1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:02,944 The following content is provided under a Creative 3 00:00:02,944 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,770 Your support will help MIT OpenCourseWare 5 00:00:05,770 --> 00:00:09,370 continue to offer high quality educational resources for free. 6 00:00:09,370 --> 00:00:12,530 To make a donation, or to view additional materials 7 00:00:12,530 --> 00:00:16,880 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,880 --> 00:00:19,390 at ocw.mit.edu. 9 00:00:19,390 --> 00:00:28,020 PROFESSOR STRANG: So today we move to a topic I really like. 10 00:00:28,020 --> 00:00:34,260 It's the beginning of the applications. 11 00:00:34,260 --> 00:00:37,210 So the particular application that comes first 12 00:00:37,210 --> 00:00:40,990 will be springs and masses, a pretty classical problem. 13 00:00:40,990 --> 00:00:47,250 But what we're looking for is how do we model it, 14 00:00:47,250 --> 00:00:51,520 what's the main framework to look 15 00:00:51,520 --> 00:00:53,850 at a whole series of problems. 16 00:00:53,850 --> 00:00:55,980 So this is number one in the series 17 00:00:55,980 --> 00:00:58,240 and it's the most straightforward. 18 00:00:58,240 --> 00:01:05,940 Let me draw it with four springs connecting three masses. 19 00:01:05,940 --> 00:01:08,080 And let me fix both ends. 20 00:01:08,080 --> 00:01:14,810 So this will be a fixed-fixed picture. 21 00:01:14,810 --> 00:01:17,890 So the masses have some weight. 22 00:01:17,890 --> 00:01:21,980 The weight pulls the springs down. 23 00:01:21,980 --> 00:01:28,270 When there was no weight acting they were not stretched. 24 00:01:28,270 --> 00:01:31,530 The masses will stretch the springs. 25 00:01:31,530 --> 00:01:34,950 And the question is how much do those, 26 00:01:34,950 --> 00:01:37,060 we're looking for the displacements. 27 00:01:37,060 --> 00:01:40,090 How much does mass one go down? 28 00:01:40,090 --> 00:01:40,590 Mass two? 29 00:01:40,590 --> 00:01:42,270 Mass three? 30 00:01:42,270 --> 00:01:45,740 And of course, essentially the displacement 31 00:01:45,740 --> 00:01:52,080 here is zero and here is zero. 32 00:01:52,080 --> 00:01:57,820 I don't know if you can imagine these masses have acted 33 00:01:57,820 --> 00:02:04,960 so that the position before gravity was turned on 34 00:02:04,960 --> 00:02:07,240 was somewhere up here and then it came here. 35 00:02:07,240 --> 00:02:11,170 So this moved down by a distance u_1. 36 00:02:11,170 --> 00:02:15,530 Let's use u for the displacements. 37 00:02:15,530 --> 00:02:18,100 So if I look at this main picture 38 00:02:18,100 --> 00:02:25,260 here I have displacements, movements, u_1, u_2, u_3. 39 00:02:25,260 --> 00:02:30,470 40 00:02:30,470 --> 00:02:33,120 Now what happens physically? 41 00:02:33,120 --> 00:02:35,710 Important in every one of these examples 42 00:02:35,710 --> 00:02:37,235 to see what's happening physically. 43 00:02:37,235 --> 00:02:41,400 Of course, this one moved down by some u_2, 44 00:02:41,400 --> 00:02:47,340 this one moved down from its total rest position to u_3. 45 00:02:47,340 --> 00:02:49,490 These are not oscillating. 46 00:02:49,490 --> 00:02:52,690 Next week they'll start moving, time will enter. 47 00:02:52,690 --> 00:02:55,710 Here I'm just looking for a steady state. 48 00:02:55,710 --> 00:02:57,950 They come to rest, they stretch. 49 00:02:57,950 --> 00:03:03,740 So what's your feeling of what's going to happen here, somehow? 50 00:03:03,740 --> 00:03:14,580 The displacements look to me like they'll all be positive. 51 00:03:14,580 --> 00:03:18,100 What's the key equation going to be? 52 00:03:18,100 --> 00:03:23,650 That when this moves down it will stretch that spring. 53 00:03:23,650 --> 00:03:29,090 Hooke's Law will say there's a force, the spring pulls back. 54 00:03:29,090 --> 00:03:31,330 The spring pulls back with a force 55 00:03:31,330 --> 00:03:34,440 proportional to the stretch. 56 00:03:34,440 --> 00:03:40,540 So u_1, u_2, and u_3 are movements. 57 00:03:40,540 --> 00:03:41,860 Here is a key question. 58 00:03:41,860 --> 00:03:44,840 What's the stretching in spring number two? 59 00:03:44,840 --> 00:03:47,790 So this is spring one, two, three and four. 60 00:03:47,790 --> 00:03:53,900 How much does spring number two stretch? u_2-u_1. 61 00:03:53,900 --> 00:03:56,020 A difference is coming in there. 62 00:03:56,020 --> 00:03:58,230 So let me put that up here. 63 00:03:58,230 --> 00:04:04,620 So stretching or elongation, I'll use two words, 64 00:04:04,620 --> 00:04:07,840 elongation I'll say sometimes because that 65 00:04:07,840 --> 00:04:09,950 starts with a letter e. 66 00:04:09,950 --> 00:04:14,990 So these are the elongations in the springs, 67 00:04:14,990 --> 00:04:22,120 in the four springs. 68 00:04:22,120 --> 00:04:24,230 It's the amount the spring stretches. 69 00:04:24,230 --> 00:04:29,830 Or what's the opposite of stretching? 70 00:04:29,830 --> 00:04:31,750 Compression somehow. 71 00:04:31,750 --> 00:04:35,230 Looks to me like this last spring, at least, 72 00:04:35,230 --> 00:04:39,170 is going to be compressed, and I'm not sure about the others. 73 00:04:39,170 --> 00:04:43,250 So we've got four springs. 74 00:04:43,250 --> 00:04:52,500 And each one has a stretching or compression, an elongation. 75 00:04:52,500 --> 00:04:56,730 And then there's a link then that you already told me. 76 00:04:56,730 --> 00:05:01,120 That e_2 is, just from the picture, 77 00:05:01,120 --> 00:05:07,000 e_2 is the difference between u_2 and u_1. 78 00:05:07,000 --> 00:05:10,250 Because the lower mass goes down by distance u_2, 79 00:05:10,250 --> 00:05:15,040 the upper mass by u_1 and spring is stretched by the difference 80 00:05:15,040 --> 00:05:17,560 u_2-u_1. 81 00:05:17,560 --> 00:05:20,570 So that's a first key fact. 82 00:05:20,570 --> 00:05:26,310 So that expresses somehow a fact of geometry. 83 00:05:26,310 --> 00:05:28,840 Of sort of the way things are connected. 84 00:05:28,840 --> 00:05:32,230 The material properties of the springs 85 00:05:32,230 --> 00:05:34,340 have not got into the picture yet. 86 00:05:34,340 --> 00:05:37,500 But now Hooke's Law brings them into the picture. 87 00:05:37,500 --> 00:05:42,250 By stretching a spring that produces 88 00:05:42,250 --> 00:05:45,980 a force that pulls back. 89 00:05:45,980 --> 00:05:54,880 So we get, can I say, forces in the spring. 90 00:05:54,880 --> 00:06:04,030 And let me give those a name w. w_1, 2, 3, and 4. 91 00:06:04,030 --> 00:06:11,590 And then the link between the stretching and the force 92 00:06:11,590 --> 00:06:18,040 that it produces is-- So that's somehow 93 00:06:18,040 --> 00:06:22,950 where the properties of the material come in. 94 00:06:22,950 --> 00:06:27,370 So I have to say, what are the properties of the springs? 95 00:06:27,370 --> 00:06:31,430 So this will be Hooke's Law, this step. 96 00:06:31,430 --> 00:06:38,190 Hooke's Law for this particular application. 97 00:06:38,190 --> 00:06:46,870 And so I have to say these springs have spring constants. 98 00:06:46,870 --> 00:06:52,020 So I haven't completed the description of the problem 99 00:06:52,020 --> 00:06:55,490 until I've told you about springs 100 00:06:55,490 --> 00:06:57,900 themselves and the masses. 101 00:06:57,900 --> 00:07:03,200 So the spring constants will be c_1, c_2, c_3, and c_4. 