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PROFESSOR: Today
we're going to talk

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00:00:27,960 --> 00:00:32,100
about this topic of
vibration isolation, which

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00:00:32,100 --> 00:00:35,430
is a very practical
use of knowing

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00:00:35,430 --> 00:00:37,320
a little bit about vibration.

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00:00:37,320 --> 00:00:41,240
So imagine a
situation you've been

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00:00:41,240 --> 00:00:44,530
in where there's an air
conditioner in a window,

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00:00:44,530 --> 00:00:48,385
and it's causing
your table to shake,

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00:00:48,385 --> 00:00:51,360
or where you're trying to
work, or your bed rattles,

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00:00:51,360 --> 00:00:52,550
or something like that.

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00:00:52,550 --> 00:00:54,341
How many of you have
ever had an experience

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00:00:54,341 --> 00:00:56,100
like that, something's
kind of annoying,

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00:00:56,100 --> 00:00:57,905
messing up a lab
experiment, or whatever?

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00:00:57,905 --> 00:01:00,180
Yeah, you've all experienced
these things, right?

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00:01:00,180 --> 00:01:04,900
So as clever
engineers, are there

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00:01:04,900 --> 00:01:08,480
simple solutions sometimes
to fixing these problems?

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00:01:08,480 --> 00:01:11,650
And that's what we're
going to talk about today.

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00:01:11,650 --> 00:01:17,200
I have a little quick demo
that I'm going to show you.

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00:01:17,200 --> 00:01:19,420
You saw this the
other day, where

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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.

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00:01:23,620 --> 00:01:25,740
And we've looked at
rotating masses a lot.

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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.

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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.

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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

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00:01:51,140 --> 00:01:52,860
dashpot system, it'll vibrate.

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00:01:52,860 --> 00:01:54,852
And so I showed you
that the other day.

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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.

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00:02:03,030 --> 00:02:06,600
This is your microscope
here, this little one.

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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,

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00:02:12,520 --> 00:02:14,650
or whatever is
causing the trouble.

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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

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00:02:39,570 --> 00:02:40,630
can see the motion.

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00:02:40,630 --> 00:02:44,180
There is the motion
of this main system.

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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

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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.

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00:03:06,290 --> 00:03:10,720
So this is your microscope
sitting on a lab bench

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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

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00:03:14,310 --> 00:03:16,980
by the original
imbalance in something,

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00:03:16,980 --> 00:03:19,820
travels through the
floor, gets to your table

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00:03:19,820 --> 00:03:20,960
with the microscope on it.

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00:03:20,960 --> 00:03:22,820
Now your microscope shakes.

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00:03:22,820 --> 00:03:25,780
So the issue is, what
can we do about it?

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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.

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00:03:34,370 --> 00:03:35,440
I'll turn it off.

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00:03:35,440 --> 00:03:39,140
And then I'm going
to have you consider.

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00:03:39,140 --> 00:03:42,691
So I want you to get in
pairs and talk about this.

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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

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00:03:48,070 --> 00:03:49,820
the vibration of
your microscope--

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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.

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00:03:56,240 --> 00:03:57,290
Talk about it.

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00:03:57,290 --> 00:03:59,850
And come up with three ways
of fixing this problem.

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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

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00:04:49,390 --> 00:04:50,510
we had the other day, OK?

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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.

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00:06:09,300 --> 00:06:10,420
OK, what's another idea?

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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.

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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?

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00:06:45,782 --> 00:06:47,320
AUDIENCE: [INAUDIBLE].

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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.

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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].

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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.