1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:06 under a Creative Commons license. 3 00:00:06 --> 00:00:07 Your support will help MIT OpenCourseWare continue to 4 00:00:07 --> 00:00:09 offer high-quality educational resources for free. 5 00:00:09 --> 00:00:13 To make a donation, or to view additional materials from 6 00:00:13 --> 00:00:19 hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19 --> 00:00:20 at ocw.mit.edu. 8 00:00:20 --> 00:00:21 PROFESSOR STRANG: So I'm hoping you will ask 9 00:00:21 --> 00:00:25 some questions today. 10 00:00:25 --> 00:00:27 So we've had the exam. 11 00:00:27 --> 00:00:31 If you have any questions about grading the two TAs 12 00:00:31 --> 00:00:33 are the people to speak to. 13 00:00:33 --> 00:00:37 Remy's graded Questions one and two, and Peter 14 00:00:37 --> 00:00:38 graded three and four. 15 00:00:38 --> 00:00:40 And they control things. 16 00:00:40 --> 00:00:45 I mean, I can help if there's an emergency, but they would be 17 00:00:45 --> 00:00:47 the right people to speak to. 18 00:00:47 --> 00:00:51 Because they know how they graded the whole test. 19 00:00:51 --> 00:00:58 Before you ask questions, can I just say why this truss, you 20 00:00:58 --> 00:01:04 remember this six-sided truss, tied me in a knot and I'm 21 00:01:04 --> 00:01:11 hoping your MATLAB solution will untie that knot. 22 00:01:11 --> 00:01:15 The knot I was in was to find the mechanisms, to find 23 00:01:15 --> 00:01:21 convenient mechanisms because we have, well, I thought 24 00:01:21 --> 00:01:23 we had six bars. 25 00:01:23 --> 00:01:26 It looks like we have six bars. 26 00:01:26 --> 00:01:30 But somebody pointed out after class that that bar six 27 00:01:30 --> 00:01:33 is not very active. 28 00:01:33 --> 00:01:38 It's connecting two supports, can't do anything, and actually 29 00:01:38 --> 00:01:44 that sixth row of A matrix will be all zeroes. 30 00:01:44 --> 00:01:48 So our matrix, if we include that, there was 31 00:01:48 --> 00:01:51 no use, it doesn't help. 32 00:01:51 --> 00:01:56 So our matrix then really comes from the five bars, so A is 33 00:01:56 --> 00:02:01 then five, by two unknowns. 34 00:02:01 --> 00:02:05 Two, two, two, making altogether eight unknowns. 35 00:02:05 --> 00:02:12 So three mechanisms and that's what I'm hoping for. 36 00:02:12 --> 00:02:17 So when I unwisely drew that picture on the board at the end 37 00:02:17 --> 00:02:21 of the truss lecture, I was only looking for two mechanisms 38 00:02:21 --> 00:02:26 because I was thinking we had six edges, six bars, but really 39 00:02:26 --> 00:02:35 this bar, when I knock out the columns that correspond this, 40 00:02:35 --> 00:02:38 that node and that node, there will only be zeroes 41 00:02:38 --> 00:02:40 left in that row. 42 00:02:40 --> 00:02:42 And it's nothing. 43 00:02:42 --> 00:02:47 It correctly tells us that the stretching of that bar is zero 44 00:02:47 --> 00:02:50 but we knew that anyway. 45 00:02:50 --> 00:02:57 OK, I don't know if you've tackled the MATLAB question, 46 00:02:57 --> 00:03:03 and I also don't know whether MATLAB would produce for us. 47 00:03:03 --> 00:03:07 I mean, you should be able to construct a with a whole lot of 48 00:03:07 --> 00:03:14 square roots of three and over two from sine of 60 degrees and 49 00:03:14 --> 00:03:20 maybe 1/2 from sine of 30 degrees. a should look pretty 50 00:03:20 --> 00:03:26 nice, but what the solutions to these mechanisms are, of course 51 00:03:26 --> 00:03:32 solutions to Au=0, and. 52 00:03:32 --> 00:03:39 Anyway, I'm hoping that we learn from this example. 53 00:03:39 --> 00:03:42 I hadn't intended so it's pretty small MATLAB, it's 54 00:03:42 --> 00:03:45 really just the creation of the matrix A. 55 00:03:45 --> 00:03:50 OK, so that was a comment on that. 56 00:03:50 --> 00:03:54 Which I added to the homework, now I'm ready for questions 57 00:03:54 --> 00:03:59 about the homework, the exam whatever, yes thank you. 58 00:03:59 --> 00:04:01 Oh, good. 59 00:04:01 --> 00:04:06 AUDIENCE: [INAUDIBLE] 60 00:04:06 --> 00:04:10 PROFESSOR STRANG: For trusses, OK. 61 00:04:10 --> 00:04:10 AUDIENCE: [INAUDIBLE] 62 00:04:10 --> 00:04:17 PROFESSOR STRANG: Yeah. a TA immediately, couldn't you? 63 00:04:17 --> 00:04:19 It's not quite so straightforward. 64 00:04:19 --> 00:04:21 And this is like more realistic. 65 00:04:21 --> 00:04:25 I mean, yeah. 66 00:04:25 --> 00:04:28 Ah. 67 00:04:28 --> 00:04:31 I guess, yeah but remind me what that question was. 68 00:04:31 --> 00:04:37 This was a 2.7 Problem one? 69 00:04:37 --> 00:04:40 OK let me just see what I'm asking for. 70 00:04:40 --> 00:04:43 So, OK, yes. 71 00:04:43 --> 00:04:46 So I only asked about A transpose A, I only asked you 72 00:04:46 --> 00:04:49 for the shape in that question. 73 00:04:49 --> 00:04:51 Oh, and then I asked you for the first row, 74 00:04:51 --> 00:04:55 good for me, yes. 75 00:04:55 --> 00:05:01 I see, the first row so I didn't put you to the agony of 76 00:05:01 --> 00:05:05 writing out the whole thing, but still how do you get the 77 00:05:05 --> 00:05:09 first row, good question. 78 00:05:09 --> 00:05:12 AUDIENCE: [INAUDIBLE] 79 00:05:12 --> 00:05:13 PROFESSOR STRANG: Oh, I see. 80 00:05:13 --> 00:05:15 Now, well it's not going to be so neat. 81 00:05:15 --> 00:05:17 Let's just think. 82 00:05:17 --> 00:05:22 We could do the first row of a, and the first column 83 00:05:22 --> 00:05:26 of a probably, yeah. 84 00:05:26 --> 00:05:31 So what am I looking for in A transpose A, because 85 00:05:31 --> 00:05:34 it didn't say row one. 86 00:05:34 --> 00:05:41 OK, so row one of A transpose A corresponds to the first, 87 00:05:41 --> 00:05:42 oh, so what is row one? 88 00:05:42 --> 00:05:44 Yeah, this is worth thinking about. 89 00:05:44 --> 00:05:49 So this is A transpose A for trusses. 90 00:05:49 --> 00:05:52 And let's just, maybe we could even take this 91 00:05:52 --> 00:05:54 one as an example. 92 00:05:54 --> 00:06:00 If I number the nodes, if that's my first node, 93 00:06:00 --> 00:06:03 A transpose A, you remember that's square. 94 00:06:03 --> 00:06:07 That tells us the edge part, the bar part is built into it 95 00:06:07 --> 00:06:13 but its size is n by n, its size is five by, no, its size 96 00:06:13 --> 00:06:15 is what, eight by eight, right? 97 00:06:15 --> 00:06:21 It's got to do with the number of, OK. 98 00:06:21 --> 00:06:29 So its first row, what will its first row be about? 99 00:06:29 --> 00:06:36 It'll be about u H 1, right? 100 00:06:36 --> 00:06:43 The first row of this a transpose a matrix will be 101 00:06:43 --> 00:06:47 about, the first node but more than that, that node has got 102 00:06:47 --> 00:06:52 two things, u H and u V, and we're putting H first. 