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00:00:21,650 --> 00:00:26,430
PROFESSOR: Are we ready
with the concept questions

9
00:00:26,430 --> 00:00:29,740
from the homework this week?

10
00:00:29,740 --> 00:00:32,284
How do we get
different-- there we go.

11
00:00:32,284 --> 00:00:33,700
I looked at one,
it was one thing.

12
00:00:33,700 --> 00:00:35,270
I looked at the
other, [INAUDIBLE].

13
00:00:35,270 --> 00:00:37,260
Does g enter into
the expression for

14
00:00:37,260 --> 00:00:39,950
the undamped natural frequency?

15
00:00:39,950 --> 00:00:45,705
And most people said no, but
about a third of you said yes.

16
00:00:45,705 --> 00:00:47,330
If you have worked
on that problem now,

17
00:00:47,330 --> 00:00:49,830
you have already
discovered the answer.

18
00:00:49,830 --> 00:00:56,520
So you'll find that g does
not come into the expression.

19
00:00:56,520 --> 00:01:02,200
When you do a pendulum,
g is in the expression.

20
00:01:02,200 --> 00:01:05,090
And there's a question
on the homework

21
00:01:05,090 --> 00:01:07,420
about what's the difference.

22
00:01:07,420 --> 00:01:09,830
How can you predict
when g is going

23
00:01:09,830 --> 00:01:12,880
to be involved in a natural
frequency expression

24
00:01:12,880 --> 00:01:14,180
and when it is not?

25
00:01:16,562 --> 00:01:18,270
I want you to think
about that one a bit,

26
00:01:18,270 --> 00:01:20,170
maybe talk about it
at-- if there's still

27
00:01:20,170 --> 00:01:21,670
questions about it,
we talk about it

28
00:01:21,670 --> 00:01:23,480
in recitation on
Thursday, Friday.

29
00:01:23,480 --> 00:01:25,020
OK.

30
00:01:25,020 --> 00:01:27,205
So does g enter into
the expression here?

31
00:01:27,205 --> 00:01:31,170
I'm sure you know this simple
pendulum, the natural frequency

32
00:01:31,170 --> 00:01:34,000
square root of g over l.

33
00:01:34,000 --> 00:01:35,910
For a simple
mass-spring dashpot,

34
00:01:35,910 --> 00:01:38,070
the natural
frequency is k over m

35
00:01:38,070 --> 00:01:41,070
whether or not it's
affected by gravity.

36
00:01:41,070 --> 00:01:44,840
So there's something
different about these two.

37
00:01:44,840 --> 00:01:46,510
OK, let's go onto the third one.

38
00:01:51,340 --> 00:01:54,180
"In an experiment, this system
is given initial velocity

39
00:01:54,180 --> 00:01:57,240
observed to decay in
amplitude of vibration

40
00:01:57,240 --> 00:01:59,620
by 50% in 10 cycles.

41
00:01:59,620 --> 00:02:01,230
You can estimate
the damping ratio

42
00:02:01,230 --> 00:02:02,742
to be approximately," what?

43
00:02:02,742 --> 00:02:05,200
Well, it gives a little rule
of thumb I gave you last week.

44
00:02:05,200 --> 00:02:06,930
I'll go over it again today.

45
00:02:06,930 --> 00:02:11,360
0.11 divided by the number
of cycles did decay 50%.

46
00:02:11,360 --> 00:02:14,290
So it took 10
cycles, 0.11 divided

47
00:02:14,290 --> 00:02:19,670
by 10, 0.01, 1.1% damping.

48
00:02:19,670 --> 00:02:22,460
OK, next.

49
00:02:22,460 --> 00:02:25,160
"At which of the three
excitation frequencies

50
00:02:25,160 --> 00:02:28,380
will the response
magnitude be greatest?"

51
00:02:28,380 --> 00:02:33,210
You've done oscillators
excited by cosine omega t

52
00:02:33,210 --> 00:02:34,820
kind of things before.

53
00:02:34,820 --> 00:02:38,142
So at the ratio 1,
most people said

54
00:02:38,142 --> 00:02:39,600
it's where it would
be the largest.

55
00:02:39,600 --> 00:02:40,880
Why at 1?

56
00:02:40,880 --> 00:02:43,870
Anybody want to
give a shout here?

57
00:02:43,870 --> 00:02:46,750
What happens when
you drive the system

58
00:02:46,750 --> 00:02:49,384
at its natural frequency?

59
00:02:49,384 --> 00:02:52,050
It's called resonance, and we're
going to talk about that today.

60
00:02:52,050 --> 00:02:54,540
So it's when the
frequency ratio is

61
00:02:54,540 --> 00:02:59,820
one that-- and for the system
being lightly damped that you

62
00:02:59,820 --> 00:03:01,540
get the largest response.

63
00:03:01,540 --> 00:03:05,749
Finally-- which
one are we on here?

64
00:03:05,749 --> 00:03:06,290
Oh, this one.

65
00:03:06,290 --> 00:03:10,680
Can all the kinetic energy be
accounted for by an expression

66
00:03:10,680 --> 00:03:12,765
of the form IZ omega squared.

67
00:03:17,040 --> 00:03:19,310
By the way, I brought this
system if you haven't.

68
00:03:22,190 --> 00:03:23,940
So there's a really
simple demo, but it

69
00:03:23,940 --> 00:03:26,180
has all sorts of--
so in this case,

70
00:03:26,180 --> 00:03:32,340
we're talking about that motion.

71
00:03:32,340 --> 00:03:36,740
It certainly has some I omega
squared kind of kinetic energy,

72
00:03:36,740 --> 00:03:40,255
but does the center of gravity
translate as it's oscillating?

73
00:03:43,841 --> 00:03:45,590
What's the potential
energy in the system?

74
00:03:48,970 --> 00:03:52,610
By the way, any time
you get an oscillation,

75
00:03:52,610 --> 00:03:57,060
energy flows from potential
to kinetic, potential kinetic.

76
00:03:57,060 --> 00:03:58,580
That's what oscillation is.

77
00:03:58,580 --> 00:04:01,790
So there has to be an exchange
going on between kinetic energy

78
00:04:01,790 --> 00:04:02,930
and potential energy.

79
00:04:02,930 --> 00:04:04,640
And if there's no
losses in the system,

80
00:04:04,640 --> 00:04:08,590
the total energy is constant.

81
00:04:08,590 --> 00:04:12,150
So the kinetic energy's
certainly in the motion,

82
00:04:12,150 --> 00:04:14,450
but when it reaches
maximum amplitude, what's

83
00:04:14,450 --> 00:04:17,459
its velocity when
it's right here?

84
00:04:17,459 --> 00:04:18,230
Zero.

85
00:04:18,230 --> 00:04:22,070
So all of its energy
must be where?

86
00:04:22,070 --> 00:04:22,880
In the potential.

87
00:04:22,880 --> 00:04:25,840
And where's the
potential in this system?

88
00:04:25,840 --> 00:04:26,960
Pardon?

89
00:04:26,960 --> 00:04:27,590
In the string.

90
00:04:27,590 --> 00:04:29,427
That's not stored in the string.

91
00:04:29,427 --> 00:04:31,260
There's only two sources
of potential energy

92
00:04:31,260 --> 00:04:33,360
we talk about in this class.

93
00:04:33,360 --> 00:04:33,920
Gravity and--

94
00:04:33,920 --> 00:04:34,670
AUDIENCE: Strings.

95
00:04:34,670 --> 00:04:35,461
PROFESSOR: Strings.

96
00:04:35,461 --> 00:04:38,700
Well, these strings
don't stretch,

97
00:04:38,700 --> 00:04:40,995
so there's no spring
kinetic energy.

98
00:04:40,995 --> 00:04:42,120
We've got potential energy.

99
00:04:42,120 --> 00:04:47,490
There must be gravitational
potential energy.

100
00:04:47,490 --> 00:04:49,215
How is it coming
into this system?

101
00:04:52,880 --> 00:04:58,180
So he says when it turns,
the center of gravity

102
00:04:58,180 --> 00:05:02,440
has to raise up a
little bit, and that's

103
00:05:02,440 --> 00:05:04,590
the potential energy
in this system.

104
00:05:04,590 --> 00:05:07,010
The center of gravity goes
up and down a tiny bit.

105
00:05:07,010 --> 00:05:10,560
So is there any velocity
in the vertical direction?

106
00:05:10,560 --> 00:05:12,270
Is there any kinetic
energy associated

107
00:05:12,270 --> 00:05:13,710
with up and down motion?

108
00:05:13,710 --> 00:05:16,480
Yeah, so that doesn't
entirely capture it,

109
00:05:16,480 --> 00:05:17,841
1/2 I omega squared.

110
00:05:17,841 --> 00:05:19,340
Is it an important
amount of energy?

111
00:05:19,340 --> 00:05:23,530
I don't know, but there is
some velocity up and down.

112
00:05:23,530 --> 00:05:26,170
My guess is that it
actually isn't important,

113
00:05:26,170 --> 00:05:28,270
that the answer is it
does move up and down.

114
00:05:28,270 --> 00:05:30,850
It has to, or you would not
have any potential energy

115
00:05:30,850 --> 00:05:32,810
exchange in the system.

116
00:05:32,810 --> 00:05:35,030
OK.

117
00:05:35,030 --> 00:05:37,130
Is that it?

118
00:05:37,130 --> 00:05:39,312
OK.

119
00:05:39,312 --> 00:05:40,270
Let's keep moving here.

120
00:05:42,960 --> 00:05:44,710
Got a lot of fun things
to show you today.

121
00:05:47,400 --> 00:05:50,862
So last time, we talked about
response to initial conditions.

122
00:05:50,862 --> 00:05:52,820
I'm going to finish up
with that and then go on

123
00:05:52,820 --> 00:05:57,360
to talking about excitation
of harmonic forces.

124
00:05:57,360 --> 00:05:59,110
So last time, we were
considering a system

125
00:05:59,110 --> 00:06:02,260
like this-- X is measured
from the zero spring

126
00:06:02,260 --> 00:06:04,510
force in this case.

127
00:06:04,510 --> 00:06:07,050
Give you an equation
of motion of that sort.

128
00:06:07,050 --> 00:06:12,320
And we've found that
you could express x of T

129
00:06:12,320 --> 00:06:19,150
as a-- I'll give you the exact
expression-- x0 square root

130
00:06:19,150 --> 00:06:20,893
of 1 minus zeta squared.

131
00:06:29,980 --> 00:06:38,050
Cosine omega damped times time
minus a little phase angle.

132
00:06:38,050 --> 00:06:50,830
And there's a second term here,
v0 over omega d sine omega dt.

133
00:06:50,830 --> 00:06:58,560
And the whole thing times e
to the minus zeta omega n t.

134
00:06:58,560 --> 00:07:02,790
So that's our response
to an initial deflection

135
00:07:02,790 --> 00:07:07,260
x0 or an initial velocity v0.

136
00:07:07,260 --> 00:07:11,000
That's the full kind
of messy expression.

137
00:07:11,000 --> 00:07:13,645
There's another way of writing
that, which I'll show you.

138
00:07:19,760 --> 00:07:34,060
Another way of saying is that
it's in x0 cosine omega dt

139
00:07:34,060 --> 00:07:53,140
plus v0 plus zeta omega n
x0, all over omega d sine

140
00:07:53,140 --> 00:07:58,110
omega d times t e to
the minus zeta omega

141
00:07:58,110 --> 00:08:01,330
nt, the same exponent.

142
00:08:01,330 --> 00:08:03,260
This is your
decaying exponential

143
00:08:03,260 --> 00:08:04,950
that makes it die out.

144
00:08:04,950 --> 00:08:07,020
And so I just rearranged
some of these things.

