1 00:00:00,090 --> 00:00:02,590 NARRATOR: The following content is provided under a Creative 2 00:00:02,590 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,070 Your support will help MIT OpenCourseWare 4 00:00:06,070 --> 00:00:10,180 continue to offer high quality educational resources for free. 5 00:00:10,180 --> 00:00:12,710 To make a donation or to view additional materials 6 00:00:12,710 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,276 at ocw.mit.edu. 8 00:00:23,266 --> 00:00:25,400 PROFESSOR: All right. 9 00:00:25,400 --> 00:00:28,950 Let's start off a quick list of important concepts 10 00:00:28,950 --> 00:00:31,000 for the week. 11 00:00:31,000 --> 00:01:02,860 Just-- and I'm going to push you right along today. 12 00:01:02,860 --> 00:01:04,097 I want to get this done. 13 00:01:04,097 --> 00:01:04,805 What do you got? 14 00:01:04,805 --> 00:01:06,570 AUDIENCE: Static Equilibrium positions? 15 00:01:06,570 --> 00:01:09,760 PROFESSOR: Static equilibrium, OK. 16 00:01:09,760 --> 00:01:18,280 So EOMs from xstatic. 17 00:01:18,280 --> 00:01:20,570 OK, good. 18 00:01:20,570 --> 00:01:22,190 Another concept for the week? 19 00:01:40,394 --> 00:01:41,870 AUDIENCE: Damping. 20 00:01:41,870 --> 00:01:45,300 PROFESSOR: Damping, yeah. 21 00:01:45,300 --> 00:01:47,440 And what have we've been talking about all week? 22 00:01:47,440 --> 00:01:48,366 AUDIENCE: Vibration. 23 00:01:48,366 --> 00:01:49,690 PROFESSOR: Vibration. 24 00:01:49,690 --> 00:01:50,560 OK. 25 00:01:50,560 --> 00:01:59,415 So vibration, damping, OK. 26 00:01:59,415 --> 00:01:59,915 More? 27 00:02:04,114 --> 00:02:06,655 What's the whole general subject that we've encased vibration 28 00:02:06,655 --> 00:02:07,110 in? 29 00:02:07,110 --> 00:02:08,318 It's an introduction to what? 30 00:02:11,274 --> 00:02:11,940 Have to make it. 31 00:02:11,940 --> 00:02:14,440 We've been making an assumption that we haven't made before. 32 00:02:20,690 --> 00:02:21,952 Yeah, linear. 33 00:02:21,952 --> 00:02:23,410 We've been studying-- this has been 34 00:02:23,410 --> 00:02:33,995 an intro to-- linear systems. 35 00:02:37,220 --> 00:02:39,230 We can only do things like transfer functions, 36 00:02:39,230 --> 00:02:42,860 that sort of thing, by assuming linearity. 37 00:02:42,860 --> 00:02:46,180 Anything else? 38 00:02:46,180 --> 00:02:49,550 What have I just spent the last two lectures deriving for you 39 00:02:49,550 --> 00:02:52,380 and showing you? 40 00:02:52,380 --> 00:02:53,335 Transfer functions. 41 00:03:00,650 --> 00:03:01,900 In fact, I've shown you three. 42 00:03:10,880 --> 00:03:11,440 All right? 43 00:03:11,440 --> 00:03:12,640 All right. 44 00:03:12,640 --> 00:03:15,600 That's beginning to get at what we've been talking about. 45 00:03:15,600 --> 00:03:17,710 So introduction to linear systems. 46 00:03:17,710 --> 00:03:23,210 We've been studying vibration as one implementation 47 00:03:23,210 --> 00:03:25,330 or application of linear systems. 48 00:03:25,330 --> 00:03:28,180 We've been looking at single degree of freedom vibration 49 00:03:28,180 --> 00:03:29,820 problems only so far. 50 00:03:29,820 --> 00:03:33,910 And we've found frequency response functions. 51 00:03:33,910 --> 00:03:36,160 These frequency response function tell you 52 00:03:36,160 --> 00:03:38,800 the response of the system to what kind of an input? 53 00:03:41,850 --> 00:03:43,935 What are the characteristics of the input 54 00:03:43,935 --> 00:03:46,807 that we've been looking at? 55 00:03:46,807 --> 00:03:47,890 What do we call harmonics? 56 00:03:47,890 --> 00:03:51,530 Single frequency cosine omega t-like input. 57 00:03:51,530 --> 00:03:54,510 Cosine omega t, sine omega t, e the i omega t. 58 00:03:54,510 --> 00:03:59,410 One frequency, repetitive, it's called a harmonic input. 59 00:03:59,410 --> 00:04:05,020 So for a linear system-- we've also we're 60 00:04:05,020 --> 00:04:08,032 talking about what we call the steady state response. 61 00:04:08,032 --> 00:04:08,740 What's that mean? 62 00:04:13,190 --> 00:04:13,742 OK. 63 00:04:13,742 --> 00:04:14,575 Put that into words. 