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PROFESSOR: This week is my
second pair of lectures.

9
00:00:24,420 --> 00:00:28,540
Last week the two lectures were
about first order differential

10
00:00:28,540 --> 00:00:32,340
equations, and this
week second order.

11
00:00:32,340 --> 00:00:38,360
Those are the two big topics
in differential equations.

12
00:00:38,360 --> 00:00:46,020
Let me start with most
basic second order equation.

13
00:00:46,020 --> 00:00:51,330
We see the second derivative
and the function itself,

14
00:00:51,330 --> 00:00:57,740
and we don't see yet the
first derivative term.

15
00:00:57,740 --> 00:01:06,030
This is the nice case, when I
just have y double prime and y.

16
00:01:06,030 --> 00:01:10,870
In general, I-- I'm taking
constant coefficients today.

17
00:01:10,870 --> 00:01:16,150
Because if the coefficients
depend on time,

18
00:01:16,150 --> 00:01:18,930
the problem gets
much, much harder now.

19
00:01:18,930 --> 00:01:21,560
So let's stay with
constant coefficients,

20
00:01:21,560 --> 00:01:28,640
meaning we have a mass, for
example, we have a spring.

21
00:01:28,640 --> 00:01:32,700
The stiffness of the
spring is k, the mass is m,

22
00:01:32,700 --> 00:01:42,310
and the y, the
unknown displacement,

23
00:01:42,310 --> 00:01:44,910
tells us the
movement of the mass.

24
00:01:44,910 --> 00:01:46,170
The classical problem.

25
00:01:46,170 --> 00:01:47,630
You will have seen it before.

26
00:01:50,240 --> 00:01:52,550
Because you have an
exam this afternoon,

27
00:01:52,550 --> 00:01:55,580
I wanted to start with
things that-- they

28
00:01:55,580 --> 00:01:57,590
are about second
order equations,

29
00:01:57,590 --> 00:02:03,550
but they're still close to
the exam idea, particularly

30
00:02:03,550 --> 00:02:07,670
the idea of exponentials.

31
00:02:07,670 --> 00:02:11,870
With constant coefficients,
that's the fundamental message.

32
00:02:11,870 --> 00:02:15,400
Exponentials in,
exponentials out.

33
00:02:15,400 --> 00:02:20,360
But it's not quite
so clear when we

34
00:02:20,360 --> 00:02:24,110
had first order,
y prime equal ay,

35
00:02:24,110 --> 00:02:28,630
we knew that the exponent was a.

36
00:02:28,630 --> 00:02:31,746
The solution was e to the at.

37
00:02:31,746 --> 00:02:36,580
Now we've got second
derivatives coming in,

38
00:02:36,580 --> 00:02:41,400
and it won't be so much
e to the at type thing.

39
00:02:41,400 --> 00:02:47,850
Either the at was growth for a
positive, decay for a negative.

40
00:02:47,850 --> 00:02:51,580
Now we're going to
see oscillation.

41
00:02:51,580 --> 00:02:55,250
It's still exponentials,
but oscillation.

42
00:02:55,250 --> 00:02:59,450
Things going up and down,
things going around.

43
00:02:59,450 --> 00:03:01,430
Harmonic motion, you call it.

44
00:03:01,430 --> 00:03:04,230
Sines and cosines.

45
00:03:04,230 --> 00:03:10,130
And sines and cosines connect
to complex exponentials.

46
00:03:10,130 --> 00:03:14,500
So that instead of e to the
at-- so now oscillations--

47
00:03:14,500 --> 00:03:21,630
they're going to be coming
from e to the i omega t.

48
00:03:21,630 --> 00:03:24,535
In other words, instead of an a,
we're going to have an i omega.

49
00:03:27,100 --> 00:03:30,940
Or, if we like to
stay real, we can

50
00:03:30,940 --> 00:03:36,320
stay with cos--
cosine-- and sine.

51
00:03:39,800 --> 00:03:43,360
And actually, I've written
two real guys there,

52
00:03:43,360 --> 00:03:45,770
so I better have
two complex ones.

53
00:03:45,770 --> 00:03:49,420
And it will turn out
to be plus or minus.

54
00:03:49,420 --> 00:03:51,650
There are two frequencies there.

55
00:03:51,650 --> 00:03:57,540
Plus i omega, minus i omega, and
they turn into cosine and sine.

56
00:03:57,540 --> 00:04:04,400
So in this case,
with no damping term,

57
00:04:04,400 --> 00:04:08,360
we can stay entirely real
without creating any problems.

58
00:04:08,360 --> 00:04:12,340
We can work with
cosines and sines.

59
00:04:12,340 --> 00:04:16,040
The first question
is, what's omega.

60
00:04:16,040 --> 00:04:22,010
What is the frequency
of oscillation.

61
00:04:22,010 --> 00:04:24,580
And of course, another
similar picture

62
00:04:24,580 --> 00:04:27,400
would be a pendulum,
a linear pendulum,

63
00:04:27,400 --> 00:04:32,030
swinging side to
side, keeping time,

64
00:04:32,030 --> 00:04:34,095
because that frequency
will stay constant.

65
00:04:38,420 --> 00:04:41,560
Always I'll start with zero
on the right hand side.

66
00:04:44,430 --> 00:04:46,660
Just look at these equations.

67
00:04:46,660 --> 00:04:47,820
Constants there.

68
00:04:47,820 --> 00:04:50,690
I'm looking for solutions.

69
00:04:50,690 --> 00:04:54,920
And I'm looking
for null solutions,

70
00:04:54,920 --> 00:05:02,570
looking for the natural motion
of this spring, the natural up

71
00:05:02,570 --> 00:05:05,780
and down motion of this spring.

72
00:05:05,780 --> 00:05:07,630
Classical problem.

73
00:05:07,630 --> 00:05:11,830
Won't be brand new, but
it's the right starting

74
00:05:11,830 --> 00:05:18,500
point for the full
second order equation.

75
00:05:18,500 --> 00:05:22,400
It'll get a little
complicated on Wednesday,

76
00:05:22,400 --> 00:05:25,050
when damping gets in there.

77
00:05:25,050 --> 00:05:29,700
The formula got a little
messy, because you've

78
00:05:29,700 --> 00:05:34,470
got a mass-- you've got
an m-- and a k, still,

79
00:05:34,470 --> 00:05:37,340
but you also will have
a damping constant.

80
00:05:40,240 --> 00:05:43,510
Then complex numbers
really come in.

81
00:05:43,510 --> 00:05:47,570
Here they're optional.

82
00:05:47,570 --> 00:05:51,420
So this is my equation to solve.

83
00:05:55,280 --> 00:06:00,230
Because we don't have
a first derivative,

84
00:06:00,230 --> 00:06:02,620
a cosine will solve that.

85
00:06:02,620 --> 00:06:06,190
So let me look for-- I
could look for exponentials.

86
00:06:06,190 --> 00:06:08,940
Maybe I should do
that first, look

87
00:06:08,940 --> 00:06:10,780
for an exponential solution.

88
00:06:10,780 --> 00:06:12,560
Yeah, that's a good idea.

89
00:06:12,560 --> 00:06:18,810
And let me not
jump ahead to know

90
00:06:18,810 --> 00:06:22,620
that the exponent has
that i omega form.

91
00:06:22,620 --> 00:06:24,430
Let me discover it.

92
00:06:24,430 --> 00:06:30,850
So I look for no
solutions-- because I

93
00:06:30,850 --> 00:06:34,900
have that zero there--
no solutions of the form

94
00:06:34,900 --> 00:06:38,995
e to the st, some exponent.

95
00:06:41,510 --> 00:06:43,500
Plug it in.

96
00:06:43,500 --> 00:06:48,740
That's the message with
constant coefficients.

97
00:06:48,740 --> 00:06:51,690
Look for exponentials,
substitute them in,

98
00:06:51,690 --> 00:06:54,010
discover what s will be.

99
00:06:54,010 --> 00:06:55,210
So let's just do that.

100
00:06:58,285 --> 00:07:00,700
This is the most basic step.

101
00:07:00,700 --> 00:07:03,430
For null solutions
will be exponentials,

102
00:07:03,430 --> 00:07:05,660
I substitute into the equation.

103
00:07:05,660 --> 00:07:09,730
I get ms squared
from two derivatives.

104
00:07:09,730 --> 00:07:11,570
It will bring s down twice.

105
00:07:14,550 --> 00:07:20,860
This is just ke to the st, and
I'm looking for null solutions.

106
00:07:20,860 --> 00:07:21,640
Zero.

107
00:07:21,640 --> 00:07:22,630
No forcing.

108
00:07:22,630 --> 00:07:26,540
So this is undamped, unforced.

109
00:07:26,540 --> 00:07:29,960
Undamped, unforced.

110
00:07:29,960 --> 00:07:32,780
Natural motion.

111
00:07:32,780 --> 00:07:35,710
What do I do now?

112
00:07:35,710 --> 00:07:40,190
Plugged in an exponential,
got this equation.

113
00:07:40,190 --> 00:07:44,290
And the beauty is that
the exponentials cancel.

114
00:07:44,290 --> 00:07:48,290
An exponential is never zero,
so I can safely divide by it.

115
00:07:48,290 --> 00:07:55,120
So I cancel those, and I get
ms squared plus k equals zero.

116
00:07:55,120 --> 00:07:57,940
The key equation-- and
it's so simple-- it's

117
00:07:57,940 --> 00:08:02,050
just we're doing algebra now.

118
00:08:02,050 --> 00:08:06,730
The calculus, the derivative
we took when we plugged it in,

119
00:08:06,730 --> 00:08:09,200
but now it's an
algebra question.

120
00:08:09,200 --> 00:08:13,210
And of course, solving
that system is easy.

121
00:08:13,210 --> 00:08:17,110
There's no s term,
no damping term.

122
00:08:17,110 --> 00:08:26,160
So the frequency, s, is-- put k
on the other side, divide by n.

123
00:08:26,160 --> 00:08:33,320
s is-- s squared, let's say--
is k over m-- is minus k over m.

124
00:08:33,320 --> 00:08:34,730
Critical point.

125
00:08:39,850 --> 00:08:42,900
That tells me, with
that minus sign there,

126
00:08:42,900 --> 00:08:48,840
that s is an imaginary number.

127
00:08:48,840 --> 00:08:54,990
A complex number has a real
part and an imaginary part.

128
00:08:54,990 --> 00:08:58,130
In this case, all imaginary.

129
00:08:58,130 --> 00:08:59,260
No real part at all.

130
00:09:03,310 --> 00:09:07,380
It's natural to think-- s
is the square root of that,

131
00:09:07,380 --> 00:09:11,870
so I'm going to write--
everybody writes-- s,

132
00:09:11,870 --> 00:09:17,080
the frequency s, is i omega.

133
00:09:17,080 --> 00:09:22,220
So if I plug that in, I have
omega squared equal k over m.

134
00:09:22,220 --> 00:09:26,030
i squared and the minus
1 deal with each other.

135
00:09:26,030 --> 00:09:30,970
So the frequency omega-- here
is the great fact-- square

136
00:09:30,970 --> 00:09:31,980
root of k over m.

137
00:09:34,520 --> 00:09:36,190
That's-- yes?

138
00:09:36,190 --> 00:09:38,146
AUDIENCE: What
difference does having

139
00:09:38,146 --> 00:09:42,058
imaginary parts to answer
affect the oscillation?

140
00:09:42,058 --> 00:09:43,560
PROFESSOR: To-- OK.

141
00:09:46,270 --> 00:09:49,430
Oscillation, just
pure oscillation--

142
00:09:49,430 --> 00:09:53,140
which is what we would
see here with no damping--

143
00:09:53,140 --> 00:10:04,370
is the frequency, e to the--
the solution-- the displacement,

144
00:10:04,370 --> 00:10:08,350
I could write-- the
displacement up and down,

145
00:10:08,350 --> 00:10:19,310
y, will involve e to the i
omega t, and e to the minus i

146
00:10:19,310 --> 00:10:19,940
omega t.

147
00:10:22,530 --> 00:10:24,280
We've got second order equation.

148
00:10:24,280 --> 00:10:29,300
Let me just go back
to that key point.