103 00:06:52 --> 00:06:57 So so I think that the first row of this, 104 00:06:57 --> 00:06:59 let's just draw it. 105 00:06:59 --> 00:07:00 Maybe I'm not seeing. 106 00:07:00 --> 00:07:07 So let that be A transpose A, I see yeah. a transpose a will be 107 00:07:07 --> 00:07:14 multiplying u H 1, u V 1, and so forth. 108 00:07:14 --> 00:07:14 Right? 109 00:07:14 --> 00:07:15 OK, good. 110 00:07:15 --> 00:07:17 So now this is better. 111 00:07:17 --> 00:07:23 All I'm asking is where do we know there are zeroes in that 112 00:07:23 --> 00:07:26 first row of A transpose A? 113 00:07:26 --> 00:07:30 Where would we know that there are zeroes for this problem 114 00:07:30 --> 00:07:33 rather than, I won't redraw the one in the book so I'm 115 00:07:33 --> 00:07:35 taking this truss. 116 00:07:35 --> 00:07:39 So if it's this truss, this is the first node, the bars are 117 00:07:39 --> 00:07:44 numbered like this, and of course your MATLAB construction 118 00:07:44 --> 00:07:47 of A, you might multiply out A transpose A. 119 00:07:47 --> 00:07:49 Because in MATLAB that would be so quick. 120 00:07:49 --> 00:07:51 And you could see what it looks like. 121 00:07:51 --> 00:07:57 And where would we expect to see zeroes or not zeroes? 122 00:07:57 --> 00:08:04 Let's see, A transpose A, so I'll have to think. 123 00:08:04 --> 00:08:09 My instinct is that it should only connect two neighbors. 124 00:08:09 --> 00:08:13 So that I would imagine, but I've got these are double 125 00:08:13 --> 00:08:17 neighbors, right, there's two people living here. 126 00:08:17 --> 00:08:19 We have to remember. 127 00:08:19 --> 00:08:22 So I see two people living here and two people at home, so I 128 00:08:22 --> 00:08:26 guess I would imagine four non-zeroes for this problem 129 00:08:26 --> 00:08:30 in in the first row. 130 00:08:30 --> 00:08:37 I would think that that would have somebody on the diagonal. 131 00:08:37 --> 00:08:43 That would be what multiplies u H 1 itself, and then maybe u V 132 00:08:43 --> 00:08:45 1 is involved, and these two guys. 133 00:08:45 --> 00:08:49 These two and those two would not be involved. 134 00:08:49 --> 00:08:52 Well, that's only a partial answer, I'm just telling you 135 00:08:52 --> 00:08:55 where the zeroes are, I think. 136 00:08:55 --> 00:09:01 And you're really asking about the non-zeroes, of course. 137 00:09:01 --> 00:09:03 So, yeah. 138 00:09:03 --> 00:09:06 Not so easy. 139 00:09:06 --> 00:09:07 Yeah -- 140 00:09:07 --> 00:09:07 AUDIENCE: [INAUDIBLE] 141 00:09:07 --> 00:09:08 PROFESSOR STRANG: -- maybe have to do it. 142 00:09:08 --> 00:09:13 OK, yeah. 143 00:09:13 --> 00:09:18 I think, the first time better to do it, yeah. 144 00:09:18 --> 00:09:21 But that's an excellent question, and maybe we will 145 00:09:21 --> 00:09:26 find a nice, and if somebody does, let me know. 146 00:09:26 --> 00:09:29 A nice way to see that. 147 00:09:29 --> 00:09:33 Somehow I must have felt that was doable when I 148 00:09:33 --> 00:09:37 created that problem. 149 00:09:37 --> 00:09:40 OK. 150 00:09:40 --> 00:09:45 The other thing to say though, that while I'm looking at A 151 00:09:45 --> 00:09:51 transpose A, is remember that it can be created by 152 00:09:51 --> 00:09:53 these element matrices. 153 00:09:53 --> 00:09:56 Yeah, yeah that's important to say. 154 00:09:56 --> 00:10:02 So there will be five element matrices, right? 155 00:10:02 --> 00:10:04 Because I've got five bars. 156 00:10:04 --> 00:10:08 And the element matrix for this bar, or say for a typical bar, 157 00:10:08 --> 00:10:14 the element matrix for this bar will involve all the u's that 158 00:10:14 --> 00:10:16 are connected to this bar. 159 00:10:16 --> 00:10:19 So two u's there and two u's there, so it'll be four by 160 00:10:19 --> 00:10:22 four, the element matrix here. 161 00:10:22 --> 00:10:30 So this will contribute four non-zeroes to that to row 162 00:10:30 --> 00:10:31 and to three other rows. 163 00:10:31 --> 00:10:36 That four by four guy from this part is, you know what 164 00:10:36 --> 00:10:37 I'm speaking about here? 165 00:10:37 --> 00:10:41 The element matrix which is just, the element matrix 166 00:10:41 --> 00:10:47 is a cosine, sine, minus cosine and minus sine. 167 00:10:47 --> 00:10:49 Column times row. 168 00:10:49 --> 00:10:54 Cosine, sine, minus cosine, minus sine. 169 00:10:54 --> 00:10:58 This is the element matrix for this bar, those are the cosine 170 00:10:58 --> 00:11:08 and sine of 60 degrees, and those positions would be 171 00:11:08 --> 00:11:10 these two and these two. 172 00:11:10 --> 00:11:14 So there would be actually four zeroes coming after, and after 173 00:11:14 --> 00:11:18 here from those two and those two. 174 00:11:18 --> 00:11:24 So this bar contributes to a transpose a, sort of 175 00:11:24 --> 00:11:26 stamps into A transpose A. 176 00:11:26 --> 00:11:32 This four by four of non zeroes plus the rest zeroes, 177 00:11:32 --> 00:11:36 stamps that in up there. 178 00:11:36 --> 00:11:43 This guy has its own element matrix. 179 00:11:43 --> 00:11:47 Now, what's different about the element matrix for this guy? 180 00:11:47 --> 00:11:52 Of course it's got its own c_1, probably, and it had a c_2, 181 00:11:52 --> 00:11:53 this difference constant. 182 00:11:53 --> 00:11:56 But bigger differences than that. 183 00:11:56 --> 00:11:59 First of all, it's a different angle, theta. 184 00:11:59 --> 00:12:05 Here the angle is 120 degrees, and also there's nobody 185 00:12:05 --> 00:12:08 home down there. 186 00:12:08 --> 00:12:13 So it would be, if we were dealing with A_0, free-free 187 00:12:13 --> 00:12:16 stuff, it would be four by four, like all the others. 188 00:12:16 --> 00:12:23 But because these two are fixed, those two displacements 189 00:12:23 --> 00:12:28 are fixed, effectively it will only contribute two by two. 190 00:12:28 --> 00:12:30 A total of four non-zeroes. 191 00:12:30 --> 00:12:34 But those will stamp in to A transpose A 192 00:12:34 --> 00:12:37 overlapping this guy. 193 00:12:37 --> 00:12:42 Then the ones from these bars will not touch the first row 194 00:12:42 --> 00:12:44 of A transpose A, so they wouldn't touch the answer 195 00:12:44 --> 00:12:47 to that homework problem. 196 00:12:47 --> 00:12:52 So you see again another way. 197 00:12:52 --> 00:12:58 By hand I don't know that you necessarily want to use these 198 00:12:58 --> 00:13:01 element matrices, I'm not sure. 199 00:13:01 --> 00:13:03 But it gives an excellent check. 200 00:13:03 --> 00:13:07 So one way to create A transpose A is 201 00:13:07 --> 00:13:09 create a multiplier. 202 00:13:09 --> 00:13:11 That's one way. 203 00:13:11 --> 00:13:16 Second way is, create A transpose A element by these 204 00:13:16 --> 00:13:23 five element matrices popped into the right places. 