 Hello, everyone. In this video, I'm going to start talking about gears. Gears are useful as power transmission devices. They've been around for a long time. There's early evidence of them as much as about 4,600 years ago of gears existing. They are rugged, which means that they can operate in dirty environments and still maintain their function. They're relatively efficient, upwards sometimes of 98% efficient, so very little loss through the use of gears. They are more costly than some other options. Because of the process of manufacturing them and the materials, they're a little bit more expensive than what we might find with other options. We're going to start with spur gears. When we get into other gear types, a lot of times we're basing what we do off of spur gears as a baseline because they're the simplest of our gear options. They give a lot of useful information. When we talk about gears, the first thing we want to be thinking about is gear geometry. My guess is that many of you are familiar with the gear tooth profile, this involute profile. The general idea behind the involute profile is that it gives us a constant rotational velocity despite the fact that these teeth are coming into contact first at the top, near the top face, and then that contact between this tooth and the corresponding tooth on the mating gear. That contact point travels down this face as it passes through that rotation. As you can imagine, because you're changing the radius of that point of contact, first from out here and then moving towards the center of the gear, you're changing the pitch line velocity, which is basically the tangential velocity of that contact point. If we didn't use this unique profile of the tooth, we'd have two gears that the drive gear would presumably be constant velocity, constant input velocity, but the output gear that's being driven would change velocity, would have a sinusoidal small variation in its velocity, which isn't really desirable for most applications. This involute profile gives us a constant velocity in that contact. Other geometry things to be aware of, we have what we call the pitch circle, which is this line here. We have the circular pitch, which is basically measured on the pitch circle, and it is the distance from one point on a tooth to the equivalent point on the next tooth, so it gives us a measure of how many teeth per inch along that pitch circle, or inches per tooth or millimeters per tooth, depending on which unit system we're talking about. We have a diameter that we often measure at that pitch circle location, we have face width, we have thickness of this top area, and you can see that the top of the tooth does not come to a point, and for the same reason that we might see that in a screw, it leads to weakness, the thinner this is, the weaker it is, and more likely it is to shear off. Also it gives us clash allowance, so it gives us clearance, and we can also see that down here. We have some clearance space so that the teeth aren't bottoming out when they mate with the pairing gear. We have fillets at the bottom for the exact reason that you might expect, that we're trying to reduce stress concentrations. If you can imagine the gears as they come into contact, and we'll talk about this in a minute, but they apply forces tangential to the gear, and that introduces stress at the bottom, so these fillets help reduce stress concentrations as a result. All right, so some of the key things that we would want to know, the pitch equal to pi d over n with units of inches, d in this case is the diameter at the pitch circle, and n is the number of teeth that the gear has, and that gives us the pitch, so it's measured again in inches. However, many times in most of our equations we use what's called the diametral pitch written with a capital P, kind of hard to distinguish here between a lower case p and a capital P, but hopefully you're kind of getting that, and it's equal to n over d with units of teeth per inch, and if we're talking about, and this is in imperial, let's mute that, we're talking about imperial engineering units here, we also have module, and this is when we're talking in si units, and m is equal to basically the inverse of the diametral pitch, so d over n, and therefore is in units of millimeters per tooth, so I actually don't know why, you know, when we're specifying in imperial units, gear gears in imperial unit systems we use pitch, diametral pitch versus in si system we're using module, and that they're the inverse of each other, I don't know, I'm sure there's a story there in some reason, but it's just something to be aware of that, you know, the units are basically the inverse of each other, teeth per inch versus millimeters per tooth, when we're looking up like a standard space for a standard basis for our options when we're trying to find appropriate gears, all right, so when we have two gears that come into contact, we have, you know, and if I just kind of draw a really bad drawing of a gear tooth, coming into contact with another gear tooth, and we can make an angle between these two, so if this was like the tangential line, we could have an angle which represents the line of contact between these two, the line of action of the force that makes, that transmits power between these two teeth, that angle we call the pressure angle, Greek letter phi, and that's again something that is standardized as part of the gear specification and designed as part of that gear profile, so we can have different standards, but pressure angle, again, representing the line of action of the force that, you know, is at that contact surface, so it's basically normal to