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BOLESLAW WYSLOUCH:
Let's get started.

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00:00:25,030 --> 00:00:27,690
So today hopefully
will be a busy day,

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00:00:27,690 --> 00:00:34,860
with lots of interesting
insights into how things work.

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00:00:34,860 --> 00:00:38,020
We talked about coupled
oscillators last time.

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00:00:38,020 --> 00:00:41,850
We developed a
formalism in which

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00:00:41,850 --> 00:00:46,860
we can find the most general
motion of oscillators.

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00:00:46,860 --> 00:00:52,650
So let's remind ourselves what
are the coupled oscillators.

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00:00:52,650 --> 00:00:55,920
Coupled oscillators, there
are many examples of them,

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00:00:55,920 --> 00:00:59,310
and they have more or less
the following features.

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00:00:59,310 --> 00:01:00,840
You have something
that oscillates--

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00:01:00,840 --> 00:01:03,320
for example, a pendulum.

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00:01:03,320 --> 00:01:08,250
You have to have more than one,
because for coupled oscillators

20
00:01:08,250 --> 00:01:10,560
you have to have at least two.

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00:01:10,560 --> 00:01:14,100
So let's say you
have two oscillators.

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00:01:14,100 --> 00:01:18,610
So each of them
is an oscillator,

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00:01:18,610 --> 00:01:23,670
which in, for example, in
the limit of small angles,

24
00:01:23,670 --> 00:01:26,250
small displacement
angles, undergoes

25
00:01:26,250 --> 00:01:29,640
a pure harmonic motion
with some frequencies.

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00:01:29,640 --> 00:01:32,610
And then you couple them
through various means.

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00:01:32,610 --> 00:01:36,150
So for example, two masses
connected by a spring

28
00:01:36,150 --> 00:01:38,760
is an example of a
coupled oscillator.

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00:01:38,760 --> 00:01:43,770
We could have two masses on
a track and another track,

30
00:01:43,770 --> 00:01:47,070
also connected by
several springs.

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00:01:47,070 --> 00:01:49,740
This is also an example
of a coupled oscillator.

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00:01:49,740 --> 00:01:54,122
Each of those masses
undergoes harmonic motion,

33
00:01:54,122 --> 00:01:56,580
and they are connected together
such that the motion of one

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00:01:56,580 --> 00:01:58,560
affects motion of the other.

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00:01:58,560 --> 00:02:00,750
You can have slightly
more complicated pendula.

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00:02:00,750 --> 00:02:07,120
For example, you can hang
one pendula from the other.

37
00:02:07,120 --> 00:02:09,639
Each of them-- again, in the
limit of small oscillations--

38
00:02:09,639 --> 00:02:11,610
will undergo harmonic motion.

39
00:02:11,610 --> 00:02:14,890
And they are coupled together
because they are supported one

40
00:02:14,890 --> 00:02:17,020
on top of each other.

41
00:02:17,020 --> 00:02:18,840
And you can have--

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00:02:18,840 --> 00:02:25,900
we have another example
of two tuning forks

43
00:02:25,900 --> 00:02:28,240
sitting on some sort of boxes.

44
00:02:28,240 --> 00:02:31,840
Each of them was an oscillator,
with audible oscillating

45
00:02:31,840 --> 00:02:38,350
frequency, and by putting them
next to each other they coupled

46
00:02:38,350 --> 00:02:41,590
through the sound waves
transmitted through the air.

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00:02:41,590 --> 00:02:45,760
So one of them felt the
oscillations in the other one.

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00:02:45,760 --> 00:02:47,700
This was an example of
coupled oscillation.

49
00:02:47,700 --> 00:02:51,100
Two masses and the thing.

50
00:02:51,100 --> 00:02:54,535
You can build oscillators
out of electronics.

51
00:02:54,535 --> 00:02:57,160
Some capacitor and
inductor together,

52
00:02:57,160 --> 00:02:58,270
with a little bit of--

53
00:02:58,270 --> 00:02:59,410
maybe without resistors.

54
00:02:59,410 --> 00:03:00,760
You have two of those.

55
00:03:00,760 --> 00:03:02,680
They constitute a
coupled oscillator

56
00:03:02,680 --> 00:03:04,690
if you put a wire between them.

57
00:03:04,690 --> 00:03:06,329
So there are many,
many examples.

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00:03:06,329 --> 00:03:07,870
And of course, these
are all examples

59
00:03:07,870 --> 00:03:10,480
in which you have two
oscillating bodies,

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00:03:10,480 --> 00:03:14,290
but it's very easy to have three
or more oscillating bodies.

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00:03:14,290 --> 00:03:17,530
Then basically the
features of the system

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00:03:17,530 --> 00:03:20,400
are the same, except the math
becomes more complicated,

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00:03:20,400 --> 00:03:23,480
and we have more types of
oscillations you can have.

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00:03:23,480 --> 00:03:27,100
And there's a couple
of characteristics

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00:03:27,100 --> 00:03:30,220
which are the same for
all oscillating systems.

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00:03:30,220 --> 00:03:31,690
And it's very
important to remember

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00:03:31,690 --> 00:03:33,790
that we are learning
on one example,

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00:03:33,790 --> 00:03:35,740
but it applies to very many.

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00:03:35,740 --> 00:03:38,170
Number one, any motion--

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00:03:38,170 --> 00:03:41,520
I can maybe summarize it here.

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00:03:41,520 --> 00:03:49,070
So if you look at the motion of
an oscillator, you can have--

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00:03:49,070 --> 00:03:52,290
let's say arbitrary oscillation.

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00:03:52,290 --> 00:03:55,130
Arbitrary excitation.

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00:03:58,300 --> 00:04:01,150
Excitation means I--

75
00:04:01,150 --> 00:04:05,630
I kick it in some sort
of arbitrary mode.

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00:04:05,630 --> 00:04:08,480
I just come in and set up
some initial condition such

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00:04:08,480 --> 00:04:10,270
that things are moving.

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00:04:10,270 --> 00:04:14,140
And motion in this
arbitrary assertion

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00:04:14,140 --> 00:04:16,930
is actually-- looks
pretty chaotic.

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00:04:16,930 --> 00:04:21,040
It looks pretty
variable, changing.

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00:04:21,040 --> 00:04:23,860
It's difficult to
understand what's going on.

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00:04:23,860 --> 00:04:26,380
So And it clearly
doesn't look harmonic.

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00:04:26,380 --> 00:04:27,400
Non-harmonic.

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00:04:30,740 --> 00:04:33,110
There is no obvious
single frequency

85
00:04:33,110 --> 00:04:36,780
that is driving the system.

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00:04:36,780 --> 00:04:40,640
If you look at
amplitude of the objects

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00:04:40,640 --> 00:04:43,070
here-- for example, two
pendula, pendulum one and two.

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00:04:43,070 --> 00:04:46,040
At any given moment of
time they are oscillating,

89
00:04:46,040 --> 00:04:48,020
there's a characteristic
amplitude.

90
00:04:48,020 --> 00:04:50,570
But what we saw is
that motion changes,

91
00:04:50,570 --> 00:04:53,300
looks like things are flowing
from one to the other.

92
00:04:53,300 --> 00:04:54,740
One of them has
a high amplitude.

93
00:04:54,740 --> 00:04:58,170
After some time, it cools
down, the other one grows.

94
00:04:58,170 --> 00:05:01,965
So the amplitudes
are changing in time.

95
00:05:01,965 --> 00:05:02,840
So they are variable.

96
00:05:10,570 --> 00:05:11,185
Are variable.

97
00:05:14,060 --> 00:05:17,540
And also, we didn't
calculate things exactly,

98
00:05:17,540 --> 00:05:21,302
but you know from study of
a single oscillator that

99
00:05:21,302 --> 00:05:23,510
if the things are moving,
it has a certain amplitude,

100
00:05:23,510 --> 00:05:26,180
there's certain energy
involved-- with some potential,

101
00:05:26,180 --> 00:05:27,350
some kinetic--

102
00:05:27,350 --> 00:05:29,820
and it's proportional to
the square of amplitude.

103
00:05:29,820 --> 00:05:32,690
So it's clear that energy
is moving from one pendulum

104
00:05:32,690 --> 00:05:34,100
to the other.

105
00:05:34,100 --> 00:05:36,470
This one was
oscillating like crazy.

106
00:05:36,470 --> 00:05:38,630
So all energy was sitting here.

107
00:05:38,630 --> 00:05:40,660
After some time,
this one stopped.

108
00:05:40,660 --> 00:05:42,080
So its energy is zero.

109
00:05:42,080 --> 00:05:44,870
And the other one was
oscillating like crazy.

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00:05:44,870 --> 00:05:47,330
So the energy's flowing
from one to another.

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00:05:47,330 --> 00:05:50,330
It's not sitting in one
place, but it's flowing.

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00:05:50,330 --> 00:05:53,420
This one has lots of
energy right now, but now

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00:05:53,420 --> 00:05:55,020
that one is picking up.

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00:05:55,020 --> 00:05:59,430
So the energy-- you see
the energy flowing here.

115
00:05:59,430 --> 00:06:01,320
And this one will
eventually stop--

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00:06:01,320 --> 00:06:03,030
well, this is a pretty
crappy oscillator,

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00:06:03,030 --> 00:06:05,040
but it will eventually
stop, and this one

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00:06:05,040 --> 00:06:07,950
will have all the energy.

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00:06:07,950 --> 00:06:10,080
And this is, again,
characteristic in every system.

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00:06:10,080 --> 00:06:13,650
We can see energy flowing
around from one to the other,

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00:06:13,650 --> 00:06:15,270
growing, stopping.

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00:06:15,270 --> 00:06:19,986
So it's-- in general, in
the most general case,

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00:06:19,986 --> 00:06:23,840
it's a complicated system.

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00:06:23,840 --> 00:06:31,100
Energy is migrating
between different masses.

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00:06:31,100 --> 00:06:34,400
However, every single one
of those coupled oscillating

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00:06:34,400 --> 00:06:35,990
systems has a magic.

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00:06:35,990 --> 00:06:39,980
There's a magic involved, namely
the existence of normal modes.

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00:06:39,980 --> 00:06:43,400
Every single coupled oscillator
system has normal modes,

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00:06:43,400 --> 00:06:45,260
and those modes are beautiful.

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00:06:45,260 --> 00:06:50,630
Those modes are-- everything
is moving in sync.

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00:06:50,630 --> 00:06:59,540
So this is normal
mode excitation.

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00:06:59,540 --> 00:07:02,440
There's a very special
way, a special setting

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00:07:02,440 --> 00:07:07,090
of initial conditions, that
leads to the-- that results

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00:07:07,090 --> 00:07:10,090
in a pure harmonic motion.

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00:07:10,090 --> 00:07:19,640
So this is a harmonic motion,
with a certain frequency omega,

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00:07:19,640 --> 00:07:22,640
characteristic frequency
for this particular motion.

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00:07:22,640 --> 00:07:26,130
The amplitudes remain fixed.

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00:07:26,130 --> 00:07:29,070
Once you set initial
conditions, you get it moving,

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00:07:29,070 --> 00:07:31,500
everything is moving,
simple harmonic motion

140
00:07:31,500 --> 00:07:33,970
means its amplitude is constant.

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00:07:33,970 --> 00:07:36,750
So if I-- and remember,
for example, this system.

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00:07:36,750 --> 00:07:39,750
It was something like this.

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00:07:39,750 --> 00:07:44,610
Symmetric or
antisymmetric motion.

144
00:07:44,610 --> 00:07:46,710
And if not for the
friction, the amplitudes

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00:07:46,710 --> 00:07:49,080
would remain constant
forever, if it

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00:07:49,080 --> 00:07:50,310
will be a perfect oscillator.

147
00:07:50,310 --> 00:07:53,580
So amplitudes-- in fact, it's
not amplitudes themselves,

148
00:07:53,580 --> 00:07:55,260
but amplitude ratio.

149
00:07:55,260 --> 00:07:59,670
The ratio of amplitude
between the different elements

150
00:07:59,670 --> 00:08:02,110
in the system is constant.

151
00:08:07,910 --> 00:08:10,220
So in a sense, every
harmonic motion

152
00:08:10,220 --> 00:08:13,010
has a characteristic shape.

153
00:08:13,010 --> 00:08:17,060
And then by-- since everything
is constant, nothing changes,

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00:08:17,060 --> 00:08:20,800
this energy stays
in the place it is.

155
00:08:25,830 --> 00:08:28,439
So energy is--
once you put energy

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00:08:28,439 --> 00:08:30,730
to mass number one, mass
number two, mass number three,

157
00:08:30,730 --> 00:08:32,159
the energy sits there.

158
00:08:32,159 --> 00:08:34,799
The energies are
constant, as the system

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00:08:34,799 --> 00:08:36,510
undergoes harmonic motion.

160
00:08:36,510 --> 00:08:39,330
Energy does not migrate.

161
00:08:39,330 --> 00:08:42,539
So this is a very nice-- and
there is another beautiful

162
00:08:42,539 --> 00:08:48,960
feature, that any arbitrary
excitation can be made out

163
00:08:48,960 --> 00:08:53,227
of some linear sum--

164
00:08:56,149 --> 00:09:00,250
sum of normal modes.

165
00:09:02,980 --> 00:09:05,780
Linear sum, of
superposition of normal.

166
00:09:05,780 --> 00:09:09,410
Any arbitrary excitation with
all its complicated motion

167
00:09:09,410 --> 00:09:12,220
can be made into
some of normal modes.

168
00:09:12,220 --> 00:09:15,450
So since normal modes are
easy and simple and beautiful,

169
00:09:15,450 --> 00:09:20,520
the description of motion of
any coupled oscillator, the best

170
00:09:20,520 --> 00:09:23,070
way to approach it
is to decompose it,

171
00:09:23,070 --> 00:09:25,740
to find all possible
normal modes,

172
00:09:25,740 --> 00:09:29,010
and then decompose the initial
condition to correspond

173
00:09:29,010 --> 00:09:31,140
to this linear sum
of normal modes.

174
00:09:31,140 --> 00:09:33,660
Once you know the normal
modes, you add them up,

175
00:09:33,660 --> 00:09:37,590
and then you can predict
exactly the motion.

176
00:09:37,590 --> 00:09:39,900
And this is what we've done.

177
00:09:39,900 --> 00:09:42,530
So we have a--

178
00:09:42,530 --> 00:09:46,590
we have introduced a
mathematic mechanism

179
00:09:46,590 --> 00:09:50,910
in which we put all the
information about forces

180
00:09:50,910 --> 00:09:54,120
and masses in the system in
some sort of matrix form.

181
00:09:54,120 --> 00:09:56,430
In our example, it was
a two by two matrix,

182
00:09:56,430 --> 00:09:58,740
but if we have three
masses or four masses,

183
00:09:58,740 --> 00:10:01,320
the dimensionality of the
matrix will have to grow.

184
00:10:01,320 --> 00:10:03,730
But the equation
will remain the same.

185
00:10:03,730 --> 00:10:08,070
So this equation of motion,
we rework it a little bit.

186
00:10:08,070 --> 00:10:10,530
Since we are looking
for normal modes,

187
00:10:10,530 --> 00:10:14,230
we know that normal modes occur
with this one single frequency.

188
00:10:14,230 --> 00:10:19,260
So we postulate an
oscillation with a frequency.

189
00:10:19,260 --> 00:10:20,220
We plug it in.

190
00:10:20,220 --> 00:10:23,870
We obtain a simple
algebraic equation.

