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How to Build a Paradox by action pig on November 15, 2011

Table of Contents How to Build a Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Intro: How to Build a Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 1: Things You Need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 2: Gear Theory for Winners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 3: Explaining the paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 4: Turning the Paradox into an Orrery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 5: Options for Cutting Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 6: Generating Gears in Sketchup and Exporting to Inkscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 7: Digital Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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File Downloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 8: Preparing Laser Cut Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 9: Assembling the Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Step 10: Assembling the Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Step 11: Marking the Ecliptic and the Lunar Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Step 12: Assembling the Ecliptic and the Moon's Inclined Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Step 13: Assembling the Sun and Crescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Step 14: Finishing Touches and Reading the Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Related Instructables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Advertisements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

http://www.instructables.com/id/How-to-Build-a-Paradox/

Author:action pig Action pig loves you. Also truth and justice.

Intro: How to Build a Paradox "...learn from this not for the future to reckon every thing absurd and impossible that you cannot comprehend" -James Ferguson Historical Notes, Plus a Paradox James Ferguson was a self taught builder of astronomical clocks circa 18th century. He was also a painter and philosopher and all around amazing guy who built this machine to resolve a religious dispute in a tavern. The paradox is that three gears (F,G,H) are driven by a single gear (D) and turn three different ways. That's right same driver, three different results. Gear F remains stationary, gear G turns in the same direction as gear D, and gear H turns in the opposite direction. The guys in the tavern couldn't believe it either! How? It is really a reference frame issue, as gear D is a fixed gear around which the other gears rotate. For more information, see step 3 - Gear Theory for Winners. Credit I built this project using Ferguson's original work, "Select Mechanical Exercises: Shewing How to Construct Different Clocks, Orreries, and Sun-Dials, on Plain and Easy Principles." . If you enjoy manuals on building clocks and like miscellanea from the 18th century, I highly recommend it. For instance, Mr. Ferguson will tell you all about inflation rates from 1100 to 1750, how many guineas would settle the debt of the British Empire, and also solves the mystery of who really invented logarithms (and why they are so great). Ferguson is really quite delightful. If you want a fantastic manual on orrery bulding, see Ferguson's biography which includes his many clock and orrery plans. Also, Ivan Law (Gears and Gear Cutting ) taught me everything I know about designing with gears. His book is simple, straightforward, and best of all, thin. Highly recommended for those interested in designing or machining gears. Also see the following: http://www.horo-logical.co.uk/ferguson.html http://www.craftsmanshipmuseum.com/gould3.htm http://armstrongmetalcrafts.com/ParadoxOrrery.aspx http://www.lisaboyer.com/Claytonsite/mechanicalparadoxpage.htm

Image Notes 1. Sketchup model (all hail sketchup!)

Image Notes 1. D 2. H G F 3. E

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Step 1: Things You Need Materials Plywood , 1/4" thick (supplied by laser cutting service) Wooden Dowels - 1/4" thick Wire (I used 1/16" aluminum tubing) Spheres (Earth and Sun) Wood Glue Epoxy Paint/Mineral Spirits (optional) Tools A Gear Cutter - I'll go over how to cut gears in step 5. I used a laser cutter as I do not have a desktop CNC, vertical mill, lathe, or scroll saw. I ordered these parts from Outfab, a laser cutting service that charges a flat rate rather than charging for time plus materials (for instance, a 12" x 12" cut costs around 30-35 bucks, depending on the material.) Due to the small size and numerous teeth of my gears, getting charged for cutting time would have been prohibitively expensive. I highly recommend Outfab for your complex laser cutting needs. Oh, and they have free shipping to the US and free sample cuts. A Saw - for cutting the rods Sketchup and Inkscape - or equivalent

