[BLANK_AUDIO]. This is module four of three dimensional dynamics. In today's learning outcome is to take the theory that we developed with the previous two modules and solve for the angular velocity going three-dimensional motion. Here's the theory we came up with. This was the angular velocity with 3D which now can have not only a K component but an I, J and K component. And we're going to, I like to apply these, this theory to real world props, and so here's a clip of a a, a, a, of a plane taking off with a landing gear reject, retracting into the plane. And that's a rather complex system. But let's look at a little simpler demo over here. So instead of. For, for the, for the one we're going to actually analyze, it, it, in this module, instead of having the gear retract like this, what I'm going to do is, we're going to look at a problem where it's a little bit simpler system where we're going to just go ahead and retract the gear up into the body of the aircraft like that. So, I'm going to have a spinning of my wheel like this. And then it's going to turn with an angular velocity up into the body, which we'll call a mega two. Okay, so here's a graphic of the problem. Here's my wheel spinning with an angular velocity of mega one. Here's my arms, the gear itself moving up into the body of the plane with an angular velocity of mega two. And so the first thing I'm going to have to do is attach my reference frames to the various bodies. I'll attach my fixed frame to the body of the aircraft itself. And then I'm going to go ahead and a, attach a, a B frame to the landing gear that's, that's retracting up into the aircraft. And finally, I'm going to attach a C frame to the spinning wheel itself. Okay, and so that's the set up of the problem uh; we want to now solve for the angular velocity of the C, the wheels, body C with respect to the ground, F. And so, try to do that on your own, and then come on back, and we'll do it together. And so the way you should have approached this problem, is to use the addition theorem that we came up with last time. And so we said that the angular velocity of C with respect of F will be the angular velocity C with respect to B plus the angular velocity of B with respect to F. Now, we have three different reference frames here. And so we can express this angular velocity in any one of the three. But we really should want to be consistent and express it, all the terms in, in a single frame. And so I'm going to go ahead and choose to express the angle of velocity in the B frame or the little IJK, and so the first thing I want to say is, okay, the angular velocity of C with respect to F is equal to the angular velocity of c with respect to B. And so if I look at the spinning wheel's angular velocity with respect to the landing gear arm B, then it spins in the I prime direction by the right-hand rule. But we notice that the I prime direction and the I direction always are parallel to one, one another, even as the B frame moves. And so, I can say that omega C with respect to B is omega one. And I'll express it in the I direction, little i direction for the B frame. Then I've got plus omega B with respect to F, okay, B with respect to F, it retracts up into the body of the aircraft with an angular velocity omega two by the right hand rule, that's in the big J direction or the little j direction, so I want it to be consistent all in the B frame. So we're going to say plus omega two, in the little j direction, and that's our answer. So it's as simple as that. So here is our expression for the angular velocity of frame C or body C, with respect to frame F, written in terms of the coordinates in the B frame. But as you analyze or design this situation for this aircraft, you may say, well, I'm more interested in knowing what the angular velocity is in terms of the, the body of the aircraft itself or frame F, and so we would like to switch. So let's, now put this angular velocity in terms of big I, J and K instead of little i, j and k. And to that, let's look at the relationship between the big F frame, here's a graphical depiction of the big F frame and the, and the B frame. And so as this B frame undergoes the angle of velocity at omega two, it, it turns after some period of time. And to analyze this, let's recall back some simple dynamics from your physics days or my earlier course. Remember if we wanted to relate the linear distance to linear velocity, we have X equals V times T. So this says the linear velocity times time equals the distance traveled, the linear distance traveled. Well, analogously, we can use that for angular motion. And so we say that the angular displacement, which we'll call theta, so this is theta here, this is the, the distance that the frames move relative to each other is equal to not just the linear velocity, but now, it's the angular velocity times time. So this angle theta is equal to omega two times time. The angular velocity at B with respect to F times time, and now we can use geometry to relate little i, j k, with respect to big IJK. So here's my triangle. I have the angle theta here. This is the opposite side. So this distance. Opposite side is going to be sign of a omega two T. This distance here is the adjacent side so that is going to be cosine of omega two T. And we see now that if I want to express little i in terms of big IJK, I have little i is equal to okay, by vector addition, I go out a distance cosine omega two T in the big I direction and then sine omega two T in the big K direction, but it's not in the positive big K direction; we see that it's in the negative big K direction. So we have I is equal to cosine omega two T in the big I direction minus sin omega two T in the big K direction. Now let's do the same thing for relating little j to big IJK. And you do that on your own and then let come on back, and we'll do it together. So as frame B rotates up into the body of the aircraft, we see that little j and big J always, for all time stay in the same direction. So little j is always going to equal big J. And finally on your own, I'd like you to relate little k to big I, J and K. And this is the result that you should have come up with because for little j by vector addition, I go a distance cosign omega two T in a big K direction plus sin omega two T in the big I direction, and so now we can use those results. Substitute in here for i. And over here for j. To come up with an expression for the angular velocity of C with respect to F in the big IJK coordinate system. So I've got omega C with respect to F is equal to I've got omega one times little i with cosine omega two T in the big I direction minus sin omega two T in the big K direction. Okay, and that takes care of that term that I have plus omega two and little j is the same as big J. And so what I end up with is omega C with respect to F is equal to omega cosine omega two T in the big I direction. Plus omega two in the big J direction and then minus omega one sin omega two T in the big K direction. And that completes the problem. I now have the angular velocity of C with respect to F, expressed in the frame attached to the aircraft body itself. Okay, let's go ahead and allow you to do a problem on your own. Here's a, a video of a satellite dish antenna turning, and so I've got a little, simple model of that here and so the antenna A is oriented, and with, with various ro, rotations as it moves, and so I give you some information here about the azimuth and the elevation, et cetera. And, I want you to go ahead and solve this problem and find the results. And so I have put the solution in the module handouts, and if you can do that, you're you've got a pretty good grasp on angular velocity in three-dimensions. And we'll go on to angular acceleration in the next set of modules.