205 00:13:23 --> 00:13:26 I think this is, so this like the computational science 206 00:13:26 --> 00:13:31 part of the course. 207 00:13:31 --> 00:13:34 It's the way you would, I'm really speaking about the way 208 00:13:34 --> 00:13:37 you would write the code, not the final result. 209 00:13:37 --> 00:13:43 So much of the course has been devoted to understanding A 210 00:13:43 --> 00:13:47 transpose A, and the fact that it's positive definite, or in 211 00:13:47 --> 00:13:51 this case only positive semi definite because we have an 212 00:13:51 --> 00:13:55 eight by eight matrix whose rank is only going to be five. 213 00:13:55 --> 00:14:03 So this matrix will have these same mechanisms, and I'm hoping 214 00:14:03 --> 00:14:06 to learn what they are. 215 00:14:06 --> 00:14:08 Well I hope that's a little help with that question. 216 00:14:08 --> 00:14:13 But basically, though, it's not help because I'm sort of saying 217 00:14:13 --> 00:14:19 you're on your own to actually, I don't have a superfragilistic 218 00:14:19 --> 00:14:25 way to construct a matrix and then you kind of have to do it. 219 00:14:25 --> 00:14:28 But I'm glad you asked. 220 00:14:28 --> 00:14:29 Yeah 221 00:14:29 --> 00:14:32 AUDIENCE: [INAUDIBLE] 222 00:14:32 --> 00:14:35 PROFESSOR STRANG: More on this problem, OK. 223 00:14:35 --> 00:14:40 The sign convention. 224 00:14:40 --> 00:14:40 AUDIENCE: [INAUDIBLE] 225 00:14:40 --> 00:14:41 PROFESSOR STRANG: Yes. 226 00:14:41 --> 00:14:43 So let's see. 227 00:14:43 --> 00:14:46 So I made a speech about sign conventions, right? 228 00:14:46 --> 00:14:54 And which was that I am too old for this. 229 00:14:54 --> 00:14:58 And the justification for that is the fact that they 230 00:14:58 --> 00:15:01 wash out in A transpose. 231 00:15:01 --> 00:15:04 So if you were to use the wrong sign convention in A, you 232 00:15:04 --> 00:15:06 won't see it in A transpose. 233 00:15:06 --> 00:15:09 So if I change the whole thing. 234 00:15:09 --> 00:15:15 So sign convention is not really a convention. 235 00:15:15 --> 00:15:20 Our convention is to decide that that movement 236 00:15:20 --> 00:15:22 that way is plus. 237 00:15:22 --> 00:15:25 And movement this way is plus. 238 00:15:25 --> 00:15:26 And here, similarly. 239 00:15:26 --> 00:15:30 That's a plus movement and that's a plus movement. 240 00:15:30 --> 00:15:32 That's our sign convention. 241 00:15:32 --> 00:15:40 If we always do that, then the A matrix is telling us how 242 00:15:40 --> 00:15:42 much this bar stretches. 243 00:15:42 --> 00:15:45 It's the matrix that gives us stretching from these 244 00:15:45 --> 00:15:46 four displacements. 245 00:15:46 --> 00:15:56 And then, these, I think in this picture, if this is the 246 00:15:56 --> 00:16:01 later node and this is the thing, then that angle from 247 00:16:01 --> 00:16:03 the horizontal is the right guy to put there. 248 00:16:03 --> 00:16:06 AUDIENCE: [INAUDIBLE] 249 00:16:06 --> 00:16:06 PROFESSOR STRANG: Here? 250 00:16:06 --> 00:16:12 AUDIENCE: [INAUDIBLE] 251 00:16:12 --> 00:16:19 PROFESSOR STRANG: Well, I guess I'm, I see, the question is 252 00:16:19 --> 00:16:22 what's the sign convention with this particular picture? 253 00:16:22 --> 00:16:30 Yeah, if this is node one and that's two, then 254 00:16:30 --> 00:16:31 question is good. 255 00:16:31 --> 00:16:35 It looks, I think, so I think I've got it not right. 256 00:16:35 --> 00:16:43 If this is node one down below, and two up above, then moving 257 00:16:43 --> 00:16:46 two positively would stretch the bar. 258 00:16:46 --> 00:16:48 And I've got minuses there. 259 00:16:48 --> 00:16:53 So I've got the opposite sides. 260 00:16:53 --> 00:16:57 I've got there, the signs that would go with this number. 261 00:16:57 --> 00:17:00 If that was node two and this was node one. 262 00:17:00 --> 00:17:03 Yeah, you're right, good. 263 00:17:03 --> 00:17:05 You have to, and that's the only way I do it, is to 264 00:17:05 --> 00:17:10 think will the movement stretch to the bar. 265 00:17:10 --> 00:17:13 Then that's got a positive entry in there. 266 00:17:13 --> 00:17:18 Will the movement like that compress the bar, that'll 267 00:17:18 --> 00:17:19 be a negative entry. 268 00:17:19 --> 00:17:26 And our conventions are if it stretches that's positive. e 269 00:17:26 --> 00:17:29 positive means the stretching; e negative means 270 00:17:29 --> 00:17:32 the compression. 271 00:17:32 --> 00:17:35 So it takes a little patience. 272 00:17:35 --> 00:17:39 Or it takes writing the code correctly once and then 273 00:17:39 --> 00:17:46 of course it would do it for all these problems. 274 00:17:46 --> 00:17:49 Yeah, I think the TA in an earlier year actually wrote 275 00:17:49 --> 00:17:52 a code to a truss code. 276 00:17:52 --> 00:17:59 I don't know what happened to it but I could look. 277 00:17:59 --> 00:18:01 AUDIENCE: [INAUDIBLE] 278 00:18:01 --> 00:18:08 PROFESSOR STRANG: Yes, yeah. 279 00:18:08 --> 00:18:11 That's right, and it will come out consistent. 280 00:18:11 --> 00:18:16 Yeah, if I take horizontal, if I take the same convention that 281 00:18:16 --> 00:18:23 forces that way our plus, that's a plus f H 1 and it's a 282 00:18:23 --> 00:18:29 plus u H 1, then I'll get a will go to A transpose. 283 00:18:29 --> 00:18:33 Yeah, so if I'm consistent that's plus for the u's, so 284 00:18:33 --> 00:18:38 this displacement is plus for the f forces and the correctly 285 00:18:38 --> 00:18:41 created a will give me the correct A transpose. 286 00:18:41 --> 00:18:48 But A transpose A has the ability to vary some 287 00:18:48 --> 00:18:51 of those conventions. 288 00:18:51 --> 00:18:52 OK. 289 00:18:52 --> 00:18:53 Good. 290 00:18:53 --> 00:18:53 Yes. 291 00:18:53 --> 00:18:55 AUDIENCE: [INAUDIBLE] 292 00:18:55 --> 00:18:56 PROFESSOR STRANG: Different. 293 00:18:56 --> 00:18:57 More. 294 00:18:57 --> 00:18:58 Oh my God. 295 00:18:58 --> 00:18:59 OK. 296 00:18:59 --> 00:18:59 Yes. 297 00:18:59 --> 00:19:08 AUDIENCE: [INAUDIBLE] 298 00:19:08 --> 00:19:09 PROFESSOR STRANG: This one, yeah. 299 00:19:09 --> 00:19:14 AUDIENCE: [INAUDIBLE] 300 00:19:14 --> 00:19:14 PROFESSOR STRANG: Yeah. 301 00:19:14 --> 00:19:22 AUDIENCE: [INAUDIBLE] 302 00:19:22 --> 00:19:27 PROFESSOR STRANG: I wish it would ask about bar three. 303 00:19:27 --> 00:19:29 Yeah, yeah bar three was good. 304 00:19:29 --> 00:19:31 Yeah right. 305 00:19:31 --> 00:19:34 Yeah, I guess here, what's here, well, the angle is, that 306 00:19:34 --> 00:19:40 must be theta, and then this'll be right provided this 307 00:19:40 --> 00:19:45 number is the, is what? 308 00:19:45 --> 00:19:50 Well, I've got to think it out again. 309 00:19:50 --> 00:19:56 I'll just think it out. 310 00:19:56 --> 00:19:59 Maybe, yeah I won't try to, yeah I won't. 311 00:19:59 --> 00:19:59 I won't. 312 00:19:59 --> 00:20:06 So I think you, but it's really, you learn the point 313 00:20:06 --> 00:20:14 by thinking that through. 