WEBVTT 00:00:00.500 --> 00:00:02.944 The following content is provided under a Creative 00:00:02.944 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.770 Your support will help MIT OpenCourseWare 00:00:05.770 --> 00:00:09.370 continue to offer high quality educational resources for free. 00:00:09.370 --> 00:00:12.530 To make a donation, or to view additional materials 00:00:12.530 --> 00:00:16.880 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.880 --> 00:00:19.390 at ocw.mit.edu. 00:00:19.390 --> 00:00:28.020 PROFESSOR STRANG: So today we move to a topic I really like. 00:00:28.020 --> 00:00:34.260 It's the beginning of the applications. 00:00:34.260 --> 00:00:37.210 So the particular application that comes first 00:00:37.210 --> 00:00:40.990 will be springs and masses, a pretty classical problem. 00:00:40.990 --> 00:00:47.250 But what we're looking for is how do we model it, 00:00:47.250 --> 00:00:51.520 what's the main framework to look 00:00:51.520 --> 00:00:53.850 at a whole series of problems. 00:00:53.850 --> 00:00:55.980 So this is number one in the series 00:00:55.980 --> 00:00:58.240 and it's the most straightforward. 00:00:58.240 --> 00:01:05.940 Let me draw it with four springs connecting three masses. 00:01:05.940 --> 00:01:08.080 And let me fix both ends. 00:01:08.080 --> 00:01:14.810 So this will be a fixed-fixed picture. 00:01:14.810 --> 00:01:17.890 So the masses have some weight. 00:01:17.890 --> 00:01:21.980 The weight pulls the springs down. 00:01:21.980 --> 00:01:28.270 When there was no weight acting they were not stretched. 00:01:28.270 --> 00:01:31.530 The masses will stretch the springs. 00:01:31.530 --> 00:01:34.950 And the question is how much do those, 00:01:34.950 --> 00:01:37.060 we're looking for the displacements. 00:01:37.060 --> 00:01:40.090 How much does mass one go down? 00:01:40.090 --> 00:01:40.590 Mass two? 00:01:40.590 --> 00:01:42.270 Mass three? 00:01:42.270 --> 00:01:45.740 And of course, essentially the displacement 00:01:45.740 --> 00:01:52.080 here is zero and here is zero. 00:01:52.080 --> 00:01:57.820 I don't know if you can imagine these masses have acted 00:01:57.820 --> 00:02:04.960 so that the position before gravity was turned on 00:02:04.960 --> 00:02:07.240 was somewhere up here and then it came here. 00:02:07.240 --> 00:02:11.170 So this moved down by a distance u_1. 00:02:11.170 --> 00:02:15.530 Let's use u for the displacements. 00:02:15.530 --> 00:02:18.100 So if I look at this main picture 00:02:18.100 --> 00:02:25.260 here I have displacements, movements, u_1, u_2, u_3. 00:02:30.470 --> 00:02:33.120 Now what happens physically? 00:02:33.120 --> 00:02:35.710 Important in every one of these examples 00:02:35.710 --> 00:02:37.235 to see what's happening physically. 00:02:37.235 --> 00:02:41.400 Of course, this one moved down by some u_2, 00:02:41.400 --> 00:02:47.340 this one moved down from its total rest position to u_3. 00:02:47.340 --> 00:02:49.490 These are not oscillating. 00:02:49.490 --> 00:02:52.690 Next week they'll start moving, time will enter. 00:02:52.690 --> 00:02:55.710 Here I'm just looking for a steady state. 00:02:55.710 --> 00:02:57.950 They come to rest, they stretch. 00:02:57.950 --> 00:03:03.740 So what's your feeling of what's going to happen here, somehow? 00:03:03.740 --> 00:03:14.580 The displacements look to me like they'll all be positive. 00:03:14.580 --> 00:03:18.100 What's the key equation going to be? 00:03:18.100 --> 00:03:23.650 That when this moves down it will stretch that spring. 00:03:23.650 --> 00:03:29.090 Hooke's Law will say there's a force, the spring pulls back. 00:03:29.090 --> 00:03:31.330 The spring pulls back with a force 00:03:31.330 --> 00:03:34.440 proportional to the stretch. 00:03:34.440 --> 00:03:40.540 So u_1, u_2, and u_3 are movements. 00:03:40.540 --> 00:03:41.860 Here is a key question. 00:03:41.860 --> 00:03:44.840 What's the stretching in spring number two? 00:03:44.840 --> 00:03:47.790 So this is spring one, two, three and four. 00:03:47.790 --> 00:03:53.900 How much does spring number two stretch? u_2-u_1. 00:03:53.900 --> 00:03:56.020 A difference is coming in there. 00:03:56.020 --> 00:03:58.230 So let me put that up here. 00:03:58.230 --> 00:04:04.620 So stretching or elongation, I'll use two words, 00:04:04.620 --> 00:04:07.840 elongation I'll say sometimes because that 00:04:07.840 --> 00:04:09.950 starts with a letter e. 00:04:09.950 --> 00:04:14.990 So these are the elongations in the springs, 00:04:14.990 --> 00:04:22.120 in the four springs. 00:04:22.120 --> 00:04:24.230 It's the amount the spring stretches. 00:04:24.230 --> 00:04:29.830 Or what's the opposite of stretching? 00:04:29.830 --> 00:04:31.750 Compression somehow. 00:04:31.750 --> 00:04:35.230 Looks to me like this last spring, at least, 00:04:35.230 --> 00:04:39.170 is going to be compressed, and I'm not sure about the others. 00:04:39.170 --> 00:04:43.250 So we've got four springs. 00:04:43.250 --> 00:04:52.500 And each one has a stretching or compression, an elongation. 00:04:52.500 --> 00:04:56.730 And then there's a link then that you already told me. 00:04:56.730 --> 00:05:01.120 That e_2 is, just from the picture, 00:05:01.120 --> 00:05:07.000 e_2 is the difference between u_2 and u_1. 00:05:07.000 --> 00:05:10.250 Because the lower mass goes down by distance u_2, 00:05:10.250 --> 00:05:15.040 the upper mass by u_1 and spring is stretched by the difference 00:05:15.040 --> 00:05:17.560 u_2-u_1. 00:05:17.560 --> 00:05:20.570 So that's a first key fact. 00:05:20.570 --> 00:05:26.310 So that expresses somehow a fact of geometry. 00:05:26.310 --> 00:05:28.840 Of sort of the way things are connected. 00:05:28.840 --> 00:05:32.230 The material properties of the springs 00:05:32.230 --> 00:05:34.340 have not got into the picture yet. 00:05:34.340 --> 00:05:37.500 But now Hooke's Law brings them into the picture. 00:05:37.500 --> 00:05:42.250 By stretching a spring that produces 00:05:42.250 --> 00:05:45.980 a force that pulls back. 00:05:45.980 --> 00:05:54.880 So we get, can I say, forces in the spring. 00:05:54.880 --> 00:06:04.030 And let me give those a name w. w_1, 2, 3, and 4. 00:06:04.030 --> 00:06:11.590 And then the link between the stretching and the force 00:06:11.590 --> 00:06:18.040 that it produces is-- So that's somehow 00:06:18.040 --> 00:06:22.950 where the properties of the material come in. 00:06:22.950 --> 00:06:27.370 So I have to say, what are the properties of the springs? 00:06:27.370 --> 00:06:31.430 So this will be Hooke's Law, this step. 00:06:31.430 --> 00:06:38.190 Hooke's Law for this particular application. 00:06:38.190 --> 00:06:46.870 And so I have to say these springs have spring constants. 00:06:46.870 --> 00:06:52.020 So I haven't completed the description of the problem 00:06:52.020 --> 00:06:55.490 until I've told you about springs 00:06:55.490 --> 00:06:57.900 themselves and the masses. 