WEBVTT 00:00:00.030 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.710 Commons license. 00:00:03.710 --> 00:00:05.450 Your support will help MIT OpenCourseWare 00:00:05.450 --> 00:00:09.960 continue to offer high quality educational resources for free. 00:00:09.960 --> 00:00:12.590 To make a donation or to view additional materials 00:00:12.590 --> 00:00:15.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.210 --> 00:00:19.847 at ocw.mit.edu. 00:00:19.847 --> 00:00:20.930 PROFESSOR STRANG: Alright. 00:00:20.930 --> 00:00:25.290 So this is Lecture 15. 00:00:25.290 --> 00:00:30.530 It's the last topic, today and Friday, like just 15 and 16. 00:00:30.530 --> 00:00:33.630 Trusses within Chapter 2. 00:00:33.630 --> 00:00:38.480 The last topic we'll do for discrete systems. 00:00:38.480 --> 00:00:41.810 Then it's a lot of fun. 00:00:41.810 --> 00:00:48.620 I wanted to say a few words first about last night's exam. 00:00:48.620 --> 00:00:50.720 Several words first. 00:00:50.720 --> 00:00:52.920 Overall, I'm sure it's going to be fine. 00:00:52.920 --> 00:00:56.430 Ramis is grading the first two problems, 00:00:56.430 --> 00:00:58.590 he'll pass them to Peter for the next two. 00:00:58.590 --> 00:01:00.140 And I'll get them back. 00:01:00.140 --> 00:01:05.180 I'm pretty sure it'll be next week. 00:01:05.180 --> 00:01:10.490 I felt it was a fair exam, except I should 00:01:10.490 --> 00:01:18.220 have done a better job in helping you with the matrix A. 00:01:18.220 --> 00:01:20.700 Especially in problem one. 00:01:20.700 --> 00:01:27.040 I'm glad that hint was there, the matrix A_0, that 00:01:27.040 --> 00:01:34.150 goes with free-free, to sort of say what kind of matrix 00:01:34.150 --> 00:01:35.570 to be looking for. 00:01:35.570 --> 00:01:42.450 And I thought I'd just repeat, make the connections that I 00:01:42.450 --> 00:01:47.700 should have made earlier. 00:01:47.700 --> 00:01:53.800 So we all see the point about these. 00:01:53.800 --> 00:01:58.350 These A's and the A transpose A's. 00:01:58.350 --> 00:02:04.050 So if I take, for that A_0, free-free one. 00:02:04.050 --> 00:02:07.130 Everybody sees that this is-- And this connects of course 00:02:07.130 --> 00:02:08.650 with our graphs. 00:02:08.650 --> 00:02:12.650 Our graph is just this simple graph with well 00:02:12.650 --> 00:02:15.840 actually is that how many, is it five nodes? 00:02:15.840 --> 00:02:19.810 I guess there are five. 00:02:19.810 --> 00:02:23.760 Because as it stands I have one, two, three, four, five columns, 00:02:23.760 --> 00:02:32.180 I've got five u's, u_0 down to u_4. 00:02:32.180 --> 00:02:36.030 And if I take A_0 transpose A_0, that 00:02:36.030 --> 00:02:38.640 would be the free-free matrix. 00:02:38.640 --> 00:02:40.150 What size would it be? 00:02:40.150 --> 00:02:42.320 And what matrix would it be? 00:02:42.320 --> 00:02:46.580 Just if we do that multiplication, 00:02:46.580 --> 00:02:49.170 this is a first difference matrix. 00:02:49.170 --> 00:02:51.670 When I do A_0 transpose A_0 I'll get 00:02:51.670 --> 00:02:53.740 one of our second difference matrices. 00:02:53.740 --> 00:02:56.120 So it'll be one of our special ones. 00:02:56.120 --> 00:02:59.980 Which special one would it be? 00:02:59.980 --> 00:03:04.480 B. It'll be the matrix B that has both ends free. 00:03:04.480 --> 00:03:06.110 And what size will it be? 00:03:06.110 --> 00:03:14.230 I guess it'll be five by five. 00:03:14.230 --> 00:03:19.380 That's right; that would be five by four times A_0, which 00:03:19.380 --> 00:03:20.520 is four by five. 00:03:20.520 --> 00:03:26.380 So it'll be the five by five matrix B. Can I call it B_5? 00:03:26.380 --> 00:03:27.580 OK. 00:03:27.580 --> 00:03:31.020 So that was there as a hint. 00:03:31.020 --> 00:03:34.480 That isn't the correct matrix for problem one 00:03:34.480 --> 00:03:37.100 because problem one was fixed-fixed. 00:03:37.100 --> 00:03:38.930 Let's get there in two steps. 00:03:38.930 --> 00:03:41.240 Suppose it's fixed-free. 00:03:41.240 --> 00:03:46.600 So suppose I make u_0 equal 0. 00:03:46.600 --> 00:03:51.340 So I ground the top node, I support the top node-- 00:03:51.340 --> 00:03:53.110 Oh no, shall I do u_0? 00:03:53.110 --> 00:03:53.670 Yeah. 00:03:53.670 --> 00:03:54.790 I'll do u_0. 00:03:54.790 --> 00:03:57.700 So that would knock out this one. 00:03:57.700 --> 00:04:02.870 If I say fix u_0, say, at zero or whatever. 00:04:02.870 --> 00:04:09.910 Now I've got, the next A, I won't call it A_0 anymore. 00:04:09.910 --> 00:04:16.550 So now four by four, now if I do A transpose A, 00:04:16.550 --> 00:04:20.510 which of our special matrices am I going to get? 00:04:20.510 --> 00:04:26.170 T. It'll be T. It'll be the one that has the first, 00:04:26.170 --> 00:04:31.110 the (1, 1) entry will only be a one. 00:04:31.110 --> 00:04:34.490 So that'll be the fixed-free matrix. 00:04:34.490 --> 00:04:35.900 It'll be of what size? 00:04:35.900 --> 00:04:38.010 Four. 00:04:38.010 --> 00:04:42.380 Now I've only got four unknowns. u_1 to u_4. 00:04:42.380 --> 00:04:45.650 OK, that still is not what problem one is. 00:04:45.650 --> 00:04:47.910 Problem one was fixed-fixed. 00:04:47.910 --> 00:04:50.850 So as I did in the review, that would knock out 00:04:50.850 --> 00:04:52.840 both of these columns. 00:04:52.840 --> 00:04:59.850 So this now is the matrix that I was looking for in problem one, 00:04:59.850 --> 00:05:05.750 and I wish I had emphasized these steps in advance. 00:05:05.750 --> 00:05:07.130 I apologize. 00:05:07.130 --> 00:05:13.330 OK, so what fixed-fixed if, without the C part in it. 00:05:13.330 --> 00:05:18.120 Just focusing on the A, what fixed-fixed matrix would I get? 00:05:18.120 --> 00:05:20.330 Which one of our guys would it be? 00:05:20.330 --> 00:05:23.570 K of size three. 00:05:23.570 --> 00:05:29.880 And while we're at it, what would be the story, how would 00:05:29.880 --> 00:05:33.390 I get one of these circular ones, which 00:05:33.390 --> 00:05:36.640 is sort of on our special list. 00:05:36.640 --> 00:05:40.040 For Fourier it's the really special guy. 00:05:40.040 --> 00:05:47.750 So a circular one I'm going to connect u_4 back to u_0. 00:05:47.750 --> 00:05:50.190 So I'm going to put these guys back in. 00:05:50.190 --> 00:05:53.780 And what else would change, if u_4 was connected back to u_0, 00:05:53.780 --> 00:05:57.960 now I'm aiming for this circulant. 00:05:57.960 --> 00:06:01.450 What matrix A is going to give me the circulant? 