WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.040 Your support will help MIT OpenCourseWare 00:00:06.040 --> 00:00:10.130 continue to offer high quality educational resources for free. 00:00:10.130 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.297 at ocw.mit.edu. 00:00:21.343 --> 00:00:22.218 PROFESSOR: All right. 00:00:22.218 --> 00:00:25.270 I'm a little slow getting started today, 00:00:25.270 --> 00:00:26.100 better get going. 00:00:32.030 --> 00:00:38.590 What we're going to talk about today is a technique-- 00:00:38.590 --> 00:00:40.130 you guys done? 00:00:40.130 --> 00:00:41.550 OK, thanks. 00:00:41.550 --> 00:00:43.760 We're going to talk about today a technique known 00:00:43.760 --> 00:00:47.840 as modal analysis, and it's a way 00:00:47.840 --> 00:00:52.290 of analyzing things that vibrate, essentially thinking 00:00:52.290 --> 00:00:54.860 about them one mode and a time. 00:00:54.860 --> 00:00:58.220 Though you might not make a lot of use 00:00:58.220 --> 00:01:01.280 of the actual calculations, doing the math, 00:01:01.280 --> 00:01:04.959 throughout your careers, I think if you understand it 00:01:04.959 --> 00:01:09.370 conceptually it'll help you just have a better 00:01:09.370 --> 00:01:12.170 understanding of what vibration is all about, 00:01:12.170 --> 00:01:14.640 just give you some insight to it that you otherwise 00:01:14.640 --> 00:01:16.480 wouldn't have. 00:01:16.480 --> 00:01:18.960 So the basic concept is that you can model just 00:01:18.960 --> 00:01:23.080 about any structural vibration as the summation 00:01:23.080 --> 00:01:25.970 of the individual contributions of each 00:01:25.970 --> 00:01:28.370 what we call natural mode. 00:01:28.370 --> 00:01:30.440 So what we mean by that is, let's 00:01:30.440 --> 00:01:32.800 start by thinking-- actually, let 00:01:32.800 --> 00:01:36.150 me say that this applies to both continuous systems 00:01:36.150 --> 00:01:41.010 like vibrating strings or beams or buildings as it 00:01:41.010 --> 00:01:45.510 does to finite degree of freedom rigid body systems. 00:01:45.510 --> 00:01:47.450 We haven't talked about continuous systems. 00:01:47.450 --> 00:01:51.060 I'll do a lecture on it as the last lecture 00:01:51.060 --> 00:01:54.230 of the term, just kind of an enrichment sort of lecture. 00:01:54.230 --> 00:01:57.590 But everything I say about finite degree of freedom 00:01:57.590 --> 00:02:01.360 systems can be extended to continuous systems. 00:02:01.360 --> 00:02:04.610 But since we've been studying rigid bodies and systems 00:02:04.610 --> 00:02:07.270 with finite numbers of degrees of freedom, 00:02:07.270 --> 00:02:10.430 I'll explain-- I'll go through this analysis 00:02:10.430 --> 00:02:14.330 in the context of rigid body finite degree of freedom 00:02:14.330 --> 00:02:16.700 systems. 00:02:16.700 --> 00:02:23.020 So in general, we can write the equations of motion 00:02:23.020 --> 00:02:28.380 for finite degree of freedom systems as a mass matrix. 00:02:28.380 --> 00:02:30.020 And to keep the kind of writing down, 00:02:30.020 --> 00:02:35.080 I'm just going to underline matrices and a squiggle 00:02:35.080 --> 00:02:36.525 under vectors so we have them. 00:02:36.525 --> 00:02:39.300 In general, we can write the equations of motion 00:02:39.300 --> 00:02:44.880 as a mass matrix times an acceleration vector 00:02:44.880 --> 00:02:50.030 plus a damping matrix times a velocity vector, 00:02:50.030 --> 00:02:54.430 stiffness matrix times a displacement vector, 00:02:54.430 --> 00:03:00.970 all equal to some external vector of excitations. 00:03:00.970 --> 00:03:03.100 And I'm writing these as if these are translations, 00:03:03.100 --> 00:03:07.570 but you know, like from doing the pendulum on the cart 00:03:07.570 --> 00:03:09.860 problem, that the equations of motion 00:03:09.860 --> 00:03:12.700 might involve rotations and displacements. 00:03:12.700 --> 00:03:16.250 And we let them-- they mix together here however they 00:03:16.250 --> 00:03:17.440 fall out. 00:03:17.440 --> 00:03:20.000 But just to write them symbolically, 00:03:20.000 --> 00:03:23.990 I'm just going to refer to all of those coordinates 00:03:23.990 --> 00:03:26.650 with an x vector. 00:03:26.650 --> 00:03:33.880 OK, now the basic premise of modal analysis 00:03:33.880 --> 00:03:43.825 is a thing called the modal expansion theorem. 00:03:48.040 --> 00:03:51.330 It's basically the assertion that you 00:03:51.330 --> 00:03:56.460 can represent any motion set of vectors-- 00:03:56.460 --> 00:04:03.420 I'll write them kind of as a vector here for a moment-- x, 00:04:03.420 --> 00:04:10.270 as the superposition of each contributing mode. 00:04:10.270 --> 00:04:14.100 Now each mode has a mode shape to it, 00:04:14.100 --> 00:04:19.490 which I'm going to call u, and up here I'll 00:04:19.490 --> 00:04:21.315 put a superscript for what mode it 00:04:21.315 --> 00:04:27.980 is, the first mode, times its time-dependent behavior. 00:04:27.980 --> 00:04:31.010 And this is called, what they call in textbooks, 00:04:31.010 --> 00:04:33.700 the natural coordinates. 00:04:33.700 --> 00:04:35.430 And we'll see what those are in a second. 00:04:35.430 --> 00:04:37.640 So mode shape one. 00:04:37.640 --> 00:04:41.430 This is the time-dependent and amplitude part 00:04:41.430 --> 00:04:45.630 that says how much the contribution of mode one 00:04:45.630 --> 00:04:51.000 is to this motion and what its time dependence is, this is. 00:04:51.000 --> 00:04:57.200 And then you'd have mode two's contribution, q2. 00:05:00.460 --> 00:05:10.000 And this goes out to the nth mode's contribution, qn of t. 00:05:10.000 --> 00:05:12.150 And that's the proposition, that you 00:05:12.150 --> 00:05:15.300 can represent the total response of the system 00:05:15.300 --> 00:05:18.230 as a superposition of the response of each 00:05:18.230 --> 00:05:20.250 of the natural modes of the system. 00:05:20.250 --> 00:05:22.790 And if it's an n degree of freedom system, 00:05:22.790 --> 00:05:24.590 there will be n natural modes, so. 00:05:42.850 --> 00:05:45.710 Now something I didn't say here. 00:05:45.710 --> 00:05:50.250 This all assumes that the system vibrates. 00:05:50.250 --> 00:05:52.820 So this is all in the discussion of things 00:05:52.820 --> 00:05:55.160 that exhibit vibratory motion. 00:05:55.160 --> 00:06:08.243 So this is all, it should say here, of vibrating systems, OK? 00:06:14.470 --> 00:06:17.190 So this kind of a long and cumbersome way 00:06:17.190 --> 00:06:18.830 of writing this out. 00:06:18.830 --> 00:06:24.475 So if you notice, each one of these is the mode shape vector. 00:06:24.475 --> 00:06:31.010 And if I put them together in a matrix just side by side, 00:06:31.010 --> 00:06:41.920 here's a u1 over to un and multiply it by this vector, 00:06:41.920 --> 00:06:45.815 q1 of t down to qn. 00:06:48.750 --> 00:06:50.800 That's the same statement but said 00:06:50.800 --> 00:06:53.670 in a much more compact way. 00:06:53.670 --> 00:06:56.070 So this statement, this modal expansion theorem, 00:06:56.070 --> 00:07:00.230 basically says the vector of-- these 00:07:00.230 --> 00:07:09.720 are your generalized coordinates, which 00:07:09.720 --> 00:07:11.269 we've been using all term long. 00:07:11.269 --> 00:07:12.810 These are the generalized coordinates 00:07:12.810 --> 00:07:15.820 that you choose to derive the equations of motion around. 00:07:15.820 --> 00:07:18.300 The vector of generalized coordinates 00:07:18.300 --> 00:07:20.320 can be written as uq. 00:07:23.990 --> 00:07:32.190 And these are often called the modal coordinates 00:07:32.190 --> 00:07:40.450 or sometimes called the natural coordinates, OK? 00:07:51.690 --> 00:07:58.820 So if we can say that x is uq, then x dot, 00:07:58.820 --> 00:08:02.260 you take the derivative of each one of those expressions. 00:08:02.260 --> 00:08:07.450 You'll find that's going to be uq dot. 00:08:07.450 --> 00:08:12.800 And x double dot equals uq double 00:08:12.800 --> 00:08:15.000 dot because these are just constants. 00:08:15.000 --> 00:08:18.270 The mode shape vectors are just a fixed set of numbers 00:08:18.270 --> 00:08:21.620 that represent the mode shape to the system. 00:08:21.620 --> 00:08:27.132 Now just to-- I think maybe this is a good time to do this. 00:08:27.132 --> 00:08:28.070 You grab one end. 00:08:31.430 --> 00:08:34.960 So this is a-- and it's hard to see black against black. 00:08:34.960 --> 00:08:36.919 My apologies for that. 00:08:36.919 --> 00:08:42.770 So this is a guitar string or any stringed instrument. 00:08:42.770 --> 00:08:48.200 In fact, any long, slender thing under tension will vibrate. 00:08:48.200 --> 00:08:51.620 And it has, if I do this carefully, 00:08:51.620 --> 00:08:56.290 that's called the first mode of vibration. 00:08:56.290 --> 00:08:59.240 And that's when you pluck your guitar string or violin 00:08:59.240 --> 00:09:00.060 in the middle. 00:09:00.060 --> 00:09:01.280 You mostly hear that. 00:09:04.150 --> 00:09:09.300 But at twice the frequency, if I can get it going here, 00:09:09.300 --> 00:09:11.400 there's a second mode of vibration. 00:09:11.400 --> 00:09:13.460 And for a taut string, it happens 00:09:13.460 --> 00:09:15.900 to be at twice the frequency of the first. 00:09:15.900 --> 00:09:19.530 And if my hand is well calibrated-- 00:09:19.530 --> 00:09:22.060 it may be easier if it's a little longer-- if I 00:09:22.060 --> 00:09:32.580 get this going right, there's a third mode, OK? 00:09:32.580 --> 00:09:38.020 So that's what we're calling-- oh, 00:09:38.020 --> 00:09:39.660 what I meant to say when I was doing 00:09:39.660 --> 00:09:44.070 this is these shapes for the vibrating string, 00:09:44.070 --> 00:09:47.933 that second mode shape happens to be one full sine wave. 00:09:47.933 --> 00:09:54.160 And the mode shape has the form sine n pi x over l, 00:09:54.160 --> 00:09:57.290 where l's the length of the string. 00:09:57.290 --> 00:09:58.430 n's the mode number. 00:09:58.430 --> 00:09:59.540 So first mode. 00:09:59.540 --> 00:10:02.060 Second mode is this, when n is 2. 00:10:02.060 --> 00:10:04.530 First mode, n is 1. 00:10:04.530 --> 00:10:08.300 nth mode or something high, you get higher modes like that. 00:10:08.300 --> 00:10:11.780 So these are the mode shapes for a vibrating string. 