[MUSIC] Okay, in this video we are going to study how we can describe the three dimension of rotational kinematics when there are several Omegas. They are of the origin. Okay, for example, for the two dimensional, your Omegas are usually the z directions. Either as coming out of the screen or into the screen, but when it turns to 3D now your Omega could be either the directions or X or Y direction. And also there are some secondary Omega which is objected to the primary Omega, which also has or idk components. But note that here they are our same origin point. So that actually easy to describe the relative motion because this point has 0 velocity and 0 acceleration. Now, what if your omega the secondary omega has a different the fixed point so that you have to now then consider the motion of that point as well,that's what we are going to cover in this section. Okay, here are the examples where we have worked on and while you're we're going to work on. So ultimately you are interested in define the kinematics like a velocity or acceleration of the point A. And why this is rotating on the desk the two axis like a primary through the axis. The blue arrow and the secondary for the, sorry the green arrow and the blue arrow. Same for the other examples like you have your interest in worship of the a and then you want to use the point B as reference point for the relative coordinate. But note that your B point is actually the intersection of the primary and secondary right? So you'd actually share the same origin for the 2 rotation vector axis right? How about the of examples on the bottom row. What do you think? Can you do kit? Did you catch it the difference between them? Yes, so here you're have a set of primary axis this way and the secondary axis that way so in that case they never met. It's parallel, right? And for this one, even though if you actually extend it, you might be able to have a intersection here, but when you want to describe a risk for the B, B is not definitely the point for along the axis where the primary rotation or vector. Same for this one, so the B is far away, certain distance away from his primary axis and this one to access never met Now in that case how we can change our formulation about the kinematics before. So previous set up is like your B is actually along the primary rotation or angle or velocity vector. But if you we have changed the system so the B is not any longer the point of the primary axis. So in that case that when you actually set the coordinate for capital omega and omega, those origin doesn't match right? And then your point B has certain distance from the primary omega. So the point B has actually showing the circular motion. So the velocity of the B is not zero and acceleration of the B is not zero. That's what you have to consider. So whenever you have a general circular motion, the velocity will be Omega cross R omega cross R and acceleration is going to be xXalpha cross or and centripetal acceleration. And then for the rigid body you can ignore all the time delivery time for the displacement. This is actually the similar form when you are when we are thinking about the appoint a with respect to the relative coordinate frame. So just point A is moving with a small omega the velocity was omega cross or an acceleration was alpha cross or plus centripetal acceleration. So this is pretty much the same format except that the relative one has all smaller omega small R and the real B is or capital mega and capital R. Just in case if you you may know it but The time that everybody will v rel is not actually a rel, right? Timed everybody with a v rel, okay, which is a definition for the a relative coordinate. Since this is moving with the omega, so should have omega cross v rel plus a rel, right? So this is what I've just mentioned here. This is a real part out of this. Now so when we want to describe the point A velocity and acceleration with respect to the B as a relative co-ordinate, this was a formula where we have been on so far right? So in that case VB and VA has gone up. AB is not going to be zero anymore and then you have to consider the capital omega and capital R across product for that and those formulas are pretty similar to the one we will use for the v rel except that for the v rel, you are handling small omega and small R. The v(b) being a(b), it's all capital. Okay with that, let's solve the example. So this is the disk rotating with more omega concent one and there's a shaft connected to the drum or motor and more has hinge fixed to the hinge. And there is a certain distance here s and the central axis is rotating. So this is a primary rotation and this is a secondary rotation. So to describe the primary one you will set the coordinated O, while you're describing the point A you're going to set the coordinate A, P right? So it definitely doesn't match, so in this case of we're still going to use a kinematics relationships of the VA and the acceleration of A, we respected reference coordinate frame it B. So I'm asking like is this B? Same is just one. Yes, and this velocity v and acceleration A is going to be zero know, since this one has a certain distance away from the primary axis v(B) A(B) is not going to be zero anymore. You have to consider those parameters here. Now, let's solve the problem. So, for capital omega, you have a capital mega, capital K vector and you have a small omegas, more j vector. And then when you take the derivative, this is a case more IJK those are cross product of the capital omega right? And then at this point, you