149
00:10:29,300 --> 00:10:30,980
When we have second
order equations,

150
00:10:30,980 --> 00:10:33,070
we look for two-- we
expect and we want

151
00:10:33,070 --> 00:10:36,160
and we need two-- solutions.

152
00:10:36,160 --> 00:10:38,730
There will be two.

153
00:10:38,730 --> 00:10:42,250
I didn't put it
here, and I should.

154
00:10:42,250 --> 00:10:48,140
s, the frequency, is
plus or minus i omega,

155
00:10:48,140 --> 00:10:53,500
because in both cases when we
square, it comes out right.

156
00:10:53,500 --> 00:10:57,440
So we get two frequencies,
and here they are.

157
00:11:01,480 --> 00:11:05,370
So let's see how to
answer your question.

158
00:11:05,370 --> 00:11:11,110
The presence of this
i is only telling me

159
00:11:11,110 --> 00:11:18,110
that, essentially, I've
got sines and cosines.

160
00:11:18,110 --> 00:11:23,720
That's really what-- when it's a
pure imaginary number-- I would

161
00:11:23,720 --> 00:11:26,090
call that a pure
imaginary number,

162
00:11:26,090 --> 00:11:34,450
there's no real part at
all-- then equally cos omega

163
00:11:34,450 --> 00:11:37,250
t and sine omega t.

164
00:11:40,490 --> 00:11:43,910
I can now, if I want, go real.

165
00:11:43,910 --> 00:11:50,860
I can say, OK, these were
the general null solutions.

166
00:11:50,860 --> 00:11:52,750
Let me put this down, then.

167
00:11:52,750 --> 00:11:55,930
The null solution-- I'm
looking only right now

168
00:11:55,930 --> 00:12:00,690
at null solutions-- is
some combination of e

169
00:12:00,690 --> 00:12:07,950
to the i omega t and e
to the minus i omega t.

170
00:12:07,950 --> 00:12:12,260
That's what we got from
plugging in e to the st,

171
00:12:12,260 --> 00:12:15,550
discovering that s was
an imaginary number,

172
00:12:15,550 --> 00:12:17,150
and we got these guys.

173
00:12:17,150 --> 00:12:26,150
But equally-- equally--
yn is a combination

174
00:12:26,150 --> 00:12:33,010
of cos omega t and sine omega t.

175
00:12:36,300 --> 00:12:38,490
And maybe you'll
like those better.

176
00:12:38,490 --> 00:12:42,200
I think everybody practically
likes those better.

177
00:12:42,200 --> 00:12:47,050
Do you see that these guys
are the same as these guys?

178
00:12:47,050 --> 00:12:51,210
The c's are a little
different because, well, we

179
00:12:51,210 --> 00:12:55,650
know that we can switch
from one to the other.

180
00:12:55,650 --> 00:13:02,210
We remember that basic fact
that e to the i omega t

181
00:13:02,210 --> 00:13:09,630
is cos omega t plus
i times sine omega t.

182
00:13:09,630 --> 00:13:13,370
You're used to maybe seeing that
omega t as theta, e to the i

183
00:13:13,370 --> 00:13:17,470
theta is cos theta
plus i sine theta.

184
00:13:17,470 --> 00:13:21,900
And e to the minus i
omega t, of course,

185
00:13:21,900 --> 00:13:28,157
is cos omega t minus
i sine omega t.

186
00:13:31,079 --> 00:13:34,140
I hope you won't think
I'm filling the blackboard

187
00:13:34,140 --> 00:13:39,900
with formulas, because I'm
really just writing down--

188
00:13:39,900 --> 00:13:41,570
well anyway, they're
beautiful formulas.

189
00:13:44,120 --> 00:13:50,550
So if I have these guys, then
I have these and vice versa.

190
00:13:50,550 --> 00:13:53,200
If I have cos--
how would you write

191
00:13:53,200 --> 00:13:58,910
cos omega t using
the exponentials?

192
00:13:58,910 --> 00:14:01,750
I want to just see
totally clearly

193
00:14:01,750 --> 00:14:07,600
that I can go back and forth
between complex imaginary

194
00:14:07,600 --> 00:14:10,860
exponentials and
cosines and sines.

195
00:14:10,860 --> 00:14:17,210
So how would I, I want to
go in the opposite direction

196
00:14:17,210 --> 00:14:24,820
and write the cosine and the
sine as combinations of these,

197
00:14:24,820 --> 00:14:28,040
just to show if I've
got combinations of one,

198
00:14:28,040 --> 00:14:29,870
I've got combinations
of the other.

199
00:14:29,870 --> 00:14:33,980
Combinations of these are the
same as combinations of those.

200
00:14:33,980 --> 00:14:37,271
So what is cos omega t
in terms of these guys?

201
00:14:37,271 --> 00:14:39,270
AUDIENCE: [INAUDIBLE]
some of them divided by 2?

202
00:14:39,270 --> 00:14:40,700
PROFESSOR: Exactly.

203
00:14:40,700 --> 00:14:44,630
If I add those two,
this part cancels.

204
00:14:44,630 --> 00:14:48,130
I've got two of these, so I
have to divide by 2, as you say.

205
00:14:48,130 --> 00:14:52,395
It's a half of the first
plus a half of the second.

206
00:14:55,890 --> 00:14:57,770
And how about sine omega t?

207
00:14:57,770 --> 00:15:00,510
Sine omega t is always
slightly more annoying,

208
00:15:00,510 --> 00:15:04,600
because it's the one-- it's
the imaginary part that

209
00:15:04,600 --> 00:15:06,430
brings in an i.

210
00:15:06,430 --> 00:15:08,010
What would be the same formula?

211
00:15:08,010 --> 00:15:10,880
How could I produce sine
omega t out of that?

212
00:15:14,361 --> 00:15:14,860
Yes?

213
00:15:14,860 --> 00:15:16,772
AUDIENCE: The difference
divided by 2i.

214
00:15:16,772 --> 00:15:17,670
PROFESSOR: Yes.

215
00:15:17,670 --> 00:15:21,160
If I take the difference,
that'll cancel the cosines.

216
00:15:21,160 --> 00:15:23,340
So I'm going to take
e to the i omega

217
00:15:23,340 --> 00:15:29,290
t minus e-- minus e to
the minus i omega t.

218
00:15:29,290 --> 00:15:30,710
Take the difference.

219
00:15:30,710 --> 00:15:36,240
But then I've got 2i
multiplying this sine.

220
00:15:36,240 --> 00:15:39,880
Up here I had a 2, but now I've
got-- when I take the subtract,

221
00:15:39,880 --> 00:15:42,790
these i's are in there,
so I divide by 2i.

222
00:15:48,630 --> 00:15:52,880
So this just tells me
that I can go either way.

223
00:15:57,530 --> 00:16:00,930
Next time, we'll see what
happens when there is damping

224
00:16:00,930 --> 00:16:09,680
and there are complex numbers
instead of pure i omegas.

225
00:16:12,480 --> 00:16:14,180
We're golden here.

226
00:16:14,180 --> 00:16:21,745
We've found the great
quantity with the right units.

227
00:16:21,745 --> 00:16:26,210
The right units of
omega are 1 over time.

228
00:16:26,210 --> 00:16:31,610
Actually the units are
radians per second,

229
00:16:31,610 --> 00:16:35,390
would be the typical
appropriate unit.

230
00:16:35,390 --> 00:16:36,470
Radians per second.

231
00:16:45,760 --> 00:16:48,720
I'll use the word
frequency for that,

232
00:16:48,720 --> 00:16:52,950
but there's another
definition of frequency,

233
00:16:52,950 --> 00:16:54,130
cycles per second.

234
00:16:57,760 --> 00:17:02,260
I just want to think about
steady motion around a circle.

235
00:17:02,260 --> 00:17:04,859
So this tells me how
many radians per second.

236
00:17:04,859 --> 00:17:12,420
And if this is 2pi-- if
omega happened to be 2pi--

237
00:17:12,420 --> 00:17:15,099
then I would go once
around the circle.

238
00:17:15,099 --> 00:17:19,940
If omega was 2pi, then
when t reached one,

239
00:17:19,940 --> 00:17:22,950
I would be around the circle.

240
00:17:22,950 --> 00:17:25,920
Let me draw a
circle in a minute.

241
00:17:25,920 --> 00:17:32,160
So there's a 2pi here hiding
behind the word radians.

242
00:17:32,160 --> 00:17:38,450
And in many cases,
you'll want also

243
00:17:38,450 --> 00:17:41,680
a definition in
cycles per second.

244
00:17:41,680 --> 00:17:50,700
So f is omega
divided by the 2pi,

245
00:17:50,700 --> 00:17:53,000
and that's in cycles per second.

246
00:17:55,940 --> 00:17:58,270
Full revolutions per second.

247
00:17:58,270 --> 00:18:00,950
And that's hertz.

248
00:18:00,950 --> 00:18:09,310
I think I misspoke last
time in confusing these two,

249
00:18:09,310 --> 00:18:10,910
so let's get them straight here.

250
00:18:14,450 --> 00:18:20,040
There's no complicated math in
here, it's just a factor 2pi,

251
00:18:20,040 --> 00:18:23,200
but of course that
factor is important.

252
00:18:23,200 --> 00:18:31,520
So a typical frequency
in everyday life

253
00:18:31,520 --> 00:18:38,110
would be like f, 60 cycles
per second, 120pi radians

254
00:18:38,110 --> 00:18:39,400
per second.

255
00:18:39,400 --> 00:18:41,080
So I'm going around in a circle.

256
00:18:43,700 --> 00:18:50,750
Now I'm ready to have
initial conditions.

257
00:18:50,750 --> 00:18:55,860
This connects, again,
to the afternoon exam.

258
00:18:55,860 --> 00:19:01,310
We found the general solution
with some constants, like here.

259
00:19:03,910 --> 00:19:06,120
Let's keep that real form.

260
00:19:06,120 --> 00:19:09,240
And now those constants
get determined

261
00:19:09,240 --> 00:19:11,660
by the initial conditions.

262
00:19:11,660 --> 00:19:18,240
Conditions plural, because
we have an initial position,

263
00:19:18,240 --> 00:19:24,310
like I stretch it-- maybe
I stretch it and let go.

264
00:19:24,310 --> 00:19:26,990
Maybe I stretch the
spring and then I let go.

265
00:19:26,990 --> 00:19:29,040
What happens?

266
00:19:29,040 --> 00:19:33,010
By stretching it, I'm giving
it an initial displacement.

267
00:19:33,010 --> 00:19:35,300
And I'm giving it
zero initial velocity,

268
00:19:35,300 --> 00:19:38,050
because I stretched
it and just let go.

269
00:19:38,050 --> 00:19:41,250
Another possibility
would be to strike it.

270
00:19:41,250 --> 00:19:43,860
If I hit that
mass, that would be

271
00:19:43,860 --> 00:19:46,050
a different initial condition.

272
00:19:46,050 --> 00:19:47,570
What would be the
initial condition

273
00:19:47,570 --> 00:19:50,850
if it's sitting there in
equilibrium quietly minding

274
00:19:50,850 --> 00:19:54,210
its own business and I hit it?

275
00:19:54,210 --> 00:19:59,090
Then I've given it
an initial velocity,

276
00:19:59,090 --> 00:20:01,630
with initial displacement zero.

277
00:20:01,630 --> 00:20:04,360
So those would be the two
extreme possibilities.

278
00:20:04,360 --> 00:20:10,380
Pull it down, let
go, or strike it

279
00:20:10,380 --> 00:20:14,470
when it's sitting
in equilibrium.

280
00:20:14,470 --> 00:20:16,195
Anyway, we've got two
initial conditions.

281
00:20:19,520 --> 00:20:23,970
You see why-- y double
prime is showing up

282
00:20:23,970 --> 00:20:28,750
because essentially
we've got Newton's Law.

283
00:20:28,750 --> 00:20:31,610
This is Newton's Law.

284
00:20:31,610 --> 00:20:37,690
Mass times acceleration is
equal to minus ky-- that's

285
00:20:37,690 --> 00:20:42,580
the, with the minus sign,
and the all-important minus

286
00:20:42,580 --> 00:20:45,780
sign, that's the acceleration.

287
00:20:45,780 --> 00:20:48,240
That's a force, sorry.

