Hi, this is module 16 of three dimensional dynamics. Today's learning outcome is to use the rotational transformation matrices that we developed last time, and actually solve a problem. And so here's the rotation transformation matrix. We're going to rotate from frame F to a frame B as we rotate about any particular axis, we use a rotational transformation matrix about that axis. We found that this was the rotation transformation matrix about an x-axis rotation. This was for the y-axis rotation and this is for the z-axis rotation. So now we want to use that in a problem. And so we're going to look at the Eulerian angles that we did a, a couple of modules back. When we used Eulerian angles to describe the rotation of a body, we went from, first, with the first Eulerian angle through an angle phi from a space fixed frame to an intermediate frame F1. And that was called procession. So we went from big I, big J, big K coordinates to coordinates in the intermediate frame F1. And then we rotate through a second Eulerian angle which was theta, and that was called nutation and we went to the intermediate frame F2. And then finally we rotated through a third Eulerian angle, psi, and that rotated us to a body fixed frame that was called spin, and the coordinates for the body fixed frame are little i, j, k. And for my example now, I want you to recall, we did this two modules ago, the angular velocity expressed in terms of the space-fixed coordinates F, and this was the result we came up with. And what we're going to do now, is we're going to use the, rotation, transformation matrices to express that angular velocity in terms of the body fixed coordinates B. And so here again is the angular velocity of B with respect to F, expressed in the space fixed frame F. And, we want to take that now and we want to transform it from F over to the frame B. And so we're going to go through a rotation around the K axis first, through an angle phi, and then we're going to go through a rotation about the intermediate Y axis, through an angle theta, and then finally around the Z axis again, with the angle psi. And when I get through that entire rotation, 1, 2, 3, I should end up with my angular velocity of B with respect to F expressed in the B frame. Now, I'm going to go through, some, mathematics here rather quickly. The advantage you're going to have with this video is that you're going to be able to stop it, and make sure you can follow each step. But I'll go through it rather quickly since it's straight forward linear algebra but you're going to have to remember your matrix multiplication to be able to do this, these calculations. And so here again, I've got my angular velocity expressed in the F frame, I go through my 3 rotational matrix, matrices to express it in the B frame. And so first of all, I'm going to put my angular velocity expressed in the f frame in vector form, so this is the I component this is the J component and this is the K component. Now our first rotation is about the z-axis by an angle phi. You'll recall the general rotation about a z-axis inters my generic angle gamma sub Z, so I'm just going to replace gamma sub Z with the angle C, and here is my rotational matrix. Now my rotational transformation matrix about the angle phi and then I'm going to have to also do angle theta and angle psi, so let's carry on. And so here's my, again, my angular velocity in the B frame, B frame, excuse me, the F frame and then I rotate it about the z-axis and then I'm going to rotate about the y-axis by an angle theta. Recall this was the transformation matrix rotation transformation matrix around the y-axis. So I substitute angle theta for my generic angles gamma sub y here, and then finally I'm going to rotate again about the z-axis. Here's the form for the rotation about the z-axis. I'm just going to replace the angle phi with the angle psi. And so there's the expression as in its entirety. And so here's where I'm going to start going a little fast on the math. Because I just go through this matrix multiplication. If I multiply this 3 by 3 matrix with this 3 by 1 vector, I get this result and you can do that on your own. Multiplied by these next two rotations. And here again, that, is that result and so I'm going to multiple these two matrices together, again a 3 by 3 by a 3 by 1 and I get this result. And then finally I'm going to multiply that by the transformation matrix around the angle psi around the z-axis again. And when I do that multiplication, here's what I end up with. So I've taken my angular velocity expressed in the F frame, and after doing each of those rotation transformations calculations, I've ended up with the angular velocity expressed in the B frame which is what we wanted to do. And so here it again, as a recap, is my angular velocity expressed in the F frame. I went through these rotations, I ended up with my angular velocity expressed in the B frame. And so, I do want you to recall back to module 14, because in that module, we went through step by step. The rotations with the Eulerian angles, and if you go back to that module and you look, you'll see that we had expressed the angular velocity in terms of the body-fixed coordinates as this, and you'll notice that these are the same result, which we should obviously arrive at. I just have a little, a little different order, you've got sines psi theta dot minus sine theta cosine psi P dot and then this term matches with this term and here's my K component, and so I have been successful in this example, and moving from the angular velocity express in the F frame, the angular velocity expressed in the body fixed frame. And I've done it rather quickly, if I just use the transformation, rotation transformation, the rotation transformation matrices and matrix algebra. And so that's it, and I'll see you next time.