1 00:00:01,210 --> 00:00:03,580 The following content is provided under a Creative 2 00:00:03,580 --> 00:00:04,970 Commons license. 3 00:00:04,970 --> 00:00:07,180 Your support will help MIT OpenCourseWare 4 00:00:07,180 --> 00:00:11,270 continue to offer high quality educational resources for free. 5 00:00:11,270 --> 00:00:13,810 To make a donation, or to view additional materials 6 00:00:13,810 --> 00:00:17,770 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,770 --> 00:00:18,640 at ocw.mit.edu. 8 00:00:23,489 --> 00:00:25,030 BOLESLAW WYSLOUCH: Let's get started. 9 00:00:25,030 --> 00:00:27,690 So today hopefully will be a busy day, 10 00:00:27,690 --> 00:00:34,860 with lots of interesting insights into how things work. 11 00:00:34,860 --> 00:00:38,020 We talked about coupled oscillators last time. 12 00:00:38,020 --> 00:00:41,850 We developed a formalism in which 13 00:00:41,850 --> 00:00:46,860 we can find the most general motion of oscillators. 14 00:00:46,860 --> 00:00:52,650 So let's remind ourselves what are the coupled oscillators. 15 00:00:52,650 --> 00:00:55,920 Coupled oscillators, there are many examples of them, 16 00:00:55,920 --> 00:00:59,310 and they have more or less the following features. 17 00:00:59,310 --> 00:01:00,840 You have something that oscillates-- 18 00:01:00,840 --> 00:01:03,320 for example, a pendulum. 19 00:01:03,320 --> 00:01:08,250 You have to have more than one, because for coupled oscillators 20 00:01:08,250 --> 00:01:10,560 you have to have at least two. 21 00:01:10,560 --> 00:01:14,100 So let's say you have two oscillators. 22 00:01:14,100 --> 00:01:18,610 So each of them is an oscillator, 23 00:01:18,610 --> 00:01:23,670 which in, for example, in the limit of small angles, 24 00:01:23,670 --> 00:01:26,250 small displacement angles, undergoes 25 00:01:26,250 --> 00:01:29,640 a pure harmonic motion with some frequencies. 26 00:01:29,640 --> 00:01:32,610 And then you couple them through various means. 27 00:01:32,610 --> 00:01:36,150 So for example, two masses connected by a spring 28 00:01:36,150 --> 00:01:38,760 is an example of a coupled oscillator. 29 00:01:38,760 --> 00:01:43,770 We could have two masses on a track and another track, 30 00:01:43,770 --> 00:01:47,070 also connected by several springs. 31 00:01:47,070 --> 00:01:49,740 This is also an example of a coupled oscillator. 32 00:01:49,740 --> 00:01:54,122 Each of those masses undergoes harmonic motion, 33 00:01:54,122 --> 00:01:56,580 and they are connected together such that the motion of one 34 00:01:56,580 --> 00:01:58,560 affects motion of the other. 35 00:01:58,560 --> 00:02:00,750 You can have slightly more complicated pendula. 36 00:02:00,750 --> 00:02:07,120 For example, you can hang one pendula from the other. 37 00:02:07,120 --> 00:02:09,639 Each of them-- again, in the limit of small oscillations-- 38 00:02:09,639 --> 00:02:11,610 will undergo harmonic motion. 39 00:02:11,610 --> 00:02:14,890 And they are coupled together because they are supported one 40 00:02:14,890 --> 00:02:17,020 on top of each other. 41 00:02:17,020 --> 00:02:18,840 And you can have-- 42 00:02:18,840 --> 00:02:25,900 we have another example of two tuning forks 43 00:02:25,900 --> 00:02:28,240 sitting on some sort of boxes. 44 00:02:28,240 --> 00:02:31,840 Each of them was an oscillator, with audible oscillating 45 00:02:31,840 --> 00:02:38,350 frequency, and by putting them next to each other they coupled 46 00:02:38,350 --> 00:02:41,590 through the sound waves transmitted through the air. 47 00:02:41,590 --> 00:02:45,760 So one of them felt the oscillations in the other one. 48 00:02:45,760 --> 00:02:47,700 This was an example of coupled oscillation. 49 00:02:47,700 --> 00:02:51,100 Two masses and the thing. 50 00:02:51,100 --> 00:02:54,535 You can build oscillators out of electronics. 51 00:02:54,535 --> 00:02:57,160 Some capacitor and inductor together, 52 00:02:57,160 --> 00:02:58,270 with a little bit of-- 53 00:02:58,270 --> 00:02:59,410 maybe without resistors. 54 00:02:59,410 --> 00:03:00,760 You have two of those. 55 00:03:00,760 --> 00:03:02,680 They constitute a coupled oscillator 56 00:03:02,680 --> 00:03:04,690 if you put a wire between them. 57 00:03:04,690 --> 00:03:06,329 So there are many, many examples. 58 00:03:06,329 --> 00:03:07,870 And of course, these are all examples 59 00:03:07,870 --> 00:03:10,480 in which you have two oscillating bodies, 60 00:03:10,480 --> 00:03:14,290 but it's very easy to have three or more oscillating bodies. 61 00:03:14,290 --> 00:03:17,530 Then basically the features of the system 62 00:03:17,530 --> 00:03:20,400 are the same, except the math becomes more complicated, 63 00:03:20,400 --> 00:03:23,480 and we have more types of oscillations you can have. 64 00:03:23,480 --> 00:03:27,100 And there's a couple of characteristics 65 00:03:27,100 --> 00:03:30,220 which are the same for all oscillating systems. 66 00:03:30,220 --> 00:03:31,690 And it's very important to remember 67 00:03:31,690 --> 00:03:33,790 that we are learning on one example, 68 00:03:33,790 --> 00:03:35,740 but it applies to very many. 69 00:03:35,740 --> 00:03:38,170 Number one, any motion-- 70 00:03:38,170 --> 00:03:41,520 I can maybe summarize it here. 71 00:03:41,520 --> 00:03:49,070 So if you look at the motion of an oscillator, you can have-- 72 00:03:49,070 --> 00:03:52,290 let's say arbitrary oscillation. 73 00:03:52,290 --> 00:03:55,130 Arbitrary excitation. 74 00:03:58,300 --> 00:04:01,150 Excitation means I-- 75 00:04:01,150 --> 00:04:05,630 I kick it in some sort of arbitrary mode. 76 00:04:05,630 --> 00:04:08,480 I just come in and set up some initial condition such 77 00:04:08,480 --> 00:04:10,270 that things are moving. 78 00:04:10,270 --> 00:04:14,140 And motion in this arbitrary assertion 79 00:04:14,140 --> 00:04:16,930 is actually-- looks pretty chaotic. 80 00:04:16,930 --> 00:04:21,040 It looks pretty variable, changing. 81 00:04:21,040 --> 00:04:23,860 It's difficult to understand what's going on. 82 00:04:23,860 --> 00:04:26,380 So And it clearly doesn't look harmonic. 83 00:04:26,380 --> 00:04:27,400 Non-harmonic. 84 00:04:30,740 --> 00:04:33,110 There is no obvious single frequency 85 00:04:33,110 --> 00:04:36,780 that is driving the system. 86 00:04:36,780 --> 00:04:40,640 If you look at amplitude of the objects 87 00:04:40,640 --> 00:04:43,070 here-- for example, two pendula, pendulum one and two. 88 00:04:43,070 --> 00:04:46,040 At any given moment of time they are oscillating, 89 00:04:46,040 --> 00:04:48,020 there's a characteristic amplitude. 90 00:04:48,020 --> 00:04:50,570 But what we saw is that motion changes, 91 00:04:50,570 --> 00:04:53,300 looks like things are flowing from one to the other. 92 00:04:53,300 --> 00:04:54,740 One of them has a high amplitude. 93 00:04:54,740 --> 00:04:58,170 After some time, it cools down, the other one grows. 94 00:04:58,170 --> 00:05:01,965 So the amplitudes are changing in time. 95 00:05:01,965 --> 00:05:02,840 So they are variable. 96 00:05:10,570 --> 00:05:11,185 Are variable. 97 00:05:14,060 --> 00:05:17,540 And also, we didn't calculate things exactly, 98 00:05:17,540 --> 00:05:21,302 but you know from study of a single oscillator that 99 00:05:21,302 --> 00:05:23,510 if the things are moving, it has a certain amplitude, 100 00:05:23,510 --> 00:05:26,180 there's certain energy involved-- with some potential, 101 00:05:26,180 --> 00:05:27,350 some kinetic-- 102 00:05:27,350 --> 00:05:29,820 and it's proportional to the square of amplitude. 103 00:05:29,820 --> 00:05:32,690 So it's clear that energy is moving from one pendulum 104 00:05:32,690 --> 00:05:34,100 to the other. 105 00:05:34,100 --> 00:05:36,470 This one was oscillating like crazy. 106 00:05:36,470 --> 00:05:38,630 So all energy was sitting here. 107 00:05:38,630 --> 00:05:40,660 After some time, this one stopped. 108 00:05:40,660 --> 00:05:42,080 So its energy is zero. 109 00:05:42,080 --> 00:05:44,870 And the other one was oscillating like crazy. 110 00:05:44,870 --> 00:05:47,330 So the energy's flowing from one to another. 111 00:05:47,330 --> 00:05:50,330 It's not sitting in one place, but it's flowing. 112 00:05:50,330 --> 00:05:53,420 This one has lots of energy right now, but now 113 00:05:53,420 --> 00:05:55,020 that one is picking up. 114 00:05:55,020 --> 00:05:59,430 So the energy-- you see the energy flowing here. 115 00:05:59,430 --> 00:06:01,320 And this one will eventually stop-- 116 00:06:01,320 --> 00:06:03,030 well, this is a pretty crappy oscillator, 117 00:06:03,030 --> 00:06:05,040 but it will eventually stop, and this one 118 00:06:05,040 --> 00:06:07,950 will have all the energy. 119 00:06:07,950 --> 00:06:10,080 And this is, again, characteristic in every system. 120 00:06:10,080 --> 00:06:13,650 We can see energy flowing around from one to the other, 121 00:06:13,650 --> 00:06:15,270 growing, stopping. 122 00:06:15,270 --> 00:06:19,986 So it's-- in general, in the most general case, 123 00:06:19,986 --> 00:06:23,840 it's a complicated system. 124 00:06:23,840 --> 00:06:31,100 Energy is migrating between different masses. 125 00:06:31,100 --> 00:06:34,400 However, every single one of those coupled oscillating 126 00:06:34,400 --> 00:06:35,990 systems has a magic. 127 00:06:35,990 --> 00:06:39,980 There's a magic involved, namely the existence of normal modes. 128 00:06:39,980 --> 00:06:43,400 Every single coupled oscillator system has normal modes, 129 00:06:43,400 --> 00:06:45,260 and those modes are beautiful. 130 00:06:45,260 --> 00:06:50,630 Those modes are-- everything is moving in sync. 131 00:06:50,630 --> 00:06:59,540 So this is normal mode excitation. 132 00:06:59,540 --> 00:07:02,440 There's a very special way, a special setting 133 00:07:02,440 --> 00:07:07,090 of initial conditions, that leads to the-- that results 134 00:07:07,090 --> 00:07:10,090 in a pure harmonic motion. 135 00:07:10,090 --> 00:07:19,640 So this is a harmonic motion, with a certain frequency omega, 136 00:07:19,640 --> 00:07:22,640 characteristic frequency for this particular motion. 137 00:07:22,640 --> 00:07:26,130 The amplitudes remain fixed. 138 00:07:26,130 --> 00:07:29,070 Once you set initial conditions, you get it moving, 139 00:07:29,070 --> 00:07:31,500 everything is moving, simple harmonic motion 140 00:07:31,500 --> 00:07:33,970 means its amplitude is constant. 141 00:07:33,970 --> 00:07:36,750 So if I-- and remember, for example, this system. 142 00:07:36,750 --> 00:07:39,750 It was something like this. 143 00:07:39,750 --> 00:07:44,610 Symmetric or antisymmetric motion. 144 00:07:44,610 --> 00:07:46,710 And if not for the friction, the amplitudes 145 00:07:46,710 --> 00:07:49,080 would remain constant forever, if it 146 00:07:49,080 --> 00:07:50,310 will be a perfect oscillator. 147 00:07:50,310 --> 00:07:53,580 So amplitudes-- in fact, it's not amplitudes themselves, 148 00:07:53,580 --> 00:07:55,260 but amplitude ratio. 149 00:07:55,260 --> 00:07:59,670 The ratio of amplitude between the different elements 150 00:07:59,670 --> 00:08:02,110 in the system is constant. 151 00:08:07,910 --> 00:08:10,220 So in a sense, every harmonic motion 152 00:08:10,220 --> 00:08:13,010 has a characteristic shape. 153 00:08:13,010 --> 00:08:17,060 And then by-- since everything is constant, nothing changes, 154 00:08:17,060 --> 00:08:20,800 this energy stays in the place it is. 155 00:08:25,830 --> 00:08:28,439 So energy is-- once you put energy 156 00:08:28,439 --> 00:08:30,730 to mass number one, mass number two, mass number three, 157 00:08:30,730 --> 00:08:32,159 the energy sits there. 158 00:08:32,159 --> 00:08:34,799 The energies are constant, as the system 159 00:08:34,799 --> 00:08:36,510 undergoes harmonic motion. 160 00:08:36,510 --> 00:08:39,330 Energy does not migrate. 161 00:08:39,330 --> 00:08:42,539 So this is a very nice-- and there is another beautiful 162 00:08:42,539 --> 00:08:48,960 feature, that any arbitrary excitation can be made out 163 00:08:48,960 --> 00:08:53,227 of some linear sum-- 164 00:08:56,149 --> 00:09:00,250 sum of normal modes. 165 00:09:02,980 --> 00:09:05,780 Linear sum, of superposition of normal. 166 00:09:05,780 --> 00:09:09,410 Any arbitrary excitation with all its complicated motion 167 00:09:09,410 --> 00:09:12,220 can be made into some of normal modes. 168 00:09:12,220 --> 00:09:15,450 So since normal modes are easy and simple and beautiful, 169 00:09:15,450 --> 00:09:20,520 the description of motion of any coupled oscillator, the best 170 00:09:20,520 --> 00:09:23,070 way to approach it is to decompose it, 171 00:09:23,070 --> 00:09:25,740 to find all possible normal modes, 172 00:09:25,740 --> 00:09:29,010 and then decompose the initial condition to correspond 173 00:09:29,010 --> 00:09:31,140 to this linear sum of normal modes. 174 00:09:31,140 --> 00:09:33,660 Once you know the normal modes, you add them up, 175 00:09:33,660 --> 00:09:37,590 and then you can predict exactly the motion. 