314 00:20:14 --> 00:20:15 I think we've got it here. 315 00:20:15 --> 00:20:22 That if that's number one then we like plus sine here because 316 00:20:22 --> 00:20:27 positive displacements will stretch the bar, right. 317 00:20:27 --> 00:20:32 And here this sign will depend on the angle. 318 00:20:32 --> 00:20:38 And it will depend on the number, which which numbers, 319 00:20:38 --> 00:20:40 what number that one is and what number that one is. 320 00:20:40 --> 00:20:45 If I reverse the numbers, then I reverse the side. 321 00:20:45 --> 00:20:47 Yeah, good. 322 00:20:47 --> 00:20:51 Well, that's sort of part of computational 323 00:20:51 --> 00:20:53 engineering isn't it? 324 00:20:53 --> 00:20:56 Like, keep track of the details. 325 00:20:56 --> 00:21:00 But of course this 18.085 course is also about 326 00:21:00 --> 00:21:02 big picture things. 327 00:21:02 --> 00:21:09 And that's what I'm hoping comes through even better. 328 00:21:09 --> 00:21:12 OK, ready for another question on any topic. 329 00:21:12 --> 00:21:17 Thank you. 330 00:21:17 --> 00:21:18 AUDIENCE: [INAUDIBLE] 331 00:21:18 --> 00:21:28 PROFESSOR STRANG: Yes. 332 00:21:28 --> 00:21:30 Well, no, but let's just try it. 333 00:21:30 --> 00:21:32 Shall we do Problem one? 334 00:21:32 --> 00:21:36 Was it Problem one? in 2.7? 335 00:21:36 --> 00:21:38 OK, let me just draw that truss. 336 00:21:38 --> 00:21:40 And let's just talk about it. 337 00:21:40 --> 00:21:44 So, boy I've put a lot of bars in there. 338 00:21:44 --> 00:21:50 OK, so this is just one bar down to this guy. 339 00:21:50 --> 00:21:53 And this is one up to here. 340 00:21:53 --> 00:21:56 And was that everything? 341 00:21:56 --> 00:21:58 OK, and then I've numbered them. 342 00:21:58 --> 00:22:02 Let me just copy the number here, so that we can 343 00:22:02 --> 00:22:05 talk about that. 344 00:22:05 --> 00:22:09 And I numbered the nodes one, two, one. 345 00:22:09 --> 00:22:12 Joints, I should say really. 346 00:22:12 --> 00:22:12 Four. 347 00:22:12 --> 00:22:15 OK, all right. 348 00:22:15 --> 00:22:17 Let's just start as I always do by, what's the 349 00:22:17 --> 00:22:20 shape of the matrix A? 350 00:22:20 --> 00:22:26 A is what, how many bars have I got? 351 00:22:26 --> 00:22:29 So help me through this now. 352 00:22:29 --> 00:22:34 Six bars, and how many unknown displacements have I got? 353 00:22:34 --> 00:22:36 Eight, OK, good. 354 00:22:36 --> 00:22:38 And have I got any rigid motions here? 355 00:22:38 --> 00:22:40 No. 356 00:22:40 --> 00:22:46 I've got two, these are going to prevent all 357 00:22:46 --> 00:22:47 the rigid motion. 358 00:22:47 --> 00:22:59 So if this matrix, if A has rank six, which we'd probably 359 00:22:59 --> 00:23:08 guess it has, you know that there's an if there. 360 00:23:08 --> 00:23:10 Then two mechanisms, right? 361 00:23:10 --> 00:23:14 Eight minus six gives two mechanisms. 362 00:23:14 --> 00:23:19 OK, and your question is how to find them. 363 00:23:19 --> 00:23:19 You've got one. 364 00:23:19 --> 00:23:21 Alright, which one have you got? 365 00:23:21 --> 00:23:24 AUDIENCE: [INAUDIBLE] 366 00:23:24 --> 00:23:26 PROFESSOR STRANG: These guys. 367 00:23:26 --> 00:23:33 So tilt, so that's a mechanism that only involves u H 1 and u 368 00:23:33 --> 00:23:36 H 2 and none of the other u's. 369 00:23:36 --> 00:23:39 OK, that looks good. 370 00:23:39 --> 00:23:41 So that's certainly one. 371 00:23:41 --> 00:23:44 Now we're looking for a second one. 372 00:23:44 --> 00:23:46 What if, anybody suggest one? 373 00:23:46 --> 00:23:49 Let me just give everybody a thought. 374 00:23:49 --> 00:23:52 If you haven't already started on this homework, 375 00:23:52 --> 00:23:54 you're starting now. 376 00:23:54 --> 00:24:01 So I'm looking for another mechanism, and of course you 377 00:24:01 --> 00:24:06 might say well, suppose these bars go this way. 378 00:24:06 --> 00:24:10 Suppose they go to the left, and you know that that's 379 00:24:10 --> 00:24:12 not going to do it right. 380 00:24:12 --> 00:24:14 And why not? 381 00:24:14 --> 00:24:17 Why isn't that an OK second answer? 382 00:24:17 --> 00:24:20 Because it's effectively the same as the first; in fact the 383 00:24:20 --> 00:24:26 mechanism u, how would the u, the one I've drawn and the u 384 00:24:26 --> 00:24:33 for this way, what would we see? 385 00:24:33 --> 00:24:34 Opposite signs. 386 00:24:34 --> 00:24:39 If this describes a u 1 H and a u 2 H positive, the other way 387 00:24:39 --> 00:24:41 they're negative, it would just be the opposite 388 00:24:41 --> 00:24:43 sign; nothing new. 389 00:24:43 --> 00:24:48 OK, so now you've had a look, so tell me. 390 00:24:48 --> 00:24:54 If somebody sees, to answer your question here, 391 00:24:54 --> 00:24:55 how do you see them? 392 00:24:55 --> 00:24:59 I don't know. 393 00:24:59 --> 00:25:03 Just look harder. 394 00:25:03 --> 00:25:06 Like you know, speaking French. 395 00:25:06 --> 00:25:12 Just say it louder and maybe it'll work. 396 00:25:12 --> 00:25:15 Of course we do have the possibility to create the 397 00:25:15 --> 00:25:18 matrix and look for solutions. 398 00:25:18 --> 00:25:21 But it's more fun to do it this way. 399 00:25:21 --> 00:25:25 And now who's going to suggest another 400 00:25:25 --> 00:25:26 mechanism, another view? 401 00:25:26 --> 00:25:30 AUDIENCE: [INAUDIBLE] 402 00:25:30 --> 00:25:36 PROFESSOR STRANG: Three and four. 403 00:25:36 --> 00:25:39 Wait a minute; what was that three? 404 00:25:39 --> 00:25:42 Oh, these nodes, OK. 405 00:25:42 --> 00:25:46 These nodes, three and four do what? 406 00:25:46 --> 00:25:47 Oh yes, sorry. 407 00:25:47 --> 00:25:52 Three go up this way, ah. 408 00:25:52 --> 00:25:56 I see you're rotating. 409 00:25:56 --> 00:26:03 So this guy goes sort of along this way, you're taking that 410 00:26:03 --> 00:26:06 top square and turning it. 411 00:26:06 --> 00:26:10 OK, so these guys will also turn, right? 412 00:26:10 --> 00:26:16 Yeah, one and two will move on your mechanism. 413 00:26:16 --> 00:26:22 So the idea is take that top thing, and I'm allowed to turn 414 00:26:22 --> 00:26:27 it, I'd say I have to turn it, I have to keep this bar, 415 00:26:27 --> 00:26:29 yeah I cannot stretch it. 416 00:26:29 --> 00:26:31 I can't stretch any bars. 417 00:26:31 --> 00:26:36 So when I do this turn, I can't just keep it in place and turn. 418 00:26:36 --> 00:26:37 No. 419 00:26:37 --> 00:26:41 But I'm going to turn it so that this one goes this 420 00:26:41 --> 00:26:47 way and this one goes this way, and now. 421 00:26:47 --> 00:26:51 Alright you're, you're responsible for telling me 422 00:26:51 --> 00:26:52 about the other two now. 423 00:26:52 --> 00:26:55 What do they do? 424 00:26:55 --> 00:26:55 AUDIENCE: [INAUDIBLE] 425 00:26:55 --> 00:26:57 PROFESSOR STRANG: Bar four, Will bar four keep its length? 