64 00:04:14,575 --> 00:04:15,535 AUDIENCE: [INAUDIBLE] 65 00:04:21,475 --> 00:04:23,270 PROFESSOR: So you turn the motor on, 66 00:04:23,270 --> 00:04:25,600 the whole building shakes, settles down, 67 00:04:25,600 --> 00:04:29,010 and just has a steady vibration you're dealing with. 68 00:04:29,010 --> 00:04:31,460 So after any startup transients have died out, 69 00:04:31,460 --> 00:04:34,880 it's the steady state response. 70 00:04:34,880 --> 00:04:42,960 If you have a linear system and you're putting in some F0 71 00:04:42,960 --> 00:04:50,820 e to the i omega t force, a real part of steady state response, 72 00:04:50,820 --> 00:04:54,842 what is the frequency of the response? 73 00:04:54,842 --> 00:04:58,300 AUDIENCE: [INAUDIBLE] 74 00:04:58,300 --> 00:04:59,130 PROFESSOR: Right. 75 00:04:59,130 --> 00:05:02,050 So for linear systems, the steady state response, 76 00:05:02,050 --> 00:05:10,930 the frequency of the output always 77 00:05:10,930 --> 00:05:13,710 it's a property of a linear system. 78 00:05:13,710 --> 00:05:16,450 So a couple really important things to remember here. 79 00:05:16,450 --> 00:05:17,510 Good. 80 00:05:17,510 --> 00:05:23,970 Now we're going to move onto a problem-- Oh. 81 00:05:23,970 --> 00:05:26,390 And then one other thing I usually do-- ask of you, 82 00:05:26,390 --> 00:05:28,010 is are there any major questions? 83 00:05:28,010 --> 00:05:29,520 Something left over from lecture. 84 00:05:29,520 --> 00:05:31,061 Something we've done in the last week 85 00:05:31,061 --> 00:05:32,520 that just isn't going down. 86 00:05:35,845 --> 00:05:40,506 AUDIENCE: [INAUDIBLE] gravity [INAUDIBLE]? 87 00:05:40,506 --> 00:05:41,880 PROFESSOR: How do you get gravity 88 00:05:41,880 --> 00:05:43,080 to go away in the equation? 89 00:05:43,080 --> 00:05:44,680 Actually, this is the most common question 90 00:05:44,680 --> 00:05:46,888 that's been asked to the TA's this week, of Professor 91 00:05:46,888 --> 00:05:48,490 Gossard, of me. 92 00:05:48,490 --> 00:05:52,090 So there still seems to be a little bit of misunderstanding 93 00:05:52,090 --> 00:05:55,600 about-- and this is the subject of static equilibrium, 94 00:05:55,600 --> 00:06:00,937 verses unstretched spring kind of positions. 95 00:06:00,937 --> 00:06:02,770 I think I'll just deal with that right away. 96 00:06:07,510 --> 00:06:10,670 You're familiar with this problem? 97 00:06:10,670 --> 00:06:15,550 K, C, M. And most of you are comfortable with the notion 98 00:06:15,550 --> 00:06:18,570 of measuring x from the unstretched spring 99 00:06:18,570 --> 00:06:19,620 position, right? 100 00:06:19,620 --> 00:06:22,310 So that's what I've done right here. 101 00:06:22,310 --> 00:06:25,874 Now you add gravity though, and let it find its equilibrium 102 00:06:25,874 --> 00:06:27,290 position under gravity, it's going 103 00:06:27,290 --> 00:06:29,550 to be at some other location. 104 00:06:29,550 --> 00:06:32,510 And I'm just going to call that xstatic. 105 00:06:32,510 --> 00:06:35,840 Static displacement. 106 00:06:35,840 --> 00:06:40,650 Now when you turn on the harmonic excitation, 107 00:06:40,650 --> 00:06:44,690 or even grab it and let it go from initial condition, 108 00:06:44,690 --> 00:06:46,410 where will it vibrate around? 109 00:06:49,290 --> 00:06:53,135 It's going to vibrate around the static equilibrium position. 110 00:06:55,900 --> 00:06:58,230 Here's my system. 111 00:06:58,230 --> 00:07:00,130 Here's my free body diagram. 112 00:07:00,130 --> 00:07:04,100 Got all the gravities and Cx's and Kx's and x dots and so 113 00:07:04,100 --> 00:07:07,410 forth in it complete, right? 114 00:07:07,410 --> 00:07:09,229 I can write an equation of motion 115 00:07:09,229 --> 00:07:11,270 for that system, which you've done lots of times. 116 00:07:11,270 --> 00:07:12,756 And it has a gravity in it. 117 00:07:15,880 --> 00:07:21,280 But now I'm going to say look, here's my x-coordinate, 118 00:07:21,280 --> 00:07:23,410 it stretches down xstatic. 119 00:07:23,410 --> 00:07:28,240 And on top of that I'm going to add a little dynamic motion. 120 00:07:28,240 --> 00:07:33,270 And I'm going to describe that as x equals xstatic a constant, 121 00:07:33,270 --> 00:07:36,515 plus the dynamic movement with respect to xstatic. 122 00:07:39,650 --> 00:07:42,270 Take the time derivative of it, because this is constant. 123 00:07:42,270 --> 00:07:43,890 It goes away. 124 00:07:43,890 --> 00:07:44,940 And two time derivative. 