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 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.305 at ocw.mit.edu. 00:00:21.735 --> 00:00:24.990 PROFESSOR: Well let's see if we can't get started. 00:00:24.990 --> 00:00:28.250 Everyone I trust can hear me adequately. 00:00:30.840 --> 00:00:33.220 Welcome back. 00:00:33.220 --> 00:00:35.990 It's Tuesday. 00:00:35.990 --> 00:00:39.020 For those of you who are not in my recitation section, 00:00:39.020 --> 00:00:44.520 I'm Dave Gossard, and I'll be your lecturer for the day. 00:00:44.520 --> 00:00:48.680 Professor Vandiver is out of town. 00:00:48.680 --> 00:00:51.060 It looks like some of you may be as well. 00:00:51.060 --> 00:00:55.575 We probably could have held this at the gate at Logan Airport 00:00:55.575 --> 00:00:57.940 and done a little better. 00:00:57.940 --> 00:01:00.990 But be that as it may, glad you came. 00:01:00.990 --> 00:01:02.030 This should be fun. 00:01:02.030 --> 00:01:08.412 Today we have a new topic and a demonstration, 00:01:08.412 --> 00:01:11.230 a real physical system. 00:01:11.230 --> 00:01:15.890 So unless there are any outstanding questions? 00:01:15.890 --> 00:01:20.690 Anybody have any questions or complaints 00:01:20.690 --> 00:01:22.356 to address to Vicente? 00:01:22.356 --> 00:01:23.348 No. 00:01:23.348 --> 00:01:28.920 All right, hearing none let's go ahead and get started then. 00:01:28.920 --> 00:01:34.990 Today the topic is multiple degree of freedom systems. 00:01:34.990 --> 00:01:37.810 Now to date, with a couple of exceptions, 00:01:37.810 --> 00:01:41.240 all of the systems that you've dealt with 00:01:41.240 --> 00:01:46.360 had a single degree of freedom, either a linear displacement 00:01:46.360 --> 00:01:50.550 x or an angular displacement theta. 00:01:50.550 --> 00:01:55.160 You know the concept of equations of motion, 00:01:55.160 --> 00:01:57.420 or I should say the equation of motion 00:01:57.420 --> 00:02:06.750 and the notion of undamped natural frequency. 00:02:06.750 --> 00:02:11.920 Well, today we're going to generalize, if you will, 00:02:11.920 --> 00:02:17.640 to systems that have not one but multiple degrees of freedom 00:02:17.640 --> 00:02:22.030 and see how those notions generalize. 00:02:22.030 --> 00:02:26.260 In particular, as you might expect, 00:02:26.260 --> 00:02:30.750 the system that has multiple degrees of freedom 00:02:30.750 --> 00:02:36.460 has multiple natural frequencies, also 00:02:36.460 --> 00:02:41.000 known as eigenvalues as we will explain here shortly. 00:02:41.000 --> 00:02:44.390 Multiple degrees of freedom systems 00:02:44.390 --> 00:02:48.200 have a new property, a new characteristic 00:02:48.200 --> 00:02:49.670 you haven't seen before. 00:02:49.670 --> 00:02:51.850 And that's what this lecture is all about, 00:02:51.850 --> 00:02:56.350 is to illustrate that to you and demonstrate it. 00:02:56.350 --> 00:02:59.880 It's the notion of natural modes, 00:02:59.880 --> 00:03:02.590 also called eigenvectors. 00:03:02.590 --> 00:03:07.710 And then the general response to initial conditions. 00:03:07.710 --> 00:03:11.370 So that is the plan for the day. 00:03:11.370 --> 00:03:14.220 And we'll start with this. 00:03:14.220 --> 00:03:17.290 This is kind of a classic textbook 00:03:17.290 --> 00:03:19.910 case, two springs, two masses. 00:03:19.910 --> 00:03:23.730 A straightforward extrapolation of what you've done before. 00:03:23.730 --> 00:03:30.040 You've got a spring K1, mass M1, spring K2, mass M2. 00:03:30.040 --> 00:03:40.420 And the displacements are indicated 00:03:40.420 --> 00:03:47.370 as shown there, X1 and X2. 00:03:47.370 --> 00:03:53.800 I want to hasten to point out that the displacements we speak 00:03:53.800 --> 00:04:01.890 of here are defined with respect to the static equilibrium 00:04:01.890 --> 00:04:02.520 position. 00:04:02.520 --> 00:04:08.560 This is a notion that Professor Vandiver went over at least 00:04:08.560 --> 00:04:09.140 once. 00:04:09.140 --> 00:04:15.020 And for those of you who've forgotten it or weren't there 00:04:15.020 --> 00:04:20.519 that day, I have for you a reference, essentially reprised 00:04:20.519 --> 00:04:24.530 that notion over there. 00:04:24.530 --> 00:04:27.215 So in the meantime, let me press on. 00:04:27.215 --> 00:04:31.320 If you have any questions, we can go back and cover that. 00:04:31.320 --> 00:04:41.760 But assuming you agree, let me simply say you've got 00:04:41.760 --> 00:04:43.915 two springs, two masses. 00:04:43.915 --> 00:04:46.720 The typical way we've taught you to do 00:04:46.720 --> 00:04:48.520 it is if you're going to generate 00:04:48.520 --> 00:04:51.270 the equations of motion by the direct method, 00:04:51.270 --> 00:04:53.890 you generate two free body diagrams, the sum 00:04:53.890 --> 00:05:00.140 forces in the x direction for each of the masses, 00:05:00.140 --> 00:05:03.210 get f equals MA and you'd get these. 00:05:03.210 --> 00:05:05.360 Conversely, you could also do it by Legrange. 00:05:05.360 --> 00:05:08.690 You could generate the expression 00:05:08.690 --> 00:05:12.460 for the kinetic energy, for the potential energy, 00:05:12.460 --> 00:05:16.960 for the Legrangian, do this Lagrange equation business, 00:05:16.960 --> 00:05:19.050 and you'd get the same thing. 00:05:19.050 --> 00:05:23.310 But either way you do it, what comes out the other side 00:05:23.310 --> 00:05:25.830 looks like this. 00:05:25.830 --> 00:05:32.830 And it's not a bad exercise for you to offline convince 00:05:32.830 --> 00:05:35.610 yourself that this is right. 00:05:35.610 --> 00:05:39.930 Not right now, but in the comfort and leisure 00:05:39.930 --> 00:05:40.820 of another time. 00:05:58.540 --> 00:06:00.540 So there you have it. 00:06:00.540 --> 00:06:03.940 That's what the two equations of motion would look like. 00:06:03.940 --> 00:06:09.740 Again, either by the direct method or by Legrange, 00:06:09.740 --> 00:06:13.850 you end up in the same place, so to speak. 00:06:13.850 --> 00:06:29.730 And now for today, we haven't asked you to do this much, 00:06:29.730 --> 00:06:33.740 but let me simply say the weapon of choice 00:06:33.740 --> 00:06:36.230 for multiple degrees of freedom system, 00:06:36.230 --> 00:06:41.790 because there's a certain repetitive quality to it, 00:06:41.790 --> 00:06:45.240 matrix notation is preferred. 00:06:45.240 --> 00:06:51.450 In these equations over here, written in matrix form 00:06:51.450 --> 00:06:52.550 would look like this. 00:07:46.980 --> 00:07:48.410 And that looks like this. 00:07:48.410 --> 00:07:51.490 There's two matrices, and let me hasten 00:07:51.490 --> 00:07:57.490 to point out that this is exactly this and nothing more. 00:07:57.490 --> 00:08:00.240 There's no magic, no additional derivation. 00:08:00.240 --> 00:08:04.580 This is simply a restructuring, and reorganization 00:08:04.580 --> 00:08:05.810 of these equations. 00:08:05.810 --> 00:08:08.550 And this may seem foreign to those 00:08:08.550 --> 00:08:13.760 of you who have not had any or very much linear algebra. 00:08:13.760 --> 00:08:15.150 Do not be dismayed. 00:08:15.150 --> 00:08:17.920 It is not a difficult thing to learn. 00:08:17.920 --> 00:08:24.210 As you probably know, a matrix multiplies by a vector-- 00:08:24.210 --> 00:08:27.290 or multiplying a vector by a matrix-- 00:08:27.290 --> 00:08:28.660 is done with two hands. 00:08:28.660 --> 00:08:35.990 The first item for example is M1 X1 dot plus 0. 00:08:35.990 --> 00:08:39.549 That gives you this term right here. 00:08:39.549 --> 00:08:44.630 Over here you get X1 times K1 plus K2. 00:08:44.630 --> 00:08:46.530 That's this one. 00:08:46.530 --> 00:08:52.690 And here you have minus K2 X2, that's that term. 00:08:52.690 --> 00:08:57.370 So matrix notation, this becomes that. 00:08:57.370 --> 00:08:57.920 No problem. 00:09:07.530 --> 00:09:09.230 It's like a model train set. 00:09:09.230 --> 00:09:11.410 It's great. 00:09:11.410 --> 00:09:14.020 Everyone should have one of these. 00:09:14.020 --> 00:09:24.230 All right, so what I would like to do for our example here 00:09:24.230 --> 00:09:39.240 is because we're going to be doing some algebra, 00:09:39.240 --> 00:09:43.895 for the express purpose of simplifying the algebra let 00:09:43.895 --> 00:09:51.330 me consider a special case where the masses are identical. 00:09:51.330 --> 00:09:56.040 And we can simply call them M. And similarly, 00:09:56.040 --> 00:10:02.800 the springs are identical, and we'll simply call them K. 00:10:02.800 --> 00:10:08.026 At that point, these equations become simplified. 00:10:11.010 --> 00:10:41.610 That's just M, that's just M. So this 00:10:41.610 --> 00:10:43.330 is the problem we're going to-- we're 00:10:43.330 --> 00:10:47.680 going to tackle this problem first, because as I say, 00:10:47.680 --> 00:10:50.750 it simplifies the algebra. 00:10:50.750 --> 00:10:54.400 Now here is-- this is not an assumption. 00:10:54.400 --> 00:11:07.800 This is a-- I would call this more a mechanism 00:11:07.800 --> 00:11:11.315 to get this job done here. 00:11:15.990 --> 00:11:23.222 Harmonic motion is one where we assume 00:11:23.222 --> 00:11:41.300 that the masses oscillate at the same frequency. 00:12:23.750 --> 00:12:28.490 So what this looks like is this. 00:12:28.490 --> 00:12:34.260 What we're basically saying is that X1 is actually 00:12:34.260 --> 00:12:43.330 equal-- X1 has amplitude A-- whoa. 00:12:43.330 --> 00:12:45.930 I'm being attacked here by the second board. 