WEBVTT 00:00:00.110 --> 00:00:02.450 The following content is provided under a Creative 00:00:02.450 --> 00:00:03.830 Commons license. 00:00:03.830 --> 00:00:06.080 Your support will help MIT OpenCourseWare 00:00:06.080 --> 00:00:10.170 continue to offer high quality educational resources for free. 00:00:10.170 --> 00:00:12.710 To make a donation or to view additional materials 00:00:12.710 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.325 at ocw.mit.edu. 00:00:21.490 --> 00:00:23.390 PROFESSOR: Review model analysis. 00:00:23.390 --> 00:00:26.469 I figured most-- I figured lots of students have had questions 00:00:26.469 --> 00:00:28.260 and we could use a little practice on this. 00:00:28.260 --> 00:00:29.778 So that's what we're going to do. 00:00:29.778 --> 00:00:31.122 AUDIENCE: Solving the q thing? 00:00:31.122 --> 00:00:31.570 PROFESSOR: Huh? 00:00:31.570 --> 00:00:32.945 AUDIENCE: Solving the q function? 00:00:32.945 --> 00:00:34.890 PROFESSOR: Yeah, solving the whole thing. 00:00:34.890 --> 00:00:36.560 So but let's start. 00:00:36.560 --> 00:00:38.810 I'd still start with the usual concepts. 00:00:38.810 --> 00:00:40.650 I haven't even erased what we had last time. 00:00:40.650 --> 00:00:43.520 But this is what the students in the last group 00:00:43.520 --> 00:00:45.430 said about key concepts for the week. 00:00:45.430 --> 00:00:48.405 If you want to add something to it, speak up. 00:00:48.405 --> 00:00:53.840 But modal analysis, multi-degree of freedom, transfer functions, 00:00:53.840 --> 00:00:56.510 and then model analysis-- more specifically, 00:00:56.510 --> 00:01:00.300 response to initial conditions, response to steady state 00:01:00.300 --> 00:01:01.940 harmonic inputs. 00:01:01.940 --> 00:01:07.160 Anything else that was significant, conceptual, 00:01:07.160 --> 00:01:08.600 and new in the last week? 00:01:12.790 --> 00:01:13.730 Add to that. 00:01:13.730 --> 00:01:16.970 And then the second thing, what are 00:01:16.970 --> 00:01:19.950 issues that are muddy for you? 00:01:19.950 --> 00:01:23.370 Not quite clear, things you want to learn more about, 00:01:23.370 --> 00:01:24.550 have questions about. 00:01:24.550 --> 00:01:26.220 Last class. 00:01:26.220 --> 00:01:28.970 was more on transfer functions for multi-degree of freedom 00:01:28.970 --> 00:01:32.610 systems and solving initial condition problems 00:01:32.610 --> 00:01:35.360 is what one student was interested in about 00:01:35.360 --> 00:01:37.345 from using modal analysis. 00:01:37.345 --> 00:01:39.880 But do any of you have questions about things 00:01:39.880 --> 00:01:42.080 that you want to practiced on? 00:01:42.080 --> 00:01:46.880 AUDIENCE: So I understand how to get the new mass 00:01:46.880 --> 00:01:51.650 and spring whatever matricies for the q system, 00:01:51.650 --> 00:01:57.680 so I can set it up as the mq bq kq, but from there, 00:01:57.680 --> 00:02:00.362 I don't know how to go back and solve for x, 00:02:00.362 --> 00:02:01.820 or I don't know how to solve for x. 00:02:01.820 --> 00:02:02.340 PROFESSOR: Right. 00:02:02.340 --> 00:02:03.520 So we'll go through that. 00:02:03.520 --> 00:02:06.095 We're going to run through a complete modal analysis 00:02:06.095 --> 00:02:09.125 today, all the steps that you need to do to make it happen. 00:02:09.125 --> 00:02:11.760 So anything else that you've got a question about that I 00:02:11.760 --> 00:02:12.810 might be able to get to? 00:02:15.720 --> 00:02:17.350 OK, let's get rolling. 00:02:17.350 --> 00:02:19.760 This will take a little while. 00:02:19.760 --> 00:02:26.940 So problem for the day is, if you recall last time, 00:02:26.940 --> 00:02:30.470 we had this demo of this double pendulum. 00:02:30.470 --> 00:02:32.470 But now we're going to take that double pendulum 00:02:32.470 --> 00:02:35.610 and make the masses unequal. 00:02:35.610 --> 00:02:37.170 So well, the masses are equal here, 00:02:37.170 --> 00:02:38.962 but they could be unequal-- M1, M2, 00:02:38.962 --> 00:02:40.170 they're each half a kilogram. 00:02:40.170 --> 00:02:44.360 But we're changing the lengths a little bit, 1.1 for L1 00:02:44.360 --> 00:02:49.550 and 1.0 meters for L2. 00:02:49.550 --> 00:02:51.600 So slightly different lengths, and that'll 00:02:51.600 --> 00:02:54.510 make this system not symmetric so it won't have 1, 1 00:02:54.510 --> 00:02:57.470 and 1 minus 1 mode shapes. 00:02:57.470 --> 00:03:00.770 A little weak spring in the middle, possibility 00:03:00.770 --> 00:03:07.740 of having some dashpot here connected to a non-moving wall, 00:03:07.740 --> 00:03:09.640 another dashpot here. 00:03:09.640 --> 00:03:12.530 And the possibility of having harmonic 00:03:12.530 --> 00:03:16.320 excitations-- F1 on this one, F2 on that one. 00:03:16.320 --> 00:03:18.510 The whole system's been linearized. 00:03:18.510 --> 00:03:20.270 The equations in motion look something 00:03:20.270 --> 00:03:25.410 like this, mass damping matrix, stiffness matrix. 00:03:25.410 --> 00:03:27.620 Of course, it has gravity terms in it 00:03:27.620 --> 00:03:31.120 as well as the spring terms. 00:03:31.120 --> 00:03:33.250 And it's been linearized. 00:03:37.140 --> 00:03:41.500 This equation in motion, is it a force equation or a moment 00:03:41.500 --> 00:03:42.670 equation? 00:03:42.670 --> 00:03:44.960 Are they force or moment or both? 00:03:44.960 --> 00:03:46.790 We can have mixed ones like the cart. 00:03:46.790 --> 00:03:49.010 Problem with the pendulum has one force equation 00:03:49.010 --> 00:03:51.740 and one equation with units of torque, 00:03:51.740 --> 00:03:53.363 this one has units of what? 00:03:53.363 --> 00:03:54.204 AUDIENCE: Torque. 00:03:54.204 --> 00:03:54.870 PROFESSOR: Yeah. 00:03:54.870 --> 00:03:58.339 This is moments about point A and moments about point B 00:03:58.339 --> 00:03:59.630 and give you the two equations. 00:03:59.630 --> 00:04:01.590 So this is a torque equation. 00:04:01.590 --> 00:04:06.290 So you need to look and see if things inside here make sense. 00:04:06.290 --> 00:04:10.250 So the MGL, the non-linear equation, the restoring torque, 00:04:10.250 --> 00:04:12.680 is MGL sine theta. 00:04:12.680 --> 00:04:14.340 Linearized, it's just theta. 00:04:14.340 --> 00:04:18.970 So MGL 1 theta is the torque on the first mass. 00:04:18.970 --> 00:04:25.720 M2 GL 2 theta is the torque on the second mass, and so forth. 00:04:25.720 --> 00:04:30.920 And K1 L1 squared, why the L1 squared? 00:04:30.920 --> 00:04:34.930 This term here gets multiplied by theta 1, so what is that? 00:04:34.930 --> 00:04:39.360 What's K1 L1 squared theta 1? 00:04:39.360 --> 00:04:43.139 What kind of a torque-- what is that? 00:04:43.139 --> 00:04:43.930 Does it make sense? 00:04:43.930 --> 00:04:46.300 Are its units correct? 00:04:46.300 --> 00:04:50.335 So first of all, what's K1 L1 times theta? 00:04:52.840 --> 00:04:53.710 AUDIENCE: Force. 00:04:53.710 --> 00:04:54.960 PROFESSOR: Yeah, it's a force. 00:04:54.960 --> 00:04:58.012 What's L1 times theta? 00:04:58.012 --> 00:05:00.990 AUDIENCE: [INAUDIBLE] 00:05:00.990 --> 00:05:02.860 PROFESSOR: Physically, that is what? 00:05:02.860 --> 00:05:03.840 AUDIENCE: Distance. 00:05:03.840 --> 00:05:07.130 PROFESSOR: That's the distance that that thing moves, right? 00:05:07.130 --> 00:05:10.840 And a displacement times a spring constant 00:05:10.840 --> 00:05:17.070 gives you a force, and then that force times a moment arm 00:05:17.070 --> 00:05:19.700 gives you a torque. 00:05:19.700 --> 00:05:21.630 So it makes sense. 00:05:21.630 --> 00:05:28.560 OK, so L1 squared theta 1 is the torque about point A 00:05:28.560 --> 00:05:35.890 caused by a displacement theta 1, 00:05:35.890 --> 00:05:39.490 assuming this one is 0 when you do it. 00:05:39.490 --> 00:05:45.490 That's the torque caused by a displacement of theta 1 only. 00:05:45.490 --> 00:05:47.180 OK, all right. 