1 00:00:00,090 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,150 to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:21,210 --> 00:00:23,150 PROFESSOR: So morning. 9 00:00:23,150 --> 00:00:29,810 This is my fourth lecture on differential equations, 10 00:00:29,810 --> 00:00:31,230 that part of the course. 11 00:00:31,230 --> 00:00:35,690 And I haven't said anything about the textbook. 12 00:00:35,690 --> 00:00:38,810 That's Differential Equations and Linear Algebra. 13 00:00:38,810 --> 00:00:44,310 I wrote it because so many courses like this one 14 00:00:44,310 --> 00:00:47,930 want to combine those two topics. 15 00:00:47,930 --> 00:00:51,760 They're the two major topics of undergraduate math, 16 00:00:51,760 --> 00:00:56,220 after calculus, the two major directions, and the book 17 00:00:56,220 --> 00:00:57,735 connects those two directions. 18 00:00:57,735 --> 00:01:05,300 And I just wanted to give you the website for lots 19 00:01:05,300 --> 00:01:07,110 of things connected with the book. 20 00:01:07,110 --> 00:01:12,340 So it's the math website, dela for differential equations 21 00:01:12,340 --> 00:01:14,270 and linear algebra. 22 00:01:14,270 --> 00:01:23,370 And so today is differential equations, second order, 23 00:01:23,370 --> 00:01:29,610 with a damping term, with a first derivative term. 24 00:01:29,610 --> 00:01:37,020 So that in many engineering problems, those coefficients 25 00:01:37,020 --> 00:01:41,420 A, B, C would have the meaning of mass, 26 00:01:41,420 --> 00:01:45,880 damping, and stiffness. 27 00:01:45,880 --> 00:01:47,410 Mass, damping, and stiffness. 28 00:01:47,410 --> 00:01:52,910 And physically, we know what the mass comes from. 29 00:01:52,910 --> 00:01:57,640 The stiffness comes from a spring, as I drew before. 30 00:01:57,640 --> 00:02:02,660 And traditionally we describe the source 31 00:02:02,660 --> 00:02:05,920 of damping as a dashpot. 32 00:02:05,920 --> 00:02:09,320 I guess in my whole life I've never seen a dashpot, 33 00:02:09,320 --> 00:02:15,770 but maybe it's-- think of a piston going up and down within 34 00:02:15,770 --> 00:02:20,770 a cylinder of oil or something, with resistance. 35 00:02:20,770 --> 00:02:25,230 OK, so that's the left-hand side. 36 00:02:25,230 --> 00:02:28,610 Linear constant coefficients still. 37 00:02:28,610 --> 00:02:35,910 We don't have formulas, it's not easy to see what's 38 00:02:35,910 --> 00:02:38,620 happening when things are varying, 39 00:02:38,620 --> 00:02:43,100 when the equations non-linear, so this is the starting place. 40 00:02:43,100 --> 00:02:45,380 OK. 41 00:02:45,380 --> 00:02:46,650 With some forcing. 42 00:02:46,650 --> 00:02:50,650 So this is damped forced motion. 43 00:02:50,650 --> 00:02:54,230 This is the ultimate within linear equations. 44 00:02:54,230 --> 00:02:54,980 OK. 45 00:02:54,980 --> 00:02:56,900 And now, what's the forcing? 46 00:02:56,900 --> 00:03:04,770 So now, always we solve first for the null solution. 47 00:03:04,770 --> 00:03:09,200 With no force, what are the natural motions? 48 00:03:09,200 --> 00:03:12,080 And we'll find a formula for those. 49 00:03:12,080 --> 00:03:16,760 Then the big one, the special right-hand side 50 00:03:16,760 --> 00:03:18,165 is always an exponential. 51 00:03:20,670 --> 00:03:24,590 So this is going to be y is going to be the null solution 52 00:03:24,590 --> 00:03:28,190 yn, and we'll get a formula for that. 53 00:03:28,190 --> 00:03:31,000 This, the exponentials. 54 00:03:31,000 --> 00:03:36,410 Always, the response to an exponential is an exponential. 55 00:03:36,410 --> 00:03:40,890 That's like the most important fact in getting solutions. 56 00:03:40,890 --> 00:03:46,050 So the response will be some ye to the st 57 00:03:46,050 --> 00:03:50,960 at the same frequency-- well, I say frequency. 58 00:03:50,960 --> 00:03:55,470 If s is a real number that would be a growth or a decay, 59 00:03:55,470 --> 00:03:58,540 but very frequently s is an imaginary number, 60 00:03:58,540 --> 00:04:00,120 like last time. 61 00:04:00,120 --> 00:04:06,080 And this is comes from a rotation or an oscillation. 62 00:04:06,080 --> 00:04:11,210 And then, the complete picture comes 63 00:04:11,210 --> 00:04:14,590 from being able to solve it with an impulse. 64 00:04:14,590 --> 00:04:17,560 So then y is the impulse response, 65 00:04:17,560 --> 00:04:23,320 which I write as g of t. 66 00:04:23,320 --> 00:04:25,940 Let me introduce just that letter g. 67 00:04:25,940 --> 00:04:30,120 That stands for growth factor in first order equations 68 00:04:30,120 --> 00:04:35,150 stands for Green's function, so that word Green 69 00:04:35,150 --> 00:04:37,310 is getting in here, his name. 70 00:04:37,310 --> 00:04:43,170 And it represents the impulse response. 71 00:04:43,170 --> 00:04:46,610 OK, one, two, three. 72 00:04:46,610 --> 00:04:52,250 And then the point of doing this one is that then we can do it, 73 00:04:52,250 --> 00:04:55,270 then we can get a formula for any f of t. 74 00:04:55,270 --> 00:04:57,710 So this is the pattern that we followed 75 00:04:57,710 --> 00:05:00,620 for first order equations. 76 00:05:00,620 --> 00:05:02,980 We followed it for second order equations 77 00:05:02,980 --> 00:05:04,790 that didn't have damping. 78 00:05:04,790 --> 00:05:10,630 And now we're doing the big one with damping. 79 00:05:10,630 --> 00:05:12,910 So what's the damping going to do? 80 00:05:12,910 --> 00:05:16,110 What's going to be effect of damping? 81 00:05:16,110 --> 00:05:23,820 Say if I had a right side of one, just a unit force? 82 00:05:23,820 --> 00:05:25,300 The damping. 83 00:05:25,300 --> 00:05:29,240 So I'll still have oscillation-- you'll 84 00:05:29,240 --> 00:05:31,260 see this-- I'll still have oscillation, 85 00:05:31,260 --> 00:05:34,310 but its amplitude damps out. 86 00:05:34,310 --> 00:05:37,240 Like that like everything we know, like something 87 00:05:37,240 --> 00:05:42,090 swinging back and forth, but friction 88 00:05:42,090 --> 00:05:45,390 is it is eventually damping out that motion. 89 00:05:45,390 --> 00:05:50,570 So it'll be oscillating with exponentially 90 00:05:50,570 --> 00:05:52,160 decaying amplitude. 91 00:05:52,160 --> 00:05:53,730 Let's find this solution. 92 00:05:53,730 --> 00:05:56,430 OK. 93 00:05:56,430 --> 00:06:01,210 How do we find the null solution? 94 00:06:01,210 --> 00:06:04,480 I've got constant coefficients here. 95 00:06:04,480 --> 00:06:11,900 So the nice, the right thing to look at is exponentials. 96 00:06:11,900 --> 00:06:15,110 In first order equations it was e to the at. 97 00:06:15,110 --> 00:06:17,110 Here, we'll have to see what it is. 98 00:06:17,110 --> 00:06:22,330 So I'm going to try a particular-- I'm 99 00:06:22,330 --> 00:06:25,310 going to look for the right exponentials. 100 00:06:25,310 --> 00:06:29,770 Certain exponentials will be null solution. 101 00:06:29,770 --> 00:06:30,460 Can I do it? 102 00:06:30,460 --> 00:06:37,040 So I take that equation and put in y 103 00:06:37,040 --> 00:06:40,550 equal e to the st. Remember now, I'm doing 0 here 104 00:06:40,550 --> 00:06:43,740 so it's not the same s as the right-hand side. 105 00:06:43,740 --> 00:06:44,500 OK. 106 00:06:44,500 --> 00:06:47,550 What happens if I put in-- so I'm 107 00:06:47,550 --> 00:06:49,560 looking for the null solution. 108 00:06:49,560 --> 00:06:54,950 I'll try y equal e to the st. Plug it in. 109 00:06:54,950 --> 00:06:57,510 I get m. 110 00:06:57,510 --> 00:07:05,660 Two derivatives gives me s squared e to the st, b. 111 00:07:05,660 --> 00:07:09,565 One derivative gives me an s, e to the st, 112 00:07:09,565 --> 00:07:14,430 and k gives me k e to the st. And that's 113 00:07:14,430 --> 00:07:18,440 supposed to equal to 0 for the null solution. 114 00:07:18,440 --> 00:07:20,840 OK. 115 00:07:20,840 --> 00:07:27,640 Have we made a good guess at what will work? 116 00:07:27,640 --> 00:07:31,224 Yes, because I can cancel e to the st, 117 00:07:31,224 --> 00:07:34,600 and I come to the most important equation. 118 00:07:34,600 --> 00:07:36,840 The equation that governs everything 119 00:07:36,840 --> 00:07:43,430 in this whole lecture is ms squared 120 00:07:43,430 --> 00:07:49,500 plus bs plus k equals 0. 121 00:07:49,500 --> 00:07:53,950 That's called the characteristic equation, or it has many names. 122 00:07:53,950 --> 00:07:56,890 It's obviously the big deal. 123 00:07:56,890 --> 00:07:58,970 It's the thing. 124 00:07:58,970 --> 00:08:09,590 It's an equation for s, for the special frequencies that 125 00:08:09,590 --> 00:08:13,920 solve the null equation with no force. 126 00:08:13,920 --> 00:08:14,930 OK. 127 00:08:14,930 --> 00:08:18,900 And how many values of s do we expect? 128 00:08:18,900 --> 00:08:20,230 Two. 129 00:08:20,230 --> 00:08:33,080 So I expect solutions s1 and s2, and then my null solution 130 00:08:33,080 --> 00:08:41,400 is e to the s1t is a null solution. 131 00:08:41,400 --> 00:08:46,700 E to the s2t is another one. 132 00:08:46,700 --> 00:08:48,920 My equation is linear. 133 00:08:48,920 --> 00:08:51,840 I can multiply that by any constant. 134 00:08:51,840 --> 00:08:54,770 I can multiply this by any constant. 135 00:08:54,770 --> 00:09:00,410 And I can superimpose, i can add, I can combine, 136 00:09:00,410 --> 00:09:04,930 because I can do linear algebra here. 137 00:09:04,930 --> 00:09:08,260 This is the most important operation in linear algebra, 138 00:09:08,260 --> 00:09:12,050 multiply things by constants and add. 139 00:09:14,900 --> 00:09:16,920 That's called a linear combination. 140 00:09:16,920 --> 00:09:20,525 It's the basic operation in linear algebra 141 00:09:20,525 --> 00:09:24,140 and it's a basic operation here, because we're 142 00:09:24,140 --> 00:09:27,170 doing linear algebra with functions. 143 00:09:27,170 --> 00:09:29,530 Do you see that that's it? 144 00:09:29,530 --> 00:09:32,520 We've got it, except we should really write a formula 145 00:09:32,520 --> 00:09:36,220 and draw some pictures to show s1 and s2. 146 00:09:39,820 --> 00:09:45,170 What would be the formula for s1 and s2, to the two solutions? 147 00:09:45,170 --> 00:09:53,420 We remember that from school, it's the quadratic formula. 