145
00:08:07,020 --> 00:08:08,978
There's another little
phase angle in here now.

146
00:08:08,978 --> 00:08:13,560
So you have just a cosine
term, this proportional x0,

147
00:08:13,560 --> 00:08:18,070
and a sine term, which
has both v0 and x0 in it.

148
00:08:18,070 --> 00:08:19,970
The x0 term, if
damping is small,

149
00:08:19,970 --> 00:08:25,540
this term is pretty small
because it's x0 times zeta.

150
00:08:25,540 --> 00:08:28,450
When you divide by omega
d, which is almost omega n,

151
00:08:28,450 --> 00:08:30,130
that goes away.

152
00:08:30,130 --> 00:08:34,039
So this term, the
scale of it is zeta x0.

153
00:08:34,039 --> 00:08:38,360
So if this is 1% or 2%,
that's a very small number.

154
00:08:38,360 --> 00:08:40,190
I gave you an
approximation, which

155
00:08:40,190 --> 00:08:48,870
for almost all practical
examples that you

156
00:08:48,870 --> 00:08:55,650
might want to do, make it
approximately sine here.

157
00:08:55,650 --> 00:09:14,080
So this is x0 cosine omega dt
plus v0 over omega d sine omega

158
00:09:14,080 --> 00:09:21,340
dte to the minus zeta omega nt.

159
00:09:21,340 --> 00:09:23,230
And this is the practical one.

160
00:09:23,230 --> 00:09:27,160
For any reasonable system that
has relatively low damping,

161
00:09:27,160 --> 00:09:31,940
even 10% or 15% damping, you
get part of the transient decay

162
00:09:31,940 --> 00:09:38,930
comes from x0 cosine, the other
part v0 over omega d sine.

163
00:09:38,930 --> 00:09:42,280
That's what I can remember in my
head when I'm trying to do it.

164
00:09:42,280 --> 00:09:44,930
Now the question, the thing
I want to address today

165
00:09:44,930 --> 00:09:48,140
is what's this useful for?

166
00:09:48,140 --> 00:09:50,410
My approach to
teaching you vibration

167
00:09:50,410 --> 00:09:54,160
is I want you to go
away with a few simple,

168
00:09:54,160 --> 00:09:57,010
practical understandings
so that you can actually

169
00:09:57,010 --> 00:10:01,820
solve some vibration
problems, and one of them

170
00:10:01,820 --> 00:10:05,412
is just knowing this allows
you to do a couple things,

171
00:10:05,412 --> 00:10:07,370
and we'll do a couple of
examples this morning.

172
00:10:11,120 --> 00:10:23,365
By the way, this form, this is
A cosine plus B sine expression.

173
00:10:26,570 --> 00:10:36,770
And I label them
A and B. A and B,

174
00:10:36,770 --> 00:10:44,830
they're both of the
form A1 cosine omega

175
00:10:44,830 --> 00:10:50,090
t plus B1 sine omega t.

176
00:10:50,090 --> 00:10:53,020
And you can always add
a sine and a cosine

177
00:10:53,020 --> 00:10:56,290
at the same frequency.

178
00:10:56,290 --> 00:10:57,670
If I put just any
frequency here,

179
00:10:57,670 --> 00:10:59,620
they just have to be the same.

180
00:10:59,620 --> 00:11:02,570
You can always take an
expression like that

181
00:11:02,570 --> 00:11:07,520
and rewrite it as some
magnitude cosine omega

182
00:11:07,520 --> 00:11:11,090
t minus a phase angle.

183
00:11:11,090 --> 00:11:14,100
And the magnitude is
just a square root

184
00:11:14,100 --> 00:11:21,060
of the sum of the squares--
A1 squared plus B1 squared.

185
00:11:21,060 --> 00:11:25,830
And the phase angle
is the tangent inverse

186
00:11:25,830 --> 00:11:28,600
of the sine term
over the cosine term.

187
00:11:28,600 --> 00:11:32,000
So you can always
rewrite sine plus cosine

188
00:11:32,000 --> 00:11:36,080
as a cosine omega t minus
v. We use that a lot,

189
00:11:36,080 --> 00:11:37,950
and that'll be used
a lot in this course.

190
00:11:43,629 --> 00:11:45,420
And then, of course,
if this whole thing is

191
00:11:45,420 --> 00:11:50,010
multiplied by an e to the minus
8 omega mt, then so is this.

192
00:11:55,010 --> 00:11:57,420
OK.

193
00:11:57,420 --> 00:11:59,340
OK, what are these
things useful for?

194
00:12:03,480 --> 00:12:08,470
And we've derived this all
for a mass spring system.

195
00:12:08,470 --> 00:12:12,810
Is that equation
applicable to a pendulum?

196
00:12:12,810 --> 00:12:20,280
So this expression is applicable
to any single degree of freedom

197
00:12:20,280 --> 00:12:23,430
system that oscillates.

198
00:12:23,430 --> 00:12:25,861
You just have to
exchange a couple things.

199
00:12:25,861 --> 00:12:27,485
So let's think about
a simple pendulum.

200
00:12:43,050 --> 00:12:49,770
So our common massless
string and a bob on the end,

201
00:12:49,770 --> 00:12:58,215
some length L,
equation of motion.

202
00:13:03,460 --> 00:13:05,810
And this is point A up here.

203
00:13:05,810 --> 00:13:20,930
IZZ with respect to A. Theta
double dot plus MgL sine theta

204
00:13:20,930 --> 00:13:22,580
equals 0.

205
00:13:22,580 --> 00:13:23,830
That's the equation of motion.

206
00:13:23,830 --> 00:13:25,329
With no damping,
that's the equation

207
00:13:25,329 --> 00:13:27,080
of motion in this system.

208
00:13:27,080 --> 00:13:30,060
Is it a linear
differential equation?

209
00:13:33,500 --> 00:13:35,790
And to do the
things that we want

210
00:13:35,790 --> 00:13:40,210
to be able to do in this
course, like vibration

211
00:13:40,210 --> 00:13:43,240
with harmonic inputs
and so forth, we

212
00:13:43,240 --> 00:13:46,040
want to deal with
linear equations.

213
00:13:46,040 --> 00:13:48,940
So one of the topics for
today is linearization.

214
00:13:48,940 --> 00:13:52,550
So this is one of the simplest
examples of linearization.

215
00:13:52,550 --> 00:13:57,470
We need a linearized
equation, and we

216
00:13:57,470 --> 00:14:01,790
need to remember in a
couple of approximations.

217
00:14:01,790 --> 00:14:06,410
So sine of theta, you can
do Taylor series expansion.

218
00:14:06,410 --> 00:14:13,780
It's theta minus theta cubed
over 3 factorial plus theta

219
00:14:13,780 --> 00:14:19,220
to the fifth over 5 factorial
plus minus and so forth.

220
00:14:19,220 --> 00:14:26,970
And cosine of theta
is 1 minus theta

221
00:14:26,970 --> 00:14:32,030
squared over 2 factorial plus
theta to the fourth over 4

222
00:14:32,030 --> 00:14:35,620
factorial, and so forth.

223
00:14:39,110 --> 00:14:42,480
So what we say, what do we mean
when we linearize something?

224
00:14:42,480 --> 00:14:44,410
So linearization means
that we're essentially

225
00:14:44,410 --> 00:14:51,940
assuming the variable that we're
working with is small enough

226
00:14:51,940 --> 00:14:55,710
that the right hand side,
an adequate approximation

227
00:14:55,710 --> 00:15:00,200
of this function, is to keep
only up to the linear terms

228
00:15:00,200 --> 00:15:03,220
on the right hand side.

229
00:15:03,220 --> 00:15:08,580
For sine, the term
raised to the 1 power,

230
00:15:08,580 --> 00:15:09,850
that's the linear term.

231
00:15:09,850 --> 00:15:12,720
This is a cubic term,
a fifth order term.

232
00:15:12,720 --> 00:15:15,790
We're going to throw those away
and say, this is close enough.

233
00:15:15,790 --> 00:15:18,320
For cosine, it's 1
minus theta squared.

234
00:15:18,320 --> 00:15:19,810
We throw these away.

235
00:15:19,810 --> 00:15:25,440
The small angle approximation
for cosine as it's equal to 1.

236
00:15:25,440 --> 00:15:27,940
That's the simplest
example of linearization,

237
00:15:27,940 --> 00:15:32,350
of a non-linear term.

238
00:15:32,350 --> 00:15:37,120
So when you linearize
this equation of motion,

239
00:15:37,120 --> 00:15:47,150
we end up with IZZ with respect
to A theta double dot plus MGL

240
00:15:47,150 --> 00:15:51,660
theta equals 0, and we get
our familiar natural frequency

241
00:15:51,660 --> 00:15:54,820
for Bob as square
root of g over L.

242
00:15:54,820 --> 00:15:57,745
So we need linearization to be
able to do pendulum problems.

243
00:16:04,865 --> 00:16:05,365
Hmm.

244
00:16:08,790 --> 00:16:10,690
OK.

245
00:16:10,690 --> 00:16:12,530
Or maybe let's do
an example here.

246
00:16:12,530 --> 00:16:15,060
I've got a pendulum that
we'll do an experiment

247
00:16:15,060 --> 00:16:16,230
with this morning.

248
00:16:16,230 --> 00:16:20,820
But 30 degrees is
about like that.

249
00:16:20,820 --> 00:16:23,220
17 degrees is about like that.

250
00:16:23,220 --> 00:16:25,380
That's quite a bit of angle.

251
00:16:25,380 --> 00:16:32,720
Is that small in the sense that
I'm linearizing this equation?

252
00:16:32,720 --> 00:16:39,520
So 17 degrees happens to be--
I'll have to use this here.

253
00:16:47,918 --> 00:16:54,550
That's actually 17.2
degrees equals 0.3 radians.

254
00:16:58,750 --> 00:17:10,140
Sine of 0.3 is 0.2955.

255
00:17:10,140 --> 00:17:14,240
And if we fill out and
look at these terms,

256
00:17:14,240 --> 00:17:22,220
the lead term here is 0.3--
so plug in the 0.3-- minus--

257
00:17:22,220 --> 00:17:29,530
and the second term when you
cube 0.3 and divide it by 6,

258
00:17:29,530 --> 00:17:40,750
the second term is minus-- what
is my number here-- 0.0045.

259
00:17:40,750 --> 00:17:44,130
And if you subtract that from
this, you get exactly this.

260
00:17:44,130 --> 00:17:48,110
So to four decimal
places, you only

261
00:17:48,110 --> 00:17:51,210
need two terms in this series
to get exactly the right answer.

262
00:17:51,210 --> 00:17:54,750
This thing out here, this fifth
order term, is really tiny.

263
00:17:54,750 --> 00:17:58,180
But the approximation, if
we say, OK, let's skip this,

264
00:17:58,180 --> 00:18:02,620
we're saying that 0.3
is approximately 0.2955.

265
00:18:02,620 --> 00:18:05,670
Pretty close.

266
00:18:05,670 --> 00:18:08,872
So up to 17 degrees,
0.3 radians,

267
00:18:08,872 --> 00:18:10,080
that's a great approximation.

268
00:18:14,060 --> 00:18:17,871
So it's a little high by about,
I think it's about 1 and 1/2%

269
00:18:17,871 --> 00:18:18,370
high.

270
00:18:26,250 --> 00:18:30,070
So for pretty large
angles for pendula,

271
00:18:30,070 --> 00:18:32,040
that simple linearization
works just fine.

272
00:18:34,700 --> 00:18:40,370
OK, once we get it linearized,
that equation of motion

273
00:18:40,370 --> 00:18:45,420
is of exactly the same
form as the one up there.