102 00:07:03,200 --> 00:07:06,320 And now what does Hooke's Law say? 103 00:07:06,320 --> 00:07:12,410 Usually this physical law in the middle we keep it linear. 104 00:07:12,410 --> 00:07:19,580 Of course, we all understand that if these springs were 105 00:07:19,580 --> 00:07:24,080 enormously stretched the elastic property 106 00:07:24,080 --> 00:07:25,510 could become non-linear. 107 00:07:25,510 --> 00:07:29,150 It could become plastic. 108 00:07:29,150 --> 00:07:33,820 The first law always has somebody's name. 109 00:07:33,820 --> 00:07:37,280 Was the person to see that in some range 110 00:07:37,280 --> 00:07:41,530 of small displacements, so I guess that's the answer. 111 00:07:41,530 --> 00:07:44,230 We're speaking here about small displacements, 112 00:07:44,230 --> 00:07:46,910 small stretching up to the point where 113 00:07:46,910 --> 00:07:50,770 Hooke's Law continues to hold. 114 00:07:50,770 --> 00:07:53,560 And now what does Hooke's Law say? 115 00:07:53,560 --> 00:08:00,100 It says that each force in the spring 116 00:08:00,100 --> 00:08:07,370 is proportional to the stretching of the spring. 117 00:08:07,370 --> 00:08:11,570 You could say it's-- a diagonal matrix is showing up here. 118 00:08:11,570 --> 00:08:19,380 The vector of w's, the vector of forces in the spring is 119 00:08:19,380 --> 00:08:24,560 a diagonal matrix C, which just has these numbers 120 00:08:24,560 --> 00:08:32,080 on the diagonal, ...c_4, times the e's. 121 00:08:32,080 --> 00:08:38,430 So of course I'm going to write that in matrix notation as W 122 00:08:38,430 --> 00:08:45,100 equals a matrix C times e. 123 00:08:45,100 --> 00:08:49,670 So there in the middle is the physics. 124 00:08:49,670 --> 00:08:53,140 The material properties, the constitutive law. 125 00:08:53,140 --> 00:08:59,950 C can stand for constants, for constitutive law, 126 00:08:59,950 --> 00:09:05,130 later for conductances. 127 00:09:05,130 --> 00:09:08,680 It's the place where the material enters. 128 00:09:08,680 --> 00:09:14,860 And now how do we complete this picture? 129 00:09:14,860 --> 00:09:21,730 In the end we have to bring in the masses. 130 00:09:21,730 --> 00:09:27,910 Gravity is the external force that's making things happen. 131 00:09:27,910 --> 00:09:33,030 We need a force term from outside 132 00:09:33,030 --> 00:09:36,110 to move us away from zeroes. 133 00:09:36,110 --> 00:09:43,860 And that will be the downward forces f_1, f_2, 134 00:09:43,860 --> 00:09:48,210 f_3 on the three masses. 135 00:09:48,210 --> 00:09:53,790 So I plan to complete this picture with a force balance 136 00:09:53,790 --> 00:10:12,070 equation on the masses, on each mass. 137 00:10:12,070 --> 00:10:14,360 When I use the word framework there, 138 00:10:14,360 --> 00:10:17,320 this is what I was talking about. 139 00:10:17,320 --> 00:10:20,840 I guess what I want to say is I really 140 00:10:20,840 --> 00:10:29,820 have found that this way of describing, modeling 141 00:10:29,820 --> 00:10:35,930 the problem is successful for so many applications. 142 00:10:35,930 --> 00:10:43,310 You have somehow a geometry, a step which'll involve a matrix 143 00:10:43,310 --> 00:10:50,790 A. Then you have a physical step which involves a matrix C. 144 00:10:50,790 --> 00:10:53,940 And then finally you have a force balance. 145 00:10:53,940 --> 00:11:02,910 In a way this force balance or its analog, 146 00:11:02,910 --> 00:11:05,650 the analog would be Kirchhoff's current law. 147 00:11:05,650 --> 00:11:07,990 We'll see that for networks. 148 00:11:07,990 --> 00:11:10,140 Flow in equals flow out. 149 00:11:10,140 --> 00:11:13,120 Force on one side equals force on the other. 150 00:11:13,120 --> 00:11:15,800 If we're talking about equilibrium 151 00:11:15,800 --> 00:11:20,130 we can expect our model to have an equation like that. 152 00:11:20,130 --> 00:11:25,560 And for me it really helps to know when a new model comes in. 153 00:11:25,560 --> 00:11:27,310 Like somebody'll come into my office 154 00:11:27,310 --> 00:11:33,310 with a problem in chemistry or biology. 155 00:11:33,310 --> 00:11:39,920 But if it fits in this framework I'll 156 00:11:39,920 --> 00:11:46,380 be looking for a balance equation, a continuity 157 00:11:46,380 --> 00:11:52,130 equation at the end. 158 00:11:52,130 --> 00:11:55,170 This part was easy and it's these two parts 159 00:11:55,170 --> 00:11:58,520 that I want to pin down. 160 00:11:58,520 --> 00:12:01,430 Well you told me how to start here. 161 00:12:01,430 --> 00:12:07,340 So the elongation, so I want to take this step again. 162 00:12:07,340 --> 00:12:14,310 I want to find the elongations from some matrix that 163 00:12:14,310 --> 00:12:20,240 multiplies the displacements. 164 00:12:20,240 --> 00:12:23,000 So I'm just completing this step. 165 00:12:23,000 --> 00:12:30,010 And you told me what is the stretching in spring two. 166 00:12:30,010 --> 00:12:32,250 Again, do you mind just saying it again? 167 00:12:32,250 --> 00:12:35,630 The stretching in that second spring, the amount, 168 00:12:35,630 --> 00:12:45,120 it's made longer by the action of gravity was? u_2-u_1. 169 00:12:45,120 --> 00:12:46,850 u_2-u_1. 170 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. 171 00:12:55,210 --> 00:13:04,090 That will be a typical row of this matrix, the displacement 172 00:13:04,090 --> 00:13:06,270 stretching matrix, you could say. 173 00:13:06,270 --> 00:13:08,860 Now what about the stretching in e_1? 174 00:13:08,860 --> 00:13:10,190 What's the stretching in e_1? 175 00:13:10,190 --> 00:13:15,210 176 00:13:15,210 --> 00:13:16,200 Only u_1. 177 00:13:16,200 --> 00:13:20,870 Because essentially it's u_1-u_0 but u_0 178 00:13:20,870 --> 00:13:24,220 is set to zero by the support. 179 00:13:24,220 --> 00:13:27,970 So we only have u_1. 180 00:13:27,970 --> 00:13:32,130 Because that multiplication just gives us-- So e_1 is u_1. 181 00:13:32,130 --> 00:13:35,210 e_2 is u_2-u_1. 182 00:13:35,210 --> 00:13:37,790 e_3 is what? 183 00:13:37,790 --> 00:13:43,790 The stretching in the third spring. 184 00:13:43,790 --> 00:13:47,790 What is it? u_3-u_2. 185 00:13:47,790 --> 00:13:52,890 So I need a one for u_3 and a minus one for u_2. 186 00:13:52,890 --> 00:13:56,690 And the stretching in the fourth spring? 187 00:13:56,690 --> 00:14:01,660 What's the stretching in the fourth spring? 188 00:14:01,660 --> 00:14:07,980 I've sort of, and you have too, mentally given a plus sign 189 00:14:07,980 --> 00:14:11,690 when the spring is extended and a minus 190 00:14:11,690 --> 00:14:13,810 sign when it's compressed. 191 00:14:13,810 --> 00:14:18,280 Plus for tension, minus for compression. 192 00:14:18,280 --> 00:14:22,900 So since I fixed that one, u_4 was zero, so what 193 00:14:22,900 --> 00:14:25,600 do I have in this last row? 194 00:14:25,600 --> 00:14:26,900 Just minus u_3. 195 00:14:26,900 --> 00:14:31,450 196 00:14:31,450 --> 00:14:36,640 I guess what I'm saying here is that if we 197 00:14:36,640 --> 00:14:40,290 get a systematic approach to problems 198 00:14:40,290 --> 00:14:46,130 then we know we're looking for a matrix that connects these. 