125 00:07:44,940 --> 00:07:48,390 So x dot is x dot d, and x double dot 126 00:07:48,390 --> 00:07:51,550 is x double dot, the dynamic quantity. 127 00:07:51,550 --> 00:07:55,580 Take these three and plug them into here. 128 00:08:00,040 --> 00:08:02,960 Mxd double dot, Cxd dot. 129 00:08:02,960 --> 00:08:07,690 But the K term, you still have xstatic in it. 130 00:08:07,690 --> 00:08:11,690 The original Kx breaks into two pieces. 131 00:08:11,690 --> 00:08:16,720 And on the right hand side is my Mg plus f of t, like before. 132 00:08:16,720 --> 00:08:22,700 But K xstatic must be equal to Mg 133 00:08:22,700 --> 00:08:25,585 in order for static equilibrium to have existed. 134 00:08:28,410 --> 00:08:30,460 Everybody agree with that? 135 00:08:30,460 --> 00:08:33,620 Another way say that is let's say there's no dynamic motion. 136 00:08:33,620 --> 00:08:34,720 This term goes away. 137 00:08:34,720 --> 00:08:35,970 This term goes away. 138 00:08:35,970 --> 00:08:37,390 This term goes away. 139 00:08:37,390 --> 00:08:38,690 Turn off the excitation. 140 00:08:38,690 --> 00:08:41,159 You're left with that equals this. 141 00:08:41,159 --> 00:08:43,760 And that's what says it sits in the static equilibrium 142 00:08:43,760 --> 00:08:44,320 position. 143 00:08:44,320 --> 00:08:49,220 If those two are equal, they cancel, leaving you with this. 144 00:08:53,760 --> 00:08:57,520 So if you had understood this in the beginning, 145 00:08:57,520 --> 00:09:04,830 you could have said let's make our inertial coordinate 146 00:09:04,830 --> 00:09:07,060 from the static equilibrium position. 147 00:09:09,940 --> 00:09:14,240 Just let the thing find its static equilibrium position, 148 00:09:14,240 --> 00:09:18,660 set the inertia coordinate measured from there, 149 00:09:18,660 --> 00:09:23,190 and you would get to this equation, 150 00:09:23,190 --> 00:09:27,270 with the possible confusion of having to deal 151 00:09:27,270 --> 00:09:29,440 with the free body diagram. 152 00:09:29,440 --> 00:09:31,120 Because the free body diagram still 153 00:09:31,120 --> 00:09:35,180 has on it Mg and the spring force that 154 00:09:35,180 --> 00:09:38,430 comes from the static position. 155 00:09:38,430 --> 00:09:41,440 So if I were doing this, and just 156 00:09:41,440 --> 00:09:44,030 say I want to go directly to this equation, 157 00:09:44,030 --> 00:09:47,330 I would draw the free body diagram that 158 00:09:47,330 --> 00:10:04,610 would have Kxd Cxd dot, F MG, but then I would also just 159 00:10:04,610 --> 00:10:07,150 put in here-- there's one more force. 160 00:10:07,150 --> 00:10:11,900 K times xstatic or delta or whatever you want to call it. 161 00:10:11,900 --> 00:10:14,090 You know there's some xstatic, and you 162 00:10:14,090 --> 00:10:16,174 know-- because you're doing this-- you 163 00:10:16,174 --> 00:10:17,340 know this about the problem. 164 00:10:17,340 --> 00:10:19,645 You know this cancels that, and it won't appear 165 00:10:19,645 --> 00:10:20,728 in the equation of motion. 166 00:10:23,870 --> 00:10:26,050 So to be able to do-- anytime you're confronted 167 00:10:26,050 --> 00:10:30,920 with a problem that has gravity, which causes 168 00:10:30,920 --> 00:10:36,960 a static deflection, or results in a static equilibrium 169 00:10:36,960 --> 00:10:41,560 position, it might be 0 in your coordinate-like pendulum. 170 00:10:41,560 --> 00:10:44,470 You look at it, and you say, now in the equation of motion 171 00:10:44,470 --> 00:10:47,080 that you derived from the unstretched spring force 172 00:10:47,080 --> 00:10:50,670 position is the Mg term a function 173 00:10:50,670 --> 00:10:54,250 of the motion coordinate? 174 00:10:54,250 --> 00:10:57,880 And in this case it is not. 175 00:10:57,880 --> 00:11:00,470 So it doesn't matter what x becomes, 176 00:11:00,470 --> 00:11:02,030 this force stays constant. 177 00:11:02,030 --> 00:11:04,720 It doesn't enter into the dynamics of the problem. 178 00:11:08,704 --> 00:11:10,370 If you ever write an equation of motion, 179 00:11:10,370 --> 00:11:13,240 and you come to an Mg term that doesn't have a theta or an x 180 00:11:13,240 --> 00:11:16,180 or any variable in it, you know that there's another way 181 00:11:16,180 --> 00:11:19,170 to write that equation about static equilibrium, 182 00:11:19,170 --> 00:11:21,660 and you'll be able to get rid of it, 183 00:11:21,660 --> 00:11:24,450 and simplify the math basically. 