00:06:57.900 --> 00:07:03.200 So the spring constants will be c_1, c_2, c_3, and c_4. 00:07:03.200 --> 00:07:06.320 And now what does Hooke's Law say? 00:07:06.320 --> 00:07:12.410 Usually this physical law in the middle we keep it linear. 00:07:12.410 --> 00:07:19.580 Of course, we all understand that if these springs were 00:07:19.580 --> 00:07:24.080 enormously stretched the elastic property 00:07:24.080 --> 00:07:25.510 could become non-linear. 00:07:25.510 --> 00:07:29.150 It could become plastic. 00:07:29.150 --> 00:07:33.820 The first law always has somebody's name. 00:07:33.820 --> 00:07:37.280 Was the person to see that in some range 00:07:37.280 --> 00:07:41.530 of small displacements, so I guess that's the answer. 00:07:41.530 --> 00:07:44.230 We're speaking here about small displacements, 00:07:44.230 --> 00:07:46.910 small stretching up to the point where 00:07:46.910 --> 00:07:50.770 Hooke's Law continues to hold. 00:07:50.770 --> 00:07:53.560 And now what does Hooke's Law say? 00:07:53.560 --> 00:08:00.100 It says that each force in the spring 00:08:00.100 --> 00:08:07.370 is proportional to the stretching of the spring. 00:08:07.370 --> 00:08:11.570 You could say it's-- a diagonal matrix is showing up here. 00:08:11.570 --> 00:08:19.380 The vector of w's, the vector of forces in the spring is 00:08:19.380 --> 00:08:24.560 a diagonal matrix C, which just has these numbers 00:08:24.560 --> 00:08:32.080 on the diagonal, ...c_4, times the e's. 00:08:32.080 --> 00:08:38.430 So of course I'm going to write that in matrix notation as W 00:08:38.430 --> 00:08:45.100 equals a matrix C times e. 00:08:45.100 --> 00:08:49.670 So there in the middle is the physics. 00:08:49.670 --> 00:08:53.140 The material properties, the constitutive law. 00:08:53.140 --> 00:08:59.950 C can stand for constants, for constitutive law, 00:08:59.950 --> 00:09:05.130 later for conductances. 00:09:05.130 --> 00:09:08.680 It's the place where the material enters. 00:09:08.680 --> 00:09:14.860 And now how do we complete this picture? 00:09:14.860 --> 00:09:21.730 In the end we have to bring in the masses. 00:09:21.730 --> 00:09:27.910 Gravity is the external force that's making things happen. 00:09:27.910 --> 00:09:33.030 We need a force term from outside 00:09:33.030 --> 00:09:36.110 to move us away from zeroes. 00:09:36.110 --> 00:09:43.860 And that will be the downward forces f_1, f_2, 00:09:43.860 --> 00:09:48.210 f_3 on the three masses. 00:09:48.210 --> 00:09:53.790 So I plan to complete this picture with a force balance 00:09:53.790 --> 00:10:12.070 equation on the masses, on each mass. 00:10:12.070 --> 00:10:14.360 When I use the word framework there, 00:10:14.360 --> 00:10:17.320 this is what I was talking about. 00:10:17.320 --> 00:10:20.840 I guess what I want to say is I really 00:10:20.840 --> 00:10:29.820 have found that this way of describing, modeling 00:10:29.820 --> 00:10:35.930 the problem is successful for so many applications. 00:10:35.930 --> 00:10:43.310 You have somehow a geometry, a step which'll involve a matrix 00:10:43.310 --> 00:10:50.790 A. Then you have a physical step which involves a matrix C. 00:10:50.790 --> 00:10:53.940 And then finally you have a force balance. 00:10:53.940 --> 00:11:02.910 In a way this force balance or its analog, 00:11:02.910 --> 00:11:05.650 the analog would be Kirchhoff's current law. 00:11:05.650 --> 00:11:07.990 We'll see that for networks. 00:11:07.990 --> 00:11:10.140 Flow in equals flow out. 00:11:10.140 --> 00:11:13.120 Force on one side equals force on the other. 00:11:13.120 --> 00:11:15.800 If we're talking about equilibrium 00:11:15.800 --> 00:11:20.130 we can expect our model to have an equation like that. 00:11:20.130 --> 00:11:25.560 And for me it really helps to know when a new model comes in. 00:11:25.560 --> 00:11:27.310 Like somebody'll come into my office 00:11:27.310 --> 00:11:33.310 with a problem in chemistry or biology. 00:11:33.310 --> 00:11:39.920 But if it fits in this framework I'll 00:11:39.920 --> 00:11:46.380 be looking for a balance equation, a continuity 00:11:46.380 --> 00:11:52.130 equation at the end. 00:11:52.130 --> 00:11:55.170 This part was easy and it's these two parts 00:11:55.170 --> 00:11:58.520 that I want to pin down. 00:11:58.520 --> 00:12:01.430 Well you told me how to start here. 00:12:01.430 --> 00:12:07.340 So the elongation, so I want to take this step again. 00:12:07.340 --> 00:12:14.310 I want to find the elongations from some matrix that 00:12:14.310 --> 00:12:20.240 multiplies the displacements. 00:12:20.240 --> 00:12:23.000 So I'm just completing this step. 00:12:23.000 --> 00:12:30.010 And you told me what is the stretching in spring two. 00:12:30.010 --> 00:12:32.250 Again, do you mind just saying it again? 00:12:32.250 --> 00:12:35.630 The stretching in that second spring, the amount, 00:12:35.630 --> 00:12:45.120 it's made longer by the action of gravity was? u_2-u_1. 00:12:45.120 --> 00:12:46.850 u_2-u_1. 00:12:46.850 --> 00:12:55.210 So e_2 will be, a minus one here for u_1, a plus one and a zero. 00:12:55.210 --> 00:13:04.090 That will be a typical row of this matrix, the displacement 00:13:04.090 --> 00:13:06.270 stretching matrix, you could say. 00:13:06.270 --> 00:13:08.860 Now what about the stretching in e_1? 00:13:08.860 --> 00:13:10.190 What's the stretching in e_1? 00:13:15.210 --> 00:13:16.200 Only u_1. 00:13:16.200 --> 00:13:20.870 Because essentially it's u_1-u_0 but u_0 00:13:20.870 --> 00:13:24.220 is set to zero by the support. 00:13:24.220 --> 00:13:27.970 So we only have u_1. 00:13:27.970 --> 00:13:32.130 Because that multiplication just gives us-- So e_1 is u_1. 00:13:32.130 --> 00:13:35.210 e_2 is u_2-u_1. 00:13:35.210 --> 00:13:37.790 e_3 is what? 00:13:37.790 --> 00:13:43.790 The stretching in the third spring. 00:13:43.790 --> 00:13:47.790 What is it? u_3-u_2. 00:13:47.790 --> 00:13:52.890 So I need a one for u_3 and a minus one for u_2. 00:13:52.890 --> 00:13:56.690 And the stretching in the fourth spring? 00:13:56.690 --> 00:14:01.660 What's the stretching in the fourth spring? 00:14:01.660 --> 00:14:07.980 I've sort of, and you have too, mentally given a plus sign 00:14:07.980 --> 00:14:11.690 when the spring is extended and a minus 00:14:11.690 --> 00:14:13.810 sign when it's compressed. 00:14:13.810 --> 00:14:18.280 Plus for tension, minus for compression. 00:14:18.280 --> 00:14:22.900 So since I fixed that one, u_4 was zero, so what 00:14:22.900 --> 00:14:25.600 do I have in this last row? 00:14:25.600 --> 00:14:26.900 Just minus u_3. 00:14:31.450 --> 00:14:36.640 I guess what I'm saying here is that if we 00:14:36.640 --> 00:14:40.290 get a systematic approach to problems 00:14:40.290 --> 00:14:46.130 then we know we're looking for a matrix that connects these. 