00:12:57.640 --> 00:13:00.340 Basically, the situation here is that we're 00:13:00.340 --> 00:13:05.590 assuming that both of these masses 00:13:05.590 --> 00:13:10.360 move-- I wouldn't call it exactly together. 00:13:10.360 --> 00:13:14.149 They're not in complete synchrony, 00:13:14.149 --> 00:13:15.440 as you'll see here in a moment. 00:13:15.440 --> 00:13:18.580 But what they are is they're going 00:13:18.580 --> 00:13:20.870 through a sinusoidal motion. 00:13:20.870 --> 00:13:25.880 And it is an oscillation at the same frequency. 00:13:25.880 --> 00:13:31.260 Both of them are oscillating at the same frequency. 00:13:31.260 --> 00:13:34.760 However, they differ in their magnitudes. 00:13:34.760 --> 00:13:36.260 They are not the same magnitude. 00:13:43.390 --> 00:13:48.900 But that assumption right there allows us to say this. 00:14:08.220 --> 00:14:19.500 If we differentiate those twice, we get the following. 00:14:31.170 --> 00:14:34.920 That comes back out. 00:14:34.920 --> 00:14:37.331 Douglas, where's that minus sign come? 00:14:37.331 --> 00:14:39.330 Can you-- first of all, can everybody read that? 00:14:39.330 --> 00:14:41.163 Can you guys read that in the back row here? 00:14:44.000 --> 00:14:50.850 For example, this says that X1 double dot, if x1 is A1 cosine, 00:14:50.850 --> 00:14:55.440 then X1 double dot is A1 cosine preceded 00:14:55.440 --> 00:14:58.080 by a minus omega squared. 00:14:58.080 --> 00:14:59.164 Where does that come from? 00:14:59.164 --> 00:15:01.579 AUDIENCE: The minus sign came from when you differentiated 00:15:01.579 --> 00:15:02.850 the cosine in the first one. 00:15:02.850 --> 00:15:03.720 PROFESSOR: Exactly. 00:15:03.720 --> 00:15:07.040 And the cosine returns. 00:15:07.040 --> 00:15:09.160 And that's what you get. 00:15:09.160 --> 00:15:12.950 Well, here is, shall we say, the heart of the matter. 00:15:12.950 --> 00:15:17.650 When you substitute this into this-- 00:15:17.650 --> 00:15:27.970 let me call this-- I'll try not to get too obsessive over this. 00:15:27.970 --> 00:15:33.590 But these are our equations of motion. 00:15:33.590 --> 00:16:22.760 So when we-- you get this, equals 0. 00:16:25.450 --> 00:16:26.330 Excuse me. 00:16:26.330 --> 00:16:30.580 Many people, I think, simply put a big 0 there, 00:16:30.580 --> 00:16:32.565 but I'll do it properly. 00:16:36.470 --> 00:16:39.280 It's two zeros, if you will. 00:16:39.280 --> 00:16:55.480 And forgive me for writing this out, 00:16:55.480 --> 00:16:58.430 but I would like you to be able to do this 00:16:58.430 --> 00:17:04.930 by yourself, to recreate this after the fact. 00:17:04.930 --> 00:17:09.784 This becomes-- dividing and collecting terms. 00:17:41.250 --> 00:17:49.360 OK, anybody unclear about how this is obtained? 00:17:49.360 --> 00:17:49.860 Yes, ma'am. 00:17:49.860 --> 00:17:51.248 Emma. 00:17:51.248 --> 00:17:53.372 AUDIENCE: I have a question about the previous one. 00:17:53.372 --> 00:17:53.844 PROFESSOR: Yeah? 00:17:53.844 --> 00:17:55.969 AUDIENCE: If the second term on the left hand side, 00:17:55.969 --> 00:17:58.559 should it also be multiplied by A1 A2? 00:17:58.559 --> 00:17:59.600 PROFESSOR: Yes it should. 00:17:59.600 --> 00:18:02.720 Thank you very much. 00:18:02.720 --> 00:18:05.570 Oh, hang on. 00:18:05.570 --> 00:18:09.496 Yes, thank you, that's exactly right. 00:18:09.496 --> 00:18:10.412 AUDIENCE: [INAUDIBLE]. 00:18:15.200 --> 00:18:17.780 PROFESSOR: Hang on a second. 00:18:17.780 --> 00:18:20.580 We're fighting the boards here. 00:18:20.580 --> 00:18:23.330 Let's see, one thing at a time. 00:18:23.330 --> 00:18:34.640 We've got A1, A2 cosine [INAUDIBLE] minus phi equals. 00:18:34.640 --> 00:18:36.880 Now Emma, does that take care of you? 00:18:36.880 --> 00:18:37.380 Yeah? 00:18:37.380 --> 00:18:39.020 And you said, Vicente? 00:18:39.020 --> 00:18:41.270 AUDIENCE: Diagonal terms-- shouldn't there be a minus? 00:18:41.270 --> 00:18:43.269 PROFESSOR: I'm sorry, that's absolutely correct. 00:18:46.556 --> 00:18:47.055 Wonderful. 00:18:53.440 --> 00:18:56.446 So there we have it. 00:18:56.446 --> 00:18:57.720 Any other questions? 00:18:57.720 --> 00:18:59.760 I hope I got it right. 00:18:59.760 --> 00:19:00.447 Yes sir. 00:19:00.447 --> 00:19:01.530 AUDIENCE: Are they plus K? 00:19:01.530 --> 00:19:02.954 PROFESSOR: I'm sorry? 00:19:02.954 --> 00:19:04.190 AUDIENCE: Are they plus K? 00:19:04.190 --> 00:19:06.150 PROFESSOR: Plus K? 00:19:06.150 --> 00:19:06.800 No. 00:19:06.800 --> 00:19:09.510 They're minus. 00:19:09.510 --> 00:19:11.295 Yeah, why is that? 00:19:11.295 --> 00:19:12.610 Everybody see that? 00:19:18.520 --> 00:19:23.590 This is a straight-- there's less here than meets the eye. 00:19:23.590 --> 00:19:29.420 There's a straight segregation collecting of terms. 00:19:29.420 --> 00:19:34.990 The minus got added to the elements of the mass matrix, 00:19:34.990 --> 00:19:36.990 but not to the K metrics. 00:19:36.990 --> 00:19:41.880 The K metrics goes shows through as is. 00:19:41.880 --> 00:19:48.960 Now, the question on the floor is what we do with this? 00:19:48.960 --> 00:19:52.820 Can everybody appreciate that-- get out 00:19:52.820 --> 00:19:56.580 of the spring mass business and look at this 00:19:56.580 --> 00:19:58.760 from a math point of view? 00:19:58.760 --> 00:20:01.050 Does everyone appreciate that this 00:20:01.050 --> 00:20:06.060 is a set of linear equations? 00:20:06.060 --> 00:20:09.850 There's the old AX equal B kind of thing. 00:20:09.850 --> 00:20:21.860 And if you recall, to solve, what we're basically going 00:20:21.860 --> 00:20:25.150 to do is solve for A1 and A2. 00:20:25.150 --> 00:20:26.830 That's the game we're playing here. 00:20:30.690 --> 00:20:39.290 And if you recall from your math course, the determinant of this 00:20:39.290 --> 00:20:41.680 has got to equal 0. 00:20:41.680 --> 00:20:44.950 So let me simply repeat it here. 00:20:44.950 --> 00:21:06.940 The determinant of has got to equal 0. 00:21:06.940 --> 00:21:10.370 And you recall, the determinant is for at least the two 00:21:10.370 --> 00:21:12.340 by two you can do it by hand more or less. 00:21:12.340 --> 00:21:17.075 It's the cross products with appropriate sign. 00:21:26.550 --> 00:21:29.390 And I'm sparing you some algebra here, 00:21:29.390 --> 00:21:33.150 but trust me when you do this, this is what you get. 00:21:42.440 --> 00:21:52.780 M squared omega 4 minus 3KM omega squared, plus K2. 00:21:52.780 --> 00:21:54.760 That's it right here. 00:21:54.760 --> 00:22:02.040 This little guy-- oh, all right. 00:22:02.040 --> 00:22:03.690 I can't do that anymore. 00:22:03.690 --> 00:22:04.670 I know, it's this one. 00:22:24.870 --> 00:22:27.213 That is called the characteristic equation. 00:22:43.800 --> 00:22:45.660 So here's the first answer. 00:23:10.162 --> 00:23:12.120 AUDIENCE: [INAUDIBLE] should that be K squared? 00:23:12.120 --> 00:23:12.995 PROFESSOR: I'm sorry. 00:23:12.995 --> 00:23:15.520 That's a typo. 00:23:15.520 --> 00:23:20.697 That's simply K. 00:23:20.697 --> 00:23:22.030 AUDIENCE: [INAUDIBLE] K squared? 00:23:22.030 --> 00:23:23.571 PROFESSOR: Or it's K squared, rather. 00:23:23.571 --> 00:23:24.710 Sorry. 00:23:24.710 --> 00:23:25.210 Thank you. 00:23:32.473 --> 00:23:34.437 AUDIENCE: I think in the above line, 00:23:34.437 --> 00:23:38.717 the determinant-- the upper left-- should had a 2K. 00:23:38.717 --> 00:23:40.300 PROFESSOR: Oh, this is 2K, absolutely. 00:23:44.110 --> 00:23:45.950 All right, 2K. 00:23:45.950 --> 00:23:48.550 Good enough? 00:23:48.550 --> 00:23:50.250 All right, thank you. 00:23:50.250 --> 00:23:52.070 So let's send this to the top. 00:23:54.780 --> 00:24:05.070 So the roots of the characteristic 00:24:05.070 --> 00:24:06.846 are the natural frequencies. 00:24:34.040 --> 00:24:35.717 Let's do this this way. 00:24:58.247 --> 00:24:59.080 Did I do that right? 00:24:59.080 --> 00:25:00.020 Yeah, plus or minus. 00:25:00.020 --> 00:25:05.450 So the situation is that when you apply the quadratic formula 00:25:05.450 --> 00:25:10.040 to that characteristic equation to find the values of omega 00:25:10.040 --> 00:25:12.440 for which that equation is satisfied, 00:25:12.440 --> 00:25:15.970 those omegas that come out are the natural frequencies. 00:25:15.970 --> 00:25:20.540 They are the quantities we seek. 00:25:20.540 --> 00:25:26.160 And what that yields, as you can see from the plus or minus 00:25:26.160 --> 00:25:28.056 here, there are two of them. 00:25:32.200 --> 00:25:33.810 I'll write the whole thing out here. 00:25:57.600 --> 00:25:58.880 And these are numerically. 00:26:15.300 --> 00:26:17.630 OK, everybody see that? 00:26:17.630 --> 00:26:22.100 So here are our two natural frequencies. 00:26:22.100 --> 00:26:26.485 Here's the first one-- excuse me, that's not right either. 00:26:40.115 --> 00:26:40.865 That's the square. 00:26:54.540 --> 00:27:02.540 OK So these are our natural frequencies, once again 00:27:02.540 --> 00:27:06.940 for this special case where the masses are equal 00:27:06.940 --> 00:27:08.800 and the springs are equal. 00:27:08.800 --> 00:27:11.530 Anybody recognize that number, 0.618, 00:27:11.530 --> 00:27:13.920 for all you fuss budgets? 00:27:16.790 --> 00:27:18.760 Ring any bells? 