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 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:26,460 --> 00:00:27,960 PROFESSOR: Today we're going to talk 9 00:00:27,960 --> 00:00:32,100 about this topic of vibration isolation, which 10 00:00:32,100 --> 00:00:35,430 is a very practical use of knowing 11 00:00:35,430 --> 00:00:37,320 a little bit about vibration. 12 00:00:37,320 --> 00:00:41,240 So imagine a situation you've been 13 00:00:41,240 --> 00:00:44,530 in where there's an air conditioner in a window, 14 00:00:44,530 --> 00:00:48,385 and it's causing your table to shake, 15 00:00:48,385 --> 00:00:51,360 or where you're trying to work, or your bed rattles, 16 00:00:51,360 --> 00:00:52,550 or something like that. 17 00:00:52,550 --> 00:00:54,341 How many of you have ever had an experience 18 00:00:54,341 --> 00:00:56,100 like that, something's kind of annoying, 19 00:00:56,100 --> 00:00:57,905 messing up a lab experiment, or whatever? 20 00:00:57,905 --> 00:01:00,180 Yeah, you've all experienced these things, right? 21 00:01:00,180 --> 00:01:04,900 So as clever engineers, are there 22 00:01:04,900 --> 00:01:08,480 simple solutions sometimes to fixing these problems? 23 00:01:08,480 --> 00:01:11,650 And that's what we're going to talk about today. 24 00:01:11,650 --> 00:01:17,200 I have a little quick demo that I'm going to show you. 25 00:01:17,200 --> 00:01:19,420 You saw this the other day, where 26 00:01:19,420 --> 00:01:23,010 this is my little squiggle pen, and it's got a rotating mass 27 00:01:23,010 --> 00:01:23,620 inside. 28 00:01:23,620 --> 00:01:25,740 And we've looked at rotating masses a lot. 29 00:01:25,740 --> 00:01:32,030 It has some unbalanced, statically unbalanced rotating 30 00:01:32,030 --> 00:01:32,560 mass. 31 00:01:32,560 --> 00:01:36,190 Could be a fan blade with a hunk of chewing gum or something 32 00:01:36,190 --> 00:01:39,540 stuck on a blade or broken piece out of it. 33 00:01:39,540 --> 00:01:42,770 Puts in a force me omega squared cosine omega t, basically 34 00:01:42,770 --> 00:01:46,220 an F0 cosine omega t kind of excitation. 35 00:01:46,220 --> 00:01:51,140 And if it happens to be a flexibly-mounted, mass-spring 36 00:01:51,140 --> 00:01:52,860 dashpot system, it'll vibrate. 37 00:01:52,860 --> 00:01:54,852 And so I showed you that the other day. 38 00:01:54,852 --> 00:01:57,310 We'll do this, and we'll need to lower the lights a little. 39 00:02:00,850 --> 00:02:03,030 But today, I've set your bed. 40 00:02:03,030 --> 00:02:06,600 This is your microscope here, this little one. 41 00:02:06,600 --> 00:02:08,250 Think you can see it in the foreground. 42 00:02:08,250 --> 00:02:12,520 And this is the air conditioner, or the water pump, 43 00:02:12,520 --> 00:02:14,650 or whatever is causing the trouble. 44 00:02:14,650 --> 00:02:20,390 So this is running at about 28 hertz. 45 00:02:20,390 --> 00:02:22,570 If I set the strobe just right, I 46 00:02:22,570 --> 00:02:26,420 can absolutely stop the motion. 47 00:02:30,390 --> 00:02:32,660 It doesn't look like it's moving at all. 48 00:02:32,660 --> 00:02:34,950 That's because the strobe is at exactly the same rate 49 00:02:34,950 --> 00:02:36,540 as the squiggle pen. 50 00:02:36,540 --> 00:02:39,570 Now I'm going to de-tune the strobe a little bit so you 51 00:02:39,570 --> 00:02:40,630 can see the motion. 52 00:02:40,630 --> 00:02:44,180 There is the motion of this main system. 53 00:02:44,180 --> 00:02:46,160 That's causing the problem. 54 00:02:46,160 --> 00:02:48,950 But it actually puts vibration into the tabletop. 55 00:02:48,950 --> 00:02:51,120 And next door over here, I have a little beam. 56 00:02:51,120 --> 00:02:52,494 And you can see that little piece 57 00:02:52,494 --> 00:02:54,390 of white moving up and down. 58 00:02:54,390 --> 00:02:57,440 It's just a little flat piece of spring steel 59 00:02:57,440 --> 00:03:00,150 with a magnet on the end as a mass and a piece of white tape 60 00:03:00,150 --> 00:03:01,480 so you can see it. 61 00:03:01,480 --> 00:03:03,510 But notice it's going up and down in synchrony 62 00:03:03,510 --> 00:03:06,290 with the other one. 63 00:03:06,290 --> 00:03:10,720 So this is your microscope sitting on a lab bench 64 00:03:10,720 --> 00:03:12,280 some distance away. 65 00:03:12,280 --> 00:03:14,310 The problem vibrations are being created 66 00:03:14,310 --> 00:03:16,980 by the original imbalance in something, 67 00:03:16,980 --> 00:03:19,820 travels through the floor, gets to your table 68 00:03:19,820 --> 00:03:20,960 with the microscope on it. 69 00:03:20,960 --> 00:03:22,820 Now your microscope shakes. 70 00:03:22,820 --> 00:03:25,780 So the issue is, what can we do about it? 71 00:03:30,300 --> 00:03:32,190 OK. 72 00:03:32,190 --> 00:03:34,370 Oh, I do-- yeah, no, I'll leave this for a second. 73 00:03:34,370 --> 00:03:35,440 I'll turn it off. 74 00:03:35,440 --> 00:03:39,140 And then I'm going to have you consider. 75 00:03:39,140 --> 00:03:42,691 So I want you to get in pairs and talk about this. 76 00:03:42,691 --> 00:03:44,440 I want, as a group, we're going to come up 77 00:03:44,440 --> 00:03:48,070 with at least three ways to reduce 78 00:03:48,070 --> 00:03:49,820 the vibration of your microscope-- 79 00:03:49,820 --> 00:03:53,330 relatively simple ways to fix it. 80 00:03:53,330 --> 00:03:55,410 How would you do it? 81 00:03:55,410 --> 00:03:56,240 So think about. 82 00:03:56,240 --> 00:03:57,290 Talk about it. 83 00:03:57,290 --> 00:03:59,850 And come up with three ways of fixing this problem. 84 00:04:40,620 --> 00:04:44,160 I'm going to do one more demo on this in a minute 85 00:04:44,160 --> 00:04:49,390 and then put up the transfer function for the force, the one 86 00:04:49,390 --> 00:04:50,510 we had the other day, OK? 87 00:04:50,510 --> 00:04:51,220 The picture. 88 00:05:16,100 --> 00:05:17,520 All right. 89 00:05:17,520 --> 00:05:18,720 Let's have some suggestions. 90 00:05:18,720 --> 00:05:20,510 How would you go about fixing this? 91 00:05:24,312 --> 00:05:24,812 All right. 92 00:05:24,812 --> 00:05:26,246 You had your hand up first. 93 00:05:26,246 --> 00:05:27,441 AUDIENCE: We have it on suspension systems, 94 00:05:27,441 --> 00:05:28,125 and springs. 95 00:05:28,125 --> 00:05:28,791 PROFESSOR: Yeah. 96 00:05:28,791 --> 00:05:31,875 So put some springs on what? 97 00:05:31,875 --> 00:05:34,125 AUDIENCE: Like, have a table where your microscope is, 98 00:05:34,125 --> 00:05:35,538 and then have [INAUDIBLE]. 99 00:05:39,310 --> 00:05:42,270 PROFESSOR: So springs support the microscope. 100 00:05:42,270 --> 00:05:43,000 All right. 101 00:05:43,000 --> 00:05:48,257 So I had this little magnet here sitting on this beam. 102 00:05:48,257 --> 00:05:49,965 This is that spring-supported microscope. 103 00:05:49,965 --> 00:05:51,560 And I have done a heck of a lousy job. 104 00:05:51,560 --> 00:05:53,410 This thing shakes like crazy. 105 00:05:53,410 --> 00:05:54,730 So what do you mean? 106 00:05:58,820 --> 00:06:00,070 Not like that. 107 00:06:00,070 --> 00:06:01,150 How might you-- OK. 108 00:06:01,150 --> 00:06:04,660 So you think you could change the properties of this system 109 00:06:04,660 --> 00:06:08,074 so that it might do better? 110 00:06:08,074 --> 00:06:09,300 Let's think about that. 111 00:06:09,300 --> 00:06:10,420 OK, what's another idea? 112 00:06:15,200 --> 00:06:17,112 AUDIENCE: Change the length of that spring 113 00:06:17,112 --> 00:06:19,520 could change its natural frequency. 114 00:06:19,520 --> 00:06:20,609 PROFESSOR: Yeah. 115 00:06:20,609 --> 00:06:22,900 Are you talking about the microscope one, the receiving 116 00:06:22,900 --> 00:06:23,400 one? 117 00:06:23,400 --> 00:06:25,740 If you change the length of it to make it longer, 118 00:06:25,740 --> 00:06:27,220 it makes it softer, actually. 119 00:06:27,220 --> 00:06:29,120 Make it shorter, it makes it stiffer. 120 00:06:29,120 --> 00:06:32,620 So you would change its natural frequency. 121 00:06:32,620 --> 00:06:36,370 Now, the two ideas together, if you set the natural frequency 122 00:06:36,370 --> 00:06:41,100 correctly, the system on the receiving end, 123 00:06:41,100 --> 00:06:42,324 you can reduce its vibration. 124 00:06:42,324 --> 00:06:43,990 And I will demonstrate that in a second. 125 00:06:43,990 --> 00:06:45,782 What's another idea? 126 00:06:45,782 --> 00:06:47,320 AUDIENCE: [INAUDIBLE]. 127 00:06:47,320 --> 00:06:48,320 PROFESSOR: Acid damping. 128 00:06:48,320 --> 00:06:48,820 OK. 129 00:06:48,820 --> 00:06:51,830 Well, that's a very interesting suggestion. 130 00:06:51,830 --> 00:06:54,667 Damping, it helps under one circumstance, 131 00:06:54,667 --> 00:06:55,750 but not under some others. 132 00:06:55,750 --> 00:06:58,500 And we're going to explore that today. 133 00:06:58,500 --> 00:07:00,960 This is all basically-- we've come up 134 00:07:00,960 --> 00:07:02,181 with one of the three ideas. 135 00:07:02,181 --> 00:07:02,680 What else? 136 00:07:05,914 --> 00:07:07,580 AUDIENCE: Attach something that vibrates 137 00:07:07,580 --> 00:07:09,224 180 degrees [INAUDIBLE]. 138 00:07:09,224 --> 00:07:10,640 PROFESSOR: Oh, that's interesting. 139 00:07:10,640 --> 00:07:11,556 That's the fourth one. 140 00:07:11,556 --> 00:07:13,350 I mean, that can be a little expensive, 141 00:07:13,350 --> 00:07:15,870 but these are like noise-canceling headsets, 142 00:07:15,870 --> 00:07:16,545 right? 143 00:07:16,545 --> 00:07:18,420 Could we put in something else somewhere else 144 00:07:18,420 --> 00:07:20,710 on the table that cancels the vibration 145 00:07:20,710 --> 00:07:22,271 out where your microscope is? 146 00:07:22,271 --> 00:07:23,270 Yeah, you could do that. 147 00:07:23,270 --> 00:07:24,980 A little expensive. 148 00:07:24,980 --> 00:07:25,550 What else? 149 00:07:25,550 --> 00:07:25,850 Yeah. 150 00:07:25,850 --> 00:07:27,516 AUDIENCE: You could cushion [INAUDIBLE]. 151 00:07:33,732 --> 00:07:35,190 PROFESSOR: Where would we put that? 152 00:07:35,190 --> 00:07:38,794 AUDIENCE: You could do it underneath [INAUDIBLE]. 153 00:07:38,794 --> 00:07:39,460 PROFESSOR: Yeah. 154 00:07:39,460 --> 00:07:41,043 And so that's generally the same idea. 155 00:07:41,043 --> 00:07:42,640 So you're all treating the microscope. 156 00:07:42,640 --> 00:07:45,090 Can you treat something else in the system? 157 00:07:45,090 --> 00:07:47,360 So yeah, we'll fix the microscope end. 158 00:07:47,360 --> 00:07:49,350 But that might knock it down by a factor of 10. 