Step 2: Gear Theory for Winners I found gears an intimidating prospect until I read Mr. Law's "Gears and Gear Cutting" book. I will go over some brief concepts here, but really nothing can substitute a manual like his, which explains why gears are the way they are. I know many of you are old hats at this, but for any curious amateurs read on. Basics Gears are used to transmit power between one shaft and another. If Gear A is moving clockwise and it turns Gear B, Gear B will turn counter-clockwise (the same movement is passed on but in the opposite direction). If you need gear B to turn in the same direction as A, a 3rd gear, or idler (pinion) , is needed. The diameter of the gear (and the number of teeth) are directly related to the speed of the gears. Tooth form I'm not going to go into the specifics on tooth form - it involves a lot of exciting math - but the take home point is that there are two types of gears: cycloidal and involute. Both types of teeth allow the teeth to engage without locking. Improperly formed teeth will result in noise and a loss of efficiency in the gear train. I am almost certain that Ferguson used the cycloidal form, but involute is the standard today (and is most likely what your gear generator will be making). Designing with Gears: Here are a few definitions to get us started: Pitch Circle (or Pitch Circle Diameter, PCD): The imaginary diameter of the gear teeth and bearings in peripheral contact (where the two gears will actually mesh). Imagine that you have replaced your gears with two circles that turn each other with friction - the diameter of these circles is the pitch circle diameter. Pitch Diameter (PD): the PCD in inches (a lot of definitions depend on whether you are using metric or imperial units. Swayed by common practice, I go with imperial. Once you pick a system be sure to be consistent). Pressure Angle: The angle on an involute gear between the acute angle formed between the line of action and the common tangent to the two pitch circles. You don't need to worry about the formal definition much - you just need to be consistent. A pressure angle of 20 is common. Diametrical Pitch (DP): The number of teeth divided by the pitch diameter. Two correctly meshing gears will have the same DP (and pressure angle, see above). For instance, if you need two gears, one with 20 teeth and one with 10 teeth, and you choose a DP of 20 than the 20 tooth gear will have a PD of 1", and the 10 tooth gear will have a PD of 0.5". Center Distance: The center distance is equal to the sum of the pitch diameters of the two gears divided by two. In the above example the center distance is equal to (1"+0.5")/2 = 0.75" Gear Ratio: The ratio of gear teeth required to reduce or increase the speed of the shaft. For instance, for every revolution of the 20 tooth gear, the 10 tooth gear will turn twice, and so the speed is increased by a factor of two.

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Step 3: Explaining the paradox So, back to the paradox. The gear frame is rotated around the immovable gear D. Gear E is a thick wheel that takes equally deep into the teeth of the three wheels F, G, and H. As the frame turns gear H turns the same way as gear E, gear G turns in the opposite direction, and gear F remains stationary. This is, of course, a question of relative motion. The motion of the gears with respect to E depends entirely on the relative numbers of teeth with respect to D. Gear H has more teeth than gear D, gear G has less teeth than gear D, and gear F has the same number of teeth as gear D. As a result, when the frame is rotated around the center axis, the effect on F is counteracted entirely by the movement of the wheel E, so it does not appear to move at all. But H, having more teeth than gear D, is not counteracted entirely by E, and so turns in the same direction as the motion of the frame. The opposite is true for gear G. We have solved the paradox! Now, I know what your thinking. How can the middle gear have a set number of teeth and engage three wheels with different numbers of teeth at a set center distance? What about the diametrical pitch? The truth is, Ferguson is using a fudge factor - by offsetting the number of teeth by a small amount, the DP of the center gear is close enough to work. The paradox also works if, instead of using a single gear in the center, you use three gears (each at the correct DP) that move as one. This eliminates unnecessary noise (but it does lose a bit of style).

http://www.instructables.com/id/How-to-Build-a-Paradox/

Step 4: Turning the Paradox into an Orrery By choosing the right numbers of teeth, the paradox can show the different lengths of days and nights, the change of the seasons, the retrograde motion of the nodes of the moon's orbit, the direct motion of the apogeal point of the moon's orbit, and the months in which the sun and moon must be eclipsed. The numbers of teeth are as follows: D - 39 teeth E1 - 39 teeth E2 - 41 teeth E3 - 34 teeth F - 39 teeth G - 37 teeth H - 44 teeth If you choose to have a single large gear for gear E, the number of teeth should be 39.

Step 5: Options for Cutting Gears Lacking a CNC, I ordered my gears laser cut. I have tried 3D printing, but have found that for DP > 20 the plastic tends to thicken around the edges are the gears are noisy (I use a DP of 24). If you design your gears large enough it is possible to cut them out with a scroll saw (I don't have one of those either). Proper gear cutting requires a lathe or vertical mill, a dividing head (a fancy way of rotating a circle in precise divisions, so the teeth are evenly spaced), and a hob (a specially shaped cutter that forms the correct tooth shape). Of course, you might ask how Ferguson cut gears without any of the above equipment. Automatic gear cutters (like the one above, pictured) were capable of turning out relatively precise gears and were turned by hand. Some of these cutters used a pantograph to trace the correct tooth shape onto the blank (would that be a great instructable or what?) See the next step for how to create gears in sketchup and transfer the files from sketchup to inkscape for laser cutting. I should note that after this build I have started designing with larger gears (DP ~16) to strengthen the teeth.

