1 00:00:02,090 --> 00:00:04,430 The following content is provided under a Creative 2 00:00:04,430 --> 00:00:05,850 Commons license. 3 00:00:05,850 --> 00:00:08,060 Your support will help MIT OpenCourseWare 4 00:00:08,060 --> 00:00:12,150 continue to offer high quality educational resources for free. 5 00:00:12,150 --> 00:00:14,690 To make a donation or to view additional materials 6 00:00:14,690 --> 00:00:18,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,620 --> 00:00:19,976 at ocw.mit.edu. 8 00:00:22,781 --> 00:00:25,270 YEN-JIE LEE: And welcome back, everybody, 9 00:00:25,270 --> 00:00:31,960 to this fun class, 8.03. 10 00:00:31,960 --> 00:00:35,150 Let's get started. 11 00:00:35,150 --> 00:00:37,960 So the first thing which we will do 12 00:00:37,960 --> 00:00:41,650 is to review a bit what we have learned last time. 13 00:00:41,650 --> 00:00:44,440 And then we'll go to the next level 14 00:00:44,440 --> 00:00:48,780 to study coupled oscillators. 15 00:00:48,780 --> 00:00:50,440 OK. 16 00:00:50,440 --> 00:00:55,550 Last time, we had learned a lot on damped driven oscillators. 17 00:00:55,550 --> 00:00:59,710 So as far as the course we've been going, 18 00:00:59,710 --> 00:01:04,510 actually, we only study a single object, 19 00:01:04,510 --> 00:01:09,040 and then we introduce more and more force 20 00:01:09,040 --> 00:01:10,560 acting on this object. 21 00:01:10,560 --> 00:01:14,710 We introduce damping force, we introduce 22 00:01:14,710 --> 00:01:17,300 a driving force last time. 23 00:01:17,300 --> 00:01:24,330 And we see that the system becomes 24 00:01:24,330 --> 00:01:28,720 more and more difficult to understand because of the added 25 00:01:28,720 --> 00:01:30,150 component. 26 00:01:30,150 --> 00:01:33,930 But after the class last time, I hope 27 00:01:33,930 --> 00:01:38,980 I convinced you that we can understand driven oscillators. 28 00:01:38,980 --> 00:01:42,920 And there are two very important things we learned last time. 29 00:01:42,920 --> 00:01:46,610 The first one is the transient behavior, 30 00:01:46,610 --> 00:01:48,970 which is actually a superposition 31 00:01:48,970 --> 00:01:53,560 of the homogeneous solution and the steady state solution. 32 00:01:53,560 --> 00:01:55,130 OK. 33 00:01:55,130 --> 00:01:59,060 One very good news is that if you are patient enough, 34 00:01:59,060 --> 00:02:03,530 you shake the system continuously, 35 00:02:03,530 --> 00:02:08,030 and if you wait long enough, then the homogeneous solution 36 00:02:08,030 --> 00:02:11,009 contribution goes away. 37 00:02:11,009 --> 00:02:16,200 And what is actually left over is the steady state solution, 38 00:02:16,200 --> 00:02:20,690 which is actually much simpler than what we saw beforehand. 39 00:02:20,690 --> 00:02:24,170 It's actually just harmonic oscillation 40 00:02:24,170 --> 00:02:26,600 at driving frequency. 41 00:02:26,600 --> 00:02:32,060 Also, I hope that we also have learned a very interesting 42 00:02:32,060 --> 00:02:34,190 phenomenon, which is resonance. 43 00:02:34,190 --> 00:02:39,710 When the driving frequency is close to the natural frequency 44 00:02:39,710 --> 00:02:46,280 of the system, then the system apparently likes it. 45 00:02:46,280 --> 00:02:51,350 Then it would respond with larger amplitude 46 00:02:51,350 --> 00:02:55,790 and oscillating up and down at driving frequency. 47 00:02:55,790 --> 00:02:59,510 So that, we call it resonance. 48 00:02:59,510 --> 00:03:01,250 This is the equation of motion, which 49 00:03:01,250 --> 00:03:03,140 we have learned last time. 50 00:03:03,140 --> 00:03:07,280 You can see is theta double dot plus Gamma theta dot, 51 00:03:07,280 --> 00:03:10,010 is a contribution from the drag force, 52 00:03:10,010 --> 00:03:14,780 and omega 0 squared theta is the contribution 53 00:03:14,780 --> 00:03:17,270 from the so-called spring force. 54 00:03:17,270 --> 00:03:23,810 And finally, that is equal to f0 cosine omega d t, the driving 55 00:03:23,810 --> 00:03:24,950 force. 56 00:03:24,950 --> 00:03:27,170 And as we mentioned in the beginning, 57 00:03:27,170 --> 00:03:34,110 if we prepare this system and under-damp the situation, 58 00:03:34,110 --> 00:03:38,820 then the full solution is a superposition 59 00:03:38,820 --> 00:03:41,130 of the steady state solution, which 60 00:03:41,130 --> 00:03:46,050 is the left-hand side, the red thing I'm pointing to, 61 00:03:46,050 --> 00:03:48,600 this steady state solution. 62 00:03:48,600 --> 00:03:54,390 There's no free parameter in the steady state solution. 63 00:03:54,390 --> 00:03:57,290 So A, the amplitude, is determined by omega d. 64 00:04:00,260 --> 00:04:03,820 Delta, which is the phase is also determined by omega d. 65 00:04:03,820 --> 00:04:06,940 There's no free parameter. 66 00:04:06,940 --> 00:04:08,180 OK. 67 00:04:08,180 --> 00:04:13,180 And in order to make the solution a full solution, 68 00:04:13,180 --> 00:04:16,630 we actually have to add in this homogeneous solution 69 00:04:16,630 --> 00:04:18,310 back into this again. 70 00:04:18,310 --> 00:04:21,339 And basically, you have B and alpha, 71 00:04:21,339 --> 00:04:23,320 those are the free parameters, which 72 00:04:23,320 --> 00:04:28,780 can be determined by the given initial conditions. 73 00:04:28,780 --> 00:04:29,590 OK. 74 00:04:29,590 --> 00:04:32,650 So if we go ahead and plot some of the examples 75 00:04:32,650 --> 00:04:36,910 as a function of time, so the y-axis 76 00:04:36,910 --> 00:04:38,860 is actually the amplitude. 77 00:04:38,860 --> 00:04:41,290 And the x-axis is time. 78 00:04:41,290 --> 00:04:43,570 And what is actually plotted here 79 00:04:43,570 --> 00:04:46,840 is a combination or the superposition 80 00:04:46,840 --> 00:04:51,640 of the steady state solution and the homogeneous solution. 81 00:04:51,640 --> 00:04:55,690 And you can see that the individual components are also 82 00:04:55,690 --> 00:04:57,610 shown in this slide. 83 00:04:57,610 --> 00:05:01,960 You can see the red thing oscillating up and down 84 00:05:01,960 --> 00:05:06,540 harmonically, is steady state solution contribution. 85 00:05:06,540 --> 00:05:10,870 And also, you have the blue curve, 86 00:05:10,870 --> 00:05:14,180 which is decaying away as a function of time. 87 00:05:14,180 --> 00:05:17,990 And you can see that if you add these two curves together, 88 00:05:17,990 --> 00:05:20,740 you get something rather complicated. 89 00:05:20,740 --> 00:05:25,720 You will get some kind of motion like, do, do, do, do. 90 00:05:25,720 --> 00:05:30,710 Then the homogeneous solution actually dies out. 91 00:05:30,710 --> 00:05:34,960 Then what is actually left over is just the steady state 92 00:05:34,960 --> 00:05:38,030 solution, harmonic oscillation. 93 00:05:38,030 --> 00:05:42,320 And in this case, omega d is actually 10 times larger 94 00:05:42,320 --> 00:05:45,290 than the natural frequency. 95 00:05:45,290 --> 00:05:48,890 And there's another example which is also very interesting. 96 00:05:48,890 --> 00:05:57,470 It's that if I make the omega d closer to omega 0-- 97 00:05:57,470 --> 00:06:00,740 OK, in this case it's actually omega d equal to 2 times omega 98 00:06:00,740 --> 00:06:04,070 0, then you can produce some kind of a motion, which 99 00:06:04,070 --> 00:06:04,910 is like this. 100 00:06:04,910 --> 00:06:06,920 So you have the oscillation. 101 00:06:06,920 --> 00:06:09,080 And they stayed there for a while, 102 00:06:09,080 --> 00:06:11,440 then goes back, and oscillates down, 103 00:06:11,440 --> 00:06:14,930 and stay there then goes back. 104 00:06:14,930 --> 00:06:17,300 OK, as you can see on there. 105 00:06:17,300 --> 00:06:22,250 The homogeneous solution part and steady state solution part 106 00:06:22,250 --> 00:06:27,170 work together and produce this kind of strange behavior. 107 00:06:27,170 --> 00:06:27,980 OK. 108 00:06:27,980 --> 00:06:30,860 And that's just another example. 109 00:06:30,860 --> 00:06:33,010 And if you wait long enough, again 110 00:06:33,010 --> 00:06:37,400 what is actually left over is the steady state solution. 111 00:06:37,400 --> 00:06:38,760 OK. 112 00:06:38,760 --> 00:06:41,440 So what are we going to do today? 113 00:06:41,440 --> 00:06:45,660 So today, we are going to investigate 114 00:06:45,660 --> 00:06:52,260 what will happen if we try to put together multiple objects 115 00:06:52,260 --> 00:06:55,170 and also allow them to talk to each other. 116 00:06:55,170 --> 00:06:55,770 OK. 117 00:06:55,770 --> 00:06:58,740 So if we have two objects, and they don't talk to each other, 118 00:06:58,740 --> 00:07:01,050 then they are still like a single object. 119 00:07:01,050 --> 00:07:04,800 They are still like simple harmonic motion on their own. 120 00:07:04,800 --> 00:07:08,130 But if you allow them to talk to each other, 121 00:07:08,130 --> 00:07:11,820 this is the so-called coupled oscillator, then 122 00:07:11,820 --> 00:07:15,130 interesting thing happen. 123 00:07:15,130 --> 00:07:23,250 So in general, coupled systems are super, super complicated. 124 00:07:23,250 --> 00:07:24,420 OK. 125 00:07:24,420 --> 00:07:27,660 So let me give you one example here. 126 00:07:27,660 --> 00:07:33,690 This is actually two pendulums that 127 00:07:33,690 --> 00:07:36,720 are a coupled to each other, they are actually 128 00:07:36,720 --> 00:07:39,950 connected to each other, one pendulum, 129 00:07:39,950 --> 00:07:42,270 the second one is here, OK. 130 00:07:42,270 --> 00:07:48,270 And for example, I can actually give it an initial velocity 131 00:07:48,270 --> 00:07:49,695 and see what is going to happen. 132 00:07:52,680 --> 00:07:56,130 You can see that the resulting motion-- 133 00:07:56,130 --> 00:07:57,150 OK. 134 00:07:57,150 --> 00:08:00,980 Remember, we are just talking about two pendulums that 135 00:08:00,980 --> 00:08:02,620 are connected to each other. 136 00:08:02,620 --> 00:08:08,530 The resulting motion is super complicated. 137 00:08:08,530 --> 00:08:11,200 This is one of my favorite demonstrations. 138 00:08:11,200 --> 00:08:17,291 You can actually stare at this machine the whole time. 139 00:08:17,291 --> 00:08:18,790 And you can see that, huh, sometimes 140 00:08:18,790 --> 00:08:20,370 it does this rotation. 141 00:08:20,370 --> 00:08:22,190 Sometimes it doesn't do that. 142 00:08:22,190 --> 00:08:25,750 And it's almost like a living creature. 143 00:08:28,840 --> 00:08:31,310 So we are going to solve this system. 144 00:08:31,310 --> 00:08:33,606 No, probably not, he knows. 