Oblong Wheel Bicycle
I fantasized one day about a peculiar illusion – a bicycle with oblong wheels that rides normally like a bicycle with round wheels. Why? I’m not sure honestly, but after sitting down and thinking hard about how to achieve such a thing, I realized it was a great critical thinking/math problem.
It turns out that graphing the height of the center of an ellipse as a function of time produces an interesting waveform. My assumption was that it would be a sinusoid related to the eccentricity of the ellipse, but it turns out to look like this:
Understanding this, I pursued a scotch yoke, (reciprocating mechanism) whose input was the oscillation of an axle attached to the center of the ellipse and whose output would be attached to the handlebars/frame in order to keep it steady.
The idea was to have the axle bob up and down (as does the rod in the GIF) and steady rest would be attached to the handlebars. I struggled however to develop a scotch yoke mechanism with a certain dwell that was mathematically related to the eccentricity of the wheel, (and also keep the wheel in phase with the center mechanism on which the scotch was attached), so I changed the design that incorporated a bearing on the end of the bike’s axle that would roll along an inner ellipse 90 degrees out of phase with the big ellipse touching the ground.
This design wasn’t smooth and definitely wouldn’t scale well because the bearing had a tough time climbing the inner ellipse (who’s eccentricity was inversely proportional to its size).
The best solution came with the idea to put a bearing at the end of the axle beyond the center of the ellipse and have it roll along an inner track along the edge of the ellipse. The axle is held upright with a pin through the center of the ellipse and the axle. The key here is that, with the axle upright, the height of the edge of the track is at a constant height from the ground, so, the axle never oscillates up and down!
Pictured below is the mathematical proof that relates the inner bearing track to the eccentricity of the oval, graciously given by Dr. David Jacobs.
Below is a video detailing the actual solution. You can also add some rake to the forks by rotating the inner track proportional to the amount of rake.