00:05:49.830 --> 00:05:52.120 So that's our equations of motion linearized. 00:05:52.120 --> 00:05:54.780 This is the mass matrix. 00:05:54.780 --> 00:05:57.040 This is the stiffness matrix that you 00:05:57.040 --> 00:06:02.510 would get if you go into there and substitute in these values. 00:06:02.510 --> 00:06:03.205 Yeah? 00:06:03.205 --> 00:06:04.579 AUDIENCE: The matrix on the right 00:06:04.579 --> 00:06:07.830 would be to one of these [INAUDIBLE] on the bottom? 00:06:10.289 --> 00:06:11.330 PROFESSOR: This one here? 00:06:11.330 --> 00:06:12.710 This is a two by two matrix. 00:06:12.710 --> 00:06:13.750 AUDIENCE: Oh, OK. 00:06:13.750 --> 00:06:14.470 PROFESSOR: And it has to be. 00:06:14.470 --> 00:06:15.845 It's a two degree freedom system. 00:06:15.845 --> 00:06:20.020 This is K11, the K11 term. 00:06:20.020 --> 00:06:26.370 This is the K12 term, the K21 term, and the K2 term. 00:06:26.370 --> 00:06:28.290 AUDIENCE: So there's a space between-- 00:06:28.290 --> 00:06:30.248 PROFESSOR: Yeah, it got a little squeezed here. 00:06:30.248 --> 00:06:34.360 This space, there's an M2GL2-- and that shouldn't 00:06:34.360 --> 00:06:38.613 be squared-- plus KL2 squared. 00:06:38.613 --> 00:06:39.738 AUDIENCE: Plus [INAUDIBLE]? 00:06:42.714 --> 00:06:44.937 PROFESSOR: All right, pretty sure 00:06:44.937 --> 00:06:46.270 that's what it out to read like. 00:06:46.270 --> 00:06:48.280 MGL2 KL2 squared. 00:06:48.280 --> 00:06:54.290 And then each of these are minus KL1L2 minus KL1L2. 00:06:54.290 --> 00:06:59.080 All of this, this is all in a handout which will be put up 00:06:59.080 --> 00:07:01.110 on the Stellar website. 00:07:01.110 --> 00:07:04.890 And just so in terms of reviewing for final and things 00:07:04.890 --> 00:07:08.584 to go look at, for almost every recitation we've done this 00:07:08.584 --> 00:07:11.000 during Professor Gossard, the other recitation instructor, 00:07:11.000 --> 00:07:13.240 has written up the complete problem 00:07:13.240 --> 00:07:15.557 that was discussed with its solution 00:07:15.557 --> 00:07:16.640 and put it on the website. 00:07:16.640 --> 00:07:18.931 So you don't even need to have to copy this stuff down. 00:07:18.931 --> 00:07:21.969 It's all posted. 00:07:21.969 --> 00:07:23.510 In each recitation, we've essentially 00:07:23.510 --> 00:07:29.110 done the problem that is sort of the objective lesson 00:07:29.110 --> 00:07:30.440 for the week. 00:07:30.440 --> 00:07:32.390 So they're a good place to go review. 00:07:32.390 --> 00:07:35.270 So this whole problem will be up on there. 00:07:35.270 --> 00:07:38.210 And we found a mistake, actually. 00:07:38.210 --> 00:07:40.410 And I think it will get fixed before it gets put up, 00:07:40.410 --> 00:07:41.910 but I think this had a zero in here, 00:07:41.910 --> 00:07:43.860 and that's actually wrong, or if something 00:07:43.860 --> 00:07:45.630 got transposed in writing. 00:07:45.630 --> 00:07:48.600 That's the correct mass. 00:07:48.600 --> 00:07:52.894 OK, so you have these numbers. 00:07:52.894 --> 00:07:54.560 I don't want you to work this stuff out, 00:07:54.560 --> 00:07:58.090 because I want to focus on the model analysis. 00:07:58.090 --> 00:07:59.480 So let's see here. 00:07:59.480 --> 00:08:01.660 I'm going to ask you a question first. 00:08:01.660 --> 00:08:02.920 So we're going to begin this. 00:08:02.920 --> 00:08:03.559 Yeah? 00:08:03.559 --> 00:08:06.100 AUDIENCE: Is there a torsional damper at the top of the page? 00:08:06.100 --> 00:08:09.095 PROFESSOR: Oh, somebody asked a question in the last class 00:08:09.095 --> 00:08:11.000 and I drew that up there. 00:08:11.000 --> 00:08:14.400 But if you put a torsional damper up there, 00:08:14.400 --> 00:08:19.640 some CT value, how would it appear 00:08:19.640 --> 00:08:22.940 in the equation of motion? 00:08:22.940 --> 00:08:26.630 Would it have L1, L2 squareds in it? 00:08:26.630 --> 00:08:28.100 No, it would just simply disappear 00:08:28.100 --> 00:08:31.380 as a plus CT theta 1 or a theta 2, 00:08:31.380 --> 00:08:34.941 wherever it's applied directly, theta 1 dot. 00:08:39.000 --> 00:08:42.309 OK, so question. 00:08:42.309 --> 00:08:44.344 Write down your piece of paper-- so I'm not 00:08:44.344 --> 00:08:46.010 going to make you go to the board today, 00:08:46.010 --> 00:08:49.910 but I want you to take a minute and write things down. 00:08:49.910 --> 00:08:53.750 And then we'll check and see if everybody agrees. 00:08:53.750 --> 00:08:57.870 The reason we can do modal analysis 00:08:57.870 --> 00:09:01.830 is because of something we called the modal expansion 00:09:01.830 --> 00:09:02.480 theorem. 00:09:02.480 --> 00:09:04.620 It's basically the fundamental statement 00:09:04.620 --> 00:09:05.830 that says we can do this. 00:09:05.830 --> 00:09:07.930 So what is the modal expansion theorem? 00:09:07.930 --> 00:09:09.740 You can write it down mathematically 00:09:09.740 --> 00:09:12.770 if you want this, just as a little linear algebraic 00:09:12.770 --> 00:09:15.240 expression, or you could write it out in words. 00:09:15.240 --> 00:09:18.780 So take 30 seconds and write down 00:09:18.780 --> 00:09:23.290 what makes modal analysis work, what basic proposition. 00:09:50.480 --> 00:09:52.550 All right, somebody help me out. 00:09:52.550 --> 00:09:55.890 What is the modal expansion theorem? 00:09:55.890 --> 00:09:57.608 What's it say? 00:09:57.608 --> 00:09:59.604 AUDIENCE: Any motion of the system 00:09:59.604 --> 00:10:03.300 can be described as a weighted sum of the natural modes? 00:10:03.300 --> 00:10:06.110 PROFESSOR: Weighted sum of the motions of each 00:10:06.110 --> 00:10:08.320 of the natural modes, OK. 00:10:08.320 --> 00:10:14.580 So that's the statement, the most succinct way 00:10:14.580 --> 00:10:20.150 to say it is in the original generalized coordinates, 00:10:20.150 --> 00:10:21.990 you can express them as uq. 00:10:26.150 --> 00:10:31.150 u is the matrix of what? 00:10:31.150 --> 00:10:33.370 AUDIENCE: [INAUDIBLE] 00:10:33.370 --> 00:10:34.370 PROFESSOR: No, this is-- 00:10:34.370 --> 00:10:37.280 AUDIENCE: Oh no, of the mode shapes. 00:10:37.280 --> 00:10:38.380 PROFESSOR: Mode shapes. 00:10:38.380 --> 00:10:43.320 And the q's are the individual modal coordinates, right? 00:10:43.320 --> 00:10:46.970 And so this, expanded, says that this two degree freedom 00:10:46.970 --> 00:10:49.510 system has two generalized coordinates-- 00:10:49.510 --> 00:10:51.130 theta 1 and theta 2. 00:10:51.130 --> 00:10:55.040 And the response of either of the actual motion 00:10:55.040 --> 00:10:57.870 of the system expressed in the generalized coordinates 00:10:57.870 --> 00:11:05.220 can be made up as the sum of each of the modal coordinates, 00:11:05.220 --> 00:11:11.450 the q sub i's, scaled to this shape 00:11:11.450 --> 00:11:14.660 the mode shape for that mode. 00:11:14.660 --> 00:11:16.770 So this multiplies the mode shape, 00:11:16.770 --> 00:11:19.640 and that-- so everything contributed by mode 00:11:19.640 --> 00:11:23.240 one will move in the shape of mode one, 00:11:23.240 --> 00:11:26.100 and that will be reflected in the motion 00:11:26.100 --> 00:11:27.810 of the generalized coordinates. 00:11:27.810 --> 00:11:30.110 And in this case, it's a two by two. 00:11:30.110 --> 00:11:32.190 It's a two-degree freedom system. 00:11:32.190 --> 00:11:35.670 Here's the mode shape of mode 1 times q1, 00:11:35.670 --> 00:11:41.070 which we're going to solve for, plus the mode shape of mode two 00:11:41.070 --> 00:11:44.755 times its model motion. 