sun/earth/moon interaction that is being some kind of periodic based phenomenon of multiple composants of different periods
sombody from London can confirm this ? i rememenr being blown seing this a a student engeneer 25 years ago .
I;m no Londoner but that's really cool.
en.wikipedia.org/wiki/Tide-predicting_machine
i have a bunch of synthesizers. pretty neat to see this.
Fantastic. In awe
why wouldn't you just use a sheet of paper or something similar set into the notches to align the gears instead of awkwardly eyeballing it? kinda seems obvious that that's what the notches are actually for.
thanks a lot, gave me ideas for an analog computer using OPamps
This is amazing.
Awesome video! Thank you!
thanks a lot!!👍
никогда не испытывал проблем с рядами Фурье)
видать поэтому все просто и понятно в видео)
Welllll over my head but fascinating and beutiful none the less!
Congratulations for the video!
I wonder if they could have used a fountain pen setup such that the nib barely did not touch the paper, but transferred ink through a connecting droplet that wicked ink to the paper via surface tension.
great design using cone & cylinder to explore circular & circus, arc . The secret hole has something to do with the rest of the other 2 set. How I wish to see 3 of them to crash the secret code
Its like a mechanical version of that multiple pendulums with decreasing length demonstration. Raise the weights then let them swing, the differing lengths makes them oscillate at different rates so they start closely in phase then drift into a series of stable nodes eventually to essentially chaos. Then begin working itself back to in phase.
......yeah, that thing. This looks like that to me, well, at least the gear cone bits at the bottom and levers, but like.... mechanical. Which means its like that pendulum whatsit but can be frozen in time.... [Mindblown.gif]
So it's basically an additive synthesiser, similar to an electric organ, but much below audio rates
What's impressive is that it's entirely mechanical.
How beautiful...!
Nifty AF !
Somebody really needs to make something like this in algodoo... so I can download it and play with it. lol
So amazing 🤓
Is a very good presentation ❤️
For the Rocker Arms of min 27:00, Could be that the misterious holes on them are to hook the springs in a different configuration of the machine??
you would also see these on a oscillascope . lovely machine .
If you'd covered the four videos into just this one, I'd have watched it.
? That's what was done ...
So cool!
My brain has burned
Can't have spectroscopy without a container!
The displacement of the nth rocker bar (as determined by the coefficient lever positions and cog inputs) translates into a force .component F_n = k x_n on the common summing bar. The summing bar’s position x_s is linearly related to the summed force F_s = k_s x_s. I am not clear why the “counter spring” for the summing bar has to be so long given the motions are very small. Is it to ensure linearity in the spring (ensure k_s constant) or it may be to allow very small adjustments in the equilibrium force that the spring exerts? Any insights?
I've never seen one of these but i get it
Well, I'd love to have that book on my coffee table, but wow - a hundred bucks????? That seems kind of excessive.
grande) thank you!
wow amazing machine
Didn't he invent the interferometer?
Neat stuff. There is a USAF major that re-did Michelson-Morley with present state of the art and his result was that the Ether exists.
how to change the phase?
So Cool ! Thanks for the post . Anybody out there making a FLOWDAC , patent 3190554 , that can say compute log tables?
CAN LEGO DO--THATé
seeing math in action.
WOW!
What real-world problems was this machine used for?
Wow! This is really interesting, beautiful! The perfection of creation, the ingenuity of man, all reflecting the Supremacy of The Creator, The Most Beneficent! SubhanAllah!
holy sh!+ that is cool
Wowww
The poster of this video is dishonest- this is NOT his material. Please look for the engineer guy on CZcams to find the original.
off topic:
why did they used to use vacuum tubes as transistors? u can make a transistor easier than tubes that break all the time! why use vacuum tubes at all?
Because there were no transistors back then
ive seen modules doing what transistors do, made out of knex or legos. they couldve done it lots of ways, WHY LIGHTBULBS?!
alot of things you could make with legos, could be made with steel. what advantages do vacuum tubes have over a metal transistor? because it wont be reliability. you know what i think? i think you just googled it to sound smart, but didnt find anything in there to specify why they used vacuum tubes, so you gave me a summary of what you read. am i right?
Dgs... How old are you, kid? Also as a preemptive edit: Your little idea was tried, they were called relays. The problem was they required lots of power to energize and de-energize. The vacuum tube required less energy to do the same job, and did it faster. Tubes also had a longer life and lower failure rate. That lead to faster computers and by extension more overall interest in the things computers could feasibly do. Without those tubes you strangely want to shit on, there wouldn't have been the computer you just typed on.
you could just say "yes, i googled it.. i have no idea why they use tubes im sorry" also, give source cuz i dont know wich wiki page ur getting this off of
But can it run Crysis?
Yes - obviously - lol
Why are you sitting in the dark?
Wait. I've seen this. So this channel just reposted TheEngineerGuy video and slapped a crypto ad on it? Gross.
i thought it was neil degrasse tyson
Interesting video but I can't handle the pause between sentences, I ended up watching the video at 1.5x speed to get through it
dude i can see the agressive wear on the teeth, please clean the old grease/grit off before using that anymore!!!
Riveting. Seriously.
This generic talker, generic hand motion guy is awful.