148 00:09:53,420 --> 00:09:55,500 The two solutions to this. 149 00:09:55,500 --> 00:09:58,254 Everybody remember this one? 150 00:09:58,254 --> 00:10:01,330 Let's see if I do. 151 00:10:01,330 --> 00:10:05,670 Does it start with-- a minus b. 152 00:10:05,670 --> 00:10:08,900 And then there's going to be a denominator that I'll remember, 153 00:10:08,900 --> 00:10:09,615 which is 2a. 154 00:10:12,150 --> 00:10:12,650 No. 155 00:10:12,650 --> 00:10:16,530 I said a but I mean m. 156 00:10:16,530 --> 00:10:18,730 2m. 157 00:10:18,730 --> 00:10:25,590 And now this is plus or minus-- what goes into here? 158 00:10:25,590 --> 00:10:28,800 B squared minus 4ac. 159 00:10:28,800 --> 00:10:34,065 The key quantity in this whole business, b squared minus 4ac. 160 00:10:37,340 --> 00:10:40,820 Ah, what is it? 161 00:10:40,820 --> 00:10:41,890 Mk, thank you. 162 00:10:41,890 --> 00:10:43,661 4mk. 163 00:10:43,661 --> 00:10:44,160 Great. 164 00:10:46,990 --> 00:10:53,440 And can I just remark, a little remark about units. 165 00:10:53,440 --> 00:10:55,950 B squared has the same units as 4mk. 166 00:10:55,950 --> 00:11:01,010 It has to or such a formula would be crazy. 167 00:11:01,010 --> 00:11:04,390 So we will see that actually there's 168 00:11:04,390 --> 00:11:07,220 something called the damping ratio that 169 00:11:07,220 --> 00:11:10,900 involves the ratio of these guys. 170 00:11:10,900 --> 00:11:13,100 OK, that's the formula. 171 00:11:13,100 --> 00:11:19,720 But it's not like-- OK it's got a square root, 172 00:11:19,720 --> 00:11:21,790 and what's the point of-- the thing we 173 00:11:21,790 --> 00:11:26,500 have to remember with this square root is that if it's 174 00:11:26,500 --> 00:11:30,080 the square root of a positive number 175 00:11:30,080 --> 00:11:34,790 then we have a plus or minus, ordinary real numbers. 176 00:11:34,790 --> 00:11:38,860 If this thing is negative, then what? 177 00:11:38,860 --> 00:11:44,970 What's up if b squared is smaller than 4mk? 178 00:11:44,970 --> 00:11:49,730 So if b squared is-- if there's not much damping, if it's 179 00:11:49,730 --> 00:11:53,380 underdamped, b squared would be smaller than 4mk. 180 00:11:53,380 --> 00:11:59,120 And what then? 181 00:11:59,120 --> 00:12:01,670 We've got a negative number here. 182 00:12:01,670 --> 00:12:05,890 When we take its square root we have an imaginary number. 183 00:12:05,890 --> 00:12:08,470 That's oscillation. 184 00:12:08,470 --> 00:12:11,720 Under-damping is going to show oscillation. 185 00:12:11,720 --> 00:12:14,920 Let me draw this. 186 00:12:14,920 --> 00:12:19,520 Let me draw that curve for different choices of m, b, 187 00:12:19,520 --> 00:12:20,490 and k. 188 00:12:20,490 --> 00:12:24,530 This is a good picture to see. 189 00:12:24,530 --> 00:12:35,004 So let me draw first of all one that has b equals 0. 190 00:12:35,004 --> 00:12:37,360 OK. 191 00:12:37,360 --> 00:12:40,670 So there's a curve. 192 00:12:40,670 --> 00:12:44,950 What I'm drawing is, up on this curve 193 00:12:44,950 --> 00:12:49,930 is this ms squared plus bs plus k. 194 00:12:49,930 --> 00:12:54,790 Ms squared and bs and k. 195 00:12:54,790 --> 00:13:00,760 And here, this one is one with no damping at all. 196 00:13:00,760 --> 00:13:08,110 This is just s squared plus 0 s plus 1. 197 00:13:08,110 --> 00:13:13,530 That's what that curve is That's the example we saw before. 198 00:13:13,530 --> 00:13:15,686 Y double prime plus y. 199 00:13:15,686 --> 00:13:18,010 Y double prime plus y equals 0. 200 00:13:18,010 --> 00:13:22,000 This is coming from Y double prime plus y equals 0. 201 00:13:22,000 --> 00:13:25,170 This is pure oscillation. 202 00:13:29,970 --> 00:13:30,740 OK. 203 00:13:30,740 --> 00:13:33,540 Now let me bring in some damping. 204 00:13:33,540 --> 00:13:38,170 Now, as I bring in damping the curve will move. 205 00:13:38,170 --> 00:13:42,880 And it takes a little patience to see where it moves. 206 00:13:42,880 --> 00:13:50,830 Let me have a little damping a little damping, 207 00:13:50,830 --> 00:13:57,210 so s squared plus 1 s plus 1. 208 00:13:57,210 --> 00:14:03,230 I think the curve-- so this is s here. 209 00:14:03,230 --> 00:14:07,250 And it's a parabola. 210 00:14:07,250 --> 00:14:09,600 Everything I draw here is a parabola, 211 00:14:09,600 --> 00:14:11,970 is just that different parabolas come 212 00:14:11,970 --> 00:14:14,470 from different choices of b. 213 00:14:14,470 --> 00:14:17,810 So I think that this choice, I think 214 00:14:17,810 --> 00:14:22,220 it goes a little bit like that. 215 00:14:22,220 --> 00:14:25,090 That's the next one. 216 00:14:25,090 --> 00:14:28,160 It goes down a bit. 217 00:14:28,160 --> 00:14:35,156 Now, let me do s squared plus 2s plus 1. 218 00:14:35,156 --> 00:14:41,040 Now, let's see, I really should stop for a moment 219 00:14:41,040 --> 00:14:48,390 and solve the equation, find the roots for each of these guys. 220 00:14:48,390 --> 00:14:52,340 And then I'm going to have an s squared plus 3s plus 1. 221 00:14:54,980 --> 00:15:04,310 So we're doing something very straightforward, parabolas. 222 00:15:04,310 --> 00:15:07,640 But it shows us the different possibilities. 223 00:15:07,640 --> 00:15:10,360 And we could give them names. 224 00:15:10,360 --> 00:15:11,870 OK. 225 00:15:11,870 --> 00:15:13,740 So I would call this one undamped. 226 00:15:18,500 --> 00:15:21,480 And what are the roots of that equation? 227 00:15:21,480 --> 00:15:25,170 S squared plus 0 s plus 1 equals 0. 228 00:15:25,170 --> 00:15:30,410 What are the s1 and s2 for that guy. 229 00:15:30,410 --> 00:15:35,420 So the roots of s squared plus 1 equals 0 are? 230 00:15:35,420 --> 00:15:38,230 Stay with me here. 231 00:15:38,230 --> 00:15:41,200 S squared plus 1 equals 0, that's 232 00:15:41,200 --> 00:15:42,485 the equation that has roots. 233 00:15:42,485 --> 00:15:44,150 STUDENT: I and minus i 234 00:15:44,150 --> 00:15:46,500 PROFESSOR: I and minus i. 235 00:15:46,500 --> 00:15:47,780 I and minus i. 236 00:15:47,780 --> 00:15:50,760 So this is undamped. 237 00:15:50,760 --> 00:15:55,120 S1 is i and s2 is minus i. 238 00:15:55,120 --> 00:15:57,870 That's the pure oscillation. 239 00:15:57,870 --> 00:16:05,280 Pure oscillation, that's the case where b is 0 here, 240 00:16:05,280 --> 00:16:09,860 I have a square root of a negative number, 241 00:16:09,860 --> 00:16:12,370 and it gives me plus or minus 2i, 242 00:16:12,370 --> 00:16:15,710 and then the 2s cancel and I get plus or minus i. 243 00:16:15,710 --> 00:16:19,530 So I can always go back to this but I'll 244 00:16:19,530 --> 00:16:23,010 try to choose numbers that come out nicely. 245 00:16:23,010 --> 00:16:26,250 Now, what happens with some damping? 246 00:16:26,250 --> 00:16:27,530 This guy. 247 00:16:27,530 --> 00:16:30,480 What are the roots for this one? 248 00:16:30,480 --> 00:16:33,190 Well, I better use this formula. 249 00:16:33,190 --> 00:16:37,880 Now, I'm always keeping m equal 1 and k equal 1, 250 00:16:37,880 --> 00:16:40,630 in all those samples. 251 00:16:40,630 --> 00:16:43,690 But now b has increased to 1. 252 00:16:47,070 --> 00:16:52,720 So if b is 1, what do I have? 253 00:16:52,720 --> 00:16:56,660 I have s is-- the roots are minus 1, 254 00:16:56,660 --> 00:17:02,350 plus or minus the square root-- And it's going to be just 2 255 00:17:02,350 --> 00:17:03,860 down below. 256 00:17:03,860 --> 00:17:07,650 What's in the square root, the all important square root. 257 00:17:07,650 --> 00:17:13,420 When m and k and b are all 1. 258 00:17:13,420 --> 00:17:16,230 Just do the calculation with me, so you see it. 259 00:17:16,230 --> 00:17:17,329 It's negative? 260 00:17:17,329 --> 00:17:18,183 STUDENT: Three. 261 00:17:18,183 --> 00:17:19,099 PROFESSOR: Negative 3. 262 00:17:19,099 --> 00:17:20,359 So what's the point there? 263 00:17:23,810 --> 00:17:28,470 The point is, this is going to be square root of 3i or minus 264 00:17:28,470 --> 00:17:29,250 i. 265 00:17:29,250 --> 00:17:34,200 We have oscillation at a frequency square root of 3, 266 00:17:34,200 --> 00:17:38,880 and we have decay from s minus one-half. 267 00:17:38,880 --> 00:17:44,900 The real part of this is giving us the drop off. 268 00:17:44,900 --> 00:17:49,000 We didn't have any drop off at all in this case. 269 00:17:49,000 --> 00:17:50,810 They were pure imaginary. 270 00:17:50,810 --> 00:17:57,960 Now the s1 and s2 are whatever I have a minus 1, 271 00:17:57,960 --> 00:18:02,330 plus or minus square root of 3i over 2. 272 00:18:05,000 --> 00:18:08,220 Those are the roots. 273 00:18:08,220 --> 00:18:09,790 All right, I'm ready for this guy, 274 00:18:09,790 --> 00:18:12,090 and it's particularly nice. 275 00:18:12,090 --> 00:18:13,270 It's particularly nice. 276 00:18:13,270 --> 00:18:18,030 What do you see for s squared plus 2s plus 1? 277 00:18:18,030 --> 00:18:24,220 When you see this parabola, now b has moved up to 2. 278 00:18:24,220 --> 00:18:27,410 What's up with that one? 279 00:18:27,410 --> 00:18:31,980 It's going to be-- let's see-- that looks 280 00:18:31,980 --> 00:18:34,420 like a perfect square to me. 281 00:18:34,420 --> 00:18:38,520 Right, inside the square root is 0. 282 00:18:38,520 --> 00:18:39,820 Right, exactly. 283 00:18:39,820 --> 00:18:44,430 Inside the square root, b squared is 4, and 4mk is 4, 284 00:18:44,430 --> 00:18:46,910 so I have 0 inside this square root. 285 00:18:46,910 --> 00:18:56,270 So now I have s equals minus 1 plus or minus 0 over 2. 286 00:18:56,270 --> 00:19:00,644 That's when this was the case, when b moved up to 2. 287 00:19:00,644 --> 00:19:03,370 STUDENT: [INAUDIBLE] minus 2. 288 00:19:03,370 --> 00:19:05,046 PROFESSOR: I'm messing it up? 289 00:19:05,046 --> 00:19:08,240 STUDENT: [INAUDIBLE] equals minus 2, plus or minus 0. 290 00:19:08,240 --> 00:19:10,180 PROFESSOR: Minus 2 plus or minus 0. 291 00:19:10,180 --> 00:19:12,020 Thank you. 292 00:19:12,020 --> 00:19:13,130 So what do I have? 293 00:19:16,620 --> 00:19:19,450 What are the roots of this guy? 294 00:19:22,600 --> 00:19:24,080 Negative 1. 295 00:19:24,080 --> 00:19:26,690 What's the other one? 296 00:19:26,690 --> 00:19:28,000 Negative 1. 297 00:19:28,000 --> 00:19:29,420 A double root. 298 00:19:29,420 --> 00:19:31,220 It's critical damping. 299 00:19:31,220 --> 00:19:33,440 Critical damping-- it's not underdamped. 300 00:19:33,440 --> 00:19:34,540 It's not overdamped. 301 00:19:34,540 --> 00:19:37,210 It's right on the borderline. 302 00:19:37,210 --> 00:19:43,060 And I see that, when you first saw quadratics, 303 00:19:43,060 --> 00:19:47,140 before anybody brought up that awful formula, 304 00:19:47,140 --> 00:19:52,990 you would have factored this into s plus 1 squared equals 0, 305 00:19:52,990 --> 00:19:57,610 and you would have discovered that s was minus 1 twice. 