1 00:00:00,070 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,140 continue to offer high quality educational resources for free. 5 00:00:10,140 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:21,735 --> 00:00:24,990 PROFESSOR: Well let's see if we can't get started. 9 00:00:24,990 --> 00:00:28,250 Everyone I trust can hear me adequately. 10 00:00:30,840 --> 00:00:33,220 Welcome back. 11 00:00:33,220 --> 00:00:35,990 It's Tuesday. 12 00:00:35,990 --> 00:00:39,020 For those of you who are not in my recitation section, 13 00:00:39,020 --> 00:00:44,520 I'm Dave Gossard, and I'll be your lecturer for the day. 14 00:00:44,520 --> 00:00:48,680 Professor Vandiver is out of town. 15 00:00:48,680 --> 00:00:51,060 It looks like some of you may be as well. 16 00:00:51,060 --> 00:00:55,575 We probably could have held this at the gate at Logan Airport 17 00:00:55,575 --> 00:00:57,940 and done a little better. 18 00:00:57,940 --> 00:01:00,990 But be that as it may, glad you came. 19 00:01:00,990 --> 00:01:02,030 This should be fun. 20 00:01:02,030 --> 00:01:08,412 Today we have a new topic and a demonstration, 21 00:01:08,412 --> 00:01:11,230 a real physical system. 22 00:01:11,230 --> 00:01:15,890 So unless there are any outstanding questions? 23 00:01:15,890 --> 00:01:20,690 Anybody have any questions or complaints 24 00:01:20,690 --> 00:01:22,356 to address to Vicente? 25 00:01:22,356 --> 00:01:23,348 No. 26 00:01:23,348 --> 00:01:28,920 All right, hearing none let's go ahead and get started then. 27 00:01:28,920 --> 00:01:34,990 Today the topic is multiple degree of freedom systems. 28 00:01:34,990 --> 00:01:37,810 Now to date, with a couple of exceptions, 29 00:01:37,810 --> 00:01:41,240 all of the systems that you've dealt with 30 00:01:41,240 --> 00:01:46,360 had a single degree of freedom, either a linear displacement 31 00:01:46,360 --> 00:01:50,550 x or an angular displacement theta. 32 00:01:50,550 --> 00:01:55,160 You know the concept of equations of motion, 33 00:01:55,160 --> 00:01:57,420 or I should say the equation of motion 34 00:01:57,420 --> 00:02:06,750 and the notion of undamped natural frequency. 35 00:02:06,750 --> 00:02:11,920 Well, today we're going to generalize, if you will, 36 00:02:11,920 --> 00:02:17,640 to systems that have not one but multiple degrees of freedom 37 00:02:17,640 --> 00:02:22,030 and see how those notions generalize. 38 00:02:22,030 --> 00:02:26,260 In particular, as you might expect, 39 00:02:26,260 --> 00:02:30,750 the system that has multiple degrees of freedom 40 00:02:30,750 --> 00:02:36,460 has multiple natural frequencies, also 41 00:02:36,460 --> 00:02:41,000 known as eigenvalues as we will explain here shortly. 42 00:02:41,000 --> 00:02:44,390 Multiple degrees of freedom systems 43 00:02:44,390 --> 00:02:48,200 have a new property, a new characteristic 44 00:02:48,200 --> 00:02:49,670 you haven't seen before. 45 00:02:49,670 --> 00:02:51,850 And that's what this lecture is all about, 46 00:02:51,850 --> 00:02:56,350 is to illustrate that to you and demonstrate it. 47 00:02:56,350 --> 00:02:59,880 It's the notion of natural modes, 48 00:02:59,880 --> 00:03:02,590 also called eigenvectors. 49 00:03:02,590 --> 00:03:07,710 And then the general response to initial conditions. 50 00:03:07,710 --> 00:03:11,370 So that is the plan for the day. 51 00:03:11,370 --> 00:03:14,220 And we'll start with this. 52 00:03:14,220 --> 00:03:17,290 This is kind of a classic textbook 53 00:03:17,290 --> 00:03:19,910 case, two springs, two masses. 54 00:03:19,910 --> 00:03:23,730 A straightforward extrapolation of what you've done before. 55 00:03:23,730 --> 00:03:30,040 You've got a spring K1, mass M1, spring K2, mass M2. 56 00:03:30,040 --> 00:03:40,420 And the displacements are indicated 57 00:03:40,420 --> 00:03:47,370 as shown there, X1 and X2. 58 00:03:47,370 --> 00:03:53,800 I want to hasten to point out that the displacements we speak 59 00:03:53,800 --> 00:04:01,890 of here are defined with respect to the static equilibrium 60 00:04:01,890 --> 00:04:02,520 position. 61 00:04:02,520 --> 00:04:08,560 This is a notion that Professor Vandiver went over at least 62 00:04:08,560 --> 00:04:09,140 once. 63 00:04:09,140 --> 00:04:15,020 And for those of you who've forgotten it or weren't there 64 00:04:15,020 --> 00:04:20,519 that day, I have for you a reference, essentially reprised 65 00:04:20,519 --> 00:04:24,530 that notion over there. 66 00:04:24,530 --> 00:04:27,215 So in the meantime, let me press on. 67 00:04:27,215 --> 00:04:31,320 If you have any questions, we can go back and cover that. 68 00:04:31,320 --> 00:04:41,760 But assuming you agree, let me simply say you've got 69 00:04:41,760 --> 00:04:43,915 two springs, two masses. 70 00:04:43,915 --> 00:04:46,720 The typical way we've taught you to do 71 00:04:46,720 --> 00:04:48,520 it is if you're going to generate 72 00:04:48,520 --> 00:04:51,270 the equations of motion by the direct method, 73 00:04:51,270 --> 00:04:53,890 you generate two free body diagrams, the sum 74 00:04:53,890 --> 00:05:00,140 forces in the x direction for each of the masses, 75 00:05:00,140 --> 00:05:03,210 get f equals MA and you'd get these. 76 00:05:03,210 --> 00:05:05,360 Conversely, you could also do it by Legrange. 77 00:05:05,360 --> 00:05:08,690 You could generate the expression 78 00:05:08,690 --> 00:05:12,460 for the kinetic energy, for the potential energy, 79 00:05:12,460 --> 00:05:16,960 for the Legrangian, do this Lagrange equation business, 80 00:05:16,960 --> 00:05:19,050 and you'd get the same thing. 81 00:05:19,050 --> 00:05:23,310 But either way you do it, what comes out the other side 82 00:05:23,310 --> 00:05:25,830 looks like this. 83 00:05:25,830 --> 00:05:32,830 And it's not a bad exercise for you to offline convince 84 00:05:32,830 --> 00:05:35,610 yourself that this is right. 85 00:05:35,610 --> 00:05:39,930 Not right now, but in the comfort and leisure 86 00:05:39,930 --> 00:05:40,820 of another time. 87 00:05:58,540 --> 00:06:00,540 So there you have it. 88 00:06:00,540 --> 00:06:03,940 That's what the two equations of motion would look like. 89 00:06:03,940 --> 00:06:09,740 Again, either by the direct method or by Legrange, 90 00:06:09,740 --> 00:06:13,850 you end up in the same place, so to speak. 91 00:06:13,850 --> 00:06:29,730 And now for today, we haven't asked you to do this much, 92 00:06:29,730 --> 00:06:33,740 but let me simply say the weapon of choice 93 00:06:33,740 --> 00:06:36,230 for multiple degrees of freedom system, 94 00:06:36,230 --> 00:06:41,790 because there's a certain repetitive quality to it, 95 00:06:41,790 --> 00:06:45,240 matrix notation is preferred. 96 00:06:45,240 --> 00:06:51,450 In these equations over here, written in matrix form 97 00:06:51,450 --> 00:06:52,550 would look like this. 98 00:07:46,980 --> 00:07:48,410 And that looks like this. 99 00:07:48,410 --> 00:07:51,490 There's two matrices, and let me hasten 100 00:07:51,490 --> 00:07:57,490 to point out that this is exactly this and nothing more. 101 00:07:57,490 --> 00:08:00,240 There's no magic, no additional derivation. 102 00:08:00,240 --> 00:08:04,580 This is simply a restructuring, and reorganization 103 00:08:04,580 --> 00:08:05,810 of these equations. 104 00:08:05,810 --> 00:08:08,550 And this may seem foreign to those 105 00:08:08,550 --> 00:08:13,760 of you who have not had any or very much linear algebra. 106 00:08:13,760 --> 00:08:15,150 Do not be dismayed. 107 00:08:15,150 --> 00:08:17,920 It is not a difficult thing to learn. 108 00:08:17,920 --> 00:08:24,210 As you probably know, a matrix multiplies by a vector-- 109 00:08:24,210 --> 00:08:27,290 or multiplying a vector by a matrix-- 110 00:08:27,290 --> 00:08:28,660 is done with two hands. 111 00:08:28,660 --> 00:08:35,990 The first item for example is M1 X1 dot plus 0. 112 00:08:35,990 --> 00:08:39,549 That gives you this term right here. 113 00:08:39,549 --> 00:08:44,630 Over here you get X1 times K1 plus K2. 114 00:08:44,630 --> 00:08:46,530 That's this one. 115 00:08:46,530 --> 00:08:52,690 And here you have minus K2 X2, that's that term. 116 00:08:52,690 --> 00:08:57,370 So matrix notation, this becomes that. 117 00:08:57,370 --> 00:08:57,920 No problem. 118 00:09:07,530 --> 00:09:09,230 It's like a model train set. 119 00:09:09,230 --> 00:09:11,410 It's great. 120 00:09:11,410 --> 00:09:14,020 Everyone should have one of these. 121 00:09:14,020 --> 00:09:24,230 All right, so what I would like to do for our example here 122 00:09:24,230 --> 00:09:39,240 is because we're going to be doing some algebra, 123 00:09:39,240 --> 00:09:43,895 for the express purpose of simplifying the algebra let 124 00:09:43,895 --> 00:09:51,330 me consider a special case where the masses are identical. 125 00:09:51,330 --> 00:09:56,040 And we can simply call them M. And similarly, 126 00:09:56,040 --> 00:10:02,800 the springs are identical, and we'll simply call them K. 127 00:10:02,800 --> 00:10:08,026 At that point, these equations become simplified. 128 00:10:11,010 --> 00:10:41,610 That's just M, that's just M. So this 129 00:10:41,610 --> 00:10:43,330 is the problem we're going to-- we're 130 00:10:43,330 --> 00:10:47,680 going to tackle this problem first, because as I say, 131 00:10:47,680 --> 00:10:50,750 it simplifies the algebra. 132 00:10:50,750 --> 00:10:54,400 Now here is-- this is not an assumption. 133 00:10:54,400 --> 00:11:07,800 This is a-- I would call this more a mechanism 134 00:11:07,800 --> 00:11:11,315 to get this job done here. 135 00:11:15,990 --> 00:11:23,222 Harmonic motion is one where we assume 136 00:11:23,222 --> 00:11:41,300 that the masses oscillate at the same frequency. 137 00:12:23,750 --> 00:12:28,490 So what this looks like is this. 138 00:12:28,490 --> 00:12:34,260 What we're basically saying is that X1 is actually 139 00:12:34,260 --> 00:12:43,330 equal-- X1 has amplitude A-- whoa. 140 00:12:43,330 --> 00:12:45,930 I'm being attacked here by the second board. 