1 00:00:00,000 --> 00:00:00,247 2 00:00:00,247 --> 00:00:02,330 The following content is provided under a Creative 3 00:00:02,330 --> 00:00:03,710 Commons license. 4 00:00:03,710 --> 00:00:05,450 Your support will help MIT OpenCourseWare 5 00:00:05,450 --> 00:00:09,756 continue to offer high quality educational resources for free. 6 00:00:09,756 --> 00:00:11,880 To make a donation, or to view additional materials 7 00:00:11,880 --> 00:00:14,940 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:14,940 --> 00:00:20,380 at ocw.mit.edu. 9 00:00:20,380 --> 00:00:23,880 PROFESSOR STRANG: OK, so this is the second, last lecture 10 00:00:23,880 --> 00:00:25,800 on trusses. 11 00:00:25,800 --> 00:00:28,440 Then we've got a holiday on Monday. 12 00:00:28,440 --> 00:00:32,230 And then after that we'll be into Chapter 3. 13 00:00:32,230 --> 00:00:37,030 I thought I'd write down just in case it's any use to you, 14 00:00:37,030 --> 00:00:40,060 the four problems that I intend to include 15 00:00:40,060 --> 00:00:42,100 with the next homework. 16 00:00:42,100 --> 00:00:46,560 That won't be due for quite a while, a week from Monday. 17 00:00:46,560 --> 00:00:48,430 So these will be the problems on trusses 18 00:00:48,430 --> 00:00:52,570 that come from particular trusses drawn in the book. 19 00:00:52,570 --> 00:00:56,650 And then there'll be some problems from the new material, 20 00:00:56,650 --> 00:00:58,510 that we do next week. 21 00:00:58,510 --> 00:01:04,480 So trusses and really, there's two main jobs for today. 22 00:01:04,480 --> 00:01:08,220 One is to identify this matrix A, 23 00:01:08,220 --> 00:01:13,080 the strain-displacement matrix or the stretching matrix. 24 00:01:13,080 --> 00:01:16,400 How far do the bars stretch? 25 00:01:16,400 --> 00:01:20,870 Everybody remembers A is going to come in this step. 26 00:01:20,870 --> 00:01:25,450 If we have displacements, then, of the nodes like, 27 00:01:25,450 --> 00:01:32,320 this would be like a u_1, this would be a u_1^H and a u_1^V, 28 00:01:32,320 --> 00:01:35,070 this would be a u_2^H and a u_2^V, 29 00:01:35,070 --> 00:01:42,070 so there are four movements of the ends of the truss 30 00:01:42,070 --> 00:01:47,060 and of one particular bar, and then we'll stretch that bar. 31 00:01:47,060 --> 00:01:49,220 And the question, is how much? 32 00:01:49,220 --> 00:01:57,470 So that will be one row of A. So if we follow one bar, 33 00:01:57,470 --> 00:01:59,910 you remember in the matrix A, there's 34 00:01:59,910 --> 00:02:02,790 going to be a row for every bar. 35 00:02:02,790 --> 00:02:09,090 So a row for each, row of A. For each bar. 36 00:02:09,090 --> 00:02:12,560 And if we track down one of those rows, 37 00:02:12,560 --> 00:02:15,310 we'll have the idea. 38 00:02:15,310 --> 00:02:17,570 And then of course at the end we'd 39 00:02:17,570 --> 00:02:22,860 maybe be constructing A without, sort of a free-free A. 40 00:02:22,860 --> 00:02:27,090 And then at the end, any fixed displacements 41 00:02:27,090 --> 00:02:33,350 that will knock out columns of A. So that's one job. 42 00:02:33,350 --> 00:02:35,835 And then to see, so the A is going 43 00:02:35,835 --> 00:02:39,430 to be a little more messy. 44 00:02:39,430 --> 00:02:41,930 It's because we're in two dimensions. 45 00:02:41,930 --> 00:02:46,380 So compared to the network problems, and 46 00:02:46,380 --> 00:02:51,320 and the line of springs, now we have more happening. 47 00:02:51,320 --> 00:02:57,990 We've got more columns because every node has now two 48 00:02:57,990 --> 00:02:58,590 unknowns. 49 00:02:58,590 --> 00:03:00,140 A horizontal and vertical. 50 00:03:00,140 --> 00:03:02,990 So A is kind of bigger. 51 00:03:02,990 --> 00:03:07,410 And therefore A transpose C A, you might think, 52 00:03:07,410 --> 00:03:09,480 it's going to be hard to see what's going on. 53 00:03:09,480 --> 00:03:11,600 But you'll see the right way to look 54 00:03:11,600 --> 00:03:15,880 at A transpose C A is a bar at a time. 55 00:03:15,880 --> 00:03:20,700 That's the nice fact about A transpose C A, 56 00:03:20,700 --> 00:03:23,330 I might focus on that first. 57 00:03:23,330 --> 00:03:28,820 And then comes the fun part. 58 00:03:28,820 --> 00:03:32,760 I'll draw some more trusses, that may or may not 59 00:03:32,760 --> 00:03:35,410 have mechanisms. 60 00:03:35,410 --> 00:03:37,510 They may or may not be stable. 61 00:03:37,510 --> 00:03:41,820 And we can try to identify the mechanisms. 62 00:03:41,820 --> 00:03:43,370 Actually, as before. 63 00:03:43,370 --> 00:03:47,390 We'll do it by engineering instinct 64 00:03:47,390 --> 00:03:51,320 rather than by solving. 65 00:03:51,320 --> 00:03:55,840 I mean, in principle, we could always use elimination. 66 00:03:55,840 --> 00:03:59,830 Or ask MATLAB or any other system to do it, 67 00:03:59,830 --> 00:04:03,030 and look for the solutions to Au=0. 68 00:04:03,030 --> 00:04:06,860 And decide are the columns of A independent. 69 00:04:06,860 --> 00:04:09,280 In that case the truss is stable. 70 00:04:09,280 --> 00:04:11,610 This matrix is invertible, we know 71 00:04:11,610 --> 00:04:13,690 all the good possibilities. 72 00:04:13,690 --> 00:04:17,140 And then there's the more interesting possibility, 73 00:04:17,140 --> 00:04:20,850 of having some solutions to that. 74 00:04:20,850 --> 00:04:24,520 In which case that matrix will be singular. 75 00:04:24,520 --> 00:04:28,310 There'll be some modes in our big system 76 00:04:28,310 --> 00:04:31,990 that will cause it to fail. 77 00:04:31,990 --> 00:04:34,040 But it's kind of fun to find those. 78 00:04:34,040 --> 00:04:37,730 OK, while I've written A transpose C A, 79 00:04:37,730 --> 00:04:43,560 may I remind you about a good way to do that multiplication. 80 00:04:43,560 --> 00:04:48,310 OK, so imagine I'm just putting a number. 81 00:04:48,310 --> 00:04:52,420 Here's going to be the matrix A. So the matrix A 82 00:04:52,420 --> 00:04:57,930 will have a bunch of rows, row one, row two, so on. 83 00:04:57,930 --> 00:05:02,410 These rows will correspond to bar one, bar two, and bar 84 00:05:02,410 --> 00:05:03,620 three. 85 00:05:03,620 --> 00:05:05,390 OK. 86 00:05:05,390 --> 00:05:09,750 OK, then we have C, so that's a square matrix. 87 00:05:09,750 --> 00:05:15,100 That each bar has a spring constant, so c_1, c_2, c_3, 88 00:05:15,100 --> 00:05:17,210 and then we have A transpose. 89 00:05:17,210 --> 00:05:20,710 And those rows, or columns of A transpose. 90 00:05:20,710 --> 00:05:26,270 So that's the sort of picture of A transpose C A, 91 00:05:26,270 --> 00:05:30,140 for a three-bar, three bars only. 92 00:05:30,140 --> 00:05:33,870 But the point is made right here. 93 00:05:33,870 --> 00:05:36,470 There's a row of A for every bar. 94 00:05:36,470 --> 00:05:37,210 Right? 95 00:05:37,210 --> 00:05:40,970 Because our matrix A is m by n. 96 00:05:40,970 --> 00:05:47,110 If there are m bars, a row for every bar, and it 97 00:05:47,110 --> 00:05:50,670 tells us how far that bar is stretched. 