191
00:10:23,870 --> 00:10:25,290
Doesn't have any
time dependence,

192
00:10:25,290 --> 00:10:27,030
doesn't have any exponents.

193
00:10:27,030 --> 00:10:30,600
It's a simple algebraic
equation, basically a set

194
00:10:30,600 --> 00:10:35,930
of linear equations, which
we can solve and find

195
00:10:35,930 --> 00:10:39,170
the eigenvalue, or the
characteristic frequency

196
00:10:39,170 --> 00:10:40,510
for normal modes.

197
00:10:40,510 --> 00:10:43,190
And you can show that the number
of those frequencies in general

198
00:10:43,190 --> 00:10:46,400
is equal to the number of
masses involved in the system.

199
00:10:46,400 --> 00:10:49,130
And you solve it,
and then once you

200
00:10:49,130 --> 00:10:51,590
know the characteristic
frequencies,

201
00:10:51,590 --> 00:10:55,250
then you can find shape, you
can find the eigenvectors.

202
00:10:55,250 --> 00:10:59,510
What is the ratio of amplitudes
which corresponds to the mode.

203
00:10:59,510 --> 00:11:02,970
And in case of our two pendula,
there are two of such things.

204
00:11:02,970 --> 00:11:06,500
One is where both
amplitudes are equal,

205
00:11:06,500 --> 00:11:08,870
and this corresponds
to oscillation

206
00:11:08,870 --> 00:11:13,670
in which two pendola are
moving parallel to each other,

207
00:11:13,670 --> 00:11:15,860
with a spring being--

208
00:11:15,860 --> 00:11:18,350
not paying any roll.

209
00:11:18,350 --> 00:11:19,490
So this is one mode.

210
00:11:19,490 --> 00:11:21,570
And then amplitude is--

211
00:11:21,570 --> 00:11:25,070
as I said, any given moment is
the same, so the ratio is 1.

212
00:11:25,070 --> 00:11:28,800
And then you have a motion
in which the two pendula

213
00:11:28,800 --> 00:11:30,350
are going against each other.

214
00:11:30,350 --> 00:11:32,420
So any given moment
of time, they're

215
00:11:32,420 --> 00:11:35,103
in their negative position,
so the ratio is minus 1.

216
00:11:38,430 --> 00:11:41,650
The motion of one of
them can be obtained

217
00:11:41,650 --> 00:11:43,525
by looking at where
the first one is

218
00:11:43,525 --> 00:11:45,880
and multiplying by minus 1.

219
00:11:45,880 --> 00:11:48,190
So these are the two modes,
and any arbitrary-- any

220
00:11:48,190 --> 00:11:52,420
complicated, nasty excitation
with things moving around

221
00:11:52,420 --> 00:11:56,780
is a linear sum of
the oscillation.

222
00:11:56,780 --> 00:11:57,660
So we know that.

223
00:11:57,660 --> 00:11:59,380
We've worked it out.

224
00:11:59,380 --> 00:12:02,820
We used this example.

225
00:12:02,820 --> 00:12:06,430
And by the way, today, we'll
be using two examples--

226
00:12:06,430 --> 00:12:09,940
one which is the same thing
with two pendula and the spring,

227
00:12:09,940 --> 00:12:11,510
and the other one
with two masses,

228
00:12:11,510 --> 00:12:14,020
or maybe later three masses.

229
00:12:14,020 --> 00:12:20,730
And the exact values of
coefficients in matrix k

230
00:12:20,730 --> 00:12:23,870
are different in
two different cases.

231
00:12:23,870 --> 00:12:29,510
But in all types of other
motion, the shape of motion,

232
00:12:29,510 --> 00:12:32,340
the behavior of the
system is identical.

233
00:12:32,340 --> 00:12:36,390
So the solutions to the
two cases are identical.

234
00:12:36,390 --> 00:12:40,130
The difference is basically
numerical in how the spring

235
00:12:40,130 --> 00:12:41,960
constants and masses come in.

236
00:12:41,960 --> 00:12:45,080
So we can in fact
treat those two systems

237
00:12:45,080 --> 00:12:46,190
completely the same.

238
00:12:46,190 --> 00:12:48,050
So I'll be jumping
from one to another,

239
00:12:48,050 --> 00:12:49,190
but we don't have to worry.

240
00:12:49,190 --> 00:12:52,700
But let's now look
on the system.

241
00:12:52,700 --> 00:12:54,470
So what we are trying
to do today is,

242
00:12:54,470 --> 00:13:00,040
we are trying to
apply external force

243
00:13:00,040 --> 00:13:02,609
so we'll have a driven
coupled oscillator.

244
00:13:02,609 --> 00:13:04,150
And I assume that
you know everything

245
00:13:04,150 --> 00:13:06,040
about driven oscillators.

246
00:13:06,040 --> 00:13:09,300
So the idea was that you
come with an external .

247
00:13:09,300 --> 00:13:11,980
In 8.03, we assumed
that this external force

248
00:13:11,980 --> 00:13:13,160
is harmonic force.

249
00:13:13,160 --> 00:13:15,490
So there's a
characteristic frequency

250
00:13:15,490 --> 00:13:18,364
which is given by external--

251
00:13:18,364 --> 00:13:19,030
let's say by me.

252
00:13:19,030 --> 00:13:22,450
It has nothing to do with normal
frequencies of the system.

253
00:13:22,450 --> 00:13:25,000
It's an external frequency,
omega d, which I apply.

254
00:13:25,000 --> 00:13:26,740
Driven frequency.

255
00:13:26,740 --> 00:13:29,990
And then I look at how
the system responds.

256
00:13:29,990 --> 00:13:32,090
And I look for steady
state oscillations--

257
00:13:32,090 --> 00:13:35,540
the ones where everything
oscillates with the same driven

258
00:13:35,540 --> 00:13:36,680
frequency--

259
00:13:36,680 --> 00:13:38,270
trying to look for solutions.

260
00:13:38,270 --> 00:13:42,110
And as you know from a
single oscillator, what

261
00:13:42,110 --> 00:13:45,550
we were calculating is what is
the the response of the system?

262
00:13:45,550 --> 00:13:46,790
What is the amplitude?

263
00:13:46,790 --> 00:13:49,410
And the certain
frequencies that--

264
00:13:49,410 --> 00:13:52,310
you wiggle it and the
system doesn't do anything,

265
00:13:52,310 --> 00:13:55,507
but if you apply a certain
resonant frequency,

266
00:13:55,507 --> 00:13:56,840
then the response is very large.

267
00:13:56,840 --> 00:13:59,240
The system starts moving
like crazy, et cetera.

268
00:13:59,240 --> 00:14:01,550
And the same type of
thing will happen here,

269
00:14:01,550 --> 00:14:04,480
except that we have
multiple frequencies.

270
00:14:04,480 --> 00:14:06,950
So there will be a
possibility of a resonance

271
00:14:06,950 --> 00:14:08,640
for several frequencies.

272
00:14:08,640 --> 00:14:10,820
All right?

273
00:14:10,820 --> 00:14:16,090
So let me quickly set this up.

274
00:14:16,090 --> 00:14:17,320
Just-- yeah.

275
00:14:17,320 --> 00:14:18,165
Doesn't matter.

276
00:14:18,165 --> 00:14:20,790
So there were some--

277
00:14:20,790 --> 00:14:23,940
let's just start
working on the example.

278
00:14:23,940 --> 00:14:26,820
So just a reminder,
this is our system.

279
00:14:26,820 --> 00:14:29,500
A pendula of some
length L. There

280
00:14:29,500 --> 00:14:33,930
are two identical
masses, M. There

281
00:14:33,930 --> 00:14:37,320
is a spring of constant k.

282
00:14:37,320 --> 00:14:41,000
They are all-- and
for simplicity, we

283
00:14:41,000 --> 00:14:44,630
assume that we are all in
Earth's gravitational field.

284
00:14:44,630 --> 00:14:46,460
So we don't have to
worry about traveling

285
00:14:46,460 --> 00:14:49,730
to Jupiter or the moon.

286
00:14:49,730 --> 00:14:53,720
And-- except that
the difference will

287
00:14:53,720 --> 00:14:58,490
be that we apply an external
force to one of those masses.

288
00:14:58,490 --> 00:15:02,962
How, it doesn't matter, but
there is an external force F--

289
00:15:02,962 --> 00:15:08,960
F with subscript d,
which is equal to some--

290
00:15:08,960 --> 00:15:13,540
it has some amplitude
F0 cosine omega

291
00:15:13,540 --> 00:15:19,610
d times t, along
the x direction.

292
00:15:19,610 --> 00:15:23,570
And this is applied to mass one.

293
00:15:23,570 --> 00:15:25,000
OK.

294
00:15:25,000 --> 00:15:28,225
And there is a little
bit of just a warning.

295
00:15:30,880 --> 00:15:37,069
We will be assuming that there
is no damping in the system.

296
00:15:37,069 --> 00:15:38,860
For the single oscillator,
there was always

297
00:15:38,860 --> 00:15:40,370
a little bit of damping.

298
00:15:40,370 --> 00:15:43,360
So between you and me, remember
there's always a little

299
00:15:43,360 --> 00:15:43,870
damping.

300
00:15:43,870 --> 00:15:46,270
So in case we need damping--

301
00:15:46,270 --> 00:15:48,580
it will come in
and will help us,

302
00:15:48,580 --> 00:15:50,860
but if we try to use
damping in calculations,

303
00:15:50,860 --> 00:15:52,600
calculations become horrendous.

304
00:15:52,600 --> 00:15:54,730
So for the purpose
of calculations,

305
00:15:54,730 --> 00:15:56,710
we will ignore damping.

306
00:15:56,710 --> 00:15:57,430
It'll get some.

307
00:15:57,430 --> 00:16:01,780
But if things go bad with the
results, like dividing by 0,

308
00:16:01,780 --> 00:16:04,510
then we will bring in damping
and say no no, it's not so bad.

309
00:16:04,510 --> 00:16:05,630
Damping helps you.

310
00:16:05,630 --> 00:16:06,990
We are not dividing by 0.

311
00:16:06,990 --> 00:16:09,340
OK?

312
00:16:09,340 --> 00:16:12,250
So let's write those
equations of motions.

313
00:16:12,250 --> 00:16:13,375
Equations of motion.

314
00:16:16,370 --> 00:16:21,810
So we have-- so the forces
and accelerations on mass one

315
00:16:21,810 --> 00:16:23,650
is the same as before.

316
00:16:23,650 --> 00:16:25,750
There was a spring.

317
00:16:25,750 --> 00:16:29,170
There is mg over l.

318
00:16:29,170 --> 00:16:33,580
That's the pendulum by itself.

319
00:16:33,580 --> 00:16:36,220
Depending on position x1.

320
00:16:36,220 --> 00:16:40,090
There is the influence of
a spring, which depends

321
00:16:40,090 --> 00:16:43,090
on where spring number two is.

322
00:16:43,090 --> 00:16:49,540
And, plus, there is this new
driven term, F0 cosine omega

323
00:16:49,540 --> 00:16:53,870
d times t, where omega
d is fixed, arbitrary,

324
00:16:53,870 --> 00:16:54,730
externally given.

325
00:16:54,730 --> 00:17:00,130
So both F0 and omega d are
decided by somebody outside

326
00:17:00,130 --> 00:17:01,630
of the system.

327
00:17:01,630 --> 00:17:08,138
Now, the second mass
M X2 dot dot, is--

328
00:17:08,138 --> 00:17:14,079
actually has feels position
of x1, through the spring.

329
00:17:14,079 --> 00:17:21,730
And there is this-- its own
pendulum effect plus a string,

330
00:17:21,730 --> 00:17:23,560
depending on position x2.

331
00:17:23,560 --> 00:17:25,720
Interestingly, there
is no force here,

332
00:17:25,720 --> 00:17:29,360
because the force is
applied to mass one.

333
00:17:29,360 --> 00:17:32,290
So mass two a priori doesn't
know anything about the force.

334
00:17:32,290 --> 00:17:35,110
But of course it will
know through the coupling.

335
00:17:35,110 --> 00:17:35,610
Yes?

336
00:17:35,610 --> 00:17:37,882
Questions?

337
00:17:37,882 --> 00:17:40,350
Anybody have questions so far?

338
00:17:40,350 --> 00:17:42,990
So it's the same as
before, with the addition

339
00:17:42,990 --> 00:17:45,430
of this external force.

340
00:17:45,430 --> 00:17:51,820
Again, this is writing all
coordinates one by one.

341
00:17:51,820 --> 00:17:54,940
We immediately switch
to matrix form.

342
00:17:54,940 --> 00:17:59,170
We write it MX double
dot, where X is the same

343
00:17:59,170 --> 00:18:02,185
as we defined before, minus KX.

344
00:18:05,210 --> 00:18:09,590
I think I will stop writing
these kind of thick lines.

345
00:18:09,590 --> 00:18:12,160
But for now, let me--

346
00:18:12,160 --> 00:18:17,350
F cosine omega d times t.

347
00:18:17,350 --> 00:18:21,370
So this is now a matrix
equation for the vector XD.

348
00:18:21,370 --> 00:18:24,000
And let's remind ourselves
what those matrices are.

349
00:18:24,000 --> 00:18:30,120
Matrix M is M 0
0 M. This is just

350
00:18:30,120 --> 00:18:32,740
mass of the individual systems.

351
00:18:32,740 --> 00:18:38,650
We use M minus 1, which
is 1 over M, 1 over M,

352
00:18:38,650 --> 00:18:41,830
and diagonal 0 and 0.

353
00:18:41,830 --> 00:18:43,720
So this carries
information about masses,

354
00:18:43,720 --> 00:18:45,670
inertia of the system.

355
00:18:45,670 --> 00:18:50,680
Matrix K contains information
about all the springs

356
00:18:50,680 --> 00:18:53,860
in the system, and
some pendula effects.

357
00:18:53,860 --> 00:19:01,480
So we have a k plus
mg over l, minus k,

358
00:19:01,480 --> 00:19:09,220
minus k, k plus mg over l.

359
00:19:09,220 --> 00:19:11,500
And now there is
this new thing, which

360
00:19:11,500 --> 00:19:24,540
is this vector F. Vector F
is equal to F0 0 cosine omega

361
00:19:24,540 --> 00:19:27,110
d times t.

362
00:19:27,110 --> 00:19:30,290
So this is in a vector
form, this external force,

363
00:19:30,290 --> 00:19:33,900
which is applied only
to mass number one.

364
00:19:33,900 --> 00:19:34,400
OK?

365
00:19:37,990 --> 00:19:40,570
So these are the elements
which are plugged in.

366
00:19:40,570 --> 00:19:43,060
So now the question is, what
do you want to do with this?

367
00:19:43,060 --> 00:19:45,720
So we have the
equation of motion.

368
00:19:45,720 --> 00:19:48,690
And so what do we do with this?

369
00:19:48,690 --> 00:19:56,320
So there are two steps
that we have to do.

370
00:19:56,320 --> 00:20:00,090
Number one, we have
to remind ourselves

371
00:20:00,090 --> 00:20:04,200
what are the normal modes
of the system, in case--

372
00:20:04,200 --> 00:20:05,280
because we will need--

373
00:20:05,280 --> 00:20:10,792
the information about normal
modes will come in as--

374
00:20:10,792 --> 00:20:15,120
into solutions for
a driven motion.

375
00:20:15,120 --> 00:20:18,030
So let's remind
ourselves what this was.