184 00:11:24,450 --> 00:11:31,710 OK, now we want to move on to the problem of the day. 185 00:11:31,710 --> 00:11:34,430 And the problem of the day, I've written up here. 186 00:11:38,160 --> 00:11:41,370 That's the last class's list. 187 00:11:41,370 --> 00:11:43,540 We've got a motor, fan or something. 188 00:11:43,540 --> 00:11:46,480 It's got a rotating imbalance. 189 00:11:46,480 --> 00:11:47,625 It's got a flexible base. 190 00:11:57,530 --> 00:12:00,020 It's got a lot of vertical vibration. 191 00:12:00,020 --> 00:12:01,830 Driving you crazy. 192 00:12:01,830 --> 00:12:04,350 The rotation rate-- this is a pump 193 00:12:04,350 --> 00:12:08,540 or something with a rotating-- some rotating mass inside. 194 00:12:08,540 --> 00:12:10,350 Has some eccentricity. 195 00:12:10,350 --> 00:12:13,120 So it's causing some external force. 196 00:12:13,120 --> 00:12:16,690 Causing a force that results in vertical vibration. 197 00:12:16,690 --> 00:12:23,320 The rotation rate of that fan is 1750 RPM. 198 00:12:23,320 --> 00:12:26,910 The weight of this motor sitting on this frame is 500 pounds. 199 00:12:29,910 --> 00:12:32,970 When you set this motor and bolt it down on that frame, 200 00:12:32,970 --> 00:12:41,060 the frame deflects 0.026 inches. 201 00:12:41,060 --> 00:12:43,230 And this thing's vibrating like crazy. 202 00:12:43,230 --> 00:12:44,700 Your assistant comes in and says, 203 00:12:44,700 --> 00:12:47,090 I got a solution for this. 204 00:12:47,090 --> 00:12:54,690 We're going to put braces in here, 205 00:12:54,690 --> 00:12:57,360 and those braces will double the K. The spring 206 00:12:57,360 --> 00:12:59,900 constant of the thing. 207 00:12:59,900 --> 00:13:04,160 And he or she asserts that he thinks 208 00:13:04,160 --> 00:13:08,200 it will cut the amplitude of response in half. 209 00:13:08,200 --> 00:13:11,240 Cut the vibration amplitude in half. 210 00:13:11,240 --> 00:13:13,080 So you're going to do the calculation today 211 00:13:13,080 --> 00:13:16,260 to see whether or not this is a good idea. 212 00:13:16,260 --> 00:13:17,440 And we're going to begin. 213 00:13:17,440 --> 00:13:25,780 So assignment 1 is come up with a simple lump parameter model-- 214 00:13:25,780 --> 00:13:29,430 masses, springs, dash, pots-- of this system 215 00:13:29,430 --> 00:13:32,604 so that you can model it. 216 00:13:32,604 --> 00:13:35,020 So we've only been doing single degree of freedom systems, 217 00:13:35,020 --> 00:13:37,280 it's probably going to look something like that. 218 00:13:37,280 --> 00:13:40,320 So come up with a lump parameter model of that, 219 00:13:40,320 --> 00:13:41,880 and we've got what? 220 00:13:41,880 --> 00:13:44,910 4, 8, 12. 221 00:13:44,910 --> 00:13:47,930 We could do four groups easily here of four or five each. 222 00:13:47,930 --> 00:13:50,040 So break into four groups. 223 00:13:50,040 --> 00:13:53,430 Come up first with the-- and group 1-- 224 00:13:53,430 --> 00:13:55,600 we'll put group 1 right here. 225 00:13:55,600 --> 00:13:58,317 And group 2 here, and 3 and 4 there. 226 00:13:58,317 --> 00:14:00,650 So come when you get-- when your group figures this out. 227 00:14:00,650 --> 00:14:02,810 And we got to move right along because there's 228 00:14:02,810 --> 00:14:04,579 several parts to this. 229 00:14:04,579 --> 00:14:06,370 Just come up, sketch your model that you're 230 00:14:06,370 --> 00:14:09,630 going to do it with, and then part 2 right 231 00:14:09,630 --> 00:14:13,110 after that is to compute the natural frequency 232 00:14:13,110 --> 00:14:14,871 of the system. 233 00:14:14,871 --> 00:14:17,120 We're going to eventually have to compute the transfer 234 00:14:17,120 --> 00:14:19,370 functions and then try out the fix. 235 00:14:19,370 --> 00:14:23,320 And see if the fix makes things better or worse. 236 00:14:23,320 --> 00:14:26,560 First assignment, come up with a lump parameter model 237 00:14:26,560 --> 00:14:28,925 and find the natural frequency. 238 00:14:28,925 --> 00:14:30,300 Form your groups and get talking. 239 00:14:38,976 --> 00:14:41,390 AUDIENCE: Who's is this? 240 00:14:41,390 --> 00:14:42,790 OK, guys, what about the force? 