145 00:08:33,606 --> 00:08:36,190 [LAUGHTER] 146 00:08:36,190 --> 00:08:41,080 But as I mentioned before, you can always write down 147 00:08:41,080 --> 00:08:43,409 the equation of motion. 148 00:08:43,409 --> 00:08:47,350 And you can solve it by computer. 149 00:08:47,350 --> 00:08:49,560 Maybe some of the course 6 people 150 00:08:49,560 --> 00:08:52,980 can actually try and write the program to solve this thing 151 00:08:52,980 --> 00:08:56,820 and to simulate what is going to happen. 152 00:08:56,820 --> 00:09:02,730 So let's take a look at this complicated motion again. 153 00:09:02,730 --> 00:09:06,480 So you can see that the good news is that there are only two 154 00:09:06,480 --> 00:09:07,470 objects. 155 00:09:07,470 --> 00:09:10,260 And you can see-- 156 00:09:10,260 --> 00:09:14,280 look at the green, sorry, the orange dot. 157 00:09:14,280 --> 00:09:21,460 The orange dot is always moving along a semi circle. 158 00:09:21,460 --> 00:09:27,680 But if you focus on the yellow dot, 159 00:09:27,680 --> 00:09:30,940 the yellow is doing all kinds of different things. 160 00:09:30,940 --> 00:09:35,640 It's very hard to predict what is going to happen. 161 00:09:35,640 --> 00:09:40,130 So what I want to say is, those are interesting examples 162 00:09:40,130 --> 00:09:41,960 of coupled systems. 163 00:09:41,960 --> 00:09:45,800 But they are actually far more complicated than what 164 00:09:45,800 --> 00:09:51,050 we thought, because they are not smooth oscillation 165 00:09:51,050 --> 00:09:54,260 around equilibrium position. 166 00:09:54,260 --> 00:09:56,920 So you can see that now if I stop this machine 167 00:09:56,920 --> 00:10:02,940 and just perturb it slightly, giving it a small angle 168 00:10:02,940 --> 00:10:08,370 displacement, then you can see that the motion is 169 00:10:08,370 --> 00:10:12,560 much more easier to understand. 170 00:10:12,560 --> 00:10:14,030 You see. 171 00:10:14,030 --> 00:10:17,530 You even get one of these questions in your p-set. 172 00:10:17,530 --> 00:10:18,795 OK, that's good news. 173 00:10:21,390 --> 00:10:25,920 So our job today is to understand 174 00:10:25,920 --> 00:10:30,150 what is going to happen to those coupled oscillators. 175 00:10:30,150 --> 00:10:34,560 Let me give you a few examples before we start 176 00:10:34,560 --> 00:10:38,130 to work on a specific question. 177 00:10:38,130 --> 00:10:43,170 The second example I would like to show you is a saw 178 00:10:43,170 --> 00:10:47,220 and you actually connect it to two - actually 179 00:10:47,220 --> 00:10:50,970 a ruler, a metal ruler, which is connected 180 00:10:50,970 --> 00:10:54,750 to two massive objects. 181 00:10:54,750 --> 00:11:00,090 Now I can actually give it the initial velocity 182 00:11:00,090 --> 00:11:02,470 and see what happens. 183 00:11:02,470 --> 00:11:04,560 And you can see that they do talk 184 00:11:04,560 --> 00:11:09,490 to each other through this ruler, this metal ruler. 185 00:11:09,490 --> 00:11:10,020 Can you see? 186 00:11:10,020 --> 00:11:11,820 I hope you can see. 187 00:11:11,820 --> 00:11:13,530 It's a bit small. 188 00:11:13,530 --> 00:11:16,630 But it's really interesting that you can see-- originally, 189 00:11:16,630 --> 00:11:20,440 I just introduced some displacement 190 00:11:20,440 --> 00:11:23,100 in the left-hand side mass. 191 00:11:23,100 --> 00:11:27,270 And the left-had side thing start to move or so 192 00:11:27,270 --> 00:11:30,260 after a while. 193 00:11:30,260 --> 00:11:36,690 There are two more examples which I would like to give you, 194 00:11:36,690 --> 00:11:38,920 introduce to you. 195 00:11:38,920 --> 00:11:42,390 There are two kinds of pendulums, 196 00:11:42,390 --> 00:11:44,380 which I prepared here. 197 00:11:44,380 --> 00:11:49,050 The first one is there are two pendulums that are connected 198 00:11:49,050 --> 00:11:52,530 to each other by a spring. 199 00:11:52,530 --> 00:11:57,850 And if I try to introduce displacement, 200 00:11:57,850 --> 00:12:03,340 I move both masses slightly and see what is going to happen. 201 00:12:03,340 --> 00:12:07,400 And we see that the motion is still complicated. 202 00:12:07,400 --> 00:12:12,250 Although, if you stare at one objects, it looks more 203 00:12:12,250 --> 00:12:16,870 like harmonic oscillation, but not quite. 204 00:12:16,870 --> 00:12:19,360 For example, this guy is slowing down, 205 00:12:19,360 --> 00:12:21,680 and this is actually moving faster. 206 00:12:21,680 --> 00:12:26,500 And now the right hand side guy is actually moving faster. 207 00:12:26,500 --> 00:12:29,590 Motion seems to be completed. 208 00:12:29,590 --> 00:12:32,810 Also, you can look at this one. 209 00:12:32,810 --> 00:12:34,900 Those are the two pendulums. 210 00:12:34,900 --> 00:12:39,670 They are connected to each other by this rod here. 211 00:12:39,670 --> 00:12:42,340 And of course, you can displace the mass 212 00:12:42,340 --> 00:12:44,964 from the equilibrium position. 213 00:12:44,964 --> 00:12:45,880 I'm not going to hit-- 214 00:12:45,880 --> 00:12:47,080 not hitting each other. 215 00:12:47,080 --> 00:12:49,730 So you can displace the masses from each other. 216 00:12:49,730 --> 00:12:54,040 And you can see that they do complicated things 217 00:12:54,040 --> 00:12:55,640 as a function the time. 218 00:12:55,640 --> 00:12:58,280 How are we going to understand this? 219 00:12:58,280 --> 00:13:02,890 And I hope that by the end of this lecture 220 00:13:02,890 --> 00:13:06,940 you are convinced that you can as you solve this really easy, 221 00:13:06,940 --> 00:13:10,390 following a fixed procedure. 222 00:13:10,390 --> 00:13:15,130 In those examples, we have two objects that 223 00:13:15,130 --> 00:13:17,120 are connected to each other. 224 00:13:17,120 --> 00:13:19,450 And therefore, they talk to each other 225 00:13:19,450 --> 00:13:23,800 and produce coupled motion. 226 00:13:23,800 --> 00:13:27,650 Those are a couple oscillator examples. 227 00:13:27,650 --> 00:13:31,000 There's another very interesting example, which 228 00:13:31,000 --> 00:13:36,370 is called Wilberforce pendulum. 229 00:13:36,370 --> 00:13:39,220 So this is actually a pendulum. 230 00:13:39,220 --> 00:13:42,820 You can rotate like this. 231 00:13:42,820 --> 00:13:44,900 And it can also move up and down. 232 00:13:44,900 --> 00:13:48,310 It's connected to a spring. 233 00:13:48,310 --> 00:13:51,200 The interesting thing is that if I just 234 00:13:51,200 --> 00:13:56,270 start with some rotation, you can 235 00:13:56,270 --> 00:14:01,910 see that it starts to also oscillate up and down. 236 00:14:01,910 --> 00:14:02,590 You see? 237 00:14:02,590 --> 00:14:08,330 So initially I just introduced a rotation. 238 00:14:08,330 --> 00:14:10,120 Now it's actually fully rotating. 239 00:14:10,120 --> 00:14:13,150 And now it starts to move up and down. 240 00:14:13,150 --> 00:14:18,600 And you can see that the energy stored in the pendulum 241 00:14:18,600 --> 00:14:27,260 is going back and forth between the gravitational potential, 242 00:14:27,260 --> 00:14:31,360 between the potential of the spring, 243 00:14:31,360 --> 00:14:38,920 and also between the kinetic energy of up and down motion 244 00:14:38,920 --> 00:14:40,320 and the rotation. 245 00:14:40,320 --> 00:14:46,840 They're actually doing all those transitions all the time. 246 00:14:46,840 --> 00:14:50,280 So you can see-- 247 00:14:50,280 --> 00:14:53,830 so initially it's just rotating. 248 00:14:53,830 --> 00:14:56,440 And then it starts to move up and down. 249 00:14:56,440 --> 00:14:58,840 And this one is also very similar. 250 00:14:58,840 --> 00:15:02,110 But now the mass is much more displaced. 251 00:15:02,110 --> 00:15:07,340 And if I try to rotate this system without introducing 252 00:15:07,340 --> 00:15:10,360 a horizontal direction displacement, 253 00:15:10,360 --> 00:15:17,170 it still does is up and down motion, like a simple spring 254 00:15:17,170 --> 00:15:18,620 mass system. 255 00:15:18,620 --> 00:15:23,481 So what causes this kind of motion? 256 00:15:23,481 --> 00:15:29,590 That is because when we move this pendulum up and down, 257 00:15:29,590 --> 00:15:34,180 we also slightly unwind the spring. 258 00:15:34,180 --> 00:15:39,070 That can generate some kind of torque to this mass 259 00:15:39,070 --> 00:15:44,900 and produce rotational behavior. 260 00:15:44,900 --> 00:15:47,755 And you can see that this is just 261 00:15:47,755 --> 00:15:49,480 involving one single object. 262 00:15:49,480 --> 00:15:53,650 But there's a coupling between the rotation 263 00:15:53,650 --> 00:15:56,890 and the horizontal direction motion. 264 00:15:56,890 --> 00:16:04,240 So that's also special kind of coupled oscillator. 265 00:16:04,240 --> 00:16:10,580 So after all this, before we get started, 266 00:16:10,580 --> 00:16:13,150 I would like to say that what we are going to do 267 00:16:13,150 --> 00:16:17,800 is to assume all those things are ideal, 268 00:16:17,800 --> 00:16:21,670 without them being forced, without a driving force. 269 00:16:21,670 --> 00:16:24,410 We may introduce that later in the class. 270 00:16:24,410 --> 00:16:28,870 But for simplicity, we'll just stay with this idea case, 271 00:16:28,870 --> 00:16:31,390 before the mass becomes super complicated 272 00:16:31,390 --> 00:16:34,430 to solve it in front of you. 273 00:16:34,430 --> 00:16:43,020 And also, we can see that all those complicated motion 274 00:16:43,020 --> 00:16:43,955 are just illusion. 275 00:16:46,590 --> 00:16:49,900 Actually, the reality is that all of those 276 00:16:49,900 --> 00:16:54,250 are just superposition of harmonic motions. 277 00:16:54,250 --> 00:16:57,680 You will see that by the end of this class. 278 00:16:57,680 --> 00:16:59,961 So that is really amazing. 279 00:16:59,961 --> 00:17:00,460 OK. 280 00:17:00,460 --> 00:17:02,870 Let's immediately get started. 281 00:17:02,870 --> 00:17:06,609 So let's take a look at this system 282 00:17:06,609 --> 00:17:08,980 together and see if we can actually 283 00:17:08,980 --> 00:17:12,670 figure out the motion of this system together. 284 00:17:12,670 --> 00:17:17,109 So I have a system with three little masses. 285 00:17:17,109 --> 00:17:20,609 So there are three little masses in this system. 286 00:17:20,609 --> 00:17:25,240 They are connected to each other by spring. 287 00:17:25,240 --> 00:17:29,590 Those springs are highly idealized, the springs. 