00:10:14.900 --> 00:10:15.930 That's good for now. 00:10:18.530 --> 00:10:23.114 This two degree of freedom system with the two lump 00:10:23.114 --> 00:10:24.530 masses-- and it's going to show up 00:10:24.530 --> 00:10:28.570 there, yeah-- this is basically two lump masses. 00:10:28.570 --> 00:10:31.960 And we idealize the springs as being massless, 00:10:31.960 --> 00:10:34.610 but it's a pretty good approximation. 00:10:34.610 --> 00:10:36.920 This has two modes of vibration. 00:10:36.920 --> 00:10:46.060 And Professor Gossard made these neat little things that 00:10:46.060 --> 00:10:48.810 can make it so-- and I'm going to come back to this, 00:10:48.810 --> 00:10:52.130 but there's mode one. 00:10:52.130 --> 00:10:54.160 And the mode shape is as this goes down 00:10:54.160 --> 00:10:58.310 one unit, that goes down about two times as much. 00:10:58.310 --> 00:11:00.340 I'll give you the exact numbers in a minute. 00:11:00.340 --> 00:11:10.504 And the other mode shape of the system-- 00:11:10.504 --> 00:11:12.920 we're going to talk about this today and why this happens. 00:11:12.920 --> 00:11:15.010 But if I give it the right initial conditions, 00:11:15.010 --> 00:11:18.360 I can make it vibrate only in the second mode shape. 00:11:18.360 --> 00:11:21.780 And so it's now deflected with the right conditions 00:11:21.780 --> 00:11:23.922 so that it'll respond only in second mode. 00:11:26.520 --> 00:11:28.190 This mass goes up and down a lot. 00:11:28.190 --> 00:11:31.130 That mass goes up in that little, opposite to it, 00:11:31.130 --> 00:11:31.710 actually. 00:11:31.710 --> 00:11:34.420 The frequency is different from the first. 00:11:34.420 --> 00:11:37.640 But if this is moving one unit, then this down here 00:11:37.640 --> 00:11:40.220 is moving minus 0.3 or something like that. 00:11:40.220 --> 00:11:42.740 And that ratio is constant. 00:11:42.740 --> 00:11:45.660 And that's called the mode shape. 00:11:45.660 --> 00:11:49.740 So if you just pick one of them and say, let its motion be one, 00:11:49.740 --> 00:11:52.090 then all of the other masses in the system 00:11:52.090 --> 00:11:56.760 will move in a particular ratio to the motion of that one 00:11:56.760 --> 00:11:58.830 that you arbitrarily set to one. 00:11:58.830 --> 00:12:00.620 So this is what we mean by mode shapes 00:12:00.620 --> 00:12:04.150 and their natural frequencies. 00:12:04.150 --> 00:12:05.900 There's the natural frequency associated 00:12:05.900 --> 00:12:07.030 with that first mode. 00:12:07.030 --> 00:12:09.360 And we can solve these things mathematically, 00:12:09.360 --> 00:12:11.499 and we've been doing that a little bit 00:12:11.499 --> 00:12:12.790 in the last couple of lectures. 00:12:15.500 --> 00:12:16.040 All right. 00:12:18.830 --> 00:12:21.670 So this is the relationship between these things, 00:12:21.670 --> 00:12:25.200 the generalized coordinates and the modal coordinates. 00:12:25.200 --> 00:12:34.140 And we now need to see how we're going to use these. 00:12:40.330 --> 00:12:42.710 So in general, we have our equations of motion. 00:12:57.950 --> 00:13:01.310 And I'm going to substitute for x, x dot, 00:13:01.310 --> 00:13:27.870 and x double dot, these and pre multiply by u transpose. 00:13:27.870 --> 00:13:29.545 Remember the transpose of a matrix. 00:13:29.545 --> 00:13:32.410 You just take the first column, make it the first row. 00:13:32.410 --> 00:13:34.580 Second column, make it the second row. 00:13:34.580 --> 00:13:38.540 So if I plug in these up here, I get 00:13:38.540 --> 00:13:47.213 muq-- I'm going to leave some space here 00:13:47.213 --> 00:13:55.980 because I'm going to pre multiply in a second-- plus cuq 00:13:55.980 --> 00:14:10.740 dot plus kuq equals the external exciting forces. 00:14:10.740 --> 00:14:25.150 Now I'm going to pre multiply by u transpose, OK? 00:14:25.150 --> 00:14:30.300 Now a remarkable thing happens. 00:14:30.300 --> 00:14:35.350 It happens that when you do this calculation, when you multiply 00:14:35.350 --> 00:14:39.640 this matrix times that, one row at a time-- so this 00:14:39.640 --> 00:14:42.430 has vectors in it, 1 through n. 00:14:42.430 --> 00:14:44.860 I'm going to pick vector r, the rth one. 00:14:44.860 --> 00:14:49.160 If I take that rth vector and multiply it one 00:14:49.160 --> 00:14:53.700 at a time by row by row by row, then I 00:14:53.700 --> 00:15:00.350 get a new vector that results, which 00:15:00.350 --> 00:15:03.240 I'm going to multiply by this. 00:15:03.240 --> 00:15:07.190 And so if I'm going to pick out one of the vectors, 00:15:07.190 --> 00:15:13.850 multiply it through times one of the rows 00:15:13.850 --> 00:15:18.290 here-- when you transpose them, the rows are now the vectors. 00:15:18.290 --> 00:15:19.230 So I'm going to pick. 00:15:19.230 --> 00:15:26.640 If I do the calculation-- lost my right piece of paper here. 00:15:41.690 --> 00:15:46.670 So I'm going to just pull out one of the calculations 00:15:46.670 --> 00:15:51.600 that you end up doing if you do this whole triple matrix 00:15:51.600 --> 00:16:00.610 multiplication, you need to know the following fact. 00:16:00.610 --> 00:16:04.280 So for the mode s transpose-- that's one 00:16:04.280 --> 00:16:10.410 of the rows out of here-- times m times 00:16:10.410 --> 00:16:17.340 one of the columns, the rth one from here, 00:16:17.340 --> 00:16:25.165 and I do this calculation, this is 0 for r not equal to s. 00:16:27.670 --> 00:16:32.240 What that statement says is, the only non-zero result from this 00:16:32.240 --> 00:16:36.810 is when you multiply-- when you take the rth column from here 00:16:36.810 --> 00:16:39.380 and you use the rth from here. 00:16:39.380 --> 00:16:42.780 All the other combinations of this thing go to 0. 00:16:42.780 --> 00:16:45.580 And the net result of that is that this 00:16:45.580 --> 00:16:53.820 implies that u transpose mu always 00:16:53.820 --> 00:17:01.080 equals to a diagonal vector, which I'll call this like that. 00:17:01.080 --> 00:17:03.590 Sometimes a mass matrix is diagonal to start with. 00:17:03.590 --> 00:17:06.050 But even if it isn't, you do this calculation, 00:17:06.050 --> 00:17:08.310 it will produce a diagonal matrix. 00:17:08.310 --> 00:17:12.680 And that's because these multiplications are always 00:17:12.680 --> 00:17:17.160 0 unless r is the same as s. 00:17:17.160 --> 00:17:23.619 And the same is true for u transpose Ku will give you 00:17:23.619 --> 00:17:31.090 a K matrix that is diagonal. 00:17:31.090 --> 00:17:32.640 And you know, normally the stiffness 00:17:32.640 --> 00:17:34.930 matrix we've come up with, they've 00:17:34.930 --> 00:17:37.330 generally been full matrices oftentimes. 00:17:37.330 --> 00:17:42.640 But you do uKu, you will get a diagonal stiffness matrix. 00:17:45.600 --> 00:17:48.430 And there the little problem comes 00:17:48.430 --> 00:18:04.510 because u transpose cu, well, sometimes, this one is diagonal 00:18:04.510 --> 00:18:14.110 only for ideal conditions of damping. 00:18:14.110 --> 00:18:16.205 So that's something you just have to address. 00:18:29.110 --> 00:18:31.470 So only for ideal conditions, and that's 00:18:31.470 --> 00:18:33.250 just something you have to deal with. 00:18:40.180 --> 00:18:42.720 So why is this? 00:18:42.720 --> 00:18:44.720 Why is there this special, wonderful thing? 00:18:44.720 --> 00:18:49.050 The natural modes of a system-- this one is 00:18:49.050 --> 00:18:54.510 a two degree of freedom system-- form a complete and independent 00:18:54.510 --> 00:18:58.430 set of vectors. 00:18:58.430 --> 00:19:01.810 And in this case of this two degree of freedom system, 00:19:01.810 --> 00:19:07.440 I can pick any kinematically allowable position, 00:19:07.440 --> 00:19:13.000 like this-- stationary, static is one of the solutions, 00:19:13.000 --> 00:19:15.150 right, to this two degree of freedom system-- 00:19:15.150 --> 00:19:19.720 so any possible allowable position of these two things, 00:19:19.720 --> 00:19:23.270 static or moving, can be described 00:19:23.270 --> 00:19:28.950 by a linear combination of the mode shapes of the system, 00:19:28.950 --> 00:19:32.041 a weighted sum of the mode shapes of the system. 00:19:32.041 --> 00:19:33.040 And that's all it takes. 00:19:33.040 --> 00:19:34.990 So this one has two mode shapes, one 00:19:34.990 --> 00:19:37.050 that looks like that, one that looks 00:19:37.050 --> 00:19:40.490 like this one's going down, this one's going up, 00:19:40.490 --> 00:19:44.990 their particular ratios. 00:19:44.990 --> 00:19:49.000 And I can take a weighted amount of that first mode, so much 00:19:49.000 --> 00:19:51.000 of it, and a weighted amount of the second mode 00:19:51.000 --> 00:19:55.500 and add them together and describe any possible position 00:19:55.500 --> 00:19:56.670 of the system. 00:19:56.670 --> 00:19:59.590 The same thing is true of that string. 00:19:59.590 --> 00:20:01.790 It has mode shapes that are sine waves, 00:20:01.790 --> 00:20:08.100 but they're sine 1 pi x, sine pi 2x, over and so forth. 00:20:08.100 --> 00:20:11.190 Any possible allowable shape of that guitar string 00:20:11.190 --> 00:20:13.980 can be made up of a weighted sum of the mode 00:20:13.980 --> 00:20:15.670 shapes of the system. 00:20:15.670 --> 00:20:19.940 And moreover, the mode shapes, the reason 00:20:19.940 --> 00:20:34.500 this works is because the mode shapes 00:20:34.500 --> 00:20:36.540 are orthogonal to one another. 00:20:45.812 --> 00:20:50.860 Now, you know that if you take 2 sine waves like that string 00:20:50.860 --> 00:20:55.140 and you take first mode sine pi x over l, 00:20:55.140 --> 00:20:57.560 and second mode say sine 2 pi x over l 00:20:57.560 --> 00:20:59.984 and you multiply them together and integrate from 0 00:20:59.984 --> 00:21:00.900 to l, what do you get? 00:21:05.040 --> 00:21:08.950 You'll always get 0 if the two sines are-- 00:21:08.950 --> 00:21:11.530 if they're full wavelengths, they go to nodes at the end, 00:21:11.530 --> 00:21:15.890 you will always get 0 if the wavelengths are different, 00:21:15.890 --> 00:21:18.610 always, right? 