00:06:01.450 --> 00:06:05.980 So again, these guys are the A transpose A's. 00:06:05.980 --> 00:06:10.500 This over here was the A, and over here is the A transpose A. 00:06:10.500 --> 00:06:13.800 And now I want to fix A, and then 00:06:13.800 --> 00:06:17.550 I want to see that A transpose A. So suppose 00:06:17.550 --> 00:06:19.310 I give you that graph, then. 00:06:19.310 --> 00:06:23.520 Oops, I should have, well. 00:06:23.520 --> 00:06:30.770 Just connect the whole guy. 00:06:30.770 --> 00:06:36.790 So fifth node coming back to the first. 00:06:36.790 --> 00:06:39.890 So that's my circle of nodes. 00:06:39.890 --> 00:06:42.140 That's a simple graph. 00:06:42.140 --> 00:06:49.240 What's the A_circulant now? 00:06:49.240 --> 00:06:54.500 So this would be the A for the circulant case. 00:06:54.500 --> 00:06:57.590 So it's got that back in. 00:06:57.590 --> 00:07:00.020 That shouldn't be erased, that shouldn't be erased. 00:07:00.020 --> 00:07:04.930 And what else has it got? 00:07:04.930 --> 00:07:09.610 If I ask you for the incidence matrix, now I'm in Section 2.4, 00:07:09.610 --> 00:07:12.530 like I've given you a graph, or you 00:07:12.530 --> 00:07:16.480 can think of masses and springs in a circle. 00:07:16.480 --> 00:07:19.660 So I've got five masses, five springs. 00:07:19.660 --> 00:07:25.140 What's my matrix A missing? 00:07:25.140 --> 00:07:29.070 It needs another row. 00:07:29.070 --> 00:07:32.690 We just put in another edge, it needs another row. 00:07:32.690 --> 00:07:39.350 That edge went from the last node back to the first node. 00:07:39.350 --> 00:07:45.510 So we've got a fifth row. 00:07:45.510 --> 00:07:48.090 So you see, now it really is circulant. 00:07:48.090 --> 00:07:53.430 I would call this one also a circulant matrix. 00:07:53.430 --> 00:07:58.750 The diagonals are constant. 00:07:58.750 --> 00:08:02.850 That's what I and MATLAB and everybody else 00:08:02.850 --> 00:08:06.620 would call a Toeplitz matrix, and the command toeplitz 00:08:06.620 --> 00:08:08.740 could create this. 00:08:08.740 --> 00:08:10.460 That diagonal is constant. 00:08:10.460 --> 00:08:12.620 That diagonal is constant. 00:08:12.620 --> 00:08:14.590 The other diagonals are constant. 00:08:14.590 --> 00:08:17.600 But more than that, what's additional 00:08:17.600 --> 00:08:20.080 here in the circulant, which is the thing that 00:08:20.080 --> 00:08:22.500 makes Fourier happy? 00:08:22.500 --> 00:08:26.930 The diagonal circles around. 00:08:26.930 --> 00:08:30.430 That diagonal has only got four entries in it, 00:08:30.430 --> 00:08:35.190 but it circles around sort of periodically 00:08:35.190 --> 00:08:38.030 to its fifth entry. 00:08:38.030 --> 00:08:40.210 So that's more than Toeplitz. 00:08:40.210 --> 00:08:43.820 It's circulant because it's coming around again. 00:08:43.820 --> 00:08:47.570 This we'll see in the discrete Fourier transform. 00:08:47.570 --> 00:08:50.270 It's really all good stuff. 00:08:50.270 --> 00:08:53.240 And now there is my a circulant. 00:08:53.240 --> 00:08:57.620 And what would be my A transpose circulant A_circulant? 00:08:57.620 --> 00:09:03.770 What would be A transpose A if I take that five by five matrix? 00:09:03.770 --> 00:09:09.790 C. Finally I've created, so I've already got B, I've got T, 00:09:09.790 --> 00:09:12.780 I've got K, all those three special guys. 00:09:12.780 --> 00:09:16.220 And now the A transpose A for this circulant, 00:09:16.220 --> 00:09:20.210 so that's a first difference matrix for a periodic problem. 00:09:20.210 --> 00:09:22.790 And A transpose A will be a second difference 00:09:22.790 --> 00:09:27.200 matrix for a periodic problem, C_5, I guess. 00:09:27.200 --> 00:09:29.305 It'll be five. 00:09:29.305 --> 00:09:29.805 OK. 00:09:29.805 --> 00:09:34.130 I hope that brings together what I, if I was on the ball, 00:09:34.130 --> 00:09:38.370 I would have brought it together before the quiz. 00:09:38.370 --> 00:09:45.030 Can I just say a few words about the quiz and grades? 00:09:45.030 --> 00:09:47.650 They come out fine. 00:09:47.650 --> 00:09:49.070 Really they do. 00:09:49.070 --> 00:09:51.130 I've been doing this a long time. 00:09:51.130 --> 00:09:59.050 And just, enjoy October. 00:09:59.050 --> 00:10:01.760 I'm sorry to give you any exam at all, 00:10:01.760 --> 00:10:08.230 but it's a chance for you yeah, I'm working on this stuff. 00:10:08.230 --> 00:10:09.590 I'm learning it. 00:10:09.590 --> 00:10:11.940 Everybody didn't learn it first time. 00:10:11.940 --> 00:10:13.820 I don't learn it first time, every time 00:10:13.820 --> 00:10:16.080 I teach the course I learn something more. 00:10:16.080 --> 00:10:21.400 And if you're learning from this course then I'm totally happy. 00:10:21.400 --> 00:10:23.430 And I believe that's the case. 00:10:23.430 --> 00:10:25.380 So I am entirely happy. 00:10:25.380 --> 00:10:32.120 And I hope the quiz, some points of it 00:10:32.120 --> 00:10:34.030 I wish I'd prepared better. 00:10:34.030 --> 00:10:39.740 But I feel pretty good about it. 00:10:39.740 --> 00:10:42.680 I feel good about it, let me just say. 00:10:42.680 --> 00:10:48.340 So, and I'm happy to have any comments, email or in person. 00:10:48.340 --> 00:10:50.740 But allow me to go forward with trusses. 00:10:50.740 --> 00:10:53.980 However, I'm ready always for a comment. 00:10:53.980 --> 00:10:54.740 Yeah. 00:10:54.740 --> 00:10:56.100 OK. 00:10:56.100 --> 00:10:58.680 Anyway, enjoy trusses. 00:10:58.680 --> 00:10:59.990 Enjoy life. 00:10:59.990 --> 00:11:02.150 Yeah. 00:11:02.150 --> 00:11:05.740 And this should have been in the book, this page. 00:11:05.740 --> 00:11:11.170 So if it wasn't too late I would paste it in. 00:11:11.170 --> 00:11:14.120 Because this connects A transpose A 00:11:14.120 --> 00:11:18.680 to special matrices, in the way I had in my mind, 00:11:18.680 --> 00:11:22.250 but I didn't put it on the board until just now. 00:11:22.250 --> 00:11:29.020 OK, so I'll cover that up and ready to go with trusses. 00:11:29.020 --> 00:11:31.180 OK. 00:11:31.180 --> 00:11:34.840 So trusses, we want to know what's up. 00:11:34.840 --> 00:11:36.500 We want to get the setup right. 00:11:36.500 --> 00:11:45.000 Once we get the setup we'll know we're looking for. 00:11:45.000 --> 00:11:52.100 OK, so a truss is a bunch of elastic bars with pin joints 00:11:52.100 --> 00:11:52.900 connecting them. 00:11:52.900 --> 00:11:55.110 Now, what do I mean by a pin joint? 00:11:55.110 --> 00:12:05.570 I mean that stretching the bars takes force. 00:12:05.570 --> 00:12:09.810 Turning around the pin joint doesn't take force. 