159 00:07:49,350 --> 00:07:51,516 I want to knock it down by at least a factor of 100, 160 00:07:51,516 --> 00:07:53,794 if not a factor of 1,000. 161 00:07:53,794 --> 00:07:55,210 AUDIENCE: Fix the air conditioner. 162 00:07:55,210 --> 00:07:57,607 PROFESSOR: Ah-ha-ha-ha-ha! 163 00:07:57,607 --> 00:08:01,910 You know, put another piece of gum on the other blade. 164 00:08:01,910 --> 00:08:06,760 Or clean it up, or balance the rotor, in effect. 165 00:08:06,760 --> 00:08:09,650 Rotors are not manufactured defective usually. 166 00:08:09,650 --> 00:08:15,650 They get rejected at QC before it goes out the door. 167 00:08:15,650 --> 00:08:17,910 So fix the rotor. 168 00:08:17,910 --> 00:08:20,020 Can be a little expensive sometimes, but worth it. 169 00:08:20,020 --> 00:08:22,500 They do maintenance checks. 170 00:08:22,500 --> 00:08:24,650 Actually, they have accelerometers 171 00:08:24,650 --> 00:08:28,620 built in to all expensive rotating equipment these days. 172 00:08:28,620 --> 00:08:30,557 And it's called condition monitoring. 173 00:08:30,557 --> 00:08:32,390 And when they get outside of certain limits, 174 00:08:32,390 --> 00:08:35,161 they shut the thing down and rebuild it. 175 00:08:35,161 --> 00:08:38,110 In electric generator sets, gas turbines, 176 00:08:38,110 --> 00:08:42,000 jet engines on all aircraft are all incredibly carefully 177 00:08:42,000 --> 00:08:42,750 balanced in a way. 178 00:08:42,750 --> 00:08:43,890 They start getting out of balance, 179 00:08:43,890 --> 00:08:46,260 they stop and fix them before the things blow up on them 180 00:08:46,260 --> 00:08:48,140 and you have a $20 million problem instead of 181 00:08:48,140 --> 00:08:51,337 maybe a $20,000 tear-down. 182 00:08:51,337 --> 00:08:52,710 All right? 183 00:08:52,710 --> 00:08:55,930 So yeah, you fix the rotor. 184 00:08:55,930 --> 00:08:56,730 Third idea. 185 00:09:03,260 --> 00:09:07,840 Well, could you-- this thing's shaking like crazy. 186 00:09:10,490 --> 00:09:12,980 What if you change the length of this beam? 187 00:09:12,980 --> 00:09:16,016 Could you stop this thing from shaking so much? 188 00:09:16,016 --> 00:09:18,050 And if you stop this from shaking so much, 189 00:09:18,050 --> 00:09:21,170 would it put so much excitation into the table? 190 00:09:21,170 --> 00:09:21,670 No. 191 00:09:21,670 --> 00:09:27,855 So fix the rotor, isolate the source, isolate the receiver. 192 00:09:27,855 --> 00:09:29,659 Now you get at least three ways. 193 00:09:29,659 --> 00:09:31,950 And the gentleman up here came up with the fourth way-- 194 00:09:31,950 --> 00:09:33,770 active cancellation. 195 00:09:33,770 --> 00:09:34,660 OK. 196 00:09:34,660 --> 00:09:35,690 Great. 197 00:09:35,690 --> 00:09:39,224 I have a demo of one of these. 198 00:09:39,224 --> 00:09:40,890 Sometimes if you're desperate and you're 199 00:09:40,890 --> 00:09:42,560 trying to get some sleep-- it's your bed 200 00:09:42,560 --> 00:09:44,090 that's rattling or something-- you 201 00:09:44,090 --> 00:09:47,650 might want to try the following that you can do very quickly. 202 00:09:47,650 --> 00:09:48,920 So we'll dim the lights again. 203 00:09:48,920 --> 00:09:50,220 We'll turn it back on. 204 00:09:50,220 --> 00:09:52,890 [VIBRATING] 205 00:09:54,130 --> 00:09:56,830 Both are vibrating like crazy. 206 00:09:56,830 --> 00:09:57,370 All right. 207 00:09:57,370 --> 00:10:00,330 So one way to detune the receiver 208 00:10:00,330 --> 00:10:06,920 is to put a big weight on it and change its natural frequency 209 00:10:06,920 --> 00:10:10,170 by changing the mass. 210 00:10:10,170 --> 00:10:15,080 And now the little thing is hardly shaken at all, 211 00:10:15,080 --> 00:10:18,370 because I've detuned just by changing the mass. 212 00:10:18,370 --> 00:10:21,060 Accomplished the same thing as switching the length. 213 00:10:21,060 --> 00:10:22,800 Instead of messing with the stiffness, 214 00:10:22,800 --> 00:10:25,160 I've changed the mass of the system. 215 00:10:25,160 --> 00:10:27,850 This is still shaking like crazy. 216 00:10:27,850 --> 00:10:30,420 OK. 217 00:10:30,420 --> 00:10:33,200 So now it's back to vibrating again. 218 00:10:33,200 --> 00:10:34,852 But over here is my source. 219 00:10:34,852 --> 00:10:36,810 I don't know if it'll work so well in this one, 220 00:10:36,810 --> 00:10:39,052 because the source is a lot more massive. 221 00:10:39,052 --> 00:10:40,760 But I'm going to put the same mass on it. 222 00:10:43,500 --> 00:10:45,996 Ooh, it made it worse. 223 00:10:45,996 --> 00:10:46,800 Ah. 224 00:10:46,800 --> 00:10:51,140 And that's another good demonstration. 225 00:10:51,140 --> 00:10:51,950 Excellent. 226 00:10:51,950 --> 00:10:59,120 So we could come back up with the lights. 227 00:10:59,120 --> 00:11:01,510 You got to be careful of that. 228 00:11:01,510 --> 00:11:06,090 You go to mess with one of these systems, if you do it wrong, 229 00:11:06,090 --> 00:11:08,180 you make the matters worse. 230 00:11:08,180 --> 00:11:12,940 First consulting job I ever had back in 1977 or something 231 00:11:12,940 --> 00:11:16,390 like that, they had a vibration problem on a ship. 232 00:11:16,390 --> 00:11:18,830 And the first consultant in said, stiffen up. 233 00:11:18,830 --> 00:11:21,152 It was actually the exhaust stacks on these 5,000 234 00:11:21,152 --> 00:11:23,360 horsepower diesel engines, and they were 30 feet tall 235 00:11:23,360 --> 00:11:24,870 and shaking like crazy. 236 00:11:24,870 --> 00:11:28,112 And the first guy said, stiffen up those exhaust stacks. 237 00:11:28,112 --> 00:11:29,570 And he did exactly the wrong thing. 238 00:11:29,570 --> 00:11:33,171 And it just shook worse than ever. 239 00:11:33,171 --> 00:11:33,670 OK. 240 00:11:37,310 --> 00:11:39,490 So now what I'm going to show you-- 241 00:11:39,490 --> 00:11:40,970 what we'll put on the board today 242 00:11:40,970 --> 00:11:43,350 is a little bit of mathematics to back up how you 243 00:11:43,350 --> 00:11:46,880 go about doing the two things. 244 00:11:46,880 --> 00:11:50,400 One is isolating the receiver, or the other one's 245 00:11:50,400 --> 00:11:51,970 isolating the source. 246 00:11:51,970 --> 00:11:54,260 I'm going to start with isolating the receiver. 247 00:11:54,260 --> 00:11:58,535 But we're going to start with a little bit of math, 248 00:11:58,535 --> 00:12:00,520 a little math tool that we need that will 249 00:12:00,520 --> 00:12:04,520 make life a lot easier for us. 250 00:12:04,520 --> 00:12:05,470 Yeah, I'll work here. 251 00:12:08,040 --> 00:12:11,860 If you recall last time, now would be the time to do it, 252 00:12:11,860 --> 00:12:25,420 we derived the transfer function for essentially this system, 253 00:12:25,420 --> 00:12:29,970 where we had an F0 cosine omega t. 254 00:12:29,970 --> 00:12:34,610 And we computed the response, x of t, 255 00:12:34,610 --> 00:12:40,580 as some x0 cosine omega t minus the phase angle. 256 00:12:40,580 --> 00:12:42,770 And we worked it all down to where 257 00:12:42,770 --> 00:12:45,240 we could plot it like that. 258 00:12:45,240 --> 00:12:48,910 But quite frankly, it was kind of a lot of lines of math. 259 00:12:48,910 --> 00:12:50,035 And it was sort of painful. 260 00:12:50,035 --> 00:12:52,230 I actually hated doing it on the board. 261 00:12:52,230 --> 00:12:54,086 But it was easy, because it was familiar. 262 00:12:54,086 --> 00:12:54,960 It's just trig stuff. 263 00:12:54,960 --> 00:12:57,600 It was all trig and a little bit of calculus. 264 00:12:57,600 --> 00:13:00,610 So we do that first, because it all makes sense to you 265 00:13:00,610 --> 00:13:01,490 mathematically. 266 00:13:01,490 --> 00:13:04,320 But there's a vastly easier and quicker way 267 00:13:04,320 --> 00:13:07,240 to do this, which we'll address right now. 268 00:13:07,240 --> 00:13:08,900 And that's using complex numbers. 269 00:13:11,830 --> 00:13:16,300 So we need a couple bits of information here. 270 00:13:16,300 --> 00:13:17,935 One is Euler's formula. 271 00:13:20,480 --> 00:13:25,560 So if you have e raised to the power i theta, 272 00:13:25,560 --> 00:13:30,630 you can show that that is the same thing as cosine theta 273 00:13:30,630 --> 00:13:35,910 plus i sine of theta. 274 00:13:35,910 --> 00:13:39,380 That breaks into a real part and an imaginary part. 275 00:13:42,780 --> 00:13:53,630 So if we wanted to express this excitation, F0 cosine omega t, 276 00:13:53,630 --> 00:13:56,380 in complex notation, we would say 277 00:13:56,380 --> 00:14:00,220 it is the real part, which I'll denote 278 00:14:00,220 --> 00:14:09,530 as Re, the real part of F0 e to the i omega t. 279 00:14:09,530 --> 00:14:16,060 And I'm going to just specify that F0 itself here, this 280 00:14:16,060 --> 00:14:18,725 is real and positive. 281 00:14:23,030 --> 00:14:24,580 So this is real and positive. 282 00:14:24,580 --> 00:14:27,250 So the real part's going to be F0 times-- 283 00:14:27,250 --> 00:14:31,780 and if you break down co ee to the i omega t into its-- 284 00:14:31,780 --> 00:14:34,250 by Euler's formula, it gives you cosine omega t 285 00:14:34,250 --> 00:14:35,340 plus i sine omega t. 286 00:14:35,340 --> 00:14:38,970 And the real part is the cosine part just according to this. 287 00:14:38,970 --> 00:14:39,470 OK. 288 00:14:44,160 --> 00:14:52,800 Now, another little fact that I want to show you 289 00:14:52,800 --> 00:15:00,930 is if you have a complex number, a plus bi, 290 00:15:00,930 --> 00:15:06,030 I want to express it as some ce to the i theta. 291 00:15:06,030 --> 00:15:08,120 So you want to use Euler's formula 292 00:15:08,120 --> 00:15:12,170 to express a complex number. 293 00:15:12,170 --> 00:15:15,950 And if we draw it, the answer becomes pretty obvious. 294 00:15:15,950 --> 00:15:19,190 This is a point up here, a, bi. 295 00:15:19,190 --> 00:15:23,530 And this is the imaginary axis here and the real. 296 00:15:28,320 --> 00:15:36,460 And this point, this side is a, and this side here is b. 297 00:15:36,460 --> 00:15:40,310 And this side here, the length of this triangle, is c. 298 00:15:40,310 --> 00:15:43,870 And the angle in here is theta. 299 00:15:43,870 --> 00:15:46,160 So now if I ask you, what's c, well, you say, 300 00:15:46,160 --> 00:15:50,610 oh, well, c is obviously the square root of a squared 301 00:15:50,610 --> 00:15:53,210 plus b squared. 