00:21:18.610 --> 00:21:20.510 That's a statement of orthogonality 00:21:20.510 --> 00:21:23.500 of sine functions. 00:21:23.500 --> 00:21:24.000 All right. 00:21:24.000 --> 00:21:25.958 The same thing is true of these simple vectors. 00:21:25.958 --> 00:21:28.140 They are orthogonal to one another such 00:21:28.140 --> 00:21:30.940 that if you do this multiplication, 00:21:30.940 --> 00:21:34.930 you transpose mu, you only get contributions 00:21:34.930 --> 00:21:39.475 when you are using mode r transpose m mode r. 00:21:39.475 --> 00:21:41.602 You only get a contribution of each of those. 00:21:41.602 --> 00:21:42.810 That gives you the diagonals. 00:21:42.810 --> 00:21:45.600 The same is true when you do u transpose ku. 00:21:45.600 --> 00:21:48.620 Because of orthogonality, you only get a diagonal matrix 00:21:48.620 --> 00:21:49.610 at the end. 00:21:49.610 --> 00:21:52.450 And under the right conditions, u transpose cu 00:21:52.450 --> 00:21:55.950 gives you a diagonal matrix. 00:21:55.950 --> 00:21:57.370 So what's that good for? 00:22:11.130 --> 00:22:13.470 Well, here was the set of equations 00:22:13.470 --> 00:22:16.240 that we get when we make that substitution. 00:22:16.240 --> 00:22:23.730 This is going to give us a diagonal mass 00:22:23.730 --> 00:22:29.590 matrix times q double dot plus, when conditions are right, 00:22:29.590 --> 00:22:34.518 a diagonal damping matrix times q dot, 00:22:34.518 --> 00:22:41.990 plus a diagonal stiffness matrix times q 00:22:41.990 --> 00:22:49.970 equals u transpose F, which as a vector times a matrix 00:22:49.970 --> 00:22:52.390 gives you back a vector, which we call capital 00:22:52.390 --> 00:22:54.795 Q. It's a function of time. 00:22:54.795 --> 00:22:56.465 And this is called the modal force. 00:23:02.140 --> 00:23:04.240 But if you look carefully at these, 00:23:04.240 --> 00:23:13.150 if I pick the rth one, mode r out of this whole thing-- 00:23:13.150 --> 00:23:16.540 if I just pick any mode out of this, any part of this vector, 00:23:16.540 --> 00:23:18.100 and complete this multiplication, 00:23:18.100 --> 00:23:22.750 I will find that I get an Mr, which is the rth entry here. 00:23:22.750 --> 00:23:26.125 And now I'm going to refer to these as the modal masses, 00:23:26.125 --> 00:23:28.080 and I'll write them with capitals 00:23:28.080 --> 00:23:30.690 and I'll give a subscript to tell you what the mode is. 00:23:30.690 --> 00:23:32.250 This is a number. 00:23:32.250 --> 00:23:35.140 This is the modal mass for mode r. 00:23:35.140 --> 00:23:37.070 This gives me an equation that looks 00:23:37.070 --> 00:23:50.161 like Mrqr double dot plus crqr dot plus Krqr equals Qr of t. 00:23:50.161 --> 00:23:55.410 And what does that remind you of that we've done a lot of work 00:23:55.410 --> 00:23:56.517 with? 00:23:56.517 --> 00:23:58.271 AUDIENCE: [INAUDIBLE] 00:23:58.271 --> 00:24:00.145 PROFESSOR: How many degree of freedom system? 00:24:03.490 --> 00:24:07.150 That's the equation of motion, the generic equation of motion, 00:24:07.150 --> 00:24:09.130 of a single degree of freedom oscillator. 00:24:09.130 --> 00:24:11.450 And you know how to calculate the response 00:24:11.450 --> 00:24:13.160 to initial conditions for that. 00:24:13.160 --> 00:24:14.920 You know how to calculate the steady state 00:24:14.920 --> 00:24:18.600 response for that when you have a harmonic input, right? 00:24:18.600 --> 00:24:23.360 What I said at the beginning of the discussion about vibration 00:24:23.360 --> 00:24:25.540 is it's really important to understand 00:24:25.540 --> 00:24:27.410 the single degree of freedom oscillator 00:24:27.410 --> 00:24:31.290 because it'll give you insight as to the behavior 00:24:31.290 --> 00:24:33.920 of complicated multiple degree of freedom systems. 00:24:33.920 --> 00:24:36.723 And here's the proof of this. 00:24:40.347 --> 00:24:47.430 This is now n uncoupled single degree of freedom systems. 00:24:47.430 --> 00:25:01.000 This is n independent single, one degree of freedom systems 00:25:01.000 --> 00:25:02.600 which you can solve one at a time. 00:25:19.100 --> 00:25:25.610 Now, lots of times a vibrating system, a complicated one, 00:25:25.610 --> 00:25:28.440 might be this thing. 00:25:28.440 --> 00:25:30.110 If I hit this, it's vibrating. 00:25:30.110 --> 00:25:31.930 And actually, it's pretty much vibrating 00:25:31.930 --> 00:25:32.846 at a single frequency. 00:25:36.260 --> 00:25:41.320 And once I've hit it, are there any external forces driving it? 00:25:41.320 --> 00:25:43.060 So what kind of response are you seeing? 00:25:45.950 --> 00:25:47.032 Response to? 00:25:47.032 --> 00:25:48.240 AUDIENCE: Initial conditions. 00:25:48.240 --> 00:25:50.230 PROFESSOR: Initial conditions, right? 00:25:50.230 --> 00:25:54.720 Now in general, each one of the natural modes of a system 00:25:54.720 --> 00:25:58.730 has a different natural frequency, right? 00:25:58.730 --> 00:26:00.979 So if I hit this thing and I look at it, 00:26:00.979 --> 00:26:03.020 really, I can just see it wiggling back and forth 00:26:03.020 --> 00:26:06.140 basically at one frequency So if you 00:26:06.140 --> 00:26:09.580 wanted to come up with a simple model of this system, 00:26:09.580 --> 00:26:11.630 how many natural modes you think you'd 00:26:11.630 --> 00:26:15.320 have to include to describe the motion of this system? 00:26:15.320 --> 00:26:16.220 AUDIENCE: One. 00:26:16.220 --> 00:26:17.150 PROFESSOR: One. 00:26:17.150 --> 00:26:19.570 Now is that a lot easier than having 00:26:19.570 --> 00:26:22.120 to do the full general equation of motion 00:26:22.120 --> 00:26:25.780 for all the possible modes that this thing has? 00:26:25.780 --> 00:26:27.654 And it turns out a lot, right, you 00:26:27.654 --> 00:26:29.320 have to deal with the equation of motion 00:26:29.320 --> 00:26:31.890 of a single degree of freedom system to describe this. 00:26:31.890 --> 00:26:34.050 And that's the real point. 00:26:34.050 --> 00:26:36.440 You know they built the Hancock building 00:26:36.440 --> 00:26:40.350 across the river 35 years ago. 00:26:40.350 --> 00:26:42.510 It was losing windows like crazy. 00:26:42.510 --> 00:26:44.250 It was a brand new building. 00:26:44.250 --> 00:26:47.800 And when the wind would get above 40 miles an hour, 00:26:47.800 --> 00:26:49.180 the windows started falling out. 00:26:49.180 --> 00:26:53.900 60 stories high, 60, 61 stories high, and the wind was blowing. 00:26:53.900 --> 00:26:56.440 Where do you suppose the windows fall out? 00:26:56.440 --> 00:26:58.072 What part of the building? 00:26:58.072 --> 00:26:59.425 AUDIENCE: [INAUDIBLE] 00:26:59.425 --> 00:27:00.359 PROFESSOR: Huh? 00:27:00.359 --> 00:27:02.400 I mean, you'd think that when the wind is blowing 00:27:02.400 --> 00:27:03.650 it get stronger as it goes up. 00:27:03.650 --> 00:27:06.860 It was probably blowing out the windows at the top, right? 00:27:06.860 --> 00:27:10.010 But the windows were breaking-- as time went by, 00:27:10.010 --> 00:27:13.530 every time a window would break, they replaced this five foot 00:27:13.530 --> 00:27:16.580 by nine foot sheet of glass with a piece of plywood. 00:27:16.580 --> 00:27:18.250 And so you get this statistical sampling 00:27:18.250 --> 00:27:20.190 after a while of where the breakage was. 00:27:20.190 --> 00:27:24.060 So you had no windows broken at the top and a few 00:27:24.060 --> 00:27:27.380 as you got further down and more and lots of them broken out 00:27:27.380 --> 00:27:29.860 at the bottom. 00:27:29.860 --> 00:27:34.750 It turns out that that building was vibrating mostly 00:27:34.750 --> 00:27:35.925 in its first bending mode. 00:27:35.925 --> 00:27:39.315 It was going back and forth like this. 00:27:39.315 --> 00:27:40.940 Also happened to have a torsional mode. 00:27:40.940 --> 00:27:43.120 Its first torsional mode was kind of twisting around 00:27:43.120 --> 00:27:44.400 the base like that. 00:27:44.400 --> 00:27:46.415 So in fact the moment when the wind 00:27:46.415 --> 00:27:47.790 would get about 40 miles an hour, 00:27:47.790 --> 00:27:50.270 this building would start rocking and rolling, 00:27:50.270 --> 00:27:54.590 mostly like this with a little of this thrown in, OK? 00:27:54.590 --> 00:27:57.420 But you can basically model that complicated building that 00:27:57.420 --> 00:28:01.200 has millions of possible natural modes in it by one 00:28:01.200 --> 00:28:05.520 or possibly two single degree of freedom oscillators. 00:28:05.520 --> 00:28:07.240 So that's the power of modal analysis. 00:28:07.240 --> 00:28:09.580 But I think the real power of understanding 00:28:09.580 --> 00:28:12.080 that you can do this is that it gives you 00:28:12.080 --> 00:28:13.779 this immediate insight as to what 00:28:13.779 --> 00:28:15.070 might be going on in something. 00:28:15.070 --> 00:28:16.090 So I look at this. 00:28:16.090 --> 00:28:17.860 I don't see a complicated thing that I 00:28:17.860 --> 00:28:19.900 have to model with a big finite element model. 00:28:19.900 --> 00:28:22.570 I see something that's vibrating at one frequency. 00:28:22.570 --> 00:28:26.260 And I know it has a little decay. 00:28:26.260 --> 00:28:27.390 It'll have damping. 00:28:27.390 --> 00:28:29.480 It'll have a natural frequency. 00:28:29.480 --> 00:28:31.970 And I get immediate insight as to its behavior 00:28:31.970 --> 00:28:34.238 by knowing this, OK. 00:28:34.238 --> 00:28:38.060 And that's the real reason why I wanted to show this to you. 00:28:38.060 --> 00:28:46.595 So today we'll do-- there are sort of two directions we 00:28:46.595 --> 00:28:47.640 can go with this. 00:28:47.640 --> 00:28:51.340 One is to talk about response to initial conditions, 00:28:51.340 --> 00:28:53.190 and the other is to talk about the response 00:28:53.190 --> 00:28:54.700 to force excitation. 00:28:54.700 --> 00:29:08.810 So we're going to begin by doing response to ICs, OK? 00:29:08.810 --> 00:29:11.350 And then we'll move on probably next time 00:29:11.350 --> 00:29:13.960 and talk about response to harmonic excitations. 00:29:17.420 --> 00:29:21.310 And we're going to use that as the example. 