00:12:09.810 --> 00:12:18.130 So the pin just lets them turn, so we'll 00:12:18.130 --> 00:12:20.740 have forces in those bars. 00:12:20.740 --> 00:12:24.210 So it's like masses and springs. 00:12:24.210 --> 00:12:26.300 Exactly like masses and springs. 00:12:26.300 --> 00:12:30.140 But yet we have a 2-D problem. 00:12:30.140 --> 00:12:31.800 So it's a two dimensional problem 00:12:31.800 --> 00:12:33.340 with masses and springs. 00:12:33.340 --> 00:12:35.550 And we could certainly have a 3-D truss, 00:12:35.550 --> 00:12:39.680 but 2-D makes all the important points. 00:12:39.680 --> 00:12:41.650 And then I can count the bars. 00:12:41.650 --> 00:12:44.360 One, two, three, four, five. 00:12:44.360 --> 00:12:46.800 And I can count the nodes. 00:12:46.800 --> 00:12:48.720 There happen to be five here. 00:12:48.720 --> 00:12:50.710 But now comes the moment. 00:12:50.710 --> 00:12:53.370 I have to tell you, what are the unknowns? 00:12:53.370 --> 00:12:54.380 What are the u's. 00:12:54.380 --> 00:12:56.090 Because of course, you know that I'm 00:12:56.090 --> 00:13:05.220 going to go from u's to e's to w's, to forces, f. 00:13:05.220 --> 00:13:08.780 And you know that a matrix A is going to do that. 00:13:08.780 --> 00:13:11.080 A matrix C is going to do that. 00:13:11.080 --> 00:13:13.650 A matrix A transpose is going to do that. 00:13:13.650 --> 00:13:17.090 You're all ready, we need to know what's the setup. 00:13:17.090 --> 00:13:18.890 What are these matrices. 00:13:18.890 --> 00:13:22.490 OK, and how many-- So let me explain the setup. 00:13:22.490 --> 00:13:31.630 Typical node, node one. we have forces in these bars, 00:13:31.630 --> 00:13:38.250 so that node one could have a force. 00:13:38.250 --> 00:13:39.960 We're in the plane. 00:13:39.960 --> 00:13:44.200 So we have a horizontal force and a vertical force. 00:13:44.200 --> 00:13:48.060 Together, that would give, produces a force 00:13:48.060 --> 00:13:50.460 in any direction whatever. 00:13:50.460 --> 00:13:51.880 So this is the key point. 00:13:51.880 --> 00:13:59.340 That there is a horizontal force. f horizontal one. 00:13:59.340 --> 00:14:01.440 The one being the force on node one. 00:14:01.440 --> 00:14:04.700 But there's also a vertical force. 00:14:04.700 --> 00:14:07.940 And let me take horizontal to the right, 00:14:07.940 --> 00:14:09.510 positive to the right. 00:14:09.510 --> 00:14:11.320 Vertical positive upwards. 00:14:11.320 --> 00:14:13.550 Just to have a convention. 00:14:13.550 --> 00:14:17.180 So how many f's have I got? 00:14:17.180 --> 00:14:21.140 Well, the point is I now have two per node. 00:14:21.140 --> 00:14:22.610 That's the difference. 00:14:22.610 --> 00:14:24.960 I have two per node, two forces. 00:14:24.960 --> 00:14:27.480 And I have two displacements per node. 00:14:27.480 --> 00:14:32.410 Because that point under, there will be more forces. 00:14:32.410 --> 00:14:38.020 Some maybe pulling this way, whatever. 00:14:38.020 --> 00:14:40.030 Maybe let's look at node two. 00:14:40.030 --> 00:14:42.710 So node two could have a couple of forces on it. 00:14:42.710 --> 00:14:45.420 f h two, and f v two. 00:14:45.420 --> 00:14:49.000 And it moves like the other nodes. 00:14:49.000 --> 00:14:53.140 So now I'm introducing the unknowns. u 00:14:53.140 --> 00:14:57.680 is the movement. u, again horizontal, 00:14:57.680 --> 00:15:01.240 and again, now we're talking about node two. 00:15:01.240 --> 00:15:04.080 And it moves up. 00:15:04.080 --> 00:15:05.870 Or doesn't, or moves down. 00:15:05.870 --> 00:15:09.470 But that's an unknown. u, a displacement, 00:15:09.470 --> 00:15:12.130 a vertical displacement of node two. 00:15:12.130 --> 00:15:14.330 Do you see the setup? 00:15:14.330 --> 00:15:17.350 Two forces per node. 00:15:17.350 --> 00:15:20.210 Two displacements per node. 00:15:20.210 --> 00:15:27.650 So that's like, the number of unknown is like doubled. 00:15:27.650 --> 00:15:33.220 Like, doubled, and that produces an interesting situation. 00:15:33.220 --> 00:15:36.180 I've marked supports here. 00:15:36.180 --> 00:15:38.860 So let's just speak about supports. 00:15:38.860 --> 00:15:41.310 So what's happening at the supports? 00:15:41.310 --> 00:15:49.110 At the support there's no movement. 00:15:49.110 --> 00:15:51.940 The whole-- That point is pinned. 00:15:51.940 --> 00:15:57.120 So this is telling me that u horizontal five 00:15:57.120 --> 00:16:02.920 is zero and u vertical, sorry that was four 00:16:02.920 --> 00:16:06.830 and it'll be the same for five. u vertical five is zero. 00:16:06.830 --> 00:16:10.760 It's like grounding a node in the electrical case. 00:16:10.760 --> 00:16:13.580 We just see this pattern over and over. 00:16:13.580 --> 00:16:17.090 And we want to see OK, what does it look like for trusses? 00:16:17.090 --> 00:16:20.140 So here's a support that fixes those. 00:16:20.140 --> 00:16:22.090 So those are not unknowns. 00:16:22.090 --> 00:16:24.480 And similarly, they're not unknowns there. 00:16:24.480 --> 00:16:27.420 Still saying five when I mean four. 00:16:27.420 --> 00:16:34.300 So those are boundary conditions, n conditions, 00:16:34.300 --> 00:16:34.870 whatever. 00:16:34.870 --> 00:16:36.330 And similarly here. 00:16:36.330 --> 00:16:39.500 So how many unknowns are there? 00:16:39.500 --> 00:16:45.120 Now look at this picture, how many unknown displacements 00:16:45.120 --> 00:16:48.830 are there in this truss? 00:16:48.830 --> 00:16:49.380 Six. 00:16:49.380 --> 00:16:50.320 Six, right? 00:16:50.320 --> 00:16:53.180 Two here, two here, two here and none there. 00:16:53.180 --> 00:17:00.520 So the number of actual unknowns is six. 00:17:00.520 --> 00:17:06.340 My idea would be that it's twice the number of nodes 00:17:06.340 --> 00:17:13.980 minus the number of fixed things, reaction, whatever. 00:17:13.980 --> 00:17:16.250 That R would be four in this case. 00:17:16.250 --> 00:17:20.490 I've got two fixed here and two fixed here, 00:17:20.490 --> 00:17:23.190 so this would be two times five. 00:17:23.190 --> 00:17:26.980 Ten possible displacements but R counts 00:17:26.980 --> 00:17:29.580 the number of fixed displacements, four, 00:17:29.580 --> 00:17:32.580 and leaves us with six. 00:17:32.580 --> 00:17:34.320 OK. 00:17:34.320 --> 00:17:38.590 So my matrix A will now be, it's always m by n. 00:17:38.590 --> 00:17:42.880 My matrix A will be five by six. 00:17:42.880 --> 00:17:46.060 OK. 00:17:46.060 --> 00:17:49.340 Now you're going to ask what is that matrix. 00:17:49.340 --> 00:17:53.940 But let me hold that off for a little moment. 