274
00:18:45,420 --> 00:18:47,380
We don't have any damping in it.

275
00:18:47,380 --> 00:18:50,510
We could add some damping.

276
00:18:50,510 --> 00:18:54,510
We can put a damping in here
with a torsional damper-- ct

277
00:18:54,510 --> 00:18:56,110
theta dot.

278
00:18:56,110 --> 00:18:59,570
And now that equation is
of exactly the same form

279
00:18:59,570 --> 00:19:04,690
as the linear oscillator,
linear meaning translational

280
00:19:04,690 --> 00:19:06,270
oscillator.

281
00:19:06,270 --> 00:19:09,480
Have that inertia term, a
damping term, a stiffness term.

282
00:19:09,480 --> 00:19:12,700
It's a second order linear
differential equation,

283
00:19:12,700 --> 00:19:14,499
homogeneous linear
differential equation,

284
00:19:14,499 --> 00:19:15,790
nothing on the right hand side.

285
00:19:22,730 --> 00:19:25,400
Because they're
exactly the same form,

286
00:19:25,400 --> 00:19:31,940
then the solution for
decay, transient decay

287
00:19:31,940 --> 00:19:35,670
from initial conditions,
takes on exactly the same form

288
00:19:35,670 --> 00:19:40,590
except that it has an
initial angle, theta 0,

289
00:19:40,590 --> 00:19:42,430
and I use the
approximation here.

290
00:19:42,430 --> 00:19:50,940
Cosine omega dt plus theta
0 dot, the initial velocity,

291
00:19:50,940 --> 00:20:01,070
over omega d sine omega dt all
times e to the minus zeta omega

292
00:20:01,070 --> 00:20:02,960
nt.

293
00:20:02,960 --> 00:20:06,030
So that's the exact same
transient decay equation,

294
00:20:06,030 --> 00:20:08,850
but now cast in angular terms.

295
00:20:15,310 --> 00:20:20,460
And if you wanted to express
it as a cosine omega t minus v,

296
00:20:20,460 --> 00:20:24,080
then A would be this squared
plus this squared square root

297
00:20:24,080 --> 00:20:30,570
and the phi would be a
similar calculation as we

298
00:20:30,570 --> 00:20:32,690
have up there someone here.

299
00:20:35,330 --> 00:20:38,900
Just the B term over the
A term, tangent numbers.

300
00:20:38,900 --> 00:20:41,790
All right.

301
00:20:41,790 --> 00:20:42,610
What's it good for?

302
00:20:42,610 --> 00:20:45,350
So I use this, this equation
gets used quite a lot.

303
00:20:45,350 --> 00:20:46,695
It has some practical uses.

304
00:20:51,140 --> 00:20:57,000
Let's do an example, a little
more complicated pendulum.

305
00:20:57,000 --> 00:20:57,840
Draw a stick maybe.

306
00:21:01,470 --> 00:21:02,580
Center of mass there.

307
00:21:07,295 --> 00:21:15,020
A, IZZ with respect
to A. We'll call it

308
00:21:15,020 --> 00:21:18,155
ML cubed over 3
for a slender rod.

309
00:21:23,310 --> 00:21:26,340
And now, what I
want to do is I have

310
00:21:26,340 --> 00:21:34,490
coming along here a mass,
a bullet, that has mass m.

311
00:21:34,490 --> 00:21:39,940
Has velocity vi
for initial here,

312
00:21:39,940 --> 00:21:46,990
and that's its linear
momentum, p initial.

313
00:21:49,540 --> 00:21:52,532
And it's going to hit
this stick and bed in it.

314
00:21:52,532 --> 00:21:53,990
So you've done this
problem before.

315
00:21:53,990 --> 00:21:54,490
Yeah.

316
00:21:54,490 --> 00:21:56,720
AUDIENCE: [INAUDIBLE].

317
00:21:56,720 --> 00:21:57,470
PROFESSOR: Pardon?

318
00:21:57,470 --> 00:21:59,730
AUDIENCE: ML cubed [INAUDIBLE].

319
00:21:59,730 --> 00:22:01,450
PROFESSOR: ML cubed over 3.

320
00:22:05,850 --> 00:22:06,470
Good.

321
00:22:06,470 --> 00:22:08,620
I don't know why I was
thinking cubed this morning.

322
00:22:08,620 --> 00:22:10,090
ML squared over 3.

323
00:22:10,090 --> 00:22:10,990
Good catch there.

324
00:22:10,990 --> 00:22:12,360
OK, so we have it.

325
00:22:12,360 --> 00:22:14,250
This is mass moment of inertia.

326
00:22:14,250 --> 00:22:15,764
This is a pendulum.

327
00:22:15,764 --> 00:22:17,180
This bullet's going
to come along.

328
00:22:20,870 --> 00:22:22,550
So this is exactly
what I've got here.

329
00:22:22,550 --> 00:22:24,100
I'll get it in the picture.

330
00:22:24,100 --> 00:22:25,240
Yeah.

331
00:22:25,240 --> 00:22:27,140
So it's initially at rest.

332
00:22:27,140 --> 00:22:29,690
Coming along, this
paper, this clip here,

333
00:22:29,690 --> 00:22:31,005
it represents the bullet.

334
00:22:31,005 --> 00:22:33,530
So it's swimming along.

335
00:22:33,530 --> 00:22:35,800
Hits this thing, sticks to
it, and when it hits it,

336
00:22:35,800 --> 00:22:37,870
it does that.

337
00:22:37,870 --> 00:22:40,900
So then this thing after it
hits swings back and forth.

338
00:22:40,900 --> 00:22:43,420
So what's the response
of this system

339
00:22:43,420 --> 00:22:44,915
to being hit by the bullet?

340
00:22:44,915 --> 00:22:49,330
Well, I claim you can do it
entirely by evaluating response

341
00:22:49,330 --> 00:22:51,620
to initial conditions.

342
00:22:51,620 --> 00:22:55,625
But we need to use one
conservation law to get there.

343
00:22:59,790 --> 00:23:03,260
So what's conserved on impact?

344
00:23:03,260 --> 00:23:05,640
Is linear momentum
conserved on impact?

345
00:23:09,930 --> 00:23:12,580
How many think yes?

346
00:23:12,580 --> 00:23:14,270
Linear momentum conserved.

347
00:23:14,270 --> 00:23:16,330
How many think angular
momentum's conserved?

348
00:23:16,330 --> 00:23:16,830
Hm.

349
00:23:16,830 --> 00:23:19,780
Good. you guys have learned
something this year.

350
00:23:19,780 --> 00:23:20,600
That's great.

351
00:23:20,600 --> 00:23:22,255
Why is linear momentum
not conserved?

352
00:23:27,560 --> 00:23:31,130
Because are there any
possible other external forces

353
00:23:31,130 --> 00:23:33,320
on the system?

354
00:23:33,320 --> 00:23:34,270
At the pin joint.

355
00:23:34,270 --> 00:23:36,860
You can have reaction
forces here and there.

356
00:23:36,860 --> 00:23:38,910
You have no control of them.

357
00:23:38,910 --> 00:23:41,350
But the moments
about this point,

358
00:23:41,350 --> 00:23:44,640
are there any external
moments about that point

359
00:23:44,640 --> 00:23:45,780
during the impact?

360
00:23:45,780 --> 00:23:46,280
No.

361
00:23:46,280 --> 00:23:48,405
They're reaction forces,
but there's no moment arm.

362
00:23:48,405 --> 00:23:49,830
So there's no moments.

363
00:23:49,830 --> 00:23:52,780
So you can use conservation
of angular momentum.

364
00:23:52,780 --> 00:23:56,620
So H1 I'll call it
here with respect to A

365
00:23:56,620 --> 00:24:01,410
is just R cross Pi.

366
00:24:01,410 --> 00:24:07,030
And the R is the length
in the I direction.

367
00:24:07,030 --> 00:24:10,730
P is in the j direction, so
the momentum is in the k.

368
00:24:10,730 --> 00:24:17,970
So this should be
mv initial times L,

369
00:24:17,970 --> 00:24:20,830
and its direction is
in the k hat direction.

370
00:24:20,830 --> 00:24:23,960
So that's the initial
angular momentum

371
00:24:23,960 --> 00:24:25,760
of the system with
respect to this.

372
00:24:25,760 --> 00:24:27,520
This has no initial
angular momentum

373
00:24:27,520 --> 00:24:29,550
because it's motionless.

374
00:24:29,550 --> 00:24:32,010
And since angular
momentum is conserved,

375
00:24:32,010 --> 00:24:35,785
that H2 we'll call
it with respect to A

376
00:24:35,785 --> 00:24:40,910
has got to be equal to
H1 with respect to A,

377
00:24:40,910 --> 00:24:49,880
and that will then be IZZ
with respect to A theta dot.

378
00:24:49,880 --> 00:24:54,040
But I need to account
for the mass, this thing.

379
00:24:54,040 --> 00:24:58,215
So the total mass moment of
inertia with respect to A

380
00:24:58,215 --> 00:25:01,780
is IZZ with respect
to A plus M-- what?

381
00:25:06,580 --> 00:25:09,250
Now I've got the total mass
moment of inertia with respect

382
00:25:09,250 --> 00:25:13,335
to this point, that of the stick
plus that of the initial mass

383
00:25:13,335 --> 00:25:15,140
that I've stuck on there.

384
00:25:15,140 --> 00:25:18,380
And this must be
equal to theta dot.

385
00:25:18,380 --> 00:25:19,980
And I put a not down
here because I'm

386
00:25:19,980 --> 00:25:23,540
looking for my equivalent
initial condition.

387
00:25:23,540 --> 00:25:31,810
And this then is
mv initial times

388
00:25:31,810 --> 00:25:37,730
L. Then I can solve
for theta dot 0,

389
00:25:37,730 --> 00:25:50,520
and that looks like mv initial
L over IZZ A plus mL squared.

390
00:25:50,520 --> 00:25:55,081
And everything on the
right hand side you know.

391
00:25:55,081 --> 00:25:57,330
You know the initial velocity,
the mass of the bullet,

392
00:25:57,330 --> 00:26:01,470
the length of the distance
from the pivot, mass moment

393
00:26:01,470 --> 00:26:05,240
of inertia, and the additional
mass moment of inertia.

394
00:26:05,240 --> 00:26:06,892
These are all
numbers you plug in,

395
00:26:06,892 --> 00:26:08,100
and you get a value for this.

396
00:26:08,100 --> 00:26:11,970
And once you have a value
for this, you can use that.

397
00:26:14,750 --> 00:26:17,110
In this problem, what's theta 0?

398
00:26:21,320 --> 00:26:25,460
The initial angular
deflection at time t0

399
00:26:25,460 --> 00:26:29,550
plus right after
the bullets hit it.

400
00:26:29,550 --> 00:26:35,600
And it hasn't moved because it
hasn't had time to move yet.

401
00:26:35,600 --> 00:26:37,710
At some velocity,
it takes finite time

402
00:26:37,710 --> 00:26:39,500
to get a deflection.

403
00:26:39,500 --> 00:26:42,700
So there's zero initial
angular deflection,

404
00:26:42,700 --> 00:26:48,020
but you get a step up in
initial angular velocity.

405
00:26:48,020 --> 00:26:51,180
And so the response
of this system

406
00:26:51,180 --> 00:27:05,210
is theta t is theta 0 dot
over omega d sine omega dt.

407
00:27:05,210 --> 00:27:06,060
So what's omega d?

408
00:27:15,950 --> 00:27:18,930
Remember, I'll
define a few things.