want to express everything in terms of a small ijk, you have to express a capital K vector into smaller ijk. So if you stand the access and think about the angle that vertical line is make with the blue coordinate It's going to be sine J and cosine K, right? Okay, now your omega then is going to be some of the capital and the small omega. And then if you plug that in, you can express everything in terms of small ijk. Now let's calculate the angular acceleration. Angular acceleration time derivative this and then you know that you're small omega is under the influence of cap primary omega. So you have to consider omega cross Small omega term. Now your velocity now is going to be vB omega cross r + vrel. So you have from origin to the point A, you have a capital R to the point B, and small r from point B to the point A. So your vB is not zero anymore. It's going to be capital omega cross capital R. And your vrel is going to be small omega cross small r. So if you plug that in, here, your R is going to be what? The s capital J and h capital K, okay? That's your R vector. And your small r vector is going to be lj and dk. So your small r is going to be lj and dk. So this one is this one, this one is that one. And then since you want to express everything as a same coordinate, you can actually transfer capital J, K into small j, k components using the coordinate transform. And that's what you can get for the velocity of the point A. Now let's calculate the acceleration of the point A. Well, a little bit longer formula, but pretty much the same concept or the one that you apply for the velocity. So a of A is going to be a of B plus all four terms, right? Now the a of B is not going to be zero anymore because it's rotating, where we use the capital omega. So you have a four terms for the acceleration. But due to the rigid body constraints and then constants, you only have a centripetal acceleration in terms of capital omega and capital R. Now when you calculate the vrel, since you're only describing this secondary omega, it's going to be small omega small r. And your acceleration is going to be small omega related centripetal acceleration. So note that this one is a capital and this one is small letter. So if you plug that in, you have a long equations about the acceleration term. And then if you plug that in, all the parameters from the problem, you can have everything here. And then, finally, there are some mixture for the small k and capital K. So you have to substitute capital I, J, K into small i, j, k using the coordinate transform relationship. And that's the final step that you can get the acceleration of point A. Okay, now solve another problem. Now there's a disk and you want to know the acceleration of point A. And this one has been clamped and the clamp axis is rotating. This one is rotating here. And then the whole assembly is rotating to the vertical at the axis here. So this is the problem they are involved with three omegas, small omega for the disk, and it will make capital omega 2 for the clamp shaft, and the capital omega for the whole assembly. So there is a primary omega in the final small omega term, but in between, there is a second primary, secondary omega 2 exist. How we can handle? Pretty much the same, but since you have more omegas, there are more steps. Okay, set the x, y, z coordinate at point O. And set the x, y, z coordinate at point B to describe the omega 2. And set another x, y, z coordinate at point here, at the center of the disk to describe the motion here. Now, we still want to apply, well, the relative coordinate approach. So try to describe the vA. We are going to find the absolute velocity and the relative velocity and acceleration, okay? Now is the B what we have memorized for the kinematic formula is going to be same as this B here? Is it got the same B? What do you think? No, well, yes and no. You can actually set any point as a reference point for the relative coordinate, right? But the point for the relative coordinate is to make the rest of the part easy so that when you can calculate the vrel, you just have to consider small omega cross r, right? And for the arel, you just have to consider the centripetal acceleration for the small omega and small r, right? So you have to find the B which is only make your relative velocity and acceleration simple. However, if this point B is going to be a relative coordinate to describe A, there is a primary and secondary rotation axis exist. So it'll be complicated. At that point, you should any way to another step for the relative coordinates. So I would suggest let's define point C at the center of the disk so that at the center of the disk, it'll be really easy to describe the velocity and the acceleration of point A relative to the C, okay? So that's the point for using the relative coordinate approach. So instead of the B, find the point where you can make the relative velocity and acceleration part become simple. So once you set up coordinate at three different way. Capital omega is going to be capital I. And capital omega 2 is going to be the j vector, which is defined in the black coordinate. And then when you take the derivative, of course, it's just under the influence of capital omega, primary omega, so you have to do the cross product of capital omega. And at this instant, the capital X, Y, Z is going to be same as black x, y, z. So your capital I is going to be the same as black i. And the small omega, the third one, third omega is going to be omega k. And when you take the derivative of this vector, what