1 00:00:00,000 --> 00:00:00,030 2 00:00:00,030 --> 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 --> 00:00:09,960 continue to offer high quality educational resources for free. 6 00:00:09,960 --> 00:00:12,590 To make a donation or to view additional materials 7 00:00:12,590 --> 00:00:15,210 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,210 --> 00:00:19,847 at ocw.mit.edu. 9 00:00:19,847 --> 00:00:20,930 PROFESSOR STRANG: Alright. 10 00:00:20,930 --> 00:00:25,290 So this is Lecture 15. 11 00:00:25,290 --> 00:00:30,530 It's the last topic, today and Friday, like just 15 and 16. 12 00:00:30,530 --> 00:00:33,630 Trusses within Chapter 2. 13 00:00:33,630 --> 00:00:38,480 The last topic we'll do for discrete systems. 14 00:00:38,480 --> 00:00:41,810 Then it's a lot of fun. 15 00:00:41,810 --> 00:00:48,620 I wanted to say a few words first about last night's exam. 16 00:00:48,620 --> 00:00:50,720 Several words first. 17 00:00:50,720 --> 00:00:52,920 Overall, I'm sure it's going to be fine. 18 00:00:52,920 --> 00:00:56,430 Ramis is grading the first two problems, 19 00:00:56,430 --> 00:00:58,590 he'll pass them to Peter for the next two. 20 00:00:58,590 --> 00:01:00,140 And I'll get them back. 21 00:01:00,140 --> 00:01:05,180 I'm pretty sure it'll be next week. 22 00:01:05,180 --> 00:01:10,490 I felt it was a fair exam, except I should 23 00:01:10,490 --> 00:01:18,220 have done a better job in helping you with the matrix A. 24 00:01:18,220 --> 00:01:20,700 Especially in problem one. 25 00:01:20,700 --> 00:01:27,040 I'm glad that hint was there, the matrix A_0, that 26 00:01:27,040 --> 00:01:34,150 goes with free-free, to sort of say what kind of matrix 27 00:01:34,150 --> 00:01:35,570 to be looking for. 28 00:01:35,570 --> 00:01:42,450 And I thought I'd just repeat, make the connections that I 29 00:01:42,450 --> 00:01:47,700 should have made earlier. 30 00:01:47,700 --> 00:01:53,800 So we all see the point about these. 31 00:01:53,800 --> 00:01:58,350 These A's and the A transpose A's. 32 00:01:58,350 --> 00:02:04,050 So if I take, for that A_0, free-free one. 33 00:02:04,050 --> 00:02:07,130 Everybody sees that this is-- And this connects of course 34 00:02:07,130 --> 00:02:08,650 with our graphs. 35 00:02:08,650 --> 00:02:12,650 Our graph is just this simple graph with well 36 00:02:12,650 --> 00:02:15,840 actually is that how many, is it five nodes? 37 00:02:15,840 --> 00:02:19,810 I guess there are five. 38 00:02:19,810 --> 00:02:23,760 Because as it stands I have one, two, three, four, five columns, 39 00:02:23,760 --> 00:02:32,180 I've got five u's, u_0 down to u_4. 40 00:02:32,180 --> 00:02:36,030 And if I take A_0 transpose A_0, that 41 00:02:36,030 --> 00:02:38,640 would be the free-free matrix. 42 00:02:38,640 --> 00:02:40,150 What size would it be? 43 00:02:40,150 --> 00:02:42,320 And what matrix would it be? 44 00:02:42,320 --> 00:02:46,580 Just if we do that multiplication, 45 00:02:46,580 --> 00:02:49,170 this is a first difference matrix. 46 00:02:49,170 --> 00:02:51,670 When I do A_0 transpose A_0 I'll get 47 00:02:51,670 --> 00:02:53,740 one of our second difference matrices. 48 00:02:53,740 --> 00:02:56,120 So it'll be one of our special ones. 49 00:02:56,120 --> 00:02:59,980 Which special one would it be? 50 00:02:59,980 --> 00:03:04,480 B. It'll be the matrix B that has both ends free. 51 00:03:04,480 --> 00:03:06,110 And what size will it be? 52 00:03:06,110 --> 00:03:14,230 I guess it'll be five by five. 53 00:03:14,230 --> 00:03:19,380 That's right; that would be five by four times A_0, which 54 00:03:19,380 --> 00:03:20,520 is four by five. 55 00:03:20,520 --> 00:03:26,380 So it'll be the five by five matrix B. Can I call it B_5? 56 00:03:26,380 --> 00:03:27,580 OK. 57 00:03:27,580 --> 00:03:31,020 So that was there as a hint. 58 00:03:31,020 --> 00:03:34,480 That isn't the correct matrix for problem one 59 00:03:34,480 --> 00:03:37,100 because problem one was fixed-fixed. 60 00:03:37,100 --> 00:03:38,930 Let's get there in two steps. 61 00:03:38,930 --> 00:03:41,240 Suppose it's fixed-free. 62 00:03:41,240 --> 00:03:46,600 So suppose I make u_0 equal 0. 63 00:03:46,600 --> 00:03:51,340 So I ground the top node, I support the top node-- 64 00:03:51,340 --> 00:03:53,110 Oh no, shall I do u_0? 65 00:03:53,110 --> 00:03:53,670 Yeah. 66 00:03:53,670 --> 00:03:54,790 I'll do u_0. 67 00:03:54,790 --> 00:03:57,700 So that would knock out this one. 68 00:03:57,700 --> 00:04:02,870 If I say fix u_0, say, at zero or whatever. 69 00:04:02,870 --> 00:04:09,910 Now I've got, the next A, I won't call it A_0 anymore. 70 00:04:09,910 --> 00:04:16,550 So now four by four, now if I do A transpose A, 71 00:04:16,550 --> 00:04:20,510 which of our special matrices am I going to get? 72 00:04:20,510 --> 00:04:26,170 T. It'll be T. It'll be the one that has the first, 73 00:04:26,170 --> 00:04:31,110 the (1, 1) entry will only be a one. 74 00:04:31,110 --> 00:04:34,490 So that'll be the fixed-free matrix. 75 00:04:34,490 --> 00:04:35,900 It'll be of what size? 76 00:04:35,900 --> 00:04:38,010 Four. 77 00:04:38,010 --> 00:04:42,380 Now I've only got four unknowns. u_1 to u_4. 78 00:04:42,380 --> 00:04:45,650 OK, that still is not what problem one is. 79 00:04:45,650 --> 00:04:47,910 Problem one was fixed-fixed. 80 00:04:47,910 --> 00:04:50,850 So as I did in the review, that would knock out 81 00:04:50,850 --> 00:04:52,840 both of these columns. 82 00:04:52,840 --> 00:04:59,850 So this now is the matrix that I was looking for in problem one, 83 00:04:59,850 --> 00:05:05,750 and I wish I had emphasized these steps in advance. 84 00:05:05,750 --> 00:05:07,130 I apologize. 85 00:05:07,130 --> 00:05:13,330 OK, so what fixed-fixed if, without the C part in it. 86 00:05:13,330 --> 00:05:18,120 Just focusing on the A, what fixed-fixed matrix would I get? 87 00:05:18,120 --> 00:05:20,330 Which one of our guys would it be? 88 00:05:20,330 --> 00:05:23,570 K of size three. 89 00:05:23,570 --> 00:05:29,880 And while we're at it, what would be the story, how would 90 00:05:29,880 --> 00:05:33,390 I get one of these circular ones, which 91 00:05:33,390 --> 00:05:36,640 is sort of on our special list. 92 00:05:36,640 --> 00:05:40,040 For Fourier it's the really special guy. 93 00:05:40,040 --> 00:05:47,750 So a circular one I'm going to connect u_4 back to u_0. 94 00:05:47,750 --> 00:05:50,190 So I'm going to put these guys back in. 95 00:05:50,190 --> 00:05:53,780 And what else would change, if u_4 was connected back to u_0, 96 00:05:53,780 --> 00:05:57,960 now I'm aiming for this circulant. 97 00:05:57,960 --> 00:06:01,450 What matrix A is going to give me the circulant? 