00:14:46.130 --> 00:14:50.140 We're looking for the material constitutive law that does this 00:14:50.140 --> 00:14:51.710 and now we're looking for this one. 00:14:51.710 --> 00:14:53.570 We kind of know where we are. 00:14:53.570 --> 00:14:55.450 What to look for. 00:14:55.450 --> 00:14:59.190 And so this matrix is the matrix I'm going to call A. 00:14:59.190 --> 00:15:01.010 So this is e=Au. 00:15:10.000 --> 00:15:12.980 Well one more step to go. 00:15:12.980 --> 00:15:16.400 And that will be the force balance step. 00:15:16.400 --> 00:15:22.010 So now, what's the equation for balance? 00:15:22.010 --> 00:15:25.790 The external forces are the masses. 00:15:25.790 --> 00:15:28.900 Well, I guess to get the units right, 00:15:28.900 --> 00:15:34.160 it should be mass times g, the gravitational constant. 00:15:34.160 --> 00:15:43.220 So let me put external forces f_1, f_2, and f_3. 00:15:43.220 --> 00:15:51.390 The three masses will be m_1*g, m_2*g, and m_3*g. 00:15:51.390 --> 00:15:55.350 So those are the forces from outside. 00:15:55.350 --> 00:15:58.800 Now it's the balance equation I'm after. 00:15:58.800 --> 00:16:01.570 So this is in this position. 00:16:01.570 --> 00:16:03.910 It's in equilibrium. 00:16:03.910 --> 00:16:05.970 And what does that tell us? 00:16:05.970 --> 00:16:09.020 That tells us that the total force on this mass, 00:16:09.020 --> 00:16:12.300 so I'm going to take each mass, it's 00:16:12.300 --> 00:16:15.440 like a free body force diagram here. 00:16:15.440 --> 00:16:17.770 I'm looking now at that mass. 00:16:17.770 --> 00:16:20.870 I'm saying what forces are acting on it 00:16:20.870 --> 00:16:23.380 and I'm making them balance. 00:16:23.380 --> 00:16:25.800 So what equation will that give me? 00:16:25.800 --> 00:16:27.010 So let me write that. 00:16:27.010 --> 00:16:30.290 This is now the force balance equation. 00:16:30.290 --> 00:16:37.830 Force balance at each mass. 00:16:37.830 --> 00:16:41.280 How much force is pulling up? 00:16:41.280 --> 00:16:44.320 What's the force pulling up on? 00:16:44.320 --> 00:16:47.580 So this spring is pulling upwards. 00:16:47.580 --> 00:16:51.750 And it's pulling upwards by w_1, right? 00:16:51.750 --> 00:16:54.160 Just getting these letters right. 00:16:54.160 --> 00:17:00.090 The w's were the internal resisting force, 00:17:00.090 --> 00:17:03.930 reacting force in the spring. w_1 is pulling up. 00:17:03.930 --> 00:17:09.570 What other forces are acting? w_2 is pulling down. 00:17:09.570 --> 00:17:14.550 And also pulling down is? 00:17:14.550 --> 00:17:16.760 Gravity, m_1*g. 00:17:16.760 --> 00:17:22.480 So the balance of forces there says that w_1, the force up, 00:17:22.480 --> 00:17:27.980 is w_2, the force down, and m_1*g. 00:17:27.980 --> 00:17:31.990 And similarly the next one will have, 00:17:31.990 --> 00:17:35.740 the next mass if I look just at that I see a force up, 00:17:35.740 --> 00:17:38.080 a force down and gravity down. 00:17:38.080 --> 00:17:45.990 So w_2 will be, well, that's the pull up, will be w_3+m_2*g. 00:17:45.990 --> 00:17:49.410 And the third one, the force up on the third one 00:17:49.410 --> 00:17:53.600 will be the force down on the third one. 00:17:53.600 --> 00:17:56.920 so I think those are the equations of force 00:17:56.920 --> 00:18:00.980 balance written one at a time. 00:18:00.980 --> 00:18:08.080 And now, of course I'm going to write that-- 00:18:08.080 --> 00:18:14.990 So that's three equations with four w's. 00:18:14.990 --> 00:18:20.010 So I want to write that as, I want to bring the w's 00:18:20.010 --> 00:18:21.640 all to the left-hand side. 00:18:21.640 --> 00:18:25.310 Can I do that? 00:18:25.310 --> 00:18:29.910 Can I just bring those over with minus signs? 00:18:29.910 --> 00:18:34.710 And make these equal signs. 00:18:34.710 --> 00:18:41.620 So now we've got internal force balancing external force. 00:18:41.620 --> 00:18:45.670 This vector of external forces is the f's and this 00:18:45.670 --> 00:18:47.620 is the internal forces. 00:18:47.620 --> 00:18:51.350 Now somewhere there we're going to see a matrix. 00:18:51.350 --> 00:18:54.670 So I'm going to write this equation as some matrix. 00:18:54.670 --> 00:18:57.750 Well, let's figure out what that matrix is. 00:18:57.750 --> 00:19:01.180 So its shape is what? 00:19:01.180 --> 00:19:05.470 I've got three equations, so I need three rows in the matrix. 00:19:05.470 --> 00:19:09.060 I've got four w's so I need four columns. 00:19:09.060 --> 00:19:14.790 So it's going to multiply w_1, w_2, w_3, w_4 00:19:14.790 --> 00:19:21.940 to give these three masses, can I call them f_1, f_2, f_3, 00:19:21.940 --> 00:19:26.890 just to have a good letter. 00:19:26.890 --> 00:19:29.350 We're almost there. 00:19:29.350 --> 00:19:30.900 What's the matrix? 00:19:30.900 --> 00:19:37.290 What's the matrix for this final step, the force balance 00:19:37.290 --> 00:19:38.540 equation? 00:19:38.540 --> 00:19:40.700 I just read it off. 00:19:40.700 --> 00:19:42.810 w_1-w_2. 00:19:42.810 --> 00:19:47.280 I think I've got that. 00:19:47.280 --> 00:19:49.940 w_2-w_3. 00:19:49.940 --> 00:19:52.030 Tell me what the second row of the matrix 00:19:52.030 --> 00:19:56.390 looks like to give me w_2-w_3. 00:19:56.390 --> 00:20:01.250 zero, one for the w_2, minus one. 00:20:01.250 --> 00:20:02.280 Good. 00:20:02.280 --> 00:20:13.260 And for the third, the final row? [0, 0, 1, -1]. 00:20:13.260 --> 00:20:17.240 So that completes the third piece. 00:20:17.240 --> 00:20:21.900 If I'd given you the problem as I did, drawn the problem, 00:20:21.900 --> 00:20:26.270 described it, you know that there's 00:20:26.270 --> 00:20:29.450 going to be a connection between the external forces 00:20:29.450 --> 00:20:32.270 and the displacements. 00:20:32.270 --> 00:20:34.700 But what I'm trying to say is a good way 00:20:34.700 --> 00:20:39.650 to see the connection is to see it in three simple steps. 00:20:39.650 --> 00:20:42.770 The simple step that gets you from the displacements 00:20:42.770 --> 00:20:44.730 to the springs. 00:20:44.730 --> 00:20:47.140 A second step within the springs. 00:20:47.140 --> 00:20:51.310 A third step back to the nodes, you 00:20:51.310 --> 00:20:53.990 could say, back to the masses. 00:20:53.990 --> 00:20:59.430 And of course, the key question is, what's that matrix? 00:20:59.430 --> 00:21:02.940 And do you recognize it? 00:21:02.940 --> 00:21:05.770 Do we need a new name for that matrix? 00:21:05.770 --> 00:21:08.230 The matrix in the third step? 00:21:08.230 --> 00:21:13.470 So this third step is going to be that some matrix times 00:21:13.470 --> 00:21:19.070 w is f and what's that matrix? 00:21:19.070 --> 00:21:22.290 What's the good name for us to give it? 00:21:22.290 --> 00:21:25.170 A transpose is the best possible name. 00:21:25.170 --> 00:21:28.010 If we've given this matrix the name 00:21:28.010 --> 00:21:33.500 A, the stretching displacement matrix, 00:21:33.500 --> 00:21:38.650 the strain in elasticity, this becomes the strains, 00:21:38.650 --> 00:21:40.930 these become the stresses. 