426 00:26:57 --> 00:27:01 Ha. 427 00:27:01 --> 00:27:03 OK, well we're not allowed to use fancy words 428 00:27:03 --> 00:27:07 like four-bar linkage. 429 00:27:07 --> 00:27:09 Because somebody might say what does that mean. 430 00:27:09 --> 00:27:11 OK, does it, do you see that? 431 00:27:11 --> 00:27:12 I think it does. 432 00:27:12 --> 00:27:15 But it's wonderful. 433 00:27:15 --> 00:27:19 If that's the original bar and I bring this up a little; a 434 00:27:19 --> 00:27:24 little, remember and this down a little, I think that the 435 00:27:24 --> 00:27:30 new bar has the same length. 436 00:27:30 --> 00:27:32 You believe that? 437 00:27:32 --> 00:27:33 No. 438 00:27:33 --> 00:27:38 Some, yeah, but what happened there? 439 00:27:38 --> 00:27:41 AUDIENCE: [INAUDIBLE] 440 00:27:41 --> 00:27:43 PROFESSOR STRANG: Yeah, this is a good example. 441 00:27:43 --> 00:27:46 AUDIENCE: [INAUDIBLE] 442 00:27:46 --> 00:27:49 PROFESSOR STRANG: Sorry, yeah, certainly bringing that up 443 00:27:49 --> 00:27:53 tended to make the bar shorter, but then moving this down 444 00:27:53 --> 00:27:55 tended to make it a little longer. 445 00:27:55 --> 00:27:57 AUDIENCE: [INAUDIBLE] 446 00:27:57 --> 00:27:58 PROFESSOR STRANG: Right 447 00:27:58 --> 00:28:00 AUDIENCE: [INAUDIBLE] 448 00:28:00 --> 00:28:06 PROFESSOR STRANG: Yeah, I think so. 449 00:28:06 --> 00:28:08 See, that was the key. 450 00:28:08 --> 00:28:12 We went at 90 degrees to five. 451 00:28:12 --> 00:28:17 And by going at 90 degrees, then the stretch in there was 452 00:28:17 --> 00:28:21 only that one minus cosine that was higher order. 453 00:28:21 --> 00:28:22 That's the key. 454 00:28:22 --> 00:28:26 Similarly, here, all these so we're avoiding stretching 455 00:28:26 --> 00:28:28 by this trick. 456 00:28:28 --> 00:28:31 So maybe this guy is going down this same way. 457 00:28:31 --> 00:28:34 And this guy is going up that same way. 458 00:28:34 --> 00:28:40 Yeah, because then that bar is just translating, and this bar 459 00:28:40 --> 00:28:44 is translating and these two are doing the same thing, which 460 00:28:44 --> 00:28:49 I believe is not stretching. 461 00:28:49 --> 00:28:53 Is this the answer that maybe somebody already got? 462 00:28:53 --> 00:28:55 And believed in? 463 00:28:55 --> 00:28:55 Yeah. 464 00:28:55 --> 00:29:00 So to answer your original question, it wasn't 465 00:29:00 --> 00:29:02 obvious, was it? 466 00:29:02 --> 00:29:06 But I think that is the right thing. 467 00:29:06 --> 00:29:11 And actually, you could create the A for this problem. 468 00:29:11 --> 00:29:17 Well, it's a bit large, 48 entries, but because four of 469 00:29:17 --> 00:29:20 the bars are horizontal or vertical, you will have many, 470 00:29:20 --> 00:29:24 many, many, zeroes in the A matrix. 471 00:29:24 --> 00:29:26 It would be sort of fun to create the A 472 00:29:26 --> 00:29:28 matrix and then this. 473 00:29:28 --> 00:29:31 So what is this displacement that I believe in? 474 00:29:31 --> 00:29:35 This u mechanism that we've talked about. 475 00:29:35 --> 00:29:40 Let's see, if I look at one that was positive positive. 476 00:29:40 --> 00:29:46 If I looked at node two, that was positive over but down. 477 00:29:46 --> 00:29:49 So horizontal but down. 478 00:29:49 --> 00:29:53 If I look at node three, that's positive positive. 479 00:29:53 --> 00:29:56 Yeah, one and three is hanging on. 480 00:29:56 --> 00:30:00 And number four is like number two, one, and minus one. 481 00:30:00 --> 00:30:05 I think that's the u which should solve Au=0. 482 00:30:05 --> 00:30:09 483 00:30:09 --> 00:30:11 Yeah. 484 00:30:11 --> 00:30:11 OK. 485 00:30:11 --> 00:30:16 But that's a good example. 486 00:30:16 --> 00:30:19 OK, so that's trusses. 487 00:30:19 --> 00:30:21 What else? 488 00:30:21 --> 00:30:22 AUDIENCE: [INAUDIBLE] 489 00:30:22 --> 00:30:22 PROFESSOR STRANG: Thank you. 490 00:30:22 --> 00:30:25 AUDIENCE: [INAUDIBLE] 491 00:30:25 --> 00:30:28 PROFESSOR STRANG: Oh, well, let's see. 492 00:30:28 --> 00:30:33 That's a good question and now I guess we're, ah. 493 00:30:33 --> 00:30:37 Yeah well, the way I've drawn it at 45 degrees, which is what 494 00:30:37 --> 00:30:43 I wrote here, then I did build in, I did make that a 40, I 495 00:30:43 --> 00:30:47 did make this 45 or 135 or something. 496 00:30:47 --> 00:30:51 And 45. 497 00:30:51 --> 00:30:55 To get that, if that goes at 45 degrees, then this had 498 00:30:55 --> 00:30:58 better go at 135, right? 499 00:30:58 --> 00:31:01 So that has to be a right angle. 500 00:31:01 --> 00:31:07 If the support was over here then the angle that would have 501 00:31:07 --> 00:31:11 gone off with was changed. 502 00:31:11 --> 00:31:15 Yeah, so very good point, that the numbers I've written 503 00:31:15 --> 00:31:19 down on the picture I drew required these angles to 504 00:31:19 --> 00:31:25 be those nice numbers. 505 00:31:25 --> 00:31:27 I hope you like these trusses a little. 506 00:31:27 --> 00:31:40 I mean, you get some freedom to visualize a little. 507 00:31:40 --> 00:31:41 Good, yes. 508 00:31:41 --> 00:32:02 AUDIENCE: [INAUDIBLE] 509 00:32:02 --> 00:32:02 PROFESSOR STRANG: Maybe. 510 00:32:02 --> 00:32:03 Let me see. 511 00:32:03 --> 00:32:07 I thought you were going to say, could I have two and 512 00:32:07 --> 00:32:14 three go inward and one and four go outwards? 513 00:32:14 --> 00:32:15 You don't like that. 514 00:32:15 --> 00:32:21 I would go with that. 515 00:32:21 --> 00:32:22 Is that any good? 516 00:32:22 --> 00:32:24 Well, OK. 517 00:32:24 --> 00:32:26 Yeah, linear algebra spoke there. 518 00:32:26 --> 00:32:29 It's a combination of the other two, yes. 519 00:32:29 --> 00:32:32 If we only got a two dimensional. 520 00:32:32 --> 00:32:34 AUDIENCE: [INAUDIBLE] 521 00:32:34 --> 00:32:34 PROFESSOR STRANG: That one. 522 00:32:34 --> 00:32:36 AUDIENCE: [INAUDIBLE] 523 00:32:36 --> 00:32:36 PROFESSOR STRANG: OK. 524 00:32:36 --> 00:32:38 AUDIENCE: [INAUDIBLE] 525 00:32:38 --> 00:32:39 PROFESSOR STRANG: Yeah. 526 00:32:39 --> 00:32:42 AUDIENCE: [INAUDIBLE] 527 00:32:42 --> 00:32:43 PROFESSOR STRANG: Yeah. 528 00:32:43 --> 00:32:46 AUDIENCE: [INAUDIBLE] 529 00:32:46 --> 00:32:46 PROFESSOR STRANG: Yeah. 530 00:32:46 --> 00:32:51 AUDIENCE: [INAUDIBLE] 531 00:32:51 --> 00:32:54 PROFESSOR STRANG: OK. 532 00:32:54 --> 00:33:00 But let me, when you suggested one, I overwrote it. 533 00:33:00 --> 00:33:07 Now, the one you suggested, when you told me, OK, bring 534 00:33:07 --> 00:33:10 these in, I thought, OK, these have to go out. 535 00:33:10 --> 00:33:19 AUDIENCE: [INAUDIBLE] 536 00:33:19 --> 00:33:21 PROFESSOR STRANG: They're not allowed to change lengths, 537 00:33:21 --> 00:33:25 so they can only swing around these pin joints. 