1 00:00:00,100 --> 00:00:02,450 The following content is provided under a Creative 2 00:00:02,450 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,070 Your support will help MIT OpenCourseWare 4 00:00:06,070 --> 00:00:10,170 continue to offer high quality educational resources for free. 5 00:00:10,170 --> 00:00:12,710 To make a donation or to view additional materials 6 00:00:12,710 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,325 at ocw.mit.edu. 8 00:00:20,372 --> 00:00:22,580 J. KIM VANDIVER: We're waiting for people to come in. 9 00:00:22,580 --> 00:00:24,496 And what are the important concepts this week? 10 00:00:24,496 --> 00:00:27,270 There's only the one lecture, and the lecture yesterday 11 00:00:27,270 --> 00:00:32,180 was pretty dense in concepts. 12 00:00:32,180 --> 00:00:34,290 Let's build a quick list here. 13 00:00:36,970 --> 00:00:39,130 Who's got the first offering? 14 00:00:39,130 --> 00:00:41,371 AUDIENCE: Rotational dynamics of rigid bodies. 15 00:00:41,371 --> 00:00:43,620 J. KIM VANDIVER: OK, that was a general overall topic, 16 00:00:43,620 --> 00:00:47,095 so how about a concept though. 17 00:00:47,095 --> 00:00:49,350 AUDIENCE: The mass moment of inertia matrix. 18 00:00:49,350 --> 00:00:53,630 J. KIM VANDIVER: OK, yep, we introduced the concept 19 00:00:53,630 --> 00:00:55,984 of the mass moment of inertia matrix. 20 00:00:55,984 --> 00:00:58,150 And I'll give it a context, so you can, for example, 21 00:00:58,150 --> 00:01:04,310 write H as I times omega. 22 00:01:04,310 --> 00:01:06,930 So we introduced that, didn't get very far with it. 23 00:01:06,930 --> 00:01:09,468 Another one from yesterday? 24 00:01:09,468 --> 00:01:10,926 AUDIENCE: I'm not sure if this fits 25 00:01:10,926 --> 00:01:12,960 but talking about how the body can 26 00:01:12,960 --> 00:01:15,210 be rotating around the point opposite the [INAUDIBLE]. 27 00:01:15,210 --> 00:01:16,251 J. KIM VANDIVER: Ah yeah. 28 00:01:16,251 --> 00:01:21,170 Yep So we talked about rotation. 29 00:01:29,590 --> 00:01:36,850 The subject yesterday was really rigid body rotation 30 00:01:36,850 --> 00:01:47,420 about fixed point on or off the body. 31 00:01:53,580 --> 00:01:55,550 So here's our axis of rotation. 32 00:01:55,550 --> 00:01:59,800 Call it A. And it's spinning about that point at some omega. 33 00:01:59,800 --> 00:02:01,800 But that point A could be over here too. 34 00:02:01,800 --> 00:02:04,100 And then the whole body is just going, 35 00:02:04,100 --> 00:02:07,150 so an example of rigid body rotation 36 00:02:07,150 --> 00:02:10,830 about a point not on the body would be something doing that. 37 00:02:15,010 --> 00:02:17,670 OK, something else. 38 00:02:29,124 --> 00:02:32,996 AUDIENCE: The angler velocity [INAUDIBLE]. 39 00:02:32,996 --> 00:02:33,870 J. KIM VANDIVER: Yes. 40 00:02:33,870 --> 00:02:44,990 Yeah, what does it means when H and omega may not align? 41 00:02:49,470 --> 00:02:51,660 What's that mean? 42 00:02:51,660 --> 00:02:53,215 So we got into that a bit yesterday. 43 00:02:56,150 --> 00:03:00,880 And anything else that struck you? 44 00:03:06,700 --> 00:03:10,560 OK, well that's a good start. 45 00:03:10,560 --> 00:03:11,920 Let's just work with that. 46 00:03:11,920 --> 00:03:14,900 Now I will do that second. 47 00:03:14,900 --> 00:03:16,700 We're going to come work a little bit, 48 00:03:16,700 --> 00:03:20,950 talking about this system in those contexts. 49 00:03:20,950 --> 00:03:25,220 Before I do that, I want to-- the staff 50 00:03:25,220 --> 00:03:28,360 decided we hadn't done something in lecture yet very carefully 51 00:03:28,360 --> 00:03:32,430 so we wanted to do it once carefully within recitation. 52 00:03:32,430 --> 00:03:38,050 And that's a methodical way of doing free body diagrams. 53 00:03:42,900 --> 00:03:48,770 So we have a system, two masses, springs, dashpots. 54 00:03:48,770 --> 00:03:50,940 In the last class, there were a few people 55 00:03:50,940 --> 00:03:55,820 who didn't know really what we meant by k's and c's and how 56 00:03:55,820 --> 00:03:56,360 they work. 57 00:03:56,360 --> 00:03:59,900 So I will give you a quick definition. 58 00:03:59,900 --> 00:04:02,420 So here we have a spring. 59 00:04:02,420 --> 00:04:06,090 And these are called linear springs, so Hooke's law. 60 00:04:06,090 --> 00:04:09,420 If you cause that spring to move, 61 00:04:09,420 --> 00:04:12,220 and I'll call it here the plus x direction-- you grab it 62 00:04:12,220 --> 00:04:15,530 and you pull on it-- you have to apply a force. 63 00:04:15,530 --> 00:04:19,560 So the force to pull that spring is positive kx 64 00:04:19,560 --> 00:04:21,589 in the same direction as x. 65 00:04:21,589 --> 00:04:24,960 But if that spring's attached to a mass what 66 00:04:24,960 --> 00:04:29,790 direction is the force that the spring applies on the mass? 67 00:04:29,790 --> 00:04:32,910 So I've given you a hint here. 68 00:04:32,910 --> 00:04:34,000 It pulls back, right? 69 00:04:34,000 --> 00:04:36,200 Because this is Newton's third law. 70 00:04:36,200 --> 00:04:37,860 So we are doing free body diagrams, 71 00:04:37,860 --> 00:04:42,000 and we're concerned with forces on the rigid bodies. 72 00:04:42,000 --> 00:04:46,390 So, this spring, if this mass went that way, 73 00:04:46,390 --> 00:04:48,240 the spring would pull this way. 74 00:04:48,240 --> 00:04:51,930 And we indicated the direction with an arrow and its value 75 00:04:51,930 --> 00:04:53,220 as a kx. 76 00:04:53,220 --> 00:04:55,120 Dashpots are similar. 77 00:04:55,120 --> 00:04:59,400 Except dashpots are activated by velocity. 78 00:04:59,400 --> 00:05:01,940 So this is a linear dashpot, such that it 79 00:05:01,940 --> 00:05:04,830 has a dashpot constant c. 80 00:05:04,830 --> 00:05:10,000 And if I give this a positive value of velocity, 81 00:05:10,000 --> 00:05:14,966 the force required to pull that dashpot is cx dot. 82 00:05:14,966 --> 00:05:16,310 It's linearly proportional. 83 00:05:16,310 --> 00:05:20,160 Velocity and the force that if it were attached to a mass 84 00:05:20,160 --> 00:05:22,180 that the dashpot puts on the mass 85 00:05:22,180 --> 00:05:26,790 would be opposite direction but same value, 86 00:05:26,790 --> 00:05:29,450 cx dot, but is resisting the motion. 87 00:05:29,450 --> 00:05:30,490 OK. 88 00:05:30,490 --> 00:05:34,100 So I'm going to just give you general method 89 00:05:34,100 --> 00:05:35,430 for doing free body diagrams. 90 00:05:35,430 --> 00:05:37,220 And the reason we choose this problem is 91 00:05:37,220 --> 00:05:42,150 there's two bodies, and depending on which 92 00:05:42,150 --> 00:05:45,210 is doing what, this intermediate spring in here 93 00:05:45,210 --> 00:05:47,185 particularly can push or pull on either body 94 00:05:47,185 --> 00:05:52,170 and having a method so you don't mess up the signs 95 00:05:52,170 --> 00:05:55,100 is what this is all about. 96 00:05:55,100 --> 00:06:02,680 So let's just start by building a free body 97 00:06:02,680 --> 00:06:05,770 diagram of mass one. 98 00:06:11,250 --> 00:06:13,920 And the assignment I'm going to give you to work on, 99 00:06:13,920 --> 00:06:15,730 just seat work here for a minute, 100 00:06:15,730 --> 00:06:18,780 is only consider the springs for now. 101 00:06:18,780 --> 00:06:21,800 We'll just deal with the springs. 