[BLANK_AUDIO]. Hi and welcome to module 14 of three dimensional dynamics. Today's outcome is to Express the Angular Velocity of bodies undergoing 3D Rotational Motion using Eulerian Angles. And so, as a review, here's the Eulerian angles. We went through a rotation from the space-fixed f frame to an intermediate frame, f1. First to an Eulerian angle, fee, which we called procession. And that went from the capital I, capital J, capital K frame, [INAUDIBLE] the capital X, Y, Z frame. To frame F1, which was n 11, n 12, n 13. And then we went through a rotation about the n 12 axis by an angle phi dot or angular velocity phi dot, which is through an angle phi, which we called the nutation. And there we moved to an intermediate frame f2. Which was coordinates n 2 1, n 2 2, n 2 3. And then finally we went through one more angular rotation with psi. Through an angle psi. Which was the spin angle, to get to our final frame, which was the body fixed frame, frame B. And its coordinates are little i, little j, little k. So, now let's look at the angular velocity total, as we go through these, these these Eulerian angle rotations, and, so we, you need to recall the addition thereom. Which said, okay, the angular velocity of frame B, with respect to frame F, F, which is your space fixed frame back here, is a, an addition of all of the the angular velocities as we move back. So it's going to be the angular velocity of B with respect to F2. Plus the angular velocity of F2 with respect to F1, and finally the angular velocity of frame F1 with respect to the fixed frame itself, [COUGH] and, so we can write this another way, we can say the angular velocity of frame B with respect to F. Is psi., and decided it didn't come out so well, so I'm going to put a dot up here. That's psi.k, little k, about the little k axis. And then we've got theta dot about the n 1 2 axis, and then finally phi dot about the capital z or big k axis. And, so that's an expression. Of our angular velocity. So here is my angular velocity again. And what I want to do now is write that angular. I can see that I've got it in three different frames which is not real convenient. I've got this coordinate in this, body fixed frame. I've got N 12. In the intermediate frame F1, and then I've got big K in the in the the space-fixed frame. And I can write the angular velocity in any one of those three frames. But let's go ahead and write it. In the body fix frame. And, so in the body fix frame, we're going to have to replace N 12 here, and for N 12, here's N 12, we see that N 12 is the same as N 1 uh,22, and so N 22 or n 12 in terms of. The coordinate psi is cosine psi in the little J direction plus sign sigh in the little I direction. So we'll now be able to replace. That coordinate in terms of little I and little J. The next one we need to replace is the big K coordinate in terms of the body-fixed coordinates, and, so here now the big K, we see that the big K, axis is the same as the N13 axis. And so here's the N13 axis here. And, and doing that, we go cosine theta in the N23 direction, but that, we know that the N23 direction is the same as the K direction. So that's cosine theta K. And then we've got minus sign theta in the N21 direction, and, so we're still going to have to replace the N21 direction. So I've got minus sine theta n two one, plus cosine theta k. This is the expression for the N 21 axis. We see N21, whoops, here's N21, here. There's N21 there. That is cosine psi in the little I direction, and then minus sign psi. In the little J direction. And so now we have an expression, since we can now substitute this in here, we've got an expression for omega B with respect to F, in the little i, j, k coordinate system. And, so here's, here's what I just came up with. I'm again, going to substitute. Here, and I'm going to substitute here, remember this is a psi. And so, when I do that I'm moving from my space-fixed coordinate system to my body-fixed rotation. And we're just looking at the rotation part, okay, we're not looking at the translations since that's not. That's not arbitrary we're, we're just looking at the rotation itself, and so when I substitute those in, here is the expression I get for the Omega of the B frame with respect to the F frame. And I've used the little shorthand here, I've for sign psi I'm just saying F sub psi, for cosine theta I'm just saying C sub theta etcetera. So this is psi sub theta, this is cosign of psi, et cetera. And so, that's a way of expressing the angular velocity, and body fixed coordinate system. What I would like you to do as an exercise on your own. Is to find the angular velocity in terms of the space fix coordinates, and once you've done that, come on back and see if you've got your answer and this is what you should have come out, come up with as a result of that.