302 00:15:53,210 --> 00:16:01,790 And theta is a tangent inverse of b/a, 303 00:16:01,790 --> 00:16:05,400 which is the imaginary part over the real part. 304 00:16:18,900 --> 00:16:21,610 If you want to express a complex number this way, 305 00:16:21,610 --> 00:16:23,770 well, the magnitude is square root of a squared 306 00:16:23,770 --> 00:16:24,890 plus b squared. 307 00:16:24,890 --> 00:16:26,820 And the phase angle that you put up here 308 00:16:26,820 --> 00:16:31,310 is tangent inverse of the imaginary part over the real. 309 00:16:31,310 --> 00:16:31,810 OK. 310 00:16:35,500 --> 00:16:42,450 So now we have the basic tools we 311 00:16:42,450 --> 00:16:46,540 need to take on the vibration problem. 312 00:16:46,540 --> 00:16:48,295 And so we have that system up there. 313 00:16:53,690 --> 00:16:59,330 And our output, from the way we derived it last time, 314 00:16:59,330 --> 00:17:05,724 the output is some x0 cosine omega t minus phi. 315 00:17:05,724 --> 00:17:08,140 And one of the reasons it was so painful doing it this way 316 00:17:08,140 --> 00:17:10,140 last time is you have to-- this is 317 00:17:10,140 --> 00:17:12,630 cosine is a function of both time and phase. 318 00:17:12,630 --> 00:17:16,670 And to break it apart takes a lot of work. 319 00:17:16,670 --> 00:17:19,520 So we want to do the same thing, but with complex variables 320 00:17:19,520 --> 00:17:20,630 this time. 321 00:17:20,630 --> 00:17:28,620 So I want to express this then as the real part of-- I 322 00:17:28,620 --> 00:17:34,780 could say it's the real part of x0 e to the i omega 323 00:17:34,780 --> 00:17:36,200 t minus phi. 324 00:17:39,010 --> 00:17:42,970 And despite Euler's formula if this is real, just the number. 325 00:17:42,970 --> 00:17:46,180 Than this breaks down into cosine omega t minus phi and i 326 00:17:46,180 --> 00:17:48,480 sine omega t minus phi. 327 00:17:52,260 --> 00:17:56,150 But here's the beauty of using complex notation 328 00:17:56,150 --> 00:17:58,230 and exponentials. 329 00:17:58,230 --> 00:18:09,030 This now becomes the real part of x0 e to the minus i phi 330 00:18:09,030 --> 00:18:11,945 times e to the i omega t. 331 00:18:11,945 --> 00:18:13,740 If I could separate these two. 332 00:18:13,740 --> 00:18:15,290 And this is what makes it so much 333 00:18:15,290 --> 00:18:19,890 easier to use this approach. 334 00:18:19,890 --> 00:18:26,370 And I'm going to call this part of it just some capital X. 335 00:18:26,370 --> 00:18:30,180 And it is a complex number. 336 00:18:30,180 --> 00:18:30,730 For sure. 337 00:18:30,730 --> 00:18:39,140 If I break this up into cosine of minus phi and minus i sine 338 00:18:39,140 --> 00:18:40,810 phi, it's got an i sine phi. 339 00:18:40,810 --> 00:18:44,830 So this is a complex number-- a plus bi kind of form. 340 00:18:44,830 --> 00:18:46,650 So in general, this thing is complex. 341 00:18:49,420 --> 00:18:52,630 So this whole thing is the real part 342 00:18:52,630 --> 00:18:57,650 of some X, which I don't know now, e to the i omega t. 343 00:19:05,980 --> 00:19:10,000 So now we can quite quickly do the derivation 344 00:19:10,000 --> 00:19:11,650 we did last time. 345 00:19:11,650 --> 00:19:15,760 We're talking about representing linear systems by some kind 346 00:19:15,760 --> 00:19:19,750 of black box-- has a transfer function in it, which we call 347 00:19:19,750 --> 00:19:26,700 H, in this case, x/F. Response x per unit input F. And remember, 348 00:19:26,700 --> 00:19:35,360 we're talking about steady state response only. 349 00:19:39,590 --> 00:19:47,730 And we have, as our input here, some F0 e to the i omega t. 350 00:19:47,730 --> 00:19:53,130 And we have, as an output, some X e to the i omega t. 351 00:19:53,130 --> 00:19:55,620 And we know that we're going to use the convention that we 352 00:19:55,620 --> 00:20:01,650 care about we have to have real number answers. 353 00:20:01,650 --> 00:20:04,640 So we'll be eventually actually using 354 00:20:04,640 --> 00:20:07,970 the real part of the input and the real part of the output. 355 00:20:07,970 --> 00:20:11,350 But to get there, we're going to use complex notation first 356 00:20:11,350 --> 00:20:15,560 and then separate out the real and imaginary parts at the end. 357 00:20:21,660 --> 00:20:36,900 So for our system, we know the equation of motion, 358 00:20:36,900 --> 00:20:51,110 so now it's some F0 e to the i omega t. 359 00:20:51,110 --> 00:20:53,190 There's our equation of motion. 360 00:20:53,190 --> 00:21:02,900 And I'm going to let x here be this unknown capital X e 361 00:21:02,900 --> 00:21:04,890 to the i omega t. 362 00:21:04,890 --> 00:21:06,260 And I'm going to plug it in. 363 00:21:09,870 --> 00:21:12,040 And the exponentials are particularly 364 00:21:12,040 --> 00:21:14,245 easy to deal with when you're taking derivatives. 365 00:21:18,470 --> 00:21:23,790 So upon doing that, we immediately get minus omega 366 00:21:23,790 --> 00:21:35,450 squared M plus i omega c plus k. 367 00:21:38,100 --> 00:21:43,050 All of that times Xe to the i omega t 368 00:21:43,050 --> 00:21:47,190 equals F0 e to the i omega t. 369 00:21:51,650 --> 00:21:54,935 Immediately, I can get rid of the time-dependent parts. 370 00:21:58,680 --> 00:22:03,420 And I can solve for x/F, which is what we set out 371 00:22:03,420 --> 00:22:07,160 to do the other day to find this transfer function between input 372 00:22:07,160 --> 00:22:07,830 and output. 373 00:22:21,360 --> 00:22:25,370 So if I solve for x divided by F, 374 00:22:25,370 --> 00:22:28,200 I'm going to get all this stuff and the denominator 375 00:22:28,200 --> 00:22:29,660 on one side. 376 00:22:29,660 --> 00:22:31,290 And I'll write it out here. 377 00:22:53,610 --> 00:22:55,730 It simply looks like that. 378 00:22:55,730 --> 00:22:58,730 And now remember, I can substitute in some things. 379 00:22:58,730 --> 00:23:04,790 I remember k/m is omega n squared. 380 00:23:04,790 --> 00:23:10,920 And zeta in c over 2 omega n M. 381 00:23:10,920 --> 00:23:15,780 And I plug those things in and just 382 00:23:15,780 --> 00:23:18,250 rearrange it a little tiny bit. 383 00:23:18,250 --> 00:23:21,400 We should come up with something like we 384 00:23:21,400 --> 00:23:32,560 found before, so that x/F, 1/k, and the denominator, 385 00:23:32,560 --> 00:23:40,570 1 minus omega squared over omega n squared-- not quite yet 386 00:23:40,570 --> 00:23:47,105 here-- plus 2 i zeta omega over omega n. 387 00:23:51,327 --> 00:23:52,410 That's what it looks like. 388 00:23:55,420 --> 00:23:57,260 So you still have a complex denominator. 389 00:24:02,150 --> 00:24:07,220 And this basically looks like a number 1 over k times 1 390 00:24:07,220 --> 00:24:08,960 over some a plus bi. 391 00:24:12,830 --> 00:24:13,810 There's your a term. 392 00:24:13,810 --> 00:24:14,840 Here's your bi term. 393 00:24:26,892 --> 00:24:28,850 And the way you deal with something like this-- 394 00:24:28,850 --> 00:24:32,090 you have an a plus bi in the denominator-- you multiply 395 00:24:32,090 --> 00:24:34,900 the numerator and denominator by the complex conjugate 396 00:24:34,900 --> 00:24:38,250 in order to get this into actually standard a plus bi 397 00:24:38,250 --> 00:24:41,150 form. 398 00:24:41,150 --> 00:24:46,630 If I do that symbolically here, it 399 00:24:46,630 --> 00:24:51,190 comes out looking like a minus bi over a squared 400 00:24:51,190 --> 00:24:52,050 plus b squared. 401 00:24:56,108 --> 00:25:06,710 And that's 1/k e to the minus i phi 402 00:25:06,710 --> 00:25:12,240 over square root of a squared plus b squared. 403 00:25:15,600 --> 00:25:18,125 Because now, see, the denominator's 404 00:25:18,125 --> 00:25:20,120 just a real number. 405 00:25:20,120 --> 00:25:25,270 So this whole thing is, in some form, c plus a di. 406 00:25:25,270 --> 00:25:29,410 You could break this into a real part, complex part. 407 00:25:29,410 --> 00:25:31,680 We could say that's equal to some magnitude 408 00:25:31,680 --> 00:25:36,050 times e to the i phi. 409 00:25:36,050 --> 00:25:41,160 To get the magnitude, you take a squared 410 00:25:41,160 --> 00:25:42,840 plus b squared square root. 411 00:25:42,840 --> 00:25:44,100 It cancels. 412 00:25:44,100 --> 00:25:46,380 This is squared the denominator, so you 413 00:25:46,380 --> 00:25:50,760 end up with this part, square root, in the denominator. 414 00:25:54,690 --> 00:25:59,470 This is what the-- we need to know what phi looks like. 415 00:25:59,470 --> 00:26:03,040 Well, phi had better come out like before, 416 00:26:03,040 --> 00:26:13,530 where now phi is minus tangent inverse of the imaginary part 417 00:26:13,530 --> 00:26:16,370 over the real part. 418 00:26:16,370 --> 00:26:20,250 And the imaginary part has a minus here. 419 00:26:20,250 --> 00:26:22,710 That's why a minus pops up here. 420 00:26:22,710 --> 00:26:24,780 Imaginary part comes from this. 421 00:26:24,780 --> 00:26:27,410 The real part comes from there. 422 00:26:27,410 --> 00:26:29,680 The common denominator stuff all cancels out 423 00:26:29,680 --> 00:26:31,060 when you take the ratio. 424 00:26:31,060 --> 00:26:41,820 So this is tangent inverse of two zeta omega over omega n 425 00:26:41,820 --> 00:26:47,280 all over 1 minus omega squared over omega n squared, 426 00:26:47,280 --> 00:26:49,220 as before. 427 00:26:49,220 --> 00:26:52,460 I've skipped a couple of steps, but we cranked this whole thing 428 00:26:52,460 --> 00:26:53,030 out before. 429 00:26:55,690 --> 00:27:02,160 This is the same steps that you would go through to do that. 430 00:27:02,160 --> 00:27:05,850 We're just doing this to get to the phase angle. 431 00:27:08,530 --> 00:27:20,550 But this now is exactly the same thing we got before, 432 00:27:20,550 --> 00:27:22,310 which we have plotted up there. 433 00:27:28,560 --> 00:27:30,380 We work with magnitude and phase angle. 434 00:27:30,380 --> 00:27:36,650 So the magnitude of x/F is the same thing 435 00:27:36,650 --> 00:27:39,795 as saying the magnitude of the transfer function. 436 00:27:43,500 --> 00:27:48,510 And that transfer function looks like 1/k, the magnitude, 437 00:27:48,510 --> 00:27:56,060 all divided by 1 minus omega squared over omega n squared 438 00:27:56,060 --> 00:28:06,020 squared plus 2 zeta omega over omega n squared square root. 439 00:28:06,020 --> 00:28:09,970 That is the same transfer function magnitude 440 00:28:09,970 --> 00:28:15,060 that we derived last time, with a lot more work. 441 00:28:15,060 --> 00:28:18,200 And this approach, using complex variables, 442 00:28:18,200 --> 00:28:23,470 you can use for any single input, single output 443 00:28:23,470 --> 00:28:24,660 linear system. 