1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high-quality educational resources for free. 6 00:00:09 --> 00:00:12 To make a donation or to view additional materials from 7 00:00:12 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:20 PROFESSOR STRANG: Alright. 10 00:00:20 --> 00:00:25 So this is Lecture 15. 11 00:00:25 --> 00:00:30 It's the last topic, today and Friday, like just 15 and 16, 12 00:00:30 --> 00:00:33 trusses within Chapter 2. 13 00:00:33 --> 00:00:38 The last topic we'll do for discrete systems. 14 00:00:38 --> 00:00:41 Then it's a lot of fun. 15 00:00:41 --> 00:00:48 I wanted to say a few words first about last night's exam. 16 00:00:48 --> 00:00:50 Several words first. 17 00:00:50 --> 00:00:52 Overall, I'm sure it's going to be fine. 18 00:00:52 --> 00:00:57 Ramis is grading the first two problems, he'll pass them 19 00:00:57 --> 00:00:58 to Peter for the next two. 20 00:00:58 --> 00:01:00 And I'll get them back. 21 00:01:00 --> 00:01:05 I'm pretty sure it'll be next week. 22 00:01:05 --> 00:01:10 I felt it was a fair exam, except I should have done 23 00:01:10 --> 00:01:18 a better job in helping you with the matrix A. 24 00:01:18 --> 00:01:20 Especially in problem one. 25 00:01:20 --> 00:01:27 I'm glad that hint was there, the matrix A_0, that goes with 26 00:01:27 --> 00:01:34 free-free to sort of say what kind of matrix to 27 00:01:34 --> 00:01:35 be looking for. 28 00:01:35 --> 00:01:42 And I thought I'd just repeat, make the connections that I 29 00:01:42 --> 00:01:47 should have made earlier. 30 00:01:47 --> 00:01:53 So we all see the point about these. 31 00:01:53 --> 00:01:58 These A's and the A transpose A's. 32 00:01:58 --> 00:02:04 So if I take, for that A_0, free-free one. 33 00:02:04 --> 00:02:06 Everybody sees that this, and this connects of 34 00:02:06 --> 00:02:08 course with our graphs. 35 00:02:08 --> 00:02:13 Our graph is just this simple graph with well actually is 36 00:02:13 --> 00:02:15 that how many, is it five nodes? 37 00:02:15 --> 00:02:19 I guess there are five. 38 00:02:19 --> 00:02:23 Because as it stands I have one, two, three, four, five 39 00:02:23 --> 00:02:32 columns, I've got five u's, u_0 down to u_4. 40 00:02:32 --> 00:02:36 And if I take A_0 transpose A_0, that would be 41 00:02:36 --> 00:02:38 the free-free matrix. 42 00:02:38 --> 00:02:40 What size would it be? 43 00:02:40 --> 00:02:42 And what matrix would it be? 44 00:02:42 --> 00:02:47 Just if we do that multiplication, this is a 45 00:02:47 --> 00:02:49 first difference matrix. 46 00:02:49 --> 00:02:52 When I do A_0 transpose A_0 I'll get one of our second 47 00:02:52 --> 00:02:53 difference matrices. 48 00:02:53 --> 00:02:56 So it'll be one of our special ones. 49 00:02:56 --> 00:02:59 Which special one would it be? 50 00:02:59 --> 00:03:00 B. 51 00:03:00 --> 00:03:04 It'll be the matrix B that has both ends free. 52 00:03:04 --> 00:03:06 And what size will it be? 53 00:03:06 --> 00:03:14 I guess it'll be five by five. 54 00:03:14 --> 00:03:19 That's right; that would be five by four times A_0, 55 00:03:19 --> 00:03:20 which is four by five. 56 00:03:20 --> 00:03:24 So it'll be the five by five matrix B. 57 00:03:24 --> 00:03:26 Can I call it B_5? 58 00:03:26 --> 00:03:27 OK. 59 00:03:27 --> 00:03:31 So that was there as a hint. 60 00:03:31 --> 00:03:34 That isn't the correct matrix for problem one because 61 00:03:34 --> 00:03:37 problem one was fixed fixed. 62 00:03:37 --> 00:03:38 Let's get there in two steps. 63 00:03:38 --> 00:03:41 Suppose it's fixed free. 64 00:03:41 --> 00:03:45 So suppose I make u_0=0. 65 00:03:46 --> 00:03:51 So I ground the top node, I support the top node - 66 00:03:51 --> 00:03:53 oh no, shall I do u_0? 67 00:03:53 --> 00:03:53 Yeah. 68 00:03:53 --> 00:03:54 I'll do u_0. 69 00:03:54 --> 00:03:57 So that would knock out this one. 70 00:03:57 --> 00:04:02 If I say fix u_0, say, at zero or whatever. 71 00:04:02 --> 00:04:09 Now I've got, the next a, I won't call it A_0 anymore. 72 00:04:09 --> 00:04:17 So now four by four, now if I do A transpose A, which of our 73 00:04:17 --> 00:04:21 special matrices am I going to get? t. 74 00:04:21 --> 00:04:22 It'll be T. 75 00:04:22 --> 00:04:28 It'll be the one that has the first 1, 1 entry 76 00:04:28 --> 00:04:31 will only be a one. 77 00:04:31 --> 00:04:34 So that'll be the fixed free matrix. 78 00:04:34 --> 00:04:35 It'll be of what size? 79 00:04:35 --> 00:04:38 Four. 80 00:04:38 --> 00:04:42 Now I've only got four unknowns. u_1 to u_4. 81 00:04:42 --> 00:04:45 OK, that still is not what problem one is. 82 00:04:45 --> 00:04:47 Problem one was fixed fixed. 83 00:04:47 --> 00:04:50 So as I did in the review, that would knock out 84 00:04:50 --> 00:04:52 both of these columns. 85 00:04:52 --> 00:04:59 So this now is the matrix that I was looking for in problem 86 00:04:59 --> 00:05:04 one, and I wish I had emphasized these 87 00:05:04 --> 00:05:05 steps in advance. 88 00:05:05 --> 00:05:07 I apologize. 89 00:05:07 --> 00:05:13 OK, so what fixed-fixed if, without the C part in it. 90 00:05:13 --> 00:05:18 Just focusing on the A, what fixed-fixed matrix would I get? 91 00:05:18 --> 00:05:20 Which one of our guys would it be? 92 00:05:20 --> 00:05:23 K of size three. 93 00:05:23 --> 00:05:29 And while we're at it, what would be the story, how would I 94 00:05:29 --> 00:05:34 get one of these circular ones, which is sort of on 95 00:05:34 --> 00:05:36 our special list. 96 00:05:36 --> 00:05:40 For Fourier it's the really special guy. 97 00:05:40 --> 00:05:47 So a circular one I'm going to connect u_4 back to u_0. 98 00:05:47 --> 00:05:50 So I'm going to put these guys back in. 99 00:05:50 --> 00:05:53 And what else would change, if u_4 was connected back to u_0, 100 00:05:53 --> 00:05:57 now I'm aiming for this circulant. 101 00:05:57 --> 00:06:01 What matrix A is going to give me the circulant? 