Need explanation on the points on a rolling wheel

[moorepants]

The idea that you can select the point that is always at the contact location but it fixed on the disk, is complex. A student asked about it. It needs a little video explaining it.

[Peter230655]

As I see this issue: As seen from the street's and from the wheel's point of view, the contact point is a 'new' point at each new time instance. This new point is at rest relative to the street, but gone and replaced by a new point an instance later. So, the life time of the contact point is only an instance long, but it is at rest during it's infinitesimally short live.

No idea, whether this helps understanding the issue - at least it helped me.

[Peter230655]

I wrote a small program to model a wheel running down an uneven street, using exactly this fixed contact point. Results seem to look as one would expect.

[moorepants]

but gone and replaced by a new point an instance later

I like this for sure. I'll use some of your wording when I write it.

[moorepants]

I added this in the 2023 update to the nonholonomic chapter.

[Peter230655]

I think, eq(108) may also be arrived at in a slightly different way, which lends itself to generalization. This is how I finally got my "ellipse on an uneven street" to work (the uneven street is modeled by a function f(x(t)) )

When the ellipse is rotated from 0 to $q_2$, the arc length is $\int_{0}^{q_2(t)} \sqrt{a^2sin(k)^2 + b^2cos(k)^2}\ dk \ $.

The arc length of a function f(k(t)) from 0 to $q_1(t)$ is: $\int_{0}^{q_1(t)} \sqrt{1 + \left(\frac{d}{dx}(f(k(t))\right) ^2} \ dk $

The non slip condition requires that these two arc lengths are the same. Differentiating both sides leads to the speed constraint $\dot q_1(t) - f(\dot q_2(t), params) = 0.$ (Pretty obvious, but it took me six months to come up with it!)