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I was trying to figure out the motion of an axle in simple dry contact with the inner race of a wheel (see attachment). Even with simplifying assumptions there seemed to be no solution with steady motion except for very special cases (a particular location of the axle in the wheel hole and a special ratio of wheel race radius to its outer radius).

For a wheel radius that does not lead to a steady solution, it seems easy to envision chaotic behavior because the axle can lose contact with the wheel (or maybe suddenly change the way it is sliding).

I tired to apply equations of motion (allowing acceleration) for the case in which the axle stays in contact with the wheel race. It seemed complicated and I’m not sure how to tell if I have the right number of equations (if they are all independent) and I’m not sure how to solve them.

I am interested in any insights or corrections. Will the motion be chaotic and if so what kind of approach can be used to figure out the motion.

Also it seemed to me that some of the simplifying assumptions might be unrealistic for this system. I tried to describe some of the ideas in the attachment.

I also saw a simple approach to this problem in

http://www.pinewoodderbyphysics.com/pdf files/Lecture 4.pdf

but I questioned this approach for a few reasons, for example, it seems to show a steady solution with unbalanced forces on the axle.

For a wheel radius that does not lead to a steady solution, it seems easy to envision chaotic behavior because the axle can lose contact with the wheel (or maybe suddenly change the way it is sliding).

I tired to apply equations of motion (allowing acceleration) for the case in which the axle stays in contact with the wheel race. It seemed complicated and I’m not sure how to tell if I have the right number of equations (if they are all independent) and I’m not sure how to solve them.

I am interested in any insights or corrections. Will the motion be chaotic and if so what kind of approach can be used to figure out the motion.

Also it seemed to me that some of the simplifying assumptions might be unrealistic for this system. I tried to describe some of the ideas in the attachment.

I also saw a simple approach to this problem in

http://www.pinewoodderbyphysics.com/pdf files/Lecture 4.pdf

but I questioned this approach for a few reasons, for example, it seems to show a steady solution with unbalanced forces on the axle.