1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,800 Commons license. 3 00:00:03,800 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,140 continue to offer high quality educational resources for free. 5 00:00:10,140 --> 00:00:12,680 To make a donation or to view additional materials 6 00:00:12,680 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:22,100 --> 00:00:26,835 PROFESSOR: So what's important the last couple of lectures? 9 00:00:26,835 --> 00:00:28,232 Christina. 10 00:00:28,232 --> 00:00:29,440 AUDIENCE: Transfer functions. 11 00:00:29,440 --> 00:00:31,364 PROFESSOR: Transfer functions, all right. 12 00:00:40,503 --> 00:00:42,154 How about something else? 13 00:00:42,154 --> 00:00:43,320 AUDIENCE: Modal coordinates. 14 00:00:43,320 --> 00:00:44,528 PROFESSOR: Modal coordinates. 15 00:00:47,020 --> 00:00:54,910 And I'll expand that to call this modal analysis in general. 16 00:00:54,910 --> 00:00:55,520 Anything else? 17 00:01:15,220 --> 00:01:17,320 Well, think more about that. 18 00:01:17,320 --> 00:01:21,660 And are there any questions for the week, anything muddy, 19 00:01:21,660 --> 00:01:23,710 fuzzy, not quite clear to you that if I 20 00:01:23,710 --> 00:01:26,294 have time to say a few words about today 21 00:01:26,294 --> 00:01:27,460 you'd like me to talk about? 22 00:01:30,412 --> 00:01:32,380 AUDIENCE: So I guess I'm a little bit confused 23 00:01:32,380 --> 00:01:37,220 with what we did in class, the modal analysis. 24 00:01:37,220 --> 00:01:44,505 Is that just like-- so when we did originally Newton's law, 25 00:01:44,505 --> 00:01:47,900 and then we did the [INAUDIBLE]. 26 00:01:50,810 --> 00:01:52,426 Is that sort of similar here where 27 00:01:52,426 --> 00:01:55,478 we can do either the things in motion, 28 00:01:55,478 --> 00:02:00,160 or we can do [INAUDIBLE] analysis? 29 00:02:00,160 --> 00:02:02,925 PROFESSOR: Is modal analysis an alternative way of-- 30 00:02:02,925 --> 00:02:03,800 AUDIENCE: [INAUDIBLE] 31 00:02:07,150 --> 00:02:10,389 PROFESSOR: The analogy isn't too helpful. 32 00:02:10,389 --> 00:02:15,010 But modal analysis is one way of attacking 33 00:02:15,010 --> 00:02:18,150 the equations of motion that describe 34 00:02:18,150 --> 00:02:20,450 vibration of linear systems. 35 00:02:20,450 --> 00:02:23,570 You can work through the entire analysis 36 00:02:23,570 --> 00:02:31,680 and figure out the solution to the equations of motion. 37 00:02:31,680 --> 00:02:35,820 And usually with these multiple degree of freedom vibration 38 00:02:35,820 --> 00:02:42,680 problems, you can cast them as a mass matrix 39 00:02:42,680 --> 00:02:46,430 times an acceleration vector plus a stiffness matrix 40 00:02:46,430 --> 00:02:51,670 times a displacement vector equals some forcing function. 41 00:02:51,670 --> 00:02:53,570 So these are-- and we linearize them. 42 00:02:53,570 --> 00:02:55,920 And we know they're for vibration problems. 43 00:02:55,920 --> 00:02:59,410 We can solve these equations using modal analysis. 44 00:02:59,410 --> 00:03:03,440 Or we can solve them brute force directly 45 00:03:03,440 --> 00:03:09,350 without breaking the results into their modal contributions. 46 00:03:09,350 --> 00:03:12,490 And so there really are two kinds of approaches 47 00:03:12,490 --> 00:03:14,884 you can use to doing it. 48 00:03:14,884 --> 00:03:16,050 Any other kind of questions? 49 00:03:16,050 --> 00:03:16,550 Yeah. 50 00:03:16,550 --> 00:03:19,620 AUDIENCE: So for a definition of mode shape, 51 00:03:19,620 --> 00:03:22,840 is mode shape just the ratio of the two amplitudes? 52 00:03:22,840 --> 00:03:25,970 PROFESSOR: In the case of a two degree of freedom system, 53 00:03:25,970 --> 00:03:42,010 so mode shapes for mode n of an n degree of freedom system-- 54 00:03:42,010 --> 00:03:44,860 like let's say n equals 3 here. 55 00:03:44,860 --> 00:03:53,730 So for mode n, the mode shape you always have 56 00:03:53,730 --> 00:03:55,960 to present some normalization. 57 00:03:55,960 --> 00:03:58,420 I do the normalization oftentimes by just saying, OK, 58 00:03:58,420 --> 00:04:00,420 I'm going to make the top coordinate-- I'm going 59 00:04:00,420 --> 00:04:01,810 to give it unit amplitude. 60 00:04:01,810 --> 00:04:04,760 So I'm going to take a1 and divide it by itself. 61 00:04:04,760 --> 00:04:06,020 And that'll give me a 1. 62 00:04:06,020 --> 00:04:12,180 Then every other one, this becomes a2 over a1, a3 over a1. 63 00:04:12,180 --> 00:04:23,570 The mode shape says if this is then 1 minus 1/2 and a 1/4, 64 00:04:23,570 --> 00:04:28,270 it says that if generalized coordinate 65 00:04:28,270 --> 00:04:33,120 1 moves a unit amount, generalized coordinate 2 66 00:04:33,120 --> 00:04:40,050 will move half of that in its negative direction. 67 00:04:40,050 --> 00:04:41,630 This could be an x, a displacement, 68 00:04:41,630 --> 00:04:43,820 and that could be a rotation. 69 00:04:43,820 --> 00:04:47,130 But they all have positively defined directions. 70 00:04:47,130 --> 00:04:51,150 So if there's a positive 1 at x1, there's a minus 1/2 71 00:04:51,150 --> 00:04:54,350 at generalized coordinate 2, and there's a plus 1/4 at 72 00:04:54,350 --> 00:04:55,870 generalized coordinate 3. 73 00:04:55,870 --> 00:05:00,570 And that ratio stays constant for the mode. 74 00:05:00,570 --> 00:05:04,910 And so if it's an initial condition problem, 75 00:05:04,910 --> 00:05:08,120 and you set it up so that you actually give it 76 00:05:08,120 --> 00:05:12,110 an exhibit, just an initial displacement, 77 00:05:12,110 --> 00:05:14,770 of exactly that shape, it'll sit there and vibrate only 78 00:05:14,770 --> 00:05:15,900 in that mode. 79 00:05:15,900 --> 00:05:19,150 And you'll notice that proportion, 80 00:05:19,150 --> 00:05:22,590 even as it dies out with damping, of the first 81 00:05:22,590 --> 00:05:25,210 to the other two stays exactly constant. 82 00:05:25,210 --> 00:05:26,560 That's what a mode is. 83 00:05:26,560 --> 00:05:28,210 And it's a character. 