1 00:00:00,070 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,140 continue to offer high quality educational resources for free. 5 00:00:10,140 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:21,650 --> 00:00:26,430 PROFESSOR: Are we ready with the concept questions 9 00:00:26,430 --> 00:00:29,740 from the homework this week? 10 00:00:29,740 --> 00:00:32,284 How do we get different-- there we go. 11 00:00:32,284 --> 00:00:33,700 I looked at one, it was one thing. 12 00:00:33,700 --> 00:00:35,270 I looked at the other, [INAUDIBLE]. 13 00:00:35,270 --> 00:00:37,260 Does g enter into the expression for 14 00:00:37,260 --> 00:00:39,950 the undamped natural frequency? 15 00:00:39,950 --> 00:00:45,705 And most people said no, but about a third of you said yes. 16 00:00:45,705 --> 00:00:47,330 If you have worked on that problem now, 17 00:00:47,330 --> 00:00:49,830 you have already discovered the answer. 18 00:00:49,830 --> 00:00:56,520 So you'll find that g does not come into the expression. 19 00:00:56,520 --> 00:01:02,200 When you do a pendulum, g is in the expression. 20 00:01:02,200 --> 00:01:05,090 And there's a question on the homework 21 00:01:05,090 --> 00:01:07,420 about what's the difference. 22 00:01:07,420 --> 00:01:09,830 How can you predict when g is going 23 00:01:09,830 --> 00:01:12,880 to be involved in a natural frequency expression 24 00:01:12,880 --> 00:01:14,180 and when it is not? 25 00:01:16,562 --> 00:01:18,270 I want you to think about that one a bit, 26 00:01:18,270 --> 00:01:20,170 maybe talk about it at-- if there's still 27 00:01:20,170 --> 00:01:21,670 questions about it, we talk about it 28 00:01:21,670 --> 00:01:23,480 in recitation on Thursday, Friday. 29 00:01:23,480 --> 00:01:25,020 OK. 30 00:01:25,020 --> 00:01:27,205 So does g enter into the expression here? 31 00:01:27,205 --> 00:01:31,170 I'm sure you know this simple pendulum, the natural frequency 32 00:01:31,170 --> 00:01:34,000 square root of g over l. 33 00:01:34,000 --> 00:01:35,910 For a simple mass-spring dashpot, 34 00:01:35,910 --> 00:01:38,070 the natural frequency is k over m 35 00:01:38,070 --> 00:01:41,070 whether or not it's affected by gravity. 36 00:01:41,070 --> 00:01:44,840 So there's something different about these two. 37 00:01:44,840 --> 00:01:46,510 OK, let's go onto the third one. 38 00:01:51,340 --> 00:01:54,180 "In an experiment, this system is given initial velocity 39 00:01:54,180 --> 00:01:57,240 observed to decay in amplitude of vibration 40 00:01:57,240 --> 00:01:59,620 by 50% in 10 cycles. 41 00:01:59,620 --> 00:02:01,230 You can estimate the damping ratio 42 00:02:01,230 --> 00:02:02,742 to be approximately," what? 43 00:02:02,742 --> 00:02:05,200 Well, it gives a little rule of thumb I gave you last week. 44 00:02:05,200 --> 00:02:06,930 I'll go over it again today. 45 00:02:06,930 --> 00:02:11,360 0.11 divided by the number of cycles did decay 50%. 46 00:02:11,360 --> 00:02:14,290 So it took 10 cycles, 0.11 divided 47 00:02:14,290 --> 00:02:19,670 by 10, 0.01, 1.1% damping. 48 00:02:19,670 --> 00:02:22,460 OK, next. 49 00:02:22,460 --> 00:02:25,160 "At which of the three excitation frequencies 50 00:02:25,160 --> 00:02:28,380 will the response magnitude be greatest?" 51 00:02:28,380 --> 00:02:33,210 You've done oscillators excited by cosine omega t 52 00:02:33,210 --> 00:02:34,820 kind of things before. 53 00:02:34,820 --> 00:02:38,142 So at the ratio 1, most people said 54 00:02:38,142 --> 00:02:39,600 it's where it would be the largest. 55 00:02:39,600 --> 00:02:40,880 Why at 1? 56 00:02:40,880 --> 00:02:43,870 Anybody want to give a shout here? 57 00:02:43,870 --> 00:02:46,750 What happens when you drive the system 58 00:02:46,750 --> 00:02:49,384 at its natural frequency? 59 00:02:49,384 --> 00:02:52,050 It's called resonance, and we're going to talk about that today. 60 00:02:52,050 --> 00:02:54,540 So it's when the frequency ratio is 61 00:02:54,540 --> 00:02:59,820 one that-- and for the system being lightly damped that you 62 00:02:59,820 --> 00:03:01,540 get the largest response. 63 00:03:01,540 --> 00:03:05,749 Finally-- which one are we on here? 64 00:03:05,749 --> 00:03:06,290 Oh, this one. 65 00:03:06,290 --> 00:03:10,680 Can all the kinetic energy be accounted for by an expression 66 00:03:10,680 --> 00:03:12,765 of the form IZ omega squared. 67 00:03:17,040 --> 00:03:19,310 By the way, I brought this system if you haven't. 68 00:03:22,190 --> 00:03:23,940 So there's a really simple demo, but it 69 00:03:23,940 --> 00:03:26,180 has all sorts of-- so in this case, 70 00:03:26,180 --> 00:03:32,340 we're talking about that motion. 71 00:03:32,340 --> 00:03:36,740 It certainly has some I omega squared kind of kinetic energy, 72 00:03:36,740 --> 00:03:40,255 but does the center of gravity translate as it's oscillating? 73 00:03:43,841 --> 00:03:45,590 What's the potential energy in the system? 74 00:03:48,970 --> 00:03:52,610 By the way, any time you get an oscillation, 75 00:03:52,610 --> 00:03:57,060 energy flows from potential to kinetic, potential kinetic. 76 00:03:57,060 --> 00:03:58,580 That's what oscillation is. 77 00:03:58,580 --> 00:04:01,790 So there has to be an exchange going on between kinetic energy 78 00:04:01,790 --> 00:04:02,930 and potential energy. 79 00:04:02,930 --> 00:04:04,640 And if there's no losses in the system, 80 00:04:04,640 --> 00:04:08,590 the total energy is constant. 81 00:04:08,590 --> 00:04:12,150 So the kinetic energy's certainly in the motion, 82 00:04:12,150 --> 00:04:14,450 but when it reaches maximum amplitude, what's 83 00:04:14,450 --> 00:04:17,459 its velocity when it's right here? 84 00:04:17,459 --> 00:04:18,230 Zero. 85 00:04:18,230 --> 00:04:22,070 So all of its energy must be where? 86 00:04:22,070 --> 00:04:22,880 In the potential. 87 00:04:22,880 --> 00:04:25,840 And where's the potential in this system? 88 00:04:25,840 --> 00:04:26,960 Pardon? 89 00:04:26,960 --> 00:04:27,590 In the string. 90 00:04:27,590 --> 00:04:29,427 That's not stored in the string. 91 00:04:29,427 --> 00:04:31,260 There's only two sources of potential energy 92 00:04:31,260 --> 00:04:33,360 we talk about in this class. 93 00:04:33,360 --> 00:04:33,920 Gravity and-- 94 00:04:33,920 --> 00:04:34,670 AUDIENCE: Strings. 95 00:04:34,670 --> 00:04:35,461 PROFESSOR: Strings. 96 00:04:35,461 --> 00:04:38,700 Well, these strings don't stretch, 97 00:04:38,700 --> 00:04:40,995 so there's no spring kinetic energy. 98 00:04:40,995 --> 00:04:42,120 We've got potential energy. 99 00:04:42,120 --> 00:04:47,490 There must be gravitational potential energy. 100 00:04:47,490 --> 00:04:49,215 How is it coming into this system? 101 00:04:52,880 --> 00:04:58,180 So he says when it turns, the center of gravity 102 00:04:58,180 --> 00:05:02,440 has to raise up a little bit, and that's 103 00:05:02,440 --> 00:05:04,590 the potential energy in this system. 104 00:05:04,590 --> 00:05:07,010 The center of gravity goes up and down a tiny bit. 105 00:05:07,010 --> 00:05:10,560 So is there any velocity in the vertical direction? 106 00:05:10,560 --> 00:05:12,270 Is there any kinetic energy associated 107 00:05:12,270 --> 00:05:13,710 with up and down motion? 108 00:05:13,710 --> 00:05:16,480 Yeah, so that doesn't entirely capture it, 109 00:05:16,480 --> 00:05:17,841 1/2 I omega squared. 110 00:05:17,841 --> 00:05:19,340 Is it an important amount of energy? 111 00:05:19,340 --> 00:05:23,530 I don't know, but there is some velocity up and down. 112 00:05:23,530 --> 00:05:26,170 My guess is that it actually isn't important, 113 00:05:26,170 --> 00:05:28,270 that the answer is it does move up and down. 114 00:05:28,270 --> 00:05:30,850 It has to, or you would not have any potential energy 115 00:05:30,850 --> 00:05:32,810 exchange in the system. 116 00:05:32,810 --> 00:05:35,030 OK. 117 00:05:35,030 --> 00:05:37,130 Is that it? 118 00:05:37,130 --> 00:05:39,312 OK. 119 00:05:39,312 --> 00:05:40,270 Let's keep moving here. 120 00:05:42,960 --> 00:05:44,710 Got a lot of fun things to show you today. 121 00:05:47,400 --> 00:05:50,862 So last time, we talked about response to initial conditions. 122 00:05:50,862 --> 00:05:52,820 I'm going to finish up with that and then go on 123 00:05:52,820 --> 00:05:57,360 to talking about excitation of harmonic forces. 124 00:05:57,360 --> 00:05:59,110 So last time, we were considering a system 125 00:05:59,110 --> 00:06:02,260 like this-- X is measured from the zero spring 126 00:06:02,260 --> 00:06:04,510 force in this case. 127 00:06:04,510 --> 00:06:07,050 Give you an equation of motion of that sort. 128 00:06:07,050 --> 00:06:12,320 And we've found that you could express x of T 129 00:06:12,320 --> 00:06:19,150 as a-- I'll give you the exact expression-- x0 square root 130 00:06:19,150 --> 00:06:20,893 of 1 minus zeta squared. 131 00:06:29,980 --> 00:06:38,050 Cosine omega damped times time minus a little phase angle. 132 00:06:38,050 --> 00:06:50,830 And there's a second term here, v0 over omega d sine omega dt. 133 00:06:50,830 --> 00:06:58,560 And the whole thing times e to the minus zeta omega n t. 134 00:06:58,560 --> 00:07:02,790 So that's our response to an initial deflection 135 00:07:02,790 --> 00:07:07,260 x0 or an initial velocity v0. 136 00:07:07,260 --> 00:07:11,000 That's the full kind of messy expression. 137 00:07:11,000 --> 00:07:13,645 There's another way of writing that, which I'll show you. 138 00:07:19,760 --> 00:07:34,060 Another way of saying is that it's in x0 cosine omega dt 139 00:07:34,060 --> 00:07:53,140 plus v0 plus zeta omega n x0, all over omega d sine 140 00:07:53,140 --> 00:07:58,110 omega d times t e to the minus zeta omega 141 00:07:58,110 --> 00:08:01,330 nt, the same exponent. 