WEBVTT 00:00:02.090 --> 00:00:04.430 The following content is provided under a Creative 00:00:04.430 --> 00:00:05.850 Commons license. 00:00:05.850 --> 00:00:08.060 Your support will help MIT OpenCourseWare 00:00:08.060 --> 00:00:12.150 continue to offer high quality educational resources for free. 00:00:12.150 --> 00:00:14.690 To make a donation or to view additional materials 00:00:14.690 --> 00:00:18.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.620 --> 00:00:19.976 at ocw.mit.edu. 00:00:22.781 --> 00:00:25.270 YEN-JIE LEE: And welcome back, everybody, 00:00:25.270 --> 00:00:31.960 to this fun class, 8.03. 00:00:31.960 --> 00:00:35.150 Let's get started. 00:00:35.150 --> 00:00:37.960 So the first thing which we will do 00:00:37.960 --> 00:00:41.650 is to review a bit what we have learned last time. 00:00:41.650 --> 00:00:44.440 And then we'll go to the next level 00:00:44.440 --> 00:00:48.780 to study coupled oscillators. 00:00:48.780 --> 00:00:50.440 OK. 00:00:50.440 --> 00:00:55.550 Last time, we had learned a lot on damped driven oscillators. 00:00:55.550 --> 00:00:59.710 So as far as the course we've been going, 00:00:59.710 --> 00:01:04.510 actually, we only study a single object, 00:01:04.510 --> 00:01:09.040 and then we introduce more and more force 00:01:09.040 --> 00:01:10.560 acting on this object. 00:01:10.560 --> 00:01:14.710 We introduce damping force, we introduce 00:01:14.710 --> 00:01:17.300 a driving force last time. 00:01:17.300 --> 00:01:24.330 And we see that the system becomes 00:01:24.330 --> 00:01:28.720 more and more difficult to understand because of the added 00:01:28.720 --> 00:01:30.150 component. 00:01:30.150 --> 00:01:33.930 But after the class last time, I hope 00:01:33.930 --> 00:01:38.980 I convinced you that we can understand driven oscillators. 00:01:38.980 --> 00:01:42.920 And there are two very important things we learned last time. 00:01:42.920 --> 00:01:46.610 The first one is the transient behavior, 00:01:46.610 --> 00:01:48.970 which is actually a superposition 00:01:48.970 --> 00:01:53.560 of the homogeneous solution and the steady state solution. 00:01:53.560 --> 00:01:55.130 OK. 00:01:55.130 --> 00:01:59.060 One very good news is that if you are patient enough, 00:01:59.060 --> 00:02:03.530 you shake the system continuously, 00:02:03.530 --> 00:02:08.030 and if you wait long enough, then the homogeneous solution 00:02:08.030 --> 00:02:11.009 contribution goes away. 00:02:11.009 --> 00:02:16.200 And what is actually left over is the steady state solution, 00:02:16.200 --> 00:02:20.690 which is actually much simpler than what we saw beforehand. 00:02:20.690 --> 00:02:24.170 It's actually just harmonic oscillation 00:02:24.170 --> 00:02:26.600 at driving frequency. 00:02:26.600 --> 00:02:32.060 Also, I hope that we also have learned a very interesting 00:02:32.060 --> 00:02:34.190 phenomenon, which is resonance. 00:02:34.190 --> 00:02:39.710 When the driving frequency is close to the natural frequency 00:02:39.710 --> 00:02:46.280 of the system, then the system apparently likes it. 00:02:46.280 --> 00:02:51.350 Then it would respond with larger amplitude 00:02:51.350 --> 00:02:55.790 and oscillating up and down at driving frequency. 00:02:55.790 --> 00:02:59.510 So that, we call it resonance. 00:02:59.510 --> 00:03:01.250 This is the equation of motion, which 00:03:01.250 --> 00:03:03.140 we have learned last time. 00:03:03.140 --> 00:03:07.280 You can see is theta double dot plus Gamma theta dot, 00:03:07.280 --> 00:03:10.010 is a contribution from the drag force, 00:03:10.010 --> 00:03:14.780 and omega 0 squared theta is the contribution 00:03:14.780 --> 00:03:17.270 from the so-called spring force. 00:03:17.270 --> 00:03:23.810 And finally, that is equal to f0 cosine omega d t, the driving 00:03:23.810 --> 00:03:24.950 force. 00:03:24.950 --> 00:03:27.170 And as we mentioned in the beginning, 00:03:27.170 --> 00:03:34.110 if we prepare this system and under-damp the situation, 00:03:34.110 --> 00:03:38.820 then the full solution is a superposition 00:03:38.820 --> 00:03:41.130 of the steady state solution, which 00:03:41.130 --> 00:03:46.050 is the left-hand side, the red thing I'm pointing to, 00:03:46.050 --> 00:03:48.600 this steady state solution. 00:03:48.600 --> 00:03:54.390 There's no free parameter in the steady state solution. 00:03:54.390 --> 00:03:57.290 So A, the amplitude, is determined by omega d. 00:04:00.260 --> 00:04:03.820 Delta, which is the phase is also determined by omega d. 00:04:03.820 --> 00:04:06.940 There's no free parameter. 00:04:06.940 --> 00:04:08.180 OK. 00:04:08.180 --> 00:04:13.180 And in order to make the solution a full solution, 00:04:13.180 --> 00:04:16.630 we actually have to add in this homogeneous solution 00:04:16.630 --> 00:04:18.310 back into this again. 00:04:18.310 --> 00:04:21.339 And basically, you have B and alpha, 00:04:21.339 --> 00:04:23.320 those are the free parameters, which 00:04:23.320 --> 00:04:28.780 can be determined by the given initial conditions. 00:04:28.780 --> 00:04:29.590 OK. 00:04:29.590 --> 00:04:32.650 So if we go ahead and plot some of the examples 00:04:32.650 --> 00:04:36.910 as a function of time, so the y-axis 00:04:36.910 --> 00:04:38.860 is actually the amplitude. 00:04:38.860 --> 00:04:41.290 And the x-axis is time. 00:04:41.290 --> 00:04:43.570 And what is actually plotted here 00:04:43.570 --> 00:04:46.840 is a combination or the superposition 00:04:46.840 --> 00:04:51.640 of the steady state solution and the homogeneous solution. 00:04:51.640 --> 00:04:55.690 And you can see that the individual components are also 00:04:55.690 --> 00:04:57.610 shown in this slide. 00:04:57.610 --> 00:05:01.960 You can see the red thing oscillating up and down 00:05:01.960 --> 00:05:06.540 harmonically, is steady state solution contribution. 00:05:06.540 --> 00:05:10.870 And also, you have the blue curve, 00:05:10.870 --> 00:05:14.180 which is decaying away as a function of time. 00:05:14.180 --> 00:05:17.990 And you can see that if you add these two curves together, 00:05:17.990 --> 00:05:20.740 you get something rather complicated. 00:05:20.740 --> 00:05:25.720 You will get some kind of motion like, do, do, do, do. 00:05:25.720 --> 00:05:30.710 Then the homogeneous solution actually dies out. 00:05:30.710 --> 00:05:34.960 Then what is actually left over is just the steady state 00:05:34.960 --> 00:05:38.030 solution, harmonic oscillation. 00:05:38.030 --> 00:05:42.320 And in this case, omega d is actually 10 times larger 00:05:42.320 --> 00:05:45.290 than the natural frequency. 00:05:45.290 --> 00:05:48.890 And there's another example which is also very interesting. 00:05:48.890 --> 00:05:57.470 It's that if I make the omega d closer to omega 0-- 00:05:57.470 --> 00:06:00.740 OK, in this case it's actually omega d equal to 2 times omega 00:06:00.740 --> 00:06:04.070 0, then you can produce some kind of a motion, which 00:06:04.070 --> 00:06:04.910 is like this. 00:06:04.910 --> 00:06:06.920 So you have the oscillation. 00:06:06.920 --> 00:06:09.080 And they stayed there for a while, 00:06:09.080 --> 00:06:11.440 then goes back, and oscillates down, 00:06:11.440 --> 00:06:14.930 and stay there then goes back. 00:06:14.930 --> 00:06:17.300 OK, as you can see on there. 00:06:17.300 --> 00:06:22.250 The homogeneous solution part and steady state solution part 00:06:22.250 --> 00:06:27.170 work together and produce this kind of strange behavior. 00:06:27.170 --> 00:06:27.980 OK. 00:06:27.980 --> 00:06:30.860 And that's just another example. 00:06:30.860 --> 00:06:33.010 And if you wait long enough, again 00:06:33.010 --> 00:06:37.400 what is actually left over is the steady state solution. 00:06:37.400 --> 00:06:38.760 OK. 00:06:38.760 --> 00:06:41.440 So what are we going to do today? 00:06:41.440 --> 00:06:45.660 So today, we are going to investigate 00:06:45.660 --> 00:06:52.260 what will happen if we try to put together multiple objects 00:06:52.260 --> 00:06:55.170 and also allow them to talk to each other. 00:06:55.170 --> 00:06:55.770 OK. 00:06:55.770 --> 00:06:58.740 So if we have two objects, and they don't talk to each other, 00:06:58.740 --> 00:07:01.050 then they are still like a single object. 00:07:01.050 --> 00:07:04.800 They are still like simple harmonic motion on their own. 00:07:04.800 --> 00:07:08.130 But if you allow them to talk to each other, 00:07:08.130 --> 00:07:11.820 this is the so-called coupled oscillator, then 00:07:11.820 --> 00:07:15.130 interesting thing happen. 00:07:15.130 --> 00:07:23.250 So in general, coupled systems are super, super complicated. 00:07:23.250 --> 00:07:24.420 OK. 00:07:24.420 --> 00:07:27.660 So let me give you one example here. 00:07:27.660 --> 00:07:33.690 This is actually two pendulums that 00:07:33.690 --> 00:07:36.720 are a coupled to each other, they are actually 00:07:36.720 --> 00:07:39.950 connected to each other, one pendulum, 00:07:39.950 --> 00:07:42.270 the second one is here, OK. 00:07:42.270 --> 00:07:48.270 And for example, I can actually give it an initial velocity 00:07:48.270 --> 00:07:49.695 and see what is going to happen. 00:07:52.680 --> 00:07:56.130 You can see that the resulting motion-- 00:07:56.130 --> 00:07:57.150 OK. 00:07:57.150 --> 00:08:00.980 Remember, we are just talking about two pendulums that 00:08:00.980 --> 00:08:02.620 are connected to each other. 00:08:02.620 --> 00:08:08.530 The resulting motion is super complicated. 00:08:08.530 --> 00:08:11.200 This is one of my favorite demonstrations. 00:08:11.200 --> 00:08:17.291 You can actually stare at this machine the whole time. 00:08:17.291 --> 00:08:18.790 And you can see that, huh, sometimes 00:08:18.790 --> 00:08:20.370 it does this rotation. 00:08:20.370 --> 00:08:22.190 Sometimes it doesn't do that. 00:08:22.190 --> 00:08:25.750 And it's almost like a living creature. 