[BLANK_AUDIO]. Hi, and welcome to module 20 of Three Dimensional Dynamics. Today we're going to review the concepts of mass moments of inertia and products of inertia. We're going to look at my earlier module, modules from my 2D Dynamics course to get a physical feel for what mass moment of inertia and products of inertia are. And then I'm going to look at again the inertial property matrix for a 3D body, and we're, this is going to be an ability for us to determine what those inertial property values are. Either mass moments of inertia or products of inertia. And so here's where we left off last time, this was the, what we came up with for the angular momentum in three dimensions. We put the second part of the equation into vector matrix form, so I have an inertia matrix I sub P times the angular velocity. And this inertia property matrix again we said was symmetric, and it includes information about mass shape and geometry. So at this point, let's look back at my earlier 2D dynamics course to see how we determine these products mass moments of inertia and products of inertia. How can you determine first of all these products of inertia and these mass moments of inertia for different bodies. And one way you can do it, is you can do the integral mathematically. because you can see that you have, an integral that you can, do over the body. And so, as an example, I just showed a calculation of the mass moment of inertia for a homogeneous solid cylinder about the z-axis. And so this ends up being a triple integral. You can go through the math. This is one standard type sized shape that you can calculate. This, as I say, is the calculation, the mass moments of inertia. You can do the same sort of thing for other bodies, other shapes, and also for the product of inertia. But, actually since we have most of our engineering systems are combination of either a standard body shape or a combination or a compilation of standard body shapes we have tables which you can find on the internet or a text book reference for different types of shapes. And so, as an example, here's the values for a rectangular solid, solid, cylinder and a thin right triangle. So this is a rectangular, solid. We've got our x, y, z axes welded at the mass center and oriented as shown. In this particular case, we only have mass moments of inertia and these are their values. Here's a solid cylinder. again, the the coordinate axis is welded as the mass center oriented as shown, and we have the XY, IXX and the IYY mass moments inertia are the same, and the IZZ moment of inertia, mass moment of inertia is different. And then finally, I've got a thin rectangular. Our thin right triangular plate. And here I, I included this example because we have an IXX mass moment inertia, a Y, IYY mass moment inertia about the Y axis and IZZ mass moment inertia about the Z axis and this is a case where we actually have a product of inertia as well. And I would encourage you to go in and find a text reference, or on the internet, look at and get a feel for how you would find these different mass moments of inertia and products of inertia for various shapes. Okay, so now I want to explain the concept of the mass moment of inertia. It is the integral of about the Z axis, it is the integral over the body of x squared plus y squared dm. And so the z-axis is in the center of the rotation, and how far we go out and the x and the y-axis to any little piece of mass and then integrate it over the entire body gives the mass moment inertia. So we can say that this is how much mass is located how far from the axis of rotation? And let's look at a quick demo here. Here I have an XYZ axis. I'm going to rotate this body about the z-axis. It, we would be able to calculate the mass moment of inertia for this body. And then, as a feel for what the mass moment inertia is, if I moved those masses further out, away from the z-axis, so they're further out in the x and y directions, then I would have a larger mass moment of inertia. Okay, so that give you a, a physical feel. [SOUND] How about the products of inertia? Well, this is the mathematical expressions for the product of inertia. Products of inertia measure a lack of symmetry. And so if the products, you can't have the products of inertia equal to zero. And so, if there's, there's a symmetry with the z-axis or if there's symmetry in the xy plane, then we get the products of inertia as being zero. And so here's an example of a cone, this is, this cone shape is the z-axis is an axis of symmetry here. Here is my z-axis. Or the X, Y plane is also a plane of symmetry. So, here's my X, Y plane. With my X, Y planes, if I do a mirror image above and below, it's the same so it's symmetric. In both of those cases, you would find that the product of inertia would be equal to zero. All right, so, I want to show you one more example, physical example. So if we'll come over here. So here's, here's a wheel. It's in planar motion, okay? And so it's spinning. This would have products of inertia that are equal to zero, because it's symmetric about the z-axis. And as well, it's symmetric about the xy plane. So a, a good example of a case where we would only have a product, mass moment inertia and no products of inertia. Thank you. [BLANK_AUDIO]. Okay. Just because a body does not belong to either two of, either of these two examples, doesn't mean it cannot have zero product inertia. You can't always see when the products of inertia, you can't see the symmetry when the products of inertia are equal to zero. But these are common cases that are worthy of note, and give you some physical feel for what the products of inertia are. And its common to choose coordinates, you can see if I chose coordinates that were not welded in these shapes, then I would not have the symmetry and the product's inertia for those coordinate axis would include a product of inertia. And so its often common for us to choose coordinates such that the products of inertia do equal zero, and we have only mass moment of inertia. Okay, lets do an example now for a three dimensional body where we're going to find the mass moments of inertia and products of inertia with a respect to a coordinate frame that's shown here with it's origin at the mass center C and this is a uniform rectangular solid mass of mass what we'll say m. Okay, and so do that on your own using the information that I gave you in the earlier module and other references to, to use these dimensions and find the mass moments of inertia and products of inertia and come on back and see how you did. And so here is the mass moment of inertia about the x-axis about the y-axis and about the z-axis. And next, I'm going to look at the products of inertia. And they're all zero, in this case because the system is, symmetric about those planes. And so, here's the result again. I can now put this in, a matrix form. So this is my intertial matrix for this particular body, for that particular frame. And we said again, you'll see that this it's symmetric, in this case there's just zeroes on the off diagonals, and this includes information about mass, you got mass here, the shape, B and D, and the geometry of our body. And so, we call a scalar a zero order tensor, because it has it's a physical property, a tensor would be something like mass or temperature, it has one piece of information. Now a vector is what we call a first order tension, the tensor. So for instance a force is a vector, it has two unique pieces of information. It has a magnitude and it has a direction, so that's a first-order tensor. This inertia property matrix is called a second-order tensor and in this case it actually has six independent pieces of information, the three mass moments of inertia. And then these three products of inertia, these are the same products of inertia that, so that's not new information. And so a tensor you'll hear this in other courses, engineering course that you take. It's a mathematical concept. It represents a physical, geometric property or quantity by a mathematical idealization of an array of numbers which we've shown in matrix form. Okay, so lets look at a couple of properties of the inertial matrix. First of all there is, in this case our, our matrix is diagonal. It may not be, it's not the case for other orientations of the axis, but there is always going to be a particular frame such that the matrix, the inertial matrix is going to be a diagonal, and we'll learn this in a future lesson. The other thing is, is that in general, the mass moment of inertia is, time dependent. But if the axes are body fixed then it is independent of time and we often fix the axes in the body because it makes it easier if the mass moments of inertia and products of inertia are not time dependent. And so lets look at an example of that. So here again is a body. If I have a fixed coordinate system here, xyz, and my body is rotating or moving around, then the products of inertia, and the mass moments in inertia for that body with respect to this axis, is going to change with respect to time. However, if I now weld the axis to the body, at any point, I could weld it here, I could weld it here, over here. But i, if it's welded to the body, as the body moves the mass moments of inertia and products of inertia do not change and we're going to find that useful when we actually solve problems. And so that's it for todays module and I'll see you next time.