All right, everything's done! Here are some photos from the mounting session:
Wow, what a mess!
Those dowls were not meant for pounding nails into. In the end, I had
to drill holes in them and glue them to the nails. The fried gears
turned out to be great for testing drilling. The nylon discs made the
gears too unstable, so I decided not to use them.
After some difficulties with the FDM machine, I finally finished the other set of oval gears. The period of rotation is 2 (40 teeth on one, 80 on the other), eccentricity is 0.15, and distance between the two holes is 2.87. This signals the end of the production phase of the project. I just need to write a user manual for the software (and probably clean the code up a little) and assemble this log into full report. They pass the nail-down test as well.
I went down to Ace Hardware today and bought the necessary parts for the mounting of the gears: a plank, some 1/8" diameter nails, some nylon washers, and some small wooden dowls to use as handles. Finding good nails was difficult: they had to be short (ideally less than an inch), but have a 1/8" diameter. Apparently, this isn't a standard nail size. I found some copper nails for marine use that are just the right diameter and 3/4" long, that should do the trick though.
The final gears:
First attempt at the gears... Something happened with the FDM machine (notice the right part of the right gear) during fabrication. Furthermore, the teeth on the conjugate gear (the right one) used the wrong radius of curvature. I had forgotten to change an experimental setting back to its normal value.
It's not just the photo, somehow the bottom surface warped up from the platter during fabrication:
The software now supports production of pairs of conjugate gears.
There are quiet a few bugs, but it does work for a number of cases.
Here's a sample that I'm going to try to run on the FDM tomorrow:
Ok, so I finally have the oval gears done as well.
Elliptical gears are now interlockable!
Oval gears are being made right now on the FDM. I'll pick them up tomorrow morning.
These have a=1.0, e=0.65, and 21 teeth.
So, the FDM machine ran out of filament while running my part, which really wasn't so bad in and of itself; they're just only about 1/6" high instead of the desired 1/4". However, the machine kept running as normal and something went wrong. I tried to change the fillament, so that I could run the 21-tooth ellipses. But, after an hour the machine would simply not load the new fillament. Later that morning, Dan O. came and fixed the machine. The ellipses are running now (in yellow). However, here's a photo of the oval gears. They work fine, but I'll run them again to get the full depth.
Oval gears (two lobes) with e = 0.15, major radius = 1.0, 30 teeth
Here's a new photo of the 20-tooth elliptical gears.
I started the FDM run for the oval gears at 17:15. They are thinner (1/4" instead of the 1/3" used with the black ellipses), but have a larger volume, so the estimated run time is 5.2 hours. After they're done, I'm going to run the 21-tooth ellipticals. Estimated time for those is about 4 hours. Then, it's just a matter of designing and fabricating the last pair, where one gear will have rotated three times for each rotation of the other.
The first FDM run of the elliptical gears is done. These gears have an uknown eccentricity (I forgot), a major radius of 1.0, 20 teeth, and an axis point hole of radius 1/16". They do roll over each other smoothly. The run time for the simultaneous production of these gears was 5.6 hours on the Stratasys FDM machine.
There is only one known imperfection (diagramed in the right image): the gears cannot be perfectly aligned due to the fact that at the necessary contact points there are two teeth as opposed to a tooth and a pit (shown by the red arrow). This results from the fact that gears that have an even number of teeth inherently have 2 axes of mirror symmetry (roughly shown by the blue lines). Identical gears need to only have 1 axes of mirror symmetry (running along the major axis of the ellipse), which can be produced by choosing an odd number of teeth. The other alternative is to make two distinct gears, where the teeth of one of the gears are "offset" by half of the circular pitch. This is not a problem in standard circular gears, because they can simply be rotated that extra 1/2 circular pitch without distorting the meshing geometry.
I intend to remedy this problem by producing a 21-tooth elliptical gear instead.
|e = 0.1, nodes = 2, 35 teeth||e = 0.1, nodes = 3, 35 teeth|
The dedendum distance has been added into the shape. Previously, the pitch path was acting as the base path.
Fun with BMRT.
|20 teeth, e = 0.65||40 teeth, e = 0.65|
The elliptical gears are just about finished. Only two major tweaks remain: creation of an accurate base circle and dealing with proper rotation of the gears. For ellipses, these won't be a problem, but for other shapes they are. A quick circle example has problems rotating all of the teeth in the proper orientation.
Here are some more images:
|20 teeth, e = 0.5||20 teeth, e = 0.7|
Placement of the teeth has proven to be a non-trivial problem. This flows from the fact that ellipses do not have a closed form for their perimeters or for the arc length over a given theta. I think I have a reasonable approximation algorithm built, that uses closed form estimates for perimeter and progressive refinement to get tooth locations within a certain tolerance.
The current design of the software should enable me to simply give polar descriptions of the inner and outer radii, along with a function to compute curvature from theta, for any new shapes.
Eric Weisstein's world of Mathemics, hosted by Wolfram Research, makers of Mathematica, has proven to be a VERY useful source of information for mathematical description and history of ellipses, involutes, as well as all those things I forgot from multivariable calculus (curvature, radius of curvature, and partial differentiation).
Some images done in BMRT of geometry produced by the custom software:
I've chosen to write modeling software that generates SLIDE files in Python. Originally, I had planned on writing the software in C/C++, but since speed is not really an issue here (and quick development is), Python should work nicely. The above gear "blank" (an ellipse with a = 1.0, eccentricity = 0.8) required only about 245 lines of code, of which 190 are reusable for other gear shapes.
25. April After meeting with Prof. Séquin again, I've decided to scrap number two below (the elliptical rack and pinion). I'll just go with the oval gears of different periods. It could still happen though if I find lots of extra time a the end of the semester (not likely though).
I finally found a book that talks about non-circular gears. Gear Geometry and Applied Theory by Faydor L. Litvin, 1994, published by Prentice Hall.
Gears I'm going to build:
The evolute for non-circular gears is dependent upon the circumference of the gear. Left and right sides of the tooth may have different evolutes. These could be generated, but I may just do an approximation to save on the integrals. The evolute for a circle of corresponding curvature radius at point on the gear where the tooth is to be added, can be used to approximate the actual evolute.
Proposal Presentation (April 10) was a success!
My presentation/project along with Young Shon's was voted the best by members of the class.
More results to come as they are available!