288 00:17:29,590 --> 00:17:32,320 And they have spring constant k. 289 00:17:32,320 --> 00:17:35,950 And the natural length's l0. 290 00:17:35,950 --> 00:17:40,120 And they are placed on Earth. 291 00:17:40,120 --> 00:17:44,800 And I carefully design the lab so 292 00:17:44,800 --> 00:17:50,150 that there's no friction between the desk and all 293 00:17:50,150 --> 00:17:52,450 those little masses. 294 00:17:52,450 --> 00:17:56,370 So once you get started and look at this system, 295 00:17:56,370 --> 00:17:58,540 you can imagine that there can be 296 00:17:58,540 --> 00:18:02,860 all kinds of different complicated motions. 297 00:18:02,860 --> 00:18:06,950 You can actually, for example, just move this mass 298 00:18:06,950 --> 00:18:12,220 and put the other two on hold. 299 00:18:12,220 --> 00:18:14,770 And they can oscillate like crazy. 300 00:18:14,770 --> 00:18:20,540 They can do very similar kind of motion. 301 00:18:20,540 --> 00:18:23,950 There are many, many possibilities. 302 00:18:23,950 --> 00:18:30,100 But if you stare at this system long enough, 303 00:18:30,100 --> 00:18:35,560 you will be able to identify special kinds of motion which 304 00:18:35,560 --> 00:18:39,670 are easier to understand. 305 00:18:39,670 --> 00:18:42,490 So what I would like to introduce to you 306 00:18:42,490 --> 00:18:45,190 is a special kind of motion which 307 00:18:45,190 --> 00:18:51,720 you can identify from the symmetry of this system. 308 00:18:51,720 --> 00:18:53,930 That is your so-called normal mode. 309 00:19:01,310 --> 00:19:04,970 So what is a normal mode, a special kind of motion 310 00:19:04,970 --> 00:19:07,700 we are trying to identify? 311 00:19:07,700 --> 00:19:10,430 That is actually the kind of motion 312 00:19:10,430 --> 00:19:23,100 which every part of the system is 313 00:19:23,100 --> 00:19:40,975 oscillating at the same frequency and the same phase. 314 00:19:49,770 --> 00:19:52,700 So that is your so-called normal mode, 315 00:19:52,700 --> 00:19:56,420 and is a special kind of motion, which I would 316 00:19:56,420 --> 00:20:00,620 like you to identify with me. 317 00:20:00,620 --> 00:20:06,170 And we would later realize that those special kinds of motions, 318 00:20:06,170 --> 00:20:09,830 which are easier to understand, actually 319 00:20:09,830 --> 00:20:14,660 helps us to understand the general motion of the system. 320 00:20:14,660 --> 00:20:19,820 You will realize that the most general motion of the system 321 00:20:19,820 --> 00:20:26,860 is just a superposition of all the identified normal modes. 322 00:20:26,860 --> 00:20:28,580 And then we are done, because we have 323 00:20:28,580 --> 00:20:31,300 a general solution already. 324 00:20:31,300 --> 00:20:34,100 So that's very good news. 325 00:20:34,100 --> 00:20:38,020 That tells us that we can understand 326 00:20:38,020 --> 00:20:43,190 the system systematically, and step by step. 327 00:20:43,190 --> 00:20:47,710 And then we can write the general motion of the system 328 00:20:47,710 --> 00:20:51,700 as a superposition of all the normal modes. 329 00:20:51,700 --> 00:20:54,440 So let's get started. 330 00:20:54,440 --> 00:21:00,040 So can you guess what are the possible normal modes 331 00:21:00,040 --> 00:21:02,330 of this system? 332 00:21:02,330 --> 00:21:05,230 So that means each part of the system 333 00:21:05,230 --> 00:21:09,670 is oscillating at the same frequency and the same phase. 334 00:21:09,670 --> 00:21:15,970 Can anybody and any one of you guess what can happen, 335 00:21:15,970 --> 00:21:18,550 each part of this is an oscillating 336 00:21:18,550 --> 00:21:21,390 at the same frequency? 337 00:21:21,390 --> 00:21:22,845 Yeah? 338 00:21:22,845 --> 00:21:23,870 AUDIENCE: If the two masses on that side are displaced 339 00:21:23,870 --> 00:21:25,328 the same amount and then they're -- 340 00:21:29,150 --> 00:21:30,610 YEN-JIE LEE: Very good. 341 00:21:30,610 --> 00:21:36,460 So he was saying that now I displace the right hand 342 00:21:36,460 --> 00:21:40,600 side two masses all together by a fixed amount, and also 343 00:21:40,600 --> 00:21:47,000 the left hand side, right, by a fixed amount and then let go. 344 00:21:47,000 --> 00:21:49,011 So that's what you're saying, right? 345 00:21:49,011 --> 00:21:49,510 OK. 346 00:21:49,510 --> 00:21:56,500 So the first mode we have identified is like this. 347 00:21:56,500 --> 00:22:03,490 So you have left hand side mass displace by delta x. 348 00:22:03,490 --> 00:22:07,670 And the right hand side two masses 349 00:22:07,670 --> 00:22:09,965 are also displaced by delta x. 350 00:22:15,440 --> 00:22:20,590 So basically you hold this three little masses 351 00:22:20,590 --> 00:22:25,530 and stretch it by the same-- 352 00:22:25,530 --> 00:22:29,470 introduce the same amplitude to all those three little masses, 353 00:22:29,470 --> 00:22:31,000 and let go. 354 00:22:31,000 --> 00:22:34,830 So that is actually one possible mode. 355 00:22:34,830 --> 00:22:41,090 And if we do this, then basically 356 00:22:41,090 --> 00:22:43,930 what you are going to see is that this is actually 357 00:22:43,930 --> 00:22:50,130 roughly equal to this system. 358 00:22:52,920 --> 00:22:57,630 They're connected to each other by two springs. 359 00:22:57,630 --> 00:23:02,250 And the right hand side part of the system, both masses 360 00:23:02,250 --> 00:23:06,150 are oscillating at the same amplitude and the same phase. 361 00:23:06,150 --> 00:23:14,040 They look like as if they are just single mass with mass 362 00:23:14,040 --> 00:23:15,130 equal to 2m. 363 00:23:17,830 --> 00:23:22,490 And if you introduce a displacement of delta x, 364 00:23:22,490 --> 00:23:25,510 then what is going to happen is that if I 365 00:23:25,510 --> 00:23:31,210 take a look at the mass, left hand side mass, and the force 366 00:23:31,210 --> 00:23:34,300 acting on this mass, the force will 367 00:23:34,300 --> 00:23:42,580 be equal to minus 2k times 2 delta x, 368 00:23:42,580 --> 00:23:45,520 because that's the amount of stretch you 369 00:23:45,520 --> 00:23:48,970 introduce to the spring. 370 00:23:48,970 --> 00:23:52,765 And that will give you minus 4k delta x. 371 00:23:57,390 --> 00:24:00,300 And we have already solved this kind of problem 372 00:24:00,300 --> 00:24:02,310 in the first lecture. 373 00:24:02,310 --> 00:24:05,430 So therefore you can immediately identify 374 00:24:05,430 --> 00:24:10,690 omega, in this case, omega a squared will 375 00:24:10,690 --> 00:24:16,500 be equal to 4k divided by 2m. 376 00:24:16,500 --> 00:24:20,880 This is actually the effective spring constant, 377 00:24:20,880 --> 00:24:24,210 and this is actually the mass. 378 00:24:24,210 --> 00:24:31,440 So that is actually the frequency of mode A. 379 00:24:31,440 --> 00:24:38,020 Can you identify a second kind of motion which does that? 380 00:24:38,020 --> 00:24:39,990 So in this case, what is going to happen 381 00:24:39,990 --> 00:24:42,600 is that the three masses will-- 382 00:24:42,600 --> 00:24:45,120 OK, one, two, and three. 383 00:24:45,120 --> 00:24:50,560 The three masses will oscillate as a function of time 384 00:24:50,560 --> 00:25:01,610 like this with angular frequency of square root of 4k over 2m. 385 00:25:01,610 --> 00:25:05,586 What is actually a second possible motion? 386 00:25:05,586 --> 00:25:07,410 Yes? 387 00:25:07,410 --> 00:25:10,570 AUDIENCE: All masses being stretched [INAUDIBLE] 388 00:25:10,570 --> 00:25:11,641 YEN-JIE LEE: Compressed. 389 00:25:11,641 --> 00:25:14,082 AUDIENCE: Compressed the same-- 390 00:25:14,082 --> 00:25:15,040 YEN-JIE LEE: Very good. 391 00:25:18,100 --> 00:25:24,910 I'm very lucky that I'm in front of such a smart crowd today. 392 00:25:24,910 --> 00:25:28,330 And we have successfully identified 393 00:25:28,330 --> 00:25:34,060 the second mode, mode B. So what is going to happen 394 00:25:34,060 --> 00:25:41,470 is that the left hand side mass is not moving. 395 00:25:41,470 --> 00:25:49,400 And you compress the upper one slightly 396 00:25:49,400 --> 00:25:56,570 and you stretch the lower one, the lower little mass 397 00:25:56,570 --> 00:25:59,700 to the opposite direction. 398 00:25:59,700 --> 00:26:03,140 The displacement is delta x, and the displacement 399 00:26:03,140 --> 00:26:07,210 of the second mass is delta x. 400 00:26:07,210 --> 00:26:09,980 So what is going to happen? 401 00:26:09,980 --> 00:26:13,220 What is going to happen is that the left hand side 402 00:26:13,220 --> 00:26:19,400 mass will not move at all because the force, the spring 403 00:26:19,400 --> 00:26:25,490 force, acting on this mass is going to cancel. 404 00:26:25,490 --> 00:26:27,650 And apparently, these two little masses 405 00:26:27,650 --> 00:26:31,370 are going to be doing harmonic motion. 406 00:26:34,160 --> 00:26:38,136 Since this left hand side mass is not moving, 407 00:26:38,136 --> 00:26:47,900 it's as if this is a wall and this were a single spring, k, 408 00:26:47,900 --> 00:26:51,130 that's connected to a little mass. 409 00:26:51,130 --> 00:26:54,265 And it got displaced by delta x. 410 00:26:56,890 --> 00:27:02,690 So what will happen is that this mass will experience a spring 411 00:27:02,690 --> 00:27:09,700 force, which is F equal to minus k delta x. 412 00:27:09,700 --> 00:27:12,860 Therefore, we can immediately identify 413 00:27:12,860 --> 00:27:17,803 omega b squared will be equal to k over m. 414 00:27:20,380 --> 00:27:27,350 So you can see that we have identified two kinds of modes, 415 00:27:27,350 --> 00:27:30,760 which every part of the system is oscillating 416 00:27:30,760 --> 00:27:36,040 at the same frequency and the same phase. 417 00:27:36,040 --> 00:27:38,100 Everybody agree? 418 00:27:38,100 --> 00:27:41,440 No not everybody agree. 419 00:27:41,440 --> 00:27:45,870 Look at this guy this guy is not moving. 420 00:27:45,870 --> 00:27:46,830 How could this be? 421 00:27:46,830 --> 00:27:51,340 This is not the normal mode. 422 00:27:51,340 --> 00:27:51,840 Isn't it? 423 00:27:54,640 --> 00:27:55,320 OK. 424 00:27:55,320 --> 00:27:57,510 I hope that will wake you up a bit. 425 00:28:00,030 --> 00:28:02,220 I can be very tricky here. 426 00:28:02,220 --> 00:28:08,100 I can say that this mass is also oscillating, 427 00:28:08,100 --> 00:28:12,065 but with what amplitude? 428 00:28:12,065 --> 00:28:12,690 AUDIENCE: Zero. 429 00:28:12,690 --> 00:28:13,856 YEN-JIE LEE: Zero amplitude. 430 00:28:13,856 --> 00:28:18,870 Right So the conclusion is that, aha, everybody is actually 431 00:28:18,870 --> 00:28:21,250 oscillating at the same frequency, 432 00:28:21,250 --> 00:28:23,380 but these guy with zero amplitude. 