Hello. Today, we are going to start the last chapter, Chapter 7, Three-dimensional Dynamics of Rigid bodies. Finally, we are in the last chapter. Let's review the table of contents of the textbook. First half of the text is covering the particle dynamics, and second half was the rigid body dynamics. Ultimately, we wanted to teach you the kinetics, the force and motion relationship, F equals ma, and to do so, we have learned a kinematics ahead about how we can formulate the acceleration for the rotational motion or the thing that when you are in the rigid body, how we can separate it out as a translation and rotation. Same for the 3D dynamics. So we are ultimately later want to teach you how you can formulate the force and motion relationship, but to do so we have to learn about how we can handle the acceleration and the velocity relationship for the 3Ds. So that's the kinematics part. So the first part of the Chapter 7 is going to be about the kinematics, especially we are going to handle the sequence of rotational motion in 3Ds, and later once we know how we can handle those kinematics, we're going to learn the kinetics, which is F equals ma, but there are pretty quite different form of equation. So we're going to cover that later. That actually ends the dynamics chapter. First, I want the relationship for the many Omegas, many rotations in 3D. In three-dimensional motion, there are couple of the situation that there are many Omegas, which is also dependent on to the other. So for example, you have a major primary capital Omega is rotating, and under those structures, you have small Omega, which is moving subjective to the capital Omega. So this blue line is making a route like this, depending on the motion of the capital Omega. So in that case, even though the constant, the magnitude of these blue Omega, small Omega is constant, what will be the time rate of this one? So the Alpha T Omega dt, which is Omega dot, is what? So think about that in 2D motion. In the previous chapter, when we want to take a derivative of some certain value which is under rotation or coordinate, not only its time derivative of this magnitude, we also did Omega cross product of that value. Similar to that, when your Omega is subjected to the another Omega's, then when you take the derivative with those Omega value, you have to do the Omega capital, Omega cross, Omega term, and its magnitude change. That's the big difference that we are going to handle in 3D rotation motion. So even though the Omega is constant value, its time derivative, if it's under certain primary rotational motion, is not going to be zero anymore. So note that. When you take the derivative, will be Omega which is on the primary rotational motion. Its angular acceleration is capital Omega cross-product Omega. So most of the problems in 3Ds are combining many components for the Omegas. For example, in this case you have a primary Omega in the vertical direction, while this disk is rotating the other axis. Then as you can see, this capital Omega determines the whole structure, the angular motion, while this is small Omega is only affecting the disk rotation. So we are going to handle this one as a kinetic metrics and kinetics problem. So to do so, you have to actually formulate the Omega first, so that's what we are going to learn the first part of Chapter 7. So usually, you're given primary and secondary rotation. It's sequential rotation. Then you're supposed to find the velocity or acceleration, or angular velocity or acceleration in one coordinate frame so that you can actually formulate further. Suppose that you are establishing F equals ma, F is coordinate capital XYZ, acceleration is coordinates small xyz, that doesn't make sense. So you want to actually make all the coordinate in the same frame. That's what we are going to do. So there are several steps that we're going to go. First, you should specify which one is a primary and which one is secondary rotation, and formulate those, set the coordinate for primary and secondary as well. So primary as a capital IJK and secondary as a small ijk, and you want to relate the capital IJK and small ijk by considering what's going to be the time derivative of the secondary coordinates. So what's going to be the time derivative is small ijk. Here, we are going to use this relationship, capital Omega, cross Omega when small Omega is under capital Omega rotation, and finally, we are going to formulate all the capital IJK coordinate into small ijk coordinate, and this is the example. This is a 3D structure different from the 2D which contains many Omega which one belongs to the other, the other belongs to the other one. So suppose that you have a small Omega which will rotate this disk, up right, this way, while this whole structure with a shaft are pivoted to the other structure and then this is now rotating the other way in the horizontal direction. So think about it. If only small Omega rotates, that doesn't affect the rotation for the blue one. But if your blue one keep rotating, that actually affects to the whole structure Omega