Welcome to module 14 of two-dimensional dynamics. We're moving right along in the course. We have now completed particles, systems of particles. We've done all of these topics, and we're starting on bodies in rigid planar two-dimensional motion. And today we're gonna start the kinematics of bodies in rigid planar (2D) motion. So, the learning outcomes for today are to define rigid body kinematics, to identify the three types of planar rigid body motion, and to derive what's called the relative velocity equation. So first of all, I have a generic body here. And I've got two points annotated on the body, P and Q, and planar rigid body kinematics relates the linear motion of the points on the body to the angular motion of that body through the geometry of the body and its motion, and so that's a good definition of planar rigid body kinematics. So we can have three types of planar rigid body motion, the first is translation, and so if I look over here. If I'm working in a single plane, I can have rectilinear translation, which we saw was translation along a single direction or line, or we can have curvilinear translation, which is along a curved path. The other two types of planar rigid body motion are rotation about a fixed axis and general planar motion, and let's look at a demo of that. So here's a mock up or a model of a piston cylinder, and it actually has three bodies that are undergoing the three different types of two dimensional rigid planar motion. We have the piston itself, which is going back and forth in rectilinear translation. We have our cylinder here that is going around a fixed axis, where I show the pen, okay, so it's rotating around a fixed axis. And we have a connecting bar here, that is undergoing general planar motion, which is a combination of translation and rotation. So here's our body undergoing generic two dimensional motion and I have two points on that body. And I can express those points in terms of vector addition of the position vector. So, I have R from O to Q, is equal to R from O to P, plus R from P to Q. Now if I want to get velocities, I just take the derivatives of those positions and we've got velocity of point Q, with respect to what I'm gonna call a fixed reference frame. It doesn't we're just talking about the geometrical aspects of the motion, so it doesn't necessarily need to be a inertial frame, but we'll call it our fixed, or our reference frame. The only time we would need to make sure that it's an inertial frame is later when we look at kinetics, and we need to apply Newton, and Euler equations, because those equations are laws only applied in an inertial reference frame. So I've got V of Q again, we'll call that the absolute velocity of Q with respect to my reference frame, plus V of P, which is my absolute velocity of P, with respect to the reference frame, and then I need to be more careful about this velocity of Q with respect to P, because point P and point Q can both move with respect to my reference frame, and so we're gonna call that a relative velocity term, so again, R of O of Q, R of O of P are fixed in the reference frame. Therefore, we'll call V of Q and V of P absolute velocities, with respect to that reference frame. Whereas, points P and Q are not fixed in the reference frame, and so when I take this derivative of this last term, we refer to that as a relative velocity, and we're gonna need to be more careful when taking the derivative. So here's where we left off. We have the absolute velocities of Q equals the absolute velocity of P, plus the relative velocity of Q with respect to P. Let's look more closely at this derivative that we need to be careful of. It's the derivative of the position vector RPQ. And so here's my position vector RPQ, I'm gonna pull it out here to the side, and we're gonna call this angle theta. And so let's express RPQ as a vector having some magnitude RPQ and then we're gonna go some distance in the i direction and some distance in the j direction, and I'd like you to go ahead and complete that equation and then come on back. So we go RPQ times cosine theta, since this is the adjacent side in the i direction, plus sine theta in the j direction, since that's the opposite side, so adjacent and opposite, and so we can call this value in parenthesis u or a unit vector in the direction of RPQ. Okay, and so, this is an expression of the vector, it has a magnitude RPQ, and it's direction, it's unit vector is what we call u. Now let's do a derivative of that, because we want the velocity of Q with respect to P, and so I have RPQ dot is equal to RPQ. We have theta dot, minus sine theta i plus cosine theta j. And that's just a simple derivative. Then I'm gonna do a little bit of a mathematical manipulation. I'm not gonna do any magic, I'll show you that it's the same thing. But I have RPQ theta dot, and I can say k crossed with cosine theta i plus sine theta j, is equivalent. And let's just make sure that that's equivalent. We have k cross i is j. So that gives me my RPQ theta dot cosine theta j term, and then I have k cross j is minus i, so I have my RPQ theta dot minus sine theta i term. And so that is, RPQ theta dot k crossed with, okay, we see that the information in parentheses here is my unit vector u. So this has just crossed with u, and since RPQ is just a magnitude, just a scalar, I can pull it over to the other term and so I've got theta dot k crossed with RPQ u and now we have RPQ dot is equal to theta dot k crossed with RPQ in the u direction is just the vector RPQ, itself. So this is just RPQ. Okay, and so we've done the derivative of this last relative velocity term, for this last relative velocity term. Okay, and so here's my, what I call my relative velocity equation, my absolute velocities, my relative velocity term, I can now substitute this in here and I will get my relative velocity equation, which is the absolute velocity of q, is equal to the absolute velocity of p, plus theta dot k crossed with RPQ. And that's just what we call my relative velocity equation. It relates the linear velocity of two points on the same body, with respect to the motion of the body, itself. I can write it, I can say that theta dot k now, that's an, what we call an angular velocity. And since we're talking about planar or two dimensional motion, by the right hand rule, it's in the k direction, as the body rotates. And so I can also write this as V of Q is equal to V of P plus a mega cross RPQ. So you'll remember from my intro slide that I actually taught at the military academy at West Point prior to teaching at Georgia Tech, and you'll also recall from my earlier courses, that I was in the U.S. army for 24 years and retired as a colonel. While I was in the Army, I was a US Army, in fact, I still am an Army ranger. And so back at the military academy when we talked about one of these types of rigid planar body motion rotation about a fixed axis, we used the acronym RAFA, kind of compared it to rangers, so you'd be a RAFA ranger if you successfully completed this course. And we even made up a RAFA tab similar to the ranger tab that you'll find, although it's a lot larger than the ranger tab, I have smaller versions, as well. And so, if you guys successfully, or you folks successfully complete this course, you can also be a RAFA ranger. And so, let's look particularly at this rotation about a fixed point. Here's my relative velocity equation. The velocity of Q is equal to the velocity of P, plus the mega cross R from P to Q. The body, in this case, is a bar from P to Q, for rotation about a fixed axis, we see that the velocity of P is equal to zero, and so I just get the velocity of Q is equal to a mega cross R, and here's a demonstration of that. I've got a RAFA bar here, here would be point Q down here, excuse me, point P down here, which is fixed, point Q would be out here and so I have an angular velocity of a body and I have a linear velocity of point Q. One thing that's really important to mention is that the body itself, the bar itself, has one angular velocity. However, each point in the body can have a different linear velocity. So the velocity at, out here at the end, will be different from the velocity at the middle or any other point along the bar. So here is a depiction of that, again, I've got my angular velocity, I've got the linear velocity of point Q, it's equal to the distance from P to Q times the angular velocity. One thing you need to be careful of is consistency of directions, in this case, I have a mega counterclockwise, so the velocity of Q is up and to the left. And so, the point velocity, Q, of any point in the body, has to be consistent with the body velocity. And there's one for each body, angular velocity omega. And so, that's just a special case of a rigid planar body motion, and we'll come back next module and start solving some problems.