433 00:28:25,545 --> 00:28:27,920 AUDIENCE: Are they oscillating at the same phase as well? 434 00:28:27,920 --> 00:28:29,130 YEN-JIE LEE: Yeah. 435 00:28:29,130 --> 00:28:32,400 Oh very good question. 436 00:28:32,400 --> 00:28:34,770 Another objection I receive. 437 00:28:34,770 --> 00:28:38,130 So life is hard for me today. 438 00:28:38,130 --> 00:28:39,300 Hey. 439 00:28:39,300 --> 00:28:43,530 This guy is oscillating out of phase. 440 00:28:43,530 --> 00:28:45,660 These two guys are out of phase. 441 00:28:45,660 --> 00:28:50,310 But I can argue that the amplitude of the first mass 442 00:28:50,310 --> 00:28:56,760 is actually has a minus sign compared to the second mass. 443 00:28:56,760 --> 00:29:00,750 Then they are again in phase. 444 00:29:03,970 --> 00:29:05,440 So very good. 445 00:29:05,440 --> 00:29:07,540 I like those questions. 446 00:29:07,540 --> 00:29:13,220 And I hope I have convinced you that everybody is oscillating, 447 00:29:13,220 --> 00:29:17,000 although you cannot see it, because the amplitude is small, 448 00:29:17,000 --> 00:29:18,130 is zero. 449 00:29:18,130 --> 00:29:23,990 And they are all oscillating at the same phase. 450 00:29:23,990 --> 00:29:24,872 Yes. 451 00:29:24,872 --> 00:29:26,689 AUDIENCE: How come there's only one mass? 452 00:29:26,689 --> 00:29:28,230 YEN-JIE LEE: Oh, the right hand side? 453 00:29:28,230 --> 00:29:28,874 AUDIENCE: Yeah. 454 00:29:28,874 --> 00:29:29,790 YEN-JIE LEE: Oh, yeah. 455 00:29:29,790 --> 00:29:33,780 That is because the left hand side mass, the 2m one, 456 00:29:33,780 --> 00:29:35,490 is actually not moving. 457 00:29:35,490 --> 00:29:37,980 Because they are two spring forces, 458 00:29:37,980 --> 00:29:39,800 one is actually pushing the mass, 459 00:29:39,800 --> 00:29:42,330 the other one's pulling the mass. 460 00:29:42,330 --> 00:29:45,620 And they cancel perfectly. 461 00:29:45,620 --> 00:29:52,140 Therefore, it's as if those two guys are not-- 462 00:29:52,140 --> 00:29:54,480 they don't find each other. 463 00:29:54,480 --> 00:29:57,510 And then it's like, they are just tools 464 00:29:57,510 --> 00:30:01,890 mass connected to a wall along. 465 00:30:01,890 --> 00:30:05,950 And then you can now identify what is the frequency. 466 00:30:05,950 --> 00:30:06,450 OK. 467 00:30:06,450 --> 00:30:08,760 Very good. 468 00:30:08,760 --> 00:30:15,630 So we make the made a lot of the progress from the discussion. 469 00:30:15,630 --> 00:30:21,220 And now I would like to ask you for help. 470 00:30:21,220 --> 00:30:24,797 What is the third oscillation? 471 00:30:24,797 --> 00:30:25,731 Yes? 472 00:30:25,731 --> 00:30:28,540 AUDIENCE: There's no third normal mode. 473 00:30:28,540 --> 00:30:30,365 YEN-JIE LEE: There's no third normal mode. 474 00:30:30,365 --> 00:30:31,948 AUDIENCE: There's no third normal mode 475 00:30:31,948 --> 00:30:34,906 because there are restricted to one dimension. 476 00:30:34,906 --> 00:30:37,371 I can not imagine another mode that would not 477 00:30:37,371 --> 00:30:39,965 displace the central mass. 478 00:30:39,965 --> 00:30:41,130 YEN-JIE LEE: Very good. 479 00:30:41,130 --> 00:30:42,680 That's very good. 480 00:30:42,680 --> 00:30:46,100 On the other hand, you can also say, 481 00:30:46,100 --> 00:30:48,960 I also take the center mass motion 482 00:30:48,960 --> 00:30:50,410 as one of the normal mode. 483 00:30:50,410 --> 00:30:54,260 I think that's also fair to do that. 484 00:30:54,260 --> 00:30:55,640 Very good observation. 485 00:30:55,640 --> 00:31:00,770 You can see that the whole can move simultaneously. 486 00:31:06,120 --> 00:31:10,400 I can also argue that they are oscillating 487 00:31:10,400 --> 00:31:12,710 at the same frequency and the same phase, 488 00:31:12,710 --> 00:31:15,200 because they are all moving together. 489 00:31:21,560 --> 00:31:25,630 So these are the 2m connected to mass one. 490 00:31:31,160 --> 00:31:35,690 All of them are moving in the same direction. 491 00:31:35,690 --> 00:31:38,350 So now I can calculate the force. 492 00:31:38,350 --> 00:31:40,100 What is the force? 493 00:31:40,100 --> 00:31:43,050 F is 0. 494 00:31:43,050 --> 00:31:49,490 Therefore, omega c is 0. 495 00:31:49,490 --> 00:31:53,630 So you can the small limit of omega. 496 00:31:53,630 --> 00:31:59,120 So of course, I can pretend that those mass are connected 497 00:31:59,120 --> 00:32:05,000 to a really, really small spring to the wall with is a spring 498 00:32:05,000 --> 00:32:06,350 constant k'. 499 00:32:06,350 --> 00:32:08,600 And I have k' goes to zero. 500 00:32:08,600 --> 00:32:11,450 And they are actually going to oscillate 501 00:32:11,450 --> 00:32:16,880 with omega c goes to zero. 502 00:32:16,880 --> 00:32:20,640 So in this case, the amplitude is 503 00:32:20,640 --> 00:32:27,240 going to increase forever, because you have A sin omega c 504 00:32:27,240 --> 00:32:28,890 t. 505 00:32:28,890 --> 00:32:33,260 And this roughly A omega c t. 506 00:32:33,260 --> 00:32:37,620 And this is just vt. 507 00:32:37,620 --> 00:32:41,970 So what I want to argue is that this is actually 508 00:32:41,970 --> 00:32:47,640 also oscillation, but with angular frequency zero. 509 00:32:47,640 --> 00:32:58,470 And the general motion can be in written as vt times c, 510 00:32:58,470 --> 00:33:02,240 for example, some constant. 511 00:33:02,240 --> 00:33:05,770 Any questions? 512 00:33:05,770 --> 00:33:09,310 So what I'm going to do next may amaze you. 513 00:33:11,890 --> 00:33:12,550 Very good. 514 00:33:12,550 --> 00:33:18,790 So we have identified three different kinds of modes. 515 00:33:18,790 --> 00:33:30,301 We have mode A, which is with omega a squared equal to omega 516 00:33:30,301 --> 00:33:30,800 a squared. 517 00:33:30,800 --> 00:33:32,740 Where is omega a squared. 518 00:33:32,740 --> 00:33:33,610 There. 519 00:33:33,610 --> 00:33:36,410 It's 4k over 2m. 520 00:33:36,410 --> 00:33:40,870 And also, the motion is like this. 521 00:33:40,870 --> 00:33:51,720 x1 equal t A cosine omega a t plus phi a. 522 00:33:51,720 --> 00:34:00,420 x2 is equal to minus A, because they have different sine. 523 00:34:00,420 --> 00:34:05,550 So if the motion is in the left hand side direction, 524 00:34:05,550 --> 00:34:07,530 then the two masses are oscillating 525 00:34:07,530 --> 00:34:09,420 in the opposite direction. 526 00:34:09,420 --> 00:34:14,719 So therefore, I get a minus sign in front of A. Cosine omega 527 00:34:14,719 --> 00:34:17,310 a t plus phi a. 528 00:34:19,889 --> 00:34:28,590 x3 will be also equal to minus A cosine omega a t plus phi a. 529 00:34:28,590 --> 00:34:32,730 Of course, I need to define what this x1, x2, x3. 530 00:34:32,730 --> 00:34:37,770 That's why most of you got super confused. 531 00:34:37,770 --> 00:34:43,690 So the x1, what I mean is that is 532 00:34:43,690 --> 00:34:50,719 that the displacement of the mass 2m, I call it x1. 533 00:34:50,719 --> 00:34:56,590 The displacement of the upper mass, the upper little mass, 534 00:34:56,590 --> 00:34:59,110 I call it x2. 535 00:34:59,110 --> 00:35:05,410 And finally, the displacement of the third mass, I call it x3. 536 00:35:05,410 --> 00:35:09,160 Therefore, you can see that mode A, 537 00:35:09,160 --> 00:35:12,340 you have this kind of motion. 538 00:35:12,340 --> 00:35:17,500 The amplitude of the first mass is A. Therefore, 539 00:35:17,500 --> 00:35:21,320 if I define that to be A, then the second and third one, 540 00:35:21,320 --> 00:35:24,220 or the amplitude will be defined as minus A. 541 00:35:24,220 --> 00:35:27,640 And you can see that all of them are oscillating 542 00:35:27,640 --> 00:35:31,090 at fixed angular frequency, omega 543 00:35:31,090 --> 00:35:37,156 a, omega a, omega a; and also fixed phase, phi a, phi a, phi 544 00:35:37,156 --> 00:35:37,655 a. 545 00:35:42,260 --> 00:35:44,570 Of course, we can also write down 546 00:35:44,570 --> 00:35:49,210 what we get for mode B. For mode B, 547 00:35:49,210 --> 00:35:53,780 the left hand side mass is not moving, stay put. 548 00:35:53,780 --> 00:35:56,930 And the other two masses are oscillating 549 00:35:56,930 --> 00:36:00,470 at the frequency of omega b. 550 00:36:00,470 --> 00:36:06,680 And amplitude, they differ by a minus sign. 551 00:36:06,680 --> 00:36:08,580 OK. 552 00:36:08,580 --> 00:36:12,820 Omega b squared is equal to k over m 553 00:36:12,820 --> 00:36:18,320 from that logical argument. 554 00:36:18,320 --> 00:36:26,495 And then we get x1 equal to 0 times cosine omega 555 00:36:26,495 --> 00:36:31,020 b t plus phi b. 556 00:36:31,020 --> 00:36:40,460 x2, I get B cosine omega b t plus phi b. 557 00:36:40,460 --> 00:36:49,329 x3, I get minus B cosine omega b t plus phi b. 558 00:36:49,329 --> 00:36:50,120 Any questions here? 559 00:36:53,120 --> 00:36:58,480 Finally, mode C. All the mass, x1 560 00:36:58,480 --> 00:37:06,460 is equal to x2 is equal to x3, is equal to C plus vt. 561 00:37:10,450 --> 00:37:17,470 So you can see that we have identified three modes, mode 562 00:37:17,470 --> 00:37:25,660 A, mode B, and the mode C. And there are three angular 563 00:37:25,660 --> 00:37:31,840 frequencies which we identified for all of those normal modes, 564 00:37:31,840 --> 00:37:35,950 omega a, omega b, and omega c. 565 00:37:35,950 --> 00:37:38,380 And you can see that we also identified 566 00:37:38,380 --> 00:37:40,920 how many free parameters. 567 00:37:40,920 --> 00:37:51,515 One free parameter, two, three, four, five, and six. 568 00:37:54,470 --> 00:37:58,220 If you careful, you write down the equation 569 00:37:58,220 --> 00:38:04,850 of motion of this system, you will have three 570 00:38:04,850 --> 00:38:08,900 coupled differential equations. 571 00:38:08,900 --> 00:38:12,470 And those are second order differential equations. 572 00:38:12,470 --> 00:38:17,570 If you have three variables, three second order differential 573 00:38:17,570 --> 00:38:19,310 equations. 574 00:38:19,310 --> 00:38:23,660 If you manage it magically, with the help from a computer 575 00:38:23,660 --> 00:38:30,800 or from math department people, how many free parameter 576 00:38:30,800 --> 00:38:35,420 would you expect in you a general solution? 577 00:38:35,420 --> 00:38:38,680 Can anybody tell me well how many? 578 00:38:38,680 --> 00:38:44,131 I have three second order differential equations. 579 00:38:44,131 --> 00:38:44,630 Yes? 580 00:38:44,630 --> 00:38:45,670 AUDIENCE: 6? 