376
00:20:18,030 --> 00:20:19,282
Well, this was a solution.

377
00:20:19,282 --> 00:20:20,865
I'll just rewrite
it very quickly such

378
00:20:20,865 --> 00:20:22,156
that we have it for the record.

379
00:20:24,535 --> 00:20:26,200
It should fit here.

380
00:20:26,200 --> 00:20:27,390
Now let's try.

381
00:20:27,390 --> 00:20:29,280
So there were two solutions.

382
00:20:29,280 --> 00:20:33,510
There was omega 1 squared,
which was equal to g over l.

383
00:20:33,510 --> 00:20:39,320
And the corresponding normal
mode was a symmetric one.

384
00:20:39,320 --> 00:20:41,570
It was 1, 1.

385
00:20:41,570 --> 00:20:42,070
OK.

386
00:20:42,070 --> 00:20:46,360
So this was one
type of solution,

387
00:20:46,360 --> 00:20:48,950
where the two masses
were moving together.

388
00:20:48,950 --> 00:20:54,650
There was a second frequency
which was equal to g over l.

389
00:20:54,650 --> 00:20:59,410
The square of it was
equal plus 2k over m.

390
00:20:59,410 --> 00:21:01,960
And this was the
characteristic normal frequency

391
00:21:01,960 --> 00:21:05,230
for the second type
of oscillation, which

392
00:21:05,230 --> 00:21:08,290
you can write it 1, minus 1.

393
00:21:08,290 --> 00:21:12,580
And the criterion for when we
were looking for solutions,

394
00:21:12,580 --> 00:21:17,110
we would find them by
calculating the determinant

395
00:21:17,110 --> 00:21:19,230
of this two by two matrix.

396
00:21:19,230 --> 00:21:29,360
It was the determinant of m
minus 1 k minus omega squared

397
00:21:29,360 --> 00:21:33,650
times unit matrix
was equal to 0.

398
00:21:33,650 --> 00:21:37,590
So this was the
equation that had

399
00:21:37,590 --> 00:21:41,730
to be satisfied for frequencies
corresponding to normal modes

400
00:21:41,730 --> 00:21:44,610
with zero external force.

401
00:21:44,610 --> 00:21:46,800
Interestingly, if you
do the calculations,

402
00:21:46,800 --> 00:21:48,110
it turns out you can--

403
00:21:48,110 --> 00:21:51,110
algebraically, you can write--

404
00:21:51,110 --> 00:21:53,020
after you know the
solution itself,

405
00:21:53,020 --> 00:21:56,420
you can write it in
a very compact way.

406
00:21:56,420 --> 00:21:59,910
So this determinant can be
written in the following way--

407
00:21:59,910 --> 00:22:03,790
omega squared minus
omega 1 squared,

408
00:22:03,790 --> 00:22:09,830
times omega squared
minus omega 2 squared.

409
00:22:09,830 --> 00:22:13,620
And this is-- the
condition was zero.

410
00:22:13,620 --> 00:22:20,840
And you see explicitly that this
is a fourth order in frequency

411
00:22:20,840 --> 00:22:23,270
equation, fourth order
frequency, which is

412
00:22:23,270 --> 00:22:27,890
0 for omega 1 and for omega 2.

413
00:22:27,890 --> 00:22:29,260
In a very explicit way.

414
00:22:29,260 --> 00:22:35,360
So this is a nice,
compact form of writing

415
00:22:35,360 --> 00:22:38,005
this particular
eigenvalue equation.

416
00:22:41,120 --> 00:22:46,420
And again, as a reminder,
the motion of the system--

417
00:22:46,420 --> 00:22:49,930
the most general motion of the
system with no external force

418
00:22:49,930 --> 00:22:54,790
was a superposition of
those two oscillations,

419
00:22:54,790 --> 00:22:57,320
which we can write as
some sort of amplitude--

420
00:22:57,320 --> 00:23:04,990
1, 1 cosine omega
1 t plus phi 1,

421
00:23:04,990 --> 00:23:14,690
plus beta 1, minus 1 cosine
omega 2 t plus phi 2.

422
00:23:14,690 --> 00:23:20,070
So this is oscillations of
two different frequencies.

423
00:23:20,070 --> 00:23:21,990
This is the shape
of oscillations,

424
00:23:21,990 --> 00:23:25,660
the relative amplitude
of one versus the other.

425
00:23:25,660 --> 00:23:28,680
And then there's the overall
amplitude alpha and beta,

426
00:23:28,680 --> 00:23:30,260
which has to be determined.

427
00:23:30,260 --> 00:23:32,880
And then there are
arbitrary phases.

428
00:23:32,880 --> 00:23:38,900
So there are in
fact four numbers,

429
00:23:38,900 --> 00:23:43,270
which can be determined from
four initial conditions.

430
00:23:43,270 --> 00:23:45,980
So typically two positions
for the two masses,

431
00:23:45,980 --> 00:23:48,230
and two initial
velocities for two masses.

432
00:23:48,230 --> 00:23:51,360
So everything matches.

433
00:23:51,360 --> 00:23:53,420
So this a so-called
homogeneous equation.

434
00:23:59,246 --> 00:24:00,320
Homogeneous solution.

435
00:24:06,350 --> 00:24:08,280
What about driven solution?

436
00:24:08,280 --> 00:24:14,310
Driven solution, as we remember
from a single oscillator,

437
00:24:14,310 --> 00:24:20,030
results in a motion in which
all the elements in the system

438
00:24:20,030 --> 00:24:22,590
are oscillating at
the same frequency,

439
00:24:22,590 --> 00:24:24,930
and that's the driven frequency.

440
00:24:24,930 --> 00:24:25,500
It's a fact.

441
00:24:25,500 --> 00:24:28,950
I come in, I apply
100 Hertz frequency,

442
00:24:28,950 --> 00:24:30,960
and everybody oscillates
on the 100 frequency.

443
00:24:30,960 --> 00:24:35,110
That's the solution for a
driven oscillating system.

444
00:24:35,110 --> 00:24:37,230
And we saw it for a
one-dimensional oscillator,

445
00:24:37,230 --> 00:24:38,563
and we will see it here as well.

446
00:24:38,563 --> 00:24:40,640
There's one frequency, omega d.

447
00:24:40,640 --> 00:24:44,310
So we will be now looking for
a solution which corresponds

448
00:24:44,310 --> 00:24:48,210
to the oscillation of the system
with this external frequency,

449
00:24:48,210 --> 00:24:50,700
which a priori is
not the same as one

450
00:24:50,700 --> 00:24:53,000
of the normal frequencies.

451
00:24:53,000 --> 00:24:56,600
So the complete motion of the
system consists of two parts.

452
00:24:56,600 --> 00:25:01,150
One is this homogeneous
self-oscillating motion

453
00:25:01,150 --> 00:25:03,330
with two characteristic
frequencies.

454
00:25:03,330 --> 00:25:06,050
And there will be a
second type of motion,

455
00:25:06,050 --> 00:25:08,130
which is a driven one.

456
00:25:08,130 --> 00:25:12,650
So how do we go
about solving that?

457
00:25:12,650 --> 00:25:16,200
So equations of motions of
course will be the same.

458
00:25:16,200 --> 00:25:21,810
The solution, the way that we
solve it will be very similar.

459
00:25:21,810 --> 00:25:23,840
So lets try-- start working.

460
00:25:23,840 --> 00:25:29,790
Maybe we can work on
those blackboards here.

461
00:25:29,790 --> 00:25:31,350
So what is going on?

462
00:25:34,830 --> 00:25:43,100
So we know that if we apply
external frequency omega d,

463
00:25:43,100 --> 00:25:49,250
everybody in the system, all
the elements will be oscillating

464
00:25:49,250 --> 00:25:51,700
with the same frequency.

465
00:25:55,130 --> 00:26:01,420
So we can then
introduce a variable Z,

466
00:26:01,420 --> 00:26:08,130
which will be defined
B e to the i omega d t.

467
00:26:08,130 --> 00:26:09,980
This will be the
oscillating term.

468
00:26:09,980 --> 00:26:14,120
And this will be the amplitude
of oscillation, which we'll try

469
00:26:14,120 --> 00:26:16,550
to make real for simplicity.

470
00:26:16,550 --> 00:26:19,640
And then we plug this
into the equation

471
00:26:19,640 --> 00:26:23,090
of motion, which is listed
up there on the screen.

472
00:26:23,090 --> 00:26:29,990
So the equation of motion
is Z dot dot plus M minus 1

473
00:26:29,990 --> 00:26:43,400
K times Z is equal to now M
minus 1 force e to i omega d t.

474
00:26:43,400 --> 00:26:48,990
You see our external
force is F cosine

475
00:26:48,990 --> 00:26:52,920
omega d t, with a vector 1, 0.

476
00:26:52,920 --> 00:26:57,310
But of course, in the complex
notation, this is exponent.

477
00:26:57,310 --> 00:27:00,280
So this is the challenge,
what we would like to have.

478
00:27:00,280 --> 00:27:04,460
And we assume that all the
elements in the system--

479
00:27:04,460 --> 00:27:06,700
position,
acceleration-- oscillate

480
00:27:06,700 --> 00:27:09,610
at the same frequency omega d.

481
00:27:09,610 --> 00:27:15,130
If you do that,
then the equations

482
00:27:15,130 --> 00:27:19,780
become somewhat simpler, because
the oscillating term drops out.

483
00:27:19,780 --> 00:27:23,620
So when you plug this type
of solution into here,

484
00:27:23,620 --> 00:27:27,310
what you get is minus
omega d squared--

485
00:27:27,310 --> 00:27:30,660
that's from second
differentiation with respect

486
00:27:30,660 --> 00:27:32,170
to time--

487
00:27:32,170 --> 00:27:45,010
plus M minus 1 K, multiplying
vector B e to i omega d t.

488
00:27:45,010 --> 00:27:54,570
This must be equal to M
minus one F e to i omega d t.

489
00:27:54,570 --> 00:27:57,540
This is vector B,
this is vector F.

490
00:27:57,540 --> 00:28:00,000
And there is this
oscillating term.

491
00:28:00,000 --> 00:28:02,670
But both sides oscillate
at the same frequency.

492
00:28:02,670 --> 00:28:03,750
That's what we assume.

493
00:28:03,750 --> 00:28:09,340
So we can simply
divide by this, and we

494
00:28:09,340 --> 00:28:14,590
are left with an
equation that equates

495
00:28:14,590 --> 00:28:16,360
what's going on
in the oscillating

496
00:28:16,360 --> 00:28:20,440
system with the external force.

497
00:28:20,440 --> 00:28:24,820
So now, let's see
here what is known

498
00:28:24,820 --> 00:28:30,490
and what is unknown
in this equation.

499
00:28:30,490 --> 00:28:35,140
M minus 1 K carries information
about the construction built

500
00:28:35,140 --> 00:28:37,360
of the system of accelerators.

501
00:28:37,360 --> 00:28:42,280
Strength of springs, masses,
gravitational field, et cetera.

502
00:28:42,280 --> 00:28:43,150
So this is fixed.

503
00:28:43,150 --> 00:28:45,370
This is given.

504
00:28:45,370 --> 00:28:49,490
Omega d is the external driving
frequency, and it's also given.

505
00:28:49,490 --> 00:28:51,090
It's a number.

506
00:28:51,090 --> 00:28:52,800
I said this is externally given.

507
00:28:52,800 --> 00:28:55,020
I just set it at some computer.

508
00:28:55,020 --> 00:28:56,940
Say 100 Hertz, and it's
driven at 100 Hertz.

509
00:28:56,940 --> 00:28:57,690
So we know that.

510
00:28:57,690 --> 00:29:00,660
We know exactly
what this number is.

511
00:29:00,660 --> 00:29:03,270
External force, we
know what it is.

512
00:29:03,270 --> 00:29:03,970
We defined it.

513
00:29:03,970 --> 00:29:05,040
It's F0.

514
00:29:05,040 --> 00:29:07,090
We know what its magnitude--

515
00:29:07,090 --> 00:29:15,570
so everything is known except
for vector B. And vector B are

516
00:29:15,570 --> 00:29:18,300
the amplitudes of oscillation--

517
00:29:18,300 --> 00:29:21,790
remember, everything
oscillates at omega d--

518
00:29:21,790 --> 00:29:25,650
of mass one and mass two.

519
00:29:25,650 --> 00:29:29,480
So in general, if I
apply external force,

520
00:29:29,480 --> 00:29:33,230
this guy will oscillate
with some amplitude.

521
00:29:33,230 --> 00:29:36,230
That guy with some amplitude,
a priori different.

522
00:29:36,230 --> 00:29:39,630
And this will be
B1, this will be B2.

523
00:29:39,630 --> 00:29:41,460
And we don't know
that at this stage.

524
00:29:41,460 --> 00:29:44,820
So this equation will
allow us to find it.

525
00:29:47,420 --> 00:29:50,120
And it is possible because--

526
00:29:50,120 --> 00:29:53,960
this is actually a very
straightforward equation.

527
00:29:53,960 --> 00:30:04,950
It contains-- actually, to
be very precise, I have to--

528
00:30:04,950 --> 00:30:06,720
this is a number,
this is a matrix.

529
00:30:06,720 --> 00:30:11,200
So I have to put a
unit matrix right here.

530
00:30:11,200 --> 00:30:13,700
So it's omega d
times unit matrix

531
00:30:13,700 --> 00:30:20,400
plus this matrix that carries
information about the system.

532
00:30:20,400 --> 00:30:23,480
And so we can write
this down again

533
00:30:23,480 --> 00:30:26,820
in some sort of more open
way, for our specific case.

534
00:30:26,820 --> 00:30:32,390
So this will be k
over m plus g over l,

535
00:30:32,390 --> 00:30:42,360
minus omega d squared, minus
k over m, minus k over m,

536
00:30:42,360 --> 00:30:50,770
k over m, plus g over l,
minus omega d squared.

537
00:30:50,770 --> 00:30:53,770
So this is this matrix here.

538
00:30:53,770 --> 00:30:57,820
This matrix is applied to
vector B, which is our unknown.

539
00:30:57,820 --> 00:31:01,175
Let's call it B1 and B2.

540
00:31:01,175 --> 00:31:04,600
These are the amplitudes
of oscillations

541
00:31:04,600 --> 00:31:07,900
of individual elements
in our system.

542
00:31:07,900 --> 00:31:11,465
And this is equal to m--

543
00:31:11,465 --> 00:31:16,720
the inverted mass matrix
times vector F, which--

544
00:31:16,720 --> 00:31:18,760
without its
oscillating part, which

545
00:31:18,760 --> 00:31:22,830
is simply F0 over m and 0.

546
00:31:25,080 --> 00:31:25,580
All right.

547
00:31:25,580 --> 00:31:28,820
So this is the task
in question, and we

548
00:31:28,820 --> 00:31:33,740
have to find out those
two values depending

549
00:31:33,740 --> 00:31:39,490
on these parameters and the
strength of force, et cetera.

550
00:31:39,490 --> 00:31:41,240
So this is actually
not a big deal.

551
00:31:41,240 --> 00:31:45,422
It's a two by two equation, two
equations with two unknowns.

552
00:31:45,422 --> 00:31:46,630
We solve it, and we are done.

553
00:31:49,720 --> 00:31:55,390
However, we want to
learn a little bit

554
00:31:55,390 --> 00:32:00,980
about slightly more general
ways of calculating things.

555
00:32:00,980 --> 00:32:05,851
So let's call this one matrix
E, with some funny double vector

556
00:32:05,851 --> 00:32:06,350
sign.