00:17:53.940 --> 00:17:56.910 I want to just see its shape first. 00:17:56.910 --> 00:18:02.160 So you could now do this for a large truss, right? 00:18:02.160 --> 00:18:06.500 You count the bars, and you could count the nodes. 00:18:06.500 --> 00:18:09.990 And then you could count the unknown displacements, u. 00:18:09.990 --> 00:18:17.190 So there are six u's here. 00:18:17.190 --> 00:18:19.460 And there are five e's. 00:18:19.460 --> 00:18:22.530 And there are five bar forces. 00:18:22.530 --> 00:18:29.700 And there are six equilibrium, balance, force balances. 00:18:29.700 --> 00:18:33.920 Six, six for the node count, for the unknowns count, 00:18:33.920 --> 00:18:36.480 five, five for the bars count. 00:18:36.480 --> 00:18:44.520 OK, now here's a point about this particular truss. 00:18:44.520 --> 00:18:46.440 It's not safe to get on it. 00:18:46.440 --> 00:18:47.570 Right? 00:18:47.570 --> 00:18:49.650 And I want to say why is it not safe. 00:18:49.650 --> 00:18:54.860 So this is a feature that comes into the truss question that 00:18:54.860 --> 00:18:58.040 makes it a little new and more interesting. 00:18:58.040 --> 00:19:03.850 A little twist compared to the previous examples. 00:19:03.850 --> 00:19:07.450 That bar, that truss, I wouldn't stand on it. 00:19:07.450 --> 00:19:09.150 Now, why not? 00:19:09.150 --> 00:19:11.980 Well, purely for linear algebra reasons. 00:19:11.980 --> 00:19:13.230 Of course. 00:19:13.230 --> 00:19:15.650 The matrix A is five by six. 00:19:15.650 --> 00:19:18.840 So now what do we know about a matrix that's five by six? 00:19:18.840 --> 00:19:23.790 So A is five by six. 00:19:23.790 --> 00:19:28.420 Five rows, as always the m; six columns, because now we 00:19:28.420 --> 00:19:29.850 have six unknowns. 00:19:29.850 --> 00:19:36.240 And what do I know about any five by six matrix? 00:19:36.240 --> 00:19:39.400 I want to ask about the equation Au=0. 00:19:42.320 --> 00:19:44.810 So I want to ask about it in linear algebra language 00:19:44.810 --> 00:19:47.700 and then I want to ask about it in physical language. 00:19:47.700 --> 00:19:52.610 And the beauty is the thing that makes trusses sort of fun 00:19:52.610 --> 00:19:57.070 is, these matrices, A, get pretty big fast. 00:19:57.070 --> 00:19:59.590 Because when I put a few more nodes on, 00:19:59.590 --> 00:20:04.460 the book has a picture of a sort of treehouse. 00:20:04.460 --> 00:20:08.220 Then A is growing. 00:20:08.220 --> 00:20:12.030 And I don't, all the time, write down the matrix A. 00:20:12.030 --> 00:20:13.950 I haven't written it down here. 00:20:13.950 --> 00:20:16.330 What I've written down is just its size, 00:20:16.330 --> 00:20:21.540 because that's enough to tell us something about this set 00:20:21.540 --> 00:20:24.290 of equations Au=0. 00:20:24.290 --> 00:20:27.200 What's the story on Au=0? 00:20:27.200 --> 00:20:29.900 Well of course it has the solution u=0. 00:20:29.900 --> 00:20:31.940 Nothing moving. 00:20:31.940 --> 00:20:35.710 If I have no displacement, if the u's are all zero, 00:20:35.710 --> 00:20:37.840 then I have no stretching. 00:20:37.840 --> 00:20:39.290 The e's are stretching. 00:20:39.290 --> 00:20:41.890 Elongation, as before. 00:20:41.890 --> 00:20:44.250 How far does the bar stretch? 00:20:44.250 --> 00:20:44.940 OK. 00:20:44.940 --> 00:20:48.760 So if I have zero u's and zero e's. 00:20:48.760 --> 00:20:55.700 But, what other possibility am I going to have here? 00:20:55.700 --> 00:21:00.520 I'm going to have probably one solution 00:21:00.520 --> 00:21:03.800 to this system that isn't zero. 00:21:03.800 --> 00:21:08.910 I'm probably going to have one set of displacements u, look 00:21:08.910 --> 00:21:10.140 what's happening here. 00:21:10.140 --> 00:21:13.750 This is Au is the e. 00:21:13.750 --> 00:21:18.900 So I'm going to have at least one and probably in this case 00:21:18.900 --> 00:21:26.530 it will be one, there will be one, the neat word for it 00:21:26.530 --> 00:21:30.930 is a mechanism. 00:21:30.930 --> 00:21:32.850 And what does that mean? 00:21:32.850 --> 00:21:37.000 A mechanism is a solution to Au=0. 00:21:37.000 --> 00:21:40.370 So that, a mechanism is a movement of the bar. 00:21:40.370 --> 00:21:42.410 So it's going to be non-zero. 00:21:42.410 --> 00:21:45.080 The bars are going to move a little. 00:21:45.080 --> 00:21:47.060 Sorry, the nodes are going to move a little. 00:21:47.060 --> 00:21:50.180 The nodes will move a little bit. 00:21:50.180 --> 00:21:52.660 In this u, because it isn't zero. 00:21:52.660 --> 00:21:56.420 But the bars won't stretch. 00:21:56.420 --> 00:22:04.060 So that tells us we've got instability here. 00:22:04.060 --> 00:22:05.880 If there's a solution to that, that's 00:22:05.880 --> 00:22:09.020 always telling us that A transpose A is singular. 00:22:09.020 --> 00:22:13.020 So let me just put that A transpose A -- 00:22:13.020 --> 00:22:18.950 or A transpose C A, C couldn't save it -- will be singular. 00:22:18.950 --> 00:22:24.910 It's just like our free-free thing in being singular, 00:22:24.910 --> 00:22:28.450 but the picture doesn't look free-free, does it? 00:22:28.450 --> 00:22:33.000 It's got supports in here, just not good enough. 00:22:33.000 --> 00:22:38.190 And I believe that if you look at this truss, 00:22:38.190 --> 00:22:40.430 you could describe, you could tell me, 00:22:40.430 --> 00:22:47.620 and you could draw, a movement of that truss in which 00:22:47.620 --> 00:22:51.040 there is displacement but no stretching. 00:22:51.040 --> 00:22:54.560 Let me ask you how to draw that. 00:22:54.560 --> 00:22:57.410 And I believe-- Everybody understood, 00:22:57.410 --> 00:22:59.240 why was there a solution? 00:22:59.240 --> 00:23:03.070 It was because we have six unknowns 00:23:03.070 --> 00:23:05.320 and we only have five equations. 00:23:05.320 --> 00:23:07.720 So this was five equations. 00:23:07.720 --> 00:23:10.640 Any time you have five equations with a zero 00:23:10.640 --> 00:23:14.540 on the right-hand side, so five homogeneous 00:23:14.540 --> 00:23:17.315 equations, whatever you want to say when that's zero 00:23:17.315 --> 00:23:20.190 on the right and six unknowns. 00:23:20.190 --> 00:23:22.160 Six u's. 00:23:22.160 --> 00:23:25.340 Then you're going to have solution. 00:23:25.340 --> 00:23:27.460 You can't help it. 00:23:27.460 --> 00:23:29.990 You've got that many degrees of freedom, 00:23:29.990 --> 00:23:31.940 you've only got that many constraints, there's 00:23:31.940 --> 00:23:33.220 going to be solution. 