288
00:20:48,240 --> 00:20:52,860
Mass, acceleration, this thing
with a minus sign is the force,

289
00:20:52,860 --> 00:20:54,510
and the force is pulling back.

290
00:20:54,510 --> 00:20:59,425
If y is stretching,
the force is restoring.

291
00:21:04,890 --> 00:21:08,780
Let me just go ahead
with what you know.

292
00:21:08,780 --> 00:21:10,100
The initial conditions.

293
00:21:10,100 --> 00:21:17,380
And I want to solve my double
prime plus ky equals 0.

294
00:21:17,380 --> 00:21:28,340
So I'm still talking about
the unforced with given

295
00:21:28,340 --> 00:21:32,254
y of 0 and y prime of 0.

296
00:21:36,840 --> 00:21:38,800
Just think for a moment.

297
00:21:38,800 --> 00:21:41,730
Could you do that?

298
00:21:41,730 --> 00:21:44,710
This is the most basic
second order equation.

299
00:21:44,710 --> 00:21:48,030
We know what the
solutions look like.

300
00:21:48,030 --> 00:21:52,120
Let's do this one in a
box, cosines and sines.

301
00:21:52,120 --> 00:21:54,400
We know what omega is.

302
00:21:54,400 --> 00:21:57,740
Omega had to be square
root of k over m.

303
00:21:57,740 --> 00:22:01,830
Then the equation was solved.

304
00:22:01,830 --> 00:22:04,800
All I've got left
is to get c1 and c2.

305
00:22:07,440 --> 00:22:10,900
All I have left is to
match-- choose c1 and c2

306
00:22:10,900 --> 00:22:13,650
to match the two
initial conditions.

307
00:22:13,650 --> 00:22:14,950
So let me just do that.

308
00:22:17,700 --> 00:22:19,260
What are c2 and c2?

309
00:22:22,880 --> 00:22:27,470
At time zero, I have to match
an initial displacement.

310
00:22:27,470 --> 00:22:33,440
So at time zero, this is a
1, cosine of zero is a one,

311
00:22:33,440 --> 00:22:34,760
and that's a zero.

312
00:22:34,760 --> 00:22:41,450
So at t equals 0, I have y of 0.

313
00:22:44,180 --> 00:22:48,160
The displacement
matches c1 times cosine

314
00:22:48,160 --> 00:22:53,540
of omega t, which is
a 1, plus c2 times 0.

315
00:22:53,540 --> 00:22:55,901
I'll put it in there
plus c2 times 0.

316
00:22:58,490 --> 00:23:01,770
C1 cos 0 plus c2 sine 0.

317
00:23:04,750 --> 00:23:08,320
I've learned c1.

318
00:23:08,320 --> 00:23:10,620
And also-- what do I do next?

319
00:23:14,980 --> 00:23:18,130
I want to get c2.

320
00:23:18,130 --> 00:23:22,340
And where is c2 coming from?

321
00:23:22,340 --> 00:23:25,130
Now I would like to know
what's the coefficient

322
00:23:25,130 --> 00:23:29,100
of the-- the initial
conditions are supposed

323
00:23:29,100 --> 00:23:31,450
to determine that coefficient.

324
00:23:31,450 --> 00:23:34,630
It'll be that initial
condition that determines it.

325
00:23:34,630 --> 00:23:37,570
y prime of 0.

326
00:23:37,570 --> 00:23:42,140
The initial velocity should
match the derivative.

327
00:23:42,140 --> 00:23:43,525
OK, so what's the derivative?

328
00:23:48,200 --> 00:23:50,440
y prime.

329
00:23:50,440 --> 00:23:54,460
So the derivative of the
cosine will be a sine.

330
00:23:54,460 --> 00:23:57,320
And that will disappear
at t equals 0.

331
00:23:57,320 --> 00:23:59,230
The derivative of
this sine will be

332
00:23:59,230 --> 00:24:03,170
a cosine with a factor omega.

333
00:24:03,170 --> 00:24:11,650
So I'll have y prime of 0
will be the c2 omega cos of 0.

334
00:24:17,220 --> 00:24:17,975
Which is what?

335
00:24:26,100 --> 00:24:27,010
That tells me c2.

336
00:24:31,550 --> 00:24:39,590
You could do all this
without my pointing the way.

337
00:24:39,590 --> 00:24:42,160
I'm solving this equation.

338
00:24:42,160 --> 00:24:47,750
I have the solution in general
form with two constants.

339
00:24:47,750 --> 00:24:49,870
Now I'm determining
those constants,

340
00:24:49,870 --> 00:24:54,100
and cosine and sine just
determine them perfectly

341
00:24:54,100 --> 00:25:02,870
because cosine is 1 and
sine is 0 at the start.

342
00:25:02,870 --> 00:25:07,960
So we've got the answer.

343
00:25:07,960 --> 00:25:17,920
The solution is y of t is
c1 is y of 0 cos omega t,

344
00:25:17,920 --> 00:25:21,320
and c2 is-- now
you'll notice, c2

345
00:25:21,320 --> 00:25:25,720
is y prime at 0
divided by omega.

346
00:25:25,720 --> 00:25:30,850
y prime of 0 divided
by omega sine omega t.

347
00:25:30,850 --> 00:25:32,570
There we go.

348
00:25:32,570 --> 00:25:34,030
Finished.

349
00:25:34,030 --> 00:25:35,420
Finished.

350
00:25:35,420 --> 00:25:37,850
Unforced problem solved.

351
00:25:42,300 --> 00:25:44,520
Everybody in this room
could get to that point.

352
00:25:49,370 --> 00:25:52,280
Let me make some
comments about that.

353
00:25:52,280 --> 00:25:56,440
It's a combination
of cosine and sine.

354
00:25:56,440 --> 00:26:00,780
They're both running at
the same frequency, omega.

355
00:26:00,780 --> 00:26:04,780
I'm going to give a special
name to that frequency, omega,

356
00:26:04,780 --> 00:26:09,140
this famous formula,
all-important.

357
00:26:09,140 --> 00:26:10,950
Lots of physics in that formula.

358
00:26:14,070 --> 00:26:18,080
I call that the
natural frequency,

359
00:26:18,080 --> 00:26:27,170
because the next step will be
to drive the system by a driving

360
00:26:27,170 --> 00:26:31,540
frequency, which would
be different from omega.

361
00:26:31,540 --> 00:26:34,150
So we need to--
we've got 2 omegas.

362
00:26:34,150 --> 00:26:37,630
Actually when I first
wrote the book I thought,

363
00:26:37,630 --> 00:26:40,580
we've got to keep
these two separate.

364
00:26:40,580 --> 00:26:42,270
Everybody has to
keep them separate.

365
00:26:42,270 --> 00:26:46,570
My first attempt was to use
little omega and big omega

366
00:26:46,570 --> 00:26:49,310
for the two.

367
00:26:49,310 --> 00:26:53,890
I concluded after
looking at it for a while

368
00:26:53,890 --> 00:26:57,500
that it was better to
be more conventional.

369
00:26:57,500 --> 00:27:00,320
People had figured out
a good way to do it.

370
00:27:00,320 --> 00:27:04,980
And the good way is to call
this the natural frequency

371
00:27:04,980 --> 00:27:07,450
and put a subscript, m.

372
00:27:07,450 --> 00:27:11,670
So all the omegas that
you see on this board

373
00:27:11,670 --> 00:27:14,640
should be omega n.

374
00:27:14,640 --> 00:27:20,520
I can change them all, but
let me just change it here.

375
00:27:20,520 --> 00:27:22,490
I'll change omega n.

376
00:27:25,990 --> 00:27:27,240
So that's omega n.

377
00:27:27,240 --> 00:27:31,240
We've only got that
one omega right now,

378
00:27:31,240 --> 00:27:34,700
because we don't have
a driving term yet.

379
00:27:34,700 --> 00:27:37,850
So natural frequency
has the advantage,

380
00:27:37,850 --> 00:27:45,130
which kind of made me smile,
that the n stands for natural,

381
00:27:45,130 --> 00:27:47,910
and everybody calls it
the natural frequency.

382
00:27:47,910 --> 00:27:51,780
And n also stands
for null, and we're

383
00:27:51,780 --> 00:27:55,130
talking here about
the null solution,

384
00:27:55,130 --> 00:27:56,950
because there's no forcing.

385
00:27:56,950 --> 00:28:00,510
So I could have
subscripts on all the y's.

386
00:28:00,510 --> 00:28:02,930
Eventually I'll need
subscripts on the y

387
00:28:02,930 --> 00:28:07,640
to separate what we've done.

388
00:28:07,640 --> 00:28:11,785
This is really yn of
t, the null solution.

389
00:28:15,220 --> 00:28:17,670
Good?

390
00:28:17,670 --> 00:28:20,180
Now we could take one more step.

391
00:28:23,110 --> 00:28:26,730
This is a combination
of cosine and sine.

392
00:28:26,730 --> 00:28:29,590
And we learned
last time that that

393
00:28:29,590 --> 00:28:35,160
could be put in a polar form,
but I don't plan to do this.

394
00:28:35,160 --> 00:28:37,590
Let me just say I could do it.

395
00:28:37,590 --> 00:28:43,180
This would be some
amplitude, some gain--

396
00:28:43,180 --> 00:28:46,925
maybe g for gain--
no, a for amplitude

397
00:28:46,925 --> 00:28:53,620
is good-- times-- what is this
second optional form, which I'm

398
00:28:53,620 --> 00:28:57,220
just going to write here,
say that we could do it,

399
00:28:57,220 --> 00:29:01,200
remember a little about it,
but not make a big deal-- what

400
00:29:01,200 --> 00:29:02,400
is it I'm after here?

401
00:29:05,170 --> 00:29:09,620
I'm looking to write this
combination of cosine and sine,

402
00:29:09,620 --> 00:29:14,450
which is two oscillations, a
cosine curve and a sine curve,

403
00:29:14,450 --> 00:29:17,080
but with the same frequency.

404
00:29:17,080 --> 00:29:22,180
Then I can combine them
into a single cosine,

405
00:29:22,180 --> 00:29:27,200
a single cosine of omega nt.

406
00:29:27,200 --> 00:29:31,590
But now what else have
I got in this form?

407
00:29:31,590 --> 00:29:35,900
There's a phase
shift, minus phi.

408
00:29:35,900 --> 00:29:38,670
Thanks.

409
00:29:38,670 --> 00:29:43,400
So there's an a phi, two
constants, or there's y of 0,

410
00:29:43,400 --> 00:29:47,060
y prime of 0, two constants.

411
00:29:47,060 --> 00:29:50,940
Let me not write again the
formula for a or for phi,

412
00:29:50,940 --> 00:29:52,640
I don't plan to do
anything with it.

413
00:29:52,640 --> 00:29:55,210
It just could be done.

414
00:29:55,210 --> 00:29:59,280
In other words, what
we've done so far

415
00:29:59,280 --> 00:30:08,690
is just to see that the
single spring oscillates

416
00:30:08,690 --> 00:30:10,500
with the frequency omega n.

417
00:30:10,500 --> 00:30:12,370
That's really what we've done.

418
00:30:12,370 --> 00:30:16,023
A single spring oscillates
with a frequency omega n.

419
00:30:21,220 --> 00:30:26,500
Saying that makes me
think, let me look ahead

420
00:30:26,500 --> 00:30:31,790
to the linear algebra
part of the course.

421
00:30:31,790 --> 00:30:35,730
So where is linear
algebra going to come in?

422
00:30:35,730 --> 00:30:38,880
It's going to come in
for a system of springs.

423
00:30:38,880 --> 00:30:40,990
When I have another spring.

424
00:30:40,990 --> 00:30:44,580
Can I draw another
spring and another mass

425
00:30:44,580 --> 00:30:45,750
and another spring?

426
00:30:45,750 --> 00:30:50,550
Say six springs, six masses.

427
00:30:50,550 --> 00:30:55,630
Then-- and they could be
different k's, different m's,

428
00:30:55,630 --> 00:30:56,260
or not.

429
00:30:59,730 --> 00:31:03,040
Then we've got six
displacements--

430
00:31:03,040 --> 00:31:07,490
six differential equations--
coupled together,

431
00:31:07,490 --> 00:31:10,610
because the whole system
is coupled together.

