WEBVTT 00:00:00.100 --> 00:00:02.450 The following content is provided under a Creative 00:00:02.450 --> 00:00:03.830 Commons license. 00:00:03.830 --> 00:00:06.070 Your support will help MIT OpenCourseWare 00:00:06.070 --> 00:00:10.170 continue to offer high quality educational resources for free. 00:00:10.170 --> 00:00:12.710 To make a donation or to view additional materials 00:00:12.710 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.325 at ocw.mit.edu. 00:00:20.372 --> 00:00:22.580 J. KIM VANDIVER: We're waiting for people to come in. 00:00:22.580 --> 00:00:24.496 And what are the important concepts this week? 00:00:24.496 --> 00:00:27.270 There's only the one lecture, and the lecture yesterday 00:00:27.270 --> 00:00:32.180 was pretty dense in concepts. 00:00:32.180 --> 00:00:34.290 Let's build a quick list here. 00:00:36.970 --> 00:00:39.130 Who's got the first offering? 00:00:39.130 --> 00:00:41.371 AUDIENCE: Rotational dynamics of rigid bodies. 00:00:41.371 --> 00:00:43.620 J. KIM VANDIVER: OK, that was a general overall topic, 00:00:43.620 --> 00:00:47.095 so how about a concept though. 00:00:47.095 --> 00:00:49.350 AUDIENCE: The mass moment of inertia matrix. 00:00:49.350 --> 00:00:53.630 J. KIM VANDIVER: OK, yep, we introduced the concept 00:00:53.630 --> 00:00:55.984 of the mass moment of inertia matrix. 00:00:55.984 --> 00:00:58.150 And I'll give it a context, so you can, for example, 00:00:58.150 --> 00:01:04.310 write H as I times omega. 00:01:04.310 --> 00:01:06.930 So we introduced that, didn't get very far with it. 00:01:06.930 --> 00:01:09.468 Another one from yesterday? 00:01:09.468 --> 00:01:10.926 AUDIENCE: I'm not sure if this fits 00:01:10.926 --> 00:01:12.960 but talking about how the body can 00:01:12.960 --> 00:01:15.210 be rotating around the point opposite the [INAUDIBLE]. 00:01:15.210 --> 00:01:16.251 J. KIM VANDIVER: Ah yeah. 00:01:16.251 --> 00:01:21.170 Yep So we talked about rotation. 00:01:29.590 --> 00:01:36.850 The subject yesterday was really rigid body rotation 00:01:36.850 --> 00:01:47.420 about fixed point on or off the body. 00:01:53.580 --> 00:01:55.550 So here's our axis of rotation. 00:01:55.550 --> 00:01:59.800 Call it A. And it's spinning about that point at some omega. 00:01:59.800 --> 00:02:01.800 But that point A could be over here too. 00:02:01.800 --> 00:02:04.100 And then the whole body is just going, 00:02:04.100 --> 00:02:07.150 so an example of rigid body rotation 00:02:07.150 --> 00:02:10.830 about a point not on the body would be something doing that. 00:02:15.010 --> 00:02:17.670 OK, something else. 00:02:29.124 --> 00:02:32.996 AUDIENCE: The angler velocity [INAUDIBLE]. 00:02:32.996 --> 00:02:33.870 J. KIM VANDIVER: Yes. 00:02:33.870 --> 00:02:44.990 Yeah, what does it means when H and omega may not align? 00:02:49.470 --> 00:02:51.660 What's that mean? 00:02:51.660 --> 00:02:53.215 So we got into that a bit yesterday. 00:02:56.150 --> 00:03:00.880 And anything else that struck you? 00:03:06.700 --> 00:03:10.560 OK, well that's a good start. 00:03:10.560 --> 00:03:11.920 Let's just work with that. 00:03:11.920 --> 00:03:14.900 Now I will do that second. 00:03:14.900 --> 00:03:16.700 We're going to come work a little bit, 00:03:16.700 --> 00:03:20.950 talking about this system in those contexts. 00:03:20.950 --> 00:03:25.220 Before I do that, I want to-- the staff 00:03:25.220 --> 00:03:28.360 decided we hadn't done something in lecture yet very carefully 00:03:28.360 --> 00:03:32.430 so we wanted to do it once carefully within recitation. 00:03:32.430 --> 00:03:38.050 And that's a methodical way of doing free body diagrams. 00:03:42.900 --> 00:03:48.770 So we have a system, two masses, springs, dashpots. 00:03:48.770 --> 00:03:50.940 In the last class, there were a few people 00:03:50.940 --> 00:03:55.820 who didn't know really what we meant by k's and c's and how 00:03:55.820 --> 00:03:56.360 they work. 00:03:56.360 --> 00:03:59.900 So I will give you a quick definition. 00:03:59.900 --> 00:04:02.420 So here we have a spring. 00:04:02.420 --> 00:04:06.090 And these are called linear springs, so Hooke's law. 00:04:06.090 --> 00:04:09.420 If you cause that spring to move, 00:04:09.420 --> 00:04:12.220 and I'll call it here the plus x direction-- you grab it 00:04:12.220 --> 00:04:15.530 and you pull on it-- you have to apply a force. 00:04:15.530 --> 00:04:19.560 So the force to pull that spring is positive kx 00:04:19.560 --> 00:04:21.589 in the same direction as x. 00:04:21.589 --> 00:04:24.960 But if that spring's attached to a mass what 00:04:24.960 --> 00:04:29.790 direction is the force that the spring applies on the mass? 00:04:29.790 --> 00:04:32.910 So I've given you a hint here. 00:04:32.910 --> 00:04:34.000 It pulls back, right? 00:04:34.000 --> 00:04:36.200 Because this is Newton's third law. 00:04:36.200 --> 00:04:37.860 So we are doing free body diagrams, 00:04:37.860 --> 00:04:42.000 and we're concerned with forces on the rigid bodies. 00:04:42.000 --> 00:04:46.390 So, this spring, if this mass went that way, 00:04:46.390 --> 00:04:48.240 the spring would pull this way. 00:04:48.240 --> 00:04:51.930 And we indicated the direction with an arrow and its value 00:04:51.930 --> 00:04:53.220 as a kx. 00:04:53.220 --> 00:04:55.120 Dashpots are similar. 00:04:55.120 --> 00:04:59.400 Except dashpots are activated by velocity. 00:04:59.400 --> 00:05:01.940 So this is a linear dashpot, such that it 00:05:01.940 --> 00:05:04.830 has a dashpot constant c. 00:05:04.830 --> 00:05:10.000 And if I give this a positive value of velocity, 00:05:10.000 --> 00:05:14.966 the force required to pull that dashpot is cx dot. 00:05:14.966 --> 00:05:16.310 It's linearly proportional. 00:05:16.310 --> 00:05:20.160 Velocity and the force that if it were attached to a mass 00:05:20.160 --> 00:05:22.180 that the dashpot puts on the mass 00:05:22.180 --> 00:05:26.790 would be opposite direction but same value, 00:05:26.790 --> 00:05:29.450 cx dot, but is resisting the motion. 00:05:29.450 --> 00:05:30.490 OK. 00:05:30.490 --> 00:05:34.100 So I'm going to just give you general method 00:05:34.100 --> 00:05:35.430 for doing free body diagrams. 00:05:35.430 --> 00:05:37.220 And the reason we choose this problem is 00:05:37.220 --> 00:05:42.150 there's two bodies, and depending on which 00:05:42.150 --> 00:05:45.210 is doing what, this intermediate spring in here 00:05:45.210 --> 00:05:47.185 particularly can push or pull on either body 00:05:47.185 --> 00:05:52.170 and having a method so you don't mess up the signs 00:05:52.170 --> 00:05:55.100 is what this is all about. 