98 00:05:50,670 --> 00:05:53,510 And we'll figure out what its entries are. 99 00:05:53,510 --> 00:05:55,210 That's our main job. 100 00:05:55,210 --> 00:05:56,440 I'm just looking ahead. 101 00:05:56,440 --> 00:05:58,820 Suppose we've got that row. 102 00:05:58,820 --> 00:06:00,320 And that row, and that row. 103 00:06:00,320 --> 00:06:02,790 So a row for every bar. 104 00:06:02,790 --> 00:06:04,960 Now, here I've taken three bars. 105 00:06:04,960 --> 00:06:08,150 Now, how do I multiply those matrices? 106 00:06:08,150 --> 00:06:11,380 Well, I can do it different ways. 107 00:06:11,380 --> 00:06:13,800 But here's a cool way to do it. 108 00:06:13,800 --> 00:06:20,400 Just-- The way I want to point out is column times row. 109 00:06:20,400 --> 00:06:24,750 If you multiply matrices you're allowed to-- The effective c_1, 110 00:06:24,750 --> 00:06:26,950 c_2, c_3, is going to be very simple. 111 00:06:26,950 --> 00:06:29,850 So I'm really paying attention here 112 00:06:29,850 --> 00:06:34,740 to A transpose A. If I want to multiply A transpose A, 113 00:06:34,740 --> 00:06:38,680 I can do row times column as usual and get one number. 114 00:06:38,680 --> 00:06:46,140 Or I can do column times row and get a whole little matrix. 115 00:06:46,140 --> 00:06:48,720 And that's the bar one matrix. 116 00:06:48,720 --> 00:06:50,620 It's the element matrix, and that's 117 00:06:50,620 --> 00:06:53,350 how finite elements will be assembled, 118 00:06:53,350 --> 00:06:58,710 and that's why I should keep mentioning this point. 119 00:06:58,710 --> 00:07:03,880 So the way to do that column times 120 00:07:03,880 --> 00:07:07,170 row thing, and then of course that c_1 just multiplies 121 00:07:07,170 --> 00:07:10,750 that row, that'll be c_1. 122 00:07:10,750 --> 00:07:18,360 Row one, transpose, that's the column, times row one. 123 00:07:18,360 --> 00:07:21,580 That's what's coming from bar one. 124 00:07:21,580 --> 00:07:25,150 That column multiplies the c_1 and that row. 125 00:07:25,150 --> 00:07:30,390 You see how nice, that's the element matrix associated 126 00:07:30,390 --> 00:07:31,720 with the first bar. 127 00:07:31,720 --> 00:07:35,850 And then there'll be a second column times the c_2, 128 00:07:35,850 --> 00:07:37,050 times the second row. 129 00:07:37,050 --> 00:07:43,710 So plus c_2, row two transpose, row two. 130 00:07:43,710 --> 00:07:46,230 That's a matrix again. 131 00:07:46,230 --> 00:07:52,840 Plus c_3, row three transpose, row three. 132 00:07:52,840 --> 00:07:56,320 I focus on that because you don't think of this as a way 133 00:07:56,320 --> 00:08:00,980 to multiply matrices, but it's really a nice thing to notice. 134 00:08:00,980 --> 00:08:04,220 And it's better to notice it now when 135 00:08:04,220 --> 00:08:10,420 we have three bars or something, than in a big finite element 136 00:08:10,420 --> 00:08:11,210 code. 137 00:08:11,210 --> 00:08:12,020 Yeah. 138 00:08:12,020 --> 00:08:18,840 So this is, just if I complete it, complete this thought, 139 00:08:18,840 --> 00:08:22,490 this I would call a one bar matrix. 140 00:08:22,490 --> 00:08:26,640 That's the matrix A transpose A if there's only one bar. 141 00:08:26,640 --> 00:08:31,790 Actually, one of the problems at the end of this section 142 00:08:31,790 --> 00:08:35,770 is find the element matrix for one bar. 143 00:08:35,770 --> 00:08:37,810 And I guess it's about what we're 144 00:08:37,810 --> 00:08:41,970 going to get to when we do that one bar. 145 00:08:41,970 --> 00:08:45,030 Do you remember what it was in the, 146 00:08:45,030 --> 00:08:48,480 just to connect this thought, what 147 00:08:48,480 --> 00:08:53,740 was the little matrix in the case of networks? 148 00:08:53,740 --> 00:08:55,790 So in the case of networks, there 149 00:08:55,790 --> 00:09:00,220 was just one unknown for each, not two. 150 00:09:00,220 --> 00:09:04,920 So for networks, just, I'm just going to put down the, 151 00:09:04,920 --> 00:09:07,010 and you'll recognize it immediately. 152 00:09:07,010 --> 00:09:11,480 The little element matrix was the c for that, 153 00:09:11,480 --> 00:09:17,010 and there was a [1, -1; -1, 1] what? 154 00:09:17,010 --> 00:09:22,280 Do you remember that guy that was the [-1, 1] from a row? 155 00:09:22,280 --> 00:09:23,680 [-1, 1] from a column? 156 00:09:23,680 --> 00:09:29,590 So this was exactly c times the [-1, 1] 157 00:09:29,590 --> 00:09:34,800 from the column times the [-1, 1] from the row. 158 00:09:34,800 --> 00:09:39,000 That's where this simple little matrix came from. 159 00:09:39,000 --> 00:09:42,290 And you remember that the, so what's 160 00:09:42,290 --> 00:09:45,870 involved in creating this big A transpose 161 00:09:45,870 --> 00:09:52,020 C A is just create all these little pieces, which 162 00:09:52,020 --> 00:09:55,042 are like this, but they're going to be a little bigger. 163 00:09:55,042 --> 00:09:56,750 In fact, in a minute I'm going to ask you 164 00:09:56,750 --> 00:09:58,860 what size they'll be. 165 00:09:58,860 --> 00:10:00,770 Well, they're really big matrices. 166 00:10:00,770 --> 00:10:03,950 There are a whole lot of zeroes there that I didn't even put. 167 00:10:03,950 --> 00:10:07,210 Zeroes are there for rows and columns that aren't 168 00:10:07,210 --> 00:10:11,270 touching this particular edge. 169 00:10:11,270 --> 00:10:13,300 And again, this matrix. 170 00:10:13,300 --> 00:10:16,440 There'll be all kinds of zeroes in A. 171 00:10:16,440 --> 00:10:21,070 Because a typical row of A, bar one, 172 00:10:21,070 --> 00:10:27,450 is going to have non-zeroes only for the, yeah what's the size? 173 00:10:27,450 --> 00:10:32,350 How many non-zeroes in a typical row of A? 174 00:10:32,350 --> 00:10:36,390 Getting the count right first is like half the battle. 175 00:10:36,390 --> 00:10:39,620 How many non-zeroes in a typical row of A? 176 00:10:39,620 --> 00:10:43,430 This was the network case where we had a couple of nodes. 177 00:10:43,430 --> 00:10:44,830 And they were connected. 178 00:10:44,830 --> 00:10:47,730 And we had an unknown at each end. 179 00:10:47,730 --> 00:10:51,110 So two unknowns were involved. 180 00:10:51,110 --> 00:10:53,590 The little matrix was two by two. 181 00:10:53,590 --> 00:10:58,040 It properly has lots of zeroes for all the other nodes that 182 00:10:58,040 --> 00:10:59,470 are not involved. 183 00:10:59,470 --> 00:11:04,080 And then that matrix kind of gets, assembled 184 00:11:04,080 --> 00:11:06,840 is the word I think usually used. 185 00:11:06,840 --> 00:11:09,190 All these little guys get assembled, you 186 00:11:09,190 --> 00:11:15,330 know, pasted, stamped, I hear as the verb sometimes now. 187 00:11:15,330 --> 00:11:19,980 Take these little matrices for this little element. 188 00:11:19,980 --> 00:11:27,290 And stamp them into the big A transpose C A. This is the c_1, 189 00:11:27,290 --> 00:11:29,630 so this gives the c_1's in the matrix. 