432
00:31:10,610 --> 00:31:14,160
So what happens at that point?

433
00:31:14,160 --> 00:31:18,030
That's the point where linear
algebra, where matrices

434
00:31:18,030 --> 00:31:20,210
are coming in.

435
00:31:20,210 --> 00:31:23,030
You want to see what's
the point of matrices.

436
00:31:23,030 --> 00:31:27,160
It's not a separate
course by any means.

437
00:31:27,160 --> 00:31:33,350
It's a most necessary part,
because a single spring happens

438
00:31:33,350 --> 00:31:38,380
in reality but also
systems today are coupled.

439
00:31:38,380 --> 00:31:40,960
Big, actually, there
are many, many things.

440
00:31:40,960 --> 00:31:44,580
You have an electric
circuit with thousands

441
00:31:44,580 --> 00:31:47,610
or tens of thousands
of elements.

442
00:31:47,610 --> 00:31:51,640
You have a coupled
system with many gears,

443
00:31:51,640 --> 00:31:54,190
many oscillations going on.

444
00:31:56,780 --> 00:32:01,190
So we need matrices
at that point.

445
00:32:01,190 --> 00:32:07,270
Can I even just add one more
word about the language?

446
00:32:07,270 --> 00:32:11,730
When we had-- here
we have a frequency

447
00:32:11,730 --> 00:32:14,150
of motion for one spring.

448
00:32:14,150 --> 00:32:18,910
What are we going to have for
two springs or six springs?

449
00:32:18,910 --> 00:32:24,005
The motion will be a combination
of six different frequencies.

450
00:32:26,950 --> 00:32:30,500
And so you'll see that it's
a much more interesting,

451
00:32:30,500 --> 00:32:34,800
much more not so simple motion.

452
00:32:34,800 --> 00:32:38,960
A combination of six
pure frequencies.

453
00:32:38,960 --> 00:32:41,680
And those frequencies
are determined

454
00:32:41,680 --> 00:32:45,810
from the six eigenvalues
of the matrix.

455
00:32:45,810 --> 00:32:49,490
I'm just using that
word looking ahead.

456
00:32:49,490 --> 00:32:55,260
We will have a 6x6 matrix to
describe the coupled system.

457
00:32:55,260 --> 00:32:57,820
That matrix will
have six eigenvalues.

458
00:32:57,820 --> 00:33:00,910
It will tell us six
natural frequencies,

459
00:33:00,910 --> 00:33:05,460
and our solution will be
a combination of all six

460
00:33:05,460 --> 00:33:07,370
oscillations.

461
00:33:07,370 --> 00:33:10,210
Here, it's 1.

462
00:33:10,210 --> 00:33:11,080
Here it's 1.

463
00:33:11,080 --> 00:33:12,290
That spring is not there.

464
00:33:20,130 --> 00:33:23,070
So the problem
we've solved now is

465
00:33:23,070 --> 00:33:29,070
the fundamental, basic problem,
and I have to-- next step

466
00:33:29,070 --> 00:33:33,000
is forcing.

467
00:33:33,000 --> 00:33:38,640
I now want to add a force
that drives the motion.

468
00:33:43,710 --> 00:33:46,660
In general, it could be
any function of time.

469
00:33:50,840 --> 00:33:51,980
Calling it f of t.

470
00:33:51,980 --> 00:33:54,500
So that's what I'm
going to put in now.

471
00:33:54,500 --> 00:33:59,350
But in reality, very,
very, very often f of t

472
00:33:59,350 --> 00:34:04,630
is also a simple
harmonic motion.

473
00:34:04,630 --> 00:34:08,409
It's also a cosine.

474
00:34:08,409 --> 00:34:11,810
But at a different frequency,
at a driving frequency.

475
00:34:11,810 --> 00:34:13,840
So I'm going to--
the next equation

476
00:34:13,840 --> 00:34:24,300
to solve is to put in cosine--
let's stay real for now--

477
00:34:24,300 --> 00:34:30,210
at another, driving frequency.

478
00:34:30,210 --> 00:34:31,494
At a driving frequency.

479
00:34:34,300 --> 00:34:36,360
And of course, it could
have an amplitude.

480
00:34:36,360 --> 00:34:41,270
But let me take that amplitude
as 1 to keep things simple.

481
00:34:41,270 --> 00:34:46,920
So now I'm talking
about forced motion.

482
00:34:46,920 --> 00:34:49,830
Can we solve it?

483
00:34:49,830 --> 00:34:52,040
How can we solve this equation?

484
00:34:52,040 --> 00:35:00,020
Let me take out the 0 or-- take
out the 0-- equals cosine omega

485
00:35:00,020 --> 00:35:00,520
t.

486
00:35:04,352 --> 00:35:05,310
With a different omega.

487
00:35:08,170 --> 00:35:11,970
If the two omegas were the same,
if the driving frequency is

488
00:35:11,970 --> 00:35:18,270
the same as the
natural frequency,

489
00:35:18,270 --> 00:35:22,500
the formulas have to
be slightly adjusted.

490
00:35:22,500 --> 00:35:28,250
There's still an answer,
but it's a case of resonance

491
00:35:28,250 --> 00:35:30,100
and you have to look separately.

492
00:35:30,100 --> 00:35:32,290
But let's say, no.

493
00:35:32,290 --> 00:35:35,230
Let's say omega d is
different from omega n.

494
00:35:35,230 --> 00:35:36,680
How are you going to solve this?

495
00:35:40,480 --> 00:35:42,520
I have to think myself.

496
00:35:42,520 --> 00:35:47,030
How do I solve that.

497
00:35:47,030 --> 00:35:48,604
Let's start a fresh board.

498
00:35:52,400 --> 00:36:00,070
my double prime plus
ky equals cosine

499
00:36:00,070 --> 00:36:05,210
of omega dt-- or often,
I won't put the d.

500
00:36:05,210 --> 00:36:07,910
I don't have to
put the d anymore.

501
00:36:07,910 --> 00:36:11,320
Omega will now represent
the driving frequency,

502
00:36:11,320 --> 00:36:19,290
because I've got omega
n, the natural frequency,

503
00:36:19,290 --> 00:36:22,926
as the square root of k over m.

504
00:36:29,080 --> 00:36:31,550
What am I looking for now?

505
00:36:31,550 --> 00:36:33,490
I found the null solution.

506
00:36:33,490 --> 00:36:36,830
I'm looking for a
particular solution.

507
00:36:36,830 --> 00:36:39,920
I'm trying to keep the
whole thing systematic.

508
00:36:39,920 --> 00:36:42,630
Null solutions are
now dealt with.

509
00:36:42,630 --> 00:36:44,945
Took a little more
time than just ce

510
00:36:44,945 --> 00:36:48,540
to the at for first
order equations,

511
00:36:48,540 --> 00:36:54,100
because we've now got a
two-dimensional collection

512
00:36:54,100 --> 00:36:56,580
of null solutions,
but we've got them.

513
00:36:56,580 --> 00:37:01,090
Now I'm taking a forcing term.

514
00:37:01,090 --> 00:37:06,480
So I'm looking for a
particular solution.

515
00:37:06,480 --> 00:37:09,230
I'm looking for any
solution to this equation.

516
00:37:09,230 --> 00:37:12,930
I'm looking for
a particular guy.

517
00:37:12,930 --> 00:37:14,855
What do you suggest?

518
00:37:18,230 --> 00:37:22,830
Again, it's a neat
problem because

519
00:37:22,830 --> 00:37:29,130
of that particular forcing
term, a cosine, an oscillation.

520
00:37:29,130 --> 00:37:50,730
So I'm going to look for yp is
some gain times [INAUDIBLE].

521
00:37:50,730 --> 00:37:53,470
This is the next and,
fortunately, a highly, highly

522
00:37:53,470 --> 00:37:59,620
important case, in which
the particular solution has

523
00:37:59,620 --> 00:38:03,280
the same form as
the forcing term.

524
00:38:03,280 --> 00:38:05,430
It's just a multiple
of the forcing term.

525
00:38:05,430 --> 00:38:07,400
That's best possible.

526
00:38:07,400 --> 00:38:15,150
That's best possible, is to have
the forcing term reveal to me--

527
00:38:15,150 --> 00:38:21,200
the forcing term immediately
reveals a particular solution.

528
00:38:23,890 --> 00:38:26,460
Once I know what I'm
looking for, what do I do?

529
00:38:26,460 --> 00:38:28,220
Substitute it in.

530
00:38:28,220 --> 00:38:33,090
So I substitute that
particular solution in here.

531
00:38:33,090 --> 00:38:36,580
And notice everything
is going to be

532
00:38:36,580 --> 00:38:45,030
a cosine, mgy double prime.

533
00:38:45,030 --> 00:38:52,200
So what do I get when I
plug this in for that guy?

534
00:38:52,200 --> 00:38:57,340
I want to-- you can do
it quickly, but let's

535
00:38:57,340 --> 00:39:00,290
stay together and do it
together, because we can

536
00:39:00,290 --> 00:39:01,540
with this case.

537
00:39:01,540 --> 00:39:04,740
What happens when I plug that in
and take its second derivative?

538
00:39:07,410 --> 00:39:09,030
I get the g.

539
00:39:09,030 --> 00:39:10,810
And then what's the
second derivative?

540
00:39:10,810 --> 00:39:12,960
AUDIENCE: [INAUDIBLE].

541
00:39:12,960 --> 00:39:16,960
PROFESSOR: We have a negative,
because two derivatives

542
00:39:16,960 --> 00:39:22,880
of the cosine bring out a minus
omega d, will come out twice.

543
00:39:22,880 --> 00:39:25,480
And I'll keep writing
omega d for a moment,

544
00:39:25,480 --> 00:39:27,710
but then I'll give up on the d.

545
00:39:27,710 --> 00:39:32,100
Cosine of omega dt.

546
00:39:32,100 --> 00:39:39,930
And then k times this,
g cosine of omega dt,

547
00:39:39,930 --> 00:39:43,640
equals the forcing term,
cosine of omega dt.

548
00:39:46,420 --> 00:39:49,270
It worked.

549
00:39:49,270 --> 00:39:55,290
This is one of that small
family of nice functions

550
00:39:55,290 --> 00:40:04,230
where the solution has the
same form as the function.

551
00:40:04,230 --> 00:40:11,720
Actually that list of what
you could call best possible

552
00:40:11,720 --> 00:40:17,490
forcing functions, where the
form of the forcing function

553
00:40:17,490 --> 00:40:23,820
tells you the form
of the solution.

554
00:40:23,820 --> 00:40:25,085
That's a small family.

555
00:40:27,930 --> 00:40:30,640
But it's fortunately
a very important one.

556
00:40:30,640 --> 00:40:34,070
Cosines, sines are
included, and we'll

557
00:40:34,070 --> 00:40:37,730
see all the other guys
that are included.

558
00:40:37,730 --> 00:40:41,510
Most forcing functions
we couldn't just

559
00:40:41,510 --> 00:40:47,380
assume that the solution
had the same form.

560
00:40:47,380 --> 00:40:48,910
It's only these nice ones.

561
00:40:48,910 --> 00:40:51,250
But cosines are nice.

562
00:40:51,250 --> 00:40:52,300
So what do I do now?

563
00:40:55,160 --> 00:40:57,150
Everything is
multiplying cosines,

564
00:40:57,150 --> 00:41:01,970
so I just look at--
I have minus m omega

565
00:41:01,970 --> 00:41:08,410
squared g-- g is going to factor
out-- minus m omega squared

566
00:41:08,410 --> 00:41:13,300
and a k times g.

567
00:41:13,300 --> 00:41:18,020
Let me remove that
off for the moment.

568
00:41:18,020 --> 00:41:24,990
I'm canceling cosine omega,
so my right hand side is 1.

569
00:41:28,910 --> 00:41:29,960
That's it.

570
00:41:38,110 --> 00:41:42,190
We looked for a solution
with that simple format,

571
00:41:42,190 --> 00:41:43,820
and we found it.

572
00:41:43,820 --> 00:41:45,940
Now we know g, the gain.

573
00:41:45,940 --> 00:41:53,800
So the solution is-- this
is g is 1 over k minus m

574
00:41:53,800 --> 00:41:59,270
omega squared times
cosine of omega t.

