1 00:00:00,110 --> 00:00:02,450 The following content is provided under a Creative 2 00:00:02,450 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,080 Your support will help MIT OpenCourseWare 4 00:00:06,080 --> 00:00:10,170 continue to offer high quality educational resources for free. 5 00:00:10,170 --> 00:00:12,710 To make a donation or to view additional materials 6 00:00:12,710 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,325 at ocw.mit.edu. 8 00:00:21,490 --> 00:00:23,390 PROFESSOR: Review model analysis. 9 00:00:23,390 --> 00:00:26,469 I figured most-- I figured lots of students have had questions 10 00:00:26,469 --> 00:00:28,260 and we could use a little practice on this. 11 00:00:28,260 --> 00:00:29,778 So that's what we're going to do. 12 00:00:29,778 --> 00:00:31,122 AUDIENCE: Solving the q thing? 13 00:00:31,122 --> 00:00:31,570 PROFESSOR: Huh? 14 00:00:31,570 --> 00:00:32,945 AUDIENCE: Solving the q function? 15 00:00:32,945 --> 00:00:34,890 PROFESSOR: Yeah, solving the whole thing. 16 00:00:34,890 --> 00:00:36,560 So but let's start. 17 00:00:36,560 --> 00:00:38,810 I'd still start with the usual concepts. 18 00:00:38,810 --> 00:00:40,650 I haven't even erased what we had last time. 19 00:00:40,650 --> 00:00:43,520 But this is what the students in the last group 20 00:00:43,520 --> 00:00:45,430 said about key concepts for the week. 21 00:00:45,430 --> 00:00:48,405 If you want to add something to it, speak up. 22 00:00:48,405 --> 00:00:53,840 But modal analysis, multi-degree of freedom, transfer functions, 23 00:00:53,840 --> 00:00:56,510 and then model analysis-- more specifically, 24 00:00:56,510 --> 00:01:00,300 response to initial conditions, response to steady state 25 00:01:00,300 --> 00:01:01,940 harmonic inputs. 26 00:01:01,940 --> 00:01:07,160 Anything else that was significant, conceptual, 27 00:01:07,160 --> 00:01:08,600 and new in the last week? 28 00:01:12,790 --> 00:01:13,730 Add to that. 29 00:01:13,730 --> 00:01:16,970 And then the second thing, what are 30 00:01:16,970 --> 00:01:19,950 issues that are muddy for you? 31 00:01:19,950 --> 00:01:23,370 Not quite clear, things you want to learn more about, 32 00:01:23,370 --> 00:01:24,550 have questions about. 33 00:01:24,550 --> 00:01:26,220 Last class. 34 00:01:26,220 --> 00:01:28,970 was more on transfer functions for multi-degree of freedom 35 00:01:28,970 --> 00:01:32,610 systems and solving initial condition problems 36 00:01:32,610 --> 00:01:35,360 is what one student was interested in about 37 00:01:35,360 --> 00:01:37,345 from using modal analysis. 38 00:01:37,345 --> 00:01:39,880 But do any of you have questions about things 39 00:01:39,880 --> 00:01:42,080 that you want to practiced on? 40 00:01:42,080 --> 00:01:46,880 AUDIENCE: So I understand how to get the new mass 41 00:01:46,880 --> 00:01:51,650 and spring whatever matricies for the q system, 42 00:01:51,650 --> 00:01:57,680 so I can set it up as the mq bq kq, but from there, 43 00:01:57,680 --> 00:02:00,362 I don't know how to go back and solve for x, 44 00:02:00,362 --> 00:02:01,820 or I don't know how to solve for x. 45 00:02:01,820 --> 00:02:02,340 PROFESSOR: Right. 46 00:02:02,340 --> 00:02:03,520 So we'll go through that. 47 00:02:03,520 --> 00:02:06,095 We're going to run through a complete modal analysis 48 00:02:06,095 --> 00:02:09,125 today, all the steps that you need to do to make it happen. 49 00:02:09,125 --> 00:02:11,760 So anything else that you've got a question about that I 50 00:02:11,760 --> 00:02:12,810 might be able to get to? 51 00:02:15,720 --> 00:02:17,350 OK, let's get rolling. 52 00:02:17,350 --> 00:02:19,760 This will take a little while. 53 00:02:19,760 --> 00:02:26,940 So problem for the day is, if you recall last time, 54 00:02:26,940 --> 00:02:30,470 we had this demo of this double pendulum. 55 00:02:30,470 --> 00:02:32,470 But now we're going to take that double pendulum 56 00:02:32,470 --> 00:02:35,610 and make the masses unequal. 57 00:02:35,610 --> 00:02:37,170 So well, the masses are equal here, 58 00:02:37,170 --> 00:02:38,962 but they could be unequal-- M1, M2, 59 00:02:38,962 --> 00:02:40,170 they're each half a kilogram. 60 00:02:40,170 --> 00:02:44,360 But we're changing the lengths a little bit, 1.1 for L1 61 00:02:44,360 --> 00:02:49,550 and 1.0 meters for L2. 62 00:02:49,550 --> 00:02:51,600 So slightly different lengths, and that'll 63 00:02:51,600 --> 00:02:54,510 make this system not symmetric so it won't have 1, 1 64 00:02:54,510 --> 00:02:57,470 and 1 minus 1 mode shapes. 65 00:02:57,470 --> 00:03:00,770 A little weak spring in the middle, possibility 66 00:03:00,770 --> 00:03:07,740 of having some dashpot here connected to a non-moving wall, 67 00:03:07,740 --> 00:03:09,640 another dashpot here. 68 00:03:09,640 --> 00:03:12,530 And the possibility of having harmonic 69 00:03:12,530 --> 00:03:16,320 excitations-- F1 on this one, F2 on that one. 70 00:03:16,320 --> 00:03:18,510 The whole system's been linearized. 71 00:03:18,510 --> 00:03:20,270 The equations in motion look something 72 00:03:20,270 --> 00:03:25,410 like this, mass damping matrix, stiffness matrix. 73 00:03:25,410 --> 00:03:27,620 Of course, it has gravity terms in it 74 00:03:27,620 --> 00:03:31,120 as well as the spring terms. 75 00:03:31,120 --> 00:03:33,250 And it's been linearized. 76 00:03:37,140 --> 00:03:41,500 This equation in motion, is it a force equation or a moment 77 00:03:41,500 --> 00:03:42,670 equation? 78 00:03:42,670 --> 00:03:44,960 Are they force or moment or both? 79 00:03:44,960 --> 00:03:46,790 We can have mixed ones like the cart. 80 00:03:46,790 --> 00:03:49,010 Problem with the pendulum has one force equation 81 00:03:49,010 --> 00:03:51,740 and one equation with units of torque, 82 00:03:51,740 --> 00:03:53,363 this one has units of what? 83 00:03:53,363 --> 00:03:54,204 AUDIENCE: Torque. 84 00:03:54,204 --> 00:03:54,870 PROFESSOR: Yeah. 85 00:03:54,870 --> 00:03:58,339 This is moments about point A and moments about point B 86 00:03:58,339 --> 00:03:59,630 and give you the two equations. 87 00:03:59,630 --> 00:04:01,590 So this is a torque equation. 88 00:04:01,590 --> 00:04:06,290 So you need to look and see if things inside here make sense. 89 00:04:06,290 --> 00:04:10,250 So the MGL, the non-linear equation, the restoring torque, 90 00:04:10,250 --> 00:04:12,680 is MGL sine theta. 91 00:04:12,680 --> 00:04:14,340 Linearized, it's just theta. 92 00:04:14,340 --> 00:04:18,970 So MGL 1 theta is the torque on the first mass. 93 00:04:18,970 --> 00:04:25,720 M2 GL 2 theta is the torque on the second mass, and so forth. 94 00:04:25,720 --> 00:04:30,920 And K1 L1 squared, why the L1 squared? 95 00:04:30,920 --> 00:04:34,930 This term here gets multiplied by theta 1, so what is that? 96 00:04:34,930 --> 00:04:39,360 What's K1 L1 squared theta 1? 97 00:04:39,360 --> 00:04:43,139 What kind of a torque-- what is that? 98 00:04:43,139 --> 00:04:43,930 Does it make sense? 99 00:04:43,930 --> 00:04:46,300 Are its units correct? 100 00:04:46,300 --> 00:04:50,335 So first of all, what's K1 L1 times theta? 101 00:04:52,840 --> 00:04:53,710 AUDIENCE: Force. 102 00:04:53,710 --> 00:04:54,960 PROFESSOR: Yeah, it's a force. 103 00:04:54,960 --> 00:04:58,012 What's L1 times theta? 104 00:04:58,012 --> 00:05:00,990 AUDIENCE: [INAUDIBLE] 105 00:05:00,990 --> 00:05:02,860 PROFESSOR: Physically, that is what? 106 00:05:02,860 --> 00:05:03,840 AUDIENCE: Distance. 107 00:05:03,840 --> 00:05:07,130 PROFESSOR: That's the distance that that thing moves, right? 108 00:05:07,130 --> 00:05:10,840 And a displacement times a spring constant 109 00:05:10,840 --> 00:05:17,070 gives you a force, and then that force times a moment arm 110 00:05:17,070 --> 00:05:19,700 gives you a torque. 111 00:05:19,700 --> 00:05:21,630 So it makes sense. 112 00:05:21,630 --> 00:05:28,560 OK, so L1 squared theta 1 is the torque about point A 113 00:05:28,560 --> 00:05:35,890 caused by a displacement theta 1, 114 00:05:35,890 --> 00:05:39,490 assuming this one is 0 when you do it. 115 00:05:39,490 --> 00:05:45,490 That's the torque caused by a displacement of theta 1 only. 116 00:05:45,490 --> 00:05:47,180 OK, all right. 