98 00:06:01,450 --> 00:06:05,980 So again, these guys are the A transpose A's. 99 00:06:05,980 --> 00:06:10,500 This over here was the A, and over here is the A transpose A. 100 00:06:10,500 --> 00:06:13,800 And now I want to fix A, and then 101 00:06:13,800 --> 00:06:17,550 I want to see that A transpose A. So suppose 102 00:06:17,550 --> 00:06:19,310 I give you that graph, then. 103 00:06:19,310 --> 00:06:23,520 Oops, I should have, well. 104 00:06:23,520 --> 00:06:30,770 Just connect the whole guy. 105 00:06:30,770 --> 00:06:36,790 So fifth node coming back to the first. 106 00:06:36,790 --> 00:06:39,890 So that's my circle of nodes. 107 00:06:39,890 --> 00:06:42,140 That's a simple graph. 108 00:06:42,140 --> 00:06:49,240 What's the A_circulant now? 109 00:06:49,240 --> 00:06:54,500 So this would be the A for the circulant case. 110 00:06:54,500 --> 00:06:57,590 So it's got that back in. 111 00:06:57,590 --> 00:07:00,020 That shouldn't be erased, that shouldn't be erased. 112 00:07:00,020 --> 00:07:04,930 And what else has it got? 113 00:07:04,930 --> 00:07:09,610 If I ask you for the incidence matrix, now I'm in Section 2.4, 114 00:07:09,610 --> 00:07:12,530 like I've given you a graph, or you 115 00:07:12,530 --> 00:07:16,480 can think of masses and springs in a circle. 116 00:07:16,480 --> 00:07:19,660 So I've got five masses, five springs. 117 00:07:19,660 --> 00:07:25,140 What's my matrix A missing? 118 00:07:25,140 --> 00:07:29,070 It needs another row. 119 00:07:29,070 --> 00:07:32,690 We just put in another edge, it needs another row. 120 00:07:32,690 --> 00:07:39,350 That edge went from the last node back to the first node. 121 00:07:39,350 --> 00:07:45,510 So we've got a fifth row. 122 00:07:45,510 --> 00:07:48,090 So you see, now it really is circulant. 123 00:07:48,090 --> 00:07:53,430 I would call this one also a circulant matrix. 124 00:07:53,430 --> 00:07:58,750 The diagonals are constant. 125 00:07:58,750 --> 00:08:02,850 That's what I and MATLAB and everybody else 126 00:08:02,850 --> 00:08:06,620 would call a Toeplitz matrix, and the command toeplitz 127 00:08:06,620 --> 00:08:08,740 could create this. 128 00:08:08,740 --> 00:08:10,460 That diagonal is constant. 129 00:08:10,460 --> 00:08:12,620 That diagonal is constant. 130 00:08:12,620 --> 00:08:14,590 The other diagonals are constant. 131 00:08:14,590 --> 00:08:17,600 But more than that, what's additional 132 00:08:17,600 --> 00:08:20,080 here in the circulant, which is the thing that 133 00:08:20,080 --> 00:08:22,500 makes Fourier happy? 134 00:08:22,500 --> 00:08:26,930 The diagonal circles around. 135 00:08:26,930 --> 00:08:30,430 That diagonal has only got four entries in it, 136 00:08:30,430 --> 00:08:35,190 but it circles around sort of periodically 137 00:08:35,190 --> 00:08:38,030 to its fifth entry. 138 00:08:38,030 --> 00:08:40,210 So that's more than Toeplitz. 139 00:08:40,210 --> 00:08:43,820 It's circulant because it's coming around again. 140 00:08:43,820 --> 00:08:47,570 This we'll see in the discrete Fourier transform. 141 00:08:47,570 --> 00:08:50,270 It's really all good stuff. 142 00:08:50,270 --> 00:08:53,240 And now there is my a circulant. 143 00:08:53,240 --> 00:08:57,620 And what would be my A transpose circulant A_circulant? 144 00:08:57,620 --> 00:09:03,770 What would be A transpose A if I take that five by five matrix? 145 00:09:03,770 --> 00:09:09,790 C. Finally I've created, so I've already got B, I've got T, 146 00:09:09,790 --> 00:09:12,780 I've got K, all those three special guys. 147 00:09:12,780 --> 00:09:16,220 And now the A transpose A for this circulant, 148 00:09:16,220 --> 00:09:20,210 so that's a first difference matrix for a periodic problem. 149 00:09:20,210 --> 00:09:22,790 And A transpose A will be a second difference 150 00:09:22,790 --> 00:09:27,200 matrix for a periodic problem, C_5, I guess. 151 00:09:27,200 --> 00:09:29,305 It'll be five. 152 00:09:29,305 --> 00:09:29,805 OK. 153 00:09:29,805 --> 00:09:34,130 I hope that brings together what I, if I was on the ball, 154 00:09:34,130 --> 00:09:38,370 I would have brought it together before the quiz. 155 00:09:38,370 --> 00:09:45,030 Can I just say a few words about the quiz and grades? 156 00:09:45,030 --> 00:09:47,650 They come out fine. 157 00:09:47,650 --> 00:09:49,070 Really they do. 158 00:09:49,070 --> 00:09:51,130 I've been doing this a long time. 159 00:09:51,130 --> 00:09:59,050 And just, enjoy October. 160 00:09:59,050 --> 00:10:01,760 I'm sorry to give you any exam at all, 161 00:10:01,760 --> 00:10:08,230 but it's a chance for you yeah, I'm working on this stuff. 162 00:10:08,230 --> 00:10:09,590 I'm learning it. 163 00:10:09,590 --> 00:10:11,940 Everybody didn't learn it first time. 164 00:10:11,940 --> 00:10:13,820 I don't learn it first time, every time 165 00:10:13,820 --> 00:10:16,080 I teach the course I learn something more. 166 00:10:16,080 --> 00:10:21,400 And if you're learning from this course then I'm totally happy. 167 00:10:21,400 --> 00:10:23,430 And I believe that's the case. 168 00:10:23,430 --> 00:10:25,380 So I am entirely happy. 169 00:10:25,380 --> 00:10:32,120 And I hope the quiz, some points of it 170 00:10:32,120 --> 00:10:34,030 I wish I'd prepared better. 171 00:10:34,030 --> 00:10:39,740 But I feel pretty good about it. 172 00:10:39,740 --> 00:10:42,680 I feel good about it, let me just say. 173 00:10:42,680 --> 00:10:48,340 So, and I'm happy to have any comments, email or in person. 174 00:10:48,340 --> 00:10:50,740 But allow me to go forward with trusses. 175 00:10:50,740 --> 00:10:53,980 However, I'm ready always for a comment. 176 00:10:53,980 --> 00:10:54,740 Yeah. 177 00:10:54,740 --> 00:10:56,100 OK. 178 00:10:56,100 --> 00:10:58,680 Anyway, enjoy trusses. 179 00:10:58,680 --> 00:10:59,990 Enjoy life. 180 00:10:59,990 --> 00:11:02,150 Yeah. 181 00:11:02,150 --> 00:11:05,740 And this should have been in the book, this page. 182 00:11:05,740 --> 00:11:11,170 So if it wasn't too late I would paste it in. 183 00:11:11,170 --> 00:11:14,120 Because this connects A transpose A 184 00:11:14,120 --> 00:11:18,680 to special matrices, in the way I had in my mind, 185 00:11:18,680 --> 00:11:22,250 but I didn't put it on the board until just now. 186 00:11:22,250 --> 00:11:29,020 OK, so I'll cover that up and ready to go with trusses. 187 00:11:29,020 --> 00:11:31,180 OK. 188 00:11:31,180 --> 00:11:34,840 So trusses, we want to know what's up. 189 00:11:34,840 --> 00:11:36,500 We want to get the setup right. 190 00:11:36,500 --> 00:11:45,000 Once we get the setup we'll know we're looking for. 191 00:11:45,000 --> 00:11:52,100 OK, so a truss is a bunch of elastic bars with pin joints 192 00:11:52,100 --> 00:11:52,900 connecting them. 193 00:11:52,900 --> 00:11:55,110 Now, what do I mean by a pin joint? 194 00:11:55,110 --> 00:12:05,570 I mean that stretching the bars takes force. 195 00:12:05,570 --> 00:12:09,810 Turning around the pin joint doesn't take force. 196 00:12:09,810 --> 00:12:18,130 So the pin just lets them turn, so we'll 197 00:12:18,130 --> 00:12:20,740 have forces in those bars. 198 00:12:20,740 --> 00:12:24,210 So it's like masses and springs. 