At a fairly late stage in my professional life (age 57) I started a new master course at the University of Utrecht. A master of education in physics. The physics part is a lot of fun, but it is hard to enjoy the math. Our teacher (Martijn Koops) started a new course 8 weeks ago by showing a stunning video of a Fourier Analysis machine of more then 100 years old. ( https://youtu.be/NAsM30MAHLg ) He told us that this machine is capable of doing the same thing as we were going to do in his class with calculus, pen, paper and graphic calculator: Taking apart wave patterns and reconstruct the wave with a number of sinusoids. In fact this machine is an analog computer!
I have been struggling on the Fourier-math these past 8 weeks, but working odd hours in my wood-workshop kept me motivated. My 'wooden analog computer' is working now, and I am about to turn it over to some of my my own high-school students. Their task will be to research the inn's and out's as their graduation project for high school physics class (PWS). They will also experiment with the many possible settings of the machine. I hope they will come up with which springs work best in the addition lever. (see step 6) They can also compare theoretical outcomes with machine outputs. I am really curious about the results. The original machine matches theory within 1 percent!
Anyway, I am sure that understanding the machine is easier then the real math (my math exam is in two days! help?!?) And I like to share with Instructable readers what I have been able to make. The machine is working, but there is still some finetuning to do. However next two days, I'll be studying for the exam, plus I would like to go for the Laser competition which is closing tonight. So I have to publish now, here it is...
Step 1: Fourier Introduction
I guess some of you have heard the term Fourier without knowing what it really is. Google will tell you everything. The key to remember is that the Frenchmen Jean-Baptiste Joseph Fourier discovered some 200 years ago that any (mathematical) function (think of sound waves and such signals) can be reduced to a combination of simple sine and cosine functions. There is beautiful math involved. But more importantly most modern gadgets would not have seen the light of day without practical use of Fourier transformation. The fact that with JPG you can reduce a photograph file size drastically is all thanks to Fourier.
So some appreciation and insight in the workings of Fourier is worthwhile.
Around 1898 American Albert Michelson** made a machine to do the same thing mechanically as Fourier did with pure math. I can really recommend a book by Bill Hammock on this subject. Title Albert Michelson's Harmonic Analyzer. (available on Amazon or e-bay). He is the "engineer guy" in http://www.engineerguy.com
The book has wonderfully detailed pictures of a restored machine that can analyze and construct math functions with a system of 20 gears (a 80 gears machine also existed). I have reconstructed this machine in plywood with 8 gears just by using pictures in the book. I would like to think of this instructable as a means of inspiration. Go ahead and try to understand the inner workings and build your own variation. Design and build its part based on the equipment you own.
The building of this machine was limited and slowed down because of problems with my CNC mill. It would have helped a lot if I had a 3D printer for making all these gears in smaller sizes (more gear stages would fit). I expect that using a laser cutter will enable me to make make smaller and more accurate gears. I am coaching a FirstTechChallenge robotics team at school. These students are often limited in their robot designs because of the tools that we have at school. Having a laser cutter available in our tech-room at school is our big wish.
(** Michelson together with Morley are Nobel prize winners for confirming that speed of light is a constant)
Step 2: What You Need:
In the picture you see the gears and other parts I made in birch plywood.
- Really useful is a simple home-brew CNC machine (to be found on many instructables see Benne de Bakkers: https://www.instructables.com/id/Building-a-CNC-ro...
- three sheets of 9 mm plywood (120x60 cm).
- Not in this picture are some parts build with 18 mm plywood. One sheet of 120 x 60 cm
- 2 pcs 2,5 meter dowel 30mm diameter
- 8 pushrods 80 cm (I started with wooden dowel, but switched to re-used carbon kite spars of 6 mm)
- lots of M3, M4, M6 crews
- 2 meter each steel rod of 6 and 8 mm for the various axles.
all the parts made on my CNC are available in de 'achief.zip' file. The gears are drawn in a software package called "gearotic". It is best to download this really great package yourself. You can set up the gear and tool sizes exactly the way you would like. (I can provide my files in case you need them...)
In the pdf file you will find all the parts on one page. Just enlarge by 400% for full size. You can play around and multiply some parts by 8. Then export to your preferred CNC file builder. I have used CAMBAM to create the G-code files.
Step 3: Building the Gear Train
I started by building the multiple gears. A crank drives a cone with 8 gears.
The smallest that I can build is 6 teeth. That determines all the rest. They have to be multiples of 6:
Size was fixed by the gearotic gear program. I need a 3 mm cutter to cut the 9 mm plywood. The smallest gear number for this cutter is T2.5. I think I can probably go smaller in the future. With smaller teeth I can get more gears on the cone.
The crank drives a 24 teeth gearwheel.
The second image shows how the cone-shape gear system drives 8 separate 54 teeth gears. Each of these gears drives its own crankshaft.
Step 4: Cam Gears
The 'hart' of the Fourier machine are the 8 camshafts. The up and down motion of these cams is what creates 8 frequency's of sine functions.
Picture-1 shows the lay-out of camshaft-gear-pushrod system.
picture-2 shows how a jig is uses to drill 3 hole in the large gears. Each gears receives three M3 screws and 3 nylon bushings that allows a tight fit of the pushrod without too much friction.
With 8 gears stacked together the friction still adds up. I am planning to take the machine apart soon, and wax and polish all the surfaces in order to reduce the friction.
Step 5: The Amplitude Balance Arm and Amplitude Rods
The image shows a sideview of the 8 arms. Each arm wil rock back and forth in tune with it;s own camshaft.
Thes arme curved in the exact shape of a circle. The radius of the circle is fixed by the length of the pushrods. 80 cm in my machine which seems to be similar to the original machine.
The shape and size of all the frame's carrying gears and arms are to be found in the pdf file (step-3)
Obviously one need to scan to fit the chosen gear size.
The Amplitude arm's I had first in 6mm wooden dowel. Way to soft and flexible! I happend to have some 6 mm carbon tubing lying arround (I used te be a kite-builder) that fitted in the small plywood holders.
The size of these holders is critical. I think they should be slightly narrower to avoid the risk of them hitting each other.
Step 6: Rockers, Main Balancing Arm and Springs
This is where the Fourier magic happens.
We have eight amplitude rods that are moving in their own frequency and amplitude. They are all connected via a spring to a strong balancing arm. These eight different forces are counterbalanced by a single longer and stiffer spring on the other side of the balancing arm. This is where all the positive and negative amplitudes are added up.
As you can see in the third image the main spring is replaceable and adjustable. This is where you can do most of the finetuning in order to get an acceptable output.
At the moment I am still trying out a bunch of different springs. Working with a 5 euro box of all purpose springs from the 'Aldi' (Dutch version of Target in the US).
Step 7: Drawing the Sine Functions
The slight movement of the balancing arm is what represents the desired curve. The original machine is equipped with pulleys to enlarge the up and down motion with a factor 5. The wooden machine has more pronounced movement in the balancing arm. I found that the lever shown in the images was sufficient.
I have a marker hanging from a string that draws the vertical movement on a piece of paper. Very primitive, but pretty much the same af Michelson's machine.
If you look carefully a thin strip is visible that runs from the drivable ov the cone-gears tot the drawing board.
The string pulls the board horizontally, (takes about 15 turns of the cone-gears). You will almost see two full cycles of the function that you chose to generate.