102 00:06:01 --> 00:06:05 So again, these guys are the A transpose A's. 103 00:06:05 --> 00:06:10 This over here was the A, and over here is the A transpose A. 104 00:06:10 --> 00:06:14 And now I want to fix A, and then I want to see 105 00:06:14 --> 00:06:16 that A transpose A. 106 00:06:16 --> 00:06:19 So suppose I give you that graph, then. 107 00:06:19 --> 00:06:23 Oops, I should have, well. 108 00:06:23 --> 00:06:30 Just connect the whole guy. 109 00:06:30 --> 00:06:36 So fifth node coming back to the first. 110 00:06:36 --> 00:06:39 So that's my circle of nodes. 111 00:06:39 --> 00:06:42 That's a simple graph. 112 00:06:42 --> 00:06:49 What's the a circulant now? 113 00:06:49 --> 00:06:54 So this would be the A for the circulant case. 114 00:06:54 --> 00:06:57 So it's got that back in. 115 00:06:57 --> 00:07:00 That shouldn't be erased, that shouldn't be erased. 116 00:07:00 --> 00:07:04 And what else is it got? 117 00:07:04 --> 00:07:09 If I ask you for the incidence matrix, now I'm in Section 2.4, 118 00:07:09 --> 00:07:13 like I've given you a graph, or you can think of masses 119 00:07:13 --> 00:07:16 and springs in a circle. 120 00:07:16 --> 00:07:19 So I've got five masses, five springs. 121 00:07:19 --> 00:07:25 What's my matrix A missing? 122 00:07:25 --> 00:07:29 It needs another row. 123 00:07:29 --> 00:07:32 We just put in another edge, it needs another row. 124 00:07:32 --> 00:07:39 That edge went the node back to the first node. 125 00:07:39 --> 00:07:45 So we've got a fifth row. 126 00:07:45 --> 00:07:48 So you see, now it really is circulant. 127 00:07:48 --> 00:07:53 I would call this one also a circulant matrix. 128 00:07:53 --> 00:07:58 The diagonals are constant. 129 00:07:58 --> 00:08:03 That what I and MATLAB and everybody else would call a 130 00:08:03 --> 00:08:06 templates matrix, and the command templates 131 00:08:06 --> 00:08:08 could create this. 132 00:08:08 --> 00:08:10 That diagonal is constant. 133 00:08:10 --> 00:08:12 That diagonal is constant. 134 00:08:12 --> 00:08:14 The other diagonals are constant. 135 00:08:14 --> 00:08:18 But more than that, what's additional here in the 136 00:08:18 --> 00:08:22 circulant, which is the thing that makes Fourier happy? 137 00:08:22 --> 00:08:26 The diagonal circles around. 138 00:08:26 --> 00:08:31 That diagonal has only got four entries in it, but it circles 139 00:08:31 --> 00:08:38 around sort of periodically to its fifth entry. 140 00:08:38 --> 00:08:40 So that's more than templates. 141 00:08:40 --> 00:08:43 It's circulant because it's coming around again. 142 00:08:43 --> 00:08:47 This we'll see in the discrete Fourier transform. 143 00:08:47 --> 00:08:50 It's really all good stuff. 144 00:08:50 --> 00:08:53 And now there is my a circulant. 145 00:08:53 --> 00:08:56 And what would be my A transpose circulant 146 00:08:56 --> 00:08:57 A circulant? 147 00:08:57 --> 00:09:00 What would be A transpose A if I take that 148 00:09:00 --> 00:09:03 five by five matrix? 149 00:09:03 --> 00:09:05 C. 150 00:09:05 --> 00:09:09 Finally I've created, so I've already got B, I've got T, 151 00:09:09 --> 00:09:12 I've got K, all those three special guys. 152 00:09:12 --> 00:09:16 And now the A transpose A for this circulant, so that's a 153 00:09:16 --> 00:09:20 first difference matrix for a periodic problem. 154 00:09:20 --> 00:09:24 And A transpose A will be a second difference matrix for a 155 00:09:24 --> 00:09:27 periodic problem; c_5, I guess. 156 00:09:27 --> 00:09:29 It'll be five. 157 00:09:29 --> 00:09:29 OK. 158 00:09:29 --> 00:09:33 I hope that brings together what I, if I was on the 159 00:09:33 --> 00:09:38 ball, I would of brought it together before the quiz. 160 00:09:38 --> 00:09:45 Can I just say a few words about the quiz and grades? 161 00:09:45 --> 00:09:47 They come out fine. 162 00:09:47 --> 00:09:49 Really they do. 163 00:09:49 --> 00:09:51 I've been doing this a long time. 164 00:09:51 --> 00:09:59 And just, enjoy October. 165 00:09:59 --> 00:10:05 I'm sorry to give you any exam at all, but it's a chance for 166 00:10:05 --> 00:10:08 you yeah, I'm working on this stuff. 167 00:10:08 --> 00:10:09 I'm learning it. 168 00:10:09 --> 00:10:11 Everybody doesn't it first time. 169 00:10:11 --> 00:10:14 I don't learn it first time, every time I teach the course 170 00:10:14 --> 00:10:16 I learn something more. 171 00:10:16 --> 00:10:19 And if you're learning from this course then 172 00:10:19 --> 00:10:21 I'm totally happy. 173 00:10:21 --> 00:10:23 And I believe that's the case. 174 00:10:23 --> 00:10:25 So I am entirely happy. 175 00:10:25 --> 00:10:32 And I hope the quiz, some points of it I wish 176 00:10:32 --> 00:10:34 I'd prepared better. 177 00:10:34 --> 00:10:39 But I feel pretty good about it. 178 00:10:39 --> 00:10:42 I feel good about it, let me just say. 179 00:10:42 --> 00:10:48 So, and I'm happy to have any comments, email or in person. 180 00:10:48 --> 00:10:50 But allow me to go forward with trusses. 181 00:10:50 --> 00:10:53 However, I'm ready always for a comment. 182 00:10:53 --> 00:10:54 Yeah. 183 00:10:54 --> 00:10:56 OK. 184 00:10:56 --> 00:10:58 Anyway, enjoy trusses. 185 00:10:58 --> 00:10:59 Enjoy life. 186 00:10:59 --> 00:11:02 Yeah. 187 00:11:02 --> 00:11:05 And this should have been in the book, this page. 188 00:11:05 --> 00:11:11 So if I wasn't too late I would paste it in. 189 00:11:11 --> 00:11:15 Because this connects A transpose A to a special 190 00:11:15 --> 00:11:19 matrices, and the way I had in my mind but I didn't put it on 191 00:11:19 --> 00:11:22 the board until just now. 192 00:11:22 --> 00:11:29 OK, so I'll cover that up and ready to go with trusses. 193 00:11:29 --> 00:11:31 OK. 194 00:11:31 --> 00:11:34 So trusses, we want to know what's up. 195 00:11:34 --> 00:11:36 We want to get the setup right. 196 00:11:36 --> 00:11:45 Once we get the setup we'll know we're looking for. 197 00:11:45 --> 00:11:51 OK, so a truss is a bunch of elastic bars with pin 198 00:11:51 --> 00:11:52 joints connecting them. 199 00:11:52 --> 00:11:55 Now, what do I mean by a pin joint? 200 00:11:55 --> 00:12:05 I mean that stretching the bars takes force. 201 00:12:05 --> 00:12:09 Turning around the pin joint doesn't take force. 202 00:12:09 --> 00:12:18 So the pin just lets them turn, so we'll have 203 00:12:18 --> 00:12:20 forces in those bars. 204 00:12:20 --> 00:12:24 So it's like masses and springs. 