Welcome to module 13 of Three Dimensional Dynamics. Today's learning outcome will be to describe Eulerian angles for 3D rotational motion. In studying kinematics of rigid bodies undergoing three dimensional motion, we need to be able to precisely describe that motion. The general motion of a rigid body is the super position of a translation and a pure rotation. So I'm gonna take this computer, this laptop computer is my rigid body. So that's a translation and a pure rotation to get to another point. So in translation, all the points on the body follow the movement of an arbitrary point a, the rotational portion is such that the arbitrary point a remains at rest. The arbitrariness of the point selected for the translation means that the only unique part of the kinematics is the rotation itself. And so in today's module, we're gonna use a concept called Eulerian angles as a unique way to describe that rotational motion. So in describing the orientation of a rigid body, the motion consists of a translation and a pure rotation. So here is my body or reference frame in position 1. It goes through a translation and that body could be a laptop or any other body. Once it goes through that translation, we then rotate about a point a, and that's our unique rotation. And so we're going to look at a description of the unique rotation of the body using something which we call Eulerian Angles. So for the I, Eulerian angles, we will start again with a space-fixed coordinate system. And then we're gonna use three successive rotations to come to what we're gonna call a body-fixed coordinate system. Sorta welded in the body itself. And so the first Eulerian angle, here's my space fixed frame. We're gonna rotate about the capital Z or the big K axis, and we're gonna rotate with an angular velocity phi dot. The angle phi is gonna be our first Eulerian angles. And so over some period of time, you can see that the black reference frame here has rotated through a, a certain angle phi. This next axis, we're gonna label as an intermediate frame, n11, n12, n13, and we're gonna call that frame 1. And so, this first Eulerian Angle or this first rotation about that capital Z or the big K-axis by the angle theta is what we call precession. And so let's go ahead and look at a couple of examples. So here is my space-fixed coordinate system. I have my x-axis in this direction, my y-axis in this direction, and this is my z direction. Let's take my body here, which is a gyroscope. We spin it up. And as it goes around this z-axis here, okay, not the spin, but as it rotates around this z-axis, that's called the precession angle. So you can see here, it's precessing around and that's the precession angle phi. So now, let's look at the second Eulerian angle. So here is my intermediate frame I just rotated to, frame F1 with the intermediate axis n11, n12, n13. So you can think of those as like the x, y, and z-axis, intermediate x, y, z-axis. And we're gonna now rotate about the n12 axis. And we're gonna rotate through an angle, theta dot, which is our second, theta is our second Eulerian angle. Theta dot is that angular velocity. And so we have moved through an angle theta, after some period of time. And we're gonna give the labels to this coordinate system as n21, n22, n23. The, the, the number 2 being for the second intermediate frame. And we're gonna call that frame F2. Now, this Eulerian angle, the rotation about the n12 axis by the angle theta is called nutation. And so again, let's look at a demonstration. So, here is my space-fixed coordinate systems again, and I'm gonna take my body and I'm gonna spin it up. And now, as it leans, that's the nutation angle. So, you can see how it leans. It's precessing around. It's also spinning, and we're gonna see the spin angle in just a second here. But, the leaning angle is called the nutation angle. So now, let's look at the third Eulerian angle. So, here is frame 2 that we just rotated to, n21, n22, n23. Think of it as an intermediate x, y, z frame. And if you recall now, for the first Eulerian angle, we rotated about the z-axis. For the second Eulerian angle, we rotated about the y-axis. So obviously you would say, okay, the third time, we're gonna rotate about the x-axis, and actually that's not the case. We're gonna rotate about the z-axis again, for this description of the Eulerian angles. And so we're gonna rotate through an angular velocity side dot, so there is a dot on top of here. And it's going through an angle psi. And we now are going to, after these three rotations, we're gonna be at what we're gonna call our body-fixed coordinate system inside the body itself. And so that's my little i j k frame. And so the rotation about this last axis, this little k-axis or the n23-axis by the angle psi is called the spin. And so again, let's look at a demonstration. And so here is my gyroscope. And now we can see all three angles. We've got the rotation, excuse me, the precession. We've got the lean, which is the nutation. And then we have the spinning of the top itself, which is the spin angle. And so let's recap here. Here's where we started. We started with a space-fixed frame. Okay, we rotated through the first Eulerian angle, which was call the precession. And we went from the space-fixed frame, Capital XYZ or capital or a Capital IJK hat. And we went to the intermediate frame F1. And then from that intermediate frame F1, we rotated about the new y-axis and we rotated to a new frame F2. And that was called a nutation. So we move to n21, n22, n23. And then finally our third Eulerian angle, was back about the z-axis again, or the n23 axis, or the little k [INAUDIBLE]. And it ends up being the little k-axis, the body-fixed axis. And that's through an angle psi, which we call the spin angle. And so that frame is annotated with the coordinates little i, little j, little k. And so we've moved from again space fix coordinates to a set of body fixed coordinates that are welded into the body. So let's go through a simple demonstration here. For the demonstration I'm going to use our book here that I'm using in the course An Introduction to Dynamics that my colleagues Dave McGill and Wilton King have been kind enough to allow me to use their figures and examples in this course. And so I want to demonstrate that the angle or the order of the rotations is, is important and so let's look at a rotation here. This is my x-axis, this is my y-axis and my z-axis. Let's say that we rotate 90 degrees about the y-axis first. And then 90 degrees about the z-axis. So when I rotate 90 degrees about the y-axis and 90 degrees about the z-axis, this is the orientation that I end up with, with the book. However, if I then switch and rotate about the z-axis first, and then about the Y axis, you can see that I come up with a completely different orientation. And so the order of the rotations is, is important, as well as the, the angles themselves. And so re, recall now that the angles phi, theta and psi are known as Eulerian angles. They represent just one way of orienting a rigid body in space. Unfortunately the Eulerian angles do not carry the same symbol from one textbook to the next. And we're still the order and even the directions of the rotations can vary from writer to writer. So obviously then it's important that you set, you choose a set to work with, and then be consistent. And the set I've chosen are the ones that are in this textbook and we'll use those through the remainder of the course and we'll continue with the next module.