117 00:05:49,830 --> 00:05:52,120 So that's our equations of motion linearized. 118 00:05:52,120 --> 00:05:54,780 This is the mass matrix. 119 00:05:54,780 --> 00:05:57,040 This is the stiffness matrix that you 120 00:05:57,040 --> 00:06:02,510 would get if you go into there and substitute in these values. 121 00:06:02,510 --> 00:06:03,205 Yeah? 122 00:06:03,205 --> 00:06:04,579 AUDIENCE: The matrix on the right 123 00:06:04,579 --> 00:06:07,830 would be to one of these [INAUDIBLE] on the bottom? 124 00:06:10,289 --> 00:06:11,330 PROFESSOR: This one here? 125 00:06:11,330 --> 00:06:12,710 This is a two by two matrix. 126 00:06:12,710 --> 00:06:13,750 AUDIENCE: Oh, OK. 127 00:06:13,750 --> 00:06:14,470 PROFESSOR: And it has to be. 128 00:06:14,470 --> 00:06:15,845 It's a two degree freedom system. 129 00:06:15,845 --> 00:06:20,020 This is K11, the K11 term. 130 00:06:20,020 --> 00:06:26,370 This is the K12 term, the K21 term, and the K2 term. 131 00:06:26,370 --> 00:06:28,290 AUDIENCE: So there's a space between-- 132 00:06:28,290 --> 00:06:30,248 PROFESSOR: Yeah, it got a little squeezed here. 133 00:06:30,248 --> 00:06:34,360 This space, there's an M2GL2-- and that shouldn't 134 00:06:34,360 --> 00:06:38,613 be squared-- plus KL2 squared. 135 00:06:38,613 --> 00:06:39,738 AUDIENCE: Plus [INAUDIBLE]? 136 00:06:42,714 --> 00:06:44,937 PROFESSOR: All right, pretty sure 137 00:06:44,937 --> 00:06:46,270 that's what it out to read like. 138 00:06:46,270 --> 00:06:48,280 MGL2 KL2 squared. 139 00:06:48,280 --> 00:06:54,290 And then each of these are minus KL1L2 minus KL1L2. 140 00:06:54,290 --> 00:06:59,080 All of this, this is all in a handout which will be put up 141 00:06:59,080 --> 00:07:01,110 on the Stellar website. 142 00:07:01,110 --> 00:07:04,890 And just so in terms of reviewing for final and things 143 00:07:04,890 --> 00:07:08,584 to go look at, for almost every recitation we've done this 144 00:07:08,584 --> 00:07:11,000 during Professor Gossard, the other recitation instructor, 145 00:07:11,000 --> 00:07:13,240 has written up the complete problem 146 00:07:13,240 --> 00:07:15,557 that was discussed with its solution 147 00:07:15,557 --> 00:07:16,640 and put it on the website. 148 00:07:16,640 --> 00:07:18,931 So you don't even need to have to copy this stuff down. 149 00:07:18,931 --> 00:07:21,969 It's all posted. 150 00:07:21,969 --> 00:07:23,510 In each recitation, we've essentially 151 00:07:23,510 --> 00:07:29,110 done the problem that is sort of the objective lesson 152 00:07:29,110 --> 00:07:30,440 for the week. 153 00:07:30,440 --> 00:07:32,390 So they're a good place to go review. 154 00:07:32,390 --> 00:07:35,270 So this whole problem will be up on there. 155 00:07:35,270 --> 00:07:38,210 And we found a mistake, actually. 156 00:07:38,210 --> 00:07:40,410 And I think it will get fixed before it gets put up, 157 00:07:40,410 --> 00:07:41,910 but I think this had a zero in here, 158 00:07:41,910 --> 00:07:43,860 and that's actually wrong, or if something 159 00:07:43,860 --> 00:07:45,630 got transposed in writing. 160 00:07:45,630 --> 00:07:48,600 That's the correct mass. 161 00:07:48,600 --> 00:07:52,894 OK, so you have these numbers. 162 00:07:52,894 --> 00:07:54,560 I don't want you to work this stuff out, 163 00:07:54,560 --> 00:07:58,090 because I want to focus on the model analysis. 164 00:07:58,090 --> 00:07:59,480 So let's see here. 165 00:07:59,480 --> 00:08:01,660 I'm going to ask you a question first. 166 00:08:01,660 --> 00:08:02,920 So we're going to begin this. 167 00:08:02,920 --> 00:08:03,559 Yeah? 168 00:08:03,559 --> 00:08:06,100 AUDIENCE: Is there a torsional damper at the top of the page? 169 00:08:06,100 --> 00:08:09,095 PROFESSOR: Oh, somebody asked a question in the last class 170 00:08:09,095 --> 00:08:11,000 and I drew that up there. 171 00:08:11,000 --> 00:08:14,400 But if you put a torsional damper up there, 172 00:08:14,400 --> 00:08:19,640 some CT value, how would it appear 173 00:08:19,640 --> 00:08:22,940 in the equation of motion? 174 00:08:22,940 --> 00:08:26,630 Would it have L1, L2 squareds in it? 175 00:08:26,630 --> 00:08:28,100 No, it would just simply disappear 176 00:08:28,100 --> 00:08:31,380 as a plus CT theta 1 or a theta 2, 177 00:08:31,380 --> 00:08:34,941 wherever it's applied directly, theta 1 dot. 178 00:08:39,000 --> 00:08:42,309 OK, so question. 179 00:08:42,309 --> 00:08:44,344 Write down your piece of paper-- so I'm not 180 00:08:44,344 --> 00:08:46,010 going to make you go to the board today, 181 00:08:46,010 --> 00:08:49,910 but I want you to take a minute and write things down. 182 00:08:49,910 --> 00:08:53,750 And then we'll check and see if everybody agrees. 183 00:08:53,750 --> 00:08:57,870 The reason we can do modal analysis 184 00:08:57,870 --> 00:09:01,830 is because of something we called the modal expansion 185 00:09:01,830 --> 00:09:02,480 theorem. 186 00:09:02,480 --> 00:09:04,620 It's basically the fundamental statement 187 00:09:04,620 --> 00:09:05,830 that says we can do this. 188 00:09:05,830 --> 00:09:07,930 So what is the modal expansion theorem? 189 00:09:07,930 --> 00:09:09,740 You can write it down mathematically 190 00:09:09,740 --> 00:09:12,770 if you want this, just as a little linear algebraic 191 00:09:12,770 --> 00:09:15,240 expression, or you could write it out in words. 192 00:09:15,240 --> 00:09:18,780 So take 30 seconds and write down 193 00:09:18,780 --> 00:09:23,290 what makes modal analysis work, what basic proposition. 194 00:09:50,480 --> 00:09:52,550 All right, somebody help me out. 195 00:09:52,550 --> 00:09:55,890 What is the modal expansion theorem? 196 00:09:55,890 --> 00:09:57,608 What's it say? 197 00:09:57,608 --> 00:09:59,604 AUDIENCE: Any motion of the system 198 00:09:59,604 --> 00:10:03,300 can be described as a weighted sum of the natural modes? 199 00:10:03,300 --> 00:10:06,110 PROFESSOR: Weighted sum of the motions of each 200 00:10:06,110 --> 00:10:08,320 of the natural modes, OK. 201 00:10:08,320 --> 00:10:14,580 So that's the statement, the most succinct way 202 00:10:14,580 --> 00:10:20,150 to say it is in the original generalized coordinates, 203 00:10:20,150 --> 00:10:21,990 you can express them as uq. 204 00:10:26,150 --> 00:10:31,150 u is the matrix of what? 205 00:10:31,150 --> 00:10:33,370 AUDIENCE: [INAUDIBLE] 206 00:10:33,370 --> 00:10:34,370 PROFESSOR: No, this is-- 207 00:10:34,370 --> 00:10:37,280 AUDIENCE: Oh no, of the mode shapes. 208 00:10:37,280 --> 00:10:38,380 PROFESSOR: Mode shapes. 209 00:10:38,380 --> 00:10:43,320 And the q's are the individual modal coordinates, right? 210 00:10:43,320 --> 00:10:46,970 And so this, expanded, says that this two degree freedom 211 00:10:46,970 --> 00:10:49,510 system has two generalized coordinates-- 212 00:10:49,510 --> 00:10:51,130 theta 1 and theta 2. 213 00:10:51,130 --> 00:10:55,040 And the response of either of the actual motion 214 00:10:55,040 --> 00:10:57,870 of the system expressed in the generalized coordinates 215 00:10:57,870 --> 00:11:05,220 can be made up as the sum of each of the modal coordinates, 216 00:11:05,220 --> 00:11:11,450 the q sub i's, scaled to this shape 217 00:11:11,450 --> 00:11:14,660 the mode shape for that mode. 218 00:11:14,660 --> 00:11:16,770 So this multiplies the mode shape, 219 00:11:16,770 --> 00:11:19,640 and that-- so everything contributed by mode 220 00:11:19,640 --> 00:11:23,240 one will move in the shape of mode one, 221 00:11:23,240 --> 00:11:26,100 and that will be reflected in the motion 222 00:11:26,100 --> 00:11:27,810 of the generalized coordinates. 223 00:11:27,810 --> 00:11:30,110 And in this case, it's a two by two. 224 00:11:30,110 --> 00:11:32,190 It's a two-degree freedom system. 225 00:11:32,190 --> 00:11:35,670 Here's the mode shape of mode 1 times q1, 226 00:11:35,670 --> 00:11:41,070 which we're going to solve for, plus the mode shape of mode two 227 00:11:41,070 --> 00:11:44,755 times its model motion. 228 00:11:44,755 --> 00:11:46,790 OK, that's the model expansion theorem. 