 In this video, I want to start talking about stress analysis for gears. So when we're talking about stress analysis, I mentioned before that usually what we're doing is saying that there is a load applied somewhere out near the gear tip, it's a tangential force, and it leads to stress at the base of that gear tooth. If we look at this image, which is a photoelastic stress image, so produced by basically making some gears out of a like a curlic, like a like a polycarbonate type material, and then looking at them using some special filters. When we stress those materials, we can see these bands, and the bands in this photoelastic image indicate stress, and the closer together that the bands are, the higher the stress. And the key things that we can see here are that we see stresses here, which is at the contact point where the two gear teeth are touching from the mating gear and then the gear we're looking at. So those are contact stresses, important to consider, but not really what we're talking about here today. The other location of high stresses is down here in those fillet regions. So that's kind of what we would expect, right? If we have a force applied up here, we have the highest moment, as if this was like a cantilevered beam, we have the highest moment down here at the base, and that's where we see these stress concentrations. So these stresses are really the ones that we're going to look at. And there's a couple of methods that you can use for for analyzing gear stress. The first one we're going to talk about is called the Lewis equation. So the Lewis equation uses or makes a few assumptions. First, it assumes that the full load being transmitted between the gears is applied at the the tooth tip of the gear. So the the tip of the tooth, I should say. And that's, you know, not really that accurate of an assumption. It typically doesn't happen that the full load would be applied at the tip. It's usually somewhere down from the tip and moving lower as those teeth engage. It neglects any radial force. So radial force, again, is is force pushing down towards the center of the gear. That makes this assumption a little bit or this analysis a little bit conservative, because radial force generally cancels some of the bending force or bending stress and reduces, therefore, the overall stress. So neglecting that makes it a little conservative. It assumes uniform force distribution across the width of the gear. That's, you know, in a perfect world, great. But it doesn't account for any sort of gear misalignment or anything like that. So a little bit misleading there. It neglects any sliding friction. So as these teeth move past each other, there's friction, doesn't take that into account. And it neglects stress concentrations, which could potentially be problematic. Because as we know, in particular, with things like fatigue failure, we know that stress concentrations generally are what lead to the failure itself. So that can be problematic. And, you know, we'll address that later. But broadly speaking, we have this first method that I want to talk about, which is the Lewis equation. So a basic setup for this, looking at a gear tooth profile, the Lewis equation makes use of a geometry assumption, which is that you have a parabola that you can inscribe within this gear tooth. And we call it the constant strength parabola, which basically means that it is uniform. The stress would be uniform throughout the entirety of that parabola. So as the stress increases, as you move closer to the base, the parabola gets wider, and therefore increases its strength. So as a result, you end up with this thing that's constant strength. So primarily with this, we're worrying about bending stress. So we would start effectively with our normal bending stress equation. But then we have to figure out how to apply the geometry of the gear tooth to this. So I'm not going to go through derivations using that gear tooth geometry. It's kind of complicated using this parabola, parabolic geometry. But effectively what we end up with is an equation that looks like this. Once I take into account the geometry and substitute it in for this area moment of inertia, and I have my y value based on this height h. But I'm going to take it further and say, well, we can define this value x here, which again is kind of an interesting geometric property. The distance between this horizontal line at the fillet, and if we inscribed a line that was normal to that surface, passed it out to the center line, and then drew it across there, the distance along that center line between those two points. Not critically important for us to fully understand that, but if we take that and substitute it in, we end up with an equation that looks like this. Great. And that's just a step along the way to what we're doing. So then we define the Lewis form factor with the value lowercase y as equal to 2x over 3 times the pitch. Now our stress equation can be rewritten like this, d over b p y. And that's a useful equation again. But we're going to take it a step further because most of the time our gears aren't specified in terms of pitch, but rather diametral pitch. So we'll take p pitch is related to diametral pitch like this. And then we're going to also come up with an alternative version of the Lewis form factor using an uppercase y by a similar relation. Now again, substituting this in distress, we get f t p over b. So this is useful in that it has force transmission, or excuse me, tangential force, which is our powertrain's mission force, diametral pitch, which is often our gear specification, b, which is the face width. So it's how wide the tooth is, or the gear itself is, and y, which is this modified form of the Lewis form factor. Forgot an M here I just saw. And this is great. We can actually pull values for y from the textbook. There's a figure 15.21 in the textbook gives us values of this Lewis form factor y on the basis of how many teeth our gear has, as well as the pressure, whatever standard pressure angle we're using. This is an imperial units in Si. We'd write this equation as f t over m b y. Great. So that's a useful equation. And this gives us the ability to calculate stress in the gear tooth and determine our failure. Now, I already mentioned that fatigue is going to be a primary consideration. And that's, you know, really true of any cyclically loaded thing. And a gear is probably, you know, a perfect example of something that's loaded cyclically, you know, every rotation, each tooth is subject to the same bending stress on and off, on and off, over and over and over again. So this gives us the ability to calculate stress. The next step, of course, would be to compare it against a limit, right? And you may recall that for fatigue, we have an equation that looks like this, 0.5 SU specifically for steel. And then we have a bunch of correction factors that we would factor in. And we can apply our same fatigue analysis that we've already talked about previously in determining the limit here and comparing it against this stress that we've now calculated for our, for our gear tooth. All right, I'm going to stop there. Thanks.