84 00:05:28,210 --> 00:05:32,440 It's a property of the system. 85 00:05:32,440 --> 00:05:34,050 The natural frequency is a property. 86 00:05:34,050 --> 00:05:38,790 And the shape of every mode is a property of that system. 87 00:05:38,790 --> 00:05:42,100 OK, anything else, questions? 88 00:05:42,100 --> 00:05:44,380 Next lecture, we'll do more modal analysis. 89 00:05:44,380 --> 00:05:47,460 We did the response to initial conditions yesterday. 90 00:05:47,460 --> 00:05:51,480 Tuesday, we'll talk about the response to external forces. 91 00:05:51,480 --> 00:05:54,360 And that will give us a chance to review it and post it 92 00:05:54,360 --> 00:05:55,700 on Stellar. 93 00:05:55,700 --> 00:05:57,970 Last night, I put up a little two page sheet 94 00:05:57,970 --> 00:06:00,375 that is a cookbook, how to do the procedure 95 00:06:00,375 --> 00:06:02,070 of a modal analysis. 96 00:06:02,070 --> 00:06:04,770 It's very cookbook, just step by step, bang, bang, bang. 97 00:06:04,770 --> 00:06:07,837 And everything falls out. 98 00:06:07,837 --> 00:06:08,920 Any other questions, yeah? 99 00:06:08,920 --> 00:06:12,340 AUDIENCE: So is that where we get the mode shape from? 100 00:06:12,340 --> 00:06:14,220 PROFESSOR: You get the mode shape 101 00:06:14,220 --> 00:06:19,430 by solving the characteristic-- if you're 102 00:06:19,430 --> 00:06:22,671 doing it by hand, by solving the characteristic equation, which 103 00:06:22,671 --> 00:06:24,420 we're going to get some practice at today. 104 00:06:24,420 --> 00:06:26,200 So I think this is the exercise of today, 105 00:06:26,200 --> 00:06:28,725 is how to get natural frequencies and mode shapes. 106 00:06:28,725 --> 00:06:32,256 AUDIENCE: Then you'd be actually controlling 107 00:06:32,256 --> 00:06:36,594 the amount that-- isn't the mode shape actually drawing 108 00:06:36,594 --> 00:06:38,827 some sort of graph? 109 00:06:38,827 --> 00:06:41,160 PROFESSOR: As a graph that shows the displacement of it, 110 00:06:41,160 --> 00:06:42,970 or it shows the time history? 111 00:06:42,970 --> 00:06:45,761 AUDIENCE: Either, I guess. 112 00:06:45,761 --> 00:06:49,058 Because in the book, they have mode shape. 113 00:06:49,058 --> 00:06:51,510 But actually using this, they draw a certain graph. 114 00:06:51,510 --> 00:06:52,718 PROFESSOR: Sometimes they do. 115 00:06:52,718 --> 00:06:55,480 If it's easy, like for a vibrating string, 116 00:06:55,480 --> 00:06:57,414 it's easy to draw the mode shape. 117 00:06:57,414 --> 00:06:59,580 AUDIENCE: And the piece that they ask us to do that. 118 00:06:59,580 --> 00:07:01,746 PROFESSOR: Yeah, and so vibrating string, first mode 119 00:07:01,746 --> 00:07:03,562 vibration looks like half a sign wave. 120 00:07:03,562 --> 00:07:05,270 The mode shape is just half the sine wave 121 00:07:05,270 --> 00:07:08,280 going up and down in space. 122 00:07:08,280 --> 00:07:13,140 For this system, the mode shape has two different amplitudes 123 00:07:13,140 --> 00:07:15,160 at these two bobs. 124 00:07:15,160 --> 00:07:18,190 And their relative motion we're going to figure that out today. 125 00:07:18,190 --> 00:07:20,350 OK, we better get going or we won't finish. 126 00:07:20,350 --> 00:07:23,430 So good questions. 127 00:07:23,430 --> 00:07:34,590 And here's this system simplified, just into two 128 00:07:34,590 --> 00:07:37,410 bobs and massless strings. 129 00:07:37,410 --> 00:07:40,100 Here's the equations of motion for that. 130 00:07:40,100 --> 00:07:41,680 They've been linearized already. 131 00:07:41,680 --> 00:07:43,970 There's no sine thetas or anything. 132 00:07:43,970 --> 00:07:46,890 And part of the linearization-- so this assumes, 133 00:07:46,890 --> 00:07:50,050 then, small amplitudes, right? 134 00:07:50,050 --> 00:07:53,530 Not only did the sine-- the torque for a pendulum, 135 00:07:53,530 --> 00:07:56,460 you get this mgL sine theta term. 136 00:07:56,460 --> 00:07:59,080 And so the linearization, you say small angles, 137 00:07:59,080 --> 00:08:00,920 sine theta becomes theta. 138 00:08:00,920 --> 00:08:06,846 So that's where this g/L sine theta has become theta term. 139 00:08:06,846 --> 00:08:08,220 There's another assumption that's 140 00:08:08,220 --> 00:08:09,610 been made to model this system. 141 00:08:09,610 --> 00:08:11,892 We're essentially modeling the system. 142 00:08:11,892 --> 00:08:13,350 And that is for small angles, we're 143 00:08:13,350 --> 00:08:16,700 assuming the spring remains horizontal. 144 00:08:16,700 --> 00:08:20,230 Because the spring puts a force on the rod. 145 00:08:20,230 --> 00:08:22,440 And what you want is torque about that point. 146 00:08:22,440 --> 00:08:24,150 These two equations are the equations 147 00:08:24,150 --> 00:08:28,190 for torque about that point and about that point. 148 00:08:28,190 --> 00:08:32,300 And the torque is the spring force times that distance. 149 00:08:32,300 --> 00:08:35,020 And if they're perpendicular, that's just one times 150 00:08:35,020 --> 00:08:35,520 the other. 151 00:08:35,520 --> 00:08:37,144 But if it doesn't remain perpendicular, 152 00:08:37,144 --> 00:08:39,570 then you'd have to take a cosine. 153 00:08:39,570 --> 00:08:41,260 And it gets really messy. 154 00:08:41,260 --> 00:08:43,419 So another part of this small angle approximation 155 00:08:43,419 --> 00:08:47,010 is that spring stays horizontal. 156 00:08:47,010 --> 00:08:48,520 There's your equation of motion. 157 00:08:48,520 --> 00:08:52,270 And you can see what's happened here is that this 158 00:08:52,270 --> 00:08:58,070 used to have an m1 L1 squared. 159 00:08:58,070 --> 00:09:01,050 This is the mass moment of inertia of that mass. 160 00:09:01,050 --> 00:09:05,080 We've divided through the equation by m1 L1 squared. 