102 00:06:21,800 --> 00:06:23,500 So assume the system is moving. 103 00:06:23,500 --> 00:06:25,700 You have to do free body diagrams of it. 104 00:06:25,700 --> 00:06:30,830 Put the spring forces on the mass in your free body 105 00:06:30,830 --> 00:06:31,330 diagrams. 106 00:06:31,330 --> 00:06:32,880 So spend a minute or so. 107 00:06:32,880 --> 00:06:35,340 Talk to your neighbors after you get a little ways along 108 00:06:35,340 --> 00:06:37,890 with it. 109 00:06:37,890 --> 00:06:40,330 And I'm going to add-- there's also 110 00:06:40,330 --> 00:06:44,510 on this whole system external force F1, external force F2. 111 00:06:44,510 --> 00:06:47,320 That's your excitation. 112 00:06:47,320 --> 00:06:49,550 But don't worry about that for now. 113 00:06:49,550 --> 00:06:53,830 Just figure out the spring forces, and I'll be quiet. 114 00:06:53,830 --> 00:06:57,180 I missed a step, my apologies. 115 00:06:57,180 --> 00:06:59,610 We should do something else first. 116 00:06:59,610 --> 00:07:03,400 The very first step is to assign the coordinates. 117 00:07:03,400 --> 00:07:06,410 Let's all agree on the coordinate system here. 118 00:07:06,410 --> 00:07:09,685 So the question really is the first thing 119 00:07:09,685 --> 00:07:12,060 you have to do is you have to decide how many independent 120 00:07:12,060 --> 00:07:15,000 coordinates you need to completely describe the motion. 121 00:07:15,000 --> 00:07:16,500 So how many coordinates do you think 122 00:07:16,500 --> 00:07:19,320 are needed for this problem? 123 00:07:19,320 --> 00:07:22,410 I see a one. 124 00:07:22,410 --> 00:07:23,770 AUDIENCE: I was just-- 125 00:07:23,770 --> 00:07:25,560 J. KIM VANDIVER: That's a second one. 126 00:07:25,560 --> 00:07:27,120 Anybody else have a suggestion? 127 00:07:27,120 --> 00:07:29,784 So where would you put the one? 128 00:07:29,784 --> 00:07:31,780 AUDIENCE: The x term. 129 00:07:31,780 --> 00:07:36,556 J. KIM VANDIVER: And measuring the displacement of mass one? 130 00:07:36,556 --> 00:07:37,180 AUDIENCE: Sure. 131 00:07:37,180 --> 00:07:38,150 J. KIM VANDIVER: OK. 132 00:07:38,150 --> 00:07:39,460 And is that all we need? 133 00:07:42,270 --> 00:07:43,529 Anybody have a further-- 134 00:07:43,529 --> 00:07:45,570 AUDIENCE: Oh, there's a displacement at mass two. 135 00:07:45,570 --> 00:07:47,486 J. KIM VANDIVER: And it's independence, right? 136 00:07:47,486 --> 00:07:49,270 They can move-- how many coordinates now 137 00:07:49,270 --> 00:07:50,450 do you think you need? 138 00:07:50,450 --> 00:07:50,990 Uh oh. 139 00:07:50,990 --> 00:07:54,200 She changed her mind to two. 140 00:07:54,200 --> 00:07:57,040 And I've got them labeled up here for you already. 141 00:07:57,040 --> 00:07:58,495 So it's going to take two. 142 00:07:58,495 --> 00:08:00,120 And one of the ways you can test if you 143 00:08:00,120 --> 00:08:03,590 have the right number, if you think it's one, assign it. 144 00:08:03,590 --> 00:08:05,000 Freeze it. 145 00:08:05,000 --> 00:08:06,660 Is there still independent movement 146 00:08:06,660 --> 00:08:07,780 of a part of the system? 147 00:08:07,780 --> 00:08:12,140 If there is, you have yet another degree of freedom. 148 00:08:12,140 --> 00:08:14,780 If we freeze both x1 and x2, and can anything 149 00:08:14,780 --> 00:08:15,680 move in the system? 150 00:08:15,680 --> 00:08:16,180 Nope. 151 00:08:16,180 --> 00:08:17,640 So we've got it nailed down. 152 00:08:17,640 --> 00:08:19,310 All right. 153 00:08:19,310 --> 00:08:23,730 But where do you measure? 154 00:08:23,730 --> 00:08:26,420 There is something else you have to determine. 155 00:08:26,420 --> 00:08:30,270 And that is, where do you make zero? 156 00:08:30,270 --> 00:08:31,610 You also have to know that. 157 00:08:31,610 --> 00:08:35,049 So where is x is 0 in your picture? 158 00:08:35,049 --> 00:08:37,860 We'll just talk about both of them, x1 and x2. 159 00:08:37,860 --> 00:08:39,730 How would you pick the 0 point? 160 00:08:42,970 --> 00:08:43,682 Christina? 161 00:08:43,682 --> 00:08:45,182 AUDIENCE: Probably when the spring's 162 00:08:45,182 --> 00:08:48,659 at equilibrium so when it's not exerting force. [INAUDIBLE]. 163 00:08:48,659 --> 00:08:50,950 J. KIM VANDIVER: OK, she says when there's not exerting 164 00:08:50,950 --> 00:08:52,033 force in either direction. 165 00:08:52,033 --> 00:08:54,815 I would call that the static equilibrium position. 166 00:08:59,430 --> 00:09:02,270 There is a conceptual problem with that. 167 00:09:02,270 --> 00:09:04,550 We'll assume that that will work if you've 168 00:09:04,550 --> 00:09:05,990 set the system up so that there's 169 00:09:05,990 --> 00:09:07,280 no forces in the springs. 170 00:09:07,280 --> 00:09:10,460 But you know, if I move this wall in a little bit 171 00:09:10,460 --> 00:09:12,910 and let it reach static equilibrium, 172 00:09:12,910 --> 00:09:14,940 it has a static equilibrium position. 173 00:09:14,940 --> 00:09:17,210 All the springs have forces in them. 174 00:09:17,210 --> 00:09:19,810 Now, you pre-compress them. 175 00:09:19,810 --> 00:09:21,894 So there's kind of a problem-- you 176 00:09:21,894 --> 00:09:23,810 could run into a conceptual problem with that. 177 00:09:23,810 --> 00:09:25,130 It will still work. 178 00:09:25,130 --> 00:09:26,760 Picking a static equilibrium position 179 00:09:26,760 --> 00:09:29,331 is a good thing to do usually. 180 00:09:29,331 --> 00:09:29,830 OK. 181 00:09:33,050 --> 00:09:35,340 There are other positions. 182 00:09:35,340 --> 00:09:39,740 For example, this system, it's a simpler system, just 183 00:09:39,740 --> 00:09:41,620 an ordinary little mass spring. 184 00:09:41,620 --> 00:09:45,870 But where do you make zero when you pick your coordinate? 185 00:09:45,870 --> 00:09:47,870 AUDIENCE: In this case, you could zero it out 186 00:09:47,870 --> 00:09:52,555 at the rest length of the spring plus the length 187 00:09:52,555 --> 00:09:54,930 of this spring as a result of the force of gravity. 188 00:09:54,930 --> 00:09:56,430 J. KIM VANDIVER: So right now you're 189 00:09:56,430 --> 00:09:58,290 seeing the static equilibrium position. 190 00:09:58,290 --> 00:10:01,750 The spring has tension in it, the weight of the mass, right? 191 00:10:01,750 --> 00:10:04,120 So you could either make x right here. 192 00:10:04,120 --> 00:10:06,564 Or the other obvious choice would be? 193 00:10:06,564 --> 00:10:07,980 AUDIENCE: Where you're holding it. 194 00:10:07,980 --> 00:10:10,656 AUDIENCE: You're holding it. 195 00:10:10,656 --> 00:10:12,280 J. KIM VANDIVER: I heard a couple voice 196 00:10:12,280 --> 00:10:13,155 but I couldn't hear what you said. 197 00:10:13,155 --> 00:10:14,730 AUDIENCE: Where you're holding it? 198 00:10:14,730 --> 00:10:17,440 J. KIM VANDIVER: Well, yeah, but it turns out 199 00:10:17,440 --> 00:10:18,899 that actually is a terrible choice. 200 00:10:18,899 --> 00:10:20,440 Because then you have to get involved 201 00:10:20,440 --> 00:10:21,729 with the length of the spring. 