00:23:33.220 --> 00:23:37.540 OK, tell me how to draw that. 00:23:37.540 --> 00:23:41.570 Let me put in the truss now. 00:23:41.570 --> 00:23:43.940 What's the solution? 00:23:43.940 --> 00:23:49.570 So this is the fun part in a particular example 00:23:49.570 --> 00:23:52.140 at the start. 00:23:52.140 --> 00:24:04.790 How could that move without stretching bars? 00:24:04.790 --> 00:24:08.340 Let me see. 00:24:08.340 --> 00:24:11.690 What could happen? 00:24:11.690 --> 00:24:16.400 What do you mean now, who's going to move where? 00:24:16.400 --> 00:24:18.280 What's the movement here? 00:24:18.280 --> 00:24:19.830 And I want to draw it over there. 00:24:19.830 --> 00:24:23.480 So you give the answer by drawing it as well 00:24:23.480 --> 00:24:27.160 as by telling me the six unknown u's. 00:24:27.160 --> 00:24:33.300 So what can happen at this thing? 00:24:33.300 --> 00:24:40.200 So you're going to say the truss could, these bars could, 00:24:40.200 --> 00:24:42.630 turn a little? 00:24:42.630 --> 00:24:44.750 And notice that word a little. 00:24:44.750 --> 00:24:47.050 We're talking small displacement, 00:24:47.050 --> 00:24:49.980 small stretches all the time here. 00:24:49.980 --> 00:24:53.240 I'll show you why we're always making 00:24:53.240 --> 00:24:58.430 that linearity assumption, or small assumption. 00:24:58.430 --> 00:25:00.290 OK, those move a little. 00:25:00.290 --> 00:25:05.090 And what happens to that triangle at the top? 00:25:05.090 --> 00:25:07.630 It sort of just moves along, right? 00:25:07.630 --> 00:25:10.190 So the picture you would draw would 00:25:10.190 --> 00:25:16.210 be that you started there. 00:25:16.210 --> 00:25:18.820 And it moved along a little, I'll 00:25:18.820 --> 00:25:24.030 make it a larger displacement than I really have in mind. 00:25:24.030 --> 00:25:26.380 These guys of course are here. 00:25:26.380 --> 00:25:31.180 So they come out and the rest of the truss, 00:25:31.180 --> 00:25:34.470 the top of the truss, just kind of goes with it. 00:25:34.470 --> 00:25:36.720 Goes with the flow. 00:25:36.720 --> 00:25:40.970 That would be the answer that I would be looking for, 00:25:40.970 --> 00:25:42.720 to draw the mechanism. 00:25:42.720 --> 00:25:44.230 That would show it. 00:25:44.230 --> 00:25:46.970 And if I wanted to write down the u that 00:25:46.970 --> 00:25:50.450 goes with it, what would it be? 00:25:50.450 --> 00:25:53.230 Let me again number these guys; this is one, two, 00:25:53.230 --> 00:25:54.780 and this is three. 00:25:54.780 --> 00:26:02.880 So what are the displacements of nodes one, two, and three? 00:26:02.880 --> 00:26:07.940 I'll always write u_1 horizontal before vertical. 00:26:07.940 --> 00:26:09.620 Can we make an agreement? 00:26:09.620 --> 00:26:12.440 So I want to know about the horizontal movement 00:26:12.440 --> 00:26:14.780 then the vertical movement of node one, 00:26:14.780 --> 00:26:16.860 then node two, then node three. 00:26:16.860 --> 00:26:19.080 So I'll have six numbers there. 00:26:19.080 --> 00:26:22.710 And what could I put in for those six numbers? 00:26:22.710 --> 00:26:27.120 So the horizontal, let me suppose that that first guy, 00:26:27.120 --> 00:26:30.140 I'll put a one. 00:26:30.140 --> 00:26:32.370 Really that's a bigger number than I should put, 00:26:32.370 --> 00:26:33.980 but it's a convenient number. 00:26:33.980 --> 00:26:36.860 So I'll just take it to be one even though I really 00:26:36.860 --> 00:26:41.500 have in mind-- Let's say that's one angstrom, or one 00:26:41.500 --> 00:26:43.320 tiny little person. 00:26:43.320 --> 00:26:51.820 OK, so what about the rest? 00:26:51.820 --> 00:26:58.630 What's the vertical-- Oh yeah, this is a key point here, 00:26:58.630 --> 00:27:01.600 what's the vertical movement? 00:27:01.600 --> 00:27:05.110 This movement to me is horizontal. 00:27:05.110 --> 00:27:09.910 I'm going to say that the vertical moment is zero. 00:27:09.910 --> 00:27:11.830 Of node one, just moves over. 00:27:11.830 --> 00:27:13.780 And node two does the same. 00:27:13.780 --> 00:27:15.440 And node three does the same. 00:27:15.440 --> 00:27:19.840 So that's my solution. [1, 0, 1, 0, 1, 0]. 00:27:19.840 --> 00:27:25.370 That's a simple movement, a simple set of displacements, 00:27:25.370 --> 00:27:30.820 think most to the right. 00:27:30.820 --> 00:27:33.340 I have not written down the matrix A, 00:27:33.340 --> 00:27:37.610 but probably won't even do it until next time. 00:27:37.610 --> 00:27:41.320 But you will see that when we do the matrix A 00:27:41.320 --> 00:27:46.490 for this particular truss will have this particular u 00:27:46.490 --> 00:27:49.210 as a mechanism. 00:27:49.210 --> 00:27:56.130 In linear algebra, u is in the null space of A. Au=0, 00:27:56.130 --> 00:27:57.410 that's all that means. 00:27:57.410 --> 00:28:04.260 OK, do you see more or less what's up? 00:28:04.260 --> 00:28:09.610 But now there's one little thing that may be bothering you. 00:28:09.610 --> 00:28:13.070 Which is what? 00:28:13.070 --> 00:28:18.650 If I come back to the zero, zero, zero there, 00:28:18.650 --> 00:28:23.520 you could correctly say wait a minute, 00:28:23.520 --> 00:28:25.590 if those bars didn't stretch, if they just 00:28:25.590 --> 00:28:29.170 rotated as you told me to do, then 00:28:29.170 --> 00:28:35.130 this was mostly across but a little bit down, right? 00:28:35.130 --> 00:28:36.570 And I'm saying no. 00:28:36.570 --> 00:28:38.120 I'm saying zero. 00:28:38.120 --> 00:28:42.930 OK, how do I get away with that? 00:28:42.930 --> 00:28:47.520 So I'm saying in 18.085 it's a zero. 00:28:47.520 --> 00:28:49.480 And why? 00:28:49.480 --> 00:28:55.080 So this is like a little time-out just to focus in on, 00:28:55.080 --> 00:28:57.490 let me focus in on node two. 00:28:57.490 --> 00:29:02.420 So here's the bottom node four, so it 00:29:02.420 --> 00:29:04.060 used to be vertical up to two. 00:29:04.060 --> 00:29:06.920 This was node two and this was number four. 00:29:06.920 --> 00:29:11.620 And then it rotated a little, to there. 00:29:11.620 --> 00:29:13.690 To this position. 00:29:13.690 --> 00:29:18.370 So it went, if this angle was, let's say, 00:29:18.370 --> 00:29:26.560 theta, then what is that actual position? 00:29:26.560 --> 00:29:30.760 So let's say this was, let's say the bar had length one. 00:29:30.760 --> 00:29:33.450 This is the origin, (0, 0). 00:29:33.450 --> 00:29:37.760 This is the point (0, 1) above it, OK? 00:29:37.760 --> 00:29:41.280 And now, that's before it moved. 00:29:41.280 --> 00:29:42.780 Then it moved a little bit. 00:29:42.780 --> 00:29:45.470 It moved to an angle theta. 