205 00:12:24 --> 00:12:26 Exactly like masses and springs. 206 00:12:26 --> 00:12:30 But yet we have a 2-D problem. 207 00:12:30 --> 00:12:32 So it's a two dimensional problem with masses 208 00:12:32 --> 00:12:33 and springs. 209 00:12:33 --> 00:12:38 And we could certainly have a 3-D truss, but 2-D makes 210 00:12:38 --> 00:12:39 all the important points. 211 00:12:39 --> 00:12:41 And then I can count the bars. 212 00:12:41 --> 00:12:44 One, two, three, four, five. 213 00:12:44 --> 00:12:46 And I can count the nodes. 214 00:12:46 --> 00:12:48 There happen to be five here. 215 00:12:48 --> 00:12:50 But now comes the moment. 216 00:12:50 --> 00:12:53 I have to tell you, what are the unknowns? 217 00:12:53 --> 00:12:54 What are the u's. 218 00:12:54 --> 00:12:58 Because of course, you know that I'm going to go from u's 219 00:12:58 --> 00:13:05 to e's to w's, to forces. f. 220 00:13:05 --> 00:13:08 And you know that a matrix A is going to do that. 221 00:13:08 --> 00:13:11 A matrix C is going to do that. 222 00:13:11 --> 00:13:13 A matrix A transpose is going to do that. 223 00:13:13 --> 00:13:17 You're all ready, we need to know what's the setup. 224 00:13:17 --> 00:13:18 What are these matrices. 225 00:13:18 --> 00:13:20 OK, and how many. 226 00:13:20 --> 00:13:22 So let me explain the setup. 227 00:13:22 --> 00:13:32 Typical node, node one. we have forces in these bars, so that 228 00:13:32 --> 00:13:38 node one could have a force. 229 00:13:38 --> 00:13:39 We're in the plane. 230 00:13:39 --> 00:13:44 So we have a horizontal force and a vertical force. 231 00:13:44 --> 00:13:49 Together, that would produces a force in any 232 00:13:49 --> 00:13:50 direction whatever. 233 00:13:50 --> 00:13:51 So this is the key point. 234 00:13:51 --> 00:13:59 That there is a horizontal force. f, horizontal one. 235 00:13:59 --> 00:14:01 The one being the force on node one. 236 00:14:01 --> 00:14:04 But there's also a vertical force. 237 00:14:04 --> 00:14:08 And let me take horizontal to the right, positive 238 00:14:08 --> 00:14:09 to the right. 239 00:14:09 --> 00:14:11 Vertical positive upwards. 240 00:14:11 --> 00:14:13 Just to have a convention. 241 00:14:13 --> 00:14:17 So how many f's have I got? 242 00:14:17 --> 00:14:21 Well, the point is I now have two per node. 243 00:14:21 --> 00:14:22 That's the difference. 244 00:14:22 --> 00:14:25 I have two per node, two forces. and I have two 245 00:14:25 --> 00:14:27 displacements per node. 246 00:14:27 --> 00:14:32 Because that point under, there will be more forces. 247 00:14:32 --> 00:14:38 Some maybe pulling this way, whatever. 248 00:14:38 --> 00:14:40 Maybe let's look at node two. 249 00:14:40 --> 00:14:42 So node two could have a couple of forces on it. 250 00:14:42 --> 00:14:45 F H two, and F V two. 251 00:14:45 --> 00:14:49 And it moves like the other nodes. 252 00:14:49 --> 00:14:55 So now I'm introducing the unknowns. u is the movement. u, 253 00:14:55 --> 00:14:59 again horizontal, and again, now we're talking 254 00:14:59 --> 00:15:01 about node two. 255 00:15:01 --> 00:15:04 And it moves up. 256 00:15:04 --> 00:15:05 Or doesn't, or moves down. 257 00:15:05 --> 00:15:10 But that's an unknown. u, a displacement, a vertical 258 00:15:10 --> 00:15:12 displacement of node two. 259 00:15:12 --> 00:15:14 Do you see the setup? 260 00:15:14 --> 00:15:17 Two forces per node. 261 00:15:17 --> 00:15:20 Two displacements per node. 262 00:15:20 --> 00:15:27 So that's like, the number of unknown is like double. 263 00:15:27 --> 00:15:31 Like, doubled, and that produces an interesting 264 00:15:31 --> 00:15:33 situation. 265 00:15:33 --> 00:15:36 I've marked supports here. 266 00:15:36 --> 00:15:38 So let's just speak about supports. 267 00:15:38 --> 00:15:41 So what's happening at the supports? 268 00:15:41 --> 00:15:49 At the support there's no movement. 269 00:15:49 --> 00:15:51 The whole that point is pinned. 270 00:15:51 --> 00:15:59 So this is telling me that u horizontal five is zero and u 271 00:15:59 --> 00:16:04 vertical, sorry that was four and it'll be the same for five. 272 00:16:04 --> 00:16:06 u vertical five is zero. 273 00:16:06 --> 00:16:10 It's like grounding a node in the electrical case. 274 00:16:10 --> 00:16:13 We just see this pattern over and over. 275 00:16:13 --> 00:16:17 And we want to see OK, what does it look like for trusses? 276 00:16:17 --> 00:16:20 So here's a support that fixes those. 277 00:16:20 --> 00:16:22 So those are not unknowns. 278 00:16:22 --> 00:16:24 And similarly, they're not unknowns there. 279 00:16:24 --> 00:16:27 Still saying five when I mean four. 280 00:16:27 --> 00:16:33 So those are boundary conditions, n 281 00:16:33 --> 00:16:34 conditions, whatever. 282 00:16:34 --> 00:16:36 And similarly here. 283 00:16:36 --> 00:16:39 So how many unknowns are there? 284 00:16:39 --> 00:16:45 Now look at this picture, how many unknown displacements 285 00:16:45 --> 00:16:48 are there in this truss? 286 00:16:48 --> 00:16:49 Six. 287 00:16:49 --> 00:16:50 Six, right? 288 00:16:50 --> 00:16:53 Two here, two here, two here and none there. 289 00:16:53 --> 00:17:00 So the number of actual unknowns is six. 290 00:17:00 --> 00:17:06 My idea would be that it's twice the number of nodes 291 00:17:06 --> 00:17:14 minus the number of fixed things, that R would 292 00:17:14 --> 00:17:16 be four in this case. 293 00:17:16 --> 00:17:20 I've got two fixed here and two fixed here, so this 294 00:17:20 --> 00:17:23 would be two times five. 295 00:17:23 --> 00:17:27 Ten possible displacements but R counts the number of fixed 296 00:17:27 --> 00:17:32 displacements, four, and leaves us with six. 297 00:17:32 --> 00:17:34 OK. 298 00:17:34 --> 00:17:38 So my matrix A will now be, it's always m by n. 299 00:17:38 --> 00:17:42 My matrix A will be five by six. 300 00:17:42 --> 00:17:46 OK. 301 00:17:46 --> 00:17:49 Now you're going to ask what is that matrix. 302 00:17:49 --> 00:17:53 But let me hold that off for a little moment. 303 00:17:53 --> 00:17:56 I want to just see its shape first. 