161 00:09:05,080 --> 00:09:08,100 And that's put things in the bottom here like that. 162 00:09:08,100 --> 00:09:12,260 And it just makes this actually a little easier to work with. 163 00:09:12,260 --> 00:09:17,590 The mass matrix turns out to be 1, 1 when you do this. 164 00:09:17,590 --> 00:09:19,600 But it's still the same two equations of motion. 165 00:09:19,600 --> 00:09:25,150 I've just divided through by m1 L squared, and this one by m2 L 166 00:09:25,150 --> 00:09:26,960 squared. 167 00:09:26,960 --> 00:09:29,130 So these are your equations of motion. 168 00:09:29,130 --> 00:09:34,180 And your job is to come up-- first of all, 169 00:09:34,180 --> 00:09:36,527 just put them in matrix form. 170 00:09:36,527 --> 00:09:38,110 And you're going to do this in groups. 171 00:09:38,110 --> 00:09:46,200 Today we've got five, 10, and 18 people. 172 00:09:46,200 --> 00:09:48,740 So that's probably four groups of four. 173 00:09:48,740 --> 00:09:51,730 Work together in groups quickly. 174 00:09:51,730 --> 00:09:54,040 Let's put up the equations of motion 175 00:09:54,040 --> 00:09:56,536 from this in matrix form. 176 00:09:56,536 --> 00:10:00,022 AUDIENCE: [INAUDIBLE] 177 00:11:06,330 --> 00:11:09,200 PROFESSOR: In your lower left matrix? 178 00:11:09,200 --> 00:11:14,020 Is the-- oh, the k. 179 00:11:14,020 --> 00:11:16,290 I'm worried about the m. 180 00:11:16,290 --> 00:11:19,290 AUDIENCE: The k's don't have any symbols. it's just m2. 181 00:11:19,290 --> 00:11:21,453 PROFESSOR: There you go. 182 00:11:21,453 --> 00:11:22,974 Yeah, it's just one. 183 00:11:22,974 --> 00:11:23,765 There's a single k. 184 00:11:23,765 --> 00:11:25,920 There's only one spring, so we just call it k. 185 00:11:53,860 --> 00:11:56,610 All right, you're all looking pretty good to me. 186 00:11:56,610 --> 00:11:59,534 And the last hour, we did this, and it went up, 187 00:11:59,534 --> 00:12:00,950 and I looked at it, and I just had 188 00:12:00,950 --> 00:12:03,930 one of these moments of just, uhh, cognitive dissonance. 189 00:12:03,930 --> 00:12:08,570 It just was, how can that possibly be? 190 00:12:08,570 --> 00:12:16,000 And the reason was this stiffness matrix. 191 00:12:16,000 --> 00:12:20,820 This matrix is not diagonal-- excuse me, wrong word. 192 00:12:20,820 --> 00:12:24,580 This matrix is not symmetric. 193 00:12:24,580 --> 00:12:29,740 And linear stiffness matrices are always symmetric. 194 00:12:29,740 --> 00:12:32,690 So I saw this, and I said, what has gone wrong? 195 00:12:35,670 --> 00:12:37,730 Every one of you had the same answer. 196 00:12:37,730 --> 00:12:41,330 Every one of you, your stiffness matrices are not symmetric. 197 00:12:41,330 --> 00:12:46,470 It's minus k over m1 and minus k over m2. 198 00:12:46,470 --> 00:12:49,260 And I about had heart failure, and I've 199 00:12:49,260 --> 00:12:51,019 been doing this for 40 years. 200 00:12:51,019 --> 00:12:52,310 And it's always been symmetric. 201 00:12:52,310 --> 00:12:57,320 And all of a sudden, you guys unanimously get asymmetric. 202 00:12:57,320 --> 00:13:02,020 So I figured it out at great relief. 203 00:13:02,020 --> 00:13:08,380 We divided through by m1 L1 squared in the first equation, 204 00:13:08,380 --> 00:13:12,380 and m2 L2 squared in the second equation. 205 00:13:12,380 --> 00:13:16,500 The stiffness matrix, the true stiffness matrix, 206 00:13:16,500 --> 00:13:19,770 hasn't been divided through by the mL squareds. 207 00:13:19,770 --> 00:13:22,762 And it is indeed symmetric. 208 00:13:22,762 --> 00:13:24,970 It's one of the things that's really helpful to know. 209 00:13:24,970 --> 00:13:27,020 Because when you're working these things out, 210 00:13:27,020 --> 00:13:27,940 you know they're symmetric. 211 00:13:27,940 --> 00:13:29,648 You don't have to evaluate all the terms. 212 00:13:29,648 --> 00:13:32,710 You only have to do the diagonals and half of the ones. 213 00:13:32,710 --> 00:13:34,700 The mass matrices are usually symmetric, too. 214 00:13:34,700 --> 00:13:36,780 There's maybe some exceptions if you 215 00:13:36,780 --> 00:13:38,780 start putting in gyroscopes and stuff like that. 216 00:13:38,780 --> 00:13:42,260 But then it's not very linear. 217 00:13:42,260 --> 00:13:45,220 So your next task is to-- the way 218 00:13:45,220 --> 00:13:47,500 we find natural frequencies and mode 219 00:13:47,500 --> 00:13:51,020 shapes is kind of the hard way, using algebra, 220 00:13:51,020 --> 00:13:54,730 is to find what we call the characteristic equation. 221 00:13:54,730 --> 00:13:57,120 It turns out to be a fourth order 222 00:13:57,120 --> 00:13:59,450 equation, omega to the fourth. 223 00:13:59,450 --> 00:14:00,850 Find it. 224 00:14:00,850 --> 00:14:03,470 Don't solve it, just write down the characteristic equation 225 00:14:03,470 --> 00:14:05,917 for this thing. 226 00:14:05,917 --> 00:14:08,250 And as soon as you have that, come up and write it down. 227 00:14:12,710 --> 00:14:14,530 So remember, you basically have matrices. 228 00:14:14,530 --> 00:14:15,904 This is what you're working with. 229 00:14:15,904 --> 00:14:17,720 You've got something of this form. 230 00:14:17,720 --> 00:14:20,630 We've divided through by some stuff to collect terms. 231 00:14:20,630 --> 00:14:24,880 But how do you go about going from the equations of motion 232 00:14:24,880 --> 00:14:29,415 to your algebraic equation in omega to the fourth? 233 00:14:41,070 --> 00:14:44,019 I just want you to write up the characteristic equation. 234 00:14:44,019 --> 00:14:46,310 You're going to get a quadratic in omega to the fourth. 235 00:14:46,310 --> 00:14:47,934 Just go up and write down the equation. 236 00:14:52,060 --> 00:14:55,350 AUDIENCE: So do you want us to expand? 237 00:14:55,350 --> 00:14:57,440 PROFESSOR: By expand, what do you mean? 238 00:14:57,440 --> 00:15:00,230 AUDIENCE: [INAUDIBLE] multiply this by this? 239 00:15:00,230 --> 00:15:02,560 But we could also expand that out. 240 00:15:02,560 --> 00:15:04,640 PROFESSOR: Well, I want an equation in omega 241 00:15:04,640 --> 00:15:05,900 to the fourth. 242 00:15:05,900 --> 00:15:07,100 Just write one up there. 