WEBVTT 00:00:00.070 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.140 continue to offer high-quality educational resources for free. 00:00:10.140 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.260 at ocw.mit.edu. 00:00:26.010 --> 00:00:32.310 PROFESSOR: Today we are going to talk about the vibration 00:00:32.310 --> 00:00:34.030 of continuous systems. 00:00:34.030 --> 00:00:37.590 Not covered on the quiz, but it's 00:00:37.590 --> 00:00:41.680 a really important part of real-world vibration 00:00:41.680 --> 00:00:51.080 and the most-- one of the easiest ones to demonstrate, 00:00:51.080 --> 00:00:54.290 I've shown you this one before, is the taut string. 00:00:54.290 --> 00:01:00.040 But I want to show you something unusual about-- something 00:01:00.040 --> 00:01:01.830 you may not know about strings. 00:01:01.830 --> 00:01:03.959 Wait until it calms down here a little bit. 00:01:06.940 --> 00:01:11.930 OK, so this is your guitar string or a piano string. 00:01:11.930 --> 00:01:13.360 It's under tension. 00:01:13.360 --> 00:01:17.640 We've already seen that it exhibits 00:01:17.640 --> 00:01:19.640 natural frequencies in mode shape, 00:01:19.640 --> 00:01:21.420 so there's the first mode. 00:01:21.420 --> 00:01:23.310 Looks like half a sine wave. 00:01:23.310 --> 00:01:27.770 Has a particular frequency associated with it. 00:01:27.770 --> 00:01:31.690 Get it to stop doing that-- but if I excite it 00:01:31.690 --> 00:01:35.040 at twice the frequency-- I don't know if I can do this. 00:01:35.040 --> 00:01:36.484 There we go. 00:01:40.260 --> 00:01:42.330 That turns out to be exactly twice 00:01:42.330 --> 00:01:43.990 the frequency of the first. 00:01:43.990 --> 00:01:45.820 The mode shaped one full sine wave. 00:01:45.820 --> 00:01:50.550 The mode shapes for a taut string are sine n pi x over L. 00:01:50.550 --> 00:01:52.860 But strings can do something else kind of neat. 00:01:56.300 --> 00:02:01.281 And that is if I hit this thing-- 00:02:01.281 --> 00:02:03.030 I'm going to wait till it calms down here. 00:02:05.950 --> 00:02:09.520 If I give this thing just a pulse, 00:02:09.520 --> 00:02:12.790 what do you expect to see? 00:02:12.790 --> 00:02:15.620 Are you going to see vibration? 00:02:15.620 --> 00:02:16.750 Tell me what you see. 00:02:25.620 --> 00:02:28.310 What do you see happening? 00:02:28.310 --> 00:02:29.930 Something running back and forth. 00:02:29.930 --> 00:02:30.920 Right? 00:02:30.920 --> 00:02:33.710 What you're seeing is wave propagation. 00:02:33.710 --> 00:02:34.850 It's not really vibration. 00:02:34.850 --> 00:02:38.131 Vibration we see of its modes and standing waves and things 00:02:38.131 --> 00:02:38.630 like that. 00:02:38.630 --> 00:02:39.180 Right? 00:02:39.180 --> 00:02:45.250 So the taut string satisfies an equation of motion that's 00:02:45.250 --> 00:02:46.660 called the wave equation. 00:02:46.660 --> 00:02:49.040 We're going to talk quite a bit about that this morning. 00:02:49.040 --> 00:02:53.020 And the wave equation has its name give something away. 00:02:53.020 --> 00:02:56.530 The wave equation describes continuous systems 00:02:56.530 --> 00:03:00.690 of a particular kind that support travelling waves. 00:03:00.690 --> 00:03:03.870 And so the string will both support-- 00:03:03.870 --> 00:03:05.875 I can give it a little pluck. 00:03:05.875 --> 00:03:10.300 I'll try to just place it in a particular shape and let go. 00:03:10.300 --> 00:03:10.959 There it is. 00:03:10.959 --> 00:03:13.500 And that little pluck just goes back and forth back and forth 00:03:13.500 --> 00:03:15.420 at a particular speed. 00:03:15.420 --> 00:03:18.190 So is there a relationship between the speed 00:03:18.190 --> 00:03:20.950 at which things can travel in a string 00:03:20.950 --> 00:03:25.040 and the natural frequencies of the string? 00:03:25.040 --> 00:03:26.910 Well, we'll get into that today. 00:03:30.370 --> 00:03:34.280 And I'm going to start by just showing you 00:03:34.280 --> 00:03:37.940 a little something that comes from my research 00:03:37.940 --> 00:03:41.680 and-- let's see. 00:03:41.680 --> 00:03:42.990 Let me do this. 00:03:42.990 --> 00:03:44.060 I think this will work. 00:03:53.310 --> 00:03:55.750 Hear that? 00:03:55.750 --> 00:04:00.380 As I go slower, does frequency go up or down? 00:04:00.380 --> 00:04:02.360 It's kind of slow, and I'm going to speed up. 00:04:05.680 --> 00:04:06.570 Right? 00:04:06.570 --> 00:04:08.980 Goes up as the speed goes up. 00:04:08.980 --> 00:04:11.520 So that's the result of the phenomenon 00:04:11.520 --> 00:04:14.440 called flow-induced vibration. 00:04:14.440 --> 00:04:20.240 And I'll give you a very brief intro 00:04:20.240 --> 00:04:21.619 to flow-induced vibration. 00:04:21.619 --> 00:04:25.980 You have a cylinder sitting still, flow coming by it-- 00:04:25.980 --> 00:04:29.050 water or air. 00:04:29.050 --> 00:04:35.070 The cylinder is diameter D, velocity U, for the flow. 00:04:35.070 --> 00:04:40.310 What happens in the wake of that cylinder, vortices are formed. 00:04:40.310 --> 00:04:42.916 And just like if you're paddling a canoe or something 00:04:42.916 --> 00:04:44.540 and stick a paddle in the water, you'll 00:04:44.540 --> 00:04:48.730 see vortices shed off the side. 00:04:48.730 --> 00:04:50.240 First you get one that's positive 00:04:50.240 --> 00:04:54.300 and then one that's negative And so one full cycle of this 00:04:54.300 --> 00:04:57.800 is from here to here. 00:04:57.800 --> 00:05:00.765 There's a frequency to this shedding. 00:05:00.765 --> 00:05:04.750 And the shedding frequency, FS, in hertz, 00:05:04.750 --> 00:05:08.530 can be predicted by a simple dimensionless parameter called 00:05:08.530 --> 00:05:15.600 the Strouhal number, St U over D. And that's approximately 0.2 00:05:15.600 --> 00:05:18.360 U/D for stationary cylinders. 00:05:18.360 --> 00:05:22.030 You can predict the frequency at which these vortices are shed. 00:05:22.030 --> 00:05:24.830 Now, associated with the shedding of vortices 00:05:24.830 --> 00:05:26.640 is a lift force. 00:05:26.640 --> 00:05:32.080 I'll call it some FL cosine omega 00:05:32.080 --> 00:05:35.340 t, which is 2 pi FS, times t. 00:05:35.340 --> 00:05:38.420 So at this frequency of vortex shedding 00:05:38.420 --> 00:05:40.780 there is a transverse force. 00:05:40.780 --> 00:05:42.860 There's actually an inline force also, 00:05:42.860 --> 00:05:45.220 which I'll call FD for drag. 00:05:45.220 --> 00:05:51.360 And it goes like cosine 2 omega s times t. 00:05:51.360 --> 00:05:53.780 It's twice the frequency of that. 00:05:53.780 --> 00:05:57.990 And so you'll get some inline oscillatory excitation 00:05:57.990 --> 00:06:01.580 and what we call cross-flow oscillatory excitation. 00:06:01.580 --> 00:06:08.250 And this is the cause of lots of things 00:06:08.250 --> 00:06:12.670 that the people who work on it call flow-induced vibration. 00:06:12.670 --> 00:06:18.030 Now, an amazing thing happens is if this cylinder is elastically 00:06:18.030 --> 00:06:22.760 mounted or is flexible, and that force starts to act on it, 00:06:22.760 --> 00:06:25.620 it will begin to vibrate. 00:06:25.620 --> 00:06:28.570 And the amazing thing, as it begins to vibrate, 00:06:28.570 --> 00:06:33.770 it correlates the shedding of these vortices 00:06:33.770 --> 00:06:35.720 all along the cylinder. 00:06:35.720 --> 00:06:38.390 So it's like soldiers marching in step 00:06:38.390 --> 00:06:40.360 going across the bridge. 00:06:40.360 --> 00:06:43.349 If everybody's walking randomly, then the bridge 00:06:43.349 --> 00:06:44.390 doesn't respond too much. 00:06:44.390 --> 00:06:45.960 But if everybody marches together, 00:06:45.960 --> 00:06:47.980 you can put a pretty good excitation into it. 00:06:47.980 --> 00:06:50.250 Well, the motion of the cylinder itself 00:06:50.250 --> 00:06:54.820 organizes these vortex shedding all along the cylinder, 00:06:54.820 --> 00:06:56.540 so they're all marching in step. 00:06:56.540 --> 00:07:00.110 And that means the force is all correlated on the length. 00:07:00.110 --> 00:07:02.640 And you can get some pretty substantial response. 00:07:02.640 --> 00:07:05.720 So that's the subject called flow-induced vibration. 00:07:05.720 --> 00:07:10.800 And with that, I'm going to show you a few slides. 00:07:10.800 --> 00:07:15.000 Let's dim the lights a little bit, if you could, to see this. 00:07:15.000 --> 00:07:17.761 There's some pictures I just want you to see better. 00:07:17.761 --> 00:07:18.260 All right. 00:07:18.260 --> 00:07:19.690 So I do flow-induced vibration. 00:07:19.690 --> 00:07:21.630 I've been doing this-- working on this 00:07:21.630 --> 00:07:23.970 for all my professional career. 00:07:23.970 --> 00:07:29.060 And it's applied, primarily, to big, flexible cylinders 00:07:29.060 --> 00:07:29.710 in the ocean. 00:07:29.710 --> 00:07:33.490 Particularly associated with the things that the US Navy does. 00:07:33.490 --> 00:07:36.990 Long cables and things and also the offshore oil industry. 00:07:36.990 --> 00:07:38.174 Next slide. 00:07:38.174 --> 00:07:39.090 Can we dim the lights? 00:07:41.870 --> 00:07:43.195 Can we dim the lights? 00:07:48.752 --> 00:07:49.960 I want you to be able to see. 00:07:49.960 --> 00:07:52.100 This is a tension leg platform. 00:07:52.100 --> 00:07:54.080 It's one of the structures that's 00:07:54.080 --> 00:07:58.070 used in the offshore industry to produce oil. 00:07:58.070 --> 00:08:03.220 And one of these might be moored in 3,000 feet of water, 00:08:03.220 --> 00:08:04.840 1,000 meters of water. 00:08:04.840 --> 00:08:07.330 Might weigh 20,000 tons. 00:08:07.330 --> 00:08:10.160 And what's connecting it-- what holds in place-- 00:08:10.160 --> 00:08:12.230 are steel cylinders a half a meter 00:08:12.230 --> 00:08:15.710 in diameter, 3,000 feet long, going vertically 00:08:15.710 --> 00:08:19.840 down from each of those three pontoon legs sticking out. 00:08:19.840 --> 00:08:21.630 And they're under a lot of tension. 