00:29:45.470 --> 00:29:48.090 What's the position of that bar? 00:29:48.090 --> 00:29:48.890 Of that node? 00:29:48.890 --> 00:29:51.030 What's the new position of the node, 00:29:51.030 --> 00:29:52.700 and then we'll look at the difference 00:29:52.700 --> 00:29:56.760 and we'll see the movement u, the displacement. 00:29:56.760 --> 00:30:06.820 So how far did it move? 00:30:06.820 --> 00:30:08.390 What's the x-coordinate? 00:30:08.390 --> 00:30:10.400 How far did it go across? 00:30:10.400 --> 00:30:13.470 If I put in that line you'll know. 00:30:13.470 --> 00:30:16.760 So the movement across was, sin(theta)? 00:30:16.760 --> 00:30:20.220 Good. sin(theta). 00:30:20.220 --> 00:30:28.630 And the movement down, well, yes, so let's find its position 00:30:28.630 --> 00:30:30.550 and then we'll take the difference. 00:30:30.550 --> 00:30:41.530 So what's the vertical new position for that guy? 00:30:41.530 --> 00:30:47.470 It moved by, it moved across by sin(theta). 00:30:50.470 --> 00:30:56.810 And down by 1-cos(theta). 00:30:56.810 --> 00:30:58.940 Are you agreed with that? 00:30:58.940 --> 00:31:04.680 Because here is cosine theta, right there. 00:31:04.680 --> 00:31:08.150 And here's the little bit it moved down. 00:31:08.150 --> 00:31:09.490 OK. 00:31:09.490 --> 00:31:12.410 So these are exactly correct. 00:31:12.410 --> 00:31:16.550 Yeah this is in the position of sin(theta), cos(theta), 00:31:16.550 --> 00:31:18.540 and the difference was the 1-cos(theta). 00:31:21.150 --> 00:31:23.970 OK, so now here comes the key point. 00:31:23.970 --> 00:31:27.590 Approximately, sin(theta) is approximately, 00:31:27.590 --> 00:31:31.680 if theta is small and now here comes the smallness, sin(theta) 00:31:31.680 --> 00:31:35.480 is approximately theta. sin(theta)'s 00:31:35.480 --> 00:31:36.840 approximately theta. 00:31:36.840 --> 00:31:40.440 And 1-cos(theta) is approximately what? 00:31:40.440 --> 00:31:48.180 Now, this is the important point. 00:31:48.180 --> 00:31:51.690 So theta is like the first term. 00:31:51.690 --> 00:31:54.450 If I expand, I mean, the exact term 00:31:54.450 --> 00:31:59.380 would be theta minus theta cubed over six, dot dot dot. 00:31:59.380 --> 00:32:03.490 But I'm only keeping that term. 00:32:03.490 --> 00:32:08.740 And 1-cos(theta), now what's the formula for cos(theta)? 00:32:08.740 --> 00:32:10.350 This is like worth, just should-- 00:32:10.350 --> 00:32:15.070 It's a one, because of course cos(0) is one, 00:32:15.070 --> 00:32:17.570 and then you subtract what? 00:32:17.570 --> 00:32:19.420 Theta squared over two. 00:32:19.420 --> 00:32:21.710 And so on. 00:32:21.710 --> 00:32:25.540 And then plus theta fourth over 24 or whatever. 00:32:25.540 --> 00:32:29.640 OK, so the ones cancel as we expect. 00:32:29.640 --> 00:32:35.580 And I'm getting theta squared over two. 00:32:35.580 --> 00:32:40.100 And this, here was theta to the first power. 00:32:40.100 --> 00:32:42.810 Theta cubed was, we didn't care. 00:32:42.810 --> 00:32:45.440 And we don't care about theta squared. 00:32:45.440 --> 00:32:48.870 So that's why it's zero. 00:32:48.870 --> 00:32:53.050 Because it's a second order movement. 00:32:53.050 --> 00:32:57.330 If theta is small, as I'm going to assume, small displacement, 00:32:57.330 --> 00:33:01.690 theta squared would, if I allowed theta squared 00:33:01.690 --> 00:33:05.910 and cos(theta) in here, I'd have a non-linear problem. 00:33:05.910 --> 00:33:07.120 And I don't want that. 00:33:07.120 --> 00:33:08.630 And I don't need it. 00:33:08.630 --> 00:33:14.740 I mean, finite elements, structures, bridges, whatever. 00:33:14.740 --> 00:33:18.300 Your first hope and expectation and calculation 00:33:18.300 --> 00:33:22.030 is small theta, linear problem. 00:33:22.030 --> 00:33:27.990 So to a linear person theta squared is zero. 00:33:27.990 --> 00:33:31.490 That's why those guys are zero. 00:33:31.490 --> 00:33:36.890 OK, so that's an assumption we'll often see, 00:33:36.890 --> 00:33:41.990 so it kind of was. 00:33:41.990 --> 00:33:45.750 There are two kinds of non-linearities 00:33:45.750 --> 00:33:49.650 in structures and elasticity. 00:33:49.650 --> 00:33:54.980 One would be to allow this geometric non-linearity. 00:33:54.980 --> 00:33:57.670 Thetas, large displacements, theta large 00:33:57.670 --> 00:34:01.200 enough so that you can't neglect theta squared. 00:34:01.200 --> 00:34:02.330 That's a tough one. 00:34:02.330 --> 00:34:05.960 If you allow geometric non-linearity in, 00:34:05.960 --> 00:34:08.320 as finite element codes have to do. 00:34:08.320 --> 00:34:14.660 If you-- ABAQUS is a code that does major finite element 00:34:14.660 --> 00:34:17.420 calculations, nonlinear ones. 00:34:17.420 --> 00:34:20.060 I mean they, at the beginning they were studying 00:34:20.060 --> 00:34:22.410 what happens, what are the stresses on cables 00:34:22.410 --> 00:34:23.800 under the Atlantic. 00:34:23.800 --> 00:34:26.910 I mean, those are fascinating problems. 00:34:26.910 --> 00:34:29.690 Or I mention car crashes. 00:34:29.690 --> 00:34:32.170 I mean, car crashes, the geometry changes, 00:34:32.170 --> 00:34:33.950 you have big displacements. 00:34:33.950 --> 00:34:38.340 But we're talking here about linear small displacement 00:34:38.340 --> 00:34:39.060 cases. 00:34:39.060 --> 00:34:44.200 OK, so don't forget that part. 00:34:44.200 --> 00:34:49.120 That when the truss gets more complicated, 00:34:49.120 --> 00:34:51.090 the principle stays the same. 00:34:51.090 --> 00:34:55.070 That we distinguish between the thetas that matter 00:34:55.070 --> 00:34:57.220 and the theta squareds that don't. 00:34:57.220 --> 00:35:01.220 OK, so now what? 00:35:01.220 --> 00:35:04.960 Now I guess I'm ready to complete 00:35:04.960 --> 00:35:06.661 this picture a little more. 00:35:06.661 --> 00:35:07.160 OK. 00:35:07.160 --> 00:35:11.740 So let me, so we've understood what's the idea of a mechanism. 00:35:11.740 --> 00:35:17.220 Oh, how could I prevent a mechanism? 00:35:17.220 --> 00:35:22.450 In other words, if I stood on this truss, 00:35:22.450 --> 00:35:29.410 the slightest bit of wind would crash it down, right? 00:35:29.410 --> 00:35:32.150 So that's unstable. 00:35:32.150 --> 00:35:34.780 That's an unstable truss. 00:35:34.780 --> 00:35:36.310 How could I make it stable? 00:35:36.310 --> 00:35:43.630 I mean if you were designing this thing, what would you do? 00:35:43.630 --> 00:35:44.790 Add another edge. 00:35:44.790 --> 00:35:46.370 You'd stick in another bar. 00:35:46.370 --> 00:35:49.370 Maybe stick in a bar there. 