199 00:12:24,210 --> 00:12:26,300 Exactly like masses and springs. 200 00:12:26,300 --> 00:12:30,140 But yet we have a 2-D problem. 201 00:12:30,140 --> 00:12:31,800 So it's a two dimensional problem 202 00:12:31,800 --> 00:12:33,340 with masses and springs. 203 00:12:33,340 --> 00:12:35,550 And we could certainly have a 3-D truss, 204 00:12:35,550 --> 00:12:39,680 but 2-D makes all the important points. 205 00:12:39,680 --> 00:12:41,650 And then I can count the bars. 206 00:12:41,650 --> 00:12:44,360 One, two, three, four, five. 207 00:12:44,360 --> 00:12:46,800 And I can count the nodes. 208 00:12:46,800 --> 00:12:48,720 There happen to be five here. 209 00:12:48,720 --> 00:12:50,710 But now comes the moment. 210 00:12:50,710 --> 00:12:53,370 I have to tell you, what are the unknowns? 211 00:12:53,370 --> 00:12:54,380 What are the u's. 212 00:12:54,380 --> 00:12:56,090 Because of course, you know that I'm 213 00:12:56,090 --> 00:13:05,220 going to go from u's to e's to w's, to forces, f. 214 00:13:05,220 --> 00:13:08,780 And you know that a matrix A is going to do that. 215 00:13:08,780 --> 00:13:11,080 A matrix C is going to do that. 216 00:13:11,080 --> 00:13:13,650 A matrix A transpose is going to do that. 217 00:13:13,650 --> 00:13:17,090 You're all ready, we need to know what's the setup. 218 00:13:17,090 --> 00:13:18,890 What are these matrices. 219 00:13:18,890 --> 00:13:22,490 OK, and how many-- So let me explain the setup. 220 00:13:22,490 --> 00:13:31,630 Typical node, node one. we have forces in these bars, 221 00:13:31,630 --> 00:13:38,250 so that node one could have a force. 222 00:13:38,250 --> 00:13:39,960 We're in the plane. 223 00:13:39,960 --> 00:13:44,200 So we have a horizontal force and a vertical force. 224 00:13:44,200 --> 00:13:48,060 Together, that would give, produces a force 225 00:13:48,060 --> 00:13:50,460 in any direction whatever. 226 00:13:50,460 --> 00:13:51,880 So this is the key point. 227 00:13:51,880 --> 00:13:59,340 That there is a horizontal force. f horizontal one. 228 00:13:59,340 --> 00:14:01,440 The one being the force on node one. 229 00:14:01,440 --> 00:14:04,700 But there's also a vertical force. 230 00:14:04,700 --> 00:14:07,940 And let me take horizontal to the right, 231 00:14:07,940 --> 00:14:09,510 positive to the right. 232 00:14:09,510 --> 00:14:11,320 Vertical positive upwards. 233 00:14:11,320 --> 00:14:13,550 Just to have a convention. 234 00:14:13,550 --> 00:14:17,180 So how many f's have I got? 235 00:14:17,180 --> 00:14:21,140 Well, the point is I now have two per node. 236 00:14:21,140 --> 00:14:22,610 That's the difference. 237 00:14:22,610 --> 00:14:24,960 I have two per node, two forces. 238 00:14:24,960 --> 00:14:27,480 And I have two displacements per node. 239 00:14:27,480 --> 00:14:32,410 Because that point under, there will be more forces. 240 00:14:32,410 --> 00:14:38,020 Some maybe pulling this way, whatever. 241 00:14:38,020 --> 00:14:40,030 Maybe let's look at node two. 242 00:14:40,030 --> 00:14:42,710 So node two could have a couple of forces on it. 243 00:14:42,710 --> 00:14:45,420 f h two, and f v two. 244 00:14:45,420 --> 00:14:49,000 And it moves like the other nodes. 245 00:14:49,000 --> 00:14:53,140 So now I'm introducing the unknowns. u 246 00:14:53,140 --> 00:14:57,680 is the movement. u, again horizontal, 247 00:14:57,680 --> 00:15:01,240 and again, now we're talking about node two. 248 00:15:01,240 --> 00:15:04,080 And it moves up. 249 00:15:04,080 --> 00:15:05,870 Or doesn't, or moves down. 250 00:15:05,870 --> 00:15:09,470 But that's an unknown. u, a displacement, 251 00:15:09,470 --> 00:15:12,130 a vertical displacement of node two. 252 00:15:12,130 --> 00:15:14,330 Do you see the setup? 253 00:15:14,330 --> 00:15:17,350 Two forces per node. 254 00:15:17,350 --> 00:15:20,210 Two displacements per node. 255 00:15:20,210 --> 00:15:27,650 So that's like, the number of unknown is like doubled. 256 00:15:27,650 --> 00:15:33,220 Like, doubled, and that produces an interesting situation. 257 00:15:33,220 --> 00:15:36,180 I've marked supports here. 258 00:15:36,180 --> 00:15:38,860 So let's just speak about supports. 259 00:15:38,860 --> 00:15:41,310 So what's happening at the supports? 260 00:15:41,310 --> 00:15:49,110 At the support there's no movement. 261 00:15:49,110 --> 00:15:51,940 The whole-- That point is pinned. 262 00:15:51,940 --> 00:15:57,120 So this is telling me that u horizontal five 263 00:15:57,120 --> 00:16:02,920 is zero and u vertical, sorry that was four 264 00:16:02,920 --> 00:16:06,830 and it'll be the same for five. u vertical five is zero. 265 00:16:06,830 --> 00:16:10,760 It's like grounding a node in the electrical case. 266 00:16:10,760 --> 00:16:13,580 We just see this pattern over and over. 267 00:16:13,580 --> 00:16:17,090 And we want to see OK, what does it look like for trusses? 268 00:16:17,090 --> 00:16:20,140 So here's a support that fixes those. 269 00:16:20,140 --> 00:16:22,090 So those are not unknowns. 270 00:16:22,090 --> 00:16:24,480 And similarly, they're not unknowns there. 271 00:16:24,480 --> 00:16:27,420 Still saying five when I mean four. 272 00:16:27,420 --> 00:16:34,300 So those are boundary conditions, n conditions, 273 00:16:34,300 --> 00:16:34,870 whatever. 274 00:16:34,870 --> 00:16:36,330 And similarly here. 275 00:16:36,330 --> 00:16:39,500 So how many unknowns are there? 276 00:16:39,500 --> 00:16:45,120 Now look at this picture, how many unknown displacements 277 00:16:45,120 --> 00:16:48,830 are there in this truss? 278 00:16:48,830 --> 00:16:49,380 Six. 279 00:16:49,380 --> 00:16:50,320 Six, right? 280 00:16:50,320 --> 00:16:53,180 Two here, two here, two here and none there. 281 00:16:53,180 --> 00:17:00,520 So the number of actual unknowns is six. 282 00:17:00,520 --> 00:17:06,340 My idea would be that it's twice the number of nodes 283 00:17:06,340 --> 00:17:13,980 minus the number of fixed things, reaction, whatever. 284 00:17:13,980 --> 00:17:16,250 That R would be four in this case. 285 00:17:16,250 --> 00:17:20,490 I've got two fixed here and two fixed here, 286 00:17:20,490 --> 00:17:23,190 so this would be two times five. 287 00:17:23,190 --> 00:17:26,980 Ten possible displacements but R counts 288 00:17:26,980 --> 00:17:29,580 the number of fixed displacements, four, 289 00:17:29,580 --> 00:17:32,580 and leaves us with six. 290 00:17:32,580 --> 00:17:34,320 OK. 291 00:17:34,320 --> 00:17:38,590 So my matrix A will now be, it's always m by n. 292 00:17:38,590 --> 00:17:42,880 My matrix A will be five by six. 293 00:17:42,880 --> 00:17:46,060 OK. 294 00:17:46,060 --> 00:17:49,340 Now you're going to ask what is that matrix. 295 00:17:49,340 --> 00:17:53,940 But let me hold that off for a little moment. 296 00:17:53,940 --> 00:17:56,910 I want to just see its shape first. 297 00:17:56,910 --> 00:18:02,160 So you could now do this for a large truss, right? 