304 00:17:56 --> 00:18:02 So you could now do this for a large truss, right? 305 00:18:02 --> 00:18:06 You count the bars, and you could count the nodes. 306 00:18:06 --> 00:18:09 And then you could count the unknown displacements, u. 307 00:18:09 --> 00:18:17 So there are six u's here. 308 00:18:17 --> 00:18:19 And there are five e's. 309 00:18:19 --> 00:18:22 And there are five bar forces. 310 00:18:22 --> 00:18:29 And there are six equilibrium, balance, force balances. 311 00:18:29 --> 00:18:34 Six, six for the node count, for the unknowns count, five, 312 00:18:34 --> 00:18:36 five for the bars count. 313 00:18:36 --> 00:18:44 OK, now here's a point about this particular truss. 314 00:18:44 --> 00:18:46 It's not safe to get on it. 315 00:18:46 --> 00:18:47 Right? 316 00:18:47 --> 00:18:49 And I want to say why is it not safe. 317 00:18:49 --> 00:18:54 So this is a feature that comes into the truss question that 318 00:18:54 --> 00:18:58 makes it a little new and more interesting. 319 00:18:58 --> 00:19:03 A little twist compared to the previous examples. 320 00:19:03 --> 00:19:07 That bar, that truss, I wouldn't stand on it. 321 00:19:07 --> 00:19:09 Now, why not? 322 00:19:09 --> 00:19:11 Well, purely for linear algebra reasons. 323 00:19:11 --> 00:19:13 Of course. 324 00:19:13 --> 00:19:15 The matrix A is five by six. 325 00:19:15 --> 00:19:18 So now what do we know about a matrix that's five by six? 326 00:19:18 --> 00:19:23 So A is five by six. 327 00:19:23 --> 00:19:28 Five rows, as always the m; six columns, because now 328 00:19:28 --> 00:19:29 we have six unknowns. 329 00:19:29 --> 00:19:36 And what do I know about any five by six matrix? 330 00:19:36 --> 00:19:39 I want to ask about the equation Au=0. 331 00:19:39 --> 00:19:42 332 00:19:42 --> 00:19:44 So I want to ask about it in linear algebra language and 333 00:19:44 --> 00:19:47 then I want to ask about it in physical language. 334 00:19:47 --> 00:19:51 And the beauty is the thing that makes trusses sort of 335 00:19:51 --> 00:19:57 fun is, these matrices, A, get pretty big fast. 336 00:19:57 --> 00:20:01 Because when I put a few more nodes on, the book has a 337 00:20:01 --> 00:20:04 picture of a sort of treehouse. 338 00:20:04 --> 00:20:08 Then A is growing. 339 00:20:08 --> 00:20:12 And I don't, all the time, write down the matrix A. 340 00:20:12 --> 00:20:13 I haven't written it down here. 341 00:20:13 --> 00:20:16 What I've written down it's just its size, because that's 342 00:20:16 --> 00:20:21 enough to tell us something about this set of 343 00:20:21 --> 00:20:22 equations Au=0. 344 00:20:24 --> 00:20:25 What's the story Au=0? 345 00:20:27 --> 00:20:29 Well of course it has the solution u=0. 346 00:20:29 --> 00:20:31 Nothing moving. 347 00:20:31 --> 00:20:35 If I have no displacement, if the u's are all zero, then 348 00:20:35 --> 00:20:37 I have no stretching. 349 00:20:37 --> 00:20:39 The e's are stretching. 350 00:20:39 --> 00:20:41 Elongation, as before. 351 00:20:41 --> 00:20:44 How far does the bar stretch? 352 00:20:44 --> 00:20:44 OK. 353 00:20:44 --> 00:20:51 So if I have zero u's and zero e's, but what other possibility 354 00:20:51 --> 00:20:55 am I going to have here? 355 00:20:55 --> 00:21:00 I'm going to have probably one solution to this 356 00:21:00 --> 00:21:03 system that isn't zero. 357 00:21:03 --> 00:21:08 I'm probably going to have one set of displacements u, 358 00:21:08 --> 00:21:10 look what's happening here. 359 00:21:10 --> 00:21:13 This is Au is the e. 360 00:21:13 --> 00:21:18 So I'm going to have at least one and probably in this case 361 00:21:18 --> 00:21:26 it will be one, there will be one, the neat word for 362 00:21:26 --> 00:21:30 it is a mechanism. 363 00:21:30 --> 00:21:32 And what does that mean? 364 00:21:32 --> 00:21:35 A mechanism is a solution to Au=0. 365 00:21:37 --> 00:21:40 So that, a mechanism is a movement of the bar. 366 00:21:40 --> 00:21:42 So it's going to be non-zero. 367 00:21:42 --> 00:21:45 The bars are going to move a little. 368 00:21:45 --> 00:21:47 Sorry, the nodes are going to move a little. 369 00:21:47 --> 00:21:50 The nodes will move a little bit. 370 00:21:50 --> 00:21:52 In this u, because it isn't zero. 371 00:21:52 --> 00:21:56 But the bars won't stretch. 372 00:21:56 --> 00:22:04 So tells us we've got instability here. 373 00:22:04 --> 00:22:07 If there's a solution to that, that's always telling us that 374 00:22:07 --> 00:22:09 A transpose A is singular. 375 00:22:09 --> 00:22:14 So let me just put that A transpose A, or A transpose 376 00:22:14 --> 00:22:16 C A, C couldn't save it. 377 00:22:16 --> 00:22:18 Will be singular. 378 00:22:18 --> 00:22:25 It's just like our free-free thing in being singular, but 379 00:22:25 --> 00:22:28 the picture doesn't look free free, does it? 380 00:22:28 --> 00:22:33 It's got supports in here, just not good enough. 381 00:22:33 --> 00:22:38 And I believe that if you look at this truss, you could 382 00:22:38 --> 00:22:42 describe, you could tell me, and you could draw, a movement 383 00:22:42 --> 00:22:51 of that truss in which there is displacement but no stretching. 384 00:22:51 --> 00:22:54 Let me ask you how to draw that. 385 00:22:54 --> 00:22:58 And I believe, everybody understood why was 386 00:22:58 --> 00:22:59 there a solution. 387 00:22:59 --> 00:23:03 It was because we have six unknowns and we only 388 00:23:03 --> 00:23:05 have five equations. 389 00:23:05 --> 00:23:07 So this was five equations. 390 00:23:07 --> 00:23:10 Any time you have five equations with a zero on the 391 00:23:10 --> 00:23:15 right hand side, so five homogeneous equations, whatever 392 00:23:15 --> 00:23:18 you want to say when that's zero on the right 393 00:23:18 --> 00:23:20 and six unknowns. 394 00:23:20 --> 00:23:22 Six u's. 395 00:23:22 --> 00:23:25 Then you're going to have solution. 396 00:23:25 --> 00:23:27 You can't help it. 397 00:23:27 --> 00:23:30 You've got that many degrees of freedom, you've only got that 398 00:23:30 --> 00:23:33 many constraints, there's going to be solution. 399 00:23:33 --> 00:23:37 OK, tell me how to draw that. 400 00:23:37 --> 00:23:41 Let me put in the truss now. 401 00:23:41 --> 00:23:43 What's the solution? 