142 00:08:01,330 --> 00:08:03,260 This is your decaying exponential 143 00:08:03,260 --> 00:08:04,950 that makes it die out. 144 00:08:04,950 --> 00:08:07,020 And so I just rearranged some of these things. 145 00:08:07,020 --> 00:08:08,978 There's another little phase angle in here now. 146 00:08:08,978 --> 00:08:13,560 So you have just a cosine term, this proportional x0, 147 00:08:13,560 --> 00:08:18,070 and a sine term, which has both v0 and x0 in it. 148 00:08:18,070 --> 00:08:19,970 The x0 term, if damping is small, 149 00:08:19,970 --> 00:08:25,540 this term is pretty small because it's x0 times zeta. 150 00:08:25,540 --> 00:08:28,450 When you divide by omega d, which is almost omega n, 151 00:08:28,450 --> 00:08:30,130 that goes away. 152 00:08:30,130 --> 00:08:34,039 So this term, the scale of it is zeta x0. 153 00:08:34,039 --> 00:08:38,360 So if this is 1% or 2%, that's a very small number. 154 00:08:38,360 --> 00:08:40,190 I gave you an approximation, which 155 00:08:40,190 --> 00:08:48,870 for almost all practical examples that you 156 00:08:48,870 --> 00:08:55,650 might want to do, make it approximately sine here. 157 00:08:55,650 --> 00:09:14,080 So this is x0 cosine omega dt plus v0 over omega d sine omega 158 00:09:14,080 --> 00:09:21,340 dte to the minus zeta omega nt. 159 00:09:21,340 --> 00:09:23,230 And this is the practical one. 160 00:09:23,230 --> 00:09:27,160 For any reasonable system that has relatively low damping, 161 00:09:27,160 --> 00:09:31,940 even 10% or 15% damping, you get part of the transient decay 162 00:09:31,940 --> 00:09:38,930 comes from x0 cosine, the other part v0 over omega d sine. 163 00:09:38,930 --> 00:09:42,280 That's what I can remember in my head when I'm trying to do it. 164 00:09:42,280 --> 00:09:44,930 Now the question, the thing I want to address today 165 00:09:44,930 --> 00:09:48,140 is what's this useful for? 166 00:09:48,140 --> 00:09:50,410 My approach to teaching you vibration 167 00:09:50,410 --> 00:09:54,160 is I want you to go away with a few simple, 168 00:09:54,160 --> 00:09:57,010 practical understandings so that you can actually 169 00:09:57,010 --> 00:10:01,820 solve some vibration problems, and one of them 170 00:10:01,820 --> 00:10:05,412 is just knowing this allows you to do a couple things, 171 00:10:05,412 --> 00:10:07,370 and we'll do a couple of examples this morning. 172 00:10:11,120 --> 00:10:23,365 By the way, this form, this is A cosine plus B sine expression. 173 00:10:26,570 --> 00:10:36,770 And I label them A and B. A and B, 174 00:10:36,770 --> 00:10:44,830 they're both of the form A1 cosine omega 175 00:10:44,830 --> 00:10:50,090 t plus B1 sine omega t. 176 00:10:50,090 --> 00:10:53,020 And you can always add a sine and a cosine 177 00:10:53,020 --> 00:10:56,290 at the same frequency. 178 00:10:56,290 --> 00:10:57,670 If I put just any frequency here, 179 00:10:57,670 --> 00:10:59,620 they just have to be the same. 180 00:10:59,620 --> 00:11:02,570 You can always take an expression like that 181 00:11:02,570 --> 00:11:07,520 and rewrite it as some magnitude cosine omega 182 00:11:07,520 --> 00:11:11,090 t minus a phase angle. 183 00:11:11,090 --> 00:11:14,100 And the magnitude is just a square root 184 00:11:14,100 --> 00:11:21,060 of the sum of the squares-- A1 squared plus B1 squared. 185 00:11:21,060 --> 00:11:25,830 And the phase angle is the tangent inverse 186 00:11:25,830 --> 00:11:28,600 of the sine term over the cosine term. 187 00:11:28,600 --> 00:11:32,000 So you can always rewrite sine plus cosine 188 00:11:32,000 --> 00:11:36,080 as a cosine omega t minus v. We use that a lot, 189 00:11:36,080 --> 00:11:37,950 and that'll be used a lot in this course. 190 00:11:43,629 --> 00:11:45,420 And then, of course, if this whole thing is 191 00:11:45,420 --> 00:11:50,010 multiplied by an e to the minus 8 omega mt, then so is this. 192 00:11:55,010 --> 00:11:57,420 OK. 193 00:11:57,420 --> 00:11:59,340 OK, what are these things useful for? 194 00:12:03,480 --> 00:12:08,470 And we've derived this all for a mass spring system. 195 00:12:08,470 --> 00:12:12,810 Is that equation applicable to a pendulum? 196 00:12:12,810 --> 00:12:20,280 So this expression is applicable to any single degree of freedom 197 00:12:20,280 --> 00:12:23,430 system that oscillates. 198 00:12:23,430 --> 00:12:25,861 You just have to exchange a couple things. 199 00:12:25,861 --> 00:12:27,485 So let's think about a simple pendulum. 200 00:12:43,050 --> 00:12:49,770 So our common massless string and a bob on the end, 201 00:12:49,770 --> 00:12:58,215 some length L, equation of motion. 202 00:13:03,460 --> 00:13:05,810 And this is point A up here. 203 00:13:05,810 --> 00:13:20,930 IZZ with respect to A. Theta double dot plus MgL sine theta 204 00:13:20,930 --> 00:13:22,580 equals 0. 205 00:13:22,580 --> 00:13:23,830 That's the equation of motion. 206 00:13:23,830 --> 00:13:25,329 With no damping, that's the equation 207 00:13:25,329 --> 00:13:27,080 of motion in this system. 208 00:13:27,080 --> 00:13:30,060 Is it a linear differential equation? 209 00:13:33,500 --> 00:13:35,790 And to do the things that we want 210 00:13:35,790 --> 00:13:40,210 to be able to do in this course, like vibration 211 00:13:40,210 --> 00:13:43,240 with harmonic inputs and so forth, we 212 00:13:43,240 --> 00:13:46,040 want to deal with linear equations. 213 00:13:46,040 --> 00:13:48,940 So one of the topics for today is linearization. 214 00:13:48,940 --> 00:13:52,550 So this is one of the simplest examples of linearization. 215 00:13:52,550 --> 00:13:57,470 We need a linearized equation, and we 216 00:13:57,470 --> 00:14:01,790 need to remember in a couple of approximations. 217 00:14:01,790 --> 00:14:06,410 So sine of theta, you can do Taylor series expansion. 218 00:14:06,410 --> 00:14:13,780 It's theta minus theta cubed over 3 factorial plus theta 219 00:14:13,780 --> 00:14:19,220 to the fifth over 5 factorial plus minus and so forth. 220 00:14:19,220 --> 00:14:26,970 And cosine of theta is 1 minus theta 221 00:14:26,970 --> 00:14:32,030 squared over 2 factorial plus theta to the fourth over 4 222 00:14:32,030 --> 00:14:35,620 factorial, and so forth. 223 00:14:39,110 --> 00:14:42,480 So what we say, what do we mean when we linearize something? 224 00:14:42,480 --> 00:14:44,410 So linearization means that we're essentially 225 00:14:44,410 --> 00:14:51,940 assuming the variable that we're working with is small enough 226 00:14:51,940 --> 00:14:55,710 that the right hand side, an adequate approximation 227 00:14:55,710 --> 00:15:00,200 of this function, is to keep only up to the linear terms 228 00:15:00,200 --> 00:15:03,220 on the right hand side. 229 00:15:03,220 --> 00:15:08,580 For sine, the term raised to the 1 power, 230 00:15:08,580 --> 00:15:09,850 that's the linear term. 231 00:15:09,850 --> 00:15:12,720 This is a cubic term, a fifth order term. 232 00:15:12,720 --> 00:15:15,790 We're going to throw those away and say, this is close enough. 233 00:15:15,790 --> 00:15:18,320 For cosine, it's 1 minus theta squared. 234 00:15:18,320 --> 00:15:19,810 We throw these away. 235 00:15:19,810 --> 00:15:25,440 The small angle approximation for cosine as it's equal to 1. 236 00:15:25,440 --> 00:15:27,940 That's the simplest example of linearization, 237 00:15:27,940 --> 00:15:32,350 of a non-linear term. 238 00:15:32,350 --> 00:15:37,120 So when you linearize this equation of motion, 239 00:15:37,120 --> 00:15:47,150 we end up with IZZ with respect to A theta double dot plus MGL 240 00:15:47,150 --> 00:15:51,660 theta equals 0, and we get our familiar natural frequency 241 00:15:51,660 --> 00:15:54,820 for Bob as square root of g over L. 242 00:15:54,820 --> 00:15:57,745 So we need linearization to be able to do pendulum problems. 243 00:16:04,865 --> 00:16:05,365 Hmm. 244 00:16:08,790 --> 00:16:10,690 OK. 245 00:16:10,690 --> 00:16:12,530 Or maybe let's do an example here. 246 00:16:12,530 --> 00:16:15,060 I've got a pendulum that we'll do an experiment 247 00:16:15,060 --> 00:16:16,230 with this morning. 248 00:16:16,230 --> 00:16:20,820 But 30 degrees is about like that. 249 00:16:20,820 --> 00:16:23,220 17 degrees is about like that. 250 00:16:23,220 --> 00:16:25,380 That's quite a bit of angle. 251 00:16:25,380 --> 00:16:32,720 Is that small in the sense that I'm linearizing this equation? 252 00:16:32,720 --> 00:16:39,520 So 17 degrees happens to be-- I'll have to use this here. 253 00:16:47,918 --> 00:16:54,550 That's actually 17.2 degrees equals 0.3 radians. 254 00:16:58,750 --> 00:17:10,140 Sine of 0.3 is 0.2955. 255 00:17:10,140 --> 00:17:14,240 And if we fill out and look at these terms, 256 00:17:14,240 --> 00:17:22,220 the lead term here is 0.3-- so plug in the 0.3-- minus-- 257 00:17:22,220 --> 00:17:29,530 and the second term when you cube 0.3 and divide it by 6, 258 00:17:29,530 --> 00:17:40,750 the second term is minus-- what is my number here-- 0.0045. 259 00:17:40,750 --> 00:17:44,130 And if you subtract that from this, you get exactly this. 260 00:17:44,130 --> 00:17:48,110 So to four decimal places, you only 261 00:17:48,110 --> 00:17:51,210 need two terms in this series to get exactly the right answer. 262 00:17:51,210 --> 00:17:54,750 This thing out here, this fifth order term, is really tiny. 263 00:17:54,750 --> 00:17:58,180 But the approximation, if we say, OK, let's skip this, 264 00:17:58,180 --> 00:18:02,620 we're saying that 0.3 is approximately 0.2955. 265 00:18:02,620 --> 00:18:05,670 Pretty close. 266 00:18:05,670 --> 00:18:08,872 So up to 17 degrees, 0.3 radians, 267 00:18:08,872 --> 00:18:10,080 that's a great approximation. 268 00:18:14,060 --> 00:18:17,871 So it's a little high by about, I think it's about 1 and 1/2% 269 00:18:17,871 --> 00:18:18,370 high. 270 00:18:26,250 --> 00:18:30,070 So for pretty large angles for pendula, 271 00:18:30,070 --> 00:18:32,040 that simple linearization works just fine. 272 00:18:34,700 --> 00:18:40,370 OK, once we get it linearized, that equation of motion 273 00:18:40,370 --> 00:18:45,420 is of exactly the same form as the one up there. 