1 00:00:00,000 --> 00:00:02,220 2 00:00:02,220 --> 00:00:05,432 under a Creative Commons license. 3 00:00:05,432 --> 00:00:07,140 Your support will help MIT OpenCourseWare 4 00:00:07,140 --> 00:00:09,723 continue to offer high quality educational resources for free. 5 00:00:09,723 --> 00:00:13,450 To make a donation, or to view additional materials 6 00:00:13,450 --> 00:00:19,340 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,340 --> 00:00:20,062 at ocw.mit.edu. 8 00:00:20,062 --> 00:00:21,520 PROFESSOR STRANG: So I'm hoping you 9 00:00:21,520 --> 00:00:25,560 will ask some questions today. 10 00:00:25,560 --> 00:00:27,940 So we've had the exam. 11 00:00:27,940 --> 00:00:31,320 If you have any questions about grading the two TAs 12 00:00:31,320 --> 00:00:33,320 are the people to speak to. 13 00:00:33,320 --> 00:00:37,540 Ramis graded Questions one and two, and Peter 14 00:00:37,540 --> 00:00:38,820 graded three and four. 15 00:00:38,820 --> 00:00:40,980 And they control things. 16 00:00:40,980 --> 00:00:44,570 I mean, I can help if there's an emergency, 17 00:00:44,570 --> 00:00:47,370 but they would be the right people to speak to. 18 00:00:47,370 --> 00:00:51,440 Because they know how they graded the whole test. 19 00:00:51,440 --> 00:00:57,970 Before you ask questions, can I just say why this truss -- 20 00:00:57,970 --> 00:01:02,450 you remember this six-sided truss -- 21 00:01:02,450 --> 00:01:09,070 tied me in a knot and I'm hoping your MATLAB solution will untie 22 00:01:09,070 --> 00:01:11,010 that knot. 23 00:01:11,010 --> 00:01:15,380 The knot I was in was to find the mechanisms, to find 24 00:01:15,380 --> 00:01:19,490 convenient mechanisms because we have, 25 00:01:19,490 --> 00:01:23,960 well, I thought we had six bars. 26 00:01:23,960 --> 00:01:26,510 It looks like we have six bars. 27 00:01:26,510 --> 00:01:28,640 But somebody pointed out after class 28 00:01:28,640 --> 00:01:33,350 that that bar six is not very active. 29 00:01:33,350 --> 00:01:37,940 It's connecting two supports, can't do anything, 30 00:01:37,940 --> 00:01:42,550 and actually that sixth row of the A matrix 31 00:01:42,550 --> 00:01:44,570 will be all zeroes. 32 00:01:44,570 --> 00:01:49,540 So our matrix, if we include that, there was no use, 33 00:01:49,540 --> 00:01:51,250 it doesn't help. 34 00:01:51,250 --> 00:01:55,230 So our matrix then really comes from the five bars, 35 00:01:55,230 --> 00:02:03,030 so A is then five by, two unknowns, two, two, two, 36 00:02:03,030 --> 00:02:05,370 making altogether eight unknowns. 37 00:02:05,370 --> 00:02:12,120 So three mechanisms and that's what I'm hoping for. 38 00:02:12,120 --> 00:02:17,480 So when I unwisely drew that picture on the board 39 00:02:17,480 --> 00:02:20,200 at the end of the truss lecture, I was only 40 00:02:20,200 --> 00:02:22,100 looking for two mechanisms because I 41 00:02:22,100 --> 00:02:25,140 was thinking we had six edges, six bars, 42 00:02:25,140 --> 00:02:31,260 but really this bar, when I knock out 43 00:02:31,260 --> 00:02:37,000 the columns that correspond to that node and that node, 44 00:02:37,000 --> 00:02:40,230 there will only be zeroes left in that row. 45 00:02:40,230 --> 00:02:42,310 And it's nothing. 46 00:02:42,310 --> 00:02:47,880 It correctly tells us that the stretching of that bar is zero 47 00:02:47,880 --> 00:02:50,990 but we knew that anyway. 48 00:02:50,990 --> 00:02:57,510 OK, I don't know if you've tackled the MATLAB question, 49 00:02:57,510 --> 00:03:03,620 and I also don't know whether MATLAB would produce for us-- 50 00:03:03,620 --> 00:03:06,240 I mean, you should be able to construct 51 00:03:06,240 --> 00:03:09,650 A with a whole lot of square roots of three 52 00:03:09,650 --> 00:03:16,290 over two from sine of 60 degrees and maybe 1/2's 53 00:03:16,290 --> 00:03:19,890 from sine of 30 degrees. 54 00:03:19,890 --> 00:03:25,250 A should look pretty nice, but what the solutions to these 55 00:03:25,250 --> 00:03:30,000 mechanisms are, of course solutions to Au=0, 56 00:03:30,000 --> 00:03:39,350 and-- Anyway, I'm hoping that we learn from this example. 57 00:03:39,350 --> 00:03:42,490 I hadn't intended-- So it's pretty small MATLAB, 58 00:03:42,490 --> 00:03:45,900 it's really just the creation of the matrix A. 59 00:03:45,900 --> 00:03:50,500 OK, so that was a comment on that, 60 00:03:50,500 --> 00:03:52,190 which I added to the homework. 61 00:03:52,190 --> 00:03:57,340 Now, I'm ready for questions about the homework, the exam 62 00:03:57,340 --> 00:03:59,070 whatever, yes thank you. 63 00:03:59,070 --> 00:04:01,430 Oh, good. 64 00:04:01,430 --> 00:04:06,570 AUDIENCE: [INAUDIBLE] 65 00:04:06,570 --> 00:04:09,596 PROFESSOR STRANG: For trusses, OK. 66 00:04:09,596 --> 00:04:10,470 AUDIENCE: [INAUDIBLE] 67 00:04:10,470 --> 00:04:14,250 PROFESSOR STRANG: Yeah. 68 00:04:14,250 --> 00:04:17,600 A^T A immediately, couldn't you? 69 00:04:17,600 --> 00:04:19,780 It's not quite so straightforward. 70 00:04:19,780 --> 00:04:21,650 And this is like more realistic. 71 00:04:21,650 --> 00:04:25,080 I mean, yeah. 72 00:04:25,080 --> 00:04:28,080 Ah. 73 00:04:28,080 --> 00:04:31,630 I guess, yeah, but remind me what that question was. 74 00:04:31,630 --> 00:04:37,170 This was a 2.7 problem one? 75 00:04:37,170 --> 00:04:40,050 OK, let me just see what I'm asking for. 76 00:04:40,050 --> 00:04:43,020 So, OK, yes. 77 00:04:43,020 --> 00:04:45,540 So I only asked about A transpose A, 78 00:04:45,540 --> 00:04:49,750 I only asked you for the shape in that question. 79 00:04:49,750 --> 00:04:55,820 Oh, and then I asked you for the first row, good for me, yes. 80 00:04:55,820 --> 00:04:58,985 I see, the first row so I didn't put you 81 00:04:58,985 --> 00:05:03,210 to the agony of writing out the whole thing, 82 00:05:03,210 --> 00:05:09,550 but still how do you get the first row, good question. 83 00:05:09,550 --> 00:05:12,334 AUDIENCE: [INAUDIBLE] 84 00:05:12,334 --> 00:05:13,500 PROFESSOR STRANG: Oh, I see. 85 00:05:13,500 --> 00:05:15,790 Now, well it's not going to be so neat. 86 00:05:15,790 --> 00:05:17,320 Let's just think. 87 00:05:17,320 --> 00:05:23,090 We could do the first row of A, and the first column of A 88 00:05:23,090 --> 00:05:26,750 probably, yeah. 89 00:05:26,750 --> 00:05:30,800 So what am I looking for in A transpose A, 90 00:05:30,800 --> 00:05:34,140 because it didn't say row one. 91 00:05:34,140 --> 00:05:41,710 OK, so row one of A transpose A corresponds to the first-- Oh, 92 00:05:41,710 --> 00:05:42,800 so what is row one? 93 00:05:42,800 --> 00:05:44,380 Yeah, this is worth thinking about. 94 00:05:44,380 --> 00:05:49,850 So this is A transpose A for trusses. 95 00:05:49,850 --> 00:05:51,830 And let's just, maybe we could even 96 00:05:51,830 --> 00:05:54,830 take this one as an example. 97 00:05:54,830 --> 00:06:01,330 If I number the nodes, if that's my first node, A transpose A, 98 00:06:01,330 --> 00:06:03,460 you remember that's square. 99 00:06:03,460 --> 00:06:07,960 That tells us the edge part, the bar part is built into it. 100 00:06:07,960 --> 00:06:13,540 But its size is n by n, its size is five by, no, its size 101 00:06:13,540 --> 00:06:15,740 is what, eight by eight, right? 102 00:06:15,740 --> 00:06:21,670 It's got to do with the number of-- OK. 103 00:06:21,670 --> 00:06:29,830 So its first row, what will its first row be about? 104 00:06:29,830 --> 00:06:36,920 It'll be about u H 1, right? 105 00:06:36,920 --> 00:06:41,050 The first row of this A transpose 106 00:06:41,050 --> 00:06:46,260 A matrix will be about, the first node but more than that, 107 00:06:46,260 --> 00:06:50,690 that node has got two things, u H and u V, 108 00:06:50,690 --> 00:06:52,160 and we're putting H first. 