409
00:27:18,930 --> 00:27:25,880
In this case, this
is ct over 2 IZZ

410
00:27:25,880 --> 00:27:32,675
z A plus little ml
squared-- we have

411
00:27:32,675 --> 00:27:37,120
to deal with all the quantities
after the collision-- 2 times

412
00:27:37,120 --> 00:27:38,690
omega n.

413
00:27:38,690 --> 00:27:47,080
That's the damping ratio
for this torsional pendulum,

414
00:27:47,080 --> 00:27:48,170
with this pendulum.

415
00:27:48,170 --> 00:27:49,960
It's the damping constant.

416
00:27:49,960 --> 00:27:53,170
2 times the mass, the
inertial quantity,

417
00:27:53,170 --> 00:27:57,300
times omega n for a
translational system

418
00:27:57,300 --> 00:28:04,260
at c over 2 m omega n.

419
00:28:04,260 --> 00:28:07,840
For a pendulum system,
it's the torsional

420
00:28:07,840 --> 00:28:13,040
damping over 2 times the mass
moment of inertia times omega

421
00:28:13,040 --> 00:28:14,490
n.

422
00:28:14,490 --> 00:28:22,630
And omega n, well, it
is going to calculate

423
00:28:22,630 --> 00:28:23,930
the natural frequency.

424
00:28:23,930 --> 00:28:27,730
It's just MgL divided
by IZZ plus this.

425
00:28:27,730 --> 00:28:29,540
Maybe you ought to
write that down.

426
00:28:35,790 --> 00:28:38,630
So always for a sample singular
[INAUDIBLE] oscillator,

427
00:28:38,630 --> 00:28:40,340
you want the undamped
natural frequency.

428
00:28:40,340 --> 00:28:42,050
Ignore the damping term.

429
00:28:42,050 --> 00:28:45,880
Take the stiffness term
coefficient here and divide it

430
00:28:45,880 --> 00:28:47,570
by the inertial coefficient.

431
00:28:47,570 --> 00:28:51,510
But we care about the natural
frequency after the impact,

432
00:28:51,510 --> 00:28:58,055
so this is going to be-- ah.

433
00:28:58,055 --> 00:29:01,740
The trouble is here I
don't know for this system,

434
00:29:01,740 --> 00:29:03,910
I haven't worked out yet,
what this term looks like.

435
00:29:03,910 --> 00:29:04,698
What is it?

436
00:29:07,710 --> 00:29:11,920
This result right here
is for the simple Bob.

437
00:29:11,920 --> 00:29:18,880
For this stick, it's MgL over
2 plus the little m times l.

438
00:29:18,880 --> 00:29:20,690
Little more messy.

439
00:29:20,690 --> 00:29:27,060
So MgL over 2 plus little ml.

440
00:29:27,060 --> 00:29:30,880
That'll be the-- come from the
potential energy in this system

441
00:29:30,880 --> 00:29:37,490
all over IZZ A plus mL squared.

442
00:29:37,490 --> 00:29:39,230
So you get your
natural frequency out

443
00:29:39,230 --> 00:29:42,146
of that expression.

444
00:29:42,146 --> 00:29:42,645
OK.

445
00:29:49,450 --> 00:29:53,970
So you do this problem
sometimes before when

446
00:29:53,970 --> 00:29:56,660
you do, say, somebody asks
you how high does it swing.

447
00:29:56,660 --> 00:29:57,250
AND so forth.

448
00:29:57,250 --> 00:30:00,030
Well, you can do
it by conservation

449
00:30:00,030 --> 00:30:02,050
of energy, et cetera.

450
00:30:02,050 --> 00:30:04,520
But now, you have
actually exact expression

451
00:30:04,520 --> 00:30:10,130
for the time history of
the thing after the impact,

452
00:30:10,130 --> 00:30:12,420
including the
effects of damping.

453
00:30:12,420 --> 00:30:19,870
And if you were to draw
the result of this function

454
00:30:19,870 --> 00:30:23,970
of theta as a function
of time, this one

455
00:30:23,970 --> 00:30:30,020
starts with no initial
displacement but a velocity

456
00:30:30,020 --> 00:30:30,685
and does this.

457
00:30:37,600 --> 00:30:39,830
And that's your
exponential decay envelope,

458
00:30:39,830 --> 00:30:42,150
and this is time.

459
00:30:45,910 --> 00:30:47,970
Now, what-- yeah.

460
00:30:47,970 --> 00:30:48,470
Excuse me.

461
00:30:51,970 --> 00:30:52,970
AUDIENCE: [INAUDIBLE].

462
00:30:52,970 --> 00:30:54,465
PROFESSOR: Why this one?

463
00:30:54,465 --> 00:30:57,860
AUDIENCE: Yes. [INAUDIBLE].

464
00:30:57,860 --> 00:30:59,170
PROFESSOR: Excuse me.

465
00:30:59,170 --> 00:30:59,870
Forgot the g.

466
00:30:59,870 --> 00:31:02,996
I mean, it accounts for
the restoring moment,

467
00:31:02,996 --> 00:31:04,870
the additional little
bit of restoring moment

468
00:31:04,870 --> 00:31:08,040
that you get from having
added the mass of this thing

469
00:31:08,040 --> 00:31:09,250
to it, right.

470
00:31:09,250 --> 00:31:12,810
So it has by itself
MgL sine theta,

471
00:31:12,810 --> 00:31:13,970
and we linearize that too.

472
00:31:13,970 --> 00:31:18,650
So it's MgL theta, and those
two terms would add together.

473
00:31:18,650 --> 00:31:22,380
So you just have
a second term here

474
00:31:22,380 --> 00:31:27,030
that has MgL like behavior.

475
00:31:27,030 --> 00:31:31,470
How I most often personally
make use of expressions

476
00:31:31,470 --> 00:31:34,560
like this, or the
one for translation,

477
00:31:34,560 --> 00:31:39,140
is because
experimentally, if I'm

478
00:31:39,140 --> 00:31:42,440
trying to predict the
vibration behavior of a system,

479
00:31:42,440 --> 00:31:46,360
one of the things you want
to know is the damping.

480
00:31:46,360 --> 00:31:48,790
And one of the simplest
ways to measure damping

481
00:31:48,790 --> 00:31:53,460
is to give a system an initial
deflection or initial velocity

482
00:31:53,460 --> 00:31:56,420
and measure its decay,
and from its decay,

483
00:31:56,420 --> 00:31:58,240
calculate the damping.

484
00:31:58,240 --> 00:32:06,470
So the last time I gave you
an expression for doing that,

485
00:32:06,470 --> 00:32:13,720
and that was a damping
ratio 1 over 2 pi

486
00:32:13,720 --> 00:32:18,110
times the number of cycles that
you count, that you watch it,

487
00:32:18,110 --> 00:32:27,750
times the natural log of
x of t over x of t plus n

488
00:32:27,750 --> 00:32:31,620
periods of vibration.

489
00:32:31,620 --> 00:32:32,550
This has a name.

490
00:32:32,550 --> 00:32:35,940
It's called the logarithmic
decrement, this thing.

491
00:32:35,940 --> 00:32:38,250
So if somebody
says log decrement,

492
00:32:38,250 --> 00:32:40,990
that's where they're
referring to this expression.

493
00:32:51,265 --> 00:32:54,340
A comment about this.

494
00:32:54,340 --> 00:33:07,380
In this expression, x
of t must be zero means

495
00:33:07,380 --> 00:33:12,080
if your measurement-- we have
x of t here, or theta of t--

496
00:33:12,080 --> 00:33:14,100
they must be zero mean.

497
00:33:14,100 --> 00:33:16,670
There must be
oscillations around zero

498
00:33:16,670 --> 00:33:19,460
or you have to have subtracted
the mean to get it there

499
00:33:19,460 --> 00:33:21,730
because if this is
displaced and is oscillating

500
00:33:21,730 --> 00:33:25,480
around some offset, then this
calculates and will get really

501
00:33:25,480 --> 00:33:25,980
messed up.

502
00:33:25,980 --> 00:33:28,250
It's got an offset
plus an offset here

503
00:33:28,250 --> 00:33:29,160
plus an offset there.

504
00:33:29,160 --> 00:33:31,500
It means it's
totally meaningless.

505
00:33:31,500 --> 00:33:36,970
So you must remove the mean
value from any time history

506
00:33:36,970 --> 00:33:39,360
that you go to do this.

507
00:33:39,360 --> 00:33:42,790
So there's an easier way the
same expression-- and this is,

508
00:33:42,790 --> 00:33:45,130
in fact, the way I use this.

509
00:33:45,130 --> 00:33:48,030
A plot out like just
your data acquisition

510
00:33:48,030 --> 00:33:50,420
grabs it, plots it for you.

511
00:33:50,420 --> 00:33:53,410
I take this value
from here to here,

512
00:33:53,410 --> 00:33:57,510
and this is my peak
to peak amplitude.

513
00:33:57,510 --> 00:34:01,640
And then I go out n cycles
later and find the peak

514
00:34:01,640 --> 00:34:03,940
to peak amplitude.

515
00:34:03,940 --> 00:34:06,920
And so this is perfectly,
this is just the same as 1

516
00:34:06,920 --> 00:34:15,530
over 2 pi n, but now you
do natural log of x peak

517
00:34:15,530 --> 00:34:22,969
to peak t over x peak to
peak at t plus n periods.

518
00:34:25,670 --> 00:34:27,844
And that now, peak
to peak measurement,

519
00:34:27,844 --> 00:34:29,010
you totally ignore the mean.

520
00:34:29,010 --> 00:34:30,870
Doesn't matter where you are.

521
00:34:30,870 --> 00:34:35,010
You want the here to here,
here to here, plug it in there,

522
00:34:35,010 --> 00:34:37,320
and you're done.

523
00:34:37,320 --> 00:34:49,219
OK, so let's-- I got 1, 2, 3, 4.

524
00:34:49,219 --> 00:35:01,060
Let's let n equal 4, and let's
assume this expression here--

525
00:35:01,060 --> 00:35:05,830
n is 1 over 2 pi times 4.

526
00:35:05,830 --> 00:35:10,090
And let's assume that
in these four periods

527
00:35:10,090 --> 00:35:15,230
from-- that's 1 period,
2, 3, 4 getting out here

528
00:35:15,230 --> 00:35:21,710
to this fourth, four periods
away, that this is one fifth

529
00:35:21,710 --> 00:35:23,610
the initial.

530
00:35:23,610 --> 00:35:25,945
So this would be the
natural log of 5.

531
00:35:28,980 --> 00:35:32,290
So 1 over 2 pi times
4, natural log of 5,

532
00:35:32,290 --> 00:35:36,895
and you run the
numbers, you get 0.064,

533
00:35:36,895 --> 00:35:42,575
or what we call 6.4% damping.

534
00:35:42,575 --> 00:35:43,450
That's how you do it.

535
00:35:43,450 --> 00:35:46,630
That's the way you do a
calculation like that.

536
00:35:46,630 --> 00:35:52,150
Now, I gave you a quick rule of
thumb for estimating damping,

537
00:35:52,150 --> 00:35:54,910
and this is what
I-- I can't work.

538
00:35:54,910 --> 00:35:58,220
I don't do logs
in my head, but I

539
00:35:58,220 --> 00:36:01,050
can do damping
estimates without that

540
00:36:01,050 --> 00:36:06,400
because I know
that zeta is also--

541
00:36:06,400 --> 00:36:09,040
if I just plug in
some numbers here

542
00:36:09,040 --> 00:36:13,680
and run them all
in advance is 0.11

543
00:36:13,680 --> 00:36:16,695
divided by the number
of cycles to decay 50%.

544
00:36:20,132 --> 00:36:21,590
So we're going to
do an experiment.

545
00:36:26,990 --> 00:36:29,620
And I guess it can be
seen with the camera.