00:08:21.630 --> 00:08:25.140 And in fact, it pulls the thing down into the water 00:08:25.140 --> 00:08:28.060 so the buoyancy of the whole thing 00:08:28.060 --> 00:08:30.240 puts tension on these cylinders. 00:08:30.240 --> 00:08:32.669 But now, what happens if an ocean current 00:08:32.669 --> 00:08:34.179 comes by those cylinders? 00:08:34.179 --> 00:08:37.330 Vortex shedding, and the cylinders vibrate. 00:08:37.330 --> 00:08:40.289 And if they vibrate, over time they will fatigue and fail. 00:08:40.289 --> 00:08:40.789 OK. 00:08:40.789 --> 00:08:41.617 Next slide. 00:08:41.617 --> 00:08:42.950 There's a picture of a real one. 00:08:42.950 --> 00:08:44.730 That's a bigger one called Marco Polo. 00:08:44.730 --> 00:08:47.476 It's on a launch ship that'll take it out to the site 00:08:47.476 --> 00:08:47.975 that it is. 00:08:47.975 --> 00:08:52.780 And the ship will lower and it will slide off. 00:08:52.780 --> 00:08:54.610 So these are big. 00:08:54.610 --> 00:08:56.200 Next slide. 00:08:56.200 --> 00:08:59.319 This is a diagram of the Gulf of Mexico. 00:08:59.319 --> 00:09:00.610 South America is at the bottom. 00:09:00.610 --> 00:09:02.609 The Yucatan Peninsula is sticking up there right 00:09:02.609 --> 00:09:04.010 in the middle of the bottom. 00:09:04.010 --> 00:09:06.930 This is a picture of satellite imagery of currents 00:09:06.930 --> 00:09:08.200 in the Gulf of Mexico. 00:09:08.200 --> 00:09:11.110 And there's a current that flows up off of South America 00:09:11.110 --> 00:09:14.530 into the Gulf of Mexico, goes around in a loop, 00:09:14.530 --> 00:09:15.437 and then comes out. 00:09:15.437 --> 00:09:17.770 You can see Florida sticking down in there on the right. 00:09:17.770 --> 00:09:19.890 That current comes out of the Gulf, 00:09:19.890 --> 00:09:22.805 goes around the tip of Florida, and goes up the Atlantic Coast, 00:09:22.805 --> 00:09:25.190 and is known as the Gulf Stream. 00:09:25.190 --> 00:09:27.060 But it starts as a big current that 00:09:27.060 --> 00:09:28.350 comes into the Gulf of Mexic. 00:09:28.350 --> 00:09:31.060 And, every now and then, that current pinches off an eddy. 00:09:31.060 --> 00:09:33.530 And that's what that red circle is in the middle. 00:09:33.530 --> 00:09:37.510 And it's an eddy that's many, many kilometers in diameter 00:09:37.510 --> 00:09:40.880 with surface currents on the order of a meter per second 00:09:40.880 --> 00:09:41.790 or more. 00:09:41.790 --> 00:09:48.360 And those are the biggest threat for causing 00:09:48.360 --> 00:09:51.880 flow-induced vibration failures of long members 00:09:51.880 --> 00:09:55.340 from hanging off of offshore structures. 00:09:55.340 --> 00:09:56.660 Next. 00:09:56.660 --> 00:09:59.500 So I've been doing research in this area for a long time. 00:09:59.500 --> 00:10:03.120 This is a picture taken in the summer of 1981. 00:10:03.120 --> 00:10:06.920 It is a piece of steel pipe about 2 inches 00:10:06.920 --> 00:10:09.390 in diameter and 75 feet long. 00:10:09.390 --> 00:10:12.130 It's under 750 pounds of tension, 00:10:12.130 --> 00:10:13.630 and it's pinned at each end. 00:10:13.630 --> 00:10:18.650 It behaves almost exactly like my rubber cord here. 00:10:18.650 --> 00:10:22.440 It has natural frequencies, and it will vibrate 00:10:22.440 --> 00:10:23.910 if a current comes by it. 00:10:23.910 --> 00:10:26.100 So this is actually a sandbar. 00:10:26.100 --> 00:10:29.040 And at low tide, we'd do all the work putting it up. 00:10:29.040 --> 00:10:31.280 Then, as the tide comes in, the flow 00:10:31.280 --> 00:10:33.670 is perpendicular to the cylinder, 00:10:33.670 --> 00:10:37.590 and vortices start shedding. 00:10:37.590 --> 00:10:39.100 And as the pipe begins to move, they 00:10:39.100 --> 00:10:41.150 get organized all along the length. 00:10:41.150 --> 00:10:45.300 And a typical response mode was when 00:10:45.300 --> 00:10:48.720 the vortex shedding frequency, therefore the lift force 00:10:48.720 --> 00:10:52.470 frequency, coincided with the natural frequency. 00:10:52.470 --> 00:10:56.210 Then you'd expect it to give quite a bit of response. 00:10:56.210 --> 00:11:00.740 The diagram on the left is if you cut the cylinder 00:11:00.740 --> 00:11:03.750 and looked down its axis, this is the trajectory 00:11:03.750 --> 00:11:05.290 that you'd see the cylinder make. 00:11:05.290 --> 00:11:08.030 It would sit there and just make big figure eights. 00:11:08.030 --> 00:11:11.100 So up and down vertical is its vertical motion. 00:11:11.100 --> 00:11:13.445 Flow's coming from, say, left to right. 00:11:13.445 --> 00:11:15.340 Its vertical motion is up and down. 00:11:15.340 --> 00:11:16.770 In-line motion's like this. 00:11:16.770 --> 00:11:20.500 And exactly such a phase it just makes big beautiful figure 00:11:20.500 --> 00:11:21.180 eights. 00:11:21.180 --> 00:11:23.140 That's the kind of motion you'd see. 00:11:23.140 --> 00:11:24.170 OK? 00:11:24.170 --> 00:11:28.780 So then, very much what I was talking about a minute ago, 00:11:28.780 --> 00:11:32.700 very much behavior dominated by vibration. 00:11:32.700 --> 00:11:36.560 Vibration in the third mode, cross flow, was a typical one. 00:11:36.560 --> 00:11:39.660 And fifth mode, inline, was typical. 00:11:39.660 --> 00:11:42.740 But as cylinders go in the ocean, 00:11:42.740 --> 00:11:43.900 that one's kind of short. 00:11:43.900 --> 00:11:46.840 Third mode vibration is sort of low. 00:11:46.840 --> 00:11:51.670 So as years have gone by and oil is 00:11:51.670 --> 00:11:54.000 being produced in deeper and deeper and deeper water, 00:11:54.000 --> 00:11:55.541 the cylinders we're putting out there 00:11:55.541 --> 00:11:57.420 get longer and longer and longer and longer. 00:11:57.420 --> 00:12:00.240 And the modes that are excited by currents coming by 00:12:00.240 --> 00:12:01.887 get quite high. 00:12:01.887 --> 00:12:03.220 So this is an experiment we did. 00:12:03.220 --> 00:12:05.580 It was roughly a 1/10 scale model. 00:12:05.580 --> 00:12:09.280 Model is almost 2 inches in diameter, 500 feet long. 00:12:09.280 --> 00:12:10.950 Scale that up by a factor of 10, you're 00:12:10.950 --> 00:12:13.350 up around 20 inches in diameter and 5,000 feet 00:12:13.350 --> 00:12:17.520 long, which is exactly the size of the drilling riser 00:12:17.520 --> 00:12:20.960 that BP had hung off the drilling 00:12:20.960 --> 00:12:24.800 ship when the blowout occurred. 00:12:24.800 --> 00:12:27.780 It's a piece of steel pipe, 21 inches in diameter, 00:12:27.780 --> 00:12:30.720 3/4 of an inch wall thickness, 5,000 feet long, 00:12:30.720 --> 00:12:32.320 under a lot of tension. 00:12:32.320 --> 00:12:34.340 And when ocean currents come by, it 00:12:34.340 --> 00:12:37.424 behaves just like this string. 00:12:37.424 --> 00:12:39.340 And so we're out-- this is a 1/10 scale model. 00:12:39.340 --> 00:12:42.040 So we put a big weight on the bottom of the cylinder, 00:12:42.040 --> 00:12:45.060 put it behind a boat, and towed it in the Gulf Stream. 00:12:45.060 --> 00:12:47.130 Next picture. 00:12:47.130 --> 00:12:47.992 So there's the boat. 00:12:47.992 --> 00:12:49.200 It's an oceanographic vessel. 00:12:49.200 --> 00:12:50.830 It's actually a catamaran. 00:12:50.830 --> 00:12:52.140 Next. 00:12:52.140 --> 00:12:55.860 This is a spool that had our test cylinder on it. 00:12:55.860 --> 00:12:59.340 There's a reddish object down on the bottom which 00:12:59.340 --> 00:13:08.500 is-- that's a 750-pound piece of railroad wheel, 00:13:08.500 --> 00:13:09.970 and it's the weight on the bottom. 00:13:09.970 --> 00:13:13.510 And so you'd spool this thing off, lower it down, 00:13:13.510 --> 00:13:14.770 and then do your tests. 00:13:14.770 --> 00:13:16.280 Next. 00:13:16.280 --> 00:13:18.203 Top, we measured tension inclination. 00:13:21.140 --> 00:13:25.630 And then we also had-- it's a pin joint at the top, 00:13:25.630 --> 00:13:27.550 so it would vibrate freely. 00:13:27.550 --> 00:13:30.430 Inside, though, was fiber optics. 00:13:30.430 --> 00:13:32.090 Next. 00:13:32.090 --> 00:13:34.030 We had eight optical fibers. 00:13:34.030 --> 00:13:36.070 And in those optical fibers were what 00:13:36.070 --> 00:13:38.260 we call optical strain gauges. 00:13:38.260 --> 00:13:42.850 So we had 280 optical strain gauges instrumented up and down 00:13:42.850 --> 00:13:44.932 that pipe so we could measure its vibration. 00:13:44.932 --> 00:13:47.140 And so you're looking at a cross section of the pipe. 00:13:47.140 --> 00:13:50.060 There were two optical fibers in each quadrant, 00:13:50.060 --> 00:13:55.300 and each one of those fibers had 35 sensors on it. 00:13:55.300 --> 00:13:56.470 Next. 00:13:56.470 --> 00:14:00.330 This is typical experimental case. 00:14:00.330 --> 00:14:01.370 This is the surface. 00:14:01.370 --> 00:14:03.200 This is 500 feet down. 00:14:03.200 --> 00:14:04.730 This is the current profile. 00:14:04.730 --> 00:14:07.630 So the flow velocity is about 2 feet 00:14:07.630 --> 00:14:09.860 per second near the surface, up to 4 feet 00:14:09.860 --> 00:14:11.600 per second down on the bottom. 00:14:11.600 --> 00:14:14.700 And this is the region where most of the excitation 00:14:14.700 --> 00:14:18.570 was coming from that would drive the flow-induced vibration. 00:14:18.570 --> 00:14:25.270 This is measured RMS strain caused by the bending vibration 00:14:25.270 --> 00:14:25.980 in the cylinders. 00:14:25.980 --> 00:14:28.950 And peak-- the maximum strain-- is about right there. 00:14:28.950 --> 00:14:30.200 Next. 00:14:30.200 --> 00:14:32.060 Typical response spectrum. 00:14:32.060 --> 00:14:35.090 Basically, the frequency content at three different locations. 