199 00:14:46,130 --> 00:14:50,140 We're looking for the material constitutive law that does this 200 00:14:50,140 --> 00:14:51,710 and now we're looking for this one. 201 00:14:51,710 --> 00:14:53,570 We kind of know where we are. 202 00:14:53,570 --> 00:14:55,450 What to look for. 203 00:14:55,450 --> 00:14:59,190 And so this matrix is the matrix I'm going to call A. 204 00:14:59,190 --> 00:15:01,010 So this is e=Au. 205 00:15:01,010 --> 00:15:10,000 206 00:15:10,000 --> 00:15:12,980 Well one more step to go. 207 00:15:12,980 --> 00:15:16,400 And that will be the force balance step. 208 00:15:16,400 --> 00:15:22,010 So now, what's the equation for balance? 209 00:15:22,010 --> 00:15:25,790 The external forces are the masses. 210 00:15:25,790 --> 00:15:28,900 Well, I guess to get the units right, 211 00:15:28,900 --> 00:15:34,160 it should be mass times g, the gravitational constant. 212 00:15:34,160 --> 00:15:43,220 So let me put external forces f_1, f_2, and f_3. 213 00:15:43,220 --> 00:15:51,390 The three masses will be m_1*g, m_2*g, and m_3*g. 214 00:15:51,390 --> 00:15:55,350 So those are the forces from outside. 215 00:15:55,350 --> 00:15:58,800 Now it's the balance equation I'm after. 216 00:15:58,800 --> 00:16:01,570 So this is in this position. 217 00:16:01,570 --> 00:16:03,910 It's in equilibrium. 218 00:16:03,910 --> 00:16:05,970 And what does that tell us? 219 00:16:05,970 --> 00:16:09,020 That tells us that the total force on this mass, 220 00:16:09,020 --> 00:16:12,300 so I'm going to take each mass, it's 221 00:16:12,300 --> 00:16:15,440 like a free body force diagram here. 222 00:16:15,440 --> 00:16:17,770 I'm looking now at that mass. 223 00:16:17,770 --> 00:16:20,870 I'm saying what forces are acting on it 224 00:16:20,870 --> 00:16:23,380 and I'm making them balance. 225 00:16:23,380 --> 00:16:25,800 So what equation will that give me? 226 00:16:25,800 --> 00:16:27,010 So let me write that. 227 00:16:27,010 --> 00:16:30,290 This is now the force balance equation. 228 00:16:30,290 --> 00:16:37,830 Force balance at each mass. 229 00:16:37,830 --> 00:16:41,280 How much force is pulling up? 230 00:16:41,280 --> 00:16:44,320 What's the force pulling up on? 231 00:16:44,320 --> 00:16:47,580 So this spring is pulling upwards. 232 00:16:47,580 --> 00:16:51,750 And it's pulling upwards by w_1, right? 233 00:16:51,750 --> 00:16:54,160 Just getting these letters right. 234 00:16:54,160 --> 00:17:00,090 The w's were the internal resisting force, 235 00:17:00,090 --> 00:17:03,930 reacting force in the spring. w_1 is pulling up. 236 00:17:03,930 --> 00:17:09,570 What other forces are acting? w_2 is pulling down. 237 00:17:09,570 --> 00:17:14,550 And also pulling down is? 238 00:17:14,550 --> 00:17:16,760 Gravity, m_1*g. 239 00:17:16,760 --> 00:17:22,480 So the balance of forces there says that w_1, the force up, 240 00:17:22,480 --> 00:17:27,980 is w_2, the force down, and m_1*g. 241 00:17:27,980 --> 00:17:31,990 And similarly the next one will have, 242 00:17:31,990 --> 00:17:35,740 the next mass if I look just at that I see a force up, 243 00:17:35,740 --> 00:17:38,080 a force down and gravity down. 244 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. 245 00:17:45,990 --> 00:17:49,410 And the third one, the force up on the third one 246 00:17:49,410 --> 00:17:53,600 will be the force down on the third one. 247 00:17:53,600 --> 00:17:56,920 so I think those are the equations of force 248 00:17:56,920 --> 00:18:00,980 balance written one at a time. 249 00:18:00,980 --> 00:18:08,080 And now, of course I'm going to write that-- 250 00:18:08,080 --> 00:18:14,990 So that's three equations with four w's. 251 00:18:14,990 --> 00:18:20,010 So I want to write that as, I want to bring the w's 252 00:18:20,010 --> 00:18:21,640 all to the left-hand side. 253 00:18:21,640 --> 00:18:25,310 Can I do that? 254 00:18:25,310 --> 00:18:29,910 Can I just bring those over with minus signs? 255 00:18:29,910 --> 00:18:34,710 And make these equal signs. 256 00:18:34,710 --> 00:18:41,620 So now we've got internal force balancing external force. 257 00:18:41,620 --> 00:18:45,670 This vector of external forces is the f's and this 258 00:18:45,670 --> 00:18:47,620 is the internal forces. 259 00:18:47,620 --> 00:18:51,350 Now somewhere there we're going to see a matrix. 260 00:18:51,350 --> 00:18:54,670 So I'm going to write this equation as some matrix. 261 00:18:54,670 --> 00:18:57,750 Well, let's figure out what that matrix is. 262 00:18:57,750 --> 00:19:01,180 So its shape is what? 263 00:19:01,180 --> 00:19:05,470 I've got three equations, so I need three rows in the matrix. 264 00:19:05,470 --> 00:19:09,060 I've got four w's so I need four columns. 265 00:19:09,060 --> 00:19:14,790 So it's going to multiply w_1, w_2, w_3, w_4 266 00:19:14,790 --> 00:19:21,940 to give these three masses, can I call them f_1, f_2, f_3, 267 00:19:21,940 --> 00:19:26,890 just to have a good letter. 268 00:19:26,890 --> 00:19:29,350 We're almost there. 269 00:19:29,350 --> 00:19:30,900 What's the matrix? 270 00:19:30,900 --> 00:19:37,290 What's the matrix for this final step, the force balance 271 00:19:37,290 --> 00:19:38,540 equation? 272 00:19:38,540 --> 00:19:40,700 I just read it off. 273 00:19:40,700 --> 00:19:42,810 w_1-w_2. 274 00:19:42,810 --> 00:19:47,280 I think I've got that. 275 00:19:47,280 --> 00:19:49,940 w_2-w_3. 276 00:19:49,940 --> 00:19:52,030 Tell me what the second row of the matrix 277 00:19:52,030 --> 00:19:56,390 looks like to give me w_2-w_3. 278 00:19:56,390 --> 00:20:01,250 zero, one for the w_2, minus one. 279 00:20:01,250 --> 00:20:02,280 Good. 280 00:20:02,280 --> 00:20:13,260 And for the third, the final row? [0, 0, 1, -1]. 281 00:20:13,260 --> 00:20:17,240 So that completes the third piece. 282 00:20:17,240 --> 00:20:21,900 If I'd given you the problem as I did, drawn the problem, 283 00:20:21,900 --> 00:20:26,270 described it, you know that there's 284 00:20:26,270 --> 00:20:29,450 going to be a connection between the external forces 285 00:20:29,450 --> 00:20:32,270 and the displacements. 286 00:20:32,270 --> 00:20:34,700 But what I'm trying to say is a good way 287 00:20:34,700 --> 00:20:39,650 to see the connection is to see it in three simple steps. 288 00:20:39,650 --> 00:20:42,770 The simple step that gets you from the displacements 289 00:20:42,770 --> 00:20:44,730 to the springs. 290 00:20:44,730 --> 00:20:47,140 A second step within the springs. 291 00:20:47,140 --> 00:20:51,310 A third step back to the nodes, you 292 00:20:51,310 --> 00:20:53,990 could say, back to the masses. 293 00:20:53,990 --> 00:20:59,430 And of course, the key question is, what's that matrix? 294 00:20:59,430 --> 00:21:02,940 And do you recognize it? 295 00:21:02,940 --> 00:21:05,770 Do we need a new name for that matrix? 296 00:21:05,770 --> 00:21:08,230 The matrix in the third step? 297 00:21:08,230 --> 00:21:13,470 So this third step is going to be that some matrix times 298 00:21:13,470 --> 00:21:19,070 w is f and what's that matrix? 299 00:21:19,070 --> 00:21:22,290 What's the good name for us to give it? 300 00:21:22,290 --> 00:21:25,170 A transpose is the best possible name. 301 00:21:25,170 --> 00:21:28,010 If we've given this matrix the name 302 00:21:28,010 --> 00:21:33,500 A, the stretching displacement matrix, 303 00:21:33,500 --> 00:21:38,650 the strain in elasticity, this becomes the strains, 304 00:21:38,650 --> 00:21:40,930 these become the stresses. 305 00:21:40,930 --> 00:21:43,720 But the beauty is, just beautiful, 306 00:21:43,720 --> 00:21:49,070 that the matrix in this law is the transpose of this one. 307 00:21:49,070 --> 00:21:51,830 So it's A transpose. 