202 00:10:21,729 --> 00:10:23,270 And the length of the spring actually 203 00:10:23,270 --> 00:10:25,870 really doesn't enter into the equation in motion 204 00:10:25,870 --> 00:10:28,230 unless you're really silly about where 205 00:10:28,230 --> 00:10:30,976 you pick the measure from. 206 00:10:30,976 --> 00:10:32,850 Static equilibrium is good, but the other one 207 00:10:32,850 --> 00:10:35,800 is the zero spring force position 208 00:10:35,800 --> 00:10:37,150 is the other natural one to do. 209 00:10:37,150 --> 00:10:41,560 So at zero spring force, that's no extension of the spring. 210 00:10:41,560 --> 00:10:44,400 Because this is preloaded when you do this, and have to figure 211 00:10:44,400 --> 00:10:46,250 out what that does to you. 212 00:10:46,250 --> 00:10:47,860 OK. 213 00:10:47,860 --> 00:10:52,093 Again, static equilibrium has some real advantages here, 214 00:10:52,093 --> 00:10:53,426 but we'll talk about that later. 215 00:10:53,426 --> 00:10:54,120 Yeah? 216 00:10:54,120 --> 00:10:56,526 AUDIENCE: So why does zero spring force 217 00:10:56,526 --> 00:10:57,847 work there but not-- 218 00:10:57,847 --> 00:11:00,430 J. KIM VANDIVER: I'm just saying there are actually situations 219 00:11:00,430 --> 00:11:02,320 where there is no zero force. 220 00:11:02,320 --> 00:11:03,860 AUDIENCE: Oh, OK. 221 00:11:03,860 --> 00:11:06,930 J. KIM VANDIVER: And you have a conception-- if you hadn't ever 222 00:11:06,930 --> 00:11:11,300 thought of that before, that could give you pause when 223 00:11:11,300 --> 00:11:13,090 you go to solve the problem. 224 00:11:13,090 --> 00:11:14,210 The springs are preloaded. 225 00:11:17,380 --> 00:11:21,380 Are your equations of motion valid? 226 00:11:21,380 --> 00:11:24,880 So you preloaded this a little bit, 227 00:11:24,880 --> 00:11:29,302 squeezed it in, this system has natural frequencies, right? 228 00:11:29,302 --> 00:11:30,260 Do you agree with that? 229 00:11:30,260 --> 00:11:35,400 It's very similar to this system except it's just now-- 230 00:11:35,400 --> 00:11:37,500 it doesn't happen to have a third spring 231 00:11:37,500 --> 00:11:38,627 and have gravity involved. 232 00:11:38,627 --> 00:11:40,460 But this system has two natural frequencies, 233 00:11:40,460 --> 00:11:44,660 that one and one that's a little harder for me 234 00:11:44,660 --> 00:11:49,440 to do but that one. 235 00:11:49,440 --> 00:11:51,730 So if I've preloaded this a little bit, 236 00:11:51,730 --> 00:11:57,000 squeezed it in some, would the natural frequencies change? 237 00:11:57,000 --> 00:11:58,800 Yes-- how many think yes? 238 00:11:58,800 --> 00:11:59,550 Get your hands up. 239 00:11:59,550 --> 00:12:00,050 Come on. 240 00:12:00,050 --> 00:12:01,310 You're gamblers. 241 00:12:01,310 --> 00:12:02,590 How many think yes? 242 00:12:02,590 --> 00:12:04,560 How many think no? 243 00:12:04,560 --> 00:12:06,790 How many just don't want to raise their hand? 244 00:12:06,790 --> 00:12:09,670 [LAUGHS] Like you. 245 00:12:09,670 --> 00:12:10,460 OK. 246 00:12:10,460 --> 00:12:13,980 The natural frequency doesn't change. 247 00:12:13,980 --> 00:12:16,060 It turns out to preload doesn't matter 248 00:12:16,060 --> 00:12:18,350 in a linear system like this. 249 00:12:18,350 --> 00:12:20,360 So there's some nuances in here. 250 00:12:20,360 --> 00:12:23,810 A simple system like this has some little traps in it. 251 00:12:23,810 --> 00:12:26,590 It will take them time to learn your way through. 252 00:12:26,590 --> 00:12:28,760 But let's keep it simple. 253 00:12:28,760 --> 00:12:33,292 No preload, the static equilibrium position, 254 00:12:33,292 --> 00:12:36,640 draw the free body diagram for mass one, springs only. 255 00:12:36,640 --> 00:12:38,542 Only deal with the springs. 256 00:12:38,542 --> 00:12:40,500 Take a couple minutes and talk to your neighbor 257 00:12:40,500 --> 00:12:43,180 if you need to. 258 00:12:43,180 --> 00:12:48,620 OK, so I won't come look at each of your things. 259 00:12:48,620 --> 00:12:51,020 I want to have different groups kind of-- they 260 00:12:51,020 --> 00:12:52,440 should take one force at a time. 261 00:12:52,440 --> 00:12:57,480 So you guys, give me a force on this mass, due to a spring. 262 00:12:57,480 --> 00:13:01,094 Give me a spring force and tell me what direction it's in. 263 00:13:01,094 --> 00:13:02,864 AUDIENCE: I guess it's [INAUDIBLE]. 264 00:13:02,864 --> 00:13:04,780 J. KIM VANDIVER: So tell me which spring we're 265 00:13:04,780 --> 00:13:05,868 talking about. 266 00:13:05,868 --> 00:13:07,850 AUDIENCE: So [? I didn't hear. ?] So is this 267 00:13:07,850 --> 00:13:10,140 being impressed, like are these springs? 268 00:13:10,140 --> 00:13:11,840 J. KIM VANDIVER: No, this is static equilibrium position. 269 00:13:11,840 --> 00:13:14,090 But now if you give it initial [INAUDIBLE] and let go, 270 00:13:14,090 --> 00:13:15,900 it's going to sit there and do something. 271 00:13:15,900 --> 00:13:17,950 We're trying to derive the equations of motion. 272 00:13:17,950 --> 00:13:21,170 And to do so, we have to start with a free body diagram. 273 00:13:21,170 --> 00:13:22,990 So it's moving, and there are forces 274 00:13:22,990 --> 00:13:25,400 on it caused by springs and by dashpots 275 00:13:25,400 --> 00:13:27,684 and by the external forces. 276 00:13:27,684 --> 00:13:30,100 So right now, we're going to be free body diagram but only 277 00:13:30,100 --> 00:13:32,050 the components through the spring. 278 00:13:32,050 --> 00:13:33,900 Springs. 279 00:13:33,900 --> 00:13:36,870 So tell me what happens. 280 00:13:36,870 --> 00:13:42,150 What does that spring-- how does this spring appear 281 00:13:42,150 --> 00:13:44,661 on that free body diagram. 282 00:13:44,661 --> 00:13:47,487 AUDIENCE: [? I have an idea. ?] [INAUDIBLE] it's air force 283 00:13:47,487 --> 00:13:49,380 is 11 [INAUDIBLE]. 284 00:13:49,380 --> 00:13:50,976 J. KIM VANDIVER: OK, force of. 285 00:13:50,976 --> 00:13:52,760 AUDIENCE: Force of the spring on the-- 286 00:13:52,760 --> 00:13:54,988 J. KIM VANDIVER: Which direction? 287 00:13:54,988 --> 00:13:58,429 AUDIENCE: Left. [LAUGHS] I don't understand how you know which-- 288 00:13:58,429 --> 00:13:59,970 J. KIM VANDIVER: OK, well that's kind 289 00:13:59,970 --> 00:14:01,178 of the point of the exercise. 290 00:14:01,178 --> 00:14:04,020 And I expect many of you to have some confusion about what 291 00:14:04,020 --> 00:14:07,420 direction to go because you don't have a standard method 292 00:14:07,420 --> 00:14:08,580 by which you approach this. 293 00:14:08,580 --> 00:14:11,664 AUDIENCE: So both springs and cause of force 294 00:14:11,664 --> 00:14:14,580 goes in the direction of the displacement. 295 00:14:14,580 --> 00:14:17,140 J. KIM VANDIVER: OK, so what's the displacement, though? 296 00:14:17,140 --> 00:14:20,184 This system has two possible displacements. 297 00:14:20,184 --> 00:14:21,630 AUDIENCE: [INAUDIBLE]. 298 00:14:21,630 --> 00:14:26,487 We're assuming m1 is going in the positive x1 [INAUDIBLE]. 299 00:14:26,487 --> 00:14:27,320 J. KIM VANDIVER: OK. 300 00:14:27,320 --> 00:14:30,560 So if you said if you move it a positive x1, 301 00:14:30,560 --> 00:14:33,360 then you will get a spring force. 