546
00:36:29,620 --> 00:36:33,000
So here's my pendulum.

547
00:36:33,000 --> 00:36:38,030
This is my initial amplitude,
and this is about half.

548
00:36:38,030 --> 00:36:45,190
So if I take this thing over
here, like that, then let go,

549
00:36:45,190 --> 00:36:49,780
and count the cycles that
it takes to decay halfway,

550
00:36:49,780 --> 00:36:50,889
we can do this experiment.

551
00:36:50,889 --> 00:36:51,930
So let's do it carefully.

552
00:36:51,930 --> 00:36:54,160
So line it up like
that, and you're

553
00:36:54,160 --> 00:36:57,480
going to help me tell-- you
count how many cycles it

554
00:36:57,480 --> 00:36:59,250
takes till it gets to here.

555
00:36:59,250 --> 00:37:15,330
So 1, 2, 3, 4, 5, 6, 7, 8.

556
00:37:15,330 --> 00:37:16,920
About eight cycles.

557
00:37:16,920 --> 00:37:20,170
So it decayed halfway
in eight cycles.

558
00:37:20,170 --> 00:37:27,016
So zeta is approximately 0.11/8.

559
00:37:27,016 --> 00:37:30,240
It's 1 and 1/2%, 1.4%,
something like that.

560
00:37:33,042 --> 00:37:34,500
Perfectly good
estimate of damping.

561
00:37:34,500 --> 00:37:44,780
Now, the stopwatch here, we
can do this experiment again.

562
00:37:47,660 --> 00:37:49,100
I want you to count.

563
00:37:49,100 --> 00:37:52,000
You're doing the counting.

564
00:37:52,000 --> 00:37:57,390
And I'm going to say start,
and I want you to count cycles

565
00:37:57,390 --> 00:37:59,570
until I say stop.

566
00:37:59,570 --> 00:38:04,040
Now, I'll probably stop on 10
to make the calculation easy,

567
00:38:04,040 --> 00:38:08,090
so quietly to yourself
count the number of cycles

568
00:38:08,090 --> 00:38:11,950
from the time I release
it until the time I stop.

569
00:38:11,950 --> 00:38:13,060
Come back here.

570
00:38:19,110 --> 00:38:22,150
So this time, the
backdrop doesn't matter.

571
00:38:22,150 --> 00:38:23,870
I just want you to count cycles.

572
00:38:27,200 --> 00:38:29,220
And I'll start-- I'll
let it get going,

573
00:38:29,220 --> 00:38:30,730
and when it comes
back to me is when

574
00:38:30,730 --> 00:38:33,705
I'm going to start the
stopwatch because I

575
00:38:33,705 --> 00:38:35,580
have a hard time doing
both at the same time.

576
00:38:35,580 --> 00:38:38,190
So start.

577
00:38:57,810 --> 00:38:59,320
How many cycles?

578
00:38:59,320 --> 00:39:01,320
AUDIENCE: [INAUDIBLE].

579
00:39:01,320 --> 00:39:11,390
PROFESSOR: So I
got 17.84 for 10.

580
00:39:11,390 --> 00:39:25,010
So 10 divided by 10 is
1.784 seconds per cycle.

581
00:39:25,010 --> 00:39:26,960
Can't write like that.

582
00:39:26,960 --> 00:39:29,450
1.784 seconds per cycle.

583
00:39:29,450 --> 00:39:33,159
The frequency would
be 1 over that, right.

584
00:39:33,159 --> 00:39:34,950
The thing you have to
be careful about when

585
00:39:34,950 --> 00:39:44,400
you're counting cycles is if
I start here, that's 0, 1, 2.

586
00:39:44,400 --> 00:39:47,160
A very common human mistake is
when you're counting something

587
00:39:47,160 --> 00:39:49,380
like this is to say
one when you start,

588
00:39:49,380 --> 00:39:52,610
and then you're going
to be off by one count.

589
00:39:52,610 --> 00:39:54,020
Follow me?

590
00:39:54,020 --> 00:39:59,320
If I start 0, 1, 2.

591
00:39:59,320 --> 00:40:01,960
So I start the clock on
zero, but the first cycle

592
00:40:01,960 --> 00:40:04,030
isn't completed till
one whole cycle later.

593
00:40:04,030 --> 00:40:06,985
So be careful how you count.

594
00:40:06,985 --> 00:40:07,485
OK.

595
00:40:23,670 --> 00:40:35,380
Now we're going to shift
gears and take on a new topic,

596
00:40:35,380 --> 00:40:40,450
and that's the response
to a harmonic input,

597
00:40:40,450 --> 00:40:43,823
some cosine omega t excitation.

598
00:40:43,823 --> 00:40:44,769
Yeah.

599
00:40:44,769 --> 00:40:46,065
AUDIENCE: [INAUDIBLE].

600
00:40:46,065 --> 00:40:47,190
PROFESSOR: What is omega d?

601
00:41:02,340 --> 00:41:05,224
So it's the damped
natural frequency.

602
00:41:05,224 --> 00:41:06,390
That's how it's referred to.

603
00:41:06,390 --> 00:41:09,560
It is the frequency
you observe when you do

604
00:41:09,560 --> 00:41:11,782
an experiment like we just did.

605
00:41:11,782 --> 00:41:14,430
The actual oscillation
frequency when

606
00:41:14,430 --> 00:41:16,760
it's responding to
initial conditions

607
00:41:16,760 --> 00:41:18,930
is slightly different
from the undamped,

608
00:41:18,930 --> 00:41:23,590
but if you have light damping,
if you have even 10% damping,

609
00:41:23,590 --> 00:41:27,550
0.1 squared is 0.01.

610
00:41:27,550 --> 00:41:29,370
That's 0.99 square root.

611
00:41:29,370 --> 00:41:32,090
It's 0.995.

612
00:41:32,090 --> 00:41:33,660
So you're only off by half.

613
00:41:33,660 --> 00:41:35,600
They're only half a
percent difference.

614
00:41:35,600 --> 00:41:38,450
So for lightly damped systems,
for all intents and purposes,

615
00:41:38,450 --> 00:41:41,250
mega n and mega d are
almost exactly the same.

616
00:41:43,920 --> 00:41:44,420
OK.

617
00:41:47,020 --> 00:41:53,890
We now want to think about--
we have a linear system putting

618
00:41:53,890 --> 00:41:56,360
a force into it.

619
00:41:56,360 --> 00:42:01,080
It looks like some
F0 cosine omega t.

620
00:42:01,080 --> 00:42:05,430
And out of that system, we
measure a response, x of t.

621
00:42:09,380 --> 00:42:14,300
And inside this box here is
my system transfer function.

622
00:42:14,300 --> 00:42:17,545
It's the mathematics that
tells me I can take F of t

623
00:42:17,545 --> 00:42:21,687
in and predict
what x of t out is.

624
00:42:21,687 --> 00:42:23,270
So I need to know
the information that

625
00:42:23,270 --> 00:42:25,720
goes into this box, and of
course, the real system-- this

626
00:42:25,720 --> 00:42:27,280
is just the mechanical system.

627
00:42:27,280 --> 00:42:29,880
Force in, measured output out.

628
00:42:29,880 --> 00:42:33,660
This is what we call a
single input single output

629
00:42:33,660 --> 00:42:37,820
system, SISO, single input
single output linear system.

630
00:42:42,515 --> 00:42:44,140
And there's all sorts
of linear systems

631
00:42:44,140 --> 00:42:46,265
that you're going to study
as mechanical engineers,

632
00:42:46,265 --> 00:42:49,380
and you've already begun, I'm
sure, studying some of them.

633
00:42:49,380 --> 00:42:51,720
One of the properties
of a linear system

634
00:42:51,720 --> 00:42:57,220
is that you put a force in, F1,
and measure a response out, x1.

635
00:42:57,220 --> 00:42:58,970
And then you try a
different force, F2,

636
00:42:58,970 --> 00:43:01,540
and you measure a
response out, x2.

637
00:43:01,540 --> 00:43:03,600
What's the response if
you put them both in

638
00:43:03,600 --> 00:43:05,300
at the same time, F1 and F2?

639
00:43:09,550 --> 00:43:13,050
You just add the responses
to them individually.

640
00:43:13,050 --> 00:43:15,360
So F1 gives you x1.

641
00:43:15,360 --> 00:43:16,690
F2 give you x2.

642
00:43:16,690 --> 00:43:20,010
F1 plus F2 gives you
x1 plus x2, and that's

643
00:43:20,010 --> 00:43:25,020
one of the characteristics
of a linear system.

644
00:43:25,020 --> 00:43:30,240
We use that concept to be
able to separate the response.

645
00:43:30,240 --> 00:43:33,330
Our calculation's about
the response of a system,

646
00:43:33,330 --> 00:43:37,100
like our oscillator here,
separate its response

647
00:43:37,100 --> 00:43:42,070
to transient effects, transience
being initial conditions.

648
00:43:42,070 --> 00:43:45,750
They die out over time-- that's
why we call them transients--

649
00:43:45,750 --> 00:43:48,690
and steady state effects.

650
00:43:48,690 --> 00:43:50,450
So cosine omega t,
you can leave it

651
00:43:50,450 --> 00:43:52,950
running for a long, long time,
and pretty soon, the system

652
00:43:52,950 --> 00:43:58,930
will settle down to responding
just to that cosine omega t.

653
00:43:58,930 --> 00:44:00,900
And that we call steady state.

654
00:44:00,900 --> 00:44:02,800
And we use them separately.

655
00:44:02,800 --> 00:44:04,640
So we've done
initial conditions.

656
00:44:04,640 --> 00:44:06,920
Now we're going to look
at the steady state

657
00:44:06,920 --> 00:44:13,490
response of a-- say our
oscillator, our mass spring

658
00:44:13,490 --> 00:44:17,920
dashpot, to a harmonic
input, F0 cosine omega t.

659
00:44:20,850 --> 00:44:22,260
Another brief word.

660
00:44:22,260 --> 00:44:33,080
If I have a force, F0 cosine
omega t would look like that.

661
00:44:35,810 --> 00:44:42,770
And the response that I
measure to start off with my--

662
00:44:42,770 --> 00:44:47,090
it's sitting here at zero
when you turn this on.

663
00:44:47,090 --> 00:44:52,780
And it's going to do some odd
things initially, and then

664
00:44:52,780 --> 00:45:02,850
eventually settle down to some
long term steady response.

665
00:45:02,850 --> 00:45:04,900
The amplitude stays constant.

666
00:45:04,900 --> 00:45:08,360
It stays angle with
respect to the input

667
00:45:08,360 --> 00:45:09,840
isn't necessarily the same.

668
00:45:09,840 --> 00:45:18,547
There's some
possibly phase shift.

669
00:45:18,547 --> 00:45:20,880
And that's so the two, if
you're plotting them together,

670
00:45:20,880 --> 00:45:21,671
they won't line up.

671
00:45:21,671 --> 00:45:24,090
But see this messy
stuff at the beginning?

672
00:45:24,090 --> 00:45:27,600
When you first turn this
on, it jumps from here

673
00:45:27,600 --> 00:45:31,740
to here, that force, and it
gives it a kick to begin with.

674
00:45:31,740 --> 00:45:36,390
And this will have some
response initially due

675
00:45:36,390 --> 00:45:38,750
to that transient start up.

676
00:45:38,750 --> 00:45:42,910
And this response is all
modeled by the response

677
00:45:42,910 --> 00:45:44,770
to initial conditions.

678
00:45:44,770 --> 00:45:47,310
And it'll die out after a
while, this messy stuff.

679
00:45:47,310 --> 00:45:49,280
What's the frequency?