141 00:12:57,640 --> 00:13:00,340 Basically, the situation here is that we're 142 00:13:00,340 --> 00:13:05,590 assuming that both of these masses 143 00:13:05,590 --> 00:13:10,360 move-- I wouldn't call it exactly together. 144 00:13:10,360 --> 00:13:14,149 They're not in complete synchrony, 145 00:13:14,149 --> 00:13:15,440 as you'll see here in a moment. 146 00:13:15,440 --> 00:13:18,580 But what they are is they're going 147 00:13:18,580 --> 00:13:20,870 through a sinusoidal motion. 148 00:13:20,870 --> 00:13:25,880 And it is an oscillation at the same frequency. 149 00:13:25,880 --> 00:13:31,260 Both of them are oscillating at the same frequency. 150 00:13:31,260 --> 00:13:34,760 However, they differ in their magnitudes. 151 00:13:34,760 --> 00:13:36,260 They are not the same magnitude. 152 00:13:43,390 --> 00:13:48,900 But that assumption right there allows us to say this. 153 00:14:08,220 --> 00:14:19,500 If we differentiate those twice, we get the following. 154 00:14:31,170 --> 00:14:34,920 That comes back out. 155 00:14:34,920 --> 00:14:37,331 Douglas, where's that minus sign come? 156 00:14:37,331 --> 00:14:39,330 Can you-- first of all, can everybody read that? 157 00:14:39,330 --> 00:14:41,163 Can you guys read that in the back row here? 158 00:14:44,000 --> 00:14:50,850 For example, this says that X1 double dot, if x1 is A1 cosine, 159 00:14:50,850 --> 00:14:55,440 then X1 double dot is A1 cosine preceded 160 00:14:55,440 --> 00:14:58,080 by a minus omega squared. 161 00:14:58,080 --> 00:14:59,164 Where does that come from? 162 00:14:59,164 --> 00:15:01,579 AUDIENCE: The minus sign came from when you differentiated 163 00:15:01,579 --> 00:15:02,850 the cosine in the first one. 164 00:15:02,850 --> 00:15:03,720 PROFESSOR: Exactly. 165 00:15:03,720 --> 00:15:07,040 And the cosine returns. 166 00:15:07,040 --> 00:15:09,160 And that's what you get. 167 00:15:09,160 --> 00:15:12,950 Well, here is, shall we say, the heart of the matter. 168 00:15:12,950 --> 00:15:17,650 When you substitute this into this-- 169 00:15:17,650 --> 00:15:27,970 let me call this-- I'll try not to get too obsessive over this. 170 00:15:27,970 --> 00:15:33,590 But these are our equations of motion. 171 00:15:33,590 --> 00:16:22,760 So when we-- you get this, equals 0. 172 00:16:25,450 --> 00:16:26,330 Excuse me. 173 00:16:26,330 --> 00:16:30,580 Many people, I think, simply put a big 0 there, 174 00:16:30,580 --> 00:16:32,565 but I'll do it properly. 175 00:16:36,470 --> 00:16:39,280 It's two zeros, if you will. 176 00:16:39,280 --> 00:16:55,480 And forgive me for writing this out, 177 00:16:55,480 --> 00:16:58,430 but I would like you to be able to do this 178 00:16:58,430 --> 00:17:04,930 by yourself, to recreate this after the fact. 179 00:17:04,930 --> 00:17:09,784 This becomes-- dividing and collecting terms. 180 00:17:41,250 --> 00:17:49,360 OK, anybody unclear about how this is obtained? 181 00:17:49,360 --> 00:17:49,860 Yes, ma'am. 182 00:17:49,860 --> 00:17:51,248 Emma. 183 00:17:51,248 --> 00:17:53,372 AUDIENCE: I have a question about the previous one. 184 00:17:53,372 --> 00:17:53,844 PROFESSOR: Yeah? 185 00:17:53,844 --> 00:17:55,969 AUDIENCE: If the second term on the left hand side, 186 00:17:55,969 --> 00:17:58,559 should it also be multiplied by A1 A2? 187 00:17:58,559 --> 00:17:59,600 PROFESSOR: Yes it should. 188 00:17:59,600 --> 00:18:02,720 Thank you very much. 189 00:18:02,720 --> 00:18:05,570 Oh, hang on. 190 00:18:05,570 --> 00:18:09,496 Yes, thank you, that's exactly right. 191 00:18:09,496 --> 00:18:10,412 AUDIENCE: [INAUDIBLE]. 192 00:18:15,200 --> 00:18:17,780 PROFESSOR: Hang on a second. 193 00:18:17,780 --> 00:18:20,580 We're fighting the boards here. 194 00:18:20,580 --> 00:18:23,330 Let's see, one thing at a time. 195 00:18:23,330 --> 00:18:34,640 We've got A1, A2 cosine [INAUDIBLE] minus phi equals. 196 00:18:34,640 --> 00:18:36,880 Now Emma, does that take care of you? 197 00:18:36,880 --> 00:18:37,380 Yeah? 198 00:18:37,380 --> 00:18:39,020 And you said, Vicente? 199 00:18:39,020 --> 00:18:41,270 AUDIENCE: Diagonal terms-- shouldn't there be a minus? 200 00:18:41,270 --> 00:18:43,269 PROFESSOR: I'm sorry, that's absolutely correct. 201 00:18:46,556 --> 00:18:47,055 Wonderful. 202 00:18:53,440 --> 00:18:56,446 So there we have it. 203 00:18:56,446 --> 00:18:57,720 Any other questions? 204 00:18:57,720 --> 00:18:59,760 I hope I got it right. 205 00:18:59,760 --> 00:19:00,447 Yes sir. 206 00:19:00,447 --> 00:19:01,530 AUDIENCE: Are they plus K? 207 00:19:01,530 --> 00:19:02,954 PROFESSOR: I'm sorry? 208 00:19:02,954 --> 00:19:04,190 AUDIENCE: Are they plus K? 209 00:19:04,190 --> 00:19:06,150 PROFESSOR: Plus K? 210 00:19:06,150 --> 00:19:06,800 No. 211 00:19:06,800 --> 00:19:09,510 They're minus. 212 00:19:09,510 --> 00:19:11,295 Yeah, why is that? 213 00:19:11,295 --> 00:19:12,610 Everybody see that? 214 00:19:18,520 --> 00:19:23,590 This is a straight-- there's less here than meets the eye. 215 00:19:23,590 --> 00:19:29,420 There's a straight segregation collecting of terms. 216 00:19:29,420 --> 00:19:34,990 The minus got added to the elements of the mass matrix, 217 00:19:34,990 --> 00:19:36,990 but not to the K metrics. 218 00:19:36,990 --> 00:19:41,880 The K metrics goes shows through as is. 219 00:19:41,880 --> 00:19:48,960 Now, the question on the floor is what we do with this? 220 00:19:48,960 --> 00:19:52,820 Can everybody appreciate that-- get out 221 00:19:52,820 --> 00:19:56,580 of the spring mass business and look at this 222 00:19:56,580 --> 00:19:58,760 from a math point of view? 223 00:19:58,760 --> 00:20:01,050 Does everyone appreciate that this 224 00:20:01,050 --> 00:20:06,060 is a set of linear equations? 225 00:20:06,060 --> 00:20:09,850 There's the old AX equal B kind of thing. 226 00:20:09,850 --> 00:20:21,860 And if you recall, to solve, what we're basically going 227 00:20:21,860 --> 00:20:25,150 to do is solve for A1 and A2. 228 00:20:25,150 --> 00:20:26,830 That's the game we're playing here. 229 00:20:30,690 --> 00:20:39,290 And if you recall from your math course, the determinant of this 230 00:20:39,290 --> 00:20:41,680 has got to equal 0. 231 00:20:41,680 --> 00:20:44,950 So let me simply repeat it here. 232 00:20:44,950 --> 00:21:06,940 The determinant of has got to equal 0. 233 00:21:06,940 --> 00:21:10,370 And you recall, the determinant is for at least the two 234 00:21:10,370 --> 00:21:12,340 by two you can do it by hand more or less. 235 00:21:12,340 --> 00:21:17,075 It's the cross products with appropriate sign. 236 00:21:26,550 --> 00:21:29,390 And I'm sparing you some algebra here, 237 00:21:29,390 --> 00:21:33,150 but trust me when you do this, this is what you get. 238 00:21:42,440 --> 00:21:52,780 M squared omega 4 minus 3KM omega squared, plus K2. 239 00:21:52,780 --> 00:21:54,760 That's it right here. 240 00:21:54,760 --> 00:22:02,040 This little guy-- oh, all right. 241 00:22:02,040 --> 00:22:03,690 I can't do that anymore. 242 00:22:03,690 --> 00:22:04,670 I know, it's this one. 243 00:22:24,870 --> 00:22:27,213 That is called the characteristic equation. 244 00:22:43,800 --> 00:22:45,660 So here's the first answer. 245 00:23:10,162 --> 00:23:12,120 AUDIENCE: [INAUDIBLE] should that be K squared? 246 00:23:12,120 --> 00:23:12,995 PROFESSOR: I'm sorry. 247 00:23:12,995 --> 00:23:15,520 That's a typo. 248 00:23:15,520 --> 00:23:20,697 That's simply K. 249 00:23:20,697 --> 00:23:22,030 AUDIENCE: [INAUDIBLE] K squared? 250 00:23:22,030 --> 00:23:23,571 PROFESSOR: Or it's K squared, rather. 251 00:23:23,571 --> 00:23:24,710 Sorry. 252 00:23:24,710 --> 00:23:25,210 Thank you. 253 00:23:32,473 --> 00:23:34,437 AUDIENCE: I think in the above line, 254 00:23:34,437 --> 00:23:38,717 the determinant-- the upper left-- should had a 2K. 255 00:23:38,717 --> 00:23:40,300 PROFESSOR: Oh, this is 2K, absolutely. 256 00:23:44,110 --> 00:23:45,950 All right, 2K. 257 00:23:45,950 --> 00:23:48,550 Good enough? 258 00:23:48,550 --> 00:23:50,250 All right, thank you. 259 00:23:50,250 --> 00:23:52,070 So let's send this to the top. 260 00:23:54,780 --> 00:24:05,070 So the roots of the characteristic 261 00:24:05,070 --> 00:24:06,846 are the natural frequencies. 262 00:24:34,040 --> 00:24:35,717 Let's do this this way. 263 00:24:58,247 --> 00:24:59,080 Did I do that right? 264 00:24:59,080 --> 00:25:00,020 Yeah, plus or minus. 265 00:25:00,020 --> 00:25:05,450 So the situation is that when you apply the quadratic formula 266 00:25:05,450 --> 00:25:10,040 to that characteristic equation to find the values of omega 267 00:25:10,040 --> 00:25:12,440 for which that equation is satisfied, 268 00:25:12,440 --> 00:25:15,970 those omegas that come out are the natural frequencies. 269 00:25:15,970 --> 00:25:20,540 They are the quantities we seek. 270 00:25:20,540 --> 00:25:26,160 And what that yields, as you can see from the plus or minus 271 00:25:26,160 --> 00:25:28,056 here, there are two of them. 272 00:25:32,200 --> 00:25:33,810 I'll write the whole thing out here. 273 00:25:57,600 --> 00:25:58,880 And these are numerically. 274 00:26:15,300 --> 00:26:17,630 OK, everybody see that? 275 00:26:17,630 --> 00:26:22,100 So here are our two natural frequencies. 276 00:26:22,100 --> 00:26:26,485 Here's the first one-- excuse me, that's not right either. 277 00:26:40,115 --> 00:26:40,865 That's the square. 278 00:26:54,540 --> 00:27:02,540 OK So these are our natural frequencies, once again 279 00:27:02,540 --> 00:27:06,940 for this special case where the masses are equal 280 00:27:06,940 --> 00:27:08,800 and the springs are equal. 281 00:27:08,800 --> 00:27:11,530 Anybody recognize that number, 0.618, 282 00:27:11,530 --> 00:27:13,920 for all you fuss budgets? 283 00:27:16,790 --> 00:27:18,760 Ring any bells? 284 00:27:18,760 --> 00:27:21,880 Any number freaks here? 285 00:27:21,880 --> 00:27:23,960 No? 286 00:27:23,960 --> 00:27:25,490 I heard it. 287 00:27:25,490 --> 00:27:26,370 That's it. 288 00:27:26,370 --> 00:27:28,640 Exactly, nice job. 289 00:27:28,640 --> 00:27:32,000 The golden mean, the golden ratio. 