581 00:38:45,670 --> 00:38:48,180 YEN-JIE LEE: 6. 582 00:38:48,180 --> 00:38:53,220 So look at what we have done we identified already 583 00:38:53,220 --> 00:38:56,190 1, 2, 3, three normal modes. 584 00:38:56,190 --> 00:39:03,030 By there are 1, 2, 3, 4, 5 6, six free parameters. 585 00:39:03,030 --> 00:39:05,985 That tells me I am done. 586 00:39:09,640 --> 00:39:10,280 I'm done. 587 00:39:10,280 --> 00:39:15,140 Because what is the general solution? 588 00:39:15,140 --> 00:39:20,960 The general solution is just a superposition of mode A mode B 589 00:39:20,960 --> 00:39:27,620 and mode C. You have six free meters to be determined 590 00:39:27,620 --> 00:39:31,790 by six initial conditions, which I would like-- 591 00:39:31,790 --> 00:39:36,650 I have to tell you what are those initial conditions. 592 00:39:36,650 --> 00:39:40,730 So isn't this amazing to you? 593 00:39:40,730 --> 00:39:44,390 I didn't even solve the differential equation, 594 00:39:44,390 --> 00:39:48,050 and I already get the solution. 595 00:39:48,050 --> 00:39:50,540 And you can see another lesson we learned 596 00:39:50,540 --> 00:39:53,750 from here is that, oh no, you can 597 00:39:53,750 --> 00:39:56,180 imagine that the motion of the system 598 00:39:56,180 --> 00:39:58,310 can be super complicated. 599 00:39:58,310 --> 00:40:02,180 This whole thing can do this, all the crazy things 600 00:40:02,180 --> 00:40:04,190 are all displaced, and the center of mass 601 00:40:04,190 --> 00:40:06,860 can move, as you said. 602 00:40:06,860 --> 00:40:12,830 But the result is actually very easy to understand. 603 00:40:12,830 --> 00:40:17,360 It's just three kinds of motion, the displacement, and two kinds 604 00:40:17,360 --> 00:40:21,410 of simple harmonic motion. 605 00:40:21,410 --> 00:40:22,760 We add them together. 606 00:40:22,760 --> 00:40:27,590 And then you get the general description of that system. 607 00:40:27,590 --> 00:40:31,010 So everything is so nice. 608 00:40:31,010 --> 00:40:34,690 We understand the motion of that system. 609 00:40:34,690 --> 00:40:41,180 But in general, life is very hard. 610 00:40:41,180 --> 00:40:46,010 For example, now I do something crazy here. 611 00:40:46,010 --> 00:40:51,070 I change this to 3. 612 00:40:51,070 --> 00:40:53,350 What are the normal modes? 613 00:40:53,350 --> 00:40:55,550 Can anybody tell me? 614 00:40:55,550 --> 00:40:59,050 It becomes very, very difficult, because there's 615 00:40:59,050 --> 00:41:03,370 no general symmetry of that kind of system. 616 00:41:03,370 --> 00:41:05,890 So we are in trouble. 617 00:41:05,890 --> 00:41:09,250 One of the modes maybe still there, 618 00:41:09,250 --> 00:41:13,770 which is actually mode B. But the other modes 619 00:41:13,770 --> 00:41:16,900 are harder to actually guess. 620 00:41:16,900 --> 00:41:22,790 So you can see that that already brings you a lot of trouble. 621 00:41:22,790 --> 00:41:29,330 And you can see that I can now couple not just two objects, 622 00:41:29,330 --> 00:41:32,750 I can couple three objects, four objects, five objects, 623 00:41:32,750 --> 00:41:33,920 10 objects. 624 00:41:33,920 --> 00:41:37,020 Maybe I will put that in your p set and see what happens. 625 00:41:37,020 --> 00:41:39,980 And you can see that this becomes 626 00:41:39,980 --> 00:41:42,590 very difficult to manage. 627 00:41:42,590 --> 00:41:47,030 So what I'm going to do in the rest of this lecture 628 00:41:47,030 --> 00:41:49,760 is to introduce you a method which 629 00:41:49,760 --> 00:41:54,800 you can follow in general to solve the question 630 00:41:54,800 --> 00:41:59,120 and get the normal mode frequencies and normal modes. 631 00:41:59,120 --> 00:42:01,950 So we will take a four minute break. 632 00:42:01,950 --> 00:42:04,940 And we come back at 12:20. 633 00:42:04,940 --> 00:42:07,160 So if you have any questions, let me know. 634 00:42:12,790 --> 00:42:17,380 What we are going to do in the following exercise 635 00:42:17,380 --> 00:42:22,750 is to try to understand a general strategy to solve 636 00:42:22,750 --> 00:42:26,980 the normal mode frequencies and the normal mode amplitudes, 637 00:42:26,980 --> 00:42:30,640 so that you can apply this technique to all kinds 638 00:42:30,640 --> 00:42:32,320 of different systems. 639 00:42:32,320 --> 00:42:36,590 So what I am going to do today is to take these three mass 640 00:42:36,590 --> 00:42:41,595 system, and of course as usual, I try to define what 641 00:42:41,595 --> 00:42:43,600 is this coordinate system? 642 00:42:43,600 --> 00:42:45,520 The coordinate system I'm going to use 643 00:42:45,520 --> 00:42:51,790 is x1 and x2 an x3 describing the displacement of the mass 644 00:42:51,790 --> 00:42:53,750 from the equilibrium position. 645 00:42:53,750 --> 00:42:55,420 And the equilibrium means that there's 646 00:42:55,420 --> 00:42:57,970 no stretch on the spring. 647 00:42:57,970 --> 00:43:00,310 The string is unstretched. 648 00:43:00,310 --> 00:43:05,780 It's at their own natural length, l0. 649 00:43:05,780 --> 00:43:12,910 So once I define that, I can do a force diagram analysis. 650 00:43:12,910 --> 00:43:17,140 So that starts from the left hand side mass 651 00:43:17,140 --> 00:43:19,870 with mass equal to 2m. 652 00:43:19,870 --> 00:43:23,350 I can write down the equation of motion, 2m x1 double dot. 653 00:43:26,290 --> 00:43:37,730 And this is going to be equal to k x2 minus x1 plus k x3 654 00:43:37,730 --> 00:43:38,230 minus x1. 655 00:43:40,870 --> 00:43:45,850 So there are two spring forces acting on this mass, 656 00:43:45,850 --> 00:43:47,140 the left hand side mass. 657 00:43:47,140 --> 00:43:49,990 The first one is the upper spring. 658 00:43:49,990 --> 00:43:55,720 The second one is coming from the lower one. 659 00:43:55,720 --> 00:43:58,340 And you can see that both of them 660 00:43:58,340 --> 00:44:00,940 are proportional to spring constant k, 661 00:44:00,940 --> 00:44:05,650 and also proportional to the relative displacement. 662 00:44:05,650 --> 00:44:09,610 And you can see that the two relative displacement, which 663 00:44:09,610 --> 00:44:14,130 is the amount of stretch to the spring, 664 00:44:14,130 --> 00:44:20,540 is actually x2 minus x1, and the x3 minus x1. 665 00:44:20,540 --> 00:44:22,720 Am I going too fast? 666 00:44:22,720 --> 00:44:23,220 OK. 667 00:44:23,220 --> 00:44:24,410 Everybody's following. 668 00:44:24,410 --> 00:44:26,420 And you can actually check the sign. 669 00:44:26,420 --> 00:44:27,620 So you may not be sure. 670 00:44:27,620 --> 00:44:30,470 Maybe this is your x1 minus x2. 671 00:44:30,470 --> 00:44:33,200 But you can check that, because if you increase 672 00:44:33,200 --> 00:44:36,200 x1, what is going to happen? 673 00:44:36,200 --> 00:44:38,780 This term will become more negative. 674 00:44:38,780 --> 00:44:42,820 More negative in this coordinate system 675 00:44:42,820 --> 00:44:45,410 is pointing to the left hand side. 676 00:44:45,410 --> 00:44:46,670 So that makes sense. 677 00:44:46,670 --> 00:44:49,890 Because if I move this mass to the right side, 678 00:44:49,890 --> 00:44:53,380 then I am compressing the springs. 679 00:44:53,380 --> 00:44:55,520 Therefore, they are pushing it back. 680 00:44:55,520 --> 00:45:02,210 Therefore, this is actually the correct sign, x2 minus x1. 681 00:45:02,210 --> 00:45:07,130 The same thing also applies to the second spring force. 682 00:45:07,130 --> 00:45:10,655 So that's a way I double check if I make a mistake. 683 00:45:13,220 --> 00:45:16,450 Now, this is actually the first equations of motion. 684 00:45:16,450 --> 00:45:20,300 And I can now also work on a second mass. 685 00:45:20,300 --> 00:45:23,540 Now I focus on a mass number two. 686 00:45:23,540 --> 00:45:26,020 The displacement is x2. 687 00:45:26,020 --> 00:45:28,760 Therefore the left hand side of Newton's Law 688 00:45:28,760 --> 00:45:32,780 is m x2 double dot. 689 00:45:32,780 --> 00:45:36,920 And that is equal to the spring force. 690 00:45:36,920 --> 00:45:42,650 The spring force, there's only one spring force 691 00:45:42,650 --> 00:45:44,510 acting on the mass. 692 00:45:44,510 --> 00:45:48,790 Therefore, what I am going to get is k x1 minus x2. 693 00:45:53,850 --> 00:45:55,060 Everybody's following? 694 00:45:57,830 --> 00:46:02,730 You can actually check the sign carefully, also. 695 00:46:02,730 --> 00:46:05,780 And finally, I have the third mass, very similar 696 00:46:05,780 --> 00:46:07,850 to mass number two. 697 00:46:07,850 --> 00:46:10,380 I can write down the equation of motion, 698 00:46:10,380 --> 00:46:16,120 which should k x1 minus x3. 699 00:46:16,120 --> 00:46:21,740 So that is my coupled second order differential equations. 700 00:46:21,740 --> 00:46:23,060 There are three equations. 701 00:46:23,060 --> 00:46:27,260 And all of them are second order equations. 702 00:46:27,260 --> 00:46:30,050 So this looks a bit messy. 703 00:46:30,050 --> 00:46:32,810 So what I'm going to do Is no magic. 704 00:46:32,810 --> 00:46:37,550 I'm just collecting all the terms belonging to x1, 705 00:46:37,550 --> 00:46:41,480 and put them together, all the terms belonging to x2, 706 00:46:41,480 --> 00:46:45,470 putting all together, and just rearranging things. 707 00:46:45,470 --> 00:46:47,300 So no magic. 708 00:46:47,300 --> 00:46:50,340 So I copied this thing, left hand side. 709 00:46:50,340 --> 00:46:52,790 2m x1 double dot. 710 00:46:52,790 --> 00:47:00,260 And the dot will be equal to minus 2k x1 plus, 711 00:47:00,260 --> 00:47:05,930 I collect all the times related to x2, plus k x2, 712 00:47:05,930 --> 00:47:13,070 there's only one term here, then plus k x3. 713 00:47:13,070 --> 00:47:17,065 I'm just trying to organize my question. 714 00:47:17,065 --> 00:47:19,190 So you can see that I collect all the terms related 715 00:47:19,190 --> 00:47:21,800 to x1 to here. 716 00:47:21,800 --> 00:47:24,560 Minus k, minus k, I get minus 2k. 717 00:47:24,560 --> 00:47:28,850 And the plus k for x2, plus k for the x3. 718 00:47:28,850 --> 00:47:32,480 And I can also do that for m x2 double dot. 719 00:47:32,480 --> 00:47:41,930 That will be equal to k x1 minus k x2 plus zero x3, 720 00:47:41,930 --> 00:47:44,900 just for completeness. 721 00:47:44,900 --> 00:47:50,207 I can also do the same thing for the third mass, m x3 double 722 00:47:50,207 --> 00:47:50,940 dot. 723 00:47:50,940 --> 00:47:58,920 This is equal to k x1 plus 0 x2. 