241 00:14:46,270 --> 00:14:47,660 And what is it? 242 00:14:53,602 --> 00:15:02,760 PROFESSOR: And it's really some me omega squared cos omega t. 243 00:15:02,760 --> 00:15:05,600 And this comes from the equivalent little mass 244 00:15:05,600 --> 00:15:08,600 on some arm, e going around and around inside. 245 00:15:08,600 --> 00:15:13,510 This is the equivalent unbalanced mass here. 246 00:15:13,510 --> 00:15:14,630 So keep coming up. 247 00:15:14,630 --> 00:15:15,630 Put your models up here. 248 00:15:15,630 --> 00:15:16,130 Yeah? 249 00:15:16,130 --> 00:15:18,462 AUDIENCE: You get the [? mpd ?] [INAUDIBLE]. 250 00:15:18,462 --> 00:15:20,010 PROFESSOR: Well, let's go back. 251 00:15:20,010 --> 00:15:21,690 We've done this problem about four times 252 00:15:21,690 --> 00:15:22,856 over the course of homework. 253 00:15:22,856 --> 00:15:26,730 And the term block with an arm with a mass on the end. 254 00:15:26,730 --> 00:15:30,940 The e is the length of the arm, and the mass on the end 255 00:15:30,940 --> 00:15:34,170 is essentially the location of the center 256 00:15:34,170 --> 00:15:36,330 of mass of the rotating part. 257 00:15:41,760 --> 00:15:44,765 So the other three groups need to draw a model up and come up 258 00:15:44,765 --> 00:15:45,890 with the natural frequency. 259 00:15:51,649 --> 00:15:53,190 One of the annoyances of this problem 260 00:15:53,190 --> 00:15:54,955 is it has nasty English units. 261 00:15:59,910 --> 00:16:03,145 You all remember what g is in English units? 262 00:16:21,060 --> 00:16:22,150 I want radians per second. 263 00:16:22,150 --> 00:16:23,120 AUDIENCE: [INAUDIBLE] 264 00:16:25,760 --> 00:16:28,939 PROFESSOR: Yeah, but why do-- convert English inches to feet 265 00:16:28,939 --> 00:16:29,605 and you're done. 266 00:16:29,605 --> 00:16:33,420 Because you know g in feet per second squared. 267 00:16:33,420 --> 00:16:35,805 So rather than do three conversions, do one. 268 00:16:35,805 --> 00:16:38,595 AUDIENCE: [INAUDIBLE] 269 00:16:38,595 --> 00:16:42,100 PROFESSOR: Well that's one, and then you know g. 270 00:16:42,100 --> 00:16:43,380 So you can do it that way. 271 00:16:43,380 --> 00:16:43,920 Sure. 272 00:16:43,920 --> 00:16:44,461 I don't care. 273 00:16:44,461 --> 00:16:58,730 I just want the result. 274 00:16:58,730 --> 00:17:00,940 Which is group 1? 275 00:17:00,940 --> 00:17:01,910 You guys are 1? 276 00:17:01,910 --> 00:17:03,650 You got a number you can give me? 277 00:17:06,270 --> 00:17:07,859 How many MIT students does it take 278 00:17:07,859 --> 00:17:09,774 to convert from inches to feet? 279 00:17:09,774 --> 00:17:11,482 AUDIENCE: We got that, but we're not sure 280 00:17:11,482 --> 00:17:14,779 if it's giving hertz or radians afterwards. 281 00:17:14,779 --> 00:17:16,888 PROFESSOR: Well, in this equation, 282 00:17:16,888 --> 00:17:18,304 does it give you Hertz or radians? 283 00:17:25,140 --> 00:17:25,640 Sure. 284 00:17:25,640 --> 00:17:27,500 And you just made this substitution in here. 285 00:17:27,500 --> 00:17:29,425 So it hasn't done anything to change to Hertz. 286 00:17:38,460 --> 00:17:39,220 Write that down. 287 00:17:45,420 --> 00:17:46,660 What a struggle! 288 00:17:58,940 --> 00:17:59,440 OK. 289 00:18:02,850 --> 00:18:03,430 Good. 290 00:18:03,430 --> 00:18:03,930 That's good. 291 00:18:03,930 --> 00:18:05,330 You can leave it. 292 00:18:05,330 --> 00:18:05,830 All right. 293 00:18:05,830 --> 00:18:08,310 We've all converged to about 121. 294 00:18:08,310 --> 00:18:09,730 That was quite a struggle. 295 00:18:09,730 --> 00:18:12,560 Was it the units, or was it-- partly was Hertz? 296 00:18:12,560 --> 00:18:14,920 And you guys haven't had-- you're dealing with nothing 297 00:18:14,920 --> 00:18:17,380 but equations for weeks. 298 00:18:17,380 --> 00:18:20,180 And to actually have to do real engineering numbers 299 00:18:20,180 --> 00:18:22,900 has thrown you a little bit. 300 00:18:22,900 --> 00:18:25,600 But so it's about 121. 301 00:18:25,600 --> 00:18:27,600 And that g over delta thing is really handy. 302 00:18:30,990 --> 00:18:34,560 Now I want to know what is the response 303 00:18:34,560 --> 00:18:39,480 of this system to this input given 304 00:18:39,480 --> 00:18:43,516 the frequency of the input is 1750 RPM? 305 00:18:43,516 --> 00:18:49,830 And RPM, N is often written, how is 306 00:18:49,830 --> 00:18:53,380 it related to frequency in Hertz or cycles per second? 