00:14:35.090 --> 00:14:37.190 Down deep, in the middle, near the top. 00:14:37.190 --> 00:14:38.830 This is frequency. 00:14:38.830 --> 00:14:42.660 So this would be the peak that describes 00:14:42.660 --> 00:14:46.300 the principal cross-flow vibration at the vortex 00:14:46.300 --> 00:14:47.640 shedding frequency. 00:14:47.640 --> 00:14:48.140 Next. 00:14:51.140 --> 00:14:54.990 This is position, bottom to top. 00:14:54.990 --> 00:14:59.310 This is time, and these are strain records from all 00:14:59.310 --> 00:15:00.620 of those strain sensors. 00:15:00.620 --> 00:15:05.770 There's a strain sensor about every 2 meters along here. 00:15:05.770 --> 00:15:08.140 But what you're seeing is-- this is evidence. 00:15:08.140 --> 00:15:10.720 The red is the amplitude and red-- 00:15:10.720 --> 00:15:15.790 let's say red is positive strain and blue is negative strain. 00:15:15.790 --> 00:15:19.190 And so at any location on the pipe where it's vibrating, 00:15:19.190 --> 00:15:22.660 it's going to go from red to blue, red to blue, red to blue. 00:15:22.660 --> 00:15:25.284 But it's showing you that they're highly correlated 00:15:25.284 --> 00:15:26.700 all along the length, that there's 00:15:26.700 --> 00:15:31.240 a red streak all lined up, but it's not parallel to the pipe. 00:15:31.240 --> 00:15:32.320 It's inclined. 00:15:32.320 --> 00:15:34.400 This is showing you wave propagation. 00:15:34.400 --> 00:15:36.170 The behavior of the pipe is completely 00:15:36.170 --> 00:15:40.350 dominated by wave propagation, not by standing wave vibration. 00:15:40.350 --> 00:15:45.930 So totally different than that short pipe in 1981. 00:15:45.930 --> 00:15:46.716 The wave equation. 00:15:57.500 --> 00:16:02.660 Let's imagine we have a long pipe or a string like that, 00:16:02.660 --> 00:16:08.760 and it can carry waves traveling along it. 00:16:08.760 --> 00:16:13.760 The position at any location on here-- here's a coordinate x. 00:16:13.760 --> 00:16:16.040 We describe the motion at a point 00:16:16.040 --> 00:16:20.250 by a coordinate w of x and t. 00:16:20.250 --> 00:16:25.640 So it's a function of where it is and time. 00:16:25.640 --> 00:16:28.790 What describes the motion of something 00:16:28.790 --> 00:16:32.800 which obeys the wave equation is the following equation. 00:16:32.800 --> 00:16:38.400 Partial squared w with respect to x squared 00:16:38.400 --> 00:16:44.640 equals 1/c squared partial squared w with respect 00:16:44.640 --> 00:16:46.220 to t squared. 00:16:46.220 --> 00:16:49.310 That's what's known as the one-dimensional wave equation. 00:16:49.310 --> 00:16:51.540 And the one-dimensional wave equation 00:16:51.540 --> 00:16:57.240 governs an incredibly broad category of physical phenomena. 00:16:57.240 --> 00:17:02.700 Light behaves according to the wave equation. 00:17:02.700 --> 00:17:04.958 Sound propagating across the room to you 00:17:04.958 --> 00:17:07.069 is governed by the wave equation. 00:17:07.069 --> 00:17:11.859 Longitudinal vibration of rods, torsional vibration of rods-- 00:17:11.859 --> 00:17:13.609 all governed by the wave equation. 00:17:13.609 --> 00:17:15.859 So it's worthwhile to know a little bit about the wave 00:17:15.859 --> 00:17:17.609 equation. 00:17:17.609 --> 00:17:19.650 And what I showed you this morning, 00:17:19.650 --> 00:17:21.550 it has this kind of duality to it. 00:17:21.550 --> 00:17:24.940 You can have things that vibrate with standing waves and mode 00:17:24.940 --> 00:17:28.400 shapes, but the same system can support 00:17:28.400 --> 00:17:30.390 waves that travel along it. 00:17:30.390 --> 00:17:33.370 So let's figure out why that is. 00:17:36.250 --> 00:17:41.060 So I'm going to do the derivation for you of the wave 00:17:41.060 --> 00:17:42.539 equation for a string, just so you 00:17:42.539 --> 00:17:44.080 know where it comes from because then 00:17:44.080 --> 00:17:48.500 that general derivation applies to all these different things. 00:17:48.500 --> 00:17:56.540 So imagine you've got now-- we're interested in eventually 00:17:56.540 --> 00:17:57.510 getting to vibration. 00:17:57.510 --> 00:18:01.150 So I'm going to make this a finite length string. 00:18:01.150 --> 00:18:11.340 And it has this position we'll describe as a w of x and t. 00:18:11.340 --> 00:18:17.920 It has a tension, T, a mass per unit length, m. 00:18:17.920 --> 00:18:22.630 So this is like kilograms per meter 00:18:22.630 --> 00:18:28.510 is the mass per unit length of this thing which can vibrate. 00:18:28.510 --> 00:18:29.500 So tension. 00:18:29.500 --> 00:18:31.970 Mass per unit length. 00:18:31.970 --> 00:18:35.480 L, the length of it. 00:18:35.480 --> 00:18:37.820 What other parameters do we need? 00:18:37.820 --> 00:18:39.980 That'll do for the moment. 00:18:39.980 --> 00:18:43.255 Now-- so let's draw it again without. 00:18:46.730 --> 00:18:52.010 In some displaced position and what's exciting 00:18:52.010 --> 00:18:54.060 it may be my vortex shedding, and so I'm going 00:18:54.060 --> 00:18:57.560 to draw that excitation here. 00:18:57.560 --> 00:19:00.980 And that we'll describe as F of x and t, 00:19:00.980 --> 00:19:05.310 some force per unit length. 00:19:05.310 --> 00:19:07.665 So this has units of newtons per meter. 00:19:11.750 --> 00:19:15.190 Now, in that little-- there may also 00:19:15.190 --> 00:19:17.105 be drag forces, the fluid damping. 00:19:19.880 --> 00:19:24.950 So I'm going to cut out a little piece of this cylinder 00:19:24.950 --> 00:19:33.810 and do a force balance on that piece of cylinder. 00:19:33.810 --> 00:19:35.350 So basically, F equals ma. 00:19:35.350 --> 00:19:37.990 We're just applying Newton to this piece of cylinder. 00:19:37.990 --> 00:19:42.320 And I'll draw it right here. 00:19:42.320 --> 00:19:45.580 A little section of it is curved. 00:19:45.580 --> 00:19:47.910 Here's horizontal. 00:19:47.910 --> 00:19:48.770 There's horizontal. 00:19:48.770 --> 00:19:53.130 We need to evaluate all the forces on it. 00:19:53.130 --> 00:19:59.020 So the tension on this end-- so like that. 00:19:59.020 --> 00:20:01.475 And the tension on this end is some different angle. 00:20:04.070 --> 00:20:07.440 This we'll call theta 1. 00:20:07.440 --> 00:20:10.290 This we'll call theta 2. 00:20:10.290 --> 00:20:19.150 And along here are my excitation forces, F of x and t. 00:20:19.150 --> 00:20:23.360 There may be some resistance-- drag forces, damping. 00:20:23.360 --> 00:20:27.380 That'll be a damping constant, R of x, 00:20:27.380 --> 00:20:30.940 which is force per unit length per unit velocity, 00:20:30.940 --> 00:20:36.120 times-- the force on this would have 00:20:36.120 --> 00:20:37.970 to be multiplied by the velocity, so 00:20:37.970 --> 00:20:42.600 the derivative of this displacement with respect 00:20:42.600 --> 00:20:43.280 to time. 00:20:43.280 --> 00:20:49.040 That's the force along here, and it can vary with position. 00:20:49.040 --> 00:20:50.420 Have we accounted for everything? 00:20:50.420 --> 00:20:58.840 Ah, well, this is position x, and this is at x plus dx. 00:20:58.840 --> 00:21:01.685 So this little element is dx in length. 00:21:04.290 --> 00:21:07.265 And this is all for small motions. 00:21:33.470 --> 00:21:35.820 And if you assume small motions, then you 00:21:35.820 --> 00:21:43.630 can say theta 1 is approximately equal to sine theta 1. 00:21:43.630 --> 00:21:48.400 That's also approximately equal to tan theta 1. 00:21:48.400 --> 00:21:51.480 And that's equal to the derivative of w with respect 00:21:51.480 --> 00:21:53.870 to x, just the slope. 00:21:53.870 --> 00:21:56.070 We're going to take advantage of that. 00:21:56.070 --> 00:21:59.205 Theta 2, same thing. 00:21:59.205 --> 00:22:05.660 It's approximately equal to tan theta 2 here, and sin 00:22:05.660 --> 00:22:07.130 and all those things. 00:22:07.130 --> 00:22:11.100 But that, then-- the slope has changed a little bit 00:22:11.100 --> 00:22:13.270 when you go through dx. 00:22:13.270 --> 00:22:18.950 And this is equal to the slope on the left-hand side 00:22:18.950 --> 00:22:28.860 plus the rate of change of the slope times dx. 00:22:31.930 --> 00:22:34.340 So the slope on the left, this is now the slope 00:22:34.340 --> 00:22:36.490 on the right-hand side. 00:22:36.490 --> 00:22:38.290 And so now, all that's left to do 00:22:38.290 --> 00:22:54.510 is to write a force balance for that little piece 00:22:54.510 --> 00:22:55.410 on the element dx. 00:22:58.570 --> 00:23:01.510 So if positive, upward. 00:23:01.510 --> 00:23:07.240 We have a T sine theta. 00:23:07.240 --> 00:23:09.340 But because sine theta is approximately tan 00:23:09.340 --> 00:23:12.310 theta is equal to dw dx, then there's 00:23:12.310 --> 00:23:29.900 an upward force on the right-hand side, which is T. 00:23:29.900 --> 00:23:33.660 And this turns into partial squared w with respect 00:23:33.660 --> 00:23:37.180 to x squared dx. 00:23:37.180 --> 00:23:39.530 So on the right-hand side-- positive 00:23:39.530 --> 00:23:41.710 upwards-- you have T times the partial 00:23:41.710 --> 00:23:49.666 of w with respect to x, plus partial square w with respect 00:23:49.666 --> 00:23:53.300 to x squared dx. 00:23:53.300 --> 00:23:57.100 That's the upward force on the right-hand side. 00:23:57.100 --> 00:24:01.460 On the left-hand side, we have a downward force, minus T partial 00:24:01.460 --> 00:24:04.240 of w with respect to x. 00:24:04.240 --> 00:24:07.840 And you notice that this one's going to cancel that one. 00:24:07.840 --> 00:24:15.600 We have minus R of x partial w with respect to t-- 00:24:15.600 --> 00:24:17.940 that's the velocity-- dx long. 00:24:17.940 --> 00:24:21.840 Because that's force per unit length. 00:24:21.840 --> 00:24:24.020 And have we missed anything? 00:24:24.020 --> 00:24:26.660 So that's the sum of the external forces 00:24:26.660 --> 00:24:28.530 on this little slice. 