00:08:28.840 --> 00:08:31.310 So we are going to solve this system. 00:08:31.310 --> 00:08:33.606 No, probably not, he knows. 00:08:33.606 --> 00:08:36.190 [LAUGHTER] 00:08:36.190 --> 00:08:41.080 But as I mentioned before, you can always write down 00:08:41.080 --> 00:08:43.409 the equation of motion. 00:08:43.409 --> 00:08:47.350 And you can solve it by computer. 00:08:47.350 --> 00:08:49.560 Maybe some of the course 6 people 00:08:49.560 --> 00:08:52.980 can actually try and write the program to solve this thing 00:08:52.980 --> 00:08:56.820 and to simulate what is going to happen. 00:08:56.820 --> 00:09:02.730 So let's take a look at this complicated motion again. 00:09:02.730 --> 00:09:06.480 So you can see that the good news is that there are only two 00:09:06.480 --> 00:09:07.470 objects. 00:09:07.470 --> 00:09:10.260 And you can see-- 00:09:10.260 --> 00:09:14.280 look at the green, sorry, the orange dot. 00:09:14.280 --> 00:09:21.460 The orange dot is always moving along a semi circle. 00:09:21.460 --> 00:09:27.680 But if you focus on the yellow dot, 00:09:27.680 --> 00:09:30.940 the yellow is doing all kinds of different things. 00:09:30.940 --> 00:09:35.640 It's very hard to predict what is going to happen. 00:09:35.640 --> 00:09:40.130 So what I want to say is, those are interesting examples 00:09:40.130 --> 00:09:41.960 of coupled systems. 00:09:41.960 --> 00:09:45.800 But they are actually far more complicated than what 00:09:45.800 --> 00:09:51.050 we thought, because they are not smooth oscillation 00:09:51.050 --> 00:09:54.260 around equilibrium position. 00:09:54.260 --> 00:09:56.920 So you can see that now if I stop this machine 00:09:56.920 --> 00:10:02.940 and just perturb it slightly, giving it a small angle 00:10:02.940 --> 00:10:08.370 displacement, then you can see that the motion is 00:10:08.370 --> 00:10:12.560 much more easier to understand. 00:10:12.560 --> 00:10:14.030 You see. 00:10:14.030 --> 00:10:17.530 You even get one of these questions in your p-set. 00:10:17.530 --> 00:10:18.795 OK, that's good news. 00:10:21.390 --> 00:10:25.920 So our job today is to understand 00:10:25.920 --> 00:10:30.150 what is going to happen to those coupled oscillators. 00:10:30.150 --> 00:10:34.560 Let me give you a few examples before we start 00:10:34.560 --> 00:10:38.130 to work on a specific question. 00:10:38.130 --> 00:10:43.170 The second example I would like to show you is a saw 00:10:43.170 --> 00:10:47.220 and you actually connect it to two - actually 00:10:47.220 --> 00:10:50.970 a ruler, a metal ruler, which is connected 00:10:50.970 --> 00:10:54.750 to two massive objects. 00:10:54.750 --> 00:11:00.090 Now I can actually give it the initial velocity 00:11:00.090 --> 00:11:02.470 and see what happens. 00:11:02.470 --> 00:11:04.560 And you can see that they do talk 00:11:04.560 --> 00:11:09.490 to each other through this ruler, this metal ruler. 00:11:09.490 --> 00:11:10.020 Can you see? 00:11:10.020 --> 00:11:11.820 I hope you can see. 00:11:11.820 --> 00:11:13.530 It's a bit small. 00:11:13.530 --> 00:11:16.630 But it's really interesting that you can see-- originally, 00:11:16.630 --> 00:11:20.440 I just introduced some displacement 00:11:20.440 --> 00:11:23.100 in the left-hand side mass. 00:11:23.100 --> 00:11:27.270 And the left-had side thing start to move or so 00:11:27.270 --> 00:11:30.260 after a while. 00:11:30.260 --> 00:11:36.690 There are two more examples which I would like to give you, 00:11:36.690 --> 00:11:38.920 introduce to you. 00:11:38.920 --> 00:11:42.390 There are two kinds of pendulums, 00:11:42.390 --> 00:11:44.380 which I prepared here. 00:11:44.380 --> 00:11:49.050 The first one is there are two pendulums that are connected 00:11:49.050 --> 00:11:52.530 to each other by a spring. 00:11:52.530 --> 00:11:57.850 And if I try to introduce displacement, 00:11:57.850 --> 00:12:03.340 I move both masses slightly and see what is going to happen. 00:12:03.340 --> 00:12:07.400 And we see that the motion is still complicated. 00:12:07.400 --> 00:12:12.250 Although, if you stare at one objects, it looks more 00:12:12.250 --> 00:12:16.870 like harmonic oscillation, but not quite. 00:12:16.870 --> 00:12:19.360 For example, this guy is slowing down, 00:12:19.360 --> 00:12:21.680 and this is actually moving faster. 00:12:21.680 --> 00:12:26.500 And now the right hand side guy is actually moving faster. 00:12:26.500 --> 00:12:29.590 Motion seems to be completed. 00:12:29.590 --> 00:12:32.810 Also, you can look at this one. 00:12:32.810 --> 00:12:34.900 Those are the two pendulums. 00:12:34.900 --> 00:12:39.670 They are connected to each other by this rod here. 00:12:39.670 --> 00:12:42.340 And of course, you can displace the mass 00:12:42.340 --> 00:12:44.964 from the equilibrium position. 00:12:44.964 --> 00:12:45.880 I'm not going to hit-- 00:12:45.880 --> 00:12:47.080 not hitting each other. 00:12:47.080 --> 00:12:49.730 So you can displace the masses from each other. 00:12:49.730 --> 00:12:54.040 And you can see that they do complicated things 00:12:54.040 --> 00:12:55.640 as a function the time. 00:12:55.640 --> 00:12:58.280 How are we going to understand this? 00:12:58.280 --> 00:13:02.890 And I hope that by the end of this lecture 00:13:02.890 --> 00:13:06.940 you are convinced that you can as you solve this really easy, 00:13:06.940 --> 00:13:10.390 following a fixed procedure. 00:13:10.390 --> 00:13:15.130 In those examples, we have two objects that 00:13:15.130 --> 00:13:17.120 are connected to each other. 00:13:17.120 --> 00:13:19.450 And therefore, they talk to each other 00:13:19.450 --> 00:13:23.800 and produce coupled motion. 00:13:23.800 --> 00:13:27.650 Those are a couple oscillator examples. 00:13:27.650 --> 00:13:31.000 There's another very interesting example, which 00:13:31.000 --> 00:13:36.370 is called Wilberforce pendulum. 00:13:36.370 --> 00:13:39.220 So this is actually a pendulum. 00:13:39.220 --> 00:13:42.820 You can rotate like this. 00:13:42.820 --> 00:13:44.900 And it can also move up and down. 00:13:44.900 --> 00:13:48.310 It's connected to a spring. 00:13:48.310 --> 00:13:51.200 The interesting thing is that if I just 00:13:51.200 --> 00:13:56.270 start with some rotation, you can 00:13:56.270 --> 00:14:01.910 see that it starts to also oscillate up and down. 00:14:01.910 --> 00:14:02.590 You see? 00:14:02.590 --> 00:14:08.330 So initially I just introduced a rotation. 00:14:08.330 --> 00:14:10.120 Now it's actually fully rotating. 00:14:10.120 --> 00:14:13.150 And now it starts to move up and down. 00:14:13.150 --> 00:14:18.600 And you can see that the energy stored in the pendulum 00:14:18.600 --> 00:14:27.260 is going back and forth between the gravitational potential, 00:14:27.260 --> 00:14:31.360 between the potential of the spring, 00:14:31.360 --> 00:14:38.920 and also between the kinetic energy of up and down motion 00:14:38.920 --> 00:14:40.320 and the rotation. 00:14:40.320 --> 00:14:46.840 They're actually doing all those transitions all the time. 00:14:46.840 --> 00:14:50.280 So you can see-- 00:14:50.280 --> 00:14:53.830 so initially it's just rotating. 00:14:53.830 --> 00:14:56.440 And then it starts to move up and down. 00:14:56.440 --> 00:14:58.840 And this one is also very similar. 00:14:58.840 --> 00:15:02.110 But now the mass is much more displaced. 00:15:02.110 --> 00:15:07.340 And if I try to rotate this system without introducing 00:15:07.340 --> 00:15:10.360 a horizontal direction displacement, 00:15:10.360 --> 00:15:17.170 it still does is up and down motion, like a simple spring 00:15:17.170 --> 00:15:18.620 mass system. 00:15:18.620 --> 00:15:23.481 So what causes this kind of motion? 00:15:23.481 --> 00:15:29.590 That is because when we move this pendulum up and down, 00:15:29.590 --> 00:15:34.180 we also slightly unwind the spring. 00:15:34.180 --> 00:15:39.070 That can generate some kind of torque to this mass 00:15:39.070 --> 00:15:44.900 and produce rotational behavior. 00:15:44.900 --> 00:15:47.755 And you can see that this is just 00:15:47.755 --> 00:15:49.480 involving one single object. 00:15:49.480 --> 00:15:53.650 But there's a coupling between the rotation 00:15:53.650 --> 00:15:56.890 and the horizontal direction motion. 00:15:56.890 --> 00:16:04.240 So that's also special kind of coupled oscillator. 00:16:04.240 --> 00:16:10.580 So after all this, before we get started, 00:16:10.580 --> 00:16:13.150 I would like to say that what we are going to do 00:16:13.150 --> 00:16:17.800 is to assume all those things are ideal, 00:16:17.800 --> 00:16:21.670 without them being forced, without a driving force. 00:16:21.670 --> 00:16:24.410 We may introduce that later in the class. 00:16:24.410 --> 00:16:28.870 But for simplicity, we'll just stay with this idea case, 00:16:28.870 --> 00:16:31.390 before the mass becomes super complicated 00:16:31.390 --> 00:16:34.430 to solve it in front of you. 00:16:34.430 --> 00:16:43.020 And also, we can see that all those complicated motion 00:16:43.020 --> 00:16:43.955 are just illusion. 00:16:46.590 --> 00:16:49.900 Actually, the reality is that all of those 00:16:49.900 --> 00:16:54.250 are just superposition of harmonic motions. 00:16:54.250 --> 00:16:57.680 You will see that by the end of this class. 00:16:57.680 --> 00:16:59.961 So that is really amazing. 00:16:59.961 --> 00:17:00.460 OK. 00:17:00.460 --> 00:17:02.870 Let's immediately get started. 00:17:02.870 --> 00:17:06.609 So let's take a look at this system 00:17:06.609 --> 00:17:08.980 together and see if we can actually 00:17:08.980 --> 00:17:12.670 figure out the motion of this system together. 00:17:12.670 --> 00:17:17.109 So I have a system with three little masses. 00:17:17.109 --> 00:17:20.609 So there are three little masses in this system. 00:17:20.609 --> 00:17:25.240 They are connected to each other by spring. 