00:11:44.755 --> 00:11:46.790 OK, that's the model expansion theorem. 00:11:50.990 --> 00:11:57.330 This allows you to-- in order to do this, 00:11:57.330 --> 00:12:00.120 you have to solve for these qi's. 00:12:00.120 --> 00:12:03.280 So what is the equation of motion 00:12:03.280 --> 00:12:11.820 that governs the behavior of each of the modal coordinates? 00:12:11.820 --> 00:12:13.700 Write it down. 00:12:13.700 --> 00:12:15.720 What's the whole reason we do this? 00:12:15.720 --> 00:12:19.345 There's one particular equation that every one of them 00:12:19.345 --> 00:12:19.845 satisfies. 00:12:38.440 --> 00:12:41.130 So equation of motion, not asking for solution. 00:12:41.130 --> 00:12:43.270 I just want equation of motion that 00:12:43.270 --> 00:12:47.975 governs these modal motions. 00:12:57.767 --> 00:12:59.980 I think I'll give you a minute to think about it. 00:13:11.490 --> 00:13:13.000 OK, somebody help me out. 00:13:13.000 --> 00:13:15.590 What's the equation of motion that I 00:13:15.590 --> 00:13:18.120 can write that will describe the motion of any one 00:13:18.120 --> 00:13:19.730 of these modal coordinates, qi? 00:13:23.390 --> 00:13:24.000 Christina? 00:13:24.000 --> 00:13:26.370 AUDIENCE: With the fancy M's and C's and K? 00:13:26.370 --> 00:13:27.120 PROFESSOR: Pardon? 00:13:27.120 --> 00:13:29.290 AUDIENCE: With the fancy M's and C's and K? 00:13:29.290 --> 00:13:30.980 PROFESSOR: Well yeah, but it's basically 00:13:30.980 --> 00:13:32.880 a very simple equation of motion, which you 00:13:32.880 --> 00:13:34.180 should be familiar with by now. 00:13:34.180 --> 00:13:35.300 What's it look like? 00:13:35.300 --> 00:13:38.400 For any one of these modal coordinates, 00:13:38.400 --> 00:13:40.780 what is the equation of motion that governs it? 00:13:45.820 --> 00:13:48.200 Why do we go to all this trouble? 00:13:48.200 --> 00:13:49.580 There's a reason for doing this. 00:13:49.580 --> 00:13:50.220 It's because-- 00:13:50.220 --> 00:13:51.160 AUDIENCE: Single degree of freedom. 00:13:51.160 --> 00:13:53.640 PROFESSOR: Ahh, the single degree of freedom oscillator 00:13:53.640 --> 00:13:54.590 equation, right? 00:14:00.040 --> 00:14:02.200 This is the reason we do this. 00:14:02.200 --> 00:14:07.590 Single degree of freedom systems are mathematically simple. 00:14:07.590 --> 00:14:10.150 You've seen them since high school. 00:14:10.150 --> 00:14:12.220 You've seen them in 1803. 00:14:12.220 --> 00:14:14.430 It's the second order linear differential 00:14:14.430 --> 00:14:17.119 equation that looks like this. 00:14:17.119 --> 00:14:18.660 And you already know everything there 00:14:18.660 --> 00:14:21.010 is to be known about that equation. 00:14:21.010 --> 00:14:23.130 And that's why one of the reasons why we do this. 00:14:23.130 --> 00:14:24.820 You don't have to solve a complex set 00:14:24.820 --> 00:14:26.720 of simultaneous differential equations, 00:14:26.720 --> 00:14:28.780 you just have to know one. 00:14:28.780 --> 00:14:31.755 And so the ith one, you need to know the ith modal mass, 00:14:31.755 --> 00:14:34.970 the ith modal damping, the ith modal stiffness, 00:14:34.970 --> 00:14:37.140 and the modal force. 00:14:37.140 --> 00:14:40.030 And we've-- in this course, we've taught you how to solve 00:14:40.030 --> 00:14:45.740 two kinds of single degree of freedom system problems. 00:14:45.740 --> 00:14:48.020 One is response to initial conditions 00:14:48.020 --> 00:14:52.090 when the forces on the right hand side is zero, 00:14:52.090 --> 00:14:54.000 and the other is this steady state 00:14:54.000 --> 00:14:58.420 response to a harmonic input, so a cosine omega t kind of input. 00:14:58.420 --> 00:15:00.870 So that's what we focused on in this course, 00:15:00.870 --> 00:15:03.150 because it's vibration we're interested in. 00:15:03.150 --> 00:15:06.355 So we've solve this equation for two kinds of problems. 00:15:09.880 --> 00:15:12.660 Now to do modal analysis, you need 00:15:12.660 --> 00:15:16.730 to be able to find these quantities. 00:15:16.730 --> 00:15:18.970 These we called the modal masses, 00:15:18.970 --> 00:15:23.360 the modal damping coefficients, the modal stiffnesses. 00:15:23.360 --> 00:15:25.940 How do you get those? 00:15:25.940 --> 00:15:26.980 How, for example? 00:15:26.980 --> 00:15:29.440 Write down on your paper, what equation, 00:15:29.440 --> 00:15:32.390 what linear algebra thing do you have 00:15:32.390 --> 00:15:35.384 to work out to get the modal masses for this system? 00:15:35.384 --> 00:15:36.300 Let's say all of them. 00:15:36.300 --> 00:15:38.642 I want a two-degree system. 00:15:38.642 --> 00:15:39.350 What's the state? 00:15:39.350 --> 00:15:40.820 What's the mass, the linear algebra 00:15:40.820 --> 00:15:42.694 you have to work out to get the modal masses? 00:16:02.222 --> 00:16:02.930 Somebody help me. 00:16:02.930 --> 00:16:03.545 What is it? 00:16:03.545 --> 00:16:05.030 Yeah? 00:16:05.030 --> 00:16:10.475 AUDIENCE: Transpose of the mode vector-- 00:16:10.475 --> 00:16:13.445 sorry, the mode matrix multiplied by the mass vector? 00:16:13.445 --> 00:16:15.920 PROFESSOR: Yeah? 00:16:15.920 --> 00:16:18.535 AUDIENCE: And then multiplied by the mode matrix. 00:16:18.535 --> 00:16:19.410 PROFESSOR: All right. 00:16:26.172 --> 00:16:28.110 All right. 00:16:28.110 --> 00:16:33.680 The modal forces are u transpose f. 00:16:33.680 --> 00:16:35.140 The modal masses-- and I've drawn 00:16:35.140 --> 00:16:37.056 these little diagonal marks in here to remind, 00:16:37.056 --> 00:16:39.619 you these matrices become all diagonal 00:16:39.619 --> 00:16:40.910 when you do the model analysis. 00:16:45.190 --> 00:16:47.930 Coordinate transformation, I'll call it. 00:16:47.930 --> 00:16:50.290 So the modal masses are u transpose mu. 00:16:50.290 --> 00:16:54.270 And this m is a matrix, and it's the original mass 00:16:54.270 --> 00:16:56.200 matrix of the system. 00:16:56.200 --> 00:16:59.270 Modal stiffness, matrix u transpose ku. 00:16:59.270 --> 00:17:02.400 And the modal damping matrix, u transpose cu. 00:17:02.400 --> 00:17:05.359 But this one can be problematic. 00:17:05.359 --> 00:17:08.990 You have to force this one to behave. 00:17:08.990 --> 00:17:13.890 These are guaranteed to behave, all right? 00:17:13.890 --> 00:17:18.569 So for this problem, and I know some of you 00:17:18.569 --> 00:17:22.300 are a little rusty calculating these things. 00:17:22.300 --> 00:17:26.000 So there's the model mass matrix. 00:17:26.000 --> 00:17:32.910 And here is the modal matrix of eigenvectors or mode 00:17:32.910 --> 00:17:34.500 shapes of the system. 00:17:34.500 --> 00:17:39.790 u is made up of columns, and each column 00:17:39.790 --> 00:17:41.990 is one of the mode shapes. 00:17:41.990 --> 00:17:48.270 The convention is to order them from the first mode to the nth 00:17:48.270 --> 00:17:49.740 mode where the order is established 00:17:49.740 --> 00:17:50.865 by the natural frequencies. 00:17:50.865 --> 00:17:53.750 The lowest natural frequency is first-- second, second, 00:17:53.750 --> 00:17:55.520 up to the highest natural frequency. 00:17:55.520 --> 00:17:58.110 Anyway, here's mode one, here's mode two for this system. 00:18:03.900 --> 00:18:05.880 You have to choose a way in which 00:18:05.880 --> 00:18:10.850 to normalize the mode shapes. 00:18:10.850 --> 00:18:13.300 I choose to normalize them usually. 00:18:13.300 --> 00:18:15.950 I say, I'm going to make the top element of them one. 00:18:15.950 --> 00:18:19.780 And I do that whatever-- if do MATLAB like you do, 00:18:19.780 --> 00:18:22.190 there's a function called Eig, which means eigenvalue. 