for the pink disk. Therefore, we would say that this is a capital Omega, the primary rotation and this one is a secondary rotation. So you have to specify capital and the small Omega. Now, you have to then formulate, set the coordinate to specify this one. So if you want to just square the capital Omega, your coordinate will be, may be fixed at O and then capital XYZ. Now, if you want to describe the small Omega where will you just at the coordinate? Maybe set the coordinate at disk or even maybe you can set the discrete coordinate at point O. Like overlapping coordinated. Yet that doesn't make sense. So you have a blue coordinate for the small Omega and green coordinate for the capital Omega, which is the same time. That's the part that you specify the IJK. Now then, with the coordinate you can specify. Capital Omega is something, something capital I coordinate. Your small Omega is something, something small k coordinate because this is coordinate that you're going to use. Now, when you take the derivative of small Omega, then you have to take the derivative of the IJK unit vector as well. So think about what would be the time derivative of small k-vector. Small k-vector, initially this one, and then when you are rotating, primary, those k-vector wrote it together. So what would be the k dot that capital Omega crossproduct k? Now that we went to the third part. Now, finally you can relate everything in terms of a single coordinate. So let's make it everything as a small ijk. So small Omega is small k vector, capital Omega is capital I vector here, which is, what? Like here. At this instant, this capital I coordinate is coincide with the small I coordinate, so you have a small I coordinate for the capital Omega. So by doing so, you can relate all the Omega information and its time derivative information ready. Now, let's work on another problem. Now, your primary Omega is this vertical, and secondary is this horizontal. So think about that while you're rotating the pink one that doesn't really affect the motion for the blue one. But if you're rotating the blue one, that actually moves a pure pink structure, the disk around. So this one is a primary coordinate Omega, and this one is a secondary Omega. This was done. Let's specify the coordinate. So to describe the capital Omega, cross Omega as a coordinate, maybe we can set the coordinate here at O here, like IJK, XYZ, To describe the small Omega, we will just set the coordinate. Well, maybe I can also set the coordinator here. So I could say another small xyz coordinate here. So this one has been done. So to bind the relationship between the primary and secondary, I have to know what would be the coordinate differentiation. So what it's going to be mall ijk time derivative. So once you set their coordinate, capital Omega is capital K, and then small Omega is going to be small i. Now, take a derivative of this i vector because when you want to find the acceleration, you have to anyway take the derivative of this one. In that case you have to take a derivative i vector. This I vector initially like this, but as this coordinate is under primary rotation, i vetor varies. So you have the time derivative component of the i, which is what? Capital Omega cross product i. So you got the information about this unit vector time derivative and finally write them down everything in terms of small ijk. So here, what's going to be this k-vector? K-vector at this instant as same as what? The small k-vector. So this is the preprocessed that you have to solve the kinematics problem. Okay, let's do another example. Now you have a disk still rotating for small Omega, and the whole structure of the shaft and the mortar of the drums are pivoted to another clamp structure and the whole structure is rotating vertically. So here, this is the vertical one is going to be your primary Omega, and this one is going to be your secondary Omega. Then how we can set the coordinate for the capital Omega? Maybe I, J, K at O. So since this system is under primary Omega and the secondary Omega, you might say, well, instead of just splitting this out, can I just write down its total Omega this way? Well, yes, the resultant motion is right, its resultant motion is capital Omega and small Omega. But note that this small Omega is dependent to the capital Omega depending on the motion for the capital Omega, this blue one varies. So even though the total is the vector sum is right, you should note that one belongs to the other, one is dependent to the other. Now let's find the coordinate here. So I might set the coordinate with respect to the O. So I have X, Y, Z coordinate, and to describe this blue one, the small Omega where we're just at the coordinate. Well, maybe I could share the origin here and then may be the same with the capital X, Y, J? Yes, you can do that. Or you can do the different way, is having this component as a Z components. Maybe you might have another plane to describe the disk motion like something like parallel to the disk. Well, this doesn't look like a parallel to the disk, but I intended