00:05:55.100 --> 00:06:02.680 So let's just start by building a free body 00:06:02.680 --> 00:06:05.770 diagram of mass one. 00:06:11.250 --> 00:06:13.920 And the assignment I'm going to give you to work on, 00:06:13.920 --> 00:06:15.730 just seat work here for a minute, 00:06:15.730 --> 00:06:18.780 is only consider the springs for now. 00:06:18.780 --> 00:06:21.800 We'll just deal with the springs. 00:06:21.800 --> 00:06:23.500 So assume the system is moving. 00:06:23.500 --> 00:06:25.700 You have to do free body diagrams of it. 00:06:25.700 --> 00:06:30.830 Put the spring forces on the mass in your free body 00:06:30.830 --> 00:06:31.330 diagrams. 00:06:31.330 --> 00:06:32.880 So spend a minute or so. 00:06:32.880 --> 00:06:35.340 Talk to your neighbors after you get a little ways along 00:06:35.340 --> 00:06:37.890 with it. 00:06:37.890 --> 00:06:40.330 And I'm going to add-- there's also 00:06:40.330 --> 00:06:44.510 on this whole system external force F1, external force F2. 00:06:44.510 --> 00:06:47.320 That's your excitation. 00:06:47.320 --> 00:06:49.550 But don't worry about that for now. 00:06:49.550 --> 00:06:53.830 Just figure out the spring forces, and I'll be quiet. 00:06:53.830 --> 00:06:57.180 I missed a step, my apologies. 00:06:57.180 --> 00:06:59.610 We should do something else first. 00:06:59.610 --> 00:07:03.400 The very first step is to assign the coordinates. 00:07:03.400 --> 00:07:06.410 Let's all agree on the coordinate system here. 00:07:06.410 --> 00:07:09.685 So the question really is the first thing 00:07:09.685 --> 00:07:12.060 you have to do is you have to decide how many independent 00:07:12.060 --> 00:07:15.000 coordinates you need to completely describe the motion. 00:07:15.000 --> 00:07:16.500 So how many coordinates do you think 00:07:16.500 --> 00:07:19.320 are needed for this problem? 00:07:19.320 --> 00:07:22.410 I see a one. 00:07:22.410 --> 00:07:23.770 AUDIENCE: I was just-- 00:07:23.770 --> 00:07:25.560 J. KIM VANDIVER: That's a second one. 00:07:25.560 --> 00:07:27.120 Anybody else have a suggestion? 00:07:27.120 --> 00:07:29.784 So where would you put the one? 00:07:29.784 --> 00:07:31.780 AUDIENCE: The x term. 00:07:31.780 --> 00:07:36.556 J. KIM VANDIVER: And measuring the displacement of mass one? 00:07:36.556 --> 00:07:37.180 AUDIENCE: Sure. 00:07:37.180 --> 00:07:38.150 J. KIM VANDIVER: OK. 00:07:38.150 --> 00:07:39.460 And is that all we need? 00:07:42.270 --> 00:07:43.529 Anybody have a further-- 00:07:43.529 --> 00:07:45.570 AUDIENCE: Oh, there's a displacement at mass two. 00:07:45.570 --> 00:07:47.486 J. KIM VANDIVER: And it's independence, right? 00:07:47.486 --> 00:07:49.270 They can move-- how many coordinates now 00:07:49.270 --> 00:07:50.450 do you think you need? 00:07:50.450 --> 00:07:50.990 Uh oh. 00:07:50.990 --> 00:07:54.200 She changed her mind to two. 00:07:54.200 --> 00:07:57.040 And I've got them labeled up here for you already. 00:07:57.040 --> 00:07:58.495 So it's going to take two. 00:07:58.495 --> 00:08:00.120 And one of the ways you can test if you 00:08:00.120 --> 00:08:03.590 have the right number, if you think it's one, assign it. 00:08:03.590 --> 00:08:05.000 Freeze it. 00:08:05.000 --> 00:08:06.660 Is there still independent movement 00:08:06.660 --> 00:08:07.780 of a part of the system? 00:08:07.780 --> 00:08:12.140 If there is, you have yet another degree of freedom. 00:08:12.140 --> 00:08:14.780 If we freeze both x1 and x2, and can anything 00:08:14.780 --> 00:08:15.680 move in the system? 00:08:15.680 --> 00:08:16.180 Nope. 00:08:16.180 --> 00:08:17.640 So we've got it nailed down. 00:08:17.640 --> 00:08:19.310 All right. 00:08:19.310 --> 00:08:23.730 But where do you measure? 00:08:23.730 --> 00:08:26.420 There is something else you have to determine. 00:08:26.420 --> 00:08:30.270 And that is, where do you make zero? 00:08:30.270 --> 00:08:31.610 You also have to know that. 00:08:31.610 --> 00:08:35.049 So where is x is 0 in your picture? 00:08:35.049 --> 00:08:37.860 We'll just talk about both of them, x1 and x2. 00:08:37.860 --> 00:08:39.730 How would you pick the 0 point? 00:08:42.970 --> 00:08:43.682 Christina? 00:08:43.682 --> 00:08:45.182 AUDIENCE: Probably when the spring's 00:08:45.182 --> 00:08:48.659 at equilibrium so when it's not exerting force. [INAUDIBLE]. 00:08:48.659 --> 00:08:50.950 J. KIM VANDIVER: OK, she says when there's not exerting 00:08:50.950 --> 00:08:52.033 force in either direction. 00:08:52.033 --> 00:08:54.815 I would call that the static equilibrium position. 00:08:59.430 --> 00:09:02.270 There is a conceptual problem with that. 00:09:02.270 --> 00:09:04.550 We'll assume that that will work if you've 00:09:04.550 --> 00:09:05.990 set the system up so that there's 00:09:05.990 --> 00:09:07.280 no forces in the springs. 00:09:07.280 --> 00:09:10.460 But you know, if I move this wall in a little bit 00:09:10.460 --> 00:09:12.910 and let it reach static equilibrium, 00:09:12.910 --> 00:09:14.940 it has a static equilibrium position. 00:09:14.940 --> 00:09:17.210 All the springs have forces in them. 00:09:17.210 --> 00:09:19.810 Now, you pre-compress them. 00:09:19.810 --> 00:09:21.894 So there's kind of a problem-- you 00:09:21.894 --> 00:09:23.810 could run into a conceptual problem with that. 00:09:23.810 --> 00:09:25.130 It will still work. 00:09:25.130 --> 00:09:26.760 Picking a static equilibrium position 00:09:26.760 --> 00:09:29.331 is a good thing to do usually. 00:09:29.331 --> 00:09:29.830 OK. 00:09:33.050 --> 00:09:35.340 There are other positions. 00:09:35.340 --> 00:09:39.740 For example, this system, it's a simpler system, just 00:09:39.740 --> 00:09:41.620 an ordinary little mass spring. 00:09:41.620 --> 00:09:45.870 But where do you make zero when you pick your coordinate? 00:09:45.870 --> 00:09:47.870 AUDIENCE: In this case, you could zero it out 00:09:47.870 --> 00:09:52.555 at the rest length of the spring plus the length 00:09:52.555 --> 00:09:54.930 of this spring as a result of the force of gravity. 00:09:54.930 --> 00:09:56.430 J. KIM VANDIVER: So right now you're 00:09:56.430 --> 00:09:58.290 seeing the static equilibrium position. 00:09:58.290 --> 00:10:01.750 The spring has tension in it, the weight of the mass, right? 00:10:01.750 --> 00:10:04.120 So you could either make x right here. 00:10:04.120 --> 00:10:06.564 Or the other obvious choice would be? 00:10:06.564 --> 00:10:07.980 AUDIENCE: Where you're holding it. 00:10:07.980 --> 00:10:10.656 AUDIENCE: You're holding it. 00:10:10.656 --> 00:10:12.280 J. KIM VANDIVER: I heard a couple voice 00:10:12.280 --> 00:10:13.155 but I couldn't hear what you said. 00:10:13.155 --> 00:10:14.730 AUDIENCE: Where you're holding it? 00:10:14.730 --> 00:10:17.440 J. KIM VANDIVER: Well, yeah, but it turns out 00:10:17.440 --> 00:10:18.899 that actually is a terrible choice. 00:10:18.899 --> 00:10:20.440 Because then you have to get involved 00:10:20.440 --> 00:10:21.729 with the length of the spring. 