557
00:32:06,350 --> 00:32:09,890
Let's call this one vector B,
and let's call this one vector

558
00:32:09,890 --> 00:32:13,620
D, because we will use this--

559
00:32:13,620 --> 00:32:15,000
use it later.

560
00:32:15,000 --> 00:32:17,970
And what we are
trying to do is, we

561
00:32:17,970 --> 00:32:27,510
are trying to use the
so-called Cramer's rule to find

562
00:32:27,510 --> 00:32:29,580
those coefficients B1 and B2.

563
00:32:29,580 --> 00:32:32,170
And for some historical
reasons, 8.03 really

564
00:32:32,170 --> 00:32:33,840
likes Cramer's rule.

565
00:32:33,840 --> 00:32:36,390
I like MATLAB or Mathematica.

566
00:32:36,390 --> 00:32:41,340
I just plug things in, and it
crunches out and calculates.

567
00:32:41,340 --> 00:32:43,800
But it turns out
that for two by two,

568
00:32:43,800 --> 00:32:45,570
you can always do it quickly.

569
00:32:45,570 --> 00:32:48,500
Even for three by three, if
you just sit down and do it,

570
00:32:48,500 --> 00:32:49,740
you can actually work it out.

571
00:32:49,740 --> 00:32:50,920
It's not scary.

572
00:32:50,920 --> 00:32:53,640
By five by five--

573
00:32:53,640 --> 00:32:57,180
but even four by four, I'm sure
you are mighty students who

574
00:32:57,180 --> 00:33:01,040
can just do it in the exam.

575
00:33:01,040 --> 00:33:04,830
I have never seen an 8.03
exam with four masses,

576
00:33:04,830 --> 00:33:06,880
unless they're
general questions.

577
00:33:06,880 --> 00:33:08,790
But three-- well...

578
00:33:08,790 --> 00:33:10,140
All right.

579
00:33:10,140 --> 00:33:13,620
So do we go about
finding this B1 and B2?

580
00:33:13,620 --> 00:33:17,485
Because, again, this is a
simple two by two question.

581
00:33:23,600 --> 00:33:27,630
So maybe just to again
bring it even closer

582
00:33:27,630 --> 00:33:29,730
to what we are used
to, let me just quickly

583
00:33:29,730 --> 00:33:32,740
write this down as a set
of two by two equations.

584
00:33:32,740 --> 00:33:39,300
So there is a coefficient here,
k over m plus g over l minus

585
00:33:39,300 --> 00:33:42,330
omega d squared, which
is-- this is a number,

586
00:33:42,330 --> 00:33:53,190
times B1 minus k over m times
B2 is equal to F0 over m minus k

587
00:33:53,190 --> 00:34:03,720
over m B1 plus k over m plus g
over l minus omega d squared is

588
00:34:03,720 --> 00:34:05,190
equal to 0--

589
00:34:05,190 --> 00:34:08,280
times B2 is equal to 0.

590
00:34:08,280 --> 00:34:11,310
So you see two equations
with two unknowns.

591
00:34:11,310 --> 00:34:13,699
Couple of coefficients,
all fixed.

592
00:34:13,699 --> 00:34:16,489
You can eliminate variables.

593
00:34:16,489 --> 00:34:19,010
You can calculate B2
from here, plug it into--

594
00:34:19,010 --> 00:34:21,600
you can work it
out if you want to.

595
00:34:21,600 --> 00:34:24,740
However, there is,
again, a better way.

596
00:34:24,740 --> 00:34:30,170
It's Cramer's rule or method.

597
00:34:34,650 --> 00:34:36,880
Should have known if
it's method or rule.

598
00:34:36,880 --> 00:34:37,409
Rule.

599
00:34:37,409 --> 00:34:38,560
Right.

600
00:34:38,560 --> 00:34:41,090
And so the way you do
it is the following.

601
00:34:41,090 --> 00:34:47,320
So you look at those questions--
you calculate all kinds

602
00:34:47,320 --> 00:34:52,840
of determinants, and by taking
the set of two equations

603
00:34:52,840 --> 00:34:54,190
and plugging into--

604
00:34:54,190 --> 00:34:56,890
replacing columns in the matrix.

605
00:34:56,890 --> 00:35:03,700
So B1, what you do is you take
the original matrix, which

606
00:35:03,700 --> 00:35:08,810
is here, and you replace the
first column of the matrix

607
00:35:08,810 --> 00:35:12,400
with vector B. So you--

608
00:35:12,400 --> 00:35:16,730
no wait, with-- sorry, with
vector D. Take this matrix,

609
00:35:16,730 --> 00:35:18,160
and you plug in this.

610
00:35:18,160 --> 00:35:19,180
So what you do is--

611
00:35:19,180 --> 00:35:19,990
so it turns out--

612
00:35:22,620 --> 00:35:25,050
so B1 can be
explicitly calculated,

613
00:35:25,050 --> 00:35:29,430
but taking the determinant
of the first column replaced,

614
00:35:29,430 --> 00:35:35,310
F0 over M0, and keeping the
second column, which is minus

615
00:35:35,310 --> 00:35:38,485
k over m.

616
00:35:38,485 --> 00:35:47,910
m and then k over m plus g
over l minus omega d squared.

617
00:35:47,910 --> 00:35:51,680
So this is-- you
calculate the determinant

618
00:35:51,680 --> 00:35:54,601
of this thing, where-- original
matrix with the first column

619
00:35:54,601 --> 00:35:55,100
replaced.

620
00:35:55,100 --> 00:35:58,940
And you divide it
by the determinant

621
00:35:58,940 --> 00:36:00,090
of the original matrix.

622
00:36:00,090 --> 00:36:05,310
Let's call it E. So you
calculate this determinant

623
00:36:05,310 --> 00:36:08,850
again for the frequency omega d.

624
00:36:08,850 --> 00:36:12,630
So this can be written very
nicely, in a very compact way.

625
00:36:12,630 --> 00:36:13,830
This determinant is easy.

626
00:36:13,830 --> 00:36:15,720
It's just this times that.

627
00:36:15,720 --> 00:36:25,500
So have 0 over m multiplying
k over n plus g over l

628
00:36:25,500 --> 00:36:28,950
minus on I got the squared
remember this is a given

629
00:36:28,950 --> 00:36:33,720
number divided by n Here
comes this nice compact form

630
00:36:33,720 --> 00:36:38,360
for the determinant, which is
omega d squared minus omega 1

631
00:36:38,360 --> 00:36:45,492
squared, times omega d
squared minus omega 2 squared,

632
00:36:45,492 --> 00:36:54,350
where omega 1 and omega 2 were
the normal mode frequencies.

633
00:36:54,350 --> 00:36:54,850
Yes?

634
00:36:54,850 --> 00:36:57,516
AUDIENCE: Where are you getting
the minus k in the [INAUDIBLE]??

635
00:37:01,022 --> 00:37:03,225
BOLESLAW WYSLOUCH: This one?

636
00:37:03,225 --> 00:37:04,215
AUDIENCE: Yeah.

637
00:37:04,215 --> 00:37:04,734
[INAUDIBLE]

638
00:37:04,734 --> 00:37:05,900
BOLESLAW WYSLOUCH: This one?

639
00:37:05,900 --> 00:37:07,530
This is the second column.

640
00:37:07,530 --> 00:37:09,141
See?

641
00:37:09,141 --> 00:37:16,020
I'm taking-- so this is the
first column, second column.

642
00:37:16,020 --> 00:37:18,870
I take the first
column, I replace it

643
00:37:18,870 --> 00:37:22,080
with driven equation--
with a solution.

644
00:37:22,080 --> 00:37:23,430
I plug it here.

645
00:37:23,430 --> 00:37:24,820
So I have F0 for M0.

646
00:37:27,350 --> 00:37:29,570
And I keep the second column.

647
00:37:29,570 --> 00:37:30,070
All right?

648
00:37:30,070 --> 00:37:31,500
That's for the
first coefficient.

649
00:37:31,500 --> 00:37:33,250
For the second coefficient
what you do is,

650
00:37:33,250 --> 00:37:38,276
you put a driving term here
and you keep the first column.

651
00:37:38,276 --> 00:37:40,230
All right?

652
00:37:40,230 --> 00:37:43,290
So this is actually an
explicit solution for B1.

653
00:37:43,290 --> 00:37:48,350
This is magnitude
of oscillations

654
00:37:48,350 --> 00:37:52,020
of the first element.

655
00:37:52,020 --> 00:37:53,610
And you can do the
same thing for B2.

656
00:38:02,020 --> 00:38:04,080
And I'm not trying
to prove anything,

657
00:38:04,080 --> 00:38:06,381
I'm not trying to
derive anything.

658
00:38:06,381 --> 00:38:07,130
I'm just using it.

659
00:38:07,130 --> 00:38:09,730
And I'll show you a nice
slide with this to summarize.

660
00:38:09,730 --> 00:38:16,810
So B2 is the determinant of--

661
00:38:16,810 --> 00:38:19,240
I keep the first column.

662
00:38:19,240 --> 00:38:24,790
It's k over m plus g
over l, minus omega d

663
00:38:24,790 --> 00:38:28,590
squared, minus k over m.

664
00:38:28,590 --> 00:38:29,690
That's the first column.

665
00:38:29,690 --> 00:38:36,340
And I'm plugging in F0
over M here, and 0 here.

666
00:38:36,340 --> 00:38:42,520
So this is-- and divided by
omega d squared minus omega 1

667
00:38:42,520 --> 00:38:47,940
squared times omega d squared
minus omega 2 squared.

668
00:38:47,940 --> 00:38:51,100
That's the determinant
of the original matrix.

669
00:38:51,100 --> 00:38:56,650
And this one is also very
simple It's this time this is 0.

670
00:38:56,650 --> 00:38:57,550
I have minus that.

671
00:38:57,550 --> 00:39:04,360
So I simply have F0 k
over m squared divided

672
00:39:04,360 --> 00:39:11,320
by omega d squared minus
omega 1 squared, omega d

673
00:39:11,320 --> 00:39:15,475
squared minus omega 2 squared.

674
00:39:15,475 --> 00:39:17,070
All right.

675
00:39:17,070 --> 00:39:19,170
So we have those
things, and also what?

676
00:39:19,170 --> 00:39:21,950
Do you see anything
happening here?

677
00:39:21,950 --> 00:39:25,780
Yeah, there are some numbers,
but what do they mean?

678
00:39:25,780 --> 00:39:26,530
What does it mean?

679
00:39:26,530 --> 00:39:28,545
Yes, we can calculate it.

680
00:39:28,545 --> 00:39:29,270
You can trust me.

681
00:39:29,270 --> 00:39:30,392
These are the--

682
00:39:30,392 --> 00:39:31,850
I'm not sure that
you can trust it,

683
00:39:31,850 --> 00:39:33,960
but most likely these
are good results.

684
00:39:33,960 --> 00:39:38,270
And so we know the oscillation
of the first mass, oscillation

685
00:39:38,270 --> 00:39:43,580
of the second mass as they are
driven by the external force.

686
00:39:43,580 --> 00:39:47,640
Now, one of the
interesting things to do

687
00:39:47,640 --> 00:39:49,530
is to try to see
what's going on.

688
00:39:49,530 --> 00:39:53,460
One of the-- when we
talked about normal modes,

689
00:39:53,460 --> 00:39:56,820
the ratio of amplitudes
carried information.

690
00:39:56,820 --> 00:39:58,865
Remember, we had those
two different modes.

691
00:39:58,865 --> 00:40:02,180
Either amplitudes were the same,
or they were opposite sign.

692
00:40:02,180 --> 00:40:05,860
So let's ask ourselves, what
is the ratio of B1 and B2?

693
00:40:05,860 --> 00:40:07,490
So let's just divide
one by the other.

694
00:40:11,800 --> 00:40:14,790
So let's do B1 over B2.

695
00:40:14,790 --> 00:40:17,550
Let's see if we learn
anything from this.

696
00:40:17,550 --> 00:40:23,370
If you divide B1 over B2,
this bottom cancels out,

697
00:40:23,370 --> 00:40:31,650
and I have k over m plus
g over l minus omega

698
00:40:31,650 --> 00:40:36,910
d squared over k over m.

699
00:40:40,190 --> 00:40:42,470
And-- yeah.

700
00:40:42,470 --> 00:40:47,270
So now comes the
interesting question.

701
00:40:47,270 --> 00:40:51,495
This omega d can be anything.

702
00:40:54,340 --> 00:41:01,490
So let's say omega d is-- so we
can analyze it different ways.

703
00:41:01,490 --> 00:41:03,911
So for example,
when omega d is--

704
00:41:03,911 --> 00:41:06,160
you can look at small, large,
and so I can compare it.

705
00:41:06,160 --> 00:41:08,150
But one of the
interesting places to look

706
00:41:08,150 --> 00:41:13,060
is, what happens when omega
is very close to one of the--

707
00:41:13,060 --> 00:41:16,360
to the characteristic
frequencies?

708
00:41:16,360 --> 00:41:19,600
Because, remember, when we
analyzed a single driven

709
00:41:19,600 --> 00:41:21,880
oscillator, the
real cool stuff was

710
00:41:21,880 --> 00:41:25,430
happening when you are near
the resonant frequency.

711
00:41:25,430 --> 00:41:28,450
Things, you know, the bridges
broke down, et cetera.

712
00:41:28,450 --> 00:41:31,120
So let's see if we can do
something similar here.

713
00:41:31,120 --> 00:41:32,260
Now we have two choices.

714
00:41:32,260 --> 00:41:34,700
We have omega 1, omega 2.

715
00:41:34,700 --> 00:41:40,230
So let's see what happens
if I plug in omega 1.

716
00:41:40,230 --> 00:41:44,550
Omega d being very,
very close to omega 1.

717
00:41:44,550 --> 00:41:46,350
Let's say equal to omega 1.

718
00:41:46,350 --> 00:41:54,510
Omega 1 is-- omega 1
squared was g over l.

719
00:41:54,510 --> 00:41:59,840
So if I plug omega 1 here, I
have k over m plus g over l.

720
00:41:59,840 --> 00:42:07,410
So I have k over m plus
g over l, minus g over l,

721
00:42:07,410 --> 00:42:13,120
divide by k over m,
which is equal to what?

722
00:42:13,120 --> 00:42:14,460
Those two terms cancels.

723
00:42:14,460 --> 00:42:17,110
k over m, it's plus 1.

724
00:42:17,110 --> 00:42:17,990
That's interesting.

725
00:42:17,990 --> 00:42:24,000
So if I drive at a frequency
which corresponds to omega 1--

726
00:42:24,000 --> 00:42:28,440
and omega 1 was the
oscillation where both masses

727
00:42:28,440 --> 00:42:31,150
were going together.

728
00:42:31,150 --> 00:42:33,270
So the characteristic
normal mode

729
00:42:33,270 --> 00:42:36,520
had the ratio of two
masses equal to one.

730
00:42:36,520 --> 00:42:41,580
And here I'm getting the system
to drive at this type of mode.

731
00:42:41,580 --> 00:42:44,880
Again, I have-- the
driven amplitudes

732
00:42:44,880 --> 00:42:47,390
are the ratio is equal to one.

733
00:42:52,150 --> 00:42:58,630
So what happens if I drive
at omega d close to omega 2?

734
00:42:58,630 --> 00:43:09,690
Omega 2 squared was equal
to g over l plus 2k over m.