298 00:18:02,160 --> 00:18:06,500 You count the bars, and you could count the nodes. 299 00:18:06,500 --> 00:18:09,990 And then you could count the unknown displacements, u. 300 00:18:09,990 --> 00:18:17,190 So there are six u's here. 301 00:18:17,190 --> 00:18:19,460 And there are five e's. 302 00:18:19,460 --> 00:18:22,530 And there are five bar forces. 303 00:18:22,530 --> 00:18:29,700 And there are six equilibrium, balance, force balances. 304 00:18:29,700 --> 00:18:33,920 Six, six for the node count, for the unknowns count, 305 00:18:33,920 --> 00:18:36,480 five, five for the bars count. 306 00:18:36,480 --> 00:18:44,520 OK, now here's a point about this particular truss. 307 00:18:44,520 --> 00:18:46,440 It's not safe to get on it. 308 00:18:46,440 --> 00:18:47,570 Right? 309 00:18:47,570 --> 00:18:49,650 And I want to say why is it not safe. 310 00:18:49,650 --> 00:18:54,860 So this is a feature that comes into the truss question that 311 00:18:54,860 --> 00:18:58,040 makes it a little new and more interesting. 312 00:18:58,040 --> 00:19:03,850 A little twist compared to the previous examples. 313 00:19:03,850 --> 00:19:07,450 That bar, that truss, I wouldn't stand on it. 314 00:19:07,450 --> 00:19:09,150 Now, why not? 315 00:19:09,150 --> 00:19:11,980 Well, purely for linear algebra reasons. 316 00:19:11,980 --> 00:19:13,230 Of course. 317 00:19:13,230 --> 00:19:15,650 The matrix A is five by six. 318 00:19:15,650 --> 00:19:18,840 So now what do we know about a matrix that's five by six? 319 00:19:18,840 --> 00:19:23,790 So A is five by six. 320 00:19:23,790 --> 00:19:28,420 Five rows, as always the m; six columns, because now we 321 00:19:28,420 --> 00:19:29,850 have six unknowns. 322 00:19:29,850 --> 00:19:36,240 And what do I know about any five by six matrix? 323 00:19:36,240 --> 00:19:39,400 I want to ask about the equation Au=0. 324 00:19:39,400 --> 00:19:42,320 325 00:19:42,320 --> 00:19:44,810 So I want to ask about it in linear algebra language 326 00:19:44,810 --> 00:19:47,700 and then I want to ask about it in physical language. 327 00:19:47,700 --> 00:19:52,610 And the beauty is the thing that makes trusses sort of fun 328 00:19:52,610 --> 00:19:57,070 is, these matrices, A, get pretty big fast. 329 00:19:57,070 --> 00:19:59,590 Because when I put a few more nodes on, 330 00:19:59,590 --> 00:20:04,460 the book has a picture of a sort of treehouse. 331 00:20:04,460 --> 00:20:08,220 Then A is growing. 332 00:20:08,220 --> 00:20:12,030 And I don't, all the time, write down the matrix A. 333 00:20:12,030 --> 00:20:13,950 I haven't written it down here. 334 00:20:13,950 --> 00:20:16,330 What I've written down is just its size, 335 00:20:16,330 --> 00:20:21,540 because that's enough to tell us something about this set 336 00:20:21,540 --> 00:20:24,290 of equations Au=0. 337 00:20:24,290 --> 00:20:27,200 What's the story on Au=0? 338 00:20:27,200 --> 00:20:29,900 Well of course it has the solution u=0. 339 00:20:29,900 --> 00:20:31,940 Nothing moving. 340 00:20:31,940 --> 00:20:35,710 If I have no displacement, if the u's are all zero, 341 00:20:35,710 --> 00:20:37,840 then I have no stretching. 342 00:20:37,840 --> 00:20:39,290 The e's are stretching. 343 00:20:39,290 --> 00:20:41,890 Elongation, as before. 344 00:20:41,890 --> 00:20:44,250 How far does the bar stretch? 345 00:20:44,250 --> 00:20:44,940 OK. 346 00:20:44,940 --> 00:20:48,760 So if I have zero u's and zero e's. 347 00:20:48,760 --> 00:20:55,700 But, what other possibility am I going to have here? 348 00:20:55,700 --> 00:21:00,520 I'm going to have probably one solution 349 00:21:00,520 --> 00:21:03,800 to this system that isn't zero. 350 00:21:03,800 --> 00:21:08,910 I'm probably going to have one set of displacements u, look 351 00:21:08,910 --> 00:21:10,140 what's happening here. 352 00:21:10,140 --> 00:21:13,750 This is Au is the e. 353 00:21:13,750 --> 00:21:18,900 So I'm going to have at least one and probably in this case 354 00:21:18,900 --> 00:21:26,530 it will be one, there will be one, the neat word for it 355 00:21:26,530 --> 00:21:30,930 is a mechanism. 356 00:21:30,930 --> 00:21:32,850 And what does that mean? 357 00:21:32,850 --> 00:21:37,000 A mechanism is a solution to Au=0. 358 00:21:37,000 --> 00:21:40,370 So that, a mechanism is a movement of the bar. 359 00:21:40,370 --> 00:21:42,410 So it's going to be non-zero. 360 00:21:42,410 --> 00:21:45,080 The bars are going to move a little. 361 00:21:45,080 --> 00:21:47,060 Sorry, the nodes are going to move a little. 362 00:21:47,060 --> 00:21:50,180 The nodes will move a little bit. 363 00:21:50,180 --> 00:21:52,660 In this u, because it isn't zero. 364 00:21:52,660 --> 00:21:56,420 But the bars won't stretch. 365 00:21:56,420 --> 00:22:04,060 So that tells us we've got instability here. 366 00:22:04,060 --> 00:22:05,880 If there's a solution to that, that's 367 00:22:05,880 --> 00:22:09,020 always telling us that A transpose A is singular. 368 00:22:09,020 --> 00:22:13,020 So let me just put that A transpose A -- 369 00:22:13,020 --> 00:22:18,950 or A transpose C A, C couldn't save it -- will be singular. 370 00:22:18,950 --> 00:22:24,910 It's just like our free-free thing in being singular, 371 00:22:24,910 --> 00:22:28,450 but the picture doesn't look free-free, does it? 372 00:22:28,450 --> 00:22:33,000 It's got supports in here, just not good enough. 373 00:22:33,000 --> 00:22:38,190 And I believe that if you look at this truss, 374 00:22:38,190 --> 00:22:40,430 you could describe, you could tell me, 375 00:22:40,430 --> 00:22:47,620 and you could draw, a movement of that truss in which 376 00:22:47,620 --> 00:22:51,040 there is displacement but no stretching. 377 00:22:51,040 --> 00:22:54,560 Let me ask you how to draw that. 378 00:22:54,560 --> 00:22:57,410 And I believe-- Everybody understood, 379 00:22:57,410 --> 00:22:59,240 why was there a solution? 380 00:22:59,240 --> 00:23:03,070 It was because we have six unknowns 381 00:23:03,070 --> 00:23:05,320 and we only have five equations. 382 00:23:05,320 --> 00:23:07,720 So this was five equations. 383 00:23:07,720 --> 00:23:10,640 Any time you have five equations with a zero 384 00:23:10,640 --> 00:23:14,540 on the right-hand side, so five homogeneous 385 00:23:14,540 --> 00:23:17,315 equations, whatever you want to say when that's zero 386 00:23:17,315 --> 00:23:20,190 on the right and six unknowns. 387 00:23:20,190 --> 00:23:22,160 Six u's. 388 00:23:22,160 --> 00:23:25,340 Then you're going to have solution. 389 00:23:25,340 --> 00:23:27,460 You can't help it. 390 00:23:27,460 --> 00:23:29,990 You've got that many degrees of freedom, 391 00:23:29,990 --> 00:23:31,940 you've only got that many constraints, there's 392 00:23:31,940 --> 00:23:33,220 going to be solution. 393 00:23:33,220 --> 00:23:37,540 OK, tell me how to draw that. 394 00:23:37,540 --> 00:23:41,570 Let me put in the truss now. 395 00:23:41,570 --> 00:23:43,940 What's the solution? 