to draw it the parallel to the disk. So yeah, that's another option. You don't really necessarily have the small x, y, z and capital X, Y, Z overlaps. So I have, this is a perpendicular, and this is perpendicular this way, small x, y, z coordinate. Now next step is, what's going to be the time derivative of those small x, y, z, the secondary coordinate subjected to the primary one? So think about it, if you have a vector, and then once you set the coordinate, you should write them. Capital Omega is something, something capital K, and small Omega is something, something small k. Now if you take the time derivative, what will be the change for the k-vector? k-vector initially this way, but as you're rotating under the primary rotation, this k varies. So you have a k-vector time derivative as Omega cross product k. So finally, you have to write them everything in terms of some single coordinate frame which is small i, j, k. So this one is a small k already. So how this small k-vector could be transferred to the smaller i, j, k. This one is now capital Z or as capital K of product and then if you make some projection of this capital Omega vector into the blue one, the small i, j, k small x, y, z, then this one is Gamma, with respect to this value this one is Gamma. So what you can have is, this capital K is going to be cosine Gamma j and sine Gamma k. So by doing so, you have everything about all the rotation in 3D in terms of small i, j, k and plus you even have their informations ready, when you take the derivative for the small Omega. Let's do another problem. This one has a rod attached to this drum and drum has been clipped by this clip and then this one has been connected to the shaft and shaft is rotating. So initially, the primary rotation will be a vertical one, and the secondary to that is a drums rotation, a counterclockwise rotation. So set the coordinated to describe the capital Omega and small Omega. Here maybe you can put it as an overlapping coordinate. So your capital Omega is something, something capital K. Your small Omega is something, something small i. Now take the derivative of this i vector. What would happen? So this i is rotating with respect to capital Omega, so your i vector time derivative will be capital Omega cross-product i. Now, let's write down everything in terms of single I, J, K coordinate. So your capital K coordinate is what? At this instant, it is overlapping with the small k coordinate, so it's going to be small k. So this is the preprocess that you have to handle the kinematics problem, the 3D rotation. Let's open another problem. So this is a disk, is rotating this way and which is attached to the shaft to the main assembly and the main assembly is rotating with capital Omega. This is a primary Omega, and this is a secondary Omega, such that the coordinate for primary Omega at O capital X, Y, Z and the secondary coordinate as small x, y, z as well. So I can make it as overlap. That's one option. In that case, your capital omega is going to be something, something capital K and your small Omega will be something, something. Okay. So in this case, since you made it as overlap, this one is going to be a projection here, so like this and like that. So your Omega is going to be sine that X components and Z components. So it's going to be sine Beta i and cosine beta k. When you take the time derivative i and k, what would happen? Both are rotating, i is going to be rotating with the capital Omega, but the j component is just parallel. So you can have Omega cross i term only and then you can finally rewrite everything in terms of small i, j, k coordinate. So your capital K is going to be coincide with the small k vector at this instance. So this is the preprocess that you have to use when you're handling the kinematics problem. One other option when you set that small coordinate for the small Omega, the secondary Omega, I would like to make this one as a Z parallel to the Z vector so you can set the coordinate differently. That's perfectly fine. So in this case, your capital Omega is going to be a capital K, your small Omega is going to be now small k and then when you take the derivative, this k vector is subjective to their rotation, over the capital Omega this way. So it's going to be capital Omega cross k product. Now, finally you have to rewrite everything in terms of small i, j, k coordinate to actually handle the problem. So how would you change this k-vector? K vector with the Beta value has what? Y component and Z component. So what you can have is something, something j component and cosine Beta k components. So you can either preprocess your three-dimensional rotational motion into this way or the other way that's matter. So in this chapter, we briefly go over how we can formulate the many Omega cases in 3D. In 3D, there are some perpendicular Omegas or some Omegas with the different angle, but primary and subjected to one to the other. So we call it as a primary Omega or secondary Omega, and we learned how we can formulate them in one coordinate system. Thank you for listening.