00:35:49.370 --> 00:35:51.650 What would happen now? 00:35:51.650 --> 00:35:53.392 Would it now be stable? 00:35:53.392 --> 00:35:54.850 You'd have to answer that question. 00:35:54.850 --> 00:35:57.580 You couldn't just put in bars, whatever. 00:35:57.580 --> 00:36:01.120 You want to put bars that do the job. 00:36:01.120 --> 00:36:02.560 OK, now how many bars? 00:36:02.560 --> 00:36:04.940 We've now got six bars. 00:36:04.940 --> 00:36:07.500 So m is now up to six. 00:36:07.500 --> 00:36:12.720 The matrix A is six by six now, whatever that matrix may be. 00:36:12.720 --> 00:36:15.940 We have six bars, six displacements. 00:36:15.940 --> 00:36:19.280 We can hope that we now have a six by six, well, 00:36:19.280 --> 00:36:23.220 we do have a six by six matrix, whatever it looks like. 00:36:23.220 --> 00:36:27.300 And we can hope that it's not singular. 00:36:27.300 --> 00:36:29.180 We can hope it's invertible, we can hope 00:36:29.180 --> 00:36:30.820 that that mechanism is killed. 00:36:30.820 --> 00:36:33.040 And you see it is killed. 00:36:33.040 --> 00:36:36.410 The six by six, that truss is now stable. 00:36:36.410 --> 00:36:42.350 No mechanism there, right? 00:36:42.350 --> 00:36:44.780 Again I haven't written down the matrix, 00:36:44.780 --> 00:36:49.920 but I'm really calling for engineering intuition here. 00:36:49.920 --> 00:36:53.630 That this truss is now stable, and of course I 00:36:53.630 --> 00:36:57.120 can make it even more stable by adding a seventh edge. 00:36:57.120 --> 00:36:59.690 A seventh bar. 00:36:59.690 --> 00:37:03.960 So when it was six I had square matrices, A transpose and then 00:37:03.960 --> 00:37:06.250 C and then A would have been square. 00:37:06.250 --> 00:37:12.630 Now I've got seven bars and, so now I've put in a seventh guy. 00:37:12.630 --> 00:37:15.400 m is now up to seven. 00:37:15.400 --> 00:37:24.010 My matrix a would now be seven by six. 00:37:24.010 --> 00:37:27.150 Mechanism will be gone because I've 00:37:27.150 --> 00:37:32.270 now got, what, seven equations. 00:37:32.270 --> 00:37:33.990 Same six u's. 00:37:33.990 --> 00:37:39.350 So we begin to get a feel of, are there solutions or not? 00:37:39.350 --> 00:37:44.750 What I'm saying is, I can't tell just from the count 00:37:44.750 --> 00:37:46.730 that A is not singular. 00:37:46.730 --> 00:37:51.880 I could have a lot of bars and still be unstable. 00:37:51.880 --> 00:37:53.300 Invent a truss for me. 00:37:53.300 --> 00:37:58.450 Just because, how could you invent a truss that had, maybe 00:37:58.450 --> 00:38:00.850 it has seven bars. 00:38:00.850 --> 00:38:05.000 With those seven bars, those diagonal guys, that did it. 00:38:05.000 --> 00:38:06.870 That made it stable. 00:38:06.870 --> 00:38:10.640 Our eye tells us that before we do any linear algebra. 00:38:10.640 --> 00:38:15.330 Tell me a seven by-- A thing. 00:38:15.330 --> 00:38:17.460 Well, yeah. 00:38:17.460 --> 00:38:23.300 OK, so here would be, shall we support both of these? 00:38:23.300 --> 00:38:26.860 I'll start out the same, OK. 00:38:26.860 --> 00:38:32.950 Now, yeah, how could I, let's see, I've haven't prepared. 00:38:32.950 --> 00:38:36.230 How could I get a whole lot of bars. 00:38:36.230 --> 00:38:39.340 I might not get seven by six exactly, but how could 00:38:39.340 --> 00:38:44.610 I have plenty of bars and still unstable? 00:38:44.610 --> 00:38:48.000 Well, suppose I do this. 00:38:48.000 --> 00:38:51.640 Oh yeah, that's a good example. 00:38:51.640 --> 00:38:54.350 That's not stable, right? 00:38:54.350 --> 00:38:54.860 OK. 00:38:54.860 --> 00:38:57.610 Every let's practice with that one. 00:38:57.610 --> 00:39:01.140 That's just my idea, and problems in the book 00:39:01.140 --> 00:39:04.410 just ask you to practice with things like that. 00:39:04.410 --> 00:39:06.070 Tell me the count, first. 00:39:06.070 --> 00:39:10.220 What is m, the number of bars? 00:39:10.220 --> 00:39:11.260 Six. 00:39:11.260 --> 00:39:14.480 What is n, the number of unknowns, little n, 00:39:14.480 --> 00:39:15.560 the number of unknowns? 00:39:15.560 --> 00:39:18.200 What's the shape of my matrix here? 00:39:18.200 --> 00:39:25.220 A is, it's got six bars and how many unknowns? 00:39:25.220 --> 00:39:26.650 Eight. 00:39:26.650 --> 00:39:28.700 Eight, right? 00:39:28.700 --> 00:39:30.580 Two here, two, two, two. 00:39:30.580 --> 00:39:31.650 None here. 00:39:31.650 --> 00:39:33.310 Six by eight. 00:39:33.310 --> 00:39:34.010 OK. 00:39:34.010 --> 00:39:38.900 And how many mechanisms am I now expecting? 00:39:38.900 --> 00:39:40.200 Probably two. 00:39:40.200 --> 00:39:45.850 Probably two, there would be two independent mechanisms here. 00:39:45.850 --> 00:39:47.350 Can you tell me what they look like? 00:39:47.350 --> 00:39:48.790 Draw them. 00:39:48.790 --> 00:39:50.600 What would they look like? 00:39:50.600 --> 00:39:53.430 What would be two different things that could happen, 00:39:53.430 --> 00:39:58.870 could go wrong with that truss? 00:39:58.870 --> 00:40:00.210 You see it, right? 00:40:00.210 --> 00:40:02.400 This could turn. 00:40:02.400 --> 00:40:08.100 As in our example with the top part moving with it. 00:40:08.100 --> 00:40:13.190 Or, a second one possibility would be the top part goes. 00:40:13.190 --> 00:40:15.260 And the bottom part stays. 00:40:15.260 --> 00:40:17.620 Or any combination. 00:40:17.620 --> 00:40:20.540 So the whole thing could go like that. 00:40:20.540 --> 00:40:21.540 That would be one. 00:40:21.540 --> 00:40:24.230 But that wouldn't be the only one, of course. 00:40:24.230 --> 00:40:27.020 So in other words, we have a two-dimensional space 00:40:27.020 --> 00:40:30.690 of mechanisms and you could give me two different, 00:40:30.690 --> 00:40:35.590 and there are not just two guys, all their combinations 00:40:35.590 --> 00:40:36.250 are there. 00:40:36.250 --> 00:40:40.436 So this would have two mechanisms. 00:40:40.436 --> 00:40:41.060 Two mechanisms. 00:40:41.060 --> 00:40:45.040 And I could put in bars, of course, 00:40:45.040 --> 00:40:46.940 that would try to save it. 00:40:46.940 --> 00:40:50.870 Well, how many bars, what's the minimum number of bars 00:40:50.870 --> 00:40:54.910 I absolutely need to make this thing stable again? 00:40:54.910 --> 00:40:55.990 Two. 00:40:55.990 --> 00:41:00.790 Well, now suppose I put in these two bars. 00:41:00.790 --> 00:41:01.480 Right? 00:41:01.480 --> 00:41:04.540 I've got enough bars, I've got an eight by eight matrix, 00:41:04.540 --> 00:41:06.250 but I haven't saved it. 00:41:06.250 --> 00:41:07.340 Right? 