00:21:40.930 --> 00:21:43.720 But the beauty is, just beautiful, 00:21:43.720 --> 00:21:49.070 that the matrix in this law is the transpose of this one. 00:21:49.070 --> 00:21:51.830 So it's A transpose. 00:21:51.830 --> 00:21:59.510 So that's the framework seen now here for the first time. 00:21:59.510 --> 00:22:03.640 So the key point was that A and A transpose both 00:22:03.640 --> 00:22:08.060 appeared but with physical material properties, 00:22:08.060 --> 00:22:10.190 constitutive matrix in between. 00:22:10.190 --> 00:22:15.720 So if we put the pieces together, then we're golden. 00:22:15.720 --> 00:22:19.380 And then, let's do an example to see what actually happened. 00:22:19.380 --> 00:22:31.240 So the equations were e=Aw-- e=Au, then w=Ce, 00:22:31.240 --> 00:22:34.010 that's Hooke's Law, and then A transpose-- 00:22:34.010 --> 00:22:40.020 or maybe I'll write it as f=A transpose*w. 00:22:42.900 --> 00:22:45.530 That's the three steps. 00:22:45.530 --> 00:22:50.770 So in this problem the source term showed up at that point. 00:22:50.770 --> 00:22:55.700 The source term came from external forces. 00:22:55.700 --> 00:22:57.780 I've got three equations. 00:22:57.780 --> 00:23:00.570 Now I'm going to put them together into one. 00:23:00.570 --> 00:23:02.560 I'll put them into one equation. 00:23:02.560 --> 00:23:05.510 So this w I'll just substitute. 00:23:05.510 --> 00:23:11.740 So it's A transpose w is Ce, and e is Au. 00:23:11.740 --> 00:23:16.170 So I have A transpose C Au. 00:23:16.170 --> 00:23:20.790 So that's the ultimate. 00:23:20.790 --> 00:23:23.810 That's put the whole structure together. 00:23:23.810 --> 00:23:27.850 That's the equation you have to solve. 00:23:27.850 --> 00:23:35.780 This would be called the stiffness matrix. 00:23:35.780 --> 00:23:42.930 And I use the letter K for that one. 00:23:42.930 --> 00:23:46.800 So our equation is Ku=f. 00:23:46.800 --> 00:23:56.590 This is our final equation. 00:23:56.590 --> 00:24:03.140 Well, we didn't know w. 00:24:03.140 --> 00:24:05.520 There are two unknowns here. 00:24:05.520 --> 00:24:08.460 Two physical things that you want to find. 00:24:08.460 --> 00:24:11.460 If you're designing a bridge or a structure 00:24:11.460 --> 00:24:14.640 you want to know the displacements 00:24:14.640 --> 00:24:21.720 and then you want to know the internal forces w. 00:24:21.720 --> 00:24:22.690 It's really beautiful. 00:24:22.690 --> 00:24:28.860 The two unknowns of u and w are somehow dual, 00:24:28.860 --> 00:24:36.350 we can work with one, work with the other, work with both. 00:24:36.350 --> 00:24:40.390 Oh let me just mention that the finite element method will 00:24:40.390 --> 00:24:46.590 fit this framework and somehow this name stiffness matrix has 00:24:46.590 --> 00:24:50.970 become famous for finite elements in structures 00:24:50.970 --> 00:25:01.630 and then it's just exploded to appear all over the place. 00:25:01.630 --> 00:25:03.273 I guess we should look at A transpose 00:25:03.273 --> 00:25:09.710 C A. We can see what it looks like. 00:25:09.710 --> 00:25:14.010 And also just from the way it looks there. 00:25:14.010 --> 00:25:16.440 So I can write it out explicitly. 00:25:16.440 --> 00:25:18.040 I think we want to. 00:25:18.040 --> 00:25:20.990 But at the same time I can learn something 00:25:20.990 --> 00:25:25.190 from just seeing how it's put together. 00:25:25.190 --> 00:25:28.190 What can you tell me about A transpose C A? 00:25:28.190 --> 00:25:30.380 Let's get the shape first. 00:25:30.380 --> 00:25:33.400 Just to see the shape of these things. 00:25:33.400 --> 00:25:36.510 The matrix A is what? 00:25:36.510 --> 00:25:39.400 What's the shape of A? 00:25:39.400 --> 00:25:40.960 It's over here. 00:25:40.960 --> 00:25:43.180 Four by three. 00:25:43.180 --> 00:25:45.880 Four by three. 00:25:45.880 --> 00:25:52.400 And the shape of C was, three by three is it? 00:25:52.400 --> 00:25:55.870 Where have I got, that C matrix better be here somewhere. 00:25:55.870 --> 00:25:58.090 Oh, no, it's four by four. 00:25:58.090 --> 00:25:59.110 Four springs. 00:25:59.110 --> 00:26:02.440 Of course, it had to be four by four to do that multiplication. 00:26:02.440 --> 00:26:04.310 There's the C matrix. 00:26:04.310 --> 00:26:06.320 Four by four, thanks. 00:26:06.320 --> 00:26:09.280 And the A transpose matrix? 00:26:09.280 --> 00:26:11.200 Three by four, thanks. 00:26:11.200 --> 00:26:15.470 So the net result is three by three. 00:26:15.470 --> 00:26:15.970 Good. 00:26:15.970 --> 00:26:20.220 So it's a square matrix. 00:26:20.220 --> 00:26:22.260 K is a square matrix. 00:26:22.260 --> 00:26:27.550 What else can you tell me about it? 00:26:27.550 --> 00:26:31.480 Now we're going to begin to use some of the, sort of the matrix 00:26:31.480 --> 00:26:34.120 preparation. 00:26:34.120 --> 00:26:37.430 These matrices are kind of friends by now. 00:26:37.430 --> 00:26:42.081 This is a difference matrix, somehow. 00:26:42.081 --> 00:26:42.580 Right? 00:26:42.580 --> 00:26:47.430 The stretchings are differences and displacements. 00:26:47.430 --> 00:26:49.230 That's its transpose. 00:26:49.230 --> 00:26:53.130 And then the C matrix, which is the new thing, sort 00:26:53.130 --> 00:26:59.570 of the new guy to appear today, is diagonal. 00:26:59.570 --> 00:27:03.340 Well if I asked you now, without writing out the matrix, 00:27:03.340 --> 00:27:06.030 for one more property, it's square, 00:27:06.030 --> 00:27:08.300 what else could you tell me about it? 00:27:08.300 --> 00:27:12.810 Symmetric is going to be a very good guess and let's see why. 00:27:12.810 --> 00:27:15.150 Why is it symmetric? 00:27:15.150 --> 00:27:19.470 How do we show that that? 00:27:19.470 --> 00:27:21.020 What do I do? 00:27:21.020 --> 00:27:25.300 I take the transpose. 00:27:25.300 --> 00:27:32.530 If I take my K transpose, now I write it as, what do I do? 00:27:32.530 --> 00:27:34.890 It's a product of things. 00:27:34.890 --> 00:27:39.210 So when I transpose a product I have the individual transposes 00:27:39.210 --> 00:27:40.430 in the opposite order. 00:27:40.430 --> 00:27:43.900 So A, its transpose comes first. 00:27:43.900 --> 00:27:46.760 C, its transpose comes next. 00:27:46.760 --> 00:27:51.730 A transpose, its transpose comes last. 00:27:51.730 --> 00:27:57.270 So that's just the rules of matrix transposes. 00:27:57.270 --> 00:27:58.430 Now what? 00:27:58.430 --> 00:28:02.670 Now I'm ready to use the wonderful fact of what 00:28:02.670 --> 00:28:03.800 we've got here. 00:28:03.800 --> 00:28:06.630 So what is C transpose? 00:28:06.630 --> 00:28:14.080 So notice we wanted a symmetric matrix in the middle 00:28:14.080 --> 00:28:16.450 to be able to knock that T out. 00:28:16.450 --> 00:28:19.200 And what is A transpose transpose? 00:28:19.200 --> 00:28:24.900 That's A. We've learned that the thing is symmetric, 00:28:24.900 --> 00:28:30.540 that if I transpose it I get it back again. 00:28:30.540 --> 00:28:33.160 We're going to see more about that. 00:28:33.160 --> 00:28:38.370 But let me do the multiplication. 