109 00:06:52,160 --> 00:06:59,140 So so I think that the first row of this, let's just draw it. 110 00:06:59,140 --> 00:07:00,660 Maybe I'm not saying-- 111 00:07:00,660 --> 00:07:05,860 So let that be A transpose A, I see, yeah. 112 00:07:05,860 --> 00:07:12,820 A transpose A will be multiplying u H 1, u V 1, 113 00:07:12,820 --> 00:07:13,991 and so forth. 114 00:07:13,991 --> 00:07:14,490 Right? 115 00:07:14,490 --> 00:07:15,950 OK, good. 116 00:07:15,950 --> 00:07:17,800 So now this is better. 117 00:07:17,800 --> 00:07:22,260 All I'm asking is where do we know 118 00:07:22,260 --> 00:07:26,040 there are zeroes in that first row of A transpose A? 119 00:07:26,040 --> 00:07:30,480 Where would we know that there are zeroes for this problem 120 00:07:30,480 --> 00:07:33,210 rather than-- I won't redraw the one in the book 121 00:07:33,210 --> 00:07:35,260 so I'm taking this truss. 122 00:07:35,260 --> 00:07:38,600 So if it's this truss, this is the first node, 123 00:07:38,600 --> 00:07:44,130 the bars are numbered like this, and of course your MATLAB 124 00:07:44,130 --> 00:07:47,630 construction of A, you might multiply out A transpose A. 125 00:07:47,630 --> 00:07:49,700 Because in MATLAB that would be so quick. 126 00:07:49,700 --> 00:07:51,780 And you could see what it looks like. 127 00:07:51,780 --> 00:07:57,980 And where would we expect to see zeroes or not zeroes? 128 00:07:57,980 --> 00:08:04,580 Let's see, A transpose A, so I'll have to think. 129 00:08:04,580 --> 00:08:09,780 My instinct is that it should only connect two neighbors. 130 00:08:09,780 --> 00:08:12,750 So that I would imagine, but I've 131 00:08:12,750 --> 00:08:14,490 got-- These are double neighbors, 132 00:08:14,490 --> 00:08:17,670 right, there's two people living here. 133 00:08:17,670 --> 00:08:19,290 We have to remember. 134 00:08:19,290 --> 00:08:22,370 So I see two people living here and two people at home, 135 00:08:22,370 --> 00:08:26,920 so I guess I would imagine four non-zeroes for this problem 136 00:08:26,920 --> 00:08:30,770 in in the first row. 137 00:08:30,770 --> 00:08:37,820 I would think that that would have somebody on the diagonal. 138 00:08:37,820 --> 00:08:41,450 That would be what multiplies u H 1 itself, 139 00:08:41,450 --> 00:08:45,970 and then maybe u V 1 is involved, and these two guys. 140 00:08:45,970 --> 00:08:49,750 These two and those two would not be involved. 141 00:08:49,750 --> 00:08:52,100 Well, that's only a partial answer, 142 00:08:52,100 --> 00:08:55,970 I'm just telling you where the zeroes are, I think. 143 00:08:55,970 --> 00:09:01,110 And you're really asking about the non-zeroes, of course. 144 00:09:01,110 --> 00:09:03,680 So, yeah. 145 00:09:03,680 --> 00:09:06,990 Not so easy. 146 00:09:06,990 --> 00:09:07,490 Yeah -- 147 00:09:07,490 --> 00:09:07,750 AUDIENCE: [INAUDIBLE] 148 00:09:07,750 --> 00:09:09,458 PROFESSOR STRANG: -- maybe have to do it. 149 00:09:09,458 --> 00:09:13,340 OK, yeah. 150 00:09:13,340 --> 00:09:18,140 I think, the first time better to do it, yeah. 151 00:09:18,140 --> 00:09:19,900 But that's an excellent question, 152 00:09:19,900 --> 00:09:25,250 and maybe we will find a nice, and if somebody does, 153 00:09:25,250 --> 00:09:26,190 let me know. 154 00:09:26,190 --> 00:09:29,210 A nice way to see that. 155 00:09:29,210 --> 00:09:31,450 Somehow I must have felt that was doable 156 00:09:31,450 --> 00:09:37,590 when I created that problem. 157 00:09:37,590 --> 00:09:40,120 OK. 158 00:09:40,120 --> 00:09:44,520 The other thing to say though, that while I'm 159 00:09:44,520 --> 00:09:47,090 looking at A transpose A, is remember 160 00:09:47,090 --> 00:09:53,710 that it can be created by these element matrices. 161 00:09:53,710 --> 00:09:56,530 Yeah, yeah, that's important to say. 162 00:09:56,530 --> 00:10:02,230 So there will be five element matrices, right? 163 00:10:02,230 --> 00:10:04,460 Because I've got five bars. 164 00:10:04,460 --> 00:10:07,810 And the element matrix for this bar, or say 165 00:10:07,810 --> 00:10:12,310 for a typical bar, the element matrix for this bar 166 00:10:12,310 --> 00:10:16,230 will involve all the u's that are connected to this bar. 167 00:10:16,230 --> 00:10:18,550 So two u's there and two u's there, 168 00:10:18,550 --> 00:10:22,130 so it'll be four by four, the element matrix here. 169 00:10:22,130 --> 00:10:30,230 So this will contribute four non-zeroes to that to row 170 00:10:30,230 --> 00:10:31,470 and to three other rows. 171 00:10:31,470 --> 00:10:35,750 That four by four guy from this part is, 172 00:10:35,750 --> 00:10:37,640 you know what I'm speaking about here? 173 00:10:37,640 --> 00:10:41,370 The element matrix which is just-- The element matrix 174 00:10:41,370 --> 00:10:47,950 is a cosine, sine, minus cosine and minus sine. 175 00:10:47,950 --> 00:10:49,780 Column times row. 176 00:10:49,780 --> 00:10:54,440 Cosine, sine, minus cosine, minus sine. 177 00:10:54,440 --> 00:10:57,450 This is the element matrix for this bar, 178 00:10:57,450 --> 00:11:00,750 those are the cosine and sine of 60 degrees, 179 00:11:00,750 --> 00:11:10,450 and those positions would be these two and these two. 180 00:11:10,450 --> 00:11:14,640 So there would be actually four zeroes coming after, and after 181 00:11:14,640 --> 00:11:18,360 here from those two and those two. 182 00:11:18,360 --> 00:11:23,570 So this bar contributes to A transpose A, 183 00:11:23,570 --> 00:11:26,620 sort of stamps into A transpose A. 184 00:11:26,620 --> 00:11:33,190 This four by four of non-zeroes plus the rest zeroes, stamps 185 00:11:33,190 --> 00:11:36,690 that in up there. 186 00:11:36,690 --> 00:11:43,530 This guy has its own element matrix. 187 00:11:43,530 --> 00:11:47,470 Now, what's different about the element matrix for this guy? 188 00:11:47,470 --> 00:11:52,300 Of course it's got its own c_1, probably, and it had a c_2, 189 00:11:52,300 --> 00:11:53,480 this difference constant. 190 00:11:53,480 --> 00:11:56,250 But bigger differences than that. 191 00:11:56,250 --> 00:11:59,000 First of all, it's a different angle, theta. 192 00:11:59,000 --> 00:12:05,540 Here the angle is 120 degrees, and also there's 193 00:12:05,540 --> 00:12:08,000 nobody home down there. 194 00:12:08,000 --> 00:12:12,440 So it would be, if we were dealing with A_0, 195 00:12:12,440 --> 00:12:15,220 free-free stuff, it would be four by four, 196 00:12:15,220 --> 00:12:16,550 like all the others. 197 00:12:16,550 --> 00:12:23,080 But because these two are fixed, those two displacements 198 00:12:23,080 --> 00:12:28,530 are fixed, effectively it will only contribute two by two. 199 00:12:28,530 --> 00:12:30,250 A total of four non-zeroes. 200 00:12:30,250 --> 00:12:33,090 But those will stamp into A transpose 201 00:12:33,090 --> 00:12:37,270 A overlapping this guy. 202 00:12:37,270 --> 00:12:40,030 Then the ones from these bars will not 203 00:12:40,030 --> 00:12:43,160 touch the first row of A transpose A, 204 00:12:43,160 --> 00:12:44,730 so they wouldn't touch the answer 205 00:12:44,730 --> 00:12:47,810 to that homework problem. 206 00:12:47,810 --> 00:12:52,510 So you see again another way. 207 00:12:52,510 --> 00:12:57,210 By hand, I don't know that you necessarily 208 00:12:57,210 --> 00:13:01,610 want to use these element matrices, I'm not sure. 209 00:13:01,610 --> 00:13:03,720 But it gives an excellent check. 210 00:13:03,720 --> 00:13:06,320 So one way to create A transpose A 211 00:13:06,320 --> 00:13:11,520 is create a-- Multiply, that's one way. 212 00:13:11,520 --> 00:13:16,450 Second way is, create A transpose A element by-- 213 00:13:16,450 --> 00:13:23,310 By these five element matrices popped into the right places. 