00:10:21.729 --> 00:10:23.270 And the length of the spring actually 00:10:23.270 --> 00:10:25.870 really doesn't enter into the equation in motion 00:10:25.870 --> 00:10:28.230 unless you're really silly about where 00:10:28.230 --> 00:10:30.976 you pick the measure from. 00:10:30.976 --> 00:10:32.850 Static equilibrium is good, but the other one 00:10:32.850 --> 00:10:35.800 is the zero spring force position 00:10:35.800 --> 00:10:37.150 is the other natural one to do. 00:10:37.150 --> 00:10:41.560 So at zero spring force, that's no extension of the spring. 00:10:41.560 --> 00:10:44.400 Because this is preloaded when you do this, and have to figure 00:10:44.400 --> 00:10:46.250 out what that does to you. 00:10:46.250 --> 00:10:47.860 OK. 00:10:47.860 --> 00:10:52.093 Again, static equilibrium has some real advantages here, 00:10:52.093 --> 00:10:53.426 but we'll talk about that later. 00:10:53.426 --> 00:10:54.120 Yeah? 00:10:54.120 --> 00:10:56.526 AUDIENCE: So why does zero spring force 00:10:56.526 --> 00:10:57.847 work there but not-- 00:10:57.847 --> 00:11:00.430 J. KIM VANDIVER: I'm just saying there are actually situations 00:11:00.430 --> 00:11:02.320 where there is no zero force. 00:11:02.320 --> 00:11:03.860 AUDIENCE: Oh, OK. 00:11:03.860 --> 00:11:06.930 J. KIM VANDIVER: And you have a conception-- if you hadn't ever 00:11:06.930 --> 00:11:11.300 thought of that before, that could give you pause when 00:11:11.300 --> 00:11:13.090 you go to solve the problem. 00:11:13.090 --> 00:11:14.210 The springs are preloaded. 00:11:17.380 --> 00:11:21.380 Are your equations of motion valid? 00:11:21.380 --> 00:11:24.880 So you preloaded this a little bit, 00:11:24.880 --> 00:11:29.302 squeezed it in, this system has natural frequencies, right? 00:11:29.302 --> 00:11:30.260 Do you agree with that? 00:11:30.260 --> 00:11:35.400 It's very similar to this system except it's just now-- 00:11:35.400 --> 00:11:37.500 it doesn't happen to have a third spring 00:11:37.500 --> 00:11:38.627 and have gravity involved. 00:11:38.627 --> 00:11:40.460 But this system has two natural frequencies, 00:11:40.460 --> 00:11:44.660 that one and one that's a little harder for me 00:11:44.660 --> 00:11:49.440 to do but that one. 00:11:49.440 --> 00:11:51.730 So if I've preloaded this a little bit, 00:11:51.730 --> 00:11:57.000 squeezed it in some, would the natural frequencies change? 00:11:57.000 --> 00:11:58.800 Yes-- how many think yes? 00:11:58.800 --> 00:11:59.550 Get your hands up. 00:11:59.550 --> 00:12:00.050 Come on. 00:12:00.050 --> 00:12:01.310 You're gamblers. 00:12:01.310 --> 00:12:02.590 How many think yes? 00:12:02.590 --> 00:12:04.560 How many think no? 00:12:04.560 --> 00:12:06.790 How many just don't want to raise their hand? 00:12:06.790 --> 00:12:09.670 [LAUGHS] Like you. 00:12:09.670 --> 00:12:10.460 OK. 00:12:10.460 --> 00:12:13.980 The natural frequency doesn't change. 00:12:13.980 --> 00:12:16.060 It turns out to preload doesn't matter 00:12:16.060 --> 00:12:18.350 in a linear system like this. 00:12:18.350 --> 00:12:20.360 So there's some nuances in here. 00:12:20.360 --> 00:12:23.810 A simple system like this has some little traps in it. 00:12:23.810 --> 00:12:26.590 It will take them time to learn your way through. 00:12:26.590 --> 00:12:28.760 But let's keep it simple. 00:12:28.760 --> 00:12:33.292 No preload, the static equilibrium position, 00:12:33.292 --> 00:12:36.640 draw the free body diagram for mass one, springs only. 00:12:36.640 --> 00:12:38.542 Only deal with the springs. 00:12:38.542 --> 00:12:40.500 Take a couple minutes and talk to your neighbor 00:12:40.500 --> 00:12:43.180 if you need to. 00:12:43.180 --> 00:12:48.620 OK, so I won't come look at each of your things. 00:12:48.620 --> 00:12:51.020 I want to have different groups kind of-- they 00:12:51.020 --> 00:12:52.440 should take one force at a time. 00:12:52.440 --> 00:12:57.480 So you guys, give me a force on this mass, due to a spring. 00:12:57.480 --> 00:13:01.094 Give me a spring force and tell me what direction it's in. 00:13:01.094 --> 00:13:02.864 AUDIENCE: I guess it's [INAUDIBLE]. 00:13:02.864 --> 00:13:04.780 J. KIM VANDIVER: So tell me which spring we're 00:13:04.780 --> 00:13:05.868 talking about. 00:13:05.868 --> 00:13:07.850 AUDIENCE: So [? I didn't hear. ?] So is this 00:13:07.850 --> 00:13:10.140 being impressed, like are these springs? 00:13:10.140 --> 00:13:11.840 J. KIM VANDIVER: No, this is static equilibrium position. 00:13:11.840 --> 00:13:14.090 But now if you give it initial [INAUDIBLE] and let go, 00:13:14.090 --> 00:13:15.900 it's going to sit there and do something. 00:13:15.900 --> 00:13:17.950 We're trying to derive the equations of motion. 00:13:17.950 --> 00:13:21.170 And to do so, we have to start with a free body diagram. 00:13:21.170 --> 00:13:22.990 So it's moving, and there are forces 00:13:22.990 --> 00:13:25.400 on it caused by springs and by dashpots 00:13:25.400 --> 00:13:27.684 and by the external forces. 00:13:27.684 --> 00:13:30.100 So right now, we're going to be free body diagram but only 00:13:30.100 --> 00:13:32.050 the components through the spring. 00:13:32.050 --> 00:13:33.900 Springs. 00:13:33.900 --> 00:13:36.870 So tell me what happens. 00:13:36.870 --> 00:13:42.150 What does that spring-- how does this spring appear 00:13:42.150 --> 00:13:44.661 on that free body diagram. 00:13:44.661 --> 00:13:47.487 AUDIENCE: [? I have an idea. ?] [INAUDIBLE] it's air force 00:13:47.487 --> 00:13:49.380 is 11 [INAUDIBLE]. 00:13:49.380 --> 00:13:50.976 J. KIM VANDIVER: OK, force of. 00:13:50.976 --> 00:13:52.760 AUDIENCE: Force of the spring on the-- 00:13:52.760 --> 00:13:54.988 J. KIM VANDIVER: Which direction? 00:13:54.988 --> 00:13:58.429 AUDIENCE: Left. [LAUGHS] I don't understand how you know which-- 00:13:58.429 --> 00:13:59.970 J. KIM VANDIVER: OK, well that's kind 00:13:59.970 --> 00:14:01.178 of the point of the exercise. 00:14:01.178 --> 00:14:04.020 And I expect many of you to have some confusion about what 00:14:04.020 --> 00:14:07.420 direction to go because you don't have a standard method 00:14:07.420 --> 00:14:08.580 by which you approach this. 00:14:08.580 --> 00:14:11.664 AUDIENCE: So both springs and cause of force 00:14:11.664 --> 00:14:14.580 goes in the direction of the displacement. 00:14:14.580 --> 00:14:17.140 J. KIM VANDIVER: OK, so what's the displacement, though? 00:14:17.140 --> 00:14:20.184 This system has two possible displacements. 00:14:20.184 --> 00:14:21.630 AUDIENCE: [INAUDIBLE]. 00:14:21.630 --> 00:14:26.487 We're assuming m1 is going in the positive x1 [INAUDIBLE]. 00:14:26.487 --> 00:14:27.320 J. KIM VANDIVER: OK. 00:14:27.320 --> 00:14:30.560 So if you said if you move it a positive x1, 00:14:30.560 --> 00:14:33.360 then you will get a spring force. 