735
00:43:09,690 --> 00:43:16,274
If I plug it in here, I get
that the ratio is minus 1.

736
00:43:16,274 --> 00:43:20,470
Again, the ratio is strikingly
similar to the ratio

737
00:43:20,470 --> 00:43:25,130
of the normal mode corresponding
to frequency omega 2.

738
00:43:25,130 --> 00:43:28,240
So it's like I'm inducing
those oscillations.

739
00:43:32,070 --> 00:43:35,380
So what does this all mean?

740
00:43:35,380 --> 00:43:38,040
There's, by the way,
a little catch here

741
00:43:38,040 --> 00:43:40,770
for all of your mathematicians.

742
00:43:40,770 --> 00:43:44,730
What happens to equations if I
set omega d equal to minus 1--

743
00:43:44,730 --> 00:43:48,337
to omega 1, for example?

744
00:43:48,337 --> 00:43:50,170
I just plugged it here,
and nobody screamed.

745
00:43:50,170 --> 00:43:52,250
But there was something
fishy about what I did.

746
00:43:52,250 --> 00:43:52,750
Yes?

747
00:43:52,750 --> 00:43:56,439
AUDIENCE: --coefficient
[INAUDIBLE]

748
00:43:56,439 --> 00:43:57,772
BOLESLAW WYSLOUCH: If you took--

749
00:43:57,772 --> 00:43:58,734
AUDIENCE: Oh, sorry.

750
00:43:58,734 --> 00:44:01,035
[INAUDIBLE]

751
00:44:01,035 --> 00:44:02,160
BOLESLAW WYSLOUCH: Exactly.

752
00:44:02,160 --> 00:44:07,200
So the ratio of the two was one,
but both of them were infinite.

753
00:44:07,200 --> 00:44:09,410
So infinite divided by
infinite equals what?

754
00:44:09,410 --> 00:44:11,280
I mean, this happens.

755
00:44:11,280 --> 00:44:12,080
So what's going on?

756
00:44:12,080 --> 00:44:12,900
Why can I do it?

757
00:44:12,900 --> 00:44:17,204
One-- we should
not really scream.

758
00:44:17,204 --> 00:44:17,971
Damping.

759
00:44:17,971 --> 00:44:18,470
Exactly.

760
00:44:18,470 --> 00:44:20,233
This is where the
damping comes in.

761
00:44:20,233 --> 00:44:22,804
So the amplitude is enormous,
but it's not infinite,

762
00:44:22,804 --> 00:44:24,470
because there's always
a little damping.

763
00:44:24,470 --> 00:44:26,810
The system will
not go to infinity.

764
00:44:26,810 --> 00:44:30,200
So in real life, there's
a little term here

765
00:44:30,200 --> 00:44:32,290
that makes sure things
don't blow up completely.

766
00:44:32,290 --> 00:44:33,539
There's a little damping here.

767
00:44:33,539 --> 00:44:34,052
Yes?

768
00:44:34,052 --> 00:44:35,820
AUDIENCE: Does it
at all matter--

769
00:44:35,820 --> 00:44:37,327
also the fact that
those equations

770
00:44:37,327 --> 00:44:39,701
are inexact in the
first place, because we

771
00:44:39,701 --> 00:44:40,784
had made theta smaller--

772
00:44:40,784 --> 00:44:41,700
BOLESLAW WYSLOUCH: No.

773
00:44:41,700 --> 00:44:43,860
That's not-- no.

774
00:44:43,860 --> 00:44:45,630
This doesn't actually matter.

775
00:44:45,630 --> 00:44:53,000
It's the absence of damping that
makes things look nonphysical.

776
00:44:53,000 --> 00:44:57,320
AUDIENCE: But as the frequency--
as the amplitude increases,

777
00:44:57,320 --> 00:45:00,200
when we're in resonance,
eventually those equations

778
00:45:00,200 --> 00:45:02,120
wouldn't hold any
longer, and perhaps--

779
00:45:02,120 --> 00:45:03,703
BOLESLAW WYSLOUCH:
Yeah, that's right.

780
00:45:03,703 --> 00:45:06,850
But you could-- that's true.

781
00:45:06,850 --> 00:45:08,570
That's true.

782
00:45:08,570 --> 00:45:13,650
But you can come up with, for
example, an electronic system

783
00:45:13,650 --> 00:45:15,870
which has a huge range of--

784
00:45:15,870 --> 00:45:18,150
enormous range of possibilities.

785
00:45:18,150 --> 00:45:21,090
And then-- or of amplitudes.

786
00:45:21,090 --> 00:45:26,950
Many, many-- so the damping is
much more important in that.

787
00:45:26,950 --> 00:45:29,560
So in reality, there is some
damping here and so forth.

788
00:45:29,560 --> 00:45:30,060
All right.

789
00:45:30,060 --> 00:45:32,310
So why don't we do,
now, the following.

790
00:45:32,310 --> 00:45:40,200
So let's try to see
how this all works out.

791
00:45:40,200 --> 00:45:42,360
First of all, such
that we can get

792
00:45:42,360 --> 00:45:45,490
started, I will make
a sketch for you.

793
00:45:45,490 --> 00:45:51,860
I'll calculate these formulas--

794
00:45:51,860 --> 00:45:59,740
just a second-- and display
you as a function of frequency,

795
00:45:59,740 --> 00:46:02,604
such that we can
analyze what's going on.

796
00:46:02,604 --> 00:46:03,270
So where is it--

797
00:46:06,140 --> 00:46:06,705
OK.

798
00:46:06,705 --> 00:46:08,580
It's still slow.

799
00:46:08,580 --> 00:46:11,490
All right.

800
00:46:11,490 --> 00:46:15,610
So this is what those--

801
00:46:15,610 --> 00:46:16,930
OK, so let's say--

802
00:46:16,930 --> 00:46:22,090
I don't know which is which,
but let's say B1 is the red one,

803
00:46:22,090 --> 00:46:23,860
B2 is the blue
one, or vice versa.

804
00:46:23,860 --> 00:46:25,770
It doesn't matter.

805
00:46:25,770 --> 00:46:28,690
These are the numbers which
I plug in for some values

806
00:46:28,690 --> 00:46:30,410
for some system.

807
00:46:30,410 --> 00:46:31,320
So we see that--

808
00:46:31,320 --> 00:46:34,790
and this is as a
function of frequency.

809
00:46:34,790 --> 00:46:38,890
So first of all, you see
a characteristic frequency

810
00:46:38,890 --> 00:46:41,350
around one, characteristic
frequency around three

811
00:46:41,350 --> 00:46:42,940
on my plot.

812
00:46:42,940 --> 00:46:46,520
And in the region in the
vicinity of frequency number

813
00:46:46,520 --> 00:46:50,860
one, you see that
both the blue and red,

814
00:46:50,860 --> 00:46:53,950
the individual amplitudes
are basically close together.

815
00:46:53,950 --> 00:46:56,940
So the ratio is close to one.

816
00:46:56,940 --> 00:46:59,250
If you look at this plot,
you should believe me

817
00:46:59,250 --> 00:47:01,650
that it's plausible
that if you are

818
00:47:01,650 --> 00:47:03,870
very close to the
frequency, basically

819
00:47:03,870 --> 00:47:08,460
the red and blue
will move together.

820
00:47:08,460 --> 00:47:12,960
If you go around the
second frequency,

821
00:47:12,960 --> 00:47:17,360
you see that red goes up,
blue goes down, or vice versa

822
00:47:17,360 --> 00:47:19,260
on the other side.

823
00:47:19,260 --> 00:47:22,170
So the ratio is minus 1.

824
00:47:22,170 --> 00:47:24,920
So this plot actually
carries in formation.

825
00:47:24,920 --> 00:47:27,710
And in fact, what
you see also is

826
00:47:27,710 --> 00:47:32,580
that there is some sort
of resonant behavior.

827
00:47:32,580 --> 00:47:35,000
So the amplitudes
are enormous if you

828
00:47:35,000 --> 00:47:39,660
are close to any of those
characteristic frequencies,

829
00:47:39,660 --> 00:47:42,790
but they're much smaller
if you're further out.

830
00:47:42,790 --> 00:47:47,300
There is some motion, but
not as pronounced as when

831
00:47:47,300 --> 00:47:51,120
you're at the right
driving frequencies.

832
00:47:51,120 --> 00:47:51,620
All right.

833
00:47:51,620 --> 00:47:54,650
So let's try to see it.

834
00:47:54,650 --> 00:47:55,940
Why not?

835
00:47:55,940 --> 00:47:57,530
So let me go to another system--

836
00:47:57,530 --> 00:47:59,870
a system which
consists of two masses,

837
00:47:59,870 --> 00:48:06,500
has the same type of behavior,
slightly different parameters.

838
00:48:06,500 --> 00:48:09,960
There is no g here, but
everything looks the same.

839
00:48:09,960 --> 00:48:12,980
It's just much easier to show.

840
00:48:12,980 --> 00:48:16,320
And I can remove
most of damping.

841
00:48:19,250 --> 00:48:22,430
And you'll see there
are again two modes, one

842
00:48:22,430 --> 00:48:24,890
which is like this--

843
00:48:24,890 --> 00:48:28,160
that's number one,
that slow motion.

844
00:48:28,160 --> 00:48:30,050
They move together.

845
00:48:30,050 --> 00:48:34,160
And the other one, which
is like this, where

846
00:48:34,160 --> 00:48:35,960
the amplitudes are minus 1.

847
00:48:35,960 --> 00:48:40,240
This is the
frequency number two.

848
00:48:40,240 --> 00:48:42,080
So now let's try to drive it.

849
00:48:46,040 --> 00:48:47,030
How do I drive it?

850
00:48:47,030 --> 00:48:52,440
I have some sort of engine here
which is applying frequency.

851
00:48:52,440 --> 00:49:01,530
So let's start with some
sort of slow motion.

852
00:49:08,930 --> 00:49:12,110
So you see they are
moving a little bit.

853
00:49:12,110 --> 00:49:14,800
Very small, minimally.

854
00:49:14,800 --> 00:49:16,910
Just a tiny motion.

855
00:49:16,910 --> 00:49:21,080
But they're kind of together,
more or less, right?

856
00:49:21,080 --> 00:49:22,980
Slowly, but together.

857
00:49:22,980 --> 00:49:24,050
And this is what--

858
00:49:24,050 --> 00:49:27,040
this is this area here.

859
00:49:27,040 --> 00:49:28,970
I don't know if you see that.

860
00:49:28,970 --> 00:49:31,100
This is this area.

861
00:49:31,100 --> 00:49:34,310
I'm driving at a
very slow frequency.

862
00:49:34,310 --> 00:49:36,020
I'm somewhere here.

863
00:49:36,020 --> 00:49:41,120
The two masses kind of go
together, but very slowly.

864
00:49:41,120 --> 00:49:43,130
So let me now crank
up the frequency

865
00:49:43,130 --> 00:49:47,900
and try to be in the
region of oscillation.

866
00:49:53,732 --> 00:49:56,570
So you see?

867
00:49:56,570 --> 00:50:00,050
All I did is I
changed frequency.

868
00:50:00,050 --> 00:50:01,460
The effect is enormous.

869
00:50:01,460 --> 00:50:02,456
I'm somewhere here now.

870
00:50:06,936 --> 00:50:07,436
You see?

871
00:50:10,440 --> 00:50:12,000
Enormous resonance.

872
00:50:12,000 --> 00:50:13,740
And very soon, I
will hit the limit.

873
00:50:13,740 --> 00:50:15,790
The system will break.

874
00:50:15,790 --> 00:50:17,740
OK, so we are somewhere here.

875
00:50:17,740 --> 00:50:20,370
I'm driving it.

876
00:50:20,370 --> 00:50:23,550
Interestingly, this really
looks like a harmonic motion

877
00:50:23,550 --> 00:50:24,600
of first type.

878
00:50:24,600 --> 00:50:27,030
There is no other things.

879
00:50:27,030 --> 00:50:32,290
OK, so now let's swing
by and get to this area.

880
00:50:32,290 --> 00:50:41,559
So all I'm doing is, I quickly
change frequency to- this one.

881
00:50:48,780 --> 00:50:51,090
So now what you
see is that there

882
00:50:51,090 --> 00:50:53,910
were some random
initial conditions, so

883
00:50:53,910 --> 00:50:56,220
we have a homogeneous
equation going,

884
00:50:56,220 --> 00:50:57,900
but the driven is coming in.

885
00:50:57,900 --> 00:51:00,600
All I did is I
changed frequency.

886
00:51:00,600 --> 00:51:06,796
And suddenly the system knows
that it has to go like that.

887
00:51:06,796 --> 00:51:07,792
Isn't that cool?

888
00:51:12,590 --> 00:51:16,320
So this is the region here.

889
00:51:16,320 --> 00:51:21,150
And all I'm doing is I'm
bringing the amplitude up,

890
00:51:21,150 --> 00:51:25,790
because this is close to zero.

891
00:51:25,790 --> 00:51:28,650
And then I'm keeping
the ratios close

892
00:51:28,650 --> 00:51:31,130
to the characteristic modes.

893
00:51:31,130 --> 00:51:40,436
So I think-- to be honest,
this is one of the coolest--

894
00:51:40,436 --> 00:51:42,060
all I'm doing, just
changing frequency.

895
00:51:42,060 --> 00:51:45,060
And the system just
responds and starts

896
00:51:45,060 --> 00:51:48,840
going with a resonance
of one particular mode.

897
00:51:48,840 --> 00:51:52,440
So imagine a system
that has 1,000 masses,

898
00:51:52,440 --> 00:51:54,541
and you come in with
1,000 frequencies.

899
00:51:54,541 --> 00:51:56,790
You tune one frequency, and
suddenly everything starts

900
00:51:56,790 --> 00:51:59,970
oscillating in one go.

901
00:51:59,970 --> 00:52:02,290
And imagine you have
multiple buildings,

902
00:52:02,290 --> 00:52:05,100
each with different frequency,
and there's an earthquake.

903
00:52:05,100 --> 00:52:06,837
And the frequency is
of a certain type,

904
00:52:06,837 --> 00:52:09,420
and one building collapses, and
all the other ones are happily

905
00:52:09,420 --> 00:52:10,000
standing.

906
00:52:10,000 --> 00:52:10,500
Why?

907
00:52:10,500 --> 00:52:14,010
Because the earthquake just
happened to hit the frequency

908
00:52:14,010 --> 00:52:17,400
that corresponded to one
of the normal frequencies

909
00:52:17,400 --> 00:52:19,660
of that particular building.

910
00:52:19,660 --> 00:52:24,045
And it's an extremely
powerful trick.

911
00:52:24,045 --> 00:52:27,480
It fishes out normal modes
through this driving thing.

912
00:52:27,480 --> 00:52:30,450
And we are able to
calculate it explicitly.

913
00:52:30,450 --> 00:52:33,930
So now what I will do is,
I will modify the system

914
00:52:33,930 --> 00:52:38,120
and I will make it into
a three mass thing, which

915
00:52:38,120 --> 00:52:41,340
will have a somewhat
more complicated set

916
00:52:41,340 --> 00:52:44,250
of normal modes.

917
00:52:44,250 --> 00:52:46,800
And then I will show
you that I can in fact

918
00:52:46,800 --> 00:52:48,810
go with three
different frequencies,

919
00:52:48,810 --> 00:52:53,840
and pull out those
even complicated modes.

920
00:52:53,840 --> 00:52:55,030
So this will be it.