724 00:47:58,920 --> 00:48:04,880 There's no dependence on x2, because x1 and x2-- 725 00:48:04,880 --> 00:48:09,440 x3 and x2 are not talking to each either directly. 726 00:48:09,440 --> 00:48:13,880 Finally, I have the third, which is minus k x3. 727 00:48:19,720 --> 00:48:27,400 Now our job is to solve those coupled equations. 728 00:48:27,400 --> 00:48:29,720 Of course, you have the freedom, if you 729 00:48:29,720 --> 00:48:33,130 know how to solve it yourself, you 730 00:48:33,130 --> 00:48:35,570 can already go ahead and solve it. 731 00:48:35,570 --> 00:48:37,540 But what I am going to do here is 732 00:48:37,540 --> 00:48:41,560 to introduce technique, which can be useful for you 733 00:48:41,560 --> 00:48:44,750 and make it easier to follow. 734 00:48:44,750 --> 00:48:47,020 It's a fixed procedure. 735 00:48:47,020 --> 00:48:49,880 So what I can do is the following. 736 00:48:49,880 --> 00:48:56,450 I can write everything in them form of a matrix. 737 00:48:56,450 --> 00:49:00,480 How many of you heard the matrix for the first time? 738 00:49:00,480 --> 00:49:03,550 1, 2, 3, 4. 739 00:49:03,550 --> 00:49:04,050 OK. 740 00:49:04,050 --> 00:49:05,470 Only four. 741 00:49:05,470 --> 00:49:09,490 But if you are not being familiar with matrix, 742 00:49:09,490 --> 00:49:11,160 let me know, and I can help you. 743 00:49:11,160 --> 00:49:12,370 Let the TA know. 744 00:49:12,370 --> 00:49:17,860 And also, there's a section in the textbooks, which 745 00:49:17,860 --> 00:49:21,790 I posted on announcement, which is actually very 746 00:49:21,790 --> 00:49:24,260 helpful to understand matrices. 747 00:49:24,260 --> 00:49:26,680 But sorry to these four students, 748 00:49:26,680 --> 00:49:28,000 we are going to use that. 749 00:49:28,000 --> 00:49:32,500 And maybe, you already learn how it works from here. 750 00:49:32,500 --> 00:49:36,280 So one trick which we will use in this class 751 00:49:36,280 --> 00:49:41,500 is to convert everything into matrix format. 752 00:49:41,500 --> 00:49:44,260 What I am going to do is to write everything 753 00:49:44,260 --> 00:49:49,270 in terms of M, capital X, capital M, 754 00:49:49,270 --> 00:49:56,620 capital X double dot equal to minus capital K capital X. 755 00:49:56,620 --> 00:50:01,450 Capital M, capital X, and capital K, 756 00:50:01,450 --> 00:50:05,800 those are all matrices. 757 00:50:05,800 --> 00:50:09,130 Because I write this thing I really carefully, 758 00:50:09,130 --> 00:50:12,640 therefore we can already immediately identify 759 00:50:12,640 --> 00:50:18,010 what would be M, capital M and capital X and a K. 760 00:50:18,010 --> 00:50:28,210 So I can write down immediately will be equal to 2m, 0, 0, 0, 761 00:50:28,210 --> 00:50:35,190 m, 0, 0, 0, m. 762 00:50:35,190 --> 00:50:39,750 Because there's only one in each line, you only have one term. 763 00:50:39,750 --> 00:50:44,250 X1 double dot, x2 double dot, x3 double dot. 764 00:50:44,250 --> 00:50:46,290 And also, you can write down what 765 00:50:46,290 --> 00:50:51,030 will be the X. This is actually a vector. 766 00:50:51,030 --> 00:50:56,040 X will be equal to x1, x2, x3. 767 00:50:59,680 --> 00:51:02,310 Finally, you have the K? 768 00:51:02,310 --> 00:51:05,860 How do I read off K? 769 00:51:05,860 --> 00:51:08,010 Careful, there's a minus sign here, 770 00:51:08,010 --> 00:51:13,510 because I would like to make this matrix equation as if it's 771 00:51:13,510 --> 00:51:16,740 describing a simple harmonic motion of a one 772 00:51:16,740 --> 00:51:18,080 dimensional system. 773 00:51:18,080 --> 00:51:20,940 So it looks the same, but they are different because those 774 00:51:20,940 --> 00:51:22,810 are matrices. 775 00:51:22,810 --> 00:51:25,710 But therefore, I have in my convention 776 00:51:25,710 --> 00:51:27,780 I have this minus sign there. 777 00:51:27,780 --> 00:51:32,300 Therefore, when you read off the K, you have to be careful. 778 00:51:32,300 --> 00:51:35,124 So what is K? 779 00:51:35,124 --> 00:51:40,950 K is equal to 2k. 780 00:51:40,950 --> 00:51:44,190 You have the minus 2k here in front of x1. 781 00:51:44,190 --> 00:51:47,520 But because I have a minus sign there, therefore 782 00:51:47,520 --> 00:51:50,610 this one is actually taken out. 783 00:51:50,610 --> 00:51:53,580 So we have 2k there. 784 00:51:53,580 --> 00:52:04,570 Then you have minus k, minus k, minus k, k, 0. 785 00:52:04,570 --> 00:52:08,370 Minus k, plus k, 0. 786 00:52:08,370 --> 00:52:14,220 And finally, you can also finish the last row. 787 00:52:14,220 --> 00:52:17,618 You get minus k, 0, k. 788 00:52:21,960 --> 00:52:23,740 K becomes minus k. 789 00:52:23,740 --> 00:52:27,560 Minus k becomes k. 790 00:52:27,560 --> 00:52:32,930 So we have read off all those matrices successfully. 791 00:52:32,930 --> 00:52:38,060 So you may ask, what do they mean? 792 00:52:38,060 --> 00:52:40,040 Do they get the meaning? 793 00:52:40,040 --> 00:52:45,350 M, K, X, what those? 794 00:52:45,350 --> 00:52:52,370 M, capital M matrix, tells you the mass distribution 795 00:52:52,370 --> 00:52:54,270 inside the system. 796 00:52:54,270 --> 00:52:58,390 So that's the meaning of this matrix. 797 00:52:58,390 --> 00:53:02,180 X is actually vector, which tells 798 00:53:02,180 --> 00:53:08,800 the position of individual components in the system. 799 00:53:08,800 --> 00:53:11,140 Finally, what is K? 800 00:53:11,140 --> 00:53:16,890 K is telling you how each component in the system 801 00:53:16,890 --> 00:53:21,420 talks to the other components. 802 00:53:21,420 --> 00:53:25,470 So K is telling you the communication 803 00:53:25,470 --> 00:53:29,550 inside the system. 804 00:53:29,550 --> 00:53:33,180 So now we understand a bit what is going on. 805 00:53:33,180 --> 00:53:38,040 And as usual, I will go to the complex notation. 806 00:53:45,790 --> 00:53:53,310 So I have xj, the small xj are the position 807 00:53:53,310 --> 00:53:57,310 of the mass, x1, x2, and x3. 808 00:53:57,310 --> 00:54:04,540 xj will be real part of small zj. 809 00:54:04,540 --> 00:54:07,710 Small xj equal to real part of zj. 810 00:54:07,710 --> 00:54:14,370 Therefore, I can now write everything in terms of matrices 811 00:54:14,370 --> 00:54:15,150 again. 812 00:54:15,150 --> 00:54:20,708 So now I can write the solution to be Z, 813 00:54:20,708 --> 00:54:30,410 the capitol z is a matrix, exponential i omega t plus phi. 814 00:54:30,410 --> 00:54:33,320 This is the guess the solution I have. 815 00:54:33,320 --> 00:54:37,640 A1, A2, and A3. 816 00:54:37,640 --> 00:54:41,130 Those are the amplitudes, amplitude A 817 00:54:41,130 --> 00:54:45,620 of the first mass, amplitude of the second mass, amplitude 818 00:54:45,620 --> 00:54:52,020 of the third mass, in their normal mode. 819 00:54:52,020 --> 00:55:00,300 And all of those are oscillating at the same frequency, omega, 820 00:55:00,300 --> 00:55:03,720 and the same phase, phi. 821 00:55:03,720 --> 00:55:08,580 Does that tell you something which we learned before? 822 00:55:08,580 --> 00:55:12,150 Oh, that's the definition of the normal mode. 823 00:55:12,150 --> 00:55:15,150 I'm using the definition of the normal mode. 824 00:55:15,150 --> 00:55:17,580 Every part of the system is oscillating 825 00:55:17,580 --> 00:55:20,520 at the same frequency in the same phase. 826 00:55:20,520 --> 00:55:25,590 And we use that to construct my solution. 827 00:55:25,590 --> 00:55:29,940 The complex version is exponential i omega t plus phi, 828 00:55:29,940 --> 00:55:32,000 oscillating at the same frequency, 829 00:55:32,000 --> 00:55:34,680 oscillating at the same phase. 830 00:55:34,680 --> 00:55:37,730 And those are the amplitude, which I will solve later. 831 00:55:37,730 --> 00:55:41,280 OK, any questions? 832 00:55:41,280 --> 00:55:44,100 I hope I'm not going too fast. 833 00:55:44,100 --> 00:55:48,060 If everybody can follow, now I can 834 00:55:48,060 --> 00:55:54,540 go ahead and solve the equation in the matrix format. 835 00:55:54,540 --> 00:55:59,550 So now I go to the complex notation. 836 00:55:59,550 --> 00:56:06,510 So the equation M X double dot equal to minus KX 837 00:56:06,510 --> 00:56:12,810 becomes M Z double dot equal to minus KZ. 838 00:56:17,990 --> 00:56:22,530 And also, I can immediately get the Z double dot will 839 00:56:22,530 --> 00:56:27,920 be equal to minus omega squared Z, 840 00:56:27,920 --> 00:56:31,800 because each time I do a differentiation, 841 00:56:31,800 --> 00:56:37,470 I get i omega out of the exponential function. 842 00:56:37,470 --> 00:56:39,460 And I cannot kill that exponential function, 843 00:56:39,460 --> 00:56:40,950 so it's still there. 844 00:56:40,950 --> 00:56:43,440 Therefore, I get minus omega squared 845 00:56:43,440 --> 00:56:48,360 in front of Z. I hope that doesn't surprise you. 846 00:56:48,360 --> 00:56:50,880 So that's very nice and very good news. 847 00:56:50,880 --> 00:56:55,010 That means I can replace this Z double prime with minus omega 848 00:56:55,010 --> 00:57:01,050 squared Z. Then I get minus M omega squared Z. 849 00:57:01,050 --> 00:57:04,200 And this is equal to minus KZ. 850 00:57:04,200 --> 00:57:06,600 And I can cancel the minus sign. 851 00:57:06,600 --> 00:57:08,230 That becomes something like this. 852 00:57:12,930 --> 00:57:20,790 So, I can now cancel the exponential i omega t plus phi, 853 00:57:20,790 --> 00:57:23,430 because I have Z in the left hand side. 854 00:57:23,430 --> 00:57:28,270 And exponential i omega t plus phi is just a number. 855 00:57:28,270 --> 00:57:30,780 So therefore, I can cancel it. 856 00:57:30,780 --> 00:57:34,420 So what is going to happen if I do that? 857 00:57:34,420 --> 00:57:37,740 Basically, what I'm going get is I 858 00:57:37,740 --> 00:57:44,670 get M omega squared A equal to K A. 859 00:57:44,670 --> 00:57:48,330 I'm trying to go extremely slowly, because this 860 00:57:48,330 --> 00:57:52,200 is the first time we go through matrices. 861 00:57:52,200 --> 00:57:55,020 So now you have this expression. 862 00:57:55,020 --> 00:58:00,410 Left hand is a matrix, M, times some constant, omega squared. 863 00:58:00,410 --> 00:58:02,910 I can actually get omega squared in front of it, 864 00:58:02,910 --> 00:58:06,060 because this is actually just a number. 865 00:58:06,060 --> 00:58:12,480 A is just a vector, which is A1, A2, A3, also a matrix. 866 00:58:12,480 --> 00:58:17,340 K is actually how the individual components talks to the others. 