# Friction Model for a rolling disk

1. Nov 15, 2008

### f_nosferatu

Hey guys!

I'm doing my B.S thesis (Mech. Eng.) and I've come across the problem of a rolling disk with friction. Imagine a disk in real world rolling on a flat surface with no external force or torque applied to it except for the interaction of the surface (Normal, friction, etc.).

Now, since this is a real world problem, the disk will eventually come to a stop. However, if you write the equilibrium equations for such a disk using the single point friction model, you will come to a paradox: The friction force (let's say it's acting on the contact point in the opposite direction of moving) gives the disk's center of mass a negative linear acceleration, while the moment of the friction force about the center of mass tends to give the disk a positive angular acceleration! (Don't forget that the disk is in pure rolling, so: linear acc. = R * angular acc.)

This is why I need a friction model that doesn't have this bug! I think it should be based on a contact patch instead of a contact point, but I can't go further than that by myself. Can anyone help?

Thanks!

2. Nov 15, 2008

### rcgldr

Why would the single friction point result in a negative acceleration of the center of mass? It appears that what you've decribed is a wheel rolling along a surface with no losses, therefore no linear or angular deceleration. The only time you'd have linear deceleration and angular acceleration is if the wheel was sliding as well as rolling.

It seems to me that rolling resistance requires some type of lossy deformation, that involves a contact patch area, as opposed to an infinitely thin contact point.

3. Nov 15, 2008

### Staff: Mentor

If the disk is rolling along a horizontal surface, the required static friction to maintain the motion (in that simple model you refer to) is zero. It's in equilibrium; No force is required to maintain constant velocity.

What you need to consider is rolling friction, due to the deformation of the surfaces. Here's a link to get you started: http://webphysics.davidson.edu/faculty/dmb/PY430/Friction/rolling.html" [Broken]

Last edited by a moderator: May 3, 2017
4. Nov 15, 2008

### f_nosferatu

Thanks a lot you both!

Doc Al: This is exactly what I need, but the link you provided is just the theory. I need actual formulations, whether numerical or analytical. For example, a function for the force distribution on the contact patch (as in the 3rd diagram of the link) could prove to be really useful. I know I can use the simple formula of [Frr=Crr*N] for rolling resistance force, but that won't help me with the rolling resistance torque. Could you provide any further help?