290 00:27:32,000 --> 00:27:37,870 Also, let me simply say if there are-- 291 00:27:37,870 --> 00:27:40,670 as far as the number of things-- if there 292 00:27:40,670 --> 00:27:44,240 are n degrees of freedom. 293 00:27:44,240 --> 00:27:47,800 There are n natural frequencies. 294 00:27:59,569 --> 00:28:02,230 What else? 295 00:28:02,230 --> 00:28:03,230 So that's that. 296 00:28:09,620 --> 00:28:26,043 So now it's time to get to this notion of the natural modes. 297 00:28:38,177 --> 00:28:40,010 Let me say, we've got to go all the way back 298 00:28:40,010 --> 00:28:41,250 to this set over here. 299 00:28:41,250 --> 00:28:45,510 If you take the first row of this matrix equation-- 300 00:28:45,510 --> 00:28:53,160 that's the first of the equations of motion-- 301 00:28:53,160 --> 00:28:57,120 and you make that assumption of the harmonic motion in there. 302 00:29:55,220 --> 00:29:56,680 Does everybody see that? 303 00:29:56,680 --> 00:30:01,200 What we've done is we've taken basically the first row 304 00:30:01,200 --> 00:30:04,440 of that expression right up there 305 00:30:04,440 --> 00:30:08,620 and formed the amplitude ratio A1 over A2. 306 00:30:08,620 --> 00:30:11,350 What we're doing is we've found the omegas. 307 00:30:11,350 --> 00:30:13,500 You remember, just review the bidding. 308 00:30:13,500 --> 00:30:16,670 Our original assumption was harmonic 309 00:30:16,670 --> 00:30:20,370 motion, that is to say all the displacements are 310 00:30:20,370 --> 00:30:24,720 moving in synchrony as it were. 311 00:30:24,720 --> 00:30:27,060 The same sinusoidal frequency, we've 312 00:30:27,060 --> 00:30:31,310 just found what frequencies those are. 313 00:30:31,310 --> 00:30:34,450 There are two of them, and they're right there. 314 00:30:34,450 --> 00:30:38,710 Now we're after these guys. 315 00:30:38,710 --> 00:30:41,960 Now we're after the relative magnitudes 316 00:30:41,960 --> 00:30:46,270 or the relative amplitudes of A1 and A2. 317 00:30:46,270 --> 00:30:52,020 And we from one of the equations isolated one of those. 318 00:30:52,020 --> 00:31:01,170 And let me just say, if you plug these back in, 319 00:31:01,170 --> 00:31:42,450 plug in the first one, you'll get oddly enough 1.618, 320 00:31:42,450 --> 00:31:44,250 These amplitude ratios. 321 00:32:08,450 --> 00:32:11,400 Are the so-called natural modes. 322 00:32:11,400 --> 00:32:16,060 And I think you can appreciate that this is the first one, 323 00:32:16,060 --> 00:32:17,855 and this is the second one. 324 00:32:29,490 --> 00:32:31,300 Any questions so far? 325 00:32:31,300 --> 00:32:32,680 Wonderful. 326 00:32:32,680 --> 00:32:33,370 Hearing none. 327 00:32:33,370 --> 00:32:35,497 Yes ma'am, Sara? 328 00:32:35,497 --> 00:32:36,372 AUDIENCE: [INAUDIBLE] 329 00:32:40,540 --> 00:32:43,610 PROFESSOR: You see this amplitude ratio. 330 00:32:43,610 --> 00:32:45,170 You saw how we got that. 331 00:32:45,170 --> 00:32:48,450 You see that the right hand side has got system parameter, Ks 332 00:32:48,450 --> 00:32:50,380 and Ms, and stuff like that. 333 00:32:50,380 --> 00:32:54,530 But this is the ringer, omega. 334 00:32:54,530 --> 00:32:58,090 This amplitude ratio is expressed in part 335 00:32:58,090 --> 00:32:59,930 in terms of omega. 336 00:32:59,930 --> 00:33:04,050 So what omega-- there's no ambiguity as to Ks and Ms, 337 00:33:04,050 --> 00:33:05,890 but what omega? 338 00:33:05,890 --> 00:33:09,640 Well the answer is, when we plug in this one, 339 00:33:09,640 --> 00:33:11,040 you get this answer. 340 00:33:11,040 --> 00:33:13,530 When you plug-in this one, you get this answer. 341 00:33:13,530 --> 00:33:19,060 So while we're at it-- Sara, want to hazard a guess? 342 00:33:19,060 --> 00:33:21,730 How many natural mode do you think we've got? 343 00:33:21,730 --> 00:33:22,420 Yeah, exactly. 344 00:33:30,200 --> 00:33:34,970 So you're going to have one of these 345 00:33:34,970 --> 00:33:36,970 for each degree of freedom. 346 00:33:39,940 --> 00:33:45,420 Let me just point out a couple of elements here, 347 00:33:45,420 --> 00:33:48,070 and then I'll show you a demonstration because we 348 00:33:48,070 --> 00:33:49,295 have to have some fun today. 349 00:33:56,970 --> 00:33:58,900 These are point of informations. 350 00:33:58,900 --> 00:34:06,460 They're ratios, not absolute magnitudes. 351 00:34:12,080 --> 00:34:13,219 That's number one. 352 00:34:13,219 --> 00:34:16,770 The second is-- I already told you, they got the same number. 353 00:34:16,770 --> 00:34:24,699 OK, each natural mode is associated 354 00:34:24,699 --> 00:34:38,330 with a particular natural frequency. 355 00:34:38,330 --> 00:34:39,710 This one goes with that one. 356 00:34:39,710 --> 00:34:42,800 This one goes with that one. 357 00:34:42,800 --> 00:34:54,780 And once again, they're associated with-- yeah, 358 00:34:54,780 --> 00:34:56,139 let me say that. 359 00:35:01,080 --> 00:35:03,000 I need another board. 360 00:35:03,000 --> 00:35:04,490 Let's just go over here. 361 00:36:08,600 --> 00:37:13,900 So in a sense-- this is decouple. 362 00:37:35,450 --> 00:37:40,106 Decouple essentially into independent subsystems. 363 00:38:39,390 --> 00:38:53,020 So in general, what the system's response looks like is-- 364 00:38:53,020 --> 00:38:55,870 I'm talking about the one in front of us here. 365 00:38:55,870 --> 00:39:03,100 This special case, where the masses and springs are equal. 366 00:39:15,600 --> 00:39:17,175 I think there's a minus sign in here. 367 00:39:26,870 --> 00:39:27,932 Does this come up? 368 00:39:27,932 --> 00:39:28,432 Wonderful. 369 00:40:07,020 --> 00:40:07,798 That's it. 370 00:40:13,360 --> 00:40:14,360 Does everybody see that? 371 00:40:14,360 --> 00:40:15,070 Yes, sir. 372 00:40:15,070 --> 00:40:17,480 AUDIENCE: [INAUDIBLE]. 373 00:40:17,480 --> 00:40:18,892 PROFESSOR: I'm sorry? 374 00:40:18,892 --> 00:40:21,720 AUDIENCE: What does it say under that first bullet point? 375 00:40:21,720 --> 00:40:22,390 PROFESSOR: Here? 376 00:40:22,390 --> 00:40:23,110 AUDIENCE: These describe the situation. 377 00:40:23,110 --> 00:40:24,734 PROFESSOR: These describe the situation 378 00:40:24,734 --> 00:40:29,340 in which the entire system is oscillating at. 379 00:40:35,600 --> 00:40:37,880 It's the second bullet here. 380 00:40:37,880 --> 00:40:38,650 Thank you. 381 00:40:44,110 --> 00:40:44,860 AUDIENCE: At what? 382 00:40:44,860 --> 00:40:46,026 PROFESSOR: At one frequency. 383 00:40:48,947 --> 00:40:51,030 Sorry, I'm just getting a little tired of writing. 384 00:41:04,140 --> 00:41:10,670 So, any other questions, problems, complaints? 385 00:41:10,670 --> 00:41:11,655 All right. 386 00:41:11,655 --> 00:41:11,980 AUDIENCE: I have a question. 387 00:41:11,980 --> 00:41:13,032 PROFESSOR: Yes, sir. 388 00:41:13,032 --> 00:41:13,907 AUDIENCE: [INAUDIBLE] 389 00:41:19,300 --> 00:41:21,580 PROFESSOR: That's correct. 390 00:41:21,580 --> 00:41:23,450 Then, let's see. 391 00:41:23,450 --> 00:41:26,100 Then there's a mistake right here. 392 00:41:26,100 --> 00:41:28,370 Thank you. 393 00:41:28,370 --> 00:41:32,060 Yeah, because that's the way it came out. 394 00:41:32,060 --> 00:41:40,030 When you plug omega 2 having this value into here, 395 00:41:40,030 --> 00:41:42,455 the amplitude ratio comes out minus. 396 00:41:46,420 --> 00:41:47,216 Fair enough? 397 00:41:49,739 --> 00:41:51,030 It threw me there for a minute. 398 00:41:51,030 --> 00:41:56,000 I thought you were going to say, why is the minus sign is there, 399 00:41:56,000 --> 00:42:05,860 rather than you could have had minus 1.618 and plus 1. 400 00:42:05,860 --> 00:42:10,740 And the answer is no reason, because these are ratios. 401 00:42:10,740 --> 00:42:11,903 Yeah, Kaitlin? 402 00:42:11,903 --> 00:42:15,525 AUDIENCE: But shouldn't-- when we go back and look at what you 403 00:42:15,525 --> 00:42:18,665 wrote down, it's [INAUDIBLE]. 404 00:42:18,665 --> 00:42:21,580 I don't understand how that [INAUDIBLE]. 405 00:42:21,580 --> 00:42:23,055 PROFESSOR: I'm sorry, say again? 406 00:42:23,055 --> 00:42:24,030 AUDIENCE: Never mind. 407 00:42:24,030 --> 00:42:24,821 PROFESSOR: Find it? 408 00:42:24,821 --> 00:42:28,580 Yeah, they're ratios, It's just as simple as that. 409 00:42:28,580 --> 00:42:32,080 So you multiply them by any number and it still works. 410 00:42:32,080 --> 00:42:35,750 I'll actually show you here in a second. 411 00:42:35,750 --> 00:42:38,020 At least, I believe that's the case. 412 00:42:38,020 --> 00:42:39,510 We'll just see here in a second. 413 00:42:39,510 --> 00:42:40,580 OK, questions? 414 00:42:40,580 --> 00:42:42,130 Comments? 415 00:42:42,130 --> 00:42:43,850 All right. 416 00:42:43,850 --> 00:42:45,760 Now is the time. 417 00:42:45,760 --> 00:42:50,370 Could I bring up the side board here? 418 00:42:50,370 --> 00:42:56,180 Let me show you-- anybody here taken 2086? 419 00:42:56,180 --> 00:42:56,680 Wonderful. 420 00:42:56,680 --> 00:42:57,640 I've got one person? 421 00:42:57,640 --> 00:42:58,680 Great. 422 00:42:58,680 --> 00:43:03,597 Anyway, I believe in 2086, don't they teach you MATLAB? 423 00:43:03,597 --> 00:43:04,930 Isn't that the weapon of choice? 424 00:43:04,930 --> 00:43:09,202 OK, that's the program I'm using here, MATLAB. 425 00:43:09,202 --> 00:43:10,910 For those of you who haven't seen it yet, 426 00:43:10,910 --> 00:43:13,052 it is definitely a mixed bag. 427 00:43:13,052 --> 00:43:14,510 I don't know how you feel about it. 428 00:43:14,510 --> 00:43:17,840 It's very-- yeah-- it's very powerful. 429 00:43:17,840 --> 00:43:19,720 It stands for Matrix Laboratory. 430 00:43:19,720 --> 00:43:22,490 It was written, I don't know, 20, 30 years ago here, 431 00:43:22,490 --> 00:43:27,770 I believe, at MIT by people who were into matrices, into matrix 432 00:43:27,770 --> 00:43:28,310 algebra. 433 00:43:28,310 --> 00:43:31,310 And it's kind of command line oriented. 434 00:43:31,310 --> 00:43:33,550 The good news, it's very powerful. 