176 00:09:37,590 --> 00:09:39,900 And this is what we've done. 177 00:09:39,900 --> 00:09:42,530 So we have a-- 178 00:09:42,530 --> 00:09:46,590 we have introduced a mathematic mechanism 179 00:09:46,590 --> 00:09:50,910 in which we put all the information about forces 180 00:09:50,910 --> 00:09:54,120 and masses in the system in some sort of matrix form. 181 00:09:54,120 --> 00:09:56,430 In our example, it was a two by two matrix, 182 00:09:56,430 --> 00:09:58,740 but if we have three masses or four masses, 183 00:09:58,740 --> 00:10:01,320 the dimensionality of the matrix will have to grow. 184 00:10:01,320 --> 00:10:03,730 But the equation will remain the same. 185 00:10:03,730 --> 00:10:08,070 So this equation of motion, we rework it a little bit. 186 00:10:08,070 --> 00:10:10,530 Since we are looking for normal modes, 187 00:10:10,530 --> 00:10:14,230 we know that normal modes occur with this one single frequency. 188 00:10:14,230 --> 00:10:19,260 So we postulate an oscillation with a frequency. 189 00:10:19,260 --> 00:10:20,220 We plug it in. 190 00:10:20,220 --> 00:10:23,870 We obtain a simple algebraic equation. 191 00:10:23,870 --> 00:10:25,290 Doesn't have any time dependence, 192 00:10:25,290 --> 00:10:27,030 doesn't have any exponents. 193 00:10:27,030 --> 00:10:30,600 It's a simple algebraic equation, basically a set 194 00:10:30,600 --> 00:10:35,930 of linear equations, which we can solve and find 195 00:10:35,930 --> 00:10:39,170 the eigenvalue, or the characteristic frequency 196 00:10:39,170 --> 00:10:40,510 for normal modes. 197 00:10:40,510 --> 00:10:43,190 And you can show that the number of those frequencies in general 198 00:10:43,190 --> 00:10:46,400 is equal to the number of masses involved in the system. 199 00:10:46,400 --> 00:10:49,130 And you solve it, and then once you 200 00:10:49,130 --> 00:10:51,590 know the characteristic frequencies, 201 00:10:51,590 --> 00:10:55,250 then you can find shape, you can find the eigenvectors. 202 00:10:55,250 --> 00:10:59,510 What is the ratio of amplitudes which corresponds to the mode. 203 00:10:59,510 --> 00:11:02,970 And in case of our two pendula, there are two of such things. 204 00:11:02,970 --> 00:11:06,500 One is where both amplitudes are equal, 205 00:11:06,500 --> 00:11:08,870 and this corresponds to oscillation 206 00:11:08,870 --> 00:11:13,670 in which two pendola are moving parallel to each other, 207 00:11:13,670 --> 00:11:15,860 with a spring being-- 208 00:11:15,860 --> 00:11:18,350 not paying any roll. 209 00:11:18,350 --> 00:11:19,490 So this is one mode. 210 00:11:19,490 --> 00:11:21,570 And then amplitude is-- 211 00:11:21,570 --> 00:11:25,070 as I said, any given moment is the same, so the ratio is 1. 212 00:11:25,070 --> 00:11:28,800 And then you have a motion in which the two pendula 213 00:11:28,800 --> 00:11:30,350 are going against each other. 214 00:11:30,350 --> 00:11:32,420 So any given moment of time, they're 215 00:11:32,420 --> 00:11:35,103 in their negative position, so the ratio is minus 1. 216 00:11:38,430 --> 00:11:41,650 The motion of one of them can be obtained 217 00:11:41,650 --> 00:11:43,525 by looking at where the first one is 218 00:11:43,525 --> 00:11:45,880 and multiplying by minus 1. 219 00:11:45,880 --> 00:11:48,190 So these are the two modes, and any arbitrary-- any 220 00:11:48,190 --> 00:11:52,420 complicated, nasty excitation with things moving around 221 00:11:52,420 --> 00:11:56,780 is a linear sum of the oscillation. 222 00:11:56,780 --> 00:11:57,660 So we know that. 223 00:11:57,660 --> 00:11:59,380 We've worked it out. 224 00:11:59,380 --> 00:12:02,820 We used this example. 225 00:12:02,820 --> 00:12:06,430 And by the way, today, we'll be using two examples-- 226 00:12:06,430 --> 00:12:09,940 one which is the same thing with two pendula and the spring, 227 00:12:09,940 --> 00:12:11,510 and the other one with two masses, 228 00:12:11,510 --> 00:12:14,020 or maybe later three masses. 229 00:12:14,020 --> 00:12:20,730 And the exact values of coefficients in matrix k 230 00:12:20,730 --> 00:12:23,870 are different in two different cases. 231 00:12:23,870 --> 00:12:29,510 But in all types of other motion, the shape of motion, 232 00:12:29,510 --> 00:12:32,340 the behavior of the system is identical. 233 00:12:32,340 --> 00:12:36,390 So the solutions to the two cases are identical. 234 00:12:36,390 --> 00:12:40,130 The difference is basically numerical in how the spring 235 00:12:40,130 --> 00:12:41,960 constants and masses come in. 236 00:12:41,960 --> 00:12:45,080 So we can in fact treat those two systems 237 00:12:45,080 --> 00:12:46,190 completely the same. 238 00:12:46,190 --> 00:12:48,050 So I'll be jumping from one to another, 239 00:12:48,050 --> 00:12:49,190 but we don't have to worry. 240 00:12:49,190 --> 00:12:52,700 But let's now look on the system. 241 00:12:52,700 --> 00:12:54,470 So what we are trying to do today is, 242 00:12:54,470 --> 00:13:00,040 we are trying to apply external force 243 00:13:00,040 --> 00:13:02,609 so we'll have a driven coupled oscillator. 244 00:13:02,609 --> 00:13:04,150 And I assume that you know everything 245 00:13:04,150 --> 00:13:06,040 about driven oscillators. 246 00:13:06,040 --> 00:13:09,300 So the idea was that you come with an external . 247 00:13:09,300 --> 00:13:11,980 In 8.03, we assumed that this external force 248 00:13:11,980 --> 00:13:13,160 is harmonic force. 249 00:13:13,160 --> 00:13:15,490 So there's a characteristic frequency 250 00:13:15,490 --> 00:13:18,364 which is given by external-- 251 00:13:18,364 --> 00:13:19,030 let's say by me. 252 00:13:19,030 --> 00:13:22,450 It has nothing to do with normal frequencies of the system. 253 00:13:22,450 --> 00:13:25,000 It's an external frequency, omega d, which I apply. 254 00:13:25,000 --> 00:13:26,740 Driven frequency. 255 00:13:26,740 --> 00:13:29,990 And then I look at how the system responds. 256 00:13:29,990 --> 00:13:32,090 And I look for steady state oscillations-- 257 00:13:32,090 --> 00:13:35,540 the ones where everything oscillates with the same driven 258 00:13:35,540 --> 00:13:36,680 frequency-- 259 00:13:36,680 --> 00:13:38,270 trying to look for solutions. 260 00:13:38,270 --> 00:13:42,110 And as you know from a single oscillator, what 261 00:13:42,110 --> 00:13:45,550 we were calculating is what is the the response of the system? 262 00:13:45,550 --> 00:13:46,790 What is the amplitude? 263 00:13:46,790 --> 00:13:49,410 And the certain frequencies that-- 264 00:13:49,410 --> 00:13:52,310 you wiggle it and the system doesn't do anything, 265 00:13:52,310 --> 00:13:55,507 but if you apply a certain resonant frequency, 266 00:13:55,507 --> 00:13:56,840 then the response is very large. 267 00:13:56,840 --> 00:13:59,240 The system starts moving like crazy, et cetera. 268 00:13:59,240 --> 00:14:01,550 And the same type of thing will happen here, 269 00:14:01,550 --> 00:14:04,480 except that we have multiple frequencies. 270 00:14:04,480 --> 00:14:06,950 So there will be a possibility of a resonance 271 00:14:06,950 --> 00:14:08,640 for several frequencies. 272 00:14:08,640 --> 00:14:10,820 All right? 273 00:14:10,820 --> 00:14:16,090 So let me quickly set this up. 274 00:14:16,090 --> 00:14:17,320 Just-- yeah. 275 00:14:17,320 --> 00:14:18,165 Doesn't matter. 276 00:14:18,165 --> 00:14:20,790 So there were some-- 277 00:14:20,790 --> 00:14:23,940 let's just start working on the example. 278 00:14:23,940 --> 00:14:26,820 So just a reminder, this is our system. 279 00:14:26,820 --> 00:14:29,500 A pendula of some length L. There 280 00:14:29,500 --> 00:14:33,930 are two identical masses, M. There 281 00:14:33,930 --> 00:14:37,320 is a spring of constant k. 282 00:14:37,320 --> 00:14:41,000 They are all-- and for simplicity, we 283 00:14:41,000 --> 00:14:44,630 assume that we are all in Earth's gravitational field. 284 00:14:44,630 --> 00:14:46,460 So we don't have to worry about traveling 285 00:14:46,460 --> 00:14:49,730 to Jupiter or the moon. 286 00:14:49,730 --> 00:14:53,720 And-- except that the difference will 287 00:14:53,720 --> 00:14:58,490 be that we apply an external force to one of those masses. 288 00:14:58,490 --> 00:15:02,962 How, it doesn't matter, but there is an external force F-- 289 00:15:02,962 --> 00:15:08,960 F with subscript d, which is equal to some-- 290 00:15:08,960 --> 00:15:13,540 it has some amplitude F0 cosine omega 291 00:15:13,540 --> 00:15:19,610 d times t, along the x direction. 292 00:15:19,610 --> 00:15:23,570 And this is applied to mass one. 293 00:15:23,570 --> 00:15:25,000 OK. 294 00:15:25,000 --> 00:15:28,225 And there is a little bit of just a warning. 295 00:15:30,880 --> 00:15:37,069 We will be assuming that there is no damping in the system. 296 00:15:37,069 --> 00:15:38,860 For the single oscillator, there was always 297 00:15:38,860 --> 00:15:40,370 a little bit of damping. 298 00:15:40,370 --> 00:15:43,360 So between you and me, remember there's always a little 299 00:15:43,360 --> 00:15:43,870 damping. 300 00:15:43,870 --> 00:15:46,270 So in case we need damping-- 301 00:15:46,270 --> 00:15:48,580 it will come in and will help us, 302 00:15:48,580 --> 00:15:50,860 but if we try to use damping in calculations, 303 00:15:50,860 --> 00:15:52,600 calculations become horrendous. 304 00:15:52,600 --> 00:15:54,730 So for the purpose of calculations, 305 00:15:54,730 --> 00:15:56,710 we will ignore damping. 306 00:15:56,710 --> 00:15:57,430 It'll get some. 307 00:15:57,430 --> 00:16:01,780 But if things go bad with the results, like dividing by 0, 308 00:16:01,780 --> 00:16:04,510 then we will bring in damping and say no no, it's not so bad. 309 00:16:04,510 --> 00:16:05,630 Damping helps you. 310 00:16:05,630 --> 00:16:06,990 We are not dividing by 0. 311 00:16:06,990 --> 00:16:09,340 OK? 312 00:16:09,340 --> 00:16:12,250 So let's write those equations of motions. 313 00:16:12,250 --> 00:16:13,375 Equations of motion. 314 00:16:16,370 --> 00:16:21,810 So we have-- so the forces and accelerations on mass one 315 00:16:21,810 --> 00:16:23,650 is the same as before. 316 00:16:23,650 --> 00:16:25,750 There was a spring. 317 00:16:25,750 --> 00:16:29,170 There is mg over l. 318 00:16:29,170 --> 00:16:33,580 That's the pendulum by itself. 319 00:16:33,580 --> 00:16:36,220 Depending on position x1. 320 00:16:36,220 --> 00:16:40,090 There is the influence of a spring, which depends 321 00:16:40,090 --> 00:16:43,090 on where spring number two is. 322 00:16:43,090 --> 00:16:49,540 And, plus, there is this new driven term, F0 cosine omega 323 00:16:49,540 --> 00:16:53,870 d times t, where omega d is fixed, arbitrary, 324 00:16:53,870 --> 00:16:54,730 externally given. 325 00:16:54,730 --> 00:17:00,130 So both F0 and omega d are decided by somebody outside 326 00:17:00,130 --> 00:17:01,630 of the system. 327 00:17:01,630 --> 00:17:08,138 Now, the second mass M X2 dot dot, is-- 328 00:17:08,138 --> 00:17:14,079 actually has feels position of x1, through the spring. 329 00:17:14,079 --> 00:17:21,730 And there is this-- its own pendulum effect plus a string, 330 00:17:21,730 --> 00:17:23,560 depending on position x2. 331 00:17:23,560 --> 00:17:25,720 Interestingly, there is no force here, 332 00:17:25,720 --> 00:17:29,360 because the force is applied to mass one. 333 00:17:29,360 --> 00:17:32,290 So mass two a priori doesn't know anything about the force. 334 00:17:32,290 --> 00:17:35,110 But of course it will know through the coupling. 335 00:17:35,110 --> 00:17:35,610 Yes? 336 00:17:35,610 --> 00:17:37,882 Questions? 337 00:17:37,882 --> 00:17:40,350 Anybody have questions so far? 338 00:17:40,350 --> 00:17:42,990 So it's the same as before, with the addition 339 00:17:42,990 --> 00:17:45,430 of this external force. 340 00:17:45,430 --> 00:17:51,820 Again, this is writing all coordinates one by one. 341 00:17:51,820 --> 00:17:54,940 We immediately switch to matrix form. 342 00:17:54,940 --> 00:17:59,170 We write it MX double dot, where X is the same 343 00:17:59,170 --> 00:18:02,185 as we defined before, minus KX. 344 00:18:05,210 --> 00:18:09,590 I think I will stop writing these kind of thick lines. 345 00:18:09,590 --> 00:18:12,160 But for now, let me-- 346 00:18:12,160 --> 00:18:17,350 F cosine omega d times t. 347 00:18:17,350 --> 00:18:21,370 So this is now a matrix equation for the vector XD. 348 00:18:21,370 --> 00:18:24,000 And let's remind ourselves what those matrices are. 349 00:18:24,000 --> 00:18:30,120 Matrix M is M 0 0 M. This is just 350 00:18:30,120 --> 00:18:32,740 mass of the individual systems. 351 00:18:32,740 --> 00:18:38,650 We use M minus 1, which is 1 over M, 1 over M, 352 00:18:38,650 --> 00:18:41,830 and diagonal 0 and 0. 353 00:18:41,830 --> 00:18:43,720 So this carries information about masses, 354 00:18:43,720 --> 00:18:45,670 inertia of the system. 355 00:18:45,670 --> 00:18:50,680 Matrix K contains information about all the springs 356 00:18:50,680 --> 00:18:53,860 in the system, and some pendula effects. 