00:18:22.190 --> 00:18:28.050 You do Eig of A, it'll give you the eigenvalues of matrix A. 00:18:28.050 --> 00:18:31.915 And it'll give them back to you unnormalized and unordered. 00:18:34.480 --> 00:18:36.424 Well, so you can write a little program 00:18:36.424 --> 00:18:37.590 to put it all in nice order. 00:18:37.590 --> 00:18:44.110 But if MATLAB gave you back the mode shapes for a system 00:18:44.110 --> 00:18:46.600 and it said, well, the mode shapes of the system 00:18:46.600 --> 00:18:49.260 are-- and it's a two by two system. 00:18:49.260 --> 00:18:58.260 2 and 0.4 and 0.6 0.5, you know that the vectors are the mode 00:18:58.260 --> 00:19:01.230 shapes, the columns. 00:19:01.230 --> 00:19:03.440 How would you normalize those? 00:19:03.440 --> 00:19:07.364 How would you make the top element 1 in this first one? 00:19:07.364 --> 00:19:08.280 AUDIENCE: Divide by 2. 00:19:08.280 --> 00:19:09.446 PROFESSOR: Divide what by 2? 00:19:09.446 --> 00:19:10.750 AUDIENCE: The Entire column. 00:19:10.750 --> 00:19:12.166 PROFESSOR: The entire column by 2. 00:19:12.166 --> 00:19:13.250 Just factor 2 out. 00:19:13.250 --> 00:19:16.950 So this would become 2/2 and 0.4/2. 00:19:19.770 --> 00:19:25.260 And then that's 1 and 0.2. 00:19:25.260 --> 00:19:29.620 So you just normalize this vector so the top element is 1. 00:19:29.620 --> 00:19:31.270 So do you have to normalize it? 00:19:31.270 --> 00:19:32.740 Could you use them this way? 00:19:32.740 --> 00:19:35.350 Sure. 00:19:35.350 --> 00:19:42.030 But once chosen, once the normalization is chosen, 00:19:42.030 --> 00:19:45.680 the key to doing modal analysis is you have to stick with it. 00:19:45.680 --> 00:19:46.522 You can't move. 00:19:46.522 --> 00:19:48.230 You can't mess with that halfway through, 00:19:48.230 --> 00:19:50.625 or you totally screw up the solution. 00:19:50.625 --> 00:19:52.560 So you pick your normalization. 00:19:52.560 --> 00:19:56.140 When you calculate the natural frequencies and mode shapes, 00:19:56.140 --> 00:19:58.060 you pick a normalization, and you 00:19:58.060 --> 00:20:01.190 must ride with that all the way through, 00:20:01.190 --> 00:20:03.130 including putting it back together here 00:20:03.130 --> 00:20:06.590 at the end, this summation. 00:20:06.590 --> 00:20:10.660 OK, so let's do-- I want you to do this computation. 00:20:10.660 --> 00:20:13.450 Calculate the modal masses for this problem. 00:20:16.480 --> 00:20:18.740 That means you have to remember what a transpose is. 00:20:18.740 --> 00:20:21.390 There's the model, that's the model matrix. 00:20:21.390 --> 00:20:23.950 And the modal mass matrix is right there. 00:20:23.950 --> 00:20:25.450 So actually, just do the arithmetic. 00:20:25.450 --> 00:20:26.620 Take a few minutes. 00:20:26.620 --> 00:20:27.480 Yeah? 00:20:27.480 --> 00:20:28.980 AUDIENCE: On the exam, we won't have 00:20:28.980 --> 00:20:32.122 calculators, so like is it all going to be variables, or--? 00:20:32.122 --> 00:20:33.568 PROFESSOR: Say that again? 00:20:33.568 --> 00:20:36.109 AUDIENCE: On the exam, we won't have calculators or anything. 00:20:36.109 --> 00:20:38.590 PROFESSOR: On the exam, we'd either make it so simple 00:20:38.590 --> 00:20:42.430 that you can, in fact, do it in your head or on paper, 00:20:42.430 --> 00:20:45.280 or we won't ask a question that you have to do it that way. 00:20:52.360 --> 00:20:54.845 Or we'll accept an answer where you put it down but don't 00:20:54.845 --> 00:20:55.845 have to multiply it out. 00:20:59.620 --> 00:21:02.050 So just do this one, just to see if you 00:21:02.050 --> 00:21:06.300 remember the mechanics of doing the linear algebra to get that. 00:21:06.300 --> 00:21:09.480 OK, somebody have an answer for me 00:21:09.480 --> 00:21:12.600 here for the modal mass matrix? 00:21:12.600 --> 00:21:13.920 What's the first element? 00:21:13.920 --> 00:21:14.628 Somebody help me. 00:21:17.700 --> 00:21:20.205 Give me a number and then everybody else can check you. 00:21:25.916 --> 00:21:27.585 AUDIENCE: 0.254. 00:21:27.585 --> 00:21:28.460 PROFESSOR: Say again? 00:21:28.460 --> 00:21:30.040 AUDIENCE: 0.254. 00:21:30.040 --> 00:21:33.850 PROFESSOR: 0.254. 00:21:33.850 --> 00:21:37.340 OK, what about the second one, this element over here? 00:21:40.780 --> 00:21:41.280 Speak up. 00:21:41.280 --> 00:21:41.790 AUDIENCE: 0. 00:21:41.790 --> 00:21:43.123 PROFESSOR: Yeah, it better be 0. 00:21:43.123 --> 00:21:44.670 What about this one down here? 00:21:44.670 --> 00:21:45.990 All right, how about this one? 00:22:05.217 --> 00:22:07.057 AUDIENCE: Wait, that first one's not right. 00:22:07.057 --> 00:22:07.682 PROFESSOR: Hmm? 00:22:07.682 --> 00:22:09.200 AUDIENCE: That first one's not right. 00:22:09.200 --> 00:22:09.783 PROFESSOR: OK. 00:22:12.806 --> 00:22:14.760 AUDIENCE: [INAUDIBLE] 00:22:14.760 --> 00:22:17.550 PROFESSOR: So did you give me the first one? 00:22:17.550 --> 00:22:20.010 So you're authorized to change this. 00:22:20.010 --> 00:22:21.680 OK, you got point what? 00:22:21.680 --> 00:22:22.842 AUDIENCE: 859. 00:22:22.842 --> 00:22:30.370 PROFESSOR: 8598, And this 0, 0-- how about this second one 00:22:30.370 --> 00:22:31.518 down here now? 00:22:31.518 --> 00:22:37.520 AUDIENCE: 3.48. 00:22:37.520 --> 00:22:42.940 PROFESSOR: 3.48. 00:22:42.940 --> 00:22:44.525 Anybody else get anything different? 00:22:48.310 --> 00:22:59.130 So let's-- we have 605 00 and 0.5, 00:22:59.130 --> 00:23:00.785 and we're multiplying that. 00:23:45.100 --> 00:23:50.851 So if you're looking for just one of them, by the way-- 00:23:50.851 --> 00:23:55.790 where'd my eraser go?-- all you need is one of the modal 00:23:55.790 --> 00:23:56.290 masses. 00:24:02.330 --> 00:24:06.350 The only ones that give you non-zero results 00:24:06.350 --> 00:24:22.720 is when you compute u transpose for mode r m u for mode r. 00:24:22.720 --> 00:24:25.400 So if you're only looking for this second one, 00:24:25.400 --> 00:24:30.140 you only have to do the calculation for that mode. 00:24:30.140 --> 00:24:33.060 So this then becomes a set of matrix, matrix, matrix, 00:24:33.060 --> 00:24:34.850 you only have to do a couple of vectors. 00:24:34.850 --> 00:24:41.070 So this looks like for mode 2, it's minus what? 00:24:43.820 --> 00:25:04.640 Mode 2 is 1, and minus 1.6949 times 0.60500 0.5 times 1 00:25:04.640 --> 00:25:08.520 and minus 1.6949. 00:25:08.520 --> 00:25:12.990 So to get just one modal mass, this is M2. 00:25:12.990 --> 00:25:14.680 To get just one modal mass, now you only 00:25:14.680 --> 00:25:15.846 have to do that calculation. 00:25:15.846 --> 00:25:21.040 You only have to do the computation using 00:25:21.040 --> 00:25:22.190 one of the mode shapes. 00:25:22.190 --> 00:25:26.390 So it's this times that, and this times that 00:25:26.390 --> 00:25:28.400 gives you some numbers back. 00:25:28.400 --> 00:25:30.790 0.605 and half of this about point 00:25:30.790 --> 00:25:33.370 0.8, and then you take that and multiply again. 00:25:33.370 --> 00:25:37.150 Anyway, can somebody give me this second number? 00:25:37.150 --> 00:25:39.010 I have a 3.48. 00:25:39.010 --> 00:25:40.400 Anybody get anything different? 00:25:40.400 --> 00:25:40.900 Pardon? 00:25:40.900 --> 00:25:41.950 AUDIENCE: I messed it up. 00:25:41.950 --> 00:25:44.000 PROFESSOR: OK. 00:25:44.000 --> 00:25:49.375 There are about 18 of you and nobody can do this calculation? 00:25:52.155 --> 00:25:52.830 AUDIENCE: 2.13. 00:25:52.830 --> 00:25:53.705 PROFESSOR: Say again? 00:25:53.705 --> 00:25:55.360 AUDIENCE: 2.130. 00:25:55.360 --> 00:26:12.030 PROFESSOR: All right, I have a 2.132 and a 2.0. 