[MUSIC] Okay, in this video, let's work on the real kinematics problem for the three-dimensional rigid body which is under sequential rotational motion, like omegas, multiple omegas. Okay, suppose that we want to describe the kinematics of the point A like angular velocity of the drum or the acceleration of point A, which is under the rotation of omega. And the whole structure is also under the rotational motion of capital omega. So how we can describe it? Suppose that, so this is system where it's under primary omega and the secondary omega. And suppose that we want to describe it over the absolute coordinate frame O here with the ground. Then your point A is here rotating with respect to the axis for the drum and to the whole structure. So the overall absolute trajectory for the point A will be somewhat like this, right? It's going to be pretty up complex. So instead of just describing the point A with respect to the A, why don't we just do the relative coordinate frame so that that put the relative coordinate here at the B, and then viewing the point A with respect to B. So the B is rotating with the capital omega. So the motion of the A is only described by the pretend that there's only a small omega exist, okay? So your total absolute motion of the A is going to be described by the absolute motion of the B and its relative motion of r A over B, okay? So here, r A over B is relative displacement defined with respect to the B, which is rotate with omega. So it seems like point A is only under the effect of small omega, okay? So to describe A, I'm going to set the local coordinate frame small x, y, z, attach it to the point B, which is rotating in capital omega. So the absolute displacement of A is going to be absolute displacement of B and the relative displacement of A with respect to the B. And it's going to be xi, yj, xyz, ijk. If you take the derivative of it and then derivative of the absolute velocity of the B and the relative velocity of the point A over B, and that's going to be the total time derivative, the small ijk and time derivative of xyz. Since your i vector, the blue one, is going to rotate with capital omega, the primary rotation, i dot is not going to be zero anymore. So we'll have a capital omega cross i product. And same for the j and k. And this part is assuming that ijk is not rotating and only ijk xyz variation exist, that's the v relative, okay? So assuming that there is no capital omega exists, only small omega exists. So from the fixed ijk coordinate, you can only considering the displacement change for the x, y and z, that's vrel. So your v of A is going to be v of B plus this is the relative velocity. And then, not only just considering the v relative a change, you should also consider capital omega cross r, because the coordinate of the blue is rotating with the green, okay? So to calculate the acceleration, you take the derivative of the velocity. So you have a time derivative absolute point B and the omega dot here and then R dot here and the vrel dot here. Now, this one are the relative displacement time derivative, again, is going to this process. So you have xyz ijk dot, and then ijk fixed and xyz dot. So this one is, again, split it out as omega cross capital omega cross r term and the v relative term. This blue one is like in the red box. How about the vrel dot? Well, by definition, vrel is x dot i, y dot j, z dot k. And then you take the derivative. And then you take the derivative of the i vector and the magnitude of x dot, y dot, z dot again. So you can split them out. And then this one, again, is going to be omega cross velocity term. And this one is the acceleration, like by setting the ijk vector fixed, you just have to consider the displacement, the second derivative. So it's just a single acceleration of a single point A at the fixed xyz blue, small xyz coordinate. So generally, if you're under rotational motion, you have a four acceleration term, right? Like alpha cross r, omega cross(omega cross r), 2 omega cross r dot and then r double dot is all small omega term. And that's what we define as a relative. The a relative means there is no capital omega, only small omega exist. So your vrel dot is going to be capital omega cross the vrel and a relative. So if you sum them up, what you can have is aB, and then capital angular acceleration of the capital omega and cross r term. And omega cross(omega cross r + vrel) term. And then for the a relative, you have arel and then omega cross vrel. So if you have a two common term, omega cross vrel here, omega cross vrel there. So if you sum them up, finally, you will have an absolute acceleration of the v and then four relative acceleration terms, similar to a two-dimensional point rotational motion, like alpha cross r, centripetal acceleration, omega omega cross r. Coriolis acceleration, 2 omega cross vrel and the a relatives, okay, a relative. So these are the relationship you have to memorize. Well, I would say just keep practicing and then derive this relationship by yourself up to the point where you almost memorize this. So position is expressed by the relative with respect to the B. In that case, velocity term has absolute velocity of the B plus relative, two terms. And for the acceleration, relative acceleration, absolute acceleration of the AB and the four terms here, okay? So hope you keep practicing, almost memorize it. Okay, once you memorized them, now you're ready to actually solve the kinematic problem for the 3D. So again, through the brief review, for the second 2D problem, when you take the derivative for the rotating coordinate, you have to multiply the omega cross term. For the 3D, if your axis is rotating with the capital omega, you also have to consider omega cross term for that. So for the vA, not only have a vB and a v relative, since your axis is coordinate with the capital omega, note that you have to consider this term. For the acceleration, since you have a Rotating the capital omega not only have a VB and every relative, you have all this term like a omega dad cross product or