00:29:21.310 --> 00:29:25.920 Before I go there, once we have broken the system down 00:29:25.920 --> 00:29:29.637 and analyzed this way, how do we get back 00:29:29.637 --> 00:29:34.811 to the motion of the system in our generalized coordinates, 00:29:34.811 --> 00:29:36.560 which are the ones we're comfortable with? 00:29:36.560 --> 00:29:39.930 Because I don't know where to take a ruler and go 00:29:39.930 --> 00:29:45.090 measure this natural coordinate. 00:29:45.090 --> 00:29:46.850 So somehow I have to get back to putting 00:29:46.850 --> 00:29:49.570 in the real physical measurements 00:29:49.570 --> 00:29:50.590 that I can relate to. 00:29:50.590 --> 00:29:53.620 Well, that's easy because where did we start with this? 00:29:53.620 --> 00:29:56.960 We started by saying this whole thing began right here. 00:30:03.760 --> 00:30:09.180 And so at the end we just come back and say, oh, well, 00:30:09.180 --> 00:30:12.420 x here, our generalized coordinates, 00:30:12.420 --> 00:30:24.180 is this summation of the mode shapes ui here, summed 00:30:24.180 --> 00:30:30.880 over i of qi of t. 00:30:30.880 --> 00:30:35.750 Now the reason I wrote it here as a summation 00:30:35.750 --> 00:30:37.735 is to remind you that you do this. 00:30:50.272 --> 00:30:51.730 That's the beauty of this thing, is 00:30:51.730 --> 00:30:54.570 you only have to do it over the modes that matter. 00:30:54.570 --> 00:30:56.830 So if you've decided to approximate the motion 00:30:56.830 --> 00:31:00.020 of this complicated system, by just a couple 00:31:00.020 --> 00:31:01.550 of motile contributions because you 00:31:01.550 --> 00:31:04.210 know they're the important ones, this 00:31:04.210 --> 00:31:06.450 is a pretty short summation. 00:31:06.450 --> 00:31:09.770 This is how you get back to your original modal coordinates. 00:31:09.770 --> 00:31:15.630 Just take the modal amplitude, multiply it by the mode shape. 00:31:15.630 --> 00:31:18.360 And when you do that, it says, if this turns out 00:31:18.360 --> 00:31:23.130 to be, say, sum a sine omega t, when you multiply by the mode 00:31:23.130 --> 00:31:25.440 shape it basically tells you how much 00:31:25.440 --> 00:31:29.150 each generalized coordinate gets of the motion. 00:31:29.150 --> 00:31:31.570 The mode shape distributes thing answer 00:31:31.570 --> 00:31:34.360 out proportionally in the correct amount. 00:31:34.360 --> 00:31:37.270 So this is how you get back to the original. 00:31:37.270 --> 00:31:43.710 So let's think about that system and we'll 00:31:43.710 --> 00:31:46.190 do an initial conditions kind of problem. 00:31:46.190 --> 00:31:48.120 So I think Professor Gossard-- I think 00:31:48.120 --> 00:31:51.070 in class you sort of figured out what the approximate ks 00:31:51.070 --> 00:31:53.500 and ms and things were for that system. 00:31:53.500 --> 00:31:56.830 So I actually took it apart, weighed it, 00:31:56.830 --> 00:31:58.470 measured some natural frequencies, 00:31:58.470 --> 00:32:06.120 and have come up with a pretty good model, 00:32:06.120 --> 00:32:08.800 or at least pretty good set of numbers, 00:32:08.800 --> 00:32:11.690 characterizing this two degree of freedom system. 00:32:22.250 --> 00:32:36.210 So c1, k1, m1, k2, c2, x1, x2. 00:32:36.210 --> 00:32:39.094 So these are my generalized coordinates, measured probably 00:32:39.094 --> 00:32:39.885 from what position? 00:32:45.140 --> 00:32:46.690 Static equilibrium, right? 00:32:46.690 --> 00:32:49.980 So I don't have to mess with gravity in this. 00:32:49.980 --> 00:32:51.960 Measured from static equilibrium. 00:32:51.960 --> 00:32:58.780 And to try to help keep things understandable, 00:32:58.780 --> 00:33:01.840 I tried to write the parameters of the system 00:33:01.840 --> 00:33:04.690 as lowercase k1s, k2s, k3s because I 00:33:04.690 --> 00:33:09.490 want to write modal stiffness for mode one as a capital K1, 00:33:09.490 --> 00:33:12.200 so I try to be consistent about that. 00:33:12.200 --> 00:33:16.000 And notice where I put the dampers in the system. 00:33:16.000 --> 00:33:18.010 That's because most of the damping in this thing 00:33:18.010 --> 00:33:23.400 comes from the upper mass rubs against a stationary object, 00:33:23.400 --> 00:33:25.930 which is the bar here. 00:33:25.930 --> 00:33:28.610 The lower mass rubs against a stationary object. 00:33:28.610 --> 00:33:32.010 So I'm going to model that as a dashpot between each mass 00:33:32.010 --> 00:33:36.230 and the fixed reference frame because the bar doesn't move. 00:33:36.230 --> 00:33:38.680 So it's an approximate model of the damping. 00:33:38.680 --> 00:33:43.180 And so if we do our sum of forces on each of these masses, 00:33:43.180 --> 00:33:45.440 just do Newton's laws on the mass, 00:33:45.440 --> 00:33:48.530 we can come up with our two equations of motion. 00:33:48.530 --> 00:33:50.140 We get two equations of motion. 00:33:54.550 --> 00:33:55.740 And let's see. 00:33:59.010 --> 00:34:01.746 I think I'll give you some information here first. 00:34:01.746 --> 00:34:02.246 m1. 00:34:43.420 --> 00:34:45.150 And I really don't know the damping, 00:34:45.150 --> 00:34:49.000 but we can get that by just counting how many cycles it 00:34:49.000 --> 00:34:51.110 takes to decay and so forth. 00:34:51.110 --> 00:34:53.850 So that's basically what I come into this problem knowing. 00:34:57.580 --> 00:35:00.210 And I'm going to write my equations of motion 00:35:00.210 --> 00:35:02.300 in matrix form. 00:35:02.300 --> 00:35:04.560 So it's going to end up looking like m1. 00:35:18.790 --> 00:35:22.320 Now notice the damping in this one, the damping force, 00:35:22.320 --> 00:35:26.190 is only proportion-- it'll be c1 x1. 00:35:26.190 --> 00:35:28.640 Doesn't involve the motion of the other object. 00:35:28.640 --> 00:35:30.270 In this one, the damping force only 00:35:30.270 --> 00:35:31.990 involves the second motion. 00:35:31.990 --> 00:35:37.680 So this one happens to look like a c1, 0, c2. 00:35:45.950 --> 00:35:55.170 And the stiffness matrix, well, that's k1 plus k2, minus k2, 00:35:55.170 --> 00:36:05.520 minus k2, and k2, x1, x2. 00:36:05.520 --> 00:36:11.586 And for no external forces, this starts off this one 00:36:11.586 --> 00:36:14.380 has nothing on the right hand side. it's equal to 0. 00:36:14.380 --> 00:36:16.130 So those are my equations of motion. 00:36:16.130 --> 00:36:17.730 And you know if you multiply these out 00:36:17.730 --> 00:36:19.620 you'd get two equations. 00:36:19.620 --> 00:36:22.030 And each one would be this result 00:36:22.030 --> 00:36:25.380 that you get by apply Newton's law to mass one 00:36:25.380 --> 00:36:27.239 and Newton's law to mass two. 00:36:27.239 --> 00:36:28.780 But you we've done that enough times. 00:36:28.780 --> 00:36:31.070 I'm not going to go through that part of it. 00:36:34.014 --> 00:36:34.514 OK. 00:37:03.580 --> 00:37:17.115 And putting it in real numbers, that's our mass matrix. 00:37:20.670 --> 00:37:22.760 I don't know this. 00:37:22.760 --> 00:37:24.180 My stiffness matrix. 00:37:47.080 --> 00:37:50.160 So this is my K matrix here. 00:37:50.160 --> 00:37:55.635 And stiffness matrices, they're always symmetric. 00:37:58.190 --> 00:38:00.130 Although this one happened to be diagonal, 00:38:00.130 --> 00:38:02.660 you'll find that mass matrices and even the damping matrices 00:38:02.660 --> 00:38:05.430 for our linear systems are symmetric. 00:38:05.430 --> 00:38:06.960 So here's my stiffness matrix. 00:38:06.960 --> 00:38:09.770 Here's my mass matrix, OK? 00:38:12.280 --> 00:38:14.690 And also in this case here's my damping matrix, 00:38:14.690 --> 00:38:16.981 but I'm going to leave that because it's the one that's 00:38:16.981 --> 00:38:19.420 a little bit troublesome. 00:38:19.420 --> 00:38:24.810 So what do I need to do to this to carry out my modal analysis? 00:38:24.810 --> 00:38:31.250 So I need to go find the results of computing u transpose m 00:38:31.250 --> 00:38:34.912 and u and transpose Ku. 00:38:34.912 --> 00:38:36.130 And let's see what we get. 00:38:43.390 --> 00:38:45.630 So we need to know a couple things about this system. 00:38:50.460 --> 00:38:52.920 We need to know natural frequencies and mode shapes. 00:38:52.920 --> 00:38:55.210 So if we have this mass matrix and we 00:38:55.210 --> 00:39:00.710 have the stiffness matrix, then we know we can cast this. 00:39:00.710 --> 00:39:09.555 We want the undamped natural frequencies and our mode 00:39:09.555 --> 00:39:10.055 shapes. 00:39:28.510 --> 00:39:32.440 And we know that we can transform 00:39:32.440 --> 00:39:35.700 the equations of motion into an algebraic problem where 00:39:35.700 --> 00:39:38.390 we solve for the natural frequencies and mode shapes. 00:39:38.390 --> 00:39:43.340 So we have, just to remind you really quickly of that, 00:39:43.340 --> 00:39:46.550 remember our equations look like this undamped. 00:39:49.670 --> 00:39:58.240 And you assume that x is some form u in fact e to the i omega 00:39:58.240 --> 00:39:59.990 t. 00:39:59.990 --> 00:40:12.210 Plug it in, you get minus omega squared m plus K u e 00:40:12.210 --> 00:40:15.155 to the i omega t equals 0. 00:40:17.740 --> 00:40:20.160 And this now is your algebraic problem. 00:40:20.160 --> 00:40:24.140 e to this unknown set of amplitudes is 0. 00:40:24.140 --> 00:40:26.850 These are going to turn out to be the mode shapes. 00:40:26.850 --> 00:40:30.660 And they're not generally 0 so that means this has to be 0. 00:40:30.660 --> 00:40:33.460 That means we know the determinant of this matrix. 00:40:41.740 --> 00:40:44.670 And that'll give you in this case the two 00:40:44.670 --> 00:40:45.680 natural frequencies. 00:40:45.680 --> 00:40:48.490 This gives of you the omega ns of the system. 00:40:48.490 --> 00:40:51.190 Omega n squareds is what you solve for, OK? 00:40:51.190 --> 00:40:57.830 And then you go back and you get the mode shapes out of it. 00:40:57.830 --> 00:40:59.640 But this you can do on the computer too. 00:40:59.640 --> 00:41:03.190 You can either crank out-- for a two degree of freedom system, 00:41:03.190 --> 00:41:06.140 this gives you a quadratic omega squared. 00:41:06.140 --> 00:41:07.507 You solve it. 00:41:07.507 --> 00:41:09.340 You plug it back in and get the mode shapes. 00:41:09.340 --> 00:41:10.839 I'm not going to take the time to do 00:41:10.839 --> 00:41:13.820 that today because I want to emphasize the modal analysis 00:41:13.820 --> 00:41:14.527 part. 00:41:14.527 --> 00:41:15.735 So I'll give you the answers. 