444 00:28:24,660 --> 00:28:28,270 And we're going to do it to derive right away the transfer 445 00:28:28,270 --> 00:28:34,905 function for the response of this to motion of the base. 446 00:28:39,460 --> 00:28:42,325 So if you follow how we used this complex variables in e 447 00:28:42,325 --> 00:28:45,290 to the i omega t's to get here, we 448 00:28:45,290 --> 00:28:49,170 can now apply the same tools to do other transfer functions 449 00:28:49,170 --> 00:28:50,755 to be a lot more efficient about it. 450 00:28:57,880 --> 00:29:02,920 Before I jump to this one, remind you 451 00:29:02,920 --> 00:29:06,260 how, in practice, we use this. 452 00:29:06,260 --> 00:29:09,400 So if the statement magnitude of x/F 453 00:29:09,400 --> 00:29:11,980 equals everything on the right there. 454 00:29:11,980 --> 00:29:14,020 Then in the way we would normally 455 00:29:14,020 --> 00:29:16,830 use this is to say, well, if you want the magnitude 456 00:29:16,830 --> 00:29:22,640 of the response, you take the magnitude of the input force, 457 00:29:22,640 --> 00:29:26,160 multiply it by the magnitude of the transfer function, 458 00:29:26,160 --> 00:29:28,655 evaluate it at the correct frequency. 459 00:29:34,420 --> 00:29:35,840 That would give you the magnitude. 460 00:29:35,840 --> 00:29:39,620 If you want the time dependence, x of t, 461 00:29:39,620 --> 00:29:42,810 well, that's the magnitude of the force, 462 00:29:42,810 --> 00:29:46,400 magnitude of the transfer function 463 00:29:46,400 --> 00:29:55,680 times the real part of e to the i omega t minus phi. 464 00:29:59,050 --> 00:30:02,780 And this gets us back to when you work this out, 465 00:30:02,780 --> 00:30:05,010 this is your x0. 466 00:30:05,010 --> 00:30:11,730 And this is your cosine omega t minus phi. 467 00:30:16,400 --> 00:30:21,400 So once you know what the excitation force is 468 00:30:21,400 --> 00:30:25,640 and its frequency, you put the force in here. 469 00:30:25,640 --> 00:30:31,670 You evaluate that thing on the left at the correct frequency. 470 00:30:31,670 --> 00:30:34,170 And you write out the answer directly. 471 00:30:34,170 --> 00:30:37,320 In one of the homeworks for today, 472 00:30:37,320 --> 00:30:40,330 the question just had you go through the exercise 473 00:30:40,330 --> 00:30:43,770 of figuring this out at three different frequency 474 00:30:43,770 --> 00:30:47,700 ratios, like 1/2, 1, and 3, or something like that, 475 00:30:47,700 --> 00:30:51,950 would put you to the left of the peak at 1/2, on the peak at 1, 476 00:30:51,950 --> 00:30:55,810 and way off to the right out at the right edge at 3. 477 00:30:55,810 --> 00:30:58,170 And you'll get three different response 478 00:30:58,170 --> 00:31:01,280 amplitudes and three different phase angles that go with it. 479 00:31:05,580 --> 00:31:06,080 All right. 480 00:31:06,080 --> 00:31:08,340 So that's how you review. 481 00:31:08,340 --> 00:31:10,890 Did the same thing a different way. 482 00:31:10,890 --> 00:31:13,330 And I'm going to move on to base motion. 483 00:31:13,330 --> 00:31:14,720 But any questions about this now? 484 00:31:14,720 --> 00:31:15,220 Yeah. 485 00:31:15,220 --> 00:31:18,385 AUDIENCE: Was the e to the negative ib included 486 00:31:18,385 --> 00:31:20,965 in your F of x? 487 00:31:20,965 --> 00:31:21,760 PROFESSOR: Yes. 488 00:31:21,760 --> 00:31:25,225 AUDIENCE: OK, so why did the negative 489 00:31:25,225 --> 00:31:27,205 b appear again after your final [INAUDIBLE]? 490 00:31:34,140 --> 00:31:35,560 PROFESSOR: The very top expression 491 00:31:35,560 --> 00:31:40,120 up there, it says x/F. It says we're 492 00:31:40,120 --> 00:31:45,420 trying to cast it in the e to the minus i phi form. 493 00:31:45,420 --> 00:31:46,560 That's my goal. 494 00:31:46,560 --> 00:31:49,080 And I did that, because we started over here 495 00:31:49,080 --> 00:31:51,280 with the problem that we had done before, 496 00:31:51,280 --> 00:31:54,380 where that's the way we decided to write the answer. 497 00:31:54,380 --> 00:31:59,260 And it turns out that it's just a convention in vibration 498 00:31:59,260 --> 00:32:05,550 engineering that authors and people have adopted to express 499 00:32:05,550 --> 00:32:07,271 the phase angle as minus phi. 500 00:32:07,271 --> 00:32:08,770 They could have done it as plus phi. 501 00:32:11,590 --> 00:32:13,306 The plots like this are phi. 502 00:32:13,306 --> 00:32:13,972 AUDIENCE: Right. 503 00:32:13,972 --> 00:32:20,870 But I guess what I'm wondering, isn't [INAUDIBLE] x/F. 504 00:32:20,870 --> 00:32:23,110 PROFESSOR: Oh, I see what you mean. 505 00:32:23,110 --> 00:32:26,280 It's in there before you take its magnitude. 506 00:32:26,280 --> 00:32:39,290 So the Hx/F, when it is-- this here 507 00:32:39,290 --> 00:32:41,680 is left in complex notation. 508 00:32:41,680 --> 00:32:46,440 And this is Hx/F of omega. 509 00:32:49,880 --> 00:32:53,180 And it is complex. 510 00:32:53,180 --> 00:32:55,630 We take its magnitude. 511 00:32:55,630 --> 00:32:58,620 Then the magnitude is not complex, right? 512 00:32:58,620 --> 00:33:00,760 And so we take its magnitude. 513 00:33:00,760 --> 00:33:03,399 We get that expression. 514 00:33:03,399 --> 00:33:04,940 But when we take its magnitude, we've 515 00:33:04,940 --> 00:33:07,170 thrown away the phase information. 516 00:33:07,170 --> 00:33:09,650 So we have to keep it and put it somewhere. 517 00:33:09,650 --> 00:33:16,380 And so we put it in the e to the i phi form. 518 00:33:16,380 --> 00:33:20,880 And I guess what I should have done here 519 00:33:20,880 --> 00:33:27,160 is now this is-- I've taken-- this 520 00:33:27,160 --> 00:33:34,610 is Hx/F, same thing as x/F, in complex form. 521 00:33:34,610 --> 00:33:37,620 And I've said, OK, if I write it this way, 522 00:33:37,620 --> 00:33:39,476 I have just said it is a magnitude. 523 00:33:42,460 --> 00:33:44,930 Times its phase information. 524 00:33:44,930 --> 00:33:48,410 I've separated its phase information from its magnitude 525 00:33:48,410 --> 00:33:51,548 by writing it this way. 526 00:33:51,548 --> 00:33:53,340 OK. 527 00:33:53,340 --> 00:33:56,330 And the phase then is that. 528 00:33:56,330 --> 00:33:59,920 And its magnitude is that. 529 00:33:59,920 --> 00:34:00,639 Good question. 530 00:34:00,639 --> 00:34:01,139 All right. 531 00:34:11,679 --> 00:34:15,580 So now let's see if we can kind of pretty quickly 532 00:34:15,580 --> 00:34:17,450 do the same problem for base motion. 533 00:34:31,560 --> 00:34:35,540 So this is our microscope now, idealizes a mass spring system. 534 00:34:35,540 --> 00:34:36,844 So this is our microscope. 535 00:34:41,510 --> 00:34:50,810 Has some mass stiffness damping motion, x of t. 536 00:34:50,810 --> 00:34:55,580 And how do you suppose-- where would you measure 537 00:34:55,580 --> 00:34:59,100 that motion x of t from? 538 00:34:59,100 --> 00:35:02,590 Like, to define your coordinate here 539 00:35:02,590 --> 00:35:06,985 is a major point in the last homework. 540 00:35:06,985 --> 00:35:08,335 Is gravity involved? 541 00:35:10,980 --> 00:35:14,380 But only as a constant term, mg in the equation of motion. 542 00:35:14,380 --> 00:35:16,910 It's only there depending on if you 543 00:35:16,910 --> 00:35:20,630 write the equation of motion in the less desirable way. 544 00:35:20,630 --> 00:35:24,420 Where is this measured from do you guess? 545 00:35:24,420 --> 00:35:25,580 Equilibrium position? 546 00:35:25,580 --> 00:35:27,010 Static equilibrium position? 547 00:35:27,010 --> 00:35:28,515 Or 0 spring force position? 548 00:35:31,160 --> 00:35:34,300 How many suggest 0 spring force position? 549 00:35:34,300 --> 00:35:35,970 How many suggest static equilibrium? 550 00:35:35,970 --> 00:35:36,470 OK. 551 00:35:36,470 --> 00:35:38,440 You got the message. 552 00:35:38,440 --> 00:35:41,570 This is from equilibrium, because you don't 553 00:35:41,570 --> 00:35:43,910 have to deal with the mg term. 554 00:35:43,910 --> 00:35:46,090 So this is measured from equilibrium. 555 00:35:46,090 --> 00:35:49,900 That's the deflection of the microscope support. 556 00:35:49,900 --> 00:35:53,880 This is the deflection of the floor that's driving it. 557 00:35:53,880 --> 00:35:56,337 Then we know we've got that table shaking like crazy. 558 00:35:56,337 --> 00:35:57,920 That's what's causing this to vibrate. 559 00:36:01,940 --> 00:36:03,750 And we need a free-body diagram. 560 00:36:07,360 --> 00:36:09,740 And we approach free-body diagrams just like before. 561 00:36:09,740 --> 00:36:14,940 You imagine positive motions of x and x dot, 562 00:36:14,940 --> 00:36:18,830 positive motions of y and y dot, and deduce their forces. 563 00:36:18,830 --> 00:36:24,410 So positive x gives you a kx opposing. 564 00:36:24,410 --> 00:36:30,890 A positive x dot gives you a cx dot opposing. 565 00:36:30,890 --> 00:36:35,290 A positive y gives you what? 566 00:36:35,290 --> 00:36:37,460 A force that results on this. 567 00:36:37,460 --> 00:36:38,920 Positive motion of the floor. 568 00:36:46,510 --> 00:36:49,336 Positive or negative force? 569 00:36:49,336 --> 00:36:51,531 How many think positive? 570 00:36:51,531 --> 00:36:53,870 How many think negative? 571 00:36:53,870 --> 00:36:55,090 How many aren't sure? 572 00:36:55,090 --> 00:36:58,050 How many aren't awake? 573 00:36:58,050 --> 00:36:58,650 OK. 574 00:36:58,650 --> 00:36:59,800 Look. 575 00:36:59,800 --> 00:37:03,120 If I push up on this-- and now this 576 00:37:03,120 --> 00:37:05,640 is fixed when you do this mental experiment. 577 00:37:05,640 --> 00:37:07,120 You fix this momentarily. 578 00:37:07,120 --> 00:37:09,470 You cause a positive deflection here. 579 00:37:09,470 --> 00:37:11,250 It compresses the spring. 580 00:37:11,250 --> 00:37:14,700 Does the spring push back or not? 581 00:37:14,700 --> 00:37:18,182 So if I'm moving upwards, which way is the spring pushing? 582 00:37:18,182 --> 00:37:18,890 All right. 583 00:37:24,040 --> 00:37:26,210 But if I'm pushing upwards, which way is 584 00:37:26,210 --> 00:37:27,920 the spring pushing on the mass? 585 00:37:27,920 --> 00:37:29,740 Up. 586 00:37:29,740 --> 00:37:33,210 So this one gives me a ky up. 587 00:37:33,210 --> 00:37:38,750 And the dashpot does a similar thing-- cy dot up. 588 00:37:38,750 --> 00:37:41,870 And there's also an mg here. 589 00:37:41,870 --> 00:37:46,770 But there's also a kx static, if you will, up. 590 00:37:46,770 --> 00:37:47,530 And they cancel. 