921
00:52:55,030 --> 00:52:57,610
So this is a three mass system.

922
00:52:57,610 --> 00:53:02,890
Now before, since we didn't
calculate it, what I will do

923
00:53:02,890 --> 00:53:10,060
is, I'll go to the web and I
will pull out a nice example.

924
00:53:10,060 --> 00:53:13,330
Let me go to my bookmarks.

925
00:53:13,330 --> 00:53:15,860
Normal modes.

926
00:53:15,860 --> 00:53:21,240
So this is a nice
applet from Colorado.

927
00:53:21,240 --> 00:53:23,980
And you can--

928
00:53:23,980 --> 00:53:27,100
I suppose preso ENG will
send you links, et cetera.

929
00:53:27,100 --> 00:53:31,810
You can simulate-- you
can do everything with it.

930
00:53:31,810 --> 00:53:35,900
So it has two masses.

931
00:53:35,900 --> 00:53:40,790
It has different amplitudes,
different normal modes.

932
00:53:40,790 --> 00:53:42,930
And you can see nothing happens.

933
00:53:42,930 --> 00:53:45,230
So I have to give it
some initial condition.

934
00:53:45,230 --> 00:53:49,220
Sorry, I have to
change polarization.

935
00:53:49,220 --> 00:53:50,210
Where is polarization?

936
00:53:50,210 --> 00:53:51,920
Here.

937
00:53:51,920 --> 00:53:54,670
I give it some
initial condition.

938
00:53:54,670 --> 00:53:56,790
So this is basically
what you just saw.

939
00:53:56,790 --> 00:54:02,010
I'm just demonstrating to you
that this applet looks the same

940
00:54:02,010 --> 00:54:03,370
as our track.

941
00:54:03,370 --> 00:54:06,112
So this is you can
see normal modes.

942
00:54:06,112 --> 00:54:07,570
It's a combination
of normal modes.

943
00:54:07,570 --> 00:54:12,792
There's one which is first
frequency, second frequency.

944
00:54:12,792 --> 00:54:16,400
This is first normal mode.

945
00:54:16,400 --> 00:54:18,580
This is second normal mode.

946
00:54:18,580 --> 00:54:21,740
You can very quickly
see what happens.

947
00:54:21,740 --> 00:54:23,270
So this is what
we just looked at.

948
00:54:23,270 --> 00:54:26,310
This is what we calculated,
more or less, and so on.

949
00:54:26,310 --> 00:54:28,370
Now I want to show you
three masses where things

950
00:54:28,370 --> 00:54:29,840
are somewhat more complicated.

951
00:54:29,840 --> 00:54:32,150
In general, three normal modes.

952
00:54:32,150 --> 00:54:35,750
For the three mass elements, the
first normal mode is like that.

953
00:54:35,750 --> 00:54:38,950
All the three masses
move together.

954
00:54:38,950 --> 00:54:43,000
And slightly different--
the ratio of amplitudes

955
00:54:43,000 --> 00:54:45,090
is slightly different.

956
00:54:45,090 --> 00:54:48,300
The second mode of operation
is actually quite interesting.

957
00:54:48,300 --> 00:54:51,330
The central mass is
stationary, and those two

958
00:54:51,330 --> 00:54:56,130
are going forth and
back, like this.

959
00:54:56,130 --> 00:54:59,760
And then I have
a third frequency

960
00:54:59,760 --> 00:55:04,230
where the middle one is
going double the distance,

961
00:55:04,230 --> 00:55:07,080
and the two other
ones are going up.

962
00:55:07,080 --> 00:55:09,270
So this is the
third normal mode.

963
00:55:13,210 --> 00:55:13,710
All right.

964
00:55:13,710 --> 00:55:17,520
So this is the system which
we now have standing here.

965
00:55:17,520 --> 00:55:19,260
Let's quickly see if
it works in reality.

966
00:55:29,410 --> 00:55:32,280
So this is the first--

967
00:55:35,010 --> 00:55:36,930
so this is the first mode.

968
00:55:45,740 --> 00:55:49,230
This is the second one.

969
00:55:49,230 --> 00:55:50,110
All right.

970
00:55:50,110 --> 00:55:51,260
And the third one will be--

971
00:55:58,410 --> 00:56:02,900
Sometimes I do five of them,
and then it's really difficult.

972
00:56:02,900 --> 00:56:03,400
OK.

973
00:56:03,400 --> 00:56:10,690
But-- so we have a computer
model, we have a real model.

974
00:56:10,690 --> 00:56:14,320
Let's now do the calculation
of the frequencies, the ratios,

975
00:56:14,320 --> 00:56:16,765
such that we can
see what happens.

976
00:56:16,765 --> 00:56:20,350
So I'm coming here, I'm
changing mass to three.

977
00:56:20,350 --> 00:56:26,100
I'm running my-- the
terminal calculating thingy.

978
00:56:26,100 --> 00:56:26,855
OK.

979
00:56:26,855 --> 00:56:27,910
It's very slow.

980
00:56:27,910 --> 00:56:29,360
It's busy, busy, busy.

981
00:56:29,360 --> 00:56:32,400
Imagine-- OK.

982
00:56:32,400 --> 00:56:33,490
Spectacularly slow.

983
00:56:33,490 --> 00:56:34,530
Where is it?

984
00:56:34,530 --> 00:56:37,030
I hope it's not--
oh, here it is.

985
00:56:37,030 --> 00:56:40,250
OK, so this is
what's coming out.

986
00:56:40,250 --> 00:56:43,480
So this is the same
calculation as we

987
00:56:43,480 --> 00:56:46,810
did, except for three masses.

988
00:56:46,810 --> 00:56:48,610
So what do we have here?

989
00:56:48,610 --> 00:56:50,110
Where's my pointer?

990
00:56:50,110 --> 00:56:54,100
So we have, again, three
characteristic frequencies,

991
00:56:54,100 --> 00:56:58,130
we have three masses, and
the same type of behavior.

992
00:56:58,130 --> 00:57:00,560
See, if you are far
away from resonance,

993
00:57:00,560 --> 00:57:05,230
if you have very low frequency,
everybody goes together.

994
00:57:05,230 --> 00:57:06,770
I haven't shown
you this one here,

995
00:57:06,770 --> 00:57:08,492
which is also interesting.

996
00:57:08,492 --> 00:57:10,150
I'll show you in a second.

997
00:57:10,150 --> 00:57:12,280
And then-- so
presumably if you are

998
00:57:12,280 --> 00:57:15,550
close to the first frequency,
you see all three of them

999
00:57:15,550 --> 00:57:17,170
go together.

1000
00:57:17,170 --> 00:57:19,840
And this is the first mode.

1001
00:57:19,840 --> 00:57:22,510
So I should see, if I
set the proper frequency,

1002
00:57:22,510 --> 00:57:25,000
the thing should respond
in mode number one.

1003
00:57:25,000 --> 00:57:29,050
This is the one where two of
them go opposite to each other,

1004
00:57:29,050 --> 00:57:31,440
and the red one is stationary.

1005
00:57:31,440 --> 00:57:33,020
It doesn't move.

1006
00:57:33,020 --> 00:57:35,620
And then you have those
things where they're

1007
00:57:35,620 --> 00:57:36,700
kind of more complicated.

1008
00:57:36,700 --> 00:57:39,940
It's difficult to
read them from here.

1009
00:57:39,940 --> 00:57:42,310
And I can do it for
more masses, et cetera.

1010
00:57:42,310 --> 00:57:44,245
So generally it's calculable.

1011
00:57:44,245 --> 00:57:47,000
It can be calculated and can
be actually demonstrated.

1012
00:57:47,000 --> 00:57:49,720
So let's try it.

1013
00:57:49,720 --> 00:57:52,705
So-- 32.

1014
00:57:59,700 --> 00:58:01,730
So there's this magic
frequency number one.

1015
00:58:06,944 --> 00:58:10,060
I'm setting frequency
by turning a knob.

1016
00:58:10,060 --> 00:58:11,100
That's omega d.

1017
00:58:11,100 --> 00:58:14,230
I'm a supervisor
of this operation.

1018
00:58:14,230 --> 00:58:20,530
It stops because of other
reasons, but it will continue.

1019
00:58:20,530 --> 00:58:24,375
Then I go to 56.

1020
00:58:34,360 --> 00:58:37,370
By the way, remember
that every--

1021
00:58:37,370 --> 00:58:39,420
this is the particular solution.

1022
00:58:39,420 --> 00:58:42,280
This is a steady state
distillation with omega d.

1023
00:58:42,280 --> 00:58:44,830
But we also have all those
homogeneous solutions, which

1024
00:58:44,830 --> 00:58:46,840
have to die down with damping.

1025
00:58:46,840 --> 00:58:50,200
Remember, it's a combination
of homogeneous plus particular.

1026
00:58:50,200 --> 00:58:52,420
So the motion is actually
a little bit distorted

1027
00:58:52,420 --> 00:58:56,200
because we have this homogeneous
stuff hanging around.

1028
00:58:56,200 --> 00:58:59,320
But hopefully, if I can start it
with little homogeneous stuff,

1029
00:58:59,320 --> 00:59:01,820
it will be better.

1030
00:59:01,820 --> 00:59:02,320
So you see?

1031
00:59:05,050 --> 00:59:06,160
Pretty cool.

1032
00:59:06,160 --> 00:59:07,466
Almost there.

1033
00:59:07,466 --> 00:59:10,846
It's almost in assembly.

1034
00:59:10,846 --> 00:59:11,870
Then it kind of stops.

1035
00:59:14,636 --> 00:59:15,310
You see?

1036
00:59:15,310 --> 00:59:17,380
I get two of those
going forth and back,

1037
00:59:17,380 --> 00:59:19,790
more or less, and
this one going.

1038
00:59:19,790 --> 00:59:21,700
I could probably
tune the frequency

1039
00:59:21,700 --> 00:59:23,890
a little bit higher or lower.

1040
00:59:23,890 --> 00:59:27,520
I'm not exactly at the
right place, but I'm close.

1041
00:59:27,520 --> 00:59:34,030
And now let's go to the
last one, which is 68

1042
00:59:34,030 --> 00:59:36,896
according to my helpers here.

1043
00:59:44,640 --> 00:59:45,610
You see?

1044
00:59:45,610 --> 00:59:48,310
This one goes opposite
phase, and those two

1045
00:59:48,310 --> 00:59:51,480
more or less together.

1046
00:59:51,480 --> 00:59:54,820
Then they keep going.

1047
00:59:54,820 --> 00:59:57,750
See now, those two move a
little bit forth and back,

1048
00:59:57,750 --> 00:59:58,790
but they are in phase.

1049
00:59:58,790 --> 00:59:59,760
They move together.

1050
00:59:59,760 --> 01:00:01,490
The ratio is 1.

1051
01:00:01,490 --> 01:00:03,796
And this one-- the
ratio is minus 2.

1052
01:00:08,500 --> 01:00:09,000
Right?

1053
01:00:09,000 --> 01:00:10,371
Make sense?

1054
01:00:10,371 --> 01:00:11,120
That's the beauty.

1055
01:00:11,120 --> 01:00:15,200
You drive it at some frequency,
and those normal modes pop out.

1056
01:00:15,200 --> 01:00:17,200
It's actually very, very cool.

1057
01:00:17,200 --> 01:00:23,900
And as I said, you encounter
those type of behaviors

1058
01:00:23,900 --> 01:00:24,490
very often.

1059
01:00:24,490 --> 01:00:27,160
Sometimes you drive a car and
something starts vibrating,

1060
01:00:27,160 --> 01:00:30,940
it's just because the
car driving on the road

1061
01:00:30,940 --> 01:00:33,280
creates a frequency, provides
a driving frequency which

1062
01:00:33,280 --> 01:00:37,290
corresponds to oscillation
frequency or some piece of--

1063
01:00:37,290 --> 01:00:37,920
old car.

1064
01:00:37,920 --> 01:00:39,430
Usually it happens in old cars.

1065
01:00:42,120 --> 01:00:46,470
So I think that's
the message we can--

1066
01:00:46,470 --> 01:00:49,330
and we have all the machinery
to be able to do it.

1067
01:00:49,330 --> 01:00:52,240
We can set up any
matrix at K, which

1068
01:00:52,240 --> 01:00:55,240
has information about all the
forces acting on anything,

1069
01:00:55,240 --> 01:00:57,640
and we can set matrix
M with the masses.

1070
01:00:57,640 --> 01:01:01,870
We can put it all together,
we can find normal modes,

1071
01:01:01,870 --> 01:01:03,880
and then we can use
Cramer's equation

1072
01:01:03,880 --> 01:01:07,870
to take care of the
arbitrary external forces.

1073
01:01:07,870 --> 01:01:09,740
And what comes out, just as a--

1074
01:01:09,740 --> 01:01:14,380
for summary, for
future reference,

1075
01:01:14,380 --> 01:01:18,250
the oscillation
of the system is--

1076
01:01:18,250 --> 01:01:20,110
this is conveniently written.

1077
01:01:20,110 --> 01:01:24,250
This is vector X.
In general, this

1078
01:01:24,250 --> 01:01:34,700
homogeneous solution this plus
the particular solution, which

1079
01:01:34,700 --> 01:01:41,480
is plus vector B, which
is very important.

1080
01:01:41,480 --> 01:01:45,780
Vector B depends on
the driving frequency.

1081
01:01:45,780 --> 01:01:48,950
Those amplitudes of a particular
solution during emotions

1082
01:01:48,950 --> 01:01:51,040
are dependent on
driving frequency.

1083
01:01:51,040 --> 01:01:57,220
Cosine omega d times t.

1084
01:01:57,220 --> 01:02:01,500
So in the most
general situation,

1085
01:02:01,500 --> 01:02:04,100
we have some homogeneous
solution here,

1086
01:02:04,100 --> 01:02:07,310
and there is this
driven solution

1087
01:02:07,310 --> 01:02:11,120
which we observed in action,
with proper amplitudes.

1088
01:02:11,120 --> 01:02:13,160
So in fact, what
you've seen is the sum

1089
01:02:13,160 --> 01:02:16,230
of both, because this depends
on the initial conditions.

1090
01:02:16,230 --> 01:02:20,720
Now, in reality, as with
a single oscillator,

1091
01:02:20,720 --> 01:02:23,120
this homogeneous equation,
there's always a little

1092
01:02:23,120 --> 01:02:25,730
damping, which we ignore it.

1093
01:02:25,730 --> 01:02:28,190
And the damping
comes in, and it only

1094
01:02:28,190 --> 01:02:29,840
affects the
homogeneous solution.

1095
01:02:29,840 --> 01:02:33,530
So this part will
eventually die down,

1096
01:02:33,530 --> 01:02:36,840
whereas a driven
solution is always there.

1097
01:02:36,840 --> 01:02:40,400
There's external force that
is driving the system forever

1098
01:02:40,400 --> 01:02:41,850
and ever.

1099
01:02:41,850 --> 01:02:45,860
So this part, this steady
state or particular solution

1100
01:02:45,860 --> 01:02:49,110
will remain forever,
because there's

1101
01:02:49,110 --> 01:02:52,170
an external source of energy
which will always provide it.

1102
01:02:52,170 --> 01:02:53,940
So these guys will die down.

1103
01:02:53,940 --> 01:02:57,090
And of course, because of
damping the exact value

1104
01:02:57,090 --> 01:02:59,790
of coefficients B will
be slightly modified,

1105
01:02:59,790 --> 01:03:03,692
because as you know from the
from a one oscillator example,

1106
01:03:03,692 --> 01:03:05,400
the presence of damping
actually slightly

1107
01:03:05,400 --> 01:03:07,280
modifies the frequency.