00:26:12.030 --> 00:26:18.290 OK, there you are. 00:26:21.000 --> 00:26:24.610 OK, so if you were having trouble sorting that out, 00:26:24.610 --> 00:26:27.290 probably a good thing to go back and review a little bit 00:26:27.290 --> 00:26:29.550 of your linear algebra. 00:26:29.550 --> 00:26:33.680 OK, if you do u transpose ku, you get this. 00:26:33.680 --> 00:26:34.688 Yeah? 00:26:34.688 --> 00:26:36.560 AUDIENCE: What is it for the [INAUDIBLE] 00:26:36.560 --> 00:26:40.110 you have a multiplied by F1 L1, not just F1? 00:26:40.110 --> 00:26:42.800 PROFESSOR: OK, I'm going to guess where I'm going next. 00:26:42.800 --> 00:26:49.000 So to get the stiffness matrix, u transpose ku, you get this. 00:26:49.000 --> 00:26:51.970 We're going to leave the damping matrix for a minute. 00:26:51.970 --> 00:26:52.830 We need that. 00:26:52.830 --> 00:26:55.290 We need the modal excitations. 00:26:55.290 --> 00:26:57.870 So I'm just going to do a particular problem. 00:26:57.870 --> 00:26:59.900 I'm going to say, let's let F2 be 0. 00:26:59.900 --> 00:27:02.290 We'll only have one force. 00:27:02.290 --> 00:27:06.080 And the first force will be F1 cosine omega t. 00:27:06.080 --> 00:27:08.610 And so now I need to do u transpose 00:27:08.610 --> 00:27:11.150 F. So here is u transpose. 00:27:11.150 --> 00:27:12.902 Here's the F's, and I guess I've got 00:27:12.902 --> 00:27:17.890 to keep my cosine omega t here. 00:27:17.890 --> 00:27:19.890 So you multiply that out. 00:27:19.890 --> 00:27:21.620 What do you get? 00:27:21.620 --> 00:27:22.120 Yeah? 00:27:22.120 --> 00:27:24.070 AUDIENCE: Why is F1 L1 [INAUDIBLE]? 00:27:24.070 --> 00:27:27.290 PROFESSOR: Well, because F1 is just 00:27:27.290 --> 00:27:30.310 the applied force, but the equation in motion that we're 00:27:30.310 --> 00:27:32.860 working with, if you go back and look at it, 00:27:32.860 --> 00:27:34.770 what is the forces on the right hand side? 00:27:38.970 --> 00:27:41.150 The forces have to be moments, right? 00:27:41.150 --> 00:27:44.200 If we're putting a force down there, it's a moment equation. 00:27:44.200 --> 00:27:46.050 We need the moments about the pivot. 00:27:46.050 --> 00:27:51.074 So it's the force times L1 or the force times L2. 00:27:51.074 --> 00:27:54.726 AUDIENCE: Oh, so we don't have to use actual magnitude of 4. 00:27:54.726 --> 00:27:56.452 We have to use [INAUDIBLE]. 00:27:56.452 --> 00:27:57.410 PROFESSOR: You have to. 00:27:57.410 --> 00:28:00.830 You well eventually-- well, these kind of problems 00:28:00.830 --> 00:28:04.380 are easiest to do once you reduce them to numbers. 00:28:04.380 --> 00:28:07.350 I'm leaving the force in it as a variable at the moment 00:28:07.350 --> 00:28:10.760 just so you can see how it carries through the problem. 00:28:10.760 --> 00:28:13.490 But I'm just saying, in the real problem 00:28:13.490 --> 00:28:16.860 there, let's say there is no F2. 00:28:16.860 --> 00:28:24.490 There is an F1, and the F1 of t looks like a magnitude 00:28:24.490 --> 00:28:27.200 f1 times cosine omega t. 00:28:27.200 --> 00:28:29.520 That's the only force I have in the system. 00:28:29.520 --> 00:28:32.256 But we're working with equations of motions. 00:28:32.256 --> 00:28:35.440 An equation of motion is the right hand side-- F1, L1, 00:28:35.440 --> 00:28:36.760 and F2 L2. 00:28:36.760 --> 00:28:39.570 And you have to retain the L1's and L2's in order 00:28:39.570 --> 00:28:41.555 to have the correct equation of motion. 00:28:45.390 --> 00:28:50.530 So when we say this is kind of just a generic form, 00:28:50.530 --> 00:28:55.610 this is the modal force vector is the mode shape 00:28:55.610 --> 00:28:59.950 matrix times the modal excitations 00:28:59.950 --> 00:29:01.610 in the original coordinates. 00:29:01.610 --> 00:29:04.930 So I just wrote F here, but what this really means 00:29:04.930 --> 00:29:06.060 is this is F1, L1. 00:29:11.710 --> 00:29:13.540 F2, L2. 00:29:13.540 --> 00:29:21.000 These are the real generalized forces in the system. 00:29:21.000 --> 00:29:23.190 OK, they're the real, generalized forces. 00:29:23.190 --> 00:29:25.960 Now, I've let F2 be 0. 00:29:25.960 --> 00:29:31.710 So the only generalized force is F1, L1 cosine omega T. 00:29:31.710 --> 00:29:35.790 I multiply that by u transpose to get-- and what do I get? 00:29:35.790 --> 00:29:39.300 This is a pretty simple calculation. 00:29:39.300 --> 00:29:47.810 So this becomes F1, L1 cos and F1, L1 cos. 00:29:47.810 --> 00:29:51.220 So the two modal forces are identical. 00:29:51.220 --> 00:29:51.720 Yeah? 00:29:51.720 --> 00:29:53.148 AUDIENCE: Why do we know this one 00:29:53.148 --> 00:29:54.617 so we don't have to include a phi? 00:29:54.617 --> 00:29:56.450 PROFESSOR: Ah, well the phi doesn't come out 00:29:56.450 --> 00:29:57.158 until the answer. 00:29:59.970 --> 00:30:01.550 What does the fee mean? 00:30:01.550 --> 00:30:03.650 What's that phase angle mean? 00:30:03.650 --> 00:30:10.140 If you-- remember, we're doing steady state problems 00:30:10.140 --> 00:30:19.720 in which F1, L1 for example, if it's cosine, 00:30:19.720 --> 00:30:23.640 we're just assuming it looks like that. 00:30:23.640 --> 00:30:28.235 And we're looking for a solution of theta 1. 00:30:32.490 --> 00:30:34.720 And actually, we can't quite go there yet. 00:30:34.720 --> 00:30:37.600 We're doing single degree of freedom problems, right? 00:30:37.600 --> 00:30:42.020 We are looking for a solution for Q1. 00:30:42.020 --> 00:30:45.680 We turn this into a modal force, q, 00:30:45.680 --> 00:30:49.680 but it happens to be the capital Q1, the modal force, 00:30:49.680 --> 00:30:51.820 is F1 L1, right? 00:30:51.820 --> 00:30:53.250 Cosine omega t. 00:30:53.250 --> 00:30:55.120 So that's the input. 00:30:55.120 --> 00:30:59.200 The output is q, little q, of t, the modal coordinate. 00:30:59.200 --> 00:31:01.310 And what does it look like? 00:31:01.310 --> 00:31:04.630 We're only doing steady state, no transience. 00:31:04.630 --> 00:31:12.650 It looks like a response that looks like this, 00:31:12.650 --> 00:31:14.830 but it's shifted. 00:31:14.830 --> 00:31:16.910 I'll draw this so its peak is right here, 00:31:16.910 --> 00:31:23.010 cosine is its-- it's shifted in time by this amount, 00:31:23.010 --> 00:31:26.470 between this peak is here versus the peak being there. 00:31:26.470 --> 00:31:30.450 And that we can represent as a phase angle. 00:31:30.450 --> 00:31:34.680 Remember, one period from here to here is 2 pi radians. 00:31:34.680 --> 00:31:39.300 So some portion of that period is an angle. 00:31:39.300 --> 00:31:41.745 You can interpret it as an angle or you can interpret it 00:31:41.745 --> 00:31:44.290 as a time delay. 00:31:44.290 --> 00:31:52.320 And this then has the form of some q1 magnitude cosine omega 00:31:52.320 --> 00:31:57.050 t minus that phase shift. 00:31:57.050 --> 00:32:00.320 So it's only-- that's the only thing the phase shift means. 00:32:00.320 --> 00:32:03.570 Cosine in doesn't mean exactly the response 00:32:03.570 --> 00:32:07.797 out's going to be exactly in the same perfectly in time with it. 00:32:07.797 --> 00:32:08.630 It could be shifted. 00:32:11.475 --> 00:32:13.225 Now you know that it-- for a single degree 00:32:13.225 --> 00:32:15.555 of freedom system at resonance, what's the phase angle? 00:32:15.555 --> 00:32:16.710 Do you remember that? 00:32:16.710 --> 00:32:19.170 It's always one number. 00:32:19.170 --> 00:32:20.435 It's pi over 2. 00:32:24.140 --> 00:32:27.110 A shift of pi over 2, if you remember your trigonometry, 00:32:27.110 --> 00:32:30.845 takes you from cosine to sine or sine to cosine. 00:32:30.845 --> 00:32:34.610 It's trying to tell you that the response is 00:32:34.610 --> 00:32:40.240 shifted by exactly a quarter of a cycle, pi over 2. 