435 00:43:33,550 --> 00:43:35,890 Whatever you want to do, you can do in MATLAB. 436 00:43:35,890 --> 00:43:39,820 The bad news is, the user interface stinks. 437 00:43:39,820 --> 00:43:41,840 The language is very difficult to learn. 438 00:43:41,840 --> 00:43:44,880 It's even harder to remember. 439 00:43:44,880 --> 00:43:48,550 So with that rousing endorsement, 440 00:43:48,550 --> 00:43:53,430 let me show you what we've got here. 441 00:43:53,430 --> 00:44:01,550 This is a program I've-- is that font readable by you guys? 442 00:44:01,550 --> 00:44:02,540 No? 443 00:44:02,540 --> 00:44:03,323 No? 444 00:44:03,323 --> 00:44:05,740 AUDIENCE: [INAUDIBLE]. 445 00:44:05,740 --> 00:44:07,512 PROFESSOR: I'm sorry? 446 00:44:07,512 --> 00:44:10,430 AUDIENCE: [INAUDIBLE]. 447 00:44:10,430 --> 00:44:11,060 PROFESSOR: Yes. 448 00:44:11,060 --> 00:44:16,015 Well, I believe I can-- here we go, fonts. 449 00:44:18,620 --> 00:44:21,410 Upping the fonts is kind of a mixed bag, 450 00:44:21,410 --> 00:44:28,330 because you get bigger letters but they're. 451 00:44:28,330 --> 00:44:31,430 OK, how's that? 452 00:44:31,430 --> 00:44:34,230 So here's the situation. 453 00:44:34,230 --> 00:44:35,700 This is a MATLAB program. 454 00:44:35,700 --> 00:44:39,240 And I'll explain to you what it does as we go. 455 00:44:39,240 --> 00:44:42,060 Let me see if my little cursor-- my cursor's here, 456 00:44:42,060 --> 00:44:43,080 but I can't see it. 457 00:44:43,080 --> 00:44:45,490 All right, here's the system parameters. 458 00:44:45,490 --> 00:44:47,940 Once again, we're doing a simple spring mass-- 459 00:44:47,940 --> 00:44:52,240 this simplified spring mass system, exactly the one 460 00:44:52,240 --> 00:44:55,040 we've done here. 461 00:44:55,040 --> 00:45:02,120 When I wrote it, you'll see I generalized it to do this guy. 462 00:45:02,120 --> 00:45:04,780 So we got M1 and M2, K1 and K2. 463 00:45:04,780 --> 00:45:09,400 But if you'll notice, you see here their values are equal. 464 00:45:09,400 --> 00:45:11,560 We've got the mass at one kilogram each. 465 00:45:11,560 --> 00:45:17,240 And we've got 10 newtons per meter on each of the springs. 466 00:45:17,240 --> 00:45:20,090 Everybody appreciate that this system's numbers 467 00:45:20,090 --> 00:45:25,690 that we're putting in here match our case here, K over M? 468 00:45:25,690 --> 00:45:26,540 OK. 469 00:45:26,540 --> 00:45:28,990 And you can see here, we've defined-- 470 00:45:28,990 --> 00:45:31,390 and again, let me just say I'm not 471 00:45:31,390 --> 00:45:34,365 trying to sell you on MATLAB. 472 00:45:34,365 --> 00:45:36,240 I don't want to leave you with the impression 473 00:45:36,240 --> 00:45:39,835 that we expect you to be able to instantly become 474 00:45:39,835 --> 00:45:41,260 a user of MATLAB. 475 00:45:41,260 --> 00:45:47,580 This is simply to illustrate the point of the lecture here. 476 00:45:47,580 --> 00:45:50,110 Here is the M, the system matrix. 477 00:45:50,110 --> 00:45:51,910 There's the K matrix. 478 00:45:51,910 --> 00:45:57,850 And I'll show you the eigenvalue and eigenvector thing later. 479 00:45:57,850 --> 00:45:59,867 But let me-- take my word for it. 480 00:45:59,867 --> 00:46:00,450 See this here? 481 00:46:00,450 --> 00:46:08,910 Ode45 is a cryptic allusion to the Runge-Kutta algorithm, 482 00:46:08,910 --> 00:46:16,040 fourth order Runge-Kutta that is the workhorse for integrating 483 00:46:16,040 --> 00:46:18,380 differential equations. 484 00:46:18,380 --> 00:46:21,000 And so let me just run this. 485 00:46:21,000 --> 00:46:23,690 And what I've got here is, here's the point 486 00:46:23,690 --> 00:46:27,375 I wanted you to get here, because I'll 487 00:46:27,375 --> 00:46:29,800 bet you can't see that cursor either. 488 00:46:29,800 --> 00:46:32,410 Yes, anyway, see this right here? 489 00:46:32,410 --> 00:46:37,342 tspan is the time scale and the time step, defined up here. 490 00:46:37,342 --> 00:46:39,300 But these are basically the initial conditions. 491 00:46:39,300 --> 00:46:40,260 See it here? 492 00:46:40,260 --> 00:46:44,260 X1, X1 dot, X2, X2 dot. 493 00:46:44,260 --> 00:46:46,390 So here's the first one. 494 00:46:46,390 --> 00:46:49,770 This is a 0.618 is for the X1. 495 00:46:49,770 --> 00:46:53,720 And 1 is for X2. 496 00:46:53,720 --> 00:46:55,880 Everybody appreciate that? 497 00:46:55,880 --> 00:46:57,245 Got it? 498 00:46:57,245 --> 00:46:57,745 OK. 499 00:47:00,700 --> 00:47:04,060 If these are the initial conditions, what I've done, 500 00:47:04,060 --> 00:47:07,260 I have artfully chosen the initial conditions 501 00:47:07,260 --> 00:47:10,575 to have the same ratio. 502 00:47:14,870 --> 00:47:16,590 What do you expect is going to happen? 503 00:47:16,590 --> 00:47:18,940 When I turn this thing-- I've got a simulation here. 504 00:47:18,940 --> 00:47:20,700 I'm going to run this, and you're actually 505 00:47:20,700 --> 00:47:22,177 going to see it. 506 00:47:22,177 --> 00:47:23,760 What do you think you're going to see? 507 00:47:27,710 --> 00:47:29,880 It's a two spring, two mass system. 508 00:47:29,880 --> 00:47:32,620 What I've done is I've displaced the two masses. 509 00:47:36,805 --> 00:47:38,200 AUDIENCE: [INAUDIBLE] 510 00:47:38,200 --> 00:47:39,770 PROFESSOR: They'll certainly have an amplitude, because I'm 511 00:47:39,770 --> 00:47:40,910 putting it in there. 512 00:47:40,910 --> 00:47:43,780 That's the initial condition. 513 00:47:43,780 --> 00:47:49,255 The question is, what frequency you think they'll oscillate at? 514 00:47:49,255 --> 00:47:50,130 AUDIENCE: [INAUDIBLE] 515 00:47:50,130 --> 00:47:51,177 PROFESSOR: Pardon? 516 00:47:51,177 --> 00:47:53,037 AUDIENCE: [INAUDIBLE] 517 00:47:53,037 --> 00:47:54,620 PROFESSOR: Each of them will oscillate 518 00:47:54,620 --> 00:47:56,420 with the same frequency, for sure, but what 519 00:47:56,420 --> 00:47:57,670 do you think it's going to be? 520 00:47:57,670 --> 00:47:59,060 AUDIENCE: That one. 521 00:47:59,060 --> 00:48:01,080 PROFESSOR: It's going to be that one. 522 00:48:01,080 --> 00:48:03,550 So off we go. 523 00:48:03,550 --> 00:48:09,705 So let us hope that yours truly's program worked. 524 00:48:16,320 --> 00:48:17,540 Here we go. 525 00:48:17,540 --> 00:48:19,860 Oh, look at that. 526 00:48:19,860 --> 00:48:21,940 [INAUDIBLE], please interpret that for me. 527 00:48:21,940 --> 00:48:24,700 What do you see there? 528 00:48:24,700 --> 00:48:25,770 Hang on a second. 529 00:48:25,770 --> 00:48:27,980 Let me blow it up so you can see it. 530 00:48:27,980 --> 00:48:30,470 Ooh, isn't that pretty? 531 00:48:30,470 --> 00:48:38,483 And I believe the blue is X1 and the green is X2. 532 00:48:42,250 --> 00:48:43,994 See? 533 00:48:43,994 --> 00:48:44,660 Everybody agree? 534 00:48:44,660 --> 00:48:46,680 Everyone appreciate what's going on? 535 00:48:46,680 --> 00:48:51,440 You pull them both at slightly different-- 536 00:48:51,440 --> 00:48:57,080 you basically used the first natural mode 537 00:48:57,080 --> 00:48:58,680 as the initial condition. 538 00:48:58,680 --> 00:49:01,189 And sure enough, they oscillate together. 539 00:49:01,189 --> 00:49:02,730 They oscillate at the same frequency. 540 00:49:02,730 --> 00:49:06,030 They oscillate at that frequency. 541 00:49:06,030 --> 00:49:08,490 Let me just see-- I just want to make 542 00:49:08,490 --> 00:49:11,740 sure we get the full value out of this thing. 543 00:49:11,740 --> 00:49:14,210 Well, of course you can't see it anymore 544 00:49:14,210 --> 00:49:17,139 because our numbers are so big. 545 00:49:17,139 --> 00:49:17,930 Well, that's great. 546 00:49:17,930 --> 00:49:21,760 Anyway, take my word for it at-- oh, here it is. 547 00:49:21,760 --> 00:49:27,150 The period for the first natural frequency-- or I 548 00:49:27,150 --> 00:49:35,900 guess it's the second-- it should be like 1.2. 549 00:49:35,900 --> 00:49:36,880 Or is it 3? 550 00:49:36,880 --> 00:49:40,520 Yeah, I'm sorry, the period is 3.2. 551 00:49:40,520 --> 00:49:42,005 And sure enough, there it is. 552 00:49:42,005 --> 00:49:45,291 It's about 3. 553 00:49:45,291 --> 00:49:45,790 3.2. 554 00:49:45,790 --> 00:49:47,380 Fabulous. 555 00:49:47,380 --> 00:49:48,590 Everybody got it? 556 00:49:48,590 --> 00:49:51,570 OK, now watch closely. 557 00:49:51,570 --> 00:49:52,855 Let me see if I can do this. 558 00:49:52,855 --> 00:49:55,210 This requires a little dexterity, which 559 00:49:55,210 --> 00:49:58,770 is always a short supply here. 560 00:49:58,770 --> 00:50:01,000 I have to hit this and this. 561 00:50:08,060 --> 00:50:08,720 Make sense? 562 00:50:08,720 --> 00:50:11,190 That's what it actually looks like. 563 00:50:11,190 --> 00:50:15,360 They're both oscillating at the same natural frequency, 564 00:50:15,360 --> 00:50:17,300 going up and down together. 565 00:50:17,300 --> 00:50:20,310 But they have different amplitudes. 566 00:50:20,310 --> 00:50:23,280 So one's bigger than the other. 567 00:50:23,280 --> 00:50:24,920 So that's what it looks like. 568 00:50:24,920 --> 00:50:26,000 Questions? 569 00:50:26,000 --> 00:50:27,590 Christina, you good? 570 00:50:27,590 --> 00:50:30,480 Clear enough? 571 00:50:30,480 --> 00:50:32,190 Wonderful. 572 00:50:32,190 --> 00:50:38,800 So let's go to our program. 573 00:50:38,800 --> 00:50:46,730 And instead of that set of initial conditions, 574 00:50:46,730 --> 00:50:48,725 we'll do the other. 575 00:50:48,725 --> 00:50:49,600 Read them to us here. 576 00:50:49,600 --> 00:50:51,170 What are the initial conditions here? 577 00:50:55,590 --> 00:50:57,110 AUDIENCE: It's 1.618. 578 00:50:57,110 --> 00:50:58,940 PROFESSOR: That's right, it's this guy. 579 00:50:58,940 --> 00:50:59,731 AUDIENCE: That guy. 580 00:50:59,731 --> 00:51:02,360 PROFESSOR: It's this guy. 581 00:51:02,360 --> 00:51:04,020 It's this ratio. 582 00:51:04,020 --> 00:51:06,730 So I basically arbitrarily chose, is it 583 00:51:06,730 --> 00:51:07,585 the negative first? 584 00:51:07,585 --> 00:51:08,084 No. 585 00:51:10,750 --> 00:51:14,870 I chose that one over there, 1.618. 