229 00:11:50,990 --> 00:11:57,330 This allows you to-- in order to do this, 230 00:11:57,330 --> 00:12:00,120 you have to solve for these qi's. 231 00:12:00,120 --> 00:12:03,280 So what is the equation of motion 232 00:12:03,280 --> 00:12:11,820 that governs the behavior of each of the modal coordinates? 233 00:12:11,820 --> 00:12:13,700 Write it down. 234 00:12:13,700 --> 00:12:15,720 What's the whole reason we do this? 235 00:12:15,720 --> 00:12:19,345 There's one particular equation that every one of them 236 00:12:19,345 --> 00:12:19,845 satisfies. 237 00:12:38,440 --> 00:12:41,130 So equation of motion, not asking for solution. 238 00:12:41,130 --> 00:12:43,270 I just want equation of motion that 239 00:12:43,270 --> 00:12:47,975 governs these modal motions. 240 00:12:57,767 --> 00:12:59,980 I think I'll give you a minute to think about it. 241 00:13:11,490 --> 00:13:13,000 OK, somebody help me out. 242 00:13:13,000 --> 00:13:15,590 What's the equation of motion that I 243 00:13:15,590 --> 00:13:18,120 can write that will describe the motion of any one 244 00:13:18,120 --> 00:13:19,730 of these modal coordinates, qi? 245 00:13:23,390 --> 00:13:24,000 Christina? 246 00:13:24,000 --> 00:13:26,370 AUDIENCE: With the fancy M's and C's and K? 247 00:13:26,370 --> 00:13:27,120 PROFESSOR: Pardon? 248 00:13:27,120 --> 00:13:29,290 AUDIENCE: With the fancy M's and C's and K? 249 00:13:29,290 --> 00:13:30,980 PROFESSOR: Well yeah, but it's basically 250 00:13:30,980 --> 00:13:32,880 a very simple equation of motion, which you 251 00:13:32,880 --> 00:13:34,180 should be familiar with by now. 252 00:13:34,180 --> 00:13:35,300 What's it look like? 253 00:13:35,300 --> 00:13:38,400 For any one of these modal coordinates, 254 00:13:38,400 --> 00:13:40,780 what is the equation of motion that governs it? 255 00:13:45,820 --> 00:13:48,200 Why do we go to all this trouble? 256 00:13:48,200 --> 00:13:49,580 There's a reason for doing this. 257 00:13:49,580 --> 00:13:50,220 It's because-- 258 00:13:50,220 --> 00:13:51,160 AUDIENCE: Single degree of freedom. 259 00:13:51,160 --> 00:13:53,640 PROFESSOR: Ahh, the single degree of freedom oscillator 260 00:13:53,640 --> 00:13:54,590 equation, right? 261 00:14:00,040 --> 00:14:02,200 This is the reason we do this. 262 00:14:02,200 --> 00:14:07,590 Single degree of freedom systems are mathematically simple. 263 00:14:07,590 --> 00:14:10,150 You've seen them since high school. 264 00:14:10,150 --> 00:14:12,220 You've seen them in 1803. 265 00:14:12,220 --> 00:14:14,430 It's the second order linear differential 266 00:14:14,430 --> 00:14:17,119 equation that looks like this. 267 00:14:17,119 --> 00:14:18,660 And you already know everything there 268 00:14:18,660 --> 00:14:21,010 is to be known about that equation. 269 00:14:21,010 --> 00:14:23,130 And that's why one of the reasons why we do this. 270 00:14:23,130 --> 00:14:24,820 You don't have to solve a complex set 271 00:14:24,820 --> 00:14:26,720 of simultaneous differential equations, 272 00:14:26,720 --> 00:14:28,780 you just have to know one. 273 00:14:28,780 --> 00:14:31,755 And so the ith one, you need to know the ith modal mass, 274 00:14:31,755 --> 00:14:34,970 the ith modal damping, the ith modal stiffness, 275 00:14:34,970 --> 00:14:37,140 and the modal force. 276 00:14:37,140 --> 00:14:40,030 And we've-- in this course, we've taught you how to solve 277 00:14:40,030 --> 00:14:45,740 two kinds of single degree of freedom system problems. 278 00:14:45,740 --> 00:14:48,020 One is response to initial conditions 279 00:14:48,020 --> 00:14:52,090 when the forces on the right hand side is zero, 280 00:14:52,090 --> 00:14:54,000 and the other is this steady state 281 00:14:54,000 --> 00:14:58,420 response to a harmonic input, so a cosine omega t kind of input. 282 00:14:58,420 --> 00:15:00,870 So that's what we focused on in this course, 283 00:15:00,870 --> 00:15:03,150 because it's vibration we're interested in. 284 00:15:03,150 --> 00:15:06,355 So we've solve this equation for two kinds of problems. 285 00:15:09,880 --> 00:15:12,660 Now to do modal analysis, you need 286 00:15:12,660 --> 00:15:16,730 to be able to find these quantities. 287 00:15:16,730 --> 00:15:18,970 These we called the modal masses, 288 00:15:18,970 --> 00:15:23,360 the modal damping coefficients, the modal stiffnesses. 289 00:15:23,360 --> 00:15:25,940 How do you get those? 290 00:15:25,940 --> 00:15:26,980 How, for example? 291 00:15:26,980 --> 00:15:29,440 Write down on your paper, what equation, 292 00:15:29,440 --> 00:15:32,390 what linear algebra thing do you have 293 00:15:32,390 --> 00:15:35,384 to work out to get the modal masses for this system? 294 00:15:35,384 --> 00:15:36,300 Let's say all of them. 295 00:15:36,300 --> 00:15:38,642 I want a two-degree system. 296 00:15:38,642 --> 00:15:39,350 What's the state? 297 00:15:39,350 --> 00:15:40,820 What's the mass, the linear algebra 298 00:15:40,820 --> 00:15:42,694 you have to work out to get the modal masses? 299 00:16:02,222 --> 00:16:02,930 Somebody help me. 300 00:16:02,930 --> 00:16:03,545 What is it? 301 00:16:03,545 --> 00:16:05,030 Yeah? 302 00:16:05,030 --> 00:16:10,475 AUDIENCE: Transpose of the mode vector-- 303 00:16:10,475 --> 00:16:13,445 sorry, the mode matrix multiplied by the mass vector? 304 00:16:13,445 --> 00:16:15,920 PROFESSOR: Yeah? 305 00:16:15,920 --> 00:16:18,535 AUDIENCE: And then multiplied by the mode matrix. 306 00:16:18,535 --> 00:16:19,410 PROFESSOR: All right. 307 00:16:26,172 --> 00:16:28,110 All right. 308 00:16:28,110 --> 00:16:33,680 The modal forces are u transpose f. 309 00:16:33,680 --> 00:16:35,140 The modal masses-- and I've drawn 310 00:16:35,140 --> 00:16:37,056 these little diagonal marks in here to remind, 311 00:16:37,056 --> 00:16:39,619 you these matrices become all diagonal 312 00:16:39,619 --> 00:16:40,910 when you do the model analysis. 313 00:16:45,190 --> 00:16:47,930 Coordinate transformation, I'll call it. 314 00:16:47,930 --> 00:16:50,290 So the modal masses are u transpose mu. 315 00:16:50,290 --> 00:16:54,270 And this m is a matrix, and it's the original mass 316 00:16:54,270 --> 00:16:56,200 matrix of the system. 317 00:16:56,200 --> 00:16:59,270 Modal stiffness, matrix u transpose ku. 318 00:16:59,270 --> 00:17:02,400 And the modal damping matrix, u transpose cu. 319 00:17:02,400 --> 00:17:05,359 But this one can be problematic. 320 00:17:05,359 --> 00:17:08,990 You have to force this one to behave. 321 00:17:08,990 --> 00:17:13,890 These are guaranteed to behave, all right? 322 00:17:13,890 --> 00:17:18,569 So for this problem, and I know some of you 323 00:17:18,569 --> 00:17:22,300 are a little rusty calculating these things. 324 00:17:22,300 --> 00:17:26,000 So there's the model mass matrix. 325 00:17:26,000 --> 00:17:32,910 And here is the modal matrix of eigenvectors or mode 326 00:17:32,910 --> 00:17:34,500 shapes of the system. 327 00:17:34,500 --> 00:17:39,790 u is made up of columns, and each column 328 00:17:39,790 --> 00:17:41,990 is one of the mode shapes. 329 00:17:41,990 --> 00:17:48,270 The convention is to order them from the first mode to the nth 330 00:17:48,270 --> 00:17:49,740 mode where the order is established 331 00:17:49,740 --> 00:17:50,865 by the natural frequencies. 332 00:17:50,865 --> 00:17:53,750 The lowest natural frequency is first-- second, second, 333 00:17:53,750 --> 00:17:55,520 up to the highest natural frequency. 334 00:17:55,520 --> 00:17:58,110 Anyway, here's mode one, here's mode two for this system. 335 00:18:03,900 --> 00:18:05,880 You have to choose a way in which 336 00:18:05,880 --> 00:18:10,850 to normalize the mode shapes. 337 00:18:10,850 --> 00:18:13,300 I choose to normalize them usually. 338 00:18:13,300 --> 00:18:15,950 I say, I'm going to make the top element of them one. 339 00:18:15,950 --> 00:18:19,780 And I do that whatever-- if do MATLAB like you do, 340 00:18:19,780 --> 00:18:22,190 there's a function called Eig, which means eigenvalue. 341 00:18:22,190 --> 00:18:28,050 You do Eig of A, it'll give you the eigenvalues of matrix A. 342 00:18:28,050 --> 00:18:31,915 And it'll give them back to you unnormalized and unordered. 