243 00:15:07,100 --> 00:15:13,960 And if you want to simplify the writing, you can let h equal-- 244 00:15:13,960 --> 00:15:15,950 and let's see, I forgot the tell you something, 245 00:15:15,950 --> 00:15:17,550 a key assumption here, a key thing. 246 00:15:17,550 --> 00:15:19,880 I was going to make it easier for you. 247 00:15:19,880 --> 00:15:20,680 Let-- 248 00:15:20,680 --> 00:15:23,246 AUDIENCE: Are you going to say the m's are equal? 249 00:15:23,246 --> 00:15:26,850 PROFESSOR: Yeah, sorry, my mistake. 250 00:15:26,850 --> 00:15:27,790 Let the m's be equal. 251 00:15:27,790 --> 00:15:29,280 Then it makes it a lot easier. 252 00:15:29,280 --> 00:15:35,310 And then you can say, call h g/L plus k/m. 253 00:15:35,310 --> 00:15:39,000 And it'll kind of make the equation a whole lot easier 254 00:15:39,000 --> 00:15:39,590 to write. 255 00:15:43,076 --> 00:15:45,566 AUDIENCE: [INAUDIBLE] 256 00:16:36,880 --> 00:16:40,305 PROFESSOR: Great, OK, people are getting all the same answers 257 00:16:40,305 --> 00:16:41,720 here, good. 258 00:16:41,720 --> 00:16:47,390 So the next step is I will give you-- if you solve this, 259 00:16:47,390 --> 00:16:55,250 you get omega 1 is g/L. Omega 1 squared is g/L. 260 00:16:55,250 --> 00:17:03,900 And omega 2 squared is g/L plus 2k over m. 261 00:17:03,900 --> 00:17:05,810 Those are your two natural frequencies. 262 00:17:05,810 --> 00:17:08,989 So now, find the mode shapes. 263 00:17:11,650 --> 00:17:13,000 Find the mode shapes. 264 00:17:13,000 --> 00:17:15,335 If you know the natural frequencies, now you go back in 265 00:17:15,335 --> 00:17:19,740 and you get a-- pardon? 266 00:17:19,740 --> 00:17:21,450 Yeah, keep h. 267 00:17:21,450 --> 00:17:26,670 I think h will-- things will fall out rather quickly. 268 00:17:26,670 --> 00:17:31,410 Well, I like to normalize the mode shapes 269 00:17:31,410 --> 00:17:34,290 so the top one is 1. 270 00:17:34,290 --> 00:17:36,730 And so the top element of the mode shape 271 00:17:36,730 --> 00:17:41,000 vector I call a1, or u1, down through un. 272 00:17:41,000 --> 00:17:44,150 And so if you divide out each one by the top one, 273 00:17:44,150 --> 00:17:45,620 the top one becomes 1. 274 00:17:45,620 --> 00:17:47,570 The second one becomes a2 over a1. 275 00:17:47,570 --> 00:17:49,570 The third one, if it's three degrees of freedom, 276 00:17:49,570 --> 00:17:50,620 would be a3 over a1. 277 00:17:53,480 --> 00:17:58,350 The fact that this-- you just solve this equation. 278 00:17:58,350 --> 00:18:00,860 To get that characteristic determinant, 279 00:18:00,860 --> 00:18:04,250 you said this was true. 280 00:18:04,250 --> 00:18:07,850 And that's a particular kind of situation 281 00:18:07,850 --> 00:18:11,530 where you have a set of linear homogeneous 282 00:18:11,530 --> 00:18:13,336 equations equal to 0. 283 00:18:13,336 --> 00:18:14,335 They're not independent. 284 00:18:14,335 --> 00:18:17,550 Remember from algebra when you have little equations 285 00:18:17,550 --> 00:18:22,115 with constant coefficients, and they're equal to a constant, 286 00:18:22,115 --> 00:18:26,090 and the constants are all 0, means they are not 287 00:18:26,090 --> 00:18:27,230 independent equations. 288 00:18:27,230 --> 00:18:29,670 So this is two equations and two unknowns. 289 00:18:29,670 --> 00:18:31,350 And they're not independent. 290 00:18:31,350 --> 00:18:34,040 You won't be able to solve for unique values of a1, u1, 291 00:18:34,040 --> 00:18:34,710 and u2. 292 00:18:34,710 --> 00:18:37,840 You'd only be able to get the ratio. 293 00:18:37,840 --> 00:18:41,700 So basically you found this value. 294 00:18:41,700 --> 00:18:44,190 You're going to plug in a value for omega squared 295 00:18:44,190 --> 00:18:48,310 and find out the values of a1 and a2 that work. 296 00:18:48,310 --> 00:18:49,650 That's what the mode shapes are. 297 00:18:52,800 --> 00:18:55,970 Let me pull you back together here, and let's 298 00:18:55,970 --> 00:18:58,410 run through this kind of quickly. 299 00:18:58,410 --> 00:19:01,860 You've basically solved this equation. 300 00:19:01,860 --> 00:19:04,550 You've said, I'm going to assume this thing has 301 00:19:04,550 --> 00:19:10,320 a mode shape and a frequency, a harmonic-- it's 302 00:19:10,320 --> 00:19:11,670 going to vibrate. 303 00:19:11,670 --> 00:19:12,270 It's undamped. 304 00:19:12,270 --> 00:19:15,550 So either you could write this as cosine omega t, sine omega 305 00:19:15,550 --> 00:19:17,220 t, e to the i omega t. 306 00:19:17,220 --> 00:19:20,310 But it has some constants out here called the shape. 307 00:19:20,310 --> 00:19:26,670 You plug that into the equations of motion, 308 00:19:26,670 --> 00:19:28,805 you're going to get back this expression. 309 00:19:28,805 --> 00:19:31,080 The two derivatives, the theta double dots, 310 00:19:31,080 --> 00:19:33,720 give you the minus omega squareds. 311 00:19:33,720 --> 00:19:35,440 And you can write it like that. 312 00:19:35,440 --> 00:19:38,950 You can throw away the e to the i omega t. 313 00:19:38,950 --> 00:19:43,120 In this case, I've expand this. m is just that 1, 1 matrix. 314 00:19:43,120 --> 00:19:46,890 So this part looks like this. 315 00:19:46,890 --> 00:19:49,380 The k matrix looks like this. 316 00:19:49,380 --> 00:19:51,730 You can add them element by element. 317 00:19:51,730 --> 00:19:57,440 So this is minus omega squared plus h, minus k/m. 318 00:19:57,440 --> 00:20:00,770 This one is minus k/m, and this term is minus omega 319 00:20:00,770 --> 00:20:05,100 squared plus h again times u1 or u2, the two elements 320 00:20:05,100 --> 00:20:07,520 that we're looking for. 321 00:20:07,520 --> 00:20:08,920 That's two equations. 322 00:20:08,920 --> 00:20:10,840 This is just now an algebraic equation, 323 00:20:10,840 --> 00:20:16,840 two algebraic equations-- this time u1 minus k/m times u2. 