190 00:11:29,630 --> 00:11:31,980 And the c_2's and the c_3's. 191 00:11:31,980 --> 00:11:32,830 Alright. 192 00:11:32,830 --> 00:11:36,560 Now, just before we-- I'm like doing this preliminary, 193 00:11:36,560 --> 00:11:40,740 before I write down anything, the exact row. 194 00:11:40,740 --> 00:11:48,030 What's the size of, for trusses, how many non-zeroes 195 00:11:48,030 --> 00:11:49,390 in a row of A? 196 00:11:49,390 --> 00:11:50,930 So that's my question. 197 00:11:50,930 --> 00:11:58,460 How many non-zeroes in a typical row, like for that bar, 198 00:11:58,460 --> 00:12:05,470 non-zeroes in a row of A? 199 00:12:05,470 --> 00:12:12,620 So A is the matrix that tells us how much, that row of A 200 00:12:12,620 --> 00:12:16,990 is the row that tells us how much this bar stretched when 201 00:12:16,990 --> 00:12:24,010 this moved along by u H 1, and up by you u V 1, 202 00:12:24,010 --> 00:12:32,520 and this moved along by say, u H 2, and up by u V 2. 203 00:12:32,520 --> 00:12:35,010 Well, I've written all those in. 204 00:12:35,010 --> 00:12:36,980 So that you can tell me this number. 205 00:12:36,980 --> 00:12:38,940 How many? 206 00:12:38,940 --> 00:12:40,280 What's your guess? 207 00:12:40,280 --> 00:12:43,520 When I tell you, you'll say of course. 208 00:12:43,520 --> 00:12:48,350 How many u's are involved in the stretching of that bar? 209 00:12:48,350 --> 00:12:49,560 Four. 210 00:12:49,560 --> 00:12:50,700 Four. 211 00:12:50,700 --> 00:12:51,680 Exactly. 212 00:12:51,680 --> 00:12:54,940 Instead of one at each end, we have two at each end. 213 00:12:54,940 --> 00:12:56,700 So the answer is four. 214 00:12:56,700 --> 00:12:58,970 How many, the answer is four. 215 00:12:58,970 --> 00:13:03,800 And now the only remaining question is, what are they? 216 00:13:03,800 --> 00:13:06,730 What are those four numbers? 217 00:13:06,730 --> 00:13:08,850 The four non-zeroes in the row? 218 00:13:08,850 --> 00:13:13,320 So let me just answer that. 219 00:13:13,320 --> 00:13:17,130 They are, so here is that row. 220 00:13:17,130 --> 00:13:22,770 So we the two non-zeroes associated with it. 221 00:13:22,770 --> 00:13:26,549 Well, the way I've numbered these nodes one and two-- 222 00:13:26,549 --> 00:13:28,090 Since I've numbered them one and two, 223 00:13:28,090 --> 00:13:31,570 the non-zeroes are going to come right at the start. 224 00:13:31,570 --> 00:13:33,340 And then a whole lot, then this is all 225 00:13:33,340 --> 00:13:34,930 going to be zero after that. 226 00:13:34,930 --> 00:13:37,810 Because those will be nodes three, four, or five, whatever, 227 00:13:37,810 --> 00:13:40,280 that don't involve bar one. 228 00:13:40,280 --> 00:13:43,530 So bar one just connects node one to node two. 229 00:13:43,530 --> 00:13:47,020 Now, what do you think? 230 00:13:47,020 --> 00:13:51,570 Well, let me put in the key quantity here. 231 00:13:51,570 --> 00:13:53,460 This bar is at an angle. 232 00:13:53,460 --> 00:13:55,670 It's at an angle theta. 233 00:13:55,670 --> 00:13:56,760 So there's a theta. 234 00:13:56,760 --> 00:14:00,650 Angle theta. 235 00:14:00,650 --> 00:14:02,300 OK. 236 00:14:02,300 --> 00:14:10,840 And so that angle is going to enter these things. 237 00:14:10,840 --> 00:14:13,980 In fact, here's what you get. 238 00:14:13,980 --> 00:14:18,920 You get, I think if I put the one up there and the two down 239 00:14:18,920 --> 00:14:20,250 there, let's see. 240 00:14:20,250 --> 00:14:22,250 What am I thinking now? 241 00:14:22,250 --> 00:14:26,490 I'm saying if u 1 H is positive, that's 242 00:14:26,490 --> 00:14:28,330 going to stretch the bar. 243 00:14:28,330 --> 00:14:30,180 That's a positive stretching. 244 00:14:30,180 --> 00:14:33,370 So I'm expecting a positive u 1 H to give me, 245 00:14:33,370 --> 00:14:36,660 I'm expecting that sort of to come in with a plus sign. 246 00:14:36,660 --> 00:14:42,100 Now suppose the bar is horizontal. 247 00:14:42,100 --> 00:14:45,790 Suppose the bar is horizontal, then 248 00:14:45,790 --> 00:14:48,960 how much does the u 1 H stretch it? 249 00:14:48,960 --> 00:14:52,840 It stretches it by the whole u 1 H, right? 250 00:14:52,840 --> 00:14:57,170 If the bar was horizontal, so theta equals zero. 251 00:14:57,170 --> 00:15:02,230 I'm just doing these, we got to sort out this theta angle 252 00:15:02,230 --> 00:15:02,770 stuff. 253 00:15:02,770 --> 00:15:04,270 So here's my thing. 254 00:15:04,270 --> 00:15:06,940 If the bar happens to be horizontal, 255 00:15:06,940 --> 00:15:13,660 then that stretching by u H 1, will completely 256 00:15:13,660 --> 00:15:17,370 stretch the bar. 257 00:15:17,370 --> 00:15:20,210 If the bar happened to be-- Yeah yeah. 258 00:15:20,210 --> 00:15:22,210 And of course, this way. 259 00:15:22,210 --> 00:15:26,820 So that for a horizontal bar, I'll just be back to this step. 260 00:15:26,820 --> 00:15:32,990 I'll have a one and a minus one. u-- Oh yeah, remind me 261 00:15:32,990 --> 00:15:34,160 about that. 262 00:15:34,160 --> 00:15:39,900 Why doesn't u vertical, for a horizontal bar like this, 263 00:15:39,900 --> 00:15:46,400 why does this one not stretch the bar? 264 00:15:46,400 --> 00:15:47,960 You remember that from last time, 265 00:15:47,960 --> 00:15:53,810 that was the little bit of trig that we did when we forced 266 00:15:53,810 --> 00:15:56,300 ourselves to stay linear. 267 00:15:56,300 --> 00:16:00,680 So we dropped the second order correction, that 268 00:16:00,680 --> 00:16:03,160 would come from going this way. 269 00:16:03,160 --> 00:16:06,690 Right? 270 00:16:06,690 --> 00:16:09,710 I mean you must have noticed, like walking? 271 00:16:09,710 --> 00:16:14,050 Suppose you want to walk from here to the end of the bar, OK? 272 00:16:14,050 --> 00:16:17,690 Well, if somebody moves the end of the bar forward, 273 00:16:17,690 --> 00:16:19,520 you have to take those extra steps. 274 00:16:19,520 --> 00:16:21,520 The bar really stretches. 275 00:16:21,520 --> 00:16:24,140 But, if somebody moves the bar this way, 276 00:16:24,140 --> 00:16:28,590 then the extra bit of length is much less. 277 00:16:28,590 --> 00:16:33,860 In fact, it's zero to first order. 278 00:16:33,860 --> 00:16:35,590 This is like taking shortcuts when 279 00:16:35,590 --> 00:16:40,410 you walk across the courtyard. 280 00:16:40,410 --> 00:16:45,310 So when the angle's theta, I'm only 281 00:16:45,310 --> 00:16:49,090 expecting a one and a minus one. 282 00:16:49,090 --> 00:16:50,030 On the horizontal. 283 00:16:50,030 --> 00:16:51,260 And zeroes on the vertical. 284 00:16:51,260 --> 00:16:52,970 OK, now I'm ready to write it in. 285 00:16:52,970 --> 00:16:56,480 I think when the angle's theta, when the angle's theta, 286 00:16:56,480 --> 00:17:00,500 any theta, that was when the angle was zero, 287 00:17:00,500 --> 00:17:05,050 I think we get a cos theta. 