396 00:23:43,940 --> 00:23:49,570 So this is the fun part in a particular example 397 00:23:49,570 --> 00:23:52,140 at the start. 398 00:23:52,140 --> 00:24:04,790 How could that move without stretching bars? 399 00:24:04,790 --> 00:24:08,340 Let me see. 400 00:24:08,340 --> 00:24:11,690 What could happen? 401 00:24:11,690 --> 00:24:16,400 What do you mean now, who's going to move where? 402 00:24:16,400 --> 00:24:18,280 What's the movement here? 403 00:24:18,280 --> 00:24:19,830 And I want to draw it over there. 404 00:24:19,830 --> 00:24:23,480 So you give the answer by drawing it as well 405 00:24:23,480 --> 00:24:27,160 as by telling me the six unknown u's. 406 00:24:27,160 --> 00:24:33,300 So what can happen at this thing? 407 00:24:33,300 --> 00:24:40,200 So you're going to say the truss could, these bars could, 408 00:24:40,200 --> 00:24:42,630 turn a little? 409 00:24:42,630 --> 00:24:44,750 And notice that word a little. 410 00:24:44,750 --> 00:24:47,050 We're talking small displacement, 411 00:24:47,050 --> 00:24:49,980 small stretches all the time here. 412 00:24:49,980 --> 00:24:53,240 I'll show you why we're always making 413 00:24:53,240 --> 00:24:58,430 that linearity assumption, or small assumption. 414 00:24:58,430 --> 00:25:00,290 OK, those move a little. 415 00:25:00,290 --> 00:25:05,090 And what happens to that triangle at the top? 416 00:25:05,090 --> 00:25:07,630 It sort of just moves along, right? 417 00:25:07,630 --> 00:25:10,190 So the picture you would draw would 418 00:25:10,190 --> 00:25:16,210 be that you started there. 419 00:25:16,210 --> 00:25:18,820 And it moved along a little, I'll 420 00:25:18,820 --> 00:25:24,030 make it a larger displacement than I really have in mind. 421 00:25:24,030 --> 00:25:26,380 These guys of course are here. 422 00:25:26,380 --> 00:25:31,180 So they come out and the rest of the truss, 423 00:25:31,180 --> 00:25:34,470 the top of the truss, just kind of goes with it. 424 00:25:34,470 --> 00:25:36,720 Goes with the flow. 425 00:25:36,720 --> 00:25:40,970 That would be the answer that I would be looking for, 426 00:25:40,970 --> 00:25:42,720 to draw the mechanism. 427 00:25:42,720 --> 00:25:44,230 That would show it. 428 00:25:44,230 --> 00:25:46,970 And if I wanted to write down the u that 429 00:25:46,970 --> 00:25:50,450 goes with it, what would it be? 430 00:25:50,450 --> 00:25:53,230 Let me again number these guys; this is one, two, 431 00:25:53,230 --> 00:25:54,780 and this is three. 432 00:25:54,780 --> 00:26:02,880 So what are the displacements of nodes one, two, and three? 433 00:26:02,880 --> 00:26:07,940 I'll always write u_1 horizontal before vertical. 434 00:26:07,940 --> 00:26:09,620 Can we make an agreement? 435 00:26:09,620 --> 00:26:12,440 So I want to know about the horizontal movement 436 00:26:12,440 --> 00:26:14,780 then the vertical movement of node one, 437 00:26:14,780 --> 00:26:16,860 then node two, then node three. 438 00:26:16,860 --> 00:26:19,080 So I'll have six numbers there. 439 00:26:19,080 --> 00:26:22,710 And what could I put in for those six numbers? 440 00:26:22,710 --> 00:26:27,120 So the horizontal, let me suppose that that first guy, 441 00:26:27,120 --> 00:26:30,140 I'll put a one. 442 00:26:30,140 --> 00:26:32,370 Really that's a bigger number than I should put, 443 00:26:32,370 --> 00:26:33,980 but it's a convenient number. 444 00:26:33,980 --> 00:26:36,860 So I'll just take it to be one even though I really 445 00:26:36,860 --> 00:26:41,500 have in mind-- Let's say that's one angstrom, or one 446 00:26:41,500 --> 00:26:43,320 tiny little person. 447 00:26:43,320 --> 00:26:51,820 OK, so what about the rest? 448 00:26:51,820 --> 00:26:58,630 What's the vertical-- Oh yeah, this is a key point here, 449 00:26:58,630 --> 00:27:01,600 what's the vertical movement? 450 00:27:01,600 --> 00:27:05,110 This movement to me is horizontal. 451 00:27:05,110 --> 00:27:09,910 I'm going to say that the vertical moment is zero. 452 00:27:09,910 --> 00:27:11,830 Of node one, just moves over. 453 00:27:11,830 --> 00:27:13,780 And node two does the same. 454 00:27:13,780 --> 00:27:15,440 And node three does the same. 455 00:27:15,440 --> 00:27:19,840 So that's my solution. [1, 0, 1, 0, 1, 0]. 456 00:27:19,840 --> 00:27:25,370 That's a simple movement, a simple set of displacements, 457 00:27:25,370 --> 00:27:30,820 think most to the right. 458 00:27:30,820 --> 00:27:33,340 I have not written down the matrix A, 459 00:27:33,340 --> 00:27:37,610 but probably won't even do it until next time. 460 00:27:37,610 --> 00:27:41,320 But you will see that when we do the matrix A 461 00:27:41,320 --> 00:27:46,490 for this particular truss will have this particular u 462 00:27:46,490 --> 00:27:49,210 as a mechanism. 463 00:27:49,210 --> 00:27:56,130 In linear algebra, u is in the null space of A. Au=0, 464 00:27:56,130 --> 00:27:57,410 that's all that means. 465 00:27:57,410 --> 00:28:04,260 OK, do you see more or less what's up? 466 00:28:04,260 --> 00:28:09,610 But now there's one little thing that may be bothering you. 467 00:28:09,610 --> 00:28:13,070 Which is what? 468 00:28:13,070 --> 00:28:18,650 If I come back to the zero, zero, zero there, 469 00:28:18,650 --> 00:28:23,520 you could correctly say wait a minute, 470 00:28:23,520 --> 00:28:25,590 if those bars didn't stretch, if they just 471 00:28:25,590 --> 00:28:29,170 rotated as you told me to do, then 472 00:28:29,170 --> 00:28:35,130 this was mostly across but a little bit down, right? 473 00:28:35,130 --> 00:28:36,570 And I'm saying no. 474 00:28:36,570 --> 00:28:38,120 I'm saying zero. 475 00:28:38,120 --> 00:28:42,930 OK, how do I get away with that? 476 00:28:42,930 --> 00:28:47,520 So I'm saying in 18.085 it's a zero. 477 00:28:47,520 --> 00:28:49,480 And why? 478 00:28:49,480 --> 00:28:55,080 So this is like a little time-out just to focus in on, 479 00:28:55,080 --> 00:28:57,490 let me focus in on node two. 480 00:28:57,490 --> 00:29:02,420 So here's the bottom node four, so it 481 00:29:02,420 --> 00:29:04,060 used to be vertical up to two. 482 00:29:04,060 --> 00:29:06,920 This was node two and this was number four. 483 00:29:06,920 --> 00:29:11,620 And then it rotated a little, to there. 484 00:29:11,620 --> 00:29:13,690 To this position. 485 00:29:13,690 --> 00:29:18,370 So it went, if this angle was, let's say, 486 00:29:18,370 --> 00:29:26,560 theta, then what is that actual position? 487 00:29:26,560 --> 00:29:30,760 So let's say this was, let's say the bar had length one. 488 00:29:30,760 --> 00:29:33,450 This is the origin, (0, 0). 489 00:29:33,450 --> 00:29:37,760 This is the point (0, 1) above it, OK? 490 00:29:37,760 --> 00:29:41,280 And now, that's before it moved. 491 00:29:41,280 --> 00:29:42,780 Then it moved a little bit. 492 00:29:42,780 --> 00:29:45,470 It moved to an angle theta. 