00:41:07.340 --> 00:41:09.700 Because it still has that mechanism. 00:41:09.700 --> 00:41:14.980 So you can't assume that because the count is right 00:41:14.980 --> 00:41:17.980 you've avoided mechanisms because in that example you 00:41:17.980 --> 00:41:18.990 haven't. 00:41:18.990 --> 00:41:23.740 OK, so that would be a case of square eight by eight, 00:41:23.740 --> 00:41:29.740 but not good. 00:41:29.740 --> 00:41:33.280 So as soon as I say there's a solution to Au=0, 00:41:33.280 --> 00:41:36.270 I know that A transpose A will be singular. 00:41:36.270 --> 00:41:38.020 And unstable. 00:41:38.020 --> 00:41:44.420 OK, before I go to the framework let's just do one more thing. 00:41:44.420 --> 00:41:51.680 Suppose I take away the supports. 00:41:51.680 --> 00:41:57.360 All right, let me put in some bars, though. 00:41:57.360 --> 00:42:00.200 I'll put in some bars. 00:42:00.200 --> 00:42:03.640 OK, plenty of bars. 00:42:03.640 --> 00:42:04.840 Want another one? 00:42:04.840 --> 00:42:06.960 OK, how many bars have I got? 00:42:06.960 --> 00:42:08.720 Lots, right? 00:42:08.720 --> 00:42:14.490 OK, now the matrix A, what do you think about this? 00:42:14.490 --> 00:42:16.084 Are there solutions? 00:42:16.084 --> 00:42:18.000 You haven't even seen the matrix A, of course, 00:42:18.000 --> 00:42:21.280 but you've seen the truss, that's what matters. 00:42:21.280 --> 00:42:25.190 How many solutions, are there solutions to Au=0? 00:42:25.190 --> 00:42:31.090 Are there ways that this truss could move without stretching? 00:42:31.090 --> 00:42:35.010 Are there ways that this truss could move without stretching? 00:42:35.010 --> 00:42:38.300 And what are they, and how many are there? 00:42:38.300 --> 00:42:40.800 And what name should we use? 00:42:40.800 --> 00:42:41.640 OK, what are they? 00:42:41.640 --> 00:42:46.410 How could that move without stretching? 00:42:46.410 --> 00:42:48.570 Well, it's got no supports at all. 00:42:48.570 --> 00:42:50.660 It's just free out there in space. 00:42:50.660 --> 00:42:51.540 So it could move. 00:42:51.540 --> 00:42:54.290 How many ways could it move? 00:42:54.290 --> 00:42:55.220 Three. 00:42:55.220 --> 00:42:58.980 It could move, everybody could move this way. 00:42:58.980 --> 00:43:01.710 All ones on the horizontal guys. 00:43:01.710 --> 00:43:03.990 Everybody could move this way, all six 00:43:03.990 --> 00:43:08.660 ones on the vertical guys, it'll be 12 unknowns here. 00:43:08.660 --> 00:43:12.690 And it could also rotate, what would be the rotation? 00:43:12.690 --> 00:43:15.490 I'm not talking about this rotation. 00:43:15.490 --> 00:43:16.810 This could not happen. 00:43:16.810 --> 00:43:17.970 What could happen? 00:43:17.970 --> 00:43:21.720 What rotation could happen here? 00:43:21.720 --> 00:43:25.020 For this, there's a third rigid motion. 00:43:25.020 --> 00:43:30.300 Translation, translation, and rotation around, 00:43:30.300 --> 00:43:34.090 well take this one as an example. 00:43:34.090 --> 00:43:37.450 The whole thing could swing around this. 00:43:37.450 --> 00:43:38.830 That would be a motion. 00:43:38.830 --> 00:43:40.370 Well now you're going to say well, 00:43:40.370 --> 00:43:42.750 why didn't I swing it around that one? 00:43:42.750 --> 00:43:44.200 And of course it could. 00:43:44.200 --> 00:43:47.630 But what would be the deal? 00:43:47.630 --> 00:43:56.240 It would have to be, there are only three rigid motions, 00:43:56.240 --> 00:43:59.030 right, up and around. 00:43:59.030 --> 00:44:01.730 So if you give me another one, like, around this one, 00:44:01.730 --> 00:44:04.370 then somehow it had to be a combination of those. 00:44:04.370 --> 00:44:07.620 I don't even want to think what combination it is. 00:44:07.620 --> 00:44:09.630 But there are three rigid motions. 00:44:09.630 --> 00:44:15.610 So I sort of distinguish mechanisms, this word 00:44:15.610 --> 00:44:16.560 mechanism. 00:44:16.560 --> 00:44:21.800 So that's where the truss deforms. 00:44:21.800 --> 00:44:27.060 In these-- And rigid motions, so I'll say plus, possibly. 00:44:27.060 --> 00:44:31.610 Plus rigid motions, and rigid motions 00:44:31.610 --> 00:44:42.020 would be, you know, it doesn't deform internally, 00:44:42.020 --> 00:44:44.400 the whole thing moves. 00:44:44.400 --> 00:44:46.450 And this is of course what we get 00:44:46.450 --> 00:44:50.140 in the case of not enough supports. 00:44:50.140 --> 00:44:53.750 And this is what we get in the case of not enough bars. 00:44:53.750 --> 00:44:54.250 Yeah. 00:44:54.250 --> 00:44:57.950 So maybe it's worth separating those two. 00:44:57.950 --> 00:45:02.320 In the examples we do, we'll usually put in enough supports 00:45:02.320 --> 00:45:04.470 to kill the rigid motions. 00:45:04.470 --> 00:45:07.630 And then the question would be are there some mechanisms. 00:45:07.630 --> 00:45:08.490 OK. 00:45:08.490 --> 00:45:14.890 Now, I have to start on what this is. 00:45:14.890 --> 00:45:16.960 Well it'll be just a very quick start. 00:45:16.960 --> 00:45:19.710 So what I'll do at the beginning of Friday, 00:45:19.710 --> 00:45:22.250 so Friday's the other lecture on this topic. 00:45:22.250 --> 00:45:25.700 And then the homework will ask you to do some trusses 00:45:25.700 --> 00:45:27.490 in this section. 00:45:27.490 --> 00:45:31.260 It's probably Section 2 point something. 00:45:31.260 --> 00:45:38.220 2.7, maybe. 00:45:38.220 --> 00:45:40.640 What's the matrix C? 00:45:40.640 --> 00:45:44.560 Last second question, what's the matrix C, what size is it? 00:45:44.560 --> 00:45:49.070 What size is the matrix C for our original problem? 00:45:49.070 --> 00:45:51.360 Or no. 00:45:51.360 --> 00:45:56.730 What size is C, is C involving, if I know 00:45:56.730 --> 00:46:00.150 these numbers, what size is C? 00:46:00.150 --> 00:46:02.520 Five by five. m by m, right? 00:46:02.520 --> 00:46:07.990 C is the diagonal matrix, one entry for each bar, 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, 00:46:15.790 --> 00:46:19.040 it's just Hooke's Law on each bar. 00:46:19.040 --> 00:46:22.160 So, simple. 00:46:22.160 --> 00:46:25.140 It gets there in the middle, just the way that C 00:46:25.140 --> 00:46:29.300 in the first exam problem popped in, and other C's. 00:46:29.300 --> 00:46:32.080 That gets there in the middle. 00:46:32.080 --> 00:46:34.580 But it's very, extremely, simple. 00:46:34.580 --> 00:46:38.970 OK, so the real attention is on A, as usual. 00:46:38.970 --> 00:46:41.990 And that will come Friday morning.