00:14:33.360 --> 00:14:34.679 Spring one does what? 00:14:34.679 --> 00:14:36.470 AUDIENCE: It would oppose the [? spring. ?] 00:14:36.470 --> 00:14:38.790 J. KIM VANDIVER: Opposes, and what's it value? 00:14:38.790 --> 00:14:39.764 AUDIENCE: k1x. 00:14:39.764 --> 00:14:40.680 J. KIM VANDIVER: k1x1. 00:14:43.570 --> 00:14:48.130 So that's what a motion x1 causes-- 00:14:48.130 --> 00:14:49.980 that's a result with spring one. 00:14:49.980 --> 00:14:52.920 What's the result of spring two if you have motion x1? 00:14:56.310 --> 00:14:56.970 AUDIENCE: Same. 00:14:56.970 --> 00:14:58.803 There's going to be compression force that's 00:14:58.803 --> 00:15:00.660 also opposing displacement. 00:15:00.660 --> 00:15:01.390 J. KIM VANDIVER: Right, because it's 00:15:01.390 --> 00:15:02.848 trying to squeeze that spring down, 00:15:02.848 --> 00:15:04.130 and it's pushing back, right? 00:15:04.130 --> 00:15:11.000 But you're also going to get a spring k2x1, right? 00:15:11.000 --> 00:15:13.230 Now are there any other forces that 00:15:13.230 --> 00:15:17.820 result from the motion of that body in the springs? 00:15:17.820 --> 00:15:18.882 Just spring forces now. 00:15:18.882 --> 00:15:20.340 AUDIENCE: From that body by itself? 00:15:20.340 --> 00:15:22.350 J. KIM VANDIVER: Yeah, that body by itself. 00:15:22.350 --> 00:15:22.970 No. 00:15:22.970 --> 00:15:25.520 But is that all the spring forces? 00:15:25.520 --> 00:15:26.600 No. 00:15:26.600 --> 00:15:31.552 So now what if you let body two move? 00:15:31.552 --> 00:15:34.010 And now I'm going to give you a little rubric, a little way 00:15:34.010 --> 00:15:34.718 to go about this. 00:15:34.718 --> 00:15:36.610 We start by assigning our coordinates. 00:15:36.610 --> 00:15:58.830 Then you specify positive x1, x1 dot, x2, x2 dot, one at a time. 00:16:05.920 --> 00:16:16.040 And from that, you deduce the direction of love 00:16:16.040 --> 00:16:20.080 the forces, of the resulting forces. 00:16:20.080 --> 00:16:20.740 OK. 00:16:20.740 --> 00:16:25.590 So if what we've done there is consistent 00:16:25.590 --> 00:16:30.970 with it, a positive value of x1, spring one pulls back k1x1, 00:16:30.970 --> 00:16:34.650 spring two pushes back k1x1, then 00:16:34.650 --> 00:16:37.710 that's the end of spring forces due to x1. 00:16:37.710 --> 00:16:41.390 So now just let's say, OK, let x1 be 0 for a moment. 00:16:41.390 --> 00:16:44.240 And now let there be a positive x2. 00:16:44.240 --> 00:16:47.670 Does that cause any spring force on mass one? 00:16:52.143 --> 00:16:53.982 What do you think? 00:16:53.982 --> 00:16:57.250 AUDIENCE: We're going to pull mass one [INAUDIBLE]. 00:16:57.250 --> 00:16:59.350 J. KIM VANDIVER: That one-- a positive x2 00:16:59.350 --> 00:17:02.379 puts tension in the spring and it pulls it in that direction. 00:17:02.379 --> 00:17:03.795 So now you're going to get a k2x2. 00:17:07.599 --> 00:17:11.500 Any other spring forces caused by motion of two? 00:17:14.450 --> 00:17:16.810 It's the only connecting spring. 00:17:16.810 --> 00:17:19.350 So those are the spring forces on mass one. 00:17:22.810 --> 00:17:25.770 Now let's move on to dashpot forces. 00:17:25.770 --> 00:17:27.859 And you do exactly the same thing. 00:17:27.859 --> 00:17:30.920 Assume a positive value of x1 dot. 00:17:30.920 --> 00:17:33.630 What does it cause in dashpot forces? 00:17:33.630 --> 00:17:39.150 What's the first dashpot do when you pull positive x1 dot 00:17:39.150 --> 00:17:41.050 in that direction? 00:17:41.050 --> 00:17:42.956 Resist or not? 00:17:42.956 --> 00:17:44.210 AUDIENCE: Resist. 00:17:44.210 --> 00:17:46.200 J. KIM VANDIVER: Dashpots resist. 00:17:46.200 --> 00:17:49.960 So it's going to be in the minus direction and a value-- how 00:17:49.960 --> 00:17:51.480 big? 00:17:51.480 --> 00:17:53.260 AUDIENCE: c1 and [INAUDIBLE]. 00:17:53.260 --> 00:17:58.750 J. KIM VANDIVER: So now you get dashpot forces, c1x1 dot. 00:17:58.750 --> 00:18:02.166 And how about the c2 dashpot? 00:18:02.166 --> 00:18:03.150 AUDIENCE: Same thing. 00:18:03.150 --> 00:18:10.370 J. KIM VANDIVER: Same thing. c2x1 dot. 00:18:10.370 --> 00:18:10.980 OK. 00:18:10.980 --> 00:18:14.270 And that's the only dashpot forces caused 00:18:14.270 --> 00:18:17.320 by a velocity of mass one. 00:18:17.320 --> 00:18:22.346 So now let that be 0 and cause velocity at the other places 00:18:22.346 --> 00:18:23.720 and find out if anything happens. 00:18:23.720 --> 00:18:28.110 So now let x2 dot be positive. 00:18:28.110 --> 00:18:30.816 What do we put on free body diagram? 00:18:30.816 --> 00:18:33.160 AUDIENCE: [INAUDIBLE]. 00:18:33.160 --> 00:18:34.900 J. KIM VANDIVER: Going which direction? 00:18:34.900 --> 00:18:36.940 Positive, and what value? 00:18:36.940 --> 00:18:39.205 AUDIENCE: c2x2 dot. 00:18:39.205 --> 00:18:40.570 J. KIM VANDIVER: All right. 00:18:40.570 --> 00:18:42.450 And is our free body diagram complete? 00:18:45.515 --> 00:18:46.940 AUDIENCE: mg. 00:18:46.940 --> 00:18:49.440 J. KIM VANDIVER: Oh, yeah. 00:18:49.440 --> 00:18:50.444 And? 00:18:50.444 --> 00:18:51.716 AUDIENCE: [INAUDIBLE]. 00:18:51.716 --> 00:18:53.736 J. KIM VANDIVER: Yeah, and? 00:18:53.736 --> 00:18:56.634 AUDIENCE: The forces, the external forces. 00:19:00.020 --> 00:19:02.250 J. KIM VANDIVER: Now it's complete. 00:19:02.250 --> 00:19:05.540 All the forces in the system, it's 00:19:05.540 --> 00:19:07.020 constrained in this direction. 00:19:07.020 --> 00:19:08.490 So we just know N equals mg. 00:19:08.490 --> 00:19:11.654 There's no motion allowed, so we get no equation 00:19:11.654 --> 00:19:12.820 of motion in that direction. 00:19:12.820 --> 00:19:16.300 We're going to get one equation of motion for this mass 00:19:16.300 --> 00:19:18.950 and one more equation of motion for the second. 00:19:18.950 --> 00:19:20.450 The number of equation of motions 00:19:20.450 --> 00:19:23.750 equal the number of independent coordinates. 00:19:26.330 --> 00:19:28.040 And we found two independent coordinates. 00:19:28.040 --> 00:19:30.010 We get two equations of motion. 00:19:30.010 --> 00:19:34.920 So take this and write down the equation of motion 00:19:34.920 --> 00:19:36.340 for mass one. 00:19:36.340 --> 00:19:36.910 Sort it out. 00:19:36.910 --> 00:19:37.932 Yeah, Betsy? 00:19:37.932 --> 00:19:40.342 AUDIENCE: So I'm really confused by why c2 [INAUDIBLE] 00:19:40.342 --> 00:19:43.234 in that direction, because so [INAUDIBLE]. 00:19:43.234 --> 00:19:45.644 They oppose like the springs that [? felt ?] 00:19:45.644 --> 00:19:48.117 [? the closest ?] [INAUDIBLE]. 