WEBVTT 00:00:00.070 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.810 Commons license. 00:00:03.810 --> 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. 00:00:26.460 --> 00:00:27.960 PROFESSOR: Today we're going to talk 00:00:27.960 --> 00:00:32.100 about this topic of vibration isolation, which 00:00:32.100 --> 00:00:35.430 is a very practical use of knowing 00:00:35.430 --> 00:00:37.320 a little bit about vibration. 00:00:37.320 --> 00:00:41.240 So imagine a situation you've been 00:00:41.240 --> 00:00:44.530 in where there's an air conditioner in a window, 00:00:44.530 --> 00:00:48.385 and it's causing your table to shake, 00:00:48.385 --> 00:00:51.360 or where you're trying to work, or your bed rattles, 00:00:51.360 --> 00:00:52.550 or something like that. 00:00:52.550 --> 00:00:54.341 How many of you have ever had an experience 00:00:54.341 --> 00:00:56.100 like that, something's kind of annoying, 00:00:56.100 --> 00:00:57.905 messing up a lab experiment, or whatever? 00:00:57.905 --> 00:01:00.180 Yeah, you've all experienced these things, right? 00:01:00.180 --> 00:01:04.900 So as clever engineers, are there 00:01:04.900 --> 00:01:08.480 simple solutions sometimes to fixing these problems? 00:01:08.480 --> 00:01:11.650 And that's what we're going to talk about today. 00:01:11.650 --> 00:01:17.200 I have a little quick demo that I'm going to show you. 00:01:17.200 --> 00:01:19.420 You saw this the other day, where 00:01:19.420 --> 00:01:23.010 this is my little squiggle pen, and it's got a rotating mass 00:01:23.010 --> 00:01:23.620 inside. 00:01:23.620 --> 00:01:25.740 And we've looked at rotating masses a lot. 00:01:25.740 --> 00:01:32.030 It has some unbalanced, statically unbalanced rotating 00:01:32.030 --> 00:01:32.560 mass. 00:01:32.560 --> 00:01:36.190 Could be a fan blade with a hunk of chewing gum or something 00:01:36.190 --> 00:01:39.540 stuck on a blade or broken piece out of it. 00:01:39.540 --> 00:01:42.770 Puts in a force me omega squared cosine omega t, basically 00:01:42.770 --> 00:01:46.220 an F0 cosine omega t kind of excitation. 00:01:46.220 --> 00:01:51.140 And if it happens to be a flexibly-mounted, mass-spring 00:01:51.140 --> 00:01:52.860 dashpot system, it'll vibrate. 00:01:52.860 --> 00:01:54.852 And so I showed you that the other day. 00:01:54.852 --> 00:01:57.310 We'll do this, and we'll need to lower the lights a little. 00:02:00.850 --> 00:02:03.030 But today, I've set your bed. 00:02:03.030 --> 00:02:06.600 This is your microscope here, this little one. 00:02:06.600 --> 00:02:08.250 Think you can see it in the foreground. 00:02:08.250 --> 00:02:12.520 And this is the air conditioner, or the water pump, 00:02:12.520 --> 00:02:14.650 or whatever is causing the trouble. 00:02:14.650 --> 00:02:20.390 So this is running at about 28 hertz. 00:02:20.390 --> 00:02:22.570 If I set the strobe just right, I 00:02:22.570 --> 00:02:26.420 can absolutely stop the motion. 00:02:30.390 --> 00:02:32.660 It doesn't look like it's moving at all. 00:02:32.660 --> 00:02:34.950 That's because the strobe is at exactly the same rate 00:02:34.950 --> 00:02:36.540 as the squiggle pen. 00:02:36.540 --> 00:02:39.570 Now I'm going to de-tune the strobe a little bit so you 00:02:39.570 --> 00:02:40.630 can see the motion. 00:02:40.630 --> 00:02:44.180 There is the motion of this main system. 00:02:44.180 --> 00:02:46.160 That's causing the problem. 00:02:46.160 --> 00:02:48.950 But it actually puts vibration into the tabletop. 00:02:48.950 --> 00:02:51.120 And next door over here, I have a little beam. 00:02:51.120 --> 00:02:52.494 And you can see that little piece 00:02:52.494 --> 00:02:54.390 of white moving up and down. 00:02:54.390 --> 00:02:57.440 It's just a little flat piece of spring steel 00:02:57.440 --> 00:03:00.150 with a magnet on the end as a mass and a piece of white tape 00:03:00.150 --> 00:03:01.480 so you can see it. 00:03:01.480 --> 00:03:03.510 But notice it's going up and down in synchrony 00:03:03.510 --> 00:03:06.290 with the other one. 00:03:06.290 --> 00:03:10.720 So this is your microscope sitting on a lab bench 00:03:10.720 --> 00:03:12.280 some distance away. 00:03:12.280 --> 00:03:14.310 The problem vibrations are being created 00:03:14.310 --> 00:03:16.980 by the original imbalance in something, 00:03:16.980 --> 00:03:19.820 travels through the floor, gets to your table 00:03:19.820 --> 00:03:20.960 with the microscope on it. 00:03:20.960 --> 00:03:22.820 Now your microscope shakes. 00:03:22.820 --> 00:03:25.780 So the issue is, what can we do about it? 00:03:30.300 --> 00:03:32.190 OK. 00:03:32.190 --> 00:03:34.370 Oh, I do-- yeah, no, I'll leave this for a second. 00:03:34.370 --> 00:03:35.440 I'll turn it off. 00:03:35.440 --> 00:03:39.140 And then I'm going to have you consider. 00:03:39.140 --> 00:03:42.691 So I want you to get in pairs and talk about this. 00:03:42.691 --> 00:03:44.440 I want, as a group, we're going to come up 00:03:44.440 --> 00:03:48.070 with at least three ways to reduce 00:03:48.070 --> 00:03:49.820 the vibration of your microscope-- 00:03:49.820 --> 00:03:53.330 relatively simple ways to fix it. 00:03:53.330 --> 00:03:55.410 How would you do it? 00:03:55.410 --> 00:03:56.240 So think about. 00:03:56.240 --> 00:03:57.290 Talk about it. 00:03:57.290 --> 00:03:59.850 And come up with three ways of fixing this problem. 00:04:40.620 --> 00:04:44.160 I'm going to do one more demo on this in a minute 00:04:44.160 --> 00:04:49.390 and then put up the transfer function for the force, the one 00:04:49.390 --> 00:04:50.510 we had the other day, OK? 00:04:50.510 --> 00:04:51.220 The picture. 00:05:16.100 --> 00:05:17.520 All right. 00:05:17.520 --> 00:05:18.720 Let's have some suggestions. 00:05:18.720 --> 00:05:20.510 How would you go about fixing this? 00:05:24.312 --> 00:05:24.812 All right. 00:05:24.812 --> 00:05:26.246 You had your hand up first. 00:05:26.246 --> 00:05:27.441 AUDIENCE: We have it on suspension systems, 00:05:27.441 --> 00:05:28.125 and springs. 00:05:28.125 --> 00:05:28.791 PROFESSOR: Yeah. 00:05:28.791 --> 00:05:31.875 So put some springs on what? 00:05:31.875 --> 00:05:34.125 AUDIENCE: Like, have a table where your microscope is, 00:05:34.125 --> 00:05:35.538 and then have [INAUDIBLE]. 00:05:39.310 --> 00:05:42.270 PROFESSOR: So springs support the microscope. 00:05:42.270 --> 00:05:43.000 All right. 00:05:43.000 --> 00:05:48.257 So I had this little magnet here sitting on this beam. 00:05:48.257 --> 00:05:49.965 This is that spring-supported microscope. 00:05:49.965 --> 00:05:51.560 And I have done a heck of a lousy job. 00:05:51.560 --> 00:05:53.410 This thing shakes like crazy. 00:05:53.410 --> 00:05:54.730 So what do you mean? 00:05:58.820 --> 00:06:00.070 Not like that. 00:06:00.070 --> 00:06:01.150 How might you-- OK. 00:06:01.150 --> 00:06:04.660 So you think you could change the properties of this system 00:06:04.660 --> 00:06:08.074 so that it might do better? 00:06:08.074 --> 00:06:09.300 Let's think about that. 00:06:09.300 --> 00:06:10.420 OK, what's another idea? 00:06:15.200 --> 00:06:17.112 AUDIENCE: Change the length of that spring 00:06:17.112 --> 00:06:19.520 could change its natural frequency. 00:06:19.520 --> 00:06:20.609 PROFESSOR: Yeah. 00:06:20.609 --> 00:06:22.900 Are you talking about the microscope one, the receiving 00:06:22.900 --> 00:06:23.400 one? 00:06:23.400 --> 00:06:25.740 If you change the length of it to make it longer, 00:06:25.740 --> 00:06:27.220 it makes it softer, actually. 00:06:27.220 --> 00:06:29.120 Make it shorter, it makes it stiffer. 00:06:29.120 --> 00:06:32.620 So you would change its natural frequency. 00:06:32.620 --> 00:06:36.370 Now, the two ideas together, if you set the natural frequency 00:06:36.370 --> 00:06:41.100 correctly, the system on the receiving end, 00:06:41.100 --> 00:06:42.324 you can reduce its vibration. 00:06:42.324 --> 00:06:43.990 And I will demonstrate that in a second. 00:06:43.990 --> 00:06:45.782 What's another idea? 00:06:45.782 --> 00:06:47.320 AUDIENCE: [INAUDIBLE]. 00:06:47.320 --> 00:06:48.320 PROFESSOR: Acid damping. 00:06:48.320 --> 00:06:48.820 OK. 00:06:48.820 --> 00:06:51.830 Well, that's a very interesting suggestion. 00:06:51.830 --> 00:06:54.667 Damping, it helps under one circumstance, 00:06:54.667 --> 00:06:55.750 but not under some others. 00:06:55.750 --> 00:06:58.500 And we're going to explore that today. 00:06:58.500 --> 00:07:00.960 This is all basically-- we've come up 00:07:00.960 --> 00:07:02.181 with one of the three ideas. 00:07:02.181 --> 00:07:02.680 What else? 00:07:05.914 --> 00:07:07.580 AUDIENCE: Attach something that vibrates 00:07:07.580 --> 00:07:09.224 180 degrees [INAUDIBLE]. 00:07:09.224 --> 00:07:10.640 PROFESSOR: Oh, that's interesting. 00:07:10.640 --> 00:07:11.556 That's the fourth one. 00:07:11.556 --> 00:07:13.350 I mean, that can be a little expensive, 00:07:13.350 --> 00:07:15.870 but these are like noise-canceling headsets, 00:07:15.870 --> 00:07:16.545 right? 00:07:16.545 --> 00:07:18.420 Could we put in something else somewhere else 00:07:18.420 --> 00:07:20.710 on the table that cancels the vibration 00:07:20.710 --> 00:07:22.271 out where your microscope is? 00:07:22.271 --> 00:07:23.270 Yeah, you could do that. 00:07:23.270 --> 00:07:24.980 A little expensive. 00:07:24.980 --> 00:07:25.550 What else? 00:07:25.550 --> 00:07:25.850 Yeah. 00:07:25.850 --> 00:07:27.516 AUDIENCE: You could cushion [INAUDIBLE]. 00:07:33.732 --> 00:07:35.190 PROFESSOR: Where would we put that? 00:07:35.190 --> 00:07:38.794 AUDIENCE: You could do it underneath [INAUDIBLE]. 00:07:38.794 --> 00:07:39.460 PROFESSOR: Yeah. 00:07:39.460 --> 00:07:41.043 And so that's generally the same idea. 00:07:41.043 --> 00:07:42.640 So you're all treating the microscope. 