[BLANK_AUDIO] Hi, and welcome to Module 26 of Two Dimensional Dynamics. Today's learning outcome is to, derive the expression for the acceleration of the same point expressed relative to two different reference frames or bodies in planar motion. And last time we did velocity in two different frames. So if you recall back, this is the situation I talked about before. One of our reference frames was, perhaps, the earth, another reference frame was me flying around in my hang glider, and we're looking at the same point, maybe perhaps a bird flying from the vantage point of the two different reference frames. Now we're going to also look at acceleration. Here's another example of where these, velocities of acceleration in, with respect of two different reference frames are important robotic applications, real world robotic applications. I've drawn just a simple robotic arm here and perhaps we're solving a problem or looking at a situation where we would want to attach one of our reference frames to one of the robotic portions of the arm and another to another portion of the arm and they move relative to each other. And so lots of applications for this this types of dynamics in in mechanical and other engineering systems. Okay, let's continue on then and find the acceleration of the same point relative to two different frames. And I want you to begin by recalling the velocity of the same point, with respect to two frames, which we've already developed. And we're going to go ahead and take the derivative of that, but we're going to differentiate in frame F. And when I do that, the derivative of P with a respect to F is going to be what we call the absolute accelerations of P, the derivative of the origin of the moving frame, O prime with a respect to F, is what we're going to call the absolute, acceleration of the origin of the moving frame, O prime. When I take the derivative of P with respect to B, we've gotta be more careful because that's a vector that's expressed in the B frame and we're taking the derivative in the F frame. Then we have by the product rule, theta double k crossed with r from O Prime to P, plus theta dot k crossed with this derivative of r O prime to B with respect to the F frame. Alright, so let's go ahead and continue on with that with that development. And so I have a of p with respect to F is equal to a of O prime with respect to F. And then my question to you is, how am I going to take the derivative of this velocity of P with respect to the frame B, the moving frame B. I'm going to take the derivative back in the F frame. How do I go about doing that? And the answer you should say is, we're going to use that derivative formula that we derived before. So we're going to take the derivative of the vector with respect to the F frame. That's going to be the derivative of the vector with respect to the B frame. Plus the angular velocity of the B frame, crossed with the vector itself. So that's going to give us the velocity of P with respect to B, dot, the derivative in the B frame plus how the B frame is is rotating, theta dot k crossed with the velo vector itself, which is the velocity vector P with respect to B. And my next question is tell me what this term here is write it in a, in a simpler terminology, this term right here. And the simpler terminology you sh-, you should come up with is okay, this is the derivative of the velocity with respect to the B frame. Taking that derivative in the B frame. So that's going to be the acceleration of point P with respect to the B frame or what we often refer to as the relative acceleration. So, this is the acceleration of P with respect to B. And then we continue on. Let's go with plus theta double dot k, crossed with r, O prime to P. And then plus theta dot k, crossed with. Okay, we gotta be careful here again. Now we've got a position vector expressed in the moving frame B. We're taking the derivative in the moving frame F. How do I take that derivative? And the answer is again, use the Derivative Formula. So, we've got crossed with r O Prime to P, dot in the B frame plus theta dot k crossed with original vector itself, O Prime to P. And another question. What, how can I express this term and with different nomenclature. What, what, what is this term here and what you should come up with is, okay, that's the derivative of the position in the moving frame from the, the moving frame's perspective which is the velocity of P with respect to the moving frame or what we often refer to as the relative velocity. So this is, V of P with respect to B. Okay? So, let's go ahead and collect terms now, so I've got a of P with respect to F is equal to a of O prime with respect to F. That should have been with respect to F up here. And, and so that takes care of that term. And then we have plus we've got a of P with respect to B at relative acceleration. Plus, I'll come back to this portion. So, let's see. We've taken care of this and we've taken care of this. Now I'm going to take care of this term right here which is plus theta double dot k crossed with r from O prime to P. And then I'm going to do, I've got theta dot k crossed with theta dot k, crossed with r of O prime to P. We did that earlier in, when we developed this relative, or this velocity equation. And we found out that that's might, by, we used a, a vector identity and that was minus theta dot squared, r O prime to P. And, so that takes care of, this term with these two. And then I still have theta dot k crossed with v P with respect to b and theta dot k crossed with v P with respect to B. So that's plus 2, theta dot k crossed with V P with respect to B. Alright, so let's, I'll show that on the next slide. Here we go. And so, let's go ahead and write this in a little bit of a shorthand and define each of these terms. So this is the acceleration of P with respect to F, so that's the acceleration with respect to the frame F, and I've got the I wanted to write shorthand, so let's do shorthand here. That's, we're just going to call that the acceleration of P, because that's the absolute acceleration of P. And I'm going to say, this is the acceleration of O prime with respect to F or the absolute acceleration of O prime. The origin of the moving frame and then this is the acceleration of P with respect to the moving frame or what we call A rail. This is plus a relative acceleration, relative acceleration to the moving frame. Plus now, we've got theta dot, theta double-dot k is the angular acceleration, second derivative of angular displacements, so we'll call that alpha crossed with, and, my vector r is expressed in moving frame, and so I'm just going to call it r, for short hand. And, that becomes our tangential acceleration turn that we've seen before. [SOUND] And then I've got minus theta dot squared is the same as omega squared, since theta dot is the derivative of the angular displacement, which is the same as angular velocity. So this becomes minus omega squared r, again which is just the position vector expressed the B frame, and that term is called the, you should know by now, the normal acceleration, or radial acceleration. And then when we have the acceleration of the same point relative to two different frames or bodies, we get one more term. And that's this plus two and theta dot k is omega again, crossed with V of P with respect to B which is V rel, the relative velocity with respect to the moving frame. And that 2 omega cross v is what we're going to call the Coriolis acceleration. So, it's a new acceleration term and it arises, Coriolis acceleration. And it arises whenever we have an angular velocity of a moving frame with respect to another frame, okay? So, there's angular velocity between 2 frames and there's a relative velocity as expressed in the moving frame. And so, if I have those two terms, this Coriolis acceleration turns up. And that's, that's the theory for acceleration of the same point relative to two different bodies or, or frames and we're going to come back next time and do an example.