308 00:21:51,830 --> 00:21:59,510 So that's the framework seen now here for the first time. 309 00:21:59,510 --> 00:22:03,640 So the key point was that A and A transpose both 310 00:22:03,640 --> 00:22:08,060 appeared but with physical material properties, 311 00:22:08,060 --> 00:22:10,190 constitutive matrix in between. 312 00:22:10,190 --> 00:22:15,720 So if we put the pieces together, then we're golden. 313 00:22:15,720 --> 00:22:19,380 And then, let's do an example to see what actually happened. 314 00:22:19,380 --> 00:22:31,240 So the equations were e=Aw-- e=Au, then w=Ce, 315 00:22:31,240 --> 00:22:34,010 that's Hooke's Law, and then A transpose-- 316 00:22:34,010 --> 00:22:40,020 or maybe I'll write it as f=A transpose*w. 317 00:22:40,020 --> 00:22:42,900 318 00:22:42,900 --> 00:22:45,530 That's the three steps. 319 00:22:45,530 --> 00:22:50,770 So in this problem the source term showed up at that point. 320 00:22:50,770 --> 00:22:55,700 The source term came from external forces. 321 00:22:55,700 --> 00:22:57,780 I've got three equations. 322 00:22:57,780 --> 00:23:00,570 Now I'm going to put them together into one. 323 00:23:00,570 --> 00:23:02,560 I'll put them into one equation. 324 00:23:02,560 --> 00:23:05,510 So this w I'll just substitute. 325 00:23:05,510 --> 00:23:11,740 So it's A transpose w is Ce, and e is Au. 326 00:23:11,740 --> 00:23:16,170 So I have A transpose C Au. 327 00:23:16,170 --> 00:23:20,790 So that's the ultimate. 328 00:23:20,790 --> 00:23:23,810 That's put the whole structure together. 329 00:23:23,810 --> 00:23:27,850 That's the equation you have to solve. 330 00:23:27,850 --> 00:23:35,780 This would be called the stiffness matrix. 331 00:23:35,780 --> 00:23:42,930 And I use the letter K for that one. 332 00:23:42,930 --> 00:23:46,800 So our equation is Ku=f. 333 00:23:46,800 --> 00:23:56,590 This is our final equation. 334 00:23:56,590 --> 00:24:03,140 Well, we didn't know w. 335 00:24:03,140 --> 00:24:05,520 There are two unknowns here. 336 00:24:05,520 --> 00:24:08,460 Two physical things that you want to find. 337 00:24:08,460 --> 00:24:11,460 If you're designing a bridge or a structure 338 00:24:11,460 --> 00:24:14,640 you want to know the displacements 339 00:24:14,640 --> 00:24:21,720 and then you want to know the internal forces w. 340 00:24:21,720 --> 00:24:22,690 It's really beautiful. 341 00:24:22,690 --> 00:24:28,860 The two unknowns of u and w are somehow dual, 342 00:24:28,860 --> 00:24:36,350 we can work with one, work with the other, work with both. 343 00:24:36,350 --> 00:24:40,390 Oh let me just mention that the finite element method will 344 00:24:40,390 --> 00:24:46,590 fit this framework and somehow this name stiffness matrix has 345 00:24:46,590 --> 00:24:50,970 become famous for finite elements in structures 346 00:24:50,970 --> 00:25:01,630 and then it's just exploded to appear all over the place. 347 00:25:01,630 --> 00:25:03,273 I guess we should look at A transpose 348 00:25:03,273 --> 00:25:09,710 C A. We can see what it looks like. 349 00:25:09,710 --> 00:25:14,010 And also just from the way it looks there. 350 00:25:14,010 --> 00:25:16,440 So I can write it out explicitly. 351 00:25:16,440 --> 00:25:18,040 I think we want to. 352 00:25:18,040 --> 00:25:20,990 But at the same time I can learn something 353 00:25:20,990 --> 00:25:25,190 from just seeing how it's put together. 354 00:25:25,190 --> 00:25:28,190 What can you tell me about A transpose C A? 355 00:25:28,190 --> 00:25:30,380 Let's get the shape first. 356 00:25:30,380 --> 00:25:33,400 Just to see the shape of these things. 357 00:25:33,400 --> 00:25:36,510 The matrix A is what? 358 00:25:36,510 --> 00:25:39,400 What's the shape of A? 359 00:25:39,400 --> 00:25:40,960 It's over here. 360 00:25:40,960 --> 00:25:43,180 Four by three. 361 00:25:43,180 --> 00:25:45,880 Four by three. 362 00:25:45,880 --> 00:25:52,400 And the shape of C was, three by three is it? 363 00:25:52,400 --> 00:25:55,870 Where have I got, that C matrix better be here somewhere. 364 00:25:55,870 --> 00:25:58,090 Oh, no, it's four by four. 365 00:25:58,090 --> 00:25:59,110 Four springs. 366 00:25:59,110 --> 00:26:02,440 Of course, it had to be four by four to do that multiplication. 367 00:26:02,440 --> 00:26:04,310 There's the C matrix. 368 00:26:04,310 --> 00:26:06,320 Four by four, thanks. 369 00:26:06,320 --> 00:26:09,280 And the A transpose matrix? 370 00:26:09,280 --> 00:26:11,200 Three by four, thanks. 371 00:26:11,200 --> 00:26:15,470 So the net result is three by three. 372 00:26:15,470 --> 00:26:15,970 Good. 373 00:26:15,970 --> 00:26:20,220 So it's a square matrix. 374 00:26:20,220 --> 00:26:22,260 K is a square matrix. 375 00:26:22,260 --> 00:26:27,550 What else can you tell me about it? 376 00:26:27,550 --> 00:26:31,480 Now we're going to begin to use some of the, sort of the matrix 377 00:26:31,480 --> 00:26:34,120 preparation. 378 00:26:34,120 --> 00:26:37,430 These matrices are kind of friends by now. 379 00:26:37,430 --> 00:26:42,081 This is a difference matrix, somehow. 380 00:26:42,081 --> 00:26:42,580 Right? 381 00:26:42,580 --> 00:26:47,430 The stretchings are differences and displacements. 382 00:26:47,430 --> 00:26:49,230 That's its transpose. 383 00:26:49,230 --> 00:26:53,130 And then the C matrix, which is the new thing, sort 384 00:26:53,130 --> 00:26:59,570 of the new guy to appear today, is diagonal. 385 00:26:59,570 --> 00:27:03,340 Well if I asked you now, without writing out the matrix, 386 00:27:03,340 --> 00:27:06,030 for one more property, it's square, 387 00:27:06,030 --> 00:27:08,300 what else could you tell me about it? 388 00:27:08,300 --> 00:27:12,810 Symmetric is going to be a very good guess and let's see why. 389 00:27:12,810 --> 00:27:15,150 Why is it symmetric? 390 00:27:15,150 --> 00:27:19,470 How do we show that that? 391 00:27:19,470 --> 00:27:21,020 What do I do? 392 00:27:21,020 --> 00:27:25,300 I take the transpose. 393 00:27:25,300 --> 00:27:32,530 If I take my K transpose, now I write it as, what do I do? 394 00:27:32,530 --> 00:27:34,890 It's a product of things. 395 00:27:34,890 --> 00:27:39,210 So when I transpose a product I have the individual transposes 396 00:27:39,210 --> 00:27:40,430 in the opposite order. 397 00:27:40,430 --> 00:27:43,900 So A, its transpose comes first. 398 00:27:43,900 --> 00:27:46,760 C, its transpose comes next. 399 00:27:46,760 --> 00:27:51,730 A transpose, its transpose comes last. 400 00:27:51,730 --> 00:27:57,270 So that's just the rules of matrix transposes. 401 00:27:57,270 --> 00:27:58,430 Now what? 402 00:27:58,430 --> 00:28:02,670 Now I'm ready to use the wonderful fact of what 403 00:28:02,670 --> 00:28:03,800 we've got here. 404 00:28:03,800 --> 00:28:06,630 So what is C transpose? 405 00:28:06,630 --> 00:28:14,080 So notice we wanted a symmetric matrix in the middle 406 00:28:14,080 --> 00:28:16,450 to be able to knock that T out. 407 00:28:16,450 --> 00:28:19,200 And what is A transpose transpose? 408 00:28:19,200 --> 00:28:24,900 That's A. We've learned that the thing is symmetric, 409 00:28:24,900 --> 00:28:30,540 that if I transpose it I get it back again. 410 00:28:30,540 --> 00:28:33,160 We're going to see more about that. 411 00:28:33,160 --> 00:28:38,370 But let me do the multiplication. 