00:07:42.640 --> 00:07:45.090 Can you treat something else in the system? 00:07:45.090 --> 00:07:47.360 So yeah, we'll fix the microscope end. 00:07:47.360 --> 00:07:49.350 But that might knock it down by a factor of 10. 00:07:49.350 --> 00:07:51.516 I want to knock it down by at least a factor of 100, 00:07:51.516 --> 00:07:53.794 if not a factor of 1,000. 00:07:53.794 --> 00:07:55.210 AUDIENCE: Fix the air conditioner. 00:07:55.210 --> 00:07:57.607 PROFESSOR: Ah-ha-ha-ha-ha! 00:07:57.607 --> 00:08:01.910 You know, put another piece of gum on the other blade. 00:08:01.910 --> 00:08:06.760 Or clean it up, or balance the rotor, in effect. 00:08:06.760 --> 00:08:09.650 Rotors are not manufactured defective usually. 00:08:09.650 --> 00:08:15.650 They get rejected at QC before it goes out the door. 00:08:15.650 --> 00:08:17.910 So fix the rotor. 00:08:17.910 --> 00:08:20.020 Can be a little expensive sometimes, but worth it. 00:08:20.020 --> 00:08:22.500 They do maintenance checks. 00:08:22.500 --> 00:08:24.650 Actually, they have accelerometers 00:08:24.650 --> 00:08:28.620 built in to all expensive rotating equipment these days. 00:08:28.620 --> 00:08:30.557 And it's called condition monitoring. 00:08:30.557 --> 00:08:32.390 And when they get outside of certain limits, 00:08:32.390 --> 00:08:35.161 they shut the thing down and rebuild it. 00:08:35.161 --> 00:08:38.110 In electric generator sets, gas turbines, 00:08:38.110 --> 00:08:42.000 jet engines on all aircraft are all incredibly carefully 00:08:42.000 --> 00:08:42.750 balanced in a way. 00:08:42.750 --> 00:08:43.890 They start getting out of balance, 00:08:43.890 --> 00:08:46.260 they stop and fix them before the things blow up on them 00:08:46.260 --> 00:08:48.140 and you have a $20 million problem instead of 00:08:48.140 --> 00:08:51.337 maybe a $20,000 tear-down. 00:08:51.337 --> 00:08:52.710 All right? 00:08:52.710 --> 00:08:55.930 So yeah, you fix the rotor. 00:08:55.930 --> 00:08:56.730 Third idea. 00:09:03.260 --> 00:09:07.840 Well, could you-- this thing's shaking like crazy. 00:09:10.490 --> 00:09:12.980 What if you change the length of this beam? 00:09:12.980 --> 00:09:16.016 Could you stop this thing from shaking so much? 00:09:16.016 --> 00:09:18.050 And if you stop this from shaking so much, 00:09:18.050 --> 00:09:21.170 would it put so much excitation into the table? 00:09:21.170 --> 00:09:21.670 No. 00:09:21.670 --> 00:09:27.855 So fix the rotor, isolate the source, isolate the receiver. 00:09:27.855 --> 00:09:29.659 Now you get at least three ways. 00:09:29.659 --> 00:09:31.950 And the gentleman up here came up with the fourth way-- 00:09:31.950 --> 00:09:33.770 active cancellation. 00:09:33.770 --> 00:09:34.660 OK. 00:09:34.660 --> 00:09:35.690 Great. 00:09:35.690 --> 00:09:39.224 I have a demo of one of these. 00:09:39.224 --> 00:09:40.890 Sometimes if you're desperate and you're 00:09:40.890 --> 00:09:42.560 trying to get some sleep-- it's your bed 00:09:42.560 --> 00:09:44.090 that's rattling or something-- you 00:09:44.090 --> 00:09:47.650 might want to try the following that you can do very quickly. 00:09:47.650 --> 00:09:48.920 So we'll dim the lights again. 00:09:48.920 --> 00:09:50.220 We'll turn it back on. 00:09:50.220 --> 00:09:52.890 [VIBRATING] 00:09:54.130 --> 00:09:56.830 Both are vibrating like crazy. 00:09:56.830 --> 00:09:57.370 All right. 00:09:57.370 --> 00:10:00.330 So one way to detune the receiver 00:10:00.330 --> 00:10:06.920 is to put a big weight on it and change its natural frequency 00:10:06.920 --> 00:10:10.170 by changing the mass. 00:10:10.170 --> 00:10:15.080 And now the little thing is hardly shaken at all, 00:10:15.080 --> 00:10:18.370 because I've detuned just by changing the mass. 00:10:18.370 --> 00:10:21.060 Accomplished the same thing as switching the length. 00:10:21.060 --> 00:10:22.800 Instead of messing with the stiffness, 00:10:22.800 --> 00:10:25.160 I've changed the mass of the system. 00:10:25.160 --> 00:10:27.850 This is still shaking like crazy. 00:10:27.850 --> 00:10:30.420 OK. 00:10:30.420 --> 00:10:33.200 So now it's back to vibrating again. 00:10:33.200 --> 00:10:34.852 But over here is my source. 00:10:34.852 --> 00:10:36.810 I don't know if it'll work so well in this one, 00:10:36.810 --> 00:10:39.052 because the source is a lot more massive. 00:10:39.052 --> 00:10:40.760 But I'm going to put the same mass on it. 00:10:43.500 --> 00:10:45.996 Ooh, it made it worse. 00:10:45.996 --> 00:10:46.800 Ah. 00:10:46.800 --> 00:10:51.140 And that's another good demonstration. 00:10:51.140 --> 00:10:51.950 Excellent. 00:10:51.950 --> 00:10:59.120 So we could come back up with the lights. 00:10:59.120 --> 00:11:01.510 You got to be careful of that. 00:11:01.510 --> 00:11:06.090 You go to mess with one of these systems, if you do it wrong, 00:11:06.090 --> 00:11:08.180 you make the matters worse. 00:11:08.180 --> 00:11:12.940 First consulting job I ever had back in 1977 or something 00:11:12.940 --> 00:11:16.390 like that, they had a vibration problem on a ship. 00:11:16.390 --> 00:11:18.830 And the first consultant in said, stiffen up. 00:11:18.830 --> 00:11:21.152 It was actually the exhaust stacks on these 5,000 00:11:21.152 --> 00:11:23.360 horsepower diesel engines, and they were 30 feet tall 00:11:23.360 --> 00:11:24.870 and shaking like crazy. 00:11:24.870 --> 00:11:28.112 And the first guy said, stiffen up those exhaust stacks. 00:11:28.112 --> 00:11:29.570 And he did exactly the wrong thing. 00:11:29.570 --> 00:11:33.171 And it just shook worse than ever. 00:11:33.171 --> 00:11:33.670 OK. 00:11:37.310 --> 00:11:39.490 So now what I'm going to show you-- 00:11:39.490 --> 00:11:40.970 what we'll put on the board today 00:11:40.970 --> 00:11:43.350 is a little bit of mathematics to back up how you 00:11:43.350 --> 00:11:46.880 go about doing the two things. 00:11:46.880 --> 00:11:50.400 One is isolating the receiver, or the other one's 00:11:50.400 --> 00:11:51.970 isolating the source. 00:11:51.970 --> 00:11:54.260 I'm going to start with isolating the receiver. 00:11:54.260 --> 00:11:58.535 But we're going to start with a little bit of math, 00:11:58.535 --> 00:12:00.520 a little math tool that we need that will 00:12:00.520 --> 00:12:04.520 make life a lot easier for us. 00:12:04.520 --> 00:12:05.470 Yeah, I'll work here. 00:12:08.040 --> 00:12:11.860 If you recall last time, now would be the time to do it, 00:12:11.860 --> 00:12:25.420 we derived the transfer function for essentially this system, 00:12:25.420 --> 00:12:29.970 where we had an F0 cosine omega t. 00:12:29.970 --> 00:12:34.610 And we computed the response, x of t, 00:12:34.610 --> 00:12:40.580 as some x0 cosine omega t minus the phase angle. 00:12:40.580 --> 00:12:42.770 And we worked it all down to where 00:12:42.770 --> 00:12:45.240 we could plot it like that. 00:12:45.240 --> 00:12:48.910 But quite frankly, it was kind of a lot of lines of math. 00:12:48.910 --> 00:12:50.035 And it was sort of painful. 00:12:50.035 --> 00:12:52.230 I actually hated doing it on the board. 00:12:52.230 --> 00:12:54.086 But it was easy, because it was familiar. 00:12:54.086 --> 00:12:54.960 It's just trig stuff. 00:12:54.960 --> 00:12:57.600 It was all trig and a little bit of calculus. 00:12:57.600 --> 00:13:00.610 So we do that first, because it all makes sense to you 00:13:00.610 --> 00:13:01.490 mathematically. 00:13:01.490 --> 00:13:04.320 But there's a vastly easier and quicker way 00:13:04.320 --> 00:13:07.240 to do this, which we'll address right now. 00:13:07.240 --> 00:13:08.900 And that's using complex numbers. 00:13:11.830 --> 00:13:16.300 So we need a couple bits of information here. 00:13:16.300 --> 00:13:17.935 One is Euler's formula. 00:13:20.480 --> 00:13:25.560 So if you have e raised to the power i theta, 00:13:25.560 --> 00:13:30.630 you can show that that is the same thing as cosine theta 00:13:30.630 --> 00:13:35.910 plus i sine of theta. 00:13:35.910 --> 00:13:39.380 That breaks into a real part and an imaginary part. 00:13:42.780 --> 00:13:53.630 So if we wanted to express this excitation, F0 cosine omega t, 00:13:53.630 --> 00:13:56.380 in complex notation, we would say 00:13:56.380 --> 00:14:00.220 it is the real part, which I'll denote 00:14:00.220 --> 00:14:09.530 as Re, the real part of F0 e to the i omega t. 00:14:09.530 --> 00:14:16.060 And I'm going to just specify that F0 itself here, this 00:14:16.060 --> 00:14:18.725 is real and positive. 00:14:23.030 --> 00:14:24.580 So this is real and positive. 00:14:24.580 --> 00:14:27.250 So the real part's going to be F0 times-- 00:14:27.250 --> 00:14:31.780 and if you break down co ee to the i omega t into its-- 00:14:31.780 --> 00:14:34.250 by Euler's formula, it gives you cosine omega t 00:14:34.250 --> 00:14:35.340 plus i sine omega t. 00:14:35.340 --> 00:14:38.970 And the real part is the cosine part just according to this. 00:14:38.970 --> 00:14:39.470 OK. 00:14:44.160 --> 00:14:52.800 Now, another little fact that I want to show you 00:14:52.800 --> 00:15:00.930 is if you have a complex number, a plus bi, 00:15:00.930 --> 00:15:06.030 I want to express it as some ce to the i theta. 00:15:06.030 --> 00:15:08.120 So you want to use Euler's formula 00:15:08.120 --> 00:15:12.170 to express a complex number. 00:15:12.170 --> 00:15:15.950 And if we draw it, the answer becomes pretty obvious. 00:15:15.950 --> 00:15:19.190 This is a point up here, a, bi. 00:15:19.190 --> 00:15:23.530 And this is the imaginary axis here and the real. 00:15:28.320 --> 00:15:36.460 And this point, this side is a, and this side here is b. 00:15:36.460 --> 00:15:40.310 And this side here, the length of this triangle, is c. 00:15:40.310 --> 00:15:43.870 And the angle in here is theta. 00:15:43.870 --> 00:15:46.160 So now if I ask you, what's c, well, you say, 00:15:46.160 --> 00:15:50.610 oh, well, c is obviously the square root of a squared 00:15:50.610 --> 00:15:53.210 plus b squared. 