591 00:37:47,530 --> 00:37:48,140 We know that. 592 00:37:48,140 --> 00:37:51,180 So we don't have to deal with the mg terms. 593 00:37:51,180 --> 00:37:52,990 So now we can write our equation of motion. 594 00:37:52,990 --> 00:37:57,880 And the equation of motion for this system 595 00:37:57,880 --> 00:38:01,100 is the mass times the acceleration. 596 00:38:01,100 --> 00:38:03,380 That's got to equal to the sum of all the 597 00:38:03,380 --> 00:38:06,550 external forces-- one, two, three, four of them. 598 00:38:06,550 --> 00:38:10,160 And I'll just save a little time and board space. 599 00:38:10,160 --> 00:38:15,700 I'll put them on the correct sides of the equation. 600 00:38:15,700 --> 00:38:19,730 So these are the x-- put the x terms on the left side. 601 00:38:19,730 --> 00:38:23,250 cx dot plus kx. 602 00:38:23,250 --> 00:38:32,080 And on the right-hand side, I get ky plus cy dot. 603 00:38:32,080 --> 00:38:34,120 This is my excitation. 604 00:38:34,120 --> 00:38:35,980 That's the floor motion. 605 00:38:35,980 --> 00:38:38,170 And this is my response on the left-hand side. 606 00:38:40,690 --> 00:38:46,880 So I'm going to let y of t, the input, 607 00:38:46,880 --> 00:38:56,000 be some y0 real positive times e to the i omega t. 608 00:38:56,000 --> 00:39:01,080 And I'm going to assume that the response is some x, probably 609 00:39:01,080 --> 00:39:05,330 complex, e to the i omega t. 610 00:39:07,950 --> 00:39:10,520 So this is x of t here. 611 00:39:10,520 --> 00:39:12,931 Equals some x I don't know e to the i omega t. 612 00:39:12,931 --> 00:39:15,055 And I'm going to plug those two into this equation. 613 00:39:26,990 --> 00:39:29,470 If I just do that directly, x is on the left side. 614 00:39:29,470 --> 00:39:31,020 y is on the right side. 615 00:39:31,020 --> 00:39:41,795 Then I find minus omega squared m plus i omega c plus k, just 616 00:39:41,795 --> 00:39:57,710 like before, xe to the i omega t equals k plus i omega c y0 617 00:39:57,710 --> 00:40:01,320 e to the i omega t. 618 00:40:01,320 --> 00:40:07,340 And nicely, I can for now get rid of the time-dependent part. 619 00:40:07,340 --> 00:40:18,120 And I can solve for the response that I'm 620 00:40:18,120 --> 00:40:22,450 looking for-- x over the input is real and positive, 621 00:40:22,450 --> 00:40:24,890 amplitude of vibration of the floor. 622 00:40:24,890 --> 00:40:31,410 And that I will call Hx/y of omega, a transfer 623 00:40:31,410 --> 00:40:35,090 function, probably complex, that I can then 624 00:40:35,090 --> 00:40:38,920 deal with like I did above. 625 00:40:38,920 --> 00:40:42,820 And when I finish manipulating things, substituting 626 00:40:42,820 --> 00:40:47,560 in zetas and omega n squareds and that kind of thing, 627 00:40:47,560 --> 00:40:53,600 this becomes-- well, first, I'll write it this way. 628 00:40:53,600 --> 00:40:59,571 I can write this as a magnitude times an e to the minus i phi 629 00:40:59,571 --> 00:41:00,070 again. 630 00:41:00,070 --> 00:41:02,050 That's where I want to go. 631 00:41:09,230 --> 00:41:18,710 And when I do that, 1 plus-- a little messier-- 2 zeta 632 00:41:18,710 --> 00:41:27,260 omega over omega n squared square root. 633 00:41:27,260 --> 00:41:28,425 This is just the numerator. 634 00:41:31,340 --> 00:41:36,890 And the denominator is just the same as the other single degree 635 00:41:36,890 --> 00:41:39,390 of freedom things. 636 00:41:39,390 --> 00:41:44,780 1 minus omega squared over omega n 637 00:41:44,780 --> 00:41:50,400 squared squared plus 2 zeta omega 638 00:41:50,400 --> 00:42:00,070 over omega n squared square root e to the minus i phi. 639 00:42:00,070 --> 00:42:02,350 So now it's the transfer function 640 00:42:02,350 --> 00:42:05,940 as before except the denominator's a little messy. 641 00:42:05,940 --> 00:42:09,750 And there's no 1/k. 642 00:42:09,750 --> 00:42:14,010 And I am going to have a messier expression for phi here. 643 00:42:26,490 --> 00:42:35,490 So there is something wrong with one of the boards this morning. 644 00:43:28,400 --> 00:43:30,120 Kind of messy, complicated. 645 00:43:30,120 --> 00:43:30,930 Do I ever use it? 646 00:43:30,930 --> 00:43:31,430 Rarely. 647 00:43:34,290 --> 00:43:37,920 What's important in these things and what isn't-- really 648 00:43:37,920 --> 00:43:40,750 what's important when you're just trying to get a quick 649 00:43:40,750 --> 00:43:43,700 solution to vibration isolate something, 650 00:43:43,700 --> 00:43:45,950 you really want to know what this is going to come out 651 00:43:45,950 --> 00:43:47,910 looking like. 652 00:43:47,910 --> 00:43:52,090 You're trying to make the response x small 653 00:43:52,090 --> 00:43:53,635 compared to the input. 654 00:43:53,635 --> 00:43:54,760 That's the whole objective. 655 00:43:58,370 --> 00:44:01,775 Right now the table might be moving a half a millimeter 656 00:44:01,775 --> 00:44:03,760 or something like that, but this thing's 657 00:44:03,760 --> 00:44:07,410 moving out here five or six or seven millimeters, 658 00:44:07,410 --> 00:44:09,070 5 or 10 times that. 659 00:44:09,070 --> 00:44:13,390 And what we'd really like is if the table's moving 660 00:44:13,390 --> 00:44:15,120 a millimeter, you'd like this thing 661 00:44:15,120 --> 00:44:18,070 out here moving 1/10th of a millimeter. 662 00:44:18,070 --> 00:44:20,690 So the real objective here is to make this small. 663 00:44:20,690 --> 00:44:23,020 It's the magnitude you care about. 664 00:44:23,020 --> 00:44:27,620 Phase you rarely even want to know or need to know. 665 00:44:33,650 --> 00:44:36,200 So we're going to do a sample calculation. 666 00:44:36,200 --> 00:44:37,850 Let's give an example here. 667 00:44:42,820 --> 00:44:50,280 So the source is at 20 hertz. 668 00:44:50,280 --> 00:44:52,420 So your unbalanced pump, your unbalanced rotor. 669 00:44:52,420 --> 00:44:52,920 Yeah. 670 00:44:52,920 --> 00:44:54,836 AUDIENCE: How do we know in the previous thing 671 00:44:54,836 --> 00:45:01,420 that the frequency of oscillation has to be the same? 672 00:45:01,420 --> 00:45:03,670 Like, why wouldn't it be twice that? 673 00:45:03,670 --> 00:45:04,270 PROFESSOR: OK. 674 00:45:04,270 --> 00:45:06,761 That's a great question. 675 00:45:06,761 --> 00:45:11,340 And I haven't mentioned this before, and I intended to. 676 00:45:11,340 --> 00:45:13,210 These systems that we're looking at 677 00:45:13,210 --> 00:45:18,220 are linear systems, which is where we started the other day. 678 00:45:18,220 --> 00:45:21,840 Linear systems have some interesting and very useful 679 00:45:21,840 --> 00:45:23,250 properties that we depend upon. 680 00:45:23,250 --> 00:45:28,320 One was, I said, force one gives you output one, 681 00:45:28,320 --> 00:45:29,800 force two gives you output two. 682 00:45:29,800 --> 00:45:33,860 Force one plus two gives you the sum of the outputs. 683 00:45:33,860 --> 00:45:38,110 The other feature of a linear system 684 00:45:38,110 --> 00:45:42,450 is steady state response after the transients have died away. 685 00:45:42,450 --> 00:45:47,710 If the frequency of the input is at 21.5 Hertz, 686 00:45:47,710 --> 00:45:53,370 the frequency of the output is at 21.5 Hertz, period. 687 00:45:53,370 --> 00:45:55,290 Linear systems, the frequency of the input 688 00:45:55,290 --> 00:45:57,340 is equal to the frequency of the output. 689 00:45:57,340 --> 00:45:59,752 That's a really important little factoid to remember. 690 00:46:02,710 --> 00:46:07,035 So I turn on the pump, the pump's running at 20 Hertz. 691 00:46:10,550 --> 00:46:18,530 20 Hertz times 60 is 1,200 RPM, very common motor speed. 692 00:46:18,530 --> 00:46:20,120 So the pump's running at 20 Hertz. 693 00:46:20,120 --> 00:46:21,820 So that fan, it's got an imbalance. 694 00:46:21,820 --> 00:46:23,740 So that means you're putting excitation 695 00:46:23,740 --> 00:46:27,230 into the floor at 20 Hertz. 696 00:46:27,230 --> 00:46:42,290 And I want to reduce the vibration at the microscope 697 00:46:42,290 --> 00:46:47,700 by 90%. 698 00:46:47,700 --> 00:46:49,950 What that really means is that my goal 699 00:46:49,950 --> 00:46:57,710 is that the magnitude of x/y is 0.1. 700 00:46:57,710 --> 00:47:00,380 And that's the magnitude of this transfer function, Hx/y. 701 00:47:03,530 --> 00:47:07,170 So I want this transfer function to be 0.1. 702 00:47:07,170 --> 00:47:08,340 So just look at the picture. 703 00:47:11,340 --> 00:47:15,050 Can I get that answer to the left of the peak? 704 00:47:17,880 --> 00:47:20,450 And what this plot shows you is this magnitude 705 00:47:20,450 --> 00:47:24,209 of the transfer function, for a variety of values, a damping. 706 00:47:24,209 --> 00:47:25,750 And of course, the lower the damping, 707 00:47:25,750 --> 00:47:28,460 the higher the peak gets at resonance. 708 00:47:28,460 --> 00:47:29,100 Right? 709 00:47:29,100 --> 00:47:32,320 So no matter what the damping is, 710 00:47:32,320 --> 00:47:35,660 what is the curves all go to in the left-hand side? 711 00:47:35,660 --> 00:47:38,860 They go to 1. 712 00:47:38,860 --> 00:47:42,630 And that's really saying the static response of this system 713 00:47:42,630 --> 00:47:49,880 is if you deflect the floor an inch, the table moves with it. 714 00:47:49,880 --> 00:47:52,690 Everything has to move together when you get down 715 00:47:52,690 --> 00:47:56,360 to 0 frequency input. 716 00:47:56,360 --> 00:47:58,210 So everything goes to 1 on the left. 717 00:47:58,210 --> 00:48:00,610 You go through resonance at omega 718 00:48:00,610 --> 00:48:02,200 equals a natural frequency. 719 00:48:02,200 --> 00:48:05,550 But out to the right, as the excitation frequency 720 00:48:05,550 --> 00:48:08,090 gets higher than the natural frequency, 721 00:48:08,090 --> 00:48:10,280 the response drops off below 1. 722 00:48:10,280 --> 00:48:13,650 Which one drops the fastest? 723 00:48:13,650 --> 00:48:18,741 As you increase omega over omega n beyond 1, 724 00:48:18,741 --> 00:48:20,740 there's a whole mess of curves to the right that 725 00:48:20,740 --> 00:48:21,375 blend together. 726 00:48:21,375 --> 00:48:25,870 And they differ only in damping. 