274 00:18:45,420 --> 00:18:47,380 We don't have any damping in it. 275 00:18:47,380 --> 00:18:50,510 We could add some damping. 276 00:18:50,510 --> 00:18:54,510 We can put a damping in here with a torsional damper-- ct 277 00:18:54,510 --> 00:18:56,110 theta dot. 278 00:18:56,110 --> 00:18:59,570 And now that equation is of exactly the same form 279 00:18:59,570 --> 00:19:04,690 as the linear oscillator, linear meaning translational 280 00:19:04,690 --> 00:19:06,270 oscillator. 281 00:19:06,270 --> 00:19:09,480 Have that inertia term, a damping term, a stiffness term. 282 00:19:09,480 --> 00:19:12,700 It's a second order linear differential equation, 283 00:19:12,700 --> 00:19:14,499 homogeneous linear differential equation, 284 00:19:14,499 --> 00:19:15,790 nothing on the right hand side. 285 00:19:22,730 --> 00:19:25,400 Because they're exactly the same form, 286 00:19:25,400 --> 00:19:31,940 then the solution for decay, transient decay 287 00:19:31,940 --> 00:19:35,670 from initial conditions, takes on exactly the same form 288 00:19:35,670 --> 00:19:40,590 except that it has an initial angle, theta 0, 289 00:19:40,590 --> 00:19:42,430 and I use the approximation here. 290 00:19:42,430 --> 00:19:50,940 Cosine omega dt plus theta 0 dot, the initial velocity, 291 00:19:50,940 --> 00:20:01,070 over omega d sine omega dt all times e to the minus zeta omega 292 00:20:01,070 --> 00:20:02,960 nt. 293 00:20:02,960 --> 00:20:06,030 So that's the exact same transient decay equation, 294 00:20:06,030 --> 00:20:08,850 but now cast in angular terms. 295 00:20:15,310 --> 00:20:20,460 And if you wanted to express it as a cosine omega t minus v, 296 00:20:20,460 --> 00:20:24,080 then A would be this squared plus this squared square root 297 00:20:24,080 --> 00:20:30,570 and the phi would be a similar calculation as we 298 00:20:30,570 --> 00:20:32,690 have up there someone here. 299 00:20:35,330 --> 00:20:38,900 Just the B term over the A term, tangent numbers. 300 00:20:38,900 --> 00:20:41,790 All right. 301 00:20:41,790 --> 00:20:42,610 What's it good for? 302 00:20:42,610 --> 00:20:45,350 So I use this, this equation gets used quite a lot. 303 00:20:45,350 --> 00:20:46,695 It has some practical uses. 304 00:20:51,140 --> 00:20:57,000 Let's do an example, a little more complicated pendulum. 305 00:20:57,000 --> 00:20:57,840 Draw a stick maybe. 306 00:21:01,470 --> 00:21:02,580 Center of mass there. 307 00:21:07,295 --> 00:21:15,020 A, IZZ with respect to A. We'll call it 308 00:21:15,020 --> 00:21:18,155 ML cubed over 3 for a slender rod. 309 00:21:23,310 --> 00:21:26,340 And now, what I want to do is I have 310 00:21:26,340 --> 00:21:34,490 coming along here a mass, a bullet, that has mass m. 311 00:21:34,490 --> 00:21:39,940 Has velocity vi for initial here, 312 00:21:39,940 --> 00:21:46,990 and that's its linear momentum, p initial. 313 00:21:49,540 --> 00:21:52,532 And it's going to hit this stick and bed in it. 314 00:21:52,532 --> 00:21:53,990 So you've done this problem before. 315 00:21:53,990 --> 00:21:54,490 Yeah. 316 00:21:54,490 --> 00:21:56,720 AUDIENCE: [INAUDIBLE]. 317 00:21:56,720 --> 00:21:57,470 PROFESSOR: Pardon? 318 00:21:57,470 --> 00:21:59,730 AUDIENCE: ML cubed [INAUDIBLE]. 319 00:21:59,730 --> 00:22:01,450 PROFESSOR: ML cubed over 3. 320 00:22:05,850 --> 00:22:06,470 Good. 321 00:22:06,470 --> 00:22:08,620 I don't know why I was thinking cubed this morning. 322 00:22:08,620 --> 00:22:10,090 ML squared over 3. 323 00:22:10,090 --> 00:22:10,990 Good catch there. 324 00:22:10,990 --> 00:22:12,360 OK, so we have it. 325 00:22:12,360 --> 00:22:14,250 This is mass moment of inertia. 326 00:22:14,250 --> 00:22:15,764 This is a pendulum. 327 00:22:15,764 --> 00:22:17,180 This bullet's going to come along. 328 00:22:20,870 --> 00:22:22,550 So this is exactly what I've got here. 329 00:22:22,550 --> 00:22:24,100 I'll get it in the picture. 330 00:22:24,100 --> 00:22:25,240 Yeah. 331 00:22:25,240 --> 00:22:27,140 So it's initially at rest. 332 00:22:27,140 --> 00:22:29,690 Coming along, this paper, this clip here, 333 00:22:29,690 --> 00:22:31,005 it represents the bullet. 334 00:22:31,005 --> 00:22:33,530 So it's swimming along. 335 00:22:33,530 --> 00:22:35,800 Hits this thing, sticks to it, and when it hits it, 336 00:22:35,800 --> 00:22:37,870 it does that. 337 00:22:37,870 --> 00:22:40,900 So then this thing after it hits swings back and forth. 338 00:22:40,900 --> 00:22:43,420 So what's the response of this system 339 00:22:43,420 --> 00:22:44,915 to being hit by the bullet? 340 00:22:44,915 --> 00:22:49,330 Well, I claim you can do it entirely by evaluating response 341 00:22:49,330 --> 00:22:51,620 to initial conditions. 342 00:22:51,620 --> 00:22:55,625 But we need to use one conservation law to get there. 343 00:22:59,790 --> 00:23:03,260 So what's conserved on impact? 344 00:23:03,260 --> 00:23:05,640 Is linear momentum conserved on impact? 345 00:23:09,930 --> 00:23:12,580 How many think yes? 346 00:23:12,580 --> 00:23:14,270 Linear momentum conserved. 347 00:23:14,270 --> 00:23:16,330 How many think angular momentum's conserved? 348 00:23:16,330 --> 00:23:16,830 Hm. 349 00:23:16,830 --> 00:23:19,780 Good. you guys have learned something this year. 350 00:23:19,780 --> 00:23:20,600 That's great. 351 00:23:20,600 --> 00:23:22,255 Why is linear momentum not conserved? 352 00:23:27,560 --> 00:23:31,130 Because are there any possible other external forces 353 00:23:31,130 --> 00:23:33,320 on the system? 354 00:23:33,320 --> 00:23:34,270 At the pin joint. 355 00:23:34,270 --> 00:23:36,860 You can have reaction forces here and there. 356 00:23:36,860 --> 00:23:38,910 You have no control of them. 357 00:23:38,910 --> 00:23:41,350 But the moments about this point, 358 00:23:41,350 --> 00:23:44,640 are there any external moments about that point 359 00:23:44,640 --> 00:23:45,780 during the impact? 360 00:23:45,780 --> 00:23:46,280 No. 361 00:23:46,280 --> 00:23:48,405 They're reaction forces, but there's no moment arm. 362 00:23:48,405 --> 00:23:49,830 So there's no moments. 363 00:23:49,830 --> 00:23:52,780 So you can use conservation of angular momentum. 364 00:23:52,780 --> 00:23:56,620 So H1 I'll call it here with respect to A 365 00:23:56,620 --> 00:24:01,410 is just R cross Pi. 366 00:24:01,410 --> 00:24:07,030 And the R is the length in the I direction. 367 00:24:07,030 --> 00:24:10,730 P is in the j direction, so the momentum is in the k. 368 00:24:10,730 --> 00:24:17,970 So this should be mv initial times L, 369 00:24:17,970 --> 00:24:20,830 and its direction is in the k hat direction. 370 00:24:20,830 --> 00:24:23,960 So that's the initial angular momentum 371 00:24:23,960 --> 00:24:25,760 of the system with respect to this. 372 00:24:25,760 --> 00:24:27,520 This has no initial angular momentum 373 00:24:27,520 --> 00:24:29,550 because it's motionless. 374 00:24:29,550 --> 00:24:32,010 And since angular momentum is conserved, 375 00:24:32,010 --> 00:24:35,785 that H2 we'll call it with respect to A 376 00:24:35,785 --> 00:24:40,910 has got to be equal to H1 with respect to A, 377 00:24:40,910 --> 00:24:49,880 and that will then be IZZ with respect to A theta dot. 378 00:24:49,880 --> 00:24:54,040 But I need to account for the mass, this thing. 379 00:24:54,040 --> 00:24:58,215 So the total mass moment of inertia with respect to A 380 00:24:58,215 --> 00:25:01,780 is IZZ with respect to A plus M-- what? 381 00:25:06,580 --> 00:25:09,250 Now I've got the total mass moment of inertia with respect 382 00:25:09,250 --> 00:25:13,335 to this point, that of the stick plus that of the initial mass 383 00:25:13,335 --> 00:25:15,140 that I've stuck on there. 384 00:25:15,140 --> 00:25:18,380 And this must be equal to theta dot. 385 00:25:18,380 --> 00:25:19,980 And I put a not down here because I'm 386 00:25:19,980 --> 00:25:23,540 looking for my equivalent initial condition. 387 00:25:23,540 --> 00:25:31,810 And this then is mv initial times 388 00:25:31,810 --> 00:25:37,730 L. Then I can solve for theta dot 0, 389 00:25:37,730 --> 00:25:50,520 and that looks like mv initial L over IZZ A plus mL squared. 390 00:25:50,520 --> 00:25:55,081 And everything on the right hand side you know. 391 00:25:55,081 --> 00:25:57,330 You know the initial velocity, the mass of the bullet, 392 00:25:57,330 --> 00:26:01,470 the length of the distance from the pivot, mass moment 393 00:26:01,470 --> 00:26:05,240 of inertia, and the additional mass moment of inertia. 394 00:26:05,240 --> 00:26:06,892 These are all numbers you plug in, 395 00:26:06,892 --> 00:26:08,100 and you get a value for this. 396 00:26:08,100 --> 00:26:11,970 And once you have a value for this, you can use that. 397 00:26:14,750 --> 00:26:17,110 In this problem, what's theta 0? 398 00:26:21,320 --> 00:26:25,460 The initial angular deflection at time t0 399 00:26:25,460 --> 00:26:29,550 plus right after the bullets hit it. 400 00:26:29,550 --> 00:26:35,600 And it hasn't moved because it hasn't had time to move yet. 401 00:26:35,600 --> 00:26:37,710 At some velocity, it takes finite time 402 00:26:37,710 --> 00:26:39,500 to get a deflection. 403 00:26:39,500 --> 00:26:42,700 So there's zero initial angular deflection, 404 00:26:42,700 --> 00:26:48,020 but you get a step up in initial angular velocity. 405 00:26:48,020 --> 00:26:51,180 And so the response of this system 406 00:26:51,180 --> 00:27:05,210 is theta t is theta 0 dot over omega d sine omega dt. 407 00:27:05,210 --> 00:27:06,060 So what's omega d? 408 00:27:15,950 --> 00:27:18,930 Remember, I'll define a few things. 