586 00:51:14,870 --> 00:51:17,350 And then a minus 1 for the second one. 587 00:51:17,350 --> 00:51:18,310 Fair enough? 588 00:51:18,310 --> 00:51:19,560 OK, there it goes. 589 00:51:19,560 --> 00:51:22,880 We've got to save it and make sure we got it. 590 00:51:22,880 --> 00:51:26,440 So again, you got a clue what's going to happen here? 591 00:51:31,630 --> 00:51:32,260 Here we go. 592 00:51:32,260 --> 00:51:32,760 Boom. 593 00:51:35,190 --> 00:51:36,720 Look at that. 594 00:51:36,720 --> 00:51:39,400 What's going on there? 595 00:51:39,400 --> 00:51:42,260 Yikes. 596 00:51:42,260 --> 00:51:43,400 Explain me. 597 00:51:43,400 --> 00:51:44,870 Is that good, bad, indifferent? 598 00:51:44,870 --> 00:51:45,500 Is it right? 599 00:51:45,500 --> 00:51:46,562 Wrong? 600 00:51:46,562 --> 00:51:47,978 AUDIENCE: The way the system acts, 601 00:51:47,978 --> 00:51:49,350 it has a higher frequency. 602 00:51:49,350 --> 00:51:51,080 PROFESSOR: Yeah, exactly. 603 00:51:51,080 --> 00:51:51,640 Two things. 604 00:51:51,640 --> 00:51:53,520 One is, they're out of phase. 605 00:51:53,520 --> 00:51:54,680 They're doing this. 606 00:51:54,680 --> 00:51:56,490 One's going this way, and the other's 607 00:51:56,490 --> 00:51:58,780 going the other at different amplitudes 608 00:51:58,780 --> 00:51:59,920 but the same frequency. 609 00:51:59,920 --> 00:52:03,091 But the frequency in question is higher than the previous. 610 00:52:03,091 --> 00:52:04,590 AUDIENCE: Why are they out of phase? 611 00:52:04,590 --> 00:52:05,465 PROFESSOR: I'm sorry? 612 00:52:05,465 --> 00:52:07,420 AUDIENCE: Why are they opposite of each other? 613 00:52:07,420 --> 00:52:09,794 PROFESSOR: [INAUDIBLE], why are they opposite each other? 614 00:52:14,210 --> 00:52:15,630 Because we made them that way. 615 00:52:15,630 --> 00:52:19,545 We said, that's the initial condition. 616 00:52:19,545 --> 00:52:20,420 Does that make sense? 617 00:52:23,380 --> 00:52:24,330 That minus sign does. 618 00:52:27,480 --> 00:52:31,370 One starts out, and one starts in. 619 00:52:31,370 --> 00:52:32,659 And they do that. 620 00:52:32,659 --> 00:52:33,200 Clear enough? 621 00:52:36,070 --> 00:52:39,110 Now what's going to happen if we plain just choose 622 00:52:39,110 --> 00:52:42,470 any old initial condition? 623 00:52:42,470 --> 00:52:44,470 These were special. 624 00:52:44,470 --> 00:52:47,620 We worked like a dog to compute these, 625 00:52:47,620 --> 00:52:52,410 so that the system would decouple in that way. 626 00:52:52,410 --> 00:53:01,640 So what if we-- now let me put that back. 627 00:53:01,640 --> 00:53:04,290 Now look at this one. 628 00:53:04,290 --> 00:53:05,450 All right, look at that. 629 00:53:09,410 --> 00:53:10,263 Read that to me. 630 00:53:13,510 --> 00:53:14,550 AUDIENCE: [INAUDIBLE] 631 00:53:14,550 --> 00:53:15,216 PROFESSOR: Yeah. 632 00:53:15,216 --> 00:53:19,320 So that says the initial condition for the first mass 633 00:53:19,320 --> 00:53:21,940 is 1 and whatever that is. 634 00:53:21,940 --> 00:53:23,780 One whatever that is. 635 00:53:23,780 --> 00:53:26,450 The second masses' initial condition is half 636 00:53:26,450 --> 00:53:28,150 that in the same direction. 637 00:53:28,150 --> 00:53:29,199 Both positive. 638 00:53:29,199 --> 00:53:30,990 So Christina, they're going to go together. 639 00:53:33,530 --> 00:53:37,170 But [INAUDIBLE], at what frequency? 640 00:53:37,170 --> 00:53:40,450 Any idea what it's going to look like? 641 00:53:40,450 --> 00:53:43,140 If you do, you're a better man than I, 642 00:53:43,140 --> 00:53:48,410 because what you're going to see here is that. 643 00:53:48,410 --> 00:53:51,040 It's this thing right here. 644 00:53:51,040 --> 00:53:54,210 It's that expression right there. 645 00:53:54,210 --> 00:53:57,140 And here's what it looks like. 646 00:54:01,190 --> 00:54:04,570 Did I stop the-- oh, wait a minute. 647 00:54:04,570 --> 00:54:07,786 Did I ever show you that before? 648 00:54:07,786 --> 00:54:09,410 I think I forgot to show you the other. 649 00:54:09,410 --> 00:54:10,720 Anyway, not to worry. 650 00:54:13,300 --> 00:54:14,130 Hang on a second. 651 00:54:14,130 --> 00:54:16,162 I've got to stop this guy. 652 00:54:16,162 --> 00:54:18,560 First I have to find my finger. 653 00:54:22,240 --> 00:54:23,050 There it goes. 654 00:54:23,050 --> 00:54:24,930 That's the previous case. 655 00:54:24,930 --> 00:54:29,140 When they're out of phase, different magnitudes, 656 00:54:29,140 --> 00:54:30,660 going in opposite directions. 657 00:54:30,660 --> 00:54:34,060 And you can see, they're going at a higher frequency 658 00:54:34,060 --> 00:54:35,620 than before. 659 00:54:35,620 --> 00:54:36,230 Make sense? 660 00:54:39,340 --> 00:54:45,100 So now we are-- just to refresh your memory-- 661 00:54:45,100 --> 00:54:48,410 now we're going for the third case, in which there's 662 00:54:48,410 --> 00:54:49,150 nothing special. 663 00:54:49,150 --> 00:54:54,650 We just picked a couple of initial conditions 664 00:54:54,650 --> 00:54:57,960 out of a hat. 665 00:54:57,960 --> 00:54:59,800 And here we go. 666 00:54:59,800 --> 00:55:01,698 Oops, I think not. 667 00:55:07,180 --> 00:55:10,070 I think that's the previous case. 668 00:55:10,070 --> 00:55:12,210 So let's go here. 669 00:55:12,210 --> 00:55:15,230 This is another wonderful thing about MATLAB 670 00:55:15,230 --> 00:55:19,890 is nothing happens until you save it. 671 00:55:22,510 --> 00:55:24,435 So we were just running the previous case. 672 00:55:27,500 --> 00:55:28,130 Nasty. 673 00:55:28,130 --> 00:55:28,900 Look at this. 674 00:55:32,130 --> 00:55:33,765 All right, can everybody see that? 675 00:55:36,940 --> 00:55:40,310 If you can interpret this, you're smarter than I am. 676 00:55:40,310 --> 00:55:46,710 But what this is, this is simply this expression over here. 677 00:55:46,710 --> 00:55:49,640 It's this expression for just some arbitrary 678 00:55:49,640 --> 00:55:50,680 initial condition. 679 00:55:50,680 --> 00:55:53,240 Do you see that that behavior though? 680 00:55:53,240 --> 00:55:55,780 Each of them, they're going together kind of, 681 00:55:55,780 --> 00:56:00,954 but they-- anyway, watch this. 682 00:56:00,954 --> 00:56:02,870 Here's what the simulation of that looks like. 683 00:56:15,760 --> 00:56:18,820 What the heck is that? 684 00:56:18,820 --> 00:56:20,330 Well anyway, the point of the story 685 00:56:20,330 --> 00:56:23,210 is that multiple degrees of freedom 686 00:56:23,210 --> 00:56:27,305 system in general's response can be arbitrarily complicated. 687 00:56:27,305 --> 00:56:28,680 It's not arbitrarily complicated, 688 00:56:28,680 --> 00:56:31,090 but pretty complicated. 689 00:56:31,090 --> 00:56:34,960 You'll get, in general if it's an nth order system, if you 690 00:56:34,960 --> 00:56:37,490 don't know anything about the worst case, 691 00:56:37,490 --> 00:56:40,700 you'll see four frequencies in there. 692 00:56:40,700 --> 00:56:44,810 And they're all mixed together in some mystical way that's 693 00:56:44,810 --> 00:56:46,270 unknown to you. 694 00:56:46,270 --> 00:56:47,130 Fair enough? 695 00:56:47,130 --> 00:56:54,040 And it's only when you reach the natural modes that you actually 696 00:56:54,040 --> 00:56:58,500 find out what is going on here. 697 00:56:58,500 --> 00:57:03,570 Well now I have to turn your attention to this guy. 698 00:57:03,570 --> 00:57:11,710 This is made by Professor Vandiver's machinist, 699 00:57:11,710 --> 00:57:15,000 a perfect example of a second order system. 700 00:57:15,000 --> 00:57:17,170 And I bring it to your attention here 701 00:57:17,170 --> 00:57:20,380 for two-- at the end of the day what we're going to do 702 00:57:20,380 --> 00:57:25,000 is I'm going to demonstrate exactly what I just 703 00:57:25,000 --> 00:57:28,580 did for the textbook case, the textbook system. 704 00:57:28,580 --> 00:57:33,240 I want to demonstrate exactly the same thing for this guy, 705 00:57:33,240 --> 00:57:35,920 only this is a real system. 706 00:57:35,920 --> 00:57:37,180 Very nice. 707 00:57:37,180 --> 00:57:39,060 We have a steel rod. 708 00:57:39,060 --> 00:57:40,810 It must be a half inch in diameter. 709 00:57:40,810 --> 00:57:43,360 The whole thing weighs several pounds. 710 00:57:43,360 --> 00:57:48,715 These sliding masses are right circular cylinders with a hole 711 00:57:48,715 --> 00:57:49,590 drilled through them. 712 00:57:49,590 --> 00:57:54,002 It's ever so slightly larger than these here. 713 00:57:54,002 --> 00:57:55,210 They're of different lengths. 714 00:57:55,210 --> 00:57:56,210 They're made of brass. 715 00:57:56,210 --> 00:58:00,010 They're serious masses. 716 00:58:00,010 --> 00:58:02,350 And the springs, which extend from here 717 00:58:02,350 --> 00:58:07,815 to here, and from here to here are wound on a lathe, 718 00:58:07,815 --> 00:58:09,930 and attached, and so forth. 719 00:58:09,930 --> 00:58:10,960 Pretty, no? 720 00:58:14,340 --> 00:58:16,100 Now look right off the bat. 721 00:58:16,100 --> 00:58:18,685 Did you see how that thing operates? 722 00:58:27,320 --> 00:58:33,270 Would you agree you have some complicated behavior here? 723 00:58:33,270 --> 00:58:40,290 Now also would you agree that this is it like that? 724 00:58:40,290 --> 00:58:42,820 Everybody see that? 725 00:58:42,820 --> 00:58:45,410 Before we go too far, this is a mixed message here. 726 00:58:45,410 --> 00:58:51,120 [INAUDIBLE], is this exactly like that? 727 00:58:51,120 --> 00:58:56,665 In what way is it similar to that? 728 00:58:56,665 --> 00:58:57,540 AUDIENCE: [INAUDIBLE] 729 00:59:00,360 --> 00:59:05,070 PROFESSOR: Well, what is clear is that you've got two springs 730 00:59:05,070 --> 00:59:07,310 and you've got two masses. 731 00:59:07,310 --> 00:59:10,044 About that there is very little argument. 732 00:59:10,044 --> 00:59:10,960 AUDIENCE: It's damped. 733 00:59:10,960 --> 00:59:11,990 PROFESSOR: It's damped. 734 00:59:11,990 --> 00:59:13,410 Can everybody see that? 735 00:59:13,410 --> 00:59:15,870 How does [INAUDIBLE] know that it's damped? 