214 00:13:23,310 --> 00:13:26,670 I think this is, so this like the computational science 215 00:13:26,670 --> 00:13:31,430 part of the course. 216 00:13:31,430 --> 00:13:33,580 It's the way you would, I'm really 217 00:13:33,580 --> 00:13:36,310 speaking about the way you would write the code, not 218 00:13:36,310 --> 00:13:40,090 the final result. So, much of the course 219 00:13:40,090 --> 00:13:44,200 has been devoted to understanding A transpose A, 220 00:13:44,200 --> 00:13:46,900 and the fact that it's positive definite, 221 00:13:46,900 --> 00:13:50,990 or in this case only positive semi-definite because we 222 00:13:50,990 --> 00:13:53,560 have an eight by eight matrix whose rank is only 223 00:13:53,560 --> 00:13:55,440 going to be five. 224 00:13:55,440 --> 00:14:01,960 So this matrix will have these same mechanisms, 225 00:14:01,960 --> 00:14:06,420 and I'm hoping to learn what they are. 226 00:14:06,420 --> 00:14:08,700 Well I hope that's a little help with that question. 227 00:14:08,700 --> 00:14:11,600 But basically, though, it's non-help 228 00:14:11,600 --> 00:14:14,200 because I'm sort of saying you're on your own 229 00:14:14,200 --> 00:14:20,000 to actually-- I don't have a superfragilistic way 230 00:14:20,000 --> 00:14:25,550 to construct a matrix and then you kind of have to do it. 231 00:14:25,550 --> 00:14:28,860 But I'm glad you asked. 232 00:14:28,860 --> 00:14:29,850 Yes? 233 00:14:29,850 --> 00:14:32,460 AUDIENCE: [INAUDIBLE] 234 00:14:32,460 --> 00:14:35,770 PROFESSOR STRANG: More on this problem, OK. 235 00:14:35,770 --> 00:14:39,795 The sign convention. 236 00:14:39,795 --> 00:14:40,670 AUDIENCE: [INAUDIBLE] 237 00:14:40,670 --> 00:14:41,586 PROFESSOR STRANG: Yes. 238 00:14:41,586 --> 00:14:43,270 So let's see. 239 00:14:43,270 --> 00:14:46,310 So I made a speech about sign conventions, right? 240 00:14:46,310 --> 00:14:49,050 241 00:14:49,050 --> 00:14:54,290 Which was that I am too old for them. 242 00:14:54,290 --> 00:14:57,310 And the justification for that is the fact 243 00:14:57,310 --> 00:15:01,010 that they wash out in A transpose. 244 00:15:01,010 --> 00:15:04,630 So if you were to use the wrong sign convention in A, 245 00:15:04,630 --> 00:15:06,360 you won't see it in A transpose A. 246 00:15:06,360 --> 00:15:09,140 So if I change the whole thing. 247 00:15:09,140 --> 00:15:15,420 So sign convention is not really a convention. 248 00:15:15,420 --> 00:15:22,590 Our convention is to decide that that movement that way is plus. 249 00:15:22,590 --> 00:15:25,290 And movement this way is plus. 250 00:15:25,290 --> 00:15:26,550 And here, similarly. 251 00:15:26,550 --> 00:15:30,220 That's a plus movement and that's a plus movement. 252 00:15:30,220 --> 00:15:32,620 That's our sign convention. 253 00:15:32,620 --> 00:15:38,250 If we always do that, then the A matrix 254 00:15:38,250 --> 00:15:42,390 is telling us how much this bar stretches. 255 00:15:42,390 --> 00:15:45,480 It's the matrix that gives us stretching from these four 256 00:15:45,480 --> 00:15:46,840 displacements. 257 00:15:46,840 --> 00:15:54,500 And then, these, I think in this picture, 258 00:15:54,500 --> 00:15:58,510 if this is the later node and this is the thing, 259 00:15:58,510 --> 00:16:01,860 then that angle from the horizontal 260 00:16:01,860 --> 00:16:03,540 is the right guy to put there. 261 00:16:03,540 --> 00:16:05,902 AUDIENCE: [INAUDIBLE] 262 00:16:05,902 --> 00:16:06,860 PROFESSOR STRANG: Here? 263 00:16:06,860 --> 00:16:12,630 AUDIENCE: [INAUDIBLE] 264 00:16:12,630 --> 00:16:18,130 PROFESSOR STRANG: Well, I guess I'm, I see, 265 00:16:18,130 --> 00:16:20,360 the question is what's the sign convention 266 00:16:20,360 --> 00:16:22,200 with this particular picture? 267 00:16:22,200 --> 00:16:27,280 Yeah, if this is node one and that's two, 268 00:16:27,280 --> 00:16:31,340 then-- Your question is good. 269 00:16:31,340 --> 00:16:35,970 It looks, I think, so I think I've got it not right. 270 00:16:35,970 --> 00:16:40,890 If this is node one down below, and two up above, 271 00:16:40,890 --> 00:16:46,690 then moving two positively would stretch the bar. 272 00:16:46,690 --> 00:16:48,580 And I've got minuses there. 273 00:16:48,580 --> 00:16:53,200 So I've got the opposite signs. 274 00:16:53,200 --> 00:16:57,450 I've got there, the signs that would go with this numbering. 275 00:16:57,450 --> 00:17:00,120 If that was node two and this was node one. 276 00:17:00,120 --> 00:17:03,440 Yeah, you're right, good. 277 00:17:03,440 --> 00:17:05,440 You have to, and that's the only way I do it, 278 00:17:05,440 --> 00:17:10,260 is to think will the movement stretch the bar? 279 00:17:10,260 --> 00:17:13,620 Then that's got a positive entry in A. Will 280 00:17:13,620 --> 00:17:18,040 the movement like that compress the bar, that'll 281 00:17:18,040 --> 00:17:19,410 be a negative entry. 282 00:17:19,410 --> 00:17:25,900 And our conventions are if it stretches that's positive. 283 00:17:25,900 --> 00:17:32,180 e positive means a stretching; e negative means a compression. 284 00:17:32,180 --> 00:17:35,480 So it takes a little patience. 285 00:17:35,480 --> 00:17:38,590 Or it takes writing the code correctly once 286 00:17:38,590 --> 00:17:46,140 and then of course it would do it for all these problems. 287 00:17:46,140 --> 00:17:48,910 Yeah, I think the TA in an earlier year 288 00:17:48,910 --> 00:17:52,960 actually wrote a code to, a truss code. 289 00:17:52,960 --> 00:17:59,010 I don't know what happened to it but I could look. 290 00:17:59,010 --> 00:18:01,310 AUDIENCE: [INAUDIBLE] 291 00:18:01,310 --> 00:18:08,330 PROFESSOR STRANG: Yes, yeah. 292 00:18:08,330 --> 00:18:11,060 That's right, and it will come out consistent. 293 00:18:11,060 --> 00:18:16,500 Yeah, if I take horizontal-- if I take the same convention that 294 00:18:16,500 --> 00:18:22,920 forces that way are plus, that's a plus f H 1, 295 00:18:22,920 --> 00:18:29,410 and it's a plus u H 1, then I'll get A will go to A transpose. 296 00:18:29,410 --> 00:18:33,020 Yeah, so if I'm consistent that plus for the u's, 297 00:18:33,020 --> 00:18:36,160 for the displacements, and plus for the f's, forces, 298 00:18:36,160 --> 00:18:40,510 then the correctly created A will give me the correct A 299 00:18:40,510 --> 00:18:41,680 transpose. 300 00:18:41,680 --> 00:18:45,150 But A transpose A has the ability 301 00:18:45,150 --> 00:18:51,640 to bury some of those conventions. 302 00:18:51,640 --> 00:18:52,641 OK. 303 00:18:52,641 --> 00:18:53,140 Good. 304 00:18:53,140 --> 00:18:53,690 Yes. 305 00:18:53,690 --> 00:18:55,030 AUDIENCE: [INAUDIBLE] 306 00:18:55,030 --> 00:18:56,540 PROFESSOR STRANG: Different. 307 00:18:56,540 --> 00:18:57,480 More. 308 00:18:57,480 --> 00:18:58,230 Oh my God. 309 00:18:58,230 --> 00:18:58,920 OK. 310 00:18:58,920 --> 00:18:59,420 Yes. 311 00:18:59,420 --> 00:19:08,536 AUDIENCE: [INAUDIBLE] 312 00:19:08,536 --> 00:19:09,910 PROFESSOR STRANG: This one, yeah. 313 00:19:09,910 --> 00:19:13,492 AUDIENCE: [INAUDIBLE] 314 00:19:13,492 --> 00:19:14,450 PROFESSOR STRANG: Yeah. 315 00:19:14,450 --> 00:19:22,150 AUDIENCE: [INAUDIBLE] 316 00:19:22,150 --> 00:19:27,360 PROFESSOR STRANG: I wish it would ask about bar three. 317 00:19:27,360 --> 00:19:29,490 Yeah, yeah bar three was good. 318 00:19:29,490 --> 00:19:31,240 Yeah right. 319 00:19:31,240 --> 00:19:35,110 Yeah, I guess here, what's here, well, the angle is, that must 320 00:19:35,110 --> 00:19:40,300 be theta, and then this'll be right provided 321 00:19:40,300 --> 00:19:45,740 this number is the, is what? 322 00:19:45,740 --> 00:19:50,110 Well, I've got to think it out again. 