493 00:29:45,470 --> 00:29:48,090 What's the position of that bar? 494 00:29:48,090 --> 00:29:48,890 Of that node? 495 00:29:48,890 --> 00:29:51,030 What's the new position of the node, 496 00:29:51,030 --> 00:29:52,700 and then we'll look at the difference 497 00:29:52,700 --> 00:29:56,760 and we'll see the movement u, the displacement. 498 00:29:56,760 --> 00:30:06,820 So how far did it move? 499 00:30:06,820 --> 00:30:08,390 What's the x-coordinate? 500 00:30:08,390 --> 00:30:10,400 How far did it go across? 501 00:30:10,400 --> 00:30:13,470 If I put in that line you'll know. 502 00:30:13,470 --> 00:30:16,760 So the movement across was, sin(theta)? 503 00:30:16,760 --> 00:30:20,220 Good. sin(theta). 504 00:30:20,220 --> 00:30:28,630 And the movement down, well, yes, so let's find its position 505 00:30:28,630 --> 00:30:30,550 and then we'll take the difference. 506 00:30:30,550 --> 00:30:41,530 So what's the vertical new position for that guy? 507 00:30:41,530 --> 00:30:47,470 It moved by, it moved across by sin(theta). 508 00:30:47,470 --> 00:30:50,470 509 00:30:50,470 --> 00:30:56,810 And down by 1-cos(theta). 510 00:30:56,810 --> 00:30:58,940 Are you agreed with that? 511 00:30:58,940 --> 00:31:04,680 Because here is cosine theta, right there. 512 00:31:04,680 --> 00:31:08,150 And here's the little bit it moved down. 513 00:31:08,150 --> 00:31:09,490 OK. 514 00:31:09,490 --> 00:31:12,410 So these are exactly correct. 515 00:31:12,410 --> 00:31:16,550 Yeah this is in the position of sin(theta), cos(theta), 516 00:31:16,550 --> 00:31:18,540 and the difference was the 1-cos(theta). 517 00:31:18,540 --> 00:31:21,150 518 00:31:21,150 --> 00:31:23,970 OK, so now here comes the key point. 519 00:31:23,970 --> 00:31:27,590 Approximately, sin(theta) is approximately, 520 00:31:27,590 --> 00:31:31,680 if theta is small and now here comes the smallness, sin(theta) 521 00:31:31,680 --> 00:31:35,480 is approximately theta. sin(theta)'s 522 00:31:35,480 --> 00:31:36,840 approximately theta. 523 00:31:36,840 --> 00:31:40,440 And 1-cos(theta) is approximately what? 524 00:31:40,440 --> 00:31:48,180 Now, this is the important point. 525 00:31:48,180 --> 00:31:51,690 So theta is like the first term. 526 00:31:51,690 --> 00:31:54,450 If I expand, I mean, the exact term 527 00:31:54,450 --> 00:31:59,380 would be theta minus theta cubed over six, dot dot dot. 528 00:31:59,380 --> 00:32:03,490 But I'm only keeping that term. 529 00:32:03,490 --> 00:32:08,740 And 1-cos(theta), now what's the formula for cos(theta)? 530 00:32:08,740 --> 00:32:10,350 This is like worth, just should-- 531 00:32:10,350 --> 00:32:15,070 It's a one, because of course cos(0) is one, 532 00:32:15,070 --> 00:32:17,570 and then you subtract what? 533 00:32:17,570 --> 00:32:19,420 Theta squared over two. 534 00:32:19,420 --> 00:32:21,710 And so on. 535 00:32:21,710 --> 00:32:25,540 And then plus theta fourth over 24 or whatever. 536 00:32:25,540 --> 00:32:29,640 OK, so the ones cancel as we expect. 537 00:32:29,640 --> 00:32:35,580 And I'm getting theta squared over two. 538 00:32:35,580 --> 00:32:40,100 And this, here was theta to the first power. 539 00:32:40,100 --> 00:32:42,810 Theta cubed was, we didn't care. 540 00:32:42,810 --> 00:32:45,440 And we don't care about theta squared. 541 00:32:45,440 --> 00:32:48,870 So that's why it's zero. 542 00:32:48,870 --> 00:32:53,050 Because it's a second order movement. 543 00:32:53,050 --> 00:32:57,330 If theta is small, as I'm going to assume, small displacement, 544 00:32:57,330 --> 00:33:01,690 theta squared would, if I allowed theta squared 545 00:33:01,690 --> 00:33:05,910 and cos(theta) in here, I'd have a non-linear problem. 546 00:33:05,910 --> 00:33:07,120 And I don't want that. 547 00:33:07,120 --> 00:33:08,630 And I don't need it. 548 00:33:08,630 --> 00:33:14,740 I mean, finite elements, structures, bridges, whatever. 549 00:33:14,740 --> 00:33:18,300 Your first hope and expectation and calculation 550 00:33:18,300 --> 00:33:22,030 is small theta, linear problem. 551 00:33:22,030 --> 00:33:27,990 So to a linear person theta squared is zero. 552 00:33:27,990 --> 00:33:31,490 That's why those guys are zero. 553 00:33:31,490 --> 00:33:36,890 OK, so that's an assumption we'll often see, 554 00:33:36,890 --> 00:33:41,990 so it kind of was. 555 00:33:41,990 --> 00:33:45,750 There are two kinds of non-linearities 556 00:33:45,750 --> 00:33:49,650 in structures and elasticity. 557 00:33:49,650 --> 00:33:54,980 One would be to allow this geometric non-linearity. 558 00:33:54,980 --> 00:33:57,670 Thetas, large displacements, theta large 559 00:33:57,670 --> 00:34:01,200 enough so that you can't neglect theta squared. 560 00:34:01,200 --> 00:34:02,330 That's a tough one. 561 00:34:02,330 --> 00:34:05,960 If you allow geometric non-linearity in, 562 00:34:05,960 --> 00:34:08,320 as finite element codes have to do. 563 00:34:08,320 --> 00:34:14,660 If you-- ABAQUS is a code that does major finite element 564 00:34:14,660 --> 00:34:17,420 calculations, nonlinear ones. 565 00:34:17,420 --> 00:34:20,060 I mean they, at the beginning they were studying 566 00:34:20,060 --> 00:34:22,410 what happens, what are the stresses on cables 567 00:34:22,410 --> 00:34:23,800 under the Atlantic. 568 00:34:23,800 --> 00:34:26,910 I mean, those are fascinating problems. 569 00:34:26,910 --> 00:34:29,690 Or I mention car crashes. 570 00:34:29,690 --> 00:34:32,170 I mean, car crashes, the geometry changes, 571 00:34:32,170 --> 00:34:33,950 you have big displacements. 572 00:34:33,950 --> 00:34:38,340 But we're talking here about linear small displacement 573 00:34:38,340 --> 00:34:39,060 cases. 574 00:34:39,060 --> 00:34:44,200 OK, so don't forget that part. 575 00:34:44,200 --> 00:34:49,120 That when the truss gets more complicated, 576 00:34:49,120 --> 00:34:51,090 the principle stays the same. 577 00:34:51,090 --> 00:34:55,070 That we distinguish between the thetas that matter 578 00:34:55,070 --> 00:34:57,220 and the theta squareds that don't. 579 00:34:57,220 --> 00:35:01,220 OK, so now what? 580 00:35:01,220 --> 00:35:04,960 Now I guess I'm ready to complete 581 00:35:04,960 --> 00:35:06,661 this picture a little more. 582 00:35:06,661 --> 00:35:07,160 OK. 583 00:35:07,160 --> 00:35:11,740 So let me, so we've understood what's the idea of a mechanism. 584 00:35:11,740 --> 00:35:17,220 Oh, how could I prevent a mechanism? 585 00:35:17,220 --> 00:35:22,450 In other words, if I stood on this truss, 586 00:35:22,450 --> 00:35:29,410 the slightest bit of wind would crash it down, right? 587 00:35:29,410 --> 00:35:32,150 So that's unstable. 588 00:35:32,150 --> 00:35:34,780 That's an unstable truss. 589 00:35:34,780 --> 00:35:36,310 How could I make it stable? 590 00:35:36,310 --> 00:35:43,630 I mean if you were designing this thing, what would you do? 591 00:35:43,630 --> 00:35:44,790 Add another edge. 592 00:35:44,790 --> 00:35:46,370 You'd stick in another bar. 593 00:35:46,370 --> 00:35:49,370 Maybe stick in a bar there. 