00:28:38.370 --> 00:28:41.840 So I'm going to take that, oh, boy. 00:28:41.840 --> 00:28:45.100 How am I going to do that? 00:28:45.100 --> 00:28:48.660 I want to multiply three matrices to see what K actually 00:28:48.660 --> 00:28:52.440 looks like here. 00:28:52.440 --> 00:28:55.330 One question first. 00:28:55.330 --> 00:29:01.200 Eventually the solution, the short formula for the solution, 00:29:01.200 --> 00:29:03.880 will be u equal K inverse f. 00:29:03.880 --> 00:29:04.860 Right? 00:29:04.860 --> 00:29:11.780 So the answer will be u equal K inverse f in matrix notation 00:29:11.780 --> 00:29:16.210 but I'm looking for numbers. 00:29:16.210 --> 00:29:22.160 And then if I know u then I know the stretching. e is A times K 00:29:22.160 --> 00:29:24.190 inverse f. 00:29:24.190 --> 00:29:29.680 And w is, I'm just going down the list, is C times A times 00:29:29.680 --> 00:29:30.580 K inverse f. 00:29:30.580 --> 00:29:32.500 We've got everything. 00:29:32.500 --> 00:29:34.770 So that's the key. 00:29:34.770 --> 00:29:36.220 This is the key equation. 00:29:36.220 --> 00:29:39.470 That's the answer. 00:29:39.470 --> 00:29:44.080 Let me ask you about inverses. 00:29:44.080 --> 00:29:46.320 What about K inverse? 00:29:46.320 --> 00:29:49.050 We took three steps. 00:29:49.050 --> 00:29:54.030 Now what if I just ask you about inverses? 00:29:54.030 --> 00:29:56.840 This is K inverse that we would like to know. 00:29:56.840 --> 00:30:04.650 So again, for inverses I'm going to start this 00:30:04.650 --> 00:30:09.690 and I'm going to stop halfway and you'll tell me why. 00:30:09.690 --> 00:30:11.800 If you give me a product of matrices 00:30:11.800 --> 00:30:15.080 and I don't think particularly much 00:30:15.080 --> 00:30:18.080 I'll take the inverse of that times the inverse of that 00:30:18.080 --> 00:30:25.280 times the inverse of that. 00:30:25.280 --> 00:30:31.490 And what's the matter with that? 00:30:31.490 --> 00:30:35.360 You would say, why not just undo each step? 00:30:35.360 --> 00:30:45.620 Why not find the w's from the f's and then the e's from 00:30:45.620 --> 00:30:52.830 the w's by dividing and then the u's from the e's? 00:30:52.830 --> 00:30:56.510 Why don't we just go backwards around the loop 00:30:56.510 --> 00:31:00.570 rather than what I'm saying we have to do. 00:31:00.570 --> 00:31:05.210 We eventually get this step across with a matrix K 00:31:05.210 --> 00:31:13.840 that does all three at once. 00:31:13.840 --> 00:31:15.710 Well sometimes we might be able to, 00:31:15.710 --> 00:31:18.810 but I don't think we can in this time. 00:31:18.810 --> 00:31:21.280 What's the trouble with A, that I 00:31:21.280 --> 00:31:26.310 don't want to write A inverse? 00:31:26.310 --> 00:31:28.290 Well I don't say singular. 00:31:28.290 --> 00:31:30.900 What do I say here? 00:31:30.900 --> 00:31:37.020 Look at this matrix A here. 00:31:37.020 --> 00:31:38.650 It's not square. 00:31:38.650 --> 00:31:40.780 It's not square, that's right. 00:31:40.780 --> 00:31:46.700 So I'm not comfortable, I'm not willing to write A inverse 00:31:46.700 --> 00:31:51.070 when A is not a square matrix. 00:31:51.070 --> 00:31:55.700 And this distinction, is the matrix A square or not, 00:31:55.700 --> 00:31:57.630 is the first issue. 00:31:57.630 --> 00:31:59.090 It's just the picture. 00:31:59.090 --> 00:32:01.810 Let me show you an example of where it would be square. 00:32:01.810 --> 00:32:02.590 May I? 00:32:02.590 --> 00:32:07.060 Before I do this multiplication, can I jump to a-- I'll 00:32:07.060 --> 00:32:15.410 change the line of springs in a way that'll change A. 00:32:15.410 --> 00:32:16.940 And let me show you what happens. 00:32:16.940 --> 00:32:23.030 Suppose I take out that spring. 00:32:23.030 --> 00:32:27.800 So I've removed the fourth spring. 00:32:27.800 --> 00:32:31.120 It's a line of springs now, hanging from a support. 00:32:31.120 --> 00:32:33.040 It's a perfectly good problem. 00:32:33.040 --> 00:32:37.150 It's problem two, but it's a different problem. 00:32:37.150 --> 00:32:39.040 And what's different now? 00:32:39.040 --> 00:32:44.080 There is no fourth spring. 00:32:44.080 --> 00:32:46.990 If this was my problem, what would be different? 00:32:46.990 --> 00:32:50.190 There's no fourth spring. 00:32:50.190 --> 00:32:53.010 So that's gone. 00:32:53.010 --> 00:32:56.980 I just have three springs stretching from three masses. 00:32:56.980 --> 00:32:59.740 Then the force balance is the same. 00:32:59.740 --> 00:33:03.690 Everything looks the same except there's no force, 00:33:03.690 --> 00:33:06.680 there's no fourth spring, so there's no force there, 00:33:06.680 --> 00:33:11.330 that's gone. 00:33:11.330 --> 00:33:14.940 And of course, how does C change? 00:33:14.940 --> 00:33:22.090 So in my new picture now I have, let me write now, 00:33:22.090 --> 00:33:27.730 A transpose C A. A is now three by three, 00:33:27.730 --> 00:33:29.550 right, I've lost a row. 00:33:29.550 --> 00:33:31.810 A transpose is now three by three, 00:33:31.810 --> 00:33:34.700 I've lost a column, that fourth spring is gone. 00:33:34.700 --> 00:33:36.270 And what is C? 00:33:36.270 --> 00:33:43.370 Well of course there's no guy here anymore. 00:33:43.370 --> 00:33:50.170 What I'm trying to say is for this problem the matrices have 00:33:50.170 --> 00:33:54.280 become square. 00:33:54.280 --> 00:33:56.890 This would be correct. 00:33:56.890 --> 00:34:00.280 So this is an especially nice kind of problem. 00:34:00.280 --> 00:34:03.160 It's called statically determinate. 00:34:03.160 --> 00:34:08.830 It means I can determine the three w's from the three f's. 00:34:08.830 --> 00:34:09.850 I can go backwards. 00:34:09.850 --> 00:34:11.680 Everything is determined. 00:34:11.680 --> 00:34:16.430 The long word for the fixed-fixed one, 00:34:16.430 --> 00:34:19.830 our main example, is statically indeterminate. 00:34:19.830 --> 00:34:24.020 I cannot determine four w's from three forces. 00:34:24.020 --> 00:34:27.640 I can't determine what these internal forces are 00:34:27.640 --> 00:34:33.330 until I put the whole loop into one matrix K. 00:34:33.330 --> 00:34:36.480 So that's like a warning, and at the same time, 00:34:36.480 --> 00:34:39.690 an important separation. 00:34:39.690 --> 00:34:45.030 A few nice problems where you don't have too many springs, 00:34:45.030 --> 00:34:47.790 you don't have too many bars in a truss. 00:34:47.790 --> 00:34:51.390 You just have like, the minimum number to hold it together. 00:34:51.390 --> 00:34:55.350 Could be statically determinate and square matrices. 00:34:55.350 --> 00:34:59.840 But here we're not square. 00:34:59.840 --> 00:35:01.580 Now I go back. 00:35:01.580 --> 00:35:05.120 So that would be fixed-free. 00:35:05.120 --> 00:35:05.620 Right? 00:35:05.620 --> 00:35:09.940 That example that I just described would be fixed-free 00:35:09.940 --> 00:35:14.300 and we can kind of carry that along because we know that what 00:35:14.300 --> 00:35:19.570 happens is we lose a row and a column 00:35:19.570 --> 00:35:24.070 and a-- c_4 is just not in the picture anymore. 