# Force of friction between wheel and axle

1. Oct 26, 2009

### gunhed508

I've been working on a simulation of a Lorenz Waterwheel, and friction was killing it - I couldn't use realistic coefficients of friction. The problem, I discovered was that I was not using the correct equation to calculate the force of friction that counters the wheel's rotation. Having misplaced my physics reference book, I thought I could easily find this equation on the web - not so! After devoting hours of internet search, I cannot for the life of me find this simple equation! Can anyone help me? To clarify what I need, I've provided a basic model below:

View attachment wheel_and_axle_friction.bmp

A wheel of radius rw is centered on an axle of radius ra.
The wheel has mass, m.
There exists a coefficient of static friction between the wheel and axle, us.

What is the equation to calculate the force of friction that opposes rotation of the wheel?

Many thanks!

2. Oct 26, 2009

### Staff: Mentor

Welcome to the PF. The basic equation for frictional force is F = mu * N, where mu is the friction coefficient, and N is the normal force. For the case of the wheel on the axle, you will need to figure out the distribution of the normal force as a function of the angle. Max force on top and tapering off as you get to the sides of the axle.

That force distribution then becomes a torque distribution, which is what resists the rotating motion of the wheel.

3. Oct 26, 2009

### rcgldr

The torque produced by friction in the axle remains the same regardless of the radius of the wheel, however, the force required to produce the torque at the edge of the wheel is rw x tangental force, so the tangental force is decreased if the wheel to axle radius ratio is increased.

Last edited: Oct 26, 2009
4. Oct 26, 2009

### Staff: Mentor

BTW, the use of bearings will reduce the friction force a lot -- can you use bearings in your design?

5. Oct 26, 2009

### gunhed508

berkeman:
Thank you for the welcome on board! Yes, the basic "sliding" friction equation is the one I had incorrectly applied to my model. I have an old Statics textbook from college that I had many years ago, and I remember specifically it had an already laid-out equation for my particular model. (By the way, this is for a computer simulation, so using bearings is simply a matter of using the correct coefficients of static and kinetic friction.) What I'm needing is the equivalent equation of the sliding surfaces physical model for a wheel and axle physical model. I need this to properly sum the forces in each iteration of the simulation.

Jeff:
I had hoped that this was a commonly-needed equation and that it would have been posted somewhere, but I'm beginning to wonder... I'm not wanting to “re-invent the wheel” (no pun intended) by struggling through the calculus required to re-create that equation (I've long forgotten how). I'm hoping someone here will just recall the equation or dig it out from a textbook or reference of some kind. I do recall that the equation included the radius of the axe and the radius of either the outside of the wheel or the inside of the wheel immediately surrounding the axle, which is probably more correct.

Any ideas?

6. Oct 26, 2009

### rcgldr

Make that:

Torque anywhere on the wheel is rw x tangental force at the edge of the wheel, so the larger the radius of the wheel, the less tangental force it takes to produce the same torque.

7. Oct 26, 2009

### gunhed508

I understand about torque - the net torque on the wheel is only one of the forces on the wheel's angular motion in the simulation. Another force acting on the wheel is the resistance of friction as the wheel grinds around the central axle. It's this resistance I need to calculate, the axle friction.

8. Oct 26, 2009

### Staff: Mentor

That's a frictional torque. Communication problem?

9. Oct 27, 2009

### gunhed508

Frictional torque may be the more correct term, assuming it describes the friction between the wheel and its axle (not to be confused with, say inertia). At the very least, it will give me more keywords to search on the web. Thank you. :)

10. Oct 27, 2009

### gunhed508

Please allow me to re-state the problem.

I had searched the web again, this time for frictional torque, but I still couldn't find what I needed. Most of the examples I saw were experiments where the outcomes of the system's motion had already been measured, and the frictional torque was determined from the data points, quite the opposite of whet I'm doing. What I have is a system where I know all its attributes (mass, size, position, etc.), but I need to calculate how it will end up moving as time progresses. Below is hopefully a more clear explanation of my problem:

Please review the following physical model:

Knowns:
F = tangential force exerted on wheel
m = total mass of wheel
G = acceleration of gravity
ra = radius of axle
ro = outer radius of wheel
ri = inner radius of wheel
μk = coefficient of kinetic friction
Tw = applied torque resulting from F

Unknown (solve for this):
Tf = opposing torque due to friction

Assumptions:
- the wheel is already in motion (i.e. kinetic friction)

I need an equation expressing Tf in terms of the known variables in the system. It's quite possible I'm forgetting something there, such as inertia; it's been way too long since physics class for me to remember! Can anyone help?

11. Oct 29, 2011

### eddybob123

Last edited by a moderator: May 5, 2017
12. Oct 29, 2011

### eddybob123

Hope it helps.

13. Oct 29, 2011

### Staff: Mentor

Well, the thread is 2 years old, but that reference is pretty good, so I'll leave your necroposts for now.

Last edited by a moderator: May 5, 2017
14. Oct 31, 2011

### gunhed508

Thank you for giving this thread back some life! :) I still do have an interest in it.

FYI - I took a hard look at a copy of my Statics book a while back and put together this possible solution (see the attached PDF for review). I didn't have room to show all my calculations, but the basic answer is this:

Tw = Tf ≈ rμkmG

I will take a good look at the PDF posted by Berkeman and re-check my work. Any one else who is interested, feel free to check mine. This is a long-term project for a Lorenzean Waterwheel simulation as an "amateur scientist."

#### Attached Files:

• ###### Frictional Torque Physical Model - Waterwheel rev 2.pdf
File size:
71.9 KB
Views:
603
15. Oct 31, 2011

### Staff: Mentor

(Actually, the new PDF was posted by eddybob123, and I only quoted it in my response.)