00:19:48.117 --> 00:19:48.950 J. KIM VANDIVER: OK. 00:19:48.950 --> 00:19:52.480 So let's see if we can do something here. 00:19:52.480 --> 00:19:57.320 So, this is second mass. 00:19:57.320 --> 00:19:58.665 This is the first mass. 00:19:58.665 --> 00:20:02.540 Actually, here's the wall and the first mass. 00:20:02.540 --> 00:20:04.880 So x1 is in that direction. 00:20:04.880 --> 00:20:07.410 If I pull that way, the spring pulls back on the mass. 00:20:07.410 --> 00:20:09.160 That's obvious to you, right? 00:20:09.160 --> 00:20:13.520 And on the other side, if this is mass one now 00:20:13.520 --> 00:20:15.560 and you have a second spring over here, 00:20:15.560 --> 00:20:18.240 a second mass over here and a spring in between them, 00:20:18.240 --> 00:20:22.410 if you put a positive motion of mass one, 00:20:22.410 --> 00:20:25.170 this spring pushes back. 00:20:25.170 --> 00:20:30.940 So that counts for the minus direction of k2x1. 00:20:30.940 --> 00:20:37.060 Then if the second mass moves in that direction-- here, 00:20:37.060 --> 00:20:37.590 you move it. 00:20:37.590 --> 00:20:39.340 You're the second mass. 00:20:39.340 --> 00:20:41.140 Is there tension in this? 00:20:41.140 --> 00:20:44.370 Am I pulling back to resist you are or not? 00:20:44.370 --> 00:20:46.630 What is this spring doing to my hand? 00:20:46.630 --> 00:20:49.379 It's pulling that way on it, k2x2. 00:20:49.379 --> 00:20:49.920 AUDIENCE: OK. 00:20:53.907 --> 00:20:54.740 J. KIM VANDIVER: OK. 00:20:54.740 --> 00:20:56.720 So that's our free body diagram. 00:20:56.720 --> 00:20:59.260 Write right out an equation of motion for that system. 00:21:10.902 --> 00:21:12.235 I'll give you a little reminder. 00:21:22.270 --> 00:21:23.910 That's how you ought to begin, right? 00:21:37.370 --> 00:21:41.955 And if you're done, do the same thing for the second mass 00:21:41.955 --> 00:21:43.875 while the others are working on the first one. 00:21:43.875 --> 00:21:46.720 Draw a free body diagram of mass two 00:21:46.720 --> 00:21:50.530 and write down the equation of motion. 00:21:55.151 --> 00:21:57.060 AUDIENCE: Shouldn't that be x2 dot? 00:21:57.060 --> 00:21:58.185 J. KIM VANDIVER: Which one? 00:21:58.185 --> 00:21:59.980 AUDIENCE: The bottom left, c2x2 dot. 00:22:04.011 --> 00:22:06.010 J. KIM VANDIVER: All right, so let's build this. 00:22:12.030 --> 00:22:13.560 I'm just going to talk through it 00:22:13.560 --> 00:22:17.050 rather than have you help me build this. 00:22:17.050 --> 00:22:19.760 The sum of the external forces is 00:22:19.760 --> 00:22:23.760 what all of the external forces and only the external forces 00:22:23.760 --> 00:22:27.450 should show up on that diagram. 00:22:27.450 --> 00:22:28.830 The mass times the accelerations, 00:22:28.830 --> 00:22:31.454 the sum of the external forces-- so every arrow on that diagram 00:22:31.454 --> 00:22:33.840 ought to appear over here in the direction 00:22:33.840 --> 00:22:35.280 of whichever this equation is. 00:22:35.280 --> 00:22:37.450 So this is the equation that has to do 00:22:37.450 --> 00:22:39.230 with the motion of mass one. 00:22:39.230 --> 00:22:42.800 And I look over here, and I see, OK, I'm 00:22:42.800 --> 00:22:49.970 just going to go top to bottom, minus k1x1 minus k2x1-- 00:22:49.970 --> 00:22:52.370 and the minuses are coming from the directions 00:22:52.370 --> 00:22:58.050 of the arrows-- minus c1x1 dot. 00:22:58.050 --> 00:23:12.350 And then I have plus k2x2 plus c2x2 dot plus F1. 00:23:12.350 --> 00:23:13.474 AUDIENCE: And a minus c1x1. 00:23:13.474 --> 00:23:14.377 [INTERPOSING VOICES] 00:23:14.377 --> 00:23:15.710 J. KIM VANDIVER: Did I miss one? 00:23:15.710 --> 00:23:19.520 Oh, I missed the c-- I missed this guy, right? 00:23:19.520 --> 00:23:25.490 Minus c2x1 dot. 00:23:25.490 --> 00:23:26.590 OK. 00:23:26.590 --> 00:23:29.720 So the arrows tell me the signs. 00:23:29.720 --> 00:23:32.280 And all the forces are there. 00:23:32.280 --> 00:23:35.020 They just all add up over here on this side. 00:23:35.020 --> 00:23:37.440 And I've gone ahead and drawn what 00:23:37.440 --> 00:23:41.600 I think is the right thing for mass two. 00:23:41.600 --> 00:23:43.380 But you do the same system. 00:23:43.380 --> 00:23:46.210 You go to mass two and you say, OK, positive 00:23:46.210 --> 00:23:47.997 deflection of mass one. 00:23:47.997 --> 00:23:49.580 What is the spring force that results? 00:23:49.580 --> 00:23:54.310 Well, it pushes on it, positive velocity of mass one. 00:23:54.310 --> 00:23:56.510 This dashpot pushes on it. 00:23:56.510 --> 00:24:01.620 Positive deflection of mass two-- two springs push back. 00:24:01.620 --> 00:24:05.970 Positive velocity of mass two-- one dashpot resists. 00:24:05.970 --> 00:24:07.990 And you have an external force. 00:24:07.990 --> 00:24:10.110 And you could write this one down. 00:24:10.110 --> 00:24:12.250 You just add up all those forces. 00:24:12.250 --> 00:24:14.130 And now you have a second equation, 00:24:14.130 --> 00:24:19.380 m2x2 double dot equals a you add it up. 00:24:19.380 --> 00:24:20.250 OK. 00:24:20.250 --> 00:24:23.930 So I want to do one other thing. 00:24:23.930 --> 00:24:26.430 I'm going to ask you a question, get you to raise your hand. 00:24:26.430 --> 00:24:30.810 I want everybody to try to raise their hands, OK? 00:24:30.810 --> 00:24:34.580 Assume that I had written out that second equation. 00:24:34.580 --> 00:24:36.080 Actually, let me take one more step. 00:24:36.080 --> 00:24:39.410 Normally what we would do is to rewrite 00:24:39.410 --> 00:24:42.090 this in sort of a standard form is to move all the motion 00:24:42.090 --> 00:24:45.970 variables to one side and all the actual exciting forces 00:24:45.970 --> 00:24:46.600 to the other. 00:24:46.600 --> 00:24:54.300 And so we'd say, m1x1 double dot plus, 00:24:54.300 --> 00:24:58.080 and then normally you put in the x dot terms next. 00:24:58.080 --> 00:25:16.810 So we have a c1 plus c2 x1 dot minus c2x2 dot plus-- now 00:25:16.810 --> 00:25:29.170 I get my k terms-- k1 plus k2 x1 minus k2x2 equals F1. 00:25:29.170 --> 00:25:32.370 And that's kind of standard form now. 00:25:32.370 --> 00:25:34.957 I get a second equation from the second one 00:25:34.957 --> 00:25:36.040 that would look like that. 00:25:36.040 --> 00:25:41.170 Now once you get it in this sort of standard form, 00:25:41.170 --> 00:25:45.550 can you remember how, if you've even ever had linear algebra, 00:25:45.550 --> 00:25:46.975 write this in matrix form? 00:25:51.052 --> 00:25:52.950 AUDIENCE: It would be [INAUDIBLE]. 00:25:52.950 --> 00:25:53.580 J. KIM VANDIVER: So how many of you 00:25:53.580 --> 00:25:56.080 feel comfortable that you could just sit down and write this 00:25:56.080 --> 00:25:58.040 out in matrix form? 00:25:58.040 --> 00:25:58.890 Go on. 00:25:58.890 --> 00:26:01.860 And how many of you think it might be a bit of a challenge? 