402 00:23:43 --> 00:23:49 So this is the fun part in a particular 403 00:23:49 --> 00:23:52 example at the start. 404 00:23:52 --> 00:24:04 How could that move without stretching bars? 405 00:24:04 --> 00:24:08 Let me see. 406 00:24:08 --> 00:24:11 What could happen? 407 00:24:11 --> 00:24:16 What do you mean now, who's going to move where? 408 00:24:16 --> 00:24:18 What's the movement here? 409 00:24:18 --> 00:24:19 And I want to draw it over there. 410 00:24:19 --> 00:24:23 So you give the answer by drawing it as well as by 411 00:24:23 --> 00:24:27 telling me the six unknown u's. 412 00:24:27 --> 00:24:33 So what can happen at this thing? 413 00:24:33 --> 00:24:39 So you're going to say the truss could, these bars 414 00:24:39 --> 00:24:42 could, turn a little? 415 00:24:42 --> 00:24:44 And notice that word a little. 416 00:24:44 --> 00:24:48 We're talking small displacement, small stretches 417 00:24:48 --> 00:24:49 all the time here. 418 00:24:49 --> 00:24:53 I'll show you why we're always making that 419 00:24:53 --> 00:24:58 linearity assumption, or small assumption. 420 00:24:58 --> 00:25:00 OK, those move a little. 421 00:25:00 --> 00:25:05 And what happens to that triangle at the top? 422 00:25:05 --> 00:25:07 It sort of just moves along, right? 423 00:25:07 --> 00:25:11 So the picture you would draw would be that 424 00:25:11 --> 00:25:16 you started there. 425 00:25:16 --> 00:25:20 And it moved along a little, I'll make it a larger 426 00:25:20 --> 00:25:24 displacement than I really have in mind. 427 00:25:24 --> 00:25:26 These guys of course are here. 428 00:25:26 --> 00:25:31 So they come out and the rest of the truss, the top of 429 00:25:31 --> 00:25:34 the truss just kind of goes with it. 430 00:25:34 --> 00:25:36 Goes with the flow. 431 00:25:36 --> 00:25:40 That would be the answer that I would be looking for, 432 00:25:40 --> 00:25:42 to draw the mechanism. 433 00:25:42 --> 00:25:44 That would show it. 434 00:25:44 --> 00:25:47 And if I wanted to write down the u that goes with 435 00:25:47 --> 00:25:50 it, what would it be? 436 00:25:50 --> 00:25:53 Let me again number these guys; this is one, two, 437 00:25:53 --> 00:25:54 and this is three. 438 00:25:54 --> 00:26:02 So what are the displacements of nodes one, two, and three? 439 00:26:02 --> 00:26:07 I'll always write u_1 horizontal before vertical. 440 00:26:07 --> 00:26:09 Can we make an agreement? 441 00:26:09 --> 00:26:12 So I want to know about the horizontal movement then the 442 00:26:12 --> 00:26:16 vertical movement of node one, then node two, then node three. 443 00:26:16 --> 00:26:19 So I'll have six numbers there. 444 00:26:19 --> 00:26:22 And what could I put in for those six numbers? 445 00:26:22 --> 00:26:26 So the horizontal, let me suppose that first 446 00:26:26 --> 00:26:30 guy, I'll put a one. 447 00:26:30 --> 00:26:32 Really that's a bigger number than I should put, but 448 00:26:32 --> 00:26:33 it's a convenient number. 449 00:26:33 --> 00:26:37 So I'll just take it to be one even so I really have in mind. 450 00:26:37 --> 00:26:43 Let's say that's one angstrom, or one tiny little person. 451 00:26:43 --> 00:26:51 OK, so what about the rest? 452 00:26:51 --> 00:26:58 What's the vertical cool, oh yeah this is a key point here, 453 00:26:58 --> 00:27:01 what's the vertical movement. 454 00:27:01 --> 00:27:05 This movement to me is horizontal. 455 00:27:05 --> 00:27:09 I'm going to say that the vertical moment is zero. 456 00:27:09 --> 00:27:11 Of node one, just moves over. 457 00:27:11 --> 00:27:13 And node two does the same. 458 00:27:13 --> 00:27:15 And node three does the same. 459 00:27:15 --> 00:27:19 So that's my solution. . 460 00:27:19 --> 00:27:25 That's a simple movement, a simple set of displacements, 461 00:27:25 --> 00:27:30 think most to the right. 462 00:27:30 --> 00:27:34 I have not written down the matrix A, but probably won't 463 00:27:34 --> 00:27:37 even do it until next time. 464 00:27:37 --> 00:27:41 But you will see that when we do the matrix A for this 465 00:27:41 --> 00:27:49 particular truss will have this particular u as a mechanism. 466 00:27:49 --> 00:27:55 In linear algebra, u is in the null space of A. 467 00:27:55 --> 00:27:57 Au=0, that's all that means. 468 00:27:57 --> 00:28:04 OK, do you see more or less what's up? 469 00:28:04 --> 00:28:07 But now there's one little thing that may 470 00:28:07 --> 00:28:09 be bothering you. 471 00:28:09 --> 00:28:13 Which is what? 472 00:28:13 --> 00:28:19 If I come back to the zero, zero, zero there, you could 473 00:28:19 --> 00:28:25 correctly say wait a minute, if those bars didn't stretch, if 474 00:28:25 --> 00:28:31 they just rotated as you told me to do, then this was mostly 475 00:28:31 --> 00:28:35 across but a little bit down, right? 476 00:28:35 --> 00:28:36 And I'm saying no. 477 00:28:36 --> 00:28:38 I'm saying zero. 478 00:28:38 --> 00:28:42 OK, how do I get away with that? 479 00:28:42 --> 00:28:47 So I'm saying in 18.085 it's a zero. 480 00:28:47 --> 00:28:49 And why? 481 00:28:49 --> 00:28:55 So this is like a little time out just to focus in on, let 482 00:28:55 --> 00:28:57 me focus in on node two. 483 00:28:57 --> 00:29:02 So here's the bottom node four, so it used to be 484 00:29:02 --> 00:29:04 vertical up to two. 485 00:29:04 --> 00:29:06 This was node two and this was number four. 486 00:29:06 --> 00:29:11 And then it rotated a little, to there. 487 00:29:11 --> 00:29:13 To this position. 488 00:29:13 --> 00:29:23 So it went, if this angle was, let's say, theta, then what 489 00:29:23 --> 00:29:26 is that actual position? 490 00:29:26 --> 00:29:30 So let's say this was, let's say the bar had length one. 491 00:29:30 --> 00:29:33 This is the origin, zero, zero. 492 00:29:33 --> 00:29:37 This is the point zero, one above it, OK? 493 00:29:37 --> 00:29:41 And now, that's before it moved. 494 00:29:41 --> 00:29:42 Then it moved a little bit. 495 00:29:42 --> 00:29:45 It moved to an angle theta. 