306 00:20:01,540 --> 00:20:02,170 A double root. 307 00:20:02,170 --> 00:20:13,764 So the picture there would be-- there's minus 1. 308 00:20:13,764 --> 00:20:15,960 Yeah. 309 00:20:15,960 --> 00:20:22,600 Everybody recognize that this, we're hitting 0, height 0. 310 00:20:22,600 --> 00:20:26,490 We're hitting 0 twice at s equal minus 1. 311 00:20:29,090 --> 00:20:31,620 So this is now the case. 312 00:20:31,620 --> 00:20:34,530 This was b equal 0, no damping. 313 00:20:34,530 --> 00:20:37,590 This was b equal 1, under-damping. 314 00:20:37,590 --> 00:20:42,060 This is b equal 2, critical damping, just on the border. 315 00:20:42,060 --> 00:20:45,040 And what do you think the s squared 316 00:20:45,040 --> 00:20:48,210 plus 3s plus 1 is going to look like. 317 00:20:48,210 --> 00:20:52,370 Again, we can find it. 318 00:20:52,370 --> 00:20:58,050 No, let me do the s squared plus-- let me take b equal 3 319 00:20:58,050 --> 00:21:01,620 and find the roots and draw the picture. 320 00:21:01,620 --> 00:21:06,220 If you're with, me that picture and these formulas 321 00:21:06,220 --> 00:21:10,950 tell you the difference between these four cases. 322 00:21:10,950 --> 00:21:12,520 So what do I get? 323 00:21:12,520 --> 00:21:19,550 Minus 3 plus or minus the square root of, 9 minus 4, 324 00:21:19,550 --> 00:21:22,510 is 5, over 2. 325 00:21:26,160 --> 00:21:30,370 So I have two negative roots. 326 00:21:30,370 --> 00:21:32,520 I have decay. 327 00:21:32,520 --> 00:21:35,670 I have decay at a fast rate and a slow rate 328 00:21:35,670 --> 00:21:38,280 but both are giving decay. 329 00:21:38,280 --> 00:21:44,660 So the curve now is coming down here and back up there, 330 00:21:44,660 --> 00:21:51,350 and it hits there, and it's got the two roots. 331 00:21:51,350 --> 00:21:54,290 These two roots are x and x. 332 00:22:01,810 --> 00:22:04,380 So these are s1 and s2. 333 00:22:04,380 --> 00:22:06,510 Let me copy them over here. 334 00:22:06,510 --> 00:22:12,190 S1 and s2 are minus 3 plus or minus 335 00:22:12,190 --> 00:22:16,610 the square root of 5 over 2. 336 00:22:16,610 --> 00:22:17,110 Yeah. 337 00:22:17,110 --> 00:22:18,800 Real. 338 00:22:18,800 --> 00:22:19,800 Real roots. 339 00:22:19,800 --> 00:22:22,030 So this is two real roots. 340 00:22:22,030 --> 00:22:24,140 This is a double real root. 341 00:22:24,140 --> 00:22:26,490 This is two complex roots. 342 00:22:26,490 --> 00:22:29,650 And this is two pure imaginary roots. 343 00:22:29,650 --> 00:22:31,950 The four possibilities. 344 00:22:31,950 --> 00:22:33,850 The famous four. 345 00:22:33,850 --> 00:22:34,460 OK. 346 00:22:34,460 --> 00:22:38,460 And for me, that picture is a-- so 347 00:22:38,460 --> 00:22:44,910 this was the b equal 3 curve, more overdamped. 348 00:22:44,910 --> 00:22:47,730 It's a little interesting that overdamping 349 00:22:47,730 --> 00:22:51,110 has this root that's pretty near 0. 350 00:22:51,110 --> 00:22:58,400 So overdamping doesn't mean that you go to 0 real fast. 351 00:22:58,400 --> 00:23:01,860 Actually, the one that goes fastest to 0 352 00:23:01,860 --> 00:23:03,470 is the critical damping. 353 00:23:03,470 --> 00:23:09,450 Then, as b grows, one root gets closer to 0, 354 00:23:09,450 --> 00:23:13,570 so it's like slower decay, and another root 355 00:23:13,570 --> 00:23:15,156 is going off to fast decay. 356 00:23:21,870 --> 00:23:26,050 I think you have to know those guys, 357 00:23:26,050 --> 00:23:29,140 because the physically that's very important, 358 00:23:29,140 --> 00:23:31,320 where's the damping. 359 00:23:31,320 --> 00:23:37,500 But we've now found the null solution completely. 360 00:23:37,500 --> 00:23:41,550 The null solution completely is-- 361 00:23:41,550 --> 00:23:46,360 let me write it again here-- is anything times e 362 00:23:46,360 --> 00:23:53,370 to the s1t and anything times e to the s2t. 363 00:23:53,370 --> 00:23:56,620 And where do those constants, c1 and c2 get decided? 364 00:23:59,450 --> 00:24:00,365 By the? 365 00:24:00,365 --> 00:24:01,240 STUDENT: [INAUDIBLE]. 366 00:24:01,240 --> 00:24:03,011 PROFESSOR: Initial conditions. 367 00:24:03,011 --> 00:24:03,510 Right. 368 00:24:03,510 --> 00:24:08,150 These two conditions determine c1 and c2. 369 00:24:08,150 --> 00:24:09,820 OK. 370 00:24:09,820 --> 00:24:12,520 Are we good? 371 00:24:12,520 --> 00:24:18,790 So the null solution has already separated the different cases 372 00:24:18,790 --> 00:24:22,390 that depend on how much damping. 373 00:24:22,390 --> 00:24:26,530 I'm ready for number two. 374 00:24:26,530 --> 00:24:28,750 Number two now. 375 00:24:28,750 --> 00:24:30,140 OK. 376 00:24:30,140 --> 00:24:36,885 So the idea is I now have a forcing term, some frequency 377 00:24:36,885 --> 00:24:41,190 e to the st. And I will assume that it's not-- 378 00:24:41,190 --> 00:24:45,080 s not equal s1 or s2. 379 00:24:49,320 --> 00:24:51,610 That's to make my life easy. 380 00:24:51,610 --> 00:24:55,570 Just as last time, the formulas will break down. 381 00:24:55,570 --> 00:25:01,550 And I'll have to put in something different 382 00:25:01,550 --> 00:25:05,030 if there's resonance, if the driving force is 383 00:25:05,030 --> 00:25:09,960 at the same frequency as a natural frequency. 384 00:25:09,960 --> 00:25:11,720 In that case, there's a resonance. 385 00:25:11,720 --> 00:25:15,350 And the way you spot a resonance formula 386 00:25:15,350 --> 00:25:18,220 is there's an extra factor of t. 387 00:25:18,220 --> 00:25:19,930 There's a growth of t. 388 00:25:19,930 --> 00:25:22,940 Up here, we're seeing no factors of t. 389 00:25:22,940 --> 00:25:25,760 Over there in the exponent, of course. 390 00:25:25,760 --> 00:25:31,440 I mean, down below, there would be a t times e to the st. 391 00:25:31,440 --> 00:25:34,980 And the book does that carefully. 392 00:25:38,240 --> 00:25:43,230 I don't see that it has a place in the very first lecture 393 00:25:43,230 --> 00:25:45,930 on this topic. 394 00:25:45,930 --> 00:25:46,430 OK. 395 00:25:46,430 --> 00:25:50,180 So this is saying no resonance. 396 00:25:50,180 --> 00:25:51,310 All right. 397 00:25:51,310 --> 00:25:53,070 What's the solution then? 398 00:25:53,070 --> 00:25:57,070 The solution is a multiple of e to the st. 399 00:25:57,070 --> 00:26:00,620 The input was e to the st. The response is 400 00:26:00,620 --> 00:26:06,740 a multiple of e to st, so that's the frequency response. 401 00:26:06,740 --> 00:26:10,220 Capital Y is telling us the response to s. 402 00:26:10,220 --> 00:26:12,980 And it's a very, very important function. 403 00:26:12,980 --> 00:26:15,040 It's called the transfer function. 404 00:26:15,040 --> 00:26:18,490 It's just the key to everything. 405 00:26:18,490 --> 00:26:22,150 I probably say that so often, that this 406 00:26:22,150 --> 00:26:23,460 is the key to everything. 407 00:26:23,460 --> 00:26:26,130 Well, it's partly because I just have 408 00:26:26,130 --> 00:26:31,980 one lecture to do a big part of differential equations, 409 00:26:31,980 --> 00:26:34,590 and it's got some key ideas. 410 00:26:34,590 --> 00:26:38,590 And the first key idea is s1 and s2. 411 00:26:38,590 --> 00:26:44,170 And the second key idea is Y. And let's go for it. 412 00:26:44,170 --> 00:26:48,040 All right, so this is the s1 and s2 picture. 413 00:26:48,040 --> 00:26:56,200 I'll move that up, and now, in the equation, 414 00:26:56,200 --> 00:27:03,640 try Y e to the st. We hope it works. 415 00:27:03,640 --> 00:27:04,330 What is this? 416 00:27:04,330 --> 00:27:11,240 I'm trying, this is my solution, and it's a particular solution 417 00:27:11,240 --> 00:27:12,720 now. 418 00:27:12,720 --> 00:27:14,000 I got the null solution. 419 00:27:14,000 --> 00:27:16,870 I've moved to a particular 420 00:27:16,870 --> 00:27:24,390 And this is when the force is e to the st. In other words, 421 00:27:24,390 --> 00:27:28,690 I'll just say again, if we have an exponential force, 422 00:27:28,690 --> 00:27:31,650 try an exponential response. 423 00:27:31,650 --> 00:27:37,490 1803 would call this the exponential response formula. 424 00:27:37,490 --> 00:27:39,330 So you could use the word exponential 425 00:27:39,330 --> 00:27:42,060 response, very appropriate. 426 00:27:42,060 --> 00:27:44,810 You could use the word frequency response. 427 00:27:44,810 --> 00:27:47,650 That frequency response is kind of the right word when 428 00:27:47,650 --> 00:27:53,860 s is an imaginary number giving an oscillation frequency. 429 00:27:53,860 --> 00:27:56,500 OK, I'm going to plug that in. 430 00:27:56,500 --> 00:28:01,030 So into the differential equation, m. 431 00:28:01,030 --> 00:28:09,970 Second derivative of that is s squared Y e to the st, right? 432 00:28:09,970 --> 00:28:11,860 That's m Y double prime. 433 00:28:14,540 --> 00:28:17,360 The beauty of exponentials. 434 00:28:17,360 --> 00:28:20,520 Take every derivative, just brings down an s. 435 00:28:20,520 --> 00:28:21,750 The next term. 436 00:28:21,750 --> 00:28:22,660 What's the next term? 437 00:28:22,660 --> 00:28:25,850 Maybe do it with me, do it for me. 438 00:28:25,850 --> 00:28:33,906 What happens when I plug in this as Y into b, so there'll be a b 439 00:28:33,906 --> 00:28:34,840 Y prime. 440 00:28:34,840 --> 00:28:35,950 So what's Y prime? 441 00:28:35,950 --> 00:28:37,734 STUDENT: [INAUDIBLE]. 442 00:28:37,734 --> 00:28:45,196 PROFESSOR: s, Y, this constant, e to the st. 443 00:28:45,196 --> 00:28:50,460 And then the final term is k times this Y itself, 444 00:28:50,460 --> 00:28:53,453 no derivative, so it's just Y e to the st, 445 00:28:53,453 --> 00:28:59,480 and that's matching e to the st. That's the force. 446 00:28:59,480 --> 00:29:01,765 This is f. 447 00:29:01,765 --> 00:29:05,290 F, this is an exponential force. 448 00:29:05,290 --> 00:29:10,850 Of course, I could and should have a constant here 449 00:29:10,850 --> 00:29:16,580 to give me the units of force. 450 00:29:16,580 --> 00:29:18,520 Let me just keep the formula as clean 451 00:29:18,520 --> 00:29:22,990 as possible by taking units, so that's one. 452 00:29:22,990 --> 00:29:25,295 OK what do I do? 453 00:29:25,295 --> 00:29:26,392 STUDENT: [INAUDIBLE]. 454 00:29:26,392 --> 00:29:27,225 PROFESSOR: I cancel. 455 00:29:29,990 --> 00:29:36,604 The nice part of 287 is canceling e to the st. 456 00:29:36,604 --> 00:29:39,760 That's the most fun you get. 457 00:29:43,010 --> 00:29:47,160 And now, I have Y's on the left. 458 00:29:47,160 --> 00:29:53,170 So Y is-- can I see what Y is? 459 00:29:53,170 --> 00:30:00,760 Y is this 1, divided by the coefficient of Y. 460 00:30:00,760 --> 00:30:02,590 And what's the coefficient of Y? 461 00:30:02,590 --> 00:30:04,780 We know it. 462 00:30:04,780 --> 00:30:07,640 We've seen that coefficient. 