00:27:18.760 --> 00:27:21.880 Any number freaks here? 00:27:21.880 --> 00:27:23.960 No? 00:27:23.960 --> 00:27:25.490 I heard it. 00:27:25.490 --> 00:27:26.370 That's it. 00:27:26.370 --> 00:27:28.640 Exactly, nice job. 00:27:28.640 --> 00:27:32.000 The golden mean, the golden ratio. 00:27:32.000 --> 00:27:37.870 Also, let me simply say if there are-- 00:27:37.870 --> 00:27:40.670 as far as the number of things-- if there 00:27:40.670 --> 00:27:44.240 are n degrees of freedom. 00:27:44.240 --> 00:27:47.800 There are n natural frequencies. 00:27:59.569 --> 00:28:02.230 What else? 00:28:02.230 --> 00:28:03.230 So that's that. 00:28:09.620 --> 00:28:26.043 So now it's time to get to this notion of the natural modes. 00:28:38.177 --> 00:28:40.010 Let me say, we've got to go all the way back 00:28:40.010 --> 00:28:41.250 to this set over here. 00:28:41.250 --> 00:28:45.510 If you take the first row of this matrix equation-- 00:28:45.510 --> 00:28:53.160 that's the first of the equations of motion-- 00:28:53.160 --> 00:28:57.120 and you make that assumption of the harmonic motion in there. 00:29:55.220 --> 00:29:56.680 Does everybody see that? 00:29:56.680 --> 00:30:01.200 What we've done is we've taken basically the first row 00:30:01.200 --> 00:30:04.440 of that expression right up there 00:30:04.440 --> 00:30:08.620 and formed the amplitude ratio A1 over A2. 00:30:08.620 --> 00:30:11.350 What we're doing is we've found the omegas. 00:30:11.350 --> 00:30:13.500 You remember, just review the bidding. 00:30:13.500 --> 00:30:16.670 Our original assumption was harmonic 00:30:16.670 --> 00:30:20.370 motion, that is to say all the displacements are 00:30:20.370 --> 00:30:24.720 moving in synchrony as it were. 00:30:24.720 --> 00:30:27.060 The same sinusoidal frequency, we've 00:30:27.060 --> 00:30:31.310 just found what frequencies those are. 00:30:31.310 --> 00:30:34.450 There are two of them, and they're right there. 00:30:34.450 --> 00:30:38.710 Now we're after these guys. 00:30:38.710 --> 00:30:41.960 Now we're after the relative magnitudes 00:30:41.960 --> 00:30:46.270 or the relative amplitudes of A1 and A2. 00:30:46.270 --> 00:30:52.020 And we from one of the equations isolated one of those. 00:30:52.020 --> 00:31:01.170 And let me just say, if you plug these back in, 00:31:01.170 --> 00:31:42.450 plug in the first one, you'll get oddly enough 1.618, 00:31:42.450 --> 00:31:44.250 These amplitude ratios. 00:32:08.450 --> 00:32:11.400 Are the so-called natural modes. 00:32:11.400 --> 00:32:16.060 And I think you can appreciate that this is the first one, 00:32:16.060 --> 00:32:17.855 and this is the second one. 00:32:29.490 --> 00:32:31.300 Any questions so far? 00:32:31.300 --> 00:32:32.680 Wonderful. 00:32:32.680 --> 00:32:33.370 Hearing none. 00:32:33.370 --> 00:32:35.497 Yes ma'am, Sara? 00:32:35.497 --> 00:32:36.372 AUDIENCE: [INAUDIBLE] 00:32:40.540 --> 00:32:43.610 PROFESSOR: You see this amplitude ratio. 00:32:43.610 --> 00:32:45.170 You saw how we got that. 00:32:45.170 --> 00:32:48.450 You see that the right hand side has got system parameter, Ks 00:32:48.450 --> 00:32:50.380 and Ms, and stuff like that. 00:32:50.380 --> 00:32:54.530 But this is the ringer, omega. 00:32:54.530 --> 00:32:58.090 This amplitude ratio is expressed in part 00:32:58.090 --> 00:32:59.930 in terms of omega. 00:32:59.930 --> 00:33:04.050 So what omega-- there's no ambiguity as to Ks and Ms, 00:33:04.050 --> 00:33:05.890 but what omega? 00:33:05.890 --> 00:33:09.640 Well the answer is, when we plug in this one, 00:33:09.640 --> 00:33:11.040 you get this answer. 00:33:11.040 --> 00:33:13.530 When you plug-in this one, you get this answer. 00:33:13.530 --> 00:33:19.060 So while we're at it-- Sara, want to hazard a guess? 00:33:19.060 --> 00:33:21.730 How many natural mode do you think we've got? 00:33:21.730 --> 00:33:22.420 Yeah, exactly. 00:33:30.200 --> 00:33:34.970 So you're going to have one of these 00:33:34.970 --> 00:33:36.970 for each degree of freedom. 00:33:39.940 --> 00:33:45.420 Let me just point out a couple of elements here, 00:33:45.420 --> 00:33:48.070 and then I'll show you a demonstration because we 00:33:48.070 --> 00:33:49.295 have to have some fun today. 00:33:56.970 --> 00:33:58.900 These are point of informations. 00:33:58.900 --> 00:34:06.460 They're ratios, not absolute magnitudes. 00:34:12.080 --> 00:34:13.219 That's number one. 00:34:13.219 --> 00:34:16.770 The second is-- I already told you, they got the same number. 00:34:16.770 --> 00:34:24.699 OK, each natural mode is associated 00:34:24.699 --> 00:34:38.330 with a particular natural frequency. 00:34:38.330 --> 00:34:39.710 This one goes with that one. 00:34:39.710 --> 00:34:42.800 This one goes with that one. 00:34:42.800 --> 00:34:54.780 And once again, they're associated with-- yeah, 00:34:54.780 --> 00:34:56.139 let me say that. 00:35:01.080 --> 00:35:03.000 I need another board. 00:35:03.000 --> 00:35:04.490 Let's just go over here. 00:36:08.600 --> 00:37:13.900 So in a sense-- this is decouple. 00:37:35.450 --> 00:37:40.106 Decouple essentially into independent subsystems. 00:38:39.390 --> 00:38:53.020 So in general, what the system's response looks like is-- 00:38:53.020 --> 00:38:55.870 I'm talking about the one in front of us here. 00:38:55.870 --> 00:39:03.100 This special case, where the masses and springs are equal. 00:39:15.600 --> 00:39:17.175 I think there's a minus sign in here. 00:39:26.870 --> 00:39:27.932 Does this come up? 00:39:27.932 --> 00:39:28.432 Wonderful. 00:40:07.020 --> 00:40:07.798 That's it. 00:40:13.360 --> 00:40:14.360 Does everybody see that? 00:40:14.360 --> 00:40:15.070 Yes, sir. 00:40:15.070 --> 00:40:17.480 AUDIENCE: [INAUDIBLE]. 00:40:17.480 --> 00:40:18.892 PROFESSOR: I'm sorry? 00:40:18.892 --> 00:40:21.720 AUDIENCE: What does it say under that first bullet point? 00:40:21.720 --> 00:40:22.390 PROFESSOR: Here? 00:40:22.390 --> 00:40:23.110 AUDIENCE: These describe the situation. 00:40:23.110 --> 00:40:24.734 PROFESSOR: These describe the situation 00:40:24.734 --> 00:40:29.340 in which the entire system is oscillating at. 00:40:35.600 --> 00:40:37.880 It's the second bullet here. 00:40:37.880 --> 00:40:38.650 Thank you. 00:40:44.110 --> 00:40:44.860 AUDIENCE: At what? 00:40:44.860 --> 00:40:46.026 PROFESSOR: At one frequency. 00:40:48.947 --> 00:40:51.030 Sorry, I'm just getting a little tired of writing. 00:41:04.140 --> 00:41:10.670 So, any other questions, problems, complaints? 00:41:10.670 --> 00:41:11.655 All right. 00:41:11.655 --> 00:41:11.980 AUDIENCE: I have a question. 00:41:11.980 --> 00:41:13.032 PROFESSOR: Yes, sir. 00:41:13.032 --> 00:41:13.907 AUDIENCE: [INAUDIBLE] 00:41:19.300 --> 00:41:21.580 PROFESSOR: That's correct. 00:41:21.580 --> 00:41:23.450 Then, let's see. 00:41:23.450 --> 00:41:26.100 Then there's a mistake right here. 00:41:26.100 --> 00:41:28.370 Thank you. 00:41:28.370 --> 00:41:32.060 Yeah, because that's the way it came out. 00:41:32.060 --> 00:41:40.030 When you plug omega 2 having this value into here, 00:41:40.030 --> 00:41:42.455 the amplitude ratio comes out minus. 00:41:46.420 --> 00:41:47.216 Fair enough? 00:41:49.739 --> 00:41:51.030 It threw me there for a minute. 00:41:51.030 --> 00:41:56.000 I thought you were going to say, why is the minus sign is there, 00:41:56.000 --> 00:42:05.860 rather than you could have had minus 1.618 and plus 1. 00:42:05.860 --> 00:42:10.740 And the answer is no reason, because these are ratios. 00:42:10.740 --> 00:42:11.903 Yeah, Kaitlin? 00:42:11.903 --> 00:42:15.525 AUDIENCE: But shouldn't-- when we go back and look at what you 00:42:15.525 --> 00:42:18.665 wrote down, it's [INAUDIBLE]. 00:42:18.665 --> 00:42:21.580 I don't understand how that [INAUDIBLE]. 00:42:21.580 --> 00:42:23.055 PROFESSOR: I'm sorry, say again? 00:42:23.055 --> 00:42:24.030 AUDIENCE: Never mind. 00:42:24.030 --> 00:42:24.821 PROFESSOR: Find it? 00:42:24.821 --> 00:42:28.580 Yeah, they're ratios, It's just as simple as that. 00:42:28.580 --> 00:42:32.080 So you multiply them by any number and it still works. 00:42:32.080 --> 00:42:35.750 I'll actually show you here in a second. 00:42:35.750 --> 00:42:38.020 At least, I believe that's the case. 00:42:38.020 --> 00:42:39.510 We'll just see here in a second. 00:42:39.510 --> 00:42:40.580 OK, questions? 00:42:40.580 --> 00:42:42.130 Comments? 00:42:42.130 --> 00:42:43.850 All right. 00:42:43.850 --> 00:42:45.760 Now is the time. 00:42:45.760 --> 00:42:50.370 Could I bring up the side board here? 00:42:50.370 --> 00:42:56.180 Let me show you-- anybody here taken 2086? 00:42:56.180 --> 00:42:56.680 Wonderful. 00:42:56.680 --> 00:42:57.640 I've got one person? 00:42:57.640 --> 00:42:58.680 Great. 00:42:58.680 --> 00:43:03.597 Anyway, I believe in 2086, don't they teach you MATLAB? 00:43:03.597 --> 00:43:04.930 Isn't that the weapon of choice? 00:43:04.930 --> 00:43:09.202 OK, that's the program I'm using here, MATLAB. 00:43:09.202 --> 00:43:10.910 For those of you who haven't seen it yet, 00:43:10.910 --> 00:43:13.052 it is definitely a mixed bag. 00:43:13.052 --> 00:43:14.510 I don't know how you feel about it. 00:43:14.510 --> 00:43:17.840 It's very-- yeah-- it's very powerful. 00:43:17.840 --> 00:43:19.720 It stands for Matrix Laboratory. 00:43:19.720 --> 00:43:22.490 It was written, I don't know, 20, 30 years ago here, 00:43:22.490 --> 00:43:27.770 I believe, at MIT by people who were into matrices, into matrix 00:43:27.770 --> 00:43:28.310 algebra. 