343 00:18:34,480 --> 00:18:36,424 Well, so you can write a little program 344 00:18:36,424 --> 00:18:37,590 to put it all in nice order. 345 00:18:37,590 --> 00:18:44,110 But if MATLAB gave you back the mode shapes for a system 346 00:18:44,110 --> 00:18:46,600 and it said, well, the mode shapes of the system 347 00:18:46,600 --> 00:18:49,260 are-- and it's a two by two system. 348 00:18:49,260 --> 00:18:58,260 2 and 0.4 and 0.6 0.5, you know that the vectors are the mode 349 00:18:58,260 --> 00:19:01,230 shapes, the columns. 350 00:19:01,230 --> 00:19:03,440 How would you normalize those? 351 00:19:03,440 --> 00:19:07,364 How would you make the top element 1 in this first one? 352 00:19:07,364 --> 00:19:08,280 AUDIENCE: Divide by 2. 353 00:19:08,280 --> 00:19:09,446 PROFESSOR: Divide what by 2? 354 00:19:09,446 --> 00:19:10,750 AUDIENCE: The Entire column. 355 00:19:10,750 --> 00:19:12,166 PROFESSOR: The entire column by 2. 356 00:19:12,166 --> 00:19:13,250 Just factor 2 out. 357 00:19:13,250 --> 00:19:16,950 So this would become 2/2 and 0.4/2. 358 00:19:19,770 --> 00:19:25,260 And then that's 1 and 0.2. 359 00:19:25,260 --> 00:19:29,620 So you just normalize this vector so the top element is 1. 360 00:19:29,620 --> 00:19:31,270 So do you have to normalize it? 361 00:19:31,270 --> 00:19:32,740 Could you use them this way? 362 00:19:32,740 --> 00:19:35,350 Sure. 363 00:19:35,350 --> 00:19:42,030 But once chosen, once the normalization is chosen, 364 00:19:42,030 --> 00:19:45,680 the key to doing modal analysis is you have to stick with it. 365 00:19:45,680 --> 00:19:46,522 You can't move. 366 00:19:46,522 --> 00:19:48,230 You can't mess with that halfway through, 367 00:19:48,230 --> 00:19:50,625 or you totally screw up the solution. 368 00:19:50,625 --> 00:19:52,560 So you pick your normalization. 369 00:19:52,560 --> 00:19:56,140 When you calculate the natural frequencies and mode shapes, 370 00:19:56,140 --> 00:19:58,060 you pick a normalization, and you 371 00:19:58,060 --> 00:20:01,190 must ride with that all the way through, 372 00:20:01,190 --> 00:20:03,130 including putting it back together here 373 00:20:03,130 --> 00:20:06,590 at the end, this summation. 374 00:20:06,590 --> 00:20:10,660 OK, so let's do-- I want you to do this computation. 375 00:20:10,660 --> 00:20:13,450 Calculate the modal masses for this problem. 376 00:20:16,480 --> 00:20:18,740 That means you have to remember what a transpose is. 377 00:20:18,740 --> 00:20:21,390 There's the model, that's the model matrix. 378 00:20:21,390 --> 00:20:23,950 And the modal mass matrix is right there. 379 00:20:23,950 --> 00:20:25,450 So actually, just do the arithmetic. 380 00:20:25,450 --> 00:20:26,620 Take a few minutes. 381 00:20:26,620 --> 00:20:27,480 Yeah? 382 00:20:27,480 --> 00:20:28,980 AUDIENCE: On the exam, we won't have 383 00:20:28,980 --> 00:20:32,122 calculators, so like is it all going to be variables, or--? 384 00:20:32,122 --> 00:20:33,568 PROFESSOR: Say that again? 385 00:20:33,568 --> 00:20:36,109 AUDIENCE: On the exam, we won't have calculators or anything. 386 00:20:36,109 --> 00:20:38,590 PROFESSOR: On the exam, we'd either make it so simple 387 00:20:38,590 --> 00:20:42,430 that you can, in fact, do it in your head or on paper, 388 00:20:42,430 --> 00:20:45,280 or we won't ask a question that you have to do it that way. 389 00:20:52,360 --> 00:20:54,845 Or we'll accept an answer where you put it down but don't 390 00:20:54,845 --> 00:20:55,845 have to multiply it out. 391 00:20:59,620 --> 00:21:02,050 So just do this one, just to see if you 392 00:21:02,050 --> 00:21:06,300 remember the mechanics of doing the linear algebra to get that. 393 00:21:06,300 --> 00:21:09,480 OK, somebody have an answer for me 394 00:21:09,480 --> 00:21:12,600 here for the modal mass matrix? 395 00:21:12,600 --> 00:21:13,920 What's the first element? 396 00:21:13,920 --> 00:21:14,628 Somebody help me. 397 00:21:17,700 --> 00:21:20,205 Give me a number and then everybody else can check you. 398 00:21:25,916 --> 00:21:27,585 AUDIENCE: 0.254. 399 00:21:27,585 --> 00:21:28,460 PROFESSOR: Say again? 400 00:21:28,460 --> 00:21:30,040 AUDIENCE: 0.254. 401 00:21:30,040 --> 00:21:33,850 PROFESSOR: 0.254. 402 00:21:33,850 --> 00:21:37,340 OK, what about the second one, this element over here? 403 00:21:40,780 --> 00:21:41,280 Speak up. 404 00:21:41,280 --> 00:21:41,790 AUDIENCE: 0. 405 00:21:41,790 --> 00:21:43,123 PROFESSOR: Yeah, it better be 0. 406 00:21:43,123 --> 00:21:44,670 What about this one down here? 407 00:21:44,670 --> 00:21:45,990 All right, how about this one? 408 00:22:05,217 --> 00:22:07,057 AUDIENCE: Wait, that first one's not right. 409 00:22:07,057 --> 00:22:07,682 PROFESSOR: Hmm? 410 00:22:07,682 --> 00:22:09,200 AUDIENCE: That first one's not right. 411 00:22:09,200 --> 00:22:09,783 PROFESSOR: OK. 412 00:22:12,806 --> 00:22:14,760 AUDIENCE: [INAUDIBLE] 413 00:22:14,760 --> 00:22:17,550 PROFESSOR: So did you give me the first one? 414 00:22:17,550 --> 00:22:20,010 So you're authorized to change this. 415 00:22:20,010 --> 00:22:21,680 OK, you got point what? 416 00:22:21,680 --> 00:22:22,842 AUDIENCE: 859. 417 00:22:22,842 --> 00:22:30,370 PROFESSOR: 8598, And this 0, 0-- how about this second one 418 00:22:30,370 --> 00:22:31,518 down here now? 419 00:22:31,518 --> 00:22:37,520 AUDIENCE: 3.48. 420 00:22:37,520 --> 00:22:42,940 PROFESSOR: 3.48. 421 00:22:42,940 --> 00:22:44,525 Anybody else get anything different? 422 00:22:48,310 --> 00:22:59,130 So let's-- we have 605 00 and 0.5, 423 00:22:59,130 --> 00:23:00,785 and we're multiplying that. 424 00:23:45,100 --> 00:23:50,851 So if you're looking for just one of them, by the way-- 425 00:23:50,851 --> 00:23:55,790 where'd my eraser go?-- all you need is one of the modal 426 00:23:55,790 --> 00:23:56,290 masses. 427 00:24:02,330 --> 00:24:06,350 The only ones that give you non-zero results 428 00:24:06,350 --> 00:24:22,720 is when you compute u transpose for mode r m u for mode r. 429 00:24:22,720 --> 00:24:25,400 So if you're only looking for this second one, 430 00:24:25,400 --> 00:24:30,140 you only have to do the calculation for that mode. 431 00:24:30,140 --> 00:24:33,060 So this then becomes a set of matrix, matrix, matrix, 432 00:24:33,060 --> 00:24:34,850 you only have to do a couple of vectors. 433 00:24:34,850 --> 00:24:41,070 So this looks like for mode 2, it's minus what? 434 00:24:43,820 --> 00:25:04,640 Mode 2 is 1, and minus 1.6949 times 0.60500 0.5 times 1 435 00:25:04,640 --> 00:25:08,520 and minus 1.6949. 436 00:25:08,520 --> 00:25:12,990 So to get just one modal mass, this is M2. 437 00:25:12,990 --> 00:25:14,680 To get just one modal mass, now you only 438 00:25:14,680 --> 00:25:15,846 have to do that calculation. 439 00:25:15,846 --> 00:25:21,040 You only have to do the computation using 440 00:25:21,040 --> 00:25:22,190 one of the mode shapes. 441 00:25:22,190 --> 00:25:26,390 So it's this times that, and this times that 442 00:25:26,390 --> 00:25:28,400 gives you some numbers back. 443 00:25:28,400 --> 00:25:30,790 0.605 and half of this about point 444 00:25:30,790 --> 00:25:33,370 0.8, and then you take that and multiply again. 445 00:25:33,370 --> 00:25:37,150 Anyway, can somebody give me this second number? 446 00:25:37,150 --> 00:25:39,010 I have a 3.48. 447 00:25:39,010 --> 00:25:40,400 Anybody get anything different? 448 00:25:40,400 --> 00:25:40,900 Pardon? 449 00:25:40,900 --> 00:25:41,950 AUDIENCE: I messed it up. 450 00:25:41,950 --> 00:25:44,000 PROFESSOR: OK. 451 00:25:44,000 --> 00:25:49,375 There are about 18 of you and nobody can do this calculation? 452 00:25:52,155 --> 00:25:52,830 AUDIENCE: 2.13. 453 00:25:52,830 --> 00:25:53,705 PROFESSOR: Say again? 454 00:25:53,705 --> 00:25:55,360 AUDIENCE: 2.130. 455 00:25:55,360 --> 00:26:12,030 PROFESSOR: All right, I have a 2.132 and a 2.0. 456 00:26:12,030 --> 00:26:18,290 OK, there you are. 457 00:26:21,000 --> 00:26:24,610 OK, so if you were having trouble sorting that out, 458 00:26:24,610 --> 00:26:27,290 probably a good thing to go back and review a little bit 459 00:26:27,290 --> 00:26:29,550 of your linear algebra. 