867 00:58:17,340 --> 00:58:20,490 So that's there, times A. 868 00:58:20,490 --> 00:58:23,910 Now I would like to move everything 869 00:58:23,910 --> 00:58:26,640 to the right hand side, all the matrices in front A 870 00:58:26,640 --> 00:58:27,840 to the right hand side. 871 00:58:27,840 --> 00:58:32,760 Then I multiply both sides by M minus 1. 872 00:58:32,760 --> 00:58:39,100 So I multiply M minus 1 to the whole equation. 873 00:58:39,100 --> 00:58:43,110 M minus 1, what is M minus 1? 874 00:58:43,110 --> 00:58:46,710 The definition is that the inverse of M 875 00:58:46,710 --> 00:58:48,940 is called M minus 1. 876 00:58:48,940 --> 00:58:57,240 M minus 1 times M is equal to I, which is actually 1, 1, 1. 877 00:58:57,240 --> 00:59:02,670 Therefore, if I do this thing, then I would get omega squared. 878 00:59:02,670 --> 00:59:09,970 M minus 1 times M becomes I, unit matrix. 879 00:59:09,970 --> 00:59:17,280 And this is equal to M minus 1 K A. And be careful, 880 00:59:17,280 --> 00:59:21,140 I multiply M minus 1, the inverse of M, 881 00:59:21,140 --> 00:59:23,060 from the left hand side. 882 00:59:23,060 --> 00:59:25,800 That matters. 883 00:59:25,800 --> 00:59:30,390 So now I can move everything to the same side. 884 00:59:30,390 --> 00:59:33,970 I moved the left hand side term to the right hand side. 885 00:59:33,970 --> 00:59:40,670 Therefore, I get M minus 1 K minus omega squared 886 00:59:40,670 --> 00:59:46,430 I. Those are all matrices. 887 00:59:46,430 --> 00:59:49,330 Times A, this is equal to 0. 888 00:59:54,400 --> 00:59:55,370 Any questions? 889 01:00:00,430 --> 01:00:02,950 So a lot of manipulation. 890 01:00:02,950 --> 01:00:07,880 But if you think about it, and you are following me, 891 01:00:07,880 --> 01:00:11,840 you'll see that all those steps are exactly 892 01:00:11,840 --> 01:00:17,140 identical to what we have been doing for a single harmonic 893 01:00:17,140 --> 01:00:18,440 oscillator. 894 01:00:18,440 --> 01:00:20,000 Looks pretty familiar to you. 895 01:00:20,000 --> 01:00:24,708 But the difference is that now we are dealing with matrices. 896 01:00:24,708 --> 01:00:25,910 AUDIENCE: What is A? 897 01:00:25,910 --> 01:00:30,870 YEN-JIE LEE: Oh, A. A is actually this guy. 898 01:00:30,870 --> 01:00:33,620 I define this to be A. And that means 899 01:00:33,620 --> 01:00:40,930 Z will be exponential i omega t plus phi times A. I didn't 900 01:00:40,930 --> 01:00:42,360 actually write it explicitly. 901 01:00:42,360 --> 01:00:43,910 But that's what I mean. 902 01:00:46,500 --> 01:00:49,412 Any more questions? 903 01:00:49,412 --> 01:00:50,861 Yes? 904 01:00:50,861 --> 01:00:55,605 AUDIENCE: [INAUDIBLE] is for [INAUDIBLE]?? 905 01:00:55,605 --> 01:00:56,980 YEN-JIE LEE: Can you repeat that? 906 01:00:56,980 --> 01:01:01,230 AUDIENCE: So this whole process, this is mode A, right? 907 01:01:01,230 --> 01:01:01,980 YEN-JIE LEE: Yeah. 908 01:01:01,980 --> 01:01:05,970 So this whole process is for, not really the for mode 909 01:01:05,970 --> 01:01:09,390 A. So that A may be confusing. 910 01:01:09,390 --> 01:01:12,080 But in general, if I have a solution, 911 01:01:12,080 --> 01:01:16,830 and I assume that the amplitude can be described by a matrix. 912 01:01:16,830 --> 01:01:17,820 So it's in general. 913 01:01:17,820 --> 01:01:19,630 And you'll see that we can actually 914 01:01:19,630 --> 01:01:24,690 derive the angular frequency of mode A, mode B, and mode C 915 01:01:24,690 --> 01:01:26,290 afterward. 916 01:01:26,290 --> 01:01:28,020 I hope that answers your question. 917 01:01:28,020 --> 01:01:31,350 So you see that for in general, what I have been doing 918 01:01:31,350 --> 01:01:34,320 is that now, all those things are 919 01:01:34,320 --> 01:01:39,520 equivalent to the original equation of motion. 920 01:01:39,520 --> 01:01:44,780 What I am doing is purely cosmetic. 921 01:01:44,780 --> 01:01:47,810 You see, make it beautiful. 922 01:01:47,810 --> 01:01:52,850 So all those things, this thing is exactly the equivalent 923 01:01:52,850 --> 01:01:56,190 to that thing, up there. 924 01:01:56,190 --> 01:01:59,300 Up to M X double dot equal to minus K x. 925 01:01:59,300 --> 01:02:01,140 Cosmetics. 926 01:02:01,140 --> 01:02:02,560 Beautiful. 927 01:02:02,560 --> 01:02:06,570 Looks-- I like it. 928 01:02:06,570 --> 01:02:07,650 All right. 929 01:02:07,650 --> 01:02:10,170 Then what I have been doing is that now 930 01:02:10,170 --> 01:02:15,210 I introduce using a definition of normal mode. 931 01:02:15,210 --> 01:02:20,190 I guess the solution will have this functional form. 932 01:02:20,190 --> 01:02:25,300 Z equals to exponential i omega t plus phi, everybody 933 01:02:25,300 --> 01:02:28,830 oscillating at the same frequency, the same phase. 934 01:02:28,830 --> 01:02:32,010 Frequency omega, phase phi. 935 01:02:32,010 --> 01:02:35,130 And everybody can have different amplitude. 936 01:02:35,130 --> 01:02:39,240 You can see from this example, normal modes, 937 01:02:39,240 --> 01:02:42,710 they can have different amplitude. 938 01:02:42,710 --> 01:02:44,490 The amplitude is what? 939 01:02:44,490 --> 01:02:45,900 I don't know yet. 940 01:02:45,900 --> 01:02:48,150 But we will figure it out. 941 01:02:48,150 --> 01:02:50,520 Then that's my assumption. 942 01:02:50,520 --> 01:02:52,510 The definition of normal mode. 943 01:02:52,510 --> 01:02:56,220 And I plug in to the equation of motion. 944 01:02:56,220 --> 01:02:59,790 Then this is what we are doing to simplify 945 01:02:59,790 --> 01:03:01,350 the equation of motion. 946 01:03:01,350 --> 01:03:03,540 There's no magic here. 947 01:03:03,540 --> 01:03:09,670 If I plug in the definition on normal mode to that equation, 948 01:03:09,670 --> 01:03:15,460 this is actually going to bring you to this equation, matrix 949 01:03:15,460 --> 01:03:18,210 equation. 950 01:03:18,210 --> 01:03:24,000 So if you have learned matrices before, you have something, 951 01:03:24,000 --> 01:03:30,930 some matrix, times Z. This is equal to zero. 952 01:03:30,930 --> 01:03:33,660 A is not zero. 953 01:03:33,660 --> 01:03:34,600 I hope. 954 01:03:34,600 --> 01:03:37,780 If it's zero, then the whole system is not moving. 955 01:03:37,780 --> 01:03:39,780 Then it's not fun. 956 01:03:42,520 --> 01:03:47,040 So if A is not zero, then this thing should be-- 957 01:03:47,040 --> 01:03:52,290 this thing times A should make this equation equal to 0. 958 01:03:52,290 --> 01:03:56,980 So what is actually the required condition? 959 01:03:56,980 --> 01:03:58,080 I get stuck, 960 01:03:58,080 --> 01:04:01,440 and of course again, my friend from math department 961 01:04:01,440 --> 01:04:04,080 comes to save me. 962 01:04:04,080 --> 01:04:10,930 That means if this thing has a solution, 963 01:04:10,930 --> 01:04:12,640 this equation has a solution, that 964 01:04:12,640 --> 01:04:23,090 means that determinant of M minus 1 K minus omega 965 01:04:23,090 --> 01:04:29,750 squared I has to be equal to 0. 966 01:04:29,750 --> 01:04:34,790 So that is the condition for this equation 967 01:04:34,790 --> 01:04:41,030 to satisfy this to be equal to 0. 968 01:04:41,030 --> 01:04:45,320 And just to make sure that I don't know what is the angular 969 01:04:45,320 --> 01:04:46,820 frequency omega yet. 970 01:04:46,820 --> 01:04:49,990 I don't know what is the phi yet. 971 01:04:49,990 --> 01:04:55,730 We can actually solve the angular frequency, omega. 972 01:04:55,730 --> 01:05:01,070 So now, turn everything around. 973 01:05:01,070 --> 01:05:05,060 And basically now, using this normal mode 974 01:05:05,060 --> 01:05:11,290 definition, and some mathematical manipulation, 975 01:05:11,290 --> 01:05:16,440 the condition we need for this equation to satisfy equal to 0, 976 01:05:16,440 --> 01:05:22,190 is determinant M minus 1 K minus omega squared I. 977 01:05:22,190 --> 01:05:25,930 I can write down M minus 1 K minus omega 978 01:05:25,930 --> 01:05:32,870 squared I explicitly, just to help you with mathematics. 979 01:05:36,910 --> 01:05:50,500 M minus 1 K is equal to 1 over 2m, 0, 0, 0, 1/m, 0, 0, 0, 1/m. 980 01:05:50,500 --> 01:05:56,230 It's just the inverse matrix of the M matrix. 981 01:05:56,230 --> 01:06:07,090 Therefore, now I can write down the explicit expression 982 01:06:07,090 --> 01:06:10,350 of M minus 1 K minus omega squared 983 01:06:10,350 --> 01:06:23,730 I. This will be equal to k over m minus omega squared, 984 01:06:23,730 --> 01:06:34,370 minus k over 2m, minus k over 2, minus k over m, 985 01:06:34,370 --> 01:06:40,030 k over m minus omega squared, 0. 986 01:06:40,030 --> 01:06:42,600 I will write down all the elements first. 987 01:06:42,600 --> 01:06:46,392 Then I will explain to you how I arrived at the expression. 988 01:06:52,200 --> 01:06:53,150 OK. 989 01:06:53,150 --> 01:06:55,180 So this is M minus 1 k. 990 01:06:55,180 --> 01:06:58,790 The definition of M minus 1 is that. 991 01:06:58,790 --> 01:07:02,280 And the definition of K is in the upper right 992 01:07:02,280 --> 01:07:04,980 corner of the black board. 993 01:07:04,980 --> 01:07:08,620 Therefore, if you multiply M minus 1 K, 994 01:07:08,620 --> 01:07:14,130 basically, the first column will get-- 995 01:07:18,240 --> 01:07:18,850 wait a second. 996 01:07:18,850 --> 01:07:20,100 Did I make a mistake? 997 01:07:24,670 --> 01:07:25,170 No. 998 01:07:25,170 --> 01:07:26,130 OK. 999 01:07:26,130 --> 01:07:32,440 So basically, what you arrive at is k/m, k/m, k/m. 1000 01:07:36,430 --> 01:07:42,760 And also, the minus k over 2m for the rest part 1001 01:07:42,760 --> 01:07:44,440 of the matrix. 1002 01:07:44,440 --> 01:07:48,235 And the minus omega squared I will give you 1003 01:07:48,235 --> 01:07:51,145 the diagonal component. 1004 01:07:51,145 --> 01:07:52,227 Yes? 1005 01:07:52,227 --> 01:07:54,268 AUDIENCE: Why do you have to take the determinant 1006 01:07:54,268 --> 01:07:56,287 and set it equal to 0 instead of just 1007 01:07:56,287 --> 01:07:59,230 setting that equal to zero? 1008 01:07:59,230 --> 01:08:02,740 AUDIENCE: This is a matrix. 1009 01:08:02,740 --> 01:08:04,150 So these are the matrix. 1010 01:08:04,150 --> 01:08:08,020 So a matrix times A will be equal to zero. 1011 01:08:08,020 --> 01:08:12,670 The general condition for that to be satisfied 1012 01:08:12,670 --> 01:08:14,440 is more general. 1013 01:08:14,440 --> 01:08:19,359 It's actually the determinant of this matrix equal to zero. 1014 01:08:19,359 --> 01:08:27,790 Because this is actually multiplied by some back to A. 1015 01:08:27,790 --> 01:08:31,430 So I think there are mathematical manipulation. 