00:32:40.240 --> 00:32:45.400 And the reason for that is that at resonance, 00:32:45.400 --> 00:32:49.990 all of the excitation is going into overpowering the damper. 00:32:49.990 --> 00:32:54.020 And the damper's motion is proportional to velocity. 00:32:54.020 --> 00:32:57.490 And if velocity, if displacement looks like cosine, 00:32:57.490 --> 00:32:59.580 velocity looks like one derivative of it, 00:32:59.580 --> 00:33:02.280 which is sine. 00:33:02.280 --> 00:33:07.890 So that pi over 2 says that the response 00:33:07.890 --> 00:33:12.010 velocity is in phase with the force, and that makes sense. 00:33:12.010 --> 00:33:13.510 Something I hadn't said in lecture 00:33:13.510 --> 00:33:15.550 but I really meant to is I want you 00:33:15.550 --> 00:33:16.820 to think about something here. 00:33:16.820 --> 00:33:18.977 Single degree of freedom system, we'll 00:33:18.977 --> 00:33:20.560 even write the one we're working with. 00:33:20.560 --> 00:33:29.420 M1 Q1 double dot plus C1 Q1 dot plus K1 Q 00:33:29.420 --> 00:33:33.960 equals some F1 L1 cosine omega t, 00:33:33.960 --> 00:33:37.510 and I'm going to let omega E be at omega 1. 00:33:37.510 --> 00:33:40.240 But this is the equation of motion, right? 00:33:40.240 --> 00:33:41.560 Let's plug in. 00:33:41.560 --> 00:33:45.430 We're saying we're going to do this right at the-- 00:33:45.430 --> 00:33:52.360 and we know the response of this is Q1 is some magnitude cosine 00:33:52.360 --> 00:33:56.600 omega t minus the phasing. 00:33:56.600 --> 00:33:57.920 We know we can plug that in. 00:33:57.920 --> 00:34:21.719 We get minus M1 omega squared Q1 plus K1 Q1 minus C1 omega Q1. 00:34:21.719 --> 00:34:26.739 And this one goes like sine omega T minus phi. 00:34:26.739 --> 00:34:28.870 This one here needs to get multiplied. 00:34:28.870 --> 00:34:33.254 This term gets multiplied by cosine omega t minus phi. 00:34:33.254 --> 00:34:35.920 You plug that into this. 00:34:35.920 --> 00:34:39.570 This term and this term both behave like cosine. 00:34:39.570 --> 00:34:42.650 This term, one derivative behaves like minus sine. 00:34:42.650 --> 00:34:45.690 One derivative of cosine gives you minus sine, right? 00:34:45.690 --> 00:34:50.260 And when omega equals omega 1, so 00:34:50.260 --> 00:34:54.830 when you're right at resonance here, what happens? 00:34:54.830 --> 00:34:58.520 This is squared. 00:34:58.520 --> 00:35:02.530 And put the squared down here, there we go. 00:35:02.530 --> 00:35:03.310 So this is omega. 00:35:03.310 --> 00:35:06.200 But now I'm going to let it be right at omega 1. 00:35:06.200 --> 00:35:12.150 What is omega 1 in terms of K's and M's? 00:35:12.150 --> 00:35:14.790 K1/M1, otherwise, one of the checks 00:35:14.790 --> 00:35:18.270 you can make when you finish doing your modal-- if you take 00:35:18.270 --> 00:35:21.200 that modal mass and the modal stiffness 00:35:21.200 --> 00:35:26.400 and you divide 7.96 by 0.8598, that had better be omega 1 00:35:26.400 --> 00:35:27.497 squared. 00:35:27.497 --> 00:35:29.080 That's a good way to check that you've 00:35:29.080 --> 00:35:30.950 done all your arithmetic right. 00:35:30.950 --> 00:35:33.970 All right, I'm going to plug in omega 1 00:35:33.970 --> 00:35:35.650 squared here equals K1/M1. 00:35:38.240 --> 00:35:46.070 So I put in, this becomes minus M1 K1 over M1, which 00:35:46.070 --> 00:35:47.830 is minus K1, right? 00:35:47.830 --> 00:35:50.670 Hmm, plus K1. 00:35:53.570 --> 00:35:57.040 That resonance, this term accounts 00:35:57.040 --> 00:36:01.170 for the inertial force in the system, the force required 00:36:01.170 --> 00:36:03.340 to accelerate the mass. 00:36:03.340 --> 00:36:04.460 This is a force equation. 00:36:04.460 --> 00:36:08.130 This accounts for the force required to push the spring. 00:36:08.130 --> 00:36:11.970 The amazing thing that happens is at resonance, 00:36:11.970 --> 00:36:18.540 the inertial forces exactly cancel the spring forces. 00:36:18.540 --> 00:36:23.620 And the equation of motion reduces to minus C1 00:36:23.620 --> 00:36:33.550 omega 1 Q1 sine omega 1 t minus a phase angle equals, 00:36:33.550 --> 00:36:39.730 in this case, F1 L1 cosine omega 1 t. 00:36:39.730 --> 00:36:42.090 So how to satisfy that equation? 00:36:42.090 --> 00:36:46.290 What phase angle will satisfy that equation? 00:36:46.290 --> 00:36:47.590 Has to be pi over 2. 00:36:47.590 --> 00:36:49.650 And if you put pi over 2 in here, 00:36:49.650 --> 00:36:53.300 this minus sign turns into plus cosine. 00:36:53.300 --> 00:37:01.130 And you're left with C1 omega 1 Q1 equals F1 L1. 00:37:01.130 --> 00:37:07.170 So all of the exciting force goes into pushing the dashpot. 00:37:07.170 --> 00:37:11.100 So that's why you get the big peak in the transfer function. 00:37:11.100 --> 00:37:13.305 It takes in no force to move the spring. 00:37:13.305 --> 00:37:15.820 It takes no force to accelerate the mass. 00:37:15.820 --> 00:37:17.585 They exactly cancel. 00:37:17.585 --> 00:37:21.749 And all the force is available to drive just the dashpot, 00:37:21.749 --> 00:37:23.140 all right? 00:37:23.140 --> 00:37:25.900 So let's move on now. 00:37:25.900 --> 00:37:27.540 We need to get to our answer here. 00:37:34.160 --> 00:37:41.080 So the last piece of this is we now know the modal forces, 00:37:41.080 --> 00:37:52.916 and your assignment is to let's let omega-- let's see, 00:37:52.916 --> 00:37:57.820 where's my-- did I do this somewhere? 00:37:57.820 --> 00:37:58.535 I guess not. 00:37:58.535 --> 00:38:01.400 I guess I erased it, so we can pick anything we want. 00:38:01.400 --> 00:38:05.907 Let's let omega equal omega 2. 00:38:05.907 --> 00:38:08.240 We're going to drive this thing at the natural frequency 00:38:08.240 --> 00:38:09.031 of the second mode. 00:38:09.031 --> 00:38:10.489 That's the excitation. 00:38:10.489 --> 00:38:11.280 What do you expect? 00:38:11.280 --> 00:38:14.344 Which mode do you expect to dominate the response? 00:38:14.344 --> 00:38:15.308 AUDIENCE: The second. 00:38:15.308 --> 00:38:16.432 PROFESSOR: The second mode. 00:38:16.432 --> 00:38:17.102 Why? 00:38:17.102 --> 00:38:19.070 AUDIENCE: [INAUDIBLE] 00:38:19.070 --> 00:38:21.052 PROFESSOR: Because you are driving it. 00:38:21.052 --> 00:38:23.510 You know it's got a transfer function that looks like this, 00:38:23.510 --> 00:38:25.580 and you're driving it right here. 00:38:25.580 --> 00:38:27.260 And where are you driving it? 00:38:27.260 --> 00:38:29.310 If you're doing that, where are you 00:38:29.310 --> 00:38:33.970 on the transfer function for the first mode? 00:38:33.970 --> 00:38:41.100 Omega 2, it's a little higher than omega 1, but not a lot. 00:38:41.100 --> 00:38:44.280 So if this were the first mode's transfer function, 00:38:44.280 --> 00:38:49.710 where would you be driving it on this transfer function? 00:38:49.710 --> 00:38:58.709 If this is omega over omega 2, you're over here a little bit. 00:38:58.709 --> 00:39:00.250 You're driving it a little bit higher 00:39:00.250 --> 00:39:02.270 than the natural frequency of mode 1. 00:39:02.270 --> 00:39:06.240 This would be omega 2 over omega 1 on that. 00:39:06.240 --> 00:39:08.600 So one's sitting here, one's sitting there. 00:39:08.600 --> 00:39:10.360 Which one's going to dominate? 00:39:10.360 --> 00:39:11.360 The big one, OK? 00:39:11.360 --> 00:39:12.990 Because they have equal modal forces. 00:39:12.990 --> 00:39:16.290 They both happen to be F1 L1. 00:39:16.290 --> 00:39:19.320 OK, so how do you do that? 00:39:19.320 --> 00:39:21.490 So the last step in this thing is 00:39:21.490 --> 00:39:35.417 I want you to, for this case, find the first-- no, 00:39:35.417 --> 00:39:36.000 let's do this. 00:39:36.000 --> 00:39:46.960 Find the second mode contribution 00:39:46.960 --> 00:39:52.690 to the response in the original generalized coordinates. 