357 00:18:53,860 --> 00:19:01,480 So we have a k plus mg over l, minus k, 358 00:19:01,480 --> 00:19:09,220 minus k, k plus mg over l. 359 00:19:09,220 --> 00:19:11,500 And now there is this new thing, which 360 00:19:11,500 --> 00:19:24,540 is this vector F. Vector F is equal to F0 0 cosine omega 361 00:19:24,540 --> 00:19:27,110 d times t. 362 00:19:27,110 --> 00:19:30,290 So this is in a vector form, this external force, 363 00:19:30,290 --> 00:19:33,900 which is applied only to mass number one. 364 00:19:33,900 --> 00:19:34,400 OK? 365 00:19:37,990 --> 00:19:40,570 So these are the elements which are plugged in. 366 00:19:40,570 --> 00:19:43,060 So now the question is, what do you want to do with this? 367 00:19:43,060 --> 00:19:45,720 So we have the equation of motion. 368 00:19:45,720 --> 00:19:48,690 And so what do we do with this? 369 00:19:48,690 --> 00:19:56,320 So there are two steps that we have to do. 370 00:19:56,320 --> 00:20:00,090 Number one, we have to remind ourselves 371 00:20:00,090 --> 00:20:04,200 what are the normal modes of the system, in case-- 372 00:20:04,200 --> 00:20:05,280 because we will need-- 373 00:20:05,280 --> 00:20:10,792 the information about normal modes will come in as-- 374 00:20:10,792 --> 00:20:15,120 into solutions for a driven motion. 375 00:20:15,120 --> 00:20:18,030 So let's remind ourselves what this was. 376 00:20:18,030 --> 00:20:19,282 Well, this was a solution. 377 00:20:19,282 --> 00:20:20,865 I'll just rewrite it very quickly such 378 00:20:20,865 --> 00:20:22,156 that we have it for the record. 379 00:20:24,535 --> 00:20:26,200 It should fit here. 380 00:20:26,200 --> 00:20:27,390 Now let's try. 381 00:20:27,390 --> 00:20:29,280 So there were two solutions. 382 00:20:29,280 --> 00:20:33,510 There was omega 1 squared, which was equal to g over l. 383 00:20:33,510 --> 00:20:39,320 And the corresponding normal mode was a symmetric one. 384 00:20:39,320 --> 00:20:41,570 It was 1, 1. 385 00:20:41,570 --> 00:20:42,070 OK. 386 00:20:42,070 --> 00:20:46,360 So this was one type of solution, 387 00:20:46,360 --> 00:20:48,950 where the two masses were moving together. 388 00:20:48,950 --> 00:20:54,650 There was a second frequency which was equal to g over l. 389 00:20:54,650 --> 00:20:59,410 The square of it was equal plus 2k over m. 390 00:20:59,410 --> 00:21:01,960 And this was the characteristic normal frequency 391 00:21:01,960 --> 00:21:05,230 for the second type of oscillation, which 392 00:21:05,230 --> 00:21:08,290 you can write it 1, minus 1. 393 00:21:08,290 --> 00:21:12,580 And the criterion for when we were looking for solutions, 394 00:21:12,580 --> 00:21:17,110 we would find them by calculating the determinant 395 00:21:17,110 --> 00:21:19,230 of this two by two matrix. 396 00:21:19,230 --> 00:21:29,360 It was the determinant of m minus 1 k minus omega squared 397 00:21:29,360 --> 00:21:33,650 times unit matrix was equal to 0. 398 00:21:33,650 --> 00:21:37,590 So this was the equation that had 399 00:21:37,590 --> 00:21:41,730 to be satisfied for frequencies corresponding to normal modes 400 00:21:41,730 --> 00:21:44,610 with zero external force. 401 00:21:44,610 --> 00:21:46,800 Interestingly, if you do the calculations, 402 00:21:46,800 --> 00:21:48,110 it turns out you can-- 403 00:21:48,110 --> 00:21:51,110 algebraically, you can write-- 404 00:21:51,110 --> 00:21:53,020 after you know the solution itself, 405 00:21:53,020 --> 00:21:56,420 you can write it in a very compact way. 406 00:21:56,420 --> 00:21:59,910 So this determinant can be written in the following way-- 407 00:21:59,910 --> 00:22:03,790 omega squared minus omega 1 squared, 408 00:22:03,790 --> 00:22:09,830 times omega squared minus omega 2 squared. 409 00:22:09,830 --> 00:22:13,620 And this is-- the condition was zero. 410 00:22:13,620 --> 00:22:20,840 And you see explicitly that this is a fourth order in frequency 411 00:22:20,840 --> 00:22:23,270 equation, fourth order frequency, which is 412 00:22:23,270 --> 00:22:27,890 0 for omega 1 and for omega 2. 413 00:22:27,890 --> 00:22:29,260 In a very explicit way. 414 00:22:29,260 --> 00:22:35,360 So this is a nice, compact form of writing 415 00:22:35,360 --> 00:22:38,005 this particular eigenvalue equation. 416 00:22:41,120 --> 00:22:46,420 And again, as a reminder, the motion of the system-- 417 00:22:46,420 --> 00:22:49,930 the most general motion of the system with no external force 418 00:22:49,930 --> 00:22:54,790 was a superposition of those two oscillations, 419 00:22:54,790 --> 00:22:57,320 which we can write as some sort of amplitude-- 420 00:22:57,320 --> 00:23:04,990 1, 1 cosine omega 1 t plus phi 1, 421 00:23:04,990 --> 00:23:14,690 plus beta 1, minus 1 cosine omega 2 t plus phi 2. 422 00:23:14,690 --> 00:23:20,070 So this is oscillations of two different frequencies. 423 00:23:20,070 --> 00:23:21,990 This is the shape of oscillations, 424 00:23:21,990 --> 00:23:25,660 the relative amplitude of one versus the other. 425 00:23:25,660 --> 00:23:28,680 And then there's the overall amplitude alpha and beta, 426 00:23:28,680 --> 00:23:30,260 which has to be determined. 427 00:23:30,260 --> 00:23:32,880 And then there are arbitrary phases. 428 00:23:32,880 --> 00:23:38,900 So there are in fact four numbers, 429 00:23:38,900 --> 00:23:43,270 which can be determined from four initial conditions. 430 00:23:43,270 --> 00:23:45,980 So typically two positions for the two masses, 431 00:23:45,980 --> 00:23:48,230 and two initial velocities for two masses. 432 00:23:48,230 --> 00:23:51,360 So everything matches. 433 00:23:51,360 --> 00:23:53,420 So this a so-called homogeneous equation. 434 00:23:59,246 --> 00:24:00,320 Homogeneous solution. 435 00:24:06,350 --> 00:24:08,280 What about driven solution? 436 00:24:08,280 --> 00:24:14,310 Driven solution, as we remember from a single oscillator, 437 00:24:14,310 --> 00:24:20,030 results in a motion in which all the elements in the system 438 00:24:20,030 --> 00:24:22,590 are oscillating at the same frequency, 439 00:24:22,590 --> 00:24:24,930 and that's the driven frequency. 440 00:24:24,930 --> 00:24:25,500 It's a fact. 441 00:24:25,500 --> 00:24:28,950 I come in, I apply 100 Hertz frequency, 442 00:24:28,950 --> 00:24:30,960 and everybody oscillates on the 100 frequency. 443 00:24:30,960 --> 00:24:35,110 That's the solution for a driven oscillating system. 444 00:24:35,110 --> 00:24:37,230 And we saw it for a one-dimensional oscillator, 445 00:24:37,230 --> 00:24:38,563 and we will see it here as well. 446 00:24:38,563 --> 00:24:40,640 There's one frequency, omega d. 447 00:24:40,640 --> 00:24:44,310 So we will be now looking for a solution which corresponds 448 00:24:44,310 --> 00:24:48,210 to the oscillation of the system with this external frequency, 449 00:24:48,210 --> 00:24:50,700 which a priori is not the same as one 450 00:24:50,700 --> 00:24:53,000 of the normal frequencies. 451 00:24:53,000 --> 00:24:56,600 So the complete motion of the system consists of two parts. 452 00:24:56,600 --> 00:25:01,150 One is this homogeneous self-oscillating motion 453 00:25:01,150 --> 00:25:03,330 with two characteristic frequencies. 454 00:25:03,330 --> 00:25:06,050 And there will be a second type of motion, 455 00:25:06,050 --> 00:25:08,130 which is a driven one. 456 00:25:08,130 --> 00:25:12,650 So how do we go about solving that? 457 00:25:12,650 --> 00:25:16,200 So equations of motions of course will be the same. 458 00:25:16,200 --> 00:25:21,810 The solution, the way that we solve it will be very similar. 459 00:25:21,810 --> 00:25:23,840 So lets try-- start working. 460 00:25:23,840 --> 00:25:29,790 Maybe we can work on those blackboards here. 461 00:25:29,790 --> 00:25:31,350 So what is going on? 462 00:25:34,830 --> 00:25:43,100 So we know that if we apply external frequency omega d, 463 00:25:43,100 --> 00:25:49,250 everybody in the system, all the elements will be oscillating 464 00:25:49,250 --> 00:25:51,700 with the same frequency. 465 00:25:55,130 --> 00:26:01,420 So we can then introduce a variable Z, 466 00:26:01,420 --> 00:26:08,130 which will be defined B e to the i omega d t. 467 00:26:08,130 --> 00:26:09,980 This will be the oscillating term. 468 00:26:09,980 --> 00:26:14,120 And this will be the amplitude of oscillation, which we'll try 469 00:26:14,120 --> 00:26:16,550 to make real for simplicity. 470 00:26:16,550 --> 00:26:19,640 And then we plug this into the equation 471 00:26:19,640 --> 00:26:23,090 of motion, which is listed up there on the screen. 472 00:26:23,090 --> 00:26:29,990 So the equation of motion is Z dot dot plus M minus 1 473 00:26:29,990 --> 00:26:43,400 K times Z is equal to now M minus 1 force e to i omega d t. 474 00:26:43,400 --> 00:26:48,990 You see our external force is F cosine 475 00:26:48,990 --> 00:26:52,920 omega d t, with a vector 1, 0. 476 00:26:52,920 --> 00:26:57,310 But of course, in the complex notation, this is exponent. 477 00:26:57,310 --> 00:27:00,280 So this is the challenge, what we would like to have. 478 00:27:00,280 --> 00:27:04,460 And we assume that all the elements in the system-- 479 00:27:04,460 --> 00:27:06,700 position, acceleration-- oscillate 480 00:27:06,700 --> 00:27:09,610 at the same frequency omega d. 481 00:27:09,610 --> 00:27:15,130 If you do that, then the equations 482 00:27:15,130 --> 00:27:19,780 become somewhat simpler, because the oscillating term drops out. 483 00:27:19,780 --> 00:27:23,620 So when you plug this type of solution into here, 484 00:27:23,620 --> 00:27:27,310 what you get is minus omega d squared-- 485 00:27:27,310 --> 00:27:30,660 that's from second differentiation with respect 486 00:27:30,660 --> 00:27:32,170 to time-- 487 00:27:32,170 --> 00:27:45,010 plus M minus 1 K, multiplying vector B e to i omega d t. 488 00:27:45,010 --> 00:27:54,570 This must be equal to M minus one F e to i omega d t. 489 00:27:54,570 --> 00:27:57,540 This is vector B, this is vector F. 490 00:27:57,540 --> 00:28:00,000 And there is this oscillating term. 491 00:28:00,000 --> 00:28:02,670 But both sides oscillate at the same frequency. 492 00:28:02,670 --> 00:28:03,750 That's what we assume. 493 00:28:03,750 --> 00:28:09,340 So we can simply divide by this, and we 494 00:28:09,340 --> 00:28:14,590 are left with an equation that equates 495 00:28:14,590 --> 00:28:16,360 what's going on in the oscillating 496 00:28:16,360 --> 00:28:20,440 system with the external force. 497 00:28:20,440 --> 00:28:24,820 So now, let's see here what is known 498 00:28:24,820 --> 00:28:30,490 and what is unknown in this equation. 499 00:28:30,490 --> 00:28:35,140 M minus 1 K carries information about the construction built 500 00:28:35,140 --> 00:28:37,360 of the system of accelerators. 501 00:28:37,360 --> 00:28:42,280 Strength of springs, masses, gravitational field, et cetera. 502 00:28:42,280 --> 00:28:43,150 So this is fixed. 503 00:28:43,150 --> 00:28:45,370 This is given. 504 00:28:45,370 --> 00:28:49,490 Omega d is the external driving frequency, and it's also given. 505 00:28:49,490 --> 00:28:51,090 It's a number. 506 00:28:51,090 --> 00:28:52,800 I said this is externally given. 507 00:28:52,800 --> 00:28:55,020 I just set it at some computer. 508 00:28:55,020 --> 00:28:56,940 Say 100 Hertz, and it's driven at 100 Hertz. 509 00:28:56,940 --> 00:28:57,690 So we know that. 510 00:28:57,690 --> 00:29:00,660 We know exactly what this number is. 511 00:29:00,660 --> 00:29:03,270 External force, we know what it is. 512 00:29:03,270 --> 00:29:03,970 We defined it. 513 00:29:03,970 --> 00:29:05,040 It's F0. 514 00:29:05,040 --> 00:29:07,090 We know what its magnitude-- 515 00:29:07,090 --> 00:29:15,570 so everything is known except for vector B. And vector B are 516 00:29:15,570 --> 00:29:18,300 the amplitudes of oscillation-- 517 00:29:18,300 --> 00:29:21,790 remember, everything oscillates at omega d-- 518 00:29:21,790 --> 00:29:25,650 of mass one and mass two. 519 00:29:25,650 --> 00:29:29,480 So in general, if I apply external force, 520 00:29:29,480 --> 00:29:33,230 this guy will oscillate with some amplitude. 521 00:29:33,230 --> 00:29:36,230 That guy with some amplitude, a priori different. 522 00:29:36,230 --> 00:29:39,630 And this will be B1, this will be B2. 523 00:29:39,630 --> 00:29:41,460 And we don't know that at this stage. 524 00:29:41,460 --> 00:29:44,820 So this equation will allow us to find it. 525 00:29:47,420 --> 00:29:50,120 And it is possible because-- 526 00:29:50,120 --> 00:29:53,960 this is actually a very straightforward equation. 527 00:29:53,960 --> 00:30:04,950 It contains-- actually, to be very precise, I have to-- 528 00:30:04,950 --> 00:30:06,720 this is a number, this is a matrix. 529 00:30:06,720 --> 00:30:11,200 So I have to put a unit matrix right here. 530 00:30:11,200 --> 00:30:13,700 So it's omega d times unit matrix 531 00:30:13,700 --> 00:30:20,400 plus this matrix that carries information about the system. 532 00:30:20,400 --> 00:30:23,480 And so we can write this down again 533 00:30:23,480 --> 00:30:26,820 in some sort of more open way, for our specific case. 534 00:30:26,820 --> 00:30:32,390 So this will be k over m plus g over l, 535 00:30:32,390 --> 00:30:42,360 minus omega d squared, minus k over m, minus k over m, 536 00:30:42,360 --> 00:30:50,770 k over m, plus g over l, minus omega d squared. 