00:15:53.210 --> 00:16:01.790 And theta is a tangent inverse of b/a, 00:16:01.790 --> 00:16:05.400 which is the imaginary part over the real part. 00:16:18.900 --> 00:16:21.610 If you want to express a complex number this way, 00:16:21.610 --> 00:16:23.770 well, the magnitude is square root of a squared 00:16:23.770 --> 00:16:24.890 plus b squared. 00:16:24.890 --> 00:16:26.820 And the phase angle that you put up here 00:16:26.820 --> 00:16:31.310 is tangent inverse of the imaginary part over the real. 00:16:31.310 --> 00:16:31.810 OK. 00:16:35.500 --> 00:16:42.450 So now we have the basic tools we 00:16:42.450 --> 00:16:46.540 need to take on the vibration problem. 00:16:46.540 --> 00:16:48.295 And so we have that system up there. 00:16:53.690 --> 00:16:59.330 And our output, from the way we derived it last time, 00:16:59.330 --> 00:17:05.724 the output is some x0 cosine omega t minus phi. 00:17:05.724 --> 00:17:08.140 And one of the reasons it was so painful doing it this way 00:17:08.140 --> 00:17:10.140 last time is you have to-- this is 00:17:10.140 --> 00:17:12.630 cosine is a function of both time and phase. 00:17:12.630 --> 00:17:16.670 And to break it apart takes a lot of work. 00:17:16.670 --> 00:17:19.520 So we want to do the same thing, but with complex variables 00:17:19.520 --> 00:17:20.630 this time. 00:17:20.630 --> 00:17:28.620 So I want to express this then as the real part of-- I 00:17:28.620 --> 00:17:34.780 could say it's the real part of x0 e to the i omega 00:17:34.780 --> 00:17:36.200 t minus phi. 00:17:39.010 --> 00:17:42.970 And despite Euler's formula if this is real, just the number. 00:17:42.970 --> 00:17:46.180 Than this breaks down into cosine omega t minus phi and i 00:17:46.180 --> 00:17:48.480 sine omega t minus phi. 00:17:52.260 --> 00:17:56.150 But here's the beauty of using complex notation 00:17:56.150 --> 00:17:58.230 and exponentials. 00:17:58.230 --> 00:18:09.030 This now becomes the real part of x0 e to the minus i phi 00:18:09.030 --> 00:18:11.945 times e to the i omega t. 00:18:11.945 --> 00:18:13.740 If I could separate these two. 00:18:13.740 --> 00:18:15.290 And this is what makes it so much 00:18:15.290 --> 00:18:19.890 easier to use this approach. 00:18:19.890 --> 00:18:26.370 And I'm going to call this part of it just some capital X. 00:18:26.370 --> 00:18:30.180 And it is a complex number. 00:18:30.180 --> 00:18:30.730 For sure. 00:18:30.730 --> 00:18:39.140 If I break this up into cosine of minus phi and minus i sine 00:18:39.140 --> 00:18:40.810 phi, it's got an i sine phi. 00:18:40.810 --> 00:18:44.830 So this is a complex number-- a plus bi kind of form. 00:18:44.830 --> 00:18:46.650 So in general, this thing is complex. 00:18:49.420 --> 00:18:52.630 So this whole thing is the real part 00:18:52.630 --> 00:18:57.650 of some X, which I don't know now, e to the i omega t. 00:19:05.980 --> 00:19:10.000 So now we can quite quickly do the derivation 00:19:10.000 --> 00:19:11.650 we did last time. 00:19:11.650 --> 00:19:15.760 We're talking about representing linear systems by some kind 00:19:15.760 --> 00:19:19.750 of black box-- has a transfer function in it, which we call 00:19:19.750 --> 00:19:26.700 H, in this case, x/F. Response x per unit input F. And remember, 00:19:26.700 --> 00:19:35.360 we're talking about steady state response only. 00:19:39.590 --> 00:19:47.730 And we have, as our input here, some F0 e to the i omega t. 00:19:47.730 --> 00:19:53.130 And we have, as an output, some X e to the i omega t. 00:19:53.130 --> 00:19:55.620 And we know that we're going to use the convention that we 00:19:55.620 --> 00:20:01.650 care about we have to have real number answers. 00:20:01.650 --> 00:20:04.640 So we'll be eventually actually using 00:20:04.640 --> 00:20:07.970 the real part of the input and the real part of the output. 00:20:07.970 --> 00:20:11.350 But to get there, we're going to use complex notation first 00:20:11.350 --> 00:20:15.560 and then separate out the real and imaginary parts at the end. 00:20:21.660 --> 00:20:36.900 So for our system, we know the equation of motion, 00:20:36.900 --> 00:20:51.110 so now it's some F0 e to the i omega t. 00:20:51.110 --> 00:20:53.190 There's our equation of motion. 00:20:53.190 --> 00:21:02.900 And I'm going to let x here be this unknown capital X e 00:21:02.900 --> 00:21:04.890 to the i omega t. 00:21:04.890 --> 00:21:06.260 And I'm going to plug it in. 00:21:09.870 --> 00:21:12.040 And the exponentials are particularly 00:21:12.040 --> 00:21:14.245 easy to deal with when you're taking derivatives. 00:21:18.470 --> 00:21:23.790 So upon doing that, we immediately get minus omega 00:21:23.790 --> 00:21:35.450 squared M plus i omega c plus k. 00:21:38.100 --> 00:21:43.050 All of that times Xe to the i omega t 00:21:43.050 --> 00:21:47.190 equals F0 e to the i omega t. 00:21:51.650 --> 00:21:54.935 Immediately, I can get rid of the time-dependent parts. 00:21:58.680 --> 00:22:03.420 And I can solve for x/F, which is what we set out 00:22:03.420 --> 00:22:07.160 to do the other day to find this transfer function between input 00:22:07.160 --> 00:22:07.830 and output. 00:22:21.360 --> 00:22:25.370 So if I solve for x divided by F, 00:22:25.370 --> 00:22:28.200 I'm going to get all this stuff and the denominator 00:22:28.200 --> 00:22:29.660 on one side. 00:22:29.660 --> 00:22:31.290 And I'll write it out here. 00:22:53.610 --> 00:22:55.730 It simply looks like that. 00:22:55.730 --> 00:22:58.730 And now remember, I can substitute in some things. 00:22:58.730 --> 00:23:04.790 I remember k/m is omega n squared. 00:23:04.790 --> 00:23:10.920 And zeta in c over 2 omega n M. 00:23:10.920 --> 00:23:15.780 And I plug those things in and just 00:23:15.780 --> 00:23:18.250 rearrange it a little tiny bit. 00:23:18.250 --> 00:23:21.400 We should come up with something like we 00:23:21.400 --> 00:23:32.560 found before, so that x/F, 1/k, and the denominator, 00:23:32.560 --> 00:23:40.570 1 minus omega squared over omega n squared-- not quite yet 00:23:40.570 --> 00:23:47.105 here-- plus 2 i zeta omega over omega n. 00:23:51.327 --> 00:23:52.410 That's what it looks like. 00:23:55.420 --> 00:23:57.260 So you still have a complex denominator. 00:24:02.150 --> 00:24:07.220 And this basically looks like a number 1 over k times 1 00:24:07.220 --> 00:24:08.960 over some a plus bi. 00:24:12.830 --> 00:24:13.810 There's your a term. 00:24:13.810 --> 00:24:14.840 Here's your bi term. 00:24:26.892 --> 00:24:28.850 And the way you deal with something like this-- 00:24:28.850 --> 00:24:32.090 you have an a plus bi in the denominator-- you multiply 00:24:32.090 --> 00:24:34.900 the numerator and denominator by the complex conjugate 00:24:34.900 --> 00:24:38.250 in order to get this into actually standard a plus bi 00:24:38.250 --> 00:24:41.150 form. 00:24:41.150 --> 00:24:46.630 If I do that symbolically here, it 00:24:46.630 --> 00:24:51.190 comes out looking like a minus bi over a squared 00:24:51.190 --> 00:24:52.050 plus b squared. 00:24:56.108 --> 00:25:06.710 And that's 1/k e to the minus i phi 00:25:06.710 --> 00:25:12.240 over square root of a squared plus b squared. 00:25:15.600 --> 00:25:18.125 Because now, see, the denominator's 00:25:18.125 --> 00:25:20.120 just a real number. 00:25:20.120 --> 00:25:25.270 So this whole thing is, in some form, c plus a di. 00:25:25.270 --> 00:25:29.410 You could break this into a real part, complex part. 00:25:29.410 --> 00:25:31.680 We could say that's equal to some magnitude 00:25:31.680 --> 00:25:36.050 times e to the i phi. 00:25:36.050 --> 00:25:41.160 To get the magnitude, you take a squared 00:25:41.160 --> 00:25:42.840 plus b squared square root. 00:25:42.840 --> 00:25:44.100 It cancels. 00:25:44.100 --> 00:25:46.380 This is squared the denominator, so you 00:25:46.380 --> 00:25:50.760 end up with this part, square root, in the denominator. 00:25:54.690 --> 00:25:59.470 This is what the-- we need to know what phi looks like. 00:25:59.470 --> 00:26:03.040 Well, phi had better come out like before, 00:26:03.040 --> 00:26:13.530 where now phi is minus tangent inverse of the imaginary part 00:26:13.530 --> 00:26:16.370 over the real part. 00:26:16.370 --> 00:26:20.250 And the imaginary part has a minus here. 00:26:20.250 --> 00:26:22.710 That's why a minus pops up here. 00:26:22.710 --> 00:26:24.780 Imaginary part comes from this. 00:26:24.780 --> 00:26:27.410 The real part comes from there. 00:26:27.410 --> 00:26:29.680 The common denominator stuff all cancels out 00:26:29.680 --> 00:26:31.060 when you take the ratio. 00:26:31.060 --> 00:26:41.820 So this is tangent inverse of two zeta omega over omega n 00:26:41.820 --> 00:26:47.280 all over 1 minus omega squared over omega n squared, 00:26:47.280 --> 00:26:49.220 as before. 00:26:49.220 --> 00:26:52.460 I've skipped a couple of steps, but we cranked this whole thing 00:26:52.460 --> 00:26:53.030 out before. 00:26:55.690 --> 00:27:02.160 This is the same steps that you would go through to do that. 00:27:02.160 --> 00:27:05.850 We're just doing this to get to the phase angle. 00:27:08.530 --> 00:27:20.550 But this now is exactly the same thing we got before, 00:27:20.550 --> 00:27:22.310 which we have plotted up there. 00:27:28.560 --> 00:27:30.380 We work with magnitude and phase angle. 00:27:30.380 --> 00:27:36.650 So the magnitude of x/F is the same thing 00:27:36.650 --> 00:27:39.795 as saying the magnitude of the transfer function. 