1 00:00:00,680 --> 00:00:03,070 GILBERT STRANG: I'm coming back to the number one 2 00:00:03,070 --> 00:00:06,350 example, but not the easiest example, 3 00:00:06,350 --> 00:00:12,770 of a second order equation with an oscillating 4 00:00:12,770 --> 00:00:17,800 forcing term, cosine omega t. 5 00:00:17,800 --> 00:00:21,290 We have to know the answer to this problem. 6 00:00:21,290 --> 00:00:25,350 And it's a little messy, but the method is not messy. 7 00:00:25,350 --> 00:00:28,410 The method is straightforward. 8 00:00:28,410 --> 00:00:34,940 So let me begin by looking for the rectangular form. 9 00:00:34,940 --> 00:00:36,840 I call this the rectangular form. 10 00:00:36,840 --> 00:00:40,920 It separates the cosine with its amplitude 11 00:00:40,920 --> 00:00:45,920 and the sine with its amplitude into two separate pieces. 12 00:00:45,920 --> 00:00:48,950 So if I'm looking for that solution, 13 00:00:48,950 --> 00:00:54,770 and m and n are the numbers I want to find, how do I proceed? 14 00:00:54,770 --> 00:01:02,350 It's a case of undetermined coefficients, M and N. 15 00:01:02,350 --> 00:01:07,140 And the way to determine them is substitute this 16 00:01:07,140 --> 00:01:16,090 into the equation and match the cosine term and find M and N. 17 00:01:16,090 --> 00:01:18,830 And the way we find M and N, we need two equations 18 00:01:18,830 --> 00:01:22,990 for two quantities, M and N. 19 00:01:22,990 --> 00:01:28,010 And imagine this substituted in there. 20 00:01:28,010 --> 00:01:30,280 I'll get some cosines. 21 00:01:30,280 --> 00:01:33,130 So the cosines on one side will match the cosine 22 00:01:33,130 --> 00:01:34,440 on the other side. 23 00:01:34,440 --> 00:01:38,780 And also from the derivative, I'll get some sines 24 00:01:38,780 --> 00:01:42,850 and they should match 0 because I have no sine omega 25 00:01:42,850 --> 00:01:44,180 t on the right hand side. 26 00:01:44,180 --> 00:01:46,830 So I have two equations, matching the sines, 27 00:01:46,830 --> 00:01:48,200 matching the cosines. 28 00:01:48,200 --> 00:01:49,985 And I solve those. 29 00:01:49,985 --> 00:01:51,990 Two equations, two unknowns. 30 00:01:51,990 --> 00:01:54,020 And I just write the answer down. 31 00:01:54,020 --> 00:01:57,530 M involves a C minus omega squared. 32 00:01:57,530 --> 00:02:00,380 M is coming from the cosines. 33 00:02:00,380 --> 00:02:05,330 And we get cosines from that term and that term. 34 00:02:05,330 --> 00:02:10,229 Divided by some number, D, that I'll write down. 35 00:02:10,229 --> 00:02:16,726 And N is just B omega divided by that same D. 36 00:02:16,726 --> 00:02:22,670 And now I'll write down D. That's C minus A omega squared 37 00:02:22,670 --> 00:02:26,580 squared plus B omega squared. 38 00:02:29,920 --> 00:02:36,110 This is what comes out from the two equations for M and N. 39 00:02:36,110 --> 00:02:38,440 I just solve those equations. 40 00:02:38,440 --> 00:02:42,160 This D here is the two by two determinant 41 00:02:42,160 --> 00:02:46,300 if we think about the linear algebra behind two equations. 42 00:02:46,300 --> 00:02:48,560 And that's what it is. 43 00:02:48,560 --> 00:02:55,730 And so the answer now is in terms of A, C, B, and D, 44 00:02:55,730 --> 00:03:01,660 which is a mixture of all of A, B, and C. That's the solution. 