the surface where those two gear teeth make contact. We can simplify two gears that come into contact rather than like say we're trying to draw this out and try to understand it, typically we're not going to draw, you know, all the little gear teeth around because that would be super tedious, instead we can just represent these as circles using their pitch circles, and if we have one rotating and they make contact here, then of course the other one rotates this way, and these two pitch circles are related to each other by their velocities, rotational velocity, and their diameters, something like that, and here I'm using g subscript to describe the gear, which is the larger of the two that are mating together, and then the p here stands for pinion, which is the smaller of the two gears in a pair, and you may ask, you know, well why do we have this negative sign? Well because in this equation we're basically relating the angular velocities of these two gears, and I'm sorry I'm saying gears, but you know we call one of them a gear, and we call the other one a pinion, they're still both gears, but just kind of a terminology thing, so we have these angular velocities, and relating them to the diameters, and the angular velocities are in the opposite direction, right? If one of them is rotating clockwise, then the other is rotating counterclockwise, so that's really what this negative sign is indicating in that equation. Okay, so we have these these two gears coming together like this, now if we pull them apart, like is shown on this this figure on the left hand side, we have forces being applied, and the net force f here is applied along that pitch, or excuse me, that pressure angle, phi, so we can see that here, but we often break that net force down into a radial force and a tangential force, so tangent to the circle or in the radial direction, and these two forces are really important for us to be thinking about, so that tangential force f sub t is oriented like that, and this is really the force that gives us power transmission, so if you think about you know power being transmitted through one of these one of these gears, which is the driven, and one of them which is the driving, you know from driving to driven, the power really goes from one gear to the other by that tangential force, right? Tangential force times radius gives us torque on the gear, and torque is you know what we're passing through. On the other hand we have the radial force, and the radial force points down, right? And that arrow pointing down has a line of action that passes through the center of the gear, so if I kind of sketch that in, it passes through the center of the gear, right? And torque times radius in this case is zero, because the line of action is zero, there's no radius, which means that there's no power transmitted, there's no torque transmitted by the radial force. The only thing that this force accomplishes is gear separation. So as you can imagine, or as you can kind of see I guess from the diagram, we have one force pushing down on this gear, and one force pushing up, both of the radial forces equal and opposite, and they're attempting to push the gears apart, right? And that's probably generally not good, so that radial gear separation force needs to be accounted for in our mounting. So whatever we are mounting these gears onto, whether it's a shaft or something else, we need to know that that shaft can provide resistance to that separation force, because if it's too weak, say the shaft is too thin or it's the gears mounted at a long distance from like a grounding point, then those gears could push apart and start slipping, right? You could start jumping teeth. So that separation force is not really accomplishing anything in terms of actually efficiency or power transmission, it's actually a loss, right? It's a power loss, but it does require consideration because in our design it could lead to problems. All right, so of course we can use geometry to relate these two things, and if we do that, we find that f sub r equals f sub t tan phi, so that can be useful. You know, a lot of times we might know, for example, the power that we're intending to transmit through a gear set, and power gives us a tangential force, since that's the related force for power transmission, and then we can use that to find gear separating force f sub r, which allows us to design our mounting system. All right, so I mentioned that we have power, and we can relate that to tangential force. Well, it's related basically exactly how we might expect, in that we can say it's a force times a velocity. If I'm talking about imperial units, then I have a little unit conversion built into this equation, assuming that I'm working in units of horsepower, which would be a standard for this, and this v is, as I already mentioned, pitch line velocity. So it's the velocity of a point at that pitch circle on the gear in the tangential direction, so tangential velocity due to the rotation of that gear. So this v is equal to pi dn over 12, and again this is assuming d is in inches and n is in rpm. All right, also in here we have f sub t, which is going to have units of pounds in this case. If we're talking about si units, the equation is slightly simpler, in that we don't need that funky unit conversion built in. So we just have f sub t times v, where force is in newtons, velocity is in meters per second, which gives us watts as our power units in this case. Okay, so that's an introduction to spur gears. Next we're going to move into, you know, how we actually look at spur gears, do some some stress analysis for them, but I'm going to stop here. Thanks.