[BLANK_AUDIO]. Welcome to Module 6 of three dimensional dynamics. Today's learning outcome will be to solve for the angular acceleration for a body. Expressed in a series of multiple reference frames. And so, here is the an example of body with three different reference frames attached to different parts of the body undergoing three dimensional motion, and we said that for the angular velocities we could use the addition theorem, the angular velocity of frame C or body C with respect to frame F. Is equal to the angular velocity of frame C with respect to B plus the angular velocity of B with respect to F. However, we found for the angular acceleration the addition theorem didn't hold. For the angular acceleration, we had the angular acceleration of C with respect to F, as equal to the angular acceleration of C with respect to B. Plus the angular acceleration of B with respect to F. But then we also get this term which we call the gyroscopic term. And so let's go ahead and do an example. We're going to go ahead and look at that landing gear, that I did before, the retracticable, simple retractical, retractable landing gear. So I add my wheel here, spinning with an angular velocity, omega 1. It's being rotated up into the body of the airplane with an angular velocity omega2. I, I'm using three reference frames here. I've got body C I've got a frame welded to my wheel and then I have a frame B welded to my retracting arm, and then I have a frame F welded or attached to the [INAUDIBLE] craft itself. And, so we said that Omega C with respect to F, we'll just use the addition theorem, is Omega C with respect to B. Omega C with respect to B, is simply, Omega 1, and by the right-hand rule that's going to be in the I direction. And then, I've got, be with respect to F. It goes up into the plane, again by the right hand rule, that's a mega two in the J direction. So that's my expression for my angular velocities, we did that before. Now for the angular acceleration, I've got the angular acceleration C with respect to f is equal to this, given that, this is the problem statement, what I want you to tell me is, what are these two accelerations equal to zero. And what you should say is, these angular accelerations are equal to zero because we have, we're given that the angular velocities are constant, so these are zero, because we have constant angular, because constant angular velocities. And, but we still end up with this gyroscopic term, a mega b with respect to F cross with a mega C with respect to B. And so we get the angular acceleration of C with respect to F is equal to Omega B with respect to F, that's simply Omega 2J, and then crossed with Omega C with respect to B, was Omega 1I, and so this is, we get omega 1, omega 2, and then I have J cross I is actually minus k. So it's going to be minus omega 1 omega 2K. And so that's very interesting result, because we have constant angular velocities omega 1 and omega 2, and yet we still get an angular acceleration of the wheels C with respect to the aircraft, and we call that the gyroscopic term. And I want to jump ahead a little bit here, for, some context, and we'll learn much more about this later in the course. But, you recall now that, Newton's, Euler's law says that F equals MA. And, so we have, an acceleration, of, linear acceleration associated with linear forces. And you'll remember also from my, earlier course, in two dimensional dynamics that we ended up with an expression where M equals I alpha, where I was the, mass moment of inertia for two dimensional motion this was the angular acceleration and the angular moment. And so you can see in this case where we have constant angular velocities that we still have an angular acceleration which generates some moment. And we call that the gyroscopic moment. And we need to be careful of that, when we're designing or analyzing three dimensional motion particularly for things like, this, this, this landing gear example. Because even though again, we have constant angular velocities, we get this angular acceleration about the K axis. Okay, and that's a gyroscopic term, and it generates some, some, some moment about the K axis. And that's important because, if we have moments, that's going to generate some stresses and strains on the body. And, so there's going to be some moment of twisting with my retractable landing gear, and so, we could have failure of the landing gear if we don't design it properly due to those accelerations. And those accelerations can get rather large, because it's in multiple of the angular velocity of omega 1 and the angular velocity of omega 2. So if this wheel is spinning very, very quickly. And we're retracting it very quickly. These can get to be rather large accelerations that we, we generate, through this gyroscopic motion. Or this gyroscopic acceleration term. And so, let's look at a, the example here. And so here is my demon, I've got my wheel, my landing gear wheel, spinning in this direction, this is the front of the airplane, and we're retracting it up into the body of the aircraft, and it's spinning and as I retract it up. Later in the course I'll show you how you can actually feel this gyroscopic induced moment, or generated moment it actually as you pull that up, it, it feels like it's twisting the retractable arm this way. And so again. important, because it can generate this moment, it can generate stresses, and strains, and we have to include that when we design these types of three dimensional motion problems. Okay so, that's a simple example for finding the acceleration of Of a three dimensional body, and, so I want you to do a, a problem on your own for practice. So I always like to relate things to the real world, I was walking around the Georgia Tech campus the other day, and I saw this cherry picker machine, and I thought, oh, that would be a good example for my class. So here's my situation again, here's my cherry picker machine. And I've drawn a sketch here, of the machine. And so my body E, is the bucket itself. Body D is this portion of the arm. Body C is this portion of the arm. Body B is this portion of the cherry picker truck. And then I have the base itself which I'm calling body F. And so I, I'm giving you that the base of the cherry picker machine body F, is stationary with respect to the ground. I'm telling you that body E is undergoing a constant angular velocity with respect to body D of minus .1. K double prime radians per second, so it's minus K double prime direction, which means by the right hand rule it's into the board, or my bucket is rotating as I look at it from in the picture it's rotating clockwise. And then I have body D. With respect to C, or this arm, with respect to this arm, has a constant angular velocity of .1K prime ratings per second. So that means that body D here, K prime, is rotating counter-clockwise up at .1 radiant per second. Then I've got body C under going a constant angular velocity with respect to B of .05k radiant per second. So that means this body, B, with respect sew, excuse me, C, with respect to B is rotating up again, or counter clockwise .05K radiant per second constant. And then I've got body B, this portion of my cherry picker machine, is undergoing an angular velocity with respect to body F for the base, of minus .1J radians per second. And so minus .1J, minus J by the right hand rule means that it's, this whole machine. Portion here is turning out toward me. And it also has an angular acceleration of body B with respect to F of point zero two, and this should be radians per second squared so we'll put that in there I had the wrong units. And The last thing I want to tell you is, for the set up of the problem, is that during the duration of the motion that we're investigating, or analyzing, K, K prime, and K double prime are all parallel axis. And, so given all that information, a real world problem. We can go ahead now, and I'd like you to find the acceleration, the angular acceleration of body E, or the bucket, with respect to F, the, the F, the the base of the cherry picker machine. And I've put the solution in the module handouts, and with that, we'll see you at the next lesson, or next module.