302 00:14:33,360 --> 00:14:34,679 Spring one does what? 303 00:14:34,679 --> 00:14:36,470 AUDIENCE: It would oppose the [? spring. ?] 304 00:14:36,470 --> 00:14:38,790 J. KIM VANDIVER: Opposes, and what's it value? 305 00:14:38,790 --> 00:14:39,764 AUDIENCE: k1x. 306 00:14:39,764 --> 00:14:40,680 J. KIM VANDIVER: k1x1. 307 00:14:43,570 --> 00:14:48,130 So that's what a motion x1 causes-- 308 00:14:48,130 --> 00:14:49,980 that's a result with spring one. 309 00:14:49,980 --> 00:14:52,920 What's the result of spring two if you have motion x1? 310 00:14:56,310 --> 00:14:56,970 AUDIENCE: Same. 311 00:14:56,970 --> 00:14:58,803 There's going to be compression force that's 312 00:14:58,803 --> 00:15:00,660 also opposing displacement. 313 00:15:00,660 --> 00:15:01,390 J. KIM VANDIVER: Right, because it's 314 00:15:01,390 --> 00:15:02,848 trying to squeeze that spring down, 315 00:15:02,848 --> 00:15:04,130 and it's pushing back, right? 316 00:15:04,130 --> 00:15:11,000 But you're also going to get a spring k2x1, right? 317 00:15:11,000 --> 00:15:13,230 Now are there any other forces that 318 00:15:13,230 --> 00:15:17,820 result from the motion of that body in the springs? 319 00:15:17,820 --> 00:15:18,882 Just spring forces now. 320 00:15:18,882 --> 00:15:20,340 AUDIENCE: From that body by itself? 321 00:15:20,340 --> 00:15:22,350 J. KIM VANDIVER: Yeah, that body by itself. 322 00:15:22,350 --> 00:15:22,970 No. 323 00:15:22,970 --> 00:15:25,520 But is that all the spring forces? 324 00:15:25,520 --> 00:15:26,600 No. 325 00:15:26,600 --> 00:15:31,552 So now what if you let body two move? 326 00:15:31,552 --> 00:15:34,010 And now I'm going to give you a little rubric, a little way 327 00:15:34,010 --> 00:15:34,718 to go about this. 328 00:15:34,718 --> 00:15:36,610 We start by assigning our coordinates. 329 00:15:36,610 --> 00:15:58,830 Then you specify positive x1, x1 dot, x2, x2 dot, one at a time. 330 00:16:05,920 --> 00:16:16,040 And from that, you deduce the direction of love 331 00:16:16,040 --> 00:16:20,080 the forces, of the resulting forces. 332 00:16:20,080 --> 00:16:20,740 OK. 333 00:16:20,740 --> 00:16:25,590 So if what we've done there is consistent 334 00:16:25,590 --> 00:16:30,970 with it, a positive value of x1, spring one pulls back k1x1, 335 00:16:30,970 --> 00:16:34,650 spring two pushes back k1x1, then 336 00:16:34,650 --> 00:16:37,710 that's the end of spring forces due to x1. 337 00:16:37,710 --> 00:16:41,390 So now just let's say, OK, let x1 be 0 for a moment. 338 00:16:41,390 --> 00:16:44,240 And now let there be a positive x2. 339 00:16:44,240 --> 00:16:47,670 Does that cause any spring force on mass one? 340 00:16:52,143 --> 00:16:53,982 What do you think? 341 00:16:53,982 --> 00:16:57,250 AUDIENCE: We're going to pull mass one [INAUDIBLE]. 342 00:16:57,250 --> 00:16:59,350 J. KIM VANDIVER: That one-- a positive x2 343 00:16:59,350 --> 00:17:02,379 puts tension in the spring and it pulls it in that direction. 344 00:17:02,379 --> 00:17:03,795 So now you're going to get a k2x2. 345 00:17:07,599 --> 00:17:11,500 Any other spring forces caused by motion of two? 346 00:17:14,450 --> 00:17:16,810 It's the only connecting spring. 347 00:17:16,810 --> 00:17:19,350 So those are the spring forces on mass one. 348 00:17:22,810 --> 00:17:25,770 Now let's move on to dashpot forces. 349 00:17:25,770 --> 00:17:27,859 And you do exactly the same thing. 350 00:17:27,859 --> 00:17:30,920 Assume a positive value of x1 dot. 351 00:17:30,920 --> 00:17:33,630 What does it cause in dashpot forces? 352 00:17:33,630 --> 00:17:39,150 What's the first dashpot do when you pull positive x1 dot 353 00:17:39,150 --> 00:17:41,050 in that direction? 354 00:17:41,050 --> 00:17:42,956 Resist or not? 355 00:17:42,956 --> 00:17:44,210 AUDIENCE: Resist. 356 00:17:44,210 --> 00:17:46,200 J. KIM VANDIVER: Dashpots resist. 357 00:17:46,200 --> 00:17:49,960 So it's going to be in the minus direction and a value-- how 358 00:17:49,960 --> 00:17:51,480 big? 359 00:17:51,480 --> 00:17:53,260 AUDIENCE: c1 and [INAUDIBLE]. 360 00:17:53,260 --> 00:17:58,750 J. KIM VANDIVER: So now you get dashpot forces, c1x1 dot. 361 00:17:58,750 --> 00:18:02,166 And how about the c2 dashpot? 362 00:18:02,166 --> 00:18:03,150 AUDIENCE: Same thing. 363 00:18:03,150 --> 00:18:10,370 J. KIM VANDIVER: Same thing. c2x1 dot. 364 00:18:10,370 --> 00:18:10,980 OK. 365 00:18:10,980 --> 00:18:14,270 And that's the only dashpot forces caused 366 00:18:14,270 --> 00:18:17,320 by a velocity of mass one. 367 00:18:17,320 --> 00:18:22,346 So now let that be 0 and cause velocity at the other places 368 00:18:22,346 --> 00:18:23,720 and find out if anything happens. 369 00:18:23,720 --> 00:18:28,110 So now let x2 dot be positive. 370 00:18:28,110 --> 00:18:30,816 What do we put on free body diagram? 371 00:18:30,816 --> 00:18:33,160 AUDIENCE: [INAUDIBLE]. 372 00:18:33,160 --> 00:18:34,900 J. KIM VANDIVER: Going which direction? 373 00:18:34,900 --> 00:18:36,940 Positive, and what value? 374 00:18:36,940 --> 00:18:39,205 AUDIENCE: c2x2 dot. 375 00:18:39,205 --> 00:18:40,570 J. KIM VANDIVER: All right. 376 00:18:40,570 --> 00:18:42,450 And is our free body diagram complete? 377 00:18:45,515 --> 00:18:46,940 AUDIENCE: mg. 378 00:18:46,940 --> 00:18:49,440 J. KIM VANDIVER: Oh, yeah. 379 00:18:49,440 --> 00:18:50,444 And? 380 00:18:50,444 --> 00:18:51,716 AUDIENCE: [INAUDIBLE]. 381 00:18:51,716 --> 00:18:53,736 J. KIM VANDIVER: Yeah, and? 382 00:18:53,736 --> 00:18:56,634 AUDIENCE: The forces, the external forces. 383 00:19:00,020 --> 00:19:02,250 J. KIM VANDIVER: Now it's complete. 384 00:19:02,250 --> 00:19:05,540 All the forces in the system, it's 385 00:19:05,540 --> 00:19:07,020 constrained in this direction. 386 00:19:07,020 --> 00:19:08,490 So we just know N equals mg. 387 00:19:08,490 --> 00:19:11,654 There's no motion allowed, so we get no equation 388 00:19:11,654 --> 00:19:12,820 of motion in that direction. 389 00:19:12,820 --> 00:19:16,300 We're going to get one equation of motion for this mass 390 00:19:16,300 --> 00:19:18,950 and one more equation of motion for the second. 391 00:19:18,950 --> 00:19:20,450 The number of equation of motions 392 00:19:20,450 --> 00:19:23,750 equal the number of independent coordinates. 393 00:19:26,330 --> 00:19:28,040 And we found two independent coordinates. 394 00:19:28,040 --> 00:19:30,010 We get two equations of motion. 395 00:19:30,010 --> 00:19:34,920 So take this and write down the equation of motion 396 00:19:34,920 --> 00:19:36,340 for mass one. 397 00:19:36,340 --> 00:19:36,910 Sort it out. 398 00:19:36,910 --> 00:19:37,932 Yeah, Betsy? 399 00:19:37,932 --> 00:19:40,342 AUDIENCE: So I'm really confused by why c2 [INAUDIBLE] 400 00:19:40,342 --> 00:19:43,234 in that direction, because so [INAUDIBLE]. 401 00:19:43,234 --> 00:19:45,644 They oppose like the springs that [? felt ?] 402 00:19:45,644 --> 00:19:48,117 [? the closest ?] [INAUDIBLE]. 403 00:19:48,117 --> 00:19:48,950 J. KIM VANDIVER: OK. 404 00:19:48,950 --> 00:19:52,480 So let's see if we can do something here. 405 00:19:52,480 --> 00:19:57,320 So, this is second mass. 406 00:19:57,320 --> 00:19:58,665 This is the first mass. 407 00:19:58,665 --> 00:20:02,540 Actually, here's the wall and the first mass. 408 00:20:02,540 --> 00:20:04,880 So x1 is in that direction. 