324 00:20:16,840 --> 00:20:20,000 And the second equation is that. 325 00:20:20,000 --> 00:20:22,180 And we know there's only two equations, and not 326 00:20:22,180 --> 00:20:23,280 linearly independent. 327 00:20:23,280 --> 00:20:25,450 So you actually only have one useful equation. 328 00:20:25,450 --> 00:20:29,360 If this is a three by three, they're not independent. 329 00:20:29,360 --> 00:20:33,441 But you need to use two out of the three. 330 00:20:33,441 --> 00:20:33,940 Yeah. 331 00:20:33,940 --> 00:20:35,370 AUDIENCE: How do you get the second equation? 332 00:20:35,370 --> 00:20:37,747 PROFESSOR: The second one, I take the first one, 333 00:20:37,747 --> 00:20:38,830 and I multiply it up here. 334 00:20:38,830 --> 00:20:42,060 And that gives me an equation, this times u1 plus this times 335 00:20:42,060 --> 00:20:42,630 u2. 336 00:20:42,630 --> 00:20:43,960 That's equation one. 337 00:20:43,960 --> 00:20:47,320 I take this, and I multiply it by that. 338 00:20:47,320 --> 00:20:52,630 And I get minus k/m u2. 339 00:20:52,630 --> 00:21:00,300 And this is u1. 340 00:21:00,300 --> 00:21:01,410 This is the omega squared. 341 00:21:01,410 --> 00:21:02,990 I plugged in the natural frequency. 342 00:21:02,990 --> 00:21:04,230 So we solved for it. 343 00:21:04,230 --> 00:21:06,840 AUDIENCE: Oh, OK, OK. 344 00:21:06,840 --> 00:21:09,690 But for the top one, you didn't plug in the natural frequency. 345 00:21:09,690 --> 00:21:11,513 PROFESSOR: Oh, minus omega squared. 346 00:21:14,110 --> 00:21:17,730 OK, I'm picking one of the natural frequencies, g/L. 347 00:21:17,730 --> 00:21:24,530 So this up here is g/L plus h u1, that one. 348 00:21:24,530 --> 00:21:26,850 And down here it's g/L h u2. 349 00:21:26,850 --> 00:21:29,785 And you see, these two equations turn out to be-- 350 00:21:29,785 --> 00:21:31,790 AUDIENCE: So this is just for omega 1? 351 00:21:31,790 --> 00:21:33,830 PROFESSOR: Yeah, you only do it one at a time. 352 00:21:33,830 --> 00:21:36,900 You just plug in one of the natural frequencies. 353 00:21:36,900 --> 00:21:38,692 And you get two equations and two unknowns. 354 00:21:38,692 --> 00:21:40,108 And they're going to each give you 355 00:21:40,108 --> 00:21:41,730 exactly the same information. 356 00:21:41,730 --> 00:21:43,860 They're not independent any longer. 357 00:21:43,860 --> 00:21:45,215 So you solve this one. 358 00:21:48,560 --> 00:21:50,560 h has what in it? 359 00:21:50,560 --> 00:21:56,860 A plus g/L and a plus k/m. 360 00:21:56,860 --> 00:21:58,820 The g/L is canceled. 361 00:21:58,820 --> 00:22:06,910 And you're left with k/m u1 minus k/m u2 equals 0. 362 00:22:06,910 --> 00:22:11,390 And this implies that u1 equals u2. 363 00:22:17,550 --> 00:22:19,610 This is the g/L in this. 364 00:22:19,610 --> 00:22:22,620 I'm going to plug in the-- that is h. 365 00:22:22,620 --> 00:22:23,810 I just put it in here. 366 00:22:23,810 --> 00:22:26,140 I've plugged in one of the natural frequencies squared. 367 00:22:26,140 --> 00:22:28,360 The g/L parts cancel. 368 00:22:28,360 --> 00:22:32,610 This is plus k/m u1 minus k/m u2. 369 00:22:32,610 --> 00:22:36,710 That means k/m u1 equals k/m u2. 370 00:22:36,710 --> 00:22:38,190 Cancel out the k/m's. 371 00:22:38,190 --> 00:22:38,735 I get that. 372 00:22:38,735 --> 00:22:41,370 It just tells me that that's the answer. 373 00:22:41,370 --> 00:22:46,400 And if that's the answer, the vector looks like u1, u2. 374 00:22:46,400 --> 00:22:52,790 That's also equal to-- factor out a u1-- u1 times 1 and u2 375 00:22:52,790 --> 00:22:54,780 over u1. 376 00:22:54,780 --> 00:22:59,710 That's the mode shape there, 1 and u2 over u1. 377 00:22:59,710 --> 00:23:02,820 That's how you factor it out to put it in the normalized form. 378 00:23:02,820 --> 00:23:04,970 You just take whatever's in the top one and divide 379 00:23:04,970 --> 00:23:06,530 everything in the column by it. 380 00:23:10,700 --> 00:23:13,220 So to do the second equation, you just 381 00:23:13,220 --> 00:23:17,600 now go back to this, plug in for omega 2 squared. 382 00:23:17,600 --> 00:23:23,140 The second natural frequency is g/L plus 2k over m. 383 00:23:23,140 --> 00:23:29,200 This is g/L minus k/m. 384 00:23:29,200 --> 00:23:32,850 The k over m's cancel. 385 00:23:32,850 --> 00:23:35,322 And you're just left with g/L's. 386 00:23:35,322 --> 00:23:36,780 You work this through, you're going 387 00:23:36,780 --> 00:23:41,260 to find out that u2 equals minus u1 if you 388 00:23:41,260 --> 00:23:42,240 put in the second one. 389 00:23:46,180 --> 00:23:47,298 Yeah. 390 00:23:47,298 --> 00:23:54,110 AUDIENCE: So we have a1 over a2, b1 over b2, being equal to 1, 391 00:23:54,110 --> 00:23:54,610 obviously. 392 00:23:54,610 --> 00:23:56,400 Because b1 is equal to b2. 393 00:23:56,400 --> 00:23:57,882 But I don't understand the format 394 00:23:57,882 --> 00:23:59,215 that you're writing it up there. 395 00:23:59,215 --> 00:24:03,145 Because if you plug back in the u, then you just get u1 over u2 396 00:24:03,145 --> 00:24:04,540 is equal to u1 over u2. 397 00:24:04,540 --> 00:24:06,205 And that's self-explanatory, right? 398 00:24:06,205 --> 00:24:07,830 PROFESSOR: Well, when you solve this, 399 00:24:07,830 --> 00:24:13,520 you find that k/m u1 equals k/m u2. 400 00:24:13,520 --> 00:24:16,625 That's just saying in this case they are equal, period. 401 00:24:16,625 --> 00:24:18,730 And that's all you learn. 402 00:24:18,730 --> 00:24:20,074 You only have two equations. 403 00:24:20,074 --> 00:24:21,240 And they're not independent. 404 00:24:21,240 --> 00:24:23,300 So it means you only have one equation. 405 00:24:23,300 --> 00:24:25,280 And you don't learn-- you're not able to solve 406 00:24:25,280 --> 00:24:27,550 for numeric values of u1 and u2. 407 00:24:27,550 --> 00:24:29,860 At best, you can get u1 over u2. 408 00:24:29,860 --> 00:24:32,300 AUDIENCE: Yeah, because they're equal to [INAUDIBLE]. 409 00:24:32,300 --> 00:24:36,330 PROFESSOR: Right, and in fact, you can just solve this for u2 410 00:24:36,330 --> 00:24:38,261 over u1, and it would be 1. 