45 00:03:01,660 --> 00:03:07,020 Only I always want to show you a different form of the solution. 46 00:03:07,020 --> 00:03:09,060 And in this case, a better form. 47 00:03:09,060 --> 00:03:12,460 Because the most important physical quantity 48 00:03:12,460 --> 00:03:14,170 is the magnitude. 49 00:03:14,170 --> 00:03:16,720 How large does y get? 50 00:03:16,720 --> 00:03:18,920 What is the amplitude of this? 51 00:03:18,920 --> 00:03:21,270 This is a sinusoid. 52 00:03:21,270 --> 00:03:24,640 And we remember that every sinusoid 53 00:03:24,640 --> 00:03:28,570 can be written in a polar form. 54 00:03:31,830 --> 00:03:36,640 Says that y of t is some amplitude of G, 55 00:03:36,640 --> 00:03:45,940 the gain, times a cosine of omega t with a shift, 56 00:03:45,940 --> 00:03:49,190 with a lag, with an angle alpha. 57 00:03:49,190 --> 00:03:51,300 So I have two numbers now. 58 00:03:51,300 --> 00:03:52,730 That's the gain. 59 00:03:52,730 --> 00:03:56,835 And this is the phase shift alpha. 60 00:04:03,550 --> 00:04:08,480 And that's an attractive form because it has only one term. 61 00:04:08,480 --> 00:04:15,090 The two numbers, G and alpha, get put into a single term 62 00:04:15,090 --> 00:04:20,110 where we can see the magnitude of the oscillation. 63 00:04:20,110 --> 00:04:22,560 And what does that come out to be? 64 00:04:22,560 --> 00:04:25,480 I won't go through all the steps. 65 00:04:25,480 --> 00:04:28,570 I'll just write down what G turns out to be. 66 00:04:28,570 --> 00:04:32,820 G turns out to be-- it comes from there-- 67 00:04:32,820 --> 00:04:38,320 and it's 1 over the square root of D. Well, 68 00:04:38,320 --> 00:04:42,050 G is the square root of M squared plus N squared. 69 00:04:44,930 --> 00:04:46,960 The square root of M squared plus N squared. 70 00:04:46,960 --> 00:04:50,160 And if I put M squared and N squared, 71 00:04:50,160 --> 00:04:54,360 then I have D over D squared. 72 00:04:54,360 --> 00:04:55,660 I get that answer. 73 00:04:55,660 --> 00:04:57,150 That's the gain. 74 00:04:57,150 --> 00:04:59,300 Let me write that word, gain, again. 75 00:04:59,300 --> 00:05:04,620 Because you got it there. 76 00:05:04,620 --> 00:05:06,060 Here it is again. 77 00:05:06,060 --> 00:05:10,680 And as always, the tangent of alpha 78 00:05:10,680 --> 00:05:20,910 is the N over the M, which is just B omega over C 79 00:05:20,910 --> 00:05:24,860 minus A omega squared. 80 00:05:24,860 --> 00:05:26,305 I like that polar form. 81 00:05:29,140 --> 00:05:31,440 And I feel I should just do an example. 82 00:05:31,440 --> 00:05:35,300 I didn't do any of the algebra in this video. 83 00:05:35,300 --> 00:05:38,770 But you know where the algebra came from. 84 00:05:38,770 --> 00:05:42,500 It came from substituting the form 85 00:05:42,500 --> 00:05:44,590 we expect for the solution. 86 00:05:44,590 --> 00:05:47,850 And of course, that form that we expect 87 00:05:47,850 --> 00:05:53,100 is the form we get provided omega, the driving frequency, 88 00:05:53,100 --> 00:05:55,740 is different from omega N. 89 00:05:55,740 --> 00:05:56,390 Well, no. 90 00:05:56,390 --> 00:05:59,830 I guess we're all right even if omega is omega N, because we 91 00:05:59,830 --> 00:06:03,850 have a damping term. 92 00:06:03,850 --> 00:06:07,020 So that's the answer. 93 00:06:07,020 --> 00:06:08,400 So an example. 