Suppose I wanted to cut an epicycloid with K cusps (cusps = petals if you think of them as flowers). http://mathworld.wolfram.com/Epicycloid.html Well, to start simple, suppose we only wanted to cut a circular piece of wood. I would make a trammel with a pin extending through my trammel and the piece of wood I'm cutting. On the opposite end of my trammel, I would place my router at a fix distance from the pin, and then I would either rotate the trammel OR rotate the piece of wood I'm cutting If we want to cut an ellipse, we modify the trammel in the style of the "Trammel of Archimedes" -------------------------------- Now suppose we want to cut an epicycloid??? The first idea that comes to mind is to build a small circular trammel and pin it to the endpoint of the primary trammel. Yet, there is no way to guarantee that both trammels will rotate at their respective rates with such a simple device. For instance, suppose we want to cut 8 cusps about a circle of radius R. The primary trammel would measure in (9/8)R in length. Then we attach a second trammel of radius (1/8)R to the endpoint of the primary trammel. Each revolution of the secondary trammel will oscillate between (8/8)R and (10/8R) from the origin of the primary trammel. However, simply rotating the secondary trammel will not rotate the primary trammel. In this scenario, the secondary trammel must spin 8 times faster than the primary trammel (which will mathematically generate NINE, not EIGHT, revolutions towards a FIXED point in space). DELETED the ideas to accomplish this since they were terrible. ------------------------------------------- Any ideas?
Here is the first device I've designed that could work. The Primary Trammel is the red rectangle, where the pins are driven through the shafts of the wheel and the pinion. ***There are two identical PRIMARY trammels on both the front and backside of the wheel gear connected to the front and backside of the pinion gear, in order to prevent racking*** The Router bit is placed through a hole that is drilled a distance from the center that will allow the router to cut exactly at the demanded length of the secondary trammel. Since the router bit is placed through this hole, the router itself will be on the backside of the pinion. The wheel gear (large gear) is locked/clamped stationary, unable to move; therefore, the only way the pinion can rotate is if the pinion itself moves around the wheel. Since the trammel maintains a constant distance between the shafts of the gears, the router itself can be rotated by your hands, making the pinion move about the wheel. The router should cut a perfect epicycloid using this method. Yet I wonder, does a more elegant or simple (or both) method exist? Because if I have to cut two involute gears in order to generate an epicycloid, I may as well just cut the epicycloid (unless I'm going to make more than two copies of the epicycloid). The design is also "inflexible." For every different diameter of epicycloid (or hypocycloid if I switched to interior meshing gears), I would need to cut new gears of those same diameters.
One problem with the traditional gear solution is that cutting the cusps will also cut some of the main gear teeth. I think the obvious solution would be to use a numerically controlled cutter. For plywood I would recommend a water jet.
Can the Addendum be modified in a manner to preserve the diameter of the epicycloid? Apparently I'm not the first person to get stumped by this: http://www.iaeng.org/IJAM/issues_v38/issue_4/IJAM_38_4_06.pdf Right now, I'm trying something that would involve a scroll saw. The idea is that there would be a template for a single tooth (similar to involute gear cutting blades), and the gear blank is rotated precisely after each tooth is cut.
The requirement with gears is that; the rolling contact between the gears must have two points in contact at all times. If a link is used to set the axis separation then one point must be in contact. For a sharp cusp, the point being traced needs to be on the pitch circle of the gear, which must therefore obstruct a point of contact. The only way to overcome that coincidence of position is to have the gears in a separate plane to the pointer or cutter. An offset arm mounted on the planet gear, (or it's shaft), could move the point away from the plane of the gear. Sheet material is cut by the plastics industry using NC water jet cutters, the metals industry uses NC plasma cutters and lasers. They could cut your final article, or a template for you to cut your article. If I only ever wanted one pattern, I would first numerically generate the curve in a graphics file. I would then print out the master template and trace it with step and repeat onto the final material.
There is an alternative solution. You could simulate the two gear wheels with software. By using stepper motors with reduction gears to control the arm directions you could rotate the planet arm faster than the sun arm. The rates of rotation would be controlled by step rates generated from a computer or microcontroller using something equivalent to the Bresenham algorithm. The slope of the line sets the order of the epicyclic. http://en.wikipedia.org/wiki/Bresenham's_line_algorithm The gearboxes improve resolution, while their ratios can be allowed for in the algorithm. One stepper motor could be replaced with a shaft encoder but it would not give the accuracy achieved by a geared stepper motor.