307 00:18:53,380 --> 00:18:55,010 Anybody know? 308 00:18:55,010 --> 00:18:57,830 This is revolutions per minute. 309 00:18:57,830 --> 00:19:01,050 What would be revolution per second? 310 00:19:01,050 --> 00:19:03,600 Is that same thing as Hertz? 311 00:19:03,600 --> 00:19:05,680 So revolution is something going around. 312 00:19:05,680 --> 00:19:09,050 One full time around is 2 pi. 313 00:19:09,050 --> 00:19:11,090 It's one cycle. 314 00:19:11,090 --> 00:19:13,510 So if you do 60 cycles in a minute, 315 00:19:13,510 --> 00:19:15,450 how many do you do in a second? 316 00:19:15,450 --> 00:19:16,240 One. 317 00:19:16,240 --> 00:19:25,000 So this RPM divided by 60 equals f in cycles per second. 318 00:19:25,000 --> 00:19:25,630 Or Hertz. 319 00:19:29,530 --> 00:19:33,595 RPM divided by 60 gives you revolutions per second. 320 00:19:33,595 --> 00:19:35,375 AUDIENCE: That says RPS. 321 00:19:35,375 --> 00:19:37,250 PROFESSOR: Well, excuse me. 322 00:19:37,250 --> 00:19:42,520 I was talking about-- get rid of that. 323 00:19:42,520 --> 00:19:43,760 That's the relation you want. 324 00:19:43,760 --> 00:19:46,665 And what's the relationship between omega and f? 325 00:19:50,970 --> 00:19:55,370 For every cycle, you get 2 pi radians. 326 00:19:55,370 --> 00:19:59,950 So now compute-- I want you to give me an estimate-- find-- 327 00:19:59,950 --> 00:20:03,240 post my problem here. 328 00:20:03,240 --> 00:20:04,780 So part 1 you've done. 329 00:20:04,780 --> 00:20:11,200 Part 2 here and to get an estimate of the magnitude 330 00:20:11,200 --> 00:20:12,650 of this steady state response. 331 00:20:17,970 --> 00:20:21,090 And we've been talking about transfer functions. 332 00:20:21,090 --> 00:20:24,010 So it's probably a replica applying a transfer function 333 00:20:24,010 --> 00:20:24,510 here. 334 00:20:27,390 --> 00:20:29,030 I would put your symbolic answer up 335 00:20:29,030 --> 00:20:30,450 first would be a terrific idea. 336 00:20:32,587 --> 00:20:34,420 Think you all-- when you think you have kind 337 00:20:34,420 --> 00:20:36,700 of a symbolic approach to this. 338 00:20:36,700 --> 00:20:39,379 Go write it up, and that way I can know where 339 00:20:39,379 --> 00:20:40,670 you need a little help, if any. 340 00:21:02,670 --> 00:21:03,170 That's OK. 341 00:21:17,530 --> 00:21:21,760 So you don't know K. 342 00:21:21,760 --> 00:21:26,390 So just leave that has an undetermined quantity. 343 00:21:26,390 --> 00:21:30,789 F0-- what is-- you guys keep working for a second. 344 00:21:30,789 --> 00:21:31,830 Then we'll talk about it. 345 00:21:31,830 --> 00:21:34,165 You don't know K, so just leave it as an unknown. 346 00:21:37,670 --> 00:21:38,310 Don't do that. 347 00:21:38,310 --> 00:21:41,117 Just call it F0. 348 00:21:41,117 --> 00:21:43,200 But now plug in some numbers for what you do know. 349 00:21:52,931 --> 00:21:54,430 So one quantity you're going to need 350 00:21:54,430 --> 00:21:57,920 to make over omega n, so write that one down when you get it. 351 00:21:57,920 --> 00:22:03,510 What's omega over omega n, and then move on from there. 352 00:22:09,410 --> 00:22:10,900 Now there's a little pitfall here 353 00:22:10,900 --> 00:22:12,540 that you're all running into. 354 00:22:12,540 --> 00:22:16,001 And that is you're taking the positive square root as 355 00:22:16,001 --> 00:22:16,500 written. 356 00:22:16,500 --> 00:22:18,000 It's that whole quantity down there. 357 00:22:18,000 --> 00:22:20,505 Squared square root, and you're writing 358 00:22:20,505 --> 00:22:22,197 it 1 minus omega squared. 359 00:22:22,197 --> 00:22:24,280 And this thing's going to turn out negative on you 360 00:22:24,280 --> 00:22:24,990 if you're not careful. 361 00:22:24,990 --> 00:22:26,114 But you can flip it around. 362 00:22:26,114 --> 00:22:28,590 It's really the absolute value of that quantity, 363 00:22:28,590 --> 00:22:32,290 because it a-- quantity squared square root, so it can be 364 00:22:32,290 --> 00:22:32,990 plus or minus. 365 00:22:42,710 --> 00:22:49,540 So you got 1.54-- 1.5 what? 366 00:22:49,540 --> 00:22:50,840 1.5 is your ratio? 367 00:22:50,840 --> 00:22:51,600 OK, write it down. 368 00:22:51,600 --> 00:22:52,550 Just write the ratio. 