538 00:33:25 --> 00:33:33 So the picture of the one that I drew after your question 539 00:33:33 --> 00:33:37 would be, these guys, this guy, let me draw the 540 00:33:37 --> 00:33:39 square, as it was. 541 00:33:39 --> 00:33:41 And then these guys came in a little. 542 00:33:41 --> 00:33:44 These guys went out a little. 543 00:33:44 --> 00:33:45 And we got this. 544 00:33:45 --> 00:33:51 AUDIENCE: [INAUDIBLE] 545 00:33:51 --> 00:33:53 PROFESSOR STRANG: No, that's OK. 546 00:33:53 --> 00:33:57 I think this is legitimate, this is a legitimate one. 547 00:33:57 --> 00:34:03 And that bar, every bar now is doing the same kind of 548 00:34:03 --> 00:34:04 thing that this one did. 549 00:34:04 --> 00:34:08 This moved a little, but this compensated and kept 550 00:34:08 --> 00:34:10 the length the same. 551 00:34:10 --> 00:34:18 Your question has led us to another nifty mechanism. 552 00:34:18 --> 00:34:20 It's good, a good one to think about. 553 00:34:20 --> 00:34:34 AUDIENCE: [INAUDIBLE] 554 00:34:34 --> 00:34:34 PROFESSOR STRANG: Yeah. 555 00:34:34 --> 00:34:36 It's movement without stretching. 556 00:34:36 --> 00:34:40 Movement without stretching is the key, yes. 557 00:34:40 --> 00:34:43 And then where does collapse come? 558 00:34:43 --> 00:34:46 So I use the word collapse pretty freely. 559 00:34:46 --> 00:34:49 So the word collapse comes in because since it's can move 560 00:34:49 --> 00:34:52 without stretching there's nothing controlling it, it can 561 00:34:52 --> 00:35:00 move, well, if we stayed linear, then I could make all 562 00:35:00 --> 00:35:05 those ten and the thing would be even worse, of course. 563 00:35:05 --> 00:35:11 The truth is that if I made those even one or ten I would 564 00:35:11 --> 00:35:13 be out of the linear range. 565 00:35:13 --> 00:35:16 These all should be 0.01's or something. 566 00:35:16 --> 00:35:20 But linear we don't know the difference. 567 00:35:20 --> 00:35:24 OK, I'm glad you've got that suggestion. 568 00:35:24 --> 00:35:27 And now somebody correctly says that this one must be some 569 00:35:27 --> 00:35:31 combination of that and what was the first movement? 570 00:35:31 --> 00:35:35 Oh, the upper two. 571 00:35:35 --> 00:35:38 Which it must be. 572 00:35:38 --> 00:35:43 Amazing, how many, put a few bars up there and you got it. 573 00:35:43 --> 00:35:44 Yeah. 574 00:35:44 --> 00:35:49 And of course, by the way, and don't let me get too far into 575 00:35:49 --> 00:35:53 this discussion, but who is the artist, actually is it 576 00:35:53 --> 00:35:57 Alexander Calder who has, what are they called? 577 00:35:57 --> 00:36:03 You know, they're, What was it called again? 578 00:36:03 --> 00:36:06 AUDIENCE: [INAUDIBLE] 579 00:36:06 --> 00:36:08 PROFESSOR STRANG: Well, he has those, that wasn't the 580 00:36:08 --> 00:36:09 word I was thinking of. 581 00:36:09 --> 00:36:13 But he's created these trusses, like with lots of bars and lots 582 00:36:13 --> 00:36:15 of nodes that have some special property. 583 00:36:15 --> 00:36:17 You know, they just, anyway. 584 00:36:17 --> 00:36:26 It touches on art, actually this theory of mechanisms. 585 00:36:26 --> 00:36:30 Yeah it's really quite interesting. 586 00:36:30 --> 00:36:34 But then linear algebra somehow tells you just from these 587 00:36:34 --> 00:36:37 numbers how many mechanisms you're looking for. 588 00:36:37 --> 00:36:40 Which is pretty cool. 589 00:36:40 --> 00:36:45 OK, open for more questions. 590 00:36:45 --> 00:36:49 You probably haven't looked ahead, I mean today's lecture 591 00:36:49 --> 00:36:58 was a, I hope, and I left up there the central topics 592 00:36:58 --> 00:36:59 for the lecture. 593 00:36:59 --> 00:37:04 But I'm, I think, correct that you haven't started 594 00:37:04 --> 00:37:06 on those problems. 595 00:37:06 --> 00:37:08 To know what to ask. 596 00:37:08 --> 00:37:11 Let me ask you a question, which was not in the lecture. 597 00:37:11 --> 00:37:15 Suppose I wanted to use finite differences. 598 00:37:15 --> 00:37:18 It's a little bit like the one on the quiz. 599 00:37:18 --> 00:37:22 So on the quiz we had c=1's, and then c=2. 600 00:37:23 --> 00:37:27 And you knew c had to be four by four, so most people 601 00:37:27 --> 00:37:31 correctly got the diagonal as one, one, two, two. 602 00:37:31 --> 00:37:35 Just by sort of common sense. 603 00:37:35 --> 00:37:39 But what would be a finite difference, approximation 604 00:37:39 --> 00:37:40 to our equation? 605 00:37:40 --> 00:37:45 So suppose I didn't go to finite elements, but instead I 606 00:37:45 --> 00:37:49 stayed with finite differences, which would be completely fine 607 00:37:49 --> 00:37:52 in 1-D, completely sensible. 608 00:37:52 --> 00:37:56 How would you create a finite difference thing for that? 609 00:37:56 --> 00:38:00 Let me just bring down a blank board and ask. 610 00:38:00 --> 00:38:02 So you see the equation? 611 00:38:02 --> 00:38:04 Let me write it again here. 612 00:38:04 --> 00:38:08 So I want to replace this equation with a very, 613 00:38:08 --> 00:38:09 that has varying c(x). 614 00:38:09 --> 00:38:12 615 00:38:12 --> 00:38:14 Well, I won't worry about the right side. 616 00:38:14 --> 00:38:16 It's F. 617 00:38:16 --> 00:38:24 What's my K matrix for using finite differences for this? 618 00:38:24 --> 00:38:29 We're going to create a finite element K matrix by the 619 00:38:29 --> 00:38:34 weak form Galerkin trial function, that route. 620 00:38:34 --> 00:38:37 That's coming that very important, it's coming. 621 00:38:37 --> 00:38:41 Start it today and it'll be completed Friday. 622 00:38:41 --> 00:38:48 But now suppose we were back up to finite differences. 623 00:38:48 --> 00:38:54 What would you take for finite differences there? 624 00:38:54 --> 00:38:58 When the c wasn't there, what did we do? 625 00:38:58 --> 00:39:00 Then we just had second difference, right? 626 00:39:00 --> 00:39:02 How did we get the second difference? 627 00:39:02 --> 00:39:06 It was a first difference of a first difference. 628 00:39:06 --> 00:39:12 I guess I would probably, this, I would probably approximate 629 00:39:12 --> 00:39:15 by u_(i+1)-u_i/delta x. 630 00:39:15 --> 00:39:21 631 00:39:21 --> 00:39:26 That would be that, and so what should I put for c(x)? 632 00:39:29 --> 00:39:31 What would you suggest? 633 00:39:31 --> 00:39:41 The c subscript i, so let's mark. 634 00:39:41 --> 00:39:43 Here's i, and here's i+1, and here's i-1. 635 00:39:43 --> 00:39:46 636 00:39:46 --> 00:39:51 We're going to end up with those three guys involved. 