537 00:30:50,770 --> 00:30:53,770 So this is this matrix here. 538 00:30:53,770 --> 00:30:57,820 This matrix is applied to vector B, which is our unknown. 539 00:30:57,820 --> 00:31:01,175 Let's call it B1 and B2. 540 00:31:01,175 --> 00:31:04,600 These are the amplitudes of oscillations 541 00:31:04,600 --> 00:31:07,900 of individual elements in our system. 542 00:31:07,900 --> 00:31:11,465 And this is equal to m-- 543 00:31:11,465 --> 00:31:16,720 the inverted mass matrix times vector F, which-- 544 00:31:16,720 --> 00:31:18,760 without its oscillating part, which 545 00:31:18,760 --> 00:31:22,830 is simply F0 over m and 0. 546 00:31:25,080 --> 00:31:25,580 All right. 547 00:31:25,580 --> 00:31:28,820 So this is the task in question, and we 548 00:31:28,820 --> 00:31:33,740 have to find out those two values depending 549 00:31:33,740 --> 00:31:39,490 on these parameters and the strength of force, et cetera. 550 00:31:39,490 --> 00:31:41,240 So this is actually not a big deal. 551 00:31:41,240 --> 00:31:45,422 It's a two by two equation, two equations with two unknowns. 552 00:31:45,422 --> 00:31:46,630 We solve it, and we are done. 553 00:31:49,720 --> 00:31:55,390 However, we want to learn a little bit 554 00:31:55,390 --> 00:32:00,980 about slightly more general ways of calculating things. 555 00:32:00,980 --> 00:32:05,851 So let's call this one matrix E, with some funny double vector 556 00:32:05,851 --> 00:32:06,350 sign. 557 00:32:06,350 --> 00:32:09,890 Let's call this one vector B, and let's call this one vector 558 00:32:09,890 --> 00:32:13,620 D, because we will use this-- 559 00:32:13,620 --> 00:32:15,000 use it later. 560 00:32:15,000 --> 00:32:17,970 And what we are trying to do is, we 561 00:32:17,970 --> 00:32:27,510 are trying to use the so-called Cramer's rule to find 562 00:32:27,510 --> 00:32:29,580 those coefficients B1 and B2. 563 00:32:29,580 --> 00:32:32,170 And for some historical reasons, 8.03 really 564 00:32:32,170 --> 00:32:33,840 likes Cramer's rule. 565 00:32:33,840 --> 00:32:36,390 I like MATLAB or Mathematica. 566 00:32:36,390 --> 00:32:41,340 I just plug things in, and it crunches out and calculates. 567 00:32:41,340 --> 00:32:43,800 But it turns out that for two by two, 568 00:32:43,800 --> 00:32:45,570 you can always do it quickly. 569 00:32:45,570 --> 00:32:48,500 Even for three by three, if you just sit down and do it, 570 00:32:48,500 --> 00:32:49,740 you can actually work it out. 571 00:32:49,740 --> 00:32:50,920 It's not scary. 572 00:32:50,920 --> 00:32:53,640 By five by five-- 573 00:32:53,640 --> 00:32:57,180 but even four by four, I'm sure you are mighty students who 574 00:32:57,180 --> 00:33:01,040 can just do it in the exam. 575 00:33:01,040 --> 00:33:04,830 I have never seen an 8.03 exam with four masses, 576 00:33:04,830 --> 00:33:06,880 unless they're general questions. 577 00:33:06,880 --> 00:33:08,790 But three-- well... 578 00:33:08,790 --> 00:33:10,140 All right. 579 00:33:10,140 --> 00:33:13,620 So do we go about finding this B1 and B2? 580 00:33:13,620 --> 00:33:17,485 Because, again, this is a simple two by two question. 581 00:33:23,600 --> 00:33:27,630 So maybe just to again bring it even closer 582 00:33:27,630 --> 00:33:29,730 to what we are used to, let me just quickly 583 00:33:29,730 --> 00:33:32,740 write this down as a set of two by two equations. 584 00:33:32,740 --> 00:33:39,300 So there is a coefficient here, k over m plus g over l minus 585 00:33:39,300 --> 00:33:42,330 omega d squared, which is-- this is a number, 586 00:33:42,330 --> 00:33:53,190 times B1 minus k over m times B2 is equal to F0 over m minus k 587 00:33:53,190 --> 00:34:03,720 over m B1 plus k over m plus g over l minus omega d squared is 588 00:34:03,720 --> 00:34:05,190 equal to 0-- 589 00:34:05,190 --> 00:34:08,280 times B2 is equal to 0. 590 00:34:08,280 --> 00:34:11,310 So you see two equations with two unknowns. 591 00:34:11,310 --> 00:34:13,699 Couple of coefficients, all fixed. 592 00:34:13,699 --> 00:34:16,489 You can eliminate variables. 593 00:34:16,489 --> 00:34:19,010 You can calculate B2 from here, plug it into-- 594 00:34:19,010 --> 00:34:21,600 you can work it out if you want to. 595 00:34:21,600 --> 00:34:24,740 However, there is, again, a better way. 596 00:34:24,740 --> 00:34:30,170 It's Cramer's rule or method. 597 00:34:34,650 --> 00:34:36,880 Should have known if it's method or rule. 598 00:34:36,880 --> 00:34:37,409 Rule. 599 00:34:37,409 --> 00:34:38,560 Right. 600 00:34:38,560 --> 00:34:41,090 And so the way you do it is the following. 601 00:34:41,090 --> 00:34:47,320 So you look at those questions-- you calculate all kinds 602 00:34:47,320 --> 00:34:52,840 of determinants, and by taking the set of two equations 603 00:34:52,840 --> 00:34:54,190 and plugging into-- 604 00:34:54,190 --> 00:34:56,890 replacing columns in the matrix. 605 00:34:56,890 --> 00:35:03,700 So B1, what you do is you take the original matrix, which 606 00:35:03,700 --> 00:35:08,810 is here, and you replace the first column of the matrix 607 00:35:08,810 --> 00:35:12,400 with vector B. So you-- 608 00:35:12,400 --> 00:35:16,730 no wait, with-- sorry, with vector D. Take this matrix, 609 00:35:16,730 --> 00:35:18,160 and you plug in this. 610 00:35:18,160 --> 00:35:19,180 So what you do is-- 611 00:35:19,180 --> 00:35:19,990 so it turns out-- 612 00:35:22,620 --> 00:35:25,050 so B1 can be explicitly calculated, 613 00:35:25,050 --> 00:35:29,430 but taking the determinant of the first column replaced, 614 00:35:29,430 --> 00:35:35,310 F0 over M0, and keeping the second column, which is minus 615 00:35:35,310 --> 00:35:38,485 k over m. 616 00:35:38,485 --> 00:35:47,910 m and then k over m plus g over l minus omega d squared. 617 00:35:47,910 --> 00:35:51,680 So this is-- you calculate the determinant 618 00:35:51,680 --> 00:35:54,601 of this thing, where-- original matrix with the first column 619 00:35:54,601 --> 00:35:55,100 replaced. 620 00:35:55,100 --> 00:35:58,940 And you divide it by the determinant 621 00:35:58,940 --> 00:36:00,090 of the original matrix. 622 00:36:00,090 --> 00:36:05,310 Let's call it E. So you calculate this determinant 623 00:36:05,310 --> 00:36:08,850 again for the frequency omega d. 624 00:36:08,850 --> 00:36:12,630 So this can be written very nicely, in a very compact way. 625 00:36:12,630 --> 00:36:13,830 This determinant is easy. 626 00:36:13,830 --> 00:36:15,720 It's just this times that. 627 00:36:15,720 --> 00:36:25,500 So have 0 over m multiplying k over n plus g over l 628 00:36:25,500 --> 00:36:28,950 minus on I got the squared remember this is a given 629 00:36:28,950 --> 00:36:33,720 number divided by n Here comes this nice compact form 630 00:36:33,720 --> 00:36:38,360 for the determinant, which is omega d squared minus omega 1 631 00:36:38,360 --> 00:36:45,492 squared, times omega d squared minus omega 2 squared, 632 00:36:45,492 --> 00:36:54,350 where omega 1 and omega 2 were the normal mode frequencies. 633 00:36:54,350 --> 00:36:54,850 Yes? 634 00:36:54,850 --> 00:36:57,516 AUDIENCE: Where are you getting the minus k in the [INAUDIBLE]?? 635 00:37:01,022 --> 00:37:03,225 BOLESLAW WYSLOUCH: This one? 636 00:37:03,225 --> 00:37:04,215 AUDIENCE: Yeah. 637 00:37:04,215 --> 00:37:04,734 [INAUDIBLE] 638 00:37:04,734 --> 00:37:05,900 BOLESLAW WYSLOUCH: This one? 639 00:37:05,900 --> 00:37:07,530 This is the second column. 640 00:37:07,530 --> 00:37:09,141 See? 641 00:37:09,141 --> 00:37:16,020 I'm taking-- so this is the first column, second column. 642 00:37:16,020 --> 00:37:18,870 I take the first column, I replace it 643 00:37:18,870 --> 00:37:22,080 with driven equation-- with a solution. 644 00:37:22,080 --> 00:37:23,430 I plug it here. 645 00:37:23,430 --> 00:37:24,820 So I have F0 for M0. 646 00:37:27,350 --> 00:37:29,570 And I keep the second column. 647 00:37:29,570 --> 00:37:30,070 All right? 648 00:37:30,070 --> 00:37:31,500 That's for the first coefficient. 649 00:37:31,500 --> 00:37:33,250 For the second coefficient what you do is, 650 00:37:33,250 --> 00:37:38,276 you put a driving term here and you keep the first column. 651 00:37:38,276 --> 00:37:40,230 All right? 652 00:37:40,230 --> 00:37:43,290 So this is actually an explicit solution for B1. 653 00:37:43,290 --> 00:37:48,350 This is magnitude of oscillations 654 00:37:48,350 --> 00:37:52,020 of the first element. 655 00:37:52,020 --> 00:37:53,610 And you can do the same thing for B2. 656 00:38:02,020 --> 00:38:04,080 And I'm not trying to prove anything, 657 00:38:04,080 --> 00:38:06,381 I'm not trying to derive anything. 658 00:38:06,381 --> 00:38:07,130 I'm just using it. 659 00:38:07,130 --> 00:38:09,730 And I'll show you a nice slide with this to summarize. 660 00:38:09,730 --> 00:38:16,810 So B2 is the determinant of-- 661 00:38:16,810 --> 00:38:19,240 I keep the first column. 662 00:38:19,240 --> 00:38:24,790 It's k over m plus g over l, minus omega d 663 00:38:24,790 --> 00:38:28,590 squared, minus k over m. 664 00:38:28,590 --> 00:38:29,690 That's the first column. 665 00:38:29,690 --> 00:38:36,340 And I'm plugging in F0 over M here, and 0 here. 666 00:38:36,340 --> 00:38:42,520 So this is-- and divided by omega d squared minus omega 1 667 00:38:42,520 --> 00:38:47,940 squared times omega d squared minus omega 2 squared. 668 00:38:47,940 --> 00:38:51,100 That's the determinant of the original matrix. 669 00:38:51,100 --> 00:38:56,650 And this one is also very simple It's this time this is 0. 670 00:38:56,650 --> 00:38:57,550 I have minus that. 671 00:38:57,550 --> 00:39:04,360 So I simply have F0 k over m squared divided 672 00:39:04,360 --> 00:39:11,320 by omega d squared minus omega 1 squared, omega d 673 00:39:11,320 --> 00:39:15,475 squared minus omega 2 squared. 674 00:39:15,475 --> 00:39:17,070 All right. 675 00:39:17,070 --> 00:39:19,170 So we have those things, and also what? 676 00:39:19,170 --> 00:39:21,950 Do you see anything happening here? 677 00:39:21,950 --> 00:39:25,780 Yeah, there are some numbers, but what do they mean? 678 00:39:25,780 --> 00:39:26,530 What does it mean? 679 00:39:26,530 --> 00:39:28,545 Yes, we can calculate it. 680 00:39:28,545 --> 00:39:29,270 You can trust me. 681 00:39:29,270 --> 00:39:30,392 These are the-- 682 00:39:30,392 --> 00:39:31,850 I'm not sure that you can trust it, 683 00:39:31,850 --> 00:39:33,960 but most likely these are good results. 684 00:39:33,960 --> 00:39:38,270 And so we know the oscillation of the first mass, oscillation 685 00:39:38,270 --> 00:39:43,580 of the second mass as they are driven by the external force. 686 00:39:43,580 --> 00:39:47,640 Now, one of the interesting things to do 687 00:39:47,640 --> 00:39:49,530 is to try to see what's going on. 688 00:39:49,530 --> 00:39:53,460 One of the-- when we talked about normal modes, 689 00:39:53,460 --> 00:39:56,820 the ratio of amplitudes carried information. 690 00:39:56,820 --> 00:39:58,865 Remember, we had those two different modes. 691 00:39:58,865 --> 00:40:02,180 Either amplitudes were the same, or they were opposite sign. 692 00:40:02,180 --> 00:40:05,860 So let's ask ourselves, what is the ratio of B1 and B2? 693 00:40:05,860 --> 00:40:07,490 So let's just divide one by the other. 694 00:40:11,800 --> 00:40:14,790 So let's do B1 over B2. 695 00:40:14,790 --> 00:40:17,550 Let's see if we learn anything from this. 696 00:40:17,550 --> 00:40:23,370 If you divide B1 over B2, this bottom cancels out, 697 00:40:23,370 --> 00:40:31,650 and I have k over m plus g over l minus omega 698 00:40:31,650 --> 00:40:36,910 d squared over k over m. 699 00:40:40,190 --> 00:40:42,470 And-- yeah. 700 00:40:42,470 --> 00:40:47,270 So now comes the interesting question. 701 00:40:47,270 --> 00:40:51,495 This omega d can be anything. 702 00:40:54,340 --> 00:41:01,490 So let's say omega d is-- so we can analyze it different ways. 703 00:41:01,490 --> 00:41:03,911 So for example, when omega d is-- 704 00:41:03,911 --> 00:41:06,160 you can look at small, large, and so I can compare it. 705 00:41:06,160 --> 00:41:08,150 But one of the interesting places to look 706 00:41:08,150 --> 00:41:13,060 is, what happens when omega is very close to one of the-- 707 00:41:13,060 --> 00:41:16,360 to the characteristic frequencies? 708 00:41:16,360 --> 00:41:19,600 Because, remember, when we analyzed a single driven 709 00:41:19,600 --> 00:41:21,880 oscillator, the real cool stuff was 710 00:41:21,880 --> 00:41:25,430 happening when you are near the resonant frequency. 711 00:41:25,430 --> 00:41:28,450 Things, you know, the bridges broke down, et cetera. 712 00:41:28,450 --> 00:41:31,120 So let's see if we can do something similar here. 713 00:41:31,120 --> 00:41:32,260 Now we have two choices. 714 00:41:32,260 --> 00:41:34,700 We have omega 1, omega 2. 715 00:41:34,700 --> 00:41:40,230 So let's see what happens if I plug in omega 1. 716 00:41:40,230 --> 00:41:44,550 Omega d being very, very close to omega 1. 