1108
01:03:07,280 --> 01:03:08,880
Whereas here, we--
for simplicity--

1109
01:03:08,880 --> 01:03:11,640
if we introduce damping
here, those calculations

1110
01:03:11,640 --> 01:03:13,440
are really amazing.

1111
01:03:13,440 --> 01:03:16,410
So we don't want to do it.

1112
01:03:16,410 --> 01:03:17,100
All right.

1113
01:03:17,100 --> 01:03:18,120
Any questions about it?

1114
01:03:22,950 --> 01:03:23,782
Yes.

1115
01:03:23,782 --> 01:03:26,480
AUDIENCE: If we were doing
Cramer's rule with a three

1116
01:03:26,480 --> 01:03:30,510
by three matrix, would we
only replace the column

1117
01:03:30,510 --> 01:03:34,069
that corresponds to the B that
we're trying to find, and then

1118
01:03:34,069 --> 01:03:34,860
keep the other two?

1119
01:03:34,860 --> 01:03:36,160
BOLESLAW WYSLOUCH: Yes.

1120
01:03:36,160 --> 01:03:38,130
So it's always-- you'll
be doing always that.

1121
01:03:38,130 --> 01:03:40,980
In fact, I should have
some slides from Yen-Jie

1122
01:03:40,980 --> 01:03:43,830
on Cramer's rule.

1123
01:03:43,830 --> 01:03:45,859
Let's see.

1124
01:03:45,859 --> 01:03:46,358
OK.

1125
01:03:46,358 --> 01:03:51,597
So this is some
reminder of last time.

1126
01:03:51,597 --> 01:03:53,305
So this is Cramer's--
there's Mr. Cramer.

1127
01:03:55,840 --> 01:03:59,170
So this is an example of what--

1128
01:03:59,170 --> 01:04:02,730
this is the two by
two, three by three.

1129
01:04:02,730 --> 01:04:03,230
OK?

1130
01:04:03,230 --> 01:04:05,210
That's what you do.

1131
01:04:05,210 --> 01:04:06,200
Question?

1132
01:04:06,200 --> 01:04:10,655
AUDIENCE: So it makes sense that
the Cramer's rule [INAUDIBLE],,

1133
01:04:10,655 --> 01:04:12,635
but what does that mean
for physical system?

1134
01:04:16,610 --> 01:04:19,840
BOLESLAW WYSLOUCH:
Well, basically--

1135
01:04:19,840 --> 01:04:22,800
so the Cramer's rule
is Cramer's rule.

1136
01:04:22,800 --> 01:04:24,650
The question is
what do you plug in?

1137
01:04:24,650 --> 01:04:27,770
And what you plug in
depends on the omega d.

1138
01:04:27,770 --> 01:04:30,790
So it is true that if you
insist on plugging in omega

1139
01:04:30,790 --> 01:04:33,950
d exactly equal to one of
the normal frequencies,

1140
01:04:33,950 --> 01:04:38,290
then things blow
up mathematically.

1141
01:04:38,290 --> 01:04:41,620
In reality, there is--

1142
01:04:41,620 --> 01:04:43,960
this is the situation
of resonance.

1143
01:04:43,960 --> 01:04:46,150
So as I discussed this
before, in reality

1144
01:04:46,150 --> 01:04:49,070
there is a little
bit of damping.

1145
01:04:49,070 --> 01:04:51,500
So those equations
have to be modified.

1146
01:04:51,500 --> 01:04:54,710
There will be some small
additional terms here

1147
01:04:54,710 --> 01:04:57,680
that will prevent this from
being exactly equal to 0.

1148
01:04:57,680 --> 01:04:59,510
So this will be a
very large number.

1149
01:04:59,510 --> 01:05:02,026
The amplitude will be enormous.

1150
01:05:02,026 --> 01:05:03,650
If I would have a
little bit more time,

1151
01:05:03,650 --> 01:05:05,120
I'll fiddle with
frequency, I could actually

1152
01:05:05,120 --> 01:05:07,850
break the system, because those
masses would be just swinging

1153
01:05:07,850 --> 01:05:10,870
forth and back like crazy.

1154
01:05:10,870 --> 01:05:14,310
So you basically go out
of limit of the system.

1155
01:05:14,310 --> 01:05:16,560
So physically, there's always
a little bit of damping.

1156
01:05:16,560 --> 01:05:18,460
You do not divide by zero.

1157
01:05:18,460 --> 01:05:22,660
On the other hand, it's so
close that, for simplicity

1158
01:05:22,660 --> 01:05:26,110
and for most of the-- to get
a feeling of what's going on,

1159
01:05:26,110 --> 01:05:27,870
it's OK to ignore it.

1160
01:05:27,870 --> 01:05:31,720
Just have to make sure
you don't divide by 0.

1161
01:05:31,720 --> 01:05:34,715
So you can do this Cramer's
rule with arbitrary omega d.

1162
01:05:34,715 --> 01:05:37,900
Make sure you don't
divide by 0, you solve it,

1163
01:05:37,900 --> 01:05:41,040
and then you can
interpret what's going on.

1164
01:05:41,040 --> 01:05:43,700
Again, Cramer's rule has
nothing to do with physics.

1165
01:05:43,700 --> 01:05:46,464
It's just a way to solve
those matrix equations.

1166
01:05:46,464 --> 01:05:48,130
As I say, you can do
it anyway you want.

1167
01:05:48,130 --> 01:05:52,000
Two by two, you can do it
by elimination of variables.

1168
01:05:52,000 --> 01:05:55,240
Five by five I do by
running a MATLAB program.

1169
01:05:55,240 --> 01:05:56,450
Anything you want.

1170
01:05:56,450 --> 01:06:01,575
But for some historical reasons,
8.03 always does Cramer's rule.

1171
01:06:01,575 --> 01:06:02,220
All right?

1172
01:06:02,220 --> 01:06:06,110
And, yeah, it's useful,
especially for three by three.

1173
01:06:06,110 --> 01:06:06,610
All right?

1174
01:06:09,271 --> 01:06:09,770
OK.

1175
01:06:09,770 --> 01:06:12,832
So I have to start
a new chapter.

1176
01:06:12,832 --> 01:06:14,665
I'm much slower than
the engine, by the way.

1177
01:06:14,665 --> 01:06:15,831
I don't know if you noticed.

1178
01:06:18,270 --> 01:06:20,440
And that is the--

1179
01:06:20,440 --> 01:06:22,710
there's a very
interesting trick that you

1180
01:06:22,710 --> 01:06:29,670
can do which is of an absolutely
fundamental nature in physics,

1181
01:06:29,670 --> 01:06:32,310
which has to do with symmetry.

1182
01:06:32,310 --> 01:06:36,900
You see, many things
are symmetric.

1183
01:06:36,900 --> 01:06:39,040
There's a circular symmetry.

1184
01:06:39,040 --> 01:06:41,110
There's a left and
right symmetry.

1185
01:06:41,110 --> 01:06:44,800
Example, two little
smiley faces are

1186
01:06:44,800 --> 01:06:47,040
mirror images of each other.

1187
01:06:47,040 --> 01:06:50,250
There is some-- this
thing is symmetric

1188
01:06:50,250 --> 01:06:52,620
along this vertical axis.

1189
01:06:52,620 --> 01:06:55,665
This one is symmetric around
rotations by 30 degrees.

1190
01:06:58,230 --> 01:07:01,470
That house seems to be
symmetric along this way.

1191
01:07:01,470 --> 01:07:04,660
This is part of our
experiment in Switzerland,

1192
01:07:04,660 --> 01:07:07,155
also kind of symmetric
in the picture.

1193
01:07:09,624 --> 01:07:12,290
The rotational symmetry, there's
reflection symmetry, et cetera.

1194
01:07:12,290 --> 01:07:16,210
It turns out, if you have
a system that is symmetric,

1195
01:07:16,210 --> 01:07:19,570
then the normal modes
are also symmetric.

1196
01:07:19,570 --> 01:07:23,500
And there's a way to dig out
normal modes just by looking

1197
01:07:23,500 --> 01:07:24,610
at symmetry of the system.

1198
01:07:27,460 --> 01:07:29,870
So let me explain
exactly what this means.

1199
01:07:29,870 --> 01:07:34,520
So let's take our system here--

1200
01:07:34,520 --> 01:07:39,890
OK, so we have one
mass, the other mass.

1201
01:07:39,890 --> 01:07:43,000
There is a spring here.

1202
01:07:43,000 --> 01:07:46,520
This one is x1, this one is x2.

1203
01:07:46,520 --> 01:07:48,590
If I take a reflection
of a system--

1204
01:07:48,590 --> 01:07:53,170
let's say this mass is
displaced by some distance.

1205
01:07:53,170 --> 01:07:56,330
Some x2.

1206
01:07:56,330 --> 01:07:57,620
This one's some x1.

1207
01:07:57,620 --> 01:07:59,280
If I do the
following transform--

1208
01:07:59,280 --> 01:08:08,780
I replace x1 with minus
x2, and x2 with minus x1,

1209
01:08:08,780 --> 01:08:09,890
this is mirror symmetry.

1210
01:08:16,700 --> 01:08:20,050
I basically flip
this thing around.

1211
01:08:20,050 --> 01:08:25,149
In other words, what I do here
is I look at the system here--

1212
01:08:25,149 --> 01:08:28,850
hello-- and I go to
the other system.

1213
01:08:28,850 --> 01:08:29,540
Hello.

1214
01:08:29,540 --> 01:08:30,040
Right?

1215
01:08:30,040 --> 01:08:31,081
I did a mirror transform.

1216
01:08:31,081 --> 01:08:33,750
I looked at it from
this side, that side.

1217
01:08:33,750 --> 01:08:35,890
Now, when I look
at it I see the one

1218
01:08:35,890 --> 01:08:37,359
on the left, one on the right.

1219
01:08:37,359 --> 01:08:40,210
I call this one x1, this one x2.

1220
01:08:40,210 --> 01:08:41,950
It's oscillating.

1221
01:08:41,950 --> 01:08:46,029
You are looking at it, this
is your x1, this is your x2.

1222
01:08:46,029 --> 01:08:47,960
When I move this one, is it--

1223
01:08:47,960 --> 01:08:50,229
it's your negative x1.

1224
01:08:50,229 --> 01:08:53,399
For me this is positive x2.

1225
01:08:53,399 --> 01:08:56,920
This one is positive x2 for you.

1226
01:08:56,920 --> 01:09:00,160
It's negative x1 for me.

1227
01:09:00,160 --> 01:09:01,630
Do we see a different system?

1228
01:09:01,630 --> 01:09:03,130
Does it have different
oscillations?

1229
01:09:03,130 --> 01:09:04,250
Does it have a
different frequency?

1230
01:09:04,250 --> 01:09:04,750
No.

1231
01:09:04,750 --> 01:09:05,689
It's identical.

1232
01:09:05,689 --> 01:09:07,640
They're completely identical.

1233
01:09:07,640 --> 01:09:09,520
So the physics of
those two pendula

1234
01:09:09,520 --> 01:09:11,590
doesn't depend on if
he's working on it

1235
01:09:11,590 --> 01:09:13,000
or if I'm working on.

1236
01:09:13,000 --> 01:09:14,439
That's the whole thing.

1237
01:09:14,439 --> 01:09:17,529
And this is how you
write it mathematically.

1238
01:09:17,529 --> 01:09:20,920
And if you have a
solution which--

1239
01:09:20,920 --> 01:09:27,640
x1 of t, which consists of
some sort of x1 of t, x2 of t.

1240
01:09:27,640 --> 01:09:30,180
Let's say we find it.

1241
01:09:30,180 --> 01:09:31,240
Now it's over there.

1242
01:09:31,240 --> 01:09:34,490
We know alphas,
betas and everything.

1243
01:09:34,490 --> 01:09:38,029
Because of the symmetry, I
know that for sure the equation

1244
01:09:38,029 --> 01:09:41,180
which looks like this-- x1--

1245
01:09:41,180 --> 01:09:42,420
no, it's not x1.

1246
01:09:42,420 --> 01:09:44,270
It's x of t.

1247
01:09:44,270 --> 01:09:46,208
That's the vector x of t.

1248
01:09:46,208 --> 01:09:51,170
I have another one with a tilde,
which is identical functions,

1249
01:09:51,170 --> 01:09:56,990
everything is dependent, except
that this one is minus x2 of t

1250
01:09:56,990 --> 01:10:00,290
minus x1 of t.

1251
01:10:00,290 --> 01:10:04,520
And I know for sure that if
this is the correct solution,

1252
01:10:04,520 --> 01:10:07,230
this is also a correct solution.

1253
01:10:07,230 --> 01:10:07,980
Why?

1254
01:10:07,980 --> 01:10:14,370
Because he did x,
and I did x tilde.

1255
01:10:14,370 --> 01:10:16,200
But the system is the same.

1256
01:10:16,200 --> 01:10:17,220
Completely identical.

1257
01:10:17,220 --> 01:10:22,020
And you don't have to know
anything about masses, lengths,

1258
01:10:22,020 --> 01:10:23,790
springs, anything like that.

1259
01:10:23,790 --> 01:10:25,890
Just the symmetry.

1260
01:10:25,890 --> 01:10:26,730
All right.

1261
01:10:26,730 --> 01:10:29,940
How do you write
it in matrix form?

1262
01:10:29,940 --> 01:10:37,690
You introduce a symmetry
matrix, S, which is 0, minus 1,

1263
01:10:37,690 --> 01:10:40,400
minus 1, 0.

1264
01:10:40,400 --> 01:10:48,584
And then x tilde of t is
simply equal S, x of t.

1265
01:10:48,584 --> 01:10:49,500
And we can check that.

1266
01:10:49,500 --> 01:10:53,590
That's simple you just
multiply the vector by 0,

1267
01:10:53,590 --> 01:10:57,370
minus 1, minus 1, 0, and
you get the same thing.

1268
01:10:57,370 --> 01:11:02,200
Turns out-- and if
this is symmetry,

1269
01:11:02,200 --> 01:11:04,660
if this is a solution,
this is also a solution.

1270
01:11:04,660 --> 01:11:08,670
So we can make solutions
by multiplying by matrix S.

1271
01:11:08,670 --> 01:11:10,030
So what does it mean?

1272
01:11:10,030 --> 01:11:14,115
So let's look at
our motion equation.

1273
01:11:17,880 --> 01:11:19,970
The original motion
equation was--

1274
01:11:19,970 --> 01:11:23,810
equation of motion was minus
1 k matrix times x of t.

1275
01:11:23,810 --> 01:11:26,740
This is what we use
to find solutions.

1276
01:11:26,740 --> 01:11:28,550
Usual thing, normal
modes, et cetera.

1277
01:11:28,550 --> 01:11:32,970
Let's multiply both sides by
matrix S. I can take any matrix

1278
01:11:32,970 --> 01:11:35,230
and multiply by both sides.

1279
01:11:35,230 --> 01:11:39,391
So I get here S X
double dot of t.

1280
01:11:39,391 --> 01:11:43,300
And of course, S
is a fixed matrix,

1281
01:11:43,300 --> 01:11:45,710
so it survives differentiation.

1282
01:11:45,710 --> 01:11:51,910
And this is equal to minus
S M minus 1 k x of t.

1283
01:11:51,910 --> 01:11:54,970
Just multiply both sides by S.