00:41:22.930 --> 00:41:23.910 Where are we here? 00:41:32.240 --> 00:41:38.250 So you get omega 1 is 5.6546. 00:41:38.250 --> 00:41:42.080 And I seem to be keeping a lot of significant digits, 00:41:42.080 --> 00:41:43.330 and there's a reason for that. 00:41:53.790 --> 00:41:56.040 In both mode shapes and natural frequencies 00:41:56.040 --> 00:41:58.020 you need to carry a lot of significant digits 00:41:58.020 --> 00:42:01.070 or modal analysis doesn't work, or at least you 00:42:01.070 --> 00:42:02.710 don't get the clean results you expect. 00:42:02.710 --> 00:42:06.150 If you're sloppy about the number of significant digits 00:42:06.150 --> 00:42:09.440 and you compute u transpose mu, then the 00:42:09.440 --> 00:42:12.750 [? off ?] diagonal terms won't quite go to 0. 00:42:12.750 --> 00:42:17.040 And it's just because you're not carrying enough precision. 00:42:17.040 --> 00:42:22.040 OK, now that's the two natural frequencies. 00:42:22.040 --> 00:42:25.320 Now the u matrix, the mode shapes for this system 00:42:25.320 --> 00:42:27.270 that goes with that. 00:42:27.270 --> 00:42:40.089 u comes out to be 1.0 and 2.2667. 00:42:40.089 --> 00:42:40.755 And that's mode. 00:42:40.755 --> 00:42:43.230 I'll do this to help you. 00:42:43.230 --> 00:42:46.040 The columns are the mode shapes. 00:42:46.040 --> 00:42:47.925 That's the first mode shape. 00:42:47.925 --> 00:43:01.100 And the second mode shape is 1 and minus 0.2236. 00:43:01.100 --> 00:43:04.010 So those are the mode shapes for the first and second mode 00:43:04.010 --> 00:43:06.740 that go with these two natural frequencies. 00:43:06.740 --> 00:43:08.310 So that's for this system. 00:43:08.310 --> 00:43:09.810 The top one moves one unit. 00:43:09.810 --> 00:43:15.380 The bottom one moves 2.27 times that, same direction, 00:43:15.380 --> 00:43:16.750 positive, positive. 00:43:16.750 --> 00:43:18.396 So the upper one moves one unit. 00:43:18.396 --> 00:43:20.270 The bottom one moves the opposite direction-- 00:43:20.270 --> 00:43:22.186 that's the minus signs-- equivalent to a phase 00:43:22.186 --> 00:43:26.740 angle of 180 degrees minus 22% of the amount 00:43:26.740 --> 00:43:28.570 that the upper one moves. 00:43:28.570 --> 00:43:31.760 So first one moves one unit. 00:43:31.760 --> 00:43:35.820 The bottom one moves 2.2 times that. 00:43:35.820 --> 00:43:39.100 And then the second mode, which is much harder to get going. 00:43:42.765 --> 00:43:44.140 Guess the only way I can do it is 00:43:44.140 --> 00:43:46.470 to do it the way Professor Gossard intended here. 00:43:51.910 --> 00:43:58.510 One unit up and down, minus 0.2236, going the opposite way. 00:43:58.510 --> 00:43:59.890 So those are our mode shapes. 00:43:59.890 --> 00:44:01.570 These are the natural frequencies. 00:44:01.570 --> 00:44:04.240 I calculated this one and measured it with a stopwatch. 00:44:04.240 --> 00:44:06.650 This one I can do watching it with a stopwatch. 00:44:06.650 --> 00:44:12.856 And I came within better than 1% of getting the same number. 00:44:12.856 --> 00:44:13.355 OK. 00:44:36.360 --> 00:44:38.510 So I want my model mass matrix. 00:44:38.510 --> 00:44:40.680 I carry out this calculation. 00:44:40.680 --> 00:44:42.680 And for this system, remember, it's 00:44:42.680 --> 00:44:48.540 going to give me back a diagonal matrix looking like this. 00:44:48.540 --> 00:45:05.120 And in fact, the numbers are 3.5562, 0, 0, and 0.3508. 00:45:05.120 --> 00:45:10.430 And when I calculate u transpose Ku, 00:45:10.430 --> 00:45:12.690 gives me a diagonal stiffness matrix. 00:45:16.940 --> 00:45:38.670 And I get the numbers 113.71 and 0, 0, 109.839. 00:45:38.670 --> 00:45:43.480 And that's my diagonalized stiffness matrix. 00:45:43.480 --> 00:45:46.090 Now something had better be true. 00:45:46.090 --> 00:45:47.820 I'm saying that this is now going 00:45:47.820 --> 00:45:50.230 to give me my two independent single degree of freedom 00:45:50.230 --> 00:45:51.640 equations of motion, right? 00:46:03.219 --> 00:46:07.380 So what I'm seeking here, I want to get two equations, 00:46:07.380 --> 00:46:17.550 one that looks like m1q1 double dot plus c1q1 dot plus K1q1 00:46:17.550 --> 00:46:20.137 equals 0 for no external force. 00:46:20.137 --> 00:46:21.720 That's one of the equations I'm after. 00:46:21.720 --> 00:46:30.496 And the other one will look like m2q2 double dot plus c2q2 dot 00:46:30.496 --> 00:46:30.995 K2q2. 00:46:33.930 --> 00:46:36.730 Now one way to check that you've gotten the right thing 00:46:36.730 --> 00:46:39.700 is now these are two independent single degree of freedom 00:46:39.700 --> 00:46:40.340 systems. 00:46:40.340 --> 00:46:42.173 What's the natural frequency of this system? 00:46:52.750 --> 00:46:53.250 Yeah? 00:46:53.250 --> 00:46:55.870 Actually, I heard somebody say square root of K1 over m1. 00:46:55.870 --> 00:46:56.870 That had better be true. 00:46:56.870 --> 00:46:58.650 But numerically what's the number? 00:46:58.650 --> 00:47:02.010 What had it better be? 00:47:02.010 --> 00:47:04.510 It better be the omega 1 of the system, right? 00:47:04.510 --> 00:47:07.230 And so a check that you can perform 00:47:07.230 --> 00:47:18.560 is to check to see if the omega 1 squared equals K1/m1. 00:47:18.560 --> 00:47:20.590 You found two numbers. 00:47:20.590 --> 00:47:27.950 You've got, up here, K1 is 113.7. 00:47:27.950 --> 00:47:29.990 m1 is 3.55. 00:47:29.990 --> 00:47:33.850 Take K1/m1, and take its square root. 00:47:33.850 --> 00:47:39.530 So K1/m1, that's about 30 something. 00:47:39.530 --> 00:47:43.946 Square root of 30 something is a little less than 6. 00:47:43.946 --> 00:47:46.660 Omega 1 is 5.65. 00:47:46.660 --> 00:47:49.980 And same thing, omega 2 had better 00:47:49.980 --> 00:47:54.130 be equal to the square root of K2/m2. 00:47:54.130 --> 00:47:55.640 So one of the things you can always 00:47:55.640 --> 00:47:57.100 do when you do your modal analysis, 00:47:57.100 --> 00:48:01.480 you do your calculations, u transpose mu, u transpose Ku. 00:48:01.480 --> 00:48:04.326 If you calculate the ratios of each one of these things, 00:48:04.326 --> 00:48:05.742 you can go back and check that you 00:48:05.742 --> 00:48:07.408 can see that the natural frequencies are 00:48:07.408 --> 00:48:09.020 the ones that you started with. 00:48:09.020 --> 00:48:11.730 If they are not, then you've messed up in your arithmetic. 00:48:11.730 --> 00:48:14.550 So now we've got our two independent equations. 00:48:14.550 --> 00:48:18.310 And the natural frequencies check out. 00:48:18.310 --> 00:48:21.880 But we still have a couple of things to deal with. 00:48:21.880 --> 00:48:24.030 We have to figure out how to calculate 00:48:24.030 --> 00:48:26.469 the initial conditions, and we have to figure out 00:48:26.469 --> 00:48:27.510 how to deal with damping. 00:48:30.980 --> 00:48:34.280 Let's do ICs first. 00:48:53.690 --> 00:48:56.020 So those of you who were here last time, 00:48:56.020 --> 00:48:59.930 I ended kind of right at the end. 00:48:59.930 --> 00:49:02.550 We kind of worked our way through figuring out 00:49:02.550 --> 00:49:06.214 the initial conditions for a two degree of freedom system 00:49:06.214 --> 00:49:07.130 doing it the hard way. 00:49:07.130 --> 00:49:09.840 You end up with four equations and four unknowns 00:49:09.840 --> 00:49:12.450 for the a1, a2, phi 1, phi two. 00:49:12.450 --> 00:49:13.070 Remember that? 00:49:13.070 --> 00:49:16.150 I mean, it's really painful. 00:49:16.150 --> 00:49:18.510 This is incredibly easier. 00:49:18.510 --> 00:49:23.160 We're going to do the same thing, but extremely easily. 00:49:23.160 --> 00:49:27.000 So I would never go myself given the choice of grinding out 00:49:27.000 --> 00:49:29.850 all those phase angles and amplitudes 00:49:29.850 --> 00:49:31.260 in simultaneous equations. 00:49:31.260 --> 00:49:33.740 I'd do the following. 00:49:33.740 --> 00:49:35.960 Generally now I know the initial conditions are 00:49:35.960 --> 00:49:38.790 going to be specified not in q coordinates 00:49:38.790 --> 00:49:40.050 but in what coordinate system? 00:49:44.310 --> 00:49:47.200 In your original generalized coordinates, right? 00:49:47.200 --> 00:49:49.687 You know, your x, this one. 00:49:49.687 --> 00:49:51.520 If I'm going to set initial conditions here, 00:49:51.520 --> 00:49:54.640 I'm not going to say q1 is equal to something. 00:49:54.640 --> 00:49:57.420 I'm going to put this one down one unit and this one 00:49:57.420 --> 00:49:59.110 down two units and let go. 00:49:59.110 --> 00:50:01.070 This is in x1 and x2 coordinates. 00:50:01.070 --> 00:50:07.085 But the beautiful thing here is that we know that x equals uq. 00:50:10.130 --> 00:50:13.340 So if I know the initial conditions on I'll 00:50:13.340 --> 00:50:18.840 call it x0 here, if I know the initial deflections 00:50:18.840 --> 00:50:21.540 of the system, they're going to be u 00:50:21.540 --> 00:50:27.190 times the initial values of q. 00:50:27.190 --> 00:50:33.070 And if I know a vector of initial velocities at time 0, 00:50:33.070 --> 00:50:35.780 they're going to be uq0 dot. 00:50:39.570 --> 00:50:44.070 So if I told you values of x0 and you 00:50:44.070 --> 00:50:48.050 know that this equation's true, what we need is the q0s. 00:50:48.050 --> 00:50:50.730 We need the initial conditions in the modal 00:50:50.730 --> 00:50:53.249 coordinates in order to finish this problem. 00:50:53.249 --> 00:50:55.290 If I told you this, how would you solve for that? 00:50:59.510 --> 00:51:01.160 Just a little linear algebra here. 00:51:04.520 --> 00:51:05.960 AUDIENCE: Inverse matrix of u? 00:51:05.960 --> 00:51:07.400 PROFESSOR: Yeah, do what with it? 00:51:07.400 --> 00:51:08.900 AUDIENCE: Then you multiply x by it. 00:51:08.900 --> 00:51:13.290 PROFESSOR: Multiply it by u inverse, right? 00:51:13.290 --> 00:51:24.440 OK, so this implies that q0-- well, 00:51:24.440 --> 00:51:26.480 I'll write it out a little more fully here. 00:51:26.480 --> 00:51:37.590 So if I do u inverse x0, that's going 00:51:37.590 --> 00:51:40.885 to be equal to u inverse uq0. 00:51:45.210 --> 00:51:48.390 u inverse times u gives you? 00:51:48.390 --> 00:51:50.010 1, basically, right? 