00:26:01.860 --> 00:26:04.232 OK. 00:26:04.232 --> 00:26:05.940 How many of you would like me to show you 00:26:05.940 --> 00:26:08.476 what it looks like in matrix form and give you the answer? 00:26:08.476 --> 00:26:09.100 So most of you. 00:26:09.100 --> 00:26:10.660 OK, I'll take the time to do that. 00:26:13.960 --> 00:26:15.830 Actually, did I write it out already? 00:26:15.830 --> 00:26:17.100 Ooh, look at that. 00:26:17.100 --> 00:26:18.510 Bing! 00:26:18.510 --> 00:26:19.436 Ding, ding, ding! 00:26:29.860 --> 00:26:32.300 All you need to know in this course about vectors 00:26:32.300 --> 00:26:38.090 and matrices is how to multiply a vector times a matrix. 00:26:38.090 --> 00:26:41.040 So here's an acceleration vector. 00:26:41.040 --> 00:26:42.730 Here's the mass matrix. 00:26:42.730 --> 00:26:45.820 And you take these two elements, my two fingers here, 00:26:45.820 --> 00:26:48.680 and multiplying by those-- you go like this to that. 00:26:48.680 --> 00:26:57.330 So x1 double dot times m1 double dot gives you this term, right? 00:26:57.330 --> 00:27:01.115 And you get no m2x2 double dot, because there's 00:27:01.115 --> 00:27:04.930 a 0 in the matrix in that second position. 00:27:04.930 --> 00:27:07.200 So you get no x2 double dot term. 00:27:07.200 --> 00:27:09.110 And that's correct. 00:27:09.110 --> 00:27:11.750 Same thing with the-- here's the x2, x1. 00:27:11.750 --> 00:27:14.060 Here's this damping matrix. 00:27:14.060 --> 00:27:15.750 This is the stiffness matrix. 00:27:15.750 --> 00:27:17.930 And so if you just did these multiplies, 00:27:17.930 --> 00:27:22.110 you'd get back two equations, the ones we just derived. 00:27:22.110 --> 00:27:25.260 So you need to know how to do this kind of thing 00:27:25.260 --> 00:27:31.030 because we write that h equals I omega. 00:27:31.030 --> 00:27:34.090 And our omega has three components, omega x, omega y, 00:27:34.090 --> 00:27:35.180 and omega z. 00:27:35.180 --> 00:27:37.320 And you need to be able to do that manipulation. 00:27:40.522 --> 00:27:42.480 All right, we've got a little bit of time left. 00:27:42.480 --> 00:27:44.780 I wanted to do one little exercise that 00:27:44.780 --> 00:27:47.420 had to do more with this sort of thing, 00:27:47.420 --> 00:27:51.290 to kind of reinforce a bit what we did yesterday. 00:27:51.290 --> 00:27:55.620 And that would be-- so any final questions about this? 00:27:55.620 --> 00:27:56.346 Yeah? 00:27:56.346 --> 00:27:58.626 AUDIENCE: So setting that up looks OK. 00:27:58.626 --> 00:27:59.540 But solving it. 00:27:59.540 --> 00:28:02.456 J. KIM VANDIVER: Oh, solving it? 00:28:02.456 --> 00:28:04.330 I used to teach-- I taught for about 20 years 00:28:04.330 --> 00:28:05.860 a course called Mechanical Vibration. 00:28:05.860 --> 00:28:07.460 I just don't have time to teach it anymore. 00:28:07.460 --> 00:28:08.850 And that's what you do in there. 00:28:08.850 --> 00:28:11.660 Oh, in 803 you do a little bit of it. 00:28:11.660 --> 00:28:15.680 So this system has two degrees of freedom. 00:28:15.680 --> 00:28:20.700 It has two natural frequencies, two mode shapes to go with it. 00:28:20.700 --> 00:28:23.850 So the system vibrates. 00:28:23.850 --> 00:28:25.950 And it's not real hard to solve, but you 00:28:25.950 --> 00:28:27.910 would assume for initially, if you 00:28:27.910 --> 00:28:29.570 want to know natural frequencies, 00:28:29.570 --> 00:28:32.510 you set the forces to 0. 00:28:32.510 --> 00:28:35.170 You set the damping to 0 initially. 00:28:35.170 --> 00:28:38.600 And you just assume a solution of the form 00:28:38.600 --> 00:28:51.850 that x1 of t and x2 of t equals some unknown constants, cosine 00:28:51.850 --> 00:28:52.760 omega t. 00:28:52.760 --> 00:28:54.130 Just plug it in. 00:28:54.130 --> 00:28:55.340 You just plug that in. 00:28:55.340 --> 00:28:59.200 You'll end up with an algebraic equation. 00:28:59.200 --> 00:29:03.372 And the algebraic equation actually has eigenvalues. 00:29:03.372 --> 00:29:04.970 AUDIENCE: [INAUDIBLE]. 00:29:04.970 --> 00:29:07.090 J. KIM VANDIVER: And you find the two eigenvalues, 00:29:07.090 --> 00:29:09.260 and they're the two natural frequencies. 00:29:09.260 --> 00:29:09.760 All right. 00:29:09.760 --> 00:29:15.170 I want to do something with angular stuff. 00:29:15.170 --> 00:29:19.130 So here's my system that we were playing 00:29:19.130 --> 00:29:21.210 with at the beginning of class. 00:29:21.210 --> 00:29:27.130 And in this form, it's nice and balanced. 00:29:27.130 --> 00:29:30.790 So this is just to reinforce a couple things 00:29:30.790 --> 00:29:33.540 that I was just beginning to teach you yesterday. 00:29:33.540 --> 00:29:35.120 This is a rigid body. 00:29:35.120 --> 00:29:36.440 It's rotating. 00:29:36.440 --> 00:29:40.430 I attach to it a rotating frame. 00:29:40.430 --> 00:29:44.720 At a in the x direction is y and z. 00:29:44.720 --> 00:29:47.390 So y's into the board. 00:29:47.390 --> 00:29:51.130 And that frame rotates with my system. 00:29:51.130 --> 00:29:53.640 And the rotation rate of this system is omega, 00:29:53.640 --> 00:29:56.090 and it's k in the rotating system, which 00:29:56.090 --> 00:29:59.550 happens to line up with big K with the stationary system. 00:29:59.550 --> 00:30:03.100 But this is my omega vector. 00:30:03.100 --> 00:30:09.300 Here is 0, 0 omega. 00:30:09.300 --> 00:30:11.910 This is the z component of the rotation. 00:30:15.470 --> 00:30:20.820 Now, let's first start with this. 00:30:20.820 --> 00:30:25.960 What is P1, [? an 0, ?] the linear momentum 00:30:25.960 --> 00:30:30.970 of that mass in the system? 00:30:30.970 --> 00:30:32.910 And you've done this probably a lot of times. 00:30:32.910 --> 00:30:34.630 So what direction's it in? 00:30:37.340 --> 00:30:38.852 Momentum is mv, right? 00:30:38.852 --> 00:30:40.310 So what's the velocity of the mass? 00:30:44.090 --> 00:30:45.580 AUDIENCE: It's rotating. 00:30:45.580 --> 00:30:48.748 J. KIM VANDIVER: It's caused by rotation and only rotation. 00:30:48.748 --> 00:30:51.120 AUDIENCE: So it could be the [? c hat ?] direction. 00:30:51.120 --> 00:30:51.620 [INAUDIBLE]. 00:30:51.620 --> 00:30:54.135 J. KIM VANDIVER: No, P, just P. Linear momentum. 00:30:54.135 --> 00:30:56.551 AUDIENCE: That P itself-- linear because it's [INAUDIBLE]. 00:30:59.076 --> 00:31:01.200 J. KIM VANDIVER: And it's got to [? an 0, ?] right? 00:31:01.200 --> 00:31:03.410 I mean, reference to the 0 frame. 00:31:03.410 --> 00:31:09.230 But it can use unit vectors in the A frame. 00:31:09.230 --> 00:31:12.150 So what's v1? 00:31:12.150 --> 00:31:13.470 AUDIENCE: j hat. 00:31:13.470 --> 00:31:16.200 J. KIM VANDIVER: j hat, and how big? 00:31:16.200 --> 00:31:17.587 AUDIENCE: Omega. 00:31:17.587 --> 00:31:19.420 J. KIM VANDIVER: I hear an omega times what? 00:31:23.200 --> 00:31:29.040 So this is system-- I'm going to call this position, 00:31:29.040 --> 00:31:31.080 this is x1z1. 