460 00:26:29,550 --> 00:26:33,680 OK, if you do u transpose ku, you get this. 461 00:26:33,680 --> 00:26:34,688 Yeah? 462 00:26:34,688 --> 00:26:36,560 AUDIENCE: What is it for the [INAUDIBLE] 463 00:26:36,560 --> 00:26:40,110 you have a multiplied by F1 L1, not just F1? 464 00:26:40,110 --> 00:26:42,800 PROFESSOR: OK, I'm going to guess where I'm going next. 465 00:26:42,800 --> 00:26:49,000 So to get the stiffness matrix, u transpose ku, you get this. 466 00:26:49,000 --> 00:26:51,970 We're going to leave the damping matrix for a minute. 467 00:26:51,970 --> 00:26:52,830 We need that. 468 00:26:52,830 --> 00:26:55,290 We need the modal excitations. 469 00:26:55,290 --> 00:26:57,870 So I'm just going to do a particular problem. 470 00:26:57,870 --> 00:26:59,900 I'm going to say, let's let F2 be 0. 471 00:26:59,900 --> 00:27:02,290 We'll only have one force. 472 00:27:02,290 --> 00:27:06,080 And the first force will be F1 cosine omega t. 473 00:27:06,080 --> 00:27:08,610 And so now I need to do u transpose 474 00:27:08,610 --> 00:27:11,150 F. So here is u transpose. 475 00:27:11,150 --> 00:27:12,902 Here's the F's, and I guess I've got 476 00:27:12,902 --> 00:27:17,890 to keep my cosine omega t here. 477 00:27:17,890 --> 00:27:19,890 So you multiply that out. 478 00:27:19,890 --> 00:27:21,620 What do you get? 479 00:27:21,620 --> 00:27:22,120 Yeah? 480 00:27:22,120 --> 00:27:24,070 AUDIENCE: Why is F1 L1 [INAUDIBLE]? 481 00:27:24,070 --> 00:27:27,290 PROFESSOR: Well, because F1 is just 482 00:27:27,290 --> 00:27:30,310 the applied force, but the equation in motion that we're 483 00:27:30,310 --> 00:27:32,860 working with, if you go back and look at it, 484 00:27:32,860 --> 00:27:34,770 what is the forces on the right hand side? 485 00:27:38,970 --> 00:27:41,150 The forces have to be moments, right? 486 00:27:41,150 --> 00:27:44,200 If we're putting a force down there, it's a moment equation. 487 00:27:44,200 --> 00:27:46,050 We need the moments about the pivot. 488 00:27:46,050 --> 00:27:51,074 So it's the force times L1 or the force times L2. 489 00:27:51,074 --> 00:27:54,726 AUDIENCE: Oh, so we don't have to use actual magnitude of 4. 490 00:27:54,726 --> 00:27:56,452 We have to use [INAUDIBLE]. 491 00:27:56,452 --> 00:27:57,410 PROFESSOR: You have to. 492 00:27:57,410 --> 00:28:00,830 You well eventually-- well, these kind of problems 493 00:28:00,830 --> 00:28:04,380 are easiest to do once you reduce them to numbers. 494 00:28:04,380 --> 00:28:07,350 I'm leaving the force in it as a variable at the moment 495 00:28:07,350 --> 00:28:10,760 just so you can see how it carries through the problem. 496 00:28:10,760 --> 00:28:13,490 But I'm just saying, in the real problem 497 00:28:13,490 --> 00:28:16,860 there, let's say there is no F2. 498 00:28:16,860 --> 00:28:24,490 There is an F1, and the F1 of t looks like a magnitude 499 00:28:24,490 --> 00:28:27,200 f1 times cosine omega t. 500 00:28:27,200 --> 00:28:29,520 That's the only force I have in the system. 501 00:28:29,520 --> 00:28:32,256 But we're working with equations of motions. 502 00:28:32,256 --> 00:28:35,440 An equation of motion is the right hand side-- F1, L1, 503 00:28:35,440 --> 00:28:36,760 and F2 L2. 504 00:28:36,760 --> 00:28:39,570 And you have to retain the L1's and L2's in order 505 00:28:39,570 --> 00:28:41,555 to have the correct equation of motion. 506 00:28:45,390 --> 00:28:50,530 So when we say this is kind of just a generic form, 507 00:28:50,530 --> 00:28:55,610 this is the modal force vector is the mode shape 508 00:28:55,610 --> 00:28:59,950 matrix times the modal excitations 509 00:28:59,950 --> 00:29:01,610 in the original coordinates. 510 00:29:01,610 --> 00:29:04,930 So I just wrote F here, but what this really means 511 00:29:04,930 --> 00:29:06,060 is this is F1, L1. 512 00:29:11,710 --> 00:29:13,540 F2, L2. 513 00:29:13,540 --> 00:29:21,000 These are the real generalized forces in the system. 514 00:29:21,000 --> 00:29:23,190 OK, they're the real, generalized forces. 515 00:29:23,190 --> 00:29:25,960 Now, I've let F2 be 0. 516 00:29:25,960 --> 00:29:31,710 So the only generalized force is F1, L1 cosine omega T. 517 00:29:31,710 --> 00:29:35,790 I multiply that by u transpose to get-- and what do I get? 518 00:29:35,790 --> 00:29:39,300 This is a pretty simple calculation. 519 00:29:39,300 --> 00:29:47,810 So this becomes F1, L1 cos and F1, L1 cos. 520 00:29:47,810 --> 00:29:51,220 So the two modal forces are identical. 521 00:29:51,220 --> 00:29:51,720 Yeah? 522 00:29:51,720 --> 00:29:53,148 AUDIENCE: Why do we know this one 523 00:29:53,148 --> 00:29:54,617 so we don't have to include a phi? 524 00:29:54,617 --> 00:29:56,450 PROFESSOR: Ah, well the phi doesn't come out 525 00:29:56,450 --> 00:29:57,158 until the answer. 526 00:29:59,970 --> 00:30:01,550 What does the fee mean? 527 00:30:01,550 --> 00:30:03,650 What's that phase angle mean? 528 00:30:03,650 --> 00:30:10,140 If you-- remember, we're doing steady state problems 529 00:30:10,140 --> 00:30:19,720 in which F1, L1 for example, if it's cosine, 530 00:30:19,720 --> 00:30:23,640 we're just assuming it looks like that. 531 00:30:23,640 --> 00:30:28,235 And we're looking for a solution of theta 1. 532 00:30:32,490 --> 00:30:34,720 And actually, we can't quite go there yet. 533 00:30:34,720 --> 00:30:37,600 We're doing single degree of freedom problems, right? 534 00:30:37,600 --> 00:30:42,020 We are looking for a solution for Q1. 535 00:30:42,020 --> 00:30:45,680 We turn this into a modal force, q, 536 00:30:45,680 --> 00:30:49,680 but it happens to be the capital Q1, the modal force, 537 00:30:49,680 --> 00:30:51,820 is F1 L1, right? 538 00:30:51,820 --> 00:30:53,250 Cosine omega t. 539 00:30:53,250 --> 00:30:55,120 So that's the input. 540 00:30:55,120 --> 00:30:59,200 The output is q, little q, of t, the modal coordinate. 541 00:30:59,200 --> 00:31:01,310 And what does it look like? 542 00:31:01,310 --> 00:31:04,630 We're only doing steady state, no transience. 543 00:31:04,630 --> 00:31:12,650 It looks like a response that looks like this, 544 00:31:12,650 --> 00:31:14,830 but it's shifted. 545 00:31:14,830 --> 00:31:16,910 I'll draw this so its peak is right here, 546 00:31:16,910 --> 00:31:23,010 cosine is its-- it's shifted in time by this amount, 547 00:31:23,010 --> 00:31:26,470 between this peak is here versus the peak being there. 548 00:31:26,470 --> 00:31:30,450 And that we can represent as a phase angle. 549 00:31:30,450 --> 00:31:34,680 Remember, one period from here to here is 2 pi radians. 550 00:31:34,680 --> 00:31:39,300 So some portion of that period is an angle. 551 00:31:39,300 --> 00:31:41,745 You can interpret it as an angle or you can interpret it 552 00:31:41,745 --> 00:31:44,290 as a time delay. 553 00:31:44,290 --> 00:31:52,320 And this then has the form of some q1 magnitude cosine omega 554 00:31:52,320 --> 00:31:57,050 t minus that phase shift. 555 00:31:57,050 --> 00:32:00,320 So it's only-- that's the only thing the phase shift means. 556 00:32:00,320 --> 00:32:03,570 Cosine in doesn't mean exactly the response 557 00:32:03,570 --> 00:32:07,797 out's going to be exactly in the same perfectly in time with it. 558 00:32:07,797 --> 00:32:08,630 It could be shifted. 559 00:32:11,475 --> 00:32:13,225 Now you know that it-- for a single degree 560 00:32:13,225 --> 00:32:15,555 of freedom system at resonance, what's the phase angle? 561 00:32:15,555 --> 00:32:16,710 Do you remember that? 562 00:32:16,710 --> 00:32:19,170 It's always one number. 563 00:32:19,170 --> 00:32:20,435 It's pi over 2. 564 00:32:24,140 --> 00:32:27,110 A shift of pi over 2, if you remember your trigonometry, 565 00:32:27,110 --> 00:32:30,845 takes you from cosine to sine or sine to cosine. 566 00:32:30,845 --> 00:32:34,610 It's trying to tell you that the response is 567 00:32:34,610 --> 00:32:40,240 shifted by exactly a quarter of a cycle, pi over 2. 568 00:32:40,240 --> 00:32:45,400 And the reason for that is that at resonance, 569 00:32:45,400 --> 00:32:49,990 all of the excitation is going into overpowering the damper. 