409 00:20:04,880 --> 00:20:07,410 If I pull that way, the spring pulls back on the mass. 410 00:20:07,410 --> 00:20:09,160 That's obvious to you, right? 411 00:20:09,160 --> 00:20:13,520 And on the other side, if this is mass one now 412 00:20:13,520 --> 00:20:15,560 and you have a second spring over here, 413 00:20:15,560 --> 00:20:18,240 a second mass over here and a spring in between them, 414 00:20:18,240 --> 00:20:22,410 if you put a positive motion of mass one, 415 00:20:22,410 --> 00:20:25,170 this spring pushes back. 416 00:20:25,170 --> 00:20:30,940 So that counts for the minus direction of k2x1. 417 00:20:30,940 --> 00:20:37,060 Then if the second mass moves in that direction-- here, 418 00:20:37,060 --> 00:20:37,590 you move it. 419 00:20:37,590 --> 00:20:39,340 You're the second mass. 420 00:20:39,340 --> 00:20:41,140 Is there tension in this? 421 00:20:41,140 --> 00:20:44,370 Am I pulling back to resist you are or not? 422 00:20:44,370 --> 00:20:46,630 What is this spring doing to my hand? 423 00:20:46,630 --> 00:20:49,379 It's pulling that way on it, k2x2. 424 00:20:49,379 --> 00:20:49,920 AUDIENCE: OK. 425 00:20:53,907 --> 00:20:54,740 J. KIM VANDIVER: OK. 426 00:20:54,740 --> 00:20:56,720 So that's our free body diagram. 427 00:20:56,720 --> 00:20:59,260 Write right out an equation of motion for that system. 428 00:21:10,902 --> 00:21:12,235 I'll give you a little reminder. 429 00:21:22,270 --> 00:21:23,910 That's how you ought to begin, right? 430 00:21:37,370 --> 00:21:41,955 And if you're done, do the same thing for the second mass 431 00:21:41,955 --> 00:21:43,875 while the others are working on the first one. 432 00:21:43,875 --> 00:21:46,720 Draw a free body diagram of mass two 433 00:21:46,720 --> 00:21:50,530 and write down the equation of motion. 434 00:21:55,151 --> 00:21:57,060 AUDIENCE: Shouldn't that be x2 dot? 435 00:21:57,060 --> 00:21:58,185 J. KIM VANDIVER: Which one? 436 00:21:58,185 --> 00:21:59,980 AUDIENCE: The bottom left, c2x2 dot. 437 00:22:04,011 --> 00:22:06,010 J. KIM VANDIVER: All right, so let's build this. 438 00:22:12,030 --> 00:22:13,560 I'm just going to talk through it 439 00:22:13,560 --> 00:22:17,050 rather than have you help me build this. 440 00:22:17,050 --> 00:22:19,760 The sum of the external forces is 441 00:22:19,760 --> 00:22:23,760 what all of the external forces and only the external forces 442 00:22:23,760 --> 00:22:27,450 should show up on that diagram. 443 00:22:27,450 --> 00:22:28,830 The mass times the accelerations, 444 00:22:28,830 --> 00:22:31,454 the sum of the external forces-- so every arrow on that diagram 445 00:22:31,454 --> 00:22:33,840 ought to appear over here in the direction 446 00:22:33,840 --> 00:22:35,280 of whichever this equation is. 447 00:22:35,280 --> 00:22:37,450 So this is the equation that has to do 448 00:22:37,450 --> 00:22:39,230 with the motion of mass one. 449 00:22:39,230 --> 00:22:42,800 And I look over here, and I see, OK, I'm 450 00:22:42,800 --> 00:22:49,970 just going to go top to bottom, minus k1x1 minus k2x1-- 451 00:22:49,970 --> 00:22:52,370 and the minuses are coming from the directions 452 00:22:52,370 --> 00:22:58,050 of the arrows-- minus c1x1 dot. 453 00:22:58,050 --> 00:23:12,350 And then I have plus k2x2 plus c2x2 dot plus F1. 454 00:23:12,350 --> 00:23:13,474 AUDIENCE: And a minus c1x1. 455 00:23:13,474 --> 00:23:14,377 [INTERPOSING VOICES] 456 00:23:14,377 --> 00:23:15,710 J. KIM VANDIVER: Did I miss one? 457 00:23:15,710 --> 00:23:19,520 Oh, I missed the c-- I missed this guy, right? 458 00:23:19,520 --> 00:23:25,490 Minus c2x1 dot. 459 00:23:25,490 --> 00:23:26,590 OK. 460 00:23:26,590 --> 00:23:29,720 So the arrows tell me the signs. 461 00:23:29,720 --> 00:23:32,280 And all the forces are there. 462 00:23:32,280 --> 00:23:35,020 They just all add up over here on this side. 463 00:23:35,020 --> 00:23:37,440 And I've gone ahead and drawn what 464 00:23:37,440 --> 00:23:41,600 I think is the right thing for mass two. 465 00:23:41,600 --> 00:23:43,380 But you do the same system. 466 00:23:43,380 --> 00:23:46,210 You go to mass two and you say, OK, positive 467 00:23:46,210 --> 00:23:47,997 deflection of mass one. 468 00:23:47,997 --> 00:23:49,580 What is the spring force that results? 469 00:23:49,580 --> 00:23:54,310 Well, it pushes on it, positive velocity of mass one. 470 00:23:54,310 --> 00:23:56,510 This dashpot pushes on it. 471 00:23:56,510 --> 00:24:01,620 Positive deflection of mass two-- two springs push back. 472 00:24:01,620 --> 00:24:05,970 Positive velocity of mass two-- one dashpot resists. 473 00:24:05,970 --> 00:24:07,990 And you have an external force. 474 00:24:07,990 --> 00:24:10,110 And you could write this one down. 475 00:24:10,110 --> 00:24:12,250 You just add up all those forces. 476 00:24:12,250 --> 00:24:14,130 And now you have a second equation, 477 00:24:14,130 --> 00:24:19,380 m2x2 double dot equals a you add it up. 478 00:24:19,380 --> 00:24:20,250 OK. 479 00:24:20,250 --> 00:24:23,930 So I want to do one other thing. 480 00:24:23,930 --> 00:24:26,430 I'm going to ask you a question, get you to raise your hand. 481 00:24:26,430 --> 00:24:30,810 I want everybody to try to raise their hands, OK? 482 00:24:30,810 --> 00:24:34,580 Assume that I had written out that second equation. 483 00:24:34,580 --> 00:24:36,080 Actually, let me take one more step. 484 00:24:36,080 --> 00:24:39,410 Normally what we would do is to rewrite 485 00:24:39,410 --> 00:24:42,090 this in sort of a standard form is to move all the motion 486 00:24:42,090 --> 00:24:45,970 variables to one side and all the actual exciting forces 487 00:24:45,970 --> 00:24:46,600 to the other. 488 00:24:46,600 --> 00:24:54,300 And so we'd say, m1x1 double dot plus, 489 00:24:54,300 --> 00:24:58,080 and then normally you put in the x dot terms next. 490 00:24:58,080 --> 00:25:16,810 So we have a c1 plus c2 x1 dot minus c2x2 dot plus-- now 491 00:25:16,810 --> 00:25:29,170 I get my k terms-- k1 plus k2 x1 minus k2x2 equals F1. 492 00:25:29,170 --> 00:25:32,370 And that's kind of standard form now. 493 00:25:32,370 --> 00:25:34,957 I get a second equation from the second one 494 00:25:34,957 --> 00:25:36,040 that would look like that. 495 00:25:36,040 --> 00:25:41,170 Now once you get it in this sort of standard form, 496 00:25:41,170 --> 00:25:45,550 can you remember how, if you've even ever had linear algebra, 497 00:25:45,550 --> 00:25:46,975 write this in matrix form? 498 00:25:51,052 --> 00:25:52,950 AUDIENCE: It would be [INAUDIBLE]. 499 00:25:52,950 --> 00:25:53,580 J. KIM VANDIVER: So how many of you 500 00:25:53,580 --> 00:25:56,080 feel comfortable that you could just sit down and write this 501 00:25:56,080 --> 00:25:58,040 out in matrix form? 502 00:25:58,040 --> 00:25:58,890 Go on. 503 00:25:58,890 --> 00:26:01,860 And how many of you think it might be a bit of a challenge? 