412 00:28:38,370 --> 00:28:41,840 So I'm going to take that, oh, boy. 413 00:28:41,840 --> 00:28:45,100 How am I going to do that? 414 00:28:45,100 --> 00:28:48,660 I want to multiply three matrices to see what K actually 415 00:28:48,660 --> 00:28:52,440 looks like here. 416 00:28:52,440 --> 00:28:55,330 One question first. 417 00:28:55,330 --> 00:29:01,200 Eventually the solution, the short formula for the solution, 418 00:29:01,200 --> 00:29:03,880 will be u equal K inverse f. 419 00:29:03,880 --> 00:29:04,860 Right? 420 00:29:04,860 --> 00:29:11,780 So the answer will be u equal K inverse f in matrix notation 421 00:29:11,780 --> 00:29:16,210 but I'm looking for numbers. 422 00:29:16,210 --> 00:29:22,160 And then if I know u then I know the stretching. e is A times K 423 00:29:22,160 --> 00:29:24,190 inverse f. 424 00:29:24,190 --> 00:29:29,680 And w is, I'm just going down the list, is C times A times 425 00:29:29,680 --> 00:29:30,580 K inverse f. 426 00:29:30,580 --> 00:29:32,500 We've got everything. 427 00:29:32,500 --> 00:29:34,770 So that's the key. 428 00:29:34,770 --> 00:29:36,220 This is the key equation. 429 00:29:36,220 --> 00:29:39,470 That's the answer. 430 00:29:39,470 --> 00:29:44,080 Let me ask you about inverses. 431 00:29:44,080 --> 00:29:46,320 What about K inverse? 432 00:29:46,320 --> 00:29:49,050 We took three steps. 433 00:29:49,050 --> 00:29:54,030 Now what if I just ask you about inverses? 434 00:29:54,030 --> 00:29:56,840 This is K inverse that we would like to know. 435 00:29:56,840 --> 00:30:04,650 So again, for inverses I'm going to start this 436 00:30:04,650 --> 00:30:09,690 and I'm going to stop halfway and you'll tell me why. 437 00:30:09,690 --> 00:30:11,800 If you give me a product of matrices 438 00:30:11,800 --> 00:30:15,080 and I don't think particularly much 439 00:30:15,080 --> 00:30:18,080 I'll take the inverse of that times the inverse of that 440 00:30:18,080 --> 00:30:25,280 times the inverse of that. 441 00:30:25,280 --> 00:30:31,490 And what's the matter with that? 442 00:30:31,490 --> 00:30:35,360 You would say, why not just undo each step? 443 00:30:35,360 --> 00:30:45,620 Why not find the w's from the f's and then the e's from 444 00:30:45,620 --> 00:30:52,830 the w's by dividing and then the u's from the e's? 445 00:30:52,830 --> 00:30:56,510 Why don't we just go backwards around the loop 446 00:30:56,510 --> 00:31:00,570 rather than what I'm saying we have to do. 447 00:31:00,570 --> 00:31:05,210 We eventually get this step across with a matrix K 448 00:31:05,210 --> 00:31:13,840 that does all three at once. 449 00:31:13,840 --> 00:31:15,710 Well sometimes we might be able to, 450 00:31:15,710 --> 00:31:18,810 but I don't think we can in this time. 451 00:31:18,810 --> 00:31:21,280 What's the trouble with A, that I 452 00:31:21,280 --> 00:31:26,310 don't want to write A inverse? 453 00:31:26,310 --> 00:31:28,290 Well I don't say singular. 454 00:31:28,290 --> 00:31:30,900 What do I say here? 455 00:31:30,900 --> 00:31:37,020 Look at this matrix A here. 456 00:31:37,020 --> 00:31:38,650 It's not square. 457 00:31:38,650 --> 00:31:40,780 It's not square, that's right. 458 00:31:40,780 --> 00:31:46,700 So I'm not comfortable, I'm not willing to write A inverse 459 00:31:46,700 --> 00:31:51,070 when A is not a square matrix. 460 00:31:51,070 --> 00:31:55,700 And this distinction, is the matrix A square or not, 461 00:31:55,700 --> 00:31:57,630 is the first issue. 462 00:31:57,630 --> 00:31:59,090 It's just the picture. 463 00:31:59,090 --> 00:32:01,810 Let me show you an example of where it would be square. 464 00:32:01,810 --> 00:32:02,590 May I? 465 00:32:02,590 --> 00:32:07,060 Before I do this multiplication, can I jump to a-- I'll 466 00:32:07,060 --> 00:32:15,410 change the line of springs in a way that'll change A. 467 00:32:15,410 --> 00:32:16,940 And let me show you what happens. 468 00:32:16,940 --> 00:32:23,030 Suppose I take out that spring. 469 00:32:23,030 --> 00:32:27,800 So I've removed the fourth spring. 470 00:32:27,800 --> 00:32:31,120 It's a line of springs now, hanging from a support. 471 00:32:31,120 --> 00:32:33,040 It's a perfectly good problem. 472 00:32:33,040 --> 00:32:37,150 It's problem two, but it's a different problem. 473 00:32:37,150 --> 00:32:39,040 And what's different now? 474 00:32:39,040 --> 00:32:44,080 There is no fourth spring. 475 00:32:44,080 --> 00:32:46,990 If this was my problem, what would be different? 476 00:32:46,990 --> 00:32:50,190 There's no fourth spring. 477 00:32:50,190 --> 00:32:53,010 So that's gone. 478 00:32:53,010 --> 00:32:56,980 I just have three springs stretching from three masses. 479 00:32:56,980 --> 00:32:59,740 Then the force balance is the same. 480 00:32:59,740 --> 00:33:03,690 Everything looks the same except there's no force, 481 00:33:03,690 --> 00:33:06,680 there's no fourth spring, so there's no force there, 482 00:33:06,680 --> 00:33:11,330 that's gone. 483 00:33:11,330 --> 00:33:14,940 And of course, how does C change? 484 00:33:14,940 --> 00:33:22,090 So in my new picture now I have, let me write now, 485 00:33:22,090 --> 00:33:27,730 A transpose C A. A is now three by three, 486 00:33:27,730 --> 00:33:29,550 right, I've lost a row. 487 00:33:29,550 --> 00:33:31,810 A transpose is now three by three, 488 00:33:31,810 --> 00:33:34,700 I've lost a column, that fourth spring is gone. 489 00:33:34,700 --> 00:33:36,270 And what is C? 490 00:33:36,270 --> 00:33:43,370 Well of course there's no guy here anymore. 491 00:33:43,370 --> 00:33:50,170 What I'm trying to say is for this problem the matrices have 492 00:33:50,170 --> 00:33:54,280 become square. 493 00:33:54,280 --> 00:33:56,890 This would be correct. 494 00:33:56,890 --> 00:34:00,280 So this is an especially nice kind of problem. 495 00:34:00,280 --> 00:34:03,160 It's called statically determinate. 496 00:34:03,160 --> 00:34:08,830 It means I can determine the three w's from the three f's. 497 00:34:08,830 --> 00:34:09,850 I can go backwards. 498 00:34:09,850 --> 00:34:11,680 Everything is determined. 499 00:34:11,680 --> 00:34:16,430 The long word for the fixed-fixed one, 500 00:34:16,430 --> 00:34:19,830 our main example, is statically indeterminate. 501 00:34:19,830 --> 00:34:24,020 I cannot determine four w's from three forces. 502 00:34:24,020 --> 00:34:27,640 I can't determine what these internal forces are 503 00:34:27,640 --> 00:34:33,330 until I put the whole loop into one matrix K. 504 00:34:33,330 --> 00:34:36,480 So that's like a warning, and at the same time, 505 00:34:36,480 --> 00:34:39,690 an important separation. 506 00:34:39,690 --> 00:34:45,030 A few nice problems where you don't have too many springs, 507 00:34:45,030 --> 00:34:47,790 you don't have too many bars in a truss. 508 00:34:47,790 --> 00:34:51,390 You just have like, the minimum number to hold it together. 509 00:34:51,390 --> 00:34:55,350 Could be statically determinate and square matrices. 510 00:34:55,350 --> 00:34:59,840 But here we're not square. 511 00:34:59,840 --> 00:35:01,580 Now I go back. 512 00:35:01,580 --> 00:35:05,120 So that would be fixed-free. 513 00:35:05,120 --> 00:35:05,620 Right? 514 00:35:05,620 --> 00:35:09,940 That example that I just described would be fixed-free 515 00:35:09,940 --> 00:35:14,300 and we can kind of carry that along because we know that what 516 00:35:14,300 --> 00:35:19,570 happens is we lose a row and a column 517 00:35:19,570 --> 00:35:24,070 and a-- c_4 is just not in the picture anymore. 