00:27:43.500 --> 00:27:48.510 And that transfer function looks like 1/k, the magnitude, 00:27:48.510 --> 00:27:56.060 all divided by 1 minus omega squared over omega n squared 00:27:56.060 --> 00:28:06.020 squared plus 2 zeta omega over omega n squared square root. 00:28:06.020 --> 00:28:09.970 That is the same transfer function magnitude 00:28:09.970 --> 00:28:15.060 that we derived last time, with a lot more work. 00:28:15.060 --> 00:28:18.200 And this approach, using complex variables, 00:28:18.200 --> 00:28:23.470 you can use for any single input, single output 00:28:23.470 --> 00:28:24.660 linear system. 00:28:24.660 --> 00:28:28.270 And we're going to do it to derive right away the transfer 00:28:28.270 --> 00:28:34.905 function for the response of this to motion of the base. 00:28:39.460 --> 00:28:42.325 So if you follow how we used this complex variables in e 00:28:42.325 --> 00:28:45.290 to the i omega t's to get here, we 00:28:45.290 --> 00:28:49.170 can now apply the same tools to do other transfer functions 00:28:49.170 --> 00:28:50.755 to be a lot more efficient about it. 00:28:57.880 --> 00:29:02.920 Before I jump to this one, remind you 00:29:02.920 --> 00:29:06.260 how, in practice, we use this. 00:29:06.260 --> 00:29:09.400 So if the statement magnitude of x/F 00:29:09.400 --> 00:29:11.980 equals everything on the right there. 00:29:11.980 --> 00:29:14.020 Then in the way we would normally 00:29:14.020 --> 00:29:16.830 use this is to say, well, if you want the magnitude 00:29:16.830 --> 00:29:22.640 of the response, you take the magnitude of the input force, 00:29:22.640 --> 00:29:26.160 multiply it by the magnitude of the transfer function, 00:29:26.160 --> 00:29:28.655 evaluate it at the correct frequency. 00:29:34.420 --> 00:29:35.840 That would give you the magnitude. 00:29:35.840 --> 00:29:39.620 If you want the time dependence, x of t, 00:29:39.620 --> 00:29:42.810 well, that's the magnitude of the force, 00:29:42.810 --> 00:29:46.400 magnitude of the transfer function 00:29:46.400 --> 00:29:55.680 times the real part of e to the i omega t minus phi. 00:29:59.050 --> 00:30:02.780 And this gets us back to when you work this out, 00:30:02.780 --> 00:30:05.010 this is your x0. 00:30:05.010 --> 00:30:11.730 And this is your cosine omega t minus phi. 00:30:16.400 --> 00:30:21.400 So once you know what the excitation force is 00:30:21.400 --> 00:30:25.640 and its frequency, you put the force in here. 00:30:25.640 --> 00:30:31.670 You evaluate that thing on the left at the correct frequency. 00:30:31.670 --> 00:30:34.170 And you write out the answer directly. 00:30:34.170 --> 00:30:37.320 In one of the homeworks for today, 00:30:37.320 --> 00:30:40.330 the question just had you go through the exercise 00:30:40.330 --> 00:30:43.770 of figuring this out at three different frequency 00:30:43.770 --> 00:30:47.700 ratios, like 1/2, 1, and 3, or something like that, 00:30:47.700 --> 00:30:51.950 would put you to the left of the peak at 1/2, on the peak at 1, 00:30:51.950 --> 00:30:55.810 and way off to the right out at the right edge at 3. 00:30:55.810 --> 00:30:58.170 And you'll get three different response 00:30:58.170 --> 00:31:01.280 amplitudes and three different phase angles that go with it. 00:31:05.580 --> 00:31:06.080 All right. 00:31:06.080 --> 00:31:08.340 So that's how you review. 00:31:08.340 --> 00:31:10.890 Did the same thing a different way. 00:31:10.890 --> 00:31:13.330 And I'm going to move on to base motion. 00:31:13.330 --> 00:31:14.720 But any questions about this now? 00:31:14.720 --> 00:31:15.220 Yeah. 00:31:15.220 --> 00:31:18.385 AUDIENCE: Was the e to the negative ib included 00:31:18.385 --> 00:31:20.965 in your F of x? 00:31:20.965 --> 00:31:21.760 PROFESSOR: Yes. 00:31:21.760 --> 00:31:25.225 AUDIENCE: OK, so why did the negative 00:31:25.225 --> 00:31:27.205 b appear again after your final [INAUDIBLE]? 00:31:34.140 --> 00:31:35.560 PROFESSOR: The very top expression 00:31:35.560 --> 00:31:40.120 up there, it says x/F. It says we're 00:31:40.120 --> 00:31:45.420 trying to cast it in the e to the minus i phi form. 00:31:45.420 --> 00:31:46.560 That's my goal. 00:31:46.560 --> 00:31:49.080 And I did that, because we started over here 00:31:49.080 --> 00:31:51.280 with the problem that we had done before, 00:31:51.280 --> 00:31:54.380 where that's the way we decided to write the answer. 00:31:54.380 --> 00:31:59.260 And it turns out that it's just a convention in vibration 00:31:59.260 --> 00:32:05.550 engineering that authors and people have adopted to express 00:32:05.550 --> 00:32:07.271 the phase angle as minus phi. 00:32:07.271 --> 00:32:08.770 They could have done it as plus phi. 00:32:11.590 --> 00:32:13.306 The plots like this are phi. 00:32:13.306 --> 00:32:13.972 AUDIENCE: Right. 00:32:13.972 --> 00:32:20.870 But I guess what I'm wondering, isn't [INAUDIBLE] x/F. 00:32:20.870 --> 00:32:23.110 PROFESSOR: Oh, I see what you mean. 00:32:23.110 --> 00:32:26.280 It's in there before you take its magnitude. 00:32:26.280 --> 00:32:39.290 So the Hx/F, when it is-- this here 00:32:39.290 --> 00:32:41.680 is left in complex notation. 00:32:41.680 --> 00:32:46.440 And this is Hx/F of omega. 00:32:49.880 --> 00:32:53.180 And it is complex. 00:32:53.180 --> 00:32:55.630 We take its magnitude. 00:32:55.630 --> 00:32:58.620 Then the magnitude is not complex, right? 00:32:58.620 --> 00:33:00.760 And so we take its magnitude. 00:33:00.760 --> 00:33:03.399 We get that expression. 00:33:03.399 --> 00:33:04.940 But when we take its magnitude, we've 00:33:04.940 --> 00:33:07.170 thrown away the phase information. 00:33:07.170 --> 00:33:09.650 So we have to keep it and put it somewhere. 00:33:09.650 --> 00:33:16.380 And so we put it in the e to the i phi form. 00:33:16.380 --> 00:33:20.880 And I guess what I should have done here 00:33:20.880 --> 00:33:27.160 is now this is-- I've taken-- this 00:33:27.160 --> 00:33:34.610 is Hx/F, same thing as x/F, in complex form. 00:33:34.610 --> 00:33:37.620 And I've said, OK, if I write it this way, 00:33:37.620 --> 00:33:39.476 I have just said it is a magnitude. 00:33:42.460 --> 00:33:44.930 Times its phase information. 00:33:44.930 --> 00:33:48.410 I've separated its phase information from its magnitude 00:33:48.410 --> 00:33:51.548 by writing it this way. 00:33:51.548 --> 00:33:53.340 OK. 00:33:53.340 --> 00:33:56.330 And the phase then is that. 00:33:56.330 --> 00:33:59.920 And its magnitude is that. 00:33:59.920 --> 00:34:00.639 Good question. 00:34:00.639 --> 00:34:01.139 All right. 00:34:11.679 --> 00:34:15.580 So now let's see if we can kind of pretty quickly 00:34:15.580 --> 00:34:17.450 do the same problem for base motion. 00:34:31.560 --> 00:34:35.540 So this is our microscope now, idealizes a mass spring system. 00:34:35.540 --> 00:34:36.844 So this is our microscope. 00:34:41.510 --> 00:34:50.810 Has some mass stiffness damping motion, x of t. 00:34:50.810 --> 00:34:55.580 And how do you suppose-- where would you measure 00:34:55.580 --> 00:34:59.100 that motion x of t from? 00:34:59.100 --> 00:35:02.590 Like, to define your coordinate here 00:35:02.590 --> 00:35:06.985 is a major point in the last homework. 00:35:06.985 --> 00:35:08.335 Is gravity involved? 00:35:10.980 --> 00:35:14.380 But only as a constant term, mg in the equation of motion. 00:35:14.380 --> 00:35:16.910 It's only there depending on if you 00:35:16.910 --> 00:35:20.630 write the equation of motion in the less desirable way. 00:35:20.630 --> 00:35:24.420 Where is this measured from do you guess? 00:35:24.420 --> 00:35:25.580 Equilibrium position? 00:35:25.580 --> 00:35:27.010 Static equilibrium position? 00:35:27.010 --> 00:35:28.515 Or 0 spring force position? 00:35:31.160 --> 00:35:34.300 How many suggest 0 spring force position? 00:35:34.300 --> 00:35:35.970 How many suggest static equilibrium? 00:35:35.970 --> 00:35:36.470 OK. 00:35:36.470 --> 00:35:38.440 You got the message. 00:35:38.440 --> 00:35:41.570 This is from equilibrium, because you don't 00:35:41.570 --> 00:35:43.910 have to deal with the mg term. 00:35:43.910 --> 00:35:46.090 So this is measured from equilibrium. 00:35:46.090 --> 00:35:49.900 That's the deflection of the microscope support. 00:35:49.900 --> 00:35:53.880 This is the deflection of the floor that's driving it. 00:35:53.880 --> 00:35:56.337 Then we know we've got that table shaking like crazy. 00:35:56.337 --> 00:35:57.920 That's what's causing this to vibrate. 00:36:01.940 --> 00:36:03.750 And we need a free-body diagram. 00:36:07.360 --> 00:36:09.740 And we approach free-body diagrams just like before. 00:36:09.740 --> 00:36:14.940 You imagine positive motions of x and x dot, 00:36:14.940 --> 00:36:18.830 positive motions of y and y dot, and deduce their forces. 00:36:18.830 --> 00:36:24.410 So positive x gives you a kx opposing. 00:36:24.410 --> 00:36:30.890 A positive x dot gives you a cx dot opposing. 00:36:30.890 --> 00:36:35.290 A positive y gives you what? 00:36:35.290 --> 00:36:37.460 A force that results on this. 00:36:37.460 --> 00:36:38.920 Positive motion of the floor. 00:36:46.510 --> 00:36:49.336 Positive or negative force? 00:36:49.336 --> 00:36:51.531 How many think positive? 00:36:51.531 --> 00:36:53.870 How many think negative? 00:36:53.870 --> 00:36:55.090 How many aren't sure? 00:36:55.090 --> 00:36:58.050 How many aren't awake? 00:36:58.050 --> 00:36:58.650 OK. 00:36:58.650 --> 00:36:59.800 Look. 00:36:59.800 --> 00:37:03.120 If I push up on this-- and now this 00:37:03.120 --> 00:37:05.640 is fixed when you do this mental experiment. 00:37:05.640 --> 00:37:07.120 You fix this momentarily. 00:37:07.120 --> 00:37:09.470 You cause a positive deflection here. 00:37:09.470 --> 00:37:11.250 It compresses the spring. 00:37:11.250 --> 00:37:14.700 Does the spring push back or not? 00:37:14.700 --> 00:37:18.182 So if I'm moving upwards, which way is the spring pushing? 00:37:18.182 --> 00:37:18.890 All right. 