323 00:19:50,110 --> 00:19:56,510 I'll just think it out. 324 00:19:56,510 --> 00:19:59,120 Maybe, yeah I won't try to, yeah I won't. 325 00:19:59,120 --> 00:19:59,720 I won't. 326 00:19:59,720 --> 00:20:03,730 So I think you-- But it's really, 327 00:20:03,730 --> 00:20:14,160 you learn the point by thinking that through. 328 00:20:14,160 --> 00:20:15,760 I think we've got it here. 329 00:20:15,760 --> 00:20:21,810 That if that's number one then we like plus signs here 330 00:20:21,810 --> 00:20:27,320 because positive displacements will stretch the bar, right. 331 00:20:27,320 --> 00:20:32,220 And here this sign will depend on the angle. 332 00:20:32,220 --> 00:20:38,414 And it will depend on the number, which numbers, 333 00:20:38,414 --> 00:20:40,580 what number that one is and what number that one is. 334 00:20:40,580 --> 00:20:45,440 If I reverse the numbers, then I reverse the sign. 335 00:20:45,440 --> 00:20:47,550 Yeah, good. 336 00:20:47,550 --> 00:20:52,530 Well, that's sort of part of computational engineering 337 00:20:52,530 --> 00:20:53,090 isn't it? 338 00:20:53,090 --> 00:20:56,500 Like, keep track of the details. 339 00:20:56,500 --> 00:21:00,460 But of course this 18.085 course is also 340 00:21:00,460 --> 00:21:02,280 about big picture things. 341 00:21:02,280 --> 00:21:09,660 And that's what I'm hoping comes through even better. 342 00:21:09,660 --> 00:21:12,270 OK, ready for another question on any topic. 343 00:21:12,270 --> 00:21:17,265 Thank you. 344 00:21:17,265 --> 00:21:18,140 AUDIENCE: [INAUDIBLE] 345 00:21:18,140 --> 00:21:28,130 PROFESSOR STRANG: Yes. 346 00:21:28,130 --> 00:21:30,980 Well, no, but let's just try it. 347 00:21:30,980 --> 00:21:32,610 Shall we do Problem one? 348 00:21:32,610 --> 00:21:34,540 Was it Problem one? 349 00:21:34,540 --> 00:21:36,100 In 2.7? 350 00:21:36,100 --> 00:21:38,470 OK, let me just draw that truss. 351 00:21:38,470 --> 00:21:40,890 And let's just talk about it. 352 00:21:40,890 --> 00:21:44,780 So, boy I've put a lot of bars in there. 353 00:21:44,780 --> 00:21:50,270 OK, so this is just one bar down to this guy. 354 00:21:50,270 --> 00:21:53,830 And this is one up to here. 355 00:21:53,830 --> 00:21:56,420 And was that everything? 356 00:21:56,420 --> 00:21:58,510 OK, and then I've numbered them. 357 00:21:58,510 --> 00:22:01,190 Let me just copy the number here, 358 00:22:01,190 --> 00:22:05,520 so that we can talk about that. 359 00:22:05,520 --> 00:22:08,470 And I numbered the nodes one, two. 360 00:22:08,470 --> 00:22:09,950 One. 361 00:22:09,950 --> 00:22:12,350 Joints, I should say really. 362 00:22:12,350 --> 00:22:12,930 Four. 363 00:22:12,930 --> 00:22:15,200 OK, all right. 364 00:22:15,200 --> 00:22:17,350 Let's just start as I always do by, 365 00:22:17,350 --> 00:22:20,710 what's the shape of the matrix A? 366 00:22:20,710 --> 00:22:26,600 A is what, how many bars have I got? 367 00:22:26,600 --> 00:22:29,430 So help me through this now. 368 00:22:29,430 --> 00:22:34,390 Six bars, and how many unknown displacements have I got? 369 00:22:34,390 --> 00:22:36,290 Eight, OK, good. 370 00:22:36,290 --> 00:22:38,980 And have I got any rigid motions here? 371 00:22:38,980 --> 00:22:40,510 No. 372 00:22:40,510 --> 00:22:47,150 I've got two, these are going to prevent all the rigid motions. 373 00:22:47,150 --> 00:22:59,010 So if this matrix, if A has rank six, which we'd probably 374 00:22:59,010 --> 00:23:08,880 guess it has, you know that there's an if there. 375 00:23:08,880 --> 00:23:10,950 Then two mechanisms, right? 376 00:23:10,950 --> 00:23:14,330 Eight minus six gives two mechanisms. 377 00:23:14,330 --> 00:23:19,100 OK, and your question is how to find them. 378 00:23:19,100 --> 00:23:19,930 You've got one. 379 00:23:19,930 --> 00:23:21,290 Alright, which one have you got? 380 00:23:21,290 --> 00:23:24,832 AUDIENCE: [INAUDIBLE] 381 00:23:24,832 --> 00:23:26,040 PROFESSOR STRANG: These guys. 382 00:23:26,040 --> 00:23:33,320 So tilt, so that's a mechanism that only involves u H 1 and u 383 00:23:33,320 --> 00:23:36,300 H 2 and none of the other u's. 384 00:23:36,300 --> 00:23:39,250 OK, that looks good. 385 00:23:39,250 --> 00:23:41,500 So that's certainly one. 386 00:23:41,500 --> 00:23:44,190 Now we're looking for a second one. 387 00:23:44,190 --> 00:23:46,900 What if, anybody suggest one? 388 00:23:46,900 --> 00:23:49,420 Let me just give everybody a thought. 389 00:23:49,420 --> 00:23:52,760 If you haven't already started on this homework, 390 00:23:52,760 --> 00:23:54,570 you're starting now. 391 00:23:54,570 --> 00:23:59,010 So I'm looking for another mechanism, 392 00:23:59,010 --> 00:24:02,780 and of course you might say well, 393 00:24:02,780 --> 00:24:06,910 suppose these bars go this way. 394 00:24:06,910 --> 00:24:09,920 Suppose they go to the left, and you 395 00:24:09,920 --> 00:24:12,350 know that that's not going to do it right. 396 00:24:12,350 --> 00:24:14,200 And why not? 397 00:24:14,200 --> 00:24:17,940 Why isn't that an OK second answer? 398 00:24:17,940 --> 00:24:20,580 Because it's effectively the same as the first; in fact 399 00:24:20,580 --> 00:24:25,710 the mechanism u, how would the u, the one I've drawn 400 00:24:25,710 --> 00:24:33,300 and the u for this way, what would we see? 401 00:24:33,300 --> 00:24:34,980 Opposite signs. 402 00:24:34,980 --> 00:24:38,864 If this describes a u 1 H and a u 2 H positive, 403 00:24:38,864 --> 00:24:40,280 the other way they're negative, it 404 00:24:40,280 --> 00:24:43,590 would just be the opposite sign, nothing new. 405 00:24:43,590 --> 00:24:48,620 OK, so now you've had a look, so tell me. 406 00:24:48,620 --> 00:24:54,250 If somebody sees-- To answer your question here, 407 00:24:54,250 --> 00:24:55,880 how do you see them? 408 00:24:55,880 --> 00:24:59,700 I don't know. 409 00:24:59,700 --> 00:25:03,100 Just look harder. 410 00:25:03,100 --> 00:25:06,190 Like you know, speaking French. 411 00:25:06,190 --> 00:25:12,770 Just say it louder and maybe it'll work. 412 00:25:12,770 --> 00:25:14,970 Of course we do have the possibility 413 00:25:14,970 --> 00:25:18,670 to create the matrix and look for solutions. 414 00:25:18,670 --> 00:25:21,860 But it's more fun to do it this way. 415 00:25:21,860 --> 00:25:25,930 And now who's going to suggest another mechanism, 416 00:25:25,930 --> 00:25:26,650 another view? 417 00:25:26,650 --> 00:25:30,750 AUDIENCE: [INAUDIBLE] 418 00:25:30,750 --> 00:25:36,590 PROFESSOR STRANG: Three and four. 419 00:25:36,590 --> 00:25:39,560 Wait a minute, what was that three? 420 00:25:39,560 --> 00:25:42,340 Oh, these nodes, OK. 421 00:25:42,340 --> 00:25:46,520 These nodes, three and four do what? 422 00:25:46,520 --> 00:25:47,220 Oh yes, sorry. 423 00:25:47,220 --> 00:25:52,400 Three go up this way. 424 00:25:52,400 --> 00:25:53,360 Ah, I see. 425 00:25:53,360 --> 00:25:56,580 You're rotating. 426 00:25:56,580 --> 00:26:02,710 So this guy goes sort of along this way, 427 00:26:02,710 --> 00:26:06,470 you're taking that top square and turning it. 428 00:26:06,470 --> 00:26:10,720 OK, so these guys will also turn, right? 429 00:26:10,720 --> 00:26:16,410 Yeah, one and two will move on your mechanism. 430 00:26:16,410 --> 00:26:19,510 So the idea is take that top thing, 431 00:26:19,510 --> 00:26:24,300 and I'm allowed to turn it, I'd say I have to turn it, 432 00:26:24,300 --> 00:26:29,990 I have to keep this bar, yeah I cannot stretch it. 433 00:26:29,990 --> 00:26:31,940 I can't stretch any bars. 434 00:26:31,940 --> 00:26:36,840 So when I do this turn, I can't just keep it in place and turn. 435 00:26:36,840 --> 00:26:37,660 No. 436 00:26:37,660 --> 00:26:41,550 But I'm going to turn it so that this one goes 437 00:26:41,550 --> 00:26:47,280 this way and this one goes this way, and now. 438 00:26:47,280 --> 00:26:51,130 Alright you're, you're responsible for telling me 439 00:26:51,130 --> 00:26:52,390 about the other two now. 440 00:26:52,390 --> 00:26:54,766 What do they do? 441 00:26:54,766 --> 00:26:55,640 AUDIENCE: [INAUDIBLE] 442 00:26:55,640 --> 00:26:58,056 PROFESSOR STRANG: Bar four, Will bar four keep its length? 