00:39:52.690 --> 00:39:54.522 A quiz would often be written this way. 00:39:54.522 --> 00:39:56.105 It's trying to make it easier for you. 00:39:56.105 --> 00:39:58.870 I'm asking for only one mode's contribution. 00:39:58.870 --> 00:40:02.690 What does that actually mean in the original statement 00:40:02.690 --> 00:40:06.030 of the modal expansion theorem? 00:40:06.030 --> 00:40:09.720 We know the total response looks like this, right? 00:40:09.720 --> 00:40:13.290 I'm asking you to give me only the second mode's contribution. 00:40:13.290 --> 00:40:16.410 What am I asking for? 00:40:16.410 --> 00:40:17.824 Just this term. 00:40:17.824 --> 00:40:19.240 So I'm telling you, you don't have 00:40:19.240 --> 00:40:20.880 to bother with the other term to satisfy me. 00:40:20.880 --> 00:40:22.610 Just tell me what this one is, because I 00:40:22.610 --> 00:40:25.460 know this is one is going to be the dominant one. 00:40:25.460 --> 00:40:26.720 OK, so how do I do that? 00:40:26.720 --> 00:40:29.700 So for this problem, what is the steady state response 00:40:29.700 --> 00:40:34.650 of this due to mode 2 only? 00:40:34.650 --> 00:40:38.710 So mathematically, just in-- what's that look 00:40:38.710 --> 00:40:39.580 like over here? 00:40:44.280 --> 00:40:45.510 Just that second term, right? 00:40:45.510 --> 00:40:51.650 It's the modal, mode shape vector for mode 2 times Q2 00:40:51.650 --> 00:40:52.580 of t. 00:40:52.580 --> 00:40:55.050 So you need to tell me how to find Q1 of t. 00:41:00.540 --> 00:41:03.200 It's a single degree of freedom problem excited by steady state 00:41:03.200 --> 00:41:03.700 excitation. 00:41:18.094 --> 00:41:19.760 AUDIENCE: It's like a transfer function. 00:41:19.760 --> 00:41:21.910 PROFESSOR: Ah, magic word-- transfer function. 00:41:21.910 --> 00:41:26.290 So we need the magnitude of the-- this is a linear problem, 00:41:26.290 --> 00:41:29.384 so the response is linearly proportional to the--? 00:41:29.384 --> 00:41:30.050 AUDIENCE: Force. 00:41:30.050 --> 00:41:30.758 PROFESSOR: Force. 00:41:30.758 --> 00:41:34.430 So the magnitude of Q1 times a transfer 00:41:34.430 --> 00:41:37.600 function that looks like the hx over f transfer function. 00:41:37.600 --> 00:41:45.850 In this case, we call it the magnitude of HQ2 per unit Q2. 00:41:45.850 --> 00:41:49.810 And that multiplied by-- it looks 00:41:49.810 --> 00:41:55.390 like cosine in this case-- omega 2 t minus some phi 2. 00:41:55.390 --> 00:41:57.060 That's what we're looking for. 00:41:57.060 --> 00:41:58.172 What's this? 00:41:58.172 --> 00:41:59.380 Tell me what that looks like. 00:41:59.380 --> 00:41:59.879 Yeah? 00:41:59.879 --> 00:42:02.510 AUDIENCE: Shouldn't it be Q2? 00:42:02.510 --> 00:42:03.750 PROFESSOR: Thank you. 00:42:07.960 --> 00:42:10.270 So in effect, what's the transfer function? 00:42:10.270 --> 00:42:13.020 I want it in all its detail now. 00:42:20.610 --> 00:42:23.490 That's the magnitude of the force. 00:42:23.490 --> 00:42:25.245 What's in the numerator of this? 00:42:28.470 --> 00:42:29.552 Numerator. 00:42:29.552 --> 00:42:32.360 AUDIENCE: Well, it has parentheses around it. 00:42:32.360 --> 00:42:34.360 PROFESSOR: Yeah, it's in the denominator though. 00:42:34.360 --> 00:42:37.760 I want just the numerator part. 00:42:37.760 --> 00:42:39.167 This transfer function expression 00:42:39.167 --> 00:42:40.750 for a single degree of freedom system, 00:42:40.750 --> 00:42:41.749 what's in the numerator? 00:42:41.749 --> 00:42:42.780 AUDIENCE: 1/K. 00:42:42.780 --> 00:42:44.550 PROFESSOR: 1 over which K? 00:42:47.480 --> 00:42:48.850 K2. 00:42:48.850 --> 00:42:53.422 Modal K2, We're now in the modal system. 00:42:53.422 --> 00:42:55.880 And in the denominator of that transfer function, what's it 00:42:55.880 --> 00:42:56.845 look like? 00:42:56.845 --> 00:43:00.810 AUDIENCE: 1 plus omega. 00:43:00.810 --> 00:43:04.770 PROFESSOR: 1 minus omega squared over-- in this case, 00:43:04.770 --> 00:43:10.090 omega 2 squared, squared, plus-- 00:43:10.090 --> 00:43:14.184 AUDIENCE: 2 [INAUDIBLE]. 00:43:14.184 --> 00:43:14.850 PROFESSOR: Zeta. 00:43:14.850 --> 00:43:16.740 AUDIENCE: Zeta, yeah. 00:43:16.740 --> 00:43:18.034 PROFESSOR: 2. 00:43:18.034 --> 00:43:25.362 AUDIENCE: 2 omega over omega 2 quantity squared. 00:43:25.362 --> 00:43:26.320 PROFESSOR: There we go. 00:43:26.320 --> 00:43:27.778 Now in this problem, what is omega? 00:43:30.390 --> 00:43:35.280 So that makes this a 2, this a 2. 00:43:35.280 --> 00:43:36.770 This term, what happens to it? 00:43:39.640 --> 00:43:41.480 1 minus 1. 00:43:41.480 --> 00:43:42.950 This term, this goes to 1. 00:43:42.950 --> 00:43:47.160 This is 2 zeta quantity squared square root. 00:43:47.160 --> 00:43:48.480 Just 2 zeta. 00:43:48.480 --> 00:43:52.540 So this whole at resonance, any one of these single degree 00:43:52.540 --> 00:43:54.390 of freedom systems that are at resonance, 00:43:54.390 --> 00:43:58.220 the response is the magnitude of the force-- in this case, 00:43:58.220 --> 00:44:06.980 it's positive F1 L1-- over K2 times 1/2 zeta 2. 00:44:10.850 --> 00:44:16.080 So F1 L1 is the modal magnitude of the moral force divided 00:44:16.080 --> 00:44:20.320 by K2 gives you what we call the static displacement 00:44:20.320 --> 00:44:21.490 of the system. 00:44:21.490 --> 00:44:24.690 And this is the dynamic application. 00:44:24.690 --> 00:44:26.790 So oftentimes on quizzes, you're asked 00:44:26.790 --> 00:44:29.130 to do the response at resonance, because it makes 00:44:29.130 --> 00:44:30.600 all this algebra so simple. 00:44:30.600 --> 00:44:34.360 It boils down to 1/2 zeta. 00:44:34.360 --> 00:44:37.020 So the only thing left to do is we need the damping ratio 00:44:37.020 --> 00:44:37.690 for this system. 00:44:40.350 --> 00:44:43.320 So rather than do that, so now that, we're 00:44:43.320 --> 00:44:44.400 missing something yet. 00:44:44.400 --> 00:44:49.610 We're missing-- that whole thing gets multiplied by cosine omega 00:44:49.610 --> 00:44:52.500 2 t minus the face angel. 00:44:52.500 --> 00:44:54.490 What's the face angel? 00:44:54.490 --> 00:44:56.570 Pi over 2. 00:44:56.570 --> 00:45:10.640 So Q2 of t is F1 L1 over K1 1/2-- whoops, F1 L1 over K2, 00:45:10.640 --> 00:45:18.650 1/2 zeta 2 cosine omega 2t minus. 00:45:18.650 --> 00:45:21.880 And at the very-- how do we get back 00:45:21.880 --> 00:45:27.170 to generalized coordinates, theta 1 and theta 2? 00:45:27.170 --> 00:45:28.150 AUDIENCE: Stay here. 00:45:31.100 --> 00:45:32.590 PROFESSOR: Right here. 00:45:32.590 --> 00:45:34.930 And we've computed that. 00:45:34.930 --> 00:45:39.160 This part, you multiply it by the mode shape. 00:45:39.160 --> 00:45:42.300 The mode shape partitions out the response 00:45:42.300 --> 00:45:45.480 in the right amount to coordinate 00:45:45.480 --> 00:45:48.810 one and the correct amount to coordinate two. 00:45:54.762 --> 00:45:58.234 AUDIENCE: For the damping ratio, would you take the modal, like, 00:45:58.234 --> 00:46:01.870 mass and [INAUDIBLE]? 00:46:01.870 --> 00:46:03.660 PROFESSOR: In reality, what you do 00:46:03.660 --> 00:46:14.720 with damping ratios is you're working with real things 00:46:14.720 --> 00:46:17.650 out there in the real world. 00:46:17.650 --> 00:46:20.135 If you can, you go up and give the thing a kick 00:46:20.135 --> 00:46:23.620 and get your stopwatch out and say, how many cycles does it 00:46:23.620 --> 00:46:24.800 take to the k? 00:46:24.800 --> 00:46:27.560 And is it light damping or not? 00:46:27.560 --> 00:46:29.790 If it vibrates a lot, it's usually light damping. 00:46:29.790 --> 00:46:33.340 And if it's light damping, you can force this damping matrix. 