00:51:50.010 --> 00:51:58.337 And so if I do u inverse x0, I get q0. 00:51:58.337 --> 00:51:59.420 That's all there is to it. 00:51:59.420 --> 00:51:59.920 Yeah? 00:51:59.920 --> 00:52:02.510 AUDIENCE: [INAUDIBLE] initial conditions, what about c? 00:52:02.510 --> 00:52:04.820 PROFESSOR: All right, c's a problem, OK, 00:52:04.820 --> 00:52:07.050 and I'm leaving it to the end. 00:52:07.050 --> 00:52:09.530 We're going to deal with it as the last step. 00:52:09.530 --> 00:52:15.690 And if I have initial velocities u inverse times x 00:52:15.690 --> 00:52:20.440 initial velocities vector, I get the initial velocities vector 00:52:20.440 --> 00:52:22.700 in the natural coordinates. 00:52:22.700 --> 00:52:25.600 So that's how simple it is to get the initial conditions 00:52:25.600 --> 00:52:26.650 in modal coordinates. 00:52:26.650 --> 00:52:29.030 Boom, OK? 00:52:34.280 --> 00:52:36.830 And we'll do a numerical example in a second here. 00:52:43.290 --> 00:52:51.089 We're seeking a solution of the form to do response 00:52:51.089 --> 00:52:52.005 to initial conditions. 00:52:55.290 --> 00:53:05.094 We seek equations that we know are 00:53:05.094 --> 00:53:07.260 right for a single degree of freedom system response 00:53:07.260 --> 00:53:08.380 to initial conditions. 00:53:08.380 --> 00:53:16.430 So we know that for a single degree of freedom system, x 00:53:16.430 --> 00:53:33.940 of t-- this is for SDOF system here-- 00:53:33.940 --> 00:53:38.660 we worked out before is equal to some e to the minus zeta 00:53:38.660 --> 00:53:40.570 omega nt. 00:53:40.570 --> 00:53:50.640 This is just a transient decay problem of x0 cosine omega dt 00:53:50.640 --> 00:54:07.860 plus v0 plus zeta omega n x0 all over omega d sine omega dt. 00:54:07.860 --> 00:54:10.680 We know that that's what the response of a single degree 00:54:10.680 --> 00:54:16.000 of freedom system looks like to initial conditions x0 and v0. 00:54:16.000 --> 00:54:18.980 And for light damping, for small damping, 00:54:18.980 --> 00:54:21.280 you can usually even ignore this term. 00:54:21.280 --> 00:54:24.300 So it's just even simpler. 00:54:24.300 --> 00:54:28.700 This term is small compared to that, all right? 00:54:28.700 --> 00:54:33.040 This term, contribution from x0, is small compared to this term. 00:54:33.040 --> 00:54:36.340 So it's basically dominated by an x0 cosine and a v0 00:54:36.340 --> 00:54:38.070 over omega d sine. 00:54:38.070 --> 00:54:40.770 But we know that's the exact response 00:54:40.770 --> 00:54:43.900 for a single degree of freedom system to initial conditions. 00:54:43.900 --> 00:54:46.120 So just by analogy to that, we're 00:54:46.120 --> 00:54:49.890 looking for mode one in modal coordinates. 00:54:49.890 --> 00:54:52.360 It's going to look exactly the same way. 00:54:52.360 --> 00:55:03.120 e to the minus zeta omega nt, q0 cosine omega 1 00:55:03.120 --> 00:55:17.960 d-- this is omega dt-- plus q0 dot plus zeta 1 omega 1 q0. 00:55:17.960 --> 00:55:24.080 I guess I need to do q10 like that. 00:55:24.080 --> 00:55:27.710 This is the first mode's equation, zeta 1. 00:55:27.710 --> 00:55:29.507 And I'll call this omega 1. 00:55:29.507 --> 00:55:31.840 But now that you get multiple degree of freedom systems, 00:55:31.840 --> 00:55:34.173 you got to keep track of what mode you're talking about. 00:55:34.173 --> 00:55:36.140 Mode one, damping ratio mode one, 00:55:36.140 --> 00:55:40.800 natural frequency mode one, initial displacement mode one, 00:55:40.800 --> 00:55:48.810 initial velocity mode one, omega 1 d like that. 00:55:48.810 --> 00:55:51.670 And mode two is going to be exactly analogous. 00:55:51.670 --> 00:55:56.650 q2 equals, and it's exactly similar, 00:55:56.650 --> 00:55:59.210 except you update it with a 2 instead of a 1. 00:56:02.550 --> 00:56:05.510 And if you plug in the initial-- you over here 00:56:05.510 --> 00:56:14.990 have found the initial values for q10 and q20 and q1 dot 0 00:56:14.990 --> 00:56:15.650 and so forth. 00:56:15.650 --> 00:56:19.410 You found the initial values that plug into that equation 00:56:19.410 --> 00:56:20.560 by just doing this. 00:56:23.150 --> 00:56:25.950 And once we have this, then we can go back to saying, 00:56:25.950 --> 00:56:29.570 how do you get to the final answer? 00:56:29.570 --> 00:56:31.350 Well, you just multiply the result 00:56:31.350 --> 00:56:33.300 for q times the mode shape and add them up. 00:56:33.300 --> 00:56:35.435 And you have the answer. 00:56:35.435 --> 00:56:37.560 But we still have to deal with the damping problem. 00:56:41.224 --> 00:56:43.687 We're going to do that one next. 00:56:43.687 --> 00:56:45.770 But I see a bunch of hands and some puzzled looks, 00:56:45.770 --> 00:56:49.060 so it means it's a good time to stop and talk for a second. 00:56:49.060 --> 00:56:50.354 Yeah? 00:56:50.354 --> 00:56:53.040 AUDIENCE: [INAUDIBLE] 00:56:53.040 --> 00:56:53.872 PROFESSOR: Pardon? 00:56:53.872 --> 00:56:55.080 AUDIENCE: What if you [INAUDIBLE] 00:56:55.080 --> 00:56:56.329 PROFESSOR: I can't quite hear. 00:56:56.329 --> 00:56:58.489 AUDIENCE: The sine theta and the sine rate of g. 00:56:58.489 --> 00:56:59.780 PROFESSOR: Yeah, what about it? 00:56:59.780 --> 00:57:00.800 AUDIENCE: Why do we lose it? 00:57:00.800 --> 00:57:01.660 PROFESSOR: Why do we use it? 00:57:01.660 --> 00:57:02.409 AUDIENCE: Lose it. 00:57:02.409 --> 00:57:03.860 PROFESSOR: Oh, you don't lose it. 00:57:03.860 --> 00:57:07.450 I was saying, you see this bit, it's like that. 00:57:07.450 --> 00:57:11.860 These are two pieces that behave like sine. 00:57:11.860 --> 00:57:14.630 And see, this one depends on initial displacement 00:57:14.630 --> 00:57:17.002 but is multiplied by the damping ratio. 00:57:17.002 --> 00:57:19.210 And the damping ratio for things that are interesting 00:57:19.210 --> 00:57:21.040 is usually pretty small. 00:57:21.040 --> 00:57:29.480 So here you have a term that's x0 cosine omega 1 damped, 00:57:29.480 --> 00:57:34.030 and here you have a contribution that's x0 small times 00:57:34.030 --> 00:57:39.600 x0 sine omega 1 damped. 00:57:39.600 --> 00:57:40.910 So you multiply the same. 00:57:40.910 --> 00:57:42.620 They're operating on the same frequency. 00:57:42.620 --> 00:57:44.920 Two terms at the same frequency, you add them together, 00:57:44.920 --> 00:57:49.620 it's like a cosine omega t minus some phase angle. 00:57:49.620 --> 00:57:52.197 If this little term is small, that phase angle's almost 0. 00:57:59.560 --> 00:58:03.940 x0 cosine plus something x0 sine, 00:58:03.940 --> 00:58:07.930 it gives you a cosine term that is shifted a little bit 00:58:07.930 --> 00:58:11.562 and its magnitude is different by this little amount. 00:58:11.562 --> 00:58:13.270 I'm just saying oftentimes this is small. 00:58:13.270 --> 00:58:14.820 But if you don't want to make that approximation, 00:58:14.820 --> 00:58:15.660 just carry it along. 00:58:15.660 --> 00:58:16.321 Just do it. 00:58:16.321 --> 00:58:17.196 AUDIENCE: [INAUDIBLE] 00:58:19.540 --> 00:58:20.290 PROFESSOR: Mm-hmm. 00:58:20.290 --> 00:58:21.410 AUDIENCE: [INAUDIBLE] 00:58:21.410 --> 00:58:22.790 PROFESSOR: Oh, no. 00:58:22.790 --> 00:58:25.950 I'm just saying you can throw out this piece usually. 00:58:25.950 --> 00:58:31.930 And it makes-- I keep in my mind-- let me see. 00:58:31.930 --> 00:58:32.430 OK, now. 00:58:32.430 --> 00:58:35.990 Vibration engineering is full of lots of approximations 00:58:35.990 --> 00:58:38.570 because it's very hard oftentimes 00:58:38.570 --> 00:58:43.990 to get detailed quantitative numbers on exactly 00:58:43.990 --> 00:58:45.310 everything you need to know. 00:58:45.310 --> 00:58:47.120 So I carry around little approximations 00:58:47.120 --> 00:58:49.390 that I know is the way the world mostly behaves. 00:58:49.390 --> 00:58:50.890 And the way the world mostly behaves 00:58:50.890 --> 00:58:52.431 for a single degree of freedom system 00:58:52.431 --> 00:58:55.610 is the response to initial conditions looks like this. 00:58:58.320 --> 00:59:03.690 And this initial value here is always approximately x0. 00:59:03.690 --> 00:59:09.920 And this initial slope here is always approximately v0. 00:59:12.842 --> 00:59:13.550 That's the slope. 00:59:16.120 --> 00:59:20.590 Now, it turns out that this thing is shifted just slightly. 00:59:20.590 --> 00:59:21.090 Why? 00:59:21.090 --> 00:59:23.600 Because of this term, OK? 00:59:23.600 --> 00:59:25.430 But honest, to tell you the truth, 00:59:25.430 --> 00:59:27.420 it really rarely matters. 00:59:27.420 --> 00:59:30.590 So as a vibration engineer, I just 00:59:30.590 --> 00:59:32.860 remember I have an x0 cosine. 00:59:32.860 --> 00:59:34.650 I have a v0 over omega d sine. 00:59:34.650 --> 00:59:37.090 And the whole thing decays like that. 00:59:37.090 --> 00:59:40.080 But if you like to be mathematically precise, 00:59:40.080 --> 00:59:41.630 you carry along that a little bit. 00:59:41.630 --> 00:59:42.130 Yeah? 00:59:42.130 --> 00:59:44.510 AUDIENCE: Don't you lose [INAUDIBLE] sine wave? 00:59:44.510 --> 00:59:46.476 PROFESSOR: You're not going to lose the sine. 00:59:46.476 --> 00:59:48.474 AUDIENCE: [INAUDIBLE] 00:59:48.474 --> 00:59:49.390 PROFESSOR: Oh, oh, oh. 00:59:49.390 --> 00:59:50.500 Wait a minute. 00:59:50.500 --> 00:59:51.455 I just left it out. 00:59:54.660 --> 00:59:57.310 You guys are-- well, I'm glad you're awake. 00:59:57.310 --> 00:59:58.060 This is good. 01:00:01.990 --> 01:00:03.680 Now how's that? 01:00:03.680 --> 01:00:04.560 Ah, good. 01:00:04.560 --> 01:00:07.390 Now I know why I had so many puzzled looks. 01:00:07.390 --> 01:00:08.860 Anybody have a different question? 01:00:08.860 --> 01:00:11.677 Just anything now about this whole modal analysis thing? 01:00:11.677 --> 01:00:14.010 Because then we have to deal with this awkward part that 01:00:14.010 --> 01:00:16.620 has to do with the damping. 01:00:16.620 --> 01:00:19.326 And I've got to finish on time, OK? 01:00:23.821 --> 01:00:24.320 All right. 01:00:27.540 --> 01:00:28.760 So damping. 01:00:28.760 --> 01:00:30.460 I've gotten this far. 01:00:30.460 --> 01:00:37.080 What I need is I need estimates for the damping for mode one 01:00:37.080 --> 01:00:39.400 and damping for mode two, right? 