717 00:41:44,550 --> 00:41:46,350 Let's say equal to omega 1. 718 00:41:46,350 --> 00:41:54,510 Omega 1 is-- omega 1 squared was g over l. 719 00:41:54,510 --> 00:41:59,840 So if I plug omega 1 here, I have k over m plus g over l. 720 00:41:59,840 --> 00:42:07,410 So I have k over m plus g over l, minus g over l, 721 00:42:07,410 --> 00:42:13,120 divide by k over m, which is equal to what? 722 00:42:13,120 --> 00:42:14,460 Those two terms cancels. 723 00:42:14,460 --> 00:42:17,110 k over m, it's plus 1. 724 00:42:17,110 --> 00:42:17,990 That's interesting. 725 00:42:17,990 --> 00:42:24,000 So if I drive at a frequency which corresponds to omega 1-- 726 00:42:24,000 --> 00:42:28,440 and omega 1 was the oscillation where both masses 727 00:42:28,440 --> 00:42:31,150 were going together. 728 00:42:31,150 --> 00:42:33,270 So the characteristic normal mode 729 00:42:33,270 --> 00:42:36,520 had the ratio of two masses equal to one. 730 00:42:36,520 --> 00:42:41,580 And here I'm getting the system to drive at this type of mode. 731 00:42:41,580 --> 00:42:44,880 Again, I have-- the driven amplitudes 732 00:42:44,880 --> 00:42:47,390 are the ratio is equal to one. 733 00:42:52,150 --> 00:42:58,630 So what happens if I drive at omega d close to omega 2? 734 00:42:58,630 --> 00:43:09,690 Omega 2 squared was equal to g over l plus 2k over m. 735 00:43:09,690 --> 00:43:16,274 If I plug it in here, I get that the ratio is minus 1. 736 00:43:16,274 --> 00:43:20,470 Again, the ratio is strikingly similar to the ratio 737 00:43:20,470 --> 00:43:25,130 of the normal mode corresponding to frequency omega 2. 738 00:43:25,130 --> 00:43:28,240 So it's like I'm inducing those oscillations. 739 00:43:32,070 --> 00:43:35,380 So what does this all mean? 740 00:43:35,380 --> 00:43:38,040 There's, by the way, a little catch here 741 00:43:38,040 --> 00:43:40,770 for all of your mathematicians. 742 00:43:40,770 --> 00:43:44,730 What happens to equations if I set omega d equal to minus 1-- 743 00:43:44,730 --> 00:43:48,337 to omega 1, for example? 744 00:43:48,337 --> 00:43:50,170 I just plugged it here, and nobody screamed. 745 00:43:50,170 --> 00:43:52,250 But there was something fishy about what I did. 746 00:43:52,250 --> 00:43:52,750 Yes? 747 00:43:52,750 --> 00:43:56,439 AUDIENCE: --coefficient [INAUDIBLE] 748 00:43:56,439 --> 00:43:57,772 BOLESLAW WYSLOUCH: If you took-- 749 00:43:57,772 --> 00:43:58,734 AUDIENCE: Oh, sorry. 750 00:43:58,734 --> 00:44:01,035 [INAUDIBLE] 751 00:44:01,035 --> 00:44:02,160 BOLESLAW WYSLOUCH: Exactly. 752 00:44:02,160 --> 00:44:07,200 So the ratio of the two was one, but both of them were infinite. 753 00:44:07,200 --> 00:44:09,410 So infinite divided by infinite equals what? 754 00:44:09,410 --> 00:44:11,280 I mean, this happens. 755 00:44:11,280 --> 00:44:12,080 So what's going on? 756 00:44:12,080 --> 00:44:12,900 Why can I do it? 757 00:44:12,900 --> 00:44:17,204 One-- we should not really scream. 758 00:44:17,204 --> 00:44:17,971 Damping. 759 00:44:17,971 --> 00:44:18,470 Exactly. 760 00:44:18,470 --> 00:44:20,233 This is where the damping comes in. 761 00:44:20,233 --> 00:44:22,804 So the amplitude is enormous, but it's not infinite, 762 00:44:22,804 --> 00:44:24,470 because there's always a little damping. 763 00:44:24,470 --> 00:44:26,810 The system will not go to infinity. 764 00:44:26,810 --> 00:44:30,200 So in real life, there's a little term here 765 00:44:30,200 --> 00:44:32,290 that makes sure things don't blow up completely. 766 00:44:32,290 --> 00:44:33,539 There's a little damping here. 767 00:44:33,539 --> 00:44:34,052 Yes? 768 00:44:34,052 --> 00:44:35,820 AUDIENCE: Does it at all matter-- 769 00:44:35,820 --> 00:44:37,327 also the fact that those equations 770 00:44:37,327 --> 00:44:39,701 are inexact in the first place, because we 771 00:44:39,701 --> 00:44:40,784 had made theta smaller-- 772 00:44:40,784 --> 00:44:41,700 BOLESLAW WYSLOUCH: No. 773 00:44:41,700 --> 00:44:43,860 That's not-- no. 774 00:44:43,860 --> 00:44:45,630 This doesn't actually matter. 775 00:44:45,630 --> 00:44:53,000 It's the absence of damping that makes things look nonphysical. 776 00:44:53,000 --> 00:44:57,320 AUDIENCE: But as the frequency-- as the amplitude increases, 777 00:44:57,320 --> 00:45:00,200 when we're in resonance, eventually those equations 778 00:45:00,200 --> 00:45:02,120 wouldn't hold any longer, and perhaps-- 779 00:45:02,120 --> 00:45:03,703 BOLESLAW WYSLOUCH: Yeah, that's right. 780 00:45:03,703 --> 00:45:06,850 But you could-- that's true. 781 00:45:06,850 --> 00:45:08,570 That's true. 782 00:45:08,570 --> 00:45:13,650 But you can come up with, for example, an electronic system 783 00:45:13,650 --> 00:45:15,870 which has a huge range of-- 784 00:45:15,870 --> 00:45:18,150 enormous range of possibilities. 785 00:45:18,150 --> 00:45:21,090 And then-- or of amplitudes. 786 00:45:21,090 --> 00:45:26,950 Many, many-- so the damping is much more important in that. 787 00:45:26,950 --> 00:45:29,560 So in reality, there is some damping here and so forth. 788 00:45:29,560 --> 00:45:30,060 All right. 789 00:45:30,060 --> 00:45:32,310 So why don't we do, now, the following. 790 00:45:32,310 --> 00:45:40,200 So let's try to see how this all works out. 791 00:45:40,200 --> 00:45:42,360 First of all, such that we can get 792 00:45:42,360 --> 00:45:45,490 started, I will make a sketch for you. 793 00:45:45,490 --> 00:45:51,860 I'll calculate these formulas-- 794 00:45:51,860 --> 00:45:59,740 just a second-- and display you as a function of frequency, 795 00:45:59,740 --> 00:46:02,604 such that we can analyze what's going on. 796 00:46:02,604 --> 00:46:03,270 So where is it-- 797 00:46:06,140 --> 00:46:06,705 OK. 798 00:46:06,705 --> 00:46:08,580 It's still slow. 799 00:46:08,580 --> 00:46:11,490 All right. 800 00:46:11,490 --> 00:46:15,610 So this is what those-- 801 00:46:15,610 --> 00:46:16,930 OK, so let's say-- 802 00:46:16,930 --> 00:46:22,090 I don't know which is which, but let's say B1 is the red one, 803 00:46:22,090 --> 00:46:23,860 B2 is the blue one, or vice versa. 804 00:46:23,860 --> 00:46:25,770 It doesn't matter. 805 00:46:25,770 --> 00:46:28,690 These are the numbers which I plug in for some values 806 00:46:28,690 --> 00:46:30,410 for some system. 807 00:46:30,410 --> 00:46:31,320 So we see that-- 808 00:46:31,320 --> 00:46:34,790 and this is as a function of frequency. 809 00:46:34,790 --> 00:46:38,890 So first of all, you see a characteristic frequency 810 00:46:38,890 --> 00:46:41,350 around one, characteristic frequency around three 811 00:46:41,350 --> 00:46:42,940 on my plot. 812 00:46:42,940 --> 00:46:46,520 And in the region in the vicinity of frequency number 813 00:46:46,520 --> 00:46:50,860 one, you see that both the blue and red, 814 00:46:50,860 --> 00:46:53,950 the individual amplitudes are basically close together. 815 00:46:53,950 --> 00:46:56,940 So the ratio is close to one. 816 00:46:56,940 --> 00:46:59,250 If you look at this plot, you should believe me 817 00:46:59,250 --> 00:47:01,650 that it's plausible that if you are 818 00:47:01,650 --> 00:47:03,870 very close to the frequency, basically 819 00:47:03,870 --> 00:47:08,460 the red and blue will move together. 820 00:47:08,460 --> 00:47:12,960 If you go around the second frequency, 821 00:47:12,960 --> 00:47:17,360 you see that red goes up, blue goes down, or vice versa 822 00:47:17,360 --> 00:47:19,260 on the other side. 823 00:47:19,260 --> 00:47:22,170 So the ratio is minus 1. 824 00:47:22,170 --> 00:47:24,920 So this plot actually carries in formation. 825 00:47:24,920 --> 00:47:27,710 And in fact, what you see also is 826 00:47:27,710 --> 00:47:32,580 that there is some sort of resonant behavior. 827 00:47:32,580 --> 00:47:35,000 So the amplitudes are enormous if you 828 00:47:35,000 --> 00:47:39,660 are close to any of those characteristic frequencies, 829 00:47:39,660 --> 00:47:42,790 but they're much smaller if you're further out. 830 00:47:42,790 --> 00:47:47,300 There is some motion, but not as pronounced as when 831 00:47:47,300 --> 00:47:51,120 you're at the right driving frequencies. 832 00:47:51,120 --> 00:47:51,620 All right. 833 00:47:51,620 --> 00:47:54,650 So let's try to see it. 834 00:47:54,650 --> 00:47:55,940 Why not? 835 00:47:55,940 --> 00:47:57,530 So let me go to another system-- 836 00:47:57,530 --> 00:47:59,870 a system which consists of two masses, 837 00:47:59,870 --> 00:48:06,500 has the same type of behavior, slightly different parameters. 838 00:48:06,500 --> 00:48:09,960 There is no g here, but everything looks the same. 839 00:48:09,960 --> 00:48:12,980 It's just much easier to show. 840 00:48:12,980 --> 00:48:16,320 And I can remove most of damping. 841 00:48:19,250 --> 00:48:22,430 And you'll see there are again two modes, one 842 00:48:22,430 --> 00:48:24,890 which is like this-- 843 00:48:24,890 --> 00:48:28,160 that's number one, that slow motion. 844 00:48:28,160 --> 00:48:30,050 They move together. 845 00:48:30,050 --> 00:48:34,160 And the other one, which is like this, where 846 00:48:34,160 --> 00:48:35,960 the amplitudes are minus 1. 847 00:48:35,960 --> 00:48:40,240 This is the frequency number two. 848 00:48:40,240 --> 00:48:42,080 So now let's try to drive it. 849 00:48:46,040 --> 00:48:47,030 How do I drive it? 850 00:48:47,030 --> 00:48:52,440 I have some sort of engine here which is applying frequency. 851 00:48:52,440 --> 00:49:01,530 So let's start with some sort of slow motion. 852 00:49:08,930 --> 00:49:12,110 So you see they are moving a little bit. 853 00:49:12,110 --> 00:49:14,800 Very small, minimally. 854 00:49:14,800 --> 00:49:16,910 Just a tiny motion. 855 00:49:16,910 --> 00:49:21,080 But they're kind of together, more or less, right? 856 00:49:21,080 --> 00:49:22,980 Slowly, but together. 857 00:49:22,980 --> 00:49:24,050 And this is what-- 858 00:49:24,050 --> 00:49:27,040 this is this area here. 859 00:49:27,040 --> 00:49:28,970 I don't know if you see that. 860 00:49:28,970 --> 00:49:31,100 This is this area. 861 00:49:31,100 --> 00:49:34,310 I'm driving at a very slow frequency. 862 00:49:34,310 --> 00:49:36,020 I'm somewhere here. 863 00:49:36,020 --> 00:49:41,120 The two masses kind of go together, but very slowly. 864 00:49:41,120 --> 00:49:43,130 So let me now crank up the frequency 865 00:49:43,130 --> 00:49:47,900 and try to be in the region of oscillation. 866 00:49:53,732 --> 00:49:56,570 So you see? 867 00:49:56,570 --> 00:50:00,050 All I did is I changed frequency. 868 00:50:00,050 --> 00:50:01,460 The effect is enormous. 869 00:50:01,460 --> 00:50:02,456 I'm somewhere here now. 870 00:50:06,936 --> 00:50:07,436 You see? 871 00:50:10,440 --> 00:50:12,000 Enormous resonance. 872 00:50:12,000 --> 00:50:13,740 And very soon, I will hit the limit. 873 00:50:13,740 --> 00:50:15,790 The system will break. 874 00:50:15,790 --> 00:50:17,740 OK, so we are somewhere here. 875 00:50:17,740 --> 00:50:20,370 I'm driving it. 876 00:50:20,370 --> 00:50:23,550 Interestingly, this really looks like a harmonic motion 877 00:50:23,550 --> 00:50:24,600 of first type. 878 00:50:24,600 --> 00:50:27,030 There is no other things. 879 00:50:27,030 --> 00:50:32,290 OK, so now let's swing by and get to this area. 880 00:50:32,290 --> 00:50:41,559 So all I'm doing is, I quickly change frequency to- this one. 881 00:50:48,780 --> 00:50:51,090 So now what you see is that there 882 00:50:51,090 --> 00:50:53,910 were some random initial conditions, so 883 00:50:53,910 --> 00:50:56,220 we have a homogeneous equation going, 884 00:50:56,220 --> 00:50:57,900 but the driven is coming in. 885 00:50:57,900 --> 00:51:00,600 All I did is I changed frequency. 886 00:51:00,600 --> 00:51:06,796 And suddenly the system knows that it has to go like that. 887 00:51:06,796 --> 00:51:07,792 Isn't that cool? 888 00:51:12,590 --> 00:51:16,320 So this is the region here. 889 00:51:16,320 --> 00:51:21,150 And all I'm doing is I'm bringing the amplitude up, 890 00:51:21,150 --> 00:51:25,790 because this is close to zero. 891 00:51:25,790 --> 00:51:28,650 And then I'm keeping the ratios close 892 00:51:28,650 --> 00:51:31,130 to the characteristic modes. 893 00:51:31,130 --> 00:51:40,436 So I think-- to be honest, this is one of the coolest-- 894 00:51:40,436 --> 00:51:42,060 all I'm doing, just changing frequency. 895 00:51:42,060 --> 00:51:45,060 And the system just responds and starts 896 00:51:45,060 --> 00:51:48,840 going with a resonance of one particular mode. 897 00:51:48,840 --> 00:51:52,440 So imagine a system that has 1,000 masses, 898 00:51:52,440 --> 00:51:54,541 and you come in with 1,000 frequencies. 899 00:51:54,541 --> 00:51:56,790 You tune one frequency, and suddenly everything starts 900 00:51:56,790 --> 00:51:59,970 oscillating in one go. 901 00:51:59,970 --> 00:52:02,290 And imagine you have multiple buildings, 902 00:52:02,290 --> 00:52:05,100 each with different frequency, and there's an earthquake. 