00:24:28.530 --> 00:24:33.400 And that has to be equal to-- what did Newton say? 00:24:33.400 --> 00:24:39.030 The mass, which is the mass per unit length, times dx, 00:24:39.030 --> 00:24:43.300 is the total mass, times the acceleration, 00:24:43.300 --> 00:24:48.235 partial squared w with respect to t squared. 00:24:51.841 --> 00:24:56.130 So this cancels this term. 00:24:56.130 --> 00:24:59.690 And then you notice I'm left with everything 00:24:59.690 --> 00:25:04.960 as just something dx, something dx, something dx. 00:25:04.960 --> 00:25:14.360 Get rid of the dx's, and I can write-- oh, I 00:25:14.360 --> 00:25:16.280 left out something. 00:25:16.280 --> 00:25:25.900 I left out my distributed force, F of x and t dx. 00:25:25.900 --> 00:25:28.010 It's positive as it's drawn. 00:25:28.010 --> 00:25:29.030 It's over here also. 00:25:29.030 --> 00:25:32.190 So this, and I cancel out that dx. 00:25:32.190 --> 00:25:35.710 So I put them all together now and assemble them. 00:25:35.710 --> 00:25:44.410 I can write down the equation that governs this motion. 00:25:44.410 --> 00:25:49.840 So T partial square w with respect 00:25:49.840 --> 00:26:02.700 to x squared minus r of x times velocity plus f of x and t 00:26:02.700 --> 00:26:09.472 equals m partial square w with respect to t squared. 00:26:09.472 --> 00:26:11.930 And that just says that the sum of the forces on the object 00:26:11.930 --> 00:26:13.555 equals its mass times its acceleration. 00:26:18.570 --> 00:26:22.500 Now, if we're interested in natural frequencies and mode 00:26:22.500 --> 00:26:28.970 shapes, when we've been doing one and two degree of freedom 00:26:28.970 --> 00:26:31.480 systems, and we want to get the natural frequencies in mode 00:26:31.480 --> 00:26:35.530 shapes, we temporarily let the damping be 0 and the force 00:26:35.530 --> 00:26:36.341 be 0, right? 00:26:36.341 --> 00:26:37.840 So we want to do the same thing now. 00:26:37.840 --> 00:26:46.330 We're interested in how do you find the omega n's and what I 00:26:46.330 --> 00:26:47.880 call the psi n's. 00:26:47.880 --> 00:26:50.710 Because now the mode shapes are functions. 00:26:50.710 --> 00:26:52.590 And so this is a natural frequency 00:26:52.590 --> 00:26:55.020 and the mode shape for mode n. 00:26:55.020 --> 00:26:56.870 We know there's lots of modes. 00:26:56.870 --> 00:27:06.540 So we let r of x and f of x and t be 0. 00:27:06.540 --> 00:27:13.970 And when we do that, this term goes away. 00:27:13.970 --> 00:27:14.840 This term goes away. 00:27:14.840 --> 00:27:18.150 I'm just left with T partial squared w with respect 00:27:18.150 --> 00:27:19.450 to x squared equals this. 00:27:19.450 --> 00:27:21.340 And I'm going to divide through by t. 00:27:21.340 --> 00:27:29.330 So I get partial squared w with respect to x squared equals 1 00:27:29.330 --> 00:27:43.260 over T over m partial squared w with respect to t squared. 00:27:43.260 --> 00:27:49.390 And this T/m quantity turns out to be 00:27:49.390 --> 00:27:53.840 the speed of wave propagation in the medium. 00:28:01.390 --> 00:28:06.335 And that is the wave equation. 00:28:11.440 --> 00:28:14.620 So we've just found the wave equation for the string 00:28:14.620 --> 00:28:19.027 just by applying Newton's law to a little section of string. 00:28:19.027 --> 00:28:20.360 You can do that for the vibrate. 00:28:20.360 --> 00:28:21.870 You're going to do the same thing, 00:28:21.870 --> 00:28:24.650 cut out a little section of a beam, 00:28:24.650 --> 00:28:28.110 do the force balance on it, set it equal to the mass times 00:28:28.110 --> 00:28:28.890 acceleration. 00:28:28.890 --> 00:28:32.620 And for a beam, you'll get a fourth order differential 00:28:32.620 --> 00:28:34.450 equation. 00:28:34.450 --> 00:28:36.860 And it's not the wave equation. 00:28:36.860 --> 00:28:41.040 It still vibrates, but it's not governed by what 00:28:41.040 --> 00:28:43.550 we call the wave equation. 00:28:43.550 --> 00:28:53.030 OK, so this is the one dimensional wave equation. 00:28:53.030 --> 00:28:57.245 This quantity T/m is the phase velocity. 00:28:57.245 --> 00:28:58.370 It's called phase velocity. 00:29:09.304 --> 00:29:10.970 You know, that's a good one to remember. 00:29:10.970 --> 00:29:14.094 For a simple string, the speed of phenomena running down 00:29:14.094 --> 00:29:16.260 the string is the square root of the tension divided 00:29:16.260 --> 00:29:17.426 by the mass per unit length. 00:29:22.000 --> 00:29:27.460 And if you had a long string, I put that little pluck in it, 00:29:27.460 --> 00:29:29.910 and you can see that pluck running back and forth on it. 00:29:29.910 --> 00:29:31.201 That's the speed it's going at. 00:29:36.010 --> 00:29:39.667 Basically, it's called-- well, so if I have my string, 00:29:39.667 --> 00:29:41.500 and I put a little bump on it, and that bump 00:29:41.500 --> 00:29:43.630 goes zipping along, your eye will see 00:29:43.630 --> 00:29:45.670 this thing propagating at c. 00:29:53.530 --> 00:29:55.480 So to get natural frequencies in mode shapes, 00:29:55.480 --> 00:30:01.700 we basically need to solve this equation. 00:30:01.700 --> 00:30:03.955 And it's quite straightforward to do. 00:30:06.710 --> 00:30:08.760 And a technique known as separation 00:30:08.760 --> 00:30:14.440 of variables works, which means that all you're doing 00:30:14.440 --> 00:30:17.120 is saying, I believe that I'm going 00:30:17.120 --> 00:30:20.180 to be able to write the solution as some function of x 00:30:20.180 --> 00:30:26.310 only times some function of time only, product of two terms. 00:30:26.310 --> 00:30:41.250 And that in fact-- because we're interested in vibration. 00:30:41.250 --> 00:30:43.720 You can tell me what the function of time is. 00:30:46.444 --> 00:30:49.110 You're going to tell me half the solution just from observation. 00:30:49.110 --> 00:30:51.590 What is it? 00:30:51.590 --> 00:30:55.160 Just the time dependent part. 00:30:55.160 --> 00:30:57.275 It's the same as anything else that vibrates. 00:30:57.275 --> 00:30:59.370 So a single degree of freedom system, 00:30:59.370 --> 00:31:03.450 what is the time dependent function that we substitute in 00:31:03.450 --> 00:31:06.040 to find the natural frequency? 00:31:06.040 --> 00:31:06.915 AUDIENCE: [INAUDIBLE] 00:31:10.190 --> 00:31:11.332 PROFESSOR: Say again? 00:31:11.332 --> 00:31:12.540 AUDIENCE: e to the i omega t. 00:31:12.540 --> 00:31:14.850 PROFESSOR: e to the i omega t would be just fine. 00:31:14.850 --> 00:31:16.160 Cosine omega t works. 00:31:16.160 --> 00:31:17.230 Sine omega t works. 00:31:17.230 --> 00:31:19.840 But e to the i omega t is pretty easy to use. 00:31:19.840 --> 00:31:22.760 Because it's so simple to take the derivatives. 00:31:22.760 --> 00:31:25.140 So we can guess that this is going 00:31:25.140 --> 00:31:34.430 to be some W of x times Ae to the i omega t. 00:31:34.430 --> 00:31:35.240 And plug it in. 00:31:39.540 --> 00:31:42.180 Plug it into our wave equation over here. 00:31:42.180 --> 00:31:47.976 So I'll make sure I write it consistently. 00:31:53.310 --> 00:31:57.110 So we plug this into the first term. 00:31:57.110 --> 00:31:59.790 It's two derivatives with respect to x. 00:31:59.790 --> 00:32:09.270 So this is just-- and the time-dependent part just 00:32:09.270 --> 00:32:09.966 stays outside. 00:32:12.570 --> 00:32:18.730 And on the right-hand side, when we plug it in here, 1 over c 00:32:18.730 --> 00:32:21.520 squared, two derivatives with respect to time, 00:32:21.520 --> 00:32:24.120 it's going to give me minus omega squared, 00:32:24.120 --> 00:32:28.430 so minus omega squared over c squared. 00:32:28.430 --> 00:32:37.610 And then it gives me back W of x Ae to the i omega t. 00:32:37.610 --> 00:32:42.650 And now I can get rid of the Ae to the i omega t's. 00:32:42.650 --> 00:32:47.250 And I'm left with just an equation involving x only. 00:32:47.250 --> 00:32:50.250 And it's an ordinary differential equation 00:32:50.250 --> 00:32:51.750 in w of x. 00:32:58.940 --> 00:33:07.730 So it turns into d2W dx squared plus omega squared 00:33:07.730 --> 00:33:12.170 over c squared W equals 0. 00:33:12.170 --> 00:33:16.500 And you've seen this equation before. 00:33:16.500 --> 00:33:20.000 Does this not look like, have some similarity to, 00:33:20.000 --> 00:33:25.840 Mx double dot plus kx equals 0? 00:33:25.840 --> 00:33:28.520 They're basically the same equation. 00:33:28.520 --> 00:33:29.950 This one's a function of x. 00:33:29.950 --> 00:33:33.250 That one's a function of time. 00:33:33.250 --> 00:33:38.450 And we know the solution to this one is some x of t 00:33:38.450 --> 00:33:43.920 is some amplitude e to the i omega t. 00:33:43.920 --> 00:33:47.710 So therefore, we can guess that the solution to this one 00:33:47.710 --> 00:33:55.605 is W of x is going to be-- I'll write it as some B. 00:33:55.605 --> 00:33:57.290 Now I need a function of x. 00:33:57.290 --> 00:34:03.392 But it can be just like this-- e to the i, and I'll say kx. 00:34:03.392 --> 00:34:04.933 I know that's going to be a solution. 00:34:09.420 --> 00:34:11.870 So let's plug it in. 00:34:11.870 --> 00:34:23.850 If I plug that in, I get minus k squared 00:34:23.850 --> 00:34:42.469 Be to the ikx plus omega squared over c squared Be to the ikx 00:34:42.469 --> 00:34:44.260 equals 0. 00:34:44.260 --> 00:34:47.349 Well, now I get rid of these. 00:34:47.349 --> 00:34:51.860 And what I found out is that k squared is 00:34:51.860 --> 00:34:55.600 omega squared over c squared. 00:34:55.600 --> 00:34:59.845 And this has a name-- k. 00:35:04.200 --> 00:35:07.920 It's called the wave number. 00:35:07.920 --> 00:35:12.050 And it also happens to be 2 pi over lambda. 00:35:12.050 --> 00:35:13.100 We'll come back to that. 00:35:13.100 --> 00:35:14.141 Lambda is the wavelength. 00:35:14.141 --> 00:35:18.080 You have sinusoidal waves running through the medium. 