00:35:24.070 --> 00:35:32.000 But now I want to go back to the fixed-fixed one and finish it. 00:35:32.000 --> 00:35:38.590 So that's got a support down there, too. 00:35:38.590 --> 00:35:41.710 Key question, what's this matrix K? 00:35:41.710 --> 00:35:47.830 This A transpose C A. We know it's a square matrix, 00:35:47.830 --> 00:35:49.920 we know it's a symmetric matrix, but it 00:35:49.920 --> 00:35:53.050 would be really nice to know what does it look like. 00:35:53.050 --> 00:35:55.310 What does that matrix look like? 00:35:55.310 --> 00:35:57.410 Can I do the multiplication? 00:35:57.410 --> 00:36:03.170 So this is going to be K. So it starts with a three by four. 00:36:03.170 --> 00:36:08.560 1, -1; 1, -1; 1 -1. 00:36:08.560 --> 00:36:14.020 Then it's got the four by four, c_1, c_2, c_3, c_4. 00:36:14.020 --> 00:36:16.390 And then it's got the transpose of that, which 00:36:16.390 --> 00:36:22.290 is the 1, -1; 1, -1; 1, -1. 00:36:22.290 --> 00:36:32.380 With zeroes where I didn't write anything. 00:36:32.380 --> 00:36:38.690 We've got three matrices to multiply together. 00:36:38.690 --> 00:36:41.400 What's going to happen here? 00:36:41.400 --> 00:36:43.260 Well, let's see. 00:36:43.260 --> 00:36:45.100 I guess, why don't I multiply that by that? 00:36:45.100 --> 00:36:47.120 Can I do that? 00:36:47.120 --> 00:36:50.690 So that's like getting two steps together. 00:36:50.690 --> 00:36:53.140 It's going to be easy because of this. 00:36:53.140 --> 00:36:56.050 This is usually an easy matrix. 00:36:56.050 --> 00:36:57.570 Often diagonal. 00:36:57.570 --> 00:37:00.500 So when I do that multiplication, so let me, 00:37:00.500 --> 00:37:05.690 I'll just copy this guy. 00:37:05.690 --> 00:37:13.100 And now c_1 multiplies that row, c_2 multiplies this row, 00:37:13.100 --> 00:37:24.060 c_3 multiplies this row and c_4 multiplies the last row. 00:37:24.060 --> 00:37:26.850 c_1 in that row, c_2, c_3, and c_4. 00:37:26.850 --> 00:37:30.820 And now I'm ready to put those together into K. 00:37:30.820 --> 00:37:33.080 So K will be three by three. 00:37:33.080 --> 00:37:34.670 What does it have? 00:37:34.670 --> 00:37:37.270 It has c_1+c_2. 00:37:41.010 --> 00:37:44.730 And then next to that is going to be this row 00:37:44.730 --> 00:37:47.940 one against column two, there'll be a zero 00:37:47.940 --> 00:37:50.800 or they'll be a -c_2 here. 00:37:50.800 --> 00:37:55.060 And then when row one goes against column three 00:37:55.060 --> 00:38:02.010 there's nothing. 00:38:02.010 --> 00:38:05.790 Why nothing? 00:38:05.790 --> 00:38:10.540 When do I expect to see a zero in the overall matrix? 00:38:10.540 --> 00:38:12.110 What is it about? 00:38:12.110 --> 00:38:16.390 So that zero is in the position 1, 3. 00:38:16.390 --> 00:38:22.300 What is it about masses one and three that 00:38:22.300 --> 00:38:24.320 is putting that zero in there. 00:38:24.320 --> 00:38:30.470 We kind of expect to see that zero even before we find it. 00:38:30.470 --> 00:38:32.780 If I look at the picture, what do you 00:38:32.780 --> 00:38:35.850 notice about masses one and three that 00:38:35.850 --> 00:38:38.420 is going to produce the zero? 00:38:38.420 --> 00:38:40.200 They're not connected. 00:38:40.200 --> 00:38:41.400 They're not connected. 00:38:41.400 --> 00:38:44.620 If I had another spring, which I could have, 00:38:44.620 --> 00:38:49.050 connecting mass one to mass three that would produce, 00:38:49.050 --> 00:38:50.700 I'd have another. 00:38:50.700 --> 00:38:53.430 I'd be up to five. 00:38:53.430 --> 00:38:55.740 Instead of four, there'd be a fifth spring. 00:38:55.740 --> 00:38:57.430 It would have its own constant. 00:38:57.430 --> 00:38:59.300 It would show up. 00:38:59.300 --> 00:39:00.860 Absolutely could. 00:39:00.860 --> 00:39:03.180 Here we don't have it. 00:39:03.180 --> 00:39:04.490 Now let me keep going. 00:39:04.490 --> 00:39:07.170 I know from symmetry that the second row 00:39:07.170 --> 00:39:13.510 times this is going to be zero, is going to be -c_2. 00:39:13.510 --> 00:39:15.270 Symmetric as I expected. 00:39:15.270 --> 00:39:21.450 What are you expecting on the diagonal there? c_2+c_3. 00:39:21.450 --> 00:39:24.310 That's certainly the right pattern. 00:39:24.310 --> 00:39:27.770 Zero, c_2+c_3. 00:39:27.770 --> 00:39:30.230 c_2+c_3. 00:39:30.230 --> 00:39:34.310 And what are you expecting over here? 00:39:34.310 --> 00:39:36.880 -c_3 is a good guess. 00:39:36.880 --> 00:39:38.500 It's seeing that pattern. 00:39:38.500 --> 00:39:40.010 Let's just see it happen. 00:39:40.010 --> 00:39:43.100 That second row times this third guy 00:39:43.100 --> 00:39:51.470 will give me zero, two rows, two zeroes, and then a -c_3, good. 00:39:51.470 --> 00:39:55.200 And now we know the zeroes going to show up here, 00:39:55.200 --> 00:39:57.840 the -c_3 is going to show up here. 00:39:57.840 --> 00:40:01.590 And what will show up here? c_3+c_4. 00:40:13.480 --> 00:40:14.610 So we've got it. 00:40:14.610 --> 00:40:18.500 That's the matrix K that controls this whole problem. 00:40:18.500 --> 00:40:20.570 Now we check. 00:40:20.570 --> 00:40:22.440 It's square, yes. 00:40:22.440 --> 00:40:25.680 It's symmetric, yes. 00:40:25.680 --> 00:40:31.550 And notice also it's the kind of matrix we've seen already. 00:40:31.550 --> 00:40:35.960 In fact, it's exactly the matrix we've seen already 00:40:35.960 --> 00:40:38.550 Suppose all the c's are one. 00:40:38.550 --> 00:40:42.820 Suppose every c_1, c_2, c_3, c_4 is one. 00:40:42.820 --> 00:40:51.650 Then what's the matrix capital C in that standard case? 00:40:51.650 --> 00:40:56.490 C will just be the identity if these are all ones. 00:40:56.490 --> 00:41:00.460 And then I'm only left with A transpose A. 00:41:00.460 --> 00:41:08.040 So let me take that special case below it. 00:41:08.040 --> 00:41:15.020 Special if so this is if C is I, what matrix do we have then? 00:41:15.020 --> 00:41:18.960 Just to see that we have a matrix that we know about. 00:41:18.960 --> 00:41:24.060 So I'm copying this now here in the case when all the c's are 00:41:24.060 --> 00:41:24.560 one. 00:41:24.560 --> 00:41:28.860 So if you put all those c's to be one, what matrix do you get? 00:41:28.860 --> 00:41:30.030 You get, yes. 00:41:30.030 --> 00:41:38.260 You get the special K. Right, you get the special. 00:41:38.260 --> 00:41:43.440 So the work we did to understand that special matrix pays off 00:41:43.440 --> 00:41:44.350 here. 00:41:44.350 --> 00:41:47.130 Because we know how that matrix works. 00:41:47.130 --> 00:41:56.560 And this matrix, well, it's got four spring constants in it. 00:41:56.560 --> 00:42:03.400 But we can guess the important facts about this one 00:42:03.400 --> 00:42:04.780 from this one. 00:42:04.780 --> 00:42:11.210 So what are the important questions about that matrix? 