409 00:27:18,930 --> 00:27:25,880 In this case, this is ct over 2 IZZ 410 00:27:25,880 --> 00:27:32,675 z A plus little ml squared-- we have 411 00:27:32,675 --> 00:27:37,120 to deal with all the quantities after the collision-- 2 times 412 00:27:37,120 --> 00:27:38,690 omega n. 413 00:27:38,690 --> 00:27:47,080 That's the damping ratio for this torsional pendulum, 414 00:27:47,080 --> 00:27:48,170 with this pendulum. 415 00:27:48,170 --> 00:27:49,960 It's the damping constant. 416 00:27:49,960 --> 00:27:53,170 2 times the mass, the inertial quantity, 417 00:27:53,170 --> 00:27:57,300 times omega n for a translational system 418 00:27:57,300 --> 00:28:04,260 at c over 2 m omega n. 419 00:28:04,260 --> 00:28:07,840 For a pendulum system, it's the torsional 420 00:28:07,840 --> 00:28:13,040 damping over 2 times the mass moment of inertia times omega 421 00:28:13,040 --> 00:28:14,490 n. 422 00:28:14,490 --> 00:28:22,630 And omega n, well, it is going to calculate 423 00:28:22,630 --> 00:28:23,930 the natural frequency. 424 00:28:23,930 --> 00:28:27,730 It's just MgL divided by IZZ plus this. 425 00:28:27,730 --> 00:28:29,540 Maybe you ought to write that down. 426 00:28:35,790 --> 00:28:38,630 So always for a sample singular [INAUDIBLE] oscillator, 427 00:28:38,630 --> 00:28:40,340 you want the undamped natural frequency. 428 00:28:40,340 --> 00:28:42,050 Ignore the damping term. 429 00:28:42,050 --> 00:28:45,880 Take the stiffness term coefficient here and divide it 430 00:28:45,880 --> 00:28:47,570 by the inertial coefficient. 431 00:28:47,570 --> 00:28:51,510 But we care about the natural frequency after the impact, 432 00:28:51,510 --> 00:28:58,055 so this is going to be-- ah. 433 00:28:58,055 --> 00:29:01,740 The trouble is here I don't know for this system, 434 00:29:01,740 --> 00:29:03,910 I haven't worked out yet, what this term looks like. 435 00:29:03,910 --> 00:29:04,698 What is it? 436 00:29:07,710 --> 00:29:11,920 This result right here is for the simple Bob. 437 00:29:11,920 --> 00:29:18,880 For this stick, it's MgL over 2 plus the little m times l. 438 00:29:18,880 --> 00:29:20,690 Little more messy. 439 00:29:20,690 --> 00:29:27,060 So MgL over 2 plus little ml. 440 00:29:27,060 --> 00:29:30,880 That'll be the-- come from the potential energy in this system 441 00:29:30,880 --> 00:29:37,490 all over IZZ A plus mL squared. 442 00:29:37,490 --> 00:29:39,230 So you get your natural frequency out 443 00:29:39,230 --> 00:29:42,146 of that expression. 444 00:29:42,146 --> 00:29:42,645 OK. 445 00:29:49,450 --> 00:29:53,970 So you do this problem sometimes before when 446 00:29:53,970 --> 00:29:56,660 you do, say, somebody asks you how high does it swing. 447 00:29:56,660 --> 00:29:57,250 AND so forth. 448 00:29:57,250 --> 00:30:00,030 Well, you can do it by conservation 449 00:30:00,030 --> 00:30:02,050 of energy, et cetera. 450 00:30:02,050 --> 00:30:04,520 But now, you have actually exact expression 451 00:30:04,520 --> 00:30:10,130 for the time history of the thing after the impact, 452 00:30:10,130 --> 00:30:12,420 including the effects of damping. 453 00:30:12,420 --> 00:30:19,870 And if you were to draw the result of this function 454 00:30:19,870 --> 00:30:23,970 of theta as a function of time, this one 455 00:30:23,970 --> 00:30:30,020 starts with no initial displacement but a velocity 456 00:30:30,020 --> 00:30:30,685 and does this. 457 00:30:37,600 --> 00:30:39,830 And that's your exponential decay envelope, 458 00:30:39,830 --> 00:30:42,150 and this is time. 459 00:30:45,910 --> 00:30:47,970 Now, what-- yeah. 460 00:30:47,970 --> 00:30:48,470 Excuse me. 461 00:30:51,970 --> 00:30:52,970 AUDIENCE: [INAUDIBLE]. 462 00:30:52,970 --> 00:30:54,465 PROFESSOR: Why this one? 463 00:30:54,465 --> 00:30:57,860 AUDIENCE: Yes. [INAUDIBLE]. 464 00:30:57,860 --> 00:30:59,170 PROFESSOR: Excuse me. 465 00:30:59,170 --> 00:30:59,870 Forgot the g. 466 00:30:59,870 --> 00:31:02,996 I mean, it accounts for the restoring moment, 467 00:31:02,996 --> 00:31:04,870 the additional little bit of restoring moment 468 00:31:04,870 --> 00:31:08,040 that you get from having added the mass of this thing 469 00:31:08,040 --> 00:31:09,250 to it, right. 470 00:31:09,250 --> 00:31:12,810 So it has by itself MgL sine theta, 471 00:31:12,810 --> 00:31:13,970 and we linearize that too. 472 00:31:13,970 --> 00:31:18,650 So it's MgL theta, and those two terms would add together. 473 00:31:18,650 --> 00:31:22,380 So you just have a second term here 474 00:31:22,380 --> 00:31:27,030 that has MgL like behavior. 475 00:31:27,030 --> 00:31:31,470 How I most often personally make use of expressions 476 00:31:31,470 --> 00:31:34,560 like this, or the one for translation, 477 00:31:34,560 --> 00:31:39,140 is because experimentally, if I'm 478 00:31:39,140 --> 00:31:42,440 trying to predict the vibration behavior of a system, 479 00:31:42,440 --> 00:31:46,360 one of the things you want to know is the damping. 480 00:31:46,360 --> 00:31:48,790 And one of the simplest ways to measure damping 481 00:31:48,790 --> 00:31:53,460 is to give a system an initial deflection or initial velocity 482 00:31:53,460 --> 00:31:56,420 and measure its decay, and from its decay, 483 00:31:56,420 --> 00:31:58,240 calculate the damping. 484 00:31:58,240 --> 00:32:06,470 So the last time I gave you an expression for doing that, 485 00:32:06,470 --> 00:32:13,720 and that was a damping ratio 1 over 2 pi 486 00:32:13,720 --> 00:32:18,110 times the number of cycles that you count, that you watch it, 487 00:32:18,110 --> 00:32:27,750 times the natural log of x of t over x of t plus n 488 00:32:27,750 --> 00:32:31,620 periods of vibration. 489 00:32:31,620 --> 00:32:32,550 This has a name. 490 00:32:32,550 --> 00:32:35,940 It's called the logarithmic decrement, this thing. 491 00:32:35,940 --> 00:32:38,250 So if somebody says log decrement, 492 00:32:38,250 --> 00:32:40,990 that's where they're referring to this expression. 493 00:32:51,265 --> 00:32:54,340 A comment about this. 494 00:32:54,340 --> 00:33:07,380 In this expression, x of t must be zero means 495 00:33:07,380 --> 00:33:12,080 if your measurement-- we have x of t here, or theta of t-- 496 00:33:12,080 --> 00:33:14,100 they must be zero mean. 497 00:33:14,100 --> 00:33:16,670 There must be oscillations around zero 498 00:33:16,670 --> 00:33:19,460 or you have to have subtracted the mean to get it there 499 00:33:19,460 --> 00:33:21,730 because if this is displaced and is oscillating 500 00:33:21,730 --> 00:33:25,480 around some offset, then this calculates and will get really 501 00:33:25,480 --> 00:33:25,980 messed up. 502 00:33:25,980 --> 00:33:28,250 It's got an offset plus an offset here 503 00:33:28,250 --> 00:33:29,160 plus an offset there. 504 00:33:29,160 --> 00:33:31,500 It means it's totally meaningless. 505 00:33:31,500 --> 00:33:36,970 So you must remove the mean value from any time history 506 00:33:36,970 --> 00:33:39,360 that you go to do this. 507 00:33:39,360 --> 00:33:42,790 So there's an easier way the same expression-- and this is, 508 00:33:42,790 --> 00:33:45,130 in fact, the way I use this. 509 00:33:45,130 --> 00:33:48,030 A plot out like just your data acquisition 510 00:33:48,030 --> 00:33:50,420 grabs it, plots it for you. 511 00:33:50,420 --> 00:33:53,410 I take this value from here to here, 512 00:33:53,410 --> 00:33:57,510 and this is my peak to peak amplitude. 513 00:33:57,510 --> 00:34:01,640 And then I go out n cycles later and find the peak 514 00:34:01,640 --> 00:34:03,940 to peak amplitude. 515 00:34:03,940 --> 00:34:06,920 And so this is perfectly, this is just the same as 1 516 00:34:06,920 --> 00:34:15,530 over 2 pi n, but now you do natural log of x peak 517 00:34:15,530 --> 00:34:22,969 to peak t over x peak to peak at t plus n periods. 518 00:34:25,670 --> 00:34:27,844 And that now, peak to peak measurement, 519 00:34:27,844 --> 00:34:29,010 you totally ignore the mean. 520 00:34:29,010 --> 00:34:30,870 Doesn't matter where you are. 521 00:34:30,870 --> 00:34:35,010 You want the here to here, here to here, plug it in there, 522 00:34:35,010 --> 00:34:37,320 and you're done. 523 00:34:37,320 --> 00:34:49,219 OK, so let's-- I got 1, 2, 3, 4. 524 00:34:49,219 --> 00:35:01,060 Let's let n equal 4, and let's assume this expression here-- 525 00:35:01,060 --> 00:35:05,830 n is 1 over 2 pi times 4. 526 00:35:05,830 --> 00:35:10,090 And let's assume that in these four periods 527 00:35:10,090 --> 00:35:15,230 from-- that's 1 period, 2, 3, 4 getting out here 528 00:35:15,230 --> 00:35:21,710 to this fourth, four periods away, that this is one fifth 529 00:35:21,710 --> 00:35:23,610 the initial. 530 00:35:23,610 --> 00:35:25,945 So this would be the natural log of 5. 531 00:35:28,980 --> 00:35:32,290 So 1 over 2 pi times 4, natural log of 5, 532 00:35:32,290 --> 00:35:36,895 and you run the numbers, you get 0.064, 533 00:35:36,895 --> 00:35:42,575 or what we call 6.4% damping. 534 00:35:42,575 --> 00:35:43,450 That's how you do it. 535 00:35:43,450 --> 00:35:46,630 That's the way you do a calculation like that. 536 00:35:46,630 --> 00:35:52,150 Now, I gave you a quick rule of thumb for estimating damping, 537 00:35:52,150 --> 00:35:54,910 and this is what I-- I can't work. 538 00:35:54,910 --> 00:35:58,220 I don't do logs in my head, but I 539 00:35:58,220 --> 00:36:01,050 can do damping estimates without that 540 00:36:01,050 --> 00:36:06,400 because I know that zeta is also-- 541 00:36:06,400 --> 00:36:09,040 if I just plug in some numbers here 542 00:36:09,040 --> 00:36:13,680 and run them all in advance is 0.11 543 00:36:13,680 --> 00:36:16,695 divided by the number of cycles to decay 50%. 544 00:36:20,132 --> 00:36:21,590 So we're going to do an experiment. 545 00:36:26,990 --> 00:36:29,620 And I guess it can be seen with the camera. 546 00:36:29,620 --> 00:36:33,000 So here's my pendulum. 