727 00:48:25,870 --> 00:48:28,000 Can you tell which one is the-- let's say 728 00:48:28,000 --> 00:48:31,690 if you go to-- at three, there, the response 729 00:48:31,690 --> 00:48:36,770 is at 0.1 for the lowest curve on that curve, right? 730 00:48:36,770 --> 00:48:40,354 And that's the one with no damping. 731 00:48:40,354 --> 00:48:42,461 It's a little counter-intuitive, right? 732 00:48:42,461 --> 00:48:42,960 All right. 733 00:48:42,960 --> 00:48:44,250 Well, let's come back to it. 734 00:48:44,250 --> 00:48:49,950 Damping does help, but not at this point. 735 00:48:49,950 --> 00:48:57,250 So we need to find a value of omega 736 00:48:57,250 --> 00:49:02,290 over omega n which is greater than 1 that satisfies this. 737 00:49:19,520 --> 00:49:20,920 That's what we're after. 738 00:49:20,920 --> 00:49:23,372 And this is kind of messy to work with. 739 00:49:23,372 --> 00:49:24,830 And since I know the one that works 740 00:49:24,830 --> 00:49:27,470 the best is the one with no damping, 741 00:49:27,470 --> 00:49:29,182 we'll solve the no damping one first, 742 00:49:29,182 --> 00:49:30,890 because it makes the algebra really easy. 743 00:49:30,890 --> 00:49:32,130 And then we can go back and say, now, 744 00:49:32,130 --> 00:49:33,880 what happens if you add some damping? 745 00:49:33,880 --> 00:49:36,060 So for the case there's no damping, 746 00:49:36,060 --> 00:49:39,210 the numerator goes to 1. 747 00:49:39,210 --> 00:49:42,690 The denominator goes to just 1 over 1 748 00:49:42,690 --> 00:49:45,242 minus omega squared over omega n squared. 749 00:50:08,600 --> 00:50:10,130 So it becomes that. 750 00:50:10,130 --> 00:50:11,480 That simple. 751 00:50:11,480 --> 00:50:15,505 And because I want to work with this ratio bigger than 1, 752 00:50:15,505 --> 00:50:16,880 I don't want this to be negative. 753 00:50:16,880 --> 00:50:19,841 And I want to mess with-- keep carrying along absolute value 754 00:50:19,841 --> 00:50:20,340 signs. 755 00:50:20,340 --> 00:50:24,690 This is the same thing as 1 over omega squared 756 00:50:24,690 --> 00:50:28,590 over omega n squared minus 1. 757 00:50:28,590 --> 00:50:30,560 I just reverse this, because I know 758 00:50:30,560 --> 00:50:34,380 we're going to deal only with the ones greater than 1 here. 759 00:50:34,380 --> 00:50:38,185 And I need this to be equal to 0.1. 760 00:50:38,185 --> 00:50:39,185 And that's just algebra. 761 00:50:39,185 --> 00:50:41,570 You could solve that. 762 00:50:41,570 --> 00:50:49,940 This implies that omega over omega n equals root 11, 763 00:50:49,940 --> 00:50:50,470 I recall. 764 00:50:58,150 --> 00:51:03,340 And that is 3.31. 765 00:51:03,340 --> 00:51:05,870 So this is saying on that curve, if you go out 766 00:51:05,870 --> 00:51:09,070 to omega over omega n equals 3.31 right about where 767 00:51:09,070 --> 00:51:16,850 that arrow is, the curve for zero damping drops down to 0.1. 768 00:51:16,850 --> 00:51:19,190 And now if, at that frequency-- ah. 769 00:51:19,190 --> 00:51:27,130 So that means we have to design the spring support such 770 00:51:27,130 --> 00:51:34,430 that omega n is equal to omega over 3.31. 771 00:51:34,430 --> 00:51:39,211 But omega-- where'd we start? 772 00:51:39,211 --> 00:51:41,615 So F equals 20 Hertz. 773 00:51:44,170 --> 00:51:48,555 Omega equals 2 pi f. 774 00:51:48,555 --> 00:51:49,680 Do I have that number here? 775 00:51:54,550 --> 00:51:58,890 No, but-- so this tells me that I need a natural frequency that 776 00:51:58,890 --> 00:52:09,980 is omega over 3.31, or I need an fn that is f over 3.31 777 00:52:09,980 --> 00:52:15,430 is 20 Hertz over 3.31. 778 00:52:15,430 --> 00:52:18,100 And that number I do have. 779 00:52:18,100 --> 00:52:21,840 6.04 Hertz. 780 00:52:21,840 --> 00:52:25,480 So I need a support whose natural frequency 781 00:52:25,480 --> 00:52:30,920 is 20 Hertz divided by 3.31. 782 00:52:30,920 --> 00:52:37,350 I need a support whose natural frequency is 6.04 Hertz. 783 00:52:37,350 --> 00:52:41,660 And that's how you go about designing a flexible base 784 00:52:41,660 --> 00:52:44,090 to isolate something from vibration 785 00:52:44,090 --> 00:52:47,390 of whatever it's sitting on. 786 00:52:47,390 --> 00:52:47,900 All right. 787 00:52:50,740 --> 00:52:56,200 So my f here, 20 Hertz. 788 00:52:56,200 --> 00:53:01,610 But my fn needs to be 6.04 Hertz. 789 00:53:01,610 --> 00:53:05,110 That implies multiply by 2 pi. 790 00:53:05,110 --> 00:53:12,100 I'm looking for 37.96 radians per second. 791 00:53:12,100 --> 00:53:17,700 And that's equal to square root of k/M. So now what's the M? 792 00:53:17,700 --> 00:53:20,690 Well, it's whatever the mass of the microscope plus its base. 793 00:53:20,690 --> 00:53:25,060 Whatever is being supported by the springs 794 00:53:25,060 --> 00:53:26,800 will have that mass. 795 00:53:26,800 --> 00:53:29,810 You have to choose the k. 796 00:53:29,810 --> 00:53:37,480 So let's say that M total for this system is 20 kilograms. 797 00:53:40,760 --> 00:53:44,230 Solve this equation for k. 798 00:53:44,230 --> 00:53:56,756 And that implies that k is 28,827 Newtons per meter. 799 00:53:56,756 --> 00:53:57,256 OK? 800 00:54:02,220 --> 00:54:06,720 So if we were to design this system-- 801 00:54:06,720 --> 00:54:10,000 and it really mounts up to in the case of this. 802 00:54:10,000 --> 00:54:10,530 Let's see. 803 00:54:10,530 --> 00:54:12,780 Beams. 804 00:54:12,780 --> 00:54:17,380 The stiffness of a beam-- ah, that's a good. 805 00:54:17,380 --> 00:54:19,590 We'll do this. 806 00:54:19,590 --> 00:54:22,760 We have a cantilever here. 807 00:54:22,760 --> 00:54:25,420 And we've got a mass on the end. 808 00:54:25,420 --> 00:54:30,350 But most of you have been taking 2001. 809 00:54:30,350 --> 00:54:33,551 If you put a force out here, P, what's 810 00:54:33,551 --> 00:54:35,300 the deflection at the end of a cantilever? 811 00:54:35,300 --> 00:54:37,170 AUDIENCE: [INAUDIBLE]. 812 00:54:37,170 --> 00:54:37,880 PROFESSOR: OK. 813 00:54:37,880 --> 00:54:42,720 So delta is PL cubed over 3EI. 814 00:54:45,570 --> 00:54:54,660 And the load, this force, is equal to some k equivalent 815 00:54:54,660 --> 00:54:56,610 times delta, right? 816 00:54:56,610 --> 00:54:58,592 This is just a spring. 817 00:54:58,592 --> 00:55:03,430 And k times the displacement is the force it takes to do it. 818 00:55:03,430 --> 00:55:05,600 So P's my force. 819 00:55:05,600 --> 00:55:08,190 The spring constant is somehow associated 820 00:55:08,190 --> 00:55:10,370 with the rest of this stuff. 821 00:55:10,370 --> 00:55:22,610 So if I solve for P over delta, I get 3EI over L cubed. 822 00:55:22,610 --> 00:55:23,560 OK? 823 00:55:23,560 --> 00:55:28,590 So if I'm running right at the natural frequency here 824 00:55:28,590 --> 00:55:31,700 and I want to reduce this to a 1/10th of its motion, 825 00:55:31,700 --> 00:55:35,400 I need to change the spring constant of this cantilever 826 00:55:35,400 --> 00:55:41,430 by a factor of-- well, I need to change the natural frequency 827 00:55:41,430 --> 00:55:51,960 by a factor of 3.31. 828 00:55:51,960 --> 00:55:59,007 So my k equivalent here is 3EI over L cubed. 829 00:55:59,007 --> 00:56:00,840 And that's what would go into this equation. 830 00:56:04,880 --> 00:56:08,310 But I know that I have a natural frequency right now. 831 00:56:08,310 --> 00:56:11,900 I want it to go down by a factor of 3.31. 832 00:56:11,900 --> 00:56:15,850 So that means I need to decrease k 833 00:56:15,850 --> 00:56:22,250 such that the square root of k goes down by the factor 3.31. 834 00:56:22,250 --> 00:56:25,230 So how much do I have to change the length? 835 00:56:25,230 --> 00:56:38,060 Probably something like the square root of 3.31. 836 00:56:38,060 --> 00:56:38,880 Roughly 2. 837 00:56:43,424 --> 00:56:45,090 So if I double the length of this thing, 838 00:56:45,090 --> 00:56:46,423 do you think it's going to work? 839 00:56:48,832 --> 00:56:51,279 If I double the length of this thing and turn it back on, 840 00:56:51,279 --> 00:56:53,195 then we shouldn't see much motion out of this. 841 00:56:53,195 --> 00:56:56,402 [VIBRATING] 842 00:57:01,600 --> 00:57:02,490 That's moving a lot. 843 00:57:07,610 --> 00:57:11,210 It's moving a tiny, tiny bit. 844 00:57:11,210 --> 00:57:12,060 So it works. 845 00:57:17,170 --> 00:57:20,460 So that's one step of vibration isolation. 846 00:57:20,460 --> 00:57:25,010 Now I'm going to show you a vibration engineer trick, which 847 00:57:25,010 --> 00:57:27,009 is a very handy thing to know. 848 00:57:35,500 --> 00:57:41,917 Where's my strong magnet here? 849 00:57:41,917 --> 00:57:43,750 So I've got another beam just like this one. 850 00:57:46,330 --> 00:57:50,480 I've got a pretty massive magnet on it. 851 00:57:50,480 --> 00:57:52,850 So it makes another cantilever beam 852 00:57:52,850 --> 00:57:56,078 just like I got over there. 853 00:57:56,078 --> 00:57:57,410 OK? 854 00:57:57,410 --> 00:58:04,760 So I claim that with just a ruler, 855 00:58:04,760 --> 00:58:07,130 if I clamp this down at some length, 856 00:58:07,130 --> 00:58:10,040 I claim, with just a ruler, I can predict 857 00:58:10,040 --> 00:58:11,699 the natural frequency of that. 858 00:58:18,092 --> 00:58:19,716 Take a couple of minutes and see if you 859 00:58:19,716 --> 00:58:20,966 could figure out how to do it. 860 00:58:23,540 --> 00:58:26,080 Think about that. 861 00:58:26,080 --> 00:58:27,860 Just a ruler. 862 00:58:27,860 --> 00:58:31,030 Measurements that I can make. 863 00:58:31,030 --> 00:58:32,702 I don't know how long it actually is. 864 00:58:32,702 --> 00:58:33,910 I don't know how thick it is. 865 00:58:33,910 --> 00:58:36,490 I know it's steel, but you just don't have enough information 866 00:58:36,490 --> 00:58:40,200 to compute 3EI over L cubed. 867 00:58:40,200 --> 00:58:44,057 But simply with a ruler, I'm going to be able to do this. 868 00:58:44,057 --> 00:58:44,640 Talk about it. 869 00:58:44,640 --> 00:58:48,605 Think about that while I set up the experiment. 870 01:00:16,490 --> 01:00:17,800 OK. 871 01:00:17,800 --> 01:00:18,870 Who's got it figured out? 872 01:00:21,711 --> 01:00:23,210 Anybody want to take a shot at this? 873 01:00:31,710 --> 01:00:35,110 So there's my beam. 874 01:00:35,110 --> 01:00:36,890 I put the weight on it. 875 01:00:36,890 --> 01:00:40,430 What does the beam do statically? 876 01:00:40,430 --> 01:00:41,410 Bends a little, right? 877 01:00:44,060 --> 01:00:55,230 kx static equals Mg, right? 