omega cross product term, edit, okay. And then those rerun and a rel means you're describing the point where there is no capital omega only just small omega. So you have a formula sora right. So the vrl is what omega cross r and r the term and the acceleration is small omega related formulation, what we have been learned since the beginning like when we're learning the particles and the rigidbody. So there are four acceleration term here for the smaller omega, okay. And then when you take the angular acceleration l, note that if your small Omega is under defect of capital omega, those omega cross terms should be included. Okay, and let’s solve the problem. So there is a disc. The pink disc is rotating through omega and then it's a constant omega, and then this disc is connected, it's chapter to the drum like a motor here and then the motor is supported by the hinge. And those tour structure is rotating through the shaft vertical shaft, okay? So it's like there is a primary capital omega was just rotating the whole structure,and then there's a secondary omega which is only rotating the desk. Okay? In this case you're supposed to find angular velocity and acceleration of the disk and velocity and acceleration of the point a. Okay, so let us set the coordinate. So, you have capital XYZ coordinate to describe the capital omega. And then small xyz coordinate to describe the smaller mega and then you can actually set the blue coordinate, coincide with a capital XYZ but. Usually, you want to make it like omega as something k, to be simple. So z has ben a little bit tilted, aligned with the disk. And also, I had x y z coordinate which is parallel to the day so that you can easily describe the position of a. So why is a parallel to the disc plane, and x is something perpendicular to the y and then to be simple x is actually aligned with the capital X but the opposite direction okay? Then even though it doesn't look like a 3D is a 3D plot. Now, you can apply the relationship what you're supposed to memorize. So you have a relative coordinated approach for the absolute a. Absolute A is going to be absolutely plus the relative coordinate of the Point A with respect to the B. So usually not only for the v rel, you have to also consider omega cross r term because a blue one is rotating coordinate, same for the acceleration. A of A is going to be absolute A of B plus relative acceleration time, but not only for the absolute relative acceleration of the A. Since the axis is rotating, you have to think about all those capital omega cross term. So you have a and there are four excellent relative acceleration term. I thought angular acceleration cos R centripetal acceleration, coriolis acceleration and only the acceleration term without omega, okay. Assuming that you memorize this. Now,you memorize this kinematic relationships, so you have to then now match where it should be this reference point here. So since you memorized it, with respect to any arbitrary point V, what is, taking the role of V here? Is that kind of? Since there is no V point here, is this C point matching with the B? Well, yes you can actually describe point A with respect to the point C but if you set the reference frame at C, okay? Like this, what will happen? This point C with the primary rotation Has the circular motion right. So this C will have a circular motion so definitely the VOB, A of B is not going to be zero anymore. Okay, you can also consider those nonzero term here, but here your primary exists. The green one and the secondary omega, the blue one. Actually shares the origin, right? So this axis actually meet at one point here, the origin point O. So it will be a lot simpler if you set the relative coordinate frame at O, so that your vB and aB turns out to be 0. So I'm going to set the relative coordinate at O, And assuming those b subscribed is going to be now replaced by the o, okay? So since you set the coordinate this way, your capital mega is going to be omega K in your small omega is going to be omega small k. And when you take the derivative of the small omega, you have to consider what's going to be the k dot. And k dot is under the influence of capital omega. So it's going to be capital omega cross for a k. Now, since this is a two different coordinate, green and the blue, so you want to Unify the coordinate with respect to small i, j, k. So how those k vector expressed? So, k vector has a projection of this y axis and the z axis. So it's going to be cosine j and sine k. Right. So, your total disk rotation omega will be capital omega plus a small omega, and then you can actually unify the coordinates to small ijk. Angular acceleration of disk, you take the derivative of the omega. And not only having the omega magnitude of change you have to consider the omega capital omega cross omega term due to the axis of rotation. So that's what you can get the angular acceleration, velocity. Did you memorize it the velocity will have a is will be plus relative, okay will be plus the relative which is going to be omega cross R plus Vrel. Vrel here means you don't have to consider the capital omega rotation but the small omega rotation. So this RA/B is here. It's our OB so it's going to be LK And DJ. DJ equals LK. And the real relative is now you only consider the A motion with respect to the small omega relative small omega, this is capital, okay sorry. So though you have omega cross R term and the R change, displacement change, but due to the rigid body, this tongue goes zero, so you only have omega cross small R term. So, you can just plug that in those of value for the predefined. And for the acceleration, you have absolute excellence of the B in the or like a four x relative acceleration term so Arfa, angular acceleration cross b or centripetal acceleration, coriolis acceleration in his own acceleration, assuming capital omega is zero. So just plug in the zero point, no acceleration at the pivot point, constant capital omega. What you can have is the rest of the relationship calculated and the are here is going to be from zero