518 00:35:24,070 --> 00:35:32,000 But now I want to go back to the fixed-fixed one and finish it. 519 00:35:32,000 --> 00:35:38,590 So that's got a support down there, too. 520 00:35:38,590 --> 00:35:41,710 Key question, what's this matrix K? 521 00:35:41,710 --> 00:35:47,830 This A transpose C A. We know it's a square matrix, 522 00:35:47,830 --> 00:35:49,920 we know it's a symmetric matrix, but it 523 00:35:49,920 --> 00:35:53,050 would be really nice to know what does it look like. 524 00:35:53,050 --> 00:35:55,310 What does that matrix look like? 525 00:35:55,310 --> 00:35:57,410 Can I do the multiplication? 526 00:35:57,410 --> 00:36:03,170 So this is going to be K. So it starts with a three by four. 527 00:36:03,170 --> 00:36:08,560 1, -1; 1, -1; 1 -1. 528 00:36:08,560 --> 00:36:14,020 Then it's got the four by four, c_1, c_2, c_3, c_4. 529 00:36:14,020 --> 00:36:16,390 And then it's got the transpose of that, which 530 00:36:16,390 --> 00:36:22,290 is the 1, -1; 1, -1; 1, -1. 531 00:36:22,290 --> 00:36:32,380 With zeroes where I didn't write anything. 532 00:36:32,380 --> 00:36:38,690 We've got three matrices to multiply together. 533 00:36:38,690 --> 00:36:41,400 What's going to happen here? 534 00:36:41,400 --> 00:36:43,260 Well, let's see. 535 00:36:43,260 --> 00:36:45,100 I guess, why don't I multiply that by that? 536 00:36:45,100 --> 00:36:47,120 Can I do that? 537 00:36:47,120 --> 00:36:50,690 So that's like getting two steps together. 538 00:36:50,690 --> 00:36:53,140 It's going to be easy because of this. 539 00:36:53,140 --> 00:36:56,050 This is usually an easy matrix. 540 00:36:56,050 --> 00:36:57,570 Often diagonal. 541 00:36:57,570 --> 00:37:00,500 So when I do that multiplication, so let me, 542 00:37:00,500 --> 00:37:05,690 I'll just copy this guy. 543 00:37:05,690 --> 00:37:13,100 And now c_1 multiplies that row, c_2 multiplies this row, 544 00:37:13,100 --> 00:37:24,060 c_3 multiplies this row and c_4 multiplies the last row. 545 00:37:24,060 --> 00:37:26,850 c_1 in that row, c_2, c_3, and c_4. 546 00:37:26,850 --> 00:37:30,820 And now I'm ready to put those together into K. 547 00:37:30,820 --> 00:37:33,080 So K will be three by three. 548 00:37:33,080 --> 00:37:34,670 What does it have? 549 00:37:34,670 --> 00:37:37,270 It has c_1+c_2. 550 00:37:37,270 --> 00:37:41,010 551 00:37:41,010 --> 00:37:44,730 And then next to that is going to be this row 552 00:37:44,730 --> 00:37:47,940 one against column two, there'll be a zero 553 00:37:47,940 --> 00:37:50,800 or they'll be a -c_2 here. 554 00:37:50,800 --> 00:37:55,060 And then when row one goes against column three 555 00:37:55,060 --> 00:38:02,010 there's nothing. 556 00:38:02,010 --> 00:38:05,790 Why nothing? 557 00:38:05,790 --> 00:38:10,540 When do I expect to see a zero in the overall matrix? 558 00:38:10,540 --> 00:38:12,110 What is it about? 559 00:38:12,110 --> 00:38:16,390 So that zero is in the position 1, 3. 560 00:38:16,390 --> 00:38:22,300 What is it about masses one and three that 561 00:38:22,300 --> 00:38:24,320 is putting that zero in there. 562 00:38:24,320 --> 00:38:30,470 We kind of expect to see that zero even before we find it. 563 00:38:30,470 --> 00:38:32,780 If I look at the picture, what do you 564 00:38:32,780 --> 00:38:35,850 notice about masses one and three that 565 00:38:35,850 --> 00:38:38,420 is going to produce the zero? 566 00:38:38,420 --> 00:38:40,200 They're not connected. 567 00:38:40,200 --> 00:38:41,400 They're not connected. 568 00:38:41,400 --> 00:38:44,620 If I had another spring, which I could have, 569 00:38:44,620 --> 00:38:49,050 connecting mass one to mass three that would produce, 570 00:38:49,050 --> 00:38:50,700 I'd have another. 571 00:38:50,700 --> 00:38:53,430 I'd be up to five. 572 00:38:53,430 --> 00:38:55,740 Instead of four, there'd be a fifth spring. 573 00:38:55,740 --> 00:38:57,430 It would have its own constant. 574 00:38:57,430 --> 00:38:59,300 It would show up. 575 00:38:59,300 --> 00:39:00,860 Absolutely could. 576 00:39:00,860 --> 00:39:03,180 Here we don't have it. 577 00:39:03,180 --> 00:39:04,490 Now let me keep going. 578 00:39:04,490 --> 00:39:07,170 I know from symmetry that the second row 579 00:39:07,170 --> 00:39:13,510 times this is going to be zero, is going to be -c_2. 580 00:39:13,510 --> 00:39:15,270 Symmetric as I expected. 581 00:39:15,270 --> 00:39:21,450 What are you expecting on the diagonal there? c_2+c_3. 582 00:39:21,450 --> 00:39:24,310 That's certainly the right pattern. 583 00:39:24,310 --> 00:39:27,770 Zero, c_2+c_3. 584 00:39:27,770 --> 00:39:30,230 c_2+c_3. 585 00:39:30,230 --> 00:39:34,310 And what are you expecting over here? 586 00:39:34,310 --> 00:39:36,880 -c_3 is a good guess. 587 00:39:36,880 --> 00:39:38,500 It's seeing that pattern. 588 00:39:38,500 --> 00:39:40,010 Let's just see it happen. 589 00:39:40,010 --> 00:39:43,100 That second row times this third guy 590 00:39:43,100 --> 00:39:51,470 will give me zero, two rows, two zeroes, and then a -c_3, good. 591 00:39:51,470 --> 00:39:55,200 And now we know the zeroes going to show up here, 592 00:39:55,200 --> 00:39:57,840 the -c_3 is going to show up here. 593 00:39:57,840 --> 00:40:01,590 And what will show up here? c_3+c_4. 594 00:40:01,590 --> 00:40:13,480 595 00:40:13,480 --> 00:40:14,610 So we've got it. 596 00:40:14,610 --> 00:40:18,500 That's the matrix K that controls this whole problem. 597 00:40:18,500 --> 00:40:20,570 Now we check. 598 00:40:20,570 --> 00:40:22,440 It's square, yes. 599 00:40:22,440 --> 00:40:25,680 It's symmetric, yes. 600 00:40:25,680 --> 00:40:31,550 And notice also it's the kind of matrix we've seen already. 601 00:40:31,550 --> 00:40:35,960 In fact, it's exactly the matrix we've seen already 602 00:40:35,960 --> 00:40:38,550 Suppose all the c's are one. 603 00:40:38,550 --> 00:40:42,820 Suppose every c_1, c_2, c_3, c_4 is one. 604 00:40:42,820 --> 00:40:51,650 Then what's the matrix capital C in that standard case? 605 00:40:51,650 --> 00:40:56,490 C will just be the identity if these are all ones. 606 00:40:56,490 --> 00:41:00,460 And then I'm only left with A transpose A. 607 00:41:00,460 --> 00:41:08,040 So let me take that special case below it. 608 00:41:08,040 --> 00:41:15,020 Special if so this is if C is I, what matrix do we have then? 609 00:41:15,020 --> 00:41:18,960 Just to see that we have a matrix that we know about. 610 00:41:18,960 --> 00:41:24,060 So I'm copying this now here in the case when all the c's are 611 00:41:24,060 --> 00:41:24,560 one. 612 00:41:24,560 --> 00:41:28,860 So if you put all those c's to be one, what matrix do you get? 613 00:41:28,860 --> 00:41:30,030 You get, yes. 614 00:41:30,030 --> 00:41:38,260 You get the special K. Right, you get the special. 615 00:41:38,260 --> 00:41:43,440 So the work we did to understand that special matrix pays off 616 00:41:43,440 --> 00:41:44,350 here. 617 00:41:44,350 --> 00:41:47,130 Because we know how that matrix works. 618 00:41:47,130 --> 00:41:56,560 And this matrix, well, it's got four spring constants in it. 619 00:41:56,560 --> 00:42:03,400 But we can guess the important facts about this one 620 00:42:03,400 --> 00:42:04,780 from this one. 621 00:42:04,780 --> 00:42:11,210 So what are the important questions about that matrix? 