01:00:54.030 --> 01:01:12.110 So the problem is that utcu does not always 01:01:12.110 --> 01:01:17.210 equal some nice diagonalized matrix. 01:01:17.210 --> 01:01:29.350 You sometimes get these are not always 0, OK? 01:01:29.350 --> 01:01:31.330 The orthogonality principle just doesn't 01:01:31.330 --> 01:01:34.879 apply to the damping terms. 01:01:34.879 --> 01:01:35.420 Just doesn't. 01:01:38.000 --> 01:01:40.780 But this actually doesn't hurt you a lot. 01:01:40.780 --> 01:01:43.210 You just got to know that this is going to be a problem. 01:01:43.210 --> 01:01:45.170 And when the systems are lightly damped, 01:01:45.170 --> 01:01:49.050 the approximation, even if your true damping in the system 01:01:49.050 --> 01:01:51.700 gives you some non-zero elements here, 01:01:51.700 --> 01:01:53.974 the first order behavior of the system 01:01:53.974 --> 01:01:55.890 is basically going to be-- you can just ignore 01:01:55.890 --> 01:01:57.800 the off-diagonal elements. 01:01:57.800 --> 01:01:59.520 What practical consequence do you 01:01:59.520 --> 01:02:06.520 think it has if you have some actual non-zero numbers here? 01:02:06.520 --> 01:02:08.080 Go back and look at the equations 01:02:08.080 --> 01:02:11.170 that you're trying to derive. 01:02:11.170 --> 01:02:14.810 These were the equations that we were trying to come up with. 01:02:14.810 --> 01:02:20.970 And we wanted them to be n individual single degree 01:02:20.970 --> 01:02:22.570 of freedom systems. 01:02:22.570 --> 01:02:26.060 But if this has non-zero off-diagonal terms, 01:02:26.060 --> 01:02:29.230 you're going to find popping up in this single degree 01:02:29.230 --> 01:02:34.200 of freedom equation another term that couples it through damping 01:02:34.200 --> 01:02:35.950 to the other modes. 01:02:35.950 --> 01:02:39.160 It provides a little bit of coupling to other modes. 01:02:39.160 --> 01:02:43.170 They can talk to one another, all right? 01:02:43.170 --> 01:02:47.340 And what that means is if I-- this may be a good time 01:02:47.340 --> 01:02:50.590 to do the demonstration. 01:03:03.485 --> 01:03:04.970 How do I want to say this? 01:03:18.870 --> 01:03:22.210 If the initial displacement of the system 01:03:22.210 --> 01:03:32.260 is in the shape of one of the natural modes-- 01:03:32.260 --> 01:03:36.810 so if this is some u, this is exactly shaped like mode r. 01:03:36.810 --> 01:03:38.605 So this looks like the ur vector. 01:03:43.970 --> 01:03:47.250 When I carry out this multiplication, 01:03:47.250 --> 01:03:50.152 what do you think will happen? 01:03:56.260 --> 01:03:58.350 If this is shaped like mode r, because 01:03:58.350 --> 01:04:01.870 of orthogonality when I do u inverse, which 01:04:01.870 --> 01:04:04.450 is all about the mode shapes information and the mode 01:04:04.450 --> 01:04:08.890 shapes are these orthogonal set of independent orthogonal 01:04:08.890 --> 01:04:12.970 vectors, if this is exactly one mode 01:04:12.970 --> 01:04:17.160 and I do u inverse times that, I will get 0 over here 01:04:17.160 --> 01:04:22.200 on the right hand side for every mode except the mode 01:04:22.200 --> 01:04:24.650 that that's shaped like. 01:04:24.650 --> 01:04:28.210 So if this is shaped like a particular mode, 01:04:28.210 --> 01:04:31.310 then over here all the modal initial conditions 01:04:31.310 --> 01:04:35.140 are 0 except that mode. 01:04:35.140 --> 01:04:39.530 That means if I set this, give its initial conditions are 01:04:39.530 --> 01:04:45.370 equal to the shape exactly of mode one, 01:04:45.370 --> 01:04:49.490 it only responds in mode one. 01:04:49.490 --> 01:04:56.180 And if I give it initial conditions that are exactly 01:04:56.180 --> 01:05:01.090 shaped like that of mode two, then it only 01:05:01.090 --> 01:05:03.670 responds in mode two. 01:05:03.670 --> 01:05:10.290 And if I give it anything else, like I move just the top one 01:05:10.290 --> 01:05:13.550 but not the bottom and let it go, 01:05:13.550 --> 01:05:17.180 then there's-- maybe I better do the other one. 01:05:17.180 --> 01:05:19.610 That one had too much of one and not the other. 01:05:19.610 --> 01:05:21.590 If I hold this one, here's its reference. 01:05:21.590 --> 01:05:23.256 I'm going to hold it right there and I'm 01:05:23.256 --> 01:05:26.352 going to give this one a unit deflection and let go. 01:05:26.352 --> 01:05:30.240 Now you see a get some of both, all right? 01:05:30.240 --> 01:05:46.520 So if when I do this first one, say first mode, 01:05:46.520 --> 01:05:48.660 I could sit here and measure how many cycles 01:05:48.660 --> 01:05:51.400 it takes to decay halfway and estimate the damping ratio 01:05:51.400 --> 01:05:52.680 for that mode. 01:05:52.680 --> 01:05:54.630 If it's only moving in this mode, 01:05:54.630 --> 01:05:58.300 I can estimate its damping directly for that mode 01:05:58.300 --> 01:05:59.375 and get zeta 1. 01:05:59.375 --> 01:06:00.563 You agree? 01:06:00.563 --> 01:06:01.430 OK. 01:06:01.430 --> 01:06:04.460 And I did the same thing with mode two, it's too fast for me 01:06:04.460 --> 01:06:07.680 to catch it with a stopwatch, but I 01:06:07.680 --> 01:06:08.980 could measure its damping. 01:06:08.980 --> 01:06:11.930 And as it decays, I could get an estimate for zeta 2, 01:06:11.930 --> 01:06:15.090 for the damping ratio for mode two, all right? 01:06:15.090 --> 01:06:16.370 All right. 01:06:16.370 --> 01:06:18.710 But somehow I have to get damping 01:06:18.710 --> 01:06:21.300 ratio for mode one, zeta 1, and damping ratio 01:06:21.300 --> 01:06:22.590 for mode two, zeta 2. 01:06:22.590 --> 01:06:26.260 I have to somehow get it out of this. 01:06:26.260 --> 01:06:30.030 I have to model it somehow with these damping coefficients that 01:06:30.030 --> 01:06:35.210 come from computing this u transpose cu, OK. 01:06:38.580 --> 01:06:41.160 So I'm going to show you kind of damping 01:06:41.160 --> 01:06:53.830 called Rayleigh damping, OK? 01:06:53.830 --> 01:06:56.630 Lord Rayleigh, who did lots of things in science 01:06:56.630 --> 01:07:00.590 that you've probably run into, proposed 01:07:00.590 --> 01:07:07.940 that if you model your damping, the c matrix as-- this 01:07:07.940 --> 01:07:11.210 is just now the system damping matrix 01:07:11.210 --> 01:07:20.150 that you start with-- some alpha times the mass matrix 01:07:20.150 --> 01:07:24.520 plus beta times the stiffness matrix-- these are now 01:07:24.520 --> 01:07:27.400 the original ones in your generalized coordinates, 01:07:27.400 --> 01:07:30.740 just your mass and stiffness matrices. 01:07:30.740 --> 01:07:33.340 If you say, I'm going to approximate my damping model 01:07:33.340 --> 01:07:50.265 like this, then I want to compute u transpose cu. 01:07:52.900 --> 01:07:57.000 I'm going to get alpha u transpose 01:07:57.000 --> 01:08:02.390 mu plus beta u transpose Ku. 01:08:05.160 --> 01:08:10.020 But we know that this gives you the diagonalized mass matrix, 01:08:10.020 --> 01:08:11.630 known as the modal mass matrix. 01:08:11.630 --> 01:08:14.960 This gives you the diagonalized stiffness matrix. 01:08:14.960 --> 01:08:17.810 And so this damping model, this is 01:08:17.810 --> 01:08:23.560 guaranteed to give you a diagonalized damping matrix 01:08:23.560 --> 01:08:27.899 which we'll call, somehow, some capital C2, 0, 01:08:27.899 --> 01:08:31.920 0, C2, all right? 01:08:31.920 --> 01:08:40.810 And it's going to be alpha times the modal mass matrix 01:08:40.810 --> 01:08:46.319 plus a beta times the modal stiffness matrix. 01:08:46.319 --> 01:08:47.979 And those alphas and betas you adjust. 01:08:47.979 --> 01:08:49.395 They're just parameters you adjust 01:08:49.395 --> 01:08:54.380 to get the amount of damping you need, OK? 01:08:54.380 --> 01:09:09.450 So for a two degree of freedom system, 01:09:09.450 --> 01:09:23.240 C1 here is alpha m1 plus beta K1. 01:09:23.240 --> 01:09:25.620 Modal mass, alpha times the modal mass 01:09:25.620 --> 01:09:27.310 plus beta times the modal stiffness. 01:09:27.310 --> 01:09:29.715 That's what you get for the first one. 01:09:29.715 --> 01:09:39.288 And C2 is alpha m2 plus beta K2, OK? 01:09:46.120 --> 01:09:48.510 And the alphas and betas give you two free parameters 01:09:48.510 --> 01:09:49.950 you can play with. 01:09:49.950 --> 01:09:52.520 And for a two degree of freedom system, 01:09:52.520 --> 01:09:55.680 I can manipulate alpha and beta to get the damping 01:09:55.680 --> 01:09:58.020 that I measure. 01:09:58.020 --> 01:10:01.900 And I forced my equations of motion a couple. 01:10:01.900 --> 01:10:04.160 Now, Mother Nature may say, you know, Vandiver, 01:10:04.160 --> 01:10:06.227 they don't uncouple, and there's going 01:10:06.227 --> 01:10:07.810 to be a little crosstalk between them. 01:10:07.810 --> 01:10:10.580 But I say, yeah, but to first order 01:10:10.580 --> 01:10:13.580 I'm going to get a pretty good model of the system. 01:10:13.580 --> 01:10:15.280 So let's do that in this case. 01:10:15.280 --> 01:10:19.650 Let's maybe just to keep it-- I've got numbers here, 01:10:19.650 --> 01:10:24.840 so let my notes so I don't get completely lost here. 01:10:24.840 --> 01:10:28.020 So I'm going to just pick one for now. 01:10:28.020 --> 01:10:29.890 I'm going to model my damping with just 01:10:29.890 --> 01:10:54.640 beta K, beta times my diagonal, my stiffness matrix. 01:10:54.640 --> 01:10:57.257 And let's see what happens here. 01:11:01.550 --> 01:11:07.240 So that says my modal damping is going to be some, 01:11:07.240 --> 01:11:12.110 for mode one, beta K1. 01:11:12.110 --> 01:11:13.790 Now what's damping ratio? 01:11:13.790 --> 01:11:18.850 Zeta 1 for a single degree of freedom system 01:11:18.850 --> 01:11:27.540 is the damping constant for the system over 2 omega 1 m1. 01:11:27.540 --> 01:11:37.610 But that's going to be beta K1 over 2 omega 1 m1. 01:11:37.610 --> 01:11:41.980 But m1/K1 is omega 1 squared. 01:11:41.980 --> 01:11:45.720 So I get an omega 1 squared in the numerator. 01:11:45.720 --> 01:11:54.344 Beta omega 1 squared over 2 omega 1. 01:11:54.344 --> 01:11:56.920 Remember, the K over the m gave me the omega one squared, 01:11:56.920 --> 01:11:58.560 so the ms are gone. 01:11:58.560 --> 01:12:00.200 You can cancel one of these. 