I was wondering if a trammel consisting of very narrow grooves could be constructed at the center. Take this rough example (not meant to produce an exact cycloid). Take the polar equation: r1 = |cos(n[theta])| r2 = k So r1 is a star of consisting of 2n arms. r2 is a circle of radius k. If we combine these as: r3 = r2 + r1, we get a curve that highly resembles an epicycloid, and a hypocycloid can also be imitated by subtracting r1 from r2: r3 = r2 - r1. In order to execute this combination of the polar functions we would need to create a trammel that goes about two different circles. The interior circle would be the BASE circle of the epicycloid, of radius k. The exterior circle is ANY sufficiently large circle (about 2k in radius or greater). Call this radius L. A thin rectangle, whose length would be slightly longer than the radius of the exterior circle, would have a groove cut (all the way through) from the bottom to a distance of (L-k)/2 At the end of that groove, we measure a distance of k and drill a hole for the router. Not too far above that, we cut another groove that measures (L-k)/2 in length. At the very center, we peg a wheel to the frame and place the groove of the rectangular trammel about the wheel. Then we add a second wheel, but this wheel is not pegged to the frame. A very narrow channel (the width of the axle) is cut within the exterior circular channel (the width of the wheel), so that the wheel can rotate about the exterior circular channel within the groove of the trammel. A hole is drilled for the router at the distance k (marked with the top green dot on the trammel) A pin is inserted through the trammel in order to regulate the movement of the trammel along the Star groove (marked with the bottom green dot). Simply rotating your router will cut the epicycloid. The only problem is the interference between the regulation pin and the axle of the pegged wheel (in center) towards the end (and start) of each epicycle. This can be avoided ( and made beyond negligible) by using extremely small pins for the regulation and the axle of the wheel. After completion: A better formula for the star (which generates a combined figure almost identical to the cycloid) can be derived by calculating the length of secant from the base circle to the rolling circle in respect to THETA. If we want N cusps, then we know that the rolling circle must rotate N times by the time theta goes from 0 to 2pi. We also want the radius of the inner circle to be K/N. The length of the secant can be computed by the Law of Sines, assuming an isosceles triangle of equal side lengths (K/N). The rotation of the rolling circle (PHI) is equal to Ntheta. So according to the Law of Sines: secant / SIN(NTheta)= (K/N) / SIN ([Pi - NTHETA]/2) therefore secant = (K/N) * SIN(N*theta)/SIN([Pi-N*theta]/2) Although this will generate a star, it will stagger the arms of the star in an oscillating fashion (because it will evaluate negative values HALF of the time). To make the equation generate the star's arms consecutively, we modify the equation by taking the absolute value. r1 = |(K/N) * SIN(N*theta)/SIN([Pi-N*theta]/2)| Now if r2 = K, then r1 + r2 nearly generates a perfect epicycloid with good accuracy, rapidly approaching a true epicycloid as the value of N increases. I cannot visually see the difference between an epicycloid of 8 cusps and this approximation using N = 8.
Either you need an accurate curve or you don't. You must decide on a tolerance. A router has a non-zero diameter and so will not cut cusp slots that are deep and sharp. A router produces a side thrust when used. A bandsaw does not as it holds the work down.
As you wrote this, I was updating my post. I produced an equation that approaches the epicycloid extremely rapid (indistinguishable by the time n = 8, and practically indistinguishable (without fine measurement tools) for all values lower than 8. Using a thin router bit (1/16 or 1/8 maximum) and slightly enlarging the base circle (maybe 1/32 inch), this problem is almost entirely avoided. But yes, the cusp slot will always be noticeably rounded. ***EDIT Now that I'm interpolating at theta intervals of 0.5 degrees (instead of 5 degrees), I'm actually wondering if this equation IS an EXACT epicycloid for all values of n. I would like to see an overlay of a parametrically generated epicycloid and this polar equation.
So you have moved the problem from, that of gererating an epicycloid, to that of generating a smaller star, so you can amplify the errors while cutting the approximation to an epicycloid.
Its more for convenience. You only need to cut this star once, then you can cut as many epicycloids as you wish afterwards. Used in combination with a peg board, if you wanted to make cycloids with different cusps, you could "switch" the star template to another, assuming that the radius of the Base Circle is the same. The stars would have to be cut with ultimate precision, which is a good time to use the water jet or some other high powered and digitally highly accurate cutting tool. A computer can easily determine the shape of the star using the TRUE (not approximate) polar equation of an epicycloid. One would take this equation and subtract a constant equal to the radius of the Base Circle. http://mathworld.wolfram.com/Epicycloid.html I wonder if there is a way to modify the design such that the radius of the base circle could change without having to change the Star template. Although it's easy enough to modify the trammel (by drilling multiple holes under the wheel channels), I wonder if there is a way to create a "scaling" effect.
One of the coolest things you can get for a kid................and it's still cool even as an adult. Create drawing with this, scan drawing into computer, enlarge, print, put pages together for scale, and have at whatever pattern you want to make.