00:42:11.210 --> 00:42:14.840 This is my matrix K now. 00:42:14.840 --> 00:42:17.570 What would be, we know it's square, we know it's symmetric. 00:42:17.570 --> 00:42:20.900 What else do we ask about a matrix? 00:42:20.900 --> 00:42:25.240 Well, positive definite, that's the perfect question, right. 00:42:25.240 --> 00:42:27.960 And built into positive definiteness 00:42:27.960 --> 00:42:32.360 would be a property that we mentioned the very first day. 00:42:32.360 --> 00:42:34.850 Is it invertible? 00:42:34.850 --> 00:42:36.640 What's your guess? 00:42:36.640 --> 00:42:39.160 Is that matrix invertible? 00:42:39.160 --> 00:42:44.780 Everybody's going to guess yes because, if you guessed no, 00:42:44.780 --> 00:42:50.050 where would you be, the whole course would end. 00:42:50.050 --> 00:42:55.480 In fact, the world would end because the problem 00:42:55.480 --> 00:42:57.430 is correctly posed. 00:42:57.430 --> 00:43:00.630 Those displacements are determined by the forces 00:43:00.630 --> 00:43:03.510 and that just says K is an invertible matrix. 00:43:03.510 --> 00:43:06.820 So but how do we see that it's invertible and, even 00:43:06.820 --> 00:43:09.530 more, positive definite, because that's 00:43:09.530 --> 00:43:11.000 the property we now know. 00:43:11.000 --> 00:43:14.500 So why is that matrix positive definite? 00:43:14.500 --> 00:43:18.220 Do we want to check determinants? 00:43:18.220 --> 00:43:21.410 We could say, okay, that guy's positive. 00:43:21.410 --> 00:43:29.150 We could evaluate this product and find that it came out well. 00:43:29.150 --> 00:43:33.140 Would you want to do that one? 00:43:33.140 --> 00:43:38.000 We could probably do the two by two determinant. 00:43:38.000 --> 00:43:42.470 Could you take that times that and subtract that? 00:43:42.470 --> 00:43:46.130 Let's just write it above what we would get. 00:43:46.130 --> 00:43:50.300 Just to see it. 00:43:50.300 --> 00:43:54.610 That number times that number would be a c_1*c_2 00:43:54.610 --> 00:44:02.130 and a c_1*c_3 and a c_2*c_2 twice. 00:44:02.130 --> 00:44:04.840 And a c_2*c_3. 00:44:04.840 --> 00:44:08.180 And then I would subtract off this guy. 00:44:08.180 --> 00:44:13.910 So it would knock out that, right? 00:44:13.910 --> 00:44:19.000 And it would leave something that would be positive. 00:44:19.000 --> 00:44:21.340 All the spring constants are positive here. 00:44:21.340 --> 00:44:27.030 We're talking normal materials. 00:44:27.030 --> 00:44:30.170 I guess, actually, people are producing now 00:44:30.170 --> 00:44:34.880 really amazing materials with amazing properties. 00:44:34.880 --> 00:44:41.480 And the amazing property is a material with a negative c. 00:44:41.480 --> 00:44:47.360 But that's like-- 18.085 does not allow such a thing. 00:44:47.360 --> 00:44:49.320 Right? 00:44:49.320 --> 00:44:52.130 All these c's are positive. 00:44:52.130 --> 00:44:56.050 And you might guess that the whole determinant is positive. 00:44:56.050 --> 00:44:58.930 But now I'd like you to tell me why. 00:44:58.930 --> 00:45:07.560 So now we can use our growing familiarity 00:45:07.560 --> 00:45:17.870 with matrices to say why is this matrix positive definite. 00:45:17.870 --> 00:45:21.900 Is symmetric, of course. 00:45:21.900 --> 00:45:27.710 Positive definite. 00:45:27.710 --> 00:45:30.900 Why? 00:45:30.900 --> 00:45:36.460 So that's what the previous lecture helped us to answer. 00:45:36.460 --> 00:45:38.730 We've got these various tests, but what 00:45:38.730 --> 00:45:42.910 was the core idea of positive definiteness? 00:45:42.910 --> 00:45:46.720 The core idea was positive energy. 00:45:46.720 --> 00:45:51.190 The core idea was I looked at the energy x trans-- no, u, 00:45:51.190 --> 00:45:51.830 sorry. 00:45:51.830 --> 00:45:56.260 Have to call it u now. u transpose times 00:45:56.260 --> 00:46:00.770 that matrix times u. 00:46:00.770 --> 00:46:08.780 And there was a reason why that matrix was, why this number, 00:46:08.780 --> 00:46:13.150 it's going to be a number, right? 00:46:13.150 --> 00:46:16.220 This combination will involve all four of these 00:46:16.220 --> 00:46:19.470 c's, it'll involve three u's. 00:46:19.470 --> 00:46:22.470 I don't want to write out that quantity. 00:46:22.470 --> 00:46:27.420 It would be, I'll have some u_1 squareds and some u_1*u_2. 00:46:27.420 --> 00:46:31.500 I won't have any u_1*u_3, because that 1, 00:46:31.500 --> 00:46:33.150 3 entry is zero. 00:46:33.150 --> 00:46:36.670 But why was this positive? 00:46:36.670 --> 00:46:39.230 Where do I put the parentheses. 00:46:39.230 --> 00:46:44.030 Where do I put the parentheses to see that that's positive? 00:46:44.030 --> 00:46:50.050 I put them around where? 00:46:50.050 --> 00:46:52.340 Around that, good. 00:46:52.340 --> 00:46:57.940 And around this? 00:46:57.940 --> 00:47:01.420 This is really, since we now have a letter for Au, 00:47:01.420 --> 00:47:09.040 this is really e transpose Ce, right? 00:47:09.040 --> 00:47:09.990 That's e. 00:47:09.990 --> 00:47:13.910 This is e, Au, and this is its transpose. 00:47:13.910 --> 00:47:17.940 And now what? 00:47:17.940 --> 00:47:21.860 So now we've narrowed it down to C. 00:47:21.860 --> 00:47:26.050 Oh, we can actually see why it's an energy. 00:47:26.050 --> 00:47:28.240 Remember C is that diagonal matrix. 00:47:28.240 --> 00:47:29.950 What will this be? 00:47:29.950 --> 00:47:37.570 This is the row of stretchings, the diagonal matrix of c's, 00:47:37.570 --> 00:47:40.250 and the column of stretchings. 00:47:40.250 --> 00:47:45.460 And now if I do that multiplication, what do I get? 00:47:45.460 --> 00:47:46.640 Do you see it? 00:47:46.640 --> 00:47:49.610 Because the physics is coming in. 00:47:49.610 --> 00:47:53.560 What do I get? 00:47:53.560 --> 00:47:54.830 This will multiply that. 00:47:54.830 --> 00:47:58.930 So what's the first term I should write here? e_1? 00:48:01.830 --> 00:48:03.220 What will it be? 00:48:03.220 --> 00:48:04.390 I only have diagonal. 00:48:04.390 --> 00:48:06.470 In other words, I only have perfect squares 00:48:06.470 --> 00:48:08.420 when I look at this thing. 00:48:08.420 --> 00:48:16.700 I think I just have c_1*e_1 squared coming from that 00:48:16.700 --> 00:48:17.380 diagonal. 00:48:17.380 --> 00:48:20.720 That c_1 there, this e_1 here, this 00:48:20.720 --> 00:48:22.950 e_1 there is going to give me that c_1-- 00:48:22.950 --> 00:48:25.730 What else will I have? 00:48:25.730 --> 00:48:30.450 c_2*e_2 squared, c_3*e_3 squared and c_4*e_4 squared. 00:48:36.890 --> 00:48:40.520 And do you remember about springs and Hooke's Law 00:48:40.520 --> 00:48:42.360 and energy? 00:48:42.360 --> 00:48:44.820 What's the energy in a spring? 00:48:44.820 --> 00:48:48.420 This is a stretched spring. 00:48:48.420 --> 00:48:55.440 So the energy in a stretched spring, what I wanted to say, 00:48:55.440 --> 00:49:00.610 this is the sum of four internal energies in the four springs 00:49:00.610 --> 00:49:05.210 but it properly should have a factor 1/2. 