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Image Notes 1. automatic gear cutter

Step 6: Generating Gears in Sketchup and Exporting to Inkscape Google has a great introduction to sketchup (here ). In order to generate gears you need to install this plugin . Once the plugin is installed it can be reached from the menu draw > draw involute gears. This plugin also has a draw key involute gears option. In order to generate gears with a hole for the shaft select 0 for the keyway width and depth, and then select an appropriate shaft radius. Note that this plugin uses pitch radius instead of diameter. Note: Inkscape also has a gear generating feature (we need inkscape for the laser cutting template) but I like sketchup more for general design. In order to export faces to inkscape from sketchup you need to install this plugin . After selecting a face or edge, right click and select "Export to SVG File." From the window that pops up you can name the file and select its destination. File requirements may vary for whatever laser cutting service you use. Outfab specifies line thickness and color (see guidelines here ).

Image Notes 1. I abandoned the cut-out gear design for simplicity and added strength. 2. Click here

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Step 7: Digital Plans The digital plans are below in pdf form (the svg file seems to be too big for instructables). Please note that the pdf is not formatted for a 12 x 12 cutting space and be sure to check that the line thicknesses and colors are appropriate.

Image Notes 1. The extra hole is used so that the three middle gears move together without glue

File Downloads

Ferguson Parts.pdf (150 KB) [NOTE: When saving, if you see .tmp as the file ext, rename it to 'Ferguson Parts.pdf']

Step 8: Preparing Laser Cut Parts Except for the dial plate, which has lettering, I sanded each piece with 220 grit sandpaper and put on a single coat of 50/50 tung oil and mineral spirits (wear gloves and wipe off, let dry for at least 12 hours). I also took a dremel and sanded off the dark burned edges. There is one very important exception - ON NO ACCOUNT SAND, FINISH, OR GENERALLY ALTER THE GEAR TEETH. You want to avoid anything that adds noise to the mechanism.

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Image Notes 1. This is an early version of the dial plate (I later expanded the calendar to include the zodiac)

Step 9: Assembling the Gears The innermost gear is fixed with respect to the dial plate. This is accomplished by fixing it to a post which is glued to the square insert from the center of the dial plate (see pictures below). A washer is added to smooth the turning motion of the frame (the washer sits above the dial plate and below the frame). Assemble the three center gears around a 1/4" post. A second post is added to keep the gears moving together. I tried gluing the gears in an earlier version of this project and it turned out messy and added noise to the mechanism. The three outer gears are each attached to concentric hollow tubes (aluminum, from inventables ) using epoxy. These tubes will control three flat plates above the frame. As mentioned earlier, one of these will appear stationary, one will move clockwise, and one counterclockwise. The outermost tube rises ~0.5 inches above its gear (or about two thicknesses of the plywood), the middle tube rises ~0.75" above its gear, and the innermost tube rises ~1" above its gear. In order to keep the frame horizontal, I extended the innermost tube below the frame just enough to add a washer (see picture). Remember, the center and outer gears need to be able to turn within the frame (no gluing the posts!!!)

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Image Notes 1. All of this sits below the frame

Image Notes 1. Post to keep 3 gears moving together

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Image Notes 1. Outer gears

Step 10: Assembling the Frame I prepared four wooden posts to keep the frame together. The holes for these posts are 3/16" across. I sanded down 1/4" dowel to fit. I tried gluing the posts in to the frame, but found that it added pressure and prevented the gears from turning smoothly. The mechanism works fine if the posts are in but not glued. Also, each hole in the frame had to be sanded a little in order to accomodate the dowels and aluminum tubes.