00:37:24.040 --> 00:37:26.210 But if I'm pushing upwards, which way is 00:37:26.210 --> 00:37:27.920 the spring pushing on the mass? 00:37:27.920 --> 00:37:29.740 Up. 00:37:29.740 --> 00:37:33.210 So this one gives me a ky up. 00:37:33.210 --> 00:37:38.750 And the dashpot does a similar thing-- cy dot up. 00:37:38.750 --> 00:37:41.870 And there's also an mg here. 00:37:41.870 --> 00:37:46.770 But there's also a kx static, if you will, up. 00:37:46.770 --> 00:37:47.530 And they cancel. 00:37:47.530 --> 00:37:48.140 We know that. 00:37:48.140 --> 00:37:51.180 So we don't have to deal with the mg terms. 00:37:51.180 --> 00:37:52.990 So now we can write our equation of motion. 00:37:52.990 --> 00:37:57.880 And the equation of motion for this system 00:37:57.880 --> 00:38:01.100 is the mass times the acceleration. 00:38:01.100 --> 00:38:03.380 That's got to equal to the sum of all the 00:38:03.380 --> 00:38:06.550 external forces-- one, two, three, four of them. 00:38:06.550 --> 00:38:10.160 And I'll just save a little time and board space. 00:38:10.160 --> 00:38:15.700 I'll put them on the correct sides of the equation. 00:38:15.700 --> 00:38:19.730 So these are the x-- put the x terms on the left side. 00:38:19.730 --> 00:38:23.250 cx dot plus kx. 00:38:23.250 --> 00:38:32.080 And on the right-hand side, I get ky plus cy dot. 00:38:32.080 --> 00:38:34.120 This is my excitation. 00:38:34.120 --> 00:38:35.980 That's the floor motion. 00:38:35.980 --> 00:38:38.170 And this is my response on the left-hand side. 00:38:40.690 --> 00:38:46.880 So I'm going to let y of t, the input, 00:38:46.880 --> 00:38:56.000 be some y0 real positive times e to the i omega t. 00:38:56.000 --> 00:39:01.080 And I'm going to assume that the response is some x, probably 00:39:01.080 --> 00:39:05.330 complex, e to the i omega t. 00:39:07.950 --> 00:39:10.520 So this is x of t here. 00:39:10.520 --> 00:39:12.931 Equals some x I don't know e to the i omega t. 00:39:12.931 --> 00:39:15.055 And I'm going to plug those two into this equation. 00:39:26.990 --> 00:39:29.470 If I just do that directly, x is on the left side. 00:39:29.470 --> 00:39:31.020 y is on the right side. 00:39:31.020 --> 00:39:41.795 Then I find minus omega squared m plus i omega c plus k, just 00:39:41.795 --> 00:39:57.710 like before, xe to the i omega t equals k plus i omega c y0 00:39:57.710 --> 00:40:01.320 e to the i omega t. 00:40:01.320 --> 00:40:07.340 And nicely, I can for now get rid of the time-dependent part. 00:40:07.340 --> 00:40:18.120 And I can solve for the response that I'm 00:40:18.120 --> 00:40:22.450 looking for-- x over the input is real and positive, 00:40:22.450 --> 00:40:24.890 amplitude of vibration of the floor. 00:40:24.890 --> 00:40:31.410 And that I will call Hx/y of omega, a transfer 00:40:31.410 --> 00:40:35.090 function, probably complex, that I can then 00:40:35.090 --> 00:40:38.920 deal with like I did above. 00:40:38.920 --> 00:40:42.820 And when I finish manipulating things, substituting 00:40:42.820 --> 00:40:47.560 in zetas and omega n squareds and that kind of thing, 00:40:47.560 --> 00:40:53.600 this becomes-- well, first, I'll write it this way. 00:40:53.600 --> 00:40:59.571 I can write this as a magnitude times an e to the minus i phi 00:40:59.571 --> 00:41:00.070 again. 00:41:00.070 --> 00:41:02.050 That's where I want to go. 00:41:09.230 --> 00:41:18.710 And when I do that, 1 plus-- a little messier-- 2 zeta 00:41:18.710 --> 00:41:27.260 omega over omega n squared square root. 00:41:27.260 --> 00:41:28.425 This is just the numerator. 00:41:31.340 --> 00:41:36.890 And the denominator is just the same as the other single degree 00:41:36.890 --> 00:41:39.390 of freedom things. 00:41:39.390 --> 00:41:44.780 1 minus omega squared over omega n 00:41:44.780 --> 00:41:50.400 squared squared plus 2 zeta omega 00:41:50.400 --> 00:42:00.070 over omega n squared square root e to the minus i phi. 00:42:00.070 --> 00:42:02.350 So now it's the transfer function 00:42:02.350 --> 00:42:05.940 as before except the denominator's a little messy. 00:42:05.940 --> 00:42:09.750 And there's no 1/k. 00:42:09.750 --> 00:42:14.010 And I am going to have a messier expression for phi here. 00:42:26.490 --> 00:42:35.490 So there is something wrong with one of the boards this morning. 00:43:28.400 --> 00:43:30.120 Kind of messy, complicated. 00:43:30.120 --> 00:43:30.930 Do I ever use it? 00:43:30.930 --> 00:43:31.430 Rarely. 00:43:34.290 --> 00:43:37.920 What's important in these things and what isn't-- really 00:43:37.920 --> 00:43:40.750 what's important when you're just trying to get a quick 00:43:40.750 --> 00:43:43.700 solution to vibration isolate something, 00:43:43.700 --> 00:43:45.950 you really want to know what this is going to come out 00:43:45.950 --> 00:43:47.910 looking like. 00:43:47.910 --> 00:43:52.090 You're trying to make the response x small 00:43:52.090 --> 00:43:53.635 compared to the input. 00:43:53.635 --> 00:43:54.760 That's the whole objective. 00:43:58.370 --> 00:44:01.775 Right now the table might be moving a half a millimeter 00:44:01.775 --> 00:44:03.760 or something like that, but this thing's 00:44:03.760 --> 00:44:07.410 moving out here five or six or seven millimeters, 00:44:07.410 --> 00:44:09.070 5 or 10 times that. 00:44:09.070 --> 00:44:13.390 And what we'd really like is if the table's moving 00:44:13.390 --> 00:44:15.120 a millimeter, you'd like this thing 00:44:15.120 --> 00:44:18.070 out here moving 1/10th of a millimeter. 00:44:18.070 --> 00:44:20.690 So the real objective here is to make this small. 00:44:20.690 --> 00:44:23.020 It's the magnitude you care about. 00:44:23.020 --> 00:44:27.620 Phase you rarely even want to know or need to know. 00:44:33.650 --> 00:44:36.200 So we're going to do a sample calculation. 00:44:36.200 --> 00:44:37.850 Let's give an example here. 00:44:42.820 --> 00:44:50.280 So the source is at 20 hertz. 00:44:50.280 --> 00:44:52.420 So your unbalanced pump, your unbalanced rotor. 00:44:52.420 --> 00:44:52.920 Yeah. 00:44:52.920 --> 00:44:54.836 AUDIENCE: How do we know in the previous thing 00:44:54.836 --> 00:45:01.420 that the frequency of oscillation has to be the same? 00:45:01.420 --> 00:45:03.670 Like, why wouldn't it be twice that? 00:45:03.670 --> 00:45:04.270 PROFESSOR: OK. 00:45:04.270 --> 00:45:06.761 That's a great question. 00:45:06.761 --> 00:45:11.340 And I haven't mentioned this before, and I intended to. 00:45:11.340 --> 00:45:13.210 These systems that we're looking at 00:45:13.210 --> 00:45:18.220 are linear systems, which is where we started the other day. 00:45:18.220 --> 00:45:21.840 Linear systems have some interesting and very useful 00:45:21.840 --> 00:45:23.250 properties that we depend upon. 00:45:23.250 --> 00:45:28.320 One was, I said, force one gives you output one, 00:45:28.320 --> 00:45:29.800 force two gives you output two. 00:45:29.800 --> 00:45:33.860 Force one plus two gives you the sum of the outputs. 00:45:33.860 --> 00:45:38.110 The other feature of a linear system 00:45:38.110 --> 00:45:42.450 is steady state response after the transients have died away. 00:45:42.450 --> 00:45:47.710 If the frequency of the input is at 21.5 Hertz, 00:45:47.710 --> 00:45:53.370 the frequency of the output is at 21.5 Hertz, period. 00:45:53.370 --> 00:45:55.290 Linear systems, the frequency of the input 00:45:55.290 --> 00:45:57.340 is equal to the frequency of the output. 00:45:57.340 --> 00:45:59.752 That's a really important little factoid to remember. 00:46:02.710 --> 00:46:07.035 So I turn on the pump, the pump's running at 20 Hertz. 00:46:10.550 --> 00:46:18.530 20 Hertz times 60 is 1,200 RPM, very common motor speed. 00:46:18.530 --> 00:46:20.120 So the pump's running at 20 Hertz. 00:46:20.120 --> 00:46:21.820 So that fan, it's got an imbalance. 00:46:21.820 --> 00:46:23.740 So that means you're putting excitation 00:46:23.740 --> 00:46:27.230 into the floor at 20 Hertz. 00:46:27.230 --> 00:46:42.290 And I want to reduce the vibration at the microscope 00:46:42.290 --> 00:46:47.700 by 90%. 00:46:47.700 --> 00:46:49.950 What that really means is that my goal 00:46:49.950 --> 00:46:57.710 is that the magnitude of x/y is 0.1. 00:46:57.710 --> 00:47:00.380 And that's the magnitude of this transfer function, Hx/y. 00:47:03.530 --> 00:47:07.170 So I want this transfer function to be 0.1. 00:47:07.170 --> 00:47:08.340 So just look at the picture. 00:47:11.340 --> 00:47:15.050 Can I get that answer to the left of the peak? 00:47:17.880 --> 00:47:20.450 And what this plot shows you is this magnitude 00:47:20.450 --> 00:47:24.209 of the transfer function, for a variety of values, a damping. 00:47:24.209 --> 00:47:25.750 And of course, the lower the damping, 00:47:25.750 --> 00:47:28.460 the higher the peak gets at resonance. 00:47:28.460 --> 00:47:29.100 Right? 00:47:29.100 --> 00:47:32.320 So no matter what the damping is, 00:47:32.320 --> 00:47:35.660 what is the curves all go to in the left-hand side? 00:47:35.660 --> 00:47:38.860 They go to 1. 00:47:38.860 --> 00:47:42.630 And that's really saying the static response of this system 00:47:42.630 --> 00:47:49.880 is if you deflect the floor an inch, the table moves with it. 00:47:49.880 --> 00:47:52.690 Everything has to move together when you get down 00:47:52.690 --> 00:47:56.360 to 0 frequency input. 00:47:56.360 --> 00:47:58.210 So everything goes to 1 on the left. 00:47:58.210 --> 00:48:00.610 You go through resonance at omega 00:48:00.610 --> 00:48:02.200 equals a natural frequency. 00:48:02.200 --> 00:48:05.550 But out to the right, as the excitation frequency 00:48:05.550 --> 00:48:08.090 gets higher than the natural frequency, 00:48:08.090 --> 00:48:10.280 the response drops off below 1. 00:48:10.280 --> 00:48:13.650 Which one drops the fastest? 00:48:13.650 --> 00:48:18.741 As you increase omega over omega n beyond 1, 00:48:18.741 --> 00:48:20.740 there's a whole mess of curves to the right that 00:48:20.740 --> 00:48:21.375 blend together. 