443 00:26:58,056 --> 00:27:01,740 Ha. 444 00:27:01,740 --> 00:27:03,970 OK, well we're not allowed to use fancy words 445 00:27:03,970 --> 00:27:07,430 like four-bar linkage. 446 00:27:07,430 --> 00:27:09,550 Because somebody might say what does that mean. 447 00:27:09,550 --> 00:27:11,900 OK, does it, do you see that? 448 00:27:11,900 --> 00:27:12,980 I think it does. 449 00:27:12,980 --> 00:27:15,600 But it's wonderful. 450 00:27:15,600 --> 00:27:19,580 If that's the original bar and I bring this up a little, 451 00:27:19,580 --> 00:27:23,270 a little, remember, and this down a little, 452 00:27:23,270 --> 00:27:30,020 I think that the new bar has the same length. 453 00:27:30,020 --> 00:27:32,080 You believe that? 454 00:27:32,080 --> 00:27:33,560 No. 455 00:27:33,560 --> 00:27:38,170 Some, yeah, but what happened there? 456 00:27:38,170 --> 00:27:41,222 AUDIENCE: [INAUDIBLE] 457 00:27:41,222 --> 00:27:43,180 PROFESSOR STRANG: Yeah, this is a good example. 458 00:27:43,180 --> 00:27:46,560 AUDIENCE: [INAUDIBLE] 459 00:27:46,560 --> 00:27:49,500 PROFESSOR STRANG: Sorry, yeah, certainly bringing 460 00:27:49,500 --> 00:27:51,990 that up tended to make the bar shorter, 461 00:27:51,990 --> 00:27:55,250 but then moving this down tended to make it a little longer. 462 00:27:55,250 --> 00:27:57,322 AUDIENCE: [INAUDIBLE] 463 00:27:57,322 --> 00:27:58,280 PROFESSOR STRANG: Right 464 00:27:58,280 --> 00:28:00,000 AUDIENCE: [INAUDIBLE] 465 00:28:00,000 --> 00:28:06,680 PROFESSOR STRANG: Yeah, I think so. 466 00:28:06,680 --> 00:28:08,180 See, that was the key. 467 00:28:08,180 --> 00:28:12,560 We went at 90 degrees to five. 468 00:28:12,560 --> 00:28:16,940 And by going at 90 degrees, then the stretch in there 469 00:28:16,940 --> 00:28:21,250 was only that one minus cosine that was higher order. 470 00:28:21,250 --> 00:28:22,580 That's the key. 471 00:28:22,580 --> 00:28:25,290 Similarly, here, all these-- So we're 472 00:28:25,290 --> 00:28:28,420 avoiding stretching by this trick. 473 00:28:28,420 --> 00:28:31,630 So maybe this guy is going down this same way. 474 00:28:31,630 --> 00:28:34,390 And this guy is going up that same way. 475 00:28:34,390 --> 00:28:39,830 Yeah, because then that bar is just translating, 476 00:28:39,830 --> 00:28:42,580 and this bar is translating and these two 477 00:28:42,580 --> 00:28:49,530 are doing the same thing, which I believe is not stretching. 478 00:28:49,530 --> 00:28:53,080 Is this the answer that maybe somebody already got? 479 00:28:53,080 --> 00:28:55,000 And believed in? 480 00:28:55,000 --> 00:28:55,960 Yeah. 481 00:28:55,960 --> 00:29:00,910 So to answer your original question, it wasn't obvious, 482 00:29:00,910 --> 00:29:02,250 was it? 483 00:29:02,250 --> 00:29:06,800 But I think that is the right thing. 484 00:29:06,800 --> 00:29:11,270 And actually, you could create the A for this problem. 485 00:29:11,270 --> 00:29:13,980 Well, it's a bit large, 48 entries, 486 00:29:13,980 --> 00:29:19,600 but because four of the bars are horizontal or vertical, 487 00:29:19,600 --> 00:29:24,670 you will have many, many, many, zeroes in the A matrix. 488 00:29:24,670 --> 00:29:26,210 It would be sort of fun to create 489 00:29:26,210 --> 00:29:28,000 the A matrix and then this. 490 00:29:28,000 --> 00:29:31,970 So what is this displacement that I believe in? 491 00:29:31,970 --> 00:29:35,860 This u mechanism that we've talked about. 492 00:29:35,860 --> 00:29:40,770 Let's see, if I look at one that was positive positive. 493 00:29:40,770 --> 00:29:46,580 If I looked at node two, that was positive over but down. 494 00:29:46,580 --> 00:29:49,340 So horizontal but down. 495 00:29:49,340 --> 00:29:53,470 If I look at node three, that's positive positive. 496 00:29:53,470 --> 00:29:56,660 Yeah, one and three is hanging on. 497 00:29:56,660 --> 00:30:00,840 And number four is like number two, one and minus one. 498 00:30:00,840 --> 00:30:05,590 I think that's the u which should solve Au=0. 499 00:30:05,590 --> 00:30:09,020 500 00:30:09,020 --> 00:30:11,030 Yeah. 501 00:30:11,030 --> 00:30:11,580 OK. 502 00:30:11,580 --> 00:30:16,560 But that's a good example. 503 00:30:16,560 --> 00:30:19,600 OK, so that's trusses. 504 00:30:19,600 --> 00:30:21,475 What else? 505 00:30:21,475 --> 00:30:22,350 AUDIENCE: [INAUDIBLE] 506 00:30:22,350 --> 00:30:23,516 PROFESSOR STRANG: Thank you. 507 00:30:23,516 --> 00:30:25,720 AUDIENCE: [INAUDIBLE] 508 00:30:25,720 --> 00:30:28,790 PROFESSOR STRANG: Oh, well, let's see. 509 00:30:28,790 --> 00:30:33,400 That's a good question and now I guess we're, ah. 510 00:30:33,400 --> 00:30:37,200 Yeah well, the way I've drawn it at 45 degrees, which 511 00:30:37,200 --> 00:30:41,500 is what I wrote here, then I did build in, 512 00:30:41,500 --> 00:30:45,890 I did make that a forty-- I did make this 45 or 135 513 00:30:45,890 --> 00:30:47,460 or something. 514 00:30:47,460 --> 00:30:51,790 And 45. 515 00:30:51,790 --> 00:30:54,840 To get that, if that goes at 45 degrees, 516 00:30:54,840 --> 00:30:58,170 then this had better go at 135, right? 517 00:30:58,170 --> 00:31:01,880 So that has to be a right angle. 518 00:31:01,880 --> 00:31:06,160 If the support was over here then 519 00:31:06,160 --> 00:31:11,700 the angle that would have gone off with was changed. 520 00:31:11,700 --> 00:31:15,020 Yeah, so very good point, that the numbers I've 521 00:31:15,020 --> 00:31:18,970 written down on the picture I drew required these angles 522 00:31:18,970 --> 00:31:25,040 to be those nice numbers. 523 00:31:25,040 --> 00:31:27,040 I hope you like these trusses a little. 524 00:31:27,040 --> 00:31:40,430 I mean, you get some freedom to visualize a little. 525 00:31:40,430 --> 00:31:41,250 Good, yes. 526 00:31:41,250 --> 00:32:01,951 AUDIENCE: [INAUDIBLE] 527 00:32:01,951 --> 00:32:02,950 PROFESSOR STRANG: Maybe. 528 00:32:02,950 --> 00:32:03,560 Let me see. 529 00:32:03,560 --> 00:32:06,410 I thought you were going to say, could I 530 00:32:06,410 --> 00:32:14,070 have two and three go inwards and one and four go outwards? 531 00:32:14,070 --> 00:32:15,810 You don't like that. 532 00:32:15,810 --> 00:32:21,590 I would go with that. 533 00:32:21,590 --> 00:32:22,540 Is that any good? 534 00:32:22,540 --> 00:32:24,000 Well, OK. 535 00:32:24,000 --> 00:32:26,360 Yeah, linear algebra spoke there. 536 00:32:26,360 --> 00:32:28,825 It's a combination of the other two, yes. 537 00:32:28,825 --> 00:32:30,200 If we only got a two-dimensional. 538 00:32:30,200 --> 00:32:32,800 539 00:32:32,800 --> 00:32:33,696 AUDIENCE: [INAUDIBLE] 540 00:32:33,696 --> 00:32:34,820 PROFESSOR STRANG: That one. 541 00:32:34,820 --> 00:32:36,105 AUDIENCE: [INAUDIBLE] 542 00:32:36,105 --> 00:32:36,980 PROFESSOR STRANG: OK. 543 00:32:36,980 --> 00:32:38,092 AUDIENCE: [INAUDIBLE] 544 00:32:38,092 --> 00:32:39,050 PROFESSOR STRANG: Yeah. 545 00:32:39,050 --> 00:32:42,552 AUDIENCE: [INAUDIBLE] 546 00:32:42,552 --> 00:32:43,510 PROFESSOR STRANG: Yeah. 547 00:32:43,510 --> 00:32:45,872 AUDIENCE: [INAUDIBLE] 548 00:32:45,872 --> 00:32:46,830 PROFESSOR STRANG: Yeah. 549 00:32:46,830 --> 00:32:51,270 AUDIENCE: [INAUDIBLE] 550 00:32:51,270 --> 00:32:54,130 PROFESSOR STRANG: OK. 551 00:32:54,130 --> 00:33:00,190 But let me-- When you suggested one, I overwrote it. 552 00:33:00,190 --> 00:33:07,130 Now, the one you suggested-- When you told me, 553 00:33:07,130 --> 00:33:10,890 OK, bring these in, I thought, OK, these have to go out. 554 00:33:10,890 --> 00:33:19,347 AUDIENCE: [INAUDIBLE] 555 00:33:19,347 --> 00:33:21,680 PROFESSOR STRANG: They're not allowed to change lengths, 556 00:33:21,680 --> 00:33:25,550 so they can only swing around these pin joints. 