622 00:42:11,210 --> 00:42:14,840 This is my matrix K now. 623 00:42:14,840 --> 00:42:17,570 What would be, we know it's square, we know it's symmetric. 624 00:42:17,570 --> 00:42:20,900 What else do we ask about a matrix? 625 00:42:20,900 --> 00:42:25,240 Well, positive definite, that's the perfect question, right. 626 00:42:25,240 --> 00:42:27,960 And built into positive definiteness 627 00:42:27,960 --> 00:42:32,360 would be a property that we mentioned the very first day. 628 00:42:32,360 --> 00:42:34,850 Is it invertible? 629 00:42:34,850 --> 00:42:36,640 What's your guess? 630 00:42:36,640 --> 00:42:39,160 Is that matrix invertible? 631 00:42:39,160 --> 00:42:44,780 Everybody's going to guess yes because, if you guessed no, 632 00:42:44,780 --> 00:42:50,050 where would you be, the whole course would end. 633 00:42:50,050 --> 00:42:55,480 In fact, the world would end because the problem 634 00:42:55,480 --> 00:42:57,430 is correctly posed. 635 00:42:57,430 --> 00:43:00,630 Those displacements are determined by the forces 636 00:43:00,630 --> 00:43:03,510 and that just says K is an invertible matrix. 637 00:43:03,510 --> 00:43:06,820 So but how do we see that it's invertible and, even 638 00:43:06,820 --> 00:43:09,530 more, positive definite, because that's 639 00:43:09,530 --> 00:43:11,000 the property we now know. 640 00:43:11,000 --> 00:43:14,500 So why is that matrix positive definite? 641 00:43:14,500 --> 00:43:18,220 Do we want to check determinants? 642 00:43:18,220 --> 00:43:21,410 We could say, okay, that guy's positive. 643 00:43:21,410 --> 00:43:29,150 We could evaluate this product and find that it came out well. 644 00:43:29,150 --> 00:43:33,140 Would you want to do that one? 645 00:43:33,140 --> 00:43:38,000 We could probably do the two by two determinant. 646 00:43:38,000 --> 00:43:42,470 Could you take that times that and subtract that? 647 00:43:42,470 --> 00:43:46,130 Let's just write it above what we would get. 648 00:43:46,130 --> 00:43:50,300 Just to see it. 649 00:43:50,300 --> 00:43:54,610 That number times that number would be a c_1*c_2 650 00:43:54,610 --> 00:44:02,130 and a c_1*c_3 and a c_2*c_2 twice. 651 00:44:02,130 --> 00:44:04,840 And a c_2*c_3. 652 00:44:04,840 --> 00:44:08,180 And then I would subtract off this guy. 653 00:44:08,180 --> 00:44:13,910 So it would knock out that, right? 654 00:44:13,910 --> 00:44:19,000 And it would leave something that would be positive. 655 00:44:19,000 --> 00:44:21,340 All the spring constants are positive here. 656 00:44:21,340 --> 00:44:27,030 We're talking normal materials. 657 00:44:27,030 --> 00:44:30,170 I guess, actually, people are producing now 658 00:44:30,170 --> 00:44:34,880 really amazing materials with amazing properties. 659 00:44:34,880 --> 00:44:41,480 And the amazing property is a material with a negative c. 660 00:44:41,480 --> 00:44:47,360 But that's like-- 18.085 does not allow such a thing. 661 00:44:47,360 --> 00:44:49,320 Right? 662 00:44:49,320 --> 00:44:52,130 All these c's are positive. 663 00:44:52,130 --> 00:44:56,050 And you might guess that the whole determinant is positive. 664 00:44:56,050 --> 00:44:58,930 But now I'd like you to tell me why. 665 00:44:58,930 --> 00:45:07,560 So now we can use our growing familiarity 666 00:45:07,560 --> 00:45:17,870 with matrices to say why is this matrix positive definite. 667 00:45:17,870 --> 00:45:21,900 Is symmetric, of course. 668 00:45:21,900 --> 00:45:27,710 Positive definite. 669 00:45:27,710 --> 00:45:30,900 Why? 670 00:45:30,900 --> 00:45:36,460 So that's what the previous lecture helped us to answer. 671 00:45:36,460 --> 00:45:38,730 We've got these various tests, but what 672 00:45:38,730 --> 00:45:42,910 was the core idea of positive definiteness? 673 00:45:42,910 --> 00:45:46,720 The core idea was positive energy. 674 00:45:46,720 --> 00:45:51,190 The core idea was I looked at the energy x trans-- no, u, 675 00:45:51,190 --> 00:45:51,830 sorry. 676 00:45:51,830 --> 00:45:56,260 Have to call it u now. u transpose times 677 00:45:56,260 --> 00:46:00,770 that matrix times u. 678 00:46:00,770 --> 00:46:08,780 And there was a reason why that matrix was, why this number, 679 00:46:08,780 --> 00:46:13,150 it's going to be a number, right? 680 00:46:13,150 --> 00:46:16,220 This combination will involve all four of these 681 00:46:16,220 --> 00:46:19,470 c's, it'll involve three u's. 682 00:46:19,470 --> 00:46:22,470 I don't want to write out that quantity. 683 00:46:22,470 --> 00:46:27,420 It would be, I'll have some u_1 squareds and some u_1*u_2. 684 00:46:27,420 --> 00:46:31,500 I won't have any u_1*u_3, because that 1, 685 00:46:31,500 --> 00:46:33,150 3 entry is zero. 686 00:46:33,150 --> 00:46:36,670 But why was this positive? 687 00:46:36,670 --> 00:46:39,230 Where do I put the parentheses. 688 00:46:39,230 --> 00:46:44,030 Where do I put the parentheses to see that that's positive? 689 00:46:44,030 --> 00:46:50,050 I put them around where? 690 00:46:50,050 --> 00:46:52,340 Around that, good. 691 00:46:52,340 --> 00:46:57,940 And around this? 692 00:46:57,940 --> 00:47:01,420 This is really, since we now have a letter for Au, 693 00:47:01,420 --> 00:47:09,040 this is really e transpose Ce, right? 694 00:47:09,040 --> 00:47:09,990 That's e. 695 00:47:09,990 --> 00:47:13,910 This is e, Au, and this is its transpose. 696 00:47:13,910 --> 00:47:17,940 And now what? 697 00:47:17,940 --> 00:47:21,860 So now we've narrowed it down to C. 698 00:47:21,860 --> 00:47:26,050 Oh, we can actually see why it's an energy. 699 00:47:26,050 --> 00:47:28,240 Remember C is that diagonal matrix. 700 00:47:28,240 --> 00:47:29,950 What will this be? 701 00:47:29,950 --> 00:47:37,570 This is the row of stretchings, the diagonal matrix of c's, 702 00:47:37,570 --> 00:47:40,250 and the column of stretchings. 703 00:47:40,250 --> 00:47:45,460 And now if I do that multiplication, what do I get? 704 00:47:45,460 --> 00:47:46,640 Do you see it? 705 00:47:46,640 --> 00:47:49,610 Because the physics is coming in. 706 00:47:49,610 --> 00:47:53,560 What do I get? 707 00:47:53,560 --> 00:47:54,830 This will multiply that. 708 00:47:54,830 --> 00:47:58,930 So what's the first term I should write here? e_1? 709 00:47:58,930 --> 00:48:01,830 710 00:48:01,830 --> 00:48:03,220 What will it be? 711 00:48:03,220 --> 00:48:04,390 I only have diagonal. 712 00:48:04,390 --> 00:48:06,470 In other words, I only have perfect squares 713 00:48:06,470 --> 00:48:08,420 when I look at this thing. 714 00:48:08,420 --> 00:48:16,700 I think I just have c_1*e_1 squared coming from that 715 00:48:16,700 --> 00:48:17,380 diagonal. 716 00:48:17,380 --> 00:48:20,720 That c_1 there, this e_1 here, this 717 00:48:20,720 --> 00:48:22,950 e_1 there is going to give me that c_1-- 718 00:48:22,950 --> 00:48:25,730 What else will I have? 719 00:48:25,730 --> 00:48:30,450 c_2*e_2 squared, c_3*e_3 squared and c_4*e_4 squared. 720 00:48:30,450 --> 00:48:36,890 721 00:48:36,890 --> 00:48:40,520 And do you remember about springs and Hooke's Law 722 00:48:40,520 --> 00:48:42,360 and energy? 723 00:48:42,360 --> 00:48:44,820 What's the energy in a spring? 724 00:48:44,820 --> 00:48:48,420 This is a stretched spring. 725 00:48:48,420 --> 00:48:55,440 So the energy in a stretched spring, what I wanted to say, 726 00:48:55,440 --> 00:49:00,610 this is the sum of four internal energies in the four springs 727 00:49:00,610 --> 00:49:05,210 but it properly should have a factor 1/2. 