01:12:00.200 --> 01:12:04.010 This gives me beta omega 1 over 2. 01:12:06.760 --> 01:12:14.390 So this now gives me a way I can fit one of the dampings. 01:12:14.390 --> 01:12:17.450 I can get exactly what I want, say, for mode one 01:12:17.450 --> 01:12:21.910 if I pick beta to be the right number. 01:12:21.910 --> 01:12:27.820 OK, so in this case, I actually did some numbers. 01:12:33.772 --> 01:12:36.715 Pardon? 01:12:36.715 --> 01:12:37.340 Can't hear you. 01:12:37.340 --> 01:12:38.980 AUDIENCE: [INAUDIBLE] 01:12:38.980 --> 01:12:40.690 PROFESSOR: No. 01:12:40.690 --> 01:12:43.790 K1/m1 is omega 1 squared. 01:12:43.790 --> 01:12:45.730 Omega 1 squared takes care of the m1. 01:12:45.730 --> 01:12:47.490 I get rid of one of the omega 1s. 01:12:47.490 --> 01:12:51.531 I'm left with this, OK? 01:12:51.531 --> 01:12:52.030 OK. 01:12:54.690 --> 01:13:00.970 So let's just let beta equal 0.01. 01:13:00.970 --> 01:13:03.250 And if you let beta equal to 0.01, 01:13:03.250 --> 01:13:09.420 then zeta 1 equals 0.01 omega 1 over 2. 01:13:09.420 --> 01:13:13.250 We know omega 1 is 5.65. 01:13:13.250 --> 01:13:15.940 This when you work it out then gives you 01:13:15.940 --> 01:13:24.230 a number of 0.0283, about 3%. 01:13:24.230 --> 01:13:28.290 And that would say that this system when it vibrates 01:13:28.290 --> 01:13:30.960 in mode one is going to damp out up to 50% 01:13:30.960 --> 01:13:34.020 in about three cycles. 01:13:34.020 --> 01:13:35.282 Not bad approximations. 01:13:35.282 --> 01:13:36.740 I'm just guessing about what it is. 01:13:36.740 --> 01:13:39.910 That's a reasonable amount of damping for mode one. 01:13:39.910 --> 01:13:45.300 Now the problem is when I only use just beta K as my model. 01:13:45.300 --> 01:13:48.220 Now I'm stuck with whatever happens for mode two 01:13:48.220 --> 01:13:53.450 once I pick beta because zeta 2 is going 01:13:53.450 --> 01:13:57.840 to be beta omega 2 over 2. 01:13:57.840 --> 01:14:00.350 And omega 2 is quite a bit larger, 01:14:00.350 --> 01:14:05.770 so now I'm stuck with a greater value for the second mode. 01:14:05.770 --> 01:14:11.800 In this case, it's 0.0885. 01:14:11.800 --> 01:14:16.010 So if I just pick a one parameter model for my damping, 01:14:16.010 --> 01:14:18.160 I can make one perfect. 01:14:18.160 --> 01:14:20.050 I can match it perfectly, but then I'm 01:14:20.050 --> 01:14:22.440 stuck with whatever the other one is. 01:14:22.440 --> 01:14:25.230 So I did this because I could do it simply with one. 01:14:25.230 --> 01:14:30.710 But if I'd kept the full two-parameter model, 01:14:30.710 --> 01:14:33.260 with manipulating both alpha and beta 01:14:33.260 --> 01:14:37.050 I could actually get both of the two measured dampings 01:14:37.050 --> 01:14:38.569 exactly right. 01:14:38.569 --> 01:14:40.110 But if I have an n degree-- if I have 01:14:40.110 --> 01:14:42.220 three degree of freedom system, I only have two parameters. 01:14:42.220 --> 01:14:43.800 I can fit two of the damping ratios, 01:14:43.800 --> 01:14:46.133 but then I'm going to be stuck with whatever it gives me 01:14:46.133 --> 01:14:46.750 for the third. 01:14:46.750 --> 01:14:50.000 But oftentimes it's just one mode you really care about. 01:14:50.000 --> 01:14:50.937 It's the problem mode. 01:14:50.937 --> 01:14:52.270 You're at its natural frequency. 01:14:52.270 --> 01:14:54.670 It's going like crazy. 01:14:54.670 --> 01:14:58.320 Initial conditions make it vibrate a lot in that mode. 01:14:58.320 --> 01:15:01.440 But this is what Rayleigh damping allows you to do. 01:15:01.440 --> 01:15:05.260 It guarantees you that you will have a diagonalized set 01:15:05.260 --> 01:15:06.754 of equations of motion. 01:15:06.754 --> 01:15:08.420 And it gives you two parameters that you 01:15:08.420 --> 01:15:12.260 can play with to fit the damping model however you want. 01:15:12.260 --> 01:15:17.900 Once you have damping, now you have the complete solution 01:15:17.900 --> 01:15:19.394 for decay from initial conditions. 01:15:23.950 --> 01:15:25.130 And there's your two models. 01:15:25.130 --> 01:15:28.710 You can solve for q1, transient decay given initial conditions. 01:15:28.710 --> 01:15:33.100 You can solve for q2, transient K of the second mode. 01:15:33.100 --> 01:15:37.130 And then to get back to the initial 01:15:37.130 --> 01:15:39.900 to the response in terms of your modal coordinates, 01:15:39.900 --> 01:15:44.342 you just add the two together, OK? 01:15:48.100 --> 01:15:50.345 I got some numbers here which are just instructive. 01:15:52.950 --> 01:15:54.072 u Inverse. 01:15:54.072 --> 01:15:55.780 In order to get these initial conditions, 01:15:55.780 --> 01:15:56.988 you've got to know u inverse. 01:15:56.988 --> 01:15:58.300 Do we know u? 01:15:58.300 --> 01:15:59.430 I gave us u. 01:15:59.430 --> 01:16:03.080 Here's our set of mode shape vectors. 01:16:03.080 --> 01:16:05.530 And I've run out of boards. 01:16:30.410 --> 01:16:32.710 So we have the u matrix. 01:16:32.710 --> 01:16:36.020 We need u inverse, so u inverse for this problem. 01:16:51.030 --> 01:16:53.110 And we're going to quickly do some examples. 01:16:53.110 --> 01:16:59.320 Let's let the v0s be 0. 01:16:59.320 --> 01:17:02.260 No initial conditions on velocities. 01:17:02.260 --> 01:17:12.450 And let's do x0, the initial displacements, be 1 and 0. 01:17:12.450 --> 01:17:14.690 So the 1 and 0, what we're saying 01:17:14.690 --> 01:17:18.945 is the bottom one doesn't move, unit deflection here, let 01:17:18.945 --> 01:17:20.000 it go. 01:17:20.000 --> 01:17:23.530 What are you're going to get for the initial conditions? 01:17:23.530 --> 01:17:35.290 x0 equals 1 and 0, well, that implies that the qs are 01:17:35.290 --> 01:17:38.460 going to be u inverse x0. 01:17:38.460 --> 01:17:40.370 So by the way, if this is true, this 01:17:40.370 --> 01:17:47.820 implies that all q dot initial conditions equal 0, right? 01:17:47.820 --> 01:17:49.730 No initial velocities in generalized 01:17:49.730 --> 01:17:52.770 coordinates, no initial velocities in modal 01:17:52.770 --> 01:17:54.462 coordinates. 01:17:54.462 --> 01:17:56.420 But we are going to have an initial deflection. 01:17:59.360 --> 01:18:06.280 I want to then compute u inverse x0 and see what I get. 01:18:06.280 --> 01:18:20.320 And what I get back when I do this one is 0.0898 and 0.9102. 01:18:20.320 --> 01:18:24.280 Remember, this is q10, q20. 01:18:27.480 --> 01:18:33.640 So for that case, it says I'm going to get 0.08 or 0.09 01:18:33.640 --> 01:18:37.770 equal to q1 and 0.9 of q2. 01:18:37.770 --> 01:18:41.120 And I go back over here to my transient decay. 01:18:41.120 --> 01:18:42.850 There's no velocity. 01:18:42.850 --> 01:18:51.720 So it's basically going to look like q10 cosine omega 01:18:51.720 --> 01:18:58.190 dt, e the minus zeta omega t, decaying, cosine. 01:18:58.190 --> 01:19:03.510 But for mode one, its initial amplitude's less than 0.1. 01:19:03.510 --> 01:19:06.580 And mode two, it's got a lot of mode two. 01:19:06.580 --> 01:19:09.910 So what happens? 01:19:09.910 --> 01:19:15.650 So unit deflection here, in fact it's mostly mode two. 01:19:15.650 --> 01:19:18.820 And just quickly I'll do one other. 01:19:18.820 --> 01:19:22.990 x0 is 0, 1. 01:19:22.990 --> 01:19:30.400 That implies that q0 that you get from that 01:19:30.400 --> 01:19:40.830 is 0.4016 and minus 0.4016. 01:19:40.830 --> 01:19:44.690 Says you get about equal amounts. 01:19:44.690 --> 01:19:46.100 So that's this one. 01:19:46.100 --> 01:19:49.110 I don't move this one, but I give this one 01:19:49.110 --> 01:19:52.390 a unit deflection, let go. 01:19:52.390 --> 01:19:55.190 I get about equal amounts of each one. 01:19:55.190 --> 01:19:59.120 And of course I've told you the answer to this one. 01:19:59.120 --> 01:20:04.250 If I let x equal mode one's mode shape, 01:20:04.250 --> 01:20:16.250 1 and 2.266, that implies that q1 equals 1 and q2, when 01:20:16.250 --> 01:20:18.780 you multiply it out, is zero 0. 01:20:18.780 --> 01:20:23.260 If I deflect it in the shape of mode one and I do u inverse x0, 01:20:23.260 --> 01:20:25.620 I will get back 0 and 1. 01:20:25.620 --> 01:20:27.960 And if I make this the shape of mode two, 01:20:27.960 --> 01:20:31.676 I will get back 0 for mode one and 1 for mode two. 01:20:36.760 --> 01:20:41.780 I've out of time, but that's your intro to modal analysis. 01:20:41.780 --> 01:20:43.966 So I think it's conceptually powerful. 01:20:43.966 --> 01:20:44.466 Yes. 01:20:44.466 --> 01:20:47.868 AUDIENCE: How did you get from the 0.898 value 01:20:47.868 --> 01:20:56.616 to the 0.0898 value? 01:20:56.616 --> 01:21:00.018 AUDIENCE: The inverse should be 0.0898. 01:21:00.018 --> 01:21:01.910 PROFESSOR: Oh, is this 0.08? 01:21:01.910 --> 01:21:03.416 Yeah, OK. 01:21:03.416 --> 01:21:04.730 I may have written that down. 01:21:10.720 --> 01:21:11.870 Yeah. 01:21:11.870 --> 01:21:12.970 I'll double check that. 01:21:12.970 --> 01:21:14.040 But yeah, question? 01:21:14.040 --> 01:21:16.550 AUDIENCE: Why is it for here that we picked c 01:21:16.550 --> 01:21:18.840 to be only a function of-- 01:21:18.840 --> 01:21:19.575 PROFESSOR: Beta? 01:21:19.575 --> 01:21:19.830 AUDIENCE: A. 01:21:19.830 --> 01:21:22.205 PROFESSOR: Because I want to get done by the end of the-- 01:21:22.205 --> 01:21:22.760 AUDIENCE: OK. 01:21:22.760 --> 01:21:24.620 PROFESSOR: --60 minutes, 80 minutes. 01:21:24.620 --> 01:21:26.550 I could have put both of them in, 01:21:26.550 --> 01:21:29.120 manipulated both parameters as two equation with [? two ?] 01:21:29.120 --> 01:21:31.130 modes, two target values of dampings. 01:21:31.130 --> 01:21:33.070 I'd find an alpha and a beta that would 01:21:33.070 --> 01:21:35.410 make both work exactly right. 01:21:35.410 --> 01:21:38.870 Actually, just this one model is pretty good for this case. 01:21:38.870 --> 01:21:40.840 The damping for second mode is greater 01:21:40.840 --> 01:21:42.890 than the first mode just happens to be. 01:21:42.890 --> 01:21:44.032 This model's not bad. 01:21:44.032 --> 01:21:45.740 AUDIENCE: All right, so you try the three 01:21:45.740 --> 01:21:47.690 and see what gets you the best results? 01:21:47.690 --> 01:21:49.240 PROFESSOR: Yeah.