1
00:00:00,420 --> 00:00:01,260
‫Welcome back.

2
00:00:01,800 --> 00:00:08,910
‫In this video, we will discuss the man's moment of inertia and the inertia tensor in 3-D.

3
00:00:09,450 --> 00:00:18,420
‫If we look at the first part of the core series, then over there we only dealt on an X Y plane, which

4
00:00:18,420 --> 00:00:19,850
‫is a to the plane.

5
00:00:20,400 --> 00:00:28,620
‫So our mass moment of inertia was only measured about one axis.

6
00:00:28,920 --> 00:00:39,540
‫And that axis was about the body frame Z axis that popped out of the screen towards you like this.

7
00:00:40,290 --> 00:00:46,060
‫However, things get more complicated when we talk in 3D, in 3D.

8
00:00:46,770 --> 00:00:48,780
‫This is your body frame axis.

9
00:00:49,440 --> 00:00:59,720
‫And so you can have moments and angular accelerations about three axis X, Y and Z.

10
00:01:00,390 --> 00:01:06,840
‫And so you can also measure mass movement of inertia about these three axis.

11
00:01:07,170 --> 00:01:13,530
‫In other words, you measure the resistance to rotation about these axis.

12
00:01:14,100 --> 00:01:18,780
‫Remember, mass movement of inertia was resistance to rotation.

13
00:01:19,740 --> 00:01:28,890
‫For example, you can have this object, here is your mass moment of inertia about this axis, which

14
00:01:28,890 --> 00:01:40,500
‫is the x axis and let's call it like this I, X and X, then this one here, which would be I y y,

15
00:01:40,800 --> 00:01:49,290
‫some mass movement of inertia about the body from y axis and finally mass movement of inertia about

16
00:01:49,530 --> 00:01:50,720
‫the axis.

17
00:01:51,300 --> 00:01:53,020
‫So the body frames the axis.

18
00:01:53,580 --> 00:01:58,950
‫So you see you have three axis and you can rotate about those axis.

19
00:01:59,310 --> 00:02:05,200
‫And then the mass moments of inertia for all these axis might be different.

20
00:02:05,400 --> 00:02:14,460
‫For example, it might be that it is much easier to rotate this object about the Z axis than, for example,

21
00:02:14,460 --> 00:02:23,790
‫about the X or Y axis, because the mass movement of inertia, perhaps about the Z axis is less than

22
00:02:24,240 --> 00:02:28,140
‫I x, x or Y, Y, Y.

23
00:02:28,620 --> 00:02:39,210
‫It means that about the Z axis, you need less movement to angular really accelerate the object compared

24
00:02:39,210 --> 00:02:48,330
‫to the moment that you would need in order to accelerate angular the object about the X and Y axis at

25
00:02:48,330 --> 00:02:49,320
‫the same rate.

26
00:02:49,860 --> 00:03:00,000
‫So in other words, if I want to accelerate this object five radians per second squared above the x

27
00:03:00,000 --> 00:03:09,480
‫axis and y axis and the axis, then if mass movement of inertia for the Z axis is less than the other

28
00:03:09,480 --> 00:03:19,170
‫two, then that means that I need to apply less movement about the Z axis in order to reach that five

29
00:03:19,170 --> 00:03:20,690
‫radians per second squared.

30
00:03:21,090 --> 00:03:28,230
‫So I need to apply more movement if I want to reach five radians per second squared about the x axis

31
00:03:28,590 --> 00:03:30,270
‫and about the Y axis.

32
00:03:31,440 --> 00:03:39,330
‫And you can formulate all that in a vector matrix form like this, you can have a matrix where the diagonal

33
00:03:39,330 --> 00:03:47,010
‫elements are the mass moments of inertia about the X, Y and Z axis, respectively.

34
00:03:47,370 --> 00:03:50,040
‫And then the rest of the elements are zeros.

35
00:03:50,340 --> 00:04:01,500
‫And then you can multiply that matrix by a vector where the first element is P dot and remember P,

36
00:04:01,770 --> 00:04:12,090
‫Q and R, where the angle of velocity is about the body frame X, Y and Z axis respectively, and they

37
00:04:12,090 --> 00:04:14,490
‫were in radians per second.

38
00:04:14,970 --> 00:04:22,770
‫So if I take the time derivative of P then I will get the angular acceleration about the body frame

39
00:04:22,770 --> 00:04:23,700
‫x axis.

40
00:04:24,330 --> 00:04:32,730
‫Then the second element would be Q dot the angular acceleration about the body frame y axis and then

41
00:04:32,730 --> 00:04:41,880
‫R dot, which is the angular acceleration in both the body frame, the axis and all these variables

42
00:04:41,880 --> 00:04:42,300
‫here.

43
00:04:42,300 --> 00:04:52,320
‫They have units radians per second squared because they're angular acceleration square and all that

44
00:04:52,890 --> 00:05:03,390
‫equals moment about the x axis, moment about the Y axis and moment about the Z axis.

45
00:05:03,930 --> 00:05:07,970
‫If you write this system out, then this is what you will get.

46
00:05:08,490 --> 00:05:17,370
‫That's how you connect moments, mass moments of inertia and angular accelerations about their respective

47
00:05:17,370 --> 00:05:18,000
‫axis.