94 00:06:08,400 --> 00:06:10,160 Why not an example? 95 00:06:10,160 --> 00:06:17,280 y double prime plus y prime plus 2y equals cosine of t. 96 00:06:21,020 --> 00:06:23,240 That's a simple example. 97 00:06:23,240 --> 00:06:25,880 I took omega to be 1, you see. 98 00:06:25,880 --> 00:06:27,680 And there is omega. 99 00:06:27,680 --> 00:06:33,500 And then A is 1, B is 1, C is 2. 100 00:06:33,500 --> 00:06:36,600 We can evaluate everything. 101 00:06:36,600 --> 00:06:39,980 In fact, I think M and N are 1/2. 102 00:06:43,870 --> 00:06:48,790 D, by the way, will be 1 squared plus 1 squared. 103 00:06:48,790 --> 00:06:50,790 That's 2 square root. 104 00:06:54,251 --> 00:06:54,750 Sorry. 105 00:06:54,750 --> 00:06:55,970 D will be 2. 106 00:06:55,970 --> 00:06:57,360 1 squared plus 1 squared. 107 00:07:00,340 --> 00:07:01,450 So what do I know? 108 00:07:01,450 --> 00:07:04,000 Do I know the rectangular form? 109 00:07:04,000 --> 00:07:05,910 Yes. 110 00:07:05,910 --> 00:07:08,800 Rectangular form is 1/2. 111 00:07:08,800 --> 00:07:12,060 1/2 for both the cosine and the sine. 112 00:07:12,060 --> 00:07:19,050 1/2 of cosine t plus sine t. 113 00:07:19,050 --> 00:07:22,230 That's the rectangular form. 114 00:07:22,230 --> 00:07:24,550 Two simple things, but I have to add them. 115 00:07:24,550 --> 00:07:26,930 And in my mind, I don't necessarily 116 00:07:26,930 --> 00:07:31,060 see how the cosine adds to the sine. 117 00:07:31,060 --> 00:07:38,900 But the sinusoidal identity, the polar form, gives it to me. 118 00:07:38,900 --> 00:07:41,630 So what is it in polar form? 119 00:07:41,630 --> 00:07:47,840 So G, the gain, is going to be 1 over the square root of 2. 120 00:07:53,620 --> 00:07:57,690 At the highest point, the cosine and the sine are the same. 121 00:07:57,690 --> 00:07:59,970 They're both 1 over the square root of 2. 122 00:07:59,970 --> 00:08:01,170 I have two of them. 123 00:08:01,170 --> 00:08:03,260 So I get 1 over the square root of 2 124 00:08:03,260 --> 00:08:11,010 cosine of t minus pi over 4 is the angle, the phase lag. 125 00:08:11,010 --> 00:08:14,070 When I add the cosine and the sine, 126 00:08:14,070 --> 00:08:19,450 I get a sinusoid that's sitting over pi over 4, 45 degrees. 127 00:08:19,450 --> 00:08:22,640 So those are the two forms. 128 00:08:22,640 --> 00:08:27,450 So in a nice example, we certainly got a nice answer. 129 00:08:27,450 --> 00:08:28,530 We certainly did. 130 00:08:28,530 --> 00:08:29,990 Yes. 131 00:08:29,990 --> 00:08:38,289 So that is the-- worked out, more or less worked out, 132 00:08:38,289 --> 00:08:41,340 in principle, worked out-- is the solution 133 00:08:41,340 --> 00:08:47,870 to what I think of as the most important application when 134 00:08:47,870 --> 00:08:52,750 the forcing term is a cosine. 135 00:08:52,750 --> 00:08:55,490 So it gives oscillating motion. 136 00:08:55,490 --> 00:08:57,350 It gives a phase shift. 137 00:08:57,350 --> 00:09:00,740 And it gives these formulas. 138 00:09:00,740 --> 00:09:06,490 The only thing I would add is that I need 139 00:09:06,490 --> 00:09:09,650 to comment on better notation. 140 00:09:09,650 --> 00:09:15,870 So I have used in these formulas A, B, and C. 141 00:09:15,870 --> 00:09:23,630 But those have meaning as mass, damping constant, 142 00:09:23,630 --> 00:09:26,310 spring constant. 