557 00:33:25,550 --> 00:33:33,130 So the picture of the one that I drew after your question 558 00:33:33,130 --> 00:33:37,970 would be, these guys, this guy, let me draw the square, 559 00:33:37,970 --> 00:33:39,120 as it was. 560 00:33:39,120 --> 00:33:41,670 And then these guys came in a little. 561 00:33:41,670 --> 00:33:44,400 These guys went out a little. 562 00:33:44,400 --> 00:33:45,620 And we got this. 563 00:33:45,620 --> 00:33:51,757 AUDIENCE: [INAUDIBLE] 564 00:33:51,757 --> 00:33:53,090 PROFESSOR STRANG: No, that's OK. 565 00:33:53,090 --> 00:33:57,430 I think this is legitimate, this is a legitimate one. 566 00:33:57,430 --> 00:34:04,090 And that bar, every bar now is doing the same kind of thing 567 00:34:04,090 --> 00:34:04,950 that this one did. 568 00:34:04,950 --> 00:34:07,770 This moved a little, but this compensated 569 00:34:07,770 --> 00:34:10,680 and kept the length the same. 570 00:34:10,680 --> 00:34:18,640 Your question has led us to another nifty mechanism. 571 00:34:18,640 --> 00:34:20,880 It's good, a good one to think about. 572 00:34:20,880 --> 00:34:33,642 AUDIENCE: [INAUDIBLE] 573 00:34:33,642 --> 00:34:34,600 PROFESSOR STRANG: Yeah. 574 00:34:34,600 --> 00:34:36,880 It's movement without stretching. 575 00:34:36,880 --> 00:34:40,370 Movement without stretching is the key, yes. 576 00:34:40,370 --> 00:34:43,530 And then where does collapse come? 577 00:34:43,530 --> 00:34:46,880 So I use the word collapse pretty freely. 578 00:34:46,880 --> 00:34:49,240 So the word collapse comes in because since it 579 00:34:49,240 --> 00:34:52,650 can move without stretching there's nothing controlling it, 580 00:34:52,650 --> 00:34:59,160 it can move-- Well, if we stayed linear, 581 00:34:59,160 --> 00:35:02,640 then I could make all those ten and the thing 582 00:35:02,640 --> 00:35:05,110 would be even worse, of course. 583 00:35:05,110 --> 00:35:10,900 The truth is that if I made those even one or ten 584 00:35:10,900 --> 00:35:13,440 I would be out of the linear range. 585 00:35:13,440 --> 00:35:16,760 These all should be 0.01's or something. 586 00:35:16,760 --> 00:35:20,440 But linear we don't know the difference. 587 00:35:20,440 --> 00:35:24,040 OK, I'm glad you've got that suggestion. 588 00:35:24,040 --> 00:35:26,580 And now somebody correctly says that this one 589 00:35:26,580 --> 00:35:29,730 must be some combination of that and what 590 00:35:29,730 --> 00:35:31,060 was the first movement? 591 00:35:31,060 --> 00:35:35,890 Oh, the upper two. 592 00:35:35,890 --> 00:35:38,320 Which it must be. 593 00:35:38,320 --> 00:35:43,850 Amazing, how many-- Put a few bars up there and you got it. 594 00:35:43,850 --> 00:35:44,510 Yeah. 595 00:35:44,510 --> 00:35:47,750 And of course, by the way, and don't 596 00:35:47,750 --> 00:35:50,320 let me get too far into this discussion, 597 00:35:50,320 --> 00:35:55,460 but who is the artist, actually is it Alexander Calder 598 00:35:55,460 --> 00:35:57,800 who has, what are they called? 599 00:35:57,800 --> 00:36:03,960 You know, they're-- What was it called again? 600 00:36:03,960 --> 00:36:06,480 AUDIENCE: [INAUDIBLE] PROFESSOR STRANG: 601 00:36:06,480 --> 00:36:09,290 Well, he has those, that wasn't the word I was thinking of. 602 00:36:09,290 --> 00:36:12,490 But he's created these trusses, like with lots 603 00:36:12,490 --> 00:36:15,720 of bars and lots of nodes that have some special property. 604 00:36:15,720 --> 00:36:17,260 You know, they just, anyway. 605 00:36:17,260 --> 00:36:26,770 It touches on art, actually this theory of mechanisms. 606 00:36:26,770 --> 00:36:30,370 Yeah it's really quite interesting. 607 00:36:30,370 --> 00:36:33,880 But then linear algebra somehow tells you 608 00:36:33,880 --> 00:36:37,790 just from these numbers how many mechanisms you're looking for. 609 00:36:37,790 --> 00:36:40,890 Which is pretty cool. 610 00:36:40,890 --> 00:36:45,170 OK, open for more questions. 611 00:36:45,170 --> 00:36:48,530 You probably haven't looked ahead, 612 00:36:48,530 --> 00:36:51,970 I mean today's lecture was a-- I hope 613 00:36:51,970 --> 00:36:59,530 and I left up there the central topics for the lecture. 614 00:36:59,530 --> 00:37:03,730 But I'm, I think, correct that you haven't 615 00:37:03,730 --> 00:37:06,330 started on those problems. 616 00:37:06,330 --> 00:37:08,390 To know what to ask. 617 00:37:08,390 --> 00:37:11,930 Let me ask you a question, which was not in the lecture. 618 00:37:11,930 --> 00:37:15,500 Suppose I wanted to use finite differences. 619 00:37:15,500 --> 00:37:18,420 It's a little bit like the one on the quiz. 620 00:37:18,420 --> 00:37:23,860 So on the quiz we had c, equal ones, and then c=2. 621 00:37:23,860 --> 00:37:26,260 And you knew c had to be four by four, 622 00:37:26,260 --> 00:37:31,710 so most people correctly got the diagonal as one, one, two, two. 623 00:37:31,710 --> 00:37:35,140 Just by sort of common sense. 624 00:37:35,140 --> 00:37:38,000 But what would be a finite difference 625 00:37:38,000 --> 00:37:40,940 approximation to our equation? 626 00:37:40,940 --> 00:37:44,550 So suppose I didn't go to finite elements, 627 00:37:44,550 --> 00:37:47,590 but instead I stayed with finite differences, which 628 00:37:47,590 --> 00:37:52,200 would be completely fine in 1-D, completely sensible. 629 00:37:52,200 --> 00:37:56,180 How would you create a finite difference thing for that? 630 00:37:56,180 --> 00:38:00,500 Let me just bring down a blank board and ask. 631 00:38:00,500 --> 00:38:02,560 So you see the equation? 632 00:38:02,560 --> 00:38:04,310 Let me write it again here. 633 00:38:04,310 --> 00:38:08,700 So I want to replace this equation with a varying-- that 634 00:38:08,700 --> 00:38:09,710 has varying c(x). 635 00:38:09,710 --> 00:38:12,550 636 00:38:12,550 --> 00:38:14,400 Well, I won't worry about the right side. 637 00:38:14,400 --> 00:38:16,510 It's f. 638 00:38:16,510 --> 00:38:24,690 What's my K matrix using finite differences for this? 639 00:38:24,690 --> 00:38:27,150 We're going to create a finite element K 640 00:38:27,150 --> 00:38:32,020 matrix by the weak form, Galerkin trial 641 00:38:32,020 --> 00:38:34,240 function, that route. 642 00:38:34,240 --> 00:38:37,580 That's coming-- That's very important, that's coming, 643 00:38:37,580 --> 00:38:41,950 started today and it'll be completed Friday. 644 00:38:41,950 --> 00:38:48,150 But now suppose we were back up to finite differences. 645 00:38:48,150 --> 00:38:54,590 What would you take for finite differences there? 646 00:38:54,590 --> 00:38:58,530 When the c wasn't there, what did we do? 647 00:38:58,530 --> 00:39:00,780 Then we just had second difference, right? 648 00:39:00,780 --> 00:39:02,450 How did we get the second difference? 649 00:39:02,450 --> 00:39:06,650 It was a first difference of a first difference. 650 00:39:06,650 --> 00:39:12,950 I guess I would probably-- This I would probably approximate 651 00:39:12,950 --> 00:39:21,550 by (u_(i+1)-u_i) over delta x. 652 00:39:21,550 --> 00:39:29,080 That would be that, and so what should I put for c(x)? 653 00:39:29,080 --> 00:39:31,190 What would you suggest? 654 00:39:31,190 --> 00:39:34,550 655 00:39:34,550 --> 00:39:41,330 c subscript i, so let's mark. 656 00:39:41,330 --> 00:39:43,920 Here's i, and here's i+1, and here's i-1. 657 00:39:43,920 --> 00:39:46,640 658 00:39:46,640 --> 00:39:51,770 We're going to end up with those three guys involved. 