736 00:59:18,460 --> 00:59:20,260 How's he know it's damped? 737 00:59:20,260 --> 00:59:23,972 I mean, that's just a wild guess on his part, but. 738 00:59:23,972 --> 00:59:25,950 AUDIENCE: You can hear it, and it slows down. 739 00:59:25,950 --> 00:59:27,010 And it slows down. 740 00:59:27,010 --> 00:59:33,170 This is the most important part is that it stops. 741 00:59:33,170 --> 00:59:37,984 Eventually if you come back in a minute or two, it's done. 742 00:59:37,984 --> 00:59:39,400 PROFESSOR: All right, [INAUDIBLE]. 743 00:59:39,400 --> 00:59:40,220 You're on a roll. 744 00:59:40,220 --> 00:59:42,060 There's definitely damping there. 745 00:59:42,060 --> 00:59:44,669 What kind of damping? 746 00:59:44,669 --> 00:59:45,927 AUDIENCE: Friction. 747 00:59:45,927 --> 00:59:47,010 PROFESSOR: Friction, yeah. 748 00:59:47,010 --> 00:59:51,890 Does that have another name that you can think of? 749 00:59:51,890 --> 00:59:54,670 It's definitely friction. 750 00:59:54,670 --> 00:59:57,070 What it's not is viscous friction. 751 00:59:57,070 --> 01:00:04,040 What it is not is a damper or a dashpot which we've shown you 752 01:00:04,040 --> 01:00:13,220 before with the ideal expression that generate a force that 753 01:00:13,220 --> 01:00:17,670 opposes the-- generation of an opposing force that's linearly 754 01:00:17,670 --> 01:00:19,930 proportional to the velocity. 755 01:00:19,930 --> 01:00:21,610 What's going on here, do you think? 756 01:00:24,270 --> 01:00:27,880 What kind of damping do you think? 757 01:00:27,880 --> 01:00:29,680 It's called Coulomb. 758 01:00:29,680 --> 01:00:31,390 This is called Coulomb damping. 759 01:00:31,390 --> 01:00:33,110 And this is a digression. 760 01:00:33,110 --> 01:00:37,760 Now we're on the part where this is really-- everything that's 761 01:00:37,760 --> 01:00:40,090 on the board is what I wanted you to really come 762 01:00:40,090 --> 01:00:41,990 away from today with. 763 01:00:41,990 --> 01:00:44,375 So now we're out kind of in the, I 764 01:00:44,375 --> 01:00:47,380 would call it the winging it area 765 01:00:47,380 --> 01:00:53,750 right here, because this is the part where I simply 766 01:00:53,750 --> 01:00:54,970 had fun with the demo. 767 01:00:58,600 --> 01:00:59,920 This is viscous. 768 01:01:05,880 --> 01:01:08,160 And this has got the symbol-- well anyway, 769 01:01:08,160 --> 01:01:09,450 this is what it looks like. 770 01:01:09,450 --> 01:01:12,570 And this is the force of the damper. 771 01:01:12,570 --> 01:01:13,440 We'll call it B. 772 01:01:13,440 --> 01:01:16,640 And this is the velocity. 773 01:01:16,640 --> 01:01:20,025 And this is for constant of proportionality B. 774 01:01:20,025 --> 01:01:25,880 And it has this little symbol, like that. 775 01:01:25,880 --> 01:01:31,140 And when equations of motion are solved that contain that, 776 01:01:31,140 --> 01:01:32,765 the response looks like this. 777 01:01:48,310 --> 01:02:05,950 What we're talking about here, the force put out 778 01:02:05,950 --> 01:02:11,460 is a constant. 779 01:02:11,460 --> 01:02:14,500 That just comes from the sliding. 780 01:02:14,500 --> 01:02:18,590 And what it generates are distinctly non-linear equations 781 01:02:18,590 --> 01:02:19,930 of motion. 782 01:02:19,930 --> 01:02:24,020 And what you get here is you get this kind of behavior. 783 01:02:24,020 --> 01:02:30,250 If you really looked at it, what you'll see 784 01:02:30,250 --> 01:02:33,860 is there's definitely damping for large motions 785 01:02:33,860 --> 01:02:38,260 when the inertial forces and so forth are large compared 786 01:02:38,260 --> 01:02:40,290 to the friction forces. 787 01:02:40,290 --> 01:02:44,460 It'll look a lot like conventional viscous damping. 788 01:02:44,460 --> 01:02:47,000 It's just that when motions get really small, 789 01:02:47,000 --> 01:02:50,630 and the forces get down there to on the order of this, 790 01:02:50,630 --> 01:02:56,320 all a sudden you'll see on one cycle it'll just stop. 791 01:02:56,320 --> 01:03:01,260 And were you up here where you could see, 792 01:03:01,260 --> 01:03:04,150 or if we had a closeup of this-- you can't see it, but just 793 01:03:04,150 --> 01:03:06,770 watch this thing stop. 794 01:03:06,770 --> 01:03:07,300 Right there. 795 01:03:07,300 --> 01:03:08,370 Do you see that? 796 01:03:08,370 --> 01:03:10,870 That's a little hard for you to see from there, but watch. 797 01:03:13,520 --> 01:03:17,595 Anyway, were you up here, you'd see this. 798 01:03:17,595 --> 01:03:20,430 That's what we're looking at. 799 01:03:20,430 --> 01:03:23,370 Well here we go. 800 01:03:23,370 --> 01:03:24,920 I need some help here. 801 01:03:24,920 --> 01:03:29,760 Who's in a volunteering frame of mind? 802 01:03:29,760 --> 01:03:31,580 Amy, all right. 803 01:03:31,580 --> 01:03:34,160 I appreciate the help here. 804 01:03:34,160 --> 01:03:35,420 Here's what I want to do. 805 01:03:35,420 --> 01:03:38,990 We just blew out some wonderful theory. 806 01:03:38,990 --> 01:03:42,410 All this is just solid as a rock. 807 01:03:42,410 --> 01:03:43,656 Yes, sir. 808 01:03:43,656 --> 01:03:45,833 AUDIENCE: For the Coulomb friction, is that a linear 809 01:03:45,833 --> 01:03:46,332 [INAUDIBLE]? 810 01:03:46,332 --> 01:03:47,230 Or is it still exponential? 811 01:03:47,230 --> 01:03:48,240 PROFESSOR: I'm sorry? 812 01:03:48,240 --> 01:03:49,250 Oh, no. 813 01:03:51,850 --> 01:03:54,510 If I'm not mistaken, I didn't really look this up, 814 01:03:54,510 --> 01:03:56,960 but I believe it's linear. 815 01:03:56,960 --> 01:03:58,857 I'd have to-- take that with a grain of salt, 816 01:03:58,857 --> 01:03:59,940 but I believe it's linear. 817 01:04:03,210 --> 01:04:05,830 Yes, Amy, here's the situation. 818 01:04:05,830 --> 01:04:07,650 We have all this marvelous theory. 819 01:04:07,650 --> 01:04:13,810 My goal is to-- and we have this fabulous demo apparatus, though 820 01:04:13,810 --> 01:04:15,680 inherited. 821 01:04:15,680 --> 01:04:19,430 And what I'd like to do-- oh, and we have computational means 822 01:04:19,430 --> 01:04:21,560 to. 823 01:04:21,560 --> 01:04:24,310 And in fact, we just went through the exercise. 824 01:04:24,310 --> 01:04:26,480 We already know those same equations that 825 01:04:26,480 --> 01:04:28,340 work for this work for this. 826 01:04:28,340 --> 01:04:30,270 Those are general. 827 01:04:30,270 --> 01:04:33,570 However, it's not my piece of apparatus. 828 01:04:33,570 --> 01:04:39,980 And well, here's the deal, what are the Ms and Ks. 829 01:04:39,980 --> 01:04:46,910 What are the values of-- I need M1, K1, M2, K2 to put into the. 830 01:04:46,910 --> 01:04:48,530 AUDIENCE: [INAUDIBLE]. 831 01:04:48,530 --> 01:04:50,560 PROFESSOR: Yeah, Yeah. 832 01:04:50,560 --> 01:04:52,641 That's what I'd like to do is I'd like to. 833 01:04:52,641 --> 01:04:54,182 AUDIENCE: Do I have to just determine 834 01:04:54,182 --> 01:04:55,160 the it by looking at it? 835 01:04:55,160 --> 01:04:56,201 PROFESSOR: Oh no, no, no. 836 01:04:56,201 --> 01:04:56,940 No, no. 837 01:04:56,940 --> 01:04:59,530 AUDIENCE: I'm not that good. 838 01:04:59,530 --> 01:05:01,060 PROFESSOR: You're my assistant. 839 01:05:01,060 --> 01:05:03,170 I guess the question is, how would 840 01:05:03,170 --> 01:05:05,930 you-- and I've got to tell you, that's 841 01:05:05,930 --> 01:05:07,250 the math part of a program. 842 01:05:07,250 --> 01:05:09,640 Now we're in the engineering part of the program, 843 01:05:09,640 --> 01:05:12,930 because somebody gave you a real live demo apparatus. 844 01:05:12,930 --> 01:05:16,600 Works like crazy, or appears to. 845 01:05:16,600 --> 01:05:18,610 And I'd love to take advantage of it, 846 01:05:18,610 --> 01:05:20,680 but I don't know any of the numbers. 847 01:05:20,680 --> 01:05:22,990 AUDIENCE: [INAUDIBLE] 848 01:05:22,990 --> 01:05:24,720 PROFESSOR: No. 849 01:05:24,720 --> 01:05:27,561 That's the constraint I'm operating on. 850 01:05:27,561 --> 01:05:28,560 It doesn't belong to me. 851 01:05:28,560 --> 01:05:30,200 I mean, I could take it apart. 852 01:05:30,200 --> 01:05:32,580 That's an absolutely appropriate thing to do. 853 01:05:32,580 --> 01:05:34,020 I would have liked to. 854 01:05:34,020 --> 01:05:35,430 It would be easier if you could. 855 01:05:35,430 --> 01:05:38,690 You just go, take a screwdriver to it. 856 01:05:38,690 --> 01:05:42,630 Here, put this on there and pull this out. 857 01:05:42,630 --> 01:05:45,950 I didn't have the luxury of any of that, so what's 858 01:05:45,950 --> 01:05:49,050 your next best suggestion? 859 01:05:49,050 --> 01:05:51,270 Nice suggestion, but no cigar. 860 01:05:51,270 --> 01:05:52,325 I'm sorry? 861 01:05:52,325 --> 01:05:54,491 AUDIENCE: Take it apart anyway, put it back together 862 01:05:54,491 --> 01:05:55,812 before the person notices. 863 01:05:55,812 --> 01:05:56,170 PROFESSOR: Well yeah. 864 01:05:56,170 --> 01:05:56,930 Yeah, no. 865 01:05:56,930 --> 01:05:57,600 That's fudging. 866 01:05:57,600 --> 01:06:00,890 Yeah, they notice. 867 01:06:00,890 --> 01:06:03,860 Have you ever taken apart anything made 868 01:06:03,860 --> 01:06:07,210 in modern manufacturing method? 869 01:06:07,210 --> 01:06:10,120 Oh, it's good because you can't put them back together. 870 01:06:10,120 --> 01:06:12,240 They're assembled by machine. 871 01:06:12,240 --> 01:06:15,780 And once upon a time you could disassemble one 872 01:06:15,780 --> 01:06:19,270 and reassemble things without detection. 873 01:06:19,270 --> 01:06:21,560 But anymore, once you take them apart, 874 01:06:21,560 --> 01:06:23,380 it's wicked hard to get them back together. 875 01:06:23,380 --> 01:06:25,080 OK, the floor is open. 876 01:06:25,080 --> 01:06:26,954 I need another suggestion. 877 01:06:26,954 --> 01:06:30,670 What are you going to do? 878 01:06:30,670 --> 01:06:33,316 AUDIENCE: Do you need to know the exact K and M, 879 01:06:33,316 --> 01:06:35,040 or do you just [INAUDIBLE] another ratio? 