00:17:25.240 --> 00:17:29.590 Those springs are highly idealized, the springs. 00:17:29.590 --> 00:17:32.320 And they have spring constant k. 00:17:32.320 --> 00:17:35.950 And the natural length's l0. 00:17:35.950 --> 00:17:40.120 And they are placed on Earth. 00:17:40.120 --> 00:17:44.800 And I carefully design the lab so 00:17:44.800 --> 00:17:50.150 that there's no friction between the desk and all 00:17:50.150 --> 00:17:52.450 those little masses. 00:17:52.450 --> 00:17:56.370 So once you get started and look at this system, 00:17:56.370 --> 00:17:58.540 you can imagine that there can be 00:17:58.540 --> 00:18:02.860 all kinds of different complicated motions. 00:18:02.860 --> 00:18:06.950 You can actually, for example, just move this mass 00:18:06.950 --> 00:18:12.220 and put the other two on hold. 00:18:12.220 --> 00:18:14.770 And they can oscillate like crazy. 00:18:14.770 --> 00:18:20.540 They can do very similar kind of motion. 00:18:20.540 --> 00:18:23.950 There are many, many possibilities. 00:18:23.950 --> 00:18:30.100 But if you stare at this system long enough, 00:18:30.100 --> 00:18:35.560 you will be able to identify special kinds of motion which 00:18:35.560 --> 00:18:39.670 are easier to understand. 00:18:39.670 --> 00:18:42.490 So what I would like to introduce to you 00:18:42.490 --> 00:18:45.190 is a special kind of motion which 00:18:45.190 --> 00:18:51.720 you can identify from the symmetry of this system. 00:18:51.720 --> 00:18:53.930 That is your so-called normal mode. 00:19:01.310 --> 00:19:04.970 So what is a normal mode, a special kind of motion 00:19:04.970 --> 00:19:07.700 we are trying to identify? 00:19:07.700 --> 00:19:10.430 That is actually the kind of motion 00:19:10.430 --> 00:19:23.100 which every part of the system is 00:19:23.100 --> 00:19:40.975 oscillating at the same frequency and the same phase. 00:19:49.770 --> 00:19:52.700 So that is your so-called normal mode, 00:19:52.700 --> 00:19:56.420 and is a special kind of motion, which I would 00:19:56.420 --> 00:20:00.620 like you to identify with me. 00:20:00.620 --> 00:20:06.170 And we would later realize that those special kinds of motions, 00:20:06.170 --> 00:20:09.830 which are easier to understand, actually 00:20:09.830 --> 00:20:14.660 helps us to understand the general motion of the system. 00:20:14.660 --> 00:20:19.820 You will realize that the most general motion of the system 00:20:19.820 --> 00:20:26.860 is just a superposition of all the identified normal modes. 00:20:26.860 --> 00:20:28.580 And then we are done, because we have 00:20:28.580 --> 00:20:31.300 a general solution already. 00:20:31.300 --> 00:20:34.100 So that's very good news. 00:20:34.100 --> 00:20:38.020 That tells us that we can understand 00:20:38.020 --> 00:20:43.190 the system systematically, and step by step. 00:20:43.190 --> 00:20:47.710 And then we can write the general motion of the system 00:20:47.710 --> 00:20:51.700 as a superposition of all the normal modes. 00:20:51.700 --> 00:20:54.440 So let's get started. 00:20:54.440 --> 00:21:00.040 So can you guess what are the possible normal modes 00:21:00.040 --> 00:21:02.330 of this system? 00:21:02.330 --> 00:21:05.230 So that means each part of the system 00:21:05.230 --> 00:21:09.670 is oscillating at the same frequency and the same phase. 00:21:09.670 --> 00:21:15.970 Can anybody and any one of you guess what can happen, 00:21:15.970 --> 00:21:18.550 each part of this is an oscillating 00:21:18.550 --> 00:21:21.390 at the same frequency? 00:21:21.390 --> 00:21:22.845 Yeah? 00:21:22.845 --> 00:21:23.870 AUDIENCE: If the two masses on that side are displaced 00:21:23.870 --> 00:21:25.328 the same amount and then they're -- 00:21:29.150 --> 00:21:30.610 YEN-JIE LEE: Very good. 00:21:30.610 --> 00:21:36.460 So he was saying that now I displace the right hand 00:21:36.460 --> 00:21:40.600 side two masses all together by a fixed amount, and also 00:21:40.600 --> 00:21:47.000 the left hand side, right, by a fixed amount and then let go. 00:21:47.000 --> 00:21:49.011 So that's what you're saying, right? 00:21:49.011 --> 00:21:49.510 OK. 00:21:49.510 --> 00:21:56.500 So the first mode we have identified is like this. 00:21:56.500 --> 00:22:03.490 So you have left hand side mass displace by delta x. 00:22:03.490 --> 00:22:07.670 And the right hand side two masses 00:22:07.670 --> 00:22:09.965 are also displaced by delta x. 00:22:15.440 --> 00:22:20.590 So basically you hold this three little masses 00:22:20.590 --> 00:22:25.530 and stretch it by the same-- 00:22:25.530 --> 00:22:29.470 introduce the same amplitude to all those three little masses, 00:22:29.470 --> 00:22:31.000 and let go. 00:22:31.000 --> 00:22:34.830 So that is actually one possible mode. 00:22:34.830 --> 00:22:41.090 And if we do this, then basically 00:22:41.090 --> 00:22:43.930 what you are going to see is that this is actually 00:22:43.930 --> 00:22:50.130 roughly equal to this system. 00:22:52.920 --> 00:22:57.630 They're connected to each other by two springs. 00:22:57.630 --> 00:23:02.250 And the right hand side part of the system, both masses 00:23:02.250 --> 00:23:06.150 are oscillating at the same amplitude and the same phase. 00:23:06.150 --> 00:23:14.040 They look like as if they are just single mass with mass 00:23:14.040 --> 00:23:15.130 equal to 2m. 00:23:17.830 --> 00:23:22.490 And if you introduce a displacement of delta x, 00:23:22.490 --> 00:23:25.510 then what is going to happen is that if I 00:23:25.510 --> 00:23:31.210 take a look at the mass, left hand side mass, and the force 00:23:31.210 --> 00:23:34.300 acting on this mass, the force will 00:23:34.300 --> 00:23:42.580 be equal to minus 2k times 2 delta x, 00:23:42.580 --> 00:23:45.520 because that's the amount of stretch you 00:23:45.520 --> 00:23:48.970 introduce to the spring. 00:23:48.970 --> 00:23:52.765 And that will give you minus 4k delta x. 00:23:57.390 --> 00:24:00.300 And we have already solved this kind of problem 00:24:00.300 --> 00:24:02.310 in the first lecture. 00:24:02.310 --> 00:24:05.430 So therefore you can immediately identify 00:24:05.430 --> 00:24:10.690 omega, in this case, omega a squared will 00:24:10.690 --> 00:24:16.500 be equal to 4k divided by 2m. 00:24:16.500 --> 00:24:20.880 This is actually the effective spring constant, 00:24:20.880 --> 00:24:24.210 and this is actually the mass. 00:24:24.210 --> 00:24:31.440 So that is actually the frequency of mode A. 00:24:31.440 --> 00:24:38.020 Can you identify a second kind of motion which does that? 00:24:38.020 --> 00:24:39.990 So in this case, what is going to happen 00:24:39.990 --> 00:24:42.600 is that the three masses will-- 00:24:42.600 --> 00:24:45.120 OK, one, two, and three. 00:24:45.120 --> 00:24:50.560 The three masses will oscillate as a function of time 00:24:50.560 --> 00:25:01.610 like this with angular frequency of square root of 4k over 2m. 00:25:01.610 --> 00:25:05.586 What is actually a second possible motion? 00:25:05.586 --> 00:25:07.410 Yes? 00:25:07.410 --> 00:25:10.570 AUDIENCE: All masses being stretched [INAUDIBLE] 00:25:10.570 --> 00:25:11.641 YEN-JIE LEE: Compressed. 00:25:11.641 --> 00:25:14.082 AUDIENCE: Compressed the same-- 00:25:14.082 --> 00:25:15.040 YEN-JIE LEE: Very good. 00:25:18.100 --> 00:25:24.910 I'm very lucky that I'm in front of such a smart crowd today. 00:25:24.910 --> 00:25:28.330 And we have successfully identified 00:25:28.330 --> 00:25:34.060 the second mode, mode B. So what is going to happen 00:25:34.060 --> 00:25:41.470 is that the left hand side mass is not moving. 00:25:41.470 --> 00:25:49.400 And you compress the upper one slightly 00:25:49.400 --> 00:25:56.570 and you stretch the lower one, the lower little mass 00:25:56.570 --> 00:25:59.700 to the opposite direction. 00:25:59.700 --> 00:26:03.140 The displacement is delta x, and the displacement 00:26:03.140 --> 00:26:07.210 of the second mass is delta x. 00:26:07.210 --> 00:26:09.980 So what is going to happen? 00:26:09.980 --> 00:26:13.220 What is going to happen is that the left hand side 00:26:13.220 --> 00:26:19.400 mass will not move at all because the force, the spring 00:26:19.400 --> 00:26:25.490 force, acting on this mass is going to cancel. 00:26:25.490 --> 00:26:27.650 And apparently, these two little masses 00:26:27.650 --> 00:26:31.370 are going to be doing harmonic motion. 00:26:34.160 --> 00:26:38.136 Since this left hand side mass is not moving, 00:26:38.136 --> 00:26:47.900 it's as if this is a wall and this were a single spring, k, 00:26:47.900 --> 00:26:51.130 that's connected to a little mass. 00:26:51.130 --> 00:26:54.265 And it got displaced by delta x. 00:26:56.890 --> 00:27:02.690 So what will happen is that this mass will experience a spring 00:27:02.690 --> 00:27:09.700 force, which is F equal to minus k delta x. 00:27:09.700 --> 00:27:12.860 Therefore, we can immediately identify 00:27:12.860 --> 00:27:17.803 omega b squared will be equal to k over m. 00:27:20.380 --> 00:27:27.350 So you can see that we have identified two kinds of modes, 00:27:27.350 --> 00:27:30.760 which every part of the system is oscillating 00:27:30.760 --> 00:27:36.040 at the same frequency and the same phase. 00:27:36.040 --> 00:27:38.100 Everybody agree? 00:27:38.100 --> 00:27:41.440 No not everybody agree. 00:27:41.440 --> 00:27:45.870 Look at this guy this guy is not moving. 00:27:45.870 --> 00:27:46.830 How could this be? 00:27:46.830 --> 00:27:51.340 This is not the normal mode. 00:27:51.340 --> 00:27:51.840 Isn't it? 00:27:54.640 --> 00:27:55.320 OK. 00:27:55.320 --> 00:27:57.510 I hope that will wake you up a bit. 00:28:00.030 --> 00:28:02.220 I can be very tricky here. 00:28:02.220 --> 00:28:08.100 I can say that this mass is also oscillating, 00:28:08.100 --> 00:28:12.065 but with what amplitude? 00:28:12.065 --> 00:28:12.690 AUDIENCE: Zero. 00:28:12.690 --> 00:28:13.856 YEN-JIE LEE: Zero amplitude. 