00:43:28.310 --> 00:43:31.310 And it's kind of command line oriented. 00:43:31.310 --> 00:43:33.550 The good news, it's very powerful. 00:43:33.550 --> 00:43:35.890 Whatever you want to do, you can do in MATLAB. 00:43:35.890 --> 00:43:39.820 The bad news is, the user interface stinks. 00:43:39.820 --> 00:43:41.840 The language is very difficult to learn. 00:43:41.840 --> 00:43:44.880 It's even harder to remember. 00:43:44.880 --> 00:43:48.550 So with that rousing endorsement, 00:43:48.550 --> 00:43:53.430 let me show you what we've got here. 00:43:53.430 --> 00:44:01.550 This is a program I've-- is that font readable by you guys? 00:44:01.550 --> 00:44:02.540 No? 00:44:02.540 --> 00:44:03.323 No? 00:44:03.323 --> 00:44:05.740 AUDIENCE: [INAUDIBLE]. 00:44:05.740 --> 00:44:07.512 PROFESSOR: I'm sorry? 00:44:07.512 --> 00:44:10.430 AUDIENCE: [INAUDIBLE]. 00:44:10.430 --> 00:44:11.060 PROFESSOR: Yes. 00:44:11.060 --> 00:44:16.015 Well, I believe I can-- here we go, fonts. 00:44:18.620 --> 00:44:21.410 Upping the fonts is kind of a mixed bag, 00:44:21.410 --> 00:44:28.330 because you get bigger letters but they're. 00:44:28.330 --> 00:44:31.430 OK, how's that? 00:44:31.430 --> 00:44:34.230 So here's the situation. 00:44:34.230 --> 00:44:35.700 This is a MATLAB program. 00:44:35.700 --> 00:44:39.240 And I'll explain to you what it does as we go. 00:44:39.240 --> 00:44:42.060 Let me see if my little cursor-- my cursor's here, 00:44:42.060 --> 00:44:43.080 but I can't see it. 00:44:43.080 --> 00:44:45.490 All right, here's the system parameters. 00:44:45.490 --> 00:44:47.940 Once again, we're doing a simple spring mass-- 00:44:47.940 --> 00:44:52.240 this simplified spring mass system, exactly the one 00:44:52.240 --> 00:44:55.040 we've done here. 00:44:55.040 --> 00:45:02.120 When I wrote it, you'll see I generalized it to do this guy. 00:45:02.120 --> 00:45:04.780 So we got M1 and M2, K1 and K2. 00:45:04.780 --> 00:45:09.400 But if you'll notice, you see here their values are equal. 00:45:09.400 --> 00:45:11.560 We've got the mass at one kilogram each. 00:45:11.560 --> 00:45:17.240 And we've got 10 newtons per meter on each of the springs. 00:45:17.240 --> 00:45:20.090 Everybody appreciate that this system's numbers 00:45:20.090 --> 00:45:25.690 that we're putting in here match our case here, K over M? 00:45:25.690 --> 00:45:26.540 OK. 00:45:26.540 --> 00:45:28.990 And you can see here, we've defined-- 00:45:28.990 --> 00:45:31.390 and again, let me just say I'm not 00:45:31.390 --> 00:45:34.365 trying to sell you on MATLAB. 00:45:34.365 --> 00:45:36.240 I don't want to leave you with the impression 00:45:36.240 --> 00:45:39.835 that we expect you to be able to instantly become 00:45:39.835 --> 00:45:41.260 a user of MATLAB. 00:45:41.260 --> 00:45:47.580 This is simply to illustrate the point of the lecture here. 00:45:47.580 --> 00:45:50.110 Here is the M, the system matrix. 00:45:50.110 --> 00:45:51.910 There's the K matrix. 00:45:51.910 --> 00:45:57.850 And I'll show you the eigenvalue and eigenvector thing later. 00:45:57.850 --> 00:45:59.867 But let me-- take my word for it. 00:45:59.867 --> 00:46:00.450 See this here? 00:46:00.450 --> 00:46:08.910 Ode45 is a cryptic allusion to the Runge-Kutta algorithm, 00:46:08.910 --> 00:46:16.040 fourth order Runge-Kutta that is the workhorse for integrating 00:46:16.040 --> 00:46:18.380 differential equations. 00:46:18.380 --> 00:46:21.000 And so let me just run this. 00:46:21.000 --> 00:46:23.690 And what I've got here is, here's the point 00:46:23.690 --> 00:46:27.375 I wanted you to get here, because I'll 00:46:27.375 --> 00:46:29.800 bet you can't see that cursor either. 00:46:29.800 --> 00:46:32.410 Yes, anyway, see this right here? 00:46:32.410 --> 00:46:37.342 tspan is the time scale and the time step, defined up here. 00:46:37.342 --> 00:46:39.300 But these are basically the initial conditions. 00:46:39.300 --> 00:46:40.260 See it here? 00:46:40.260 --> 00:46:44.260 X1, X1 dot, X2, X2 dot. 00:46:44.260 --> 00:46:46.390 So here's the first one. 00:46:46.390 --> 00:46:49.770 This is a 0.618 is for the X1. 00:46:49.770 --> 00:46:53.720 And 1 is for X2. 00:46:53.720 --> 00:46:55.880 Everybody appreciate that? 00:46:55.880 --> 00:46:57.245 Got it? 00:46:57.245 --> 00:46:57.745 OK. 00:47:00.700 --> 00:47:04.060 If these are the initial conditions, what I've done, 00:47:04.060 --> 00:47:07.260 I have artfully chosen the initial conditions 00:47:07.260 --> 00:47:10.575 to have the same ratio. 00:47:14.870 --> 00:47:16.590 What do you expect is going to happen? 00:47:16.590 --> 00:47:18.940 When I turn this thing-- I've got a simulation here. 00:47:18.940 --> 00:47:20.700 I'm going to run this, and you're actually 00:47:20.700 --> 00:47:22.177 going to see it. 00:47:22.177 --> 00:47:23.760 What do you think you're going to see? 00:47:27.710 --> 00:47:29.880 It's a two spring, two mass system. 00:47:29.880 --> 00:47:32.620 What I've done is I've displaced the two masses. 00:47:36.805 --> 00:47:38.200 AUDIENCE: [INAUDIBLE] 00:47:38.200 --> 00:47:39.770 PROFESSOR: They'll certainly have an amplitude, because I'm 00:47:39.770 --> 00:47:40.910 putting it in there. 00:47:40.910 --> 00:47:43.780 That's the initial condition. 00:47:43.780 --> 00:47:49.255 The question is, what frequency you think they'll oscillate at? 00:47:49.255 --> 00:47:50.130 AUDIENCE: [INAUDIBLE] 00:47:50.130 --> 00:47:51.177 PROFESSOR: Pardon? 00:47:51.177 --> 00:47:53.037 AUDIENCE: [INAUDIBLE] 00:47:53.037 --> 00:47:54.620 PROFESSOR: Each of them will oscillate 00:47:54.620 --> 00:47:56.420 with the same frequency, for sure, but what 00:47:56.420 --> 00:47:57.670 do you think it's going to be? 00:47:57.670 --> 00:47:59.060 AUDIENCE: That one. 00:47:59.060 --> 00:48:01.080 PROFESSOR: It's going to be that one. 00:48:01.080 --> 00:48:03.550 So off we go. 00:48:03.550 --> 00:48:09.705 So let us hope that yours truly's program worked. 00:48:16.320 --> 00:48:17.540 Here we go. 00:48:17.540 --> 00:48:19.860 Oh, look at that. 00:48:19.860 --> 00:48:21.940 [INAUDIBLE], please interpret that for me. 00:48:21.940 --> 00:48:24.700 What do you see there? 00:48:24.700 --> 00:48:25.770 Hang on a second. 00:48:25.770 --> 00:48:27.980 Let me blow it up so you can see it. 00:48:27.980 --> 00:48:30.470 Ooh, isn't that pretty? 00:48:30.470 --> 00:48:38.483 And I believe the blue is X1 and the green is X2. 00:48:42.250 --> 00:48:43.994 See? 00:48:43.994 --> 00:48:44.660 Everybody agree? 00:48:44.660 --> 00:48:46.680 Everyone appreciate what's going on? 00:48:46.680 --> 00:48:51.440 You pull them both at slightly different-- 00:48:51.440 --> 00:48:57.080 you basically used the first natural mode 00:48:57.080 --> 00:48:58.680 as the initial condition. 00:48:58.680 --> 00:49:01.189 And sure enough, they oscillate together. 00:49:01.189 --> 00:49:02.730 They oscillate at the same frequency. 00:49:02.730 --> 00:49:06.030 They oscillate at that frequency. 00:49:06.030 --> 00:49:08.490 Let me just see-- I just want to make 00:49:08.490 --> 00:49:11.740 sure we get the full value out of this thing. 00:49:11.740 --> 00:49:14.210 Well, of course you can't see it anymore 00:49:14.210 --> 00:49:17.139 because our numbers are so big. 00:49:17.139 --> 00:49:17.930 Well, that's great. 00:49:17.930 --> 00:49:21.760 Anyway, take my word for it at-- oh, here it is. 00:49:21.760 --> 00:49:27.150 The period for the first natural frequency-- or I 00:49:27.150 --> 00:49:35.900 guess it's the second-- it should be like 1.2. 00:49:35.900 --> 00:49:36.880 Or is it 3? 00:49:36.880 --> 00:49:40.520 Yeah, I'm sorry, the period is 3.2. 00:49:40.520 --> 00:49:42.005 And sure enough, there it is. 00:49:42.005 --> 00:49:45.291 It's about 3. 00:49:45.291 --> 00:49:45.790 3.2. 00:49:45.790 --> 00:49:47.380 Fabulous. 00:49:47.380 --> 00:49:48.590 Everybody got it? 00:49:48.590 --> 00:49:51.570 OK, now watch closely. 00:49:51.570 --> 00:49:52.855 Let me see if I can do this. 00:49:52.855 --> 00:49:55.210 This requires a little dexterity, which 00:49:55.210 --> 00:49:58.770 is always a short supply here. 00:49:58.770 --> 00:50:01.000 I have to hit this and this. 00:50:08.060 --> 00:50:08.720 Make sense? 00:50:08.720 --> 00:50:11.190 That's what it actually looks like. 00:50:11.190 --> 00:50:15.360 They're both oscillating at the same natural frequency, 00:50:15.360 --> 00:50:17.300 going up and down together. 00:50:17.300 --> 00:50:20.310 But they have different amplitudes. 00:50:20.310 --> 00:50:23.280 So one's bigger than the other. 00:50:23.280 --> 00:50:24.920 So that's what it looks like. 00:50:24.920 --> 00:50:26.000 Questions? 00:50:26.000 --> 00:50:27.590 Christina, you good? 00:50:27.590 --> 00:50:30.480 Clear enough? 00:50:30.480 --> 00:50:32.190 Wonderful. 00:50:32.190 --> 00:50:38.800 So let's go to our program. 00:50:38.800 --> 00:50:46.730 And instead of that set of initial conditions, 00:50:46.730 --> 00:50:48.725 we'll do the other. 00:50:48.725 --> 00:50:49.600 Read them to us here. 00:50:49.600 --> 00:50:51.170 What are the initial conditions here? 00:50:55.590 --> 00:50:57.110 AUDIENCE: It's 1.618. 00:50:57.110 --> 00:50:58.940 PROFESSOR: That's right, it's this guy. 00:50:58.940 --> 00:50:59.731 AUDIENCE: That guy. 00:50:59.731 --> 00:51:02.360 PROFESSOR: It's this guy. 00:51:02.360 --> 00:51:04.020 It's this ratio. 00:51:04.020 --> 00:51:06.730 So I basically arbitrarily chose, is it 00:51:06.730 --> 00:51:07.585 the negative first? 