00:46:33.340 --> 00:46:35.720 You can just make-- force it to behave, 00:46:35.720 --> 00:46:36.980 even if it isn't perfect. 00:46:36.980 --> 00:46:40.620 And it is a perfectly adequate, useful answer. 00:46:40.620 --> 00:46:42.220 Even if it isn't perfectly diagonal, 00:46:42.220 --> 00:46:44.695 it just doesn't matter when it's light damping. 00:46:44.695 --> 00:46:46.570 And so in this problem, what you do if you go 00:46:46.570 --> 00:46:48.070 estimate the damping for the system. 00:46:48.070 --> 00:46:51.720 You say eh, it looks to me to be about 2% for mode one 00:46:51.720 --> 00:46:54.950 and 1.5% for mode 2. 00:46:54.950 --> 00:46:58.130 And you just say, how can I fit? 00:46:58.130 --> 00:47:00.880 How can I represent the damping in the system? 00:47:00.880 --> 00:47:03.050 And one of the easiest ones is to say 00:47:03.050 --> 00:47:06.480 that the original damping matrix is some alpha times 00:47:06.480 --> 00:47:10.110 the mass matrix plus beta times the stiffness matrix. 00:47:10.110 --> 00:47:14.700 And in this problem-- or you can use any part of that. 00:47:14.700 --> 00:47:19.420 And if you're only trying to match one mode, see, 00:47:19.420 --> 00:47:22.430 this problem it's-- this system is being driven at the natural 00:47:22.430 --> 00:47:24.200 frequency of one mode. 00:47:24.200 --> 00:47:28.650 That mode is dominating the response, right? 00:47:28.650 --> 00:47:31.900 So we really actually only need a good model 00:47:31.900 --> 00:47:34.760 of the damping for that mode. 00:47:34.760 --> 00:47:36.790 Even if you have the completely wrong damping 00:47:36.790 --> 00:47:40.860 for the other mode, it will have little effect on its answer 00:47:40.860 --> 00:47:43.410 because you're not at resonance. 00:47:43.410 --> 00:47:48.730 When in the transfer functions, which look like this, 00:47:48.730 --> 00:47:51.350 at resonance, this happens. 00:47:51.350 --> 00:47:55.710 Then you find out that the only force resisting the input 00:47:55.710 --> 00:47:58.220 is the dashpot. 00:47:58.220 --> 00:48:01.030 At low frequencies, you find out that over here, 00:48:01.030 --> 00:48:04.420 the dominant force is what it takes to move the spring. 00:48:04.420 --> 00:48:06.170 And the damping isn't very important. 00:48:06.170 --> 00:48:08.380 And so even if you're wrong by 50%, 00:48:08.380 --> 00:48:12.350 it just doesn't-- 50% of a little bit compared to what it 00:48:12.350 --> 00:48:15.350 takes to move the spring is not a big deal. 00:48:15.350 --> 00:48:18.100 And over here, it behaves like the mass. 00:48:18.100 --> 00:48:21.010 Out here it's called the mass controlled region. 00:48:21.010 --> 00:48:23.350 Over here is the stiffness controlled region. 00:48:23.350 --> 00:48:26.230 And in the vicinity of the peak is the damping controlled 00:48:26.230 --> 00:48:26.730 region. 00:48:26.730 --> 00:48:30.370 So at low frequency, this term dominates. 00:48:30.370 --> 00:48:32.565 At high frequency, that term dominates. 00:48:32.565 --> 00:48:36.360 And at resonance, this is the dominant term. 00:48:36.360 --> 00:48:41.150 So in this case, let's let the damping be some alpha 00:48:41.150 --> 00:48:43.410 times the mass matrix. 00:48:43.410 --> 00:48:49.890 Then when you do UTCU, you get alpha times the modal mass 00:48:49.890 --> 00:48:50.390 matrix. 00:48:53.210 --> 00:48:57.230 And therefore, C2, which is the one we care about, 00:48:57.230 --> 00:49:01.410 is equal to alpha times M2. 00:49:01.410 --> 00:49:13.420 And we have M2, our modal mass, 2.04, right? 00:49:13.420 --> 00:49:20.960 So that says this is equal-- C2 is equal to alpha times 2.04. 00:49:20.960 --> 00:49:24.060 And if I want-- that's C2. 00:49:24.060 --> 00:49:26.400 And how do I get zeta 2? 00:49:26.400 --> 00:49:32.300 Zeta 2 is C2 over 2 omega 2 M2. 00:49:32.300 --> 00:49:35.070 That's just the definition of the damping ratio. 00:49:35.070 --> 00:49:36.070 I know this. 00:49:36.070 --> 00:49:38.050 I know this. 00:49:38.050 --> 00:49:48.630 So this is going to be alpha times 2.04 over 2 omega 2 M2. 00:49:48.630 --> 00:49:50.420 And I've measured it. 00:49:50.420 --> 00:49:52.770 I've taken it, and I know this is 0.02. 00:49:52.770 --> 00:49:54.690 About 2% damping. 00:49:54.690 --> 00:49:56.350 Solve for alpha. 00:49:56.350 --> 00:50:01.130 You now have the whole thing that you need. 00:50:01.130 --> 00:50:04.240 You can now find it. 00:50:04.240 --> 00:50:06.760 That's all you need. 00:50:06.760 --> 00:50:07.260 Yeah? 00:50:07.260 --> 00:50:10.578 AUDIENCE: Where do you get the alphas and omegas from again? 00:50:10.578 --> 00:50:13.430 Alphas and omegas. 00:50:13.430 --> 00:50:17.310 PROFESSOR: This is simply-- this is called Rayleigh damping. 00:50:17.310 --> 00:50:20.786 Lord Rayleigh 150 years ago came up with this. 00:50:20.786 --> 00:50:22.160 And he just said hey, by the way, 00:50:22.160 --> 00:50:23.576 if you model damping this way, you 00:50:23.576 --> 00:50:26.970 can automatically make the equations of motion, 00:50:26.970 --> 00:50:29.730 u transpose CU, go diagonal. 00:50:29.730 --> 00:50:33.010 And you have a two parameter model 00:50:33.010 --> 00:50:37.210 with which you can juggle them to make any two 00:50:37.210 --> 00:50:39.660 damping ratios of the system be exactly what you 00:50:39.660 --> 00:50:42.550 want them to be. 00:50:42.550 --> 00:50:46.610 So it's just-- if I left this as alpha N beta, 00:50:46.610 --> 00:50:49.290 then I would have worked this problem as-- this would 00:50:49.290 --> 00:50:54.630 have been an alpha M plus a beta K. C2 would have been an alpha 00:50:54.630 --> 00:50:56.110 M2. 00:50:56.110 --> 00:51:03.680 But now it would be equal to alpha M2 plus beta K2. 00:51:03.680 --> 00:51:06.640 And then this still applies, except that it'd 00:51:06.640 --> 00:51:08.850 have this alpha and a beta. 00:51:08.850 --> 00:51:10.840 And you could do it for the other equation. 00:51:10.840 --> 00:51:12.410 You could do it for zeta 1. 00:51:12.410 --> 00:51:14.720 And you'd have two equations and two unknowns. 00:51:14.720 --> 00:51:17.349 You solve for alpha and beta. 00:51:17.349 --> 00:51:18.890 But think about what I just did here. 00:51:18.890 --> 00:51:20.750 If I made a measurement of the system, 00:51:20.750 --> 00:51:23.490 I said the damping for mode 2 is 2%. 00:51:23.490 --> 00:51:25.750 Do I have to go through all this junk? 00:51:25.750 --> 00:51:29.640 No, because I know the answer looks like that. 00:51:29.640 --> 00:51:31.610 All I have to know is what the damping is. 00:51:31.610 --> 00:51:34.620 And that's my approximate solution for the problem. 00:51:34.620 --> 00:51:36.830 This is just how to satisfy all the mathematics 00:51:36.830 --> 00:51:39.460 if you want that perfect mathematical model 00:51:39.460 --> 00:51:43.740 for which you can write out u transpose CU and get them. 00:51:43.740 --> 00:51:47.700 Well, this is one way that you force 00:51:47.700 --> 00:51:53.277 the damper matrix to have the properties that you want it to. 00:51:53.277 --> 00:51:55.110 But in reality, you just measure the damping 00:51:55.110 --> 00:52:00.050 and put it in the answer. 00:52:00.050 --> 00:52:04.890 Very good, this is our last go around at recitations. 00:52:04.890 --> 00:52:08.270 See you in class on Tuesday. 00:52:08.270 --> 00:52:14.280 We'll do something fun that's not covered on the final exam. 00:52:14.280 --> 00:52:15.800 I'll give a little review of what's 00:52:15.800 --> 00:52:18.590 going to be on the final, a list of what's on it. 00:52:18.590 --> 00:52:20.740 And we'll talk about strings and beams and things 00:52:20.740 --> 00:52:24.720 that apply to pianos and violins and so forth.