659 00:39:51,770 --> 00:40:00,780 So this takes this difference, and then this one will take 660 00:40:00,780 --> 00:40:07,300 that one, and then I'll also have a u_i-u_(i-1) over delta 661 00:40:07,300 --> 00:40:08,570 x. 662 00:40:08,570 --> 00:40:10,120 And somehow that'll be the difference 663 00:40:10,120 --> 00:40:11,440 of these differences. 664 00:40:11,440 --> 00:40:13,410 But now tell me again, what would 665 00:40:13,410 --> 00:40:18,940 be the really cool choice of c? 666 00:40:18,940 --> 00:40:24,220 What's your instinct where what value of c to take? 667 00:40:24,220 --> 00:40:27,280 As sort of average. 668 00:40:27,280 --> 00:40:29,160 I mean you could take it just halfway. 669 00:40:29,160 --> 00:40:33,940 I think I would take c_(i+1/2) there. 670 00:40:33,940 --> 00:40:37,780 Just as being sort of right. 671 00:40:37,780 --> 00:40:44,670 And here I would take c_(i-1/2), halfway along its interval. 672 00:40:44,670 --> 00:40:50,340 And then the second difference takes the difference of it. 673 00:40:50,340 --> 00:40:55,940 So now I've got-- I think that's what I would do. 674 00:40:55,940 --> 00:40:59,750 That would be my typical, one typical finite difference 675 00:40:59,750 --> 00:41:00,760 equation. 676 00:41:00,760 --> 00:41:04,430 Would be the difference times its c 677 00:41:04,430 --> 00:41:07,840 and I took its c to be symmetric. 678 00:41:07,840 --> 00:41:09,980 You could also have taken, if you'd liked, 679 00:41:09,980 --> 00:41:12,320 if you wanted to stay at these points, 680 00:41:12,320 --> 00:41:16,530 you could do twice as much work and take, let me say, 681 00:41:16,530 --> 00:41:24,270 or-- So either c_(i+1/2), or you could average the c_i 682 00:41:24,270 --> 00:41:27,370 and the c_(i+1) over two. 683 00:41:27,370 --> 00:41:32,190 And both of those would give you that extra accuracy 684 00:41:32,190 --> 00:41:35,360 that you pick up from sort of keeping symmetry 685 00:41:35,360 --> 00:41:38,530 where you should. 686 00:41:38,530 --> 00:41:42,200 So I think, I mean, this would be the quicker one to put, 687 00:41:42,200 --> 00:41:46,050 that one would be OK too. 688 00:41:46,050 --> 00:41:49,370 AUDIENCE: [INAUDIBLE] 689 00:41:49,370 --> 00:41:52,660 PROFESSOR STRANG: This was just for your entertainment not, 690 00:41:52,660 --> 00:41:55,155 not-- Yes, go ahead. 691 00:41:55,155 --> 00:41:56,030 AUDIENCE: [INAUDIBLE] 692 00:41:56,030 --> 00:41:57,863 PROFESSOR STRANG: Ah, when c was a step yes. 693 00:41:57,863 --> 00:41:58,750 When c was a step. 694 00:41:58,750 --> 00:42:05,350 AUDIENCE: [INAUDIBLE] 695 00:42:05,350 --> 00:42:14,960 PROFESSOR STRANG: c_(i+1/2) Probably the step-- Yeah, yeah. 696 00:42:14,960 --> 00:42:16,080 Yeah. 697 00:42:16,080 --> 00:42:21,600 Let me ask you, what happens in this equation when c jumps? 698 00:42:21,600 --> 00:42:23,780 It jumped on the quiz. 699 00:42:23,780 --> 00:42:27,980 But I didn't require you to solve the differential 700 00:42:27,980 --> 00:42:31,580 equation, only to create the finite difference model, 701 00:42:31,580 --> 00:42:36,410 to create the A transpose C A, and most people did it fine. 702 00:42:36,410 --> 00:42:40,570 But suppose c jumps. 703 00:42:40,570 --> 00:42:44,170 I have some simple right-hand side like one. 704 00:42:44,170 --> 00:42:46,040 It's not a jump in the right-hand side 705 00:42:46,040 --> 00:42:47,230 I'm interested in. 706 00:42:47,230 --> 00:42:50,890 It's a jump in c from one to two. 707 00:42:50,890 --> 00:42:53,610 So this is a topic that the book does discuss 708 00:42:53,610 --> 00:42:57,410 and maybe I might come back to it in the ordinary lecture. 709 00:42:57,410 --> 00:43:03,120 But, while it's in front of us now, 710 00:43:03,120 --> 00:43:09,960 the quiz was sort of intended to help you with that. 711 00:43:09,960 --> 00:43:15,800 How do I interpret this equation when c has a jump? 712 00:43:15,800 --> 00:43:18,990 Well, it's actually, if you look at it 713 00:43:18,990 --> 00:43:20,600 right there's no difficulty. 714 00:43:20,600 --> 00:43:25,450 That equation is a combination of this equation -dw/dx equals 715 00:43:25,450 --> 00:43:27,360 the one. 716 00:43:27,360 --> 00:43:35,550 And the equation of c*du/dx equalling the w. 717 00:43:35,550 --> 00:43:38,710 I split it out for you in the last part, 718 00:43:38,710 --> 00:43:41,260 Problem 4b in the quiz. 719 00:43:41,260 --> 00:43:45,520 I really helped you to say OK, take these two 720 00:43:45,520 --> 00:43:47,980 separate equations. 721 00:43:47,980 --> 00:43:50,830 And now you don't really have a problem. 722 00:43:50,830 --> 00:43:55,350 The equation for w, nothing is out of the ordinary there. 723 00:43:55,350 --> 00:43:57,460 Jump in c is not even seen. 724 00:43:57,460 --> 00:43:59,450 So then you've got the w. 725 00:43:59,450 --> 00:44:04,100 Now, you have the c in it with its little jump. 726 00:44:04,100 --> 00:44:10,560 But, so suppose I find w from the first equation. 727 00:44:10,560 --> 00:44:14,250 How do I find u? 728 00:44:14,250 --> 00:44:15,870 I just divide by the c. 729 00:44:15,870 --> 00:44:17,720 And integrate. 730 00:44:17,720 --> 00:44:19,900 Yeah. 731 00:44:19,900 --> 00:44:24,350 What I'm saying, let me say it, clearly now. 732 00:44:24,350 --> 00:44:29,250 If there's a jump in c, that's not a bad thing. 733 00:44:29,250 --> 00:44:34,780 Because the point is there's no jump in c*u'. 734 00:44:34,780 --> 00:44:39,070 c*du/dx doesn't jump. w doesn't jump. 735 00:44:39,070 --> 00:44:40,300 From a jump in c. 736 00:44:40,300 --> 00:44:43,850 If c jumps, let it. 737 00:44:43,850 --> 00:44:49,790 There's no jump in w, which is the serious unknown. 738 00:44:49,790 --> 00:44:52,560 There'll be a jump in du/dx, and that 739 00:44:52,560 --> 00:44:56,550 was that e thing that you had on the quiz. 740 00:44:56,550 --> 00:45:00,090 There'll be a jump in du/dx to compensate the jump in c, 741 00:45:00,090 --> 00:45:03,440 but w is good. 742 00:45:03,440 --> 00:45:08,040 That's the message of-- So let me write that down. 743 00:45:08,040 --> 00:45:22,700 Even if c jumps, w does not. 744 00:45:22,700 --> 00:45:26,860 So my point is when you're looking at w, 745 00:45:26,860 --> 00:45:29,240 you're looking at the right quantity. 746 00:45:29,240 --> 00:45:35,040 And it deals, this is the sort of general feature 747 00:45:35,040 --> 00:45:37,990 that if you look at it right it's not a problem. 748 00:45:37,990 --> 00:45:43,010 If you look at it wrong and try to write out that equation, 749 00:45:43,010 --> 00:45:45,930 take the second derivative of this is OK, 750 00:45:45,930 --> 00:45:48,770 but then if you just try to take the derivative of the jump, 751 00:45:48,770 --> 00:45:51,180 you think, my God there's a delta function 752 00:45:51,180 --> 00:45:55,870 sitting in my equation, what am I to do. 753 00:45:55,870 --> 00:45:59,750 Don't do it. 754 00:45:59,750 --> 00:46:07,380 That pair of equations is no trouble. 755 00:46:07,380 --> 00:46:09,460 Do you see that point? 756 00:46:09,460 --> 00:46:17,700 That a jump in c is really OK. c=0 wouldn't be so good, right? 757 00:46:17,700 --> 00:46:21,940 If c went to zero then you have got some problems here, 758 00:46:21,940 --> 00:46:25,560 because if c is zero here what's going on? 759 00:46:25,560 --> 00:46:35,440 So your material should have some stiffness, 760 00:46:35,440 --> 00:46:37,440 some positive stiffness. 761 00:46:37,440 --> 00:46:39,250 Zero stiffness would be a problem. 762 00:46:39,250 --> 00:46:42,980 Negative stiffness would be really 763 00:46:42,980 --> 00:46:45,940 a crazy material, where you add more force 764 00:46:45,940 --> 00:46:52,460 and it compresses on you. 765 00:46:52,460 --> 00:47:00,890 OK so that was a point I could make also in the lecture. 