547 00:36:33,000 --> 00:36:38,030 This is my initial amplitude, and this is about half. 548 00:36:38,030 --> 00:36:45,190 So if I take this thing over here, like that, then let go, 549 00:36:45,190 --> 00:36:49,780 and count the cycles that it takes to decay halfway, 550 00:36:49,780 --> 00:36:50,889 we can do this experiment. 551 00:36:50,889 --> 00:36:51,930 So let's do it carefully. 552 00:36:51,930 --> 00:36:54,160 So line it up like that, and you're 553 00:36:54,160 --> 00:36:57,480 going to help me tell-- you count how many cycles it 554 00:36:57,480 --> 00:36:59,250 takes till it gets to here. 555 00:36:59,250 --> 00:37:15,330 So 1, 2, 3, 4, 5, 6, 7, 8. 556 00:37:15,330 --> 00:37:16,920 About eight cycles. 557 00:37:16,920 --> 00:37:20,170 So it decayed halfway in eight cycles. 558 00:37:20,170 --> 00:37:27,016 So zeta is approximately 0.11/8. 559 00:37:27,016 --> 00:37:30,240 It's 1 and 1/2%, 1.4%, something like that. 560 00:37:33,042 --> 00:37:34,500 Perfectly good estimate of damping. 561 00:37:34,500 --> 00:37:44,780 Now, the stopwatch here, we can do this experiment again. 562 00:37:47,660 --> 00:37:49,100 I want you to count. 563 00:37:49,100 --> 00:37:52,000 You're doing the counting. 564 00:37:52,000 --> 00:37:57,390 And I'm going to say start, and I want you to count cycles 565 00:37:57,390 --> 00:37:59,570 until I say stop. 566 00:37:59,570 --> 00:38:04,040 Now, I'll probably stop on 10 to make the calculation easy, 567 00:38:04,040 --> 00:38:08,090 so quietly to yourself count the number of cycles 568 00:38:08,090 --> 00:38:11,950 from the time I release it until the time I stop. 569 00:38:11,950 --> 00:38:13,060 Come back here. 570 00:38:19,110 --> 00:38:22,150 So this time, the backdrop doesn't matter. 571 00:38:22,150 --> 00:38:23,870 I just want you to count cycles. 572 00:38:27,200 --> 00:38:29,220 And I'll start-- I'll let it get going, 573 00:38:29,220 --> 00:38:30,730 and when it comes back to me is when 574 00:38:30,730 --> 00:38:33,705 I'm going to start the stopwatch because I 575 00:38:33,705 --> 00:38:35,580 have a hard time doing both at the same time. 576 00:38:35,580 --> 00:38:38,190 So start. 577 00:38:57,810 --> 00:38:59,320 How many cycles? 578 00:38:59,320 --> 00:39:01,320 AUDIENCE: [INAUDIBLE]. 579 00:39:01,320 --> 00:39:11,390 PROFESSOR: So I got 17.84 for 10. 580 00:39:11,390 --> 00:39:25,010 So 10 divided by 10 is 1.784 seconds per cycle. 581 00:39:25,010 --> 00:39:26,960 Can't write like that. 582 00:39:26,960 --> 00:39:29,450 1.784 seconds per cycle. 583 00:39:29,450 --> 00:39:33,159 The frequency would be 1 over that, right. 584 00:39:33,159 --> 00:39:34,950 The thing you have to be careful about when 585 00:39:34,950 --> 00:39:44,400 you're counting cycles is if I start here, that's 0, 1, 2. 586 00:39:44,400 --> 00:39:47,160 A very common human mistake is when you're counting something 587 00:39:47,160 --> 00:39:49,380 like this is to say one when you start, 588 00:39:49,380 --> 00:39:52,610 and then you're going to be off by one count. 589 00:39:52,610 --> 00:39:54,020 Follow me? 590 00:39:54,020 --> 00:39:59,320 If I start 0, 1, 2. 591 00:39:59,320 --> 00:40:01,960 So I start the clock on zero, but the first cycle 592 00:40:01,960 --> 00:40:04,030 isn't completed till one whole cycle later. 593 00:40:04,030 --> 00:40:06,985 So be careful how you count. 594 00:40:06,985 --> 00:40:07,485 OK. 595 00:40:23,670 --> 00:40:35,380 Now we're going to shift gears and take on a new topic, 596 00:40:35,380 --> 00:40:40,450 and that's the response to a harmonic input, 597 00:40:40,450 --> 00:40:43,823 some cosine omega t excitation. 598 00:40:43,823 --> 00:40:44,769 Yeah. 599 00:40:44,769 --> 00:40:46,065 AUDIENCE: [INAUDIBLE]. 600 00:40:46,065 --> 00:40:47,190 PROFESSOR: What is omega d? 601 00:41:02,340 --> 00:41:05,224 So it's the damped natural frequency. 602 00:41:05,224 --> 00:41:06,390 That's how it's referred to. 603 00:41:06,390 --> 00:41:09,560 It is the frequency you observe when you do 604 00:41:09,560 --> 00:41:11,782 an experiment like we just did. 605 00:41:11,782 --> 00:41:14,430 The actual oscillation frequency when 606 00:41:14,430 --> 00:41:16,760 it's responding to initial conditions 607 00:41:16,760 --> 00:41:18,930 is slightly different from the undamped, 608 00:41:18,930 --> 00:41:23,590 but if you have light damping, if you have even 10% damping, 609 00:41:23,590 --> 00:41:27,550 0.1 squared is 0.01. 610 00:41:27,550 --> 00:41:29,370 That's 0.99 square root. 611 00:41:29,370 --> 00:41:32,090 It's 0.995. 612 00:41:32,090 --> 00:41:33,660 So you're only off by half. 613 00:41:33,660 --> 00:41:35,600 They're only half a percent difference. 614 00:41:35,600 --> 00:41:38,450 So for lightly damped systems, for all intents and purposes, 615 00:41:38,450 --> 00:41:41,250 mega n and mega d are almost exactly the same. 616 00:41:43,920 --> 00:41:44,420 OK. 617 00:41:47,020 --> 00:41:53,890 We now want to think about-- we have a linear system putting 618 00:41:53,890 --> 00:41:56,360 a force into it. 619 00:41:56,360 --> 00:42:01,080 It looks like some F0 cosine omega t. 620 00:42:01,080 --> 00:42:05,430 And out of that system, we measure a response, x of t. 621 00:42:09,380 --> 00:42:14,300 And inside this box here is my system transfer function. 622 00:42:14,300 --> 00:42:17,545 It's the mathematics that tells me I can take F of t 623 00:42:17,545 --> 00:42:21,687 in and predict what x of t out is. 624 00:42:21,687 --> 00:42:23,270 So I need to know the information that 625 00:42:23,270 --> 00:42:25,720 goes into this box, and of course, the real system-- this 626 00:42:25,720 --> 00:42:27,280 is just the mechanical system. 627 00:42:27,280 --> 00:42:29,880 Force in, measured output out. 628 00:42:29,880 --> 00:42:33,660 This is what we call a single input single output 629 00:42:33,660 --> 00:42:37,820 system, SISO, single input single output linear system. 630 00:42:42,515 --> 00:42:44,140 And there's all sorts of linear systems 631 00:42:44,140 --> 00:42:46,265 that you're going to study as mechanical engineers, 632 00:42:46,265 --> 00:42:49,380 and you've already begun, I'm sure, studying some of them. 633 00:42:49,380 --> 00:42:51,720 One of the properties of a linear system 634 00:42:51,720 --> 00:42:57,220 is that you put a force in, F1, and measure a response out, x1. 635 00:42:57,220 --> 00:42:58,970 And then you try a different force, F2, 636 00:42:58,970 --> 00:43:01,540 and you measure a response out, x2. 637 00:43:01,540 --> 00:43:03,600 What's the response if you put them both in 638 00:43:03,600 --> 00:43:05,300 at the same time, F1 and F2? 639 00:43:09,550 --> 00:43:13,050 You just add the responses to them individually. 640 00:43:13,050 --> 00:43:15,360 So F1 gives you x1. 641 00:43:15,360 --> 00:43:16,690 F2 give you x2. 642 00:43:16,690 --> 00:43:20,010 F1 plus F2 gives you x1 plus x2, and that's 643 00:43:20,010 --> 00:43:25,020 one of the characteristics of a linear system. 644 00:43:25,020 --> 00:43:30,240 We use that concept to be able to separate the response. 645 00:43:30,240 --> 00:43:33,330 Our calculation's about the response of a system, 646 00:43:33,330 --> 00:43:37,100 like our oscillator here, separate its response 647 00:43:37,100 --> 00:43:42,070 to transient effects, transience being initial conditions. 648 00:43:42,070 --> 00:43:45,750 They die out over time-- that's why we call them transients-- 649 00:43:45,750 --> 00:43:48,690 and steady state effects. 650 00:43:48,690 --> 00:43:50,450 So cosine omega t, you can leave it 651 00:43:50,450 --> 00:43:52,950 running for a long, long time, and pretty soon, the system 652 00:43:52,950 --> 00:43:58,930 will settle down to responding just to that cosine omega t. 653 00:43:58,930 --> 00:44:00,900 And that we call steady state. 654 00:44:00,900 --> 00:44:02,800 And we use them separately. 655 00:44:02,800 --> 00:44:04,640 So we've done initial conditions. 656 00:44:04,640 --> 00:44:06,920 Now we're going to look at the steady state 657 00:44:06,920 --> 00:44:13,490 response of a-- say our oscillator, our mass spring 658 00:44:13,490 --> 00:44:17,920 dashpot, to a harmonic input, F0 cosine omega t. 659 00:44:20,850 --> 00:44:22,260 Another brief word. 660 00:44:22,260 --> 00:44:33,080 If I have a force, F0 cosine omega t would look like that. 661 00:44:35,810 --> 00:44:42,770 And the response that I measure to start off with my-- 662 00:44:42,770 --> 00:44:47,090 it's sitting here at zero when you turn this on. 663 00:44:47,090 --> 00:44:52,780 And it's going to do some odd things initially, and then 664 00:44:52,780 --> 00:45:02,850 eventually settle down to some long term steady response. 665 00:45:02,850 --> 00:45:04,900 The amplitude stays constant. 666 00:45:04,900 --> 00:45:08,360 It stays angle with respect to the input 667 00:45:08,360 --> 00:45:09,840 isn't necessarily the same. 668 00:45:09,840 --> 00:45:18,547 There's some possibly phase shift. 669 00:45:18,547 --> 00:45:20,880 And that's so the two, if you're plotting them together, 670 00:45:20,880 --> 00:45:21,671 they won't line up. 671 00:45:21,671 --> 00:45:24,090 But see this messy stuff at the beginning? 672 00:45:24,090 --> 00:45:27,600 When you first turn this on, it jumps from here 673 00:45:27,600 --> 00:45:31,740 to here, that force, and it gives it a kick to begin with. 674 00:45:31,740 --> 00:45:36,390 And this will have some response initially due 675 00:45:36,390 --> 00:45:38,750 to that transient start up. 676 00:45:38,750 --> 00:45:42,910 And this response is all modeled by the response 677 00:45:42,910 --> 00:45:44,770 to initial conditions. 678 00:45:44,770 --> 00:45:47,310 And it'll die out after a while, this messy stuff. 679 00:45:47,310 --> 00:45:49,280 What's the frequency? 680 00:45:49,280 --> 00:45:54,950 What frequency do you expect this initial, erratic looking 681 00:45:54,950 --> 00:45:55,817 stuff to be at? 682 00:45:59,400 --> 00:46:03,670 Its response to initial conditions. 