1 00:00:00,210 --> 00:00:06,630 So we've just learned that four large time starting values are a numerical solution will be different 2 00:00:06,630 --> 00:00:12,750 to the analytical solution, which is just due to the fact that the analytical solution was only true 3 00:00:12,750 --> 00:00:13,980 for an approximation. 4 00:00:14,700 --> 00:00:17,670 And so in the following, we will not consider it anymore. 5 00:00:17,700 --> 00:00:20,220 We will only trust our numerical results. 6 00:00:21,540 --> 00:00:28,440 So now we really have a tool that helps us to understand the harmonic oscillator and the pendulum. 7 00:00:29,280 --> 00:00:32,940 So let's go ahead and make the problem even more difficult by adding more terms. 8 00:00:35,340 --> 00:00:44,610 So we have defined already our differential equation here so that we copy this, maybe all of this and 9 00:00:46,320 --> 00:00:48,210 paste it here and now. 10 00:00:48,210 --> 00:00:57,420 This time I will consider a damping, and the damping is the zero point five. 11 00:00:59,730 --> 00:01:02,160 And what else will I consider? 12 00:01:02,160 --> 00:01:08,280 I will stick to the larger starting value, but I will go back to the a shorter calculation time. 13 00:01:08,910 --> 00:01:09,630 Maybe we can. 14 00:01:09,630 --> 00:01:10,860 We can change this later on. 15 00:01:11,760 --> 00:01:16,050 And though, as I said, we will not compare this to the analytical solutions that we will just use 16 00:01:16,050 --> 00:01:16,680 this one here. 17 00:01:19,110 --> 00:01:27,630 So if I run this and I get the solution for the pendulum and you see what happens, we start from a 18 00:01:27,630 --> 00:01:33,180 large value theta zero of two and then we can now change this also to plot. 19 00:01:34,230 --> 00:01:38,970 And then it oscillates it has still the characteristic time periods. 20 00:01:39,390 --> 00:01:40,050 What do you see? 21 00:01:40,110 --> 00:01:42,200 The amplitude decreases? 22 00:01:42,210 --> 00:01:47,100 This is, of course, due to the damping term that we have now turned on and like in reality, when 23 00:01:47,100 --> 00:01:53,640 you leave a pendulum swinging, then it will just decrease its amplitude over time. 24 00:01:54,880 --> 00:01:55,200 It's not. 25 00:01:55,200 --> 00:02:00,300 What we can also do is checking what happens if we decrease the step size. 26 00:02:01,530 --> 00:02:04,120 But I think there wasn't that much of a difference. 27 00:02:04,140 --> 00:02:11,610 So it seems like the solution is already converged and this is the behavior now with the damping term. 28 00:02:13,320 --> 00:02:17,760 Likewise, we can also modulate a driven oscillator. 29 00:02:18,690 --> 00:02:28,170 So this is when we change the differential equation once more, so we can do this by adding here another 30 00:02:28,170 --> 00:02:28,590 term. 31 00:02:29,430 --> 00:02:44,190 So we're right on minus these times and peer sine of omega tones T and D will be essentially the amplitude 32 00:02:44,550 --> 00:02:49,080 of the gravity of the external force that we have just added. 33 00:02:50,970 --> 00:02:53,010 So let's just test 1.0. 34 00:02:53,010 --> 00:02:53,790 I don't know if it's good. 35 00:02:58,420 --> 00:03:05,740 And also, we have to define Omega, which is the frequency of the external force. 36 00:03:06,490 --> 00:03:16,030 So what we have done is we have added here a term to the to this differential equation where the right 37 00:03:16,030 --> 00:03:19,900 side of the differential equation has a linear term and theta, which is the damping. 38 00:03:20,260 --> 00:03:27,370 It has a linear or sine theentire term, which is the gravitational force, and then it has a term which 39 00:03:27,370 --> 00:03:29,190 doesn't depend on theta at all. 40 00:03:29,200 --> 00:03:33,000 It is a term that only depends on the time explicit explicitly. 41 00:03:33,730 --> 00:03:40,780 So this is now a sign omega t term, which corresponds to when you, for example, apply some kind of 42 00:03:40,780 --> 00:03:47,860 motor to your or some, some some some engine that moves the pendulum while you leave it swinging. 