463 00:30:07,640 --> 00:30:13,910 Y on the left is multiplied by ms squared plus bs plus k ms 464 00:30:13,910 --> 00:30:19,870 squared, again, ms squared, bs, and k. 465 00:30:19,870 --> 00:30:24,110 Multiplying Y, I divide by that, so I put it down here. 466 00:30:24,110 --> 00:30:30,478 Ms squared plus bs plus k. 467 00:30:36,960 --> 00:30:44,790 And because s is not one of the roots, that's not 0, 468 00:30:44,790 --> 00:30:47,640 so we're golden. 469 00:30:47,640 --> 00:30:52,170 That problem took five minutes. 470 00:30:52,170 --> 00:30:54,690 The null solution took half an hour. 471 00:30:58,030 --> 00:31:02,020 The exponential response is clear. 472 00:31:02,020 --> 00:31:05,780 And you can see what it would be. 473 00:31:05,780 --> 00:31:07,820 And let's give it a name. 474 00:31:07,820 --> 00:31:09,900 This is the transfer function. 475 00:31:19,080 --> 00:31:20,670 Widely used name. 476 00:31:20,670 --> 00:31:25,740 Other names could be given but that's the best. 477 00:31:25,740 --> 00:31:28,915 So it's a function of s. 478 00:31:28,915 --> 00:31:29,790 It's a function of s. 479 00:31:33,090 --> 00:31:37,870 And so, again, when f is e to the st, 480 00:31:37,870 --> 00:31:44,360 it sort of transfers the input into the output. 481 00:31:44,360 --> 00:31:47,840 That's the way I think of the transfer function. 482 00:31:47,840 --> 00:31:49,950 Here is the input. 483 00:31:49,950 --> 00:31:53,740 The output is just multiplied by the transfer function. 484 00:31:53,740 --> 00:31:57,630 And the transfer function is just that nice expression. 485 00:31:57,630 --> 00:31:59,780 Just that nice expression. 486 00:31:59,780 --> 00:32:06,280 So we are golden for a frequency, 487 00:32:06,280 --> 00:32:15,390 for a linear equation with an exponential forcing function. 488 00:32:15,390 --> 00:32:17,120 What would be another example? 489 00:32:17,120 --> 00:32:22,040 I'm using mass, dashpot, spring here. 490 00:32:22,040 --> 00:32:25,750 If this was in electrical engineering, 491 00:32:25,750 --> 00:32:32,750 what three things would I be using instead of mass, dashpot, 492 00:32:32,750 --> 00:32:33,750 spring. 493 00:32:33,750 --> 00:32:37,590 The three guys would be? 494 00:32:37,590 --> 00:32:39,200 What would correspond to the dashpot? 495 00:32:43,220 --> 00:32:46,610 So can I just draw here a little-- 496 00:32:46,610 --> 00:32:50,430 let me put on a low voltage and put 497 00:32:50,430 --> 00:32:56,990 on something that does this, and then something like this. 498 00:32:56,990 --> 00:33:01,130 And then there's something like this. 499 00:33:01,130 --> 00:33:07,020 And give me a break, tell me what these things are. 500 00:33:07,020 --> 00:33:08,060 This guy is? 501 00:33:08,060 --> 00:33:08,960 STUDENT: Inductor. 502 00:33:08,960 --> 00:33:10,370 PROFESSOR: An inductor. 503 00:33:10,370 --> 00:33:11,681 This guy is a? 504 00:33:11,681 --> 00:33:12,430 STUDENT: Resistor. 505 00:33:12,430 --> 00:33:13,340 PROFESSOR: Resistor. 506 00:33:13,340 --> 00:33:16,880 Now that's the one that's like damping. 507 00:33:16,880 --> 00:33:21,640 This resistor here is like damping. 508 00:33:21,640 --> 00:33:24,300 Like the damping term, or maybe 1/b. 509 00:33:24,300 --> 00:33:27,880 I'm not getting its units right because I 510 00:33:27,880 --> 00:33:30,410 haven't got any equation here at all. 511 00:33:30,410 --> 00:33:36,040 Resisting is-- there's friction in that resistor. 512 00:33:36,040 --> 00:33:38,390 It burns up heat. 513 00:33:38,390 --> 00:33:42,020 And similarly, the dashpot slows things down. 514 00:33:42,020 --> 00:33:43,380 And then this guy is a-- 515 00:33:43,380 --> 00:33:44,300 STUDENT: [INAUDIBLE]. 516 00:33:44,300 --> 00:33:46,970 PROFESSOR: Capacitor, right. 517 00:33:46,970 --> 00:33:53,170 In other words, you can do the mechanical application 518 00:33:53,170 --> 00:33:59,080 and the electrical application with exactly the same ideas, 519 00:33:59,080 --> 00:34:07,380 just a change of letter, and of course, different units, 520 00:34:07,380 --> 00:34:09,730 but same problem. 521 00:34:09,730 --> 00:34:15,389 OK, so that's a comment that you've seen before. 522 00:34:15,389 --> 00:34:16,960 What else do I want to comment on? 523 00:34:16,960 --> 00:34:21,594 Because this example was really so straightforward. 524 00:34:27,060 --> 00:34:32,560 I think what I want to mention, and this 525 00:34:32,560 --> 00:34:39,080 is important, is that this is the central starting 526 00:34:39,080 --> 00:34:43,159 point for the Laplace transform. 527 00:34:43,159 --> 00:34:51,409 So I can't do Laplace transforms all today by any means, 528 00:34:51,409 --> 00:34:58,650 and so Professor Fry will talk about the Laplace transform 529 00:34:58,650 --> 00:35:00,100 next week. 530 00:35:00,100 --> 00:35:04,570 But what is the point of the Laplace transform? 531 00:35:04,570 --> 00:35:06,490 The point of the Laplace transform 532 00:35:06,490 --> 00:35:18,890 is to get your money's worth out of the simple formula 533 00:35:18,890 --> 00:35:21,165 for exponentials. 534 00:35:26,500 --> 00:35:30,280 Having an exponential there turns the whole differential 535 00:35:30,280 --> 00:35:33,430 equation problem into an algebra problem. 536 00:35:33,430 --> 00:35:37,220 We just have quadratic equations. 537 00:35:37,220 --> 00:35:42,130 We just have a division by a quadratic. 538 00:35:42,130 --> 00:35:45,590 That's the great thing about the Laplace transform. 539 00:35:45,590 --> 00:35:48,900 It turns the t domain, the time domain, 540 00:35:48,900 --> 00:35:53,290 where we have exponentials, into the s domain, 541 00:35:53,290 --> 00:35:57,050 the exponent domain, the frequency domain, 542 00:35:57,050 --> 00:35:59,930 where we just have quadratics. 543 00:35:59,930 --> 00:36:03,060 And then first order equation's just linear. 544 00:36:03,060 --> 00:36:10,770 And we can even get from second order of the quadratic 545 00:36:10,770 --> 00:36:14,190 to linear, because I can factor that guy. 546 00:36:14,190 --> 00:36:18,360 If I factor into s minus s1 times s minus s2, 547 00:36:18,360 --> 00:36:20,520 I've two linear pieces. 548 00:36:20,520 --> 00:36:23,030 And that's the first step in the Laplace transform, 549 00:36:23,030 --> 00:36:24,310 in the algebra. 550 00:36:24,310 --> 00:36:27,320 So all of the algebra in the Laplace transform 551 00:36:27,320 --> 00:36:34,760 is this algebra for e to the st. 552 00:36:34,760 --> 00:36:38,420 And then the job of the Laplace transform-- and this 553 00:36:38,420 --> 00:36:41,120 is the tricky part. 554 00:36:41,120 --> 00:36:43,990 So let me even take a little board space on that. 555 00:36:47,070 --> 00:36:51,590 So this is like a heads up for next week. 556 00:36:51,590 --> 00:36:52,890 So the Laplace transform. 557 00:37:02,305 --> 00:37:02,805 OK. 558 00:37:07,730 --> 00:37:16,300 So for any forcing function, f of t. 559 00:37:19,556 --> 00:37:20,930 That's the thing that we're going 560 00:37:20,930 --> 00:37:25,920 to take the Laplace transform of and the response 561 00:37:25,920 --> 00:37:33,920 and its response, y of t. 562 00:37:33,920 --> 00:37:36,860 OK. 563 00:37:36,860 --> 00:37:39,110 So here's the idea of transforms in general. 564 00:37:43,220 --> 00:37:49,600 I choose some terrific functions like exponentials. 565 00:37:49,600 --> 00:37:57,550 So I want to convert my problem to exponentials. 566 00:38:00,280 --> 00:38:07,270 e to the st for all s, s between let's say 0 and infinity. 567 00:38:12,170 --> 00:38:13,450 So what do I do? 568 00:38:13,450 --> 00:38:17,000 I take my function, and I figure out 569 00:38:17,000 --> 00:38:20,580 how much of every exponential is in it. 570 00:38:20,580 --> 00:38:22,950 That's the Laplace transform. 571 00:38:22,950 --> 00:38:27,240 I take my function f of t, and I go 572 00:38:27,240 --> 00:38:33,130 to what you'll see next week, its Laplace transform. 573 00:38:33,130 --> 00:38:36,510 Let me call it capital F of s, or it's sometimes 574 00:38:36,510 --> 00:38:42,710 written the Laplace transform of F of s. 575 00:38:42,710 --> 00:38:44,700 Something like that. 576 00:38:44,700 --> 00:38:46,760 I'm not going to do that. 577 00:38:46,760 --> 00:38:48,314 I'm not going there. 578 00:38:51,100 --> 00:38:57,430 So F of s is like the amount of a particular exponential 579 00:38:57,430 --> 00:38:59,180 in my function. 580 00:38:59,180 --> 00:39:01,790 If my function is just the sum of two exponentials, 581 00:39:01,790 --> 00:39:05,990 then the Laplace transform just is a big bump on one 582 00:39:05,990 --> 00:39:07,670 and a big bump on the other. 583 00:39:07,670 --> 00:39:15,090 But most functions, like some forcing function, 584 00:39:15,090 --> 00:39:20,000 has some e to the st is for all for a whole range of s. 585 00:39:20,000 --> 00:39:22,990 So I figure out how much of this is. 586 00:39:22,990 --> 00:39:26,040 OK now second step. 587 00:39:26,040 --> 00:39:29,100 Using linearity, I can solve the problem 588 00:39:29,100 --> 00:39:33,980 for when the right hand side is just that. 589 00:39:33,980 --> 00:39:41,810 Then solve for right hand side, F 590 00:39:41,810 --> 00:39:47,450 of s, e to the st. Solve for each frequency, 591 00:39:47,450 --> 00:39:49,340 each s separately. 592 00:39:49,340 --> 00:39:51,210 And what does that mean? 593 00:39:51,210 --> 00:39:54,940 That means just dividing by this. 594 00:39:54,940 --> 00:39:57,110 So this was the easy step. 595 00:39:57,110 --> 00:40:00,440 So this is step one, is take the Laplace transform. 596 00:40:00,440 --> 00:40:05,440 The Laplace transform tells you how much of each exponential 597 00:40:05,440 --> 00:40:06,440 is in it. 598 00:40:06,440 --> 00:40:10,140 Now step two is a cinch. 599 00:40:10,140 --> 00:40:11,290 Step two is a cinch. 600 00:40:11,290 --> 00:40:15,270 I just multiply by the transfer function. 601 00:40:15,270 --> 00:40:21,160 I divide by this, bs plus k. 602 00:40:21,160 --> 00:40:23,940 And now, what's step three. 603 00:40:23,940 --> 00:40:28,710 Step three, I have the solution for each separate exponential, 604 00:40:28,710 --> 00:40:31,030 but I've got a whole lots of exponentials, 605 00:40:31,030 --> 00:40:36,020 so I have to do an inverse Laplace transform, 606 00:40:36,020 --> 00:40:44,390 add up, figure out what function has this Laplace transform. 