00:28:13.856 --> 00:28:18.870 Right So the conclusion is that, aha, everybody is actually 00:28:18.870 --> 00:28:21.250 oscillating at the same frequency, 00:28:21.250 --> 00:28:23.380 but these guy with zero amplitude. 00:28:25.545 --> 00:28:27.920 AUDIENCE: Are they oscillating at the same phase as well? 00:28:27.920 --> 00:28:29.130 YEN-JIE LEE: Yeah. 00:28:29.130 --> 00:28:32.400 Oh very good question. 00:28:32.400 --> 00:28:34.770 Another objection I receive. 00:28:34.770 --> 00:28:38.130 So life is hard for me today. 00:28:38.130 --> 00:28:39.300 Hey. 00:28:39.300 --> 00:28:43.530 This guy is oscillating out of phase. 00:28:43.530 --> 00:28:45.660 These two guys are out of phase. 00:28:45.660 --> 00:28:50.310 But I can argue that the amplitude of the first mass 00:28:50.310 --> 00:28:56.760 is actually has a minus sign compared to the second mass. 00:28:56.760 --> 00:29:00.750 Then they are again in phase. 00:29:03.970 --> 00:29:05.440 So very good. 00:29:05.440 --> 00:29:07.540 I like those questions. 00:29:07.540 --> 00:29:13.220 And I hope I have convinced you that everybody is oscillating, 00:29:13.220 --> 00:29:17.000 although you cannot see it, because the amplitude is small, 00:29:17.000 --> 00:29:18.130 is zero. 00:29:18.130 --> 00:29:23.990 And they are all oscillating at the same phase. 00:29:23.990 --> 00:29:24.872 Yes. 00:29:24.872 --> 00:29:26.689 AUDIENCE: How come there's only one mass? 00:29:26.689 --> 00:29:28.230 YEN-JIE LEE: Oh, the right hand side? 00:29:28.230 --> 00:29:28.874 AUDIENCE: Yeah. 00:29:28.874 --> 00:29:29.790 YEN-JIE LEE: Oh, yeah. 00:29:29.790 --> 00:29:33.780 That is because the left hand side mass, the 2m one, 00:29:33.780 --> 00:29:35.490 is actually not moving. 00:29:35.490 --> 00:29:37.980 Because they are two spring forces, 00:29:37.980 --> 00:29:39.800 one is actually pushing the mass, 00:29:39.800 --> 00:29:42.330 the other one's pulling the mass. 00:29:42.330 --> 00:29:45.620 And they cancel perfectly. 00:29:45.620 --> 00:29:52.140 Therefore, it's as if those two guys are not-- 00:29:52.140 --> 00:29:54.480 they don't find each other. 00:29:54.480 --> 00:29:57.510 And then it's like, they are just tools 00:29:57.510 --> 00:30:01.890 mass connected to a wall along. 00:30:01.890 --> 00:30:05.950 And then you can now identify what is the frequency. 00:30:05.950 --> 00:30:06.450 OK. 00:30:06.450 --> 00:30:08.760 Very good. 00:30:08.760 --> 00:30:15.630 So we make the made a lot of the progress from the discussion. 00:30:15.630 --> 00:30:21.220 And now I would like to ask you for help. 00:30:21.220 --> 00:30:24.797 What is the third oscillation? 00:30:24.797 --> 00:30:25.731 Yes? 00:30:25.731 --> 00:30:28.540 AUDIENCE: There's no third normal mode. 00:30:28.540 --> 00:30:30.365 YEN-JIE LEE: There's no third normal mode. 00:30:30.365 --> 00:30:31.948 AUDIENCE: There's no third normal mode 00:30:31.948 --> 00:30:34.906 because there are restricted to one dimension. 00:30:34.906 --> 00:30:37.371 I can not imagine another mode that would not 00:30:37.371 --> 00:30:39.965 displace the central mass. 00:30:39.965 --> 00:30:41.130 YEN-JIE LEE: Very good. 00:30:41.130 --> 00:30:42.680 That's very good. 00:30:42.680 --> 00:30:46.100 On the other hand, you can also say, 00:30:46.100 --> 00:30:48.960 I also take the center mass motion 00:30:48.960 --> 00:30:50.410 as one of the normal mode. 00:30:50.410 --> 00:30:54.260 I think that's also fair to do that. 00:30:54.260 --> 00:30:55.640 Very good observation. 00:30:55.640 --> 00:31:00.770 You can see that the whole can move simultaneously. 00:31:06.120 --> 00:31:10.400 I can also argue that they are oscillating 00:31:10.400 --> 00:31:12.710 at the same frequency and the same phase, 00:31:12.710 --> 00:31:15.200 because they are all moving together. 00:31:21.560 --> 00:31:25.630 So these are the 2m connected to mass one. 00:31:31.160 --> 00:31:35.690 All of them are moving in the same direction. 00:31:35.690 --> 00:31:38.350 So now I can calculate the force. 00:31:38.350 --> 00:31:40.100 What is the force? 00:31:40.100 --> 00:31:43.050 F is 0. 00:31:43.050 --> 00:31:49.490 Therefore, omega c is 0. 00:31:49.490 --> 00:31:53.630 So you can the small limit of omega. 00:31:53.630 --> 00:31:59.120 So of course, I can pretend that those mass are connected 00:31:59.120 --> 00:32:05.000 to a really, really small spring to the wall with is a spring 00:32:05.000 --> 00:32:06.350 constant k'. 00:32:06.350 --> 00:32:08.600 And I have k' goes to zero. 00:32:08.600 --> 00:32:11.450 And they are actually going to oscillate 00:32:11.450 --> 00:32:16.880 with omega c goes to zero. 00:32:16.880 --> 00:32:20.640 So in this case, the amplitude is 00:32:20.640 --> 00:32:27.240 going to increase forever, because you have A sin omega c 00:32:27.240 --> 00:32:28.890 t. 00:32:28.890 --> 00:32:33.260 And this roughly A omega c t. 00:32:33.260 --> 00:32:37.620 And this is just vt. 00:32:37.620 --> 00:32:41.970 So what I want to argue is that this is actually 00:32:41.970 --> 00:32:47.640 also oscillation, but with angular frequency zero. 00:32:47.640 --> 00:32:58.470 And the general motion can be in written as vt times c, 00:32:58.470 --> 00:33:02.240 for example, some constant. 00:33:02.240 --> 00:33:05.770 Any questions? 00:33:05.770 --> 00:33:09.310 So what I'm going to do next may amaze you. 00:33:11.890 --> 00:33:12.550 Very good. 00:33:12.550 --> 00:33:18.790 So we have identified three different kinds of modes. 00:33:18.790 --> 00:33:30.301 We have mode A, which is with omega a squared equal to omega 00:33:30.301 --> 00:33:30.800 a squared. 00:33:30.800 --> 00:33:32.740 Where is omega a squared. 00:33:32.740 --> 00:33:33.610 There. 00:33:33.610 --> 00:33:36.410 It's 4k over 2m. 00:33:36.410 --> 00:33:40.870 And also, the motion is like this. 00:33:40.870 --> 00:33:51.720 x1 equal t A cosine omega a t plus phi a. 00:33:51.720 --> 00:34:00.420 x2 is equal to minus A, because they have different sine. 00:34:00.420 --> 00:34:05.550 So if the motion is in the left hand side direction, 00:34:05.550 --> 00:34:07.530 then the two masses are oscillating 00:34:07.530 --> 00:34:09.420 in the opposite direction. 00:34:09.420 --> 00:34:14.719 So therefore, I get a minus sign in front of A. Cosine omega 00:34:14.719 --> 00:34:17.310 a t plus phi a. 00:34:19.889 --> 00:34:28.590 x3 will be also equal to minus A cosine omega a t plus phi a. 00:34:28.590 --> 00:34:32.730 Of course, I need to define what this x1, x2, x3. 00:34:32.730 --> 00:34:37.770 That's why most of you got super confused. 00:34:37.770 --> 00:34:43.690 So the x1, what I mean is that is 00:34:43.690 --> 00:34:50.719 that the displacement of the mass 2m, I call it x1. 00:34:50.719 --> 00:34:56.590 The displacement of the upper mass, the upper little mass, 00:34:56.590 --> 00:34:59.110 I call it x2. 00:34:59.110 --> 00:35:05.410 And finally, the displacement of the third mass, I call it x3. 00:35:05.410 --> 00:35:09.160 Therefore, you can see that mode A, 00:35:09.160 --> 00:35:12.340 you have this kind of motion. 00:35:12.340 --> 00:35:17.500 The amplitude of the first mass is A. Therefore, 00:35:17.500 --> 00:35:21.320 if I define that to be A, then the second and third one, 00:35:21.320 --> 00:35:24.220 or the amplitude will be defined as minus A. 00:35:24.220 --> 00:35:27.640 And you can see that all of them are oscillating 00:35:27.640 --> 00:35:31.090 at fixed angular frequency, omega 00:35:31.090 --> 00:35:37.156 a, omega a, omega a; and also fixed phase, phi a, phi a, phi 00:35:37.156 --> 00:35:37.655 a. 00:35:42.260 --> 00:35:44.570 Of course, we can also write down 00:35:44.570 --> 00:35:49.210 what we get for mode B. For mode B, 00:35:49.210 --> 00:35:53.780 the left hand side mass is not moving, stay put. 00:35:53.780 --> 00:35:56.930 And the other two masses are oscillating 00:35:56.930 --> 00:36:00.470 at the frequency of omega b. 00:36:00.470 --> 00:36:06.680 And amplitude, they differ by a minus sign. 00:36:06.680 --> 00:36:08.580 OK. 00:36:08.580 --> 00:36:12.820 Omega b squared is equal to k over m 00:36:12.820 --> 00:36:18.320 from that logical argument. 00:36:18.320 --> 00:36:26.495 And then we get x1 equal to 0 times cosine omega 00:36:26.495 --> 00:36:31.020 b t plus phi b. 00:36:31.020 --> 00:36:40.460 x2, I get B cosine omega b t plus phi b. 00:36:40.460 --> 00:36:49.329 x3, I get minus B cosine omega b t plus phi b. 00:36:49.329 --> 00:36:50.120 Any questions here? 00:36:53.120 --> 00:36:58.480 Finally, mode C. All the mass, x1 00:36:58.480 --> 00:37:06.460 is equal to x2 is equal to x3, is equal to C plus vt. 00:37:10.450 --> 00:37:17.470 So you can see that we have identified three modes, mode 00:37:17.470 --> 00:37:25.660 A, mode B, and the mode C. And there are three angular 00:37:25.660 --> 00:37:31.840 frequencies which we identified for all of those normal modes, 00:37:31.840 --> 00:37:35.950 omega a, omega b, and omega c. 00:37:35.950 --> 00:37:38.380 And you can see that we also identified 00:37:38.380 --> 00:37:40.920 how many free parameters. 00:37:40.920 --> 00:37:51.515 One free parameter, two, three, four, five, and six. 00:37:54.470 --> 00:37:58.220 If you careful, you write down the equation 00:37:58.220 --> 00:38:04.850 of motion of this system, you will have three 00:38:04.850 --> 00:38:08.900 coupled differential equations. 00:38:08.900 --> 00:38:12.470 And those are second order differential equations. 00:38:12.470 --> 00:38:17.570 If you have three variables, three second order differential 00:38:17.570 --> 00:38:19.310 equations. 00:38:19.310 --> 00:38:23.660 If you manage it magically, with the help from a computer 00:38:23.660 --> 00:38:30.800 or from math department people, how many free parameter 00:38:30.800 --> 00:38:35.420 would you expect in you a general solution? 