1016 01:08:31,430 --> 01:08:33,270 Basically, you would just collect the terms. 1017 01:08:33,270 --> 01:08:37,140 And then calculate M minus 1 K first. 1018 01:08:37,140 --> 01:08:40,930 And the minus omega squared I will give you all the diagonal 1019 01:08:40,930 --> 01:08:44,899 and terms have a minus omega square there. 1020 01:08:44,899 --> 01:08:48,540 And that is actually the matrix. 1021 01:08:48,540 --> 01:08:52,290 And of course, I can calculate the determinant. 1022 01:08:52,290 --> 01:08:54,479 So if I calculate the determinant, 1023 01:08:54,479 --> 01:08:58,319 then basically I get this times that times that. 1024 01:08:58,319 --> 01:09:07,950 So what you get is k over m minus omega squared times k 1025 01:09:07,950 --> 01:09:15,430 over m minus omega squared times k over m minus omega squared. 1026 01:09:15,430 --> 01:09:17,510 So these are all diagonal terms. 1027 01:09:17,510 --> 01:09:26,340 And the minus 1 over 2 k squared over m squared, k 1028 01:09:26,340 --> 01:09:32,879 squared over m squared. 1029 01:09:35,770 --> 01:09:36,790 sorry. 1030 01:09:36,790 --> 01:09:40,600 Minus omega squared. 1031 01:09:40,600 --> 01:09:46,649 So that's this off diagonal term, this times this times 1032 01:09:46,649 --> 01:09:47,310 that. 1033 01:09:47,310 --> 01:09:48,130 OK. 1034 01:09:48,130 --> 01:09:50,050 It will give you the second term. 1035 01:09:50,050 --> 01:09:52,390 And the third one, which survived 1036 01:09:52,390 --> 01:09:56,320 because of those zeros, many, many terms are equal to 0. 1037 01:09:56,320 --> 01:10:01,590 And then the third term, which is nonzero, is again minus 1 1038 01:10:01,590 --> 01:10:09,790 over 2k squared over m squared, k over m minus omega squared. 1039 01:10:09,790 --> 01:10:13,270 And this is actually equal to 0, because the determinant 1040 01:10:13,270 --> 01:10:17,380 of this matrix is equal to zero. 1041 01:10:17,380 --> 01:10:19,320 Everybody following? 1042 01:10:19,320 --> 01:10:20,320 A little bit of a mess. 1043 01:10:20,320 --> 01:10:23,760 Because I have been doing something very challenging. 1044 01:10:23,760 --> 01:10:32,410 I'm solving a 3 by 3 matrix problem in front of you right. 1045 01:10:32,410 --> 01:10:35,600 So the math can get a bit complicated. 1046 01:10:35,600 --> 01:10:40,135 But next time, I think we are going to go to a second order 1047 01:10:40,135 --> 01:10:42,070 one, 2 by 2 matrix. 1048 01:10:42,070 --> 01:10:45,040 And I think that will be slightly easier. 1049 01:10:45,040 --> 01:10:47,650 But the general approach is the same. 1050 01:10:47,650 --> 01:10:52,480 So basically, you calculate M minus 1 K minus omega squared. 1051 01:10:52,480 --> 01:10:57,820 Then you get what is inside, all the content inside this matrix. 1052 01:10:57,820 --> 01:11:00,640 Then you would calculate the determinant. 1053 01:11:00,640 --> 01:11:05,800 And basically, you can solve this equation. 1054 01:11:05,800 --> 01:11:11,250 Now I can define omega0 squared to be k/m. 1055 01:11:11,250 --> 01:11:15,210 And I can actually make this expression much simpler. 1056 01:11:18,840 --> 01:11:20,970 Then basically, what you are getting 1057 01:11:20,970 --> 01:11:25,660 is omega0 squared minus omega squared 1058 01:11:25,660 --> 01:11:33,480 to the third minus 1/2 omega0 to the fourth, omega0 squared 1059 01:11:33,480 --> 01:11:35,650 minus omega squared. 1060 01:11:35,650 --> 01:11:42,930 Minus 1/2 omega0 to the fourth, omega0 to the square, 1061 01:11:42,930 --> 01:11:43,980 minus omega squared. 1062 01:11:43,980 --> 01:11:47,340 And this is equal to 0. 1063 01:11:47,340 --> 01:11:51,684 And you can factor out the common components. 1064 01:11:51,684 --> 01:11:53,100 Then basically, what you are going 1065 01:11:53,100 --> 01:11:56,370 to get is, you can write this thing 1066 01:11:56,370 --> 01:12:03,270 to be omega0 squared minus omega squared, omega squared. 1067 01:12:03,270 --> 01:12:06,390 Because all of them have omega squared. 1068 01:12:06,390 --> 01:12:12,430 And omega squared minus 2 omega0 squared. 1069 01:12:12,430 --> 01:12:14,770 And that's equal to 0. 1070 01:12:14,770 --> 01:12:18,520 So I am skipping a lot of steps from this one to that one. 1071 01:12:18,520 --> 01:12:22,690 But in general, you can solve this third order equation. 1072 01:12:22,690 --> 01:12:29,530 And I can first combine all those terms together. 1073 01:12:29,530 --> 01:12:33,050 And then I factor out the common components. 1074 01:12:33,050 --> 01:12:35,500 Then basically, what you are going to arrive at 1075 01:12:35,500 --> 01:12:38,560 is something like this. 1076 01:12:38,560 --> 01:12:39,890 A lot of math here. 1077 01:12:39,890 --> 01:12:43,240 But we are close to the end. 1078 01:12:43,240 --> 01:12:48,440 So you can see now what are the possible solutions for omega. 1079 01:12:48,440 --> 01:12:53,860 That is the omega, unknown angular frequency 1080 01:12:53,860 --> 01:12:56,380 we are trying to figure out. 1081 01:12:56,380 --> 01:13:00,460 You can see that there are three possible omegas that can 1082 01:13:00,460 --> 01:13:03,220 make this equation equal to 0. 1083 01:13:03,220 --> 01:13:12,000 The first one is omega equal to omega0. 1084 01:13:12,000 --> 01:13:19,140 The second one is square root of omega 0, 1085 01:13:19,140 --> 01:13:21,450 coming from this expression, that omega 1086 01:13:21,450 --> 01:13:24,360 squared minus 2 omega zero squared. 1087 01:13:24,360 --> 01:13:26,280 If omega equal to square root 2 omega0, 1088 01:13:26,280 --> 01:13:28,110 this will be equal to zero. 1089 01:13:28,110 --> 01:13:31,180 And that will give you the whole expression equal to 0. 1090 01:13:31,180 --> 01:13:33,670 Then finally, I take this term. 1091 01:13:33,670 --> 01:13:36,030 And then you will get zero. 1092 01:13:36,030 --> 01:13:40,170 Omega squared, if omega is equal to 0, 1093 01:13:40,170 --> 01:13:43,770 then the whole expression is 0. 1094 01:13:43,770 --> 01:13:50,130 I have defined omega0 squared to be equal to k over m. 1095 01:13:50,130 --> 01:13:54,060 Therefore, I can conclude that omega squared 1096 01:13:54,060 --> 01:14:04,002 is equal to k over m, 2k over m, and 0. 1097 01:14:06,960 --> 01:14:10,390 Look at what we have done, a lot of mathematics. 1098 01:14:10,390 --> 01:14:15,240 But in the end, after you solve the eigenvalue problem, 1099 01:14:15,240 --> 01:14:18,720 or the determinant equal to zero problem, 1100 01:14:18,720 --> 01:14:20,940 you arrive at that there are only 1101 01:14:20,940 --> 01:14:25,740 three possible values of omega which 1102 01:14:25,740 --> 01:14:28,620 can make the determinant of M minus 1 1103 01:14:28,620 --> 01:14:32,600 K minus omega squared I equal to 0. 1104 01:14:32,600 --> 01:14:35,550 What are the three? 1105 01:14:35,550 --> 01:14:38,710 k/m, 2k/m, and 0. 1106 01:14:42,260 --> 01:14:45,150 If you look at this value, then we'll say, 1107 01:14:45,150 --> 01:14:49,800 this is essentially what we actually argued before, right? 1108 01:14:49,800 --> 01:14:55,000 Omega A squared is equal to 4k over 2m is 2k over m. 1109 01:14:55,000 --> 01:14:55,930 Wow. 1110 01:14:55,930 --> 01:14:56,560 We got it. 1111 01:15:00,320 --> 01:15:03,210 The second one is, we think about really 1112 01:15:03,210 --> 01:15:04,930 keep a straight question in my head 1113 01:15:04,930 --> 01:15:07,460 and understand this system. 1114 01:15:07,460 --> 01:15:12,260 The second identified normal more is having omega 1115 01:15:12,260 --> 01:15:14,090 squared be equal to k/m. 1116 01:15:14,090 --> 01:15:18,650 I got this also here magically, after all those magics. 1117 01:15:18,650 --> 01:15:24,960 And finally, the third one, the math also knows physics. 1118 01:15:24,960 --> 01:15:29,700 It also predicted that this is one mode which have oscillation 1119 01:15:29,700 --> 01:15:32,430 frequency of 0. 1120 01:15:32,430 --> 01:15:34,924 Isn't that amazing to you? 1121 01:15:39,690 --> 01:15:43,900 But that also gives us a sense of safety. 1122 01:15:43,900 --> 01:15:48,320 Because I can now add 10 pendulums, or 10 1123 01:15:48,320 --> 01:15:51,080 coupled system to your homework, and you 1124 01:15:51,080 --> 01:15:52,210 will be able to solve it. 1125 01:15:55,350 --> 01:15:56,790 So very good. 1126 01:15:56,790 --> 01:16:00,530 This example seems to be complicated. 1127 01:16:00,530 --> 01:16:03,570 But the what I want to say, I have one minute left, 1128 01:16:03,570 --> 01:16:05,790 is that what we have been doing is 1129 01:16:05,790 --> 01:16:11,370 to write the equation of motion based on force diagram. 1130 01:16:11,370 --> 01:16:15,660 Then I convert that to matrix format, 1131 01:16:15,660 --> 01:16:19,090 and X double dot equal to minus KX. 1132 01:16:19,090 --> 01:16:21,270 Then I follow the whole procedure, 1133 01:16:21,270 --> 01:16:24,110 solve the eigenvalue problem. 1134 01:16:24,110 --> 01:16:30,060 Then I will be able to figure out what are the possible omega 1135 01:16:30,060 --> 01:16:37,555 values which can satisfy this eigenvalue problem 1136 01:16:37,555 --> 01:16:38,920 or this determinant. 1137 01:16:38,920 --> 01:16:42,630 M minus 1 K minus omega squared I equal to 0 problem. 1138 01:16:42,630 --> 01:16:46,130 And after solving all those, you will 1139 01:16:46,130 --> 01:16:51,000 be able to solve the corresponding so-called normal 1140 01:16:51,000 --> 01:16:52,530 mode frequencies. 1141 01:16:52,530 --> 01:16:53,760 You can solve it. 1142 01:16:53,760 --> 01:16:57,700 And of course, you can plug those normal mode frequencies 1143 01:16:57,700 --> 01:17:01,170 back in, then you will be able to drive 1144 01:17:01,170 --> 01:17:05,430 the relative amplitude, A1, A2, and A3. 1145 01:17:05,430 --> 01:17:07,890 So what we have we learned today? 1146 01:17:07,890 --> 01:17:11,670 We have learned how to predict the motion 1147 01:17:11,670 --> 01:17:14,460 of coupled oscillators. 1148 01:17:14,460 --> 01:17:15,900 That's really cool. 1149 01:17:15,900 --> 01:17:18,540 And then next time, we are going to learn 1150 01:17:18,540 --> 01:17:22,050 a special kind of motion in coupled oscillators, which 1151 01:17:22,050 --> 01:17:23,850 is the big phenomena. 1152 01:17:23,850 --> 01:17:26,710 And also, what will happen if I start to drive 1153 01:17:26,710 --> 01:17:28,110 the coupled oscillators? 1154 01:17:28,110 --> 01:17:31,830 So I will be here if you have any questions 1155 01:17:31,830 --> 01:17:34,130 about the lecture. 1156 01:17:34,130 --> 01:17:36,080 Thank you very much.