00:51:07.585 --> 00:51:08.084 No. 00:51:10.750 --> 00:51:14.870 I chose that one over there, 1.618. 00:51:14.870 --> 00:51:17.350 And then a minus 1 for the second one. 00:51:17.350 --> 00:51:18.310 Fair enough? 00:51:18.310 --> 00:51:19.560 OK, there it goes. 00:51:19.560 --> 00:51:22.880 We've got to save it and make sure we got it. 00:51:22.880 --> 00:51:26.440 So again, you got a clue what's going to happen here? 00:51:31.630 --> 00:51:32.260 Here we go. 00:51:32.260 --> 00:51:32.760 Boom. 00:51:35.190 --> 00:51:36.720 Look at that. 00:51:36.720 --> 00:51:39.400 What's going on there? 00:51:39.400 --> 00:51:42.260 Yikes. 00:51:42.260 --> 00:51:43.400 Explain me. 00:51:43.400 --> 00:51:44.870 Is that good, bad, indifferent? 00:51:44.870 --> 00:51:45.500 Is it right? 00:51:45.500 --> 00:51:46.562 Wrong? 00:51:46.562 --> 00:51:47.978 AUDIENCE: The way the system acts, 00:51:47.978 --> 00:51:49.350 it has a higher frequency. 00:51:49.350 --> 00:51:51.080 PROFESSOR: Yeah, exactly. 00:51:51.080 --> 00:51:51.640 Two things. 00:51:51.640 --> 00:51:53.520 One is, they're out of phase. 00:51:53.520 --> 00:51:54.680 They're doing this. 00:51:54.680 --> 00:51:56.490 One's going this way, and the other's 00:51:56.490 --> 00:51:58.780 going the other at different amplitudes 00:51:58.780 --> 00:51:59.920 but the same frequency. 00:51:59.920 --> 00:52:03.091 But the frequency in question is higher than the previous. 00:52:03.091 --> 00:52:04.590 AUDIENCE: Why are they out of phase? 00:52:04.590 --> 00:52:05.465 PROFESSOR: I'm sorry? 00:52:05.465 --> 00:52:07.420 AUDIENCE: Why are they opposite of each other? 00:52:07.420 --> 00:52:09.794 PROFESSOR: [INAUDIBLE], why are they opposite each other? 00:52:14.210 --> 00:52:15.630 Because we made them that way. 00:52:15.630 --> 00:52:19.545 We said, that's the initial condition. 00:52:19.545 --> 00:52:20.420 Does that make sense? 00:52:23.380 --> 00:52:24.330 That minus sign does. 00:52:27.480 --> 00:52:31.370 One starts out, and one starts in. 00:52:31.370 --> 00:52:32.659 And they do that. 00:52:32.659 --> 00:52:33.200 Clear enough? 00:52:36.070 --> 00:52:39.110 Now what's going to happen if we plain just choose 00:52:39.110 --> 00:52:42.470 any old initial condition? 00:52:42.470 --> 00:52:44.470 These were special. 00:52:44.470 --> 00:52:47.620 We worked like a dog to compute these, 00:52:47.620 --> 00:52:52.410 so that the system would decouple in that way. 00:52:52.410 --> 00:53:01.640 So what if we-- now let me put that back. 00:53:01.640 --> 00:53:04.290 Now look at this one. 00:53:04.290 --> 00:53:05.450 All right, look at that. 00:53:09.410 --> 00:53:10.263 Read that to me. 00:53:13.510 --> 00:53:14.550 AUDIENCE: [INAUDIBLE] 00:53:14.550 --> 00:53:15.216 PROFESSOR: Yeah. 00:53:15.216 --> 00:53:19.320 So that says the initial condition for the first mass 00:53:19.320 --> 00:53:21.940 is 1 and whatever that is. 00:53:21.940 --> 00:53:23.780 One whatever that is. 00:53:23.780 --> 00:53:26.450 The second masses' initial condition is half 00:53:26.450 --> 00:53:28.150 that in the same direction. 00:53:28.150 --> 00:53:29.199 Both positive. 00:53:29.199 --> 00:53:30.990 So Christina, they're going to go together. 00:53:33.530 --> 00:53:37.170 But [INAUDIBLE], at what frequency? 00:53:37.170 --> 00:53:40.450 Any idea what it's going to look like? 00:53:40.450 --> 00:53:43.140 If you do, you're a better man than I, 00:53:43.140 --> 00:53:48.410 because what you're going to see here is that. 00:53:48.410 --> 00:53:51.040 It's this thing right here. 00:53:51.040 --> 00:53:54.210 It's that expression right there. 00:53:54.210 --> 00:53:57.140 And here's what it looks like. 00:54:01.190 --> 00:54:04.570 Did I stop the-- oh, wait a minute. 00:54:04.570 --> 00:54:07.786 Did I ever show you that before? 00:54:07.786 --> 00:54:09.410 I think I forgot to show you the other. 00:54:09.410 --> 00:54:10.720 Anyway, not to worry. 00:54:13.300 --> 00:54:14.130 Hang on a second. 00:54:14.130 --> 00:54:16.162 I've got to stop this guy. 00:54:16.162 --> 00:54:18.560 First I have to find my finger. 00:54:22.240 --> 00:54:23.050 There it goes. 00:54:23.050 --> 00:54:24.930 That's the previous case. 00:54:24.930 --> 00:54:29.140 When they're out of phase, different magnitudes, 00:54:29.140 --> 00:54:30.660 going in opposite directions. 00:54:30.660 --> 00:54:34.060 And you can see, they're going at a higher frequency 00:54:34.060 --> 00:54:35.620 than before. 00:54:35.620 --> 00:54:36.230 Make sense? 00:54:39.340 --> 00:54:45.100 So now we are-- just to refresh your memory-- 00:54:45.100 --> 00:54:48.410 now we're going for the third case, in which there's 00:54:48.410 --> 00:54:49.150 nothing special. 00:54:49.150 --> 00:54:54.650 We just picked a couple of initial conditions 00:54:54.650 --> 00:54:57.960 out of a hat. 00:54:57.960 --> 00:54:59.800 And here we go. 00:54:59.800 --> 00:55:01.698 Oops, I think not. 00:55:07.180 --> 00:55:10.070 I think that's the previous case. 00:55:10.070 --> 00:55:12.210 So let's go here. 00:55:12.210 --> 00:55:15.230 This is another wonderful thing about MATLAB 00:55:15.230 --> 00:55:19.890 is nothing happens until you save it. 00:55:22.510 --> 00:55:24.435 So we were just running the previous case. 00:55:27.500 --> 00:55:28.130 Nasty. 00:55:28.130 --> 00:55:28.900 Look at this. 00:55:32.130 --> 00:55:33.765 All right, can everybody see that? 00:55:36.940 --> 00:55:40.310 If you can interpret this, you're smarter than I am. 00:55:40.310 --> 00:55:46.710 But what this is, this is simply this expression over here. 00:55:46.710 --> 00:55:49.640 It's this expression for just some arbitrary 00:55:49.640 --> 00:55:50.680 initial condition. 00:55:50.680 --> 00:55:53.240 Do you see that that behavior though? 00:55:53.240 --> 00:55:55.780 Each of them, they're going together kind of, 00:55:55.780 --> 00:56:00.954 but they-- anyway, watch this. 00:56:00.954 --> 00:56:02.870 Here's what the simulation of that looks like. 00:56:15.760 --> 00:56:18.820 What the heck is that? 00:56:18.820 --> 00:56:20.330 Well anyway, the point of the story 00:56:20.330 --> 00:56:23.210 is that multiple degrees of freedom 00:56:23.210 --> 00:56:27.305 system in general's response can be arbitrarily complicated. 00:56:27.305 --> 00:56:28.680 It's not arbitrarily complicated, 00:56:28.680 --> 00:56:31.090 but pretty complicated. 00:56:31.090 --> 00:56:34.960 You'll get, in general if it's an nth order system, if you 00:56:34.960 --> 00:56:37.490 don't know anything about the worst case, 00:56:37.490 --> 00:56:40.700 you'll see four frequencies in there. 00:56:40.700 --> 00:56:44.810 And they're all mixed together in some mystical way that's 00:56:44.810 --> 00:56:46.270 unknown to you. 00:56:46.270 --> 00:56:47.130 Fair enough? 00:56:47.130 --> 00:56:54.040 And it's only when you reach the natural modes that you actually 00:56:54.040 --> 00:56:58.500 find out what is going on here. 00:56:58.500 --> 00:57:03.570 Well now I have to turn your attention to this guy. 00:57:03.570 --> 00:57:11.710 This is made by Professor Vandiver's machinist, 00:57:11.710 --> 00:57:15.000 a perfect example of a second order system. 00:57:15.000 --> 00:57:17.170 And I bring it to your attention here 00:57:17.170 --> 00:57:20.380 for two-- at the end of the day what we're going to do 00:57:20.380 --> 00:57:25.000 is I'm going to demonstrate exactly what I just 00:57:25.000 --> 00:57:28.580 did for the textbook case, the textbook system. 00:57:28.580 --> 00:57:33.240 I want to demonstrate exactly the same thing for this guy, 00:57:33.240 --> 00:57:35.920 only this is a real system. 00:57:35.920 --> 00:57:37.180 Very nice. 00:57:37.180 --> 00:57:39.060 We have a steel rod. 00:57:39.060 --> 00:57:40.810 It must be a half inch in diameter. 00:57:40.810 --> 00:57:43.360 The whole thing weighs several pounds. 00:57:43.360 --> 00:57:48.715 These sliding masses are right circular cylinders with a hole 00:57:48.715 --> 00:57:49.590 drilled through them. 00:57:49.590 --> 00:57:54.002 It's ever so slightly larger than these here. 00:57:54.002 --> 00:57:55.210 They're of different lengths. 00:57:55.210 --> 00:57:56.210 They're made of brass. 00:57:56.210 --> 00:58:00.010 They're serious masses. 00:58:00.010 --> 00:58:02.350 And the springs, which extend from here 00:58:02.350 --> 00:58:07.815 to here, and from here to here are wound on a lathe, 00:58:07.815 --> 00:58:09.930 and attached, and so forth. 00:58:09.930 --> 00:58:10.960 Pretty, no? 00:58:14.340 --> 00:58:16.100 Now look right off the bat. 00:58:16.100 --> 00:58:18.685 Did you see how that thing operates? 00:58:27.320 --> 00:58:33.270 Would you agree you have some complicated behavior here? 00:58:33.270 --> 00:58:40.290 Now also would you agree that this is it like that? 00:58:40.290 --> 00:58:42.820 Everybody see that? 00:58:42.820 --> 00:58:45.410 Before we go too far, this is a mixed message here. 00:58:45.410 --> 00:58:51.120 [INAUDIBLE], is this exactly like that? 00:58:51.120 --> 00:58:56.665 In what way is it similar to that? 00:58:56.665 --> 00:58:57.540 AUDIENCE: [INAUDIBLE] 00:59:00.360 --> 00:59:05.070 PROFESSOR: Well, what is clear is that you've got two springs 00:59:05.070 --> 00:59:07.310 and you've got two masses. 00:59:07.310 --> 00:59:10.044 About that there is very little argument. 00:59:10.044 --> 00:59:10.960 AUDIENCE: It's damped. 