880 01:06:36,892 --> 01:06:38,350 PROFESSOR: I thought you were going 881 01:06:38,350 --> 01:06:42,970 to-- I need to know M or K. I need to know them all. 882 01:06:42,970 --> 01:06:45,280 AUDIENCE: Do you know the density of the-- 883 01:06:45,280 --> 01:06:47,030 PROFESSOR: I was going to say, but I don't 884 01:06:47,030 --> 01:06:49,800 need to know anything exactly. 885 01:06:49,800 --> 01:06:53,280 All I need to know is as good a guess as you can come up with. 886 01:06:53,280 --> 01:06:54,689 It's all an estimate. 887 01:06:54,689 --> 01:06:57,034 AUDIENCE: If you know the density of the material, 888 01:06:57,034 --> 01:06:59,379 you can easily work up [INAUDIBLE]. 889 01:06:59,379 --> 01:07:01,730 I'm assuming you're about [INAUDIBLE]. 890 01:07:01,730 --> 01:07:04,300 PROFESSOR: Oh absolutely, absolutely. 891 01:07:04,300 --> 01:07:05,708 AUDIENCE: [INAUDIBLE]. 892 01:07:05,708 --> 01:07:06,583 PROFESSOR: Wonderful. 893 01:07:09,300 --> 01:07:11,880 Absolutely. 894 01:07:11,880 --> 01:07:14,990 She hit the jackpot, rang the magic buzzer. 895 01:07:14,990 --> 01:07:17,970 That's exactly what I did. 896 01:07:17,970 --> 01:07:21,050 Here's a little crummy sketch. 897 01:07:21,050 --> 01:07:23,770 Oh wait, you can't see that. 898 01:07:23,770 --> 01:07:27,122 Anyway, these are right circular cylinders with holes in them. 899 01:07:27,122 --> 01:07:28,830 And they've got measurements beside them. 900 01:07:28,830 --> 01:07:31,190 I can tell you, this is 75 millimeters. 901 01:07:31,190 --> 01:07:34,520 This is 35 millimeters. 902 01:07:34,520 --> 01:07:40,360 This one is 37 millimeters long. 903 01:07:40,360 --> 01:07:41,970 So I did that. 904 01:07:41,970 --> 01:07:42,650 That's great. 905 01:07:42,650 --> 01:07:44,800 That's an excellent suggestion. 906 01:07:44,800 --> 01:07:53,910 And after I did exactly that, I won't write out the formula. 907 01:07:53,910 --> 01:07:56,220 You know area, and volume, and all of that. 908 01:07:59,992 --> 01:08:01,615 Let me get you the right order here. 909 01:08:05,970 --> 01:08:25,859 M1 is 0.2929, and M2 is 0.5938. 910 01:08:25,859 --> 01:08:27,359 Everybody got that? 911 01:08:27,359 --> 01:08:31,180 This was obtained by taking a ruler to these things, 912 01:08:31,180 --> 01:08:34,580 taking diameters, lengths, and diameters of holes, 913 01:08:34,580 --> 01:08:39,682 multiply them times the density of brass taken out of the book. 914 01:08:39,682 --> 01:08:42,069 Do you believe that? 915 01:08:42,069 --> 01:08:43,766 Do you believe that number? 916 01:08:49,479 --> 01:08:51,279 Well, you're a trusting soul. 917 01:08:51,279 --> 01:08:52,399 I don't. 918 01:08:52,399 --> 01:08:55,790 To me, I believe that number. 919 01:08:55,790 --> 01:08:58,010 This was done with a ruler. 920 01:08:58,010 --> 01:09:00,800 The little millimeter thingies. 921 01:09:00,800 --> 01:09:06,510 So I just say, don't fall in the trap of false precision. 922 01:09:06,510 --> 01:09:08,069 OK Amy, you're on a roll. 923 01:09:08,069 --> 01:09:10,220 We've got the masses. 924 01:09:10,220 --> 01:09:11,413 What now? 925 01:09:11,413 --> 01:09:13,310 AUDIENCE: Free body diagram. 926 01:09:13,310 --> 01:09:14,854 PROFESSOR: Yeah, we got all that. 927 01:09:14,854 --> 01:09:16,187 AUDIENCE: Yeah, you've got that. 928 01:09:16,187 --> 01:09:18,869 But then what you can do for the spring, 929 01:09:18,869 --> 01:09:21,781 the forces of the spring when static. 930 01:09:21,781 --> 01:09:22,364 Don't move it. 931 01:09:22,364 --> 01:09:24,854 Don't move it. 932 01:09:24,854 --> 01:09:26,850 So take the top mast. 933 01:09:26,850 --> 01:09:28,505 It's not moving, which means that you 934 01:09:28,505 --> 01:09:30,880 know that the force going upwards-- which is the spring-- 935 01:09:30,880 --> 01:09:36,280 is equal to the force going downward, which is [INAUDIBLE]. 936 01:09:36,280 --> 01:09:38,670 So you can measure the displacement 937 01:09:38,670 --> 01:09:41,892 from the start of the spring to the bottom of the spring. 938 01:09:41,892 --> 01:09:44,160 Do you know the natural length of the spring? 939 01:09:44,160 --> 01:09:46,899 PROFESSOR: No Anyway, what I was going to say 940 01:09:46,899 --> 01:09:49,160 is, excellent idea. 941 01:09:49,160 --> 01:09:51,149 Can't do it. 942 01:09:51,149 --> 01:09:54,640 But what Amy was basically saying is, 943 01:09:54,640 --> 01:09:56,700 you know the masses now. 944 01:09:56,700 --> 01:10:01,220 Why not simply take from that expression 945 01:10:01,220 --> 01:10:05,252 right there, MH over K, right? 946 01:10:05,252 --> 01:10:06,460 What's the problem with that? 947 01:10:06,460 --> 01:10:07,001 How about it? 948 01:10:07,001 --> 01:10:10,562 Devin, how come I can't do that? 949 01:10:10,562 --> 01:10:12,540 AUDIENCE: [INAUDIBLE]. 950 01:10:12,540 --> 01:10:15,645 PROFESSOR: Yeah, I really don't know the no load position. 951 01:10:18,920 --> 01:10:20,610 Is that it, Amy? 952 01:10:20,610 --> 01:10:23,430 Right there? 953 01:10:23,430 --> 01:10:23,930 Maybe. 954 01:10:27,820 --> 01:10:31,460 Devin, you're the one that suggested it. 955 01:10:31,460 --> 01:10:35,630 Are those the no load positions of the masses? 956 01:10:35,630 --> 01:10:37,060 And if not, why not? 957 01:10:39,670 --> 01:10:41,590 I did mean to give you a clue there. 958 01:10:41,590 --> 01:10:42,300 Yeah, Nick? 959 01:10:42,300 --> 01:10:44,550 AUDIENCE: It can't be because there's static friction. 960 01:10:44,550 --> 01:10:47,220 PROFESSOR: Exactly. 961 01:10:47,220 --> 01:10:52,090 So Nick, you brought it up. 962 01:10:52,090 --> 01:10:53,005 What's that number? 963 01:10:57,670 --> 01:10:59,400 You don't know that either. 964 01:10:59,400 --> 01:11:01,040 No. 965 01:11:01,040 --> 01:11:03,250 Like I said, Amy, nice idea. 966 01:11:03,250 --> 01:11:06,050 No cigar. 967 01:11:06,050 --> 01:11:06,710 What else? 968 01:11:06,710 --> 01:11:08,980 We're running out of time? 969 01:11:08,980 --> 01:11:10,790 Here we go, Douglas. 970 01:11:10,790 --> 01:11:13,039 AUDIENCE: Could you displace each mast a certain 971 01:11:13,039 --> 01:11:14,622 [INAUDIBLE], and then measure the time 972 01:11:14,622 --> 01:11:17,790 it takes for them to stop and get the damping ratio? 973 01:11:17,790 --> 01:11:19,200 PROFESSOR: Hit the damping ratio. 974 01:11:19,200 --> 01:11:21,990 Well, I'll tell you what-- you want to say that again? 975 01:11:21,990 --> 01:11:26,340 He said displace one or both count oscillations 976 01:11:26,340 --> 01:11:27,650 and get the damping ratio. 977 01:11:27,650 --> 01:11:28,340 Nick? 978 01:11:28,340 --> 01:11:30,840 AUDIENCE: Do we have anything like a force gauge or a spring 979 01:11:30,840 --> 01:11:31,339 scale? 980 01:11:31,339 --> 01:11:34,060 PROFESSOR: No. 981 01:11:34,060 --> 01:11:36,790 This is my living room I'm talking-- or my study. 982 01:11:36,790 --> 01:11:40,896 Anyway, Douglas said-- I forgot what you said now. 983 01:11:40,896 --> 01:11:41,770 He said-- oh, I know. 984 01:11:41,770 --> 01:11:44,490 You said, displace it and count the oscillations. 985 01:11:44,490 --> 01:11:46,550 Get the damping ratio. 986 01:11:46,550 --> 01:11:51,250 First off, the damping ratio is no help, even if we did get it. 987 01:11:51,250 --> 01:11:55,420 And the only formula for which we've ever given you-- 988 01:11:55,420 --> 01:11:57,060 the only formula we've ever given 989 01:11:57,060 --> 01:12:00,480 you to do that with pertains to this kind of friction, which 990 01:12:00,480 --> 01:12:02,950 is not present. 991 01:12:02,950 --> 01:12:03,720 No cigar. 992 01:12:03,720 --> 01:12:05,970 Nick says, how about force gauge? 993 01:12:05,970 --> 01:12:09,000 Now, don't have it. 994 01:12:09,000 --> 01:12:11,350 AUDIENCE: [INAUDIBLE]. 995 01:12:11,350 --> 01:12:13,988 PROFESSOR: Yeah, we are. 996 01:12:13,988 --> 01:12:16,844 AUDIENCE: [INAUDIBLE] displace the other one 997 01:12:16,844 --> 01:12:18,010 and then find the frequency. 998 01:12:18,010 --> 01:12:20,130 PROFESSOR: Oh, what's your name? 999 01:12:20,130 --> 01:12:21,360 Sean? 1000 01:12:21,360 --> 01:12:22,223 Or John? 1001 01:12:22,223 --> 01:12:23,682 AUDIENCE: Sean. 1002 01:12:23,682 --> 01:12:24,890 PROFESSOR: Say that out loud. 1003 01:12:24,890 --> 01:12:28,148 Say it loud enough that Devin can hear you. 1004 01:12:28,148 --> 01:12:29,964 AUDIENCE: You hold the first mass, 1005 01:12:29,964 --> 01:12:31,780 and then you displace the second one. 1006 01:12:31,780 --> 01:12:32,571 PROFESSOR: Hang on. 1007 01:12:32,571 --> 01:12:36,210 He says, hold the first mass like this set screw right here. 1008 01:12:39,130 --> 01:12:40,070 And? 1009 01:12:40,070 --> 01:12:43,830 AUDIENCE: And then displace the other one then. 1010 01:12:43,830 --> 01:12:45,345 PROFESSOR: Like that? 1011 01:12:45,345 --> 01:12:47,470 AUDIENCE: [INAUDIBLE]. 1012 01:12:47,470 --> 01:12:51,390 PROFESSOR: Actually, to answer your question 1013 01:12:51,390 --> 01:12:55,720 Nick, the only instrument I have is a clock. 1014 01:13:00,502 --> 01:13:01,210 Hang on a second. 1015 01:13:05,300 --> 01:13:05,890 Here it is. 1016 01:13:05,890 --> 01:13:07,880 Of course, this is the big task is finding it. 1017 01:13:11,338 --> 01:13:12,254 AUDIENCE: [INAUDIBLE]. 1018 01:13:24,877 --> 01:13:27,210 PROFESSOR: Count to 10, remember like Vandiver told you. 1019 01:13:27,210 --> 01:13:29,770 Skip 1, 1, 2, 3, count to 10. 1020 01:13:29,770 --> 01:13:31,250 Stop. 1021 01:13:31,250 --> 01:13:34,190 Excellent, excellent, excellent. 1022 01:13:34,190 --> 01:13:42,730 When you do that, that's the second one. 1023 01:13:42,730 --> 01:13:47,770 I did that and right here, right here it's TP. 1024 01:13:47,770 --> 01:13:53,275 The period of 10 of them-- and then I divide to get 1-- 1025 01:13:53,275 --> 01:14:00,680 is 0.83 seconds. 1026 01:14:00,680 --> 01:14:03,394 What does that tell you, Sean? 1027 01:14:03,394 --> 01:14:04,310 AUDIENCE: [INAUDIBLE]. 1028 01:14:08,040 --> 01:14:13,020 PROFESSOR: Well actually, these two-- 1029 01:14:13,020 --> 01:14:14,590 you can get the frequency. 