607 00:40:44,390 --> 00:40:48,840 And that's often the place where the algebra gets harder. 608 00:40:51,430 --> 00:40:54,710 In principle, we can do it for any F of t. 609 00:40:54,710 --> 00:40:56,940 We can take its Laplace transform, 610 00:40:56,940 --> 00:41:00,810 we can solve for each frequency in the transform, 611 00:41:00,810 --> 00:41:02,950 we can assemble them all together. 612 00:41:02,950 --> 00:41:05,410 That's the inverse Laplace transform, 613 00:41:05,410 --> 00:41:19,310 so this is the inverse Laplace to get the solution y of t. 614 00:41:19,310 --> 00:41:26,610 So this is really the Laplace transform of the solution, 615 00:41:26,610 --> 00:41:31,130 and we have to get back to the solution itself. 616 00:41:31,130 --> 00:41:34,445 Can I just let you think about those ideas? 617 00:41:37,690 --> 00:41:47,750 I'm not up to describing the algebra here. 618 00:41:47,750 --> 00:41:52,560 The point is the Laplace transform 619 00:41:52,560 --> 00:41:55,500 takes this into separate exponentials. 620 00:41:55,500 --> 00:41:58,430 Each of those right hand sides is simple algebra, 621 00:41:58,430 --> 00:42:00,040 divide by that. 622 00:42:00,040 --> 00:42:03,310 And then you have the job of okay, 623 00:42:03,310 --> 00:42:08,000 what function has this Laplace transform, to go backwards. 624 00:42:08,000 --> 00:42:09,650 And that's usually the hard part. 625 00:42:09,650 --> 00:42:12,960 So people make tables of Laplace transforms. 626 00:42:12,960 --> 00:42:16,705 Everybody remembers the Laplace transforms of a few functions. 627 00:42:19,880 --> 00:42:24,720 You could say that those few functions are 628 00:42:24,720 --> 00:42:29,740 the golden functions of differential equations. 629 00:42:29,740 --> 00:42:32,120 The golden functions of differential equations 630 00:42:32,120 --> 00:42:35,870 are the ones where you know their Laplace transform 631 00:42:35,870 --> 00:42:38,240 and you can go back and forth easily. 632 00:42:38,240 --> 00:42:41,360 And what are those golden functions? 633 00:42:41,360 --> 00:42:45,470 Well, you might guess exponentials, 634 00:42:45,470 --> 00:42:48,640 simple polynomials, 1 t, t squared. 635 00:42:48,640 --> 00:42:51,210 You can do Laplace transform for those. 636 00:42:51,210 --> 00:42:53,650 What else do you think would be nice? 637 00:42:53,650 --> 00:42:56,840 Cosine and sine. 638 00:42:56,840 --> 00:42:59,120 And you could multiply those together. 639 00:42:59,120 --> 00:43:03,740 We could deal with t times cosine t. 640 00:43:03,740 --> 00:43:05,510 So a little bit different. 641 00:43:05,510 --> 00:43:09,430 But having said that, the reality is I've 642 00:43:09,430 --> 00:43:13,440 given you the whole list of nice functions. 643 00:43:13,440 --> 00:43:16,080 Those functions show up in every simple method 644 00:43:16,080 --> 00:43:17,830 for solving equations. 645 00:43:17,830 --> 00:43:21,980 There's a method called undetermined coefficients 646 00:43:21,980 --> 00:43:23,900 and what does it amount to? 647 00:43:23,900 --> 00:43:26,490 I'm sorry, I don't have time to say it this morning, 648 00:43:26,490 --> 00:43:28,680 but it can come later. 649 00:43:28,680 --> 00:43:29,500 Undetermined. 650 00:43:29,500 --> 00:43:31,820 It just means if the right hand side 651 00:43:31,820 --> 00:43:33,650 is one of these nice guys-- shall 652 00:43:33,650 --> 00:43:38,040 I write down again the golden functions? 653 00:43:38,040 --> 00:43:42,440 e to the st is like the platinum function. 654 00:43:42,440 --> 00:43:46,820 And then some golden functions are like t and t squared 655 00:43:46,820 --> 00:43:48,700 and so on. 656 00:43:48,700 --> 00:43:54,600 Of course, when we get that, we're close to cosine omega t, 657 00:43:54,600 --> 00:43:57,220 and sine omega t. 658 00:43:57,220 --> 00:44:01,410 All these guys, their Laplace transforms are nice. 659 00:44:01,410 --> 00:44:03,950 We can deal with them completely. 660 00:44:03,950 --> 00:44:07,680 Or multiply any of those together. 661 00:44:07,680 --> 00:44:12,420 And when the right hand side is one of these golden functions, 662 00:44:12,420 --> 00:44:14,480 you can write down the answer. 663 00:44:14,480 --> 00:44:18,350 We've focused on this one because it's the platinum one. 664 00:44:18,350 --> 00:44:21,090 And we did these two too, because they 665 00:44:21,090 --> 00:44:24,860 come from s equal i omega. 666 00:44:24,860 --> 00:44:29,680 And then these guys are a little bit of juice. 667 00:44:29,680 --> 00:44:31,970 But that's it. 668 00:44:31,970 --> 00:44:34,260 I'm sorry the list isn't longer. 669 00:44:34,260 --> 00:44:36,680 It'd be nice to have--and of course, 670 00:44:36,680 --> 00:44:44,860 people for centuries have worked with the next hardest 671 00:44:44,860 --> 00:44:46,292 functions. 672 00:44:46,292 --> 00:44:48,335 You know, the silver functions. 673 00:44:50,880 --> 00:44:54,890 Famous functions have names like Bessel's function 674 00:44:54,890 --> 00:44:56,670 or Legendre's function. 675 00:44:56,670 --> 00:45:01,200 Others where you can get pretty far. 676 00:45:01,200 --> 00:45:03,540 Those are the best. 677 00:45:03,540 --> 00:45:07,110 Then you have the famous ones where you get pretty far, 678 00:45:07,110 --> 00:45:10,710 in the web has the Laplace transforms, 679 00:45:10,710 --> 00:45:15,380 and then you get the general function, F of t. 680 00:45:15,380 --> 00:45:17,820 Oh, could you get anywhere with delta of t? 681 00:45:17,820 --> 00:45:20,310 Oh, yes. 682 00:45:20,310 --> 00:45:24,440 Does it belong on the list of golden functions? 683 00:45:24,440 --> 00:45:25,790 Yes, it does. 684 00:45:25,790 --> 00:45:28,980 I almost forgot it, and it's like-- I'll 685 00:45:28,980 --> 00:45:33,982 call it-- Yeah, delta of t. 686 00:45:33,982 --> 00:45:36,982 Yeah, that's a beauty. 687 00:45:36,982 --> 00:45:38,475 That's a beauty. 688 00:45:38,475 --> 00:45:41,030 The Laplace transform of delta of t 689 00:45:41,030 --> 00:45:43,550 happens to come out 1 over s. 690 00:45:43,550 --> 00:45:45,960 You can't ask for more than that. 691 00:45:45,960 --> 00:45:46,770 Or maybe it's one. 692 00:45:49,810 --> 00:45:51,630 Yeah, it's probably one. 693 00:45:51,630 --> 00:45:53,810 Yeah, the Laplace transform of that is one. 694 00:45:53,810 --> 00:45:54,660 Yeah. 695 00:45:54,660 --> 00:45:58,790 It's got all exponentials in sort of, you could say, 696 00:45:58,790 --> 00:45:59,560 equal amounts. 697 00:45:59,560 --> 00:46:00,060 OK. 698 00:46:03,660 --> 00:46:07,720 So that's some thoughts about that about Laplace transforms, 699 00:46:07,720 --> 00:46:12,870 just sort of the big picture that takes the differential 700 00:46:12,870 --> 00:46:16,330 equation, turns it into an algebra problem, 701 00:46:16,330 --> 00:46:18,470 and then at the end, you have to get back, 702 00:46:18,470 --> 00:46:23,280 and that's the part that's not always doable. 703 00:46:23,280 --> 00:46:24,080 OK. 704 00:46:24,080 --> 00:46:31,670 So what's left for today is this guy. 705 00:46:31,670 --> 00:46:33,990 Now this one now. 706 00:46:33,990 --> 00:46:36,380 Have I got to that point? 707 00:46:36,380 --> 00:46:42,390 So this will be the final ideas in this course, 708 00:46:42,390 --> 00:46:44,360 this four unit core elective. 709 00:46:51,310 --> 00:46:56,630 What is the impulse response when there's damping? 710 00:46:56,630 --> 00:46:59,695 What's the impulse response when there's damping? 711 00:46:59,695 --> 00:47:00,194 OK. 712 00:47:02,966 --> 00:47:03,890 OK. 713 00:47:03,890 --> 00:47:05,480 So that means that I would really 714 00:47:05,480 --> 00:47:08,750 like to solve m y double prime, b 715 00:47:08,750 --> 00:47:15,264 y prime, k Y equals an impulse, delta of t. 716 00:47:23,380 --> 00:47:27,610 Because if I can solve this, then I can solve everything. 717 00:47:27,610 --> 00:47:28,950 And I can solve this. 718 00:47:28,950 --> 00:47:31,470 It turns out to be easy. 719 00:47:31,470 --> 00:47:32,780 Now why is it easy? 720 00:47:35,900 --> 00:47:39,990 You might think, my gosh, we've got second order equations 721 00:47:39,990 --> 00:47:46,650 here, we've got a delta function there, where do we go? 722 00:47:46,650 --> 00:47:51,540 And so my advice is go this way. 723 00:47:51,540 --> 00:47:54,270 The solution to that is the same. 724 00:47:54,270 --> 00:48:00,386 This with y of 0 equals 0 and y prime of 0 equal 0. 725 00:48:03,320 --> 00:48:15,140 So you have a spring, so again we have a spring with a mass. 726 00:48:15,140 --> 00:48:18,740 And that spring is in a damper. 727 00:48:18,740 --> 00:48:21,870 So can I just, without knowing what I'm doing, 728 00:48:21,870 --> 00:48:25,240 draw a damper around it. 729 00:48:25,240 --> 00:48:32,130 So the idea is I'm striking that mass at time 0. 730 00:48:32,130 --> 00:48:35,390 Striking that mass at time 0. 731 00:48:35,390 --> 00:48:35,940 What happens? 732 00:48:38,570 --> 00:48:40,705 What happens immediately? 733 00:48:43,440 --> 00:48:45,250 And then I don't touch it again. 734 00:48:45,250 --> 00:48:47,060 I strike it and that's it. 735 00:48:47,060 --> 00:48:51,900 I've set off, and what have I-- so my point is this 736 00:48:51,900 --> 00:49:11,140 has the same solution as m y double prime plus b y prime. 737 00:49:11,140 --> 00:49:13,505 And Ky equals 0. 738 00:49:18,280 --> 00:49:21,810 Nothing happens beyond time 0. 739 00:49:21,810 --> 00:49:34,590 But what are y of 0-- Well, let me give this a letter g, 740 00:49:34,590 --> 00:49:39,370 just to emphasize it's special and deserves its own name. 741 00:49:39,370 --> 00:49:45,020 So now, what is the starting position 742 00:49:45,020 --> 00:49:51,980 and the starting velocity for this picture? 743 00:49:51,980 --> 00:49:58,090 So I'm saying that the y here is the same as the g there. 744 00:49:58,090 --> 00:50:00,770 And really if you see it physically then 745 00:50:00,770 --> 00:50:04,550 you see it best. 746 00:50:04,550 --> 00:50:09,610 So again, at the instant t equals 0, 747 00:50:09,610 --> 00:50:12,630 I'm hitting that mass with a unit impulse. 748 00:50:15,270 --> 00:50:20,230 So what is the position of that mass, instantly 749 00:50:20,230 --> 00:50:21,340 after I've hit it? 750 00:50:21,340 --> 00:50:21,899 STUDENT: 0. 751 00:50:21,899 --> 00:50:22,690 PROFESSOR: Still 0. 752 00:50:22,690 --> 00:50:24,130 Good for you. 753 00:50:24,130 --> 00:50:25,520 Good for you. 754 00:50:25,520 --> 00:50:30,740 And what is the velocity of that mass instantly 755 00:50:30,740 --> 00:50:31,777 after I've hit it? 756 00:50:31,777 --> 00:50:32,771 STUDENT: 1. 