143 00:09:26,310 --> 00:09:31,330 M, B, and K. 144 00:09:31,330 --> 00:09:37,330 And it's combinations of those that come in. 145 00:09:37,330 --> 00:09:46,950 So let me just take this moment to say better notation. 146 00:09:46,950 --> 00:09:52,130 Or maybe I should say engineering notation 147 00:09:52,130 --> 00:10:04,550 instead of A, B, C, which are mass, damping, spring constant. 148 00:10:04,550 --> 00:10:06,860 Well, that's already better to use 149 00:10:06,860 --> 00:10:08,820 letters that have a meaning. 150 00:10:08,820 --> 00:10:14,000 But the small but very important point 151 00:10:14,000 --> 00:10:22,790 is that two combinations of A, B, C, M, B, K 152 00:10:22,790 --> 00:10:24,480 are especially good. 153 00:10:24,480 --> 00:10:28,510 One is the natural frequency that we've seen already, 154 00:10:28,510 --> 00:10:32,570 square root of C over A. 155 00:10:32,570 --> 00:10:38,220 Square root of K over M. So that gives us 156 00:10:38,220 --> 00:10:41,560 one important combination of A and C. 157 00:10:41,560 --> 00:10:44,375 And the other one is the damping ratio. 158 00:10:51,700 --> 00:10:53,365 And it's called zeta. 159 00:10:56,150 --> 00:11:02,360 And that damping ratio is B over the square root of 4ac. 160 00:11:02,360 --> 00:11:03,147 Ha! 161 00:11:03,147 --> 00:11:04,730 You'll say, where does that come from? 162 00:11:04,730 --> 00:11:09,665 Or I can use these letters, B over the square root of 4mk. 163 00:11:14,460 --> 00:11:20,180 That damping ratio is, so to speak, 164 00:11:20,180 --> 00:11:23,800 it's the right dimensionless quantity. 165 00:11:23,800 --> 00:11:32,980 The dimensions of this ratio are just numbers. 166 00:11:32,980 --> 00:11:35,460 Those two quantities have the same dimension. 167 00:11:35,460 --> 00:11:39,490 And we can see that because in the quadratic formula 168 00:11:39,490 --> 00:11:42,920 comes-- you remember that in a quadratic formula 169 00:11:42,920 --> 00:11:46,010 comes the square root of b squared minus 4ac? 170 00:11:48,830 --> 00:11:55,080 Now if you see a formula that has b squared minus 4ac in it, 171 00:11:55,080 --> 00:11:59,750 you know that these must have the same units. 172 00:12:02,400 --> 00:12:06,790 Otherwise, subtraction would be a crime. 173 00:12:06,790 --> 00:12:12,950 So they have the same ratio and the same units 174 00:12:12,950 --> 00:12:15,520 and therefore the ratio is dimensionless. 175 00:12:18,110 --> 00:12:19,637 Let me write that word. 176 00:12:19,637 --> 00:12:20,220 Dimensionless. 177 00:12:23,220 --> 00:12:25,910 So conclusion. 178 00:12:25,910 --> 00:12:30,550 I could rewrite the answer in terms of these quantities omega 179 00:12:30,550 --> 00:12:34,480 n and zeta. 180 00:12:34,480 --> 00:12:36,840 I won't do that here. 181 00:12:36,840 --> 00:12:39,550 That can wait for another time. 182 00:12:39,550 --> 00:12:45,080 But just to say since we've found a solution to the most 183 00:12:45,080 --> 00:12:50,210 important application with cosine omega 184 00:12:50,210 --> 00:12:55,130 t there, since we found the solution, 185 00:12:55,130 --> 00:12:59,010 appropriate to comment that we could write the answer in terms 186 00:12:59,010 --> 00:13:04,560 of omega n, the natural frequency, and z, zeta, 187 00:13:04,560 --> 00:13:07,800 the damping ratio. 188 00:13:07,800 --> 00:13:09,620 Thank you.