594 00:35:49,370 --> 00:35:51,650 What would happen now? 595 00:35:51,650 --> 00:35:53,392 Would it now be stable? 596 00:35:53,392 --> 00:35:54,850 You'd have to answer that question. 597 00:35:54,850 --> 00:35:57,580 You couldn't just put in bars, whatever. 598 00:35:57,580 --> 00:36:01,120 You want to put bars that do the job. 599 00:36:01,120 --> 00:36:02,560 OK, now how many bars? 600 00:36:02,560 --> 00:36:04,940 We've now got six bars. 601 00:36:04,940 --> 00:36:07,500 So m is now up to six. 602 00:36:07,500 --> 00:36:12,720 The matrix A is six by six now, whatever that matrix may be. 603 00:36:12,720 --> 00:36:15,940 We have six bars, six displacements. 604 00:36:15,940 --> 00:36:19,280 We can hope that we now have a six by six, well, 605 00:36:19,280 --> 00:36:23,220 we do have a six by six matrix, whatever it looks like. 606 00:36:23,220 --> 00:36:27,300 And we can hope that it's not singular. 607 00:36:27,300 --> 00:36:29,180 We can hope it's invertible, we can hope 608 00:36:29,180 --> 00:36:30,820 that that mechanism is killed. 609 00:36:30,820 --> 00:36:33,040 And you see it is killed. 610 00:36:33,040 --> 00:36:36,410 The six by six, that truss is now stable. 611 00:36:36,410 --> 00:36:42,350 No mechanism there, right? 612 00:36:42,350 --> 00:36:44,780 Again I haven't written down the matrix, 613 00:36:44,780 --> 00:36:49,920 but I'm really calling for engineering intuition here. 614 00:36:49,920 --> 00:36:53,630 That this truss is now stable, and of course I 615 00:36:53,630 --> 00:36:57,120 can make it even more stable by adding a seventh edge. 616 00:36:57,120 --> 00:36:59,690 A seventh bar. 617 00:36:59,690 --> 00:37:03,960 So when it was six I had square matrices, A transpose and then 618 00:37:03,960 --> 00:37:06,250 C and then A would have been square. 619 00:37:06,250 --> 00:37:12,630 Now I've got seven bars and, so now I've put in a seventh guy. 620 00:37:12,630 --> 00:37:15,400 m is now up to seven. 621 00:37:15,400 --> 00:37:24,010 My matrix a would now be seven by six. 622 00:37:24,010 --> 00:37:27,150 Mechanism will be gone because I've 623 00:37:27,150 --> 00:37:32,270 now got, what, seven equations. 624 00:37:32,270 --> 00:37:33,990 Same six u's. 625 00:37:33,990 --> 00:37:39,350 So we begin to get a feel of, are there solutions or not? 626 00:37:39,350 --> 00:37:44,750 What I'm saying is, I can't tell just from the count 627 00:37:44,750 --> 00:37:46,730 that A is not singular. 628 00:37:46,730 --> 00:37:51,880 I could have a lot of bars and still be unstable. 629 00:37:51,880 --> 00:37:53,300 Invent a truss for me. 630 00:37:53,300 --> 00:37:58,450 Just because, how could you invent a truss that had, maybe 631 00:37:58,450 --> 00:38:00,850 it has seven bars. 632 00:38:00,850 --> 00:38:05,000 With those seven bars, those diagonal guys, that did it. 633 00:38:05,000 --> 00:38:06,870 That made it stable. 634 00:38:06,870 --> 00:38:10,640 Our eye tells us that before we do any linear algebra. 635 00:38:10,640 --> 00:38:15,330 Tell me a seven by-- A thing. 636 00:38:15,330 --> 00:38:17,460 Well, yeah. 637 00:38:17,460 --> 00:38:23,300 OK, so here would be, shall we support both of these? 638 00:38:23,300 --> 00:38:26,860 I'll start out the same, OK. 639 00:38:26,860 --> 00:38:32,950 Now, yeah, how could I, let's see, I've haven't prepared. 640 00:38:32,950 --> 00:38:36,230 How could I get a whole lot of bars. 641 00:38:36,230 --> 00:38:39,340 I might not get seven by six exactly, but how could 642 00:38:39,340 --> 00:38:44,610 I have plenty of bars and still unstable? 643 00:38:44,610 --> 00:38:48,000 Well, suppose I do this. 644 00:38:48,000 --> 00:38:51,640 Oh yeah, that's a good example. 645 00:38:51,640 --> 00:38:54,350 That's not stable, right? 646 00:38:54,350 --> 00:38:54,860 OK. 647 00:38:54,860 --> 00:38:57,610 Every let's practice with that one. 648 00:38:57,610 --> 00:39:01,140 That's just my idea, and problems in the book 649 00:39:01,140 --> 00:39:04,410 just ask you to practice with things like that. 650 00:39:04,410 --> 00:39:06,070 Tell me the count, first. 651 00:39:06,070 --> 00:39:10,220 What is m, the number of bars? 652 00:39:10,220 --> 00:39:11,260 Six. 653 00:39:11,260 --> 00:39:14,480 What is n, the number of unknowns, little n, 654 00:39:14,480 --> 00:39:15,560 the number of unknowns? 655 00:39:15,560 --> 00:39:18,200 What's the shape of my matrix here? 656 00:39:18,200 --> 00:39:25,220 A is, it's got six bars and how many unknowns? 657 00:39:25,220 --> 00:39:26,650 Eight. 658 00:39:26,650 --> 00:39:28,700 Eight, right? 659 00:39:28,700 --> 00:39:30,580 Two here, two, two, two. 660 00:39:30,580 --> 00:39:31,650 None here. 661 00:39:31,650 --> 00:39:33,310 Six by eight. 662 00:39:33,310 --> 00:39:34,010 OK. 663 00:39:34,010 --> 00:39:38,900 And how many mechanisms am I now expecting? 664 00:39:38,900 --> 00:39:40,200 Probably two. 665 00:39:40,200 --> 00:39:45,850 Probably two, there would be two independent mechanisms here. 666 00:39:45,850 --> 00:39:47,350 Can you tell me what they look like? 667 00:39:47,350 --> 00:39:48,790 Draw them. 668 00:39:48,790 --> 00:39:50,600 What would they look like? 669 00:39:50,600 --> 00:39:53,430 What would be two different things that could happen, 670 00:39:53,430 --> 00:39:58,870 could go wrong with that truss? 671 00:39:58,870 --> 00:40:00,210 You see it, right? 672 00:40:00,210 --> 00:40:02,400 This could turn. 673 00:40:02,400 --> 00:40:08,100 As in our example with the top part moving with it. 674 00:40:08,100 --> 00:40:13,190 Or, a second one possibility would be the top part goes. 675 00:40:13,190 --> 00:40:15,260 And the bottom part stays. 676 00:40:15,260 --> 00:40:17,620 Or any combination. 677 00:40:17,620 --> 00:40:20,540 So the whole thing could go like that. 678 00:40:20,540 --> 00:40:21,540 That would be one. 679 00:40:21,540 --> 00:40:24,230 But that wouldn't be the only one, of course. 680 00:40:24,230 --> 00:40:27,020 So in other words, we have a two-dimensional space 681 00:40:27,020 --> 00:40:30,690 of mechanisms and you could give me two different, 682 00:40:30,690 --> 00:40:35,590 and there are not just two guys, all their combinations 683 00:40:35,590 --> 00:40:36,250 are there. 684 00:40:36,250 --> 00:40:40,436 So this would have two mechanisms. 685 00:40:40,436 --> 00:40:41,060 Two mechanisms. 686 00:40:41,060 --> 00:40:45,040 And I could put in bars, of course, 687 00:40:45,040 --> 00:40:46,940 that would try to save it. 688 00:40:46,940 --> 00:40:50,870 Well, how many bars, what's the minimum number of bars 689 00:40:50,870 --> 00:40:54,910 I absolutely need to make this thing stable again? 690 00:40:54,910 --> 00:40:55,990 Two. 691 00:40:55,990 --> 00:41:00,790 Well, now suppose I put in these two bars. 692 00:41:00,790 --> 00:41:01,480 Right? 693 00:41:01,480 --> 00:41:04,540 I've got enough bars, I've got an eight by eight matrix, 694 00:41:04,540 --> 00:41:06,250 but I haven't saved it. 695 00:41:06,250 --> 00:41:07,340 Right? 