369 00:22:52,550 --> 00:22:53,424 That's what you need. 370 00:22:57,450 --> 00:23:08,080 1.5. 371 00:23:08,080 --> 00:23:10,080 And you've got-- this is 4/5 here? 372 00:23:10,080 --> 00:23:12,480 That's what you guys have worked out? 373 00:23:12,480 --> 00:23:21,498 OK, so 0.8. 374 00:23:21,498 --> 00:23:23,913 AUDIENCE: How did they get 1.5? 375 00:23:23,913 --> 00:23:26,350 PROFESSOR: They calculated omega over omega n. 376 00:23:26,350 --> 00:23:31,842 You have 183 divided by-- what was your omega n? 377 00:23:31,842 --> 00:23:33,260 AUDIENCE: Oh, 60. 378 00:23:33,260 --> 00:23:33,760 OK. 379 00:23:33,760 --> 00:23:34,509 PROFESSOR: Not 60. 380 00:23:34,509 --> 00:23:35,830 AUDIENCE: Well, no. 381 00:23:35,830 --> 00:23:39,234 I mean it's like 360 over 260. 382 00:23:39,234 --> 00:23:39,900 PROFESSOR: Yeah. 383 00:23:39,900 --> 00:23:43,160 Well, you've got to divide 1750 by 60 to get F. 384 00:23:43,160 --> 00:23:46,170 And then you got to multiply by 2 pi. 385 00:23:46,170 --> 00:23:48,768 I'm doing it for the benefit of everybody in the room here. 386 00:23:51,576 --> 00:23:54,050 0.8 F0 over K, all right. 387 00:23:54,050 --> 00:23:56,370 Now we're getting somewhere. 388 00:23:56,370 --> 00:23:57,520 0.8. 389 00:23:57,520 --> 00:23:58,280 0.8. 390 00:23:58,280 --> 00:24:00,460 What do you guys have here? 391 00:24:00,460 --> 00:24:01,370 You got 1 over 1.3. 392 00:24:01,370 --> 00:24:04,070 How does that work out? 393 00:24:04,070 --> 00:24:05,980 So about 0.8. 394 00:24:05,980 --> 00:24:08,240 OK, so you've come to the conclusion 395 00:24:08,240 --> 00:24:12,820 that the magnitude of that study state response is about 0.8 F0 396 00:24:12,820 --> 00:24:14,270 over K. 397 00:24:14,270 --> 00:24:18,080 And we call that F0 over K-- in lecture, 398 00:24:18,080 --> 00:24:20,010 I call it the static response. 399 00:24:20,010 --> 00:24:23,060 The book calls it the static response. 400 00:24:23,060 --> 00:24:26,730 Is it the same static response as we're talking about here? 401 00:24:29,300 --> 00:24:30,220 Don't confuse the two. 402 00:24:30,220 --> 00:24:32,790 This was the effect of gravity on the system. 403 00:24:32,790 --> 00:24:34,750 The static response we're talking about here 404 00:24:34,750 --> 00:24:37,550 is if the frequency of this excitation 405 00:24:37,550 --> 00:24:41,150 went to 0, then the response-- this 406 00:24:41,150 --> 00:24:44,650 would stretch that spring by the amount we're calling 407 00:24:44,650 --> 00:24:48,390 xstatic, F0 over K. All right. 408 00:24:48,390 --> 00:24:50,660 So now you know that you're operating at a frequency. 409 00:24:50,660 --> 00:24:52,410 What's the frequency ratio that you found? 410 00:24:52,410 --> 00:24:55,310 Omega over omega n. 411 00:24:55,310 --> 00:24:58,160 That was an intermediate step here. 412 00:24:58,160 --> 00:25:00,680 All right, 1.5. 413 00:25:00,680 --> 00:25:03,780 All right, now you're going to-- the proposed fixed 414 00:25:03,780 --> 00:25:07,360 is to double the stiffness. 415 00:25:07,360 --> 00:25:08,350 Double the stiffness. 416 00:25:13,710 --> 00:25:17,130 Now we want to compute this result 417 00:25:17,130 --> 00:25:18,786 if you double the stiffness. 418 00:25:18,786 --> 00:25:20,160 And the person who proposed that, 419 00:25:20,160 --> 00:25:22,180 thinks it's going to half the response. 420 00:25:22,180 --> 00:25:23,270 That's the proposal. 421 00:25:23,270 --> 00:25:25,070 So double the stiffness. 422 00:25:25,070 --> 00:25:26,900 And in order to get finished here, 423 00:25:26,900 --> 00:25:31,745 what effect does changing the stiffness have on this system? 424 00:25:31,745 --> 00:25:33,565 AUDIENCE: Decreases the [INAUDIBLE]. 425 00:25:33,565 --> 00:25:36,330 PROFESSOR: Well, you jump into an answer-- intermediate step 426 00:25:36,330 --> 00:25:36,830 here. 427 00:25:36,830 --> 00:25:40,324 What does it do to the natural frequency? 428 00:25:40,324 --> 00:25:41,740 There's a natural frequency change 429 00:25:41,740 --> 00:25:44,030 if you change the stiffness. 430 00:25:44,030 --> 00:25:52,370 OK, so the natural frequency is normally some kind of K over m. 431 00:25:52,370 --> 00:25:54,195 Now we also know it's g over delta. 432 00:25:57,134 --> 00:25:58,800 But we're going to double the stiffness, 433 00:25:58,800 --> 00:26:00,716 so what does that do to the natural frequency? 