504 00:26:01,860 --> 00:26:04,232 OK. 505 00:26:04,232 --> 00:26:05,940 How many of you would like me to show you 506 00:26:05,940 --> 00:26:08,476 what it looks like in matrix form and give you the answer? 507 00:26:08,476 --> 00:26:09,100 So most of you. 508 00:26:09,100 --> 00:26:10,660 OK, I'll take the time to do that. 509 00:26:13,960 --> 00:26:15,830 Actually, did I write it out already? 510 00:26:15,830 --> 00:26:17,100 Ooh, look at that. 511 00:26:17,100 --> 00:26:18,510 Bing! 512 00:26:18,510 --> 00:26:19,436 Ding, ding, ding! 513 00:26:29,860 --> 00:26:32,300 All you need to know in this course about vectors 514 00:26:32,300 --> 00:26:38,090 and matrices is how to multiply a vector times a matrix. 515 00:26:38,090 --> 00:26:41,040 So here's an acceleration vector. 516 00:26:41,040 --> 00:26:42,730 Here's the mass matrix. 517 00:26:42,730 --> 00:26:45,820 And you take these two elements, my two fingers here, 518 00:26:45,820 --> 00:26:48,680 and multiplying by those-- you go like this to that. 519 00:26:48,680 --> 00:26:57,330 So x1 double dot times m1 double dot gives you this term, right? 520 00:26:57,330 --> 00:27:01,115 And you get no m2x2 double dot, because there's 521 00:27:01,115 --> 00:27:04,930 a 0 in the matrix in that second position. 522 00:27:04,930 --> 00:27:07,200 So you get no x2 double dot term. 523 00:27:07,200 --> 00:27:09,110 And that's correct. 524 00:27:09,110 --> 00:27:11,750 Same thing with the-- here's the x2, x1. 525 00:27:11,750 --> 00:27:14,060 Here's this damping matrix. 526 00:27:14,060 --> 00:27:15,750 This is the stiffness matrix. 527 00:27:15,750 --> 00:27:17,930 And so if you just did these multiplies, 528 00:27:17,930 --> 00:27:22,110 you'd get back two equations, the ones we just derived. 529 00:27:22,110 --> 00:27:25,260 So you need to know how to do this kind of thing 530 00:27:25,260 --> 00:27:31,030 because we write that h equals I omega. 531 00:27:31,030 --> 00:27:34,090 And our omega has three components, omega x, omega y, 532 00:27:34,090 --> 00:27:35,180 and omega z. 533 00:27:35,180 --> 00:27:37,320 And you need to be able to do that manipulation. 534 00:27:40,522 --> 00:27:42,480 All right, we've got a little bit of time left. 535 00:27:42,480 --> 00:27:44,780 I wanted to do one little exercise that 536 00:27:44,780 --> 00:27:47,420 had to do more with this sort of thing, 537 00:27:47,420 --> 00:27:51,290 to kind of reinforce a bit what we did yesterday. 538 00:27:51,290 --> 00:27:55,620 And that would be-- so any final questions about this? 539 00:27:55,620 --> 00:27:56,346 Yeah? 540 00:27:56,346 --> 00:27:58,626 AUDIENCE: So setting that up looks OK. 541 00:27:58,626 --> 00:27:59,540 But solving it. 542 00:27:59,540 --> 00:28:02,456 J. KIM VANDIVER: Oh, solving it? 543 00:28:02,456 --> 00:28:04,330 I used to teach-- I taught for about 20 years 544 00:28:04,330 --> 00:28:05,860 a course called Mechanical Vibration. 545 00:28:05,860 --> 00:28:07,460 I just don't have time to teach it anymore. 546 00:28:07,460 --> 00:28:08,850 And that's what you do in there. 547 00:28:08,850 --> 00:28:11,660 Oh, in 803 you do a little bit of it. 548 00:28:11,660 --> 00:28:15,680 So this system has two degrees of freedom. 549 00:28:15,680 --> 00:28:20,700 It has two natural frequencies, two mode shapes to go with it. 550 00:28:20,700 --> 00:28:23,850 So the system vibrates. 551 00:28:23,850 --> 00:28:25,950 And it's not real hard to solve, but you 552 00:28:25,950 --> 00:28:27,910 would assume for initially, if you 553 00:28:27,910 --> 00:28:29,570 want to know natural frequencies, 554 00:28:29,570 --> 00:28:32,510 you set the forces to 0. 555 00:28:32,510 --> 00:28:35,170 You set the damping to 0 initially. 556 00:28:35,170 --> 00:28:38,600 And you just assume a solution of the form 557 00:28:38,600 --> 00:28:51,850 that x1 of t and x2 of t equals some unknown constants, cosine 558 00:28:51,850 --> 00:28:52,760 omega t. 559 00:28:52,760 --> 00:28:54,130 Just plug it in. 560 00:28:54,130 --> 00:28:55,340 You just plug that in. 561 00:28:55,340 --> 00:28:59,200 You'll end up with an algebraic equation. 562 00:28:59,200 --> 00:29:03,372 And the algebraic equation actually has eigenvalues. 563 00:29:03,372 --> 00:29:04,970 AUDIENCE: [INAUDIBLE]. 564 00:29:04,970 --> 00:29:07,090 J. KIM VANDIVER: And you find the two eigenvalues, 565 00:29:07,090 --> 00:29:09,260 and they're the two natural frequencies. 566 00:29:09,260 --> 00:29:09,760 All right. 567 00:29:09,760 --> 00:29:15,170 I want to do something with angular stuff. 568 00:29:15,170 --> 00:29:19,130 So here's my system that we were playing 569 00:29:19,130 --> 00:29:21,210 with at the beginning of class. 570 00:29:21,210 --> 00:29:27,130 And in this form, it's nice and balanced. 571 00:29:27,130 --> 00:29:30,790 So this is just to reinforce a couple things 572 00:29:30,790 --> 00:29:33,540 that I was just beginning to teach you yesterday. 573 00:29:33,540 --> 00:29:35,120 This is a rigid body. 574 00:29:35,120 --> 00:29:36,440 It's rotating. 575 00:29:36,440 --> 00:29:40,430 I attach to it a rotating frame. 576 00:29:40,430 --> 00:29:44,720 At a in the x direction is y and z. 577 00:29:44,720 --> 00:29:47,390 So y's into the board. 578 00:29:47,390 --> 00:29:51,130 And that frame rotates with my system. 579 00:29:51,130 --> 00:29:53,640 And the rotation rate of this system is omega, 580 00:29:53,640 --> 00:29:56,090 and it's k in the rotating system, which 581 00:29:56,090 --> 00:29:59,550 happens to line up with big K with the stationary system. 582 00:29:59,550 --> 00:30:03,100 But this is my omega vector. 583 00:30:03,100 --> 00:30:09,300 Here is 0, 0 omega. 584 00:30:09,300 --> 00:30:11,910 This is the z component of the rotation. 585 00:30:15,470 --> 00:30:20,820 Now, let's first start with this. 586 00:30:20,820 --> 00:30:25,960 What is P1, [? an 0, ?] the linear momentum 587 00:30:25,960 --> 00:30:30,970 of that mass in the system? 588 00:30:30,970 --> 00:30:32,910 And you've done this probably a lot of times. 589 00:30:32,910 --> 00:30:34,630 So what direction's it in? 590 00:30:37,340 --> 00:30:38,852 Momentum is mv, right? 591 00:30:38,852 --> 00:30:40,310 So what's the velocity of the mass? 592 00:30:44,090 --> 00:30:45,580 AUDIENCE: It's rotating. 593 00:30:45,580 --> 00:30:48,748 J. KIM VANDIVER: It's caused by rotation and only rotation. 594 00:30:48,748 --> 00:30:51,120 AUDIENCE: So it could be the [? c hat ?] direction. 595 00:30:51,120 --> 00:30:51,620 [INAUDIBLE]. 596 00:30:51,620 --> 00:30:54,135 J. KIM VANDIVER: No, P, just P. Linear momentum. 597 00:30:54,135 --> 00:30:56,551 AUDIENCE: That P itself-- linear because it's [INAUDIBLE]. 598 00:30:59,076 --> 00:31:01,200 J. KIM VANDIVER: And it's got to [? an 0, ?] right? 599 00:31:01,200 --> 00:31:03,410 I mean, reference to the 0 frame. 600 00:31:03,410 --> 00:31:09,230 But it can use unit vectors in the A frame. 601 00:31:09,230 --> 00:31:12,150 So what's v1? 602 00:31:12,150 --> 00:31:13,470 AUDIENCE: j hat. 603 00:31:13,470 --> 00:31:16,200 J. KIM VANDIVER: j hat, and how big? 604 00:31:16,200 --> 00:31:17,587 AUDIENCE: Omega. 605 00:31:17,587 --> 00:31:19,420 J. KIM VANDIVER: I hear an omega times what? 606 00:31:23,200 --> 00:31:29,040 So this is system-- I'm going to call this position, 607 00:31:29,040 --> 00:31:31,080 this is x1z1. 