00:35:18.080 --> 00:35:21.220 2 pi over lambda is the same as omega over c. 00:35:21.220 --> 00:35:28.810 And this is called the wave number-- 00:35:28.810 --> 00:35:31.120 really important quantity if you're 00:35:31.120 --> 00:35:34.500 trying to understand wave propagation in systems. 00:35:34.500 --> 00:35:36.840 And actually, this one, this definition 00:35:36.840 --> 00:35:39.150 applies to all wave bearing systems, 00:35:39.150 --> 00:35:42.830 whether or not they obey the wave equation. 00:35:42.830 --> 00:35:45.630 It'll apply to waves traveling down a beam as well. 00:35:45.630 --> 00:35:49.260 So the definition of wave number is frequency divided by speed, 00:35:49.260 --> 00:35:52.900 or 2 pi over the wavelength. 00:35:52.900 --> 00:35:55.310 Well, let's see. 00:35:55.310 --> 00:36:02.850 We can't go much further with just the wave equation itself. 00:36:02.850 --> 00:36:05.000 In order to get the natural frequencies, 00:36:05.000 --> 00:36:07.645 we have to invoke other information 00:36:07.645 --> 00:36:09.140 that we know in the problem. 00:36:09.140 --> 00:36:11.950 In particular, we know that in order 00:36:11.950 --> 00:36:20.350 to get natural frequencies, we had to create conditions 00:36:20.350 --> 00:36:21.660 where this could vibrate. 00:36:21.660 --> 00:36:24.820 In particular, I fix that end, and I fix this end, 00:36:24.820 --> 00:36:26.590 and I put some tension on it. 00:36:26.590 --> 00:36:28.574 And now it'll vibrate. 00:36:28.574 --> 00:36:29.990 But it clearly has something to do 00:36:29.990 --> 00:36:32.030 with its ends and its length. 00:36:32.030 --> 00:36:35.900 And so this is a boundary value problem. 00:36:35.900 --> 00:36:39.780 And we have to invoke the boundary conditions to actually 00:36:39.780 --> 00:36:42.550 finish finding the natural frequencies and mode shapes. 00:36:48.960 --> 00:37:00.300 Apply the boundary conditions-- so I assumed here 00:37:00.300 --> 00:37:03.690 that my W of x is going to look something like that. 00:37:03.690 --> 00:37:06.530 In order to get a little more information out of this, 00:37:06.530 --> 00:37:12.290 I'm going to write now W of x in an alternative form that's 00:37:12.290 --> 00:37:13.450 equally valid. 00:37:13.450 --> 00:37:26.050 And I'll call it B1 cosine kx plus a B2 sine kx. 00:37:26.050 --> 00:37:29.840 And I could relate that to e to the ikx, B to the ikx, 00:37:29.840 --> 00:37:31.700 by real and imaginary parts, and so forth. 00:37:31.700 --> 00:37:33.710 This is a real part. 00:37:33.710 --> 00:37:37.370 I'm saying in general it could have a cosine part and also 00:37:37.370 --> 00:37:38.630 a sine part. 00:37:38.630 --> 00:37:43.970 But now I know my boundary conditions are W at x equals 0. 00:37:43.970 --> 00:37:45.620 W of 0 is what? 00:37:45.620 --> 00:37:47.445 What's the displacement at x equals 0? 00:37:47.445 --> 00:37:48.320 AUDIENCE: [INAUDIBLE] 00:37:52.280 --> 00:37:52.955 PROFESSOR: 0. 00:37:52.955 --> 00:37:53.580 That's the pin. 00:37:53.580 --> 00:37:55.430 That's the end where it's fixed at. 00:37:55.430 --> 00:37:59.110 And we started out here with a second order 00:37:59.110 --> 00:38:00.550 partial differential equation. 00:38:00.550 --> 00:38:03.055 And a second order equation requires two boundary 00:38:03.055 --> 00:38:03.555 conditions. 00:38:03.555 --> 00:38:05.440 A fourth order equation for the beam 00:38:05.440 --> 00:38:07.560 will require four boundary conditions. 00:38:07.560 --> 00:38:08.990 We only have to find two. 00:38:08.990 --> 00:38:11.720 One of them is it has no motion on the left. 00:38:11.720 --> 00:38:15.080 So you plug in 0 for x. 00:38:15.080 --> 00:38:16.540 Cosine of 0 is 1. 00:38:16.540 --> 00:38:18.770 Sine of 0 is 0. 00:38:18.770 --> 00:38:24.670 So we find out that this is B1 times 1. 00:38:24.670 --> 00:38:26.930 But it has to be 0 as the boundary condition. 00:38:26.930 --> 00:38:30.440 So that implies B1 is 0. 00:38:30.440 --> 00:38:32.390 There's no cosines in this answer. 00:38:32.390 --> 00:38:36.990 And W at L is 0. 00:38:36.990 --> 00:38:47.160 And so that says B2 sine kL equals 0. 00:38:47.160 --> 00:38:48.680 And that's true. 00:38:48.680 --> 00:38:57.670 That's only true if kL equals n pi. 00:38:57.670 --> 00:38:59.300 So now I've found out that there's, 00:38:59.300 --> 00:39:03.430 just for vibration of a finite length string, 00:39:03.430 --> 00:39:06.950 only particular values of k that work. 00:39:06.950 --> 00:39:11.630 So that says that there are special values of k which 00:39:11.630 --> 00:39:31.560 I'll call k sub n which are equal to n pi over L. 00:39:31.560 --> 00:39:33.989 And from that, we now have our mode shapes. 00:39:33.989 --> 00:39:35.530 Because we can say, ah, well, there's 00:39:35.530 --> 00:39:41.710 special solution for this W of x that 00:39:41.710 --> 00:39:45.160 applies only when we satisfy the boundary conditions. 00:39:45.160 --> 00:39:48.345 And that will be some undetermined amplitude. 00:39:50.870 --> 00:39:53.110 B2 came from the sine term. 00:40:00.590 --> 00:40:02.900 And those are our mode shapes. 00:40:02.900 --> 00:40:04.900 And now the natural frequencies-- once 00:40:04.900 --> 00:40:07.180 you know mode shapes, natural frequencies actually 00:40:07.180 --> 00:40:09.010 become pretty trivial to find. 00:40:09.010 --> 00:40:13.930 In this case, if we know that's the mode shape, 00:40:13.930 --> 00:40:17.690 then how do we get the natural frequencies? 00:40:17.690 --> 00:40:21.260 Well, we know that-- what's the definition of k? 00:40:27.710 --> 00:40:31.880 Therefore, the particular values of k 00:40:31.880 --> 00:40:34.600 that were allowed solutions here are 00:40:34.600 --> 00:40:40.390 going to correspond to particular values of omega n. 00:40:40.390 --> 00:40:49.570 And therefore, omega n squared is just kn squared c squared. 00:40:49.570 --> 00:40:59.720 And that's n pi over L squared T/m. 00:41:04.090 --> 00:41:06.810 That's omega n squared. 00:41:06.810 --> 00:41:09.990 So the natural frequencies of a string 00:41:09.990 --> 00:41:16.020 are n pi over L root T/m. 00:41:21.210 --> 00:41:25.270 And this is in radians per second. 00:41:25.270 --> 00:41:27.670 And I like to work in hertz sometimes. 00:41:27.670 --> 00:41:33.890 So the natural frequencies in hertz-- omega n over 2 pi. 00:41:33.890 --> 00:41:39.110 And that becomes n over 2L root T/m. 00:41:42.580 --> 00:41:50.840 So the first natural frequency, f1, is 1 over 2L root T/m. 00:41:57.910 --> 00:42:00.630 Now, let's draw. 00:42:00.630 --> 00:42:02.480 What's the mode shape for the first mode? 00:42:02.480 --> 00:42:10.960 Well, it's half a sine wave, vibrates like that. 00:42:10.960 --> 00:42:11.900 It's full wavelength. 00:42:11.900 --> 00:42:14.120 I didn't leave myself quite enough room. 00:42:14.120 --> 00:42:17.350 That's half a wavelength of a sine wave. 00:42:17.350 --> 00:42:20.510 So the full wavelength would be like that. 00:42:20.510 --> 00:42:25.760 This is of length L. And so is this piece over here. 00:42:25.760 --> 00:42:30.890 So the lambda is 2L for this particular problem. 00:42:35.160 --> 00:42:38.410 Let's see, how do I want to pose this question? 00:42:45.820 --> 00:42:52.780 So how long does it take for a wave or disturbance 00:42:52.780 --> 00:43:01.600 to travel the length of this finite string? 00:43:10.991 --> 00:43:14.372 How long does it take it to go down there and back? 00:43:14.372 --> 00:43:15.580 How would you calculate that? 00:43:20.640 --> 00:43:22.190 Distance equals rate times time. 00:43:22.190 --> 00:43:24.550 What's the distance? 00:43:24.550 --> 00:43:26.530 2L. 00:43:26.530 --> 00:43:29.500 What's the speed? 00:43:29.500 --> 00:43:30.920 c. 00:43:30.920 --> 00:43:34.820 So the length of time ought to be 2L over c, right? 00:43:39.690 --> 00:44:01.710 So the time required-- and 2L divided by T over m. 00:44:01.710 --> 00:44:05.573 But f1 is T/m divided by 2L. 00:44:05.573 --> 00:44:06.072 Hmm. 00:44:11.720 --> 00:44:15.730 So the period-- so there's a direct connection 00:44:15.730 --> 00:44:22.340 between propagation speed, frequencies, wavelengths. 00:44:22.340 --> 00:44:23.750 They're very closely related. 00:44:23.750 --> 00:44:33.940 So the natural frequency of the first mode of this string, 00:44:33.940 --> 00:44:38.610 that frequency, is exactly 1 over the length of time 00:44:38.610 --> 00:44:41.572 it takes for a disturbance to travel down and back. 00:44:49.950 --> 00:44:52.070 So with that depth of understanding 00:44:52.070 --> 00:44:58.710 of how the wave equation behaves, 00:44:58.710 --> 00:45:04.990 you can guess the behavior of lots of other things 00:45:04.990 --> 00:45:07.711 that behave like that, like my rod here. 00:45:07.711 --> 00:45:09.460 I'll do a little demo with it in a second. 00:45:24.960 --> 00:45:28.780 So for example, the longitudinal vibration, 00:45:28.780 --> 00:45:32.630 stress waves running up and down this thing, 00:45:32.630 --> 00:45:35.150 obey the wave equation. 00:45:35.150 --> 00:45:40.395 So if I take this thing and drop it on the floor, 00:45:40.395 --> 00:45:42.720 it'll bounce off the floor. 00:45:42.720 --> 00:45:44.890 How long does it take to bounce off the floor? 00:45:50.060 --> 00:45:53.320 So what do you think actually-- what physics has to happen? 00:45:53.320 --> 00:45:57.800 What's required to make this thing bounce off the floor? 00:45:57.800 --> 00:46:00.650 So we're going to consider the floor infinitely rigid. 00:46:00.650 --> 00:46:01.880 It hits the floor. 00:46:01.880 --> 00:46:04.180 It actually stays there for some finite length of time, 00:46:04.180 --> 00:46:06.430 and then it leaves. 00:46:06.430 --> 00:46:09.180 So physically, when I was holding up my string, if I 00:46:09.180 --> 00:46:11.700 smacked the end, what happened? 