570 00:32:49,990 --> 00:32:54,020 And the damper's motion is proportional to velocity. 571 00:32:54,020 --> 00:32:57,490 And if velocity, if displacement looks like cosine, 572 00:32:57,490 --> 00:32:59,580 velocity looks like one derivative of it, 573 00:32:59,580 --> 00:33:02,280 which is sine. 574 00:33:02,280 --> 00:33:07,890 So that pi over 2 says that the response 575 00:33:07,890 --> 00:33:12,010 velocity is in phase with the force, and that makes sense. 576 00:33:12,010 --> 00:33:13,510 Something I hadn't said in lecture 577 00:33:13,510 --> 00:33:15,550 but I really meant to is I want you 578 00:33:15,550 --> 00:33:16,820 to think about something here. 579 00:33:16,820 --> 00:33:18,977 Single degree of freedom system, we'll 580 00:33:18,977 --> 00:33:20,560 even write the one we're working with. 581 00:33:20,560 --> 00:33:29,420 M1 Q1 double dot plus C1 Q1 dot plus K1 Q 582 00:33:29,420 --> 00:33:33,960 equals some F1 L1 cosine omega t, 583 00:33:33,960 --> 00:33:37,510 and I'm going to let omega E be at omega 1. 584 00:33:37,510 --> 00:33:40,240 But this is the equation of motion, right? 585 00:33:40,240 --> 00:33:41,560 Let's plug in. 586 00:33:41,560 --> 00:33:45,430 We're saying we're going to do this right at the-- 587 00:33:45,430 --> 00:33:52,360 and we know the response of this is Q1 is some magnitude cosine 588 00:33:52,360 --> 00:33:56,600 omega t minus the phasing. 589 00:33:56,600 --> 00:33:57,920 We know we can plug that in. 590 00:33:57,920 --> 00:34:21,719 We get minus M1 omega squared Q1 plus K1 Q1 minus C1 omega Q1. 591 00:34:21,719 --> 00:34:26,739 And this one goes like sine omega T minus phi. 592 00:34:26,739 --> 00:34:28,870 This one here needs to get multiplied. 593 00:34:28,870 --> 00:34:33,254 This term gets multiplied by cosine omega t minus phi. 594 00:34:33,254 --> 00:34:35,920 You plug that into this. 595 00:34:35,920 --> 00:34:39,570 This term and this term both behave like cosine. 596 00:34:39,570 --> 00:34:42,650 This term, one derivative behaves like minus sine. 597 00:34:42,650 --> 00:34:45,690 One derivative of cosine gives you minus sine, right? 598 00:34:45,690 --> 00:34:50,260 And when omega equals omega 1, so 599 00:34:50,260 --> 00:34:54,830 when you're right at resonance here, what happens? 600 00:34:54,830 --> 00:34:58,520 This is squared. 601 00:34:58,520 --> 00:35:02,530 And put the squared down here, there we go. 602 00:35:02,530 --> 00:35:03,310 So this is omega. 603 00:35:03,310 --> 00:35:06,200 But now I'm going to let it be right at omega 1. 604 00:35:06,200 --> 00:35:12,150 What is omega 1 in terms of K's and M's? 605 00:35:12,150 --> 00:35:14,790 K1/M1, otherwise, one of the checks 606 00:35:14,790 --> 00:35:18,270 you can make when you finish doing your modal-- if you take 607 00:35:18,270 --> 00:35:21,200 that modal mass and the modal stiffness 608 00:35:21,200 --> 00:35:26,400 and you divide 7.96 by 0.8598, that had better be omega 1 609 00:35:26,400 --> 00:35:27,497 squared. 610 00:35:27,497 --> 00:35:29,080 That's a good way to check that you've 611 00:35:29,080 --> 00:35:30,950 done all your arithmetic right. 612 00:35:30,950 --> 00:35:33,970 All right, I'm going to plug in omega 1 613 00:35:33,970 --> 00:35:35,650 squared here equals K1/M1. 614 00:35:38,240 --> 00:35:46,070 So I put in, this becomes minus M1 K1 over M1, which 615 00:35:46,070 --> 00:35:47,830 is minus K1, right? 616 00:35:47,830 --> 00:35:50,670 Hmm, plus K1. 617 00:35:53,570 --> 00:35:57,040 That resonance, this term accounts 618 00:35:57,040 --> 00:36:01,170 for the inertial force in the system, the force required 619 00:36:01,170 --> 00:36:03,340 to accelerate the mass. 620 00:36:03,340 --> 00:36:04,460 This is a force equation. 621 00:36:04,460 --> 00:36:08,130 This accounts for the force required to push the spring. 622 00:36:08,130 --> 00:36:11,970 The amazing thing that happens is at resonance, 623 00:36:11,970 --> 00:36:18,540 the inertial forces exactly cancel the spring forces. 624 00:36:18,540 --> 00:36:23,620 And the equation of motion reduces to minus C1 625 00:36:23,620 --> 00:36:33,550 omega 1 Q1 sine omega 1 t minus a phase angle equals, 626 00:36:33,550 --> 00:36:39,730 in this case, F1 L1 cosine omega 1 t. 627 00:36:39,730 --> 00:36:42,090 So how to satisfy that equation? 628 00:36:42,090 --> 00:36:46,290 What phase angle will satisfy that equation? 629 00:36:46,290 --> 00:36:47,590 Has to be pi over 2. 630 00:36:47,590 --> 00:36:49,650 And if you put pi over 2 in here, 631 00:36:49,650 --> 00:36:53,300 this minus sign turns into plus cosine. 632 00:36:53,300 --> 00:37:01,130 And you're left with C1 omega 1 Q1 equals F1 L1. 633 00:37:01,130 --> 00:37:07,170 So all of the exciting force goes into pushing the dashpot. 634 00:37:07,170 --> 00:37:11,100 So that's why you get the big peak in the transfer function. 635 00:37:11,100 --> 00:37:13,305 It takes in no force to move the spring. 636 00:37:13,305 --> 00:37:15,820 It takes no force to accelerate the mass. 637 00:37:15,820 --> 00:37:17,585 They exactly cancel. 638 00:37:17,585 --> 00:37:21,749 And all the force is available to drive just the dashpot, 639 00:37:21,749 --> 00:37:23,140 all right? 640 00:37:23,140 --> 00:37:25,900 So let's move on now. 641 00:37:25,900 --> 00:37:27,540 We need to get to our answer here. 642 00:37:34,160 --> 00:37:41,080 So the last piece of this is we now know the modal forces, 643 00:37:41,080 --> 00:37:52,916 and your assignment is to let's let omega-- let's see, 644 00:37:52,916 --> 00:37:57,820 where's my-- did I do this somewhere? 645 00:37:57,820 --> 00:37:58,535 I guess not. 646 00:37:58,535 --> 00:38:01,400 I guess I erased it, so we can pick anything we want. 647 00:38:01,400 --> 00:38:05,907 Let's let omega equal omega 2. 648 00:38:05,907 --> 00:38:08,240 We're going to drive this thing at the natural frequency 649 00:38:08,240 --> 00:38:09,031 of the second mode. 650 00:38:09,031 --> 00:38:10,489 That's the excitation. 651 00:38:10,489 --> 00:38:11,280 What do you expect? 652 00:38:11,280 --> 00:38:14,344 Which mode do you expect to dominate the response? 653 00:38:14,344 --> 00:38:15,308 AUDIENCE: The second. 654 00:38:15,308 --> 00:38:16,432 PROFESSOR: The second mode. 655 00:38:16,432 --> 00:38:17,102 Why? 656 00:38:17,102 --> 00:38:19,070 AUDIENCE: [INAUDIBLE] 657 00:38:19,070 --> 00:38:21,052 PROFESSOR: Because you are driving it. 658 00:38:21,052 --> 00:38:23,510 You know it's got a transfer function that looks like this, 659 00:38:23,510 --> 00:38:25,580 and you're driving it right here. 660 00:38:25,580 --> 00:38:27,260 And where are you driving it? 661 00:38:27,260 --> 00:38:29,310 If you're doing that, where are you 662 00:38:29,310 --> 00:38:33,970 on the transfer function for the first mode? 663 00:38:33,970 --> 00:38:41,100 Omega 2, it's a little higher than omega 1, but not a lot. 664 00:38:41,100 --> 00:38:44,280 So if this were the first mode's transfer function, 665 00:38:44,280 --> 00:38:49,710 where would you be driving it on this transfer function? 666 00:38:49,710 --> 00:38:58,709 If this is omega over omega 2, you're over here a little bit. 667 00:38:58,709 --> 00:39:00,250 You're driving it a little bit higher 668 00:39:00,250 --> 00:39:02,270 than the natural frequency of mode 1. 669 00:39:02,270 --> 00:39:06,240 This would be omega 2 over omega 1 on that. 670 00:39:06,240 --> 00:39:08,600 So one's sitting here, one's sitting there. 671 00:39:08,600 --> 00:39:10,360 Which one's going to dominate? 672 00:39:10,360 --> 00:39:11,360 The big one, OK? 673 00:39:11,360 --> 00:39:12,990 Because they have equal modal forces. 674 00:39:12,990 --> 00:39:16,290 They both happen to be F1 L1. 675 00:39:16,290 --> 00:39:19,320 OK, so how do you do that? 676 00:39:19,320 --> 00:39:21,490 So the last step in this thing is 677 00:39:21,490 --> 00:39:35,417 I want you to, for this case, find the first-- no, 678 00:39:35,417 --> 00:39:36,000 let's do this. 679 00:39:36,000 --> 00:39:46,960 Find the second mode contribution 680 00:39:46,960 --> 00:39:52,690 to the response in the original generalized coordinates. 