00:59:10.960 --> 00:59:11.990 PROFESSOR: It's damped. 00:59:11.990 --> 00:59:13.410 Can everybody see that? 00:59:13.410 --> 00:59:15.870 How does [INAUDIBLE] know that it's damped? 00:59:18.460 --> 00:59:20.260 How's he know it's damped? 00:59:20.260 --> 00:59:23.972 I mean, that's just a wild guess on his part, but. 00:59:23.972 --> 00:59:25.950 AUDIENCE: You can hear it, and it slows down. 00:59:25.950 --> 00:59:27.010 And it slows down. 00:59:27.010 --> 00:59:33.170 This is the most important part is that it stops. 00:59:33.170 --> 00:59:37.984 Eventually if you come back in a minute or two, it's done. 00:59:37.984 --> 00:59:39.400 PROFESSOR: All right, [INAUDIBLE]. 00:59:39.400 --> 00:59:40.220 You're on a roll. 00:59:40.220 --> 00:59:42.060 There's definitely damping there. 00:59:42.060 --> 00:59:44.669 What kind of damping? 00:59:44.669 --> 00:59:45.927 AUDIENCE: Friction. 00:59:45.927 --> 00:59:47.010 PROFESSOR: Friction, yeah. 00:59:47.010 --> 00:59:51.890 Does that have another name that you can think of? 00:59:51.890 --> 00:59:54.670 It's definitely friction. 00:59:54.670 --> 00:59:57.070 What it's not is viscous friction. 00:59:57.070 --> 01:00:04.040 What it is not is a damper or a dashpot which we've shown you 01:00:04.040 --> 01:00:13.220 before with the ideal expression that generate a force that 01:00:13.220 --> 01:00:17.670 opposes the-- generation of an opposing force that's linearly 01:00:17.670 --> 01:00:19.930 proportional to the velocity. 01:00:19.930 --> 01:00:21.610 What's going on here, do you think? 01:00:24.270 --> 01:00:27.880 What kind of damping do you think? 01:00:27.880 --> 01:00:29.680 It's called Coulomb. 01:00:29.680 --> 01:00:31.390 This is called Coulomb damping. 01:00:31.390 --> 01:00:33.110 And this is a digression. 01:00:33.110 --> 01:00:37.760 Now we're on the part where this is really-- everything that's 01:00:37.760 --> 01:00:40.090 on the board is what I wanted you to really come 01:00:40.090 --> 01:00:41.990 away from today with. 01:00:41.990 --> 01:00:44.375 So now we're out kind of in the, I 01:00:44.375 --> 01:00:47.380 would call it the winging it area 01:00:47.380 --> 01:00:53.750 right here, because this is the part where I simply 01:00:53.750 --> 01:00:54.970 had fun with the demo. 01:00:58.600 --> 01:00:59.920 This is viscous. 01:01:05.880 --> 01:01:08.160 And this has got the symbol-- well anyway, 01:01:08.160 --> 01:01:09.450 this is what it looks like. 01:01:09.450 --> 01:01:12.570 And this is the force of the damper. 01:01:12.570 --> 01:01:13.440 We'll call it B. 01:01:13.440 --> 01:01:16.640 And this is the velocity. 01:01:16.640 --> 01:01:20.025 And this is for constant of proportionality B. 01:01:20.025 --> 01:01:25.880 And it has this little symbol, like that. 01:01:25.880 --> 01:01:31.140 And when equations of motion are solved that contain that, 01:01:31.140 --> 01:01:32.765 the response looks like this. 01:01:48.310 --> 01:02:05.950 What we're talking about here, the force put out 01:02:05.950 --> 01:02:11.460 is a constant. 01:02:11.460 --> 01:02:14.500 That just comes from the sliding. 01:02:14.500 --> 01:02:18.590 And what it generates are distinctly non-linear equations 01:02:18.590 --> 01:02:19.930 of motion. 01:02:19.930 --> 01:02:24.020 And what you get here is you get this kind of behavior. 01:02:24.020 --> 01:02:30.250 If you really looked at it, what you'll see 01:02:30.250 --> 01:02:33.860 is there's definitely damping for large motions 01:02:33.860 --> 01:02:38.260 when the inertial forces and so forth are large compared 01:02:38.260 --> 01:02:40.290 to the friction forces. 01:02:40.290 --> 01:02:44.460 It'll look a lot like conventional viscous damping. 01:02:44.460 --> 01:02:47.000 It's just that when motions get really small, 01:02:47.000 --> 01:02:50.630 and the forces get down there to on the order of this, 01:02:50.630 --> 01:02:56.320 all a sudden you'll see on one cycle it'll just stop. 01:02:56.320 --> 01:03:01.260 And were you up here where you could see, 01:03:01.260 --> 01:03:04.150 or if we had a closeup of this-- you can't see it, but just 01:03:04.150 --> 01:03:06.770 watch this thing stop. 01:03:06.770 --> 01:03:07.300 Right there. 01:03:07.300 --> 01:03:08.370 Do you see that? 01:03:08.370 --> 01:03:10.870 That's a little hard for you to see from there, but watch. 01:03:13.520 --> 01:03:17.595 Anyway, were you up here, you'd see this. 01:03:17.595 --> 01:03:20.430 That's what we're looking at. 01:03:20.430 --> 01:03:23.370 Well here we go. 01:03:23.370 --> 01:03:24.920 I need some help here. 01:03:24.920 --> 01:03:29.760 Who's in a volunteering frame of mind? 01:03:29.760 --> 01:03:31.580 Amy, all right. 01:03:31.580 --> 01:03:34.160 I appreciate the help here. 01:03:34.160 --> 01:03:35.420 Here's what I want to do. 01:03:35.420 --> 01:03:38.990 We just blew out some wonderful theory. 01:03:38.990 --> 01:03:42.410 All this is just solid as a rock. 01:03:42.410 --> 01:03:43.656 Yes, sir. 01:03:43.656 --> 01:03:45.833 AUDIENCE: For the Coulomb friction, is that a linear 01:03:45.833 --> 01:03:46.332 [INAUDIBLE]? 01:03:46.332 --> 01:03:47.230 Or is it still exponential? 01:03:47.230 --> 01:03:48.240 PROFESSOR: I'm sorry? 01:03:48.240 --> 01:03:49.250 Oh, no. 01:03:51.850 --> 01:03:54.510 If I'm not mistaken, I didn't really look this up, 01:03:54.510 --> 01:03:56.960 but I believe it's linear. 01:03:56.960 --> 01:03:58.857 I'd have to-- take that with a grain of salt, 01:03:58.857 --> 01:03:59.940 but I believe it's linear. 01:04:03.210 --> 01:04:05.830 Yes, Amy, here's the situation. 01:04:05.830 --> 01:04:07.650 We have all this marvelous theory. 01:04:07.650 --> 01:04:13.810 My goal is to-- and we have this fabulous demo apparatus, though 01:04:13.810 --> 01:04:15.680 inherited. 01:04:15.680 --> 01:04:19.430 And what I'd like to do-- oh, and we have computational means 01:04:19.430 --> 01:04:21.560 to. 01:04:21.560 --> 01:04:24.310 And in fact, we just went through the exercise. 01:04:24.310 --> 01:04:26.480 We already know those same equations that 01:04:26.480 --> 01:04:28.340 work for this work for this. 01:04:28.340 --> 01:04:30.270 Those are general. 01:04:30.270 --> 01:04:33.570 However, it's not my piece of apparatus. 01:04:33.570 --> 01:04:39.980 And well, here's the deal, what are the Ms and Ks. 01:04:39.980 --> 01:04:46.910 What are the values of-- I need M1, K1, M2, K2 to put into the. 01:04:46.910 --> 01:04:48.530 AUDIENCE: [INAUDIBLE]. 01:04:48.530 --> 01:04:50.560 PROFESSOR: Yeah, Yeah. 01:04:50.560 --> 01:04:52.641 That's what I'd like to do is I'd like to. 01:04:52.641 --> 01:04:54.182 AUDIENCE: Do I have to just determine 01:04:54.182 --> 01:04:55.160 the it by looking at it? 01:04:55.160 --> 01:04:56.201 PROFESSOR: Oh no, no, no. 01:04:56.201 --> 01:04:56.940 No, no. 01:04:56.940 --> 01:04:59.530 AUDIENCE: I'm not that good. 01:04:59.530 --> 01:05:01.060 PROFESSOR: You're my assistant. 01:05:01.060 --> 01:05:03.170 I guess the question is, how would 01:05:03.170 --> 01:05:05.930 you-- and I've got to tell you, that's 01:05:05.930 --> 01:05:07.250 the math part of a program. 01:05:07.250 --> 01:05:09.640 Now we're in the engineering part of the program, 01:05:09.640 --> 01:05:12.930 because somebody gave you a real live demo apparatus. 01:05:12.930 --> 01:05:16.600 Works like crazy, or appears to. 01:05:16.600 --> 01:05:18.610 And I'd love to take advantage of it, 01:05:18.610 --> 01:05:20.680 but I don't know any of the numbers. 01:05:20.680 --> 01:05:22.990 AUDIENCE: [INAUDIBLE] 01:05:22.990 --> 01:05:24.720 PROFESSOR: No. 01:05:24.720 --> 01:05:27.561 That's the constraint I'm operating on. 01:05:27.561 --> 01:05:28.560 It doesn't belong to me. 01:05:28.560 --> 01:05:30.200 I mean, I could take it apart. 01:05:30.200 --> 01:05:32.580 That's an absolutely appropriate thing to do. 01:05:32.580 --> 01:05:34.020 I would have liked to. 01:05:34.020 --> 01:05:35.430 It would be easier if you could. 01:05:35.430 --> 01:05:38.690 You just go, take a screwdriver to it. 01:05:38.690 --> 01:05:42.630 Here, put this on there and pull this out. 01:05:42.630 --> 01:05:45.950 I didn't have the luxury of any of that, so what's 01:05:45.950 --> 01:05:49.050 your next best suggestion? 01:05:49.050 --> 01:05:51.270 Nice suggestion, but no cigar. 01:05:51.270 --> 01:05:52.325 I'm sorry? 01:05:52.325 --> 01:05:54.491 AUDIENCE: Take it apart anyway, put it back together 01:05:54.491 --> 01:05:55.812 before the person notices. 01:05:55.812 --> 01:05:56.170 PROFESSOR: Well yeah. 01:05:56.170 --> 01:05:56.930 Yeah, no. 01:05:56.930 --> 01:05:57.600 That's fudging. 01:05:57.600 --> 01:06:00.890 Yeah, they notice. 01:06:00.890 --> 01:06:03.860 Have you ever taken apart anything made 01:06:03.860 --> 01:06:07.210 in modern manufacturing method? 01:06:07.210 --> 01:06:10.120 Oh, it's good because you can't put them back together. 01:06:10.120 --> 01:06:12.240 They're assembled by machine. 01:06:12.240 --> 01:06:15.780 And once upon a time you could disassemble one 01:06:15.780 --> 01:06:19.270 and reassemble things without detection. 01:06:19.270 --> 01:06:21.560 But anymore, once you take them apart, 01:06:21.560 --> 01:06:23.380 it's wicked hard to get them back together. 01:06:23.380 --> 01:06:25.080 OK, the floor is open. 01:06:25.080 --> 01:06:26.954 I need another suggestion. 