683 00:46:03,670 --> 00:46:08,250 What is the model for a response to initial conditions? 684 00:46:08,250 --> 00:46:09,962 What's the frequency of the response 685 00:46:09,962 --> 00:46:12,170 to initial conditions of the single degree of freedom 686 00:46:12,170 --> 00:46:14,470 system? 687 00:46:14,470 --> 00:46:16,580 We have an equation over here, right? 688 00:46:20,300 --> 00:46:22,640 The top has a cosine term and a sine term. 689 00:46:22,640 --> 00:46:24,690 Part of it's a response to initial displacement. 690 00:46:24,690 --> 00:46:27,560 Part of it's a response to the initial velocity. 691 00:46:27,560 --> 00:46:31,880 Any of this start up stuff can be cast as initial conditions, 692 00:46:31,880 --> 00:46:33,620 and the response to initial conditions 693 00:46:33,620 --> 00:46:37,420 is always at the natural frequency period. 694 00:46:37,420 --> 00:46:41,040 No other frequencies for same degree of freedom systems. 695 00:46:41,040 --> 00:46:43,270 So you get a behavior that's oscillating 696 00:46:43,270 --> 00:46:45,010 at its natural frequency. 697 00:46:45,010 --> 00:46:49,640 Mixed in there is a response at the excitation frequency. 698 00:46:49,640 --> 00:46:52,090 And after a long time, the response 699 00:46:52,090 --> 00:46:54,180 is only excitation frequency. 700 00:46:54,180 --> 00:46:55,150 This is now out here. 701 00:46:55,150 --> 00:46:56,590 This is omega. 702 00:46:56,590 --> 00:47:04,280 In here, you have omega and omega d going on. 703 00:47:04,280 --> 00:47:06,430 So this is messy. 704 00:47:06,430 --> 00:47:08,330 Usually isn't important, but it is. 705 00:47:08,330 --> 00:47:11,690 There are ways of getting the exact solution, but mostly, 706 00:47:11,690 --> 00:47:15,910 vibration engineers, you're interested in the long term 707 00:47:15,910 --> 00:47:21,905 steady state response to what we call a harmonic input. 708 00:47:21,905 --> 00:47:22,405 OK. 709 00:47:40,510 --> 00:47:44,980 So we'll work a classic single degree 710 00:47:44,980 --> 00:47:52,360 of freedom oscillator problem-- excited by F0 cosine omega t. 711 00:47:52,360 --> 00:47:55,800 You've done this in 1803, but now we'll 712 00:47:55,800 --> 00:48:00,230 do it using engineering terminology. 713 00:48:00,230 --> 00:48:04,150 We'll look at it the way a person studying vibration 714 00:48:04,150 --> 00:48:06,160 would think about this. 715 00:48:06,160 --> 00:48:07,535 We know the equation of motion. 716 00:48:21,070 --> 00:48:24,640 And I'm interested in the steady state response. 717 00:48:24,640 --> 00:48:28,470 So this is x, and I'll do-- you just write it once like this. 718 00:48:28,470 --> 00:48:29,650 SS, steady state. 719 00:48:35,750 --> 00:48:38,520 I'm only interested in its-- after those transients 720 00:48:38,520 --> 00:48:39,400 have died out. 721 00:48:43,680 --> 00:48:46,130 And that steady state response I know 722 00:48:46,130 --> 00:48:51,100 is going to be some amplitude X0 cosine omega 723 00:48:51,100 --> 00:48:54,320 t minus some phase angle that I don't necessarily 724 00:48:54,320 --> 00:48:55,810 know to begin with. 725 00:48:55,810 --> 00:48:58,280 But that's my input. 726 00:48:58,280 --> 00:48:59,810 This is my output. 727 00:48:59,810 --> 00:49:04,820 I plug it into here and turn the crank and see what falls out. 728 00:49:11,510 --> 00:49:16,230 So you plug both of those in, and you 729 00:49:16,230 --> 00:49:20,480 get two-- you get-- this is going 730 00:49:20,480 --> 00:49:23,360 to be a little writing intensive for a few minutes. 731 00:49:38,220 --> 00:49:46,760 So you plug the X0 cosine omega t into all of these terms. 732 00:49:46,760 --> 00:49:50,070 The m term gives you minus m omega squared, 733 00:49:50,070 --> 00:49:55,920 the k term gives you a k, and the damping term, 734 00:49:55,920 --> 00:50:11,950 minus c omega sine omega t minus v. All of that 735 00:50:11,950 --> 00:50:18,620 equals the right hand side-- F0 cosine omega t. 736 00:50:18,620 --> 00:50:20,441 So this just purely from substitution 737 00:50:20,441 --> 00:50:22,065 and then gathering some terms together. 738 00:50:36,070 --> 00:50:40,450 I'm going to divide through by k, by k. 739 00:50:40,450 --> 00:50:42,347 If I divide through by k, k divided by-- this 740 00:50:42,347 --> 00:50:43,700 gives me a one. 741 00:50:43,700 --> 00:50:46,100 This gives me an m over k, which is 742 00:50:46,100 --> 00:50:49,425 1 over the natural frequency squared, for example. 743 00:50:49,425 --> 00:50:54,400 And I'm going to put this into a form that 744 00:50:54,400 --> 00:50:58,470 is the standard form for discussing vibration problems. 745 00:50:58,470 --> 00:51:04,770 So this equation can be rewritten in this form. 746 00:51:04,770 --> 00:51:15,320 1 minus omega squared over omega n squared cosine omega 747 00:51:15,320 --> 00:51:26,990 t minus v minus 2 zeta omega over omega n sine omega t 748 00:51:26,990 --> 00:51:34,740 minus v. All that's still equal to F0 cosine omega t. 749 00:51:34,740 --> 00:51:36,910 So this is getting into kind of more standard form. 750 00:51:36,910 --> 00:51:38,270 So there's 1 minus omega. 751 00:51:38,270 --> 00:51:40,100 This now, this omega over omega n, 752 00:51:40,100 --> 00:51:45,210 is called the frequency ratio, and you see a lot of that. 753 00:51:45,210 --> 00:51:48,670 And I've substituted n here. c omega over k 754 00:51:48,670 --> 00:51:53,230 turns out to be 2 zeta omega over omega n. 755 00:51:53,230 --> 00:51:55,640 So this frequency ratio appears a lot. 756 00:51:55,640 --> 00:52:02,000 in our-- let's see here. 757 00:52:10,220 --> 00:52:16,590 You need a couple of trig identities-- cosine omega t 758 00:52:16,590 --> 00:52:31,510 minus v. Cosine omega t cosine phi plus sine omega t sine phi, 759 00:52:31,510 --> 00:52:44,880 and sine omega t minus phi gives you [INAUDIBLE] sine. 760 00:52:44,880 --> 00:52:57,310 Sine omega t cosine phi minus cosine omega t sine phi. 761 00:52:57,310 --> 00:52:59,820 So that's a trig identity you actually use quite a bit 762 00:52:59,820 --> 00:53:01,990 doing vibration problems. 763 00:53:01,990 --> 00:53:07,290 We need them, so we take these, plug them 764 00:53:07,290 --> 00:53:11,980 in in all these places, and do quite a bit of cranking. 765 00:53:11,980 --> 00:53:13,190 Yep. 766 00:53:13,190 --> 00:53:15,690 AUDIENCE: [INAUDIBLE]. 767 00:53:15,690 --> 00:53:17,280 PROFESSOR: Yeah. 768 00:53:17,280 --> 00:53:19,850 Thank you. 769 00:53:19,850 --> 00:53:22,670 And that's called, that f over k, 770 00:53:22,670 --> 00:53:26,442 is how much the spring would move statically, 771 00:53:26,442 --> 00:53:28,750 at which the point would move statically. 772 00:53:28,750 --> 00:53:31,570 We'll need that term also. 773 00:53:31,570 --> 00:53:32,850 OK. 774 00:53:32,850 --> 00:53:36,120 You do all of this. 775 00:53:36,120 --> 00:53:37,375 Here, I'll call these-- 776 00:53:59,678 --> 00:54:00,680 OK. 777 00:54:00,680 --> 00:54:12,188 So this is C. That's expression C. 778 00:54:12,188 --> 00:54:13,470 I can't see that probably. 779 00:54:13,470 --> 00:54:24,960 Call this D, this E. So you plug D and E into C 780 00:54:24,960 --> 00:54:27,105 and work it through, you get two equations. 781 00:55:03,650 --> 00:55:05,750 You break it into two parts because one 782 00:55:05,750 --> 00:55:08,350 is a function of cosine omega t, and then 783 00:55:08,350 --> 00:55:10,890 you have another part after this substitution that's 784 00:55:10,890 --> 00:55:14,880 a function of sine omega t, and you can separate them. 785 00:55:38,230 --> 00:55:40,120 But there's no sine omega t force. 786 00:55:40,120 --> 00:55:41,710 On the right hand side, you get zero. 787 00:55:41,710 --> 00:55:43,410 There are two equations here. 788 00:55:43,410 --> 00:55:47,800 How many unknowns do we have? 789 00:55:47,800 --> 00:55:55,110 All we know when we start this thing is the input, 790 00:55:55,110 --> 00:56:04,700 and we have unknown response amplitude, 791 00:56:04,700 --> 00:56:07,850 and we have an unknown phase that we're looking for. 792 00:56:07,850 --> 00:56:08,730 How many equations? 793 00:56:08,730 --> 00:56:09,500 How many unknowns? 794 00:56:09,500 --> 00:56:11,440 Two and two. 795 00:56:11,440 --> 00:56:13,170 So you can do a lot of cranking, which 796 00:56:13,170 --> 00:56:14,600 I have no intention of doing here, 797 00:56:14,600 --> 00:56:20,885 and solve for the amplitude of the response and the phase. 798 00:56:40,690 --> 00:56:45,480 And every textbook-- the Williams textbook does this. 799 00:56:45,480 --> 00:56:50,920 There are two readings posted on Stellar 800 00:56:50,920 --> 00:56:54,790 by [? Row. ?] Every textbook goes through these derivations 801 00:56:54,790 --> 00:56:56,090 that I've just done. 802 00:56:56,090 --> 00:56:57,443 Nick, you've got a question. 803 00:56:57,443 --> 00:56:58,359 AUDIENCE: [INAUDIBLE]. 804 00:57:00,900 --> 00:57:01,740 PROFESSOR: Pardon? 805 00:57:01,740 --> 00:57:03,370 AUDIENCE: [INAUDIBLE]. 806 00:57:03,370 --> 00:57:06,215 PROFESSOR: Yeah, I keep forgetting it. 807 00:57:06,215 --> 00:57:07,050 You're right. 808 00:57:07,050 --> 00:57:08,650 So we got a k here. 809 00:57:08,650 --> 00:57:13,480 And notice, this equation, we throw away that for now. 810 00:57:13,480 --> 00:57:15,270 We get rid of this for now. 811 00:57:15,270 --> 00:57:17,420 We have these two equations and two unknowns 812 00:57:17,420 --> 00:57:22,400 are just algebraic equations There's not time dependent. 813 00:57:22,400 --> 00:57:24,210 We can get rid of that part. 814 00:57:24,210 --> 00:57:26,940 So we've now reduced this to algebra, 815 00:57:26,940 --> 00:57:31,330 and the answer is plotted up there. 816 00:57:31,330 --> 00:57:33,290 You've probably seen it before. 