288 00:17:05,050 --> 00:17:07,390 Doesn't your instinct say that this 289 00:17:07,390 --> 00:17:11,940 is on the u_1, u horizontal 1. 290 00:17:11,940 --> 00:17:17,970 And then the u vertical 1, tell me what these should be. 291 00:17:17,970 --> 00:17:22,870 And then we'll make, what do you suppose 292 00:17:22,870 --> 00:17:26,650 is the entry, second non-zero, the one that corresponds 293 00:17:26,650 --> 00:17:29,020 to a vertical movement. 294 00:17:29,020 --> 00:17:31,970 Here it would be, for a horizontal bar 295 00:17:31,970 --> 00:17:35,790 when theta is zero, I'm going to see a zero there. 296 00:17:35,790 --> 00:17:38,590 But if the bar is at an angle like this, 297 00:17:38,590 --> 00:17:40,280 what am I going to see? 298 00:17:40,280 --> 00:17:42,670 Everybody's going to get it right? 299 00:17:42,670 --> 00:17:44,510 What do I put in there? 300 00:17:44,510 --> 00:17:45,320 Sine theta. 301 00:17:45,320 --> 00:17:46,330 What else could it be? 302 00:17:46,330 --> 00:17:49,760 Right, OK, sin(theta). 303 00:17:49,760 --> 00:17:53,680 And now what about the next guy, the other end of the bar? 304 00:17:53,680 --> 00:17:59,790 u H 2 and u V 2, those are the other two non-zeroes. 305 00:17:59,790 --> 00:18:03,380 What your guess for u 2 H, u H 2? 306 00:18:03,380 --> 00:18:11,820 If I move this forward, what's the change 307 00:18:11,820 --> 00:18:15,030 in length of the bar? 308 00:18:15,030 --> 00:18:20,110 What would your guess be that goes into there? 309 00:18:20,110 --> 00:18:22,110 Say it again? 310 00:18:22,110 --> 00:18:24,110 -cos(theta). 311 00:18:24,110 --> 00:18:26,550 -cos(theta), right, yeah. 312 00:18:26,550 --> 00:18:30,020 The movement of the other end, like 313 00:18:30,020 --> 00:18:34,910 if I move this guy a little bit to this side. 314 00:18:34,910 --> 00:18:37,240 That will shorten the bar. 315 00:18:37,240 --> 00:18:38,340 Forget about that one. 316 00:18:38,340 --> 00:18:43,770 If I move this over, the bar becomes shorter. 317 00:18:43,770 --> 00:18:51,860 And the cosine tells me the key number there, 318 00:18:51,860 --> 00:18:53,710 how much it becomes shorter. 319 00:18:53,710 --> 00:18:57,660 If the bar was horizontal, the cos(theta) was one, 320 00:18:57,660 --> 00:19:00,300 it counts a hundred percent. 321 00:19:00,300 --> 00:19:05,570 If the bar is vertical, and I move it horizontally, 322 00:19:05,570 --> 00:19:06,940 it comes zero percent. 323 00:19:06,940 --> 00:19:11,810 Because the linearity says there was no first order change. 324 00:19:11,810 --> 00:19:14,980 And now tell me the final non-zero entry. 325 00:19:14,980 --> 00:19:18,280 And I see I didn't leave much room for all the zeroes. 326 00:19:18,280 --> 00:19:24,580 OK, what's the u 2 V entry? 327 00:19:24,580 --> 00:19:26,910 -sin(theta), of course. 328 00:19:26,910 --> 00:19:33,810 And then come all the zeroes, for whatever other joints 329 00:19:33,810 --> 00:19:35,950 are not involved with bar one. 330 00:19:35,950 --> 00:19:38,750 So let me, maybe to make this picture best, 331 00:19:38,750 --> 00:19:43,340 I should move that over to where it belongs. 332 00:19:43,340 --> 00:19:49,160 Now, if I add up along a bar, add up the four numbers there, 333 00:19:49,160 --> 00:19:49,980 what do I get? 334 00:19:49,980 --> 00:19:51,360 Zero. 335 00:19:51,360 --> 00:19:56,180 You expected that, right? 336 00:19:56,180 --> 00:19:59,260 In fact, if I add just that and that I get zero. 337 00:19:59,260 --> 00:20:01,090 If I add that and that I get zero. 338 00:20:01,090 --> 00:20:04,030 Just the way I got zero here. 339 00:20:04,030 --> 00:20:07,000 In the incidence matrices. 340 00:20:07,000 --> 00:20:14,670 The column of all ones is certainly going to solve Au=0. 341 00:20:14,670 --> 00:20:17,390 342 00:20:17,390 --> 00:20:22,300 Unless the supports remove those, of course. 343 00:20:22,300 --> 00:20:26,230 If the supports don't allow all ones 344 00:20:26,230 --> 00:20:29,040 because some have to stay at zero, 345 00:20:29,040 --> 00:20:31,840 then I could have a stable truss. 346 00:20:31,840 --> 00:20:34,160 OK, that's a typical bar. 347 00:20:34,160 --> 00:20:35,840 A typical row. 348 00:20:35,840 --> 00:20:37,910 That's a typical row. 349 00:20:37,910 --> 00:20:43,320 OK, and now maybe while I'm on the same subject, what 350 00:20:43,320 --> 00:20:52,940 is the size, what does this thing look like now? 351 00:20:52,940 --> 00:20:58,070 This is in A. This is in the matrix A, 352 00:20:58,070 --> 00:21:00,970 and now I want to ask you before I even 353 00:21:00,970 --> 00:21:05,340 come back to all this stuff, what about in A transpose A? 354 00:21:05,340 --> 00:21:10,620 In A transpose A, A transpose C A, it the whole deal. 355 00:21:10,620 --> 00:21:17,210 The element for-- The little matrix, the element matrix, 356 00:21:17,210 --> 00:21:18,640 can I call it that? 357 00:21:18,640 --> 00:21:25,790 Or the one-bar matrix, call it the one-bar matrix. 358 00:21:25,790 --> 00:21:31,200 Will be, is what? 359 00:21:31,200 --> 00:21:37,550 So I want this, it would be typical-- c_1, row one 360 00:21:37,550 --> 00:21:41,030 transpose, row one, that's the typical guy. 361 00:21:41,030 --> 00:21:48,300 And how many non-zeroes in that? 362 00:21:48,300 --> 00:21:51,160 Multiplying a row. 363 00:21:51,160 --> 00:21:54,490 Sorry, multiplying a column that has 364 00:21:54,490 --> 00:21:57,640 four non-zeroes times a row that has 365 00:21:57,640 --> 00:22:01,880 four non-zeroes times a number, which is just fine. 366 00:22:01,880 --> 00:22:07,860 How many non-zeroes are going to sit in this element 367 00:22:07,860 --> 00:22:12,610 matrix, this one-bar matrix? 368 00:22:12,610 --> 00:22:14,110 16. 369 00:22:14,110 --> 00:22:18,180 16 non-zeroes. 370 00:22:18,180 --> 00:22:24,590 And they're going to be, I have cosine, sine, minus cosine, 371 00:22:24,590 --> 00:22:30,400 minus sine, multiplying cosine, sine, minus cosine, minus sine. 372 00:22:30,400 --> 00:22:33,620 And all multiplied by c_1. 373 00:22:33,620 --> 00:22:34,850 So that's the matrix. 374 00:22:34,850 --> 00:22:40,550 And you see what it looks like. c squared, cs, so on. 375 00:22:40,550 --> 00:22:43,470 16 guys. 376 00:22:43,470 --> 00:22:50,170 So we have four squared as our element matrix, where here we 377 00:22:50,170 --> 00:22:51,920 had two squared. 378 00:22:51,920 --> 00:22:59,490 And in finite elements, when you get to elasticity, 379 00:22:59,490 --> 00:23:07,250 and you've got triangles, you've got triangular elements, 380 00:23:07,250 --> 00:23:09,280 then there are three nodes involved. 381 00:23:09,280 --> 00:23:12,120 So you're up to higher numbers. 382 00:23:12,120 --> 00:23:14,320 But this gives you the idea. 383 00:23:14,320 --> 00:23:18,480 And remember that this four by four, the way I've done it, 384 00:23:18,480 --> 00:23:21,330 the way I've numbered it, one, two, 385 00:23:21,330 --> 00:23:24,430 happens to sit up in the upper left corner. 