00:31:31.080 --> 00:31:33.350 This coordinate is x1z1. 00:31:33.350 --> 00:31:34.820 So if I give you that information, 00:31:34.820 --> 00:31:36.832 what's the velocity of that point? 00:31:36.832 --> 00:31:37.796 AUDIENCE: Omega x. 00:31:37.796 --> 00:31:38.760 AUDIENCE: x. 00:31:38.760 --> 00:31:42.146 J. KIM VANDIVER: Omega x1-- 00:31:42.146 --> 00:31:43.090 AUDIENCE: Oh, j hat. 00:31:43.090 --> 00:31:44.048 J. KIM VANDIVER: j hat. 00:31:54.020 --> 00:31:57.719 It's into the board, the way it's spinning into the board. 00:31:57.719 --> 00:31:59.260 Yeah, ought to be in the j direction. 00:31:59.260 --> 00:32:01.570 Yeah, it ought to be an r omega, and it 00:32:01.570 --> 00:32:04.650 has an m associated with it. 00:32:04.650 --> 00:32:05.150 What's P2? 00:32:10.640 --> 00:32:12.640 AUDIENCE: In the [INAUDIBLE]? 00:32:12.640 --> 00:32:14.140 AUDIENCE: Negative [INAUDIBLE]. 00:32:14.140 --> 00:32:15.275 J. KIM VANDIVER: It's coming out of the board at you. 00:32:15.275 --> 00:32:16.146 AUDIENCE: Right. 00:32:16.146 --> 00:32:19.122 Negative m2 omega xj. 00:32:28.050 --> 00:32:30.050 J. KIM VANDIVER: Right? 00:32:30.050 --> 00:32:31.190 OK. 00:32:31.190 --> 00:32:33.910 And those little j's-- the j hats are in the rotating 00:32:33.910 --> 00:32:36.570 coordinate system where I want them. 00:32:36.570 --> 00:32:37.430 So what's H1? 00:32:41.380 --> 00:32:47.110 It's r1 with respect to what? 00:32:47.110 --> 00:32:50.880 So you can't do angular momentum without picking a--? 00:32:50.880 --> 00:32:54.010 You've got to pick the point. 00:32:54.010 --> 00:32:57.390 So this is why we choose the coordinates on the reference 00:32:57.390 --> 00:33:00.550 frame that we're going to use to give us some information 00:33:00.550 --> 00:33:01.177 that we want. 00:33:01.177 --> 00:33:03.010 I want to know the torques about this point. 00:33:03.010 --> 00:33:04.750 So I'm going to put my A here. 00:33:04.750 --> 00:33:09.610 I could have put it any place, as long as the axis of rotation 00:33:09.610 --> 00:33:10.860 passed through it. 00:33:10.860 --> 00:33:20.510 So this one is r1 with respect to A cross P1. 00:33:20.510 --> 00:33:24.055 So what is r1 in this system? 00:33:24.055 --> 00:33:24.596 AUDIENCE: x1. 00:33:27.350 --> 00:33:29.040 J. KIM VANDIVER: x1i-- 00:33:29.040 --> 00:33:31.740 AUDIENCE: Plus z-- 00:33:31.740 --> 00:33:41.040 J. KIM VANDIVER: 1k cross with m1 omega x1 j. 00:33:41.040 --> 00:33:42.595 All right? 00:33:42.595 --> 00:33:44.640 Let's work that one out. 00:33:44.640 --> 00:33:48.990 So we have an i times a j. 00:33:48.990 --> 00:34:01.540 And so I get an m1x1 squared omega k. 00:34:01.540 --> 00:34:04.420 And then I do this term times that. 00:34:04.420 --> 00:34:09.969 I get a k cross j minus i. 00:34:09.969 --> 00:34:21.665 And I get m1x1z1 omega i hat. 00:34:21.665 --> 00:34:22.165 All right. 00:34:26.500 --> 00:34:32.679 Technically this is my little h, this is little h2. 00:34:32.679 --> 00:34:42.279 And if I let m1 equal m2 here, my total h for this system-- 00:34:45.897 --> 00:34:47.022 AUDIENCE: Isn't that just-- 00:34:47.022 --> 00:34:48.022 J. KIM VANDIVER: Pardon? 00:34:48.022 --> 00:34:50.293 AUDIENCE: This is [INAUDIBLE]. 00:34:50.293 --> 00:34:50.918 AUDIENCE: Yeah. 00:34:50.918 --> 00:34:51.892 AUDIENCE: The second minus-- 00:34:51.892 --> 00:34:53.360 AUDIENCE: The second minus is still h1. 00:34:53.360 --> 00:34:55.235 J. KIM VANDIVER: Oh, whoops, how'd I do that? 00:34:55.235 --> 00:34:57.350 Yeah, I hadn't moved on to h2 yet. 00:34:57.350 --> 00:34:58.580 Sorry about that. 00:34:58.580 --> 00:35:00.620 This is still h1. 00:35:00.620 --> 00:35:01.900 OK. 00:35:01.900 --> 00:35:02.820 And it's that. 00:35:02.820 --> 00:35:06.230 So what's h2? 00:35:06.230 --> 00:35:09.052 How's it differ? 00:35:09.052 --> 00:35:10.010 Just kind of look here. 00:35:10.010 --> 00:35:12.360 It only differs in one respect. 00:35:12.360 --> 00:35:13.360 AUDIENCE: Negative sign? 00:35:13.360 --> 00:35:18.510 J. KIM VANDIVER: Yeah, you get a minus-- a negative x here 00:35:18.510 --> 00:35:23.400 when you go to carry out the multiplication. 00:35:23.400 --> 00:35:26.560 So now the i cross j term gives you a-- 00:35:26.560 --> 00:35:32.100 AUDIENCE: But isn't the P2 also have a negative under it. 00:35:32.100 --> 00:35:34.640 J. KIM VANDIVER: Yeah, you're absolutely right. 00:35:34.640 --> 00:35:37.800 So you get-- you had to put a minus x here. 00:35:37.800 --> 00:35:40.250 And you have to multiply by this one. 00:35:40.250 --> 00:35:44.870 So the i, j term, you have a minus times a minus. 00:35:44.870 --> 00:35:47.010 i times j is a positive k. 00:35:47.010 --> 00:35:56.090 You end up with m2x1 squared omega k, the same direction 00:35:56.090 --> 00:35:57.240 of each component. 00:35:57.240 --> 00:35:58.642 Then what happens here? 00:35:58.642 --> 00:36:00.350 AUDIENCE: Comes from a minus [INAUDIBLE]. 00:36:09.260 --> 00:36:10.590 J. KIM VANDIVER: OK. 00:36:10.590 --> 00:36:16.060 And so the second term, the i term, is opposite direction. 00:36:16.060 --> 00:36:18.297 And if m1 equals m2 and you added these two together, 00:36:18.297 --> 00:36:19.880 what happened to those last two terms? 00:36:22.710 --> 00:36:24.454 They cancel, right? 00:36:24.454 --> 00:36:27.060 [LAUGHTER] 00:36:27.060 --> 00:36:30.230 OK, so if this is true, the sum of these two, 00:36:30.230 --> 00:36:40.710 H, with respect to A, is just 2mx1 squared omega k. 00:36:40.710 --> 00:36:41.970 All right. 00:36:41.970 --> 00:36:49.380 So if you took-- in the first system, 00:36:49.380 --> 00:36:51.890 if these two weren't equal and they didn't cancel out, 00:36:51.890 --> 00:36:54.640 if you go through here and take dh dt, 00:36:54.640 --> 00:36:59.240 you take the time derivative of these terms, 00:36:59.240 --> 00:37:02.600 then you get an omega dot k. 00:37:02.600 --> 00:37:05.610 And over here you get an omega dot term. 00:37:05.610 --> 00:37:09.640 And then you get a di dt term, all that. 00:37:09.640 --> 00:37:13.310 But you will end up with values of dh dt 00:37:13.310 --> 00:37:14.805 that are not in the k direction. 00:37:14.805 --> 00:37:15.550 You agree? 00:37:15.550 --> 00:37:17.640 For sure. 00:37:17.640 --> 00:37:20.350 And those are associated with the real torques 00:37:20.350 --> 00:37:23.170 in the system that happen. 