878 01:00:55,230 --> 01:00:56,760 Has to. 879 01:00:56,760 --> 01:01:01,660 So x static is what I'm calling delta here. 880 01:01:01,660 --> 01:01:06,605 So k delta equals Mg. 881 01:01:06,605 --> 01:01:10,205 k equals Mg over delta. 882 01:01:13,220 --> 01:01:33,600 Natural frequency equals square root of k/M. Incredibly 883 01:01:33,600 --> 01:01:35,450 simple, huh? 884 01:01:35,450 --> 01:01:38,130 So what's the experiment that I would-- 885 01:01:38,130 --> 01:01:41,780 what measurement would I make? 886 01:01:41,780 --> 01:01:44,060 Delta, right? 887 01:01:44,060 --> 01:01:45,490 Put my ruler up there. 888 01:01:45,490 --> 01:01:48,220 I measure its static position like that. 889 01:01:48,220 --> 01:01:51,520 Then I put my mass on it, and I measure the static position 890 01:01:51,520 --> 01:01:52,080 again. 891 01:01:52,080 --> 01:01:54,090 I measure the delta. 892 01:01:54,090 --> 01:01:56,260 And I get a prediction. 893 01:01:56,260 --> 01:01:59,720 And I did this in my office. 894 01:02:17,490 --> 01:02:22,230 And the delta that I measured-- I actually 895 01:02:22,230 --> 01:02:23,700 set it at a particular length. 896 01:02:23,700 --> 01:02:27,100 It was 18 centimeters. 897 01:02:27,100 --> 01:02:38,710 Delta measured, I think, 0.5 centimeters, or 0.005 meters. 898 01:02:38,710 --> 01:02:43,330 And if you compute omega n then equals the square root 899 01:02:43,330 --> 01:03:15,918 of 9.81 over 0.005. 900 01:03:15,918 --> 01:03:17,605 And I want this in Hertz. 901 01:03:17,605 --> 01:03:20,320 So I can divide by 2 pi. 902 01:03:20,320 --> 01:03:29,960 This comes out as 7.05 Hertz. 903 01:03:29,960 --> 01:03:44,100 And Fn measured was 6.57. 904 01:03:44,100 --> 01:03:46,540 Pretty good but not perfect. 905 01:03:49,960 --> 01:03:53,584 And it's because I've made an approximation that I 906 01:03:53,584 --> 01:03:54,750 glossed over pretty quickly. 907 01:03:54,750 --> 01:03:56,580 What has been left out of this system that 908 01:03:56,580 --> 01:04:00,900 would cause the measured natural frequency to be lower 909 01:04:00,900 --> 01:04:04,267 than the predicted? 910 01:04:04,267 --> 01:04:05,100 What's been ignored? 911 01:04:05,100 --> 01:04:05,599 Yes. 912 01:04:05,599 --> 01:04:06,659 AUDIENCE: Damping. 913 01:04:06,659 --> 01:04:07,450 PROFESSOR: Damping. 914 01:04:07,450 --> 01:04:08,420 Ah. 915 01:04:08,420 --> 01:04:10,330 Maybe. 916 01:04:10,330 --> 01:04:12,378 How much damping do we have in this system? 917 01:04:15,660 --> 01:04:18,890 Probably at least 10 cycles to the k halfway, right? 918 01:04:18,890 --> 01:04:22,620 Certainly less than 1%. 919 01:04:22,620 --> 01:04:26,590 The damped natural frequency is equal to the natural frequency 920 01:04:26,590 --> 01:04:29,530 of the square root of 1 minus theta squared. 921 01:04:29,530 --> 01:04:32,000 So this is something like way less than half 922 01:04:32,000 --> 01:04:32,930 a percent difference. 923 01:04:32,930 --> 01:04:34,263 So that wouldn't account for it. 924 01:04:34,263 --> 01:04:36,160 That's considerably more than half a percent. 925 01:04:36,160 --> 01:04:38,336 So damping couldn't do it. 926 01:04:38,336 --> 01:04:38,836 Yeah. 927 01:04:38,836 --> 01:04:40,570 AUDIENCE: [INAUDIBLE]. 928 01:04:40,570 --> 01:04:43,060 PROFESSOR: Ah, the mass of the bar. 929 01:04:43,060 --> 01:04:47,570 Does this flexure have mass? 930 01:04:47,570 --> 01:04:48,070 Yeah. 931 01:04:48,070 --> 01:04:50,970 It's probably on the order of if you stack them all up 932 01:04:50,970 --> 01:04:53,950 and compared to that, it might even be as much as half 933 01:04:53,950 --> 01:04:56,120 the mass of the end. 934 01:04:56,120 --> 01:04:57,590 And as it vibrates back and forth, 935 01:04:57,590 --> 01:04:59,940 does it have kinetic energy? 936 01:04:59,940 --> 01:05:00,680 Yeah. 937 01:05:00,680 --> 01:05:04,190 We've ignored the kinetic energy of the mass. 938 01:05:04,190 --> 01:05:06,940 And in fact, that's the principal error here. 939 01:05:06,940 --> 01:05:08,280 We've left out the mass. 940 01:05:08,280 --> 01:05:09,780 There's actually a pretty simple way 941 01:05:09,780 --> 01:05:13,410 to-- using energy and just thinking in Lagrange terms, 942 01:05:13,410 --> 01:05:15,340 you can account for the energy of the mass 943 01:05:15,340 --> 01:05:17,380 in this single degree of freedom system 944 01:05:17,380 --> 01:05:20,090 and get a very accurate prediction. 945 01:05:20,090 --> 01:05:21,180 We won't do that today. 946 01:05:21,180 --> 01:05:25,618 But I think we'll do that before the term's out. 947 01:05:25,618 --> 01:05:27,530 OK. 948 01:05:27,530 --> 01:05:35,540 This applies to any simple mass spring system 949 01:05:35,540 --> 01:05:39,240 in the presence of gravity. 950 01:05:39,240 --> 01:05:41,120 So here's a mass. 951 01:05:43,679 --> 01:05:45,470 And actually, we're doing the problem today 952 01:05:45,470 --> 01:05:46,680 where I'm moving the base. 953 01:05:46,680 --> 01:05:47,990 So here's its base. 954 01:05:47,990 --> 01:05:50,070 So this is the table moving. 955 01:05:50,070 --> 01:05:53,880 And if I do this, it clearly makes that move. 956 01:05:53,880 --> 01:05:57,300 If I do this really fast, it doesn't move very much. 957 01:05:57,300 --> 01:06:00,570 If I do it close to the natural frequency, it moves a lot. 958 01:06:00,570 --> 01:06:06,290 If I move it very slowly, as I go up one unit, 959 01:06:06,290 --> 01:06:08,390 this follows me exactly. 960 01:06:08,390 --> 01:06:11,570 That's why that plot goes to 1. 961 01:06:11,570 --> 01:06:15,240 At very, very low frequency, the support and the mass 962 01:06:15,240 --> 01:06:18,250 move exactly together. 963 01:06:18,250 --> 01:06:22,485 At very high frequency-- if I can stop the transient-- 964 01:06:22,485 --> 01:06:23,826 I can't do it very well. 965 01:06:23,826 --> 01:06:24,950 The mass doesn't move much. 966 01:06:24,950 --> 01:06:25,825 The base moves a lot. 967 01:06:25,825 --> 01:06:27,700 And at resonance, it goes nuts. 968 01:06:27,700 --> 01:06:28,410 OK. 969 01:06:28,410 --> 01:06:31,800 The unstretched length of this spring is about seven inches. 970 01:06:31,800 --> 01:06:33,500 The square root of g over delta, I 971 01:06:33,500 --> 01:06:36,150 ought to be able to predict this. 972 01:06:36,150 --> 01:06:39,960 So I did a quick calculation on that. 973 01:06:39,960 --> 01:06:41,310 It was like 1% error. 974 01:06:43,870 --> 01:06:49,840 I measured it at 7.36 radians a second. 975 01:06:49,840 --> 01:06:58,477 And I predicted it at 7.43 measured 7.36. 976 01:06:58,477 --> 01:07:00,060 Same kind of thing-- ignoring the mass 977 01:07:00,060 --> 01:07:01,460 of the spring a little bit. 978 01:07:01,460 --> 01:07:05,480 So g over delta is a great little thing to remember. 979 01:07:05,480 --> 01:07:05,980 OK. 980 01:07:13,030 --> 01:07:18,730 So we have done all but one. 981 01:07:18,730 --> 01:07:20,460 Everything we've started out with today, 982 01:07:20,460 --> 01:07:22,280 we've said there's three ways to fix this, 983 01:07:22,280 --> 01:07:24,040 and came up with a fourth way. 984 01:07:24,040 --> 01:07:28,500 So in this case, soften the spring support a lot, 985 01:07:28,500 --> 01:07:33,820 so that the natural frequency is way less than the excitation. 986 01:07:33,820 --> 01:07:39,350 We said, what about spring supporting, softening, flexibly 987 01:07:39,350 --> 01:07:42,585 amounting this source, so that it doesn't put vibration 988 01:07:42,585 --> 01:07:43,210 on the table? 989 01:07:43,210 --> 01:07:45,750 That's the piece we haven't addressed. 990 01:07:45,750 --> 01:07:47,664 So let's look into that problem now. 991 01:08:00,260 --> 01:08:07,220 So here's our source, some rotating mass eccentricity 992 01:08:07,220 --> 01:08:12,510 causing an excitation. 993 01:08:12,510 --> 01:08:16,810 So this has a force F0 e to the i omega 994 01:08:16,810 --> 01:08:20,270 t, which is coming from the rotating mass. 995 01:08:20,270 --> 01:08:23,819 And it applies to the floor, through the dashpot 996 01:08:23,819 --> 01:08:27,660 in the springs, some FT, I'll call it, 997 01:08:27,660 --> 01:08:33,229 F transmitted to the floor, e to the i omega t. 998 01:08:33,229 --> 01:08:40,390 And I want to know-- I need the H force transmitted per unit 999 01:08:40,390 --> 01:08:44,014 force input transfer function. 1000 01:08:44,014 --> 01:08:45,180 That's what I'm looking for. 1001 01:08:56,010 --> 01:08:59,420 So now free-body diagram. 1002 01:08:59,420 --> 01:09:01,020 Now we're going to make an assumption. 1003 01:09:01,020 --> 01:09:06,880 We're going to assume that the motion of the floor, 1004 01:09:06,880 --> 01:09:11,069 which we'll call y of t, assume that y is much, 1005 01:09:11,069 --> 01:09:12,824 much less than x. 1006 01:09:12,824 --> 01:09:14,420 It's generally true. 1007 01:09:14,420 --> 01:09:17,580 Whatever's shaking like crazy, the table's 1008 01:09:17,580 --> 01:09:19,410 not moving much underneath it. 1009 01:09:19,410 --> 01:09:22,520 So I'm going to assume, for the purposes of calculating forces, 1010 01:09:22,520 --> 01:09:25,970 that this is 0. 1011 01:09:25,970 --> 01:09:34,590 So for the motion x, what is the force applied to the floor? 1012 01:09:34,590 --> 01:09:35,760 So F of t. 1013 01:09:41,880 --> 01:09:46,319 If you have a positive displacement x, 1014 01:09:46,319 --> 01:09:49,149 the force is kx. 1015 01:09:49,149 --> 01:09:53,330 You have a positive velocity x, the force pulling up 1016 01:09:53,330 --> 01:09:57,700 on the floor through the dashpot is cx dot. 1017 01:09:57,700 --> 01:10:03,970 So the other way of saying that is here's 1018 01:10:03,970 --> 01:10:06,240 our free-body diagram. 1019 01:10:06,240 --> 01:10:11,680 Here's our F0 e to the i omega t pulling up. 1020 01:10:11,680 --> 01:10:14,800 It responds at some x. 1021 01:10:14,800 --> 01:10:20,230 And the resulting forces through the spring 1022 01:10:20,230 --> 01:10:23,540 and the dashpot we know are kx and cx 1023 01:10:23,540 --> 01:10:27,340 dot opposing the motion x. 1024 01:10:27,340 --> 01:10:31,390 Well, by third law, if these are the forces 1025 01:10:31,390 --> 01:10:34,360 on the spring and the dashpot, then down here on the floor, 1026 01:10:34,360 --> 01:10:39,890 you better have some equal and opposite forces, kx and cx dot. 1027 01:10:39,890 --> 01:10:43,540 So this force produces a motion x. 1028 01:10:43,540 --> 01:10:48,310 The motion x produces forces in the mass in the spring, which 1029 01:10:48,310 --> 01:10:53,180 make the force on the floor, the spring 1030 01:10:53,180 --> 01:10:55,610 force, and the dashpot force. 