to origin to the a, so this is going to be dj+lk. And v rel, again, its only component by the small omega, x l r, result of air. Relative is only the acceleration due to the small omega. So for the general point moment motion for the small omega just assuming that you only have an omega and that's it. And what's going to be the acceleration with request to this origin point? You have generally have four term, right? And then assuming, just applying the relative rigid body motion, you have all those r term eliminated. And assuming constant omega then r bottom goes zero so you only have left the centripetal acceleration. So if you plug that back in everything you will have this. And then if you plug in all those predefined value, you will find only ended up getting the acceleration of point A. Okay, so we are going to solve the exactly same problem except the change for the primary axis. So instead of having the vertical rotation, maybe this drum has something like this kind of rotation, this direction, okay? So if I have a counter-clockwise omega, what would have changed? Exactly same, except the primary axis turns out to be x, the capital I, right? And then, this is a primary, and this is a secondary, the disk rotation. So you actually do the same thing throughout the process. Set the reference coordinate frame at O here. And then just apply the formula you have memorized. So your capital omega is now turns out to be capital I, okay? And then you have a small k for the small omega, secondary omega. And then when you take the time derivative note that you are under the capital omega, the primary rotation. And since I set it up the small ijk, small xyz as a parallel to the capital X, but the opposite direction. Your capital omega is now turns out to be minus omega I, small i. And if you just plug that in and just do the same process for what we have just gone through just go ahead. Then you will ended up getting the acceleration like this one, okay? Now, let's solve another problem. Now, instead of the disk, we have now a particle connected by the rod. So it's a dumbbell, dumbbell is rotating, okay? And this dumbbell has been fixed to the axis which all the way go through the whole structure assembly and assembly is rotating vertical. So what is the angular velocity of the dumbbell? Okay, this is a primary axis to the vertical and this is a secondary axis. So you have to set the coordinate for each omega. So I'm going to set capital X Y Z fixed at origin O here and small x y z fixed at B here, okay? So that I can apply the formula v of A is going to be v of B plus relative velocity. And absolute acceleration of the A is going to be absolute acceleration of the B and the whole like a relative acceleration term, right? Now is the B the same as the B here? Can I just use the place, the B here, as a formula that I memorized? Yes, you can do that. However, if you set the coordinate the relative reference point B here what will happen is, due to the the primary rotation, your point B is rotating, right? So it has a non-zero velocity and non-zero acceleration. It’s okay, you can just plug that in, but it’s going to be, there must be a simpler way. If you move the coordinate for the relative coordinate frame, I mean describing the small omega at the origin, what will happen? Your omega is along with the axis. And your primary omega has along with the axis and it meets at origin O, right? It shares the origin point. So it will be a lot easier if you set the coordinate frame at O, okay? And then assuming those reference points with respect to the O. Okay, so if that's the case, your capital omega is going to be omega capital K. Your small omega, secondary omega, is going to omega small k. And take the derivative of omega, note that you have a k dot as omega cross k vector. And how you can formulate the capital K in terms of a small ijk? Now you have a projection here and there, so this is going to be gamma. So you have what? Cosine minus i and sine k. Cosine minus i and sine k vector. Once you're done with the formulating all those omega term in terms of small ijk, the rest of the step is you can just plug into the kinematic relations that you are almost memorized. Omega total is going to be just sum them up and then using the small ijk coordinate. Angular acceleration, note that you have an omega cross term and then you can just calculate them. And velocity, the velocity of A is the velocity of the B plus the relative, the omega cross r plus v rel. So the r here is from here to there, so it's going to be l k di, di lk, so you plug that in. v relative means you only consider small omega. So it's small omega cross our product, and then rigid body, r dot times zero, you can plug that in. Acceleration, memorized all those four term here, okay? So you have omega dot cross r, omega cross omega cross r, plus 2 omega cross v rel and a rel. And note that v rel a rel means that you'll only consider the small omega, not the capital omega, okay? Again this r is going to be di lk, this v rel is going to be omega cross r. a rel is going to be whatever the four acceleration point, whenever there's only small omega can exist. Due to the rigid body, this term goes out. Whenever small omega constant, you only have left centripetal acceleration. So if you plug that it all to the original equation, what you can get is this long relationship. But know where you can just plug that in or those omega terms and small omega terms in terms of small ijk. Okay, that ends the practice for three dimensional kinematics which has multiple omega, multiple 3D rotations. And note that since the rotations are under some hierarchy like you have a primary omega and secondary omega. When you take the derivative of small omega, note that you have to consider a capital omega cross term. So that's pretty much it. And then keep that in mind you have to almost memorize a relationship with a v relative coordinate, acceleration relative coordinate. Because that's what we are going to use throughout the rest of the chapter. Thank you for listening.