386 00:23:24,430 --> 00:23:29,550 Of A transpose C A. But can you sort of imagine 387 00:23:29,550 --> 00:23:31,810 how the code would be written? 388 00:23:31,810 --> 00:23:35,570 The code would be written, take each bar, 389 00:23:35,570 --> 00:23:38,410 and what do I have to know about the bar? 390 00:23:38,410 --> 00:23:41,540 Just imagine a code that would do trusses. 391 00:23:41,540 --> 00:23:44,940 Actually, the final problem that I'm not 392 00:23:44,940 --> 00:23:49,520 assigning in this section says what would the code be like? 393 00:23:49,520 --> 00:23:55,380 Can we just have a think about what the code would look 394 00:23:55,380 --> 00:23:57,780 like if we were to write it. 395 00:23:57,780 --> 00:24:03,850 What would the input have to be? 396 00:24:03,850 --> 00:24:08,180 For each bar, what input do I need? 397 00:24:08,180 --> 00:24:14,710 For this bar, I need to know-- And for the whole truss. 398 00:24:14,710 --> 00:24:18,190 What do I have to tell, what's the information that I 399 00:24:18,190 --> 00:24:20,390 need for the whole truss? 400 00:24:20,390 --> 00:24:25,610 I have to know the positions of all of the joints, right? 401 00:24:25,610 --> 00:24:28,830 So I'd have to know the coordinates of that, (x, y), 402 00:24:28,830 --> 00:24:34,930 the coordinates of this one, (x_1, y_1) for joint one, 403 00:24:34,930 --> 00:24:36,010 (x_2, y_2). 404 00:24:36,010 --> 00:24:40,130 So I'd have to have a little list of what would that be? 405 00:24:40,130 --> 00:24:42,420 m by two? 406 00:24:42,420 --> 00:24:44,260 Oh, no. 407 00:24:44,260 --> 00:24:44,980 N. N by 2. 408 00:24:44,980 --> 00:24:50,070 I have N, what do I have now? 409 00:24:50,070 --> 00:24:52,160 Think of what information do I have 410 00:24:52,160 --> 00:24:54,680 to report about this truss? 411 00:24:54,680 --> 00:24:58,090 I guess I have N, capital N, joints. 412 00:24:58,090 --> 00:25:02,560 And I need two coordinates, x, y for each position. 413 00:25:02,560 --> 00:25:04,440 So that's N by 2, this. 414 00:25:04,440 --> 00:25:09,650 OK, and then for every bar, what do I need to tell it? 415 00:25:09,650 --> 00:25:12,980 What do I need to put in the code for a typical bar? 416 00:25:12,980 --> 00:25:19,550 I certainly have to put in the c for that bar. 417 00:25:19,550 --> 00:25:21,780 And what else do I need to know? 418 00:25:21,780 --> 00:25:26,980 I need to know which joints it's connected. 419 00:25:26,980 --> 00:25:27,660 Right? 420 00:25:27,660 --> 00:25:32,360 I have to tell the system that this bar is between two 421 00:25:32,360 --> 00:25:34,770 and one, one and two. 422 00:25:34,770 --> 00:25:36,970 I have to tell it which pair. 423 00:25:36,970 --> 00:25:42,690 So I guess I have a list of m bars, and for each bar 424 00:25:42,690 --> 00:25:48,840 I must tell the system the two node numbers, 425 00:25:48,840 --> 00:25:51,460 and the c, the stiffness. 426 00:25:51,460 --> 00:25:54,860 The constant for Hooke's Law. 427 00:25:54,860 --> 00:25:56,600 Right, do you see this picture? 428 00:25:56,600 --> 00:26:02,080 Just sort of visualizing, creating a code here. 429 00:26:02,080 --> 00:26:05,580 And then the code would do all this, 430 00:26:05,580 --> 00:26:09,800 oh, have I given enough information to find theta? 431 00:26:09,800 --> 00:26:13,860 Or do I have to import theta also? 432 00:26:13,860 --> 00:26:14,720 No. 433 00:26:14,720 --> 00:26:17,120 I told you the positions, so it'll 434 00:26:17,120 --> 00:26:19,090 figure out cos(theta) theta and sin(theta). 435 00:26:19,090 --> 00:26:20,820 It actually won't figure out theta, 436 00:26:20,820 --> 00:26:22,670 that's always a dumb thing to do, 437 00:26:22,670 --> 00:26:24,990 find the actual angle. cos(theta) 438 00:26:24,990 --> 00:26:27,760 and sin(theta) is the quantities we want. 439 00:26:27,760 --> 00:26:32,950 So given that position, (x, y), and this position (x_2, y_2), 440 00:26:32,950 --> 00:26:36,460 it would know cos(theta) and sin(theta). 441 00:26:36,460 --> 00:26:39,170 And having drawn his picture allows 442 00:26:39,170 --> 00:26:42,280 me to make once more the key point 443 00:26:42,280 --> 00:26:44,530 about small displacements. 444 00:26:44,530 --> 00:26:49,360 What's the angle of the bar after it's moved? 445 00:26:49,360 --> 00:26:51,720 After it's displaced? 446 00:26:51,720 --> 00:26:54,180 It was theta before it was displaced, 447 00:26:54,180 --> 00:26:58,990 and the angle after is theta. 448 00:26:58,990 --> 00:27:02,980 To first order, the angle doesn't change. 449 00:27:02,980 --> 00:27:07,690 Because these are little tiny movements of the ends. 450 00:27:07,690 --> 00:27:10,840 I've drawn them much bigger than they should be drawn. 451 00:27:10,840 --> 00:27:12,870 They're little, tiny movements of the ends 452 00:27:12,870 --> 00:27:18,520 so that the angle is not significantly changed. 453 00:27:18,520 --> 00:27:22,200 Otherwise we're into geometric nonlinearity 454 00:27:22,200 --> 00:27:27,600 and that stuff, that makes the problem much, much harder. 455 00:27:27,600 --> 00:27:30,140 OK, are you seeing sort of the picture? 456 00:27:30,140 --> 00:27:34,660 I guess what I haven't completely-- 457 00:27:34,660 --> 00:27:38,890 I've really depended more on your intuition 458 00:27:38,890 --> 00:27:42,980 than on a calculation to say that these are the four 459 00:27:42,980 --> 00:27:45,410 non-zeroes. 460 00:27:45,410 --> 00:27:47,270 What did I ask you to do? 461 00:27:47,270 --> 00:27:51,440 I asked you to check that that was right in the extreme cases, 462 00:27:51,440 --> 00:27:54,570 like if theta is zero, the bar is horizontal, 463 00:27:54,570 --> 00:28:00,270 then we just have a 1, 0, -1, 0; vertical isn't happening. 464 00:28:00,270 --> 00:28:04,350 If the bar is vertical so that the angle is 90 degrees then 465 00:28:04,350 --> 00:28:09,350 we would have a 0, 1, 0, -1; everything's vertical. 466 00:28:09,350 --> 00:28:14,040 And the book draws a little picture, 467 00:28:14,040 --> 00:28:20,620 and computes delta l from these four small movements. 468 00:28:20,620 --> 00:28:24,100 And takes the leading term and sure enough it 469 00:28:24,100 --> 00:28:30,790 produces that row of the matrix. 470 00:28:30,790 --> 00:28:33,910 Gosh, I talk real fast. 471 00:28:33,910 --> 00:28:38,880 But do you think you could now create the stiffness? 472 00:28:38,880 --> 00:28:46,430 If you had a real truss, you could create the matrix A 473 00:28:46,430 --> 00:28:48,180 for it? 474 00:28:48,180 --> 00:28:51,390 C is simple, it's given to you. 475 00:28:51,390 --> 00:28:54,970 You could create A transpose C A? 476 00:28:54,970 --> 00:28:58,870 You might just want to write the command as A' 477 00:28:58,870 --> 00:29:03,900 * C * A or something. 478 00:29:03,900 --> 00:29:06,280 And let MATLAB do the thinking. 