575
00:41:59,270 --> 00:42:01,690
And omega is omega d.

576
00:42:01,690 --> 00:42:03,680
Omega is omega d now.

577
00:42:10,026 --> 00:42:11,150
Does that look good to you?

578
00:42:13,840 --> 00:42:26,520
This is the periodic solution
going at the driving.

579
00:42:26,520 --> 00:42:31,130
This is what the-- this g is
the gain, the driving force.

580
00:42:31,130 --> 00:42:35,710
The driving force is 1
times cosine omega d,

581
00:42:35,710 --> 00:42:38,430
then that 1 gets
multiplied by this number.

582
00:42:38,430 --> 00:42:40,990
This is, you could say,
the amplifying factor.

583
00:42:44,390 --> 00:42:49,080
I guess frequency response
would be the right word.

584
00:42:49,080 --> 00:42:52,280
Can I bring in that
word, response, again?

585
00:42:52,280 --> 00:42:54,450
Response is a word
for a solution.

586
00:42:54,450 --> 00:42:58,920
It's what comes out.

587
00:42:58,920 --> 00:43:04,700
When the input is
this, a pure frequency,

588
00:43:04,700 --> 00:43:08,850
the output, the response,
is a pure frequency--

589
00:43:08,850 --> 00:43:13,780
same frequency, of course--
multiplied by that.

590
00:43:13,780 --> 00:43:21,350
That is the frequency
response factor.

591
00:43:24,800 --> 00:43:32,670
Notice we could write that
a cool way, by remembering

592
00:43:32,670 --> 00:43:36,580
that omega squared--
that's wrong as it stands.

593
00:43:36,580 --> 00:43:39,150
What have I forgotten
in writing k

594
00:43:39,150 --> 00:43:44,580
minus m omega squared
in that denominator?

595
00:43:44,580 --> 00:43:49,630
I forgot a subscript,
which is n.

596
00:43:49,630 --> 00:43:51,111
Which is n.

597
00:43:51,111 --> 00:43:53,190
This is n.

598
00:43:53,190 --> 00:43:56,750
This is-- is that right?

599
00:43:56,750 --> 00:43:57,250
No.

600
00:43:57,250 --> 00:43:58,690
Is it?

601
00:43:58,690 --> 00:43:59,760
Or is it d?

602
00:43:59,760 --> 00:44:01,350
Maybe I didn't make a mistake.

603
00:44:01,350 --> 00:44:02,700
Is it d?

604
00:44:02,700 --> 00:44:06,370
You're seeing a kind
of critical moment.

605
00:44:06,370 --> 00:44:07,278
Which is it?

606
00:44:07,278 --> 00:44:09,718
AUDIENCE: [INAUDIBLE].

607
00:44:09,718 --> 00:44:11,011
PROFESSOR: It's d, isn't it?

608
00:44:11,011 --> 00:44:11,510
Yeah.

609
00:44:11,510 --> 00:44:12,220
It's d.

610
00:44:12,220 --> 00:44:12,720
Sorry.

611
00:44:12,720 --> 00:44:13,400
It's d.

612
00:44:15,910 --> 00:44:21,220
But when I see this and remember
what omega n squared is--

613
00:44:21,220 --> 00:44:25,370
omega n squared is
k over m-- I can

614
00:44:25,370 --> 00:44:35,390
see that I can get an omega--
I can use this in here

615
00:44:35,390 --> 00:44:38,420
to make it even
more interesting.

616
00:44:38,420 --> 00:44:45,590
So it'll be equals-- let me
get this box ready-- cosine

617
00:44:45,590 --> 00:44:53,170
of omega dt divided by-- now
I just want to rewrite that.

618
00:44:57,440 --> 00:44:58,924
I want to take out an m.

619
00:45:01,590 --> 00:45:06,060
I'm going to write this
as m times k over m.

620
00:45:06,060 --> 00:45:08,780
m times k over m.

621
00:45:08,780 --> 00:45:10,580
Safe to do that.

622
00:45:10,580 --> 00:45:13,060
Now I have a factor, m,
that I can bring out.

623
00:45:13,060 --> 00:45:15,100
And what is m multiplying?

624
00:45:15,100 --> 00:45:17,710
That's the neat thing.

625
00:45:17,710 --> 00:45:20,680
What is m multiplying?

626
00:45:20,680 --> 00:45:25,745
k over m is-- omega n squared.

627
00:45:28,600 --> 00:45:31,330
And this is minus
m omega d squared.

628
00:45:31,330 --> 00:45:34,570
Minus omega d squared.

629
00:45:34,570 --> 00:45:36,460
That's pretty terrific.

630
00:45:40,120 --> 00:45:47,250
The gain is this multiplier,
1 over m, times that.

631
00:45:47,250 --> 00:45:50,970
And we see that the gain
is bigger and bigger when

632
00:45:50,970 --> 00:45:53,760
the frequency is near
the natural frequency.

633
00:45:53,760 --> 00:45:58,000
And of course everybody has seen
the pictures of that bridge--

634
00:45:58,000 --> 00:46:00,100
wherever the heck
was that bridge?

635
00:46:00,100 --> 00:46:02,672
Somewhere in the
Northwest, I think.

636
00:46:02,672 --> 00:46:04,740
You know the bridge
I'm talking about?

637
00:46:04,740 --> 00:46:06,484
AUDIENCE: [INAUDIBLE]
Tacoma, Washington.

638
00:46:06,484 --> 00:46:08,400
PROFESSOR: Yeah, I think
Tacoma, that's right.

639
00:46:08,400 --> 00:46:09,670
The Tacoma Narrows Bridge.

640
00:46:09,670 --> 00:46:10,170
Right.

641
00:46:10,170 --> 00:46:11,690
Tacoma, Washington.

642
00:46:11,690 --> 00:46:16,950
Where the natural--
when you build a bridge,

643
00:46:16,950 --> 00:46:19,220
you've built in a
natural frequency.

644
00:46:19,220 --> 00:46:23,080
And then when traffic comes,
it's doing a driving frequency.

645
00:46:23,080 --> 00:46:25,930
And if you haven't got
those two well-separated,

646
00:46:25,930 --> 00:46:29,600
you're in trouble,
as this shows.

647
00:46:29,600 --> 00:46:38,960
Or similarly, when an
architect designs a skyscraper,

648
00:46:38,960 --> 00:46:43,250
there's going to be a
frequency of oscillation,

649
00:46:43,250 --> 00:46:48,170
a natural frequency, at
which that skyscraper swings.

650
00:46:48,170 --> 00:46:51,230
And then there's wind.

651
00:46:51,230 --> 00:46:57,870
Actually I talked yesterday
to the-- by chance,

652
00:46:57,870 --> 00:47:04,440
the math department is not a
very party-going department,

653
00:47:04,440 --> 00:47:07,510
but once a year we let it out.

654
00:47:07,510 --> 00:47:11,830
And so we had our party
at Endicott house out

655
00:47:11,830 --> 00:47:16,490
in the suburbs, and
all the usual people--

656
00:47:16,490 --> 00:47:20,660
that's all the professors
I know-- came, of course.

657
00:47:20,660 --> 00:47:23,030
But also, there was
a really cool person.

658
00:47:23,030 --> 00:47:27,590
He's the key architect
for Building 2.

659
00:47:27,590 --> 00:47:32,560
You've noticed that
Building 2 is under wraps

660
00:47:32,560 --> 00:47:34,940
and we're moved out.

661
00:47:34,940 --> 00:47:39,170
And we move back
in January 2016.

662
00:47:39,170 --> 00:47:41,730
So we've been out a
year and a quarter

663
00:47:41,730 --> 00:47:45,150
and we have another year
and a quarter to go.

664
00:47:45,150 --> 00:47:47,060
It's going to be cool.

665
00:47:47,060 --> 00:47:49,720
And you may say,
well, who cares.

666
00:47:49,720 --> 00:47:53,870
But the key point is
Building 1 is next,

667
00:47:53,870 --> 00:47:57,360
and Building 1 is going to
have the same cool addition

668
00:47:57,360 --> 00:47:59,150
of a fourth floor.

669
00:47:59,150 --> 00:48:05,730
We're putting in a fourth floor,
which all the-- Buildings 3, 4,

670
00:48:05,730 --> 00:48:11,865
5, 6 go up to four, but
Buildings 2 and 1 stopped

671
00:48:11,865 --> 00:48:12,910
at the third floor.

672
00:48:12,910 --> 00:48:15,780
But there's a lot of space
up there under the roof.

673
00:48:15,780 --> 00:48:20,170
And they've
discovered they could

674
00:48:20,170 --> 00:48:23,090
put a fourth floor up there.

675
00:48:23,090 --> 00:48:26,330
Here was one interesting
thing, though.

676
00:48:26,330 --> 00:48:30,670
These buildings that we're
sitting in are sinking.

677
00:48:30,670 --> 00:48:38,410
You know that MIT was built
on marshy land, just the way

678
00:48:38,410 --> 00:48:41,030
the Back Bay-- which
is like the greatest

679
00:48:41,030 --> 00:48:45,510
idea in the history of Boston,
the Back Bay and the dam

680
00:48:45,510 --> 00:48:50,320
that makes the Charles
River beautiful--

681
00:48:50,320 --> 00:48:53,820
was built by bringing
in trainloads of earth

682
00:48:53,820 --> 00:48:54,840
from Needham.

683
00:48:54,840 --> 00:49:00,020
So whole mountains and hills in
Needham have come into Boston

684
00:49:00,020 --> 00:49:01,190
and come here.

685
00:49:01,190 --> 00:49:02,420
So anyway, we're sinking.

686
00:49:06,160 --> 00:49:10,605
You may say something like
3/16 of an inch a year

687
00:49:10,605 --> 00:49:14,490
is not something to
worry about, but now it's

688
00:49:14,490 --> 00:49:20,620
been more than 100 years that
these buildings have been here.

689
00:49:20,620 --> 00:49:24,650
Anyway, not good to sink faster.

690
00:49:24,650 --> 00:49:28,170
So the weight had
to be controlled.

691
00:49:28,170 --> 00:49:31,890
So by putting in a
fourth floor, that

692
00:49:31,890 --> 00:49:35,950
put in a lot a new weight,
and faster sinking,

693
00:49:35,950 --> 00:49:39,290
probably by some formula here.

694
00:49:39,290 --> 00:49:42,340
Probably there.

695
00:49:42,340 --> 00:49:44,720
So the weight had to
get subtracted out.

696
00:49:44,720 --> 00:49:48,410
It turns out that
the ceiling, the roof

697
00:49:48,410 --> 00:49:52,260
to Building 1-- Building 2
and no doubt to Building 1--

698
00:49:52,260 --> 00:49:56,420
was more than a foot
thick of concrete.

699
00:49:56,420 --> 00:49:58,150
Really heavy.

700
00:49:58,150 --> 00:50:02,896
And some more asbestos
probably, which we don't want

701
00:50:02,896 --> 00:50:03,520
to think about.

702
00:50:06,040 --> 00:50:07,410
That's much reduced.

703
00:50:07,410 --> 00:50:10,700
A whole lot of weight
came out of the roof.

704
00:50:10,700 --> 00:50:14,870
I think they probably did
the calculation right,

705
00:50:14,870 --> 00:50:17,210
so we won't get
rain coming through,

706
00:50:17,210 --> 00:50:23,960
but it won't weigh as much and
the fourth floor is acceptable.

707
00:50:23,960 --> 00:50:28,540
All this was a big decision
by MIT to pay for that,

708
00:50:28,540 --> 00:50:32,270
or to raise money and pay
for the new fourth floor.

709
00:50:32,270 --> 00:50:34,030
But it's going to be fantastic.

710
00:50:34,030 --> 00:50:36,580
And it'll be fantastic
in Building 1 also.

711
00:50:39,820 --> 00:50:44,240
So all that is discussion
of that formula.

712
00:50:47,210 --> 00:50:53,650
That's the frequency response,
this factor to frequency,

713
00:50:53,650 --> 00:50:57,070
omega d, or omega,
is this factor.

714
00:51:01,510 --> 00:51:08,330
I guess I should say
something about resonance.

715
00:51:08,330 --> 00:51:13,230
What happens when that
formula breaks down?