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Step 11: Marking the Ecliptic and the Lunar Nodes The ecliptic is the belt through which the sun and planets appear to travel from our perspective on earth and is represented by the signs and degrees of the zodiac. The signs of the ecliptic on the large dial plate show where the sun is located as seen from the earth. The moon's apogee (the farthest point in the moon's orbit) is represented in this machine by a small ecliptic around the earth. I decided early on that I wanted the ecliptic and lunar nodes to be made of metal. I experimented with etching and even pasting on paper with the printed signs before deciding to use a color wheel to represent the signs of the zodiac. The paint chipped when I wrapped the bushing with wire, and I touched it up later. For the ecliptic and nodes I used bushings (like washers, but with larger openings). One is OD 2", ID 1.5", the other is OD 1", ID 0.5". The ecliptic is the small bushing. To create the pattern I used masking tape for a guide and enamel paints to represent each constellation. The small ecliptic will always be parallel to the large ecliptic, so referencing each sign to each color is relatively easy. An easy cheat is that earth is always inclined towards the beginning of cancer (which is gold in my machine). The node is the point where the moon's path crosses the ecliptic - in other words, the moon's orbit is on an angle (~ 5 degrees) with the ecliptic, and will cross it as it rotates around the earth. There is an ascending node (or north node) and a descending node (or south node) directly opposite from each other. I borrowed from the yin/yang symbology and assigned the north node to the color black (yin), and the south node to the color white (yang). Ferguson used the symbol of a dragon to mark the north and south nodes. The north, or ascending node, is meant to be the dragon's head, and the south node is the dragon's tail. This comes from the old belief that a dragon swallowed the moon during an eclipse.

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Image Notes 1. Ecliptic 2. South 3. North

Step 12: Assembling the Ecliptic and the Moon's Inclined Orbit A bent wire is used to illustrate the position of the moon's ecliptic and is set on the bottom (largest) disc. I used 1/16" hollow aluminum tubing. I would have preferred a solid wire, as it would be less prone to breaking. As shown in the second picture, this wire points to the small ecliptic, and needs to give room for the moon's orbit to turn. The moon's orbit should be inclined at about a 5 degree angle (technically 5 deg 9'). The nodes show where the orbit intersects with the ecliptic, and should not be on the inclined extremes. The earth should be inclined at an angle of 66.5 deg from the horizontal, or 23.5 deg from the vertical. Ideally the meridian and equator would be marked on the earth to show the lengths of day and night, however, my earth is a little too small for this. The ecliptic should be mounted as shown above (it is not set at an angle). All three small discs should fit tight enough to turn with the gears but loose enough to be set by hand. Because the lunar orbit is not based on the 365 day calendar, these discs need to be set by hand at the beginning of each year. In other words, no gluing the discs to the metal posts! I used epoxy to glue the wires in the discs.

Image Notes 1. Apogee wire 2. Lunar orbit 3. Earth and small ecliptic

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Image Notes 1. The angle is not shown from this perspective - it should look flat between the nodes

Step 13: Assembling the Sun and Crescent Two wires are used for the sun. One is the solar ray, which points to the center of the earth. The other supports the sun, which should sit at the center of the machine, or above the center gear. The crescent separates the enlightened half of the earth and the half that is in the dark. It is set in the frame, and must be bent to allow the apogee wire and the lunar orbit to rotate. Ideally the crescent would go all the way around the earth - because of the orientation of the apogee wire, my crescent is less than half of a circle.

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Image Notes 1. Apogee wire 2. Lunar orbit 3. Earth and small ecliptic

Step 14: Finishing Touches and Reading the Machine Finally, glue the square insert (attached to the frame) into the large dial plate. The machine is now assembled! In order to set the machine to run, you need an ephemeris. A good one can be found here . The mean apogee is used to set the apogee wire and the mean node refers to the ascending (north) node. Remember, the orrery will only be accurate for the year that you set it (if you keep rotating the machine, it will be reading for future (or past) years rather than the one you set it for). In the picture above, the moon's apogee is about 2/3 of the way through Gemini, and the ascending node is approaching the center of Scorpio. The sun is at 10 degrees of Aquarius. When either node is between the solar ray and the earth, the moon is said to be in conjunction with the sun. If a new moon occurs within 17 days of conjunction the sun will be eclipsed. If a full moon occurs within 12 days of the sun, the moon will be eclipsed. If an earth with the meridians is used, this machine can be used to demonstrate the seasons and the lengths of the day. If I were to build this machine over again, I would make it larger - mine is ~9.5" across, while Ferguson's was about 15" across, which gave him more room to add detail and allowed larger tooth gears to be used. I have added a short video showing the machine's movement. Enjoy!

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