479 00:29:06,280 --> 00:29:10,760 But I wanted to just see what these, 480 00:29:10,760 --> 00:29:18,530 how this four by four piece appears in this product, 481 00:29:18,530 --> 00:29:21,660 from each bar. 482 00:29:21,660 --> 00:29:26,170 The 16 non-zeroes will appear in different positions 483 00:29:26,170 --> 00:29:30,750 and you told the code what those positions are. 484 00:29:30,750 --> 00:29:35,190 You had to give the code a local-to-global picture. 485 00:29:35,190 --> 00:29:37,670 This is the local picture. 486 00:29:37,670 --> 00:29:39,370 Watch one bar. 487 00:29:39,370 --> 00:29:45,040 Then it has to fit in this big n by n matrix, 488 00:29:45,040 --> 00:29:50,470 and that means you have to know what 489 00:29:50,470 --> 00:29:52,700 joints was that bar connecting. 490 00:29:52,700 --> 00:30:05,560 So which positions do these 16 non-zeroes assemble into? 491 00:30:05,560 --> 00:30:09,680 That's some time devoted to a job 492 00:30:09,680 --> 00:30:14,770 that I actually don't plan to require you to do. 493 00:30:14,770 --> 00:30:18,220 Creating this truss problem. 494 00:30:18,220 --> 00:30:23,760 What I think is kind of more fun and that's, these homework 495 00:30:23,760 --> 00:30:30,050 problems would deal with it, is part two of the lecture. 496 00:30:30,050 --> 00:30:33,870 Going back to mechanisms, and now 497 00:30:33,870 --> 00:30:42,870 thinking about more complicated trusses. 498 00:30:42,870 --> 00:30:47,140 We now in principle could find the solutions to Au=0 499 00:30:47,140 --> 00:30:50,010 because we now have constructed A, 500 00:30:50,010 --> 00:30:55,000 and we could get MATLAB to do the work or Python or whoever. 501 00:30:55,000 --> 00:31:00,660 But can I go to part two now and draw a truss 502 00:31:00,660 --> 00:31:05,080 and ask you about the mechanisms? 503 00:31:05,080 --> 00:31:10,240 Let's see. 504 00:31:10,240 --> 00:31:15,600 I guess somewhere in the problem set, 505 00:31:15,600 --> 00:31:25,810 but not one of the assigned ones is-- Start with those six bars. 506 00:31:25,810 --> 00:31:33,350 And six joints, so these are six joints. 507 00:31:33,350 --> 00:31:42,400 And OK, as it stands, how many-- That's a good question. 508 00:31:42,400 --> 00:31:47,100 As it stands, what's the shape of the matrix A? 509 00:31:47,100 --> 00:31:50,720 How many rows has it got? 510 00:31:50,720 --> 00:31:55,760 So as it stands, so I'll call it A_0, 511 00:31:55,760 --> 00:31:58,560 for no supports have been added. 512 00:31:58,560 --> 00:32:01,990 A_0, just the full matrix. 513 00:32:01,990 --> 00:32:04,140 Is what shape? 514 00:32:04,140 --> 00:32:06,330 6 by 12. 515 00:32:06,330 --> 00:32:06,830 Good. 516 00:32:06,830 --> 00:32:08,260 6 by 12. 517 00:32:08,260 --> 00:32:10,490 Is 6 by 12. 518 00:32:10,490 --> 00:32:12,890 OK, of course it's not stable. 519 00:32:12,890 --> 00:32:13,770 We know that. 520 00:32:13,770 --> 00:32:16,340 We haven't supported anything. 521 00:32:16,340 --> 00:32:24,050 So in a typical case, how many solutions to A, 522 00:32:24,050 --> 00:32:28,430 so I'm going to ask you how many solutions to Au=0? 523 00:32:28,430 --> 00:32:32,470 524 00:32:32,470 --> 00:32:35,730 And what's your guess? 525 00:32:35,730 --> 00:32:37,570 Six. 526 00:32:37,570 --> 00:32:42,170 Got six equations, we've got 12 u's, 12-6, 527 00:32:42,170 --> 00:32:44,990 so this is going to be 12-6. 528 00:32:44,990 --> 00:32:49,560 And of course six solutions to that equation. 529 00:32:49,560 --> 00:32:51,300 It's what I would expect. 530 00:32:51,300 --> 00:32:54,510 There could be, it could be possible 531 00:32:54,510 --> 00:32:59,400 that the six equations are not independent. 532 00:32:59,400 --> 00:33:01,740 If they really dropped to five then this 533 00:33:01,740 --> 00:33:05,710 would bump up to seven, I don't think it's going to happen. 534 00:33:05,710 --> 00:33:09,930 Now, can you describe those six solutions, not with 535 00:33:09,930 --> 00:33:13,980 numbers, just with, tell me. 536 00:33:13,980 --> 00:33:17,640 I hope you can, because I can't right now. 537 00:33:17,640 --> 00:33:19,560 OK, three of them we know. 538 00:33:19,560 --> 00:33:31,130 So with no supports at all, what are three rigid motions? 539 00:33:31,130 --> 00:33:33,470 And what are they? 540 00:33:33,470 --> 00:33:38,190 The whole truss could move to the right, the whole hexagon. 541 00:33:38,190 --> 00:33:41,780 It could all move up, it could all rotate about one point. 542 00:33:41,780 --> 00:33:47,090 All three of those would be movements, displacements 543 00:33:47,090 --> 00:33:48,930 that don't stretch anything. 544 00:33:48,930 --> 00:33:51,150 OK, three rigid motions. 545 00:33:51,150 --> 00:33:59,200 Across, up, and rotate. 546 00:33:59,200 --> 00:34:03,110 OK, and I can get rid of those by supporting some nodes. 547 00:34:03,110 --> 00:34:07,690 But let me see, I don't know what's going to happen. 548 00:34:07,690 --> 00:34:14,290 When I describe this topic as the fun one in 18.085, 549 00:34:14,290 --> 00:34:16,420 it's more fun for you than for me. 550 00:34:16,420 --> 00:34:21,580 Because I draw something like that and I start worrying can 551 00:34:21,580 --> 00:34:26,010 I think of three-- How many mechanisms to look for? 552 00:34:26,010 --> 00:34:26,930 Three. 553 00:34:26,930 --> 00:34:31,570 That's a big number. 554 00:34:31,570 --> 00:34:36,490 I bet you I can find one but you guys have got to, alright, 555 00:34:36,490 --> 00:34:37,700 tell me some mechanisms. 556 00:34:37,700 --> 00:34:40,290 Let me try to draw them. 557 00:34:40,290 --> 00:34:44,970 What would be one mechanism? 558 00:34:44,970 --> 00:34:47,230 Collapses, yeah, somehow. 559 00:34:47,230 --> 00:34:50,720 How shall I make it collapse? 560 00:34:50,720 --> 00:34:53,980 Can squeeze in, yeah, maybe that's the first one. 561 00:34:53,980 --> 00:35:05,320 This guy comes in, this guy, what does that do? 562 00:35:05,320 --> 00:35:09,490 I've got 15 minutes here, I could pull that board down 563 00:35:09,490 --> 00:35:10,560 and draw another one. 564 00:35:10,560 --> 00:35:13,080 What is this one here? 565 00:35:13,080 --> 00:35:16,780 Let's see, if that comes in, do these guys have to go up a bit? 566 00:35:16,780 --> 00:35:20,550 Yeah, because that angle's not 90 degrees. 567 00:35:20,550 --> 00:35:24,680 So we've got a first order change in this. 568 00:35:24,680 --> 00:35:29,000 So this comes in a little, this goes up a little, 569 00:35:29,000 --> 00:35:35,480 this guy maybe stays straight. 570 00:35:35,480 --> 00:35:40,160 Would you go for this, I mean, please say yes? 571 00:35:40,160 --> 00:35:41,990 Something happens there, right? 572 00:35:41,990 --> 00:35:50,430 These things go in and those go out. 573 00:35:50,430 --> 00:35:53,330 Could you create the-- I mean if this 574 00:35:53,330 --> 00:35:57,930 was a equal-side, regular hexagon, 575 00:35:57,930 --> 00:36:03,000 you could put in all the numbers for all 12, 576 00:36:03,000 --> 00:36:05,440 I would be looking for 12 numbers. 