716
00:51:13,230 --> 00:51:20,320
When the driving force
equals the natural frequency,

717
00:51:20,320 --> 00:51:24,035
then we're dividing by 0,
and something is different.

718
00:51:24,035 --> 00:51:27,990
The formula isn't right anymore.

719
00:51:27,990 --> 00:51:32,760
What enters in the formula-- let
me just tell you what enters,

720
00:51:32,760 --> 00:51:35,600
and then we'll see it
in a simple example.

721
00:51:38,840 --> 00:51:43,340
When I have this repeated
thing, two things are equal,

722
00:51:43,340 --> 00:51:45,230
what tends to
happen is a factor,

723
00:51:45,230 --> 00:51:47,840
an extra factor, t, appears.

724
00:51:47,840 --> 00:51:54,100
So an extra factor, t, will
appear in the case omega n

725
00:51:54,100 --> 00:51:55,520
equal omega d.

726
00:51:58,730 --> 00:52:03,580
The solution, y,
will be some factor,

727
00:52:03,580 --> 00:52:06,540
I'll still call it g-- no,
I don't want to call it g,

728
00:52:06,540 --> 00:52:08,560
let me call it a.

729
00:52:08,560 --> 00:52:14,900
There'll be a factor, t,
times cosine of omega t.

730
00:52:14,900 --> 00:52:17,470
So in this case, there's
really only one frequency.

731
00:52:17,470 --> 00:52:18,590
We're driving it.

732
00:52:18,590 --> 00:52:22,990
So the oscillation grows.

733
00:52:22,990 --> 00:52:28,400
As you know, when you
push a child on a swing,

734
00:52:28,400 --> 00:52:31,090
the whole point of
pushing that child

735
00:52:31,090 --> 00:52:35,240
is to push at the
natural frequency.

736
00:52:35,240 --> 00:52:38,860
You wait for this swing
to swing back naturally

737
00:52:38,860 --> 00:52:43,110
and you drive it again
with that-- at that--

738
00:52:43,110 --> 00:52:44,600
maintain that frequency.

739
00:52:44,600 --> 00:52:47,390
And of course you
see the amplitude--

740
00:52:47,390 --> 00:52:50,110
the child swing
higher and higher.

741
00:52:50,110 --> 00:52:59,380
Presumably you stop pushing
before disaster for that child.

742
00:52:59,380 --> 00:53:03,190
But that's a case of resonance.

743
00:53:03,190 --> 00:53:06,600
And it's what happened in
the Tacoma Narrows Bridge,

744
00:53:06,600 --> 00:53:11,210
and there was nothing
to-- nobody stopped,

745
00:53:11,210 --> 00:53:12,960
traffic just kept coming.

746
00:53:12,960 --> 00:53:14,650
The movie is amazing,
because there's

747
00:53:14,650 --> 00:53:19,730
one car that shows up after
it's already swinging wildly,

748
00:53:19,730 --> 00:53:23,620
some crazy person
still driving across.

749
00:53:23,620 --> 00:53:28,310
And you might think, OK,
that's ancient history.

750
00:53:28,310 --> 00:53:30,030
But you know the
bridge in London,

751
00:53:30,030 --> 00:53:34,610
the pedestrian bridge, the
Millennium Bridge-- it's just

752
00:53:34,610 --> 00:53:39,440
a walking bridge
across the Thames--

753
00:53:39,440 --> 00:53:44,280
a big feature of modern London,
and it had the same problem.

754
00:53:44,280 --> 00:53:45,030
It was swaying.

755
00:53:45,030 --> 00:53:47,520
People could not walk across.

756
00:53:47,520 --> 00:53:48,940
They couldn't keep
their balance.

757
00:53:48,940 --> 00:53:51,010
So they had change it.

758
00:53:51,010 --> 00:53:56,845
So it's not trivial
to anticipate.

759
00:54:02,030 --> 00:54:07,060
So now we've solved it-- we've
solved the the null equation

760
00:54:07,060 --> 00:54:14,620
with no force, and
we've solved the driving

761
00:54:14,620 --> 00:54:16,580
force equal to a cosine.

762
00:54:16,580 --> 00:54:19,780
And of course, we
could do a sine.

763
00:54:19,780 --> 00:54:23,880
What other driving
force should we do?

764
00:54:23,880 --> 00:54:27,690
I think we should
do a delta function.

765
00:54:27,690 --> 00:54:35,340
I think we have to understand
the fundamental solution is

766
00:54:35,340 --> 00:54:37,890
the case when, if
we can solve it--

767
00:54:37,890 --> 00:54:40,910
there's always this general
rule, if we can solve

768
00:54:40,910 --> 00:54:43,840
with a delta function,
that will give us

769
00:54:43,840 --> 00:54:49,300
a formula for every
driving force,

770
00:54:49,300 --> 00:54:53,880
because every function is some
combination of delta functions.

771
00:54:53,880 --> 00:54:57,130
So if we could do
it with a delta--

772
00:54:57,130 --> 00:54:59,830
really the great
right hand sides

773
00:54:59,830 --> 00:55:04,660
are-- well, cosines
and sines I'll

774
00:55:04,660 --> 00:55:06,910
include as great
right hand sides.

775
00:55:06,910 --> 00:55:09,820
Those are the
exponentials in disguise.

776
00:55:09,820 --> 00:55:11,590
So the great right
hand sides are

777
00:55:11,590 --> 00:55:17,660
really exponentials at
different frequencies and delta

778
00:55:17,660 --> 00:55:19,350
functions.

779
00:55:19,350 --> 00:55:20,830
Delta of impulses.

780
00:55:20,830 --> 00:55:23,890
So now I want to find
the impulse response.

781
00:55:23,890 --> 00:55:29,006
That's the next--
that's really a job.

782
00:55:31,900 --> 00:55:35,040
At this point, in these
last 20 minutes when

783
00:55:35,040 --> 00:55:46,310
I solve my double prime plus ky
equal a delta function-- well,

784
00:55:46,310 --> 00:55:51,760
what I was going to say was
I'm now taking you to something

785
00:55:51,760 --> 00:55:56,240
that you won't see on
the exam this afternoon.

786
00:55:56,240 --> 00:55:59,680
But maybe you will.

787
00:55:59,680 --> 00:56:02,180
Delta function, right hand side.

788
00:56:02,180 --> 00:56:03,810
I haven't seen it yet.

789
00:56:03,810 --> 00:56:05,070
Or I haven't looked recently.

790
00:56:05,070 --> 00:56:09,710
You won't see second
derivatives, I guess.

791
00:56:09,710 --> 00:56:10,585
So what is it?

792
00:56:15,760 --> 00:56:20,880
So now this is of the
form with an f of t,

793
00:56:20,880 --> 00:56:23,060
a very special f of t.

794
00:56:23,060 --> 00:56:26,960
And that very special f of
t makes that extremely easy

795
00:56:26,960 --> 00:56:27,460
to solve.

796
00:56:31,670 --> 00:56:34,550
That's really my
point here, is it

797
00:56:34,550 --> 00:56:36,770
it's going to be a
cinch to solve that,

798
00:56:36,770 --> 00:56:38,770
and we practically
have done it already

799
00:56:38,770 --> 00:56:43,900
to solve that with
a delta function.

800
00:56:43,900 --> 00:56:51,570
And the reason is
sort of physical.

801
00:56:51,570 --> 00:56:55,630
We have here our spring.

802
00:56:55,630 --> 00:56:59,540
And what am I doing
with that force?

803
00:56:59,540 --> 00:57:02,755
I'm hitting the mass.

804
00:57:02,755 --> 00:57:03,970
I'm striking the mass.

805
00:57:09,170 --> 00:57:12,840
Let me say, and I'll write
it on the board, the point I

806
00:57:12,840 --> 00:57:15,920
want to make about this.

807
00:57:15,920 --> 00:57:20,710
That point is that this
equation with a delta function

808
00:57:20,710 --> 00:57:27,930
force starting from 0-- say,
y of 0 equal y prime of 0

809
00:57:27,930 --> 00:57:33,560
equals 0, let's give
it starting from rest--

810
00:57:33,560 --> 00:57:37,090
it starts from
rest by hitting it.

811
00:57:37,090 --> 00:57:46,050
And that hit, that impulse,
is in no time at all.

812
00:57:46,050 --> 00:57:47,580
It's not stretched out.

813
00:57:47,580 --> 00:57:49,650
It's hit over one second.

814
00:57:49,650 --> 00:57:51,890
So this has the same solution.

815
00:57:51,890 --> 00:57:53,860
This is the beauty.

816
00:57:53,860 --> 00:57:57,930
This is why we can
solve it so easily.

817
00:57:57,930 --> 00:58:08,110
Same solution as-- let me write
it and see what you think--

818
00:58:08,110 --> 00:58:13,420
as my double prime
plus ky equal 0.

819
00:58:13,420 --> 00:58:15,370
We know how to solve those.

820
00:58:15,370 --> 00:58:19,860
With-- it's still, when
I hit it-- when I hit it,

821
00:58:19,860 --> 00:58:23,250
what happens in
that split second?

822
00:58:23,250 --> 00:58:26,690
In that split second, it
doesn't have time to move.

823
00:58:26,690 --> 00:58:27,600
It doesn't move.

824
00:58:27,600 --> 00:58:31,420
It still has y of 0 equals 0.

825
00:58:31,420 --> 00:58:36,640
But in that split second,
we've given it a velocity.

826
00:58:36,640 --> 00:58:38,710
We've given it a velocity.

827
00:58:38,710 --> 00:58:43,120
And that velocity
will be y prime.

828
00:58:43,120 --> 00:58:48,610
The initial velocity is 1--
because here I had a 1--

829
00:58:48,610 --> 00:58:50,300
over an m.

830
00:58:50,300 --> 00:58:52,380
We have to have the units right.

831
00:58:59,740 --> 00:59:03,210
So here's a point, and
we will stay with it.

832
00:59:03,210 --> 00:59:05,860
We'll come back to
this point next time.

833
00:59:05,860 --> 00:59:09,000
Maybe the first thing
for you to take in

834
00:59:09,000 --> 00:59:14,280
is the fact that it's
such a nice thing.

835
00:59:14,280 --> 00:59:18,330
We have this equation with
this mysterious delta function,

836
00:59:18,330 --> 00:59:20,210
and I'm saying that
the solution is

837
00:59:20,210 --> 00:59:25,820
the same as this
equation with no force,

838
00:59:25,820 --> 00:59:27,880
but starting from a mass.

839
00:59:31,510 --> 00:59:36,670
I'm tempted to take an
example to make this point.

840
00:59:36,670 --> 00:59:42,010
Let me take an example where the
whole thing is a lot simpler.

841
00:59:42,010 --> 00:59:44,325
y double prime equal delta of t.

842
00:59:47,670 --> 00:59:52,230
I've taken the spring away, so
the k is gone, the mass is 1.

843
00:59:52,230 --> 00:59:55,410
What's the solution to y
double prime equal delta of t?

844
00:59:55,410 --> 01:00:02,660
If we concentrate on this
example, we're good for today.

845
01:00:02,660 --> 01:00:09,790
So my point is the
same solution as-- now,

846
01:00:09,790 --> 01:00:12,090
what's the other problem?

847
01:00:12,090 --> 01:00:16,490
I'm just repeating here, but
making it simple by taking k

848
01:00:16,490 --> 01:00:18,410
equals 0 and m equal 1.

849
01:00:18,410 --> 01:00:25,070
So the same solution as y double
prime equals 0, with y of 0

850
01:00:25,070 --> 01:00:29,005
equal what, and y
prime of 0 equal what.

851
01:00:31,830 --> 01:00:36,570
I just wanted to repeat
here what I've said there,

852
01:00:36,570 --> 01:00:40,930
and then we'll solve it and
we'll see that it's all true.

853
01:00:40,930 --> 01:00:48,190
If I look for a solution to
y double prime equal delta

854
01:00:48,190 --> 01:00:58,780
starting from 0-- this
was starting from 0--

855
01:00:58,780 --> 01:01:03,740
if I say that's the same
as this, what should y of 0

856
01:01:03,740 --> 01:01:04,240
be here?

857
01:01:04,240 --> 01:01:04,912
AUDIENCE: Zero.