681 00:39:52,690 --> 00:39:54,522 A quiz would often be written this way. 682 00:39:54,522 --> 00:39:56,105 It's trying to make it easier for you. 683 00:39:56,105 --> 00:39:58,870 I'm asking for only one mode's contribution. 684 00:39:58,870 --> 00:40:02,690 What does that actually mean in the original statement 685 00:40:02,690 --> 00:40:06,030 of the modal expansion theorem? 686 00:40:06,030 --> 00:40:09,720 We know the total response looks like this, right? 687 00:40:09,720 --> 00:40:13,290 I'm asking you to give me only the second mode's contribution. 688 00:40:13,290 --> 00:40:16,410 What am I asking for? 689 00:40:16,410 --> 00:40:17,824 Just this term. 690 00:40:17,824 --> 00:40:19,240 So I'm telling you, you don't have 691 00:40:19,240 --> 00:40:20,880 to bother with the other term to satisfy me. 692 00:40:20,880 --> 00:40:22,610 Just tell me what this one is, because I 693 00:40:22,610 --> 00:40:25,460 know this is one is going to be the dominant one. 694 00:40:25,460 --> 00:40:26,720 OK, so how do I do that? 695 00:40:26,720 --> 00:40:29,700 So for this problem, what is the steady state response 696 00:40:29,700 --> 00:40:34,650 of this due to mode 2 only? 697 00:40:34,650 --> 00:40:38,710 So mathematically, just in-- what's that look 698 00:40:38,710 --> 00:40:39,580 like over here? 699 00:40:44,280 --> 00:40:45,510 Just that second term, right? 700 00:40:45,510 --> 00:40:51,650 It's the modal, mode shape vector for mode 2 times Q2 701 00:40:51,650 --> 00:40:52,580 of t. 702 00:40:52,580 --> 00:40:55,050 So you need to tell me how to find Q1 of t. 703 00:41:00,540 --> 00:41:03,200 It's a single degree of freedom problem excited by steady state 704 00:41:03,200 --> 00:41:03,700 excitation. 705 00:41:18,094 --> 00:41:19,760 AUDIENCE: It's like a transfer function. 706 00:41:19,760 --> 00:41:21,910 PROFESSOR: Ah, magic word-- transfer function. 707 00:41:21,910 --> 00:41:26,290 So we need the magnitude of the-- this is a linear problem, 708 00:41:26,290 --> 00:41:29,384 so the response is linearly proportional to the--? 709 00:41:29,384 --> 00:41:30,050 AUDIENCE: Force. 710 00:41:30,050 --> 00:41:30,758 PROFESSOR: Force. 711 00:41:30,758 --> 00:41:34,430 So the magnitude of Q1 times a transfer 712 00:41:34,430 --> 00:41:37,600 function that looks like the hx over f transfer function. 713 00:41:37,600 --> 00:41:45,850 In this case, we call it the magnitude of HQ2 per unit Q2. 714 00:41:45,850 --> 00:41:49,810 And that multiplied by-- it looks 715 00:41:49,810 --> 00:41:55,390 like cosine in this case-- omega 2 t minus some phi 2. 716 00:41:55,390 --> 00:41:57,060 That's what we're looking for. 717 00:41:57,060 --> 00:41:58,172 What's this? 718 00:41:58,172 --> 00:41:59,380 Tell me what that looks like. 719 00:41:59,380 --> 00:41:59,879 Yeah? 720 00:41:59,879 --> 00:42:02,510 AUDIENCE: Shouldn't it be Q2? 721 00:42:02,510 --> 00:42:03,750 PROFESSOR: Thank you. 722 00:42:07,960 --> 00:42:10,270 So in effect, what's the transfer function? 723 00:42:10,270 --> 00:42:13,020 I want it in all its detail now. 724 00:42:20,610 --> 00:42:23,490 That's the magnitude of the force. 725 00:42:23,490 --> 00:42:25,245 What's in the numerator of this? 726 00:42:28,470 --> 00:42:29,552 Numerator. 727 00:42:29,552 --> 00:42:32,360 AUDIENCE: Well, it has parentheses around it. 728 00:42:32,360 --> 00:42:34,360 PROFESSOR: Yeah, it's in the denominator though. 729 00:42:34,360 --> 00:42:37,760 I want just the numerator part. 730 00:42:37,760 --> 00:42:39,167 This transfer function expression 731 00:42:39,167 --> 00:42:40,750 for a single degree of freedom system, 732 00:42:40,750 --> 00:42:41,749 what's in the numerator? 733 00:42:41,749 --> 00:42:42,780 AUDIENCE: 1/K. 734 00:42:42,780 --> 00:42:44,550 PROFESSOR: 1 over which K? 735 00:42:47,480 --> 00:42:48,850 K2. 736 00:42:48,850 --> 00:42:53,422 Modal K2, We're now in the modal system. 737 00:42:53,422 --> 00:42:55,880 And in the denominator of that transfer function, what's it 738 00:42:55,880 --> 00:42:56,845 look like? 739 00:42:56,845 --> 00:43:00,810 AUDIENCE: 1 plus omega. 740 00:43:00,810 --> 00:43:04,770 PROFESSOR: 1 minus omega squared over-- in this case, 741 00:43:04,770 --> 00:43:10,090 omega 2 squared, squared, plus-- 742 00:43:10,090 --> 00:43:14,184 AUDIENCE: 2 [INAUDIBLE]. 743 00:43:14,184 --> 00:43:14,850 PROFESSOR: Zeta. 744 00:43:14,850 --> 00:43:16,740 AUDIENCE: Zeta, yeah. 745 00:43:16,740 --> 00:43:18,034 PROFESSOR: 2. 746 00:43:18,034 --> 00:43:25,362 AUDIENCE: 2 omega over omega 2 quantity squared. 747 00:43:25,362 --> 00:43:26,320 PROFESSOR: There we go. 748 00:43:26,320 --> 00:43:27,778 Now in this problem, what is omega? 749 00:43:30,390 --> 00:43:35,280 So that makes this a 2, this a 2. 750 00:43:35,280 --> 00:43:36,770 This term, what happens to it? 751 00:43:39,640 --> 00:43:41,480 1 minus 1. 752 00:43:41,480 --> 00:43:42,950 This term, this goes to 1. 753 00:43:42,950 --> 00:43:47,160 This is 2 zeta quantity squared square root. 754 00:43:47,160 --> 00:43:48,480 Just 2 zeta. 755 00:43:48,480 --> 00:43:52,540 So this whole at resonance, any one of these single degree 756 00:43:52,540 --> 00:43:54,390 of freedom systems that are at resonance, 757 00:43:54,390 --> 00:43:58,220 the response is the magnitude of the force-- in this case, 758 00:43:58,220 --> 00:44:06,980 it's positive F1 L1-- over K2 times 1/2 zeta 2. 759 00:44:10,850 --> 00:44:16,080 So F1 L1 is the modal magnitude of the moral force divided 760 00:44:16,080 --> 00:44:20,320 by K2 gives you what we call the static displacement 761 00:44:20,320 --> 00:44:21,490 of the system. 762 00:44:21,490 --> 00:44:24,690 And this is the dynamic application. 763 00:44:24,690 --> 00:44:26,790 So oftentimes on quizzes, you're asked 764 00:44:26,790 --> 00:44:29,130 to do the response at resonance, because it makes 765 00:44:29,130 --> 00:44:30,600 all this algebra so simple. 766 00:44:30,600 --> 00:44:34,360 It boils down to 1/2 zeta. 767 00:44:34,360 --> 00:44:37,020 So the only thing left to do is we need the damping ratio 768 00:44:37,020 --> 00:44:37,690 for this system. 769 00:44:40,350 --> 00:44:43,320 So rather than do that, so now that, we're 770 00:44:43,320 --> 00:44:44,400 missing something yet. 771 00:44:44,400 --> 00:44:49,610 We're missing-- that whole thing gets multiplied by cosine omega 772 00:44:49,610 --> 00:44:52,500 2 t minus the face angel. 773 00:44:52,500 --> 00:44:54,490 What's the face angel? 774 00:44:54,490 --> 00:44:56,570 Pi over 2. 775 00:44:56,570 --> 00:45:10,640 So Q2 of t is F1 L1 over K1 1/2-- whoops, F1 L1 over K2, 776 00:45:10,640 --> 00:45:18,650 1/2 zeta 2 cosine omega 2t minus. 777 00:45:18,650 --> 00:45:21,880 And at the very-- how do we get back 778 00:45:21,880 --> 00:45:27,170 to generalized coordinates, theta 1 and theta 2? 779 00:45:27,170 --> 00:45:28,150 AUDIENCE: Stay here. 780 00:45:31,100 --> 00:45:32,590 PROFESSOR: Right here. 781 00:45:32,590 --> 00:45:34,930 And we've computed that. 782 00:45:34,930 --> 00:45:39,160 This part, you multiply it by the mode shape. 783 00:45:39,160 --> 00:45:42,300 The mode shape partitions out the response 784 00:45:42,300 --> 00:45:45,480 in the right amount to coordinate 785 00:45:45,480 --> 00:45:48,810 one and the correct amount to coordinate two. 786 00:45:54,762 --> 00:45:58,234 AUDIENCE: For the damping ratio, would you take the modal, like, 787 00:45:58,234 --> 00:46:01,870 mass and [INAUDIBLE]? 788 00:46:01,870 --> 00:46:03,660 PROFESSOR: In reality, what you do 789 00:46:03,660 --> 00:46:14,720 with damping ratios is you're working with real things 790 00:46:14,720 --> 00:46:17,650 out there in the real world. 791 00:46:17,650 --> 00:46:20,135 If you can, you go up and give the thing a kick 792 00:46:20,135 --> 00:46:23,620 and get your stopwatch out and say, how many cycles does it 793 00:46:23,620 --> 00:46:24,800 take to the k? 794 00:46:24,800 --> 00:46:27,560 And is it light damping or not? 795 00:46:27,560 --> 00:46:29,790 If it vibrates a lot, it's usually light damping. 796 00:46:29,790 --> 00:46:33,340 And if it's light damping, you can force this damping matrix. 