577 00:36:05,440 --> 00:36:09,870 Six joints and they each have two u's, 578 00:36:09,870 --> 00:36:15,180 so that wouldn't be so simple but you could do it. 579 00:36:15,180 --> 00:36:20,600 So that would be one. 580 00:36:20,600 --> 00:36:23,220 Looking for number two. 581 00:36:23,220 --> 00:36:24,820 What would be another one? 582 00:36:24,820 --> 00:36:32,460 So sort of squeezing in like whatever. 583 00:36:32,460 --> 00:36:36,360 What do you think? 584 00:36:36,360 --> 00:36:40,080 Any others? 585 00:36:40,080 --> 00:36:42,580 Maybe that's possible. 586 00:36:42,580 --> 00:36:49,690 Maybe I just look at it, you think that would work? 587 00:36:49,690 --> 00:36:51,350 We could hope those were independent, 588 00:36:51,350 --> 00:36:54,390 but I wouldn't put my life on it. 589 00:36:54,390 --> 00:36:59,696 I have this squeeze in, this squeeze in and this squeeze in, 590 00:36:59,696 --> 00:37:03,170 I would worry a little bit. 591 00:37:03,170 --> 00:37:06,860 I can see another one. 592 00:37:06,860 --> 00:37:09,350 AUDIENCE: Half, for-- 593 00:37:09,350 --> 00:37:10,780 PROFESSOR STRANG: Fold it in half. 594 00:37:10,780 --> 00:37:16,679 AUDIENCE: Along, I guess that sort of gets into 3-D, but. 595 00:37:16,679 --> 00:37:18,970 PROFESSOR STRANG: Yeah, we've got to stay in the plane, 596 00:37:18,970 --> 00:37:24,220 right. 597 00:37:24,220 --> 00:37:27,790 I have instead of what? 598 00:37:27,790 --> 00:37:28,400 OK. 599 00:37:28,400 --> 00:37:31,770 AUDIENCE: [INAUDIBLE] 600 00:37:31,770 --> 00:37:34,440 PROFESSOR STRANG: Squeeze that out. 601 00:37:34,440 --> 00:37:36,420 Yeah. 602 00:37:36,420 --> 00:37:39,170 Good. 603 00:37:39,170 --> 00:37:42,530 Number two. 604 00:37:42,530 --> 00:37:47,210 And then I can see, here's one, here's an easy one to think of. 605 00:37:47,210 --> 00:37:51,420 Leave these three alone and just rotate these guys. 606 00:37:51,420 --> 00:37:54,950 Bring this down, right? 607 00:37:54,950 --> 00:38:03,880 Just bring these three vertically down, or something. 608 00:38:03,880 --> 00:38:11,890 You see why it's sort of, did you like that one alright? 609 00:38:11,890 --> 00:38:16,440 It seems simple to me, looking at it, just leave these guys-- 610 00:38:16,440 --> 00:38:19,060 Rotation, let this turn down. 611 00:38:19,060 --> 00:38:22,490 This turn down, let's say, and this go down. 612 00:38:22,490 --> 00:38:27,640 Maybe these would all drop by the same amount. 613 00:38:27,640 --> 00:38:33,310 Maybe. 614 00:38:33,310 --> 00:38:38,650 So anyway, whatever. 615 00:38:38,650 --> 00:38:42,480 Let's put some supports on them. 616 00:38:42,480 --> 00:38:45,290 And get these numbers down. 617 00:38:45,290 --> 00:38:50,400 So let's support, as usual, the bottom guy. 618 00:38:50,400 --> 00:38:54,190 OK, so different problem now. 619 00:38:54,190 --> 00:38:58,970 I won't call that A_0, I'll call it A. I'll ask myself, 620 00:38:58,970 --> 00:39:01,620 is it stable or unstable? 621 00:39:01,620 --> 00:39:06,400 The matrix is now six by what? 622 00:39:06,400 --> 00:39:10,570 Eight, because I've taken away, I 623 00:39:10,570 --> 00:39:14,290 have four reaction forces, two at each support, 624 00:39:14,290 --> 00:39:15,930 horizontal and vertical. 625 00:39:15,930 --> 00:39:20,170 I've got four free nodes. 626 00:39:20,170 --> 00:39:23,200 And six by eight. 627 00:39:23,200 --> 00:39:28,440 Let me put in-- So six by eight, what am I expecting now? 628 00:39:28,440 --> 00:39:31,830 Any rigid motions? 629 00:39:31,830 --> 00:39:34,770 No, no rigid motions now. 630 00:39:34,770 --> 00:39:38,360 How many solutions am I expecting? 631 00:39:38,360 --> 00:39:39,380 Two, I think. 632 00:39:39,380 --> 00:39:41,350 Probably two. 633 00:39:41,350 --> 00:39:42,760 How many mechanisms? 634 00:39:42,760 --> 00:39:44,790 Well, no rigid motions. 635 00:39:44,790 --> 00:39:47,060 So probably two mechanisms. 636 00:39:47,060 --> 00:39:49,400 Now, can we find two mechanisms? 637 00:39:49,400 --> 00:39:51,840 Alright, this is like more reasonable. 638 00:39:51,840 --> 00:39:55,430 We can see whether whether we get two mechanisms 639 00:39:55,430 --> 00:39:57,370 and whether they're really different. 640 00:39:57,370 --> 00:39:59,870 OK, what are the mechanisms now? 641 00:39:59,870 --> 00:40:01,830 These guys are fixed. 642 00:40:01,830 --> 00:40:13,180 So forget my little sketch here, and think again. 643 00:40:13,180 --> 00:40:14,910 What do you see? 644 00:40:14,910 --> 00:40:18,140 Alright, let's have one mechanism. 645 00:40:18,140 --> 00:40:23,380 What would one mechanism be now? 646 00:40:23,380 --> 00:40:25,720 There have to be two. 647 00:40:25,720 --> 00:40:28,240 What do you think? 648 00:40:28,240 --> 00:40:32,950 Sit on it. 649 00:40:32,950 --> 00:40:35,680 Alright, bring these guys down, and then these guys 650 00:40:35,680 --> 00:40:38,730 will go out, is that it? 651 00:40:38,730 --> 00:40:43,340 OK, so a number, mechanism number one, sit on truss. 652 00:40:43,340 --> 00:40:45,080 OK. 653 00:40:45,080 --> 00:40:45,680 Alright. 654 00:40:45,680 --> 00:40:48,890 Now, I don't know what number two 655 00:40:48,890 --> 00:40:51,850 is, that's why I'm taking time. 656 00:40:51,850 --> 00:40:56,162 AUDIENCE: [INAUDIBLE] 657 00:40:56,162 --> 00:40:57,120 PROFESSOR STRANG: Yeah. 658 00:40:57,120 --> 00:41:00,450 Or could we do this, could we bring these guys in and let's 659 00:41:00,450 --> 00:41:02,260 go up? 660 00:41:02,260 --> 00:41:03,720 It's the same thing. 661 00:41:03,720 --> 00:41:10,220 OK, so squeeze truss. 662 00:41:10,220 --> 00:41:11,370 Squeeze sides. 663 00:41:11,370 --> 00:41:13,260 Is that the same thing that I had, 664 00:41:13,260 --> 00:41:15,610 number one the same as two. 665 00:41:15,610 --> 00:41:18,390 Oh jeez, OK. 666 00:41:18,390 --> 00:41:24,590 Is that what-- Everybody is agreeing with this? 667 00:41:24,590 --> 00:41:25,090 OK. 668 00:41:25,090 --> 00:41:28,521 So I didn't get, my number two was no good. 669 00:41:28,521 --> 00:41:29,020 OK. 670 00:41:29,020 --> 00:41:33,250 What's a better number two? 671 00:41:33,250 --> 00:41:34,490 Hold an edge? 672 00:41:34,490 --> 00:41:35,440 Like that one. 673 00:41:35,440 --> 00:41:37,690 I'm just doing what you say, I'm not. 674 00:41:37,690 --> 00:41:47,300 AUDIENCE: [INAUDIBLE] Just this guy, rotates like so. 675 00:41:47,300 --> 00:41:50,710 And this guy will rotate, and this guy. 676 00:41:50,710 --> 00:41:56,020 That looks pretty good to me. 677 00:41:56,020 --> 00:41:59,510 Good, is that correct? 678 00:41:59,510 --> 00:42:01,110 Say that one again? 679 00:42:01,110 --> 00:42:03,550 So the first one was when these two came down 680 00:42:03,550 --> 00:42:04,730 and these went out. 681 00:42:04,730 --> 00:42:07,600 Right, OK. 682 00:42:07,600 --> 00:42:10,650 And now your suggestion is, you picked on this guy 683 00:42:10,650 --> 00:42:12,570 and held it fixed. 684 00:42:12,570 --> 00:42:15,370 And then this one came down a little bit. 685 00:42:15,370 --> 00:42:17,820 It'll, of course, how will it move? 686 00:42:17,820 --> 00:42:21,190 It will move perpendicular, right? 