411 00:24:38,261 --> 00:24:40,094 AUDIENCE: I just don't understand the format 412 00:24:40,094 --> 00:24:41,500 you're writing it in up there. 413 00:24:41,500 --> 00:24:44,100 PROFESSOR: Well, that's back in my vector format. 414 00:24:44,100 --> 00:24:47,390 I want to write the mode shape as a vector. 415 00:24:47,390 --> 00:24:50,210 You solve for u1. 416 00:24:50,210 --> 00:24:51,817 So you can actually do this. 417 00:24:51,817 --> 00:24:52,900 You know that that's true. 418 00:24:52,900 --> 00:24:54,485 You could say, well, the top one is 1, 419 00:24:54,485 --> 00:24:56,110 and the bottom one is the same as that. 420 00:24:56,110 --> 00:24:58,205 So the mode shape for mode one is that. 421 00:25:03,295 --> 00:25:07,440 AUDIENCE: In the answers, it says omega 1, 422 00:25:07,440 --> 00:25:09,340 and the ratio of a1 and a2 is 1. 423 00:25:09,340 --> 00:25:12,359 What are the a's? 424 00:25:12,359 --> 00:25:13,400 PROFESSOR: I'm using u's. 425 00:25:13,400 --> 00:25:14,870 And they used a's. 426 00:25:14,870 --> 00:25:17,000 They're the elements of the mode shape vector. 427 00:25:20,600 --> 00:25:23,880 So if we had put in the other natural frequency instead 428 00:25:23,880 --> 00:25:37,770 of-- I may have just done too many steps here at once. 429 00:25:47,000 --> 00:25:50,650 So our first equation looks like minus omega 430 00:25:50,650 --> 00:25:57,620 squared plus h minus k/m. 431 00:26:02,280 --> 00:26:07,010 And the second equation is minus k/m. 432 00:26:07,010 --> 00:26:13,810 And over here is minus omega squared plus h again. 433 00:26:13,810 --> 00:26:15,020 That's our matrix. 434 00:26:15,020 --> 00:26:17,850 When we add these two matrices together, 435 00:26:17,850 --> 00:26:20,100 the minus omega squared m plus k, 436 00:26:20,100 --> 00:26:23,300 that's what it looks like when you add elements together. 437 00:26:23,300 --> 00:26:26,340 And these are two equations and two unknowns. 438 00:26:26,340 --> 00:26:31,500 We're looking for the answer for some u1, u2 multiplied by this. 439 00:26:31,500 --> 00:26:33,220 So you multiply this out. 440 00:26:33,220 --> 00:26:42,690 You get minus omega squared plus h u1 minus k/m u2. 441 00:26:42,690 --> 00:26:46,110 And now let's plug in the second natural frequency. 442 00:26:46,110 --> 00:26:50,790 So omega 2 squared is that. 443 00:26:50,790 --> 00:26:52,650 And solve for the second mode now. 444 00:26:52,650 --> 00:26:54,530 Plug it in here. 445 00:26:54,530 --> 00:27:02,520 We get minus g/L minus 2k over m plus h, 446 00:27:02,520 --> 00:27:13,945 which is g/L, plus k/m plus, and that's times u1, plus k/m u2. 447 00:27:17,230 --> 00:27:19,920 All it's equal to 0. 448 00:27:19,920 --> 00:27:24,425 This cancels this, this, and this, plus 1 minus 2. 449 00:27:24,425 --> 00:27:28,558 I get a minus k/m out of this. 450 00:27:28,558 --> 00:27:40,156 So I have this whole thing gives me minus k/m u1 plus k/m u2. 451 00:27:43,407 --> 00:27:45,073 AUDIENCE: It's supposed to be minus k/m. 452 00:27:48,264 --> 00:27:49,430 PROFESSOR: I agree with you. 453 00:27:49,430 --> 00:27:50,680 Where did I make the mistake? 454 00:27:50,680 --> 00:27:53,795 AUDIENCE: The very first line, it's minus k/m u2. 455 00:27:53,795 --> 00:27:56,680 PROFESSOR: Ah, good, all right. 456 00:28:01,440 --> 00:28:04,070 And this is equal to 0. 457 00:28:04,070 --> 00:28:06,170 The k over m's cancel out. 458 00:28:06,170 --> 00:28:13,570 And you find that this implies that u1 equals minus u2. 459 00:28:13,570 --> 00:28:20,790 And so the mode shape for mode two you could write as u1. 460 00:28:20,790 --> 00:28:24,490 And u2 is minus u1, if you will. 461 00:28:24,490 --> 00:28:26,370 And it's obvious if you factor out 462 00:28:26,370 --> 00:28:32,690 u1 you get u1 and a 1 minus 1, right? 463 00:28:32,690 --> 00:28:35,970 So now you've solved for the second mode shape. 464 00:28:35,970 --> 00:28:37,820 If this had been more than a two by two, 465 00:28:37,820 --> 00:28:43,990 like a three degree of freedom system, 466 00:28:43,990 --> 00:28:46,740 you'd have to use two of the three equations 467 00:28:46,740 --> 00:28:48,250 to get the ratio. 468 00:28:48,250 --> 00:28:51,090 You'd get two equations and three unknowns. 469 00:28:51,090 --> 00:28:54,650 And at best, you can find the two of them 470 00:28:54,650 --> 00:28:56,010 in terms of the third. 471 00:28:56,010 --> 00:28:57,400 And the third one would be u1. 472 00:28:57,400 --> 00:29:00,790 You'd just divide all the others by u1. 473 00:29:00,790 --> 00:29:02,110 So you only get their ratios. 474 00:29:02,110 --> 00:29:06,890 OK, so those are the natural frequencies. 475 00:29:06,890 --> 00:29:07,985 Those are the mode shapes. 476 00:29:10,640 --> 00:29:11,682 Let's demonstrate it. 477 00:29:11,682 --> 00:29:12,390 Let's look at it. 478 00:29:12,390 --> 00:29:14,450 This is basically the system. 479 00:29:14,450 --> 00:29:19,880 And you now know the mode shapes, the two mode 480 00:29:19,880 --> 00:29:26,600 shapes of this thing, are 1, 1 for the first mode and 1, 481 00:29:26,600 --> 00:29:29,080 minus 1 for the second mode. 482 00:29:31,820 --> 00:29:36,550 And the modal expansion theorem depends on this fact 483 00:29:36,550 --> 00:29:45,790 that the mode shapes form a complete independent set 484 00:29:45,790 --> 00:29:49,780 of vectors that are orthogonal to one another. 485 00:29:49,780 --> 00:29:53,630 A weighted sum of all the mode shades of the system 486 00:29:53,630 --> 00:29:58,130 can represent any allowable motion of the system. 487 00:29:58,130 --> 00:29:59,570 Any possible motion of the system 488 00:29:59,570 --> 00:30:04,430 can be written as a weight of-- so any way I can displace 489 00:30:04,430 --> 00:30:07,160 these things, a little bit here, a little bit there, 490 00:30:07,160 --> 00:30:10,860 I can write that position as a weighted sum of those two mode 491 00:30:10,860 --> 00:30:12,420 shapes. 