00:49:05.210 --> 00:49:10.050 There probably, to really use the word energy properly, 00:49:10.050 --> 00:49:13.410 it should be half of all this, half of all this, 00:49:13.410 --> 00:49:16.265 half of that's the energy in the first spring, 00:49:16.265 --> 00:49:19.550 the energy in the second, the energy in the third 00:49:19.550 --> 00:49:24.760 and the energy in the fourth. 00:49:24.760 --> 00:49:30.910 But of course our matrix point was, it's positive. 00:49:30.910 --> 00:49:33.730 It's a sum of squares multiplied now 00:49:33.730 --> 00:49:40.000 by these positive numbers, these elastic constants, c_1, two, 00:49:40.000 --> 00:49:42.300 three and four. 00:49:42.300 --> 00:49:53.290 So we know the main facts about that matrix. 00:49:53.290 --> 00:49:56.180 We're really at the point here of we've 00:49:56.180 --> 00:50:00.590 got some problem formulated, we've 00:50:00.590 --> 00:50:04.120 got the essential facts about the matrix, 00:50:04.120 --> 00:50:12.020 it's symmetric, positive definite, certainly invertible. 00:50:12.020 --> 00:50:20.900 Then there'd be the step of actually computing u 00:50:20.900 --> 00:50:24.760 by solving the stiffness equation. 00:50:24.760 --> 00:50:30.560 Say, for example, Professor Bathe's big finite element 00:50:30.560 --> 00:50:34.730 code, ADINA. 00:50:34.730 --> 00:50:37.390 What's the big picture for ADINA, 00:50:37.390 --> 00:50:39.540 for any big finite element code? 00:50:39.540 --> 00:50:43.880 NASTRAN, ANSYS, whatever. 00:50:43.880 --> 00:50:44.680 Abaqus. 00:50:44.680 --> 00:50:46.490 There are so many really good ones. 00:50:46.490 --> 00:50:50.610 And they've taken years and years of work to create. 00:50:50.610 --> 00:50:54.810 But if you look to see what are the elements that go in, 00:50:54.810 --> 00:51:03.240 you choose the model, and we'll see in the next chapter, 00:51:03.240 --> 00:51:07.600 in October we'll see what finite elements is about, 00:51:07.600 --> 00:51:10.960 you have the material properties, 00:51:10.960 --> 00:51:15.660 you assemble the matrix K. 00:51:15.660 --> 00:51:20.270 That's a key step, is assembling this matrix K. 00:51:20.270 --> 00:51:23.750 And then the final step is solve the system. 00:51:23.750 --> 00:51:24.810 Ku=f. 00:51:24.810 --> 00:51:28.820 But it's assembling that matrix. 00:51:28.820 --> 00:51:30.680 Now one thing popped into my head. 00:51:30.680 --> 00:51:34.560 Do I have time to mention it or not? 00:51:34.560 --> 00:51:39.440 And there's no class Monday I think, right? 00:51:39.440 --> 00:51:41.040 Can I mention? 00:51:41.040 --> 00:51:45.300 Can you hang on one more second to mention 00:51:45.300 --> 00:51:51.750 a really remarkable way to do matrix multiplication. 00:51:51.750 --> 00:51:54.250 You may say, we know matrix multiplication. 00:51:54.250 --> 00:51:55.120 We got it. 00:51:55.120 --> 00:51:55.620 Right? 00:51:55.620 --> 00:51:58.010 We did it and we got the right answer. 00:51:58.010 --> 00:51:59.500 Can I just show you another way. 00:51:59.500 --> 00:52:03.850 And you can like, see if it works. 00:52:03.850 --> 00:52:06.490 I did this multiplication by like, 00:52:06.490 --> 00:52:08.740 I'll say rows times columns. 00:52:08.740 --> 00:52:11.570 I took rows times columns. 00:52:11.570 --> 00:52:13.300 That's the usual way. 00:52:13.300 --> 00:52:17.240 But finite elements and other, often the right way 00:52:17.240 --> 00:52:18.560 is the opposite. 00:52:18.560 --> 00:52:23.430 It's columns times rows. 00:52:23.430 --> 00:52:26.290 And of course, this guy's in here too. 00:52:26.290 --> 00:52:33.260 You might say, okay, what do I get from column one times 00:52:33.260 --> 00:52:36.050 that number, times row one? 00:52:36.050 --> 00:52:44.430 Can you do that multiplication just mentally? 00:52:44.430 --> 00:52:46.860 Multiply that column by that row. 00:52:46.860 --> 00:52:51.100 First of all, what shape will the answer have? 00:52:51.100 --> 00:52:54.163 What shape will the answer have if I multiply a three 00:52:54.163 --> 00:52:57.590 by one times a one by three. 00:52:57.590 --> 00:52:58.360 Three by three. 00:52:58.360 --> 00:52:59.450 It's a full matrix. 00:52:59.450 --> 00:53:00.690 Columns times rows. 00:53:00.690 --> 00:53:05.320 And it's a totally legitimate way to multiply matrices. 00:53:05.320 --> 00:53:09.060 That column times that row will be? 00:53:09.060 --> 00:53:11.660 Well you can see what will it be. 00:53:11.660 --> 00:53:18.280 And then the c_1 is going to come into it. 00:53:18.280 --> 00:53:20.140 If I just did those multiplications, 00:53:20.140 --> 00:53:22.120 it would just be that. 00:53:22.120 --> 00:53:28.350 And then the c_1 puts that there. 00:53:28.350 --> 00:53:30.760 What do I see there? 00:53:30.760 --> 00:53:34.310 I see the element matrix. 00:53:34.310 --> 00:53:36.910 Do you see that this is the piece that 00:53:36.910 --> 00:53:40.750 involved c_1 in the answer? 00:53:40.750 --> 00:53:42.470 Well I guess you'll see it better 00:53:42.470 --> 00:53:48.750 when I do column two times c_2 times row two. 00:53:48.750 --> 00:53:50.930 So I have to add that guy on. 00:53:50.930 --> 00:53:54.160 And then I'll leave the other. 00:53:54.160 --> 00:53:58.860 What do I get if I do that column, three by one, 00:53:58.860 --> 00:54:03.610 times itself as a row times the c_2. 00:54:03.610 --> 00:54:06.070 I don't know if you see what I'm going to get. 00:54:06.070 --> 00:54:12.240 If you just do that, you'll see a c_2 will appear here. 00:54:12.240 --> 00:54:15.470 And a -c_2 will appear there. 00:54:15.470 --> 00:54:17.600 And these will be zeroes. 00:54:17.600 --> 00:54:20.830 So this was column one times row one. 00:54:20.830 --> 00:54:22.890 This is column two times row two. 00:54:22.890 --> 00:54:24.950 And third and then the fourth. 00:54:24.950 --> 00:54:28.190 But do you see that this part is telling me 00:54:28.190 --> 00:54:30.520 all about the second spring? 00:54:30.520 --> 00:54:34.450 This part is telling me, what does the first spring, the c_1, 00:54:34.450 --> 00:54:39.175 contribute to K. This part tells me what does the c_2 part, 00:54:39.175 --> 00:54:41.450 do you see the c_2 part in K? 00:54:41.450 --> 00:54:44.580 There, there, minus there and minus there. 00:54:44.580 --> 00:54:49.160 The third part from the column row would be the c_3 part. 00:54:49.160 --> 00:54:51.110 And the fourth part from the fourth spring 00:54:51.110 --> 00:54:52.860 would be the c_4 part. 00:54:52.860 --> 00:54:56.350 So that's a way you won't have thought of. 00:54:56.350 --> 00:55:00.360 But it's the way ADINA would assemble this matrix. 00:55:00.360 --> 00:55:02.290 It would not do that multiplication. 00:55:02.290 --> 00:55:05.390 It would do it this way, columns times rows. 00:55:05.390 --> 00:55:06.590 We'll see it again. 00:55:06.590 --> 00:55:10.270 So, hope you have a great weekend and a holiday Monday 00:55:10.270 --> 00:55:12.813 that we all happy about.