1031 01:10:55,610 --> 01:10:56,110 OK. 1032 01:11:02,000 --> 01:11:11,432 So this Ft is-- I want to write it here. 1033 01:11:18,800 --> 01:11:22,460 That's all that is-- positive. 1034 01:11:22,460 --> 01:11:29,080 And I'm going to assume a solution 1035 01:11:29,080 --> 01:11:35,800 that we know to work for x, which is xe to the i omega t. 1036 01:11:35,800 --> 01:11:37,790 We've plugged it in before. 1037 01:11:37,790 --> 01:11:43,510 So I plug that in here, I get a k plus i omega 1038 01:11:43,510 --> 01:11:51,720 c, xe to the i omega t. 1039 01:11:51,720 --> 01:11:54,710 So I can just express my force on the floor 1040 01:11:54,710 --> 01:11:56,715 in terms of the motion x. 1041 01:12:34,260 --> 01:12:38,500 And I'm looking for a transfer function for force transmitted 1042 01:12:38,500 --> 01:12:40,490 over force in. 1043 01:12:40,490 --> 01:12:46,760 But force transmitted is my k plus i omega 1044 01:12:46,760 --> 01:12:52,550 c, xe to the i omega t. 1045 01:12:52,550 --> 01:12:59,630 And the force in is F0 e to the i omega t. 1046 01:12:59,630 --> 01:13:02,210 Cancel out the time-dependent part. 1047 01:13:02,210 --> 01:13:05,800 And it says the transmitted force over the input force 1048 01:13:05,800 --> 01:13:09,420 is this little complex expression times the response 1049 01:13:09,420 --> 01:13:12,210 x over F. But we know what that is. 1050 01:13:12,210 --> 01:13:15,340 That's the transfer function Hx/F. 1051 01:13:15,340 --> 01:13:23,660 So this is k plus i omega c times Hx/F of omega. 1052 01:13:26,450 --> 01:13:33,360 So this gives us a slightly different transfer function. 1053 01:13:33,360 --> 01:13:34,935 Ooh, look at this. 1054 01:13:38,500 --> 01:13:42,070 Before, when we did x/y, we ended up 1055 01:13:42,070 --> 01:13:47,310 with k plus i omega c y e to the i omega t. 1056 01:13:47,310 --> 01:14:01,520 And when we did then x/y, we got the same ratio as this. 1057 01:14:01,520 --> 01:14:03,670 Exactly the same thing. 1058 01:14:03,670 --> 01:14:08,940 So I could write all this out, but-- and let's say 1059 01:14:08,940 --> 01:14:11,280 I'll do this. 1060 01:14:11,280 --> 01:14:17,712 Hx/y-- no, no, I won't do that. 1061 01:14:20,250 --> 01:14:23,270 What I'm going to tell you-- if you just work through this now, 1062 01:14:23,270 --> 01:14:28,070 you will find that H force transmitted over force in 1063 01:14:28,070 --> 01:14:30,390 is exactly the same as Hx/y. 1064 01:14:36,080 --> 01:14:43,200 And that what we really care about is what the magnitude is. 1065 01:14:43,200 --> 01:14:47,510 So the magnitude of these two things are the same. 1066 01:14:47,510 --> 01:14:52,740 And in fact, just work out to that same expression as before, 1067 01:14:52,740 --> 01:15:00,870 the 1 plus 2 zeta omega over omega n 1068 01:15:00,870 --> 01:15:08,094 squared square root all over the usual big denominator. 1069 01:15:08,094 --> 01:15:13,370 So conveniently, for vibration isolation, 1070 01:15:13,370 --> 01:15:16,820 the solution to the two problems are exactly the same. 1071 01:15:16,820 --> 01:15:19,220 So if you have that one, you have the transfer function 1072 01:15:19,220 --> 01:15:23,630 xy that was projected on the screen a minute ago, 1073 01:15:23,630 --> 01:15:28,870 it is also the force transmitted to force in transfer function. 1074 01:15:28,870 --> 01:15:31,690 So you just have to remember one. 1075 01:15:31,690 --> 01:15:35,230 And if you now want to-- we said, let's say, 1076 01:15:35,230 --> 01:15:39,180 doubling the length of this just about accomplished the reducing 1077 01:15:39,180 --> 01:15:44,120 the vibration of the microscope by this factor of 10. 1078 01:15:44,120 --> 01:15:47,930 So if I doubled the length of this one, 1079 01:15:47,930 --> 01:15:50,770 I would roughly do the same thing. 1080 01:15:50,770 --> 01:15:53,440 I would change the natural-- this thing is 1081 01:15:53,440 --> 01:15:55,300 right on the natural frequency of this beam. 1082 01:15:55,300 --> 01:15:58,375 That's why it shakes so much. 1083 01:15:58,375 --> 01:16:01,040 And so it is this system. 1084 01:16:01,040 --> 01:16:04,190 It's shaking like crazy, putting force into the table. 1085 01:16:04,190 --> 01:16:07,310 The table is vibrating, causing the other one to move. 1086 01:16:07,310 --> 01:16:13,510 So now if I change this one, then the same kind of idea. 1087 01:16:13,510 --> 01:16:17,210 Maybe roughly double its length. 1088 01:16:17,210 --> 01:16:20,880 Natural frequency diminishes by a factor of 3 or so. 1089 01:16:27,280 --> 01:16:30,195 The vibration of this ought to go way down. 1090 01:16:39,337 --> 01:16:40,920 And actually, our little beam out here 1091 01:16:40,920 --> 01:16:43,908 is picking up more than the other one. 1092 01:16:43,908 --> 01:16:44,408 Shh. 1093 01:16:50,810 --> 01:16:52,777 So this thing is hardly moving at all now. 1094 01:16:56,760 --> 01:17:00,390 So by doing that, we've essentially detuned it. 1095 01:17:00,390 --> 01:17:07,050 This is no longer running at the natural frequency of this base. 1096 01:17:07,050 --> 01:17:08,550 So it's no longer resonant. 1097 01:17:08,550 --> 01:17:11,330 You're way out on the curve to the right. 1098 01:17:11,330 --> 01:17:14,430 So the response of this isn't very much. 1099 01:17:14,430 --> 01:17:17,230 That means it doesn't transmit much force to the base, maybe 1100 01:17:17,230 --> 01:17:18,860 down by a factor of 8 or 10. 1101 01:17:18,860 --> 01:17:24,040 That means the table vibration amplitude drops by that factor. 1102 01:17:24,040 --> 01:17:27,260 Means that the base motion over here is now a factor of 10 1103 01:17:27,260 --> 01:17:29,790 smaller than it was to begin with, 1104 01:17:29,790 --> 01:17:32,210 so that we get a reduction of 10 here. 1105 01:17:32,210 --> 01:17:36,660 And we get another reduction of 10 here, because we detuned it. 1106 01:17:36,660 --> 01:17:39,750 So you might get a factor of 100 reduction by working on both, 1107 01:17:39,750 --> 01:17:40,250 you see. 1108 01:17:40,250 --> 01:17:43,970 You've treated the source and you've treated the receiver. 1109 01:17:43,970 --> 01:17:47,000 But fortunately, they use the same curve. 1110 01:17:47,000 --> 01:17:48,466 So damping. 1111 01:17:48,466 --> 01:17:50,090 When you do vibration isolation, you're 1112 01:17:50,090 --> 01:17:54,145 trying to get well out on this curve to the right. 1113 01:17:54,145 --> 01:17:56,270 So there's a couple of practical engineering things 1114 01:17:56,270 --> 01:17:58,545 that limit how far you can go. 1115 01:17:58,545 --> 01:18:00,420 To get further out on the curve to the right, 1116 01:18:00,420 --> 01:18:03,989 what do you have to do to the spring in the system to get 1117 01:18:03,989 --> 01:18:04,780 stronger or softer? 1118 01:18:08,397 --> 01:18:10,480 You're trying to make the natural frequency-- see, 1119 01:18:10,480 --> 01:18:12,320 the excitation frequency doesn't change. 1120 01:18:12,320 --> 01:18:15,960 In order to get omega over omega n to go bigger and bigger, 1121 01:18:15,960 --> 01:18:17,430 the excitation's staying the same. 1122 01:18:17,430 --> 01:18:19,950 You're having to reduce the natural frequency. 1123 01:18:19,950 --> 01:18:23,350 And so what do you have to do to the spring constant? 1124 01:18:23,350 --> 01:18:24,660 Decrease it. 1125 01:18:24,660 --> 01:18:27,340 What is the practical limit of decreasing 1126 01:18:27,340 --> 01:18:31,190 the spring that supports your pump, 1127 01:18:31,190 --> 01:18:34,340 or your washing machine, or your air conditioner? 1128 01:18:34,340 --> 01:18:36,297 AUDIENCE: [INAUDIBLE]. 1129 01:18:36,297 --> 01:18:37,880 PROFESSOR: Pretty soon it's just going 1130 01:18:37,880 --> 01:18:39,754 to-- if it's too heavy-- you put it on there, 1131 01:18:39,754 --> 01:18:41,670 it's just going to squash the springs, right? 1132 01:18:41,670 --> 01:18:44,920 So you can't-- there's limits to how soft you can make springs 1133 01:18:44,920 --> 01:18:47,652 to support heavy machines. 1134 01:18:47,652 --> 01:18:49,860 So there is a practical limit to how far to the right 1135 01:18:49,860 --> 01:18:50,570 you can go. 1136 01:18:50,570 --> 01:18:53,910 But normally, you get out there as far as you can. 1137 01:18:53,910 --> 01:18:57,410 And then if the real system has damping, 1138 01:18:57,410 --> 01:19:01,030 does it improve or degrade the performance 1139 01:19:01,030 --> 01:19:04,020 of your vibration isolation system? 1140 01:19:04,020 --> 01:19:06,340 Well, the more the damping you have, the higher up you 1141 01:19:06,340 --> 01:19:07,650 are on those curves. 1142 01:19:07,650 --> 01:19:11,320 So the damping decreases the performance. 1143 01:19:11,320 --> 01:19:17,730 But every system has to-- when you first turn on that motor, 1144 01:19:17,730 --> 01:19:19,130 the system has to spin up. 1145 01:19:19,130 --> 01:19:23,220 And you're going to have to go through that resonance, 1146 01:19:23,220 --> 01:19:25,930 so that you want some damping. 1147 01:19:25,930 --> 01:19:29,100 Because if you've got your scanning electron 1148 01:19:29,100 --> 01:19:33,380 microscope or your laser interferometry system set up 1149 01:19:33,380 --> 01:19:39,730 on a spring-supported table, if that table has no damping 1150 01:19:39,730 --> 01:19:42,550 and you walk in the door and bump it, 1151 01:19:42,550 --> 01:19:45,950 it is going to sit there and vibrate all afternoon 1152 01:19:45,950 --> 01:19:48,460 at its natural frequency due to the initial conditions. 1153 01:19:48,460 --> 01:19:52,290 So you need some damping to prevent problems, 1154 01:19:52,290 --> 01:19:54,220 either response to initial conditions, 1155 01:19:54,220 --> 01:19:55,860 or bumping it, or whatever. 1156 01:19:55,860 --> 01:19:59,660 Or even as the system turns on and speeds up, 1157 01:19:59,660 --> 01:20:01,650 it'll have to go through that resonance. 1158 01:20:01,650 --> 01:20:04,370 And it'll vibrate like crazy as it does, and then finally 1159 01:20:04,370 --> 01:20:07,370 settle down at the higher frequency. 1160 01:20:07,370 --> 01:20:08,780 So you need some damping. 1161 01:20:08,780 --> 01:20:12,090 But damping does degrade the steady state performance. 1162 01:20:12,090 --> 01:20:14,050 And I'm out of time. 1163 01:20:14,050 --> 01:20:15,925 And we'll see you in recitation. 1164 01:20:15,925 --> 01:20:17,500 Thanks.