687 00:42:21,190 --> 00:42:22,690 Small movement. 688 00:42:22,690 --> 00:42:24,460 The bar is not going to change length, 689 00:42:24,460 --> 00:42:25,830 that's the whole point, right? 690 00:42:25,830 --> 00:42:27,670 The bar is not changing length. 691 00:42:27,670 --> 00:42:32,550 So the movement must be, it must be a simple rotation. 692 00:42:32,550 --> 00:42:33,860 Around here. 693 00:42:33,860 --> 00:42:35,420 OK. 694 00:42:35,420 --> 00:42:38,340 Right, and of course again you might say well, 695 00:42:38,340 --> 00:42:42,500 the bar really did change length because that's not 696 00:42:42,500 --> 00:42:43,510 quite the same as that. 697 00:42:43,510 --> 00:42:46,890 But then again that's my second order basis. 698 00:42:46,890 --> 00:42:50,300 So that one came down, and what did this one do? 699 00:42:50,300 --> 00:42:52,740 Came down the same. 700 00:42:52,740 --> 00:42:58,530 OK, and this one it also moves. 701 00:42:58,530 --> 00:43:03,770 What is that? 702 00:43:03,770 --> 00:43:07,800 OK, so what am I going to call this one? 703 00:43:07,800 --> 00:43:09,670 Fix one. 704 00:43:09,670 --> 00:43:10,170 Yeah. 705 00:43:10,170 --> 00:43:13,520 Fix one node. 706 00:43:13,520 --> 00:43:14,530 And that makes sense. 707 00:43:14,530 --> 00:43:16,720 Fix one joint, yeah. 708 00:43:16,720 --> 00:43:19,420 And then, and rotate the rest. 709 00:43:19,420 --> 00:43:21,210 I think that would be possible. 710 00:43:21,210 --> 00:43:23,460 Yep. 711 00:43:23,460 --> 00:43:29,960 OK, a small prize for anybody who, 712 00:43:29,960 --> 00:43:37,430 maybe handwritten, a picture of two really nice mechanisms. 713 00:43:37,430 --> 00:43:40,870 Somehow this one seems a little un-symmetric 714 00:43:40,870 --> 00:43:43,340 in a problem that's so symmetric, 715 00:43:43,340 --> 00:43:45,820 so I would guess that somewhere along the line 716 00:43:45,820 --> 00:43:50,850 we could find a kind of more some symmetric one. 717 00:43:50,850 --> 00:43:55,890 But I don't see what it is right now. 718 00:43:55,890 --> 00:44:03,520 Can the whole thing rotate a little? 719 00:44:03,520 --> 00:44:10,890 Could that rotate, could the whole thing rotate? 720 00:44:10,890 --> 00:44:12,510 Yeah, maybe it could. 721 00:44:12,510 --> 00:44:15,540 This guy would go up, maybe that's possible. 722 00:44:15,540 --> 00:44:20,370 That's somehow got everybody into the action. 723 00:44:20,370 --> 00:44:23,310 So I'll put "or rotate". 724 00:44:23,310 --> 00:44:27,440 OK, so you see what questions you get into. 725 00:44:27,440 --> 00:44:30,630 May I just draw a different truss? 726 00:44:30,630 --> 00:44:38,930 So those homework questions are other trusses. 727 00:44:38,930 --> 00:44:46,090 Here's one that I drew in the book itself. 728 00:44:46,090 --> 00:44:52,290 Yeah, may I draw this, I called it a treehouse. 729 00:44:52,290 --> 00:45:01,210 OK, so I have, so here's one that's actually in the book. 730 00:45:01,210 --> 00:45:07,680 And it's got a couple of bars going up, and one over. 731 00:45:07,680 --> 00:45:09,850 So that's the start. 732 00:45:09,850 --> 00:45:14,170 Then it's got a diagonal and that one. 733 00:45:14,170 --> 00:45:17,530 And then here comes the treehouse. 734 00:45:17,530 --> 00:45:23,000 OK, right. 735 00:45:23,000 --> 00:45:28,190 Well, just to get, let's again get the count right. 736 00:45:28,190 --> 00:45:34,220 So what's the matrix A for this treehouse? 737 00:45:34,220 --> 00:45:37,920 A is how many by how many? 738 00:45:37,920 --> 00:45:41,770 How many bars are you seeing here? 739 00:45:41,770 --> 00:45:46,820 One, two, three, four, five, six, seven, eight, eight bars. 740 00:45:46,820 --> 00:45:51,720 And how many unknown displacements? 741 00:45:51,720 --> 00:45:53,530 Ten. 742 00:45:53,530 --> 00:45:56,550 We got five joints that are not supported, 743 00:45:56,550 --> 00:45:59,010 and each one has two unknowns. 744 00:45:59,010 --> 00:46:00,770 So A is eight by ten. 745 00:46:00,770 --> 00:46:06,130 So I expect two mechanisms. 746 00:46:06,130 --> 00:46:15,300 OK, so again I'm looking for two mechanisms. 747 00:46:15,300 --> 00:46:19,245 OK, what's one? 748 00:46:19,245 --> 00:46:20,120 AUDIENCE: [INAUDIBLE] 749 00:46:20,120 --> 00:46:23,970 PROFESSOR STRANG: It's what? 750 00:46:23,970 --> 00:46:29,400 This guy just falls, right. 751 00:46:29,400 --> 00:46:32,450 This looks unfortunately very much like the treehouses 752 00:46:32,450 --> 00:46:36,630 that I built for my kids. 753 00:46:36,630 --> 00:46:44,540 Well, so linear algebra sentenced them to fall, right? 754 00:46:44,540 --> 00:46:47,370 OK, that's one. 755 00:46:47,370 --> 00:46:49,830 I probably propped it up with one more 756 00:46:49,830 --> 00:46:52,340 bar, but of course that wouldn't be enough, 757 00:46:52,340 --> 00:46:56,690 because it's got two mechanisms, so if I make it nine by ten 758 00:46:56,690 --> 00:46:58,530 I haven't saved the kids. 759 00:46:58,530 --> 00:47:06,780 OK, with eight by ten, what's the other mechanisms? 760 00:47:06,780 --> 00:47:09,770 The whole thing could turn the-- Nothing 761 00:47:09,770 --> 00:47:11,320 preventing turning here. 762 00:47:11,320 --> 00:47:13,340 They can't move but they could turn. 763 00:47:13,340 --> 00:47:19,760 So the whole thing could go over, right, the whole thing 764 00:47:19,760 --> 00:47:23,720 could just-- That would be a horizontal movement of all five 765 00:47:23,720 --> 00:47:24,220 nodes. 766 00:47:24,220 --> 00:47:27,730 The horizontal of all five nodes. 767 00:47:27,730 --> 00:47:31,970 And again, slightly downwards, but that's a second order 768 00:47:31,970 --> 00:47:34,220 effect. 769 00:47:34,220 --> 00:47:38,930 OK, so that's the second truss. 770 00:47:38,930 --> 00:47:40,140 OK. 771 00:47:40,140 --> 00:47:47,250 So this is really like practice for discrete problems, 772 00:47:47,250 --> 00:47:50,310 for the problems of plane elasticity. 773 00:47:50,310 --> 00:47:57,860 And the point is that there are two unknowns for each point. 774 00:47:57,860 --> 00:47:59,347 If we have differential equations. 775 00:47:59,347 --> 00:48:00,930 So the differential equations of plane 776 00:48:00,930 --> 00:48:04,190 elasticity are not really simple. 777 00:48:04,190 --> 00:48:05,790 They're not really simple. 778 00:48:05,790 --> 00:48:08,620 And 3-D elasticity even more. 779 00:48:08,620 --> 00:48:15,100 Because the points are physical points 780 00:48:15,100 --> 00:48:22,140 and they can move three ways, and it gets quite interesting. 781 00:48:22,140 --> 00:48:28,130 And those are the major problems of computational mechanics. 782 00:48:28,130 --> 00:48:34,540 OK, let's say, holiday time, and I'll see you next Wednesday 783 00:48:34,540 --> 00:48:36,820 for Chapter 3. 784 00:48:36,820 --> 00:48:39,953 Which moves to partial differential equations. 785 00:48:39,953 --> 00:48:40,453