680
00:45:49,280 --> 00:45:54,950
What frequency do you expect
this initial, erratic looking

681
00:45:54,950 --> 00:45:55,817
stuff to be at?

682
00:45:59,400 --> 00:46:03,670
Its response to
initial conditions.

683
00:46:03,670 --> 00:46:08,250
What is the model for a
response to initial conditions?

684
00:46:08,250 --> 00:46:09,962
What's the frequency
of the response

685
00:46:09,962 --> 00:46:12,170
to initial conditions of
the single degree of freedom

686
00:46:12,170 --> 00:46:14,470
system?

687
00:46:14,470 --> 00:46:16,580
We have an equation
over here, right?

688
00:46:20,300 --> 00:46:22,640
The top has a cosine
term and a sine term.

689
00:46:22,640 --> 00:46:24,690
Part of it's a response
to initial displacement.

690
00:46:24,690 --> 00:46:27,560
Part of it's a response
to the initial velocity.

691
00:46:27,560 --> 00:46:31,880
Any of this start up stuff can
be cast as initial conditions,

692
00:46:31,880 --> 00:46:33,620
and the response to
initial conditions

693
00:46:33,620 --> 00:46:37,420
is always at the natural
frequency period.

694
00:46:37,420 --> 00:46:41,040
No other frequencies for same
degree of freedom systems.

695
00:46:41,040 --> 00:46:43,270
So you get a behavior
that's oscillating

696
00:46:43,270 --> 00:46:45,010
at its natural frequency.

697
00:46:45,010 --> 00:46:49,640
Mixed in there is a response
at the excitation frequency.

698
00:46:49,640 --> 00:46:52,090
And after a long
time, the response

699
00:46:52,090 --> 00:46:54,180
is only excitation frequency.

700
00:46:54,180 --> 00:46:55,150
This is now out here.

701
00:46:55,150 --> 00:46:56,590
This is omega.

702
00:46:56,590 --> 00:47:04,280
In here, you have omega
and omega d going on.

703
00:47:04,280 --> 00:47:06,430
So this is messy.

704
00:47:06,430 --> 00:47:08,330
Usually isn't
important, but it is.

705
00:47:08,330 --> 00:47:11,690
There are ways of getting the
exact solution, but mostly,

706
00:47:11,690 --> 00:47:15,910
vibration engineers, you're
interested in the long term

707
00:47:15,910 --> 00:47:21,905
steady state response to what
we call a harmonic input.

708
00:47:21,905 --> 00:47:22,405
OK.

709
00:47:40,510 --> 00:47:44,980
So we'll work a
classic single degree

710
00:47:44,980 --> 00:47:52,360
of freedom oscillator problem--
excited by F0 cosine omega t.

711
00:47:52,360 --> 00:47:55,800
You've done this in
1803, but now we'll

712
00:47:55,800 --> 00:48:00,230
do it using engineering
terminology.

713
00:48:00,230 --> 00:48:04,150
We'll look at it the way a
person studying vibration

714
00:48:04,150 --> 00:48:06,160
would think about this.

715
00:48:06,160 --> 00:48:07,535
We know the equation of motion.

716
00:48:21,070 --> 00:48:24,640
And I'm interested in the
steady state response.

717
00:48:24,640 --> 00:48:28,470
So this is x, and I'll do-- you
just write it once like this.

718
00:48:28,470 --> 00:48:29,650
SS, steady state.

719
00:48:35,750 --> 00:48:38,520
I'm only interested in
its-- after those transients

720
00:48:38,520 --> 00:48:39,400
have died out.

721
00:48:43,680 --> 00:48:46,130
And that steady
state response I know

722
00:48:46,130 --> 00:48:51,100
is going to be some
amplitude X0 cosine omega

723
00:48:51,100 --> 00:48:54,320
t minus some phase angle
that I don't necessarily

724
00:48:54,320 --> 00:48:55,810
know to begin with.

725
00:48:55,810 --> 00:48:58,280
But that's my input.

726
00:48:58,280 --> 00:48:59,810
This is my output.

727
00:48:59,810 --> 00:49:04,820
I plug it into here and turn the
crank and see what falls out.

728
00:49:11,510 --> 00:49:16,230
So you plug both of
those in, and you

729
00:49:16,230 --> 00:49:20,480
get two-- you
get-- this is going

730
00:49:20,480 --> 00:49:23,360
to be a little writing
intensive for a few minutes.

731
00:49:38,220 --> 00:49:46,760
So you plug the X0 cosine omega
t into all of these terms.

732
00:49:46,760 --> 00:49:50,070
The m term gives you
minus m omega squared,

733
00:49:50,070 --> 00:49:55,920
the k term gives you a
k, and the damping term,

734
00:49:55,920 --> 00:50:11,950
minus c omega sine omega
t minus v. All of that

735
00:50:11,950 --> 00:50:18,620
equals the right hand
side-- F0 cosine omega t.

736
00:50:18,620 --> 00:50:20,441
So this just purely
from substitution

737
00:50:20,441 --> 00:50:22,065
and then gathering
some terms together.

738
00:50:36,070 --> 00:50:40,450
I'm going to divide
through by k, by k.

739
00:50:40,450 --> 00:50:42,347
If I divide through by
k, k divided by-- this

740
00:50:42,347 --> 00:50:43,700
gives me a one.

741
00:50:43,700 --> 00:50:46,100
This gives me an
m over k, which is

742
00:50:46,100 --> 00:50:49,425
1 over the natural frequency
squared, for example.

743
00:50:49,425 --> 00:50:54,400
And I'm going to put
this into a form that

744
00:50:54,400 --> 00:50:58,470
is the standard form for
discussing vibration problems.

745
00:50:58,470 --> 00:51:04,770
So this equation can be
rewritten in this form.

746
00:51:04,770 --> 00:51:15,320
1 minus omega squared over
omega n squared cosine omega

747
00:51:15,320 --> 00:51:26,990
t minus v minus 2 zeta omega
over omega n sine omega t

748
00:51:26,990 --> 00:51:34,740
minus v. All that's still
equal to F0 cosine omega t.

749
00:51:34,740 --> 00:51:36,910
So this is getting into
kind of more standard form.

750
00:51:36,910 --> 00:51:38,270
So there's 1 minus omega.

751
00:51:38,270 --> 00:51:40,100
This now, this
omega over omega n,

752
00:51:40,100 --> 00:51:45,210
is called the frequency ratio,
and you see a lot of that.

753
00:51:45,210 --> 00:51:48,670
And I've substituted
n here. c omega over k

754
00:51:48,670 --> 00:51:53,230
turns out to be 2 zeta
omega over omega n.

755
00:51:53,230 --> 00:51:55,640
So this frequency
ratio appears a lot.

756
00:51:55,640 --> 00:52:02,000
in our-- let's see here.

757
00:52:10,220 --> 00:52:16,590
You need a couple of trig
identities-- cosine omega t

758
00:52:16,590 --> 00:52:31,510
minus v. Cosine omega t cosine
phi plus sine omega t sine phi,

759
00:52:31,510 --> 00:52:44,880
and sine omega t minus phi
gives you [INAUDIBLE] sine.

760
00:52:44,880 --> 00:52:57,310
Sine omega t cosine phi minus
cosine omega t sine phi.

761
00:52:57,310 --> 00:52:59,820
So that's a trig identity
you actually use quite a bit

762
00:52:59,820 --> 00:53:01,990
doing vibration problems.

763
00:53:01,990 --> 00:53:07,290
We need them, so we
take these, plug them

764
00:53:07,290 --> 00:53:11,980
in in all these places, and
do quite a bit of cranking.

765
00:53:11,980 --> 00:53:13,190
Yep.

766
00:53:13,190 --> 00:53:15,690
AUDIENCE: [INAUDIBLE].

767
00:53:15,690 --> 00:53:17,280
PROFESSOR: Yeah.

768
00:53:17,280 --> 00:53:19,850
Thank you.

769
00:53:19,850 --> 00:53:22,670
And that's called,
that f over k,

770
00:53:22,670 --> 00:53:26,442
is how much the spring
would move statically,

771
00:53:26,442 --> 00:53:28,750
at which the point
would move statically.

772
00:53:28,750 --> 00:53:31,570
We'll need that term also.

773
00:53:31,570 --> 00:53:32,850
OK.

774
00:53:32,850 --> 00:53:36,120
You do all of this.

775
00:53:36,120 --> 00:53:37,375
Here, I'll call these--

776
00:53:59,678 --> 00:54:00,680
OK.

777
00:54:00,680 --> 00:54:12,188
So this is C.
That's expression C.

778
00:54:12,188 --> 00:54:13,470
I can't see that probably.

779
00:54:13,470 --> 00:54:24,960
Call this D, this E. So
you plug D and E into C

780
00:54:24,960 --> 00:54:27,105
and work it through,
you get two equations.

781
00:55:03,650 --> 00:55:05,750
You break it into
two parts because one

782
00:55:05,750 --> 00:55:08,350
is a function of cosine
omega t, and then

783
00:55:08,350 --> 00:55:10,890
you have another part after
this substitution that's

784
00:55:10,890 --> 00:55:14,880
a function of sine omega t,
and you can separate them.

785
00:55:38,230 --> 00:55:40,120
But there's no
sine omega t force.

786
00:55:40,120 --> 00:55:41,710
On the right hand
side, you get zero.

787
00:55:41,710 --> 00:55:43,410
There are two equations here.

788
00:55:43,410 --> 00:55:47,800
How many unknowns do we have?

789
00:55:47,800 --> 00:55:55,110
All we know when we start
this thing is the input,

790
00:55:55,110 --> 00:56:04,700
and we have unknown
response amplitude,

791
00:56:04,700 --> 00:56:07,850
and we have an unknown phase
that we're looking for.

792
00:56:07,850 --> 00:56:08,730
How many equations?

793
00:56:08,730 --> 00:56:09,500
How many unknowns?

794
00:56:09,500 --> 00:56:11,440
Two and two.

795
00:56:11,440 --> 00:56:13,170
So you can do a lot
of cranking, which

796
00:56:13,170 --> 00:56:14,600
I have no intention
of doing here,

797
00:56:14,600 --> 00:56:20,885
and solve for the amplitude
of the response and the phase.

798
00:56:40,690 --> 00:56:45,480
And every textbook-- the
Williams textbook does this.

799
00:56:45,480 --> 00:56:50,920
There are two readings
posted on Stellar

800
00:56:50,920 --> 00:56:54,790
by [? Row. ?] Every textbook
goes through these derivations

801
00:56:54,790 --> 00:56:56,090
that I've just done.

802
00:56:56,090 --> 00:56:57,443
Nick, you've got a question.

803
00:56:57,443 --> 00:56:58,359
AUDIENCE: [INAUDIBLE].

804
00:57:00,900 --> 00:57:01,740
PROFESSOR: Pardon?

805
00:57:01,740 --> 00:57:03,370
AUDIENCE: [INAUDIBLE].

806
00:57:03,370 --> 00:57:06,215
PROFESSOR: Yeah, I
keep forgetting it.

807
00:57:06,215 --> 00:57:07,050
You're right.

808
00:57:07,050 --> 00:57:08,650
So we got a k here.

809
00:57:08,650 --> 00:57:13,480
And notice, this equation,
we throw away that for now.

810
00:57:13,480 --> 00:57:15,270
We get rid of this for now.

811
00:57:15,270 --> 00:57:17,420
We have these two
equations and two unknowns

812
00:57:17,420 --> 00:57:22,400
are just algebraic equations
There's not time dependent.

813
00:57:22,400 --> 00:57:24,210
We can get rid of that part.