817 00:57:33,290 --> 00:57:40,780 It says that x0 is F0 over k-- I can get it right this time 818 00:57:40,780 --> 00:57:45,030 from the get go-- over a denominator, which 819 00:57:45,030 --> 00:57:49,420 appears again and again and again in vibration. 820 00:57:49,420 --> 00:57:54,530 Omega squared over omega n squared 821 00:57:54,530 --> 00:58:02,570 squared plus 2 zeta omega over omega n 822 00:58:02,570 --> 00:58:11,161 squared square root, the whole thing, and an expression 823 00:58:11,161 --> 00:58:11,660 for phi. 824 00:58:16,050 --> 00:58:23,630 Tangent inverse of 2 zeta omega over omega n, 1 825 00:58:23,630 --> 00:58:30,000 minus omega squared over omega n squared. 826 00:58:30,000 --> 00:58:32,310 So you can solve all that-- this mess over here 827 00:58:32,310 --> 00:58:34,490 for these two quantities. 828 00:58:34,490 --> 00:58:36,690 Do you need to remember this? 829 00:58:36,690 --> 00:58:39,770 You ever going to be asked this on a quiz? 830 00:58:39,770 --> 00:58:42,550 Not by me. 831 00:58:42,550 --> 00:58:46,320 You ever going to have to use this on a quiz and in homework? 832 00:58:46,320 --> 00:58:48,820 Absolutely. 833 00:58:48,820 --> 00:58:52,470 So the takeaway is today know how 834 00:58:52,470 --> 00:58:56,200 to use those response to initial condition formulas 835 00:58:56,200 --> 00:58:59,950 and damping and these two. 836 00:58:59,950 --> 00:59:04,560 So when you plot, when you plot these, 837 00:59:04,560 --> 00:59:07,320 you get this picture up there. 838 00:59:07,320 --> 00:59:09,700 And we need to talk about the properties of this. 839 00:59:13,380 --> 00:59:15,125 Remember, this omega over omega n 840 00:59:15,125 --> 00:59:16,940 is the same called the frequency ratio. 841 00:59:16,940 --> 00:59:19,930 It's just the ratio of the excitation frequency 842 00:59:19,930 --> 00:59:22,790 to the natural frequency of the system. 843 00:59:22,790 --> 00:59:27,090 And when they're equal, for example, this ratio is one. 844 00:59:27,090 --> 00:59:30,820 This whole thing in parentheses goes to zero. 845 00:59:30,820 --> 00:59:36,090 This expression over here goes to 2 zeta, because that's one. 846 00:59:36,090 --> 00:59:39,210 2 zeta squared square root is just 2 zeta. 847 00:59:39,210 --> 00:59:41,440 When omega equals omega n, this whole expression 848 00:59:41,440 --> 00:59:45,400 is F0 over k divided by 2 zeta, for example. 849 00:59:45,400 --> 00:59:47,570 And that's called resonance, and that's 850 00:59:47,570 --> 00:59:50,795 when you're right at where that peak goes to its maximum. 851 00:59:55,840 --> 00:59:59,290 Let's talk about this expression for a moment. 852 00:59:59,290 --> 01:00:02,790 If we have our cart, our mass-spring 853 01:00:02,790 --> 01:00:04,930 dashpot we started here. 854 01:00:04,930 --> 01:00:06,860 If you apply a force, a static force 855 01:00:06,860 --> 01:00:14,640 F0 and stretch the spring by an amount F0 over k. 856 01:00:14,640 --> 01:00:21,350 So x-- what we'll call x static is just F0 over k. 857 01:00:25,390 --> 01:00:46,710 And if I want to plot, I want to-- this has a name. 858 01:00:46,710 --> 01:00:49,210 This is called, this ratio here, this gives you 859 01:00:49,210 --> 01:00:52,230 the magnitude of the response. 860 01:00:52,230 --> 01:00:53,690 It goes by a variety of names. 861 01:00:53,690 --> 01:00:56,890 Some people call it a transfer function. 862 01:00:56,890 --> 01:01:00,190 Some people call it a frequency response function. 863 01:01:04,230 --> 01:01:05,930 I write it intentionally this way. 864 01:01:05,930 --> 01:01:12,310 This is I put output over input because this expression has 865 01:01:12,310 --> 01:01:15,170 units of output over input. 866 01:01:15,170 --> 01:01:18,440 So I just write it like this, remind myself 867 01:01:18,440 --> 01:01:20,560 what this transfer function is about. 868 01:01:20,560 --> 01:01:24,850 The input is force, the output is displacement. 869 01:01:24,850 --> 01:01:27,510 This expression has units of force, 870 01:01:27,510 --> 01:01:29,024 force per unit displacement. 871 01:01:38,710 --> 01:01:46,900 If I go to here, if I try to plot this-- let me start over. 872 01:01:46,900 --> 01:01:49,340 If I try to plot this, it's going 873 01:01:49,340 --> 01:01:52,960 to be depending on the exact value of the spring constant 874 01:01:52,960 --> 01:01:55,960 and the exact value of the force every time. 875 01:01:55,960 --> 01:01:59,840 I have to get a unique plot every time I go to do this. 876 01:01:59,840 --> 01:02:02,750 So textbooks and engineers, I don't want 877 01:02:02,750 --> 01:02:04,960 to have to remember this part. 878 01:02:04,960 --> 01:02:10,140 This is where all of the content is in is in this denominator, 879 01:02:10,140 --> 01:02:11,320 and it's dimensionless. 880 01:02:11,320 --> 01:02:17,000 So what I'd really like to plot is x0 over x static. 881 01:02:19,620 --> 01:02:27,040 And if I do that, that is x0 over the quantity F0 over k. 882 01:02:27,040 --> 01:02:30,090 If I just divide-- this is x static-- it would bring this 883 01:02:30,090 --> 01:02:35,800 to this side, then this expression, this is just 1 884 01:02:35,800 --> 01:02:40,320 over that denominator. 885 01:02:40,320 --> 01:02:43,810 And sometimes, I think in the handout by [? Row ?], 886 01:02:43,810 --> 01:02:46,155 they just call this h of omega. 887 01:02:48,700 --> 01:02:52,810 It's dimensionless, frequency over frequency, 888 01:02:52,810 --> 01:02:56,690 and that's actually what's plotted up there. 889 01:02:56,690 --> 01:03:01,950 And this is called-- has different names also. 890 01:03:01,950 --> 01:03:06,670 Magnification factor, dynamic amplification factor, 891 01:03:06,670 --> 01:03:10,360 because the ratio of x to x static 892 01:03:10,360 --> 01:03:20,760 if this is the dynamic effects magnify the response compared 893 01:03:20,760 --> 01:03:21,780 to the static response. 894 01:03:21,780 --> 01:03:24,370 So it might be this over this might be 10. 895 01:03:24,370 --> 01:03:30,020 I mean, the dynamic response is 10 times the static response. 896 01:03:30,020 --> 01:03:32,140 OK, how do you-- to sum this up-- 897 01:03:32,140 --> 01:03:37,380 and we'll be kind of getting close to the end. 898 01:03:37,380 --> 01:03:39,770 We want to talk just about the properties of this. 899 01:03:39,770 --> 01:03:40,940 How do we use this? 900 01:03:52,830 --> 01:03:57,730 So in practical use, you have an input specified, 901 01:03:57,730 --> 01:04:00,260 some force cosine omega t. 902 01:04:00,260 --> 01:04:02,900 You know you have a single degree of freedom oscillator 903 01:04:02,900 --> 01:04:09,160 that is governed by equations like that one, 904 01:04:09,160 --> 01:04:10,940 and you want to predict the response. 905 01:04:10,940 --> 01:04:18,450 Well, you say x of t is equal to the magnitude 906 01:04:18,450 --> 01:04:28,437 of the force times the-- and you divide that by-- we 907 01:04:28,437 --> 01:04:29,311 could do it this way. 908 01:04:34,230 --> 01:04:36,820 The magnitude of the force divided by k, 909 01:04:36,820 --> 01:04:38,180 which is the static response. 910 01:04:40,710 --> 01:04:43,720 To predict x0, we just have to predict this quantity, 911 01:04:43,720 --> 01:04:46,490 multiply it by F0 over k. 912 01:04:46,490 --> 01:04:48,180 So you know this. 913 01:04:48,180 --> 01:04:50,110 You better know that about your system, 914 01:04:50,110 --> 01:04:55,530 and you multiply it by this quantity magnitude 915 01:04:55,530 --> 01:04:57,260 of h of omega. 916 01:04:57,260 --> 01:05:05,400 And the time dependent part is times cosine omega t 917 01:05:05,400 --> 01:05:07,110 minus the phase angle. 918 01:05:07,110 --> 01:05:10,610 And the phase angle, you get either off the plot 919 01:05:10,610 --> 01:05:15,214 or from the-- have I written it down? 920 01:05:20,540 --> 01:05:22,573 I haven't written the phase angle down yet. 921 01:05:34,180 --> 01:05:35,770 It's kind of a messy expression too. 922 01:05:35,770 --> 01:05:39,200 That's why we plot it. 923 01:05:39,200 --> 01:05:45,330 2 zeta omega over omega n over 1 minus omega squared 924 01:05:45,330 --> 01:05:48,210 over omega n squared. 925 01:05:48,210 --> 01:05:52,150 But by knowing just this plot, what you just 926 01:05:52,150 --> 01:05:54,500 put in every textbook about vibration in the world, 927 01:05:54,500 --> 01:05:57,770 by knowing this magnification factor, 928 01:05:57,770 --> 01:06:01,960 calculating the static response, multiplying the two together, 929 01:06:01,960 --> 01:06:04,820 you have the amplitude of the response, 930 01:06:04,820 --> 01:06:08,490 and its time dependence is cosine omega t minus the phase 931 01:06:08,490 --> 01:06:09,880 angle. 932 01:06:09,880 --> 01:06:11,785 And I've got a little example here. 933 01:06:21,460 --> 01:06:23,110 Actually, rather than the example, 934 01:06:23,110 --> 01:06:25,550 I've gone to all the trouble of setting this up. 935 01:06:43,110 --> 01:06:44,460 All right. 936 01:06:44,460 --> 01:06:48,210 This is just a beam. 937 01:06:48,210 --> 01:06:49,860 Where's my other little beam? 938 01:06:56,910 --> 01:06:59,225 And a beam is just a spring. 939 01:07:02,420 --> 01:07:04,230 Put a mass on the and. 940 01:07:04,230 --> 01:07:07,720 This is basically a single degree of freedom system. 941 01:07:07,720 --> 01:07:09,500 It has a natural frequency. 942 01:07:09,500 --> 01:07:13,130 The beam has a certain stiffness. 943 01:07:13,130 --> 01:07:16,520 And now, in this case, we're interested in response 944 01:07:16,520 --> 01:07:18,750 to some harmonic input. 945 01:07:18,750 --> 01:07:23,230 So any of you know what a squiggle pen is. 946 01:07:23,230 --> 01:07:25,045 This is a kid's toy. 947 01:07:25,045 --> 01:07:26,035 AUDIENCE: Excuse me. 948 01:07:26,035 --> 01:07:29,005 Can you move the camera a tad to the left so the [INAUDIBLE]? 949 01:07:32,794 --> 01:07:33,460 PROFESSOR: Yeah. 950 01:07:36,450 --> 01:07:38,920 So all throughout the term, we've 951 01:07:38,920 --> 01:07:41,920 studied rotating masses quite a bit, right. 