637 00:39:51 --> 00:40:00 So this takes this difference, and then this one will take 638 00:40:00 --> 00:40:04 that one, and then I'll also have a u_i-u_(i-1)/delta x. 639 00:40:04 --> 00:40:08 640 00:40:08 --> 00:40:10 And somehow that'll be the difference of 641 00:40:10 --> 00:40:11 these differences. 642 00:40:11 --> 00:40:14 But now tell me again, what would be the really 643 00:40:14 --> 00:40:18 cool choice of c? 644 00:40:18 --> 00:40:24 What's your instinct where what value of c to take? 645 00:40:24 --> 00:40:27 As sort of average. 646 00:40:27 --> 00:40:29 I mean you could take it just halfway. 647 00:40:29 --> 00:40:33 I think I would take c_(i+1/2) there. 648 00:40:33 --> 00:40:37 Just as being sort of right. 649 00:40:37 --> 00:40:42 And here I would take c_(i-1/2), halfway 650 00:40:42 --> 00:40:44 along its interval. 651 00:40:44 --> 00:40:50 And then the second difference takes the difference of it. 652 00:40:50 --> 00:40:55 So now I've got, I think that's what I would do. 653 00:40:55 --> 00:41:00 That would be my one typical finite difference equation. 654 00:41:00 --> 00:41:05 Would be the difference times its c and I took 655 00:41:05 --> 00:41:07 its c to be symmetric. 656 00:41:07 --> 00:41:11 You could also have taken, if you like, if you wanted to stay 657 00:41:11 --> 00:41:14 at these points, you could do twice as much work and 658 00:41:14 --> 00:41:17 take, let me say, four. 659 00:41:17 --> 00:41:24 So either c_(i+1/2), or you could average the 660 00:41:24 --> 00:41:25 c_1 and the c_(i+1/2). 661 00:41:27 --> 00:41:32 And both of those would give you that extra accuracy that 662 00:41:32 --> 00:41:35 you pick up from sort of keeping symmetry 663 00:41:35 --> 00:41:38 where you should. 664 00:41:38 --> 00:41:42 So I think, I mean, this would be the quicker one to put, 665 00:41:42 --> 00:41:46 that one would be OK too. 666 00:41:46 --> 00:41:49 AUDIENCE: [INAUDIBLE] 667 00:41:49 --> 00:41:51 PROFESSOR STRANG: This was just for your 668 00:41:51 --> 00:41:52 entertainment not, not. 669 00:41:52 --> 00:41:55 Yes, go ahead. 670 00:41:55 --> 00:41:56 AUDIENCE: [INAUDIBLE] 671 00:41:56 --> 00:41:57 PROFESSOR STRANG: Ah, when c was a step yes. 672 00:41:57 --> 00:41:58 When c was a step. 673 00:41:58 --> 00:42:05 AUDIENCE: [INAUDIBLE] 674 00:42:05 --> 00:42:09 PROFESSOR STRANG: c_i plus, I have probably the step. 675 00:42:09 --> 00:42:14 Yeah, yeah. 676 00:42:14 --> 00:42:16 Yeah. 677 00:42:16 --> 00:42:21 Let me ask you, what happens in this equation when c jumps? 678 00:42:21 --> 00:42:23 It jumped on the quiz. 679 00:42:23 --> 00:42:27 But I didn't require you to solve the differential 680 00:42:27 --> 00:42:32 equation, only to create the finite difference model, to 681 00:42:32 --> 00:42:36 create the A transpose c A, and most people did it fine. 682 00:42:36 --> 00:42:40 But suppose c jumps. 683 00:42:40 --> 00:42:44 I have some simple right hand side like one. 684 00:42:44 --> 00:42:47 It's not a jump in the right hand side I'm interested in. 685 00:42:47 --> 00:42:50 It's a jump in c from one to two. 686 00:42:50 --> 00:42:54 So this is a topic that the book does discuss and maybe 687 00:42:54 --> 00:42:57 I might come back to it in the ordinary lecture. 688 00:42:57 --> 00:43:04 But, while it's in front of us now, the quiz was sort of 689 00:43:04 --> 00:43:09 intended to help you with that. 690 00:43:09 --> 00:43:15 How do I interpret this equation when c has a jump? 691 00:43:15 --> 00:43:19 Well, it's actually, if you look at it right 692 00:43:19 --> 00:43:20 there's no difficulty. 693 00:43:20 --> 00:43:24 That equation is a combination of this equation 694 00:43:24 --> 00:43:27 -dw/dx equals the one. 695 00:43:27 --> 00:43:35 And the equation of c*du/dx equalling the w. 696 00:43:35 --> 00:43:39 I split it out for you in the last part, Problem 697 00:43:39 --> 00:43:41 4b in the quiz. 698 00:43:41 --> 00:43:45 I really helped you to say OK, take these two 699 00:43:45 --> 00:43:47 separate equations. 700 00:43:47 --> 00:43:50 And now you don't really have a problem. 701 00:43:50 --> 00:43:55 The equation for w, nothing is out of the ordinary there. 702 00:43:55 --> 00:43:57 Jump in c is not even seen. 703 00:43:57 --> 00:43:59 So then you've got the w. 704 00:43:59 --> 00:44:04 Now, you have the c in it with its little jump. 705 00:44:04 --> 00:44:10 But, so suppose I find w from the first equation. 706 00:44:10 --> 00:44:14 How do I find u? 707 00:44:14 --> 00:44:15 I just divide by the c. 708 00:44:15 --> 00:44:17 It integrates. 709 00:44:17 --> 00:44:19 Yeah. 710 00:44:19 --> 00:44:24 What I'm saying, let me say it, clearly now. 711 00:44:24 --> 00:44:29 If there's a jump in c, that's not a bad thing. 712 00:44:29 --> 00:44:33 Because the point is there's no jump in c*u'. 713 00:44:34 --> 00:44:39 c*du/dx doesn't jump. w doesn't jump. 714 00:44:39 --> 00:44:40 From a jump in c. 715 00:44:40 --> 00:44:43 If c jumps, let it. 716 00:44:43 --> 00:44:49 There's no jump in w, which is the serious unknown. 717 00:44:49 --> 00:44:54 There'll be a jump in du/dx, and that was that e thing 718 00:44:54 --> 00:44:56 that you had on the quiz. 719 00:44:56 --> 00:44:59 There'll be a jump in du/dx to compensate the jump 720 00:44:59 --> 00:45:03 in c, but w is good. 721 00:45:03 --> 00:45:08 That's the message of, so let me write that down. 722 00:45:08 --> 00:45:22 Even if c jumps, w does not. 723 00:45:22 --> 00:45:27 So my point is when you're looking at w, you're looking 724 00:45:27 --> 00:45:29 at the right quantity. 725 00:45:29 --> 00:45:35 And it deals, this is the sort of general feature that if 726 00:45:35 --> 00:45:37 you look at it right it's not a problem. 727 00:45:37 --> 00:45:43 If you look at it wrong and try to write out that equation, 728 00:45:43 --> 00:45:46 take the second derivative of this is OK, but then if you 729 00:45:46 --> 00:45:49 just try to take the derivative of the jump, you think, my God 730 00:45:49 --> 00:45:53 there's a delta function sitting in my equation, 731 00:45:53 --> 00:45:55 what am I to do. 732 00:45:55 --> 00:45:59 Don't do it. 733 00:45:59 --> 00:46:07 That pair of equations is no trouble. 734 00:46:07 --> 00:46:09 Do you see that point? 735 00:46:09 --> 00:46:17 That a jump in c is really OK. c=0 wouldn't be so good, right? 736 00:46:17 --> 00:46:22 If c went to zero then you have got some problems here, because 737 00:46:22 --> 00:46:25 if c is zero here what's going on? 738 00:46:25 --> 00:46:35 So your material should have some stiffness, 739 00:46:35 --> 00:46:37 some positive stiffness. 740 00:46:37 --> 00:46:39 Zero stiffness would be a problem. 741 00:46:39 --> 00:46:44 Negative stiffness would be really a crazy material, 742 00:46:44 --> 00:46:52 where you add more force and it compresses on you. 743 00:46:52 --> 00:47:00 OK so that was a point I could make also in the lecture. 744 00:47:00 --> 00:47:04 Other questions. 