858
01:01:04,912 --> 01:01:05,870
PROFESSOR: Zero, right.

859
01:01:05,870 --> 01:01:09,040
It hasn't had time to move.

860
01:01:09,040 --> 01:01:10,500
It hasn't had time to move.

861
01:01:10,500 --> 01:01:15,250
But in that instant,
what happened to y prime?

862
01:01:15,250 --> 01:01:17,360
It jumped to 1.

863
01:01:17,360 --> 01:01:18,499
That's right.

864
01:01:18,499 --> 01:01:19,040
That's right.

865
01:01:19,040 --> 01:01:20,570
Exactly.

866
01:01:20,570 --> 01:01:24,700
Now just solve that
equation for me.

867
01:01:24,700 --> 01:01:25,850
Solve this example for me.

868
01:01:29,340 --> 01:01:31,450
Suppose y double prime-- yeah.

869
01:01:31,450 --> 01:01:32,370
Here we go.

870
01:01:32,370 --> 01:01:35,940
What's the solution if
y double prime is 0?

871
01:01:35,940 --> 01:01:42,230
What are the solutions to
y double prime equals 0?

872
01:01:42,230 --> 01:01:43,640
AUDIENCE: [INAUDIBLE].

873
01:01:43,640 --> 01:01:47,100
PROFESSOR: Constant and linear.

874
01:01:47,100 --> 01:01:50,620
a plus bt, right, have
second derivative 0.

875
01:01:50,620 --> 01:01:57,250
Now what's the solution that
starts from 0 that kills the a

876
01:01:57,250 --> 01:01:58,600
and has slope 1?

877
01:01:58,600 --> 01:02:00,274
What's the answer
to that question?

878
01:02:00,274 --> 01:02:01,190
AUDIENCE: [INAUDIBLE].

879
01:02:01,190 --> 01:02:02,460
PROFESSOR: t.

880
01:02:02,460 --> 01:02:03,150
t.

881
01:02:03,150 --> 01:02:08,370
The solution to this
equation is a ramp.

882
01:02:08,370 --> 01:02:12,310
It's zero everything in this
course, is zero up until time

883
01:02:12,310 --> 01:02:13,780
0.

884
01:02:13,780 --> 01:02:19,290
At time 0, in this example,
all the action happens.

885
01:02:19,290 --> 01:02:20,730
Everything happens.

886
01:02:20,730 --> 01:02:25,500
And what happens is it
gets a velocity of 1,

887
01:02:25,500 --> 01:02:30,640
and the solution
is y equal to t.

888
01:02:30,640 --> 01:02:33,160
y is 0 here, of course.

889
01:02:33,160 --> 01:02:36,590
At that point, that's the
key point, t equals 0,

890
01:02:36,590 --> 01:02:42,040
right there-- it gets a slope.

891
01:02:45,430 --> 01:02:47,500
We don't have a step function.

892
01:02:47,500 --> 01:02:49,520
There's no jump in y.

893
01:02:49,520 --> 01:02:53,460
The jump is in y prime, the
y prime the velocity jumped

894
01:02:53,460 --> 01:02:55,980
from 0 to 1.

895
01:02:55,980 --> 01:03:00,030
That's exactly-- I think when
I introduced delta functions

896
01:03:00,030 --> 01:03:02,180
and drew a picture.

897
01:03:02,180 --> 01:03:05,440
What is the derivative, the
first derivative, y prime,

898
01:03:05,440 --> 01:03:06,360
for that guy?

899
01:03:06,360 --> 01:03:11,050
Let's just review, because this
is what we've seen already.

900
01:03:11,050 --> 01:03:12,245
The first derivative is--

901
01:03:12,245 --> 01:03:13,370
AUDIENCE: [INAUDIBLE] step.

902
01:03:13,370 --> 01:03:14,790
PROFESSOR: A step.

903
01:03:14,790 --> 01:03:19,930
And the second
derivative is delta.

904
01:03:19,930 --> 01:03:24,080
The second derivative of this is
the first derivative of a step.

905
01:03:24,080 --> 01:03:30,050
The derivative of a step is 0
everywhere except at the step,

906
01:03:30,050 --> 01:03:32,200
at the jump when it jumps to 1.

907
01:03:32,200 --> 01:03:36,120
So that's the solution
in this example.

908
01:03:36,120 --> 01:03:41,485
And now to end the lecture,
let's solve it in this example.

909
01:03:44,200 --> 01:03:52,940
Again, let me just say-- why
do I like this forcing term?

910
01:03:52,940 --> 01:03:55,230
Mathematically, I
like it because, if I

911
01:03:55,230 --> 01:03:58,050
can solve that guy--
as we're doing,

912
01:03:58,050 --> 01:04:00,940
we are solving it-- if
I can solve that one,

913
01:04:00,940 --> 01:04:04,140
I can solve all forces.

914
01:04:04,140 --> 01:04:10,250
Over here, I could solve when
I had a very happy f of t,

915
01:04:10,250 --> 01:04:13,965
a perfect f of t, where
I could guess the answer

916
01:04:13,965 --> 01:04:16,070
and push through.

917
01:04:16,070 --> 01:04:20,380
Now with a delta, I can
build everything out

918
01:04:20,380 --> 01:04:21,570
of delta functions.

919
01:04:21,570 --> 01:04:24,270
That's why I like
it mathematically.

920
01:04:24,270 --> 01:04:25,760
Why do I like it physically?

921
01:04:25,760 --> 01:04:32,330
Because it's a very physical
thing to have an impulse.

922
01:04:32,330 --> 01:04:35,610
That happens in real
time, in real things.

923
01:04:35,610 --> 01:04:40,010
And by the way, let's just,
before I write down any more

924
01:04:40,010 --> 01:04:46,420
formula, what
would-- I would like

925
01:04:46,420 --> 01:04:51,300
to be able to solve it
for a step function.

926
01:04:51,300 --> 01:04:55,599
[? Heavy thud. ?] I would like
to be able to do that one.

927
01:04:55,599 --> 01:04:57,140
I'm going to have
to erase something,

928
01:04:57,140 --> 01:04:59,650
or I'll write it right
above just for the moment.

929
01:04:59,650 --> 01:05:07,680
I would also like to
solve my double prime plus

930
01:05:07,680 --> 01:05:10,309
ky equal a step function.

931
01:05:18,820 --> 01:05:24,360
So I would call the solution,
y, the step response.

932
01:05:24,360 --> 01:05:26,600
And what would be a
step function start?

933
01:05:26,600 --> 01:05:30,040
A step function start would
be like turning a switch.

934
01:05:30,040 --> 01:05:32,390
Suddenly things happen.

935
01:05:32,390 --> 01:05:40,030
That's forcing by a step, so I'm
looking for the step response.

936
01:05:40,030 --> 01:05:42,940
And how do you think
these two are related?

937
01:05:47,140 --> 01:05:50,160
I look at the relation
at the right hand sides.

938
01:05:50,160 --> 01:05:56,966
What's the relation of
this step to the delta?

939
01:05:56,966 --> 01:05:57,465
Yeah?

940
01:05:57,465 --> 01:05:58,673
AUDIENCE: One's a derivative.

941
01:05:58,673 --> 01:06:00,870
PROFESSOR: One's a
derivative of the other.

942
01:06:00,870 --> 01:06:03,320
And we've got linear equations.

943
01:06:03,320 --> 01:06:06,240
So the right hand sides.

944
01:06:06,240 --> 01:06:12,310
The step response, y step,
and the delta response,

945
01:06:12,310 --> 01:06:16,320
y delta-- I'll use a
different letter for this

946
01:06:16,320 --> 01:06:19,570
because it's so important.

947
01:06:19,570 --> 01:06:21,270
One is the derivative
of the other.

948
01:06:24,280 --> 01:06:26,040
The great thing about
linear equations

949
01:06:26,040 --> 01:06:29,950
is we have linear equations,
differentiation, integration.

950
01:06:29,950 --> 01:06:31,180
Those are linear operations.

951
01:06:34,460 --> 01:06:38,990
The step function is just
like a steady-- anyway.

952
01:06:38,990 --> 01:06:42,400
I was going to-- I won't--
is the integral of the delta.

953
01:06:42,400 --> 01:06:44,810
Step function is the
integral of the delta,

954
01:06:44,810 --> 01:06:48,205
so the step response is the
integral of the delta response.

955
01:06:51,220 --> 01:06:54,660
I guess to finish the
lecture, why don't we

956
01:06:54,660 --> 01:06:57,020
solve this problem,
which looks tricky

957
01:06:57,020 --> 01:06:59,030
because it's got a delta.

958
01:06:59,030 --> 01:07:01,240
Instead, we'll
solve this problem,

959
01:07:01,240 --> 01:07:03,130
which doesn't look
tricky at all.

960
01:07:03,130 --> 01:07:06,510
It's exactly what we
started the lecture with.

961
01:07:06,510 --> 01:07:10,320
Zero forcing and some
initial conditions.

962
01:07:10,320 --> 01:07:13,840
So let me just
finally make space

963
01:07:13,840 --> 01:07:16,690
for the big deal
from today's lecture,

964
01:07:16,690 --> 01:07:25,810
which would be the fundamental
solution with a force

965
01:07:25,810 --> 01:07:27,900
by a delta.

966
01:07:27,900 --> 01:07:30,310
I'm just going to write
down the answer when

967
01:07:30,310 --> 01:07:31,270
you tell me what it is.

968
01:07:39,214 --> 01:07:41,000
What's the answer to that?

969
01:07:41,000 --> 01:07:44,120
What's the solution
to this second order

970
01:07:44,120 --> 01:07:48,130
constant coefficient
unforced equation

971
01:07:48,130 --> 01:07:50,235
with those initial conditions?

972
01:07:53,630 --> 01:07:55,160
We probably had it here.

973
01:07:55,160 --> 01:07:56,950
I may just have erased it.

974
01:07:56,950 --> 01:07:58,950
But now let's get it.

975
01:07:58,950 --> 01:08:02,350
So y is y delta.

976
01:08:02,350 --> 01:08:05,220
This is the impulse response.

977
01:08:12,350 --> 01:08:15,895
y of t-- and I'll give
it later another name.

978
01:08:18,790 --> 01:08:24,319
So here's a perfect
review question.

979
01:08:24,319 --> 01:08:28,020
What's the solution
to this problem?

980
01:08:28,020 --> 01:08:30,740
Everybody remembers-- what
are the solutions, what's

981
01:08:30,740 --> 01:08:34,370
the general form for the
solution to the equation?

982
01:08:34,370 --> 01:08:37,080
I'm reviewing today's lecture.

983
01:08:37,080 --> 01:08:40,619
The solution to that
equation looks like what?

984
01:08:40,619 --> 01:08:42,585
AUDIENCE: [INAUDIBLE].

985
01:08:42,585 --> 01:08:46,930
PROFESSOR: It's a cosine
and a sine, right.

986
01:08:46,930 --> 01:08:52,050
And then how much of a cosine do
we have and how much of a sine

987
01:08:52,050 --> 01:08:53,120
do we have?

988
01:08:53,120 --> 01:08:58,189
The initial condition will
tell me how much of a cosine

989
01:08:58,189 --> 01:08:59,590
we have.

990
01:08:59,590 --> 01:09:01,680
And what's the answer?

991
01:09:01,680 --> 01:09:03,270
None?

992
01:09:03,270 --> 01:09:05,090
No cosine.

993
01:09:05,090 --> 01:09:08,399
This condition, this
initial velocity,

994
01:09:08,399 --> 01:09:10,729
will tell me how much
of a sine we have,

995
01:09:10,729 --> 01:09:15,399
because the sines are the things
that have initial velocities.

996
01:09:15,399 --> 01:09:21,859
So it would be a sine
of-- the sine of what?

997
01:09:21,859 --> 01:09:25,670
Square root of k over m, right?

998
01:09:25,670 --> 01:09:28,359
omega nt, right?

999
01:09:28,359 --> 01:09:30,320
And what's the number?

1000
01:09:33,575 --> 01:09:40,170
What's the number so
this has the right-- let

1001
01:09:40,170 --> 01:09:41,580
me write again what I want.

1002
01:09:41,580 --> 01:09:45,500
I want y prime at
0 to be 1 over m.