01:06:26.954 --> 01:06:30.670 What are you going to do? 01:06:30.670 --> 01:06:33.316 AUDIENCE: Do you need to know the exact K and M, 01:06:33.316 --> 01:06:35.040 or do you just [INAUDIBLE] another ratio? 01:06:36.892 --> 01:06:38.350 PROFESSOR: I thought you were going 01:06:38.350 --> 01:06:42.970 to-- I need to know M or K. I need to know them all. 01:06:42.970 --> 01:06:45.280 AUDIENCE: Do you know the density of the-- 01:06:45.280 --> 01:06:47.030 PROFESSOR: I was going to say, but I don't 01:06:47.030 --> 01:06:49.800 need to know anything exactly. 01:06:49.800 --> 01:06:53.280 All I need to know is as good a guess as you can come up with. 01:06:53.280 --> 01:06:54.689 It's all an estimate. 01:06:54.689 --> 01:06:57.034 AUDIENCE: If you know the density of the material, 01:06:57.034 --> 01:06:59.379 you can easily work up [INAUDIBLE]. 01:06:59.379 --> 01:07:01.730 I'm assuming you're about [INAUDIBLE]. 01:07:01.730 --> 01:07:04.300 PROFESSOR: Oh absolutely, absolutely. 01:07:04.300 --> 01:07:05.708 AUDIENCE: [INAUDIBLE]. 01:07:05.708 --> 01:07:06.583 PROFESSOR: Wonderful. 01:07:09.300 --> 01:07:11.880 Absolutely. 01:07:11.880 --> 01:07:14.990 She hit the jackpot, rang the magic buzzer. 01:07:14.990 --> 01:07:17.970 That's exactly what I did. 01:07:17.970 --> 01:07:21.050 Here's a little crummy sketch. 01:07:21.050 --> 01:07:23.770 Oh wait, you can't see that. 01:07:23.770 --> 01:07:27.122 Anyway, these are right circular cylinders with holes in them. 01:07:27.122 --> 01:07:28.830 And they've got measurements beside them. 01:07:28.830 --> 01:07:31.190 I can tell you, this is 75 millimeters. 01:07:31.190 --> 01:07:34.520 This is 35 millimeters. 01:07:34.520 --> 01:07:40.360 This one is 37 millimeters long. 01:07:40.360 --> 01:07:41.970 So I did that. 01:07:41.970 --> 01:07:42.650 That's great. 01:07:42.650 --> 01:07:44.800 That's an excellent suggestion. 01:07:44.800 --> 01:07:53.910 And after I did exactly that, I won't write out the formula. 01:07:53.910 --> 01:07:56.220 You know area, and volume, and all of that. 01:07:59.992 --> 01:08:01.615 Let me get you the right order here. 01:08:05.970 --> 01:08:25.859 M1 is 0.2929, and M2 is 0.5938. 01:08:25.859 --> 01:08:27.359 Everybody got that? 01:08:27.359 --> 01:08:31.180 This was obtained by taking a ruler to these things, 01:08:31.180 --> 01:08:34.580 taking diameters, lengths, and diameters of holes, 01:08:34.580 --> 01:08:39.682 multiply them times the density of brass taken out of the book. 01:08:39.682 --> 01:08:42.069 Do you believe that? 01:08:42.069 --> 01:08:43.766 Do you believe that number? 01:08:49.479 --> 01:08:51.279 Well, you're a trusting soul. 01:08:51.279 --> 01:08:52.399 I don't. 01:08:52.399 --> 01:08:55.790 To me, I believe that number. 01:08:55.790 --> 01:08:58.010 This was done with a ruler. 01:08:58.010 --> 01:09:00.800 The little millimeter thingies. 01:09:00.800 --> 01:09:06.510 So I just say, don't fall in the trap of false precision. 01:09:06.510 --> 01:09:08.069 OK Amy, you're on a roll. 01:09:08.069 --> 01:09:10.220 We've got the masses. 01:09:10.220 --> 01:09:11.413 What now? 01:09:11.413 --> 01:09:13.310 AUDIENCE: Free body diagram. 01:09:13.310 --> 01:09:14.854 PROFESSOR: Yeah, we got all that. 01:09:14.854 --> 01:09:16.187 AUDIENCE: Yeah, you've got that. 01:09:16.187 --> 01:09:18.869 But then what you can do for the spring, 01:09:18.869 --> 01:09:21.781 the forces of the spring when static. 01:09:21.781 --> 01:09:22.364 Don't move it. 01:09:22.364 --> 01:09:24.854 Don't move it. 01:09:24.854 --> 01:09:26.850 So take the top mast. 01:09:26.850 --> 01:09:28.505 It's not moving, which means that you 01:09:28.505 --> 01:09:30.880 know that the force going upwards-- which is the spring-- 01:09:30.880 --> 01:09:36.280 is equal to the force going downward, which is [INAUDIBLE]. 01:09:36.280 --> 01:09:38.670 So you can measure the displacement 01:09:38.670 --> 01:09:41.892 from the start of the spring to the bottom of the spring. 01:09:41.892 --> 01:09:44.160 Do you know the natural length of the spring? 01:09:44.160 --> 01:09:46.899 PROFESSOR: No Anyway, what I was going to say 01:09:46.899 --> 01:09:49.160 is, excellent idea. 01:09:49.160 --> 01:09:51.149 Can't do it. 01:09:51.149 --> 01:09:54.640 But what Amy was basically saying is, 01:09:54.640 --> 01:09:56.700 you know the masses now. 01:09:56.700 --> 01:10:01.220 Why not simply take from that expression 01:10:01.220 --> 01:10:05.252 right there, MH over K, right? 01:10:05.252 --> 01:10:06.460 What's the problem with that? 01:10:06.460 --> 01:10:07.001 How about it? 01:10:07.001 --> 01:10:10.562 Devin, how come I can't do that? 01:10:10.562 --> 01:10:12.540 AUDIENCE: [INAUDIBLE]. 01:10:12.540 --> 01:10:15.645 PROFESSOR: Yeah, I really don't know the no load position. 01:10:18.920 --> 01:10:20.610 Is that it, Amy? 01:10:20.610 --> 01:10:23.430 Right there? 01:10:23.430 --> 01:10:23.930 Maybe. 01:10:27.820 --> 01:10:31.460 Devin, you're the one that suggested it. 01:10:31.460 --> 01:10:35.630 Are those the no load positions of the masses? 01:10:35.630 --> 01:10:37.060 And if not, why not? 01:10:39.670 --> 01:10:41.590 I did mean to give you a clue there. 01:10:41.590 --> 01:10:42.300 Yeah, Nick? 01:10:42.300 --> 01:10:44.550 AUDIENCE: It can't be because there's static friction. 01:10:44.550 --> 01:10:47.220 PROFESSOR: Exactly. 01:10:47.220 --> 01:10:52.090 So Nick, you brought it up. 01:10:52.090 --> 01:10:53.005 What's that number? 01:10:57.670 --> 01:10:59.400 You don't know that either. 01:10:59.400 --> 01:11:01.040 No. 01:11:01.040 --> 01:11:03.250 Like I said, Amy, nice idea. 01:11:03.250 --> 01:11:06.050 No cigar. 01:11:06.050 --> 01:11:06.710 What else? 01:11:06.710 --> 01:11:08.980 We're running out of time? 01:11:08.980 --> 01:11:10.790 Here we go, Douglas. 01:11:10.790 --> 01:11:13.039 AUDIENCE: Could you displace each mast a certain 01:11:13.039 --> 01:11:14.622 [INAUDIBLE], and then measure the time 01:11:14.622 --> 01:11:17.790 it takes for them to stop and get the damping ratio? 01:11:17.790 --> 01:11:19.200 PROFESSOR: Hit the damping ratio. 01:11:19.200 --> 01:11:21.990 Well, I'll tell you what-- you want to say that again? 01:11:21.990 --> 01:11:26.340 He said displace one or both count oscillations 01:11:26.340 --> 01:11:27.650 and get the damping ratio. 01:11:27.650 --> 01:11:28.340 Nick? 01:11:28.340 --> 01:11:30.840 AUDIENCE: Do we have anything like a force gauge or a spring 01:11:30.840 --> 01:11:31.339 scale? 01:11:31.339 --> 01:11:34.060 PROFESSOR: No. 01:11:34.060 --> 01:11:36.790 This is my living room I'm talking-- or my study. 01:11:36.790 --> 01:11:40.896 Anyway, Douglas said-- I forgot what you said now. 01:11:40.896 --> 01:11:41.770 He said-- oh, I know. 01:11:41.770 --> 01:11:44.490 You said, displace it and count the oscillations. 01:11:44.490 --> 01:11:46.550 Get the damping ratio. 01:11:46.550 --> 01:11:51.250 First off, the damping ratio is no help, even if we did get it. 01:11:51.250 --> 01:11:55.420 And the only formula for which we've ever given you-- 01:11:55.420 --> 01:11:57.060 the only formula we've ever given 01:11:57.060 --> 01:12:00.480 you to do that with pertains to this kind of friction, which 01:12:00.480 --> 01:12:02.950 is not present. 01:12:02.950 --> 01:12:03.720 No cigar. 01:12:03.720 --> 01:12:05.970 Nick says, how about force gauge? 01:12:05.970 --> 01:12:09.000 Now, don't have it. 01:12:09.000 --> 01:12:11.350 AUDIENCE: [INAUDIBLE]. 01:12:11.350 --> 01:12:13.988 PROFESSOR: Yeah, we are. 01:12:13.988 --> 01:12:16.844 AUDIENCE: [INAUDIBLE] displace the other one 01:12:16.844 --> 01:12:18.010 and then find the frequency. 01:12:18.010 --> 01:12:20.130 PROFESSOR: Oh, what's your name? 01:12:20.130 --> 01:12:21.360 Sean? 01:12:21.360 --> 01:12:22.223 Or John? 01:12:22.223 --> 01:12:23.682 AUDIENCE: Sean. 01:12:23.682 --> 01:12:24.890 PROFESSOR: Say that out loud. 01:12:24.890 --> 01:12:28.148 Say it loud enough that Devin can hear you. 01:12:28.148 --> 01:12:29.964 AUDIENCE: You hold the first mass, 01:12:29.964 --> 01:12:31.780 and then you displace the second one. 01:12:31.780 --> 01:12:32.571 PROFESSOR: Hang on. 01:12:32.571 --> 01:12:36.210 He says, hold the first mass like this set screw right here. 01:12:39.130 --> 01:12:40.070 And? 01:12:40.070 --> 01:12:43.830 AUDIENCE: And then displace the other one then. 01:12:43.830 --> 01:12:45.345 PROFESSOR: Like that? 01:12:45.345 --> 01:12:47.470 AUDIENCE: [INAUDIBLE]. 01:12:47.470 --> 01:12:51.390 PROFESSOR: Actually, to answer your question 01:12:51.390 --> 01:12:55.720 Nick, the only instrument I have is a clock. 01:13:00.502 --> 01:13:01.210 Hang on a second. 01:13:05.300 --> 01:13:05.890 Here it is. 01:13:05.890 --> 01:13:07.880 Of course, this is the big task is finding it. 01:13:11.338 --> 01:13:12.254 AUDIENCE: [INAUDIBLE]. 01:13:24.877 --> 01:13:27.210 PROFESSOR: Count to 10, remember like Vandiver told you. 01:13:27.210 --> 01:13:29.770 Skip 1, 1, 2, 3, count to 10. 01:13:29.770 --> 01:13:31.250 Stop. 01:13:31.250 --> 01:13:34.190 Excellent, excellent, excellent. 01:13:34.190 --> 01:13:42.730 When you do that, that's the second one. 01:13:42.730 --> 01:13:47.770 I did that and right here, right here it's TP. 01:13:47.770 --> 01:13:53.275 The period of 10 of them-- and then I divide to get 1-- 01:13:53.275 --> 01:14:00.680 is 0.83 seconds. 01:14:00.680 --> 01:14:03.394 What does that tell you, Sean? 01:14:03.394 --> 01:14:04.310 AUDIENCE: [INAUDIBLE]. 