434 00:26:00,716 --> 00:26:02,200 Does the m change? 435 00:26:02,200 --> 00:26:06,610 OK, so what's the-- this is omega n before. 436 00:26:06,610 --> 00:26:17,600 Omega n after equals right. 437 00:26:17,600 --> 00:26:18,100 OK. 438 00:26:18,100 --> 00:26:20,300 And what does that do to your frequency ratio? 439 00:26:26,910 --> 00:26:29,930 So the effect of changing the stiffness 440 00:26:29,930 --> 00:26:32,910 changes the natural frequency, and that 441 00:26:32,910 --> 00:26:35,734 changes the ratio of the input frequency 442 00:26:35,734 --> 00:26:37,150 to the natural frequency, and that 443 00:26:37,150 --> 00:26:39,450 will change this calculation here, right? 444 00:26:39,450 --> 00:26:41,400 So come up with a new value for this. 445 00:26:50,150 --> 00:26:51,540 Anybody got a number? 446 00:26:51,540 --> 00:26:54,126 What's the frequency ratio after? 447 00:26:54,126 --> 00:26:57,542 AUDIENCE: Uh, 1.1-- never mind. 448 00:27:00,470 --> 00:27:01,934 I was doing the squared term. 449 00:27:05,770 --> 00:27:07,770 PROFESSOR: I wanted to know the frequency ratio. 450 00:27:07,770 --> 00:27:10,690 This is after is equal to the frequency 451 00:27:10,690 --> 00:27:14,060 ratio that was something to do with root 2 here. 452 00:27:14,060 --> 00:27:16,090 The natural frequency is going to go up or down 453 00:27:16,090 --> 00:27:19,214 by a factor of root 2. 454 00:27:19,214 --> 00:27:21,700 If you double K, the natural frequency 455 00:27:21,700 --> 00:27:24,360 will go up by a square root of 2. 456 00:27:24,360 --> 00:27:27,960 So the frequency ratio after-- the natural frequency 457 00:27:27,960 --> 00:27:29,360 goes up by the square root of 2. 458 00:27:29,360 --> 00:27:32,500 That's the same as root 2 over 2 here. 459 00:27:32,500 --> 00:27:34,600 So the whole thing drops down. 460 00:27:34,600 --> 00:27:37,600 So 1.5 times square root of 2 over 2. 461 00:27:37,600 --> 00:27:40,040 What is it? 462 00:27:40,040 --> 00:27:40,760 All right. 463 00:27:40,760 --> 00:27:42,310 1.07. 464 00:27:42,310 --> 00:27:45,685 Now what is-- figure out this quantity. 465 00:27:48,380 --> 00:27:50,820 Pardon? 466 00:27:50,820 --> 00:27:53,350 3.4. 467 00:27:53,350 --> 00:27:55,521 And did you put in a 2 here? 468 00:27:55,521 --> 00:27:56,020 Right. 469 00:27:59,770 --> 00:28:05,690 So this is the correct expression to be using. 470 00:28:05,690 --> 00:28:09,280 But now in this new problem-- this was before. 471 00:28:09,280 --> 00:28:16,270 In this problem afterwards, x is steady state is F0 over 2K 472 00:28:16,270 --> 00:28:22,360 if double the stiffness, all over this quantity omega 473 00:28:22,360 --> 00:28:28,490 squared over omega n squared minus 1, 474 00:28:28,490 --> 00:28:30,480 but with the new frequency ratio. 475 00:28:30,480 --> 00:28:32,890 This is now 1.07. 476 00:28:32,890 --> 00:28:35,940 You do this number, what do you get? 477 00:28:35,940 --> 00:28:39,080 3.-- what did that say? 478 00:28:39,080 --> 00:28:42,640 3.4 roughly, 479 00:28:42,640 --> 00:28:47,290 F0 over K. So you have apples and apple together. 480 00:28:47,290 --> 00:28:51,070 You want to compare the F0 over K before and after. 481 00:28:51,070 --> 00:28:52,440 End it was a factor of 3.4. 482 00:28:52,440 --> 00:28:54,845 So should you make the change? 483 00:28:54,845 --> 00:28:55,345 No. 484 00:28:55,345 --> 00:28:56,315 That's a bad idea. 485 00:28:58,890 --> 00:29:00,400 You ran into a situation like this 486 00:29:00,400 --> 00:29:02,483 before, going to all that trouble of cutting steel 487 00:29:02,483 --> 00:29:04,150 and getting out the torch. 488 00:29:04,150 --> 00:29:06,260 The welding system. 489 00:29:06,260 --> 00:29:09,990 Take yourself a great big weight and walk up, and thump 490 00:29:09,990 --> 00:29:11,430 some weight on top of this thing, 491 00:29:11,430 --> 00:29:13,197 and see if it gets better or worse. 492 00:29:13,197 --> 00:29:15,780 Because if I double the mass of the system, what would happen? 493 00:29:18,522 --> 00:29:21,580 It's going to go the other direction. 494 00:29:21,580 --> 00:29:24,670 This will go up by a factor of root 2. 495 00:29:24,670 --> 00:29:29,280 And then this denominator-- you're pushing it further out. 496 00:29:29,280 --> 00:29:32,622 And instead of being at 0.8, you'll be down further.