00:38:35.420 --> 00:38:38.680 Can anybody tell me well how many? 00:38:38.680 --> 00:38:44.131 I have three second order differential equations. 00:38:44.131 --> 00:38:44.630 Yes? 00:38:44.630 --> 00:38:45.670 AUDIENCE: 6? 00:38:45.670 --> 00:38:48.180 YEN-JIE LEE: 6. 00:38:48.180 --> 00:38:53.220 So look at what we have done we identified already 00:38:53.220 --> 00:38:56.190 1, 2, 3, three normal modes. 00:38:56.190 --> 00:39:03.030 By there are 1, 2, 3, 4, 5 6, six free parameters. 00:39:03.030 --> 00:39:05.985 That tells me I am done. 00:39:09.640 --> 00:39:10.280 I'm done. 00:39:10.280 --> 00:39:15.140 Because what is the general solution? 00:39:15.140 --> 00:39:20.960 The general solution is just a superposition of mode A mode B 00:39:20.960 --> 00:39:27.620 and mode C. You have six free meters to be determined 00:39:27.620 --> 00:39:31.790 by six initial conditions, which I would like-- 00:39:31.790 --> 00:39:36.650 I have to tell you what are those initial conditions. 00:39:36.650 --> 00:39:40.730 So isn't this amazing to you? 00:39:40.730 --> 00:39:44.390 I didn't even solve the differential equation, 00:39:44.390 --> 00:39:48.050 and I already get the solution. 00:39:48.050 --> 00:39:50.540 And you can see another lesson we learned 00:39:50.540 --> 00:39:53.750 from here is that, oh no, you can 00:39:53.750 --> 00:39:56.180 imagine that the motion of the system 00:39:56.180 --> 00:39:58.310 can be super complicated. 00:39:58.310 --> 00:40:02.180 This whole thing can do this, all the crazy things 00:40:02.180 --> 00:40:04.190 are all displaced, and the center of mass 00:40:04.190 --> 00:40:06.860 can move, as you said. 00:40:06.860 --> 00:40:12.830 But the result is actually very easy to understand. 00:40:12.830 --> 00:40:17.360 It's just three kinds of motion, the displacement, and two kinds 00:40:17.360 --> 00:40:21.410 of simple harmonic motion. 00:40:21.410 --> 00:40:22.760 We add them together. 00:40:22.760 --> 00:40:27.590 And then you get the general description of that system. 00:40:27.590 --> 00:40:31.010 So everything is so nice. 00:40:31.010 --> 00:40:34.690 We understand the motion of that system. 00:40:34.690 --> 00:40:41.180 But in general, life is very hard. 00:40:41.180 --> 00:40:46.010 For example, now I do something crazy here. 00:40:46.010 --> 00:40:51.070 I change this to 3. 00:40:51.070 --> 00:40:53.350 What are the normal modes? 00:40:53.350 --> 00:40:55.550 Can anybody tell me? 00:40:55.550 --> 00:40:59.050 It becomes very, very difficult, because there's 00:40:59.050 --> 00:41:03.370 no general symmetry of that kind of system. 00:41:03.370 --> 00:41:05.890 So we are in trouble. 00:41:05.890 --> 00:41:09.250 One of the modes maybe still there, 00:41:09.250 --> 00:41:13.770 which is actually mode B. But the other modes 00:41:13.770 --> 00:41:16.900 are harder to actually guess. 00:41:16.900 --> 00:41:22.790 So you can see that that already brings you a lot of trouble. 00:41:22.790 --> 00:41:29.330 And you can see that I can now couple not just two objects, 00:41:29.330 --> 00:41:32.750 I can couple three objects, four objects, five objects, 00:41:32.750 --> 00:41:33.920 10 objects. 00:41:33.920 --> 00:41:37.020 Maybe I will put that in your p set and see what happens. 00:41:37.020 --> 00:41:39.980 And you can see that this becomes 00:41:39.980 --> 00:41:42.590 very difficult to manage. 00:41:42.590 --> 00:41:47.030 So what I'm going to do in the rest of this lecture 00:41:47.030 --> 00:41:49.760 is to introduce you a method which 00:41:49.760 --> 00:41:54.800 you can follow in general to solve the question 00:41:54.800 --> 00:41:59.120 and get the normal mode frequencies and normal modes. 00:41:59.120 --> 00:42:01.950 So we will take a four minute break. 00:42:01.950 --> 00:42:04.940 And we come back at 12:20. 00:42:04.940 --> 00:42:07.160 So if you have any questions, let me know. 00:42:12.790 --> 00:42:17.380 What we are going to do in the following exercise 00:42:17.380 --> 00:42:22.750 is to try to understand a general strategy to solve 00:42:22.750 --> 00:42:26.980 the normal mode frequencies and the normal mode amplitudes, 00:42:26.980 --> 00:42:30.640 so that you can apply this technique to all kinds 00:42:30.640 --> 00:42:32.320 of different systems. 00:42:32.320 --> 00:42:36.590 So what I am going to do today is to take these three mass 00:42:36.590 --> 00:42:41.595 system, and of course as usual, I try to define what 00:42:41.595 --> 00:42:43.600 is this coordinate system? 00:42:43.600 --> 00:42:45.520 The coordinate system I'm going to use 00:42:45.520 --> 00:42:51.790 is x1 and x2 an x3 describing the displacement of the mass 00:42:51.790 --> 00:42:53.750 from the equilibrium position. 00:42:53.750 --> 00:42:55.420 And the equilibrium means that there's 00:42:55.420 --> 00:42:57.970 no stretch on the spring. 00:42:57.970 --> 00:43:00.310 The string is unstretched. 00:43:00.310 --> 00:43:05.780 It's at their own natural length, l0. 00:43:05.780 --> 00:43:12.910 So once I define that, I can do a force diagram analysis. 00:43:12.910 --> 00:43:17.140 So that starts from the left hand side mass 00:43:17.140 --> 00:43:19.870 with mass equal to 2m. 00:43:19.870 --> 00:43:23.350 I can write down the equation of motion, 2m x1 double dot. 00:43:26.290 --> 00:43:37.730 And this is going to be equal to k x2 minus x1 plus k x3 00:43:37.730 --> 00:43:38.230 minus x1. 00:43:40.870 --> 00:43:45.850 So there are two spring forces acting on this mass, 00:43:45.850 --> 00:43:47.140 the left hand side mass. 00:43:47.140 --> 00:43:49.990 The first one is the upper spring. 00:43:49.990 --> 00:43:55.720 The second one is coming from the lower one. 00:43:55.720 --> 00:43:58.340 And you can see that both of them 00:43:58.340 --> 00:44:00.940 are proportional to spring constant k, 00:44:00.940 --> 00:44:05.650 and also proportional to the relative displacement. 00:44:05.650 --> 00:44:09.610 And you can see that the two relative displacement, which 00:44:09.610 --> 00:44:14.130 is the amount of stretch to the spring, 00:44:14.130 --> 00:44:20.540 is actually x2 minus x1, and the x3 minus x1. 00:44:20.540 --> 00:44:22.720 Am I going too fast? 00:44:22.720 --> 00:44:23.220 OK. 00:44:23.220 --> 00:44:24.410 Everybody's following. 00:44:24.410 --> 00:44:26.420 And you can actually check the sign. 00:44:26.420 --> 00:44:27.620 So you may not be sure. 00:44:27.620 --> 00:44:30.470 Maybe this is your x1 minus x2. 00:44:30.470 --> 00:44:33.200 But you can check that, because if you increase 00:44:33.200 --> 00:44:36.200 x1, what is going to happen? 00:44:36.200 --> 00:44:38.780 This term will become more negative. 00:44:38.780 --> 00:44:42.820 More negative in this coordinate system 00:44:42.820 --> 00:44:45.410 is pointing to the left hand side. 00:44:45.410 --> 00:44:46.670 So that makes sense. 00:44:46.670 --> 00:44:49.890 Because if I move this mass to the right side, 00:44:49.890 --> 00:44:53.380 then I am compressing the springs. 00:44:53.380 --> 00:44:55.520 Therefore, they are pushing it back. 00:44:55.520 --> 00:45:02.210 Therefore, this is actually the correct sign, x2 minus x1. 00:45:02.210 --> 00:45:07.130 The same thing also applies to the second spring force. 00:45:07.130 --> 00:45:10.655 So that's a way I double check if I make a mistake. 00:45:13.220 --> 00:45:16.450 Now, this is actually the first equations of motion. 00:45:16.450 --> 00:45:20.300 And I can now also work on a second mass. 00:45:20.300 --> 00:45:23.540 Now I focus on a mass number two. 00:45:23.540 --> 00:45:26.020 The displacement is x2. 00:45:26.020 --> 00:45:28.760 Therefore the left hand side of Newton's Law 00:45:28.760 --> 00:45:32.780 is m x2 double dot. 00:45:32.780 --> 00:45:36.920 And that is equal to the spring force. 00:45:36.920 --> 00:45:42.650 The spring force, there's only one spring force 00:45:42.650 --> 00:45:44.510 acting on the mass. 00:45:44.510 --> 00:45:48.790 Therefore, what I am going to get is k x1 minus x2. 00:45:53.850 --> 00:45:55.060 Everybody's following? 00:45:57.830 --> 00:46:02.730 You can actually check the sign carefully, also. 00:46:02.730 --> 00:46:05.780 And finally, I have the third mass, very similar 00:46:05.780 --> 00:46:07.850 to mass number two. 00:46:07.850 --> 00:46:10.380 I can write down the equation of motion, 00:46:10.380 --> 00:46:16.120 which should k x1 minus x3. 00:46:16.120 --> 00:46:21.740 So that is my coupled second order differential equations. 00:46:21.740 --> 00:46:23.060 There are three equations. 00:46:23.060 --> 00:46:27.260 And all of them are second order equations. 00:46:27.260 --> 00:46:30.050 So this looks a bit messy. 00:46:30.050 --> 00:46:32.810 So what I'm going to do Is no magic. 00:46:32.810 --> 00:46:37.550 I'm just collecting all the terms belonging to x1, 00:46:37.550 --> 00:46:41.480 and put them together, all the terms belonging to x2, 00:46:41.480 --> 00:46:45.470 putting all together, and just rearranging things. 00:46:45.470 --> 00:46:47.300 So no magic. 00:46:47.300 --> 00:46:50.340 So I copied this thing, left hand side. 00:46:50.340 --> 00:46:52.790 2m x1 double dot. 00:46:52.790 --> 00:47:00.260 And the dot will be equal to minus 2k x1 plus, 00:47:00.260 --> 00:47:05.930 I collect all the times related to x2, plus k x2, 00:47:05.930 --> 00:47:13.070 there's only one term here, then plus k x3. 00:47:13.070 --> 00:47:17.065 I'm just trying to organize my question. 00:47:17.065 --> 00:47:19.190 So you can see that I collect all the terms related 00:47:19.190 --> 00:47:21.800 to x1 to here. 00:47:21.800 --> 00:47:24.560 Minus k, minus k, I get minus 2k. 00:47:24.560 --> 00:47:28.850 And the plus k for x2, plus k for the x3. 00:47:28.850 --> 00:47:32.480 And I can also do that for m x2 double dot. 00:47:32.480 --> 00:47:41.930 That will be equal to k x1 minus k x2 plus zero x3, 00:47:41.930 --> 00:47:44.900 just for completeness. 