496 00:29:45 --> 00:29:48 What's the position of that bar? 497 00:29:48 --> 00:29:48 Of that node? 498 00:29:48 --> 00:29:52 What's the new position of the node, and then we'll look at 499 00:29:52 --> 00:29:55 the difference and we'll see the movement u, 500 00:29:55 --> 00:29:56 the displacement. 501 00:29:56 --> 00:30:06 So how far did it move? 502 00:30:06 --> 00:30:08 What's the x coordinate? 503 00:30:08 --> 00:30:10 How far did it go across? 504 00:30:10 --> 00:30:13 If I put in that line you'll know. 505 00:30:13 --> 00:30:16 So the movement across was, sin(theta)? 506 00:30:16 --> 00:30:18 Good. sin(theta). 507 00:30:20 --> 00:30:28 And the movement down, well, yes, so let's find its 508 00:30:28 --> 00:30:30 position and then we'll take the difference. 509 00:30:30 --> 00:30:41 So what's the vertical new position for that guy? 510 00:30:41 --> 00:30:47 It moved by, it moved across by sin(theta). 511 00:30:47 --> 00:30:50 512 00:30:50 --> 00:30:55 And down by 1-cos(theta). 513 00:30:56 --> 00:30:58 Are you agreed with that? 514 00:30:58 --> 00:31:04 Because here is cosine theta, right there. 515 00:31:04 --> 00:31:08 And here's the little bit it moved down. 516 00:31:08 --> 00:31:09 OK. 517 00:31:09 --> 00:31:12 So these are exactly correct. 518 00:31:12 --> 00:31:16 Yeah this is in the position of sin(theta), cos(theta), and the 519 00:31:16 --> 00:31:18 difference was the 1-cos(theta). 520 00:31:18 --> 00:31:21 521 00:31:21 --> 00:31:23 OK, so now here comes the key point. 522 00:31:23 --> 00:31:28 Approximately sin(theta) is approximately, if theta is 523 00:31:28 --> 00:31:31 small and now here comes the smallness, sin(theta) is 524 00:31:31 --> 00:31:35 approximately theta. sin(theta)'s 525 00:31:35 --> 00:31:36 approximately theta. 526 00:31:36 --> 00:31:40 And 1-cos(theta) is approximately what? 527 00:31:40 --> 00:31:48 Now, this is the important point. 528 00:31:48 --> 00:31:51 So theta is like the first term. 529 00:31:51 --> 00:31:56 If I expand, I mean, the exact term would be theta minus theta 530 00:31:56 --> 00:31:59 cubed over six, dot dot dot. 531 00:31:59 --> 00:32:03 But I'm only keeping that term. 532 00:32:03 --> 00:32:08 And 1-cos(theta), now what's the formula for cos(theta)? 533 00:32:08 --> 00:32:10 This is like worth, just should? 534 00:32:10 --> 00:32:15 It's a one, because of course cos(0) is one, and then 535 00:32:15 --> 00:32:17 you subtract what? 536 00:32:17 --> 00:32:19 Theta squared over two. 537 00:32:19 --> 00:32:21 And so on. 538 00:32:21 --> 00:32:25 And then plus theta fourtg over 24 or whatever. 539 00:32:25 --> 00:32:29 OK, so the ones cancel as we expect. 540 00:32:29 --> 00:32:35 And I'm getting theta squared over two. 541 00:32:35 --> 00:32:40 And this, here was theta to the first power. 542 00:32:40 --> 00:32:42 Theta cubed was, we didn't care. 543 00:32:42 --> 00:32:45 And we don't care about theta squared. 544 00:32:45 --> 00:32:48 So that's why it's zero. 545 00:32:48 --> 00:32:53 Because it's a second order movement. 546 00:32:53 --> 00:32:57 If theta is small, as I'm going to assume, small displacement, 547 00:32:57 --> 00:33:01 theta squared would, if I allowed theta squared and 548 00:33:01 --> 00:33:05 cos(theta) in here, I'd have a non linear problem. 549 00:33:05 --> 00:33:07 And I don't want that. 550 00:33:07 --> 00:33:08 And I don't need it. 551 00:33:08 --> 00:33:14 I mean, finite elements, structures, bridges, whatever. 552 00:33:14 --> 00:33:19 Your first hope and expectation and calculation is small 553 00:33:19 --> 00:33:22 theta linear problem. 554 00:33:22 --> 00:33:27 So to a linear person theta squared is zero. 555 00:33:27 --> 00:33:31 That's why those guys are zero. 556 00:33:31 --> 00:33:36 OK, so that's an assumption we'll often see, 557 00:33:36 --> 00:33:41 so it kind of was. 558 00:33:41 --> 00:33:46 There are two kinds of non-linearities in 559 00:33:46 --> 00:33:49 structures and elasticity. 560 00:33:49 --> 00:33:54 One would be to allow this geometric non-linearity. 561 00:33:54 --> 00:33:58 Theta's large displacements, theta large enough so that you 562 00:33:58 --> 00:34:01 can't neglect theta squared. 563 00:34:01 --> 00:34:02 That's a tough one. 564 00:34:02 --> 00:34:06 If you allow geometric non-linearity in, as finite 565 00:34:06 --> 00:34:08 element codes have to do. 566 00:34:08 --> 00:34:14 If you have ABAQUS is a code that does major finite element 567 00:34:14 --> 00:34:17 calculations, nonlinear ones. 568 00:34:17 --> 00:34:20 I mean they, at the beginning they were studying what 569 00:34:20 --> 00:34:23 happens, what are the stresses on cables under the Atlantic. 570 00:34:23 --> 00:34:26 I mean, those are fascinating problems. 571 00:34:26 --> 00:34:29 Or I mention car crashes. 572 00:34:29 --> 00:34:32 I mean, car crashes, the geometry changes, you 573 00:34:32 --> 00:34:33 have big displacements. 574 00:34:33 --> 00:34:37 But we're talking here about linear small 575 00:34:37 --> 00:34:39 displacement cases. 576 00:34:39 --> 00:34:44 OK, so don't forget that part. 577 00:34:44 --> 00:34:49 That when the truss gets more complicated, the 578 00:34:49 --> 00:34:51 principle stays the same. 579 00:34:51 --> 00:34:55 That we distinguish between the thetas that matter and the 580 00:34:55 --> 00:34:57 theta squareds that don't. 581 00:34:57 --> 00:35:01 OK, so now what? 582 00:35:01 --> 00:35:05 Now I guess I'm ready to complete this picture 583 00:35:05 --> 00:35:06 a little more. 584 00:35:06 --> 00:35:07 OK. 585 00:35:07 --> 00:35:11 So let me, so we've understood what the idea of a mechanism. 586 00:35:11 --> 00:35:17 Oh, how could I prevent a mechanism? 587 00:35:17 --> 00:35:24 In other words, if I stood on this truss, the slightest bit 588 00:35:24 --> 00:35:29 of wind would crash it down, right? 589 00:35:29 --> 00:35:32 So that's unstable. 590 00:35:32 --> 00:35:34 That's an unstable truss. 591 00:35:34 --> 00:35:36 How could I make it stable? 592 00:35:36 --> 00:35:43 I mean if you were designing this thing, what would you do? 593 00:35:43 --> 00:35:44 Add another edge. 594 00:35:44 --> 00:35:46 You'd stick in another bar. 595 00:35:46 --> 00:35:49 Maybe stick in a bar there. 