1284
01:11:54,970 --> 01:12:07,286
However, if MS is equal to
SM, and KS is equal to SK--

1285
01:12:11,110 --> 01:12:14,160
in general matrices, the
multiplication of matrices

1286
01:12:14,160 --> 01:12:15,990
matters.

1287
01:12:15,990 --> 01:12:18,970
But it turns out that if
the system is symmetric,

1288
01:12:18,970 --> 01:12:24,130
if you multiply mass M
by S, you just replace--

1289
01:12:24,130 --> 01:12:26,840
it will just change
position of two masses.

1290
01:12:26,840 --> 01:12:28,520
So nothing changes.

1291
01:12:28,520 --> 01:12:34,000
Also, if the forces are the
same, then multiplying mass S,

1292
01:12:34,000 --> 01:12:34,850
you flip things.

1293
01:12:34,850 --> 01:12:36,110
Nothing changes.

1294
01:12:36,110 --> 01:12:38,480
And mathematically, it
means that the order

1295
01:12:38,480 --> 01:12:41,420
of multiplication
does not matter.

1296
01:12:41,420 --> 01:12:42,870
It means that they
are commuting.

1297
01:12:42,870 --> 01:12:50,740
And of course, M minus 1
S is equal to S M minus 1.

1298
01:12:50,740 --> 01:12:55,820
If this is the case, then I
can plug it into equations

1299
01:12:55,820 --> 01:12:58,408
and see what happens.

1300
01:13:07,870 --> 01:13:16,420
So I can take this equation,
and I can take this S here

1301
01:13:16,420 --> 01:13:18,260
and I can just move it around.

1302
01:13:18,260 --> 01:13:21,790
I can flip it with M1 position,
because the order doesn't

1303
01:13:21,790 --> 01:13:22,340
matter.

1304
01:13:22,340 --> 01:13:23,650
So I can bring it here.

1305
01:13:23,650 --> 01:13:25,780
And I can flip it with K,
because the order doesn't

1306
01:13:25,780 --> 01:13:26,280
matter.

1307
01:13:26,280 --> 01:13:28,180
I can bring it here.

1308
01:13:28,180 --> 01:13:31,080
So after using those
features, I get

1309
01:13:31,080 --> 01:13:40,580
that S X dot dot is equal
to minus M minus 1 K S X,

1310
01:13:40,580 --> 01:13:44,810
which means that X dot dot--

1311
01:13:44,810 --> 01:13:47,430
remember, this was--

1312
01:13:47,430 --> 01:13:48,920
I'm using this expression.

1313
01:13:48,920 --> 01:13:53,440
I'm just-- S times a
variable x gives me X tilde.

1314
01:13:53,440 --> 01:14:00,090
X tilde dot dot is equal to
minus M minus 1 k X tilde.

1315
01:14:00,090 --> 01:14:01,770
X tilde.

1316
01:14:01,770 --> 01:14:04,415
Which basically proves--
this is a proof--

1317
01:14:04,415 --> 01:14:06,930
that x tilde is a solution.

1318
01:14:06,930 --> 01:14:12,100
So if a system is symmetric,
it means that it commutes--

1319
01:14:12,100 --> 01:14:20,010
that mass and K matrices
commute, and you can--

1320
01:14:20,010 --> 01:14:22,530
and this means that
this holds true.

1321
01:14:22,530 --> 01:14:24,870
If I have one solution,
the symmetric solution

1322
01:14:24,870 --> 01:14:27,290
is also there.

1323
01:14:27,290 --> 01:14:28,150
All right?

1324
01:14:32,000 --> 01:14:34,354
Let's say x-- yes?

1325
01:14:34,354 --> 01:14:36,764
AUDIENCE: So in the
center equation,

1326
01:14:36,764 --> 01:14:42,080
you introduced negative S.
I didn't really get that.

1327
01:14:42,080 --> 01:14:45,040
BOLESLAW WYSLOUCH: So this
negative is simply the--

1328
01:14:45,040 --> 01:14:47,200
Hooke's law.

1329
01:14:47,200 --> 01:14:49,371
This is this minus sign here.

1330
01:14:49,371 --> 01:14:52,294
AUDIENCE: Yeah, but where
did the S come from in the--

1331
01:14:52,294 --> 01:14:54,460
BOLESLAW WYSLOUCH: Oh, I
multiplied both sides by S.

1332
01:14:54,460 --> 01:14:55,600
AUDIENCE: Oh, OK.

1333
01:14:55,600 --> 01:14:57,225
BOLESLAW WYSLOUCH:
I just brought the S

1334
01:14:57,225 --> 01:14:58,500
and I put it here.

1335
01:14:58,500 --> 01:15:02,950
S, X dot dot, and S after--

1336
01:15:02,950 --> 01:15:07,360
minus commutes with S, so
I kind of shifted my minus.

1337
01:15:07,360 --> 01:15:10,540
But then I waited before
I hit the matrices,

1338
01:15:10,540 --> 01:15:12,180
because I wanted to discuss.

1339
01:15:12,180 --> 01:15:12,680
OK?

1340
01:15:15,300 --> 01:15:18,850
So now comes the
interesting question.

1341
01:15:18,850 --> 01:15:22,070
Let's say X is a normal mode.

1342
01:15:22,070 --> 01:15:22,846
Right?

1343
01:15:22,846 --> 01:15:23,720
We have normal modes.

1344
01:15:23,720 --> 01:15:25,850
Let's say X is a normal mode.

1345
01:15:25,850 --> 01:15:29,630
It oscillates with
a certain frequency.

1346
01:15:29,630 --> 01:15:31,990
So I have X of t.

1347
01:15:31,990 --> 01:15:33,560
Let's say it's equal to--

1348
01:15:33,560 --> 01:15:36,510
let's say it's a
normal mode number one.

1349
01:15:36,510 --> 01:15:39,880
Cosine omega 1 t.

1350
01:15:42,500 --> 01:15:46,390
And we know that X tilde
is also a solution.

1351
01:15:46,390 --> 01:15:54,180
So what happens to mode number
one when I apply matrix S?

1352
01:15:54,180 --> 01:15:59,140
So X tilde-- so matrix
is a constant number.

1353
01:15:59,140 --> 01:16:00,640
It's just a couple
of numbers I just

1354
01:16:00,640 --> 01:16:01,806
reshuffle things, et cetera.

1355
01:16:01,806 --> 01:16:04,630
Try So if I have X, which is
oscillating with frequency

1356
01:16:04,630 --> 01:16:08,290
omega 1, if I multiply
by some numbers

1357
01:16:08,290 --> 01:16:10,780
and reshuffle things
around, it will also

1358
01:16:10,780 --> 01:16:12,910
be oscillating in number one.

1359
01:16:12,910 --> 01:16:16,310
So it will be also
the same normal mode.

1360
01:16:16,310 --> 01:16:20,380
So if I take matrix S, I
apply it to the normal mode,

1361
01:16:20,380 --> 01:16:23,080
I will get the same
normal mode, with maybe

1362
01:16:23,080 --> 01:16:25,310
a different coefficient.

1363
01:16:25,310 --> 01:16:26,770
Linear coefficient.

1364
01:16:26,770 --> 01:16:29,020
Plus, minus, maybe some
factor, something like that.

1365
01:16:31,670 --> 01:16:34,170
So if this is the solution,
it means automatically

1366
01:16:34,170 --> 01:16:43,540
that X tilde is proportional
to A1 cosine omega 1 t.

1367
01:16:43,540 --> 01:16:46,540
And the same is
true for omega 2.

1368
01:16:46,540 --> 01:16:48,190
So the only way
this is possible,

1369
01:16:48,190 --> 01:16:52,290
since cosine is the
same in both cases--

1370
01:16:52,290 --> 01:16:54,160
matrix S to normal
solution gives me

1371
01:16:54,160 --> 01:16:56,700
normal solution with some sign.

1372
01:16:56,700 --> 01:16:59,010
So the only way this can
work, matrix S actually

1373
01:16:59,010 --> 01:17:02,010
works on vectors, on A1.

1374
01:17:02,010 --> 01:17:04,830
This is just an
oscillating factor.

1375
01:17:04,830 --> 01:17:13,110
So we know for sure that
S A1 must be proportional.

1376
01:17:13,110 --> 01:17:13,680
to A1.

1377
01:17:17,840 --> 01:17:24,238
Similarly, S times A2
is proportional to A2.

1378
01:17:28,110 --> 01:17:32,530
So let's try to see with
our own eyes if this works.

1379
01:17:32,530 --> 01:17:36,420
So let's say S is
0, minus 1, minus 1,

1380
01:17:36,420 --> 01:17:42,820
0, times 1, 1 is equal to what?

1381
01:17:42,820 --> 01:17:47,160
0 minus 1, I get minus 1 here.

1382
01:17:47,160 --> 01:17:50,340
This one, I get
minus 1 here, which

1383
01:17:50,340 --> 01:17:57,780
is equal to minus 1 times
1, 1, which is vector A. So

1384
01:17:57,780 --> 01:18:04,850
vector 1, 1, which is our
first mode of oscillation,

1385
01:18:04,850 --> 01:18:09,410
is when you apply the
matrix S, you get a minus 1

1386
01:18:09,410 --> 01:18:12,020
the same thing.

1387
01:18:12,020 --> 01:18:19,144
And similarly, if you do the
same thing with matrix S--

1388
01:18:19,144 --> 01:18:21,890
so you see, the simple
symmetric matrix

1389
01:18:21,890 --> 01:18:24,530
consisting of 0s
and minus 1s has

1390
01:18:24,530 --> 01:18:26,600
something to do with
our solutions, which

1391
01:18:26,600 --> 01:18:27,500
is kind of amazing.

1392
01:18:30,020 --> 01:18:36,980
So if I have 0, minus 1, minus
1, 0, I multiply by 1, minus 1,

1393
01:18:36,980 --> 01:18:43,740
I get 1 here, I
get minus 1 here.

1394
01:18:48,800 --> 01:18:49,740
Just a moment.

1395
01:18:49,740 --> 01:18:52,494
Something is not right.

1396
01:18:52,494 --> 01:18:53,410
Something's not right.

1397
01:18:53,410 --> 01:18:54,040
No, it should be--

1398
01:18:54,040 --> 01:18:54,978
AUDIENCE: [INAUDIBLE]

1399
01:18:54,978 --> 01:18:55,936
BOLESLAW WYSLOUCH: Hmm?

1400
01:18:55,936 --> 01:18:57,358
AUDIENCE: [INAUDIBLE]

1401
01:18:57,358 --> 01:18:59,760
AUDIENCE: It's 1, minus 1.

1402
01:18:59,760 --> 01:19:01,010
BOLESLAW WYSLOUCH: 1, minus 1.

1403
01:19:01,010 --> 01:19:01,664
Yes.

1404
01:19:01,664 --> 01:19:03,080
I don't know how
to multiply here.

1405
01:19:03,080 --> 01:19:05,280
I should be fine.

1406
01:19:05,280 --> 01:19:06,799
OK.

1407
01:19:06,799 --> 01:19:07,340
That's right.

1408
01:19:07,340 --> 01:19:09,006
This is-- sorry, this
is 1, because it's

1409
01:19:09,006 --> 01:19:11,870
minus 1 times minus 1, and
this is-- yeah, that's right.

1410
01:19:11,870 --> 01:19:17,060
Which is 1 times 1, minus 1.

1411
01:19:17,060 --> 01:19:18,590
So this is something that--

1412
01:19:18,590 --> 01:19:21,930
I get the same vector
multiplied by plus 1.

1413
01:19:21,930 --> 01:19:23,270
So this is, of course--

1414
01:19:23,270 --> 01:19:27,150
these are eigenvectors
and eigenvalues.

1415
01:19:27,150 --> 01:19:29,870
So the matrix S has
two eigenvectors,

1416
01:19:29,870 --> 01:19:32,690
one with eigenvalue of plus
one, the other one plus 2.

1417
01:19:32,690 --> 01:19:38,196
So we have an equation SA
is equal to beta times A,

1418
01:19:38,196 --> 01:19:41,250
and beta is--

1419
01:19:44,910 --> 01:19:46,310
OK.

1420
01:19:46,310 --> 01:19:48,150
So this is something--

1421
01:19:48,150 --> 01:19:51,680
so it turns out-- and I don't
think I have time to prove it,

1422
01:19:51,680 --> 01:19:53,770
but it turns out
you can prove it--

1423
01:19:53,770 --> 01:19:56,760
if I would have
another three minutes--

1424
01:19:56,760 --> 01:20:01,350
you can prove it that the
eigenvalues of matrix S--

1425
01:20:01,350 --> 01:20:06,300
eigenvectors, sorry,
eigenvectors of matrix S

1426
01:20:06,300 --> 01:20:11,310
are the same as eigenvectors
of the full motion matrix.

1427
01:20:14,120 --> 01:20:22,260
So in other words, our
motion matrix M minus 1 K--

1428
01:20:22,260 --> 01:20:23,160
this is the matrix.

1429
01:20:23,160 --> 01:20:30,930
Then we have a matrix
S. And normal modes are,

1430
01:20:30,930 --> 01:20:34,830
you have a normal frequency
and they have a shape.

1431
01:20:34,830 --> 01:20:38,400
You have a normal vector,
the ratio of amplitudes.

1432
01:20:38,400 --> 01:20:46,560
And turns out that eigenvectors
here, so the A's are the same.

1433
01:20:46,560 --> 01:20:50,780
And again, I don't
have time to show it,

1434
01:20:50,780 --> 01:20:53,620
but you can show that
this is the case.

1435
01:20:53,620 --> 01:20:57,990
So if you have a
symmetry in the system,

1436
01:20:57,990 --> 01:21:04,770
then you can simply find
eigenvectors of the thing

1437
01:21:04,770 --> 01:21:07,170
to obtain the normal modes.

1438
01:21:07,170 --> 01:21:14,870
So if I look at my two pendula
here, the symmetry is this way,

1439
01:21:14,870 --> 01:21:19,880
so I have to have one which
is fully symmetric, like this,

1440
01:21:19,880 --> 01:21:21,830
and I have another one
which is antisymmetric.

1441
01:21:21,830 --> 01:21:26,042
Plus 1, minus 1,
plus 1, minus 1.

1442
01:21:26,042 --> 01:21:28,280
Similarly, here I have--

1443
01:21:28,280 --> 01:21:30,330
let's say if I have
two masses, there

1444
01:21:30,330 --> 01:21:32,650
is one motion
which is like this,

1445
01:21:32,650 --> 01:21:34,880
and one motion which
is like that, because

1446
01:21:34,880 --> 01:21:37,860
of the mirror symmetry.

1447
01:21:37,860 --> 01:21:40,410
And you can show that if you
have some other symmetries,

1448
01:21:40,410 --> 01:21:42,360
like on a circle
et cetera, that you

1449
01:21:42,360 --> 01:21:43,800
have a similar type of fact.

1450
01:21:43,800 --> 01:21:47,040
So you can build up on
this symmetry argument.

1451
01:21:47,040 --> 01:21:53,880
And finding eigenvectors of a
matrix 0, minus 1, minus 1, 0

1452
01:21:53,880 --> 01:21:56,940
is infinitely simpler
than finding matrix

1453
01:21:56,940 --> 01:22:00,290
with G's and K's and
everything, right?

1454
01:22:00,290 --> 01:22:00,790
All right.

1455
01:22:00,790 --> 01:22:08,120
So thank you very much, and
I hope this was educational.