00:46:11.700 --> 00:46:18.420 A pulse took off, ran down the end, reflected, came back. 00:46:18.420 --> 00:46:20.710 And that was one round trip. 00:46:20.710 --> 00:46:23.390 What do you suppose happens here? 00:46:23.390 --> 00:46:26.370 I put a pulse into the end. 00:46:26.370 --> 00:46:29.555 Is it a tension or compression, the strain that's felt? 00:46:29.555 --> 00:46:30.430 AUDIENCE: Compression 00:46:30.430 --> 00:46:31.000 PROFESSOR: Compression. 00:46:31.000 --> 00:46:33.130 So a little compression pulse is put into the end. 00:46:33.130 --> 00:46:37.270 That compression pulse then, when it first hits, 00:46:37.270 --> 00:46:41.190 the compression and the speed of propagation is finite. 00:46:41.190 --> 00:46:44.490 So that compression wave starts traveling up here. 00:46:44.490 --> 00:46:49.900 Behind the compression wave, this rod has come to a stop. 00:46:49.900 --> 00:46:51.330 In front of the compression wave, 00:46:51.330 --> 00:46:53.960 the rod doesn't know it hit the ground yet. 00:46:53.960 --> 00:46:56.820 It's still moving down. 00:46:56.820 --> 00:46:59.480 So that compression wave travels up, 00:46:59.480 --> 00:47:03.390 and it is decelerating each little slice of mass 00:47:03.390 --> 00:47:04.440 as it passes through. 00:47:04.440 --> 00:47:05.760 It brings it to a stop. 00:47:05.760 --> 00:47:09.090 And so the compression reaches the top end. 00:47:09.090 --> 00:47:12.520 The cylinder has come to a stop. 00:47:12.520 --> 00:47:13.347 The end is free. 00:47:13.347 --> 00:47:14.780 It can't take any strain. 00:47:14.780 --> 00:47:16.790 So an equal and opposite tension wave 00:47:16.790 --> 00:47:20.100 has to start to make the sum of them go to 0 at the end. 00:47:20.100 --> 00:47:22.710 The boundary condition at the end is no strain. 00:47:22.710 --> 00:47:24.260 So it reflects as a tension wave. 00:47:24.260 --> 00:47:25.900 Now you have a tension wave going down. 00:47:25.900 --> 00:47:29.020 And what it does is it accelerates every atom as it 00:47:29.020 --> 00:47:31.200 goes by, as it goes past it. 00:47:31.200 --> 00:47:32.480 So everything is stopped now. 00:47:32.480 --> 00:47:36.180 Now it starts down, and this thing 00:47:36.180 --> 00:47:39.750 starts rebounding from-- the top rebounds from the floor 00:47:39.750 --> 00:47:40.850 before the bottom does. 00:47:40.850 --> 00:47:43.310 The top starts going up. 00:47:43.310 --> 00:47:44.800 All of it-- more and more goes up. 00:47:44.800 --> 00:47:45.800 And one hits the bottom. 00:47:45.800 --> 00:47:48.660 The tension wave hits the floor, and it jumps off. 00:47:48.660 --> 00:47:49.930 So how long does it take? 00:47:59.430 --> 00:48:00.500 Right? 00:48:00.500 --> 00:48:02.930 And what do you guess the natural frequency 00:48:02.930 --> 00:48:07.369 of a free-free rod is? 00:48:07.369 --> 00:48:08.660 Now, it has a funny mode shape. 00:48:08.660 --> 00:48:11.550 The mode shape is not half a sine wave like this. 00:48:11.550 --> 00:48:14.240 The displacement of the rod, it has free ends. 00:48:14.240 --> 00:48:16.740 The ends are moving a lot. 00:48:16.740 --> 00:48:19.713 But I'll give you a clue. 00:48:19.713 --> 00:48:23.700 [ROD RINGING] 00:48:23.700 --> 00:48:26.820 I can hold it in the center and not damp it. 00:48:26.820 --> 00:48:28.755 What do you think the mode shape looks like? 00:48:32.560 --> 00:48:37.560 Half a wavelength long, ends are free-- cosine, 00:48:37.560 --> 00:48:41.080 maximum displacement, goes to zero, 00:48:41.080 --> 00:48:43.100 maximum negative displacement. 00:48:43.100 --> 00:48:45.500 So it's half a wavelength long, but it's 00:48:45.500 --> 00:48:48.560 a cosine half a wavelength. 00:48:48.560 --> 00:48:50.800 And the full wavelength is 2L. 00:48:55.330 --> 00:48:57.190 So this has mode shapes. 00:48:57.190 --> 00:49:00.199 The mode shapes-- I've applied different boundary conditions. 00:49:00.199 --> 00:49:01.865 These are free-free boundary conditions. 00:49:01.865 --> 00:49:07.290 The mode shapes are cosine n pi x over L. 00:49:07.290 --> 00:49:10.120 But they have to obey a certain other law 00:49:10.120 --> 00:49:14.240 that we know about, conservation of momentum. 00:49:14.240 --> 00:49:15.830 Because I've got gravity to deal with, 00:49:15.830 --> 00:49:17.163 I have to hang on to this thing. 00:49:17.163 --> 00:49:19.730 But I've picked a place to hang onto it that you can hear it. 00:49:19.730 --> 00:49:21.970 I'm not affecting the motion. 00:49:21.970 --> 00:49:24.440 There's no motion where I'm holding it. 00:49:24.440 --> 00:49:26.720 So if I were out in space, I could do this-- 00:49:26.720 --> 00:49:27.632 [ROD RINGING] 00:49:27.632 --> 00:49:30.250 --and just let it hang there in space, right? 00:49:30.250 --> 00:49:32.020 And it would sit there and ring. 00:49:32.020 --> 00:49:35.820 What is happening to the center of mass of this system 00:49:35.820 --> 00:49:37.545 as it vibrates? 00:49:37.545 --> 00:49:39.920 AUDIENCE: [INAUDIBLE] 00:49:39.920 --> 00:49:41.420 PROFESSOR: Stationary. 00:49:41.420 --> 00:49:43.310 So half of the mass of this thing's 00:49:43.310 --> 00:49:44.579 got to be moving that way. 00:49:44.579 --> 00:49:46.120 And half of the mass has to be moving 00:49:46.120 --> 00:49:48.660 that way so that the total center of mass doesn't move. 00:49:48.660 --> 00:49:51.700 Well, cosine mode shape, positive here, 00:49:51.700 --> 00:49:54.250 negative there, perfectly symmetric, center of mass 00:49:54.250 --> 00:49:55.360 doesn't move. 00:49:55.360 --> 00:49:58.030 So there's all sorts of neat little problems 00:49:58.030 --> 00:50:02.450 that you can solve just by knowing the wave equation 00:50:02.450 --> 00:50:06.220 and figuring out boundary conditions. 00:50:06.220 --> 00:50:08.250 How many of you stand in the shower at home 00:50:08.250 --> 00:50:11.150 and sing, and every now and then, 00:50:11.150 --> 00:50:14.619 you hit a note, man, you just sound great, right? 00:50:14.619 --> 00:50:16.160 And it's just all this reverberation. 00:50:16.160 --> 00:50:17.451 How many of you have done that? 00:50:17.451 --> 00:50:20.690 OK, right, what's going on? 00:50:20.690 --> 00:50:23.866 AUDIENCE: [INAUDIBLE] Natural frequency? 00:50:23.866 --> 00:50:25.490 PROFESSOR: You've hit a-- somebody said 00:50:25.490 --> 00:50:26.260 natural frequency. 00:50:26.260 --> 00:50:27.420 Of what? 00:50:27.420 --> 00:50:28.890 AUDIENCE: [INAUDIBLE] 00:50:28.890 --> 00:50:29.975 PROFESSOR: Huh? 00:50:29.975 --> 00:50:30.850 AUDIENCE: [INAUDIBLE] 00:50:33.800 --> 00:50:39.960 PROFESSOR: You've hit the natural frequency of the shower 00:50:39.960 --> 00:50:41.020 stall itself. 00:50:41.020 --> 00:50:47.310 If the shower stall is a meter across, 00:50:47.310 --> 00:50:49.780 pressure waves-- and you plot pressure 00:50:49.780 --> 00:50:55.130 inside of the shower, the lowest mode 00:50:55.130 --> 00:50:56.530 if you're plotting pressure. 00:50:56.530 --> 00:50:58.750 Well, let's plot actually molecular movement. 00:50:58.750 --> 00:51:00.500 What's the boundary condition at the wall, 00:51:00.500 --> 00:51:03.470 the molecules at the wall? 00:51:03.470 --> 00:51:04.610 They can't move, right? 00:51:04.610 --> 00:51:05.450 0. 00:51:05.450 --> 00:51:07.810 So the molecular motion at resonance 00:51:07.810 --> 00:51:11.400 in the shower stall, the molecules, the pressures making 00:51:11.400 --> 00:51:14.400 them move back and forth, looks like back to the string again. 00:51:14.400 --> 00:51:17.970 This is L. The first natural frequency 00:51:17.970 --> 00:51:21.470 of sound waves bouncing off the walls in the stall 00:51:21.470 --> 00:51:32.030 is 1 over 2L root times c, whatever c is. 00:51:32.030 --> 00:51:36.240 And c is the speed of sound in air, 00:51:36.240 --> 00:51:38.490 which is 340 meters per second. 00:51:38.490 --> 00:51:40.985 So 340 meters per second divided by 2L-- 00:51:40.985 --> 00:51:44.240 so if it's 1 meter across here, it's 00:51:44.240 --> 00:51:48.500 340 divided by 2, 170 hertz. 00:51:48.500 --> 00:51:50.750 So that first note you can hit in the 1 meter 00:51:50.750 --> 00:51:54.784 across shower stall is about 170 hertz-- pretty low. 00:51:54.784 --> 00:51:55.950 But you can hit second mode. 00:51:55.950 --> 00:51:57.720 It'd be twice that, and so forth. 00:51:57.720 --> 00:52:02.770 OK, what about an organ pipe? 00:52:02.770 --> 00:52:05.150 This is an organ pipe, wood. 00:52:05.150 --> 00:52:08.225 It's got a stoppered end. 00:52:08.225 --> 00:52:09.974 Actually, let's do it without the stopper. 00:52:09.974 --> 00:52:12.030 Now it's an open organ pipe. 00:52:12.030 --> 00:52:14.022 [ORGAN NOTE] 00:52:16.020 --> 00:52:18.390 Basic wave equation-- how would you 00:52:18.390 --> 00:52:20.020 model its boundary conditions? 00:52:22.640 --> 00:52:26.080 So you can talk about maybe particle molecular motion. 00:52:26.080 --> 00:52:30.960 This is, now again, just sound waves, so air particles. 00:52:30.960 --> 00:52:32.410 And this is now longitudinal. 00:52:32.410 --> 00:52:33.530 Things are moving inside. 00:52:33.530 --> 00:52:37.850 So what's the boundary condition at this end, free or fixed? 00:52:37.850 --> 00:52:38.352 Free. 00:52:38.352 --> 00:52:40.810 And here it's quite open, so the boundary condition on here 00:52:40.810 --> 00:52:42.550 is free. 00:52:42.550 --> 00:52:50.416 So for the molecular motion in a free-free organ pipe, 00:52:50.416 --> 00:52:53.720 you have to get back to that half a wavelength cosine thing. 00:52:53.720 --> 00:52:55.530 And if you wanted to plot pressure instead, 00:52:55.530 --> 00:52:57.880 you can write the wave equation in terms of pressure. 00:52:57.880 --> 00:53:00.735 Pressure is-- this is pressure relief here 00:53:00.735 --> 00:53:02.440 and pressure relief there. 00:53:02.440 --> 00:53:11.510 So in fact, if is displacement of the molecules, 00:53:11.510 --> 00:53:14.800 pressure would plot like that. 00:53:14.800 --> 00:53:17.670 You'd have what's called a pressure relief boundary 00:53:17.670 --> 00:53:18.480 condition. 