Wheel and axle mechanics

[rs54]

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.

 

[LightHero]

Any thoughts? The motion of an axle in simple dry contact with the inner race of a wheel can be difficult to predict and analyze. In order for the axle to remain in contact with the wheel race, it must have enough friction to maintain contact as it moves around the circle. If the wheel radius is too large compared to the axle radius, the axle may lose contact with the wheel race and begin to move chaotically. In order to accurately model the motion of the axle, a system of equations of motion must be used. The equations should take into account the friction between the axle and the wheel race, as well as the centripetal force acting on the axle. These equations can be solved numerically to yield the position and velocity of the axle at any given time. Alternatively, a simpler approach can be used to predict the motion of the axle. This approach assumes that the axle has enough friction to remain in contact with the wheel race, and that the centripetal force acting on the axle is balanced by an equal and opposite reactive force. However, this approach may not accurately predict the motion of the axle in some cases, such as when the wheel radius is too large compared to the axle radius.