Ellip-Service

In the pre-corona times, Marika and I would go to the gym fairly regularly, and I would often wonder about the construction of the elliptical machines. It seemed like an interesting problem to create a mechanical linkage that would result in your feet tracing an ellipse. Unfortunately, the ones at the gym had things closed up, so it was difficult to see what was going on. My in-laws though, have a model where the workings are more exposed:

The blue circle is a wheel that the machine applies resistance to, and the red is an arm handle that swings back and forth in an arc. These two are connected together to the foot pedal. The main things I wondered were, does this in fact trace an ellipse, and how does the force vary over a cycle?

I made a model in Python to create an animation of a cycle:

The green trace shows the path of your foot through one cycle, and it's pretty far from an ellipse. I was especially surprised to see it has a pinched end at the start. That made me wonder about the arbitrary choice I made to start the wheel and the arc at their rightmost points. Instead, we could introduce a phase difference, and look at how the foot trace changes:

It seems we can't do much better at making an ellipse, but we can do a lot worse!

Turning to the second question, we can find the force needed by getting the acceleration, which is just the second derivative of the position. If the wheel moves at a constant rate then with the original 0 phase shift we get

This suggests that the force is far from constant, and not even a simple variation. It's possible one of the other phase differences would give a more regular shape, but I'm still surprised by how complicated these can get with what seems like a simple mechanical system!