00:48:21.375 --> 00:48:25.870 And they differ only in damping. 00:48:25.870 --> 00:48:28.000 Can you tell which one is the-- let's say 00:48:28.000 --> 00:48:31.690 if you go to-- at three, there, the response 00:48:31.690 --> 00:48:36.770 is at 0.1 for the lowest curve on that curve, right? 00:48:36.770 --> 00:48:40.354 And that's the one with no damping. 00:48:40.354 --> 00:48:42.461 It's a little counter-intuitive, right? 00:48:42.461 --> 00:48:42.960 All right. 00:48:42.960 --> 00:48:44.250 Well, let's come back to it. 00:48:44.250 --> 00:48:49.950 Damping does help, but not at this point. 00:48:49.950 --> 00:48:57.250 So we need to find a value of omega 00:48:57.250 --> 00:49:02.290 over omega n which is greater than 1 that satisfies this. 00:49:19.520 --> 00:49:20.920 That's what we're after. 00:49:20.920 --> 00:49:23.372 And this is kind of messy to work with. 00:49:23.372 --> 00:49:24.830 And since I know the one that works 00:49:24.830 --> 00:49:27.470 the best is the one with no damping, 00:49:27.470 --> 00:49:29.182 we'll solve the no damping one first, 00:49:29.182 --> 00:49:30.890 because it makes the algebra really easy. 00:49:30.890 --> 00:49:32.130 And then we can go back and say, now, 00:49:32.130 --> 00:49:33.880 what happens if you add some damping? 00:49:33.880 --> 00:49:36.060 So for the case there's no damping, 00:49:36.060 --> 00:49:39.210 the numerator goes to 1. 00:49:39.210 --> 00:49:42.690 The denominator goes to just 1 over 1 00:49:42.690 --> 00:49:45.242 minus omega squared over omega n squared. 00:50:08.600 --> 00:50:10.130 So it becomes that. 00:50:10.130 --> 00:50:11.480 That simple. 00:50:11.480 --> 00:50:15.505 And because I want to work with this ratio bigger than 1, 00:50:15.505 --> 00:50:16.880 I don't want this to be negative. 00:50:16.880 --> 00:50:19.841 And I want to mess with-- keep carrying along absolute value 00:50:19.841 --> 00:50:20.340 signs. 00:50:20.340 --> 00:50:24.690 This is the same thing as 1 over omega squared 00:50:24.690 --> 00:50:28.590 over omega n squared minus 1. 00:50:28.590 --> 00:50:30.560 I just reverse this, because I know 00:50:30.560 --> 00:50:34.380 we're going to deal only with the ones greater than 1 here. 00:50:34.380 --> 00:50:38.185 And I need this to be equal to 0.1. 00:50:38.185 --> 00:50:39.185 And that's just algebra. 00:50:39.185 --> 00:50:41.570 You could solve that. 00:50:41.570 --> 00:50:49.940 This implies that omega over omega n equals root 11, 00:50:49.940 --> 00:50:50.470 I recall. 00:50:58.150 --> 00:51:03.340 And that is 3.31. 00:51:03.340 --> 00:51:05.870 So this is saying on that curve, if you go out 00:51:05.870 --> 00:51:09.070 to omega over omega n equals 3.31 right about where 00:51:09.070 --> 00:51:16.850 that arrow is, the curve for zero damping drops down to 0.1. 00:51:16.850 --> 00:51:19.190 And now if, at that frequency-- ah. 00:51:19.190 --> 00:51:27.130 So that means we have to design the spring support such 00:51:27.130 --> 00:51:34.430 that omega n is equal to omega over 3.31. 00:51:34.430 --> 00:51:39.211 But omega-- where'd we start? 00:51:39.211 --> 00:51:41.615 So F equals 20 Hertz. 00:51:44.170 --> 00:51:48.555 Omega equals 2 pi f. 00:51:48.555 --> 00:51:49.680 Do I have that number here? 00:51:54.550 --> 00:51:58.890 No, but-- so this tells me that I need a natural frequency that 00:51:58.890 --> 00:52:09.980 is omega over 3.31, or I need an fn that is f over 3.31 00:52:09.980 --> 00:52:15.430 is 20 Hertz over 3.31. 00:52:15.430 --> 00:52:18.100 And that number I do have. 00:52:18.100 --> 00:52:21.840 6.04 Hertz. 00:52:21.840 --> 00:52:25.480 So I need a support whose natural frequency 00:52:25.480 --> 00:52:30.920 is 20 Hertz divided by 3.31. 00:52:30.920 --> 00:52:37.350 I need a support whose natural frequency is 6.04 Hertz. 00:52:37.350 --> 00:52:41.660 And that's how you go about designing a flexible base 00:52:41.660 --> 00:52:44.090 to isolate something from vibration 00:52:44.090 --> 00:52:47.390 of whatever it's sitting on. 00:52:47.390 --> 00:52:47.900 All right. 00:52:50.740 --> 00:52:56.200 So my f here, 20 Hertz. 00:52:56.200 --> 00:53:01.610 But my fn needs to be 6.04 Hertz. 00:53:01.610 --> 00:53:05.110 That implies multiply by 2 pi. 00:53:05.110 --> 00:53:12.100 I'm looking for 37.96 radians per second. 00:53:12.100 --> 00:53:17.700 And that's equal to square root of k/M. So now what's the M? 00:53:17.700 --> 00:53:20.690 Well, it's whatever the mass of the microscope plus its base. 00:53:20.690 --> 00:53:25.060 Whatever is being supported by the springs 00:53:25.060 --> 00:53:26.800 will have that mass. 00:53:26.800 --> 00:53:29.810 You have to choose the k. 00:53:29.810 --> 00:53:37.480 So let's say that M total for this system is 20 kilograms. 00:53:40.760 --> 00:53:44.230 Solve this equation for k. 00:53:44.230 --> 00:53:56.756 And that implies that k is 28,827 Newtons per meter. 00:53:56.756 --> 00:53:57.256 OK? 00:54:02.220 --> 00:54:06.720 So if we were to design this system-- 00:54:06.720 --> 00:54:10.000 and it really mounts up to in the case of this. 00:54:10.000 --> 00:54:10.530 Let's see. 00:54:10.530 --> 00:54:12.780 Beams. 00:54:12.780 --> 00:54:17.380 The stiffness of a beam-- ah, that's a good. 00:54:17.380 --> 00:54:19.590 We'll do this. 00:54:19.590 --> 00:54:22.760 We have a cantilever here. 00:54:22.760 --> 00:54:25.420 And we've got a mass on the end. 00:54:25.420 --> 00:54:30.350 But most of you have been taking 2001. 00:54:30.350 --> 00:54:33.551 If you put a force out here, P, what's 00:54:33.551 --> 00:54:35.300 the deflection at the end of a cantilever? 00:54:35.300 --> 00:54:37.170 AUDIENCE: [INAUDIBLE]. 00:54:37.170 --> 00:54:37.880 PROFESSOR: OK. 00:54:37.880 --> 00:54:42.720 So delta is PL cubed over 3EI. 00:54:45.570 --> 00:54:54.660 And the load, this force, is equal to some k equivalent 00:54:54.660 --> 00:54:56.610 times delta, right? 00:54:56.610 --> 00:54:58.592 This is just a spring. 00:54:58.592 --> 00:55:03.430 And k times the displacement is the force it takes to do it. 00:55:03.430 --> 00:55:05.600 So P's my force. 00:55:05.600 --> 00:55:08.190 The spring constant is somehow associated 00:55:08.190 --> 00:55:10.370 with the rest of this stuff. 00:55:10.370 --> 00:55:22.610 So if I solve for P over delta, I get 3EI over L cubed. 00:55:22.610 --> 00:55:23.560 OK? 00:55:23.560 --> 00:55:28.590 So if I'm running right at the natural frequency here 00:55:28.590 --> 00:55:31.700 and I want to reduce this to a 1/10th of its motion, 00:55:31.700 --> 00:55:35.400 I need to change the spring constant of this cantilever 00:55:35.400 --> 00:55:41.430 by a factor of-- well, I need to change the natural frequency 00:55:41.430 --> 00:55:51.960 by a factor of 3.31. 00:55:51.960 --> 00:55:59.007 So my k equivalent here is 3EI over L cubed. 00:55:59.007 --> 00:56:00.840 And that's what would go into this equation. 00:56:04.880 --> 00:56:08.310 But I know that I have a natural frequency right now. 00:56:08.310 --> 00:56:11.900 I want it to go down by a factor of 3.31. 00:56:11.900 --> 00:56:15.850 So that means I need to decrease k 00:56:15.850 --> 00:56:22.250 such that the square root of k goes down by the factor 3.31. 00:56:22.250 --> 00:56:25.230 So how much do I have to change the length? 00:56:25.230 --> 00:56:38.060 Probably something like the square root of 3.31. 00:56:38.060 --> 00:56:38.880 Roughly 2. 00:56:43.424 --> 00:56:45.090 So if I double the length of this thing, 00:56:45.090 --> 00:56:46.423 do you think it's going to work? 00:56:48.832 --> 00:56:51.279 If I double the length of this thing and turn it back on, 00:56:51.279 --> 00:56:53.195 then we shouldn't see much motion out of this. 00:56:53.195 --> 00:56:56.402 [VIBRATING] 00:57:01.600 --> 00:57:02.490 That's moving a lot. 00:57:07.610 --> 00:57:11.210 It's moving a tiny, tiny bit. 00:57:11.210 --> 00:57:12.060 So it works. 00:57:17.170 --> 00:57:20.460 So that's one step of vibration isolation. 00:57:20.460 --> 00:57:25.010 Now I'm going to show you a vibration engineer trick, which 00:57:25.010 --> 00:57:27.009 is a very handy thing to know. 00:57:35.500 --> 00:57:41.917 Where's my strong magnet here? 00:57:41.917 --> 00:57:43.750 So I've got another beam just like this one. 00:57:46.330 --> 00:57:50.480 I've got a pretty massive magnet on it. 00:57:50.480 --> 00:57:52.850 So it makes another cantilever beam 00:57:52.850 --> 00:57:56.078 just like I got over there. 00:57:56.078 --> 00:57:57.410 OK? 00:57:57.410 --> 00:58:04.760 So I claim that with just a ruler, 00:58:04.760 --> 00:58:07.130 if I clamp this down at some length, 00:58:07.130 --> 00:58:10.040 I claim, with just a ruler, I can predict 00:58:10.040 --> 00:58:11.699 the natural frequency of that. 00:58:18.092 --> 00:58:19.716 Take a couple of minutes and see if you 00:58:19.716 --> 00:58:20.966 could figure out how to do it. 00:58:23.540 --> 00:58:26.080 Think about that. 00:58:26.080 --> 00:58:27.860 Just a ruler. 00:58:27.860 --> 00:58:31.030 Measurements that I can make. 00:58:31.030 --> 00:58:32.702 I don't know how long it actually is. 00:58:32.702 --> 00:58:33.910 I don't know how thick it is. 00:58:33.910 --> 00:58:36.490 I know it's steel, but you just don't have enough information 00:58:36.490 --> 00:58:40.200 to compute 3EI over L cubed. 00:58:40.200 --> 00:58:44.057 But simply with a ruler, I'm going to be able to do this. 00:58:44.057 --> 00:58:44.640 Talk about it. 00:58:44.640 --> 00:58:48.605 Think about that while I set up the experiment. 01:00:16.490 --> 01:00:17.800 OK. 01:00:17.800 --> 01:00:18.870 Who's got it figured out? 01:00:21.711 --> 01:00:23.210 Anybody want to take a shot at this? 01:00:31.710 --> 01:00:35.110 So there's my beam. 01:00:35.110 --> 01:00:36.890 I put the weight on it. 01:00:36.890 --> 01:00:40.430 What does the beam do statically? 