814
00:57:24,210 --> 00:57:26,940
So we've now reduced
this to algebra,

815
00:57:26,940 --> 00:57:31,330
and the answer is
plotted up there.

816
00:57:31,330 --> 00:57:33,290
You've probably seen it before.

817
00:57:33,290 --> 00:57:40,780
It says that x0 is F0 over k--
I can get it right this time

818
00:57:40,780 --> 00:57:45,030
from the get go-- over
a denominator, which

819
00:57:45,030 --> 00:57:49,420
appears again and again
and again in vibration.

820
00:57:49,420 --> 00:57:54,530
Omega squared over
omega n squared

821
00:57:54,530 --> 00:58:02,570
squared plus 2 zeta
omega over omega n

822
00:58:02,570 --> 00:58:11,161
squared square root, the
whole thing, and an expression

823
00:58:11,161 --> 00:58:11,660
for phi.

824
00:58:16,050 --> 00:58:23,630
Tangent inverse of 2 zeta
omega over omega n, 1

825
00:58:23,630 --> 00:58:30,000
minus omega squared
over omega n squared.

826
00:58:30,000 --> 00:58:32,310
So you can solve all
that-- this mess over here

827
00:58:32,310 --> 00:58:34,490
for these two quantities.

828
00:58:34,490 --> 00:58:36,690
Do you need to remember this?

829
00:58:36,690 --> 00:58:39,770
You ever going to be
asked this on a quiz?

830
00:58:39,770 --> 00:58:42,550
Not by me.

831
00:58:42,550 --> 00:58:46,320
You ever going to have to use
this on a quiz and in homework?

832
00:58:46,320 --> 00:58:48,820
Absolutely.

833
00:58:48,820 --> 00:58:52,470
So the takeaway
is today know how

834
00:58:52,470 --> 00:58:56,200
to use those response to
initial condition formulas

835
00:58:56,200 --> 00:58:59,950
and damping and these two.

836
00:58:59,950 --> 00:59:04,560
So when you plot,
when you plot these,

837
00:59:04,560 --> 00:59:07,320
you get this picture up there.

838
00:59:07,320 --> 00:59:09,700
And we need to talk about
the properties of this.

839
00:59:13,380 --> 00:59:15,125
Remember, this
omega over omega n

840
00:59:15,125 --> 00:59:16,940
is the same called
the frequency ratio.

841
00:59:16,940 --> 00:59:19,930
It's just the ratio of
the excitation frequency

842
00:59:19,930 --> 00:59:22,790
to the natural
frequency of the system.

843
00:59:22,790 --> 00:59:27,090
And when they're equal, for
example, this ratio is one.

844
00:59:27,090 --> 00:59:30,820
This whole thing in
parentheses goes to zero.

845
00:59:30,820 --> 00:59:36,090
This expression over here goes
to 2 zeta, because that's one.

846
00:59:36,090 --> 00:59:39,210
2 zeta squared square
root is just 2 zeta.

847
00:59:39,210 --> 00:59:41,440
When omega equals omega
n, this whole expression

848
00:59:41,440 --> 00:59:45,400
is F0 over k divided
by 2 zeta, for example.

849
00:59:45,400 --> 00:59:47,570
And that's called
resonance, and that's

850
00:59:47,570 --> 00:59:50,795
when you're right at where
that peak goes to its maximum.

851
00:59:55,840 --> 00:59:59,290
Let's talk about this
expression for a moment.

852
00:59:59,290 --> 01:00:02,790
If we have our cart,
our mass-spring

853
01:00:02,790 --> 01:00:04,930
dashpot we started here.

854
01:00:04,930 --> 01:00:06,860
If you apply a
force, a static force

855
01:00:06,860 --> 01:00:14,640
F0 and stretch the spring
by an amount F0 over k.

856
01:00:14,640 --> 01:00:21,350
So x-- what we'll call x
static is just F0 over k.

857
01:00:25,390 --> 01:00:46,710
And if I want to plot, I
want to-- this has a name.

858
01:00:46,710 --> 01:00:49,210
This is called, this
ratio here, this gives you

859
01:00:49,210 --> 01:00:52,230
the magnitude of the response.

860
01:00:52,230 --> 01:00:53,690
It goes by a variety of names.

861
01:00:53,690 --> 01:00:56,890
Some people call it
a transfer function.

862
01:00:56,890 --> 01:01:00,190
Some people call it a
frequency response function.

863
01:01:04,230 --> 01:01:05,930
I write it
intentionally this way.

864
01:01:05,930 --> 01:01:12,310
This is I put output over input
because this expression has

865
01:01:12,310 --> 01:01:15,170
units of output over input.

866
01:01:15,170 --> 01:01:18,440
So I just write it like
this, remind myself

867
01:01:18,440 --> 01:01:20,560
what this transfer
function is about.

868
01:01:20,560 --> 01:01:24,850
The input is force, the
output is displacement.

869
01:01:24,850 --> 01:01:27,510
This expression
has units of force,

870
01:01:27,510 --> 01:01:29,024
force per unit displacement.

871
01:01:38,710 --> 01:01:46,900
If I go to here, if I try to
plot this-- let me start over.

872
01:01:46,900 --> 01:01:49,340
If I try to plot
this, it's going

873
01:01:49,340 --> 01:01:52,960
to be depending on the exact
value of the spring constant

874
01:01:52,960 --> 01:01:55,960
and the exact value of
the force every time.

875
01:01:55,960 --> 01:01:59,840
I have to get a unique plot
every time I go to do this.

876
01:01:59,840 --> 01:02:02,750
So textbooks and
engineers, I don't want

877
01:02:02,750 --> 01:02:04,960
to have to remember this part.

878
01:02:04,960 --> 01:02:10,140
This is where all of the content
is in is in this denominator,

879
01:02:10,140 --> 01:02:11,320
and it's dimensionless.

880
01:02:11,320 --> 01:02:17,000
So what I'd really like to
plot is x0 over x static.

881
01:02:19,620 --> 01:02:27,040
And if I do that, that is x0
over the quantity F0 over k.

882
01:02:27,040 --> 01:02:30,090
If I just divide-- this is x
static-- it would bring this

883
01:02:30,090 --> 01:02:35,800
to this side, then this
expression, this is just 1

884
01:02:35,800 --> 01:02:40,320
over that denominator.

885
01:02:40,320 --> 01:02:43,810
And sometimes, I think in
the handout by [? Row ?],

886
01:02:43,810 --> 01:02:46,155
they just call this h of omega.

887
01:02:48,700 --> 01:02:52,810
It's dimensionless,
frequency over frequency,

888
01:02:52,810 --> 01:02:56,690
and that's actually
what's plotted up there.

889
01:02:56,690 --> 01:03:01,950
And this is called-- has
different names also.

890
01:03:01,950 --> 01:03:06,670
Magnification factor,
dynamic amplification factor,

891
01:03:06,670 --> 01:03:10,360
because the ratio
of x to x static

892
01:03:10,360 --> 01:03:20,760
if this is the dynamic effects
magnify the response compared

893
01:03:20,760 --> 01:03:21,780
to the static response.

894
01:03:21,780 --> 01:03:24,370
So it might be this
over this might be 10.

895
01:03:24,370 --> 01:03:30,020
I mean, the dynamic response is
10 times the static response.

896
01:03:30,020 --> 01:03:32,140
OK, how do you--
to sum this up--

897
01:03:32,140 --> 01:03:37,380
and we'll be kind of
getting close to the end.

898
01:03:37,380 --> 01:03:39,770
We want to talk just about
the properties of this.

899
01:03:39,770 --> 01:03:40,940
How do we use this?

900
01:03:52,830 --> 01:03:57,730
So in practical use, you
have an input specified,

901
01:03:57,730 --> 01:04:00,260
some force cosine omega t.

902
01:04:00,260 --> 01:04:02,900
You know you have a single
degree of freedom oscillator

903
01:04:02,900 --> 01:04:09,160
that is governed by
equations like that one,

904
01:04:09,160 --> 01:04:10,940
and you want to
predict the response.

905
01:04:10,940 --> 01:04:18,450
Well, you say x of t is
equal to the magnitude

906
01:04:18,450 --> 01:04:28,437
of the force times the--
and you divide that by-- we

907
01:04:28,437 --> 01:04:29,311
could do it this way.

908
01:04:34,230 --> 01:04:36,820
The magnitude of the
force divided by k,

909
01:04:36,820 --> 01:04:38,180
which is the static response.

910
01:04:40,710 --> 01:04:43,720
To predict x0, we just have
to predict this quantity,

911
01:04:43,720 --> 01:04:46,490
multiply it by F0 over k.

912
01:04:46,490 --> 01:04:48,180
So you know this.

913
01:04:48,180 --> 01:04:50,110
You better know that
about your system,

914
01:04:50,110 --> 01:04:55,530
and you multiply it by
this quantity magnitude

915
01:04:55,530 --> 01:04:57,260
of h of omega.

916
01:04:57,260 --> 01:05:05,400
And the time dependent part
is times cosine omega t

917
01:05:05,400 --> 01:05:07,110
minus the phase angle.

918
01:05:07,110 --> 01:05:10,610
And the phase angle, you
get either off the plot

919
01:05:10,610 --> 01:05:15,214
or from the-- have
I written it down?

920
01:05:20,540 --> 01:05:22,573
I haven't written the
phase angle down yet.

921
01:05:34,180 --> 01:05:35,770
It's kind of a messy
expression too.

922
01:05:35,770 --> 01:05:39,200
That's why we plot it.

923
01:05:39,200 --> 01:05:45,330
2 zeta omega over omega n
over 1 minus omega squared

924
01:05:45,330 --> 01:05:48,210
over omega n squared.

925
01:05:48,210 --> 01:05:52,150
But by knowing just
this plot, what you just

926
01:05:52,150 --> 01:05:54,500
put in every textbook about
vibration in the world,

927
01:05:54,500 --> 01:05:57,770
by knowing this
magnification factor,

928
01:05:57,770 --> 01:06:01,960
calculating the static response,
multiplying the two together,

929
01:06:01,960 --> 01:06:04,820
you have the amplitude
of the response,

930
01:06:04,820 --> 01:06:08,490
and its time dependence is
cosine omega t minus the phase

931
01:06:08,490 --> 01:06:09,880
angle.

932
01:06:09,880 --> 01:06:11,785
And I've got a
little example here.

933
01:06:21,460 --> 01:06:23,110
Actually, rather
than the example,

934
01:06:23,110 --> 01:06:25,550
I've gone to all the
trouble of setting this up.

935
01:06:43,110 --> 01:06:44,460
All right.

936
01:06:44,460 --> 01:06:48,210
This is just a beam.

937
01:06:48,210 --> 01:06:49,860
Where's my other little beam?

938
01:06:56,910 --> 01:06:59,225
And a beam is just a spring.

939
01:07:02,420 --> 01:07:04,230
Put a mass on the and.

940
01:07:04,230 --> 01:07:07,720
This is basically a single
degree of freedom system.

941
01:07:07,720 --> 01:07:09,500
It has a natural frequency.

942
01:07:09,500 --> 01:07:13,130
The beam has a
certain stiffness.

943
01:07:13,130 --> 01:07:16,520
And now, in this case,
we're interested in response

944
01:07:16,520 --> 01:07:18,750
to some harmonic input.

945
01:07:18,750 --> 01:07:23,230
So any of you know
what a squiggle pen is.

946
01:07:23,230 --> 01:07:25,045
This is a kid's toy.

947
01:07:25,045 --> 01:07:26,035
AUDIENCE: Excuse me.

948
01:07:26,035 --> 01:07:29,005
Can you move the camera a tad
to the left so the [INAUDIBLE]?

949
01:07:32,794 --> 01:07:33,460
PROFESSOR: Yeah.