757 00:50:32,771 --> 00:50:36,660 PROFESSOR: 1 is essentially the right answer. 758 00:50:36,660 --> 00:50:38,250 1 is the right answer. 759 00:50:38,250 --> 00:50:43,780 But the way I've set it up is, there is an m there. 760 00:50:43,780 --> 00:50:48,310 If I put an m there, which would be nicer, 761 00:50:48,310 --> 00:50:50,040 then the answer would be 1. 762 00:50:50,040 --> 00:50:54,500 But the way I've written it here, I have an m there, 763 00:50:54,500 --> 00:50:59,870 and I haven't fixed the units. 764 00:50:59,870 --> 00:51:03,248 So that turns out to give me a 1/m there. 765 00:51:07,160 --> 00:51:09,830 I could explain why it's a 1/m, but let me 766 00:51:09,830 --> 00:51:15,870 just for the moment say it's my fault. 767 00:51:15,870 --> 00:51:20,680 It's because I didn't get any units right that we have 1/m. 768 00:51:20,680 --> 00:51:23,650 No big deal. 769 00:51:23,650 --> 00:51:28,170 But now, what's good here? 770 00:51:28,170 --> 00:51:29,240 What's good? 771 00:51:29,240 --> 00:51:32,990 This was a problem with a mysterious delta function. 772 00:51:32,990 --> 00:51:35,560 This is a problem with 0. 773 00:51:35,560 --> 00:51:39,300 And the only price we're paying is the impulse 774 00:51:39,300 --> 00:51:44,240 gave the mass a little velocity. 775 00:51:44,240 --> 00:51:46,490 And you can imagine that the velocity gives 776 00:51:46,490 --> 00:51:52,930 it is 1/m because the strike didn't tell us about that. 777 00:51:52,930 --> 00:51:59,760 So what I'm saying is we can solve that equation for g. 778 00:51:59,760 --> 00:52:03,460 We can find g, and in fact, we have. 779 00:52:03,460 --> 00:52:10,460 So this really brings the lecture full circle. 780 00:52:10,460 --> 00:52:12,600 What do I have here? 781 00:52:12,600 --> 00:52:19,170 I have a null solution. 782 00:52:19,170 --> 00:52:21,850 So this g here is a null solution. 783 00:52:21,850 --> 00:52:24,450 So what form does it have? 784 00:52:24,450 --> 00:52:28,380 What can you tell me about g of t? 785 00:52:28,380 --> 00:52:30,490 And it's the same as y of t. 786 00:52:30,490 --> 00:52:34,565 So y is the same as g, and what's the form for it? 787 00:52:39,100 --> 00:52:39,870 Yes? 788 00:52:39,870 --> 00:52:40,690 Tell me? 789 00:52:40,690 --> 00:52:42,650 I'm taking it back to the very beginning 790 00:52:42,650 --> 00:52:46,164 of the lecture where I solved it with a 0. 791 00:52:46,164 --> 00:52:47,526 STUDENT: [INAUDIBLE]. 792 00:52:47,526 --> 00:52:49,490 PROFESSOR: C1, thanks. 793 00:52:49,490 --> 00:52:51,040 STUDENT: [INAUDIBLE]. 794 00:52:51,040 --> 00:52:53,970 PROFESSOR: e to the? 795 00:52:53,970 --> 00:53:03,260 C1, e to the s1, t, the two s's, s1 and s2, the special two 796 00:53:03,260 --> 00:53:11,642 s's, the special roots of the key equation. 797 00:53:11,642 --> 00:53:19,260 That equation gives us s1 and s2, by this formula, 798 00:53:19,260 --> 00:53:24,590 or in a homework or an exam problem, 799 00:53:24,590 --> 00:53:28,420 we hope that it would come out easily. 800 00:53:28,420 --> 00:53:29,950 So this is that part of it. 801 00:53:29,950 --> 00:53:32,440 What's the rest of it? 802 00:53:32,440 --> 00:53:36,410 C2 e to the s2 t. 803 00:53:40,350 --> 00:53:46,820 The impulse response is the particular null solution 804 00:53:46,820 --> 00:53:53,200 that starts with a shot, starts with an impulse, 805 00:53:53,200 --> 00:53:55,010 starts with a strike. 806 00:53:55,010 --> 00:53:55,510 OK. 807 00:53:55,510 --> 00:53:59,250 So I just have to find c1 and c2 here. 808 00:53:59,250 --> 00:54:00,170 All right? 809 00:54:00,170 --> 00:54:03,250 We've come back to the basic problem 810 00:54:03,250 --> 00:54:05,100 in differential equations. 811 00:54:05,100 --> 00:54:06,360 We've got the solution. 812 00:54:06,360 --> 00:54:07,630 We've got two constants. 813 00:54:07,630 --> 00:54:09,590 We've got two equations. 814 00:54:09,590 --> 00:54:14,940 We just plug that function into that equation. 815 00:54:14,940 --> 00:54:18,640 It gives us one fact about c1, c2. 816 00:54:18,640 --> 00:54:20,420 This gives us the second fact. 817 00:54:20,420 --> 00:54:21,280 We solve them. 818 00:54:21,280 --> 00:54:24,740 Why don't I just write down the answer? 819 00:54:24,740 --> 00:54:28,890 It turns out to be e to the s1 t, 820 00:54:28,890 --> 00:54:32,410 and the other guy will come in with an opposite sign e 821 00:54:32,410 --> 00:54:37,625 to the s2t over s1 minus s2. 822 00:54:40,330 --> 00:54:43,110 I think that gives us the c's. 823 00:54:46,910 --> 00:54:48,430 Oh, there'd be an m. 824 00:54:48,430 --> 00:54:52,060 There'd be an m times this, because 825 00:54:52,060 --> 00:54:55,010 of my messing with units. 826 00:54:58,600 --> 00:55:01,340 So can we just check? 827 00:55:01,340 --> 00:55:04,540 At t equals 0, what do I get out of this? 828 00:55:04,540 --> 00:55:06,380 STUDENT: 0. 829 00:55:06,380 --> 00:55:07,050 PROFESSOR: 0. 830 00:55:07,050 --> 00:55:08,760 What I'm supposed to. 831 00:55:08,760 --> 00:55:12,450 What's the derivative at t equals 0? 832 00:55:12,450 --> 00:55:15,160 The derivative at t equals 0, this derivative 833 00:55:15,160 --> 00:55:17,530 is going to bring down an s1. 834 00:55:17,530 --> 00:55:20,710 The derivative here will bring down an s2. 835 00:55:20,710 --> 00:55:23,760 At t equals 0 the exponentials will all 836 00:55:23,760 --> 00:55:27,520 be one so I'll just have the s1 minus the s2. 837 00:55:27,520 --> 00:55:30,570 It'll cancel that and it'll be 1 over m. 838 00:55:30,570 --> 00:55:36,850 So this is the neat formula for the impulse response. 839 00:55:36,850 --> 00:55:40,540 That's the neat formula for the impulse response. 840 00:55:40,540 --> 00:55:46,070 And then why-- can I use this little corner of the board? 841 00:55:46,070 --> 00:55:49,380 Why do I want that impulse response? 842 00:55:49,380 --> 00:55:51,830 What can I use it for? 843 00:55:51,830 --> 00:55:54,569 It gives me the answer, not just for the impulse 844 00:55:54,569 --> 00:55:55,360 but for everything. 845 00:55:58,790 --> 00:56:08,730 The particular solution is for any forces, force, 846 00:56:08,730 --> 00:56:14,790 I multiply by whatever the-- let me write the formula 847 00:56:14,790 --> 00:56:18,210 and I'll show you what it says. 848 00:56:18,210 --> 00:56:21,436 Yes, there is the formula. 849 00:56:21,436 --> 00:56:22,435 I'm sorry it's squeezed. 850 00:56:32,480 --> 00:56:35,900 But really, the goal here was simply 851 00:56:35,900 --> 00:56:40,690 to get a handle on what is the response to any f. 852 00:56:40,690 --> 00:56:43,090 And again, I look at that this way. 853 00:56:43,090 --> 00:56:46,636 F of s is the input at time s. 854 00:56:46,636 --> 00:56:54,720 G is the growth factor over the remaining time up until time t. 855 00:56:54,720 --> 00:57:01,690 So Y at time t, I take all the inputs up to time t, 856 00:57:01,690 --> 00:57:06,080 And each input gets multiplied by its growth factor. 857 00:57:06,080 --> 00:57:10,840 It was e to the a, t minus s in the first order equation. 858 00:57:10,840 --> 00:57:14,460 Now we've got two exponentials. 859 00:57:14,460 --> 00:57:19,910 But that's the solution of the general problem. 860 00:57:19,910 --> 00:57:26,950 So we have now in one lecture completed a solution 861 00:57:26,950 --> 00:57:33,090 to the second order constant coefficient differential 862 00:57:33,090 --> 00:57:33,898 equation. 863 00:57:33,898 --> 00:57:34,682 Right. 864 00:57:34,682 --> 00:57:35,182 Yeah. 865 00:57:37,900 --> 00:57:42,422 By finding the impulse response. 866 00:57:42,422 --> 00:57:44,827 Yes? 867 00:57:44,827 --> 00:57:47,232 STUDENT: [INAUDIBLE] would we still 868 00:57:47,232 --> 00:57:50,610 be able to [INAUDIBLE] if s1 is equal to s2? 869 00:57:50,610 --> 00:57:52,730 PROFESSOR: Ah, if s1 equals s2. 870 00:57:52,730 --> 00:57:58,130 That's the case where formulas need a patch. 871 00:57:58,130 --> 00:57:59,090 They need a patch. 872 00:57:59,090 --> 00:58:05,235 If s1 equals s2, what do you think happens? 873 00:58:08,130 --> 00:58:11,083 If s1 equal s2, everybody sees, I have 0/0. 874 00:58:14,950 --> 00:58:19,170 And so this is like a technical question 875 00:58:19,170 --> 00:58:21,990 that I wasn't going to ask myself. 876 00:58:21,990 --> 00:58:23,820 You asked it. 877 00:58:23,820 --> 00:58:25,320 You're responsible. 878 00:58:25,320 --> 00:58:27,500 What do we do for 0/0? 879 00:58:32,120 --> 00:58:34,740 What did you learn in calculus? 880 00:58:34,740 --> 00:58:40,180 Who's the crazy guy who figured out how to deal with 0/0? 881 00:58:40,180 --> 00:58:44,030 In a way, calculus is all about 0/0, right? 882 00:58:44,030 --> 00:58:48,390 Delta y over delta x, they're both headed to 0. 883 00:58:48,390 --> 00:58:51,860 And suppose you have-- let me take 884 00:58:51,860 --> 00:58:55,980 the most famous example of 0/0. 885 00:58:55,980 --> 00:59:03,060 It's like sine x over x, as x goes to 0. 886 00:59:03,060 --> 00:59:05,560 So x going to 0. 887 00:59:05,560 --> 00:59:08,265 Sine x goes to 0, x goes to 0. 888 00:59:11,580 --> 00:59:13,210 What's the answer, by the way? 889 00:59:13,210 --> 00:59:17,300 What happens to that ratio as x goes to 0? 890 00:59:17,300 --> 00:59:20,360 This is maybe the most famous example. 891 00:59:20,360 --> 00:59:24,350 Sine x over x, when x gets very small, is close to? 892 00:59:24,350 --> 00:59:24,970 STUDENT: 1. 893 00:59:24,970 --> 00:59:26,530 PROFESSOR: 1, thanks. 894 00:59:26,530 --> 00:59:30,820 And now just help me out with the name of the guy. 895 00:59:30,820 --> 00:59:35,490 It's a crazy spelling name, and do you remember? 896 00:59:35,490 --> 00:59:37,630 L'Hopital. 897 00:59:37,630 --> 00:59:38,880 L'Hopital. 898 00:59:38,880 --> 00:59:40,690 Everybody despises him. 899 00:59:40,690 --> 00:59:42,980 Probably hi friends despised him. 900 00:59:42,980 --> 00:59:48,920 But anyway, L'Hopital says in a situation like that, 901 00:59:48,920 --> 00:59:52,950 when you're going to 0/0, you're allowed 902 00:59:52,950 --> 00:59:55,580 to do something a little strange. 903 00:59:55,580 --> 00:59:58,470 You're allowed to take the derivative of the top, 904 00:59:58,470 --> 01:00:01,260 so it has the same limit. 905 01:00:01,260 --> 01:00:03,810 Instead of looking at this 0/0, you 906 01:00:03,810 --> 01:00:07,210 can take the derivative of the top, cos x, 907 01:00:07,210 --> 01:00:10,600 divided by the derivative of the bottom, 1. 