903 00:52:05,100 --> 00:52:06,837 And the frequency is of a certain type, 904 00:52:06,837 --> 00:52:09,420 and one building collapses, and all the other ones are happily 905 00:52:09,420 --> 00:52:10,000 standing. 906 00:52:10,000 --> 00:52:10,500 Why? 907 00:52:10,500 --> 00:52:14,010 Because the earthquake just happened to hit the frequency 908 00:52:14,010 --> 00:52:17,400 that corresponded to one of the normal frequencies 909 00:52:17,400 --> 00:52:19,660 of that particular building. 910 00:52:19,660 --> 00:52:24,045 And it's an extremely powerful trick. 911 00:52:24,045 --> 00:52:27,480 It fishes out normal modes through this driving thing. 912 00:52:27,480 --> 00:52:30,450 And we are able to calculate it explicitly. 913 00:52:30,450 --> 00:52:33,930 So now what I will do is, I will modify the system 914 00:52:33,930 --> 00:52:38,120 and I will make it into a three mass thing, which 915 00:52:38,120 --> 00:52:41,340 will have a somewhat more complicated set 916 00:52:41,340 --> 00:52:44,250 of normal modes. 917 00:52:44,250 --> 00:52:46,800 And then I will show you that I can in fact 918 00:52:46,800 --> 00:52:48,810 go with three different frequencies, 919 00:52:48,810 --> 00:52:53,840 and pull out those even complicated modes. 920 00:52:53,840 --> 00:52:55,030 So this will be it. 921 00:52:55,030 --> 00:52:57,610 So this is a three mass system. 922 00:52:57,610 --> 00:53:02,890 Now before, since we didn't calculate it, what I will do 923 00:53:02,890 --> 00:53:10,060 is, I'll go to the web and I will pull out a nice example. 924 00:53:10,060 --> 00:53:13,330 Let me go to my bookmarks. 925 00:53:13,330 --> 00:53:15,860 Normal modes. 926 00:53:15,860 --> 00:53:21,240 So this is a nice applet from Colorado. 927 00:53:21,240 --> 00:53:23,980 And you can-- 928 00:53:23,980 --> 00:53:27,100 I suppose preso ENG will send you links, et cetera. 929 00:53:27,100 --> 00:53:31,810 You can simulate-- you can do everything with it. 930 00:53:31,810 --> 00:53:35,900 So it has two masses. 931 00:53:35,900 --> 00:53:40,790 It has different amplitudes, different normal modes. 932 00:53:40,790 --> 00:53:42,930 And you can see nothing happens. 933 00:53:42,930 --> 00:53:45,230 So I have to give it some initial condition. 934 00:53:45,230 --> 00:53:49,220 Sorry, I have to change polarization. 935 00:53:49,220 --> 00:53:50,210 Where is polarization? 936 00:53:50,210 --> 00:53:51,920 Here. 937 00:53:51,920 --> 00:53:54,670 I give it some initial condition. 938 00:53:54,670 --> 00:53:56,790 So this is basically what you just saw. 939 00:53:56,790 --> 00:54:02,010 I'm just demonstrating to you that this applet looks the same 940 00:54:02,010 --> 00:54:03,370 as our track. 941 00:54:03,370 --> 00:54:06,112 So this is you can see normal modes. 942 00:54:06,112 --> 00:54:07,570 It's a combination of normal modes. 943 00:54:07,570 --> 00:54:12,792 There's one which is first frequency, second frequency. 944 00:54:12,792 --> 00:54:16,400 This is first normal mode. 945 00:54:16,400 --> 00:54:18,580 This is second normal mode. 946 00:54:18,580 --> 00:54:21,740 You can very quickly see what happens. 947 00:54:21,740 --> 00:54:23,270 So this is what we just looked at. 948 00:54:23,270 --> 00:54:26,310 This is what we calculated, more or less, and so on. 949 00:54:26,310 --> 00:54:28,370 Now I want to show you three masses where things 950 00:54:28,370 --> 00:54:29,840 are somewhat more complicated. 951 00:54:29,840 --> 00:54:32,150 In general, three normal modes. 952 00:54:32,150 --> 00:54:35,750 For the three mass elements, the first normal mode is like that. 953 00:54:35,750 --> 00:54:38,950 All the three masses move together. 954 00:54:38,950 --> 00:54:43,000 And slightly different-- the ratio of amplitudes 955 00:54:43,000 --> 00:54:45,090 is slightly different. 956 00:54:45,090 --> 00:54:48,300 The second mode of operation is actually quite interesting. 957 00:54:48,300 --> 00:54:51,330 The central mass is stationary, and those two 958 00:54:51,330 --> 00:54:56,130 are going forth and back, like this. 959 00:54:56,130 --> 00:54:59,760 And then I have a third frequency 960 00:54:59,760 --> 00:55:04,230 where the middle one is going double the distance, 961 00:55:04,230 --> 00:55:07,080 and the two other ones are going up. 962 00:55:07,080 --> 00:55:09,270 So this is the third normal mode. 963 00:55:13,210 --> 00:55:13,710 All right. 964 00:55:13,710 --> 00:55:17,520 So this is the system which we now have standing here. 965 00:55:17,520 --> 00:55:19,260 Let's quickly see if it works in reality. 966 00:55:29,410 --> 00:55:32,280 So this is the first-- 967 00:55:35,010 --> 00:55:36,930 so this is the first mode. 968 00:55:45,740 --> 00:55:49,230 This is the second one. 969 00:55:49,230 --> 00:55:50,110 All right. 970 00:55:50,110 --> 00:55:51,260 And the third one will be-- 971 00:55:58,410 --> 00:56:02,900 Sometimes I do five of them, and then it's really difficult. 972 00:56:02,900 --> 00:56:03,400 OK. 973 00:56:03,400 --> 00:56:10,690 But-- so we have a computer model, we have a real model. 974 00:56:10,690 --> 00:56:14,320 Let's now do the calculation of the frequencies, the ratios, 975 00:56:14,320 --> 00:56:16,765 such that we can see what happens. 976 00:56:16,765 --> 00:56:20,350 So I'm coming here, I'm changing mass to three. 977 00:56:20,350 --> 00:56:26,100 I'm running my-- the terminal calculating thingy. 978 00:56:26,100 --> 00:56:26,855 OK. 979 00:56:26,855 --> 00:56:27,910 It's very slow. 980 00:56:27,910 --> 00:56:29,360 It's busy, busy, busy. 981 00:56:29,360 --> 00:56:32,400 Imagine-- OK. 982 00:56:32,400 --> 00:56:33,490 Spectacularly slow. 983 00:56:33,490 --> 00:56:34,530 Where is it? 984 00:56:34,530 --> 00:56:37,030 I hope it's not-- oh, here it is. 985 00:56:37,030 --> 00:56:40,250 OK, so this is what's coming out. 986 00:56:40,250 --> 00:56:43,480 So this is the same calculation as we 987 00:56:43,480 --> 00:56:46,810 did, except for three masses. 988 00:56:46,810 --> 00:56:48,610 So what do we have here? 989 00:56:48,610 --> 00:56:50,110 Where's my pointer? 990 00:56:50,110 --> 00:56:54,100 So we have, again, three characteristic frequencies, 991 00:56:54,100 --> 00:56:58,130 we have three masses, and the same type of behavior. 992 00:56:58,130 --> 00:57:00,560 See, if you are far away from resonance, 993 00:57:00,560 --> 00:57:05,230 if you have very low frequency, everybody goes together. 994 00:57:05,230 --> 00:57:06,770 I haven't shown you this one here, 995 00:57:06,770 --> 00:57:08,492 which is also interesting. 996 00:57:08,492 --> 00:57:10,150 I'll show you in a second. 997 00:57:10,150 --> 00:57:12,280 And then-- so presumably if you are 998 00:57:12,280 --> 00:57:15,550 close to the first frequency, you see all three of them 999 00:57:15,550 --> 00:57:17,170 go together. 1000 00:57:17,170 --> 00:57:19,840 And this is the first mode. 1001 00:57:19,840 --> 00:57:22,510 So I should see, if I set the proper frequency, 1002 00:57:22,510 --> 00:57:25,000 the thing should respond in mode number one. 1003 00:57:25,000 --> 00:57:29,050 This is the one where two of them go opposite to each other, 1004 00:57:29,050 --> 00:57:31,440 and the red one is stationary. 1005 00:57:31,440 --> 00:57:33,020 It doesn't move. 1006 00:57:33,020 --> 00:57:35,620 And then you have those things where they're 1007 00:57:35,620 --> 00:57:36,700 kind of more complicated. 1008 00:57:36,700 --> 00:57:39,940 It's difficult to read them from here. 1009 00:57:39,940 --> 00:57:42,310 And I can do it for more masses, et cetera. 1010 00:57:42,310 --> 00:57:44,245 So generally it's calculable. 1011 00:57:44,245 --> 00:57:47,000 It can be calculated and can be actually demonstrated. 1012 00:57:47,000 --> 00:57:49,720 So let's try it. 1013 00:57:49,720 --> 00:57:52,705 So-- 32. 1014 00:57:59,700 --> 00:58:01,730 So there's this magic frequency number one. 1015 00:58:06,944 --> 00:58:10,060 I'm setting frequency by turning a knob. 1016 00:58:10,060 --> 00:58:11,100 That's omega d. 1017 00:58:11,100 --> 00:58:14,230 I'm a supervisor of this operation. 1018 00:58:14,230 --> 00:58:20,530 It stops because of other reasons, but it will continue. 1019 00:58:20,530 --> 00:58:24,375 Then I go to 56. 1020 00:58:34,360 --> 00:58:37,370 By the way, remember that every-- 1021 00:58:37,370 --> 00:58:39,420 this is the particular solution. 1022 00:58:39,420 --> 00:58:42,280 This is a steady state distillation with omega d. 1023 00:58:42,280 --> 00:58:44,830 But we also have all those homogeneous solutions, which 1024 00:58:44,830 --> 00:58:46,840 have to die down with damping. 1025 00:58:46,840 --> 00:58:50,200 Remember, it's a combination of homogeneous plus particular. 1026 00:58:50,200 --> 00:58:52,420 So the motion is actually a little bit distorted 1027 00:58:52,420 --> 00:58:56,200 because we have this homogeneous stuff hanging around. 1028 00:58:56,200 --> 00:58:59,320 But hopefully, if I can start it with little homogeneous stuff, 1029 00:58:59,320 --> 00:59:01,820 it will be better. 1030 00:59:01,820 --> 00:59:02,320 So you see? 1031 00:59:05,050 --> 00:59:06,160 Pretty cool. 1032 00:59:06,160 --> 00:59:07,466 Almost there. 1033 00:59:07,466 --> 00:59:10,846 It's almost in assembly. 1034 00:59:10,846 --> 00:59:11,870 Then it kind of stops. 1035 00:59:14,636 --> 00:59:15,310 You see? 1036 00:59:15,310 --> 00:59:17,380 I get two of those going forth and back, 1037 00:59:17,380 --> 00:59:19,790 more or less, and this one going. 1038 00:59:19,790 --> 00:59:21,700 I could probably tune the frequency 1039 00:59:21,700 --> 00:59:23,890 a little bit higher or lower. 1040 00:59:23,890 --> 00:59:27,520 I'm not exactly at the right place, but I'm close. 1041 00:59:27,520 --> 00:59:34,030 And now let's go to the last one, which is 68 1042 00:59:34,030 --> 00:59:36,896 according to my helpers here. 1043 00:59:44,640 --> 00:59:45,610 You see? 1044 00:59:45,610 --> 00:59:48,310 This one goes opposite phase, and those two 1045 00:59:48,310 --> 00:59:51,480 more or less together. 1046 00:59:51,480 --> 00:59:54,820 Then they keep going. 1047 00:59:54,820 --> 00:59:57,750 See now, those two move a little bit forth and back, 1048 00:59:57,750 --> 00:59:58,790 but they are in phase. 1049 00:59:58,790 --> 00:59:59,760 They move together. 1050 00:59:59,760 --> 01:00:01,490 The ratio is 1. 1051 01:00:01,490 --> 01:00:03,796 And this one-- the ratio is minus 2. 1052 01:00:08,500 --> 01:00:09,000 Right? 1053 01:00:09,000 --> 01:00:10,371 Make sense? 1054 01:00:10,371 --> 01:00:11,120 That's the beauty. 1055 01:00:11,120 --> 01:00:15,200 You drive it at some frequency, and those normal modes pop out. 1056 01:00:15,200 --> 01:00:17,200 It's actually very, very cool. 1057 01:00:17,200 --> 01:00:23,900 And as I said, you encounter those type of behaviors 1058 01:00:23,900 --> 01:00:24,490 very often. 1059 01:00:24,490 --> 01:00:27,160 Sometimes you drive a car and something starts vibrating, 1060 01:00:27,160 --> 01:00:30,940 it's just because the car driving on the road 1061 01:00:30,940 --> 01:00:33,280 creates a frequency, provides a driving frequency which 1062 01:00:33,280 --> 01:00:37,290 corresponds to oscillation frequency or some piece of-- 1063 01:00:37,290 --> 01:00:37,920 old car. 1064 01:00:37,920 --> 01:00:39,430 Usually it happens in old cars. 1065 01:00:42,120 --> 01:00:46,470 So I think that's the message we can-- 1066 01:00:46,470 --> 01:00:49,330 and we have all the machinery to be able to do it. 1067 01:00:49,330 --> 01:00:52,240 We can set up any matrix at K, which 1068 01:00:52,240 --> 01:00:55,240 has information about all the forces acting on anything, 1069 01:00:55,240 --> 01:00:57,640 and we can set matrix M with the masses. 1070 01:00:57,640 --> 01:01:01,870 We can put it all together, we can find normal modes, 1071 01:01:01,870 --> 01:01:03,880 and then we can use Cramer's equation 1072 01:01:03,880 --> 01:01:07,870 to take care of the arbitrary external forces. 1073 01:01:07,870 --> 01:01:09,740 And what comes out, just as a-- 1074 01:01:09,740 --> 01:01:14,380 for summary, for future reference, 1075 01:01:14,380 --> 01:01:18,250 the oscillation of the system is-- 1076 01:01:18,250 --> 01:01:20,110 this is conveniently written. 1077 01:01:20,110 --> 01:01:24,250 This is vector X. In general, this 1078 01:01:24,250 --> 01:01:34,700 homogeneous solution this plus the particular solution, which 1079 01:01:34,700 --> 01:01:41,480 is plus vector B, which is very important. 1080 01:01:41,480 --> 01:01:45,780 Vector B depends on the driving frequency. 1081 01:01:45,780 --> 01:01:48,950 Those amplitudes of a particular solution during emotions 1082 01:01:48,950 --> 01:01:51,040 are dependent on driving frequency. 1083 01:01:51,040 --> 01:01:57,220 Cosine omega d times t. 1084 01:01:57,220 --> 01:02:01,500 So in the most general situation, 1085 01:02:01,500 --> 01:02:04,100 we have some homogeneous solution here, 1086 01:02:04,100 --> 01:02:07,310 and there is this driven solution 1087 01:02:07,310 --> 01:02:11,120 which we observed in action, with proper amplitudes. 1088 01:02:11,120 --> 01:02:13,160 So in fact, what you've seen is the sum 1089 01:02:13,160 --> 01:02:16,230 of both, because this depends on the initial conditions. 