1030 01:14:14,590 --> 01:14:24,220 But what this does, because this gives you the frequency, 1031 01:14:24,220 --> 01:14:28,653 you know in general that-- in particular, you 1032 01:14:28,653 --> 01:14:31,610 know that this second natural frequency, which is just 1033 01:14:31,610 --> 01:14:35,360 associated with this single spring and a mass here. 1034 01:14:35,360 --> 01:14:36,930 It's just this guy. 1035 01:14:36,930 --> 01:14:38,280 It's not both of them. 1036 01:14:40,790 --> 01:14:45,190 Anyway, this turns out to-- I didn't graph that. 1037 01:14:45,190 --> 01:14:50,860 That's square root of K over M. Trust me, 1038 01:14:50,860 --> 01:14:53,610 you can put those two together, and you get 1039 01:14:53,610 --> 01:15:04,490 K2 is equal to-- newton meters. 1040 01:15:04,490 --> 01:15:07,590 This is exactly what you said. 1041 01:15:07,590 --> 01:15:08,870 Freeze the first mass. 1042 01:15:08,870 --> 01:15:11,410 Displace the second. 1043 01:15:11,410 --> 01:15:12,830 Measure the period. 1044 01:15:12,830 --> 01:15:24,530 You get the natural frequency for basically K2 over M2. 1045 01:15:24,530 --> 01:15:26,495 So you got K2 out of it. 1046 01:15:26,495 --> 01:15:27,577 Yeah, Douglas? 1047 01:15:27,577 --> 01:15:30,118 AUDIENCE: So how come it gives you just the natural frequency 1048 01:15:30,118 --> 01:15:32,510 and a damp natural frequency? 1049 01:15:32,510 --> 01:15:34,910 PROFESSOR: Oh no. 1050 01:15:34,910 --> 01:15:36,020 It absolutely is. 1051 01:15:36,020 --> 01:15:39,900 It's all damped, no question about it. 1052 01:15:39,900 --> 01:15:41,670 But again, what we're doing is we're 1053 01:15:41,670 --> 01:15:44,660 going close enough, right? 1054 01:15:44,660 --> 01:15:46,910 Because I have nothing. 1055 01:15:46,910 --> 01:15:49,220 So even the damped natural frequency 1056 01:15:49,220 --> 01:15:51,640 is better than nothing. 1057 01:15:51,640 --> 01:15:53,385 All right, so Amy back to you. 1058 01:15:53,385 --> 01:15:55,230 You're back in business. 1059 01:15:55,230 --> 01:15:56,220 What now? 1060 01:15:56,220 --> 01:15:59,690 So now we've got M2 and K2. 1061 01:16:04,715 --> 01:16:05,840 AUDIENCE: You just need K1. 1062 01:16:05,840 --> 01:16:06,190 PROFESSOR: Yeah. 1063 01:16:06,190 --> 01:16:06,970 Now we need K1. 1064 01:16:06,970 --> 01:16:08,979 What do we do now? 1065 01:16:08,979 --> 01:16:11,104 AUDIENCE: We want to do the same thing that we just 1066 01:16:11,104 --> 01:16:12,992 did for K2. [INAUDIBLE]. 1067 01:16:15,830 --> 01:16:17,300 PROFESSOR: Exactly. 1068 01:16:17,300 --> 01:16:21,040 Well, not quite. 1069 01:16:21,040 --> 01:16:23,170 Let's see, now I'm going to turn loose-- now we're 1070 01:16:23,170 --> 01:16:26,439 back to our original system. 1071 01:16:26,439 --> 01:16:27,480 It doesn't hurt anything. 1072 01:16:27,480 --> 01:16:30,750 It's just ugly to look at. 1073 01:16:30,750 --> 01:16:32,800 Now what? 1074 01:16:32,800 --> 01:16:35,715 Sean, do that trick again. 1075 01:16:38,445 --> 01:16:40,420 AUDIENCE: [INAUDIBLE]. 1076 01:16:40,420 --> 01:16:42,065 PROFESSOR: Push this one up? 1077 01:16:42,065 --> 01:16:42,940 AUDIENCE: [INAUDIBLE] 1078 01:16:47,644 --> 01:16:49,240 PROFESSOR: I don't think so. 1079 01:16:49,240 --> 01:16:50,407 Yeah, Nick. 1080 01:16:50,407 --> 01:16:51,990 AUDIENCE: So just fix the second mass. 1081 01:16:51,990 --> 01:16:53,355 PROFESSOR: Fix the second mass. 1082 01:16:53,355 --> 01:16:54,230 AUDIENCE: [INAUDIBLE] 1083 01:17:03,060 --> 01:17:05,010 PROFESSOR: Yeah, you see this frequency here? 1084 01:17:05,010 --> 01:17:07,230 I'm going to overrule Chandler. 1085 01:17:07,230 --> 01:17:12,470 I'm going to say, this is the natural frequency of that's 1086 01:17:12,470 --> 01:17:16,570 mass and these two springs. 1087 01:17:16,570 --> 01:17:17,880 Do that same trick again. 1088 01:17:17,880 --> 01:17:20,290 You get the equivalent spring rate, 1089 01:17:20,290 --> 01:17:23,130 subtract the second from it, and you get the other one. 1090 01:17:23,130 --> 01:17:24,880 Devin, is that what you were going to say? 1091 01:17:24,880 --> 01:17:27,020 Wonderful. 1092 01:17:27,020 --> 01:17:29,420 That is exactly what I did. 1093 01:17:29,420 --> 01:17:32,310 And you get out of it, you get K1 1094 01:17:32,310 --> 01:17:40,480 is equal to 50.45 newton per meter. 1095 01:17:40,480 --> 01:17:44,350 In the interest of time, I'm going to short circuit this. 1096 01:17:44,350 --> 01:17:48,740 I took exactly these parameters, I 1097 01:17:48,740 --> 01:17:54,870 put them into that same computer program we had before, 1098 01:17:54,870 --> 01:17:58,736 and what came out-- I have to have a place to put it. 1099 01:18:02,390 --> 01:18:03,000 Ah, wonderful. 1100 01:18:08,507 --> 01:18:09,590 And now I have to find it. 1101 01:18:17,860 --> 01:18:19,890 Here we go. 1102 01:18:19,890 --> 01:18:23,147 Here are the two modes. 1103 01:18:23,147 --> 01:18:24,605 Actually, let me put it right here. 1104 01:18:28,260 --> 01:18:34,115 For this system, because here's the first one. 1105 01:18:42,040 --> 01:18:43,358 And here's the second one. 1106 01:18:47,190 --> 01:18:54,980 0.9760 and minus 0.2177. 1107 01:18:54,980 --> 01:18:56,410 Everybody appreciate that? 1108 01:18:56,410 --> 01:18:59,610 This is by the same computational procedure 1109 01:18:59,610 --> 01:19:02,240 we spoke of earlier. 1110 01:19:02,240 --> 01:19:09,110 And sparing no expense, we have here a, made fresh 1111 01:19:09,110 --> 01:19:15,680 from my basement, a custom made initial condition setting 1112 01:19:15,680 --> 01:19:22,430 device, which I can hopefully avoid killing myself with. 1113 01:19:22,430 --> 01:19:24,760 OK here's what we have. 1114 01:19:24,760 --> 01:19:29,990 So I guess I didn't show you this first. 1115 01:19:29,990 --> 01:19:31,820 If you can see it, what we have marked 1116 01:19:31,820 --> 01:19:33,630 is the reference position. 1117 01:19:33,630 --> 01:19:36,010 That's the rest position that we couldn't 1118 01:19:36,010 --> 01:19:38,320 find by laying it down. 1119 01:19:38,320 --> 01:19:45,350 Or excuse me, this is the static equilibrium position 1120 01:19:45,350 --> 01:19:47,850 of mass number one, static equilibrium 1121 01:19:47,850 --> 01:19:49,550 position of mass number two. 1122 01:19:49,550 --> 01:19:52,330 Mode one or there. 1123 01:19:52,330 --> 01:19:54,390 What they said over there, 0.4. 1124 01:19:54,390 --> 01:19:57,160 And .97 is down here. 1125 01:19:57,160 --> 01:19:59,990 Mode two is over here. 1126 01:19:59,990 --> 01:20:03,770 So Devin, while I'm doing this, tell me 1127 01:20:03,770 --> 01:20:05,830 how am I going to know if this is right 1128 01:20:05,830 --> 01:20:09,080 or if this is all just bogus? 1129 01:20:09,080 --> 01:20:14,594 What observable's going to tell me that I got it right? 1130 01:20:14,594 --> 01:20:15,469 AUDIENCE: [INAUDIBLE] 1131 01:20:20,720 --> 01:20:22,495 PROFESSOR: I'm sorry, speak up. 1132 01:20:22,495 --> 01:20:23,740 AUDIENCE: [INAUDIBLE] 1133 01:20:23,740 --> 01:20:25,342 PROFESSOR: How about it, Nick? 1134 01:20:25,342 --> 01:20:26,550 AUDIENCE: [INAUDIBLE] 1135 01:20:26,550 --> 01:20:28,100 PROFESSOR: Exactly. 1136 01:20:28,100 --> 01:20:32,070 And you ought to be able to see it from where you are. 1137 01:20:32,070 --> 01:20:36,310 Can you appreciate that they're not at the moment? 1138 01:20:36,310 --> 01:20:41,660 All right, now hang on. 1139 01:20:41,660 --> 01:20:44,320 Here comes mode number one. 1140 01:20:44,320 --> 01:20:47,890 This takes two hands to do it. 1141 01:20:47,890 --> 01:20:50,248 All right, you ready? 1142 01:20:53,170 --> 01:20:54,184 This is mode number one. 1143 01:20:54,184 --> 01:20:55,350 Again, it's a those numbers. 1144 01:21:07,739 --> 01:21:08,280 How about it? 1145 01:21:08,280 --> 01:21:10,320 Can you see it? 1146 01:21:10,320 --> 01:21:12,730 Very good. 1147 01:21:12,730 --> 01:21:14,500 How about the other? 1148 01:21:14,500 --> 01:21:21,987 And so here, this is number two. 1149 01:21:21,987 --> 01:21:23,570 And this is a little more complicated, 1150 01:21:23,570 --> 01:21:28,656 because the other one has to be done from the bottom. 1151 01:21:28,656 --> 01:21:29,470 Hang on a second. 1152 01:21:32,470 --> 01:21:36,860 Now this one we're doing is we're deflecting-- 1153 01:21:36,860 --> 01:21:39,370 this one is positive downward. 1154 01:21:39,370 --> 01:21:44,210 So X1 is down, but X2 is negative. 1155 01:21:44,210 --> 01:21:46,710 So it's displaced upward a bit. 1156 01:21:46,710 --> 01:21:48,790 Are you ready? 1157 01:21:48,790 --> 01:21:52,594 Nick, what do you expect to see this time? 1158 01:21:52,594 --> 01:21:55,184 AUDIENCE: The frequency should be higher 1159 01:21:55,184 --> 01:21:56,850 and they'll move in opposite directions. 1160 01:21:56,850 --> 01:22:00,090 PROFESSOR: That's the key. 1161 01:22:00,090 --> 01:22:02,650 Once again, they're going to move with the same frequency, 1162 01:22:02,650 --> 01:22:04,910 albeit in different directions. 1163 01:22:04,910 --> 01:22:09,010 But that new frequency is going to be higher than before. 1164 01:22:09,010 --> 01:22:11,308 And sure enough, stand back. 1165 01:22:20,670 --> 01:22:21,500 So there you go. 1166 01:22:21,500 --> 01:22:23,440 What did that tell us? 1167 01:22:23,440 --> 01:22:26,520 That told us that the first order 1168 01:22:26,520 --> 01:22:30,410 we got the system parameters identified correctly 1169 01:22:30,410 --> 01:22:33,220 and the theory holds up. 1170 01:22:33,220 --> 01:22:33,890 Questions? 1171 01:22:33,890 --> 01:22:35,880 Comments? 1172 01:22:35,880 --> 01:22:37,740 Complaints? 1173 01:22:37,740 --> 01:22:38,696 Devin? 1174 01:22:38,696 --> 01:22:39,690 AUDIENCE: [INAUDIBLE] 1175 01:22:39,690 --> 01:22:40,620 PROFESSOR: I'm sorry? 1176 01:22:40,620 --> 01:22:42,880 AUDIENCE: What was the second set of conditions? 1177 01:22:42,880 --> 01:22:45,750 PROFESSOR: The second set of initial conditions 1178 01:22:45,750 --> 01:22:48,180 were right here. 1179 01:22:48,180 --> 01:22:57,120 This is the second mode, X1, 0.97, X2, minus 0.2. 1180 01:22:57,120 --> 01:23:01,330 OK, have a great Thanksgiving holiday.