908 01:00:10,600 --> 01:00:16,810 And now you can let x go to 0, and you get the right answer. 909 01:00:16,810 --> 01:00:19,000 So this gave a 0/0. 910 01:00:19,000 --> 01:00:20,600 Unclear. 911 01:00:20,600 --> 01:00:21,850 Fuzzy. 912 01:00:21,850 --> 01:00:24,480 This gives-- what's the right answer then? 913 01:00:24,480 --> 01:00:25,780 Just tell me again. 914 01:00:25,780 --> 01:00:27,850 When x goes to 0 this becomes? 915 01:00:27,850 --> 01:00:28,630 STUDENT: 1. 916 01:00:28,630 --> 01:00:29,171 PROFESSOR: 1. 917 01:00:29,171 --> 01:00:30,370 Right. 918 01:00:30,370 --> 01:00:31,910 So that's L'Hopital. 919 01:00:31,910 --> 01:00:34,890 So that's what I would have to do here. 920 01:00:34,890 --> 01:00:37,560 I would take the derivative of this, 921 01:00:37,560 --> 01:00:40,820 and the derivative of this-- the only sort of tricky part 922 01:00:40,820 --> 01:00:42,900 is it's the s derivative. 923 01:00:42,900 --> 01:00:46,470 It's s1 going to s2. 924 01:00:46,470 --> 01:00:49,220 Let me just tell you the result. 925 01:00:49,220 --> 01:00:50,760 Since you asked. 926 01:00:50,760 --> 01:00:52,760 A factor t comes out. 927 01:00:52,760 --> 01:00:59,220 It's t e to the s1, or s1 is the same as s2, divided by the m. 928 01:01:02,680 --> 01:01:04,530 It actually looks simpler. 929 01:01:04,530 --> 01:01:05,450 There's only one. 930 01:01:05,450 --> 01:01:09,910 This is in the case s1 equal s2. 931 01:01:09,910 --> 01:01:11,840 So s1 is the same as s2. 932 01:01:11,840 --> 01:01:13,080 I just chose s1. 933 01:01:13,080 --> 01:01:13,580 Yeah. 934 01:01:16,520 --> 01:01:20,520 L'Hopital gives a simpler answer. 935 01:01:20,520 --> 01:01:27,480 And it's got this suspicious and recognizable factor t. 936 01:01:27,480 --> 01:01:29,090 That came from L'Hopital. 937 01:01:29,090 --> 01:01:31,280 OK, I won't do that stiff. 938 01:01:35,850 --> 01:01:43,800 So let me say again, we've now done the second order 939 01:01:43,800 --> 01:01:48,160 constant coefficient equation I do just 940 01:01:48,160 --> 01:01:56,090 have 10 minutes of something to make it better. 941 01:01:56,090 --> 01:02:03,290 And that is that the famous quadratic formula for s, 942 01:02:03,290 --> 01:02:08,800 for s1 and s2 is not beautiful. 943 01:02:08,800 --> 01:02:11,480 It's correct. 944 01:02:11,480 --> 01:02:13,050 It's correct. 945 01:02:13,050 --> 01:02:15,890 But it's a little bit of a mess. 946 01:02:15,890 --> 01:02:20,250 You've got three things, b and m and k playing around. 947 01:02:20,250 --> 01:02:25,630 And we saw in this picture, we saw all the differences. 948 01:02:25,630 --> 01:02:34,590 I guess in this example I kept m1 and I kept k1, 949 01:02:34,590 --> 01:02:36,860 and I increased b. 950 01:02:36,860 --> 01:02:43,990 I could do other examples where I increase the k, 951 01:02:43,990 --> 01:02:46,600 I make it stiffer and stiffer. 952 01:02:46,600 --> 01:02:48,260 All these examples. 953 01:02:48,260 --> 01:02:54,930 And engineers have worked for 100 years 954 01:02:54,930 --> 01:03:00,740 to see, out of this formula what are the important parameters, 955 01:03:00,740 --> 01:03:04,930 what are the important numbers, and hopefully, 956 01:03:04,930 --> 01:03:06,950 where possible dimensionless. 957 01:03:06,950 --> 01:03:10,970 So I just want to- the final minutes 958 01:03:10,970 --> 01:03:16,840 would be-- back to high school-- playing with this formula, 959 01:03:16,840 --> 01:03:20,330 to get better numbers in there. 960 01:03:20,330 --> 01:03:21,320 May I do that? 961 01:03:21,320 --> 01:03:23,120 I just think, because then you'll 962 01:03:23,120 --> 01:03:29,530 see-- I learned this, actually, so this is like something math 963 01:03:29,530 --> 01:03:31,910 professors have no reason to do. 964 01:03:31,910 --> 01:03:32,610 Look at that. 965 01:03:32,610 --> 01:03:33,890 That's the formula. 966 01:03:33,890 --> 01:03:35,590 End of story. 967 01:03:35,590 --> 01:03:41,330 But the Web, 1803 website, has a class 968 01:03:41,330 --> 01:03:44,570 in which Professor Miller from the math department 969 01:03:44,570 --> 01:03:52,010 was teaching this subject, doing these, exactly these, 970 01:03:52,010 --> 01:03:56,850 but also Professor Vandiver from Engineering 971 01:03:56,850 --> 01:04:03,570 was putting in his suggestion of what are the good parameters? 972 01:04:03,570 --> 01:04:08,060 What are the parameters that engineers look at? 973 01:04:08,060 --> 01:04:10,070 So that would be my final comment, 974 01:04:10,070 --> 01:04:13,340 and I won't do it as well as Professor Vandiver did. 975 01:04:13,340 --> 01:04:16,620 But can I just take that-- let me 976 01:04:16,620 --> 01:04:20,990 erase these two special examples, 977 01:04:20,990 --> 01:04:27,000 and look at this question. 978 01:04:29,900 --> 01:04:31,320 Again, the book will do it. 979 01:04:34,380 --> 01:04:49,060 So one nice-- b/2m is a pretty natural parameter to use. 980 01:04:49,060 --> 01:04:52,520 Let me introduce that as one of them. 981 01:04:52,520 --> 01:04:59,790 I'm going to, by taking ratios like b/2m, let me call that p. 982 01:04:59,790 --> 01:05:03,970 Let me call b/2m. 983 01:05:03,970 --> 01:05:10,450 So that's a ratio of damping to mass. 984 01:05:10,450 --> 01:05:21,050 And then this has got to come out simpler. 985 01:05:21,050 --> 01:05:23,120 What does that come out? 986 01:05:23,120 --> 01:05:25,200 If you'll allow me, I'm going to open the book 987 01:05:25,200 --> 01:05:27,834 so I don't write the wrong thing here. 988 01:05:30,550 --> 01:05:35,750 This is in the book, on page 99. 989 01:05:35,750 --> 01:05:39,960 The title is Better Formulas for s1 and s2. 990 01:05:39,960 --> 01:05:42,310 Better Formulas for s1 and s2. 991 01:05:42,310 --> 01:05:44,500 And here's my first better formula. 992 01:05:47,380 --> 01:05:52,180 You can see that I get a minus b plus or minus the square root 993 01:05:52,180 --> 01:05:54,800 of something. 994 01:05:54,800 --> 01:05:57,965 And that something will turn out to be p squared. 995 01:06:00,550 --> 01:06:03,570 And then it'll be a minus something. 996 01:06:03,570 --> 01:06:05,930 And that something will turn out to be 997 01:06:05,930 --> 01:06:08,140 the natural frequency squared. 998 01:06:08,140 --> 01:06:10,630 Isn't that nice? 999 01:06:10,630 --> 01:06:15,440 So what's the natural frequency? 1000 01:06:15,440 --> 01:06:19,700 Somehow, the natural frequency's coming in from this and this? 1001 01:06:19,700 --> 01:06:23,870 And just remind me what that second parameter is, 1002 01:06:23,870 --> 01:06:27,480 omega n squared, the natural frequency 1003 01:06:27,480 --> 01:06:31,070 of oscillation with no damping. 1004 01:06:31,070 --> 01:06:33,870 Tell me again what that is, because that 1005 01:06:33,870 --> 01:06:36,750 was the fundamental ratio from last time. 1006 01:06:36,750 --> 01:06:39,275 It's central to all of engineering. 1007 01:06:39,275 --> 01:06:39,775 It's? 1008 01:06:39,775 --> 01:06:41,185 STUDENT: k/m. 1009 01:06:41,185 --> 01:06:42,010 PROFESSOR: k/m. 1010 01:06:42,010 --> 01:06:42,510 Thanks, k/m. 1011 01:06:45,580 --> 01:06:48,520 So I believe-- and maybe Professor Fry 1012 01:06:48,520 --> 01:06:53,310 could make this assignment a homework question, 1013 01:06:53,310 --> 01:06:56,020 which is just algebra question. 1014 01:06:56,020 --> 01:07:01,320 Everybody sees that I have a minus p here. 1015 01:07:01,320 --> 01:07:09,230 And with a little care you get p squared, which is quite nice. 1016 01:07:09,230 --> 01:07:10,150 Which is quite nice. 1017 01:07:10,150 --> 01:07:15,000 And so we see that the decision between overdamping-- 1018 01:07:15,000 --> 01:07:16,630 remember now? 1019 01:07:16,630 --> 01:07:21,730 Overdamping is when you damped so much that this became 1020 01:07:21,730 --> 01:07:26,690 negative and you got an imaginary number in there. 1021 01:07:26,690 --> 01:07:29,370 Underdamping, it's still positive. 1022 01:07:29,370 --> 01:07:31,430 Overdamping, it's negative. 1023 01:07:31,430 --> 01:07:33,930 And so really that separation between overdamping 1024 01:07:33,930 --> 01:07:39,124 and underdamping is the ratio of p to omega n. 1025 01:07:39,124 --> 01:07:43,590 P to omega n is the damping ratio. 1026 01:07:43,590 --> 01:07:44,360 I think. 1027 01:07:44,360 --> 01:07:45,500 There may be a factor too. 1028 01:07:45,500 --> 01:07:53,200 Let me try to-- everybody sees that's the battle 1029 01:07:53,200 --> 01:07:56,490 between these, if you accept that formula, 1030 01:07:56,490 --> 01:08:00,420 and if it's in the book it's got to be right. 1031 01:08:00,420 --> 01:08:02,920 OK. 1032 01:08:02,920 --> 01:08:11,060 And so the damping ratio is just that. 1033 01:08:11,060 --> 01:08:12,620 That's the damping ratio. 1034 01:08:12,620 --> 01:08:17,890 Now that's called zeta. 1035 01:08:17,890 --> 01:08:19,260 The Greek letter zeta. 1036 01:08:19,260 --> 01:08:24,300 I'm not Greek and not good writing zeta. 1037 01:08:24,300 --> 01:08:31,350 So I have unilaterally decided to use a capital zeta, which 1038 01:08:31,350 --> 01:08:39,870 is a Z. Zeta is the Greek letter for Z. I could try to write it 1039 01:08:39,870 --> 01:08:43,550 but you wouldn't be impressed. 1040 01:08:43,550 --> 01:08:45,910 So it's that damping ratio. 1041 01:08:45,910 --> 01:08:47,689 So now what does this mean? 1042 01:08:47,689 --> 01:08:53,779 Z smaller than 1, z equal to 1, c greater than 1? 1043 01:08:53,779 --> 01:09:01,750 Tell me what those-- obviously, when it's smaller than 1, 1044 01:09:01,750 --> 01:09:05,705 p is smaller than omega n. 1045 01:09:11,410 --> 01:09:14,180 Yeah, so what's going on here? 1046 01:09:14,180 --> 01:09:16,800 Which one is underdamping, which one is critical 1047 01:09:16,800 --> 01:09:18,858 damping, which one is overdamping? 1048 01:09:23,080 --> 01:09:31,439 Because there's no difficult stuff here. 1049 01:09:31,439 --> 01:09:33,330 We're coasting in the last minutes 1050 01:09:33,330 --> 01:09:39,859 here by just choosing words and notation that have turned out 1051 01:09:39,859 --> 01:09:47,319 over a century to be more revealing than b squared 1052 01:09:47,319 --> 01:09:51,770 minus 4mk, and this Z. 1053 01:09:51,770 --> 01:09:54,900 So z less than 1 will be what? 1054 01:09:54,900 --> 01:09:58,370 Z less than 1 will be p smaller than omega n. 1055 01:09:58,370 --> 01:10:00,160 p smaller than this. 1056 01:10:00,160 --> 01:10:02,480 What's the story on that case then? 1057 01:10:02,480 --> 01:10:04,930 STUDENT: [INAUDIBLE]. 