01:00:40.430 --> 01:00:41.410 Bends a little, right? 01:00:44.060 --> 01:00:55.230 kx static equals Mg, right? 01:00:55.230 --> 01:00:56.760 Has to. 01:00:56.760 --> 01:01:01.660 So x static is what I'm calling delta here. 01:01:01.660 --> 01:01:06.605 So k delta equals Mg. 01:01:06.605 --> 01:01:10.205 k equals Mg over delta. 01:01:13.220 --> 01:01:33.600 Natural frequency equals square root of k/M. Incredibly 01:01:33.600 --> 01:01:35.450 simple, huh? 01:01:35.450 --> 01:01:38.130 So what's the experiment that I would-- 01:01:38.130 --> 01:01:41.780 what measurement would I make? 01:01:41.780 --> 01:01:44.060 Delta, right? 01:01:44.060 --> 01:01:45.490 Put my ruler up there. 01:01:45.490 --> 01:01:48.220 I measure its static position like that. 01:01:48.220 --> 01:01:51.520 Then I put my mass on it, and I measure the static position 01:01:51.520 --> 01:01:52.080 again. 01:01:52.080 --> 01:01:54.090 I measure the delta. 01:01:54.090 --> 01:01:56.260 And I get a prediction. 01:01:56.260 --> 01:01:59.720 And I did this in my office. 01:02:17.490 --> 01:02:22.230 And the delta that I measured-- I actually 01:02:22.230 --> 01:02:23.700 set it at a particular length. 01:02:23.700 --> 01:02:27.100 It was 18 centimeters. 01:02:27.100 --> 01:02:38.710 Delta measured, I think, 0.5 centimeters, or 0.005 meters. 01:02:38.710 --> 01:02:43.330 And if you compute omega n then equals the square root 01:02:43.330 --> 01:03:15.918 of 9.81 over 0.005. 01:03:15.918 --> 01:03:17.605 And I want this in Hertz. 01:03:17.605 --> 01:03:20.320 So I can divide by 2 pi. 01:03:20.320 --> 01:03:29.960 This comes out as 7.05 Hertz. 01:03:29.960 --> 01:03:44.100 And Fn measured was 6.57. 01:03:44.100 --> 01:03:46.540 Pretty good but not perfect. 01:03:49.960 --> 01:03:53.584 And it's because I've made an approximation that I 01:03:53.584 --> 01:03:54.750 glossed over pretty quickly. 01:03:54.750 --> 01:03:56.580 What has been left out of this system that 01:03:56.580 --> 01:04:00.900 would cause the measured natural frequency to be lower 01:04:00.900 --> 01:04:04.267 than the predicted? 01:04:04.267 --> 01:04:05.100 What's been ignored? 01:04:05.100 --> 01:04:05.599 Yes. 01:04:05.599 --> 01:04:06.659 AUDIENCE: Damping. 01:04:06.659 --> 01:04:07.450 PROFESSOR: Damping. 01:04:07.450 --> 01:04:08.420 Ah. 01:04:08.420 --> 01:04:10.330 Maybe. 01:04:10.330 --> 01:04:12.378 How much damping do we have in this system? 01:04:15.660 --> 01:04:18.890 Probably at least 10 cycles to the k halfway, right? 01:04:18.890 --> 01:04:22.620 Certainly less than 1%. 01:04:22.620 --> 01:04:26.590 The damped natural frequency is equal to the natural frequency 01:04:26.590 --> 01:04:29.530 of the square root of 1 minus theta squared. 01:04:29.530 --> 01:04:32.000 So this is something like way less than half 01:04:32.000 --> 01:04:32.930 a percent difference. 01:04:32.930 --> 01:04:34.263 So that wouldn't account for it. 01:04:34.263 --> 01:04:36.160 That's considerably more than half a percent. 01:04:36.160 --> 01:04:38.336 So damping couldn't do it. 01:04:38.336 --> 01:04:38.836 Yeah. 01:04:38.836 --> 01:04:40.570 AUDIENCE: [INAUDIBLE]. 01:04:40.570 --> 01:04:43.060 PROFESSOR: Ah, the mass of the bar. 01:04:43.060 --> 01:04:47.570 Does this flexure have mass? 01:04:47.570 --> 01:04:48.070 Yeah. 01:04:48.070 --> 01:04:50.970 It's probably on the order of if you stack them all up 01:04:50.970 --> 01:04:53.950 and compared to that, it might even be as much as half 01:04:53.950 --> 01:04:56.120 the mass of the end. 01:04:56.120 --> 01:04:57.590 And as it vibrates back and forth, 01:04:57.590 --> 01:04:59.940 does it have kinetic energy? 01:04:59.940 --> 01:05:00.680 Yeah. 01:05:00.680 --> 01:05:04.190 We've ignored the kinetic energy of the mass. 01:05:04.190 --> 01:05:06.940 And in fact, that's the principal error here. 01:05:06.940 --> 01:05:08.280 We've left out the mass. 01:05:08.280 --> 01:05:09.780 There's actually a pretty simple way 01:05:09.780 --> 01:05:13.410 to-- using energy and just thinking in Lagrange terms, 01:05:13.410 --> 01:05:15.340 you can account for the energy of the mass 01:05:15.340 --> 01:05:17.380 in this single degree of freedom system 01:05:17.380 --> 01:05:20.090 and get a very accurate prediction. 01:05:20.090 --> 01:05:21.180 We won't do that today. 01:05:21.180 --> 01:05:25.618 But I think we'll do that before the term's out. 01:05:25.618 --> 01:05:27.530 OK. 01:05:27.530 --> 01:05:35.540 This applies to any simple mass spring system 01:05:35.540 --> 01:05:39.240 in the presence of gravity. 01:05:39.240 --> 01:05:41.120 So here's a mass. 01:05:43.679 --> 01:05:45.470 And actually, we're doing the problem today 01:05:45.470 --> 01:05:46.680 where I'm moving the base. 01:05:46.680 --> 01:05:47.990 So here's its base. 01:05:47.990 --> 01:05:50.070 So this is the table moving. 01:05:50.070 --> 01:05:53.880 And if I do this, it clearly makes that move. 01:05:53.880 --> 01:05:57.300 If I do this really fast, it doesn't move very much. 01:05:57.300 --> 01:06:00.570 If I do it close to the natural frequency, it moves a lot. 01:06:00.570 --> 01:06:06.290 If I move it very slowly, as I go up one unit, 01:06:06.290 --> 01:06:08.390 this follows me exactly. 01:06:08.390 --> 01:06:11.570 That's why that plot goes to 1. 01:06:11.570 --> 01:06:15.240 At very, very low frequency, the support and the mass 01:06:15.240 --> 01:06:18.250 move exactly together. 01:06:18.250 --> 01:06:22.485 At very high frequency-- if I can stop the transient-- 01:06:22.485 --> 01:06:23.826 I can't do it very well. 01:06:23.826 --> 01:06:24.950 The mass doesn't move much. 01:06:24.950 --> 01:06:25.825 The base moves a lot. 01:06:25.825 --> 01:06:27.700 And at resonance, it goes nuts. 01:06:27.700 --> 01:06:28.410 OK. 01:06:28.410 --> 01:06:31.800 The unstretched length of this spring is about seven inches. 01:06:31.800 --> 01:06:33.500 The square root of g over delta, I 01:06:33.500 --> 01:06:36.150 ought to be able to predict this. 01:06:36.150 --> 01:06:39.960 So I did a quick calculation on that. 01:06:39.960 --> 01:06:41.310 It was like 1% error. 01:06:43.870 --> 01:06:49.840 I measured it at 7.36 radians a second. 01:06:49.840 --> 01:06:58.477 And I predicted it at 7.43 measured 7.36. 01:06:58.477 --> 01:07:00.060 Same kind of thing-- ignoring the mass 01:07:00.060 --> 01:07:01.460 of the spring a little bit. 01:07:01.460 --> 01:07:05.480 So g over delta is a great little thing to remember. 01:07:05.480 --> 01:07:05.980 OK. 01:07:13.030 --> 01:07:18.730 So we have done all but one. 01:07:18.730 --> 01:07:20.460 Everything we've started out with today, 01:07:20.460 --> 01:07:22.280 we've said there's three ways to fix this, 01:07:22.280 --> 01:07:24.040 and came up with a fourth way. 01:07:24.040 --> 01:07:28.500 So in this case, soften the spring support a lot, 01:07:28.500 --> 01:07:33.820 so that the natural frequency is way less than the excitation. 01:07:33.820 --> 01:07:39.350 We said, what about spring supporting, softening, flexibly 01:07:39.350 --> 01:07:42.585 amounting this source, so that it doesn't put vibration 01:07:42.585 --> 01:07:43.210 on the table? 01:07:43.210 --> 01:07:45.750 That's the piece we haven't addressed. 01:07:45.750 --> 01:07:47.664 So let's look into that problem now. 01:08:00.260 --> 01:08:07.220 So here's our source, some rotating mass eccentricity 01:08:07.220 --> 01:08:12.510 causing an excitation. 01:08:12.510 --> 01:08:16.810 So this has a force F0 e to the i omega 01:08:16.810 --> 01:08:20.270 t, which is coming from the rotating mass. 01:08:20.270 --> 01:08:23.819 And it applies to the floor, through the dashpot 01:08:23.819 --> 01:08:27.660 in the springs, some FT, I'll call it, 01:08:27.660 --> 01:08:33.229 F transmitted to the floor, e to the i omega t. 01:08:33.229 --> 01:08:40.390 And I want to know-- I need the H force transmitted per unit 01:08:40.390 --> 01:08:44.014 force input transfer function. 01:08:44.014 --> 01:08:45.180 That's what I'm looking for. 01:08:56.010 --> 01:08:59.420 So now free-body diagram. 01:08:59.420 --> 01:09:01.020 Now we're going to make an assumption. 01:09:01.020 --> 01:09:06.880 We're going to assume that the motion of the floor, 01:09:06.880 --> 01:09:11.069 which we'll call y of t, assume that y is much, 01:09:11.069 --> 01:09:12.824 much less than x. 01:09:12.824 --> 01:09:14.420 It's generally true. 01:09:14.420 --> 01:09:17.580 Whatever's shaking like crazy, the table's 01:09:17.580 --> 01:09:19.410 not moving much underneath it. 01:09:19.410 --> 01:09:22.520 So I'm going to assume, for the purposes of calculating forces, 01:09:22.520 --> 01:09:25.970 that this is 0. 01:09:25.970 --> 01:09:34.590 So for the motion x, what is the force applied to the floor? 01:09:34.590 --> 01:09:35.760 So F of t. 01:09:41.880 --> 01:09:46.319 If you have a positive displacement x, 01:09:46.319 --> 01:09:49.149 the force is kx. 01:09:49.149 --> 01:09:53.330 You have a positive velocity x, the force pulling up 01:09:53.330 --> 01:09:57.700 on the floor through the dashpot is cx dot. 01:09:57.700 --> 01:10:03.970 So the other way of saying that is here's 01:10:03.970 --> 01:10:06.240 our free-body diagram. 01:10:06.240 --> 01:10:11.680 Here's our F0 e to the i omega t pulling up. 01:10:11.680 --> 01:10:14.800 It responds at some x. 01:10:14.800 --> 01:10:20.230 And the resulting forces through the spring 01:10:20.230 --> 01:10:23.540 and the dashpot we know are kx and cx 01:10:23.540 --> 01:10:27.340 dot opposing the motion x. 01:10:27.340 --> 01:10:31.390 Well, by third law, if these are the forces 01:10:31.390 --> 01:10:34.360 on the spring and the dashpot, then down here on the floor, 01:10:34.360 --> 01:10:39.890 you better have some equal and opposite forces, kx and cx dot. 01:10:39.890 --> 01:10:43.540 So this force produces a motion x. 01:10:43.540 --> 01:10:48.310 The motion x produces forces in the mass in the spring, which 01:10:48.310 --> 01:10:53.180 make the force on the floor, the spring 01:10:53.180 --> 01:10:55.610 force, and the dashpot force. 