696 00:41:07,340 --> 00:41:09,700 Because it still has that mechanism. 697 00:41:09,700 --> 00:41:14,980 So you can't assume that because the count is right 698 00:41:14,980 --> 00:41:17,980 you've avoided mechanisms because in that example you 699 00:41:17,980 --> 00:41:18,990 haven't. 700 00:41:18,990 --> 00:41:23,740 OK, so that would be a case of square eight by eight, 701 00:41:23,740 --> 00:41:29,740 but not good. 702 00:41:29,740 --> 00:41:33,280 So as soon as I say there's a solution to Au=0, 703 00:41:33,280 --> 00:41:36,270 I know that A transpose A will be singular. 704 00:41:36,270 --> 00:41:38,020 And unstable. 705 00:41:38,020 --> 00:41:44,420 OK, before I go to the framework let's just do one more thing. 706 00:41:44,420 --> 00:41:51,680 Suppose I take away the supports. 707 00:41:51,680 --> 00:41:57,360 All right, let me put in some bars, though. 708 00:41:57,360 --> 00:42:00,200 I'll put in some bars. 709 00:42:00,200 --> 00:42:03,640 OK, plenty of bars. 710 00:42:03,640 --> 00:42:04,840 Want another one? 711 00:42:04,840 --> 00:42:06,960 OK, how many bars have I got? 712 00:42:06,960 --> 00:42:08,720 Lots, right? 713 00:42:08,720 --> 00:42:14,490 OK, now the matrix A, what do you think about this? 714 00:42:14,490 --> 00:42:16,084 Are there solutions? 715 00:42:16,084 --> 00:42:18,000 You haven't even seen the matrix A, of course, 716 00:42:18,000 --> 00:42:21,280 but you've seen the truss, that's what matters. 717 00:42:21,280 --> 00:42:25,190 How many solutions, are there solutions to Au=0? 718 00:42:25,190 --> 00:42:31,090 Are there ways that this truss could move without stretching? 719 00:42:31,090 --> 00:42:35,010 Are there ways that this truss could move without stretching? 720 00:42:35,010 --> 00:42:38,300 And what are they, and how many are there? 721 00:42:38,300 --> 00:42:40,800 And what name should we use? 722 00:42:40,800 --> 00:42:41,640 OK, what are they? 723 00:42:41,640 --> 00:42:46,410 How could that move without stretching? 724 00:42:46,410 --> 00:42:48,570 Well, it's got no supports at all. 725 00:42:48,570 --> 00:42:50,660 It's just free out there in space. 726 00:42:50,660 --> 00:42:51,540 So it could move. 727 00:42:51,540 --> 00:42:54,290 How many ways could it move? 728 00:42:54,290 --> 00:42:55,220 Three. 729 00:42:55,220 --> 00:42:58,980 It could move, everybody could move this way. 730 00:42:58,980 --> 00:43:01,710 All ones on the horizontal guys. 731 00:43:01,710 --> 00:43:03,990 Everybody could move this way, all six 732 00:43:03,990 --> 00:43:08,660 ones on the vertical guys, it'll be 12 unknowns here. 733 00:43:08,660 --> 00:43:12,690 And it could also rotate, what would be the rotation? 734 00:43:12,690 --> 00:43:15,490 I'm not talking about this rotation. 735 00:43:15,490 --> 00:43:16,810 This could not happen. 736 00:43:16,810 --> 00:43:17,970 What could happen? 737 00:43:17,970 --> 00:43:21,720 What rotation could happen here? 738 00:43:21,720 --> 00:43:25,020 For this, there's a third rigid motion. 739 00:43:25,020 --> 00:43:30,300 Translation, translation, and rotation around, 740 00:43:30,300 --> 00:43:34,090 well take this one as an example. 741 00:43:34,090 --> 00:43:37,450 The whole thing could swing around this. 742 00:43:37,450 --> 00:43:38,830 That would be a motion. 743 00:43:38,830 --> 00:43:40,370 Well now you're going to say well, 744 00:43:40,370 --> 00:43:42,750 why didn't I swing it around that one? 745 00:43:42,750 --> 00:43:44,200 And of course it could. 746 00:43:44,200 --> 00:43:47,630 But what would be the deal? 747 00:43:47,630 --> 00:43:56,240 It would have to be, there are only three rigid motions, 748 00:43:56,240 --> 00:43:59,030 right, up and around. 749 00:43:59,030 --> 00:44:01,730 So if you give me another one, like, around this one, 750 00:44:01,730 --> 00:44:04,370 then somehow it had to be a combination of those. 751 00:44:04,370 --> 00:44:07,620 I don't even want to think what combination it is. 752 00:44:07,620 --> 00:44:09,630 But there are three rigid motions. 753 00:44:09,630 --> 00:44:15,610 So I sort of distinguish mechanisms, this word 754 00:44:15,610 --> 00:44:16,560 mechanism. 755 00:44:16,560 --> 00:44:21,800 So that's where the truss deforms. 756 00:44:21,800 --> 00:44:27,060 In these-- And rigid motions, so I'll say plus, possibly. 757 00:44:27,060 --> 00:44:31,610 Plus rigid motions, and rigid motions 758 00:44:31,610 --> 00:44:42,020 would be, you know, it doesn't deform internally, 759 00:44:42,020 --> 00:44:44,400 the whole thing moves. 760 00:44:44,400 --> 00:44:46,450 And this is of course what we get 761 00:44:46,450 --> 00:44:50,140 in the case of not enough supports. 762 00:44:50,140 --> 00:44:53,750 And this is what we get in the case of not enough bars. 763 00:44:53,750 --> 00:44:54,250 Yeah. 764 00:44:54,250 --> 00:44:57,950 So maybe it's worth separating those two. 765 00:44:57,950 --> 00:45:02,320 In the examples we do, we'll usually put in enough supports 766 00:45:02,320 --> 00:45:04,470 to kill the rigid motions. 767 00:45:04,470 --> 00:45:07,630 And then the question would be are there some mechanisms. 768 00:45:07,630 --> 00:45:08,490 OK. 769 00:45:08,490 --> 00:45:14,890 Now, I have to start on what this is. 770 00:45:14,890 --> 00:45:16,960 Well it'll be just a very quick start. 771 00:45:16,960 --> 00:45:19,710 So what I'll do at the beginning of Friday, 772 00:45:19,710 --> 00:45:22,250 so Friday's the other lecture on this topic. 773 00:45:22,250 --> 00:45:25,700 And then the homework will ask you to do some trusses 774 00:45:25,700 --> 00:45:27,490 in this section. 775 00:45:27,490 --> 00:45:31,260 It's probably Section 2 point something. 776 00:45:31,260 --> 00:45:38,220 2.7, maybe. 777 00:45:38,220 --> 00:45:40,640 What's the matrix C? 778 00:45:40,640 --> 00:45:44,560 Last second question, what's the matrix C, what size is it? 779 00:45:44,560 --> 00:45:49,070 What size is the matrix C for our original problem? 780 00:45:49,070 --> 00:45:51,360 Or no. 781 00:45:51,360 --> 00:45:56,730 What size is C, is C involving, if I know 782 00:45:56,730 --> 00:46:00,150 these numbers, what size is C? 783 00:46:00,150 --> 00:46:02,520 Five by five. m by m, right? 784 00:46:02,520 --> 00:46:07,990 C is the diagonal matrix, one entry for each bar, 785 00:46:07,990 --> 00:46:15,790 C is just C. It has a c_1, c_2, c_3, and this w=Ce, 786 00:46:15,790 --> 00:46:19,040 it's just Hooke's Law on each bar. 787 00:46:19,040 --> 00:46:22,160 So, simple. 788 00:46:22,160 --> 00:46:25,140 It gets there in the middle, just the way that C 789 00:46:25,140 --> 00:46:29,300 in the first exam problem popped in, and other C's. 790 00:46:29,300 --> 00:46:32,080 That gets there in the middle. 791 00:46:32,080 --> 00:46:34,580 But it's very, extremely, simple. 792 00:46:34,580 --> 00:46:38,970 OK, so the real attention is on A, as usual. 793 00:46:38,970 --> 00:46:41,990 And that will come Friday morning.