728 00:49:05,210 --> 00:49:10,050 There probably, to really use the word energy properly, 729 00:49:10,050 --> 00:49:13,410 it should be half of all this, half of all this, 730 00:49:13,410 --> 00:49:16,265 half of that's the energy in the first spring, 731 00:49:16,265 --> 00:49:19,550 the energy in the second, the energy in the third 732 00:49:19,550 --> 00:49:24,760 and the energy in the fourth. 733 00:49:24,760 --> 00:49:30,910 But of course our matrix point was, it's positive. 734 00:49:30,910 --> 00:49:33,730 It's a sum of squares multiplied now 735 00:49:33,730 --> 00:49:40,000 by these positive numbers, these elastic constants, c_1, two, 736 00:49:40,000 --> 00:49:42,300 three and four. 737 00:49:42,300 --> 00:49:53,290 So we know the main facts about that matrix. 738 00:49:53,290 --> 00:49:56,180 We're really at the point here of we've 739 00:49:56,180 --> 00:50:00,590 got some problem formulated, we've 740 00:50:00,590 --> 00:50:04,120 got the essential facts about the matrix, 741 00:50:04,120 --> 00:50:12,020 it's symmetric, positive definite, certainly invertible. 742 00:50:12,020 --> 00:50:20,900 Then there'd be the step of actually computing u 743 00:50:20,900 --> 00:50:24,760 by solving the stiffness equation. 744 00:50:24,760 --> 00:50:30,560 Say, for example, Professor Bathe's big finite element 745 00:50:30,560 --> 00:50:34,730 code, ADINA. 746 00:50:34,730 --> 00:50:37,390 What's the big picture for ADINA, 747 00:50:37,390 --> 00:50:39,540 for any big finite element code? 748 00:50:39,540 --> 00:50:43,880 NASTRAN, ANSYS, whatever. 749 00:50:43,880 --> 00:50:44,680 Abaqus. 750 00:50:44,680 --> 00:50:46,490 There are so many really good ones. 751 00:50:46,490 --> 00:50:50,610 And they've taken years and years of work to create. 752 00:50:50,610 --> 00:50:54,810 But if you look to see what are the elements that go in, 753 00:50:54,810 --> 00:51:03,240 you choose the model, and we'll see in the next chapter, 754 00:51:03,240 --> 00:51:07,600 in October we'll see what finite elements is about, 755 00:51:07,600 --> 00:51:10,960 you have the material properties, 756 00:51:10,960 --> 00:51:15,660 you assemble the matrix K. 757 00:51:15,660 --> 00:51:20,270 That's a key step, is assembling this matrix K. 758 00:51:20,270 --> 00:51:23,750 And then the final step is solve the system. 759 00:51:23,750 --> 00:51:24,810 Ku=f. 760 00:51:24,810 --> 00:51:28,820 But it's assembling that matrix. 761 00:51:28,820 --> 00:51:30,680 Now one thing popped into my head. 762 00:51:30,680 --> 00:51:34,560 Do I have time to mention it or not? 763 00:51:34,560 --> 00:51:39,440 And there's no class Monday I think, right? 764 00:51:39,440 --> 00:51:41,040 Can I mention? 765 00:51:41,040 --> 00:51:45,300 Can you hang on one more second to mention 766 00:51:45,300 --> 00:51:51,750 a really remarkable way to do matrix multiplication. 767 00:51:51,750 --> 00:51:54,250 You may say, we know matrix multiplication. 768 00:51:54,250 --> 00:51:55,120 We got it. 769 00:51:55,120 --> 00:51:55,620 Right? 770 00:51:55,620 --> 00:51:58,010 We did it and we got the right answer. 771 00:51:58,010 --> 00:51:59,500 Can I just show you another way. 772 00:51:59,500 --> 00:52:03,850 And you can like, see if it works. 773 00:52:03,850 --> 00:52:06,490 I did this multiplication by like, 774 00:52:06,490 --> 00:52:08,740 I'll say rows times columns. 775 00:52:08,740 --> 00:52:11,570 I took rows times columns. 776 00:52:11,570 --> 00:52:13,300 That's the usual way. 777 00:52:13,300 --> 00:52:17,240 But finite elements and other, often the right way 778 00:52:17,240 --> 00:52:18,560 is the opposite. 779 00:52:18,560 --> 00:52:23,430 It's columns times rows. 780 00:52:23,430 --> 00:52:26,290 And of course, this guy's in here too. 781 00:52:26,290 --> 00:52:33,260 You might say, okay, what do I get from column one times 782 00:52:33,260 --> 00:52:36,050 that number, times row one? 783 00:52:36,050 --> 00:52:44,430 Can you do that multiplication just mentally? 784 00:52:44,430 --> 00:52:46,860 Multiply that column by that row. 785 00:52:46,860 --> 00:52:51,100 First of all, what shape will the answer have? 786 00:52:51,100 --> 00:52:54,163 What shape will the answer have if I multiply a three 787 00:52:54,163 --> 00:52:57,590 by one times a one by three. 788 00:52:57,590 --> 00:52:58,360 Three by three. 789 00:52:58,360 --> 00:52:59,450 It's a full matrix. 790 00:52:59,450 --> 00:53:00,690 Columns times rows. 791 00:53:00,690 --> 00:53:05,320 And it's a totally legitimate way to multiply matrices. 792 00:53:05,320 --> 00:53:09,060 That column times that row will be? 793 00:53:09,060 --> 00:53:11,660 Well you can see what will it be. 794 00:53:11,660 --> 00:53:18,280 And then the c_1 is going to come into it. 795 00:53:18,280 --> 00:53:20,140 If I just did those multiplications, 796 00:53:20,140 --> 00:53:22,120 it would just be that. 797 00:53:22,120 --> 00:53:28,350 And then the c_1 puts that there. 798 00:53:28,350 --> 00:53:30,760 What do I see there? 799 00:53:30,760 --> 00:53:34,310 I see the element matrix. 800 00:53:34,310 --> 00:53:36,910 Do you see that this is the piece that 801 00:53:36,910 --> 00:53:40,750 involved c_1 in the answer? 802 00:53:40,750 --> 00:53:42,470 Well I guess you'll see it better 803 00:53:42,470 --> 00:53:48,750 when I do column two times c_2 times row two. 804 00:53:48,750 --> 00:53:50,930 So I have to add that guy on. 805 00:53:50,930 --> 00:53:54,160 And then I'll leave the other. 806 00:53:54,160 --> 00:53:58,860 What do I get if I do that column, three by one, 807 00:53:58,860 --> 00:54:03,610 times itself as a row times the c_2. 808 00:54:03,610 --> 00:54:06,070 I don't know if you see what I'm going to get. 809 00:54:06,070 --> 00:54:12,240 If you just do that, you'll see a c_2 will appear here. 810 00:54:12,240 --> 00:54:15,470 And a -c_2 will appear there. 811 00:54:15,470 --> 00:54:17,600 And these will be zeroes. 812 00:54:17,600 --> 00:54:20,830 So this was column one times row one. 813 00:54:20,830 --> 00:54:22,890 This is column two times row two. 814 00:54:22,890 --> 00:54:24,950 And third and then the fourth. 815 00:54:24,950 --> 00:54:28,190 But do you see that this part is telling me 816 00:54:28,190 --> 00:54:30,520 all about the second spring? 817 00:54:30,520 --> 00:54:34,450 This part is telling me, what does the first spring, the c_1, 818 00:54:34,450 --> 00:54:39,175 contribute to K. This part tells me what does the c_2 part, 819 00:54:39,175 --> 00:54:41,450 do you see the c_2 part in K? 820 00:54:41,450 --> 00:54:44,580 There, there, minus there and minus there. 821 00:54:44,580 --> 00:54:49,160 The third part from the column row would be the c_3 part. 822 00:54:49,160 --> 00:54:51,110 And the fourth part from the fourth spring 823 00:54:51,110 --> 00:54:52,860 would be the c_4 part. 824 00:54:52,860 --> 00:54:56,350 So that's a way you won't have thought of. 825 00:54:56,350 --> 00:55:00,360 But it's the way ADINA would assemble this matrix. 826 00:55:00,360 --> 00:55:02,290 It would not do that multiplication. 827 00:55:02,290 --> 00:55:05,390 It would do it this way, columns times rows. 828 00:55:05,390 --> 00:55:06,590 We'll see it again. 829 00:55:06,590 --> 00:55:10,270 So, hope you have a great weekend and a holiday Monday 830 00:55:10,270 --> 00:55:12,813 that we all happy about. 831 00:55:12,813 --> 00:55:13,313