01:14:08.040 --> 01:14:13.020 PROFESSOR: Well actually, these two-- 01:14:13.020 --> 01:14:14.590 you can get the frequency. 01:14:14.590 --> 01:14:24.220 But what this does, because this gives you the frequency, 01:14:24.220 --> 01:14:28.653 you know in general that-- in particular, you 01:14:28.653 --> 01:14:31.610 know that this second natural frequency, which is just 01:14:31.610 --> 01:14:35.360 associated with this single spring and a mass here. 01:14:35.360 --> 01:14:36.930 It's just this guy. 01:14:36.930 --> 01:14:38.280 It's not both of them. 01:14:40.790 --> 01:14:45.190 Anyway, this turns out to-- I didn't graph that. 01:14:45.190 --> 01:14:50.860 That's square root of K over M. Trust me, 01:14:50.860 --> 01:14:53.610 you can put those two together, and you get 01:14:53.610 --> 01:15:04.490 K2 is equal to-- newton meters. 01:15:04.490 --> 01:15:07.590 This is exactly what you said. 01:15:07.590 --> 01:15:08.870 Freeze the first mass. 01:15:08.870 --> 01:15:11.410 Displace the second. 01:15:11.410 --> 01:15:12.830 Measure the period. 01:15:12.830 --> 01:15:24.530 You get the natural frequency for basically K2 over M2. 01:15:24.530 --> 01:15:26.495 So you got K2 out of it. 01:15:26.495 --> 01:15:27.577 Yeah, Douglas? 01:15:27.577 --> 01:15:30.118 AUDIENCE: So how come it gives you just the natural frequency 01:15:30.118 --> 01:15:32.510 and a damp natural frequency? 01:15:32.510 --> 01:15:34.910 PROFESSOR: Oh no. 01:15:34.910 --> 01:15:36.020 It absolutely is. 01:15:36.020 --> 01:15:39.900 It's all damped, no question about it. 01:15:39.900 --> 01:15:41.670 But again, what we're doing is we're 01:15:41.670 --> 01:15:44.660 going close enough, right? 01:15:44.660 --> 01:15:46.910 Because I have nothing. 01:15:46.910 --> 01:15:49.220 So even the damped natural frequency 01:15:49.220 --> 01:15:51.640 is better than nothing. 01:15:51.640 --> 01:15:53.385 All right, so Amy back to you. 01:15:53.385 --> 01:15:55.230 You're back in business. 01:15:55.230 --> 01:15:56.220 What now? 01:15:56.220 --> 01:15:59.690 So now we've got M2 and K2. 01:16:04.715 --> 01:16:05.840 AUDIENCE: You just need K1. 01:16:05.840 --> 01:16:06.190 PROFESSOR: Yeah. 01:16:06.190 --> 01:16:06.970 Now we need K1. 01:16:06.970 --> 01:16:08.979 What do we do now? 01:16:08.979 --> 01:16:11.104 AUDIENCE: We want to do the same thing that we just 01:16:11.104 --> 01:16:12.992 did for K2. [INAUDIBLE]. 01:16:15.830 --> 01:16:17.300 PROFESSOR: Exactly. 01:16:17.300 --> 01:16:21.040 Well, not quite. 01:16:21.040 --> 01:16:23.170 Let's see, now I'm going to turn loose-- now we're 01:16:23.170 --> 01:16:26.439 back to our original system. 01:16:26.439 --> 01:16:27.480 It doesn't hurt anything. 01:16:27.480 --> 01:16:30.750 It's just ugly to look at. 01:16:30.750 --> 01:16:32.800 Now what? 01:16:32.800 --> 01:16:35.715 Sean, do that trick again. 01:16:38.445 --> 01:16:40.420 AUDIENCE: [INAUDIBLE]. 01:16:40.420 --> 01:16:42.065 PROFESSOR: Push this one up? 01:16:42.065 --> 01:16:42.940 AUDIENCE: [INAUDIBLE] 01:16:47.644 --> 01:16:49.240 PROFESSOR: I don't think so. 01:16:49.240 --> 01:16:50.407 Yeah, Nick. 01:16:50.407 --> 01:16:51.990 AUDIENCE: So just fix the second mass. 01:16:51.990 --> 01:16:53.355 PROFESSOR: Fix the second mass. 01:16:53.355 --> 01:16:54.230 AUDIENCE: [INAUDIBLE] 01:17:03.060 --> 01:17:05.010 PROFESSOR: Yeah, you see this frequency here? 01:17:05.010 --> 01:17:07.230 I'm going to overrule Chandler. 01:17:07.230 --> 01:17:12.470 I'm going to say, this is the natural frequency of that's 01:17:12.470 --> 01:17:16.570 mass and these two springs. 01:17:16.570 --> 01:17:17.880 Do that same trick again. 01:17:17.880 --> 01:17:20.290 You get the equivalent spring rate, 01:17:20.290 --> 01:17:23.130 subtract the second from it, and you get the other one. 01:17:23.130 --> 01:17:24.880 Devin, is that what you were going to say? 01:17:24.880 --> 01:17:27.020 Wonderful. 01:17:27.020 --> 01:17:29.420 That is exactly what I did. 01:17:29.420 --> 01:17:32.310 And you get out of it, you get K1 01:17:32.310 --> 01:17:40.480 is equal to 50.45 newton per meter. 01:17:40.480 --> 01:17:44.350 In the interest of time, I'm going to short circuit this. 01:17:44.350 --> 01:17:48.740 I took exactly these parameters, I 01:17:48.740 --> 01:17:54.870 put them into that same computer program we had before, 01:17:54.870 --> 01:17:58.736 and what came out-- I have to have a place to put it. 01:18:02.390 --> 01:18:03.000 Ah, wonderful. 01:18:08.507 --> 01:18:09.590 And now I have to find it. 01:18:17.860 --> 01:18:19.890 Here we go. 01:18:19.890 --> 01:18:23.147 Here are the two modes. 01:18:23.147 --> 01:18:24.605 Actually, let me put it right here. 01:18:28.260 --> 01:18:34.115 For this system, because here's the first one. 01:18:42.040 --> 01:18:43.358 And here's the second one. 01:18:47.190 --> 01:18:54.980 0.9760 and minus 0.2177. 01:18:54.980 --> 01:18:56.410 Everybody appreciate that? 01:18:56.410 --> 01:18:59.610 This is by the same computational procedure 01:18:59.610 --> 01:19:02.240 we spoke of earlier. 01:19:02.240 --> 01:19:09.110 And sparing no expense, we have here a, made fresh 01:19:09.110 --> 01:19:15.680 from my basement, a custom made initial condition setting 01:19:15.680 --> 01:19:22.430 device, which I can hopefully avoid killing myself with. 01:19:22.430 --> 01:19:24.760 OK here's what we have. 01:19:24.760 --> 01:19:29.990 So I guess I didn't show you this first. 01:19:29.990 --> 01:19:31.820 If you can see it, what we have marked 01:19:31.820 --> 01:19:33.630 is the reference position. 01:19:33.630 --> 01:19:36.010 That's the rest position that we couldn't 01:19:36.010 --> 01:19:38.320 find by laying it down. 01:19:38.320 --> 01:19:45.350 Or excuse me, this is the static equilibrium position 01:19:45.350 --> 01:19:47.850 of mass number one, static equilibrium 01:19:47.850 --> 01:19:49.550 position of mass number two. 01:19:49.550 --> 01:19:52.330 Mode one or there. 01:19:52.330 --> 01:19:54.390 What they said over there, 0.4. 01:19:54.390 --> 01:19:57.160 And .97 is down here. 01:19:57.160 --> 01:19:59.990 Mode two is over here. 01:19:59.990 --> 01:20:03.770 So Devin, while I'm doing this, tell me 01:20:03.770 --> 01:20:05.830 how am I going to know if this is right 01:20:05.830 --> 01:20:09.080 or if this is all just bogus? 01:20:09.080 --> 01:20:14.594 What observable's going to tell me that I got it right? 01:20:14.594 --> 01:20:15.469 AUDIENCE: [INAUDIBLE] 01:20:20.720 --> 01:20:22.495 PROFESSOR: I'm sorry, speak up. 01:20:22.495 --> 01:20:23.740 AUDIENCE: [INAUDIBLE] 01:20:23.740 --> 01:20:25.342 PROFESSOR: How about it, Nick? 01:20:25.342 --> 01:20:26.550 AUDIENCE: [INAUDIBLE] 01:20:26.550 --> 01:20:28.100 PROFESSOR: Exactly. 01:20:28.100 --> 01:20:32.070 And you ought to be able to see it from where you are. 01:20:32.070 --> 01:20:36.310 Can you appreciate that they're not at the moment? 01:20:36.310 --> 01:20:41.660 All right, now hang on. 01:20:41.660 --> 01:20:44.320 Here comes mode number one. 01:20:44.320 --> 01:20:47.890 This takes two hands to do it. 01:20:47.890 --> 01:20:50.248 All right, you ready? 01:20:53.170 --> 01:20:54.184 This is mode number one. 01:20:54.184 --> 01:20:55.350 Again, it's a those numbers. 01:21:07.739 --> 01:21:08.280 How about it? 01:21:08.280 --> 01:21:10.320 Can you see it? 01:21:10.320 --> 01:21:12.730 Very good. 01:21:12.730 --> 01:21:14.500 How about the other? 01:21:14.500 --> 01:21:21.987 And so here, this is number two. 01:21:21.987 --> 01:21:23.570 And this is a little more complicated, 01:21:23.570 --> 01:21:28.656 because the other one has to be done from the bottom. 01:21:28.656 --> 01:21:29.470 Hang on a second. 01:21:32.470 --> 01:21:36.860 Now this one we're doing is we're deflecting-- 01:21:36.860 --> 01:21:39.370 this one is positive downward. 01:21:39.370 --> 01:21:44.210 So X1 is down, but X2 is negative. 01:21:44.210 --> 01:21:46.710 So it's displaced upward a bit. 01:21:46.710 --> 01:21:48.790 Are you ready? 01:21:48.790 --> 01:21:52.594 Nick, what do you expect to see this time? 01:21:52.594 --> 01:21:55.184 AUDIENCE: The frequency should be higher 01:21:55.184 --> 01:21:56.850 and they'll move in opposite directions. 01:21:56.850 --> 01:22:00.090 PROFESSOR: That's the key. 01:22:00.090 --> 01:22:02.650 Once again, they're going to move with the same frequency, 01:22:02.650 --> 01:22:04.910 albeit in different directions. 01:22:04.910 --> 01:22:09.010 But that new frequency is going to be higher than before. 01:22:09.010 --> 01:22:11.308 And sure enough, stand back. 01:22:20.670 --> 01:22:21.500 So there you go. 01:22:21.500 --> 01:22:23.440 What did that tell us? 01:22:23.440 --> 01:22:26.520 That told us that the first order 01:22:26.520 --> 01:22:30.410 we got the system parameters identified correctly 01:22:30.410 --> 01:22:33.220 and the theory holds up. 01:22:33.220 --> 01:22:33.890 Questions? 01:22:33.890 --> 01:22:35.880 Comments? 01:22:35.880 --> 01:22:37.740 Complaints? 01:22:37.740 --> 01:22:38.696 Devin? 01:22:38.696 --> 01:22:39.690 AUDIENCE: [INAUDIBLE] 01:22:39.690 --> 01:22:40.620 PROFESSOR: I'm sorry? 01:22:40.620 --> 01:22:42.880 AUDIENCE: What was the second set of conditions? 01:22:42.880 --> 01:22:45.750 PROFESSOR: The second set of initial conditions 01:22:45.750 --> 01:22:48.180 were right here. 01:22:48.180 --> 01:22:57.120 This is the second mode, X1, 0.97, X2, minus 0.2. 01:22:57.120 --> 01:23:01.330 OK, have a great Thanksgiving holiday.