596 00:35:49 --> 00:35:51 What would happen now? 597 00:35:51 --> 00:35:53 Would it now be stable? 598 00:35:53 --> 00:35:54 You'd have to answer that question. 599 00:35:54 --> 00:35:57 You couldn't just put in bars, whatever. 600 00:35:57 --> 00:36:01 You want to put bars that do the job. 601 00:36:01 --> 00:36:02 OK, now how many bars? 602 00:36:02 --> 00:36:04 We've now got six bars. 603 00:36:04 --> 00:36:07 So m is now up to six. 604 00:36:07 --> 00:36:12 The matrix A is six by six now, whatever that matrix may be. 605 00:36:12 --> 00:36:15 We have six bars, six displacements. 606 00:36:15 --> 00:36:19 We can hope that we now have a six by six, well, we do 607 00:36:19 --> 00:36:23 have a six by six matrix, whatever it looks like. 608 00:36:23 --> 00:36:27 And we can hope that it's not singular. 609 00:36:27 --> 00:36:29 We can hope it's invertible, we can hope that that 610 00:36:29 --> 00:36:30 mechanism is killed. 611 00:36:30 --> 00:36:33 And you see it is killed. 612 00:36:33 --> 00:36:36 The six by six, that truss is now stable. 613 00:36:36 --> 00:36:42 No mechanism there, right? 614 00:36:42 --> 00:36:47 Again I haven't written down the matrix, but I'm really 615 00:36:47 --> 00:36:49 calling for engineering intuition here. 616 00:36:49 --> 00:36:54 That this truss is now stable, and of course I can make 617 00:36:54 --> 00:36:57 it even more stable by adding a seventh edge. 618 00:36:57 --> 00:36:59 A seventh bar. 619 00:36:59 --> 00:37:03 So when it was six I had square matrices, A transpose and then 620 00:37:03 --> 00:37:06 C and then A would have been square. 621 00:37:06 --> 00:37:11 Now I've got seven bars and, so now I've put in a seventh 622 00:37:11 --> 00:37:15 guy. m is now up to seven. 623 00:37:15 --> 00:37:24 My matrix a would now be seven by six. 624 00:37:24 --> 00:37:29 Mechanism will be gone because I've now got, 625 00:37:29 --> 00:37:32 what, seven equations. 626 00:37:32 --> 00:37:33 Same six u's. 627 00:37:33 --> 00:37:39 So we begin to get a feel of, are there solutions or not? 628 00:37:39 --> 00:37:44 What I'm saying is I can't tell just from the count 629 00:37:44 --> 00:37:46 that a is not singular. 630 00:37:46 --> 00:37:51 I could have a lot of bars and still be unstable. 631 00:37:51 --> 00:37:53 Invent a truss for me. 632 00:37:53 --> 00:37:58 Just because, how could you invent a truss that had, 633 00:37:58 --> 00:38:00 maybe it has seven bars. 634 00:38:00 --> 00:38:05 With those seven bars, those diagonal guys, that did it. 635 00:38:05 --> 00:38:06 That made it stable. 636 00:38:06 --> 00:38:10 Our eye tells us that before we do any linear algebra. 637 00:38:10 --> 00:38:15 Tell me a seven by, a thing. 638 00:38:15 --> 00:38:17 Well, yeah. 639 00:38:17 --> 00:38:23 OK, so here would be, shall we support both of these? 640 00:38:23 --> 00:38:26 I'll start out the same, OK. 641 00:38:26 --> 00:38:32 Now, yeah, how could I, let's see, I've haven't prepared. 642 00:38:32 --> 00:38:36 How could I get a whole lot of bars. 643 00:38:36 --> 00:38:39 I might not get seven by six exactly, but how could I have 644 00:38:39 --> 00:38:44 plenty of bars and still unstable? 645 00:38:44 --> 00:38:48 Well, suppose I do this. 646 00:38:48 --> 00:38:51 Oh yeah, that's a good example. 647 00:38:51 --> 00:38:54 That's not stable, right? 648 00:38:54 --> 00:38:54 OK. 649 00:38:54 --> 00:38:57 Every let's practice with that one. 650 00:38:57 --> 00:39:01 That's just my idea, and problems in the book just 651 00:39:01 --> 00:39:04 ask you to practice with things like that. 652 00:39:04 --> 00:39:06 Tell me the count, first. 653 00:39:06 --> 00:39:10 What is m, the number of bars? 654 00:39:10 --> 00:39:11 Six. 655 00:39:11 --> 00:39:14 What is n, the number of unknowns, little n, the 656 00:39:14 --> 00:39:15 number of unknowns? 657 00:39:15 --> 00:39:18 What's the shape of my matrix here? 658 00:39:18 --> 00:39:25 A is, it's got six bars and how many unknowns? 659 00:39:25 --> 00:39:26 Eight. 660 00:39:26 --> 00:39:30 Eight, right? two here, two, two, two. 661 00:39:30 --> 00:39:31 None here. 662 00:39:31 --> 00:39:33 Six by eight. 663 00:39:33 --> 00:39:34 OK. 664 00:39:34 --> 00:39:38 And how many mechanisms am I now expecting? 665 00:39:38 --> 00:39:40 Probably two. 666 00:39:40 --> 00:39:44 Probably two, there would be two independent 667 00:39:44 --> 00:39:45 mechanisms here. 668 00:39:45 --> 00:39:47 Can you tell me what they look like? 669 00:39:47 --> 00:39:48 Draw them. 670 00:39:48 --> 00:39:50 What would they look like? 671 00:39:50 --> 00:39:53 What would be two different things that could happen, could 672 00:39:53 --> 00:39:58 go wrong with that truss? 673 00:39:58 --> 00:40:00 You see it, right? 674 00:40:00 --> 00:40:02 This could turn. 675 00:40:02 --> 00:40:08 As in our example with the top part moving with it. 676 00:40:08 --> 00:40:13 Or, second one possibility would be the top part goes. 677 00:40:13 --> 00:40:15 And the bottom part stays. 678 00:40:15 --> 00:40:17 Or any combination. 679 00:40:17 --> 00:40:20 So the whole thing could go like that. 680 00:40:20 --> 00:40:21 That would be one. 681 00:40:21 --> 00:40:24 But that wouldn't be the only one, of course. 682 00:40:24 --> 00:40:27 So in other words, we have a two dimensional space of 683 00:40:27 --> 00:40:31 mechanisms and you could give me two different, and there are 684 00:40:31 --> 00:40:36 not just two guys, all their combinations are there. 685 00:40:36 --> 00:40:40 So this would have two mechanisms. 686 00:40:40 --> 00:40:41 Two mechanisms. 687 00:40:41 --> 00:40:45 And I could put in bars, of course, that would 688 00:40:45 --> 00:40:46 try to save it. 689 00:40:46 --> 00:40:50 Well, how many bars, what's the minimum number of bars 690 00:40:50 --> 00:40:54 I absolutely need to make this thing stable again? 691 00:40:54 --> 00:40:55 Two. 692 00:40:55 --> 00:41:00 Well, now suppose I put in these two bars. 693 00:41:00 --> 00:41:01 Right? 694 00:41:01 --> 00:41:04 I've got enough bars, I've got an eight by eight matrix, 695 00:41:04 --> 00:41:06 but I haven't saved it. 696 00:41:06 --> 00:41:07 Right? 