00:47:44.900 --> 00:47:50.207 I can also do the same thing for the third mass, m x3 double 00:47:50.207 --> 00:47:50.940 dot. 00:47:50.940 --> 00:47:58.920 This is equal to k x1 plus 0 x2. 00:47:58.920 --> 00:48:04.880 There's no dependence on x2, because x1 and x2-- 00:48:04.880 --> 00:48:09.440 x3 and x2 are not talking to each either directly. 00:48:09.440 --> 00:48:13.880 Finally, I have the third, which is minus k x3. 00:48:19.720 --> 00:48:27.400 Now our job is to solve those coupled equations. 00:48:27.400 --> 00:48:29.720 Of course, you have the freedom, if you 00:48:29.720 --> 00:48:33.130 know how to solve it yourself, you 00:48:33.130 --> 00:48:35.570 can already go ahead and solve it. 00:48:35.570 --> 00:48:37.540 But what I am going to do here is 00:48:37.540 --> 00:48:41.560 to introduce technique, which can be useful for you 00:48:41.560 --> 00:48:44.750 and make it easier to follow. 00:48:44.750 --> 00:48:47.020 It's a fixed procedure. 00:48:47.020 --> 00:48:49.880 So what I can do is the following. 00:48:49.880 --> 00:48:56.450 I can write everything in them form of a matrix. 00:48:56.450 --> 00:49:00.480 How many of you heard the matrix for the first time? 00:49:00.480 --> 00:49:03.550 1, 2, 3, 4. 00:49:03.550 --> 00:49:04.050 OK. 00:49:04.050 --> 00:49:05.470 Only four. 00:49:05.470 --> 00:49:09.490 But if you are not being familiar with matrix, 00:49:09.490 --> 00:49:11.160 let me know, and I can help you. 00:49:11.160 --> 00:49:12.370 Let the TA know. 00:49:12.370 --> 00:49:17.860 And also, there's a section in the textbooks, which 00:49:17.860 --> 00:49:21.790 I posted on announcement, which is actually very 00:49:21.790 --> 00:49:24.260 helpful to understand matrices. 00:49:24.260 --> 00:49:26.680 But sorry to these four students, 00:49:26.680 --> 00:49:28.000 we are going to use that. 00:49:28.000 --> 00:49:32.500 And maybe, you already learn how it works from here. 00:49:32.500 --> 00:49:36.280 So one trick which we will use in this class 00:49:36.280 --> 00:49:41.500 is to convert everything into matrix format. 00:49:41.500 --> 00:49:44.260 What I am going to do is to write everything 00:49:44.260 --> 00:49:49.270 in terms of M, capital X, capital M, 00:49:49.270 --> 00:49:56.620 capital X double dot equal to minus capital K capital X. 00:49:56.620 --> 00:50:01.450 Capital M, capital X, and capital K, 00:50:01.450 --> 00:50:05.800 those are all matrices. 00:50:05.800 --> 00:50:09.130 Because I write this thing I really carefully, 00:50:09.130 --> 00:50:12.640 therefore we can already immediately identify 00:50:12.640 --> 00:50:18.010 what would be M, capital M and capital X and a K. 00:50:18.010 --> 00:50:28.210 So I can write down immediately will be equal to 2m, 0, 0, 0, 00:50:28.210 --> 00:50:35.190 m, 0, 0, 0, m. 00:50:35.190 --> 00:50:39.750 Because there's only one in each line, you only have one term. 00:50:39.750 --> 00:50:44.250 X1 double dot, x2 double dot, x3 double dot. 00:50:44.250 --> 00:50:46.290 And also, you can write down what 00:50:46.290 --> 00:50:51.030 will be the X. This is actually a vector. 00:50:51.030 --> 00:50:56.040 X will be equal to x1, x2, x3. 00:50:59.680 --> 00:51:02.310 Finally, you have the K? 00:51:02.310 --> 00:51:05.860 How do I read off K? 00:51:05.860 --> 00:51:08.010 Careful, there's a minus sign here, 00:51:08.010 --> 00:51:13.510 because I would like to make this matrix equation as if it's 00:51:13.510 --> 00:51:16.740 describing a simple harmonic motion of a one 00:51:16.740 --> 00:51:18.080 dimensional system. 00:51:18.080 --> 00:51:20.940 So it looks the same, but they are different because those 00:51:20.940 --> 00:51:22.810 are matrices. 00:51:22.810 --> 00:51:25.710 But therefore, I have in my convention 00:51:25.710 --> 00:51:27.780 I have this minus sign there. 00:51:27.780 --> 00:51:32.300 Therefore, when you read off the K, you have to be careful. 00:51:32.300 --> 00:51:35.124 So what is K? 00:51:35.124 --> 00:51:40.950 K is equal to 2k. 00:51:40.950 --> 00:51:44.190 You have the minus 2k here in front of x1. 00:51:44.190 --> 00:51:47.520 But because I have a minus sign there, therefore 00:51:47.520 --> 00:51:50.610 this one is actually taken out. 00:51:50.610 --> 00:51:53.580 So we have 2k there. 00:51:53.580 --> 00:52:04.570 Then you have minus k, minus k, minus k, k, 0. 00:52:04.570 --> 00:52:08.370 Minus k, plus k, 0. 00:52:08.370 --> 00:52:14.220 And finally, you can also finish the last row. 00:52:14.220 --> 00:52:17.618 You get minus k, 0, k. 00:52:21.960 --> 00:52:23.740 K becomes minus k. 00:52:23.740 --> 00:52:27.560 Minus k becomes k. 00:52:27.560 --> 00:52:32.930 So we have read off all those matrices successfully. 00:52:32.930 --> 00:52:38.060 So you may ask, what do they mean? 00:52:38.060 --> 00:52:40.040 Do they get the meaning? 00:52:40.040 --> 00:52:45.350 M, K, X, what those? 00:52:45.350 --> 00:52:52.370 M, capital M matrix, tells you the mass distribution 00:52:52.370 --> 00:52:54.270 inside the system. 00:52:54.270 --> 00:52:58.390 So that's the meaning of this matrix. 00:52:58.390 --> 00:53:02.180 X is actually vector, which tells 00:53:02.180 --> 00:53:08.800 the position of individual components in the system. 00:53:08.800 --> 00:53:11.140 Finally, what is K? 00:53:11.140 --> 00:53:16.890 K is telling you how each component in the system 00:53:16.890 --> 00:53:21.420 talks to the other components. 00:53:21.420 --> 00:53:25.470 So K is telling you the communication 00:53:25.470 --> 00:53:29.550 inside the system. 00:53:29.550 --> 00:53:33.180 So now we understand a bit what is going on. 00:53:33.180 --> 00:53:38.040 And as usual, I will go to the complex notation. 00:53:45.790 --> 00:53:53.310 So I have xj, the small xj are the position 00:53:53.310 --> 00:53:57.310 of the mass, x1, x2, and x3. 00:53:57.310 --> 00:54:04.540 xj will be real part of small zj. 00:54:04.540 --> 00:54:07.710 Small xj equal to real part of zj. 00:54:07.710 --> 00:54:14.370 Therefore, I can now write everything in terms of matrices 00:54:14.370 --> 00:54:15.150 again. 00:54:15.150 --> 00:54:20.708 So now I can write the solution to be Z, 00:54:20.708 --> 00:54:30.410 the capitol z is a matrix, exponential i omega t plus phi. 00:54:30.410 --> 00:54:33.320 This is the guess the solution I have. 00:54:33.320 --> 00:54:37.640 A1, A2, and A3. 00:54:37.640 --> 00:54:41.130 Those are the amplitudes, amplitude A 00:54:41.130 --> 00:54:45.620 of the first mass, amplitude of the second mass, amplitude 00:54:45.620 --> 00:54:52.020 of the third mass, in their normal mode. 00:54:52.020 --> 00:55:00.300 And all of those are oscillating at the same frequency, omega, 00:55:00.300 --> 00:55:03.720 and the same phase, phi. 00:55:03.720 --> 00:55:08.580 Does that tell you something which we learned before? 00:55:08.580 --> 00:55:12.150 Oh, that's the definition of the normal mode. 00:55:12.150 --> 00:55:15.150 I'm using the definition of the normal mode. 00:55:15.150 --> 00:55:17.580 Every part of the system is oscillating 00:55:17.580 --> 00:55:20.520 at the same frequency in the same phase. 00:55:20.520 --> 00:55:25.590 And we use that to construct my solution. 00:55:25.590 --> 00:55:29.940 The complex version is exponential i omega t plus phi, 00:55:29.940 --> 00:55:32.000 oscillating at the same frequency, 00:55:32.000 --> 00:55:34.680 oscillating at the same phase. 00:55:34.680 --> 00:55:37.730 And those are the amplitude, which I will solve later. 00:55:37.730 --> 00:55:41.280 OK, any questions? 00:55:41.280 --> 00:55:44.100 I hope I'm not going too fast. 00:55:44.100 --> 00:55:48.060 If everybody can follow, now I can 00:55:48.060 --> 00:55:54.540 go ahead and solve the equation in the matrix format. 00:55:54.540 --> 00:55:59.550 So now I go to the complex notation. 00:55:59.550 --> 00:56:06.510 So the equation M X double dot equal to minus KX 00:56:06.510 --> 00:56:12.810 becomes M Z double dot equal to minus KZ. 00:56:17.990 --> 00:56:22.530 And also, I can immediately get the Z double dot will 00:56:22.530 --> 00:56:27.920 be equal to minus omega squared Z, 00:56:27.920 --> 00:56:31.800 because each time I do a differentiation, 00:56:31.800 --> 00:56:37.470 I get i omega out of the exponential function. 00:56:37.470 --> 00:56:39.460 And I cannot kill that exponential function, 00:56:39.460 --> 00:56:40.950 so it's still there. 00:56:40.950 --> 00:56:43.440 Therefore, I get minus omega squared 00:56:43.440 --> 00:56:48.360 in front of Z. I hope that doesn't surprise you. 00:56:48.360 --> 00:56:50.880 So that's very nice and very good news. 00:56:50.880 --> 00:56:55.010 That means I can replace this Z double prime with minus omega 00:56:55.010 --> 00:57:01.050 squared Z. Then I get minus M omega squared Z. 00:57:01.050 --> 00:57:04.200 And this is equal to minus KZ. 00:57:04.200 --> 00:57:06.600 And I can cancel the minus sign. 00:57:06.600 --> 00:57:08.230 That becomes something like this. 00:57:12.930 --> 00:57:20.790 So, I can now cancel the exponential i omega t plus phi, 00:57:20.790 --> 00:57:23.430 because I have Z in the left hand side. 00:57:23.430 --> 00:57:28.270 And exponential i omega t plus phi is just a number. 00:57:28.270 --> 00:57:30.780 So therefore, I can cancel it. 00:57:30.780 --> 00:57:34.420 So what is going to happen if I do that? 00:57:34.420 --> 00:57:37.740 Basically, what I'm going get is I 00:57:37.740 --> 00:57:44.670 get M omega squared A equal to K A. 00:57:44.670 --> 00:57:48.330 I'm trying to go extremely slowly, because this 00:57:48.330 --> 00:57:52.200 is the first time we go through matrices. 00:57:52.200 --> 00:57:55.020 So now you have this expression. 00:57:55.020 --> 00:58:00.410 Left hand is a matrix, M, times some constant, omega squared. 00:58:00.410 --> 00:58:02.910 I can actually get omega squared in front of it, 00:58:02.910 --> 00:58:06.060 because this is actually just a number. 00:58:06.060 --> 00:58:12.480 A is just a vector, which is A1, A2, A3, also a matrix. 