 Hi everyone. So in this video, I'm going to talk about one of the methods that we have for analyzing lubricated bearings and kind of understanding them a little bit better. So the method that I'm talking about here is called Petrov's equation. And Petrov's equation is based on lightly loaded condition for bearings. And so I've got a bearing shaft here drawn with some fixed outer surface, the hole that the shaft rides in. And we want to understand what happens if this is rotating and how I determine the drag force due to that rotation in that lubricated region. So we have a bearing of or a shaft of radius r and of course diameter d. And we have a clearance gap here, which we'll call c. And the length of the whole thing is l. And so the way we would analyze this, you might actually be familiar with from fluid dynamics or something like that, if you've already taken that. But suppose I have an object and I'm talking not in rotation in this sense, just as an example. Suppose I have an object upon which I have applied a force. It has some area, which is the bottom surface here. And it's therefore then moving forward with a velocity u. And it's got a layer of fluid between it and a fixed surface. So when we do this, we typically expect to see that the velocity as we approach the moving surface from the fixed surface, you know, goes from zero to moving at the velocity of the fixed surface, you know, as we would move further away from this in y or something like that. And, you know, really we'd get a parabolic function like this, we might approximate it as a straight line just for simplicity's sake, in this case. But generally speaking, we could then figure out what our stress is induced as a result by saying, well, we have some force over some area. And it's got to be equal to a viscous stress times the change in velocity. And if we do this straight line approximation, we can say, well, du by dy, so change of velocity over change of y would be just u over h. So this would equal u over h wherever where h is the height of this. And we get something like that. So we can apply the same general principle to the problem of a shaft rotating over here. So if I take and say I have a little segment of fluid in here that I'm going to separate out, so I'm going to pull this out over here. And look at this. I have, and I've kind of rotated it. So I've rotated this, I don't know how to explain this, I've rotated it over, as I brought it over here, such that I've got this height C. And I'm applying a force to this due to the rotation of f. And it looks a lot like my problem that I just drew up here in black in that I have a fluid or a force that's shearing across this fluid surface. And it's got some thickness C like this. And so if I analyze this, I can say, well, I'm going to have a very similar situation to what I was just talking about. But I'm going to have force equal to mu a u over C. C is replaced h. I'm using this side of this equation here. So not using tau but substituting in the equivalent values. My a in this case, then is equal to 2 pi rl. Because what I have here is I have a radius. So I have a circumference times a length to give me a whole area. So rather than a flat, you know, rectangular area, I have a rotational surface that provides my area. And then I can calculate viscosity as 2 pi rn. If I say n is my rotation in revolutions per second, this would be my surface velocity that I'm experiencing. Just like I saw up in the kind of simplified example of the flat plate, I have a surface velocity here. And C is my radial clearance. So it's the gap that I have between my shaft and the hole. And in reality, it's, you know, bearing diameter minus shaft diameter divided by 2. So I can figure out what that is, right? Now, if I substitute all of this stuff into a torque equation, what I'm actually finding is then the torque, which has to exist to overcome friction in order for my shaft to rotate that this speed. So we know that that torque is, you know, the force applied at a distance. So I can just start substituting all this back in. And if I do that, she gets something that looks like this. 4 pi squared mu n l r to the third over C. So this is the torque that I lose due to the dynamic friction of this shaft rotating in oil. And I could calculate the power loss from that as 2 pi n times that torque. So that's power loss. I could calculate friction coefficient. That's funny looking F as 2 pi squared mu n over P. This is a different P here. I'm going to define that in just a second, r over C. Now, both of these quantities here are dimensionless. So r over C is dimensionless, mu n over P is dimensionless. This P is pressure or also radial load per unit projected area. So P in this sense equals W, which is my load over 2 r l, which would be like my projected area. All right. So that gives me one analysis method for a bearing. Now, I said right at the beginning of the video that this is a lightly loaded bearing. And that's an assumption that we make here. And that's actually partly why I drew my drawing on the left over here vertically. Because the presumption is that we're not really applying much load to the shaft. So we have to be careful in that this doesn't really take into account very high loads being applied to the shaft. So it's really for only lightly loaded shafts. In the next video, I'll be talking about a more robust version that we can use for this analysis, which provides better analysis and does account for loading a little bit more completely than Petrov's equation. All right. Stop there. Thanks.