950
01:07:36,450 --> 01:07:38,920
So all throughout
the term, we've

951
01:07:38,920 --> 01:07:41,920
studied rotating masses
quite a bit, right.

952
01:07:41,920 --> 01:07:44,730
This thing has a rotating mass
that you can see in the end.

953
01:07:44,730 --> 01:07:46,420
I mean, when you leave it, you
can come down and take a look.

954
01:07:46,420 --> 01:07:47,544
It has a low rotating mass.

955
01:07:47,544 --> 01:07:50,030
It's actually a pen,
but it's a kid's toy.

956
01:07:50,030 --> 01:07:50,980
Shakes like crazy.

957
01:07:55,860 --> 01:08:00,599
And now we need the lights down.

958
01:08:06,830 --> 01:08:14,550
And it happens that the--
I've got a strobe light here,

959
01:08:14,550 --> 01:08:17,149
and I've kind of
preset the frequency

960
01:08:17,149 --> 01:08:21,870
so it's very close to the
frequency of vibration

961
01:08:21,870 --> 01:08:23,090
of this beam.

962
01:08:23,090 --> 01:08:28,939
So there's a rotating mass in
this pen going round and round,

963
01:08:28,939 --> 01:08:30,810
and it puts a force
into the system

964
01:08:30,810 --> 01:08:36,000
that looks like F0 cosine omega
t in the vertical direction.

965
01:08:36,000 --> 01:08:38,000
Also does it in the
horizontal, but vertical

966
01:08:38,000 --> 01:08:40,899
is our response direction.

967
01:08:40,899 --> 01:08:43,129
So it's putting
in a force, and I

968
01:08:43,129 --> 01:08:48,220
have set the length of this beam
so that the natural frequency

969
01:08:48,220 --> 01:08:50,180
of this beam with
this mass on the end

970
01:08:50,180 --> 01:08:54,090
is exactly very close
to being the frequency

971
01:08:54,090 --> 01:08:54,840
of the excitation.

972
01:08:57,540 --> 01:09:01,430
And the flash rate is slightly
different than the vibration

973
01:09:01,430 --> 01:09:04,542
rate, so you see it
illuminated at many positions

974
01:09:04,542 --> 01:09:05,750
as it goes through the cycle.

975
01:09:05,750 --> 01:09:09,720
So you see it going up and down.

976
01:09:09,720 --> 01:09:13,090
So if I mismatch it quite a
bit, then you see it going.

977
01:09:13,090 --> 01:09:15,200
And actually, if you look
at the very right end,

978
01:09:15,200 --> 01:09:18,446
you can see a white
thing going up and down.

979
01:09:18,446 --> 01:09:20,029
That's the mass where
you can actually

980
01:09:20,029 --> 01:09:24,950
see the mass in the very
end of the-- right there.

981
01:09:24,950 --> 01:09:28,220
You can see something
going around and round.

982
01:09:28,220 --> 01:09:31,529
There, that's the rotating mass.

983
01:09:31,529 --> 01:09:33,580
So the beam is
going up and down,

984
01:09:33,580 --> 01:09:37,710
and I've got this, the vibration
frequency of that mass going

985
01:09:37,710 --> 01:09:40,771
round and round equal to the
natural frequency of the beam

986
01:09:40,771 --> 01:09:41,729
of the mass in the end.

987
01:09:47,920 --> 01:09:56,480
And it's moving quite a bit,
and I'll loosen my clamp,

988
01:09:56,480 --> 01:09:59,300
and I'm going to change
the length of the beam.

989
01:09:59,300 --> 01:10:00,399
I've shortened it.

990
01:10:08,890 --> 01:10:11,540
And now it's still
moving up and down

991
01:10:11,540 --> 01:10:17,160
but not as much because the
frequency of the rotation

992
01:10:17,160 --> 01:10:19,250
of the eccentric
mass is no longer

993
01:10:19,250 --> 01:10:21,260
close to the natural
frequency of the system.

994
01:10:24,210 --> 01:10:27,300
In fact, I've made
the natural frequency

995
01:10:27,300 --> 01:10:34,510
of the system-- you can
bring the lights back up--

996
01:10:34,510 --> 01:10:37,030
I've made the natural
frequency of the system.

997
01:10:37,030 --> 01:10:40,310
By making the beam shorter,
I've made it stiffer.

998
01:10:40,310 --> 01:10:43,235
So the natural
frequency has gone up.

999
01:10:43,235 --> 01:10:46,290
The frequency of the rotation
of the eccentric mass

1000
01:10:46,290 --> 01:10:47,920
has stayed about the same.

1001
01:10:47,920 --> 01:10:49,530
So what's happened
to that frequency

1002
01:10:49,530 --> 01:10:52,970
ratio, omega over omega n?

1003
01:10:52,970 --> 01:10:54,620
So less than one or
greater than one.

1004
01:10:57,230 --> 01:10:59,630
So the omega n has gone up.

1005
01:10:59,630 --> 01:11:00,585
Omega stayed the same.

1006
01:11:03,690 --> 01:11:06,740
The frequency ratio when
you shorten this beam

1007
01:11:06,740 --> 01:11:14,570
is less than one, and the
properties of this transfer

1008
01:11:14,570 --> 01:11:17,400
function, we call it--
this magnification factor

1009
01:11:17,400 --> 01:11:19,050
looks like this.

1010
01:11:19,050 --> 01:11:22,900
When we're exciting
it right at one-- this

1011
01:11:22,900 --> 01:11:27,580
is omega over omega n--
you write it resonance.

1012
01:11:27,580 --> 01:11:31,800
When you excite it at a
frequency ratio less than one,

1013
01:11:31,800 --> 01:11:34,750
you start dropping
off this backside,

1014
01:11:34,750 --> 01:11:38,700
and the response goes down.

1015
01:11:38,700 --> 01:11:43,370
And if you excite it at
frequencies much greater

1016
01:11:43,370 --> 01:11:46,420
than the natural frequency,
you end up way out here.

1017
01:11:46,420 --> 01:11:47,350
I can do that too.

1018
01:11:59,820 --> 01:12:05,390
So how much you think
it will vibrate now?

1019
01:12:05,390 --> 01:12:07,690
A lot?

1020
01:12:07,690 --> 01:12:08,456
A little?

1021
01:12:16,560 --> 01:12:17,289
Hardly-- oops.

1022
01:12:17,289 --> 01:12:19,330
Oh, I've brought it out
so much you can't see it.

1023
01:12:24,780 --> 01:12:32,450
Hardly moving at all, and
that's because in terms

1024
01:12:32,450 --> 01:12:36,640
of this terminology of
magnification factors, transfer

1025
01:12:36,640 --> 01:12:41,720
functions, this is a
plot of x over x static.

1026
01:12:41,720 --> 01:12:43,530
It goes right here.

1027
01:12:43,530 --> 01:12:46,300
When you go to zero
frequency, you are at static,

1028
01:12:46,300 --> 01:12:51,052
so the response at
very, very low frequency

1029
01:12:51,052 --> 01:12:53,010
goes to being the same
as the static frequency.

1030
01:12:53,010 --> 01:12:55,530
So in this plot, it goes to one.

1031
01:12:55,530 --> 01:12:59,650
At resonance, you
put in one here.

1032
01:12:59,650 --> 01:13:00,910
This goes to zero.

1033
01:13:00,910 --> 01:13:04,060
That becomes a one
2 zeta squared.

1034
01:13:04,060 --> 01:13:12,350
This height here
is 1 over 2 zeta.

1035
01:13:12,350 --> 01:13:14,770
So the dynamic
amplification at resonance

1036
01:13:14,770 --> 01:13:16,665
is just 1 over 2 times
the damping ratio.

1037
01:13:16,665 --> 01:13:19,280
You have 1% damping.

1038
01:13:19,280 --> 01:13:21,322
Twice that is o2.

1039
01:13:21,322 --> 01:13:24,720
1 over o2 is 50.

1040
01:13:24,720 --> 01:13:29,100
So if you only had 1% damping,
the dynamic amplification,

1041
01:13:29,100 --> 01:13:31,600
the amount that this
vibrates greater

1042
01:13:31,600 --> 01:13:35,190
than its static response,
is a factor of 50 greater.

1043
01:13:35,190 --> 01:13:39,420
But then as you go higher
in this omega over omega n,

1044
01:13:39,420 --> 01:13:42,460
and you get way out here, and
you get almost no vibration

1045
01:13:42,460 --> 01:13:43,530
at all.

1046
01:13:43,530 --> 01:13:46,296
And that's what's happened
when I've lengthened this.

1047
01:13:50,180 --> 01:13:50,680
OK.

1048
01:13:55,930 --> 01:14:02,090
So there's your introduction
to linear systems.

1049
01:14:02,090 --> 01:14:05,090
In this case, a single
degree of freedom system

1050
01:14:05,090 --> 01:14:10,570
that vibrates an
oscillator, we're

1051
01:14:10,570 --> 01:14:15,230
talking about steady state
response, not the part

1052
01:14:15,230 --> 01:14:18,440
of the solution, the
mathematical solution,

1053
01:14:18,440 --> 01:14:20,850
to the initial conditions.

1054
01:14:20,850 --> 01:14:24,740
So all of this has been
about steady state response

1055
01:14:24,740 --> 01:14:27,240
of our simple oscillator
to what we call

1056
01:14:27,240 --> 01:14:32,880
a harmonic input,
F0 cosine omega t.

1057
01:14:32,880 --> 01:14:34,720
So what's important
that you need

1058
01:14:34,720 --> 01:14:36,990
to remember and be able to use?

1059
01:14:39,560 --> 01:14:43,520
This concept here, this
idea of a transfer function.

1060
01:14:43,520 --> 01:14:44,720
That's really important.

1061
01:14:44,720 --> 01:14:46,570
You might want to
remember it though

1062
01:14:46,570 --> 01:14:51,960
as this dimensionless
quantity x over x static.

1063
01:14:51,960 --> 01:14:55,010
Just remember the shape
of this transfer function.

1064
01:14:55,010 --> 01:14:59,170
Magnitude of the amplification,
1 over 2 zeta at resonance.

1065
01:14:59,170 --> 01:15:03,380
And it goes to one
at low frequency.

1066
01:15:03,380 --> 01:15:06,770
The high frequency,
it drops away off.

1067
01:15:06,770 --> 01:15:10,540
Next time, we're
going to pick up

1068
01:15:10,540 --> 01:15:14,120
the topic of what we
call vibration isolation.

1069
01:15:14,120 --> 01:15:16,540
The practical thing
to know as engineers

1070
01:15:16,540 --> 01:15:20,784
is when you have a
significant vibration

1071
01:15:20,784 --> 01:15:22,200
problem, like in
a lab, and you're

1072
01:15:22,200 --> 01:15:24,750
looking through your microscope,
and the floor vibration is

1073
01:15:24,750 --> 01:15:28,510
causing trouble with
your microscope,

1074
01:15:28,510 --> 01:15:32,820
and you can't move the
subway, what can you

1075
01:15:32,820 --> 01:15:35,989
do to solve that problem?

1076
01:15:35,989 --> 01:15:37,530
Well, you might be
able-- what if you

1077
01:15:37,530 --> 01:15:40,620
put some kind of a flexible
pad under the microscope?

1078
01:15:40,620 --> 01:15:43,120
You might be able to reduce the
vibration of the microscope.

1079
01:15:45,860 --> 01:15:47,110
Things like that.

1080
01:15:47,110 --> 01:15:49,750
That is the topic of
vibration isolation,

1081
01:15:49,750 --> 01:15:52,690
so we're going to get
into that next time.

1082
01:15:52,690 --> 01:15:54,024
Thanks.