would happen? This one is actually under influence of omega 2 and omega. So you have to do the omega + omega 2 cross product k to take the derivative, the k vector. And at this instant, all the small, like blue x, y, z is same as black x, y, z and the green x, y, z. So you can just simply replace. This one is a blue i, and this one is a blue j. So you can express everything in terms of small i, j, k attached to point C, the coordinate attached to point C. Now, your omega disk is going to be capital omega + capital omega 2 + omega. There are omegas, right? And when you take the derivative to calculate the alpha, you have to calculate the omega 2 dot and then omega dot. So omega 2 dot is actually omega 2, right? This one is also under the influence of primary omega. And how about this one? This one is actually only counts the small omega as its relative motion. So what that mean is small omega is under the influence of capital omega and omega 2. So by using those relationship, we can have omega plus omega 2 here. But here, omega plus omega 2 all together should be multiplied by the cross product to the small omega, okay? Now the velocity has a V since we are going to set the references to C. It's going to be v of c and omega cross r + v rel. Okay, so here, to describe the point C, we have a vector from the origin to the B as a capital R, B to the C as a capital R2, C to the i as a small r. So the v rel is small omega and the small r. So that's the beauty of putting the reference point at Z. The relative velocity and relative acceleration because really simple. Okay, how about this one? So this one, the aC, this vector, is under the influence of Capital Omega and Omega 2. So instead of just putting the Omega here, you should also add Omega 2 for the calculation of the second part. How about vC? Is vC, okay, is under the influence of Omega and Omega 2? And hold the total distance will be from O to C, which is R + R2. Is this hold? Like a vC is simply omega cross R? That is a question. Well, no, that's not true because this is the case where your total like a petition is under the influence of the total omega here. However, if the Rc has a splitted term like R plus R2, then Omega has a to split it term, like Omega plus Omega 2. And there is a components of the R, for example like a R capital R vector, which is under the influence of capital Omega but has nothing to do with the Omega 2, okay? So if you just launched out the Omega and displacement, there are give you an error. So, what you have to do is vC is going to be vB relative and the relative velocity. For your B is actually capital omega and R and v rel is now consider this, you have a relative coordinate frame here. So you don't know if you're moving with the capital mail. You only see the local Omega 2 with the distance R. So this is Omega 2 and R2 and this Omega is what? It’s only the Omega, the capital Omega term. So what you have Omega cross R and Omega cross and then Omega 2. Both that will affect the R2 here. And then definitely, these are not equal because this one is wrong, and this one is correct. So note that you have to use a relative coordinate frame not to make a mistake. And this is the final answer for the VOB. Now, tough part, the complicated part, acceleration, you have many, many terms, okay? So A of A, A of B is now going to have some motion so this is not going to be 0 anymore. But further than that, since A of B is not a single like a reference change, you have to change a reference from O to B and B to C, okay? So to find the acceleration at p here, here is actually the c value, right? Because that's why we set the reference coordinate to describe the relative motion. We have to do A of C with respect to B and A of A and with respect to the ACA, like you have to do the two steps, okay? So to find the A of C, okay, you have to do something like A, B and C, okay? So a of B is definitely the acceleration for the capital Omega and capital R, and r CB is capital R2. And this one is constant, so this one is 0. Now you're Vrel when you're calculating the BC, your Vrel is only calculated by this term like a Omega 2 and R2. So your acceleration is also the center of your acceleration by Omega 2 and R 2, okay? So if you plug that in, you have any questions like this. Now, once you find the acceleration of the C, now that you have to plug that into the original equation to find the a of a ultimately. So if you plug that into here, okay, now what is, yeah, what is Omega here? Because your c point is under the influence of capital Omega, primary Omega and second primary Omega, Omega2. So your Omega term here, here, this one, these are all not just Omega but Omega + Omega2, okay? And when you take the derivative of them, since this one is a constant, it turns out to be 0. But, this one is under the influence of Omega, this one is going to be Omega x, Omega 2, okay? Now, Vrel and Arel or a with the small Omega and small r. So if you write them down, so these are the what we have obtained as an of C. And the rest of them is going to be a component for the relative acceleration of the A of A with respect to C. These are the relative acceleration term, okay? Okay, so we have worked on how we can describe the rotational motion in 3Ds whenever there are many Omegas which not necessarily shares the common point, okay? So I recommend you to do a practice for the different examples. And get used to it, how the way you can describe the kinematic, like values like velocity acceleration in terms of relative coordinate frame. Thank you for listening.