43 00:03:48,850 --> 00:03:54,730 And so it's really interesting to analyze what happens when this swinging frequency omega is different 44 00:03:54,730 --> 00:03:57,460 compared to the eigen frequency of the pendulum. 45 00:03:59,350 --> 00:04:02,110 So let me just show you one more thing. 46 00:04:02,380 --> 00:04:09,610 Let me go back to the theory part where we have introduced our second order differential equation here. 47 00:04:10,600 --> 00:04:15,970 We have written down that the second order differential equation in general is characterized by a second 48 00:04:15,970 --> 00:04:22,360 or the derivative that is given by a function that depends on first derivative, the function itself. 49 00:04:22,720 --> 00:04:23,920 And then also T. 50 00:04:24,580 --> 00:04:29,230 And now we really have all three of these terms in our differential equation. 51 00:04:30,520 --> 00:04:35,800 So this is like the most general case that we can solve with our new method. 52 00:04:36,910 --> 00:04:38,470 So let's see how it works. 53 00:04:39,660 --> 00:04:42,040 Um, let me think about the parameters. 54 00:04:42,040 --> 00:04:43,990 Maybe at first, let me go back here a bit. 55 00:04:46,550 --> 00:04:48,350 And let's solve this. 56 00:04:49,100 --> 00:04:54,680 So you see the EU, yeah, the time depends of SETA is no very confusing. 57 00:04:55,100 --> 00:05:01,130 In the very beginning, it looks maybe a bit similar to the UN driven and just -- oscillator. 58 00:05:01,790 --> 00:05:08,180 But then since we have this new fourth term here, it will not decay to zero, but it will still be 59 00:05:08,180 --> 00:05:08,930 oscillating. 60 00:05:09,140 --> 00:05:13,550 And it also seems like there is no clear trend after 20 seconds. 61 00:05:14,180 --> 00:05:16,280 So let's go to a larger time. 62 00:05:17,270 --> 00:05:24,200 Let's increase the step size, for example, and then we see that this solution looks like this. 63 00:05:24,770 --> 00:05:34,790 We could also just increase this one here and then it will look like this, and we can even play a bit 64 00:05:34,790 --> 00:05:35,900 with the parameters. 65 00:05:36,050 --> 00:05:40,250 For example, we could change the sign of the the amplitude. 66 00:05:40,850 --> 00:05:49,250 Then it will look more like this, or if we increase the step size again, we get such a result. 67 00:05:49,970 --> 00:05:56,930 But since in the following, I want to compare this result with this two step Euler method with more 68 00:05:56,930 --> 00:05:58,370 of the optimized methods. 69 00:05:58,640 --> 00:06:06,890 I want to have a more difficult time dependence of data and more change in this curve. 70 00:06:07,490 --> 00:06:13,130 So what I will do, I will I will decrease the damping parameter. 71 00:06:14,210 --> 00:06:21,410 So let me try this so that we have less damping and then hopefully we get more difficult looking time 72 00:06:21,410 --> 00:06:21,980 dependence. 73 00:06:22,580 --> 00:06:23,930 Yeah, I think this is pretty good. 74 00:06:24,050 --> 00:06:27,710 We have now some strange oscillations in the beginning. 75 00:06:27,710 --> 00:06:34,010 This is because the oscillator the pendulum doesn't lose enough energy in the beginning because of damping 76 00:06:34,010 --> 00:06:40,370 is now much smaller and the external force is quite large and it really is in conflict. 77 00:06:40,370 --> 00:06:47,690 It's fighting with the oscillation of the pendulum itself, so it tries to get the pendulum out of its 78 00:06:47,690 --> 00:06:56,840 eigen frequency and eigen pendulum motion, and it tries to dictate the external force with the period 79 00:06:56,870 --> 00:06:58,430 omega onto the pendulum. 80 00:06:59,210 --> 00:07:05,600 And this really takes about one minute until the process is finished and hear the motion looks very 81 00:07:05,600 --> 00:07:06,170 chaotic. 82 00:07:06,170 --> 00:07:13,670 And then after this time, it finally moves in the periodic manner, according to this omega T sine 83 00:07:13,670 --> 00:07:14,180 function.