697 00:41:07 --> 00:41:09 Because it still has that mechanism. 698 00:41:09 --> 00:41:14 So you can't assume that because the count is right 699 00:41:14 --> 00:41:17 you've avoided mechanisms because in that 700 00:41:17 --> 00:41:18 example you haven't. 701 00:41:18 --> 00:41:23 OK, so that would be a case of square eight by 702 00:41:23 --> 00:41:29 eight, but not good. 703 00:41:29 --> 00:41:34 So as soon as I say there's a solution to Au=0, I know that A 704 00:41:34 --> 00:41:36 transpose A will be singular. 705 00:41:36 --> 00:41:38 And unstable. 706 00:41:38 --> 00:41:43 OK, before I go to the framework let's just 707 00:41:43 --> 00:41:44 do one more thing. 708 00:41:44 --> 00:41:51 Suppose I take away the supports. 709 00:41:51 --> 00:41:57 All right, let me put in some bars, though. 710 00:41:57 --> 00:42:00 I'll put in some bars. 711 00:42:00 --> 00:42:03 OK, plenty of bars. 712 00:42:03 --> 00:42:04 Want another one? 713 00:42:04 --> 00:42:06 OK, how many bars have I got? 714 00:42:06 --> 00:42:08 Lots, right? 715 00:42:08 --> 00:42:14 OK, now the matrix A, what do you think about this? 716 00:42:14 --> 00:42:16 Are there solutions? 717 00:42:16 --> 00:42:18 You haven't even seen the matrix A, of course, but 718 00:42:18 --> 00:42:21 you've seen the truss, that's what matters. 719 00:42:21 --> 00:42:23 How many solutions, are there solutions to Au=0? 720 00:42:25 --> 00:42:31 Are there ways that this truss could move without stretching? 721 00:42:31 --> 00:42:35 Are there ways that this truss could move without stretching? 722 00:42:35 --> 00:42:38 And what are they, and how many are there? 723 00:42:38 --> 00:42:40 And what name should we use? 724 00:42:40 --> 00:42:41 OK, what are they? 725 00:42:41 --> 00:42:46 How could that move without stretching? 726 00:42:46 --> 00:42:48 Well, it's got no supports at all. 727 00:42:48 --> 00:42:50 It's just free out there in space. 728 00:42:50 --> 00:42:51 So it could move. 729 00:42:51 --> 00:42:54 How many ways could it move? 730 00:42:54 --> 00:42:55 Three. 731 00:42:55 --> 00:42:58 It could move, everybody could move this way. 732 00:42:58 --> 00:43:01 All ones on the horizontal guys. 733 00:43:01 --> 00:43:05 Everybody could move this way, all six ones on the vertical 734 00:43:05 --> 00:43:08 guys, or be 12 unknowns here. 735 00:43:08 --> 00:43:12 And it could also rotate, what would be the rotation? 736 00:43:12 --> 00:43:15 I'm not talking about this rotation. 737 00:43:15 --> 00:43:16 This could not happen. 738 00:43:16 --> 00:43:17 What could happen? 739 00:43:17 --> 00:43:21 What rotation could happen here? 740 00:43:21 --> 00:43:25 For this, there's a third rigid motion. 741 00:43:25 --> 00:43:31 Translation, translation, and rotation around, well take 742 00:43:31 --> 00:43:34 this one as an example. 743 00:43:34 --> 00:43:37 The whole thing could swing around this. 744 00:43:37 --> 00:43:38 That would be a motion. 745 00:43:38 --> 00:43:41 Well now you're going to say well, why didn't I swing 746 00:43:41 --> 00:43:42 it around that one? 747 00:43:42 --> 00:43:44 And of course it could. 748 00:43:44 --> 00:43:47 But what would be the deal? 749 00:43:47 --> 00:43:56 It would have to be, there are only three rigid motions, 750 00:43:56 --> 00:43:59 right, up and around. 751 00:43:59 --> 00:44:01 So if you give me another one, like, around this one, then 752 00:44:01 --> 00:44:04 somehow it had to be a combination of those. 753 00:44:04 --> 00:44:07 I don't even want to think what combination it is. 754 00:44:07 --> 00:44:09 But there are three rigid motions. 755 00:44:09 --> 00:44:15 So I sort of distinguish mechanisms, this 756 00:44:15 --> 00:44:16 word mechanism. 757 00:44:16 --> 00:44:21 So that's where the truss deforms. 758 00:44:21 --> 00:44:27 In these and rigid motions, so I'll say plus, possibly. 759 00:44:27 --> 00:44:40 Plus rigid motions, and rigid motions would be, you know, it 760 00:44:40 --> 00:44:44 doesn't deform internally, the whole thing moves. 761 00:44:44 --> 00:44:48 And this is of course what we get in the case of 762 00:44:48 --> 00:44:50 not enough supports. 763 00:44:50 --> 00:44:53 And this is what we get in the case of not enough bars. 764 00:44:53 --> 00:44:54 Yeah. 765 00:44:54 --> 00:44:57 So maybe it's worth separating those two. 766 00:44:57 --> 00:45:02 In the examples we do, we'll usually put in enough supports 767 00:45:02 --> 00:45:04 to kill the rigid motions. 768 00:45:04 --> 00:45:07 And then the question would be are there some mechanisms. 769 00:45:07 --> 00:45:08 OK. 770 00:45:08 --> 00:45:14 Now, I have to start on what this is. 771 00:45:14 --> 00:45:16 Well it'll be just a very quick start. 772 00:45:16 --> 00:45:19 So what I'll do at the beginning of Friday, 773 00:45:19 --> 00:45:22 so Friday's the other lecture on this topic. 774 00:45:22 --> 00:45:25 And then the homework will ask you to do some 775 00:45:25 --> 00:45:27 trusses in this section. 776 00:45:27 --> 00:45:31 It's probably Section 2. something. 777 00:45:31 --> 00:45:38 2.7, maybe. 778 00:45:38 --> 00:45:40 What's the matrix C? 779 00:45:40 --> 00:45:44 Last second question, what's the matrix C, what size is it? 780 00:45:44 --> 00:45:49 What size is the matrix C for our original problem? 781 00:45:49 --> 00:45:51 Or no. 782 00:45:51 --> 00:45:57 What size is C, is C involving, if I know these numbers, 783 00:45:57 --> 00:46:00 what size is C? 784 00:46:00 --> 00:46:02 Five by five. m by m, right? 785 00:46:02 --> 00:46:07 C is the diagonal matrix, one entry for each 786 00:46:07 --> 00:46:09 bar, C is just C. 787 00:46:09 --> 00:46:16 It has a c_1, c_2, c_3, and this w=Ce, it's just 788 00:46:16 --> 00:46:19 Hooke's Law on each bar. 789 00:46:19 --> 00:46:22 So, simple. 790 00:46:22 --> 00:46:25 It gets there in the middle, just the way that C in the 791 00:46:25 --> 00:46:29 first exam problem popped in, and other C's. 792 00:46:29 --> 00:46:32 That gets there in the middle. 793 00:46:32 --> 00:46:34 But it's very, extremely, simple. 794 00:46:34 --> 00:46:38 OK, so the real attention is on A, as usual. 795 00:46:38 --> 00:46:42 And that will come Friday morning.