608 00:31:31,080 --> 00:31:33,350 This coordinate is x1z1. 609 00:31:33,350 --> 00:31:34,820 So if I give you that information, 610 00:31:34,820 --> 00:31:36,832 what's the velocity of that point? 611 00:31:36,832 --> 00:31:37,796 AUDIENCE: Omega x. 612 00:31:37,796 --> 00:31:38,760 AUDIENCE: x. 613 00:31:38,760 --> 00:31:42,146 J. KIM VANDIVER: Omega x1-- 614 00:31:42,146 --> 00:31:43,090 AUDIENCE: Oh, j hat. 615 00:31:43,090 --> 00:31:44,048 J. KIM VANDIVER: j hat. 616 00:31:54,020 --> 00:31:57,719 It's into the board, the way it's spinning into the board. 617 00:31:57,719 --> 00:31:59,260 Yeah, ought to be in the j direction. 618 00:31:59,260 --> 00:32:01,570 Yeah, it ought to be an r omega, and it 619 00:32:01,570 --> 00:32:04,650 has an m associated with it. 620 00:32:04,650 --> 00:32:05,150 What's P2? 621 00:32:10,640 --> 00:32:12,640 AUDIENCE: In the [INAUDIBLE]? 622 00:32:12,640 --> 00:32:14,140 AUDIENCE: Negative [INAUDIBLE]. 623 00:32:14,140 --> 00:32:15,275 J. KIM VANDIVER: It's coming out of the board at you. 624 00:32:15,275 --> 00:32:16,146 AUDIENCE: Right. 625 00:32:16,146 --> 00:32:19,122 Negative m2 omega xj. 626 00:32:28,050 --> 00:32:30,050 J. KIM VANDIVER: Right? 627 00:32:30,050 --> 00:32:31,190 OK. 628 00:32:31,190 --> 00:32:33,910 And those little j's-- the j hats are in the rotating 629 00:32:33,910 --> 00:32:36,570 coordinate system where I want them. 630 00:32:36,570 --> 00:32:37,430 So what's H1? 631 00:32:41,380 --> 00:32:47,110 It's r1 with respect to what? 632 00:32:47,110 --> 00:32:50,880 So you can't do angular momentum without picking a--? 633 00:32:50,880 --> 00:32:54,010 You've got to pick the point. 634 00:32:54,010 --> 00:32:57,390 So this is why we choose the coordinates on the reference 635 00:32:57,390 --> 00:33:00,550 frame that we're going to use to give us some information 636 00:33:00,550 --> 00:33:01,177 that we want. 637 00:33:01,177 --> 00:33:03,010 I want to know the torques about this point. 638 00:33:03,010 --> 00:33:04,750 So I'm going to put my A here. 639 00:33:04,750 --> 00:33:09,610 I could have put it any place, as long as the axis of rotation 640 00:33:09,610 --> 00:33:10,860 passed through it. 641 00:33:10,860 --> 00:33:20,510 So this one is r1 with respect to A cross P1. 642 00:33:20,510 --> 00:33:24,055 So what is r1 in this system? 643 00:33:24,055 --> 00:33:24,596 AUDIENCE: x1. 644 00:33:27,350 --> 00:33:29,040 J. KIM VANDIVER: x1i-- 645 00:33:29,040 --> 00:33:31,740 AUDIENCE: Plus z-- 646 00:33:31,740 --> 00:33:41,040 J. KIM VANDIVER: 1k cross with m1 omega x1 j. 647 00:33:41,040 --> 00:33:42,595 All right? 648 00:33:42,595 --> 00:33:44,640 Let's work that one out. 649 00:33:44,640 --> 00:33:48,990 So we have an i times a j. 650 00:33:48,990 --> 00:34:01,540 And so I get an m1x1 squared omega k. 651 00:34:01,540 --> 00:34:04,420 And then I do this term times that. 652 00:34:04,420 --> 00:34:09,969 I get a k cross j minus i. 653 00:34:09,969 --> 00:34:21,665 And I get m1x1z1 omega i hat. 654 00:34:21,665 --> 00:34:22,165 All right. 655 00:34:26,500 --> 00:34:32,679 Technically this is my little h, this is little h2. 656 00:34:32,679 --> 00:34:42,279 And if I let m1 equal m2 here, my total h for this system-- 657 00:34:45,897 --> 00:34:47,022 AUDIENCE: Isn't that just-- 658 00:34:47,022 --> 00:34:48,022 J. KIM VANDIVER: Pardon? 659 00:34:48,022 --> 00:34:50,293 AUDIENCE: This is [INAUDIBLE]. 660 00:34:50,293 --> 00:34:50,918 AUDIENCE: Yeah. 661 00:34:50,918 --> 00:34:51,892 AUDIENCE: The second minus-- 662 00:34:51,892 --> 00:34:53,360 AUDIENCE: The second minus is still h1. 663 00:34:53,360 --> 00:34:55,235 J. KIM VANDIVER: Oh, whoops, how'd I do that? 664 00:34:55,235 --> 00:34:57,350 Yeah, I hadn't moved on to h2 yet. 665 00:34:57,350 --> 00:34:58,580 Sorry about that. 666 00:34:58,580 --> 00:35:00,620 This is still h1. 667 00:35:00,620 --> 00:35:01,900 OK. 668 00:35:01,900 --> 00:35:02,820 And it's that. 669 00:35:02,820 --> 00:35:06,230 So what's h2? 670 00:35:06,230 --> 00:35:09,052 How's it differ? 671 00:35:09,052 --> 00:35:10,010 Just kind of look here. 672 00:35:10,010 --> 00:35:12,360 It only differs in one respect. 673 00:35:12,360 --> 00:35:13,360 AUDIENCE: Negative sign? 674 00:35:13,360 --> 00:35:18,510 J. KIM VANDIVER: Yeah, you get a minus-- a negative x here 675 00:35:18,510 --> 00:35:23,400 when you go to carry out the multiplication. 676 00:35:23,400 --> 00:35:26,560 So now the i cross j term gives you a-- 677 00:35:26,560 --> 00:35:32,100 AUDIENCE: But isn't the P2 also have a negative under it. 678 00:35:32,100 --> 00:35:34,640 J. KIM VANDIVER: Yeah, you're absolutely right. 679 00:35:34,640 --> 00:35:37,800 So you get-- you had to put a minus x here. 680 00:35:37,800 --> 00:35:40,250 And you have to multiply by this one. 681 00:35:40,250 --> 00:35:44,870 So the i, j term, you have a minus times a minus. 682 00:35:44,870 --> 00:35:47,010 i times j is a positive k. 683 00:35:47,010 --> 00:35:56,090 You end up with m2x1 squared omega k, the same direction 684 00:35:56,090 --> 00:35:57,240 of each component. 685 00:35:57,240 --> 00:35:58,642 Then what happens here? 686 00:35:58,642 --> 00:36:00,350 AUDIENCE: Comes from a minus [INAUDIBLE]. 687 00:36:09,260 --> 00:36:10,590 J. KIM VANDIVER: OK. 688 00:36:10,590 --> 00:36:16,060 And so the second term, the i term, is opposite direction. 689 00:36:16,060 --> 00:36:18,297 And if m1 equals m2 and you added these two together, 690 00:36:18,297 --> 00:36:19,880 what happened to those last two terms? 691 00:36:22,710 --> 00:36:24,454 They cancel, right? 692 00:36:24,454 --> 00:36:27,060 [LAUGHTER] 693 00:36:27,060 --> 00:36:30,230 OK, so if this is true, the sum of these two, 694 00:36:30,230 --> 00:36:40,710 H, with respect to A, is just 2mx1 squared omega k. 695 00:36:40,710 --> 00:36:41,970 All right. 696 00:36:41,970 --> 00:36:49,380 So if you took-- in the first system, 697 00:36:49,380 --> 00:36:51,890 if these two weren't equal and they didn't cancel out, 698 00:36:51,890 --> 00:36:54,640 if you go through here and take dh dt, 699 00:36:54,640 --> 00:36:59,240 you take the time derivative of these terms, 700 00:36:59,240 --> 00:37:02,600 then you get an omega dot k. 701 00:37:02,600 --> 00:37:05,610 And over here you get an omega dot term. 702 00:37:05,610 --> 00:37:09,640 And then you get a di dt term, all that. 703 00:37:09,640 --> 00:37:13,310 But you will end up with values of dh dt 704 00:37:13,310 --> 00:37:14,805 that are not in the k direction. 705 00:37:14,805 --> 00:37:15,550 You agree? 706 00:37:15,550 --> 00:37:17,640 For sure. 707 00:37:17,640 --> 00:37:20,350 And those are associated with the real torques 708 00:37:20,350 --> 00:37:23,170 in the system that happen. 