Michelson's Mechanical Harmonic Analyzer (Adding Cosines and Doing Fourier Analysis)
JRetSapDoog
Posts: 954
The following introductory video (that links to 5 or so related videos) may be of interest to mechanically- or mathematically-inclined readers.
http://www.youtube.com/watch?v=NAsM30MAHLg#t=91 (Note the related videos)
In a slick set of videos, Bill Hammack (engineerguy) of the University of Illinois gives a detailed and visually-rich walk-through of Michelson's machine which can add sine/consine waves and do a kind of Fourier analysis.
Don't tell Chip about this or he'll suspend P2 development to try to figure out how to add an electronic equivalent as an addition to his CORDIC circuitry (hence my posting in the General Discussion forum), lol. Then again, it might be something to pass the time during the long, lonely compiles he faces once he resumes Verilog development after wrapping up the full-custom schematic work towards the end of this month (see his post here).
For further information on the machine, see the related book (by Bill Hammack, Steve Kranz and Bruce Carpenter) at the link below (note a page-by-page video walk-through of the book is available from the above YouTube video):
http://www.engineerguy.com/fourier/
Source Credit: I stumbled upon this presentation while perusing Gizmodo (though they gave the topic their own slant) at the following link:
http://gizmodo.com/the-century-old-machine-that-gave-us-mp3s-and-jpegs-1657267850
http://www.youtube.com/watch?v=NAsM30MAHLg#t=91 (Note the related videos)
In a slick set of videos, Bill Hammack (engineerguy) of the University of Illinois gives a detailed and visually-rich walk-through of Michelson's machine which can add sine/consine waves and do a kind of Fourier analysis.
Don't tell Chip about this or he'll suspend P2 development to try to figure out how to add an electronic equivalent as an addition to his CORDIC circuitry (hence my posting in the General Discussion forum), lol. Then again, it might be something to pass the time during the long, lonely compiles he faces once he resumes Verilog development after wrapping up the full-custom schematic work towards the end of this month (see his post here).
For further information on the machine, see the related book (by Bill Hammack, Steve Kranz and Bruce Carpenter) at the link below (note a page-by-page video walk-through of the book is available from the above YouTube video):
http://www.engineerguy.com/fourier/
Source Credit: I stumbled upon this presentation while perusing Gizmodo (though they gave the topic their own slant) at the following link:
http://gizmodo.com/the-century-old-machine-that-gave-us-mp3s-and-jpegs-1657267850
Comments
I love the way the summing springs work and that conical gear set is damn cunning.
Presumably as you can easily adjust the thing to multiply by sines or cosines you can also do complex transforms by running it twice.
It would have been nice to have one of those machines in school. It would make understanding synthesis and analysis almost immediately obvious before even getting to the maths.
http://books.google.id=jCSpiVBH5W0C&pg=PA366&lpg=PA366&dq=Michelson's+Harmonic+Analyzer&source=bl&ots=o4sghbTjPN&sig=mRUNOWCSeFrcUIPBK3rmEh6MBh4&hl=en&sa=X&eiWJoVLWAPYTTygPFhYCQCA&ved=0CDwQ6AEwBjgK#v=onepage&q= Analyzer&f=false
Interesting history.
About the link, the top-level domain (e.g., ".com/") seems to be missing. Perhaps you cut it out by accident while removing a local country code or something. The following link seems to work for me (though a country code for the country I'm in gets added during an automatic redirect, I suppose, which I've excised from the link):
http://books.google.com/books?id=3FvELn2KiUYC&pg=PA52&source=gbs_toc_r&redir_esc=y#v=onepage&q&f=false <-- Chapter 6, pg 52 (the 1st pg. of the chapter)
I'm not completely sure about which section you were referring to, but I see that the "harmonic analyzer" is mentioned on pg. 54 of chapter 6 (to which the link above points). Chapter 6 is entitled, "Michelson, Fourier Coefficients, and the Gibbs Phenomenon. Thanks for the related information. No study of the history of computing would be complete without it.
Update: Whoops! Based on closer inspection of your link (fully expanded), it appears that you were referring to pg. 366. However, I guess I previewed too many pages of the book to access that part (or that part is otherwise unavailable). Anyway, something about that link is malformed as I get a general 404 error (page cannot be displayed/webpage is not available) upon clicking on it. Nevertheless, the link led me to the book, so thanks again. --Jim
I'm not sure of where I was in the book myself.