1003
01:09:52,450 --> 01:09:54,320
What's the number
that I put in there?

1004
01:09:54,320 --> 01:09:56,990
I've got something, its
derivative, at zero.

1005
01:09:59,630 --> 01:10:04,540
This is some number-- I'll call
it little a for the moment,

1006
01:10:04,540 --> 01:10:06,110
but I want to find
out what it is.

1007
01:10:10,630 --> 01:10:11,250
Are we right?

1008
01:10:11,250 --> 01:10:12,436
Yeah?

1009
01:10:12,436 --> 01:10:13,328
I think we're right.

1010
01:10:15,996 --> 01:10:16,496
Yeah.

1011
01:10:19,600 --> 01:10:25,360
The derivative is at zero, so
I just plug that into here,

1012
01:10:25,360 --> 01:10:27,760
take the derivative at zero--
of course that makes it

1013
01:10:27,760 --> 01:10:32,400
a cosine, which will be 1-- but
it also brings out that factor.

1014
01:10:32,400 --> 01:10:38,140
So a times-- well, that
factor will be 1 over m,

1015
01:10:38,140 --> 01:10:41,700
and that tells me
what a has to be.

1016
01:10:41,700 --> 01:10:43,830
Well, this is omega.

1017
01:10:43,830 --> 01:10:47,020
So a is-- this is
omega a equal m.

1018
01:10:47,020 --> 01:10:51,460
This is 1 over m omega.

1019
01:10:51,460 --> 01:10:52,445
And that's omega.

1020
01:10:57,420 --> 01:11:03,070
Sorry I'm erasing
stuff which I-- this is

1021
01:11:03,070 --> 01:11:04,990
the formula I'm after.

1022
01:11:04,990 --> 01:11:06,790
Sine omega t over omega.

1023
01:11:12,034 --> 01:11:12,825
I think we're good.

1024
01:11:12,825 --> 01:11:13,880
Are we?

1025
01:11:13,880 --> 01:11:14,440
Yeah?

1026
01:11:14,440 --> 01:11:15,280
Yeah.

1027
01:11:15,280 --> 01:11:20,880
I'll come back to
this in-- Wednesday

1028
01:11:20,880 --> 01:11:25,310
is my day to move
to damping terms.

1029
01:11:25,310 --> 01:11:32,490
I've intentionally stayed
with undamped equations

1030
01:11:32,490 --> 01:11:37,310
here, because you're thinking
about that level of equation.

1031
01:11:37,310 --> 01:11:40,320
Damping is going to
bring in new stuff,

1032
01:11:40,320 --> 01:11:43,850
and that should
wait till Wednesday.

1033
01:11:43,850 --> 01:11:46,550
Shall I recap today?

1034
01:11:46,550 --> 01:11:50,540
I'll just recap today,
and then we're done.

1035
01:11:50,540 --> 01:11:56,770
Today started with
the unforced equation.

1036
01:12:01,370 --> 01:12:06,660
We solved it by assuming--
by not thinking ahead,

1037
01:12:06,660 --> 01:12:09,920
just assume I have
an exponential,

1038
01:12:09,920 --> 01:12:11,620
because the beauty
of exponentials

1039
01:12:11,620 --> 01:12:17,020
is, when I plug it in,
the exponential cancels.

1040
01:12:17,020 --> 01:12:20,080
And that told me that
s was pure imaginary.

1041
01:12:20,080 --> 01:12:23,920
It told me that it had this
form, e to the i omega t.

1042
01:12:23,920 --> 01:12:25,420
And there were two s's.

1043
01:12:25,420 --> 01:12:27,495
Two possible s's,
plus and minus.

1044
01:12:30,730 --> 01:12:37,072
I get to make a little comment
about this example here.

1045
01:12:42,970 --> 01:12:43,910
What was omega?

1046
01:12:43,910 --> 01:12:49,010
What's the natural
frequency in this problem?

1047
01:12:49,010 --> 01:12:50,970
What's the natural
frequency here?

1048
01:12:53,970 --> 01:13:00,510
I guess this is a case where--
what's the natural frequency?

1049
01:13:00,510 --> 01:13:05,900
I guess this is a case where m
is 1 and k is 0, is that right?

1050
01:13:05,900 --> 01:13:10,540
This does fit into that pattern,
but it's a little special.

1051
01:13:10,540 --> 01:13:13,590
This is a case where
m is 1 and k is 0.

1052
01:13:13,590 --> 01:13:15,320
So what's the natural
frequency in this?

1053
01:13:15,320 --> 01:13:16,200
AUDIENCE: Zero.

1054
01:13:16,200 --> 01:13:18,320
PROFESSOR: Zero.

1055
01:13:18,320 --> 01:13:20,100
Zero.

1056
01:13:20,100 --> 01:13:25,880
This is a crazy
case of resonance.

1057
01:13:25,880 --> 01:13:29,130
It's a case in which the natural
frequency and the driving

1058
01:13:29,130 --> 01:13:35,440
frequency, say in this--
I'll have to do it here--

1059
01:13:35,440 --> 01:13:40,720
this simplest of all equations
is, in a way, special.

1060
01:13:40,720 --> 01:13:46,910
It's a case when the
natural frequency is zero

1061
01:13:46,910 --> 01:13:51,900
and the driving frequency
is zero and they're equal.

1062
01:13:51,900 --> 01:13:56,170
And what happens with resonance?

1063
01:13:56,170 --> 01:14:00,010
What's the new
formula, the new term

1064
01:14:00,010 --> 01:14:02,550
that comes in with resonance?

1065
01:14:02,550 --> 01:14:03,250
It's t.

1066
01:14:07,410 --> 01:14:09,510
You saw it happen
for this example,

1067
01:14:09,510 --> 01:14:11,970
and we didn't have to
use the word resonance.

1068
01:14:11,970 --> 01:14:15,460
We knew that we had a ramp.

1069
01:14:15,460 --> 01:14:18,370
We just used the word
ramp, not resonance.

1070
01:14:18,370 --> 01:14:21,280
But this is a case of resonance.

1071
01:14:21,280 --> 01:14:24,880
When omega n is zero
and omega d is zero.

1072
01:14:27,570 --> 01:14:30,790
And the factor t up here.

1073
01:14:30,790 --> 01:14:31,350
Anyway.

1074
01:14:31,350 --> 01:14:34,040
Just that small comment there.

1075
01:14:34,040 --> 01:14:37,220
And now, just going
back to the recap.

1076
01:14:37,220 --> 01:14:41,070
The recap was, we
tried exponentials.

1077
01:14:41,070 --> 01:14:44,050
We learned that they
were pure oscillations.

1078
01:14:44,050 --> 01:14:49,760
We realized that we could do
cosines and sines instead,

1079
01:14:49,760 --> 01:14:50,780
and we did.

1080
01:14:50,780 --> 01:14:52,840
And we took off.

1081
01:14:52,840 --> 01:14:56,170
We got the formula.

1082
01:14:56,170 --> 01:15:02,760
Then of course the-- so this
is section 2.1 of the book.

1083
01:15:02,760 --> 01:15:08,290
And it goes through all
those steps carefully.

1084
01:15:08,290 --> 01:15:12,290
Section 2.2 of the book tells
us about complex numbers,

1085
01:15:12,290 --> 01:15:15,510
and section 2.3
brings damping in.

1086
01:15:15,510 --> 01:15:18,730
So that's what's
coming next time.

1087
01:15:18,730 --> 01:15:20,480
So the recap again.

1088
01:15:20,480 --> 01:15:26,810
We found the null solution, we
found a particular solution--

1089
01:15:26,810 --> 01:15:28,950
oh there's just one
comment I want to make,

1090
01:15:28,950 --> 01:15:29,896
and then I'm done.

1091
01:15:33,210 --> 01:15:36,160
Where was our
particular solution?

1092
01:15:40,170 --> 01:15:41,180
Yeah.

1093
01:15:41,180 --> 01:15:42,860
This was our
particular solution.

1094
01:15:47,940 --> 01:15:49,940
Here's my comment.

1095
01:15:49,940 --> 01:15:52,190
Here's my comment.

1096
01:15:52,190 --> 01:15:56,720
Suppose I want to solve
this basic equation starting

1097
01:15:56,720 --> 01:16:01,720
from a given y of 0
and a y prime of 0.

1098
01:16:01,720 --> 01:16:03,770
I'm going to do it in
two parts, I think.

1099
01:16:03,770 --> 01:16:06,430
I've got the null
solution, and I've

1100
01:16:06,430 --> 01:16:10,080
got this particular solution.

1101
01:16:10,080 --> 01:16:15,450
Now here's my point.

1102
01:16:15,450 --> 01:16:22,760
If I want to get y of
0-- how shall I say this.

1103
01:16:22,760 --> 01:16:25,390
You can't just put together--
it's an easy mistake

1104
01:16:25,390 --> 01:16:29,500
to make-- solve
the null equation

1105
01:16:29,500 --> 01:16:33,410
with the initial
conditions and then

1106
01:16:33,410 --> 01:16:35,510
add in the particular solution.

1107
01:16:35,510 --> 01:16:39,800
You'd think, I just
followed all the rules.

1108
01:16:39,800 --> 01:16:46,360
But this particular solution
that you added in has a-- at t

1109
01:16:46,360 --> 01:16:51,120
equals 0, it's not zero.

1110
01:16:51,120 --> 01:16:53,000
So you have to change.

1111
01:16:53,000 --> 01:17:00,400
So the correct thing,
the correct yp plus yn--

1112
01:17:00,400 --> 01:17:04,720
let me make that point.

1113
01:17:04,720 --> 01:17:05,390
Just a warning.

1114
01:17:10,670 --> 01:17:12,740
So in words the
warning is, remember

1115
01:17:12,740 --> 01:17:16,360
that the particular solution
has some initial condition--

1116
01:17:16,360 --> 01:17:19,380
in that case, g-- and
then that is going

1117
01:17:19,380 --> 01:17:23,690
to affect the right
null solution.

1118
01:17:23,690 --> 01:17:31,190
So again, y is y null
plus y particular-- plus y

1119
01:17:31,190 --> 01:17:40,670
of particular-- so it's some c1
cos omega nt plus some c2 sine

1120
01:17:40,670 --> 01:17:47,748
omega nt plus this particular
guy, g, cosine of omega dt.

1121
01:17:52,900 --> 01:17:53,990
All correct.

1122
01:17:53,990 --> 01:17:55,610
All correct.

1123
01:17:55,610 --> 01:17:58,190
But now, put in the
initial conditions.

1124
01:17:58,190 --> 01:17:59,612
y of 0 is given.

1125
01:18:02,930 --> 01:18:05,760
And what do I get on
the right hand side

1126
01:18:05,760 --> 01:18:08,860
when I put in t equals 0?

1127
01:18:08,860 --> 01:18:13,810
I get c1 here.

1128
01:18:13,810 --> 01:18:17,020
What do I get when I
put t equals 0 in there?

1129
01:18:17,020 --> 01:18:17,800
Nothing.

1130
01:18:17,800 --> 01:18:21,043
What do I get when I
put t equals 0 in here?

1131
01:18:21,043 --> 01:18:21,543
g.

1132
01:18:26,360 --> 01:18:28,860
So it's not c1 equal
y of 0 anymore.

1133
01:18:28,860 --> 01:18:33,440
That's the easy mistake
that I'm correcting.

1134
01:18:33,440 --> 01:18:41,640
When you put in this
particular solution,

1135
01:18:41,640 --> 01:18:44,830
it has an initial value.

1136
01:18:44,830 --> 01:18:47,980
That initial value is
going to come in here.

1137
01:18:47,980 --> 01:18:55,240
So c1, then, the correct
c1 is y of 0 minus g.

1138
01:18:55,240 --> 01:18:56,230
End of story.

1139
01:18:56,230 --> 01:19:02,140
Just don't be too quick to just
add the two pieces and think

1140
01:19:02,140 --> 01:19:04,700
you can do them
completely separately,

1141
01:19:04,700 --> 01:19:06,657
because you're
putting them together.

1142
01:19:06,657 --> 01:19:08,240
And then you have
to put them together

1143
01:19:08,240 --> 01:19:10,390
in the initial condition.