00:37:23.170 --> 00:37:30.120 When this is true, that difficult term out here 00:37:30.120 --> 00:37:31.790 cancels out. 00:37:31.790 --> 00:37:34.540 And you're left only with the k term. 00:37:34.540 --> 00:37:37.870 And when you take this time derivative, you only get dh dt. 00:37:37.870 --> 00:37:41.920 You get a theta dot or omega dot. 00:37:41.920 --> 00:37:48.160 And that says the torque is in what direction when 00:37:48.160 --> 00:37:49.880 you get just this for h? 00:37:49.880 --> 00:37:52.680 What is the direction of dh dt? 00:37:52.680 --> 00:37:54.680 It's still in k. 00:37:54.680 --> 00:37:58.620 So the torque is aligned with the spin. 00:37:58.620 --> 00:38:01.830 h is aligned with the spin. 00:38:01.830 --> 00:38:02.720 They're all aligned. 00:38:02.720 --> 00:38:06.990 And the system is actually beautifully balanced. 00:38:06.990 --> 00:38:11.660 You don't feel any torques around where you're holding it 00:38:11.660 --> 00:38:19.340 down here at A. But if I took one of these off, kind of back 00:38:19.340 --> 00:38:22.410 to that one arm system, it's terribly unbalanced. 00:38:25.110 --> 00:38:31.870 So the moral of the story here is 00:38:31.870 --> 00:38:35.960 if I don't want to have these unwanted torques, 00:38:35.960 --> 00:38:38.560 what can you say about the desired mass 00:38:38.560 --> 00:38:40.355 distribution in the system? 00:38:40.355 --> 00:38:41.690 AUDIENCE: [INAUDIBLE]. 00:38:41.690 --> 00:38:45.000 J. KIM VANDIVER: I hear a vote for symmetry. 00:38:45.000 --> 00:38:47.310 And that's the general rule. 00:38:47.310 --> 00:38:49.870 If your masses are distributed symmetrically 00:38:49.870 --> 00:38:58.219 about the axis of spin, no off axis torques. 00:38:58.219 --> 00:39:00.219 AUDIENCE: And it's [INAUDIBLE] mass [INAUDIBLE]. 00:39:00.219 --> 00:39:01.802 J. KIM VANDIVER: Yeah, it's symmetric. 00:39:01.802 --> 00:39:03.910 Symmetry means they better be the same size too, 00:39:03.910 --> 00:39:05.070 not just in the same place. 00:39:05.070 --> 00:39:08.190 If one's twice as big as the other, no go. 00:39:08.190 --> 00:39:10.420 If I put a second mass on one of those arms, 00:39:10.420 --> 00:39:13.360 the system is back to being unbalanced. 00:39:13.360 --> 00:39:21.810 So any time this H vector, the total H of the system, 00:39:21.810 --> 00:39:26.110 is not lined up with the spin, the system 00:39:26.110 --> 00:39:28.780 has an asymmetry in it. 00:39:31.480 --> 00:39:36.106 And the system will require additional torques just 00:39:36.106 --> 00:39:37.730 to hold it in place when it's spinning. 00:39:40.650 --> 00:39:43.000 So when you have these additional torques that 00:39:43.000 --> 00:39:47.670 come from being a symmetric, the system 00:39:47.670 --> 00:39:53.450 is said to be dynamically imbalanced. 00:39:53.450 --> 00:39:56.880 So if you pick up a stone, a big stone, in your car wheel 00:39:56.880 --> 00:40:00.160 or block hunk of mud or ice that's frozen on the rim, 00:40:00.160 --> 00:40:03.280 you're going down the road, what's it feel like? 00:40:03.280 --> 00:40:05.910 You ever had an unbalanced tire on your car? 00:40:05.910 --> 00:40:07.760 Boy, where have you guys lived? 00:40:10.910 --> 00:40:14.400 An unbalanced tire on a car, you're going down the road. 00:40:14.400 --> 00:40:16.140 You know? 00:40:16.140 --> 00:40:17.200 Right? 00:40:17.200 --> 00:40:18.540 That's what this is all about. 00:40:18.540 --> 00:40:21.423 You have a case of imbalance. 00:40:24.549 --> 00:40:25.590 How are we doing on time? 00:40:25.590 --> 00:40:27.730 A couple more minutes. 00:40:27.730 --> 00:40:32.240 So imbalance comes from having angular momentum that's 00:40:32.240 --> 00:40:35.777 not pointed in same direction as-- angular momentum not 00:40:35.777 --> 00:40:37.860 pointed in the same direction as the angle of spin 00:40:37.860 --> 00:40:40.710 is evidence of unbalance. 00:40:40.710 --> 00:40:44.650 And you can calculate how bad the unbalance is by doing dh dt 00:40:44.650 --> 00:40:48.150 and actually finding out how much torque is being applied 00:40:48.150 --> 00:40:51.210 to the system that ordinarily the bearings wouldn't 00:40:51.210 --> 00:40:52.790 have to resist. 00:40:52.790 --> 00:40:54.810 But this thing has overturning torques 00:40:54.810 --> 00:40:58.990 that are trying to make it wobble back and forth on you. 00:40:58.990 --> 00:41:04.766 So one last exercise, which is actually 00:41:04.766 --> 00:41:06.810 a quite important point. 00:41:06.810 --> 00:41:12.330 If I moved A up here right on the line between these two 00:41:12.330 --> 00:41:17.780 things-- so this is A. And now this is x, and this is z. 00:41:17.780 --> 00:41:19.950 The coordinates of this point become what? 00:41:22.760 --> 00:41:24.090 x1 and-- 00:41:24.090 --> 00:41:24.850 AUDIENCE: 0. 00:41:24.850 --> 00:41:26.310 J. KIM VANDIVER: 0. 00:41:26.310 --> 00:41:29.760 And all these equations still apply. 00:41:29.760 --> 00:41:34.060 So if I just move my coordinate system so the x's are the same, 00:41:34.060 --> 00:41:35.850 z's go to 0. 00:41:35.850 --> 00:41:38.740 What do you end up with for terms? 00:41:38.740 --> 00:41:41.440 What happens to this thing? 00:41:41.440 --> 00:41:42.320 That goes to 0. 00:41:42.320 --> 00:41:43.270 It goes away. 00:41:43.270 --> 00:41:49.060 And you only get this term, and so by simply moving A to here, 00:41:49.060 --> 00:41:52.780 my angular momentum vector now lines up with the spin. 00:41:55.708 --> 00:41:56.970 Kind of weird. 00:41:56.970 --> 00:42:01.660 Does the imbalance still exist? 00:42:01.660 --> 00:42:03.751 Will your car still be doing this down the road 00:42:03.751 --> 00:42:05.250 just because you chose to look at it 00:42:05.250 --> 00:42:08.430 from a different point of view? 00:42:08.430 --> 00:42:08.930 Yeah. 00:42:08.930 --> 00:42:11.960 I mean, the car's still unhappy. 00:42:11.960 --> 00:42:16.109 So there's kind of inconsistency here, sort of. 00:42:16.109 --> 00:42:16.900 What's the problem? 00:42:20.610 --> 00:42:25.730 Or maybe not a problem, but you choose 00:42:25.730 --> 00:42:29.900 to put A when you're computing angular momentum. 00:42:29.900 --> 00:42:32.990 You can choose where you put this point to give you 00:42:32.990 --> 00:42:35.780 the information you're after. 00:42:35.780 --> 00:42:38.190 And in this case, if I were designing this system 00:42:38.190 --> 00:42:42.160 and I wanted to know the bending moment in these bars sticking 00:42:42.160 --> 00:42:44.580 down here, I put A here because it'll 00:42:44.580 --> 00:42:47.590 give you the torques with respect to this point. 00:42:47.590 --> 00:42:50.370 These torques down here still exist. 00:42:50.370 --> 00:42:53.790 If you put your point at which you compute angular momentum up 00:42:53.790 --> 00:42:56.110 here, you won't be able to find them. 00:42:56.110 --> 00:42:58.470 You can't compute it because you've reduced the moment 00:42:58.470 --> 00:43:03.980 arm to 0, this moment arm, and you just won't get that torque. 00:43:03.980 --> 00:43:06.580 OK. 00:43:06.580 --> 00:43:08.770 All right, I've run out of time. 00:43:08.770 --> 00:43:12.031 But there'll be more on this subject.