Interesting part to me is that in order to do the Fourier synthesis/analysis you need to add a lot of time varying values together. The famous Lord Kelvin had a solution to this adder problem that involved a long wire running over many pulley wheels. Every other wheel could be moved contributing a change in total length of the wire. This only worked for a small number of numbers in the sum because errors (like wire stretch) accumulated. Michaelson came up with the idea of doing the addition by applying the many values as movements to springs that were all applying force against a single bigger spring. The resulting motion in the middle being a sum of the inputs. This could sum a lot more inputs without accumulating large errors. Brilliant.
Even with the spring adder, it looks like friction in the pen drive is limiting the accuracy of the harmonic analyzer to +-5% or so. I'd love to see how accurate it would be using a long scale arm and a photographic readout. (i.e. remove as many parts as practical after the summation that have friction)
Marty
That 5% is amazingly good. From what was reading the Kelvin adder with the long wire and pulleys was only god for about 5 terms before accuracy went totally out of the window. They did actually use mechanisms like that to predict the movement of the tides quite well though.
When you need 10's or 100's of terms for an FFT the spring based adder was a winner but as we see becomes very sensitive to load on the output.
I guess a laser diode bounced off a mirror moved my the adder bar an scorching a graph into paper would be cheating though
Yes that would be cheating
Interesting that the Kelvin adder design was so sloppy. Friction in the pulleys is what I'd expect to cause the bulk of the errors. I'd expect it could quite precise if large ball bearing pulleys were used with a metal tape and bias weight. (not a period accurate design though due to ball-bearings) That said, an averaging mechanism using levers is what I think would work best. Build it like a mechanical weight scale with all knife-edge pivots, it could be both extremely stiff and low friction.
Marty
I'm a bit surprised about the accuracy of the pulley technique. But clearly that sums errors whereas the spring balance averages errors out.
What an interesting picture. What is going on there?
Looks like a load is pushing or pulling on a bar, the two ends of which are pushing or pulling on shorter bars, the ends of which are pushing or pulling on even shorter bars...and so on...until the smallest bars are pushing or pulling on the wing.
At the end of the line of course is a spring, the wing itself!
It's still not clear to me how we are going to form a sum or average using just leavers.
I looked at some of Hammack's other videos. The ones on fiber optics and the transistor were also very interesting.
That is the way I see it. The spring summer/averager does a great job but it's output impedance, as it were, is very high. Any load screws it up.
To that end they used a very light weigh aluminium alloy "amplifier" wheel. As opposed to the usual brass or iron at the time.
I wonder if the reproduction pen holder that they made from brass was actually made from the original light weight material. They might have used wood or elephant task, who knows? Perhaps that accounts for not having the original pen holder?
Perhaps even there was no pen and ink. Just a scriber scratching marks some how.
I seem to remember an experiment we did in school with a pin glued to the end of a tuning fork scratching a wave form onto a tape that was pulled by a weight dropping under the force gravity. Knowing the frequency of the fork we could calculate the acceleration due to gravity by measuring the wave lengths recorded on the tape.
You can get a better Idea of the test on youtube http://youtu.be/Ai2HmvAXcU0
Basically, to accurately simulate aerodynamic loads they needed to distribute the test loads evenly over the whole wing. (well, to each wing rib at least) Instead of having over 1000 precision winches, the lever assemblies carefully divide up the force from ~20 larger precision winches. The lever dimensions are set to simulate aerodynamic loads. Scaled way down, a similar mechanism could calculate (a0 ... a20)/20 and keep the size of the input and output motions the same. (really it'd calculate { [a0 + a10]/2 + [a1 + a11]/2 } /2 etc.)
Marty
I see. I do believe you are right.
So basically you would have a pyramid of levers. In each layer of the levers their length is halved. The end of each lever in the pyramid connects to the centre of a lever below it. All the way down the bottom layer of leavers whose ends are connected to the movements of the input "signals". The centre of the longest lever at the top being the output.
Sounds like the thing to do is build it upside down. Short input levers at the top. Long output lever at the bottom. So that it all hangs in place under the force of gravity. The output lever having a bit of a weight on it's centre and driving the pen.
You will probably want the number of inputs to be a power of two. With log 2 layers of leavers. Say 32 inputs and 5 layers.
When the end of an input lever is moved vertically with it's incoming signal it will of course be moving sideways a bit as it rotates. This would introduce an error. Better have quite a long connecting rod between the input signal and the lever end to mitigate that.
That would be quite a construction. I'm sure how well it scales. What about 256 inputs say?
BTW. Did you notice that Michaelson did use a knife edge pivot in his spring averager?