1058 01:10:04,930 --> 01:10:10,820 PROFESSOR: That is underdamping, I guess. p small has to go. 1059 01:10:10,820 --> 01:10:14,880 p is like b, the damping. 1060 01:10:14,880 --> 01:10:18,100 And small damping is underdamping. 1061 01:10:18,100 --> 01:10:19,010 So this is underdamp. 1062 01:10:22,430 --> 01:10:25,110 Underdamp. 1063 01:10:25,110 --> 01:10:27,410 And that's the case, in which we're 1064 01:10:27,410 --> 01:10:29,400 going to have some imaginary stuff. 1065 01:10:29,400 --> 01:10:31,680 We're going to have some oscillation 1066 01:10:31,680 --> 01:10:34,350 with the decay coming from there. 1067 01:10:34,350 --> 01:10:36,700 Now, what about z equal to 1? 1068 01:10:36,700 --> 01:10:40,000 z equal to 1 means p equals omega m, 1069 01:10:40,000 --> 01:10:43,160 so that equals that so it's a big 0 in there. 1070 01:10:43,160 --> 01:10:44,286 What case is that? 1071 01:10:44,286 --> 01:10:45,160 STUDENT: [INAUDIBLE]. 1072 01:10:45,160 --> 01:10:46,470 PROFESSOR: Critical damping. 1073 01:10:46,470 --> 01:10:49,300 It's this case in that picture. 1074 01:10:49,300 --> 01:10:54,890 It's that case with a double 0, equal s's. 1075 01:10:54,890 --> 01:10:58,390 Formulas that have to take account of that, this 1076 01:10:58,390 --> 01:10:59,040 is critical. 1077 01:11:02,480 --> 01:11:08,530 And then finally, z greater than 1 is what? 1078 01:11:08,530 --> 01:11:09,590 Overdamped. 1079 01:11:09,590 --> 01:11:10,890 Overdamping. 1080 01:11:10,890 --> 01:11:13,800 Z bigger than 1 means p is big. 1081 01:11:13,800 --> 01:11:19,895 P big means b is big, damping is big, it's overdamped. 1082 01:11:19,895 --> 01:11:20,395 Overdamped. 1083 01:11:24,360 --> 01:11:29,530 So we've got it down to one parameter, the damping ratio, 1084 01:11:29,530 --> 01:11:32,200 to tell us these things. 1085 01:11:32,200 --> 01:11:35,700 Rather than previously we had to say 1086 01:11:35,700 --> 01:11:37,720 is b squared smaller than 4mk? 1087 01:11:37,720 --> 01:11:40,430 Is it equal to 4mk? 1088 01:11:40,430 --> 01:11:42,510 Is it bigger than 4mk? 1089 01:11:42,510 --> 01:11:47,090 Now we've got those words down to a single number z. 1090 01:11:47,090 --> 01:11:54,810 And let me just write next to us here that the z turns out 1091 01:11:54,810 --> 01:11:59,470 to be the ratio of the damping to-- I think 1092 01:11:59,470 --> 01:12:03,480 it's right-- it's the damping divided 1093 01:12:03,480 --> 01:12:08,419 by the square root of 4mk, I think. 1094 01:12:08,419 --> 01:12:09,960 Can I just put a question mark there. 1095 01:12:12,980 --> 01:12:23,260 You couldn't mess around with the letters p and z. 1096 01:12:23,260 --> 01:12:30,030 But to get some variation from some other, 1097 01:12:30,030 --> 01:12:33,110 but the point is, you see how much cleaner that 1098 01:12:33,110 --> 01:12:34,940 is compared to this? 1099 01:12:34,940 --> 01:12:39,320 You're directly comparing that number to that number. 1100 01:12:39,320 --> 01:12:42,454 And that ratio is that number. 1101 01:12:42,454 --> 01:12:43,720 Yeah. 1102 01:12:43,720 --> 01:12:47,190 So all the formulas come out nicely. 1103 01:12:47,190 --> 01:12:49,680 Yeah, the formulas come out nicely. 1104 01:12:49,680 --> 01:12:55,400 And I guess what we see here-- final comment-- what 1105 01:12:55,400 --> 01:12:59,770 we see here is what is the frequency 1106 01:12:59,770 --> 01:13:03,770 of underdamped oscillations. 1107 01:13:03,770 --> 01:13:08,430 So I want to be in this underdamping case 1108 01:13:08,430 --> 01:13:11,100 where there is oscillation. 1109 01:13:11,100 --> 01:13:16,510 There is an imaginary number coming out of that. 1110 01:13:16,510 --> 01:13:20,060 But there's also a real number. 1111 01:13:20,060 --> 01:13:23,830 Is the frequency of underdamped oscillation the same 1112 01:13:23,830 --> 01:13:27,410 as omega m, the natural frequency? 1113 01:13:27,410 --> 01:13:28,440 No. 1114 01:13:28,440 --> 01:13:35,810 The frequency of that number-- so final comment, 1115 01:13:35,810 --> 01:13:36,900 let me put it just here. 1116 01:13:42,080 --> 01:13:46,790 I would like this whole thing to be i, to give me oscillation, 1117 01:13:46,790 --> 01:13:56,720 times omega d, the damped frequency. 1118 01:13:56,720 --> 01:13:58,870 And let me just say what that is. 1119 01:13:58,870 --> 01:14:04,500 So omega d, the damped frequency, 1120 01:14:04,500 --> 01:14:09,180 squared, is this omega natural frequency 1121 01:14:09,180 --> 01:14:12,750 squared minus the p squared. 1122 01:14:12,750 --> 01:14:20,655 If I had longer and we didn't have blackboards already full 1123 01:14:20,655 --> 01:14:24,630 of formulas, I could-- it's the thing 1124 01:14:24,630 --> 01:14:26,510 whose square root we're taking here. 1125 01:14:26,510 --> 01:14:35,050 So this is minus p plus or minus i, omega damped. 1126 01:14:35,050 --> 01:14:37,320 Omega damped is this square root. 1127 01:14:40,340 --> 01:14:51,040 There, we succeeded to fit in the better ratios, 1128 01:14:51,040 --> 01:14:56,060 the good quantities to look at. 1129 01:14:56,060 --> 01:14:58,100 So again, the good quantities to look at 1130 01:14:58,100 --> 01:15:05,075 are p, z, the damping ratio, omega d the damped frequency. 1131 01:15:07,980 --> 01:15:10,490 I think in a first lecture you could say, 1132 01:15:10,490 --> 01:15:14,080 well, we already had correct formulas, 1133 01:15:14,080 --> 01:15:17,080 we should just leave it there, and that's absolutely true, 1134 01:15:17,080 --> 01:15:22,610 but anyway, this is what-- these are the letters people have 1135 01:15:22,610 --> 01:15:26,020 introduced to make the formulas easier 1136 01:15:26,020 --> 01:15:29,380 to understand in an engineering problem. 1137 01:15:29,380 --> 01:15:30,450 OK. 1138 01:15:30,450 --> 01:15:33,110 I'm all done except for questions. 1139 01:15:36,000 --> 01:15:37,670 Yes? 1140 01:15:37,670 --> 01:15:39,528 Don't ask me about resonance again. 1141 01:15:39,528 --> 01:15:41,020 Yes, OK. 1142 01:15:41,020 --> 01:15:41,520 Yes? 1143 01:15:41,520 --> 01:15:46,251 STUDENT: In the case of where we have the delta function, what 1144 01:15:46,251 --> 01:15:48,010 is the velocity [INAUDIBLE]? 1145 01:15:48,010 --> 01:15:49,526 PROFESSOR: What is the what? 1146 01:15:49,526 --> 01:15:51,609 STUDENT: Why is the velocity equal to [INAUDIBLE]? 1147 01:15:51,609 --> 01:15:53,006 PROFESSOR: A-ha. 1148 01:15:53,006 --> 01:15:55,850 Okay. 1149 01:15:55,850 --> 01:15:57,830 You're right on the ball. 1150 01:15:57,830 --> 01:16:00,990 The question is where did this come from. 1151 01:16:00,990 --> 01:16:04,060 Where did that come from? 1152 01:16:04,060 --> 01:16:07,360 Can I tell you? 1153 01:16:07,360 --> 01:16:10,250 So if I integrate everything here, 1154 01:16:10,250 --> 01:16:14,200 if I take the integral of everything, 1155 01:16:14,200 --> 01:16:20,090 between 0, a little bit left of 0-- can I call that 0 minus? 1156 01:16:20,090 --> 01:16:22,860 Just a little bit left of 0. 1157 01:16:22,860 --> 01:16:24,310 This is crazy. 1158 01:16:24,310 --> 01:16:27,700 No math professor or whatever should ever do this. 1159 01:16:27,700 --> 01:16:33,760 To a little bit right of 0, just a real short time. 1160 01:16:33,760 --> 01:16:36,065 So what am I going to call a little bit right of 0? 1161 01:16:38,920 --> 01:16:41,640 0 plus. 1162 01:16:41,640 --> 01:16:49,960 OK now what is that integral? 1163 01:16:49,960 --> 01:16:52,920 Between a little left of 0 and a little right of 0, 1164 01:16:52,920 --> 01:16:55,525 you know what the integral of the delta function is. 1165 01:16:55,525 --> 01:16:56,230 It is? 1166 01:16:56,230 --> 01:16:57,850 STUDENT: 1. 1167 01:16:57,850 --> 01:16:59,050 PROFESSOR: 1. 1168 01:16:59,050 --> 01:17:00,290 Good. 1169 01:17:00,290 --> 01:17:03,040 Now, what are these ridiculous things? 1170 01:17:03,040 --> 01:17:09,870 Well, y is not changing in this tiny, tiny time. 1171 01:17:09,870 --> 01:17:20,060 So this is something, it's not getting big. 1172 01:17:20,060 --> 01:17:22,280 I'm integrating it over this tiny little time. 1173 01:17:22,280 --> 01:17:23,370 It's nothing. 1174 01:17:23,370 --> 01:17:25,660 Forget it. 1175 01:17:25,660 --> 01:17:29,880 Similarly here, y prime, the velocity is not 1176 01:17:29,880 --> 01:17:31,910 climbing to infinity. 1177 01:17:31,910 --> 01:17:34,540 There's no-- and I'm just integrating 1178 01:17:34,540 --> 01:17:37,330 over this infinitesimal little time. 1179 01:17:37,330 --> 01:17:38,930 Nothing here. 1180 01:17:38,930 --> 01:17:45,090 So this term has to be responsible for the 1. 1181 01:17:45,090 --> 01:17:48,803 And now you can tell me the integral of m y double prime. 1182 01:17:52,750 --> 01:17:55,220 What's the integral of m y prime? 1183 01:17:55,220 --> 01:17:56,590 STUDENT: m y prime 1184 01:17:56,590 --> 01:17:58,890 PROFESSOR: m y prime. 1185 01:17:58,890 --> 01:18:09,910 So m y prime plus, at 0 plus, minus m y prime at 0 minus. 1186 01:18:09,910 --> 01:18:13,600 But at 0 minus, it's 0. 1187 01:18:13,600 --> 01:18:15,080 You see what's happening? 1188 01:18:15,080 --> 01:18:18,980 And on the right side I'm getting a 1. 1189 01:18:18,980 --> 01:18:25,140 This is-- no person who had any skill with a blackboard would 1190 01:18:25,140 --> 01:18:26,620 allow this to happen. 1191 01:18:26,620 --> 01:18:28,920 But that happened. 1192 01:18:28,920 --> 01:18:29,630 OK. 1193 01:18:29,630 --> 01:18:33,930 So these lower order terms are typical of math. 1194 01:18:33,930 --> 01:18:37,330 Lower order terms in the limit, forget them. 1195 01:18:37,330 --> 01:18:41,920 This is the top term, and it has to have something there, 1196 01:18:41,920 --> 01:18:44,300 because it has to balance the 1. 1197 01:18:44,300 --> 01:18:47,800 And what it has is the jump in y prime. 1198 01:18:47,800 --> 01:18:50,330 So this is the instant jump in y prime 1199 01:18:50,330 --> 01:18:54,580 in velocity, is times m gives 1. 1200 01:18:54,580 --> 01:19:01,980 So the instant jump jumped us from 0 to 1 over m. 1201 01:19:01,980 --> 01:19:03,670 That's where I came from. 1202 01:19:03,670 --> 01:19:08,630 Well, that was a good question, and a kind of crazy answer 1203 01:19:08,630 --> 01:19:10,250 but there it is. 1204 01:19:10,250 --> 01:19:14,980 OK, so we've got a mention of the Laplace transform 1205 01:19:14,980 --> 01:19:18,650 as the algebra tool that works when you're 1206 01:19:18,650 --> 01:19:21,580 staying with exponentials and nice functions. 1207 01:19:21,580 --> 01:19:27,172 And you'll see more of that. 1208 01:19:27,172 --> 01:19:31,804 So it's a frequently used tool to turn problems into algebra.