952 01:07:41,920 --> 01:07:44,730 This thing has a rotating mass that you can see in the end. 953 01:07:44,730 --> 01:07:46,420 I mean, when you leave it, you can come down and take a look. 954 01:07:46,420 --> 01:07:47,544 It has a low rotating mass. 955 01:07:47,544 --> 01:07:50,030 It's actually a pen, but it's a kid's toy. 956 01:07:50,030 --> 01:07:50,980 Shakes like crazy. 957 01:07:55,860 --> 01:08:00,599 And now we need the lights down. 958 01:08:06,830 --> 01:08:14,550 And it happens that the-- I've got a strobe light here, 959 01:08:14,550 --> 01:08:17,149 and I've kind of preset the frequency 960 01:08:17,149 --> 01:08:21,870 so it's very close to the frequency of vibration 961 01:08:21,870 --> 01:08:23,090 of this beam. 962 01:08:23,090 --> 01:08:28,939 So there's a rotating mass in this pen going round and round, 963 01:08:28,939 --> 01:08:30,810 and it puts a force into the system 964 01:08:30,810 --> 01:08:36,000 that looks like F0 cosine omega t in the vertical direction. 965 01:08:36,000 --> 01:08:38,000 Also does it in the horizontal, but vertical 966 01:08:38,000 --> 01:08:40,899 is our response direction. 967 01:08:40,899 --> 01:08:43,129 So it's putting in a force, and I 968 01:08:43,129 --> 01:08:48,220 have set the length of this beam so that the natural frequency 969 01:08:48,220 --> 01:08:50,180 of this beam with this mass on the end 970 01:08:50,180 --> 01:08:54,090 is exactly very close to being the frequency 971 01:08:54,090 --> 01:08:54,840 of the excitation. 972 01:08:57,540 --> 01:09:01,430 And the flash rate is slightly different than the vibration 973 01:09:01,430 --> 01:09:04,542 rate, so you see it illuminated at many positions 974 01:09:04,542 --> 01:09:05,750 as it goes through the cycle. 975 01:09:05,750 --> 01:09:09,720 So you see it going up and down. 976 01:09:09,720 --> 01:09:13,090 So if I mismatch it quite a bit, then you see it going. 977 01:09:13,090 --> 01:09:15,200 And actually, if you look at the very right end, 978 01:09:15,200 --> 01:09:18,446 you can see a white thing going up and down. 979 01:09:18,446 --> 01:09:20,029 That's the mass where you can actually 980 01:09:20,029 --> 01:09:24,950 see the mass in the very end of the-- right there. 981 01:09:24,950 --> 01:09:28,220 You can see something going around and round. 982 01:09:28,220 --> 01:09:31,529 There, that's the rotating mass. 983 01:09:31,529 --> 01:09:33,580 So the beam is going up and down, 984 01:09:33,580 --> 01:09:37,710 and I've got this, the vibration frequency of that mass going 985 01:09:37,710 --> 01:09:40,771 round and round equal to the natural frequency of the beam 986 01:09:40,771 --> 01:09:41,729 of the mass in the end. 987 01:09:47,920 --> 01:09:56,480 And it's moving quite a bit, and I'll loosen my clamp, 988 01:09:56,480 --> 01:09:59,300 and I'm going to change the length of the beam. 989 01:09:59,300 --> 01:10:00,399 I've shortened it. 990 01:10:08,890 --> 01:10:11,540 And now it's still moving up and down 991 01:10:11,540 --> 01:10:17,160 but not as much because the frequency of the rotation 992 01:10:17,160 --> 01:10:19,250 of the eccentric mass is no longer 993 01:10:19,250 --> 01:10:21,260 close to the natural frequency of the system. 994 01:10:24,210 --> 01:10:27,300 In fact, I've made the natural frequency 995 01:10:27,300 --> 01:10:34,510 of the system-- you can bring the lights back up-- 996 01:10:34,510 --> 01:10:37,030 I've made the natural frequency of the system. 997 01:10:37,030 --> 01:10:40,310 By making the beam shorter, I've made it stiffer. 998 01:10:40,310 --> 01:10:43,235 So the natural frequency has gone up. 999 01:10:43,235 --> 01:10:46,290 The frequency of the rotation of the eccentric mass 1000 01:10:46,290 --> 01:10:47,920 has stayed about the same. 1001 01:10:47,920 --> 01:10:49,530 So what's happened to that frequency 1002 01:10:49,530 --> 01:10:52,970 ratio, omega over omega n? 1003 01:10:52,970 --> 01:10:54,620 So less than one or greater than one. 1004 01:10:57,230 --> 01:10:59,630 So the omega n has gone up. 1005 01:10:59,630 --> 01:11:00,585 Omega stayed the same. 1006 01:11:03,690 --> 01:11:06,740 The frequency ratio when you shorten this beam 1007 01:11:06,740 --> 01:11:14,570 is less than one, and the properties of this transfer 1008 01:11:14,570 --> 01:11:17,400 function, we call it-- this magnification factor 1009 01:11:17,400 --> 01:11:19,050 looks like this. 1010 01:11:19,050 --> 01:11:22,900 When we're exciting it right at one-- this 1011 01:11:22,900 --> 01:11:27,580 is omega over omega n-- you write it resonance. 1012 01:11:27,580 --> 01:11:31,800 When you excite it at a frequency ratio less than one, 1013 01:11:31,800 --> 01:11:34,750 you start dropping off this backside, 1014 01:11:34,750 --> 01:11:38,700 and the response goes down. 1015 01:11:38,700 --> 01:11:43,370 And if you excite it at frequencies much greater 1016 01:11:43,370 --> 01:11:46,420 than the natural frequency, you end up way out here. 1017 01:11:46,420 --> 01:11:47,350 I can do that too. 1018 01:11:59,820 --> 01:12:05,390 So how much you think it will vibrate now? 1019 01:12:05,390 --> 01:12:07,690 A lot? 1020 01:12:07,690 --> 01:12:08,456 A little? 1021 01:12:16,560 --> 01:12:17,289 Hardly-- oops. 1022 01:12:17,289 --> 01:12:19,330 Oh, I've brought it out so much you can't see it. 1023 01:12:24,780 --> 01:12:32,450 Hardly moving at all, and that's because in terms 1024 01:12:32,450 --> 01:12:36,640 of this terminology of magnification factors, transfer 1025 01:12:36,640 --> 01:12:41,720 functions, this is a plot of x over x static. 1026 01:12:41,720 --> 01:12:43,530 It goes right here. 1027 01:12:43,530 --> 01:12:46,300 When you go to zero frequency, you are at static, 1028 01:12:46,300 --> 01:12:51,052 so the response at very, very low frequency 1029 01:12:51,052 --> 01:12:53,010 goes to being the same as the static frequency. 1030 01:12:53,010 --> 01:12:55,530 So in this plot, it goes to one. 1031 01:12:55,530 --> 01:12:59,650 At resonance, you put in one here. 1032 01:12:59,650 --> 01:13:00,910 This goes to zero. 1033 01:13:00,910 --> 01:13:04,060 That becomes a one 2 zeta squared. 1034 01:13:04,060 --> 01:13:12,350 This height here is 1 over 2 zeta. 1035 01:13:12,350 --> 01:13:14,770 So the dynamic amplification at resonance 1036 01:13:14,770 --> 01:13:16,665 is just 1 over 2 times the damping ratio. 1037 01:13:16,665 --> 01:13:19,280 You have 1% damping. 1038 01:13:19,280 --> 01:13:21,322 Twice that is o2. 1039 01:13:21,322 --> 01:13:24,720 1 over o2 is 50. 1040 01:13:24,720 --> 01:13:29,100 So if you only had 1% damping, the dynamic amplification, 1041 01:13:29,100 --> 01:13:31,600 the amount that this vibrates greater 1042 01:13:31,600 --> 01:13:35,190 than its static response, is a factor of 50 greater. 1043 01:13:35,190 --> 01:13:39,420 But then as you go higher in this omega over omega n, 1044 01:13:39,420 --> 01:13:42,460 and you get way out here, and you get almost no vibration 1045 01:13:42,460 --> 01:13:43,530 at all. 1046 01:13:43,530 --> 01:13:46,296 And that's what's happened when I've lengthened this. 1047 01:13:50,180 --> 01:13:50,680 OK. 1048 01:13:55,930 --> 01:14:02,090 So there's your introduction to linear systems. 1049 01:14:02,090 --> 01:14:05,090 In this case, a single degree of freedom system 1050 01:14:05,090 --> 01:14:10,570 that vibrates an oscillator, we're 1051 01:14:10,570 --> 01:14:15,230 talking about steady state response, not the part 1052 01:14:15,230 --> 01:14:18,440 of the solution, the mathematical solution, 1053 01:14:18,440 --> 01:14:20,850 to the initial conditions. 1054 01:14:20,850 --> 01:14:24,740 So all of this has been about steady state response 1055 01:14:24,740 --> 01:14:27,240 of our simple oscillator to what we call 1056 01:14:27,240 --> 01:14:32,880 a harmonic input, F0 cosine omega t. 1057 01:14:32,880 --> 01:14:34,720 So what's important that you need 1058 01:14:34,720 --> 01:14:36,990 to remember and be able to use? 1059 01:14:39,560 --> 01:14:43,520 This concept here, this idea of a transfer function. 1060 01:14:43,520 --> 01:14:44,720 That's really important. 1061 01:14:44,720 --> 01:14:46,570 You might want to remember it though 1062 01:14:46,570 --> 01:14:51,960 as this dimensionless quantity x over x static. 1063 01:14:51,960 --> 01:14:55,010 Just remember the shape of this transfer function. 1064 01:14:55,010 --> 01:14:59,170 Magnitude of the amplification, 1 over 2 zeta at resonance. 1065 01:14:59,170 --> 01:15:03,380 And it goes to one at low frequency. 1066 01:15:03,380 --> 01:15:06,770 The high frequency, it drops away off. 1067 01:15:06,770 --> 01:15:10,540 Next time, we're going to pick up 1068 01:15:10,540 --> 01:15:14,120 the topic of what we call vibration isolation. 1069 01:15:14,120 --> 01:15:16,540 The practical thing to know as engineers 1070 01:15:16,540 --> 01:15:20,784 is when you have a significant vibration 1071 01:15:20,784 --> 01:15:22,200 problem, like in a lab, and you're 1072 01:15:22,200 --> 01:15:24,750 looking through your microscope, and the floor vibration is 1073 01:15:24,750 --> 01:15:28,510 causing trouble with your microscope, 1074 01:15:28,510 --> 01:15:32,820 and you can't move the subway, what can you 1075 01:15:32,820 --> 01:15:35,989 do to solve that problem? 1076 01:15:35,989 --> 01:15:37,530 Well, you might be able-- what if you 1077 01:15:37,530 --> 01:15:40,620 put some kind of a flexible pad under the microscope? 1078 01:15:40,620 --> 01:15:43,120 You might be able to reduce the vibration of the microscope. 1079 01:15:45,860 --> 01:15:47,110 Things like that. 1080 01:15:47,110 --> 01:15:49,750 That is the topic of vibration isolation, 1081 01:15:49,750 --> 01:15:52,690 so we're going to get into that next time. 1082 01:15:52,690 --> 01:15:54,024 Thanks.