745 00:47:04 --> 00:47:10 I hope you find these review sessions, they give a chance 746 00:47:10 --> 00:47:16 to bring out more points. 747 00:47:16 --> 00:47:21 Let me say a little bit about Chapter 2 things that we 748 00:47:21 --> 00:47:26 abandoned without including in the lectures. 749 00:47:26 --> 00:47:30 One very important thing that we didn't touch was the 750 00:47:30 --> 00:47:32 non linear problems. 751 00:47:32 --> 00:47:40 Where in the end you have to solve a system of 752 00:47:40 --> 00:47:43 non linear equations. 753 00:47:43 --> 00:47:46 We've only had linear equations, KU=F, where 754 00:47:46 --> 00:47:47 we almost got to. 755 00:47:47 --> 00:47:53 And we've certainly found enough to discuss there. 756 00:47:53 --> 00:47:56 But how do you deal with non linear equations? 757 00:47:56 --> 00:48:03 Well so that's Section 2.6, I guess. 758 00:48:03 --> 00:48:09 And the answer is Newton's method, or some version, 759 00:48:09 --> 00:48:11 there are many versions of Newton's method. 760 00:48:11 --> 00:48:18 So all I'm saying is that's an important topic. 761 00:48:18 --> 00:48:21 How to deal with non linear equations. 762 00:48:21 --> 00:48:27 And the answer is usually some variation of Newton's method. 763 00:48:27 --> 00:48:30 Depending on how many equations you've got, how bad the 764 00:48:30 --> 00:48:34 non-linearity is, and things like that. 765 00:48:34 --> 00:48:40 It's a case where coding is not so simple. 766 00:48:40 --> 00:48:42 But important. 767 00:48:42 --> 00:48:46 So that's one thing in Chapter 2 that we've passed, because 768 00:48:46 --> 00:48:51 really this is where we want to be today. 769 00:48:51 --> 00:48:54 Other topics, questions of any kind? 770 00:48:54 --> 00:48:54 Yes, thanks. 771 00:48:54 --> 00:48:57 AUDIENCE: [INAUDIBLE] 772 00:48:57 --> 00:48:58 PROFESSOR STRANG: This problem, OK. 773 00:48:58 --> 00:49:02 AUDIENCE: [INAUDIBLE] 774 00:49:02 --> 00:49:02 PROFESSOR STRANG: Yes 775 00:49:02 --> 00:49:07 AUDIENCE: [INAUDIBLE] 776 00:49:07 --> 00:49:16 Well, OK. 777 00:49:16 --> 00:49:19 Right, good question. 778 00:49:19 --> 00:49:20 You're right. 779 00:49:20 --> 00:49:25 AUDIENCE: [INAUDIBLE] 780 00:49:25 --> 00:49:26 PROFESSOR STRANG: Yeah, OK. 781 00:49:26 --> 00:49:29 So now suppose that delta of x was in delta of x-a, 782 00:49:29 --> 00:49:31 the jump at a was there. 783 00:49:31 --> 00:49:38 What would change in there? 784 00:49:38 --> 00:49:40 Well, the one would change, right? 785 00:49:40 --> 00:49:42 The one would now be the delta of x-a. 786 00:49:45 --> 00:49:48 So what am I seeing now? 787 00:49:48 --> 00:49:51 What's the jump situation now? 788 00:49:51 --> 00:49:56 Is there a jump in, does w jump now from this delta function? 789 00:49:56 --> 00:49:58 Yes. 790 00:49:58 --> 00:50:01 It did not jump from the jump in c. 791 00:50:01 --> 00:50:04 That did not produce us a jump in w. 792 00:50:04 --> 00:50:04 Why not? 793 00:50:04 --> 00:50:13 Because again back to the jump in c, you have a bar of copper 794 00:50:13 --> 00:50:16 and you have a bar of steel. 795 00:50:16 --> 00:50:19 And they have different c's. 796 00:50:19 --> 00:50:24 But the force at that point, when it's in equilibrium, the 797 00:50:24 --> 00:50:28 bar's not falling apart, right? 798 00:50:28 --> 00:50:31 The force is the same from above and below; 799 00:50:31 --> 00:50:33 equilibrium holds. 800 00:50:33 --> 00:50:40 It's just that the force involves a c_1 there, and down 801 00:50:40 --> 00:50:45 here the force involves a c_2 multiplying the du/dx, 802 00:50:45 --> 00:50:46 and this stretching. 803 00:50:46 --> 00:50:52 So the stretching rule changes. du/dx, the stretching factor 804 00:50:52 --> 00:50:54 will be different in the two materials. 805 00:50:54 --> 00:50:57 But the force won't be. 806 00:50:57 --> 00:50:59 You really have to keep straight what. 807 00:50:59 --> 00:51:02 Now, you asked about a delta function, OK. 808 00:51:02 --> 00:51:05 So now if I hang a delta function there, that's 809 00:51:05 --> 00:51:08 like hanging a point load at a point. 810 00:51:08 --> 00:51:10 What happens at that point? 811 00:51:10 --> 00:51:14 Well, we don't have a problem with that. 812 00:51:14 --> 00:51:17 Well, w jumps. 813 00:51:17 --> 00:51:21 The force pulling up is not the same as the force pulling down 814 00:51:21 --> 00:51:25 because there's this extra point load. 815 00:51:25 --> 00:51:29 So the balance there and the balance there, 816 00:51:29 --> 00:51:30 those aren't equal. 817 00:51:30 --> 00:51:35 There's a jump there because of the point load being put there. 818 00:51:35 --> 00:51:40 So that produces a jump in w, but not a jump in u. 819 00:51:40 --> 00:51:46 Not a jump in u, the bar is still holding together. 820 00:51:46 --> 00:51:53 And since time's running out I don't have to ask myself or ask 821 00:51:53 --> 00:51:58 you about the worst possibility that occurs to me, which is 822 00:51:58 --> 00:52:01 suppose they happen at the same place. 823 00:52:01 --> 00:52:06 Suppose there's a jump in c and a point load. 824 00:52:06 --> 00:52:12 Do we want to face that, what would happen there? 825 00:52:12 --> 00:52:17 Suppose, a was the place where this jumped. 826 00:52:17 --> 00:52:20 Suppose the point load was here. 827 00:52:20 --> 00:52:21 What's up there? 828 00:52:21 --> 00:52:24 Well, maybe I will able to do that. 829 00:52:24 --> 00:52:29 OK, so I'm putting the point load at the place where the 830 00:52:29 --> 00:52:32 copper and the steel meet. 831 00:52:32 --> 00:52:34 Is w continuous? 832 00:52:34 --> 00:52:39 Is the force the same above and below that joint? 833 00:52:39 --> 00:52:39 No. 834 00:52:39 --> 00:52:44 Because there's these extra terms from the weight. 835 00:52:44 --> 00:52:49 And then once you know the forces, then above the bar 836 00:52:49 --> 00:52:54 you're solving for u with one c, and then below that 837 00:52:54 --> 00:52:56 point with the other c. 838 00:52:56 --> 00:53:02 So w has a plus and a minus and c has a plus and a minus. 839 00:53:03 --> 00:53:12 And you're solving this one, let me make them minus plus. 840 00:53:12 --> 00:53:17 So you're solving this one in one bar, one metal. 841 00:53:17 --> 00:53:20 And c plus and the w plus in the other method. 842 00:53:20 --> 00:53:25 So yeah, you can do it. 843 00:53:25 --> 00:53:30 Shall I just repeat what my overall message is? w is the 844 00:53:30 --> 00:53:33 right thing to look at. 845 00:53:33 --> 00:53:38 That combination c*du/dx is the right thing to look at. 846 00:53:38 --> 00:53:43 And it's what sits there. 847 00:53:43 --> 00:53:45 In those parentheses. 848 00:53:45 --> 00:53:50 OK, good, so that's some review topics. 849 00:53:50 --> 00:53:54 We really have a lot of fun ahead now with finite elements.