492 00:30:12,420 --> 00:30:16,290 So let's say I want to have initial conditions 493 00:30:16,290 --> 00:30:22,240 on displacement, on r theta, 1 0 and theta 494 00:30:22,240 --> 00:30:25,620 2 0, no initial velocities, just initial angles. 495 00:30:25,620 --> 00:30:33,960 And I want that initial condition vector to be 1, 0. 496 00:30:33,960 --> 00:30:38,510 I'm claiming I can write that as a sum of something 1, 497 00:30:38,510 --> 00:30:43,000 1 plus another something 1, minus 1. 498 00:30:43,000 --> 00:30:45,940 So what do c1 and c2 have to be to give me that? 499 00:30:49,209 --> 00:30:53,060 You ought to be able to kind of do that by inspection, almost. 500 00:30:53,060 --> 00:30:54,670 For example, just try them equal. 501 00:30:54,670 --> 00:30:56,975 And what happens? 502 00:30:56,975 --> 00:30:57,850 AUDIENCE: [INAUDIBLE] 503 00:31:03,140 --> 00:31:05,530 PROFESSOR: So 1/2 and 1/2, right? 504 00:31:05,530 --> 00:31:14,210 Great, so to satisfy this, it's 1/2 1, 1 plus 1/2 1, minus 1. 505 00:31:14,210 --> 00:31:15,760 And I've just illustrated the fact 506 00:31:15,760 --> 00:31:18,020 that you can represent an arbitrary deflection, 507 00:31:18,020 --> 00:31:20,100 allowable deflection of the system, 508 00:31:20,100 --> 00:31:23,200 as a weighted sum of the two mode shapes. 509 00:31:23,200 --> 00:31:30,310 So that says if I make the initial-- now, 510 00:31:30,310 --> 00:31:35,435 this says that theta 1 is 1, and theta 2 is 0. 511 00:31:35,435 --> 00:31:38,440 And what it's telling us is by modal analysis, 512 00:31:38,440 --> 00:31:40,720 that means the resulting motion should 513 00:31:40,720 --> 00:31:44,230 look like the response to initial conditions 514 00:31:44,230 --> 00:31:47,780 where I have equal amounts of each of those two modes. 515 00:31:47,780 --> 00:31:50,110 So there ought to be some vibration at omega 1, 516 00:31:50,110 --> 00:31:54,340 and there ought to be equal amount of vibration at omega 2. 517 00:31:54,340 --> 00:31:56,610 Agreed? 518 00:31:56,610 --> 00:32:00,030 OK, let's try it. 519 00:32:00,030 --> 00:32:04,290 I'll hold one of these in place, and I'll deflect the other one 520 00:32:04,290 --> 00:32:05,250 and let go. 521 00:32:12,695 --> 00:32:16,710 Now, that one's stationary, and this one's moving. 522 00:32:16,710 --> 00:32:19,100 And a few cycles later, this one will be stationary, 523 00:32:19,100 --> 00:32:21,860 and that one will be moving. 524 00:32:21,860 --> 00:32:24,140 So what you're observing, could that motion, 525 00:32:24,140 --> 00:32:27,050 what you're seeing, possibly be a mode, 526 00:32:27,050 --> 00:32:29,720 a natural mode by itself? 527 00:32:29,720 --> 00:32:31,380 No, because the different proportions 528 00:32:31,380 --> 00:32:32,940 are changing, right? 529 00:32:32,940 --> 00:32:35,620 This is the sum of two modes-- exactly 530 00:32:35,620 --> 00:32:37,880 as you said, equal amounts of each. 531 00:32:37,880 --> 00:32:40,100 And when you do that, and you have 532 00:32:40,100 --> 00:32:45,540 two things of equal amplitude, two cosine omega t like terms 533 00:32:45,540 --> 00:32:48,860 of equal amplitude and different frequencies, 534 00:32:48,860 --> 00:32:51,410 you get a phenomenon known as beating. 535 00:32:51,410 --> 00:32:53,910 That's what this is. 536 00:32:53,910 --> 00:32:58,094 Beating means that one vibrates, and then it will get small. 537 00:32:58,094 --> 00:33:00,510 And then you'll see it build up again, and then get small. 538 00:33:00,510 --> 00:33:02,190 So if you watch either one of these, 539 00:33:02,190 --> 00:33:06,360 it's doing large and stopping. 540 00:33:06,360 --> 00:33:08,700 And the other one is doing the same thing, but actually 541 00:33:08,700 --> 00:33:11,410 90 degrees out of phase. 542 00:33:11,410 --> 00:33:21,870 So what we're really seeing is that the motion of this system, 543 00:33:21,870 --> 00:33:36,240 this is 1, 1, 1/2 and 1, 1 cosine omega 1 t plus a 1/2 1, 544 00:33:36,240 --> 00:33:41,550 minus 1 cosine omega 2 t. 545 00:33:41,550 --> 00:33:44,710 That's the total response of the system. 546 00:33:44,710 --> 00:33:46,670 Let's look at the first line of this. 547 00:33:46,670 --> 00:33:56,590 This says 1/2 cosine omega 1 t plus 1/2 cosine omega 548 00:33:56,590 --> 00:34:03,270 2 t should be equal to the motion we call theta 1 of t. 549 00:34:03,270 --> 00:34:05,220 That's the first row here. 550 00:34:05,220 --> 00:34:09,250 This is theta 1, theta 2. 551 00:34:09,250 --> 00:34:13,460 So the first row of this, the first equation, is that. 552 00:34:13,460 --> 00:34:17,630 And two equal amplitude cosines of different frequencies 553 00:34:17,630 --> 00:34:24,690 you can write as cosine omega 2 minus omega 1 554 00:34:24,690 --> 00:34:36,790 over 2 times t times cosine omega 1 plus omega 2 over 2 t. 555 00:34:40,989 --> 00:34:50,300 This beat phenomena that you're seeing when you plot this-- 556 00:34:50,300 --> 00:34:54,300 and this is now for theta 1. 557 00:34:54,300 --> 00:34:55,990 It starts off at some amplitude. 558 00:35:04,680 --> 00:35:07,910 And the actual motion you see, if you watch it, it 559 00:35:07,910 --> 00:35:11,420 does what I'm drawing right now. 560 00:35:14,420 --> 00:35:19,510 This envelope is this term, the difference frequency 561 00:35:19,510 --> 00:35:21,700 divided by 2. 562 00:35:21,700 --> 00:35:26,340 What's inside is that term. 563 00:35:26,340 --> 00:35:28,490 And this is called beating. 564 00:35:28,490 --> 00:35:29,920 This is the equation for beating. 565 00:35:29,920 --> 00:35:33,755 When you add two equal amplitude cosines together, 566 00:35:33,755 --> 00:35:35,630 they give you something that looks like that. 567 00:35:35,630 --> 00:35:38,530 And one period of the beat is how long 568 00:35:38,530 --> 00:35:41,430 it takes to go through one full cycle from here to here. 569 00:35:41,430 --> 00:35:45,010 That's one period of the beat. 570 00:35:45,010 --> 00:35:45,654 Yeah. 571 00:35:45,654 --> 00:35:47,195 AUDIENCE: How did you get those terms 572 00:35:47,195 --> 00:35:49,800 with the omega 2 [INAUDIBLE]? 