709 00:37:23,170 --> 00:37:30,120 When this is true, that difficult term out here 710 00:37:30,120 --> 00:37:31,790 cancels out. 711 00:37:31,790 --> 00:37:34,540 And you're left only with the k term. 712 00:37:34,540 --> 00:37:37,870 And when you take this time derivative, you only get dh dt. 713 00:37:37,870 --> 00:37:41,920 You get a theta dot or omega dot. 714 00:37:41,920 --> 00:37:48,160 And that says the torque is in what direction when 715 00:37:48,160 --> 00:37:49,880 you get just this for h? 716 00:37:49,880 --> 00:37:52,680 What is the direction of dh dt? 717 00:37:52,680 --> 00:37:54,680 It's still in k. 718 00:37:54,680 --> 00:37:58,620 So the torque is aligned with the spin. 719 00:37:58,620 --> 00:38:01,830 h is aligned with the spin. 720 00:38:01,830 --> 00:38:02,720 They're all aligned. 721 00:38:02,720 --> 00:38:06,990 And the system is actually beautifully balanced. 722 00:38:06,990 --> 00:38:11,660 You don't feel any torques around where you're holding it 723 00:38:11,660 --> 00:38:19,340 down here at A. But if I took one of these off, kind of back 724 00:38:19,340 --> 00:38:22,410 to that one arm system, it's terribly unbalanced. 725 00:38:25,110 --> 00:38:31,870 So the moral of the story here is 726 00:38:31,870 --> 00:38:35,960 if I don't want to have these unwanted torques, 727 00:38:35,960 --> 00:38:38,560 what can you say about the desired mass 728 00:38:38,560 --> 00:38:40,355 distribution in the system? 729 00:38:40,355 --> 00:38:41,690 AUDIENCE: [INAUDIBLE]. 730 00:38:41,690 --> 00:38:45,000 J. KIM VANDIVER: I hear a vote for symmetry. 731 00:38:45,000 --> 00:38:47,310 And that's the general rule. 732 00:38:47,310 --> 00:38:49,870 If your masses are distributed symmetrically 733 00:38:49,870 --> 00:38:58,219 about the axis of spin, no off axis torques. 734 00:38:58,219 --> 00:39:00,219 AUDIENCE: And it's [INAUDIBLE] mass [INAUDIBLE]. 735 00:39:00,219 --> 00:39:01,802 J. KIM VANDIVER: Yeah, it's symmetric. 736 00:39:01,802 --> 00:39:03,910 Symmetry means they better be the same size too, 737 00:39:03,910 --> 00:39:05,070 not just in the same place. 738 00:39:05,070 --> 00:39:08,190 If one's twice as big as the other, no go. 739 00:39:08,190 --> 00:39:10,420 If I put a second mass on one of those arms, 740 00:39:10,420 --> 00:39:13,360 the system is back to being unbalanced. 741 00:39:13,360 --> 00:39:21,810 So any time this H vector, the total H of the system, 742 00:39:21,810 --> 00:39:26,110 is not lined up with the spin, the system 743 00:39:26,110 --> 00:39:28,780 has an asymmetry in it. 744 00:39:31,480 --> 00:39:36,106 And the system will require additional torques just 745 00:39:36,106 --> 00:39:37,730 to hold it in place when it's spinning. 746 00:39:40,650 --> 00:39:43,000 So when you have these additional torques that 747 00:39:43,000 --> 00:39:47,670 come from being a symmetric, the system 748 00:39:47,670 --> 00:39:53,450 is said to be dynamically imbalanced. 749 00:39:53,450 --> 00:39:56,880 So if you pick up a stone, a big stone, in your car wheel 750 00:39:56,880 --> 00:40:00,160 or block hunk of mud or ice that's frozen on the rim, 751 00:40:00,160 --> 00:40:03,280 you're going down the road, what's it feel like? 752 00:40:03,280 --> 00:40:05,910 You ever had an unbalanced tire on your car? 753 00:40:05,910 --> 00:40:07,760 Boy, where have you guys lived? 754 00:40:10,910 --> 00:40:14,400 An unbalanced tire on a car, you're going down the road. 755 00:40:14,400 --> 00:40:16,140 You know? 756 00:40:16,140 --> 00:40:17,200 Right? 757 00:40:17,200 --> 00:40:18,540 That's what this is all about. 758 00:40:18,540 --> 00:40:21,423 You have a case of imbalance. 759 00:40:24,549 --> 00:40:25,590 How are we doing on time? 760 00:40:25,590 --> 00:40:27,730 A couple more minutes. 761 00:40:27,730 --> 00:40:32,240 So imbalance comes from having angular momentum that's 762 00:40:32,240 --> 00:40:35,777 not pointed in same direction as-- angular momentum not 763 00:40:35,777 --> 00:40:37,860 pointed in the same direction as the angle of spin 764 00:40:37,860 --> 00:40:40,710 is evidence of unbalance. 765 00:40:40,710 --> 00:40:44,650 And you can calculate how bad the unbalance is by doing dh dt 766 00:40:44,650 --> 00:40:48,150 and actually finding out how much torque is being applied 767 00:40:48,150 --> 00:40:51,210 to the system that ordinarily the bearings wouldn't 768 00:40:51,210 --> 00:40:52,790 have to resist. 769 00:40:52,790 --> 00:40:54,810 But this thing has overturning torques 770 00:40:54,810 --> 00:40:58,990 that are trying to make it wobble back and forth on you. 771 00:40:58,990 --> 00:41:04,766 So one last exercise, which is actually 772 00:41:04,766 --> 00:41:06,810 a quite important point. 773 00:41:06,810 --> 00:41:12,330 If I moved A up here right on the line between these two 774 00:41:12,330 --> 00:41:17,780 things-- so this is A. And now this is x, and this is z. 775 00:41:17,780 --> 00:41:19,950 The coordinates of this point become what? 776 00:41:22,760 --> 00:41:24,090 x1 and-- 777 00:41:24,090 --> 00:41:24,850 AUDIENCE: 0. 778 00:41:24,850 --> 00:41:26,310 J. KIM VANDIVER: 0. 779 00:41:26,310 --> 00:41:29,760 And all these equations still apply. 780 00:41:29,760 --> 00:41:34,060 So if I just move my coordinate system so the x's are the same, 781 00:41:34,060 --> 00:41:35,850 z's go to 0. 782 00:41:35,850 --> 00:41:38,740 What do you end up with for terms? 783 00:41:38,740 --> 00:41:41,440 What happens to this thing? 784 00:41:41,440 --> 00:41:42,320 That goes to 0. 785 00:41:42,320 --> 00:41:43,270 It goes away. 786 00:41:43,270 --> 00:41:49,060 And you only get this term, and so by simply moving A to here, 787 00:41:49,060 --> 00:41:52,780 my angular momentum vector now lines up with the spin. 788 00:41:55,708 --> 00:41:56,970 Kind of weird. 789 00:41:56,970 --> 00:42:01,660 Does the imbalance still exist? 790 00:42:01,660 --> 00:42:03,751 Will your car still be doing this down the road 791 00:42:03,751 --> 00:42:05,250 just because you chose to look at it 792 00:42:05,250 --> 00:42:08,430 from a different point of view? 793 00:42:08,430 --> 00:42:08,930 Yeah. 794 00:42:08,930 --> 00:42:11,960 I mean, the car's still unhappy. 795 00:42:11,960 --> 00:42:16,109 So there's kind of inconsistency here, sort of. 796 00:42:16,109 --> 00:42:16,900 What's the problem? 797 00:42:20,610 --> 00:42:25,730 Or maybe not a problem, but you choose 798 00:42:25,730 --> 00:42:29,900 to put A when you're computing angular momentum. 799 00:42:29,900 --> 00:42:32,990 You can choose where you put this point to give you 800 00:42:32,990 --> 00:42:35,780 the information you're after. 801 00:42:35,780 --> 00:42:38,190 And in this case, if I were designing this system 802 00:42:38,190 --> 00:42:42,160 and I wanted to know the bending moment in these bars sticking 803 00:42:42,160 --> 00:42:44,580 down here, I put A here because it'll 804 00:42:44,580 --> 00:42:47,590 give you the torques with respect to this point. 805 00:42:47,590 --> 00:42:50,370 These torques down here still exist. 806 00:42:50,370 --> 00:42:53,790 If you put your point at which you compute angular momentum up 807 00:42:53,790 --> 00:42:56,110 here, you won't be able to find them. 808 00:42:56,110 --> 00:42:58,470 You can't compute it because you've reduced the moment 809 00:42:58,470 --> 00:43:03,980 arm to 0, this moment arm, and you just won't get that torque. 810 00:43:03,980 --> 00:43:06,580 OK. 811 00:43:06,580 --> 00:43:08,770 All right, I've run out of time. 812 00:43:08,770 --> 00:43:12,031 But there'll be more on this subject.