01:10:55.610 --> 01:10:56.110 OK. 01:11:02.000 --> 01:11:11.432 So this Ft is-- I want to write it here. 01:11:18.800 --> 01:11:22.460 That's all that is-- positive. 01:11:22.460 --> 01:11:29.080 And I'm going to assume a solution 01:11:29.080 --> 01:11:35.800 that we know to work for x, which is xe to the i omega t. 01:11:35.800 --> 01:11:37.790 We've plugged it in before. 01:11:37.790 --> 01:11:43.510 So I plug that in here, I get a k plus i omega 01:11:43.510 --> 01:11:51.720 c, xe to the i omega t. 01:11:51.720 --> 01:11:54.710 So I can just express my force on the floor 01:11:54.710 --> 01:11:56.715 in terms of the motion x. 01:12:34.260 --> 01:12:38.500 And I'm looking for a transfer function for force transmitted 01:12:38.500 --> 01:12:40.490 over force in. 01:12:40.490 --> 01:12:46.760 But force transmitted is my k plus i omega 01:12:46.760 --> 01:12:52.550 c, xe to the i omega t. 01:12:52.550 --> 01:12:59.630 And the force in is F0 e to the i omega t. 01:12:59.630 --> 01:13:02.210 Cancel out the time-dependent part. 01:13:02.210 --> 01:13:05.800 And it says the transmitted force over the input force 01:13:05.800 --> 01:13:09.420 is this little complex expression times the response 01:13:09.420 --> 01:13:12.210 x over F. But we know what that is. 01:13:12.210 --> 01:13:15.340 That's the transfer function Hx/F. 01:13:15.340 --> 01:13:23.660 So this is k plus i omega c times Hx/F of omega. 01:13:26.450 --> 01:13:33.360 So this gives us a slightly different transfer function. 01:13:33.360 --> 01:13:34.935 Ooh, look at this. 01:13:38.500 --> 01:13:42.070 Before, when we did x/y, we ended up 01:13:42.070 --> 01:13:47.310 with k plus i omega c y e to the i omega t. 01:13:47.310 --> 01:14:01.520 And when we did then x/y, we got the same ratio as this. 01:14:01.520 --> 01:14:03.670 Exactly the same thing. 01:14:03.670 --> 01:14:08.940 So I could write all this out, but-- and let's say 01:14:08.940 --> 01:14:11.280 I'll do this. 01:14:11.280 --> 01:14:17.712 Hx/y-- no, no, I won't do that. 01:14:20.250 --> 01:14:23.270 What I'm going to tell you-- if you just work through this now, 01:14:23.270 --> 01:14:28.070 you will find that H force transmitted over force in 01:14:28.070 --> 01:14:30.390 is exactly the same as Hx/y. 01:14:36.080 --> 01:14:43.200 And that what we really care about is what the magnitude is. 01:14:43.200 --> 01:14:47.510 So the magnitude of these two things are the same. 01:14:47.510 --> 01:14:52.740 And in fact, just work out to that same expression as before, 01:14:52.740 --> 01:15:00.870 the 1 plus 2 zeta omega over omega n 01:15:00.870 --> 01:15:08.094 squared square root all over the usual big denominator. 01:15:08.094 --> 01:15:13.370 So conveniently, for vibration isolation, 01:15:13.370 --> 01:15:16.820 the solution to the two problems are exactly the same. 01:15:16.820 --> 01:15:19.220 So if you have that one, you have the transfer function 01:15:19.220 --> 01:15:23.630 xy that was projected on the screen a minute ago, 01:15:23.630 --> 01:15:28.870 it is also the force transmitted to force in transfer function. 01:15:28.870 --> 01:15:31.690 So you just have to remember one. 01:15:31.690 --> 01:15:35.230 And if you now want to-- we said, let's say, 01:15:35.230 --> 01:15:39.180 doubling the length of this just about accomplished the reducing 01:15:39.180 --> 01:15:44.120 the vibration of the microscope by this factor of 10. 01:15:44.120 --> 01:15:47.930 So if I doubled the length of this one, 01:15:47.930 --> 01:15:50.770 I would roughly do the same thing. 01:15:50.770 --> 01:15:53.440 I would change the natural-- this thing is 01:15:53.440 --> 01:15:55.300 right on the natural frequency of this beam. 01:15:55.300 --> 01:15:58.375 That's why it shakes so much. 01:15:58.375 --> 01:16:01.040 And so it is this system. 01:16:01.040 --> 01:16:04.190 It's shaking like crazy, putting force into the table. 01:16:04.190 --> 01:16:07.310 The table is vibrating, causing the other one to move. 01:16:07.310 --> 01:16:13.510 So now if I change this one, then the same kind of idea. 01:16:13.510 --> 01:16:17.210 Maybe roughly double its length. 01:16:17.210 --> 01:16:20.880 Natural frequency diminishes by a factor of 3 or so. 01:16:27.280 --> 01:16:30.195 The vibration of this ought to go way down. 01:16:39.337 --> 01:16:40.920 And actually, our little beam out here 01:16:40.920 --> 01:16:43.908 is picking up more than the other one. 01:16:43.908 --> 01:16:44.408 Shh. 01:16:50.810 --> 01:16:52.777 So this thing is hardly moving at all now. 01:16:56.760 --> 01:17:00.390 So by doing that, we've essentially detuned it. 01:17:00.390 --> 01:17:07.050 This is no longer running at the natural frequency of this base. 01:17:07.050 --> 01:17:08.550 So it's no longer resonant. 01:17:08.550 --> 01:17:11.330 You're way out on the curve to the right. 01:17:11.330 --> 01:17:14.430 So the response of this isn't very much. 01:17:14.430 --> 01:17:17.230 That means it doesn't transmit much force to the base, maybe 01:17:17.230 --> 01:17:18.860 down by a factor of 8 or 10. 01:17:18.860 --> 01:17:24.040 That means the table vibration amplitude drops by that factor. 01:17:24.040 --> 01:17:27.260 Means that the base motion over here is now a factor of 10 01:17:27.260 --> 01:17:29.790 smaller than it was to begin with, 01:17:29.790 --> 01:17:32.210 so that we get a reduction of 10 here. 01:17:32.210 --> 01:17:36.660 And we get another reduction of 10 here, because we detuned it. 01:17:36.660 --> 01:17:39.750 So you might get a factor of 100 reduction by working on both, 01:17:39.750 --> 01:17:40.250 you see. 01:17:40.250 --> 01:17:43.970 You've treated the source and you've treated the receiver. 01:17:43.970 --> 01:17:47.000 But fortunately, they use the same curve. 01:17:47.000 --> 01:17:48.466 So damping. 01:17:48.466 --> 01:17:50.090 When you do vibration isolation, you're 01:17:50.090 --> 01:17:54.145 trying to get well out on this curve to the right. 01:17:54.145 --> 01:17:56.270 So there's a couple of practical engineering things 01:17:56.270 --> 01:17:58.545 that limit how far you can go. 01:17:58.545 --> 01:18:00.420 To get further out on the curve to the right, 01:18:00.420 --> 01:18:03.989 what do you have to do to the spring in the system to get 01:18:03.989 --> 01:18:04.780 stronger or softer? 01:18:08.397 --> 01:18:10.480 You're trying to make the natural frequency-- see, 01:18:10.480 --> 01:18:12.320 the excitation frequency doesn't change. 01:18:12.320 --> 01:18:15.960 In order to get omega over omega n to go bigger and bigger, 01:18:15.960 --> 01:18:17.430 the excitation's staying the same. 01:18:17.430 --> 01:18:19.950 You're having to reduce the natural frequency. 01:18:19.950 --> 01:18:23.350 And so what do you have to do to the spring constant? 01:18:23.350 --> 01:18:24.660 Decrease it. 01:18:24.660 --> 01:18:27.340 What is the practical limit of decreasing 01:18:27.340 --> 01:18:31.190 the spring that supports your pump, 01:18:31.190 --> 01:18:34.340 or your washing machine, or your air conditioner? 01:18:34.340 --> 01:18:36.297 AUDIENCE: [INAUDIBLE]. 01:18:36.297 --> 01:18:37.880 PROFESSOR: Pretty soon it's just going 01:18:37.880 --> 01:18:39.754 to-- if it's too heavy-- you put it on there, 01:18:39.754 --> 01:18:41.670 it's just going to squash the springs, right? 01:18:41.670 --> 01:18:44.920 So you can't-- there's limits to how soft you can make springs 01:18:44.920 --> 01:18:47.652 to support heavy machines. 01:18:47.652 --> 01:18:49.860 So there is a practical limit to how far to the right 01:18:49.860 --> 01:18:50.570 you can go. 01:18:50.570 --> 01:18:53.910 But normally, you get out there as far as you can. 01:18:53.910 --> 01:18:57.410 And then if the real system has damping, 01:18:57.410 --> 01:19:01.030 does it improve or degrade the performance 01:19:01.030 --> 01:19:04.020 of your vibration isolation system? 01:19:04.020 --> 01:19:06.340 Well, the more the damping you have, the higher up you 01:19:06.340 --> 01:19:07.650 are on those curves. 01:19:07.650 --> 01:19:11.320 So the damping decreases the performance. 01:19:11.320 --> 01:19:17.730 But every system has to-- when you first turn on that motor, 01:19:17.730 --> 01:19:19.130 the system has to spin up. 01:19:19.130 --> 01:19:23.220 And you're going to have to go through that resonance, 01:19:23.220 --> 01:19:25.930 so that you want some damping. 01:19:25.930 --> 01:19:29.100 Because if you've got your scanning electron 01:19:29.100 --> 01:19:33.380 microscope or your laser interferometry system set up 01:19:33.380 --> 01:19:39.730 on a spring-supported table, if that table has no damping 01:19:39.730 --> 01:19:42.550 and you walk in the door and bump it, 01:19:42.550 --> 01:19:45.950 it is going to sit there and vibrate all afternoon 01:19:45.950 --> 01:19:48.460 at its natural frequency due to the initial conditions. 01:19:48.460 --> 01:19:52.290 So you need some damping to prevent problems, 01:19:52.290 --> 01:19:54.220 either response to initial conditions, 01:19:54.220 --> 01:19:55.860 or bumping it, or whatever. 01:19:55.860 --> 01:19:59.660 Or even as the system turns on and speeds up, 01:19:59.660 --> 01:20:01.650 it'll have to go through that resonance. 01:20:01.650 --> 01:20:04.370 And it'll vibrate like crazy as it does, and then finally 01:20:04.370 --> 01:20:07.370 settle down at the higher frequency. 01:20:07.370 --> 01:20:08.780 So you need some damping. 01:20:08.780 --> 01:20:12.090 But damping does degrade the steady state performance. 01:20:12.090 --> 01:20:14.050 And I'm out of time. 01:20:14.050 --> 01:20:15.925 And we'll see you in recitation. 01:20:15.925 --> 01:20:17.500 Thanks.