Understanding Non-Lubricated Journal Bearings

[Cyrus]

For some time I have was having trouble fully understanding what was going on in a journal bearing that is non-lubricated (So that dry friction holds true). Here is what I was having trouble with. Let's say you have something like a spool of heavy cable on a journal bearing, and its being turned at a constant rate with an applied torque. Initially, the spool will rest at the bottom, and the point of contact will be directly in line with the weight vector. But as you increase the moment, the spool will start to 'walk' up the journal that's holding it in place. So its walking up the inside of a circular orifice (to help you visualize it). So as this shaft 'walks' up the journal, the normal force is decreasing. This means the amount of static friction is also decreasing. You will eventually reach a point where it is at maximum static friction. Now you have reached what is known as the 'static friction circle'. But as this moment keeps on steadily increasing to its final value, it will surpass being static friction and start to slip. Now, in the book they say,

'As the shaft rotates in the direction shown in the figure, it rolls up against the wall of the bearing to some point A where slipping occurs.'

At first, I thought it would roll up to the limiting case where static friction is maximum, and just start slipping at that same location and stay there; however, after thinking about it for a while, I have come to the conclusion that this is false. I agree fully that it will roll up until some point where it reaches the maximum static friction; however, once it starts slipping, the friction force will drop significantly, but the normal force will stay the same (at that instant). So now you just lost a big component of your force in the tangential direction to sustain equilibrium. So I argue that this thing will start to translate back down the jounral, thus allowing the normal to increase once again. As the normal increases, the kinetic friction force will increase, and it will start to translate back up. And this process should damp out over time until it finally reaches some point A, which is Below where it initially started slipping when static friction gave out. Also, I thought about the possibility of it changing from kinetic back to static friction, and 'walking' back up the jounrnal, but came to the realization that this is impossible. The reason being, in order for it to start rolling without slipping once again, the local velocity has to be zero at the point of contact. But were talking about tolerances here that are a few millimeters at most. So this thing is not going to translate back down at such a speed that it will stop slipping. The final point where it comes to rest can then be considered where the 'kinetic friction circle' occurs, because there is constant relative speed between the jounral and the shaft, and so the angle between the normal and the reaction is $$\phi_k$$. Am I wrong, or does this sound like reasonable logic to anyone?

Somehow, I'm very temped to say I am right, because I thought long and hard about this.
http://sns.chonbuk.ac.kr/manufacturing/bearing-9.jpg
The picture sucks, but its the best I can find online. Appologies.

 

[Danger]

You're way beyond me with those technicalities, Cyrus. My common-sense approach (which unfortunately sometimes breaks down under the harsh reality of math), would be that the contact point would remain constant as soon as the pull from the side equals the downward pull of gravity, assuming that the load doesn't change. In reality, though, the gravitational force would be constantly decreasing if your spool is unwinding, or increasing if more cable is being added to it. By my reasoning, then, the contact point would be continually walking either up or down in order to equalize the forces. There's probably something wrong with that, but I don't know what it is.

 

[Cyrus]

Yeah, I used a bad example, sorry. Forget about the wire being unwound. Just assume its a constant load that's spinning.

 

[Danger]

In that case, it still seems to me that it would just find a spot where it's comfortable and stay there. My only actual experience with such a thing, though, is pulling tinfoil from the box. The roll always rides up the side. I guess that really isn't a similar situation.

 

[Cyrus]

This is what I am getting at danger. Get your tin foil, and pull on it gently. It will ride up the inside of the box and stop at some point before it starts rolling. Then any additional force will start the thing turning, and it will creep back down into the box once its rolling and settle at a second lower point. (Although with a box the place of contact will be one side of the box, so the analysis would be a little different).

EDIT:I tried it with the al-foil, unfortunately the roll is too light and the friction too small to see any effect.

 

[Danger]

Hmmm... I'm going to have to think on this a mite. A couple of other aspects of the question popped up and I'm not sure what to make of them.

 

[FredGarvin]

I have to agree with your reasoning. However, I can't say that I have ever looked at a non lubricated journal bearing before. They are mostly thin film lubed.

 

[sid_galt]

IMO, your reasoning would certainly be correct if the gradient of the coefficient of friction at transition point between static and kinetic friction was infinite.
For finite gradients in which there is a gradual (sharp, maybe but gradual) decrease of coefficient of friction, I don't believe the oscillations would be very high.

 

[Cyrus]

Aha!

Physically, this is explained by the fact that, when the wheels are set in motion, the axle "climbs" in the bearings until slippage occurs. After sliding back slightly, the axle settles more or less in the position shown

So, my intuition was right, for once, good.