766 00:47:00,890 --> 00:47:04,250 Other questions. 767 00:47:04,250 --> 00:47:09,450 I hope you find these review sessions-- 768 00:47:09,450 --> 00:47:16,510 They give a chance to bring out more points. 769 00:47:16,510 --> 00:47:19,900 Let me say a little bit about Chapter 2 things 770 00:47:19,900 --> 00:47:26,270 that we abandoned without including in the lectures. 771 00:47:26,270 --> 00:47:30,130 One very important thing that we didn't touch 772 00:47:30,130 --> 00:47:32,670 was the non-linear problems. 773 00:47:32,670 --> 00:47:40,550 Where in the end you have to solve a system 774 00:47:40,550 --> 00:47:43,020 of non-linear equations. 775 00:47:43,020 --> 00:47:46,090 We've only had linear equations, Ku=f, 776 00:47:46,090 --> 00:47:52,010 where we almost got to-- And we've certainly found enough 777 00:47:52,010 --> 00:47:53,750 to discuss there. 778 00:47:53,750 --> 00:47:56,840 But how do you deal with non-linear equations? 779 00:47:56,840 --> 00:48:03,650 Well so that's Section 2.6, I guess. 780 00:48:03,650 --> 00:48:09,060 And the answer is Newton's method, or some version, 781 00:48:09,060 --> 00:48:11,620 there are many versions of Newton's method. 782 00:48:11,620 --> 00:48:18,720 So all I'm saying is that's an important topic. 783 00:48:18,720 --> 00:48:21,060 How to deal with non-linear equations. 784 00:48:21,060 --> 00:48:27,020 And the answer is usually some variation of Newton's method. 785 00:48:27,020 --> 00:48:29,110 Depending on how many equations you've 786 00:48:29,110 --> 00:48:34,230 got, how bad the non-linearity is, and things like that. 787 00:48:34,230 --> 00:48:40,720 It's a case where coding is not so simple. 788 00:48:40,720 --> 00:48:42,330 But important. 789 00:48:42,330 --> 00:48:45,380 So that's one thing in Chapter 2 that we've 790 00:48:45,380 --> 00:48:51,280 passed, because really this is where we want to be today. 791 00:48:51,280 --> 00:48:54,140 Other topics, questions of any kind? 792 00:48:54,140 --> 00:48:54,860 Yes, thanks. 793 00:48:54,860 --> 00:48:57,350 AUDIENCE: [INAUDIBLE] 794 00:48:57,350 --> 00:48:58,870 PROFESSOR STRANG: This problem, OK. 795 00:48:58,870 --> 00:49:01,825 AUDIENCE: [INAUDIBLE] 796 00:49:01,825 --> 00:49:02,700 PROFESSOR STRANG: Yes 797 00:49:02,700 --> 00:49:16,960 AUDIENCE: [INAUDIBLE] Well, OK. 798 00:49:16,960 --> 00:49:19,230 Right, good question. 799 00:49:19,230 --> 00:49:20,210 You're right. 800 00:49:20,210 --> 00:49:25,040 AUDIENCE: [INAUDIBLE] 801 00:49:25,040 --> 00:49:26,220 PROFESSOR STRANG: Yeah, OK. 802 00:49:26,220 --> 00:49:29,700 So now suppose that delta of x was in-- delta of x-a, 803 00:49:29,700 --> 00:49:31,490 the jump at a, was there. 804 00:49:31,490 --> 00:49:38,870 What would change in there? 805 00:49:38,870 --> 00:49:40,920 Well, the one would change, right? 806 00:49:40,920 --> 00:49:45,370 The one would now be the delta of x-a. 807 00:49:45,370 --> 00:49:48,300 So what am I seeing now? 808 00:49:48,300 --> 00:49:51,290 What's the jump situation now? 809 00:49:51,290 --> 00:49:56,760 Is there a jump in, does w jump now from this delta function? 810 00:49:56,760 --> 00:49:58,250 Yes. 811 00:49:58,250 --> 00:50:01,990 It did not jump from the jump in c. 812 00:50:01,990 --> 00:50:04,070 That did not produce us a jump in w. 813 00:50:04,070 --> 00:50:04,670 Why not? 814 00:50:04,670 --> 00:50:13,690 Because again, back to the jump in c, you have a bar of copper 815 00:50:13,690 --> 00:50:16,370 and you have a bar of steel. 816 00:50:16,370 --> 00:50:19,030 And they have different c's. 817 00:50:19,030 --> 00:50:24,090 But the force at that point, when it's in equilibrium, 818 00:50:24,090 --> 00:50:28,410 the bar's not falling apart, right? 819 00:50:28,410 --> 00:50:33,440 The force is the same from above and below; equilibrium holds. 820 00:50:33,440 --> 00:50:39,290 It's just that the force involves a c_1 there, 821 00:50:39,290 --> 00:50:45,260 and down here the force involves a c_2 multiplying the du/dx, 822 00:50:45,260 --> 00:50:46,790 and this stretching. 823 00:50:46,790 --> 00:50:50,980 So the stretching rule changes. du/dx, 824 00:50:50,980 --> 00:50:54,800 the stretching factor will be different in the two materials. 825 00:50:54,800 --> 00:50:57,160 But the force won't be. 826 00:50:57,160 --> 00:50:59,430 You really have to keep straight what-- 827 00:50:59,430 --> 00:51:02,740 Now, you asked about a delta function, OK. 828 00:51:02,740 --> 00:51:05,450 So now if I hang a delta function there, 829 00:51:05,450 --> 00:51:08,830 that's like hanging a point load at a point. 830 00:51:08,830 --> 00:51:10,430 What happens at that point? 831 00:51:10,430 --> 00:51:14,640 Well, we don't have a problem with that. 832 00:51:14,640 --> 00:51:17,670 Well, w jumps. 833 00:51:17,670 --> 00:51:21,190 The force pulling up is not the same as the force 834 00:51:21,190 --> 00:51:25,490 pulling down because there's this extra point load. 835 00:51:25,490 --> 00:51:29,080 So the balance there and the balance there, 836 00:51:29,080 --> 00:51:30,660 those aren't equal. 837 00:51:30,660 --> 00:51:35,730 There's a jump there because of the point load being put there. 838 00:51:35,730 --> 00:51:40,470 So that produces a jump in w, but not a jump in u. 839 00:51:40,470 --> 00:51:46,160 Not a jump in u, the bar is still holding together. 840 00:51:46,160 --> 00:51:50,340 And since time's running out I don't 841 00:51:50,340 --> 00:51:55,510 have to ask myself, or ask you, about the worst possibility 842 00:51:55,510 --> 00:51:58,940 that occurs to me, which is suppose 843 00:51:58,940 --> 00:52:01,230 they happen at the same place. 844 00:52:01,230 --> 00:52:06,500 Suppose there's a jump in c and a point load. 845 00:52:06,500 --> 00:52:12,030 Do we want to face that, what would happen there? 846 00:52:12,030 --> 00:52:17,300 Suppose, a was the place where this jumped. 847 00:52:17,300 --> 00:52:20,070 Suppose the point load was here. 848 00:52:20,070 --> 00:52:21,890 What's up there? 849 00:52:21,890 --> 00:52:24,020 Well, maybe I will able to do that. 850 00:52:24,020 --> 00:52:26,540 OK, so I'm putting the point load 851 00:52:26,540 --> 00:52:32,670 at the place where the copper and the steel meet. 852 00:52:32,670 --> 00:52:34,950 Is w continuous? 853 00:52:34,950 --> 00:52:39,040 Is the force the same above and below that joint? 854 00:52:39,040 --> 00:52:39,600 No. 855 00:52:39,600 --> 00:52:44,560 Because there's these extra terms from the weight. 856 00:52:44,560 --> 00:52:48,100 And then once you know the forces, then 857 00:52:48,100 --> 00:52:52,560 above the bar you're solving for u with one c, and then 858 00:52:52,560 --> 00:52:56,830 below that point with the other c. 859 00:52:56,830 --> 00:53:03,930 So w has a plus and a minus and c has a plus and a minus. 860 00:53:03,930 --> 00:53:12,120 And you're solving this one, let me make them minus plus. 861 00:53:12,120 --> 00:53:17,610 So you're solving this one in one bar, one metal. 862 00:53:17,610 --> 00:53:20,800 And c plus and the w plus in the other method. 863 00:53:20,800 --> 00:53:25,660 So yeah, you can do it. 864 00:53:25,660 --> 00:53:30,200 Shall I just repeat what my overall message is? 865 00:53:30,200 --> 00:53:33,690 w is the right thing to look at. 866 00:53:33,690 --> 00:53:38,090 That combination c*du/dx is the right thing to look at. 867 00:53:38,090 --> 00:53:43,050 And it's what sits there. 868 00:53:43,050 --> 00:53:45,650 In those parentheses. 869 00:53:45,650 --> 00:53:50,630 OK, good, so that's some review topics. 870 00:53:50,630 --> 00:53:54,090 We really have a lot of fun ahead now with finite elements.