1090 01:02:16,230 --> 01:02:20,720 Now, in reality, as with a single oscillator, 1091 01:02:20,720 --> 01:02:23,120 this homogeneous equation, there's always a little 1092 01:02:23,120 --> 01:02:25,730 damping, which we ignore it. 1093 01:02:25,730 --> 01:02:28,190 And the damping comes in, and it only 1094 01:02:28,190 --> 01:02:29,840 affects the homogeneous solution. 1095 01:02:29,840 --> 01:02:33,530 So this part will eventually die down, 1096 01:02:33,530 --> 01:02:36,840 whereas a driven solution is always there. 1097 01:02:36,840 --> 01:02:40,400 There's external force that is driving the system forever 1098 01:02:40,400 --> 01:02:41,850 and ever. 1099 01:02:41,850 --> 01:02:45,860 So this part, this steady state or particular solution 1100 01:02:45,860 --> 01:02:49,110 will remain forever, because there's 1101 01:02:49,110 --> 01:02:52,170 an external source of energy which will always provide it. 1102 01:02:52,170 --> 01:02:53,940 So these guys will die down. 1103 01:02:53,940 --> 01:02:57,090 And of course, because of damping the exact value 1104 01:02:57,090 --> 01:02:59,790 of coefficients B will be slightly modified, 1105 01:02:59,790 --> 01:03:03,692 because as you know from the from a one oscillator example, 1106 01:03:03,692 --> 01:03:05,400 the presence of damping actually slightly 1107 01:03:05,400 --> 01:03:07,280 modifies the frequency. 1108 01:03:07,280 --> 01:03:08,880 Whereas here, we-- for simplicity-- 1109 01:03:08,880 --> 01:03:11,640 if we introduce damping here, those calculations 1110 01:03:11,640 --> 01:03:13,440 are really amazing. 1111 01:03:13,440 --> 01:03:16,410 So we don't want to do it. 1112 01:03:16,410 --> 01:03:17,100 All right. 1113 01:03:17,100 --> 01:03:18,120 Any questions about it? 1114 01:03:22,950 --> 01:03:23,782 Yes. 1115 01:03:23,782 --> 01:03:26,480 AUDIENCE: If we were doing Cramer's rule with a three 1116 01:03:26,480 --> 01:03:30,510 by three matrix, would we only replace the column 1117 01:03:30,510 --> 01:03:34,069 that corresponds to the B that we're trying to find, and then 1118 01:03:34,069 --> 01:03:34,860 keep the other two? 1119 01:03:34,860 --> 01:03:36,160 BOLESLAW WYSLOUCH: Yes. 1120 01:03:36,160 --> 01:03:38,130 So it's always-- you'll be doing always that. 1121 01:03:38,130 --> 01:03:40,980 In fact, I should have some slides from Yen-Jie 1122 01:03:40,980 --> 01:03:43,830 on Cramer's rule. 1123 01:03:43,830 --> 01:03:45,859 Let's see. 1124 01:03:45,859 --> 01:03:46,358 OK. 1125 01:03:46,358 --> 01:03:51,597 So this is some reminder of last time. 1126 01:03:51,597 --> 01:03:53,305 So this is Cramer's-- there's Mr. Cramer. 1127 01:03:55,840 --> 01:03:59,170 So this is an example of what-- 1128 01:03:59,170 --> 01:04:02,730 this is the two by two, three by three. 1129 01:04:02,730 --> 01:04:03,230 OK? 1130 01:04:03,230 --> 01:04:05,210 That's what you do. 1131 01:04:05,210 --> 01:04:06,200 Question? 1132 01:04:06,200 --> 01:04:10,655 AUDIENCE: So it makes sense that the Cramer's rule [INAUDIBLE],, 1133 01:04:10,655 --> 01:04:12,635 but what does that mean for physical system? 1134 01:04:16,610 --> 01:04:19,840 BOLESLAW WYSLOUCH: Well, basically-- 1135 01:04:19,840 --> 01:04:22,800 so the Cramer's rule is Cramer's rule. 1136 01:04:22,800 --> 01:04:24,650 The question is what do you plug in? 1137 01:04:24,650 --> 01:04:27,770 And what you plug in depends on the omega d. 1138 01:04:27,770 --> 01:04:30,790 So it is true that if you insist on plugging in omega 1139 01:04:30,790 --> 01:04:33,950 d exactly equal to one of the normal frequencies, 1140 01:04:33,950 --> 01:04:38,290 then things blow up mathematically. 1141 01:04:38,290 --> 01:04:41,620 In reality, there is-- 1142 01:04:41,620 --> 01:04:43,960 this is the situation of resonance. 1143 01:04:43,960 --> 01:04:46,150 So as I discussed this before, in reality 1144 01:04:46,150 --> 01:04:49,070 there is a little bit of damping. 1145 01:04:49,070 --> 01:04:51,500 So those equations have to be modified. 1146 01:04:51,500 --> 01:04:54,710 There will be some small additional terms here 1147 01:04:54,710 --> 01:04:57,680 that will prevent this from being exactly equal to 0. 1148 01:04:57,680 --> 01:04:59,510 So this will be a very large number. 1149 01:04:59,510 --> 01:05:02,026 The amplitude will be enormous. 1150 01:05:02,026 --> 01:05:03,650 If I would have a little bit more time, 1151 01:05:03,650 --> 01:05:05,120 I'll fiddle with frequency, I could actually 1152 01:05:05,120 --> 01:05:07,850 break the system, because those masses would be just swinging 1153 01:05:07,850 --> 01:05:10,870 forth and back like crazy. 1154 01:05:10,870 --> 01:05:14,310 So you basically go out of limit of the system. 1155 01:05:14,310 --> 01:05:16,560 So physically, there's always a little bit of damping. 1156 01:05:16,560 --> 01:05:18,460 You do not divide by zero. 1157 01:05:18,460 --> 01:05:22,660 On the other hand, it's so close that, for simplicity 1158 01:05:22,660 --> 01:05:26,110 and for most of the-- to get a feeling of what's going on, 1159 01:05:26,110 --> 01:05:27,870 it's OK to ignore it. 1160 01:05:27,870 --> 01:05:31,720 Just have to make sure you don't divide by 0. 1161 01:05:31,720 --> 01:05:34,715 So you can do this Cramer's rule with arbitrary omega d. 1162 01:05:34,715 --> 01:05:37,900 Make sure you don't divide by 0, you solve it, 1163 01:05:37,900 --> 01:05:41,040 and then you can interpret what's going on. 1164 01:05:41,040 --> 01:05:43,700 Again, Cramer's rule has nothing to do with physics. 1165 01:05:43,700 --> 01:05:46,464 It's just a way to solve those matrix equations. 1166 01:05:46,464 --> 01:05:48,130 As I say, you can do it anyway you want. 1167 01:05:48,130 --> 01:05:52,000 Two by two, you can do it by elimination of variables. 1168 01:05:52,000 --> 01:05:55,240 Five by five I do by running a MATLAB program. 1169 01:05:55,240 --> 01:05:56,450 Anything you want. 1170 01:05:56,450 --> 01:06:01,575 But for some historical reasons, 8.03 always does Cramer's rule. 1171 01:06:01,575 --> 01:06:02,220 All right? 1172 01:06:02,220 --> 01:06:06,110 And, yeah, it's useful, especially for three by three. 1173 01:06:06,110 --> 01:06:06,610 All right? 1174 01:06:09,271 --> 01:06:09,770 OK. 1175 01:06:09,770 --> 01:06:12,832 So I have to start a new chapter. 1176 01:06:12,832 --> 01:06:14,665 I'm much slower than the engine, by the way. 1177 01:06:14,665 --> 01:06:15,831 I don't know if you noticed. 1178 01:06:18,270 --> 01:06:20,440 And that is the-- 1179 01:06:20,440 --> 01:06:22,710 there's a very interesting trick that you 1180 01:06:22,710 --> 01:06:29,670 can do which is of an absolutely fundamental nature in physics, 1181 01:06:29,670 --> 01:06:32,310 which has to do with symmetry. 1182 01:06:32,310 --> 01:06:36,900 You see, many things are symmetric. 1183 01:06:36,900 --> 01:06:39,040 There's a circular symmetry. 1184 01:06:39,040 --> 01:06:41,110 There's a left and right symmetry. 1185 01:06:41,110 --> 01:06:44,800 Example, two little smiley faces are 1186 01:06:44,800 --> 01:06:47,040 mirror images of each other. 1187 01:06:47,040 --> 01:06:50,250 There is some-- this thing is symmetric 1188 01:06:50,250 --> 01:06:52,620 along this vertical axis. 1189 01:06:52,620 --> 01:06:55,665 This one is symmetric around rotations by 30 degrees. 1190 01:06:58,230 --> 01:07:01,470 That house seems to be symmetric along this way. 1191 01:07:01,470 --> 01:07:04,660 This is part of our experiment in Switzerland, 1192 01:07:04,660 --> 01:07:07,155 also kind of symmetric in the picture. 1193 01:07:09,624 --> 01:07:12,290 The rotational symmetry, there's reflection symmetry, et cetera. 1194 01:07:12,290 --> 01:07:16,210 It turns out, if you have a system that is symmetric, 1195 01:07:16,210 --> 01:07:19,570 then the normal modes are also symmetric. 1196 01:07:19,570 --> 01:07:23,500 And there's a way to dig out normal modes just by looking 1197 01:07:23,500 --> 01:07:24,610 at symmetry of the system. 1198 01:07:27,460 --> 01:07:29,870 So let me explain exactly what this means. 1199 01:07:29,870 --> 01:07:34,520 So let's take our system here-- 1200 01:07:34,520 --> 01:07:39,890 OK, so we have one mass, the other mass. 1201 01:07:39,890 --> 01:07:43,000 There is a spring here. 1202 01:07:43,000 --> 01:07:46,520 This one is x1, this one is x2. 1203 01:07:46,520 --> 01:07:48,590 If I take a reflection of a system-- 1204 01:07:48,590 --> 01:07:53,170 let's say this mass is displaced by some distance. 1205 01:07:53,170 --> 01:07:56,330 Some x2. 1206 01:07:56,330 --> 01:07:57,620 This one's some x1. 1207 01:07:57,620 --> 01:07:59,280 If I do the following transform-- 1208 01:07:59,280 --> 01:08:08,780 I replace x1 with minus x2, and x2 with minus x1, 1209 01:08:08,780 --> 01:08:09,890 this is mirror symmetry. 1210 01:08:16,700 --> 01:08:20,050 I basically flip this thing around. 1211 01:08:20,050 --> 01:08:25,149 In other words, what I do here is I look at the system here-- 1212 01:08:25,149 --> 01:08:28,850 hello-- and I go to the other system. 1213 01:08:28,850 --> 01:08:29,540 Hello. 1214 01:08:29,540 --> 01:08:30,040 Right? 1215 01:08:30,040 --> 01:08:31,081 I did a mirror transform. 1216 01:08:31,081 --> 01:08:33,750 I looked at it from this side, that side. 1217 01:08:33,750 --> 01:08:35,890 Now, when I look at it I see the one 1218 01:08:35,890 --> 01:08:37,359 on the left, one on the right. 1219 01:08:37,359 --> 01:08:40,210 I call this one x1, this one x2. 1220 01:08:40,210 --> 01:08:41,950 It's oscillating. 1221 01:08:41,950 --> 01:08:46,029 You are looking at it, this is your x1, this is your x2. 1222 01:08:46,029 --> 01:08:47,960 When I move this one, is it-- 1223 01:08:47,960 --> 01:08:50,229 it's your negative x1. 1224 01:08:50,229 --> 01:08:53,399 For me this is positive x2. 1225 01:08:53,399 --> 01:08:56,920 This one is positive x2 for you. 1226 01:08:56,920 --> 01:09:00,160 It's negative x1 for me. 1227 01:09:00,160 --> 01:09:01,630 Do we see a different system? 1228 01:09:01,630 --> 01:09:03,130 Does it have different oscillations? 1229 01:09:03,130 --> 01:09:04,250 Does it have a different frequency? 1230 01:09:04,250 --> 01:09:04,750 No. 1231 01:09:04,750 --> 01:09:05,689 It's identical. 1232 01:09:05,689 --> 01:09:07,640 They're completely identical. 1233 01:09:07,640 --> 01:09:09,520 So the physics of those two pendula 1234 01:09:09,520 --> 01:09:11,590 doesn't depend on if he's working on it 1235 01:09:11,590 --> 01:09:13,000 or if I'm working on. 1236 01:09:13,000 --> 01:09:14,439 That's the whole thing. 1237 01:09:14,439 --> 01:09:17,529 And this is how you write it mathematically. 1238 01:09:17,529 --> 01:09:20,920 And if you have a solution which-- 1239 01:09:20,920 --> 01:09:27,640 x1 of t, which consists of some sort of x1 of t, x2 of t. 1240 01:09:27,640 --> 01:09:30,180 Let's say we find it. 1241 01:09:30,180 --> 01:09:31,240 Now it's over there. 1242 01:09:31,240 --> 01:09:34,490 We know alphas, betas and everything. 1243 01:09:34,490 --> 01:09:38,029 Because of the symmetry, I know that for sure the equation 1244 01:09:38,029 --> 01:09:41,180 which looks like this-- x1-- 1245 01:09:41,180 --> 01:09:42,420 no, it's not x1. 1246 01:09:42,420 --> 01:09:44,270 It's x of t. 1247 01:09:44,270 --> 01:09:46,208 That's the vector x of t. 1248 01:09:46,208 --> 01:09:51,170 I have another one with a tilde, which is identical functions, 1249 01:09:51,170 --> 01:09:56,990 everything is dependent, except that this one is minus x2 of t 1250 01:09:56,990 --> 01:10:00,290 minus x1 of t. 1251 01:10:00,290 --> 01:10:04,520 And I know for sure that if this is the correct solution, 1252 01:10:04,520 --> 01:10:07,230 this is also a correct solution. 1253 01:10:07,230 --> 01:10:07,980 Why? 1254 01:10:07,980 --> 01:10:14,370 Because he did x, and I did x tilde. 1255 01:10:14,370 --> 01:10:16,200 But the system is the same. 1256 01:10:16,200 --> 01:10:17,220 Completely identical. 1257 01:10:17,220 --> 01:10:22,020 And you don't have to know anything about masses, lengths, 1258 01:10:22,020 --> 01:10:23,790 springs, anything like that. 1259 01:10:23,790 --> 01:10:25,890 Just the symmetry. 1260 01:10:25,890 --> 01:10:26,730 All right. 1261 01:10:26,730 --> 01:10:29,940 How do you write it in matrix form? 1262 01:10:29,940 --> 01:10:37,690 You introduce a symmetry matrix, S, which is 0, minus 1, 1263 01:10:37,690 --> 01:10:40,400 minus 1, 0. 1264 01:10:40,400 --> 01:10:48,584 And then x tilde of t is simply equal S, x of t. 1265 01:10:48,584 --> 01:10:49,500 And we can check that. 1266 01:10:49,500 --> 01:10:53,590 That's simple you just multiply the vector by 0, 1267 01:10:53,590 --> 01:10:57,370 minus 1, minus 1, 0, and you get the same thing. 1268 01:10:57,370 --> 01:11:02,200 Turns out-- and if this is symmetry, 1269 01:11:02,200 --> 01:11:04,660 if this is a solution, this is also a solution. 1270 01:11:04,660 --> 01:11:08,670 So we can make solutions by multiplying by matrix S. 1271 01:11:08,670 --> 01:11:10,030 So what does it mean? 1272 01:11:10,030 --> 01:11:14,115 So let's look at our motion equation. 