797 00:46:33,340 --> 00:46:35,720 You can just make-- force it to behave, 798 00:46:35,720 --> 00:46:36,980 even if it isn't perfect. 799 00:46:36,980 --> 00:46:40,620 And it is a perfectly adequate, useful answer. 800 00:46:40,620 --> 00:46:42,220 Even if it isn't perfectly diagonal, 801 00:46:42,220 --> 00:46:44,695 it just doesn't matter when it's light damping. 802 00:46:44,695 --> 00:46:46,570 And so in this problem, what you do if you go 803 00:46:46,570 --> 00:46:48,070 estimate the damping for the system. 804 00:46:48,070 --> 00:46:51,720 You say eh, it looks to me to be about 2% for mode one 805 00:46:51,720 --> 00:46:54,950 and 1.5% for mode 2. 806 00:46:54,950 --> 00:46:58,130 And you just say, how can I fit? 807 00:46:58,130 --> 00:47:00,880 How can I represent the damping in the system? 808 00:47:00,880 --> 00:47:03,050 And one of the easiest ones is to say 809 00:47:03,050 --> 00:47:06,480 that the original damping matrix is some alpha times 810 00:47:06,480 --> 00:47:10,110 the mass matrix plus beta times the stiffness matrix. 811 00:47:10,110 --> 00:47:14,700 And in this problem-- or you can use any part of that. 812 00:47:14,700 --> 00:47:19,420 And if you're only trying to match one mode, see, 813 00:47:19,420 --> 00:47:22,430 this problem it's-- this system is being driven at the natural 814 00:47:22,430 --> 00:47:24,200 frequency of one mode. 815 00:47:24,200 --> 00:47:28,650 That mode is dominating the response, right? 816 00:47:28,650 --> 00:47:31,900 So we really actually only need a good model 817 00:47:31,900 --> 00:47:34,760 of the damping for that mode. 818 00:47:34,760 --> 00:47:36,790 Even if you have the completely wrong damping 819 00:47:36,790 --> 00:47:40,860 for the other mode, it will have little effect on its answer 820 00:47:40,860 --> 00:47:43,410 because you're not at resonance. 821 00:47:43,410 --> 00:47:48,730 When in the transfer functions, which look like this, 822 00:47:48,730 --> 00:47:51,350 at resonance, this happens. 823 00:47:51,350 --> 00:47:55,710 Then you find out that the only force resisting the input 824 00:47:55,710 --> 00:47:58,220 is the dashpot. 825 00:47:58,220 --> 00:48:01,030 At low frequencies, you find out that over here, 826 00:48:01,030 --> 00:48:04,420 the dominant force is what it takes to move the spring. 827 00:48:04,420 --> 00:48:06,170 And the damping isn't very important. 828 00:48:06,170 --> 00:48:08,380 And so even if you're wrong by 50%, 829 00:48:08,380 --> 00:48:12,350 it just doesn't-- 50% of a little bit compared to what it 830 00:48:12,350 --> 00:48:15,350 takes to move the spring is not a big deal. 831 00:48:15,350 --> 00:48:18,100 And over here, it behaves like the mass. 832 00:48:18,100 --> 00:48:21,010 Out here it's called the mass controlled region. 833 00:48:21,010 --> 00:48:23,350 Over here is the stiffness controlled region. 834 00:48:23,350 --> 00:48:26,230 And in the vicinity of the peak is the damping controlled 835 00:48:26,230 --> 00:48:26,730 region. 836 00:48:26,730 --> 00:48:30,370 So at low frequency, this term dominates. 837 00:48:30,370 --> 00:48:32,565 At high frequency, that term dominates. 838 00:48:32,565 --> 00:48:36,360 And at resonance, this is the dominant term. 839 00:48:36,360 --> 00:48:41,150 So in this case, let's let the damping be some alpha 840 00:48:41,150 --> 00:48:43,410 times the mass matrix. 841 00:48:43,410 --> 00:48:49,890 Then when you do UTCU, you get alpha times the modal mass 842 00:48:49,890 --> 00:48:50,390 matrix. 843 00:48:53,210 --> 00:48:57,230 And therefore, C2, which is the one we care about, 844 00:48:57,230 --> 00:49:01,410 is equal to alpha times M2. 845 00:49:01,410 --> 00:49:13,420 And we have M2, our modal mass, 2.04, right? 846 00:49:13,420 --> 00:49:20,960 So that says this is equal-- C2 is equal to alpha times 2.04. 847 00:49:20,960 --> 00:49:24,060 And if I want-- that's C2. 848 00:49:24,060 --> 00:49:26,400 And how do I get zeta 2? 849 00:49:26,400 --> 00:49:32,300 Zeta 2 is C2 over 2 omega 2 M2. 850 00:49:32,300 --> 00:49:35,070 That's just the definition of the damping ratio. 851 00:49:35,070 --> 00:49:36,070 I know this. 852 00:49:36,070 --> 00:49:38,050 I know this. 853 00:49:38,050 --> 00:49:48,630 So this is going to be alpha times 2.04 over 2 omega 2 M2. 854 00:49:48,630 --> 00:49:50,420 And I've measured it. 855 00:49:50,420 --> 00:49:52,770 I've taken it, and I know this is 0.02. 856 00:49:52,770 --> 00:49:54,690 About 2% damping. 857 00:49:54,690 --> 00:49:56,350 Solve for alpha. 858 00:49:56,350 --> 00:50:01,130 You now have the whole thing that you need. 859 00:50:01,130 --> 00:50:04,240 You can now find it. 860 00:50:04,240 --> 00:50:06,760 That's all you need. 861 00:50:06,760 --> 00:50:07,260 Yeah? 862 00:50:07,260 --> 00:50:10,578 AUDIENCE: Where do you get the alphas and omegas from again? 863 00:50:10,578 --> 00:50:13,430 Alphas and omegas. 864 00:50:13,430 --> 00:50:17,310 PROFESSOR: This is simply-- this is called Rayleigh damping. 865 00:50:17,310 --> 00:50:20,786 Lord Rayleigh 150 years ago came up with this. 866 00:50:20,786 --> 00:50:22,160 And he just said hey, by the way, 867 00:50:22,160 --> 00:50:23,576 if you model damping this way, you 868 00:50:23,576 --> 00:50:26,970 can automatically make the equations of motion, 869 00:50:26,970 --> 00:50:29,730 u transpose CU, go diagonal. 870 00:50:29,730 --> 00:50:33,010 And you have a two parameter model 871 00:50:33,010 --> 00:50:37,210 with which you can juggle them to make any two 872 00:50:37,210 --> 00:50:39,660 damping ratios of the system be exactly what you 873 00:50:39,660 --> 00:50:42,550 want them to be. 874 00:50:42,550 --> 00:50:46,610 So it's just-- if I left this as alpha N beta, 875 00:50:46,610 --> 00:50:49,290 then I would have worked this problem as-- this would 876 00:50:49,290 --> 00:50:54,630 have been an alpha M plus a beta K. C2 would have been an alpha 877 00:50:54,630 --> 00:50:56,110 M2. 878 00:50:56,110 --> 00:51:03,680 But now it would be equal to alpha M2 plus beta K2. 879 00:51:03,680 --> 00:51:06,640 And then this still applies, except that it'd 880 00:51:06,640 --> 00:51:08,850 have this alpha and a beta. 881 00:51:08,850 --> 00:51:10,840 And you could do it for the other equation. 882 00:51:10,840 --> 00:51:12,410 You could do it for zeta 1. 883 00:51:12,410 --> 00:51:14,720 And you'd have two equations and two unknowns. 884 00:51:14,720 --> 00:51:17,349 You solve for alpha and beta. 885 00:51:17,349 --> 00:51:18,890 But think about what I just did here. 886 00:51:18,890 --> 00:51:20,750 If I made a measurement of the system, 887 00:51:20,750 --> 00:51:23,490 I said the damping for mode 2 is 2%. 888 00:51:23,490 --> 00:51:25,750 Do I have to go through all this junk? 889 00:51:25,750 --> 00:51:29,640 No, because I know the answer looks like that. 890 00:51:29,640 --> 00:51:31,610 All I have to know is what the damping is. 891 00:51:31,610 --> 00:51:34,620 And that's my approximate solution for the problem. 892 00:51:34,620 --> 00:51:36,830 This is just how to satisfy all the mathematics 893 00:51:36,830 --> 00:51:39,460 if you want that perfect mathematical model 894 00:51:39,460 --> 00:51:43,740 for which you can write out u transpose CU and get them. 895 00:51:43,740 --> 00:51:47,700 Well, this is one way that you force 896 00:51:47,700 --> 00:51:53,277 the damper matrix to have the properties that you want it to. 897 00:51:53,277 --> 00:51:55,110 But in reality, you just measure the damping 898 00:51:55,110 --> 00:52:00,050 and put it in the answer. 899 00:52:00,050 --> 00:52:04,890 Very good, this is our last go around at recitations. 900 00:52:04,890 --> 00:52:08,270 See you in class on Tuesday. 901 00:52:08,270 --> 00:52:14,280 We'll do something fun that's not covered on the final exam. 902 00:52:14,280 --> 00:52:15,800 I'll give a little review of what's 903 00:52:15,800 --> 00:52:18,590 going to be on the final, a list of what's on it. 904 00:52:18,590 --> 00:52:20,740 And we'll talk about strings and beams and things 905 00:52:20,740 --> 00:52:24,720 that apply to pianos and violins and so forth.