00:53:18.480 --> 00:53:20.120 But again, it's a half wavelength long. 00:53:20.120 --> 00:53:21.540 What do you think the first natural frequency 00:53:21.540 --> 00:53:22.738 of this organ pipe is? 00:53:31.740 --> 00:53:34.220 The period would be 2L over c. 00:53:34.220 --> 00:53:35.880 The frequency would be c over 2L. 00:53:38.390 --> 00:53:53.205 So the frequency for the organ pipe open end f1 is c over 2L. 00:53:57.080 --> 00:53:59.410 [ORGAN NOTE] 00:53:59.410 --> 00:54:00.340 Check your intuition. 00:54:00.340 --> 00:54:03.300 I'm going to close the end-- still an organ pipe. 00:54:03.300 --> 00:54:07.850 Is the frequency now going to be higher or lower? 00:54:07.850 --> 00:54:08.580 Take a vote. 00:54:08.580 --> 00:54:11.430 How many think the frequency is going to go up? 00:54:11.430 --> 00:54:13.540 Raise your hands, commit. 00:54:13.540 --> 00:54:15.479 All right, down. 00:54:15.479 --> 00:54:16.770 We've got a lot of uncertainty. 00:54:16.770 --> 00:54:18.808 All right, let's do the experiment. 00:54:18.808 --> 00:54:20.720 [ORGAN NOTE] 00:54:23.588 --> 00:54:25.500 [LOWER ORGAN NOTE] 00:54:26.940 --> 00:54:28.524 How come? 00:54:28.524 --> 00:54:30.440 I find that actually kind of counterintuitive. 00:54:30.440 --> 00:54:33.290 Until I learned this, I would have guessed the opposite way. 00:54:33.290 --> 00:54:36.715 What's going on with pressure in a closed pipe? 00:54:41.610 --> 00:54:44.960 Well, here at the orifice where the sound is actually 00:54:44.960 --> 00:54:46.770 generated, it's the pressure. 00:54:46.770 --> 00:54:49.720 If we wanted to plot pressure at the opening, 00:54:49.720 --> 00:54:52.390 that's a pressure relief place. 00:54:52.390 --> 00:54:54.060 So it's 0. 00:54:54.060 --> 00:54:57.590 But at the other end where the stopper is, it's maximum. 00:55:01.070 --> 00:55:03.620 How many wavelengths is that? 00:55:03.620 --> 00:55:04.515 A quarter. 00:55:07.090 --> 00:55:12.470 And so the length of time it takes for the thing 00:55:12.470 --> 00:55:15.440 to go through one complete period 00:55:15.440 --> 00:55:31.510 is going to be 4L over c, half the frequency of the open pipe. 00:55:31.510 --> 00:55:35.260 OK, so the wave equation is really quite powerful, 00:55:35.260 --> 00:55:36.665 governs lots of things. 00:55:41.330 --> 00:55:45.810 I've got 10, 15 minutes left here. 00:55:45.810 --> 00:55:51.510 I don't want you to go away thinking that the whole world 00:55:51.510 --> 00:55:53.160 behaves like the wave equation. 00:55:53.160 --> 00:55:58.090 Because there are some important other physical systems 00:55:58.090 --> 00:55:59.900 that we care about. 00:55:59.900 --> 00:56:02.800 And I'm going to show you just one. 00:56:02.800 --> 00:56:10.500 And that's the vibration of the beam. 00:56:10.500 --> 00:56:14.030 So here's the cantilever beam. 00:56:16.630 --> 00:56:20.290 The whole table is moving. 00:56:20.290 --> 00:56:23.770 And you can see it up on the screen. 00:56:23.770 --> 00:56:26.950 OK, so its first mode vibration, tip moves maximum. 00:56:26.950 --> 00:56:30.520 It kind of looks like a quarter wavelength. 00:56:30.520 --> 00:56:32.660 It roughly is, but not exactly. 00:56:35.410 --> 00:56:36.785 So let's draw a cantilever. 00:56:43.730 --> 00:56:48.110 And most of you have had 2.001. 00:56:48.110 --> 00:56:53.310 So if you put a load P out here-- bends, 00:56:53.310 --> 00:56:55.900 goes through a displacement delta. 00:56:55.900 --> 00:57:05.170 So you know that delta equals Pl cubed over 3EI, right? 00:57:05.170 --> 00:57:07.749 And what's this I? 00:57:07.749 --> 00:57:10.120 AUDIENCE: [INAUDIBLE] 00:57:10.120 --> 00:57:12.020 PROFESSOR: Area moment of inertia. 00:57:12.020 --> 00:57:14.470 Now that you've been doing dynamics all term, 00:57:14.470 --> 00:57:16.249 we talk about mass moments of inertia. 00:57:16.249 --> 00:57:17.790 There's also area moments of inertia. 00:57:17.790 --> 00:57:20.790 So this is the area moment of inertia of a beam. 00:57:20.790 --> 00:57:23.320 In this case, our beam is a little rectangular cross 00:57:23.320 --> 00:57:24.770 section. 00:57:24.770 --> 00:57:28.550 And the neutral axis is here, a little variable y 00:57:28.550 --> 00:57:29.810 at displacement. 00:57:29.810 --> 00:57:35.130 I is the integral of y squared dA. 00:57:35.130 --> 00:57:40.575 And dA is just a little slice of area here, dA. 00:57:40.575 --> 00:57:43.830 And the integral of y squared dA is your cross sectional area 00:57:43.830 --> 00:57:48.380 moment of inertia in the direction of bending. 00:57:48.380 --> 00:57:55.980 So that is I. You can also write it as a kappa squared A. 00:57:55.980 --> 00:57:58.260 And we ran into this in dynamics. 00:57:58.260 --> 00:57:59.930 We called it the radius of gyration. 00:57:59.930 --> 00:58:03.840 You had the same thing with area moments of inertia, 00:58:03.840 --> 00:58:04.860 the radius of gyration. 00:58:04.860 --> 00:58:08.910 This is going to be really helpful in a second. 00:58:08.910 --> 00:58:15.970 So if you solve the force balance for a beam 00:58:15.970 --> 00:58:26.960 like I did for the string, take a little slice, 00:58:26.960 --> 00:58:30.270 do force balance for transverse motions-- 00:58:30.270 --> 00:58:32.550 I'm not going to grind it out. 00:58:32.550 --> 00:58:35.540 And temporarily neglect external forces and damping. 00:58:35.540 --> 00:58:38.090 I want to get to the natural frequencies and mode shapes. 00:58:38.090 --> 00:58:41.300 So the free vibration, no damping, equation 00:58:41.300 --> 00:58:48.680 looks like EI partial 4 w with respect 00:58:48.680 --> 00:58:58.970 to x to the fourth plus rho A partial squared w with respect 00:58:58.970 --> 00:59:01.330 to t squared equals 0. 00:59:01.330 --> 00:59:03.615 And now this is density, mass density. 00:59:06.440 --> 00:59:10.930 And the A, this A, is the area of the cross section. 00:59:17.290 --> 00:59:22.090 So it's just some bh with thickness times the width. 00:59:22.090 --> 00:59:26.104 So rho times A is a mass per unit length. 00:59:29.060 --> 00:59:31.190 And so mass per unit length times 00:59:31.190 --> 00:59:33.200 dx would be the little mass associated 00:59:33.200 --> 00:59:35.380 with the element times the acceleration should 00:59:35.380 --> 00:59:37.360 be the forces on the element. 00:59:37.360 --> 00:59:40.330 So that's the fourth order partial differential 00:59:40.330 --> 00:59:44.690 equation that describes the vibration of a beam. 00:59:44.690 --> 00:59:50.630 And you have to apply the boundary conditions. 00:59:50.630 --> 00:59:55.150 And for the string, it was just B1 cosine B2 sine. 00:59:55.150 --> 01:00:02.470 For the beam, it's B1 cosine plus B2 sine plus C2 cosh 01:00:02.470 --> 01:00:05.730 plus D2 sinh x. 01:00:05.730 --> 01:00:08.160 And then you have to apply four boundary conditions 01:00:08.160 --> 01:00:13.177 and solve for B1, B2, and so forth, all four of those. 01:00:13.177 --> 01:00:13.760 I won't do it. 01:00:13.760 --> 01:00:14.801 But that's how you do it. 01:00:14.801 --> 01:00:18.110 Separation-- and separation of variables works again. 01:00:18.110 --> 01:00:20.540 So we solve this, apply the boundary conditions. 01:00:26.185 --> 01:00:27.560 What are the boundary conditions? 01:00:27.560 --> 01:00:29.643 Just so you understand what I mean by the boundary 01:00:29.643 --> 01:00:31.990 conditions, what are they for a free-free beam, 01:00:31.990 --> 01:00:33.340 zero motion at the wall? 01:00:37.360 --> 01:00:43.210 No strain at the end, no bending moment at the end, 01:00:43.210 --> 01:00:44.940 no sheer force at the end-- so there's 01:00:44.940 --> 01:00:47.315 no second derivative, no third derivative. 01:00:49.990 --> 01:00:56.790 And at the wall, the slope is 0, the first derivative. 01:00:56.790 --> 01:00:59.704 No slope comes into the wall, but the slope is 0 there. 01:00:59.704 --> 01:01:01.620 So those are the different kind of boundaries. 01:01:01.620 --> 01:01:03.980 So if you have a free-free beam, you 01:01:03.980 --> 01:01:09.740 have no bending at either end and no strain at either end. 01:01:09.740 --> 01:01:11.830 Fixed-fixed beam-- no displacement, 01:01:11.830 --> 01:01:14.342 zero slopes at both at ends, and all different combinations. 01:01:14.342 --> 01:01:16.050 And every different combination gives you 01:01:16.050 --> 01:01:17.990 different natural frequencies. 01:01:17.990 --> 01:01:21.010 So you apply the boundary conditions, and for a beam, 01:01:21.010 --> 01:01:24.230 you find out that for all beams omega 01:01:24.230 --> 01:01:31.790 n can be written as some beta n, a parameter, squared, 01:01:31.790 --> 01:01:39.600 I'll call it, times the square root of EI over rho A. 01:01:39.600 --> 01:01:42.725 And this thing varies according to the boundary conditions. 01:01:45.410 --> 01:01:50.170 Now that's what you get shown in every textbook in the world. 01:01:50.170 --> 01:01:54.810 And I have a very hard time visualizing this, 01:01:54.810 --> 01:01:59.060 getting physical intuition by that. 01:01:59.060 --> 01:02:01.140 So something you never see in a textbook 01:02:01.140 --> 01:02:12.050 but I often do is let's replace I with kappa squared A. 01:02:12.050 --> 01:02:19.310 And you get a square root of E over rho and a square root of I 01:02:19.310 --> 01:02:22.740 over A. But I is kappa squared A. The A's cancel. 01:02:22.740 --> 01:02:29.040 It's the square root of kappa squared. 01:02:29.040 --> 01:02:32.346 So this, you know what E over rho is? 01:02:32.346 --> 01:02:38.960 E over rho, square root of E over rho, 01:02:38.960 --> 01:02:47.910 is the sound speed in a solid material. 01:02:47.910 --> 01:02:51.790 So the speed of stress waves traveling up and down 01:02:51.790 --> 01:02:54.409 this thing is the square root of E over rho. 01:02:54.409 --> 01:02:54.950 [ROD RINGING] 01:02:54.950 --> 01:02:55.658 This is aluminum. 