00:58:12.480 --> 00:58:17.340 K is actually how the individual components talks to the others. 00:58:17.340 --> 00:58:20.490 So that's there, times A. 00:58:20.490 --> 00:58:23.910 Now I would like to move everything 00:58:23.910 --> 00:58:26.640 to the right hand side, all the matrices in front A 00:58:26.640 --> 00:58:27.840 to the right hand side. 00:58:27.840 --> 00:58:32.760 Then I multiply both sides by M minus 1. 00:58:32.760 --> 00:58:39.100 So I multiply M minus 1 to the whole equation. 00:58:39.100 --> 00:58:43.110 M minus 1, what is M minus 1? 00:58:43.110 --> 00:58:46.710 The definition is that the inverse of M 00:58:46.710 --> 00:58:48.940 is called M minus 1. 00:58:48.940 --> 00:58:57.240 M minus 1 times M is equal to I, which is actually 1, 1, 1. 00:58:57.240 --> 00:59:02.670 Therefore, if I do this thing, then I would get omega squared. 00:59:02.670 --> 00:59:09.970 M minus 1 times M becomes I, unit matrix. 00:59:09.970 --> 00:59:17.280 And this is equal to M minus 1 K A. And be careful, 00:59:17.280 --> 00:59:21.140 I multiply M minus 1, the inverse of M, 00:59:21.140 --> 00:59:23.060 from the left hand side. 00:59:23.060 --> 00:59:25.800 That matters. 00:59:25.800 --> 00:59:30.390 So now I can move everything to the same side. 00:59:30.390 --> 00:59:33.970 I moved the left hand side term to the right hand side. 00:59:33.970 --> 00:59:40.670 Therefore, I get M minus 1 K minus omega squared 00:59:40.670 --> 00:59:46.430 I. Those are all matrices. 00:59:46.430 --> 00:59:49.330 Times A, this is equal to 0. 00:59:54.400 --> 00:59:55.370 Any questions? 01:00:00.430 --> 01:00:02.950 So a lot of manipulation. 01:00:02.950 --> 01:00:07.880 But if you think about it, and you are following me, 01:00:07.880 --> 01:00:11.840 you'll see that all those steps are exactly 01:00:11.840 --> 01:00:17.140 identical to what we have been doing for a single harmonic 01:00:17.140 --> 01:00:18.440 oscillator. 01:00:18.440 --> 01:00:20.000 Looks pretty familiar to you. 01:00:20.000 --> 01:00:24.708 But the difference is that now we are dealing with matrices. 01:00:24.708 --> 01:00:25.910 AUDIENCE: What is A? 01:00:25.910 --> 01:00:30.870 YEN-JIE LEE: Oh, A. A is actually this guy. 01:00:30.870 --> 01:00:33.620 I define this to be A. And that means 01:00:33.620 --> 01:00:40.930 Z will be exponential i omega t plus phi times A. I didn't 01:00:40.930 --> 01:00:42.360 actually write it explicitly. 01:00:42.360 --> 01:00:43.910 But that's what I mean. 01:00:46.500 --> 01:00:49.412 Any more questions? 01:00:49.412 --> 01:00:50.861 Yes? 01:00:50.861 --> 01:00:55.605 AUDIENCE: [INAUDIBLE] is for [INAUDIBLE]?? 01:00:55.605 --> 01:00:56.980 YEN-JIE LEE: Can you repeat that? 01:00:56.980 --> 01:01:01.230 AUDIENCE: So this whole process, this is mode A, right? 01:01:01.230 --> 01:01:01.980 YEN-JIE LEE: Yeah. 01:01:01.980 --> 01:01:05.970 So this whole process is for, not really the for mode 01:01:05.970 --> 01:01:09.390 A. So that A may be confusing. 01:01:09.390 --> 01:01:12.080 But in general, if I have a solution, 01:01:12.080 --> 01:01:16.830 and I assume that the amplitude can be described by a matrix. 01:01:16.830 --> 01:01:17.820 So it's in general. 01:01:17.820 --> 01:01:19.630 And you'll see that we can actually 01:01:19.630 --> 01:01:24.690 derive the angular frequency of mode A, mode B, and mode C 01:01:24.690 --> 01:01:26.290 afterward. 01:01:26.290 --> 01:01:28.020 I hope that answers your question. 01:01:28.020 --> 01:01:31.350 So you see that for in general, what I have been doing 01:01:31.350 --> 01:01:34.320 is that now, all those things are 01:01:34.320 --> 01:01:39.520 equivalent to the original equation of motion. 01:01:39.520 --> 01:01:44.780 What I am doing is purely cosmetic. 01:01:44.780 --> 01:01:47.810 You see, make it beautiful. 01:01:47.810 --> 01:01:52.850 So all those things, this thing is exactly the equivalent 01:01:52.850 --> 01:01:56.190 to that thing, up there. 01:01:56.190 --> 01:01:59.300 Up to M X double dot equal to minus K x. 01:01:59.300 --> 01:02:01.140 Cosmetics. 01:02:01.140 --> 01:02:02.560 Beautiful. 01:02:02.560 --> 01:02:06.570 Looks-- I like it. 01:02:06.570 --> 01:02:07.650 All right. 01:02:07.650 --> 01:02:10.170 Then what I have been doing is that now 01:02:10.170 --> 01:02:15.210 I introduce using a definition of normal mode. 01:02:15.210 --> 01:02:20.190 I guess the solution will have this functional form. 01:02:20.190 --> 01:02:25.300 Z equals to exponential i omega t plus phi, everybody 01:02:25.300 --> 01:02:28.830 oscillating at the same frequency, the same phase. 01:02:28.830 --> 01:02:32.010 Frequency omega, phase phi. 01:02:32.010 --> 01:02:35.130 And everybody can have different amplitude. 01:02:35.130 --> 01:02:39.240 You can see from this example, normal modes, 01:02:39.240 --> 01:02:42.710 they can have different amplitude. 01:02:42.710 --> 01:02:44.490 The amplitude is what? 01:02:44.490 --> 01:02:45.900 I don't know yet. 01:02:45.900 --> 01:02:48.150 But we will figure it out. 01:02:48.150 --> 01:02:50.520 Then that's my assumption. 01:02:50.520 --> 01:02:52.510 The definition of normal mode. 01:02:52.510 --> 01:02:56.220 And I plug in to the equation of motion. 01:02:56.220 --> 01:02:59.790 Then this is what we are doing to simplify 01:02:59.790 --> 01:03:01.350 the equation of motion. 01:03:01.350 --> 01:03:03.540 There's no magic here. 01:03:03.540 --> 01:03:09.670 If I plug in the definition on normal mode to that equation, 01:03:09.670 --> 01:03:15.460 this is actually going to bring you to this equation, matrix 01:03:15.460 --> 01:03:18.210 equation. 01:03:18.210 --> 01:03:24.000 So if you have learned matrices before, you have something, 01:03:24.000 --> 01:03:30.930 some matrix, times Z. This is equal to zero. 01:03:30.930 --> 01:03:33.660 A is not zero. 01:03:33.660 --> 01:03:34.600 I hope. 01:03:34.600 --> 01:03:37.780 If it's zero, then the whole system is not moving. 01:03:37.780 --> 01:03:39.780 Then it's not fun. 01:03:42.520 --> 01:03:47.040 So if A is not zero, then this thing should be-- 01:03:47.040 --> 01:03:52.290 this thing times A should make this equation equal to 0. 01:03:52.290 --> 01:03:56.980 So what is actually the required condition? 01:03:56.980 --> 01:03:58.080 I get stuck, 01:03:58.080 --> 01:04:01.440 and of course again, my friend from math department 01:04:01.440 --> 01:04:04.080 comes to save me. 01:04:04.080 --> 01:04:10.930 That means if this thing has a solution, 01:04:10.930 --> 01:04:12.640 this equation has a solution, that 01:04:12.640 --> 01:04:23.090 means that determinant of M minus 1 K minus omega 01:04:23.090 --> 01:04:29.750 squared I has to be equal to 0. 01:04:29.750 --> 01:04:34.790 So that is the condition for this equation 01:04:34.790 --> 01:04:41.030 to satisfy this to be equal to 0. 01:04:41.030 --> 01:04:45.320 And just to make sure that I don't know what is the angular 01:04:45.320 --> 01:04:46.820 frequency omega yet. 01:04:46.820 --> 01:04:49.990 I don't know what is the phi yet. 01:04:49.990 --> 01:04:55.730 We can actually solve the angular frequency, omega. 01:04:55.730 --> 01:05:01.070 So now, turn everything around. 01:05:01.070 --> 01:05:05.060 And basically now, using this normal mode 01:05:05.060 --> 01:05:11.290 definition, and some mathematical manipulation, 01:05:11.290 --> 01:05:16.440 the condition we need for this equation to satisfy equal to 0, 01:05:16.440 --> 01:05:22.190 is determinant M minus 1 K minus omega squared I. 01:05:22.190 --> 01:05:25.930 I can write down M minus 1 K minus omega 01:05:25.930 --> 01:05:32.870 squared I explicitly, just to help you with mathematics. 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. 01:05:50.500 --> 01:05:56.230 It's just the inverse matrix of the M matrix. 01:05:56.230 --> 01:06:07.090 Therefore, now I can write down the explicit expression 01:06:07.090 --> 01:06:10.350 of M minus 1 K minus omega squared 01:06:10.350 --> 01:06:23.730 I. This will be equal to k over m minus omega squared, 01:06:23.730 --> 01:06:34.370 minus k over 2m, minus k over 2, minus k over m, 01:06:34.370 --> 01:06:40.030 k over m minus omega squared, 0. 01:06:40.030 --> 01:06:42.600 I will write down all the elements first. 01:06:42.600 --> 01:06:46.392 Then I will explain to you how I arrived at the expression. 01:06:52.200 --> 01:06:53.150 OK. 01:06:53.150 --> 01:06:55.180 So this is M minus 1 k. 01:06:55.180 --> 01:06:58.790 The definition of M minus 1 is that. 01:06:58.790 --> 01:07:02.280 And the definition of K is in the upper right 01:07:02.280 --> 01:07:04.980 corner of the black board. 01:07:04.980 --> 01:07:08.620 Therefore, if you multiply M minus 1 K, 01:07:08.620 --> 01:07:14.130 basically, the first column will get-- 01:07:18.240 --> 01:07:18.850 wait a second. 01:07:18.850 --> 01:07:20.100 Did I make a mistake? 01:07:24.670 --> 01:07:25.170 No. 01:07:25.170 --> 01:07:26.130 OK. 01:07:26.130 --> 01:07:32.440 So basically, what you arrive at is k/m, k/m, k/m. 01:07:36.430 --> 01:07:42.760 And also, the minus k over 2m for the rest part 01:07:42.760 --> 01:07:44.440 of the matrix. 01:07:44.440 --> 01:07:48.235 And the minus omega squared I will give you 01:07:48.235 --> 01:07:51.145 the diagonal component. 01:07:51.145 --> 01:07:52.227 Yes? 01:07:52.227 --> 01:07:54.268 AUDIENCE: Why do you have to take the determinant 01:07:54.268 --> 01:07:56.287 and set it equal to 0 instead of just 01:07:56.287 --> 01:07:59.230 setting that equal to zero? 01:07:59.230 --> 01:08:02.740 AUDIENCE: This is a matrix. 01:08:02.740 --> 01:08:04.150 So these are the matrix. 01:08:04.150 --> 01:08:08.020 So a matrix times A will be equal to zero. 01:08:08.020 --> 01:08:12.670 The general condition for that to be satisfied 01:08:12.670 --> 01:08:14.440 is more general. 