573 00:35:49,800 --> 00:35:52,320 PROFESSOR: Well, that's just trig identity. 574 00:35:52,320 --> 00:35:54,740 You add-- you just go back to your trig, 575 00:35:54,740 --> 00:35:58,270 take cosine of a plus cosine of b. 576 00:35:58,270 --> 00:36:00,750 You'll find out you can do that. 577 00:36:00,750 --> 00:36:03,550 I don't have time to go through that. 578 00:36:03,550 --> 00:36:08,870 And the other one is a 1/2 cosine omega 1 579 00:36:08,870 --> 00:36:14,510 t minus 1/2 cosine omega 2 t. 580 00:36:14,510 --> 00:36:20,920 And that turns out to be sine omega 2 minus omega 1 581 00:36:20,920 --> 00:36:32,060 over 2 t times sine omega 1 plus omega 2 over 2 quantity times 582 00:36:32,060 --> 00:36:32,870 time. 583 00:36:32,870 --> 00:36:35,960 And that means that the theta 2 coordinate, 584 00:36:35,960 --> 00:36:41,140 what it looks like when you draw it, is 90 degrees out of phase. 585 00:36:41,140 --> 00:36:47,530 It's the same thing, but it starts at 0 here and beats. 586 00:36:47,530 --> 00:36:50,620 But it starts 0 where that one started here. 587 00:36:50,620 --> 00:36:52,730 And that's what-- if you look at one. 588 00:36:55,475 --> 00:37:01,900 So if that's theta 1 and this is theta 2, when one of them 589 00:37:01,900 --> 00:37:06,629 stops, that's one of these lines, one of these equations. 590 00:37:06,629 --> 00:37:08,170 And the other equation is 90 degrees. 591 00:37:08,170 --> 00:37:11,250 So that one's stopped right now. 592 00:37:11,250 --> 00:37:16,390 Pi over 2 later, this one will be stopped. 593 00:37:16,390 --> 00:37:20,990 So the beat behaves like sine in one case, 594 00:37:20,990 --> 00:37:23,780 and behaves like cosine in the other case. 595 00:37:26,740 --> 00:37:29,798 One of these needs a minus sign. 596 00:37:29,798 --> 00:37:33,458 That's a plus, minus and plus, minus and plus. 597 00:37:36,552 --> 00:37:39,590 Now what if you have unequal amounts? 598 00:37:39,590 --> 00:37:45,810 We have exactly equal amounts of the two modes. 599 00:37:45,810 --> 00:37:47,760 If you have unequal amounts of the two modes, 600 00:37:47,760 --> 00:37:50,040 then it's not going to be 1/2 and 1/2. 601 00:37:50,040 --> 00:37:53,310 What if it's like 1 and 0.1? 602 00:37:53,310 --> 00:37:55,020 Then do you see full beats? 603 00:37:55,020 --> 00:37:57,660 Does one come to a stop? 604 00:37:57,660 --> 00:37:59,710 If they're not equal, you'll find 605 00:37:59,710 --> 00:38:03,830 that the sums of the two motions, the envelope 606 00:38:03,830 --> 00:38:04,850 will look like this. 607 00:38:10,550 --> 00:38:12,820 It'll be modulated. 608 00:38:12,820 --> 00:38:14,450 Each one's motion will look like that. 609 00:38:14,450 --> 00:38:15,990 It'll never go to 0 quite. 610 00:38:18,780 --> 00:38:20,596 But that's beating. 611 00:38:20,596 --> 00:38:21,885 How are we doing on time? 612 00:38:21,885 --> 00:38:23,480 We're good. 613 00:38:23,480 --> 00:38:29,830 And let's see, we said the one natural mode is 614 00:38:29,830 --> 00:38:31,530 the two opposite one another. 615 00:38:31,530 --> 00:38:34,890 So if I start, go equal amounts in opposite directions 616 00:38:34,890 --> 00:38:37,880 and let go, it ought to just sit there and vibrate 617 00:38:37,880 --> 00:38:41,470 all day like that-- no beats. 618 00:38:41,470 --> 00:38:46,511 Because it's the 1, 0 case over there. 619 00:38:46,511 --> 00:38:48,010 And actually, this is the 0, 1 case. 620 00:38:48,010 --> 00:38:49,690 This is 1, minus 1. 621 00:38:49,690 --> 00:38:53,340 I gave an initial displacement in exactly the shape 622 00:38:53,340 --> 00:38:54,630 of the second mode. 623 00:38:54,630 --> 00:38:57,980 And that means the two contributions 624 00:38:57,980 --> 00:39:00,630 are that one of those constants is 0. 625 00:39:00,630 --> 00:39:02,220 There's no mode one in this. 626 00:39:02,220 --> 00:39:03,370 It's only mode two. 627 00:39:03,370 --> 00:39:06,080 And it'll sit here all day long and vibrate just in mode two. 628 00:39:06,080 --> 00:39:10,570 And if I displace it in the shape of mode one only, 629 00:39:10,570 --> 00:39:11,320 it'll vibrate. 630 00:39:11,320 --> 00:39:12,361 And that one's harder do. 631 00:39:12,361 --> 00:39:16,800 Because I have to move it exactly the same amount. 632 00:39:16,800 --> 00:39:18,960 That's mode one. 633 00:39:18,960 --> 00:39:20,980 And it should vibrate all day long on mode one. 634 00:39:20,980 --> 00:39:23,886 Because now there's no mode two involved in that one. 635 00:39:23,886 --> 00:39:26,010 You see it's already getting a little out of phase. 636 00:39:26,010 --> 00:39:29,640 It's hard for me to move my hands exactly the same amount. 637 00:39:29,640 --> 00:39:34,530 So I have a little bit of the other mode in there. 638 00:39:34,530 --> 00:39:37,810 But you'll never see either one of these come to a full stop. 639 00:39:37,810 --> 00:39:42,454 It's actually doing that when you get a little contamination 640 00:39:42,454 --> 00:39:43,620 of the second mode in there. 641 00:39:49,100 --> 00:39:53,730 OK, so I've posted on Stellar a little two-page sheet 642 00:39:53,730 --> 00:39:57,375 that just gives you step by step how to do a modal analysis. 643 00:39:57,375 --> 00:39:58,680 It's just cookbook. 644 00:39:58,680 --> 00:39:59,661 Modal analysis is easy. 645 00:39:59,661 --> 00:40:00,910 You do it all on the computer. 646 00:40:00,910 --> 00:40:04,020 You just put your m matrix and your k matrix, 647 00:40:04,020 --> 00:40:05,625 and you just crank stuff out. 648 00:40:08,640 --> 00:40:10,760 We did multimodal analysis yesterday 649 00:40:10,760 --> 00:40:12,230 just from initial conditions. 650 00:40:12,230 --> 00:40:13,890 Next Tuesday, we'll do modal analysis 651 00:40:13,890 --> 00:40:16,890 assuming you've got harmonic excitation and steady state 652 00:40:16,890 --> 00:40:20,600 vibration and do that kind of thing.