01:02:55.658 --> 01:03:00.650 It's about 4,000 meters a second. 01:03:00.650 --> 01:03:03.970 So if you know just the properties of the material, 01:03:03.970 --> 01:03:04.730 you have that. 01:03:04.730 --> 01:03:08.590 And that says then omega n for beams 01:03:08.590 --> 01:03:16.020 is some beta n squared a parameter times kappa CL. 01:03:16.020 --> 01:03:18.320 And this thing, this is often written CL. 01:03:18.320 --> 01:03:20.870 It's the longitudinal sound speed. 01:03:23.980 --> 01:03:28.420 This is sound speed for waves traveling through the medium. 01:03:28.420 --> 01:03:30.990 So this tells you if you make the beam twice as thick, 01:03:30.990 --> 01:03:33.790 what do you do to its natural frequencies? 01:03:33.790 --> 01:03:37.250 Doubles-- instantly you know that. 01:03:37.250 --> 01:03:40.970 So bending properties depend a lot on the radius of gyration. 01:03:40.970 --> 01:03:43.180 And I'll give you a few natural frequencies 01:03:43.180 --> 01:03:44.770 for different boundary conditions 01:03:44.770 --> 01:03:46.560 just so you see what they behave like. 01:03:56.020 --> 01:04:06.680 So a pin-pin beam looks like that. 01:04:06.680 --> 01:04:14.469 So you put a plank across the stream, rocks on both sides, 01:04:14.469 --> 01:04:16.010 you've got a pin-pin beam, basically. 01:04:16.010 --> 01:04:17.360 It's set there in rock. 01:04:17.360 --> 01:04:20.405 So some length L has properties EI. 01:04:20.405 --> 01:04:23.410 So the natural frequencies for a pin-pin beam, 01:04:23.410 --> 01:04:29.250 the beta n's, are just n pi over L. 01:04:29.250 --> 01:04:32.690 And so your natural frequencies-- omega n 01:04:32.690 --> 01:04:38.960 looks like n pi over L quantity squared kappa CL. 01:04:43.000 --> 01:04:57.590 And for the cantilever, the natural frequencies 01:04:57.590 --> 01:05:09.795 look like omega n pi squared over 4L squared, is the beta n. 01:05:12.740 --> 01:05:16.360 And I'll write it this way again-- EI over 01:05:16.360 --> 01:05:18.870 rho A. You can always go back and do that. 01:05:18.870 --> 01:05:20.800 Or you can call it kappa CL. 01:05:20.800 --> 01:05:25.310 This is also kappa CL. 01:05:25.310 --> 01:05:29.360 But then there are some numbers you've got 01:05:29.360 --> 01:05:33.070 to use here-- 1.194 squared. 01:05:33.070 --> 01:05:34.650 That's the first mode. 01:05:34.650 --> 01:05:39.300 Second mode-- 2.988 squared. 01:05:39.300 --> 01:05:45.900 And then after that-- 5 squared, 7 squared, 9 squared. 01:05:45.900 --> 01:05:50.220 So this is the natural frequency of a cantilever. 01:05:50.220 --> 01:05:53.880 Pi squared over 4L squared times 1.194 01:05:53.880 --> 01:05:57.240 squared kappa CL, that's this natural frequency. 01:06:01.840 --> 01:06:05.590 And one final case, because I can show it 01:06:05.590 --> 01:06:08.640 to you-- the free-free case. 01:06:08.640 --> 01:06:14.822 So that's a beam bending that vibrates like that. 01:06:17.770 --> 01:06:22.810 And I happen to know on a beam for the first mode-- this 01:06:22.810 --> 01:06:24.310 is the first mode of a beam. 01:06:24.310 --> 01:06:26.710 Where these nodes are, where there's no motion, 01:06:26.710 --> 01:06:29.132 I should be able to hold it there and not damp it. 01:06:29.132 --> 01:06:31.340 And that turns out to be at about the quarter points. 01:06:34.010 --> 01:06:36.382 So whack it like that. 01:06:36.382 --> 01:06:40.720 [ROD RINGING] 01:06:40.720 --> 01:06:43.470 And do it again. 01:06:43.470 --> 01:06:44.610 [ROD RINGING] 01:06:44.610 --> 01:06:50.921 All right, so I want you to hold it about right there. 01:06:50.921 --> 01:06:52.920 Nope, you can't hold it like that, though-- just 01:06:52.920 --> 01:06:53.690 got to balance it. 01:06:53.690 --> 01:06:56.258 Because you've got to be right where the node is. 01:06:56.258 --> 01:06:59.914 [ROD RINGING] 01:06:59.914 --> 01:07:01.580 You can hear that little bit lower tone. 01:07:01.580 --> 01:07:03.160 That's that free-free bending mode. 01:07:03.160 --> 01:07:03.810 And it's just sitting. 01:07:03.810 --> 01:07:06.060 You can feel it vibrating a little bit but not much. 01:07:06.060 --> 01:07:08.060 When you're right in the right spot, 01:07:08.060 --> 01:07:11.670 you're right on the mode shape. 01:07:11.670 --> 01:07:14.120 You can almost see it if you hit it hard enough. 01:07:14.120 --> 01:07:16.210 So that's the free-free beam. 01:07:16.210 --> 01:07:23.472 And the free-free beam has natural frequencies omega n, 01:07:23.472 --> 01:07:39.160 again, pi squared over 4L squared kappa CL 3.0112 01:07:39.160 --> 01:07:47.540 squared, 5 squared, 7 squared, 9 squared, so as you go up in n. 01:07:47.540 --> 01:07:49.132 So those are the natural frequencies 01:07:49.132 --> 01:07:49.965 of a free-free beam. 01:07:53.060 --> 01:08:05.360 Oh, one last fact about beams-- so this is now 01:08:05.360 --> 01:08:08.860 a steel beam under no tension. 01:08:08.860 --> 01:08:12.310 It can support its own weight, long though. 01:08:12.310 --> 01:08:16.180 So can a beam support waves traveling down 01:08:16.180 --> 01:08:18.979 the beam, transverse waves traveling down the beam? 01:08:18.979 --> 01:08:19.810 What do you think? 01:08:22.350 --> 01:08:24.330 Well, if it can support this, it can probably 01:08:24.330 --> 01:08:25.920 support waves, right? 01:08:25.920 --> 01:08:27.500 So waves will propagate in a beam 01:08:27.500 --> 01:08:30.450 even though this is fourth order partial differential equation. 01:08:30.450 --> 01:08:33.020 But how fast do they go? 01:08:33.020 --> 01:08:35.819 That's the question. 01:08:35.819 --> 01:08:36.880 So this is a beam. 01:08:36.880 --> 01:08:39.649 And I want to know about waves traveling down it. 01:08:39.649 --> 01:08:41.590 And I'm not going to go through-- this 01:08:41.590 --> 01:08:46.540 would take another hour or so to show you where this comes from. 01:08:46.540 --> 01:08:50.859 But here's my beam. 01:08:50.859 --> 01:08:54.130 Here's a disturbance traveling along it 01:08:54.130 --> 01:08:57.899 with some speed that I'm going to call CT. 01:08:57.899 --> 01:09:00.880 It's transverse wave speed. 01:09:00.880 --> 01:09:04.189 It's the speed you'd see a crest of a wave moving 01:09:04.189 --> 01:09:06.970 at running down that beam. 01:09:06.970 --> 01:09:18.899 CT for a beam-- square root of omega kappa CL. 01:09:18.899 --> 01:09:23.819 And CL, again, is the square root of E over rho. 01:09:23.819 --> 01:09:25.790 That's the speed of sound in the material. 01:09:25.790 --> 01:09:28.920 That just turns up in here. 01:09:28.920 --> 01:09:38.250 So what does this tell you about the frequency dependence 01:09:38.250 --> 01:09:40.253 of the speed? 01:09:40.253 --> 01:09:41.794 Does the speed change with frequency? 01:09:44.979 --> 01:09:47.970 Omega kappa CL-- it's proportional to frequency. 01:09:47.970 --> 01:09:51.910 High frequency waves go faster than low frequency waves 01:09:51.910 --> 01:09:54.250 in a beam. 01:09:54.250 --> 01:09:56.450 I didn't emphasize it when we were talking about it. 01:09:56.450 --> 01:10:02.170 But the wave equation, what was c for the string? 01:10:02.170 --> 01:10:06.220 For the wave equation, the speed of wave propagation 01:10:06.220 --> 01:10:07.280 was square root of T/m. 01:10:07.280 --> 01:10:10.220 Was it frequency dependent? 01:10:10.220 --> 01:10:12.130 Always traveled at the same speed. 01:10:12.130 --> 01:10:15.455 And so there's an important consequence. 01:10:19.120 --> 01:10:24.140 So for anything that obeys the wave equation, 01:10:24.140 --> 01:10:28.330 the speed of propagation is a constant and independent 01:10:28.330 --> 01:10:30.450 to frequency. 01:10:30.450 --> 01:10:35.600 So I can make any initial shape that I make in this thing 01:10:35.600 --> 01:10:36.210 and let it go. 01:10:36.210 --> 01:10:39.260 Its initial disturbance, that little shape 01:10:39.260 --> 01:10:44.559 will stay that shape and run up and down the thing forever. 01:10:44.559 --> 01:10:46.600 And that shape-- you could imagine a little pluck 01:10:46.600 --> 01:10:47.950 like this to start with. 01:10:47.950 --> 01:10:51.370 You could imagine doing a Fourier 01:10:51.370 --> 01:10:52.720 series to approximate that. 01:10:52.720 --> 01:10:55.740 It would be made up of a bunch of different Fourier 01:10:55.740 --> 01:10:58.270 components. 01:10:58.270 --> 01:11:00.957 And yet for something that bears the wave equation, 01:11:00.957 --> 01:11:03.290 that little pluck will just stay the shape of that pluck 01:11:03.290 --> 01:11:04.890 and run around forever. 01:11:04.890 --> 01:11:07.190 But not so in a beam. 01:11:07.190 --> 01:11:09.810 If you did that in a beam, if you come up and put 01:11:09.810 --> 01:11:14.640 an impulse into a beam, all that energy 01:11:14.640 --> 01:11:16.720 would start out together. 01:11:16.720 --> 01:11:20.082 But in very brief time, the high frequency information 01:11:20.082 --> 01:11:22.415 would get out in front of the low frequency information. 01:11:22.415 --> 01:11:26.570 And if you were way down this beam, and somebody up a mile 01:11:26.570 --> 01:11:29.660 away whacks one end, and you're down further along, 01:11:29.660 --> 01:11:32.550 you'll see high frequency waves past you, 01:11:32.550 --> 01:11:34.780 and then lower frequency, and finally really slow 01:11:34.780 --> 01:11:37.300 ones coming by, the really long waves. 01:11:37.300 --> 01:11:41.560 So that's called dispersion. 01:11:41.560 --> 01:11:45.125 So beam waves are dispersive. 01:11:49.860 --> 01:11:53.100 Things that obey the wave equation are non-dispersive. 01:11:53.100 --> 01:11:56.670 The energy all travels at the same speed independent 01:11:56.670 --> 01:11:58.270 of frequency. 01:11:58.270 --> 01:12:03.700 All right, so that's it for the term. 01:12:03.700 --> 01:12:07.520 I'll see you guys on next Wednesday.