01:08:14.440 --> 01:08:19.359 It's actually the determinant of this matrix equal to zero. 01:08:19.359 --> 01:08:27.790 Because this is actually multiplied by some back to A. 01:08:27.790 --> 01:08:31.430 So I think there are mathematical manipulation. 01:08:31.430 --> 01:08:33.270 Basically, you would just collect the terms. 01:08:33.270 --> 01:08:37.140 And then calculate M minus 1 K first. 01:08:37.140 --> 01:08:40.930 And the minus omega squared I will give you all the diagonal 01:08:40.930 --> 01:08:44.899 and terms have a minus omega square there. 01:08:44.899 --> 01:08:48.540 And that is actually the matrix. 01:08:48.540 --> 01:08:52.290 And of course, I can calculate the determinant. 01:08:52.290 --> 01:08:54.479 So if I calculate the determinant, 01:08:54.479 --> 01:08:58.319 then basically I get this times that times that. 01:08:58.319 --> 01:09:07.950 So what you get is k over m minus omega squared times k 01:09:07.950 --> 01:09:15.430 over m minus omega squared times k over m minus omega squared. 01:09:15.430 --> 01:09:17.510 So these are all diagonal terms. 01:09:17.510 --> 01:09:26.340 And the minus 1 over 2 k squared over m squared, k 01:09:26.340 --> 01:09:32.879 squared over m squared. 01:09:35.770 --> 01:09:36.790 sorry. 01:09:36.790 --> 01:09:40.600 Minus omega squared. 01:09:40.600 --> 01:09:46.649 So that's this off diagonal term, this times this times 01:09:46.649 --> 01:09:47.310 that. 01:09:47.310 --> 01:09:48.130 OK. 01:09:48.130 --> 01:09:50.050 It will give you the second term. 01:09:50.050 --> 01:09:52.390 And the third one, which survived 01:09:52.390 --> 01:09:56.320 because of those zeros, many, many terms are equal to 0. 01:09:56.320 --> 01:10:01.590 And then the third term, which is nonzero, is again minus 1 01:10:01.590 --> 01:10:09.790 over 2k squared over m squared, k over m minus omega squared. 01:10:09.790 --> 01:10:13.270 And this is actually equal to 0, because the determinant 01:10:13.270 --> 01:10:17.380 of this matrix is equal to zero. 01:10:17.380 --> 01:10:19.320 Everybody following? 01:10:19.320 --> 01:10:20.320 A little bit of a mess. 01:10:20.320 --> 01:10:23.760 Because I have been doing something very challenging. 01:10:23.760 --> 01:10:32.410 I'm solving a 3 by 3 matrix problem in front of you right. 01:10:32.410 --> 01:10:35.600 So the math can get a bit complicated. 01:10:35.600 --> 01:10:40.135 But next time, I think we are going to go to a second order 01:10:40.135 --> 01:10:42.070 one, 2 by 2 matrix. 01:10:42.070 --> 01:10:45.040 And I think that will be slightly easier. 01:10:45.040 --> 01:10:47.650 But the general approach is the same. 01:10:47.650 --> 01:10:52.480 So basically, you calculate M minus 1 K minus omega squared. 01:10:52.480 --> 01:10:57.820 Then you get what is inside, all the content inside this matrix. 01:10:57.820 --> 01:11:00.640 Then you would calculate the determinant. 01:11:00.640 --> 01:11:05.800 And basically, you can solve this equation. 01:11:05.800 --> 01:11:11.250 Now I can define omega0 squared to be k/m. 01:11:11.250 --> 01:11:15.210 And I can actually make this expression much simpler. 01:11:18.840 --> 01:11:20.970 Then basically, what you are getting 01:11:20.970 --> 01:11:25.660 is omega0 squared minus omega squared 01:11:25.660 --> 01:11:33.480 to the third minus 1/2 omega0 to the fourth, omega0 squared 01:11:33.480 --> 01:11:35.650 minus omega squared. 01:11:35.650 --> 01:11:42.930 Minus 1/2 omega0 to the fourth, omega0 to the square, 01:11:42.930 --> 01:11:43.980 minus omega squared. 01:11:43.980 --> 01:11:47.340 And this is equal to 0. 01:11:47.340 --> 01:11:51.684 And you can factor out the common components. 01:11:51.684 --> 01:11:53.100 Then basically, what you are going 01:11:53.100 --> 01:11:56.370 to get is, you can write this thing 01:11:56.370 --> 01:12:03.270 to be omega0 squared minus omega squared, omega squared. 01:12:03.270 --> 01:12:06.390 Because all of them have omega squared. 01:12:06.390 --> 01:12:12.430 And omega squared minus 2 omega0 squared. 01:12:12.430 --> 01:12:14.770 And that's equal to 0. 01:12:14.770 --> 01:12:18.520 So I am skipping a lot of steps from this one to that one. 01:12:18.520 --> 01:12:22.690 But in general, you can solve this third order equation. 01:12:22.690 --> 01:12:29.530 And I can first combine all those terms together. 01:12:29.530 --> 01:12:33.050 And then I factor out the common components. 01:12:33.050 --> 01:12:35.500 Then basically, what you are going to arrive at 01:12:35.500 --> 01:12:38.560 is something like this. 01:12:38.560 --> 01:12:39.890 A lot of math here. 01:12:39.890 --> 01:12:43.240 But we are close to the end. 01:12:43.240 --> 01:12:48.440 So you can see now what are the possible solutions for omega. 01:12:48.440 --> 01:12:53.860 That is the omega, unknown angular frequency 01:12:53.860 --> 01:12:56.380 we are trying to figure out. 01:12:56.380 --> 01:13:00.460 You can see that there are three possible omegas that can 01:13:00.460 --> 01:13:03.220 make this equation equal to 0. 01:13:03.220 --> 01:13:12.000 The first one is omega equal to omega0. 01:13:12.000 --> 01:13:19.140 The second one is square root of omega 0, 01:13:19.140 --> 01:13:21.450 coming from this expression, that omega 01:13:21.450 --> 01:13:24.360 squared minus 2 omega zero squared. 01:13:24.360 --> 01:13:26.280 If omega equal to square root 2 omega0, 01:13:26.280 --> 01:13:28.110 this will be equal to zero. 01:13:28.110 --> 01:13:31.180 And that will give you the whole expression equal to 0. 01:13:31.180 --> 01:13:33.670 Then finally, I take this term. 01:13:33.670 --> 01:13:36.030 And then you will get zero. 01:13:36.030 --> 01:13:40.170 Omega squared, if omega is equal to 0, 01:13:40.170 --> 01:13:43.770 then the whole expression is 0. 01:13:43.770 --> 01:13:50.130 I have defined omega0 squared to be equal to k over m. 01:13:50.130 --> 01:13:54.060 Therefore, I can conclude that omega squared 01:13:54.060 --> 01:14:04.002 is equal to k over m, 2k over m, and 0. 01:14:06.960 --> 01:14:10.390 Look at what we have done, a lot of mathematics. 01:14:10.390 --> 01:14:15.240 But in the end, after you solve the eigenvalue problem, 01:14:15.240 --> 01:14:18.720 or the determinant equal to zero problem, 01:14:18.720 --> 01:14:20.940 you arrive at that there are only 01:14:20.940 --> 01:14:25.740 three possible values of omega which 01:14:25.740 --> 01:14:28.620 can make the determinant of M minus 1 01:14:28.620 --> 01:14:32.600 K minus omega squared I equal to 0. 01:14:32.600 --> 01:14:35.550 What are the three? 01:14:35.550 --> 01:14:38.710 k/m, 2k/m, and 0. 01:14:42.260 --> 01:14:45.150 If you look at this value, then we'll say, 01:14:45.150 --> 01:14:49.800 this is essentially what we actually argued before, right? 01:14:49.800 --> 01:14:55.000 Omega A squared is equal to 4k over 2m is 2k over m. 01:14:55.000 --> 01:14:55.930 Wow. 01:14:55.930 --> 01:14:56.560 We got it. 01:15:00.320 --> 01:15:03.210 The second one is, we think about really 01:15:03.210 --> 01:15:04.930 keep a straight question in my head 01:15:04.930 --> 01:15:07.460 and understand this system. 01:15:07.460 --> 01:15:12.260 The second identified normal more is having omega 01:15:12.260 --> 01:15:14.090 squared be equal to k/m. 01:15:14.090 --> 01:15:18.650 I got this also here magically, after all those magics. 01:15:18.650 --> 01:15:24.960 And finally, the third one, the math also knows physics. 01:15:24.960 --> 01:15:29.700 It also predicted that this is one mode which have oscillation 01:15:29.700 --> 01:15:32.430 frequency of 0. 01:15:32.430 --> 01:15:34.924 Isn't that amazing to you? 01:15:39.690 --> 01:15:43.900 But that also gives us a sense of safety. 01:15:43.900 --> 01:15:48.320 Because I can now add 10 pendulums, or 10 01:15:48.320 --> 01:15:51.080 coupled system to your homework, and you 01:15:51.080 --> 01:15:52.210 will be able to solve it. 01:15:55.350 --> 01:15:56.790 So very good. 01:15:56.790 --> 01:16:00.530 This example seems to be complicated. 01:16:00.530 --> 01:16:03.570 But the what I want to say, I have one minute left, 01:16:03.570 --> 01:16:05.790 is that what we have been doing is 01:16:05.790 --> 01:16:11.370 to write the equation of motion based on force diagram. 01:16:11.370 --> 01:16:15.660 Then I convert that to matrix format, 01:16:15.660 --> 01:16:19.090 and X double dot equal to minus KX. 01:16:19.090 --> 01:16:21.270 Then I follow the whole procedure, 01:16:21.270 --> 01:16:24.110 solve the eigenvalue problem. 01:16:24.110 --> 01:16:30.060 Then I will be able to figure out what are the possible omega 01:16:30.060 --> 01:16:37.555 values which can satisfy this eigenvalue problem 01:16:37.555 --> 01:16:38.920 or this determinant. 01:16:38.920 --> 01:16:42.630 M minus 1 K minus omega squared I equal to 0 problem. 01:16:42.630 --> 01:16:46.130 And after solving all those, you will 01:16:46.130 --> 01:16:51.000 be able to solve the corresponding so-called normal 01:16:51.000 --> 01:16:52.530 mode frequencies. 01:16:52.530 --> 01:16:53.760 You can solve it. 01:16:53.760 --> 01:16:57.700 And of course, you can plug those normal mode frequencies 01:16:57.700 --> 01:17:01.170 back in, then you will be able to drive 01:17:01.170 --> 01:17:05.430 the relative amplitude, A1, A2, and A3. 01:17:05.430 --> 01:17:07.890 So what we have we learned today? 01:17:07.890 --> 01:17:11.670 We have learned how to predict the motion 01:17:11.670 --> 01:17:14.460 of coupled oscillators. 01:17:14.460 --> 01:17:15.900 That's really cool. 01:17:15.900 --> 01:17:18.540 And then next time, we are going to learn 01:17:18.540 --> 01:17:22.050 a special kind of motion in coupled oscillators, which 01:17:22.050 --> 01:17:23.850 is the big phenomena. 01:17:23.850 --> 01:17:26.710 And also, what will happen if I start to drive 01:17:26.710 --> 01:17:28.110 the coupled oscillators? 01:17:28.110 --> 01:17:31.830 So I will be here if you have any questions 01:17:31.830 --> 01:17:34.130 about the lecture. 01:17:34.130 --> 01:17:36.080 Thank you very much.