If you’re into mechanical devices or Fourier series (or both!), you’ve got some serious YouTubing to do.

[The Engineer Guy] has posted up a series of four videos (Introduction, Synthesis, Analysis, and Operation) that demonstrate the operation and theory behind a 100-year-old machine that does Fourier analysis and synthesis with gears, cams, rocker-arms, and springs.

In Synthesis, [The Engineer Guy] explains how the machine creates an arbitrary waveform from its twenty Fourier components. In retrospect, if you’re up on your Fourier synthesis, it’s pretty obvious. Gears turn at precise ratios to each other to create the relative frequencies, and circles turning trace out sine or cosine waves easily enough. But the mechanical spring-weighted summation mechanism blew our mind, and watching the machine do its thing is mesmerizing.

In Analysis everything runs in reverse. [The Engineer Guy] sets some sample points — a square wave — into the machine and it spits out the Fourier coefficients. If you don’t have a good intuitive feel for the duality implied by Fourier analysis and synthesis, go through the video from 1:50 to 2:20 again. For good measure, [The Engineer Guy] then puts the resulting coefficient estimates back into the machine, and you get to watch a bunch of gears and springs churn out a pretty good square wave. Truly amazing.

The fact that the machine was designed by [Albert Michelson], of Michelson-Morley experiment fame, adds some star power. [The Engineer Guy] is selling a book documenting the machine, and his video about the book is probably worth your time as well. And if you still haven’t gotten enough sine-wavey goodness, watch the bonus track where he runs the machine in slow-mo: pure mechano-mathematical hotness!

This is easily the coolest thing I’ve seen today!!

Very well explained. I like his Videos.

I saw the introduction teaser some days ago and in the description he was stating to release one video of the series every other week or so. But thankfully, he published all the videos on the ~next day without the tease. Imagine you see this short video and then have to wait a week and pick up the topic again for another 3 minutes. And again. And Again. Good on you Bill Hammack. That was a nice move.

about 20 seconds into the intro video, I was starting to worry that the explanation videos were going to be really dry and kinda boring. It’s the first time I’ve watched any of Bill’s work. He’s actually really good. I liked his presentation style. He manages to explain complex things simply without treating the audience like idiots. It’s a fine balance and he nails it.

full ack, yes. That he is going the extra mile of (presumably) hiring someone to even visually enhance his videos is the part where I really admire his work. No slides and a deep voice from the off with a noise microphone. All the overlays and animation make the understanding so much easier and promoting the videos for a serious educational use in schools as well.

Oh, definitely go back and see his other videos then. I like the way he adds a little bit of humor to lighten a dry topic. One of my favorite videos of his is the one on the typewriter and how they used digital inputs (2 solenoids) to create a analog linear motion to select the correct head on the “Selectric” typewriters

Thanks … We looked at the comments (I read ’em all) and thought it the right move to get everything finalized and out. We were mesmerized by this machine and I’m glad we’re able to share it with other.s

This is absolutely incredible… mega kudos to you and everyone involved in the project and documentation process! Wow!

This whole series was wonderful; I am definitely thinking about picking up a copy of the book, but thank you for the PDFs and the videos. I’ve never seen this topic explained before in such an easy to grasp manner; furthermore, seeing it done mechanically made the mathematical expression of it easier for me to visualize.

Lastly – I loved the delivery of these videos; your cadence and delivery of the topic reminded me of late-night infomercials! Throughout, there was this feeling of the delightful “How much would you pay for this wonderful technology? Only 3 easy payments of $19.99! But wait, there’s more!” patter so typical of such commercials – but you actually managed to deliver on these promises: Education, enlightenment, and entertainment all rolled into one!

Thank you!

The first person that can 3d print one of these gets a free beer…

=)

Laser cutting or CNC is more likely, laser cut wooden gears, rockers and cams, sell it as a kit. One of the best explanations of FF analysis I have ever seen. All electronic, maths and coumputing departments should have one.

Compare and contrast with the very dry Fourier Analysis Wikipedia page. http://en.wikipedia.org/wiki/Fourier_analysis … yes my aged brain can (slowly) trawl my way through the mathematics, but with this I can actually visualise what is going on.

very interesting videos.

Just stunning. After watching “Analysis” video I’ve FINALLY after all those years understood fourier analysis from my signal processing classes.

Fascinating.

How were these machines used? I’m not sure if it was here or the other article but I did catch that the original was for spectrum analysis in astronomy. I don’t get it though.. how do you get from light in a telescope to a waveform that one could set the levers of the machine to. And do it without modern electronics!

I know that Fourier Analysis is incredibly useful today for signal processing of various kinds. That takes continuous processing at a speed far greater than the machine could run not to mention a human setting the levers.

So.. what were they using it for back then?

Actually, he used it to calculate light waves recorded with an interferometer(Michelson interferometers are still used). Using optical interference, you can produce a pattern that allows you to measure the frequency of light, no electronics needed.

See http://amrita.vlab.co.in/?sub=1&brch=189&sim=1106&cnt=1 for how that works.

Harmonic analyzers were also used in experiments with sound and earlier ones were used to calculate tides and orbits.

So cool to see this machine get its due! I got my masters in Math at UIUC, and I walked by this machine almost every day – it was just sitting in a glass case in a corner of a hallway; no information about what it was or who invented it. I spent a fair bit of idle time trying to determine how it worked – it was pretty clear that it was some sort of analog Fourier Transform computer though. Yay!

I wasn’t (and didn’t expect to be) very surprised to see this machine do synthesis but I was blown away to see it invert the function to do analysis.

Download a copy of ZynAddSubFX and go four pages into it and you can play with 128 harmonics’ intensity and phase. Then you can see the waveform and hear it too.

It’s machines like this that makes me wonder, are we just really rediscovering or reinventing things that have already existed in a different form? That and how steam-punk may not have been that far off from how it could have been. I can totally see this running on steam. As for the book, the equations Michelson used in designing the machine looks suspiciously close to the equations used for addition and subtraction when applied to a slide rule. If true, does that mean log approximations are possible? Hell, Fourier’s work is based on Taylor series expansion with allow for polynomial approximation of a function, so surly it’s possible right?

Years agoe, in college, I saw an illustraited description of a Fourier Analysing Machine that used gears, a rack, and a compensating polar planimeter to mechanicaly derive Fourier Coefficients. Having the planimeter I would like to find this reference and try to recreate this device. Thank you for your assistance. Bill Williams