Möbius Gears: a Step Beyond the Involute
I don’t know what you do with your free time, but, left undisturbed, I often find myself trying to code up some geometry. Nowadays, I am typically trying to 3D print some fancy piece of abstract math that has no business being projected into our default world of 3D Euclidean geometry. Every once in a while, however, I do something that might actually turn out useful.
One particular problem led me down a path of engineering with particularly fruitful results: how can one produce a ring consisting of an odd number of turning gears? If you grew up with Lego you likely already understand the problem. You can put four gears into a loop, or six gears, and they turn just fine. That’s because there’s an even number. With an odd number of gears, they’ll lock up on you. And that makes sense. You are effectively asking all of the gears to turn both clockwise and counterclockwise at the same time. It’s a tall order.
But what if you could have a loop of many gears, each with a slight twist, which would add up to a half twist, like a Möbius strip? That should work, if you can find the right twisty gears. I never felt like I fully understood gear geometry anyway, so I dusted off Shigley and started typing up some code in my favorite prototyping tool: SpaceClaim‘s API.
The first step was to produce a basic spur gear, which is just an extrusion if you can produce the right profile. It turns out there is a lot of hand-waving with gears, as the nuances of the geometry are generated from the manufacturing process (known as “hobbing”). Shigley went over the first principles, but he only took the theory as far as one needed to make an approximate drawing. The American Gear Manufacturers Association wanted a lot of money for their reference documents. Because I’m cheap and maybe a little stubborn. I decided I was just going to work the rest out myself.
So I just copied Shigley to the best of my ability:
Although the textbook gears had functioning involute surfaces, they left something to be desired. The addendum and dedendum didn’t perfectly match up. For real gears, you want a gap down there to deal with machine crud and other detritus, but I wanted to make gears that meshed perfectly. Here’s a close up of the perfect gear geometry between two differently sized gears:
Once you have the profiles, creating helical gears and bevel gears (and helical bevel gears) is pretty straightforward. Here’s a particularly compelling result:
I tackled screw gears next. Although they look a lot like helical gears individually, they rotate the motion by both sliding and driving at the same time. Worm gears are probably the most commonly known type of screw gear. These allow twist, which is exactly what we need to achieve a Möbius configuration. Here’s an example of screw gears with different numbers of teeth.
The literature would sometimes reference an exotic gear type, the hypoid. These are rarely seen in parts catalogs, as they are difficult to manufacture, but they are sometimes seen in specialized products such as the output shafts of automotive transmissions. They also allow twist between their axes, but unlike screw gears, they maintain contact across the entire length of the tooth. After a lot of Googling, I had difficulty coming up with a good description in the literature, so I ended up working out the details myself. Here is one result. Hard to believe that it works, but it does:
At this point, I had my bases covered. I also realized that I had the technology to make a pair of gears between any two axes: parallel axes got spur gears, intersecting axes got bevel gears, and skew axes got hypoid, with screw gears as a simpler to manufacture version of the hypoid.
The next step was to try to arrange a bunch of these in a Möbius strip. Although I’m mostly a fan of top-down design, this was an excellent opportunity to play with a bottom up approach. I created a matched pair of hypoids with 45° of twist, and just started assembling. In a little while, I ended up with this:
I shared it because I thought other engineers would find it cool, but I was unsatisfied due to its lack of symmetry. It was just a gnarled mash of gears. Although I had conquered my original challenge, the results seemed pyrrhic and inelegant. Some folks in the GrabCAD Community tried 3D printing it, but had problems with consumer printers. I did produce a workable, symmetric 9-gear concept, but it felt too simple and boring, so I didn’t pursue it further.
And then, something amazing happened: I joined the GrabCAD team. I soon felt like I could do better than the original model – a feeling amplified by a not insignificant number of comments that suggested real improvements. I felt compelled to respond. Also, GrabCAD has a Makerbot Replicator 2, so I started a redesign that would be optimal for the printer’s tolerances and material.
My biggest concern was the armature, which would have required a lot of supports and would have had to hold tight tolerances over a long distance. To address it, I decided to make the armature modular so I could print lots of little parts. A quick calculation told me that there would be plenty of degrees of freedom for movement, but I couldn’t quite visualize how it would work. Here it is one its first day assembled, before being broken in:
Here’s the model on GrabCAD, with instructions to make your own:
That one works great, but it currently requires a lot of expensive little ball bearings from McMaster. I did a redesign for holiday gifts last year that uses washers and spacers, suitable for SLS printing:
And that’s about where the project rests today. What started as a modest attempt to understand involute curves became an engineering journey. Of course I’m far from complete. Here’s a rough to-do list on the project.
- Create version with easy-to-find metric fasteners
- Create version that’s 3D printable on Stratasys FDM and Polyjet printers so no assembly is required.
- Animate the CAD model, demonstrating the poloidal rotation. Replace the red churning gears on GrabCAD with the new animation.
- Create a version with a motor and controller embedded in every gear. Use genetic algorithms to make it into a remote controlled crazy machine that can actually be steered.
- Port the gear generator code to a WebGL so anyone can make and 3D print their own gears?
Want to collaborate? Reach out to me on GrabCAD!
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About the author: (Blake Courter)
Blake has dedicated his career to making engineering more efficient, fun, and team-oriented. Blake started his career at PTC, where he created new CAD tools to assist with conceptual design and components to solve interoperability problems. In 2003, Blake co-founded SpaceClaim, Inc., a CAD company whose direct modeling paradigm heralded a new generation of solid modelers. At GrabCAD, Blake is focused on turning engineering data management into an amazing, enjoyable team experience. Outside of work, Blake is usually found reading math books, producing art from code, and occasionally playing the banjo.
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