1273 01:11:17,880 --> 01:11:19,970 The original motion equation was-- 1274 01:11:19,970 --> 01:11:23,810 equation of motion was minus 1 k matrix times x of t. 1275 01:11:23,810 --> 01:11:26,740 This is what we use to find solutions. 1276 01:11:26,740 --> 01:11:28,550 Usual thing, normal modes, et cetera. 1277 01:11:28,550 --> 01:11:32,970 Let's multiply both sides by matrix S. I can take any matrix 1278 01:11:32,970 --> 01:11:35,230 and multiply by both sides. 1279 01:11:35,230 --> 01:11:39,391 So I get here S X double dot of t. 1280 01:11:39,391 --> 01:11:43,300 And of course, S is a fixed matrix, 1281 01:11:43,300 --> 01:11:45,710 so it survives differentiation. 1282 01:11:45,710 --> 01:11:51,910 And this is equal to minus S M minus 1 k x of t. 1283 01:11:51,910 --> 01:11:54,970 Just multiply both sides by S. 1284 01:11:54,970 --> 01:12:07,286 However, if MS is equal to SM, and KS is equal to SK-- 1285 01:12:11,110 --> 01:12:14,160 in general matrices, the multiplication of matrices 1286 01:12:14,160 --> 01:12:15,990 matters. 1287 01:12:15,990 --> 01:12:18,970 But it turns out that if the system is symmetric, 1288 01:12:18,970 --> 01:12:24,130 if you multiply mass M by S, you just replace-- 1289 01:12:24,130 --> 01:12:26,840 it will just change position of two masses. 1290 01:12:26,840 --> 01:12:28,520 So nothing changes. 1291 01:12:28,520 --> 01:12:34,000 Also, if the forces are the same, then multiplying mass S, 1292 01:12:34,000 --> 01:12:34,850 you flip things. 1293 01:12:34,850 --> 01:12:36,110 Nothing changes. 1294 01:12:36,110 --> 01:12:38,480 And mathematically, it means that the order 1295 01:12:38,480 --> 01:12:41,420 of multiplication does not matter. 1296 01:12:41,420 --> 01:12:42,870 It means that they are commuting. 1297 01:12:42,870 --> 01:12:50,740 And of course, M minus 1 S is equal to S M minus 1. 1298 01:12:50,740 --> 01:12:55,820 If this is the case, then I can plug it into equations 1299 01:12:55,820 --> 01:12:58,408 and see what happens. 1300 01:13:07,870 --> 01:13:16,420 So I can take this equation, and I can take this S here 1301 01:13:16,420 --> 01:13:18,260 and I can just move it around. 1302 01:13:18,260 --> 01:13:21,790 I can flip it with M1 position, because the order doesn't 1303 01:13:21,790 --> 01:13:22,340 matter. 1304 01:13:22,340 --> 01:13:23,650 So I can bring it here. 1305 01:13:23,650 --> 01:13:25,780 And I can flip it with K, because the order doesn't 1306 01:13:25,780 --> 01:13:26,280 matter. 1307 01:13:26,280 --> 01:13:28,180 I can bring it here. 1308 01:13:28,180 --> 01:13:31,080 So after using those features, I get 1309 01:13:31,080 --> 01:13:40,580 that S X dot dot is equal to minus M minus 1 K S X, 1310 01:13:40,580 --> 01:13:44,810 which means that X dot dot-- 1311 01:13:44,810 --> 01:13:47,430 remember, this was-- 1312 01:13:47,430 --> 01:13:48,920 I'm using this expression. 1313 01:13:48,920 --> 01:13:53,440 I'm just-- S times a variable x gives me X tilde. 1314 01:13:53,440 --> 01:14:00,090 X tilde dot dot is equal to minus M minus 1 k X tilde. 1315 01:14:00,090 --> 01:14:01,770 X tilde. 1316 01:14:01,770 --> 01:14:04,415 Which basically proves-- this is a proof-- 1317 01:14:04,415 --> 01:14:06,930 that x tilde is a solution. 1318 01:14:06,930 --> 01:14:12,100 So if a system is symmetric, it means that it commutes-- 1319 01:14:12,100 --> 01:14:20,010 that mass and K matrices commute, and you can-- 1320 01:14:20,010 --> 01:14:22,530 and this means that this holds true. 1321 01:14:22,530 --> 01:14:24,870 If I have one solution, the symmetric solution 1322 01:14:24,870 --> 01:14:27,290 is also there. 1323 01:14:27,290 --> 01:14:28,150 All right? 1324 01:14:32,000 --> 01:14:34,354 Let's say x-- yes? 1325 01:14:34,354 --> 01:14:36,764 AUDIENCE: So in the center equation, 1326 01:14:36,764 --> 01:14:42,080 you introduced negative S. I didn't really get that. 1327 01:14:42,080 --> 01:14:45,040 BOLESLAW WYSLOUCH: So this negative is simply the-- 1328 01:14:45,040 --> 01:14:47,200 Hooke's law. 1329 01:14:47,200 --> 01:14:49,371 This is this minus sign here. 1330 01:14:49,371 --> 01:14:52,294 AUDIENCE: Yeah, but where did the S come from in the-- 1331 01:14:52,294 --> 01:14:54,460 BOLESLAW WYSLOUCH: Oh, I multiplied both sides by S. 1332 01:14:54,460 --> 01:14:55,600 AUDIENCE: Oh, OK. 1333 01:14:55,600 --> 01:14:57,225 BOLESLAW WYSLOUCH: I just brought the S 1334 01:14:57,225 --> 01:14:58,500 and I put it here. 1335 01:14:58,500 --> 01:15:02,950 S, X dot dot, and S after-- 1336 01:15:02,950 --> 01:15:07,360 minus commutes with S, so I kind of shifted my minus. 1337 01:15:07,360 --> 01:15:10,540 But then I waited before I hit the matrices, 1338 01:15:10,540 --> 01:15:12,180 because I wanted to discuss. 1339 01:15:12,180 --> 01:15:12,680 OK? 1340 01:15:15,300 --> 01:15:18,850 So now comes the interesting question. 1341 01:15:18,850 --> 01:15:22,070 Let's say X is a normal mode. 1342 01:15:22,070 --> 01:15:22,846 Right? 1343 01:15:22,846 --> 01:15:23,720 We have normal modes. 1344 01:15:23,720 --> 01:15:25,850 Let's say X is a normal mode. 1345 01:15:25,850 --> 01:15:29,630 It oscillates with a certain frequency. 1346 01:15:29,630 --> 01:15:31,990 So I have X of t. 1347 01:15:31,990 --> 01:15:33,560 Let's say it's equal to-- 1348 01:15:33,560 --> 01:15:36,510 let's say it's a normal mode number one. 1349 01:15:36,510 --> 01:15:39,880 Cosine omega 1 t. 1350 01:15:42,500 --> 01:15:46,390 And we know that X tilde is also a solution. 1351 01:15:46,390 --> 01:15:54,180 So what happens to mode number one when I apply matrix S? 1352 01:15:54,180 --> 01:15:59,140 So X tilde-- so matrix is a constant number. 1353 01:15:59,140 --> 01:16:00,640 It's just a couple of numbers I just 1354 01:16:00,640 --> 01:16:01,806 reshuffle things, et cetera. 1355 01:16:01,806 --> 01:16:04,630 Try So if I have X, which is oscillating with frequency 1356 01:16:04,630 --> 01:16:08,290 omega 1, if I multiply by some numbers 1357 01:16:08,290 --> 01:16:10,780 and reshuffle things around, it will also 1358 01:16:10,780 --> 01:16:12,910 be oscillating in number one. 1359 01:16:12,910 --> 01:16:16,310 So it will be also the same normal mode. 1360 01:16:16,310 --> 01:16:20,380 So if I take matrix S, I apply it to the normal mode, 1361 01:16:20,380 --> 01:16:23,080 I will get the same normal mode, with maybe 1362 01:16:23,080 --> 01:16:25,310 a different coefficient. 1363 01:16:25,310 --> 01:16:26,770 Linear coefficient. 1364 01:16:26,770 --> 01:16:29,020 Plus, minus, maybe some factor, something like that. 1365 01:16:31,670 --> 01:16:34,170 So if this is the solution, it means automatically 1366 01:16:34,170 --> 01:16:43,540 that X tilde is proportional to A1 cosine omega 1 t. 1367 01:16:43,540 --> 01:16:46,540 And the same is true for omega 2. 1368 01:16:46,540 --> 01:16:48,190 So the only way this is possible, 1369 01:16:48,190 --> 01:16:52,290 since cosine is the same in both cases-- 1370 01:16:52,290 --> 01:16:54,160 matrix S to normal solution gives me 1371 01:16:54,160 --> 01:16:56,700 normal solution with some sign. 1372 01:16:56,700 --> 01:16:59,010 So the only way this can work, matrix S actually 1373 01:16:59,010 --> 01:17:02,010 works on vectors, on A1. 1374 01:17:02,010 --> 01:17:04,830 This is just an oscillating factor. 1375 01:17:04,830 --> 01:17:13,110 So we know for sure that S A1 must be proportional. 1376 01:17:13,110 --> 01:17:13,680 to A1. 1377 01:17:17,840 --> 01:17:24,238 Similarly, S times A2 is proportional to A2. 1378 01:17:28,110 --> 01:17:32,530 So let's try to see with our own eyes if this works. 1379 01:17:32,530 --> 01:17:36,420 So let's say S is 0, minus 1, minus 1, 1380 01:17:36,420 --> 01:17:42,820 0, times 1, 1 is equal to what? 1381 01:17:42,820 --> 01:17:47,160 0 minus 1, I get minus 1 here. 1382 01:17:47,160 --> 01:17:50,340 This one, I get minus 1 here, which 1383 01:17:50,340 --> 01:17:57,780 is equal to minus 1 times 1, 1, which is vector A. So 1384 01:17:57,780 --> 01:18:04,850 vector 1, 1, which is our first mode of oscillation, 1385 01:18:04,850 --> 01:18:09,410 is when you apply the matrix S, you get a minus 1 1386 01:18:09,410 --> 01:18:12,020 the same thing. 1387 01:18:12,020 --> 01:18:19,144 And similarly, if you do the same thing with matrix S-- 1388 01:18:19,144 --> 01:18:21,890 so you see, the simple symmetric matrix 1389 01:18:21,890 --> 01:18:24,530 consisting of 0s and minus 1s has 1390 01:18:24,530 --> 01:18:26,600 something to do with our solutions, which 1391 01:18:26,600 --> 01:18:27,500 is kind of amazing. 1392 01:18:30,020 --> 01:18:36,980 So if I have 0, minus 1, minus 1, 0, I multiply by 1, minus 1, 1393 01:18:36,980 --> 01:18:43,740 I get 1 here, I get minus 1 here. 1394 01:18:48,800 --> 01:18:49,740 Just a moment. 1395 01:18:49,740 --> 01:18:52,494 Something is not right. 1396 01:18:52,494 --> 01:18:53,410 Something's not right. 1397 01:18:53,410 --> 01:18:54,040 No, it should be-- 1398 01:18:54,040 --> 01:18:54,978 AUDIENCE: [INAUDIBLE] 1399 01:18:54,978 --> 01:18:55,936 BOLESLAW WYSLOUCH: Hmm? 1400 01:18:55,936 --> 01:18:57,358 AUDIENCE: [INAUDIBLE] 1401 01:18:57,358 --> 01:18:59,760 AUDIENCE: It's 1, minus 1. 1402 01:18:59,760 --> 01:19:01,010 BOLESLAW WYSLOUCH: 1, minus 1. 1403 01:19:01,010 --> 01:19:01,664 Yes. 1404 01:19:01,664 --> 01:19:03,080 I don't know how to multiply here. 1405 01:19:03,080 --> 01:19:05,280 I should be fine. 1406 01:19:05,280 --> 01:19:06,799 OK. 1407 01:19:06,799 --> 01:19:07,340 That's right. 1408 01:19:07,340 --> 01:19:09,006 This is-- sorry, this is 1, because it's 1409 01:19:09,006 --> 01:19:11,870 minus 1 times minus 1, and this is-- yeah, that's right. 1410 01:19:11,870 --> 01:19:17,060 Which is 1 times 1, minus 1. 1411 01:19:17,060 --> 01:19:18,590 So this is something that-- 1412 01:19:18,590 --> 01:19:21,930 I get the same vector multiplied by plus 1. 1413 01:19:21,930 --> 01:19:23,270 So this is, of course-- 1414 01:19:23,270 --> 01:19:27,150 these are eigenvectors and eigenvalues. 1415 01:19:27,150 --> 01:19:29,870 So the matrix S has two eigenvectors, 1416 01:19:29,870 --> 01:19:32,690 one with eigenvalue of plus one, the other one plus 2. 1417 01:19:32,690 --> 01:19:38,196 So we have an equation SA is equal to beta times A, 1418 01:19:38,196 --> 01:19:41,250 and beta is-- 1419 01:19:44,910 --> 01:19:46,310 OK. 1420 01:19:46,310 --> 01:19:48,150 So this is something-- 1421 01:19:48,150 --> 01:19:51,680 so it turns out-- and I don't think I have time to prove it, 1422 01:19:51,680 --> 01:19:53,770 but it turns out you can prove it-- 1423 01:19:53,770 --> 01:19:56,760 if I would have another three minutes-- 1424 01:19:56,760 --> 01:20:01,350 you can prove it that the eigenvalues of matrix S-- 1425 01:20:01,350 --> 01:20:06,300 eigenvectors, sorry, eigenvectors of matrix S 1426 01:20:06,300 --> 01:20:11,310 are the same as eigenvectors of the full motion matrix. 1427 01:20:14,120 --> 01:20:22,260 So in other words, our motion matrix M minus 1 K-- 1428 01:20:22,260 --> 01:20:23,160 this is the matrix. 1429 01:20:23,160 --> 01:20:30,930 Then we have a matrix S. And normal modes are, 1430 01:20:30,930 --> 01:20:34,830 you have a normal frequency and they have a shape. 1431 01:20:34,830 --> 01:20:38,400 You have a normal vector, the ratio of amplitudes. 1432 01:20:38,400 --> 01:20:46,560 And turns out that eigenvectors here, so the A's are the same. 1433 01:20:46,560 --> 01:20:50,780 And again, I don't have time to show it, 1434 01:20:50,780 --> 01:20:53,620 but you can show that this is the case. 1435 01:20:53,620 --> 01:20:57,990 So if you have a symmetry in the system, 1436 01:20:57,990 --> 01:21:04,770 then you can simply find eigenvectors of the thing 1437 01:21:04,770 --> 01:21:07,170 to obtain the normal modes. 1438 01:21:07,170 --> 01:21:14,870 So if I look at my two pendula here, the symmetry is this way, 1439 01:21:14,870 --> 01:21:19,880 so I have to have one which is fully symmetric, like this, 1440 01:21:19,880 --> 01:21:21,830 and I have another one which is antisymmetric. 1441 01:21:21,830 --> 01:21:26,042 Plus 1, minus 1, plus 1, minus 1. 1442 01:21:26,042 --> 01:21:28,280 Similarly, here I have-- 1443 01:21:28,280 --> 01:21:30,330 let's say if I have two masses, there 1444 01:21:30,330 --> 01:21:32,650 is one motion which is like this, 1445 01:21:32,650 --> 01:21:34,880 and one motion which is like that, because 1446 01:21:34,880 --> 01:21:37,860 of the mirror symmetry. 1447 01:21:37,860 --> 01:21:40,410 And you can show that if you have some other symmetries, 1448 01:21:40,410 --> 01:21:42,360 like on a circle et cetera, that you 1449 01:21:42,360 --> 01:21:43,800 have a similar type of fact. 1450 01:21:43,800 --> 01:21:47,040 So you can build up on this symmetry argument. 1451 01:21:47,040 --> 01:21:53,880 And finding eigenvectors of a matrix 0, minus 1, minus 1, 0 1452 01:21:53,880 --> 01:21:56,940 is infinitely simpler than finding matrix 1453 01:21:56,940 --> 01:22:00,290 with G's and K's and everything, right? 1454 01:22:00,290 --> 01:22:00,790 All right. 1455 01:22:00,790 --> 01:22:08,120 So thank you very much, and I hope this was educational.